Archives de Inder J. Taneja

Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Single Digit Representations of
Natural Numbers From 1 to 5000
Inder J. Taneja1
Abstract
There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single
letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors
previous work [8] on representation of natural numbers in terms of single digit. The previous work is upto 1000. This paper bring
numbers 1 to 5000 in terms of each digit. The total work up to 20000 numbers divided in four parts. For other parts refer [9, 10, 11].
Contents
1 Crazy Representations of Natural Numbers
1.1 First Type: Increasing and Decreasing . . . . . . . . . .
1.2 Second Type: Flexible Power Representations . . . . .
1.2.1 Unequal String Lengths . . . . . . . . . . . . . .
1.2.2 Equal String Lengths . . . . . . . . . . . . . . . .
1.3 Third Way: Single Digit Representations . . . . . . . . .
1.4 Forth Way: Single Letter Representations . . . . . . . .
1.5 Running Expressions . . . . . . . . . . . . . . . . . . . .
1.5.1 Running Expressions with Fibonacci Sequence
1.5.2 Running Expressions with Triangular Numbers
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1
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2
3
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8
2 Single Digit Representations From 1 to 20000
9
2.1 Single Digit Representation: 1-5000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1 Crazy Representations of Natural Numbers
In this section, we shall write different ways of writing natural numbers. These representations are divided in four different types.
1
Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil (1978-2012). Also worked at Delhi University, India (1976-1978).
E-mail: [email protected]; Web-sites: http://inderjtaneja.com; http://indertaneja.com; Twitter: @IJTANEJA.
1
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
1.1 First Type: Increasing and Decreasing
In 2014, author [1] wrote natural numbers in increasing and decreasing orders of 1 to 9 and 9 to 1. See examples below:
100 := 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9 = 9 × 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
101 := 1 + 2 + 34 + 5 + 6 × 7 + 8 + 9
= 9×8+7+6+5+4+3×2+1
102 := 12 + 3 × 4 × 5 + 6 + 7 + 8 + 9
= 9 + 8 + 7 + 6 + 5 + 43 + 2 + 1
103 := 1 × 2 × 34 + 5 + 6 + 7 + 8 + 9
= 9 + 8 + 7 × 6 + 5 × 4 + 3 + 21
104 := 1 + 23 + 4 + 5 + 6 + 7 × 8 + 9
= 9 + 8 + 7 + 65 + 4 × 3 + 2 + 1
105 := 1 + 2 × 3 × 4 + 56 + 7 + 8 + 9
= 9+8×7+6×5+4+3+2+1
106 := 12 + 3 + 4 × 5 + 6 + 7 × 8 + 9
= 9+8×7+6×5+4+3×2+1
107 := 1 × 23 + 4 + 56 + 7 + 8 + 9
= 9 + 8 + 76 + 5 + 4 + 3 + 2 × 1
108 := 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9
= 9 + 8 + 76 + 5 + 4 + 3 + 2 + 1.
See more examples,
999 := 12 × 3 × (4 + 5) + (67 + 8) × 9
2535 := 1 + 2345 + (6 + 7 + 8) × 9
= 9 + 8 + 7 + 654 + 321.
= 9 + 87 × (6 + 5 × 4 + 3) + 2 + 1.
2607 := 123 × 4 × 5 + 6 + (7 + 8) × 9
= 987 + 6 × 54 × (3 + 2) × 1.
p
p
10958 := 12 × 3 + 4 + 5! × (67 + 8 × 9) = (9 + 8 × 7 × 65 + 4) × 3 − 2 + 1.
= − 9 + 8 + 7 × (6 + 5) × (4 × 3)2 × 1.
11807 := 1 × 234 × (5 + 6 × 7) + 89
We observe that the number 10958 is the only number among 0 to 11111, where we need extra operations, such as
square-root, factorial, etc. to write in increasing case. For more details refer author’s web-site link [5]. Extension of
numbers from 11112 to 30000 refer [2, 3, 4].
1.2 Second Type: Flexible Power Representations
Let us consider two numbers, 1 and 2. Using the idea of power and the operations of addition and subtraction, we can
write following 3 numbers in terms of 1 and 2, as 1 = −12 + 21 , 3 = 12 + 21 and 5 = 11 + 22 . In this situation, we observe
that bases and exponents are of same digits. Permutations of exponent values helps in bringing different numbers. In
case of repeated values, for example, 3 = 12 + 21 = −11 + 22 , only possibilities is considered. There is only one number
having single digit, i.e., 1 = 11 . For simplicity, let us represent the above procedure as (1, 2)(1,2) , resulting in three possible
values. The above procedure is with two digits. Instead having two digits, we can work with two letters, such as,
(a, b)(a,b) , ...(a, b, c, d , e, f , g , h, i )(a,b,c,d ,e, f ,g ,h,i ) ,
where a, b, c, d , e, f , g , h, i ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, all distinct.
1.2.1 Unequal String Lengths
2
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
100 := 26 + 62
107 := −12 + 27 − 33 + 71
114 := −22 + 35 − 53
101 := 11 + 26 + 62
108 := 17 + 26 + 62 + 71
115 := 15 − 21 − 32 + 53
102 := −25 + 32 + 53
109 := 12 + 27 − 33 + 71
116 := 22 + 35 − 44 + 53
103 := 11 − 25 + 32 + 53
110 := 19 + 26 + 62 + 91
117 := −11 + 35 − 53
104 := −11 + 23 + 34 + 42
111 := −13 + 27 − 32 − 71
118 := 35 − 53
105 := 23 + 34 + 42
112 := 35 − 44 + 53
119 := 11 + 35 − 53 .
106 := 27 + 33 − 72
113 := −15 − 21 − 32 + 53
See more examples,
638 := −15 − 21 − 42 + 54
6922 := −36 − 53 + 65
666 := −25 + 32 + 43 + 54
9711 := 13 + 24 + 38 + 42 + 55 − 81
786 := −14 + 36 + 43 − 61
9777 := 19 + 21 + 47 − 72 − 94
1933 := −13 − 22 + 37 − 44 + 71
11110 := 11 + 22 + 39 − 56 + 65 − 93
1934 := 29 + 36 − 62 + 93
11111 := −11 + 27 + 38 − 42 + 73 + 84 .
3098 := −33 + 55
2280 := −11 − 26 + 45 + 52 + 64
The whole work is from 1 to 11111. For details refer [6].
1.2.2 Equal String Lengths
Based on second type still we can write natural numbers in a sequential way with uniform representations. Instead
working with unequal strings as of previous section, here we worked with equal string using the digits 0 to 9, i.e., using
all the 10 digits, {0,1,2,3,4,5,6,7,8,9}. The results obtained are symmetric, i.e., writing in 0 to 9 or 9 to 0, the resulting
number is same. See some examples below,
201 := 03 + 19 + 24 + 37 − 48 + 51 + 66 + 75 + 82 + 90
212 := 05 + 17 − 28 − 39 + 41 + 56 + 60 + 73 + 84 + 92
202 := 00 + 19 + 26 + 38 − 47 + 55 + 63 + 72 + 81 + 94
213 := 05 + 18 − 27 − 39 + 41 + 56 + 63 + 70 + 84 + 92
203 := 03 − 19 + 24 + 37 − 48 + 50 + 66 + 75 + 82 + 91
214 := 05 + 17 − 28 − 39 + 40 + 56 + 61 + 73 + 84 + 92
204 := 08 + 19 + 25 + 37 − 46 + 51 + 64 + 72 + 80 + 93
215 := 05 + 19 + 28 + 37 − 46 + 50 + 64 + 72 + 83 + 91
205 := 03 + 19 + 24 + 37 − 48 + 50 + 66 + 75 + 82 + 91
216 := 01 − 17 + 28 − 39 + 45 + 56 + 60 + 74 + 83 + 92
206 := 07 − 19 − 25 − 38 + 46 + 51 + 63 + 74 + 80 + 92
217 := 07 − 19 + 25 − 38 + 46 + 52 + 63 + 74 + 81 + 90
207 := 08 + 19 + 25 + 37 − 46 + 50 + 64 + 72 + 81 + 93
218 := 01 + 17 + 28 − 39 + 45 + 56 + 60 + 74 + 83 + 92
208 := 07 + 19 − 25 − 38 + 46 + 51 + 63 + 74 + 80 + 92
219 := 07 + 19 + 25 − 38 + 46 + 52 + 63 + 74 + 81 + 90
209 := 07 − 19 − 25 − 38 + 46 + 50 + 63 + 74 + 81 + 92
220 := 07 + 19 + 25 − 38 + 46 + 52 + 63 + 74 + 80 + 91 .
210 := 05 − 17 − 28 − 39 + 41 + 56 + 60 + 73 + 84 + 92
211 := 07 + 19 − 25 − 38 + 46 + 50 + 63 + 74 + 81 + 92
Below are more examples,
3
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
11080 := 08 + 19 + 27 + 36 + 42 + 55 + 60 + 71 + 83 + 94
11081 := 08 − 19 + 26 + 37 + 44 + 51 + 65 + 70 + 82 + 93
11082 := 08 + 19 + 26 + 37 + 41 + 54 + 65 + 73 + 80 + 92
11083 := 08 + 19 + 26 + 37 + 44 + 51 + 65 + 70 + 82 + 93
11084 := 07 + 19 + 28 + 36 + 41 + 55 + 60 + 73 + 82 + 94
11085 := 08 + 19 + 26 + 37 + 44 + 50 + 65 + 71 + 82 + 93
11086 := 07 + 19 + 28 + 36 + 40 + 55 + 61 + 73 + 82 + 94
11087 := 06 + 19 − 28 + 37 + 42 + 54 + 65 + 70 + 81 + 93 .
The whole work is from 1 to 11111. For details refer [7].
Analysing the procedures given in sections 1.1 and 1.2 , we observe that in section 1.1 , all the 9 digits are used in
increasing and decresing ways to bring natural numbers, where each digit appears only once. In this case, the operations
used are, addition, subtraction, multiplication, division, potentiation, factorial and square-root. The section 1.2
works with representations of natural numbers written in a way that we use each digit twice, where bases and exponents
are of same digits with different permutations. Subsection 1.2.1 choose the digits from 1 to 9, according to necessity,
while subsection 1.2.2 works with all the 10 digits, i.e., 0 to 9, along with the operations of addition and subtraction.
1.3 Third Way: Single Digit Representations
In [1], author wrote natural numbers 1 to 1000 using single digit in each case. For example,
995 := (11 − 1)(1+1+1) − (11 − 1)/(1 + 1)
717 := (1 + 1)11 − 11(1+1+1)
:= 222 + 222 + 22/2
:= 22 + 2 × (222 + 2) + 2/2
:= 3(3+3) − 3 − 3 × 3
:= 3 × 333 − 3 − 3/3
:= 4 × (4 × 44 + 4) − 4 + 4/4
:= 4 × (44 − 4 − 4) + 4 − 4/4
:= (55 × (55 + 5 + 5) + 5 + 5)/5
:= 5 × (5 + 5) × (5 × 5 − 5) − 5
:= (6 × 6/(6 + 6))6 − 6 − 6
:= 666 + 6 × 66 − 66 − 6/6
:= 777 − 7 × 7 − 77/7
:= (7 + 7) × (77 − 7) + 7 + 7 + 7/7
:= 8 × 88 + (88 + 8 + 8)/8
:= 888 + 88 + 8 + 88/8
:= 9 × 9 × 9 − (99 + 9)/9.
:= 999 − (9 + 9 + 9 + 9)/9.
1000 := (11 − 1)(1+1+1)
786 := ((1 + 1 + 1)(1+1+1) + 1)(1+1) + 1 + 1
:= (22 + 2 + 2 + 2)2 + 2
:= 2 × (222 + 2(2+2) )
:= 33 × (33 − 3) − 3 − 3
:= (3 × 3 + 3/3)3
:= 4 × (4 × (44 + 4) + 4) + (4 + 4)/4
:= 4 × (44 − 4) − 4 − 4
:= 5 + (55 − 5/5)/(5 − 5/5)
:= 5 × (5 + 5) × (5 × 5 − 5)
:= 66 × (6 + 6) − 6
:= ((66 − 6)/6)(6×6/(6+6))
:= 777 + 7 + (7 + 7)/7
:= (7 + 7 + 7 − 7/7) × (7 × 7 + 7/7)
:= 8 × (88 + 8) + 8 + (88 − 8)/8
:= 888 + 88 + 8 + 8 + 8
:= 9 × 99 − 99 − 9 + (9 + 9 + 9)/9
:= 999 + 9/9.
4
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Values are calculated up to 1.000.000 (.txt file), but the work is written only from 0 to 1000. For details, refer Taneja
[8]. Recent extension to 20000 in four parts refer Taneja [9, 10, 11]
1.4 Forth Way: Single Letter Representations
We observe that the numbers written in previous section 1.3 are in terms of each digit, not necessarily symmetric. But
there are numbers, that can be written in a symmetric way, see examples below:
5=
11 − 1 22 − 2 33 − 3 44 − 4 55 − 5 66 − 6 77 − 7 88 − 8 99 − 9
=
=
=
=
=
=
=
=
.
1+1
2+2
3+3
4+4
5+5
6+6
7+7
8+8
9+9
6=
11 + 1 22 + 2 33 + 3 44 + 4 55 + 5 66 + 6 77 + 7 88 + 8 99 + 9
=
=
=
=
=
=
=
=
.
1+1
2+2
3+3
4+4
5+5
6+6
7+7
8+8
9+9
55 =
111 − 1 222 − 2 333 − 3 444 − 4 555 − 5 666 − 6 777 − 7 888 − 8 999 − 9
=
=
=
=
=
=
=
=
.
1+1
2+2
3+3
4+4
5+5
6+6
7+7
8+8
9+9
56 =
111 + 1 222 + 2 333 + 3 444 + 4 555 + 5 666 + 6 777 + 7 888 + 8 999 + 9
=
=
=
=
=
=
=
=
.
1+1
2+2
3+3
4+4
5+5
6+6
7+7
8+8
9+9
Motivated by this idea, instead working for each digit separately, we can work with a single letter "a", for example,
• Running-Type
5 := (aa − a)/(a + a)
1991 := (aaaaaa/aaa × (a + a) − aa)/a
6 := (aa + a)/(a + a)
2020 := (aaaaa − a)/aa × (a + a)/a
55 := (aaa − a)/(a + a)
2035 := (aaaa − a)/(a + a + a) × aa/(a + a)
56 := (aaa + a)/(a + a)
4477 := (aaa/(a + a + a) × aa × aa)/(a × a)
561 := (aaaa + aa)/(a + a)
4999 := (aaaaa − aaaa − a − a)/(a + a)
666 := aaa × (aa + a)/((a + a) × a)
5000 := (aaaaa − aaaa)/(a + a).
925 := (aaaaa − aa)/(aa + a)
1089 := (aaaa − aa − aa)/a
• Fraction-Type
aa − a
a+a
aa + a
6 :=
a+a
aaa − a
55 :=
a+a
aaa + a
56 :=
a+a
aaaa + aa
561 :=
a+a
aaa × (aa + a)
666 :=
(a + a) × a
(aa + a) × aa
− a) × (aa + a)
a
786 :=
(a + a) × a
aaaaa − aa
925 :=
aa + a
aaaa − aa − aa
1089 :=
a
aaaaaa
× (a + a) − aa
aaa
1991 :=
a
5 :=
(
5
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
aaaaa − a
× (a + a)
aa
2020 :=
a
aaaa − a
× aa
a
+
a
+
a
2035 :=
a+a
aaa
× aa × aa
a
+
a
+
a
4477 :=
a×a
(aaaaa − aaaa − a − a)
(a + a)
(aaaaa − aaaa)
5000 :=
(a + a)
(aaaa − a − a − a) × aaa
122988 :=
.
a×a
4999 :=
where a ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, and aa = 10 × a + a, aaa = 102 × a + 10 × a + a, etc.
The full work is from 1 to 11111 numbers, written in two different ways. One running type [15] and another in
fraction-type way [16]. For previous work refer [12, 13]. The summary of author’s work on recreation of numbers in
different situations refer [22].
1.5 Running Expressions
Previous subsections, works with natural numbers in different situations using 9 or 10 digits. In this section also we shall
do similar kind of work, but in little different way. It is based on the idea of subsection 1.1. We divide the numbers in
equal parts, two or three in such a way that the results are increasing and decreasing orders 1 to 9 or 9 to 1 or 9 to 0
separated by equalities, for example,
1234 = (5 + 67)/(8 × 9)
98/7 + 6 = 54/3 + 2 × 1.
Below are more examples, written in increasing and decreasing ways:
• Increasing Order
12 = 3 + 4 + (5 × 6 + 7 + 8)/9
123 = 4 + 5 + 6 × 7 + 8 × 9
1234 = −5 + 6! + 7 + 8
12 + 3 × 4 + 5 × (6 + 7) = 89
1 + 23 + 45 + 6! = 789
• Decreasing Order
6
p
9
(1)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
98 − 7 × (6 + 5) × (4 − 3) = 21
p
9 × 87 + 6 + 54 = 321
(2)
9 − 8 + 7! − 6 × 5! = 4321
9 − 8 + 7 − 6 + 5 + 4 − 3 + 2 = 10
9 × (8 + 7) + 6 + 5 + 43 = 210
(9 − 87 + 6!) × 5! /4! = 3210
98 = (7 + 6) × 5 + 4 × 3 + 21
987 = 6! + 5! + (4 + 3) × 21
98 = 7 + 65 + 4 + 32 − 10
987 = 6! + 54 + 3 + 210
Above examples give representations separated by equality sign having the digits in either increasing and/or decreasing orders. There are numbers that can be written in increasing as well as decreasing orders at the same time with single
or double equality signs, such as
p
16 := 12/3 × 4 = 5 + 6 + (7 + 8)/ 9
:= (9 + 87)/6 = 5 + 4 + 3 × 2 + 1
p
p
18 = 12 + 3!
= 4+5×6 = 7+8+ 9
p
p
= 9 + 8 + 7 = 6 × 54 = −3 + 21 = 3! + 2 + 10
p
120 := (1 × 2 + 3)!
= 4 × 5 × 6 = ((7 + 8)/ 9)!
p
:= (( 9)! − 8 + 7)! = 6 × 5 × 4 = (3 × 2 − 1)! = 3! × 2 × 10
(3)
The above three examples divide the numbers in two and three parts respectively with equality signs using the numbers in increasing as well as decreasing orders. From the examples (1), (2) and (3), we observe that the operations used
are addition, subtraction, multiplication, division, potentiation, factorial and square-root. More details can be seen
in [22, 18, 19]. In this work, our interest is to found examples similar to (1), (2) and (3), using Fibonacci sequence values.
1.5.1 Running Expressions with Fibonacci Sequence
Fibonacci sequence numbers are well known in literature. This sequence is defined as
F (0) = 0,
F (1) = 1,
F (n + 1) = F (n) + F (n − 1),
n ≥ 1.
Similar to (1) and (2), given above, below are examples of running expressions using Fibonacci sequence numbers.
Most of the results uses basic operations, except numbers 21 and 9876, where extra operation, such as factorial is used.
7
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
• Increasing Order
12 = F (3) × F (4) × F (5) + 6 − 7 − 8 − 9
(4)
123 = −4 × 5 × (6 − F (7)) − 8 − 9
1234 = 5 × F (6) × F (7) + F (8) × F (9)
1 + F (23 + F (4)) + (5 − 6)7 = 89
1 × 2 × 34 × 5 − F (F (6)) = 789
1 + 23 + F (4 × 5) = 6789.
• Decreasing Order
9 + (−F (8)/7 + 6) × 5 − F (4)! + 3 = 21
(5)
−98 − F (7) + F (6) × 54 = 321
(F (9) × F (8) + 7) × 6 − 5 = 4321
98 = (7 − 6) × 5 + F (4) × (32 − 1)
987 = (6 − 5) × F (4 × (3 + 2 − 1))
98 = −5 − 4 − 3 + 2 × F (10)
987 = (6 − 5)4 × F (3 × 2 + 10)
p
9876 = ( 5 + 4)! + F (F (3!) × 2) × 10
More details can be seen in Taneja [20].
1.5.2 Running Expressions with Triangular Numbers
Triangular numbers are very much famous in the literature of mathematics. These are given by
1, 3, 6, 10, 15, 21, ...
The general formula to write these numbers is given by
T (n) = 1 + 2 + 3 + ... =
n +1
= C (n + 1, 2)
2
The letter "C " represents as "binomial coefficient".
In this paper our aim is to bring running expressions by use of triangle numbers. This we have done in subsequent
sections. Due to high quantity of numbers, we the work is limited to 3 digits in case of single equality. As a part of results,
see below some interesting examples,
8
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
• Increasing Order
12 = T (3) − 4 − 5 + 6 − (7 − 8) × 9
(6)
123 = (−4 + 5) × 6 + T (7) + 89
1234 = T (56 × 7/8) + 9
1 + 2 + T (3) × 4 − 5 + 67 = 89
1 + 2 + T (3) + T (45 − 6) = 789
−1 − 2 + T (3) + T (−4 + T (T (5))) = 6789.
• Decreasing Order
9 × 8 − T (7) − T (6) + 5 − 4 − 3 = 21
(7)
T (9 + 8) − 7 × 6 + T (5 × 4) = 321
(−T (9) + T (T (8))) × 7 − T (6) − 5 = 4321
98 = (7 − 6) × 54 − T (32) + 1
987 = T (6) × (5 × T (4) − 3) × (2 − 1)
9876 = T (5 × T (4 + 3)) + T (2 + 1)
98 = (7 − 6) × T (5) − 4 + 32 + T (10)
(8)
987 = (6 − 5) × 4 × (T (T (T (3))) + 2) + T (10)
9876 = (−5 + T (T (T (4)))) × T (3) + T (T (−2 + 10))
9 × 8 − T (7) − T (6) − T (5) − 4 + 3 × 2 = 10
T (9) + 87 × (6 − 5) + T (4 × 3) = 210
T (9) + 8 + 7 + T (6) × T (5) × T (4) = 3210
More details can be seen in Taneja [21].
2 Single Digit Representations From 1 to 20000
The whole work brings numbers 1 to 10000 written in terms of single digits. Since, it is not possible to put all the numbers
in single work, we divided it in four parts as given below:
• Part I : From 0001 to 5000;
• Part II : From 5001 to 10000 [9];
• Part III : From 10001 to 15000 [10];
• Part IV : From 15000 to 20000 [11].
9
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
This paper brings first part giving single digit representations of natural numbers from 1 to 5000. For other parts
refer [9, 10, 11].
Remark 2.1. Due to high quantity of numbers the there so many extra brackets. After simplifications, these unnecessary
brackets can be removed easily.
2.1 Single Digit Representation: 1-5000
This subsection bring the first part of the whole project. Here, the numbers are represented from 1 to 5000 in terms of
different digits. The work done previously up to 1000 [6] is also repeated here.
Ï 1 := 1
Ï 9 := 11 − 1 − 1
Ï 5 := 1 + 1 + 1 + 1 + 1
:= 2/2
:= 2 + 2/2 + 2
:= (2/2 + 2)2
:= 3/3
:= 3 + 3 − 3/3
:= 3 × 3
:= 4/4
:= 4 + 4/4
:= 4 + 4 + 4/4
:= 5/5
:= 5
:= 5 + 5 − 5/5
:= 6/6
:= 6 − 6/6
:= 6 + 6 × 6/(6 + 6)
:= 7/7
:= 7 − (7 + 7)/7
:= 7 + (7 + 7)/7
:= 8/8
:= 8 + 8 − 88/8
:= 8 + 8/8
:= 9/9
:= (9 × 9 + 9)/(9 + 9)
:= 9
Ï 2 := 1 + 1
Ï 6 := (1 + 1) × (1 + 1 + 1)
Ï 10 := 11 − 1
:= 2
:= 2 + 2 + 2
:= 2 + 2 × (2 + 2)
:= 3 − 3/3
:= 3 + 3
:= 3 × 3 + 3/3
:= (4 + 4)/4
:= 4 + (4 + 4)/4
:= (44 − 4)/4
:= (5 + 5)/5
:= 5 + 5/5
:= 5 + 5
:= (6 + 6)/6
:= 6
:= (66 − 6)/6
:= (7 + 7)/7
:= 7 − 7/7
:= (77 − 7)/7
:= (8 + 8)/8
:= 8 − (8 + 8)/8
:= 8 + (8 + 8)/8
:= (9 + 9)/9
:= 9 − (9 + 9 + 9)/9
:= 9 + 9/9
Ï 3 := 1 + 1 + 1
Ï 7 := 1 + (1 + 1) × (1 + 1 + 1)
Ï 11 := 11
:= 2 + 2/2
:= 2 + 2 + 2 + 2/2
:= 22/2
:= 3
:= 3 + 3 + 3/3
:= 33/3
:= 4 − 4/4
:= 4 + 4 − 4/4
:= 44/4
:= 5 − (5 + 5)/5
:= 5 + (5 + 5)/5
:= 55/5
:= 6 × 6/(6 + 6)
:= 6 + 6/6
:= 66/6
:= (7 + 7 + 7)/7
:= 7
:= 77/7
:= 88/8 − 8
:= 8 − 8/8
:= 88/8
:= (9 + 9 + 9)/9
:= 9 − (9 + 9)/9
:= 99/9
Ï 4 := 1 + 1 + 1 + 1
Ï 8 := (1 + 1)1+1+1
Ï 12 := 1 + 11
:= 2 + 2
:= 2 × (2 + 2)
:= 2 × (2 + 2 + 2)
:= 3 + 3/3
:= 3 × 3 − 3/3
:= 3 + 3 × 3
:= 4
:= 4 + 4
:= 4 + 4 + 4
:= 5 − 5/5
:= 5 + 5 − (5 + 5)/5
:= (55 + 5)/5
:= 6 − (6 + 6)/6
:= 6 + (6 + 6)/6
:= 6 + 6
:= 77/7 − 7
:= 7 + 7/7
:= (77 + 7)/7
:= 8 × 8/(8 + 8)
:= 8
:= (88 + 8)/8
:= (9 × 9 − 9)/(9 + 9)
:= 9 − 9/9
:= (99 + 9)/9
10
Inder J. Taneja
Ï 13 := 1 + 1 + 11
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 18 := (1 + 1) × (11 − 1 − 1)
2+2
Ï 23 := 1 + 11 + 11
:= 22 + 2/2
:= 2 + 22/2
:= 2 + 2
:= 3 + 3 × 3 + 3/3
:= 3 × (3 + 3)
:= 33 − 3 − 3/3
:= 4 + 4/4 + 4 + 4
:= 4 × 4 + (4 + 4)/4
:= 4 + 4 + 4 + 44/4
:= (55 + 5 + 5)/5
:= 5 + (55 + 5 + 5)/5
:= 5 × 5 − (5 + 5)/5
:= 6 + 6/6 + 6
:= 6 + 6 + 6
:= 6 + 6 + 66/6
:= 7 + 7 − 7/7
:= 7 + 77/7
:= 7 + 7 + 7 + (7 + 7)/7
:= (88 + 8 + 8)/8
:= 8 + 8 + (8 + 8)/8
:= 8 + 8 + 8 − 8/8
:= (99 + 9 + 9)/9
:= 9 + 9
:= (99 + 99 + 9)/9
Ï 14 := 1 + 1 + 1 + 11
2+2
Ï 19 := (1 + 1) × (11 − 1) − 1
Ï 24 := (1 + 1) × (1 + 11)
:= 22 − 2/2 − 2
:= 2 + 22
:= 3 + 33/3
:= 3/3 + 3 × (3 + 3)
:= 33 − 3
:= 4 + (44 − 4)/4
:= 2
−2
:= 4 + 4 + 44/4
:= 4 + (4 × 4 + 4)
:= 5 + 5 − 5/5 + 5
:= 5 × 5 − 5 − 5/5
:= 5 × 5 − 5/5
:= 6 + 6 + (6 + 6)/6
:= 6 + 6/6 + 6 + 6
:= 6 + (6 + 6 + 6)
:= 7 + 7
:= 7 + (7 + 77)/7
:= 7 + ((77 − 7)/7 + 7)
:= 8 + 8 − (8 + 8)/8
:= 8 + 88/8
:= 8 + (8 + 8)
:= 9 + (9 × 9 + 9)/(9 + 9)
:= 9 + 9/9 + 9
:= 9 + ((9 − ((9 + 9 + 9)/9)) + 9)
Ï 15 := 1 + 1 + 1 + 1 + 11
Ï 25 := 1 + (1 + 1) × (1 + 11)
Ï 20 := (1 + 1) × (11 − 1)
:= 2 + 2 + 22/2
:= 22 − 2
:= 2 + 22 + 2/2
:= 3 + 3 + 3 × 3
:= 3 × 3 + 33/3
:= 33 − 3 + 3/3
:= 4 + 44/4
:= 4 + 4 × 4
:= 4 + 4 + 4 × 4 + 4/4
:= 5 + 5 + 5
:= 5 × 5 − 5
:= 5 × 5
:= 6 + 6 + 6 × 6/(6 + 6)
:= 6 + ((6 + 6)/6 + 6 + 6)
:= 6 × 6 − 66/6
:= 7 + 7 + 7/7
:= 7 + 7 + 7 − 7/7
:= 7 + 7 + 77/7
:= 8 + 8 − 8/8
:= 8 + (88 + 8)/8
:= 8 + 8 + 8 + 8/8
:= 9 + 9 − (9 + 9 + 9)/9
:= 9 + 99/9
:= 9 + 9 + 9 − (9 + 9)/9
Ï 16 := (1 + 1)1+1+1+1
:= 22+2
Ï 26 := (1 + 1) × (1 + 1 + 11)
Ï 21 := 11 + 11 − 1
:= 22 − 2/2
:= 2 + 22 + 2
:= 3 − 33/3
:= 3 + 3 × (3 + 3)
:= 33 − 3/3
:= 4 × 4
:= 4 + 4 × 4 + 4/4
:= 4 + 44 × 4/(4 + 4)
:= 5 + 55/5
:= 5 + 55/5 + 5
:= 5 × 5 + 5/5
:= 6 + (66 − 6)/6
:= 6 × (6 × 6 + 6)/(6 + 6)
:= 6 × 6 + (6 − 66)/6
:= 7 + 7 + (7 + 7)/7
:= 7 + 7 + 7
:= 7 + 7 + (77 + 7)/7
:= 8 + 8
:= 8 + (88 + 8 + 8)/8
:= 8 + 8 + 8 + (8 + 8)/8
:= 9 + 9 − (9 + 9)/9
:= 9 + (99 + 9)/9
:= 9 + 9 + 9 − 9/9
3
Ï 17 := 1 + (1 + 1)1+1+1+1
Ï 27 := (1 + 1 + 1)1+1+1
Ï 22 := 11 + 11
:= 2/2 + 22+2
:= 22
:= 2 + 22 + 2/2 + 2
:= 3 + 33/3 + 3
:= 33 − 33/3
:= 33
:= 4 × 4 + 4/4
:= 44 × 4/(4 + 4)
:= 4 × 4 + 44/4
:= 5 + (55 + 5)/5
:= (55 + 55)/5
:= 5 × 5 + (5 + 5)/5
:= 6 + 66/6
:= (66 + 66)/6
:= 66 × 6/(6 + 6) − 6
:= 7 + (77 − 7)/7
:= 7 + 7 + 7 + 7/7
:= 7 + 7 + 7 + 7 − 7/7
:= 8 + 8 + 8/8
:= (88 + 88)/8
:= 8 + 8 + 88/8
:= 9 + 9 − 9/9
:= (99 + 99)/9
:= 9 + 9 + 9
11
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 28 := 1 + (1 + 1 + 1)1+1+1
Ï 33 := 11 × (1 + 1 + 1)
:= 2 + 22 + 2 + 2
:= 22 + 22/2
:= 2 + (2 + 2 + 2)2
:= 3 + 3/3
:= 33
:= 33 + 33/3
:= 44 − 4 × 4
:= 4/4 + 4 × (4 + 4)
3
:= 5 + 5 × 5 − (5 + 5)/5
Ï 38 := 1 + 111/(1 + 1 + 1)
:= 44 − 4 − (4 + 4)/4
5
:= 5/5 + ((5 + 5)/5)
:= 5 + 5/5 + ((5 + 5)/5)5
:= 6 + (66 + 66)/6
:= 66 × 6/(6 + 6)
:= 6 × 6 + (6 + 6)/6
:= 7 + 7 + 7 + 7
:= 7 + 7 + 7 + (77 + 7)/7
:= 7 × 7 − 77/7
:= 8 + 8 + (88 + 8)/8
:= 8 + 8 + 8 + 8 + 8/8
:= 8 + 8 + (88 + 88)/8
:= 9 + 9 + 9 + 9/9
:= 99 × 9/(9 + 9 + 9)
:= 9 + 9 + 9 + 99/9
Ï 29 := (11 − 1) × (1 + 1 + 1) − 1
Ï 34 := 1 + 11 × (1 + 1 + 1)
Ï 39 := (1 + 1 + 1) × (1 + 1 + 11)
:= 2 + 2 + 2 + 22 + 2/2
:= 2 + 2 × 22+2
:= 2 × (22 − 2) − 2/2
:= 3 + 33 − 3/3
:= 3/3 + 33
:= 3 + 3 + 33
:= 44 + 4/4 − 4 × 4
:= 44 + (4 − 44)/4
:= 44 − 4 − 4/4
:= 5 + 5 × 5 − 5/5
:= 5 + 5 + 5 × 5 − 5/5
:= 55 − 5 − 55/5
:= 6 × 6 − 6 − 6/6
:= 6 × 6 − (6 + 6)/6
:= 6 + 66 × 6/(6 + 6)
:= 7 + 7 + 7 + 7 + 7/7
:= 7 × 7 − 7 − 7 − 7/7
:= 7 × 7 + (7 − 77)/7
:= 8 + 8 + (88 + 8 + 8)/8
:= 8 + 8 + 8 + 8 + (8 + 8)/8
:= 8 + 8 + 8 + 8 + 8 − 8/8
:= 9 + 9 + 99/9
:= 9 + 9 + 9 + 9 − (9 + 9)/9
:= 9 + 9 + 9 + (99 + 9)/9
Ï 30 := (11 − 1) × (1 + 1 + 1)
:= 22 + 2 × (2 + 2)
3
Ï 35 := 1 + 1 + 11 × (1 + 1 + 1)
Ï 40 := (1 + 1) × (1 + 1) × (11 − 1)
:= 2 + 22/2 + 22
:= 2 × (22 − 2)
:= 3 + 33 − 3/3
:= 3 + 3 + 33 + 3/3
:= 4 × (4 + 4) − (4 + 4)/4
:= 4 + 4 × (4 + 4) − 4/4
:= 44 − 4
:= 5 + 5 × 5
:= 5 + 5 + 5 × 5
:= 5 + 5 + 5 + 5 × 5
:= 6 × 6 − 6
:= 6 × 6 − 6/6
:= 6 + 6 × 6 − (6 + 6)/6
:= 7 × 7 − 7 − (7 + 7)/7
:= 3 + 3
:= 7 + 7 + 7 + 7 + (7 + 7)/7
:= 7 × 7 − 7 − 7
:= 8 + (88 + 88)/8
:= 8 + 8 + 8 + 88/8
:= 8 + 8 + 8 + 8 + 8
:= 9 + 9 + (99 + 9)/9
:= 9 + 9 + 9 + 9 − 9/9
:= (9 × 9 × 9 − 9)/(9 + 9)
Ï 31 := 1 + (11 − 1) × (1 + 1 + 1)
:= 22 + (2/2 + 2)
2
3
Ï 36 := (1 + 1 + 1) × (1 + 11)
:= (2 + 2 + 2)
2
Ï 41 := 1 + (1 + 1) × (1 + 1) × (11 − 1)
:= 2/2 + 2 × (22 − 2)
:= 3 + 3 + 3/3
:= 3 + 33
:= 3 + 33/3 + 33
:= 4 × (4 + 4) − 4/4
:= 4 + 4 × (4 + 4)
:= 44 − 4 + 4/4
:= 5 + 5 × 5 + 5/5
:= 5 × 5 + 55/5
:= 5 + 5 × 5 + 55/5
:= 6 × 6 + 6/6 − 6
:= 6 × 6
:= 6 + 6 × 6 − 6/6
:= 7 × 7 − 7 − 77/7
:= 7/7 + (7 × 7 − (7 + 7))
:= 7 × 7 − 7 − 7/7
:= 8 + 8 + 8 + 8 − 8/8
:= (8 × 8 + 8) × 8/(8 + 8)
:= 8 + 8 + 8 + 8 + 8 + 8/8
:= 9 + (99 + 99)/9
:= 9 + 9 + 9 + 9
:= (9 × 9 × 9 + 9)/(9 + 9)
Ï 32 := 11 × (1 + 1 + 1) − 1
:= 2 × 2
2+2
:= 33 − 3/3
Ï 37 := 111/(1 + 1 + 1)
Ï 42 := (1 + 1) × (11 + 11 − 1)
:= 2/2 + (2 + 2 + 2)2
:= 2 × 22 − 2
:= 3 + 33 + 3/3
:= 3 × 3 + 33
:= 4 + 4 × (4 + 4) + 4/4
:= 4 × (4 + 4)
5
:= 44 − (4 + 4)/4
:= 5 + ((5 + 5)/5)5 + 5
5
:= ((5 + 5)/5)
:= 5 + ((5 + 5)/5)
:= 6 × 6 + (6 + 6)/6 − 6
:= 6 × 6 + 6/6
:= 6 + 6 × 6
:= 7 + 77/7 + 7 + 7
:= 7 × 7 − (77 + 7)/7
:= 7 × 7 − 7
:= 8 + 8 + 8 + 8
:= 888/(8 + 8 + 8)
:= 8 × 8 − (88 + 88)/8
:= 9 + (99 + 99 + 9)/9
:= 9 + 9 + 9 + 9 + 9/9
:= 9 + 99 × 9/(9 + 9 + 9)
12
Inder J. Taneja
Ï 43 := (1 + 1) × (11 + 11) − 1
:= 2 × 22 − 2/2
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 48 := (1 + 1) × (1 + 1) × (1 + 11)
Ï 53 := (111 − 1)/(1 + 1) − 1 − 1
:= 2/2 + 2 × (22 + 2 + 2)
:= 2 × (22 + 2)
:= 3 × 3 + 3/3 + 33
:= 3 × 3 − 33
:= 33 + 33 − 3/3
:= 44 − 4/4
:= 4 + 44
:= (44 − 44)/4
:= 55 − (55 + 5)/5
3
:= 55 + 5 − (5 + 5)/5
:= 55 − (5 + 5)/5
:= 6 + 6 × 6 + 6/6
:= 6 + 6 + 6 × 6
:= 6 + 6 × 6 + 66/6
:= 7 × 7 + 7/7 − 7
:= 7 × 7 − 7/7
:= 7 × 7 − 7 + 77/7
:= 8 + 8 + 8 + 8 + 88/8
:= 8 × 8 − 8 − 8
:= 8 × 8 − 88/8
:= 9 + 9 + 9 + 9 + 9 − (9 + 9)/9
:= 9 + 9 + 9 + 9 + (99 + 9)/9
:= 9 × 9 − 9 − 9 − 9 − 9/9
Ï 44 := (1 + 1) × (11 + 11)
:= 2 × 22
Ï 49 := 1 + (1 + 1) × (1 + 1) × (1 + 11)
:= 2/2 + 2 × (22 + 2)
Ï 54 := (111 − 1)/(1 + 1) − 1
:= 2 + 2 × (22 + 2 + 2)
3
:= 33 + 33/3
:= 3 + 33 − 33/3
:= 3 × 3 × (3 + 3)
:= 44
:= 4 + 44 + 4/4
:= 44 + (44 − 4)/4
:= 55 − 55/5
:= 55 − 5 − 5/5
:= 55 − 5/5
:= 6 + 6 × 6 + (6 + 6)/6
:= 6 + 6 + 6 × 6 + 6/6
:= 66 − 6 − 6
:= 7 × 7 − 7 + (7 + 7)/7
:= 7 × 7
:= 7 + 7 × 7 − (7 + 7)/7
:= 88 × 8/(8 + 8)
:= 8 × 8 − 8 − 8 + 8/8
:= 8 × 8 + (8 − 88)/8
:= 9 + 9 + 9 + 9 + 9 − 9/9
:= (9 × 99 − 9)/(9 + 9)
:= 9 × 9 − 9 − 9 − 9
Ï 45 := 1 + (1 + 1) × (11 + 11)
Ï 50 := (11 − 1)1+1 /(1 + 1)
Ï 55 := (111 − 1)/(1 + 1)
:= 2 × 22 + 2/2
:= 2 + 2 × (22 + 2)
:= 22/2 + 2 × 22
:= 3 + 3 × 3 + 33
:= 3 + 3 + 33 + 33/3
:= 33 + 33 + 3/3
:= 44 + 4/4
:= 4 + 44 + (4 + 4)/4
:= 44 + 44/4
:= 55 − 5 − 5
:= 5 × (5 + 5)
:= 55
:= 666/6 − 66
:= 6 + 6 + 6 × 6 + (6 + 6)/6
:= 66 − 66/6
:= 7 + 7 × 7 − 77/7
:= 7 × 7 + 7/7
:= 7 + 7 × 7 − 7/7
:= 8 × 8 − 8 − 88/8
:= 8 × 8 − 8 − 8 + (8 + 8)/8
:= 8 × 8 − 8 − 8/8
:= 9 + 9 + 9 + 9 + 9
:= (9 × 99 + 9)/(9 + 9)
:= (999 − 9)/(9 + 9)
Ï 46 := (1 + 1) × (1 + 11 + 11)
:= 2 + 2 × 22
Ï 51 := 1 + (11 − 1)1+1 /(1 + 1)
:= 2 + 2 × (22 + 2) + 2/2
3
3
Ï 56 := (1 + 111)/(1 + 1)
:= 2 × (22 + 2 + 2 + 2)
:= 3 + 3 × 3 + 33 + 3/3
:= 3 + 3 − 3
:= (333 + 3)/(3 + 3)
:= 44 + (4 + 4)/4
:= 4 + 4 + 44 − 4/4
:= 4 + 4 + 4 + 44
:= 55 + 5/5 − 5 − 5
:= 55 + 5/5 − 5
:= 55 + 5/5
:= 6 × 6 + (66 − 6)/6
:= 6 − 66 + 666/6
:= 66 + (6 − 66)/6
:= 7 × 7 − (7 + 7 + 7)/7
:= 7 × 7 + (7 + 7)/7
:= 7 + 7 × 7
:= 8 × 8 − 8 − (88 − 8)/8
:= 8 × 8 − (88 + 8 + 8)/8
:= 8 × 8 − 8
:= 9 + 9 + 9 + 9 + 9 + 9/9
:= 9 × (9 + 9) − 999/9
:= (999 + 9)/(9 + 9)
Ï 47 := 1 + (1 + 1) × (1 + 11 + 11)
:= 2 + 2 × 22 + 2/2
Ï 52 := (1 + 1) × (1 + 1) × (1 + 1 + 11)
:= 2 + 22/2 + 2 × 22
:= 2 × (22 + 2 + 2)
3
Ï 57 := 1 + (1 + 111)/(1 + 1)
:= 3 + 33 + 33
3
:= 3 + 33/3 + 33
:= 3 + 3/3 − 3 + 3
:= 4 + 44 − 4/4
:= 4 + 4 + 44
:= 5 + 5 + 5 + ((5 + 5)/5)5
:= 55 − 5 + (5 + 5)/5
:= 4 + (44 − 44)/4
:= 55 + (5 + 5)/5
6
:= 6 × 6 + 66/6
:= ((6 + 6)/6) − 6 − 6
:= 66 + (6 + 6 − 66)/6
:= 7 × 7 − (7 + 7)/7
:= 7 × 7 + (7 + 7 + 7)/7
:= 7 + 7/7 + 7 × 7
:= 8 × 8 − 8 − 8 − 8/8
:= 8 + 88 × 8/(8 + 8)
:= 8/8 + 8 × 8 − 8
:= 9 + 9 + 9 + 9 + 99/9
:= 9 × 9 − 9 − 9 − 99/9
:= (((9 + 9)/9)9 + 9/9)/9
13
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 58 := 1 + 1 + (1 + 111)/(1 + 1)
:= 22 + (2 + 2 + 2)
2
Ï 63 := (1 + 1 + 1) × (11 + 11 − 1)
:= 2
2+2+2
Ï 68 := (1 + 1) × (1 + 11 × (1 + 1 + 1))
:= 2 + 2 + 22+2+2
− 2/2
:= (3/3 + 3)3 − 3 − 3
:= 3 + 33 + 33
:= ((3 + 3)3 − 3)/3 − 3
:= (44 − 4 − 4)/4 − 4
:= (44 − 4)/4
:= 4 + 4 × 4 × 4
:= 5 + 55 − (5 + 5)/5
:= (55 /5 + 5)/(5 + 5)
:= 5 + (55 /5 + 5)/(5 + 5)
:= ((6 + 6)/6)6 − 6
:= 66 − 6 × 6/(6 + 6)
:= 66 + (6 + 6)/6
:= 7 + 7 × 7 + (7 + 7)/7
:= 7 + 7 + 7 × 7
:= 77 − 7 − (7 + 7)/7
:= 8 × 8 − 8 + (8 + 8)/8
:= 8 × 8 − 8/8
:= 8 × 8 + 8 × 8/(8 + 8)
:= 9 + (9 × 99 − 9)/(9 + 9)
:= 9 × 9 − 9 − 9
:= 9 × 9 − (99 + 9 + 9)/9
Ï 59 := (111+1 − 1)/(1 + 1) − 1
Ï 64 := (1 + 1)(1+1)×(1+1+1)
Ï 69 := (1 + 1 + 1) × (1 + 11 + 11)
:= 22/2 + 2 × (22 + 2)
:= 22+2+2
:= (2/2 + 2) × (22 + 2/2)
:= 33 + 33 − 3/3
:= (3/3 + 3)3
:= 3 + 33 + 33
:= (44 − 4)/4 − 4
:= 4 × 4 × 4
:= 4 + (44 + 4)/4
:= 5 + 55 − 5/5
:= 5 + 5 + 55 − 5/5
:= 5 + 5 + 5 + 55 − 5/5
:= 66 − 6/6 − 6
:= ((6 + 6)/6)
:= 66 + 6 × 6/(6 + 6)
:= 7 × 7 + (77 − 7)/7
:= 7 + 7 + 7 × 7 + 7/7
:= 77 − 7 − 7/7
:= 8 × 8 + 88/8 − 8 − 8
:= 8 × 8
:= 88 − 8 − 88/8
:= 9 + (9 × 99 + 9)/(9 + 9)
:= 9/9 + 9 × 9 − 9 − 9
:= 9 × 9 − (99 + 9)/9
6
Ï 60 := (111+1 − 1)/(1 + 1)
Ï 65 := 1 + (1 + 1)(1+1)×(1+1+1)
Ï 70 := (11 − 1 − 1)1+1 − 11
:= 2 × (2 × (2 + 2) + 22)
:= 2/2 + 22+2+2
:= 22 + 2 × (22 + 2)
:= 33 + 33
:= (3/3 + 3)3 + 3/3
:= 3 + (((3/3 + 3)3 ) + 3)
:= 4 × 4 + 44
:= (44 + 4)/4
:= 4 + (((44 + 4) + 4)/4)
:= 5 + 55
:= 5 + 5 + 55
:= 5 + (55 + 5 + 5)
:= 66 − 6
:= 66 − 6/6
:= 6 + ((6 + 6)/6)6
:= 7 × 7 + 77/7
:= 77 − (77 + 7)/7
:= 77 − 7
:= 8 × 8 − 8 × 8/(8 + 8)
:= 8/8 + 8 × 8
:= 8 + 8 × 8 − (8 + 8)/8
:= 9 × 9 − 9 − (99 + 9)/9
:= 9 + (999 + 9)/(9 + 9)
:= 9 × 9 − 99/9
Ï 61 := (1 + 111+1 )/(1 + 1)
2+2+2
Ï 66 := (1 + 1) × 11 × (1 + 1 + 1)
Ï 71 := (1 + 11)1+1 /(1 + 1) − 1
:= 2 + 22+2+2
:= 2 × (2 + 2 + 2)2 − 2/2
:= (3/3 + 3)3 − 3
:= 33 + 33
:= ((3 + 3)3 − 3)/3
:= (44 + 4)/4 − 4
:= (44 + 4 + 4)/4
:= 4 + (44 − 4)/4 + 4
:= 5 + 55 + 5/5
:= 55 + 55/5
:= 5 + 55 + 55/5
:= 66 + 6/6 − 6
:= 66
:= 6 + 66 − 6/6
:= 7 × 7 + (77 + 7)/7
:= 77 − 77/7
:= 7/7 + 77 − 7
:= 2
− 2/2 − 2
:= 8 + 8 × 8 − 88/8
:= 8 × 8 + (8 + 8)/8
:= 8 + 8 × 8 − 8/8
:= 9 × 9 − 9 − 99/9
:= (9 + 9) × 99/(9 + 9 + 9)
:= 9 × 9 − 9/9 − 9
Ï 62 := 1 + (1 + 111+1 )/(1 + 1)
Ï 67 := 1 + (1 + 1) × 11 × (1 + 1 + 1)
Ï 72 := (1 + 11)1+1 /(1 + 1)
:= 22+2+2 − 2
:= 3 + 33 − 3/3 + 3
:= 2 + 22+2+2 + 2/2
3
4
:= 3 + (3/3 + 3)
:= 2 × (2 + 2 + 2)2
3
:= 3 × (33 − 3)
:= (4 − 4 − 4)/4
:= 4 + (44 − 4)/4
:= 4 + 4 + 4 × 4 × 4
:= 5 + 55 + (5 + 5)/5
:= 55 + (55 + 5)/5
:= 5 + 55 + (55 + 5)/5
:= 66 − 6 + (6 + 6)/6
:= 66 + 6/6
:= 6 + 66
:= 7 + 7 + 7 × 7 − 7/7
:= 7 + 7 × 7 + 77/7
:= 77 + (7 + 7)/7 − 7
:= 8 × 8 − (8 + 8)/8
:= 8 × 8 − 8 + 88/8
:= 8 + 8 × 8
:= 9 × 9 − 9 − 9 − 9/9
:= 9 + 9 + (9 × 99 − 9)/(9 + 9)
:= 9 × 9 − 9
14
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 73 := 1 + (1 + 11)1+1 /(1 + 1)
:= 2/2 + 2 × (2 + 2 + 2)2
Ï 78 := 111 − 11 × (1 + 1 + 1)
:= 2 + (2/2 + 2)2+2
:= 2 × 2 × (22 − 2) − 2
3
:= ((3 + 3)3 + 3)/3
Ï 83 := 1 + 1 + (11 − 1 − 1)1+1
:= 3 + 3 × 33 − 3/3
:= 3 × 3 − 3
4
:= 4 + (44 + 44 )/4 + 4
:= 4 + (44 + 4)/4 + 4
:= 4 + (4 − 4 + 44)/4
:= 5 × (5 + 5 + 5) − (5 + 5)/5
:= 5 × 5 + 55 − (5 + 5)/5
:= 5 + 55 + 5 × 5 − (5 + 5)/5
:= 6 + 66 + 6/6
:= 6 + 66 + 6
:= 6 + 66/6 + 66
:= 7 + 77 − 77/7
:= 7/7 + 77
:= 7 + 77 − 7/7
:= 8 + 8/8 + 8 × 8
:= 88 + (8 − 88)/8
:= 8 + 8 × 8 + 88/8
:= 9/9 + 9 × 9 − 9
:= 9 × 9 − (9 + 9 + 9)/9
:= 9 × 9 + (9 + 9)/9
Ï 74 := (1 + 1) × 111/(1 + 1 + 1)
:= 2 + 2 × (2 + 2 + 2)
2
Ï 79 := (11 − 1 − 1)1+1 − 1 − 1
:= (2/2 + 2)
2+2
−2
Ï 84 := 1 + 1 + 1 + (11 − 1 − 1)1+1
:= 2 × (2 × 22 − 2)
3
:= 3 + ((3 + 3) − 3)/3
:= 3 × 3 + 3/3 − 3
:= 3 + 3 × 33
:= (44 + 44 − 4)/4
:= 4 + (44 + 44 )/4
:= 4 + 4 × (4 × 4 + 4)
:= 5 × (5 + 5 + 5) − 5/5
:= 5 × 5 + 55 − 5/5
:= 5 + (55 − 5/5) + 5 × 5
:= 6 + 66 + (6 + 6)/6
:= 6 + 6 + 66 + 6/6
:= 6 + 66 + 6 + 6
:= 77 − (7 + 7 + 7)/7
:= 77 + (7 + 7)/7
:= 7 + 77
:= 8 × 8 + 8 + (8 + 8)/8
:= 88 − 8 − 8/8
:= 88 − 8 × 8/(8 + 8)
:= 9 × 9 − 9 + (9 + 9)/9
:= 9 × 9 − (9 + 9)/9
:= 9 × 9 + (9 + 9 + 9)/9
3
Ï 75 := 1 + (1 + 1) × 111/(1 + 1 + 1)
Ï 80 := (11 − 1 − 1)1+1 − 1
Ï 85 := 111 − (1 + 1) × (1 + 1 + 11)
:= 2 × 2 × (22 − 2)
:= 2 + 2 + (2/2 + 2)2+2
:= 3 + 3 × (33 − 3)
:= 3 × 33 − 3/3
:= 3 + 3 × 33 + 3/3
:= (44 + 44 )/4
:= 4 × (4 × 4 + 4)
:= 4 + (4 − 4/4)4
:= 5 × (5 + 5 + 5)
:= 5 × 5 + 55
:= 5 + 5 × 5 + 55
:= 666/6 − 6 × 6
:= 6 + 6 + 66 + (6 + 6)/6
:= 6 + 6 + 6 + 66 + 6/6
:= 77 − (7 + 7)/7
:= 77 + (7 + 7 + 7)/7
:= 7 + 77 + 7/7
:= 8 × 8 + 88/8
:= 88 − 8
:= 8 + 88 − 88/8
:= 9 × 9 + (9 + 9 + 9)/9 − 9
:= 9 × 9 − 9/9
:= 9 × 9 + (9 × 9 − 9)/(9 + 9)
:= 22/2 + 2
2+2+2
Ï 76 := (1 + 1) × (1 + 111/(1 + 1 + 1))
2
:= 2 × ((2 + 2 + 2) + 2)
3
:= 3 + ((3 + 3) + 3)/3
Ï 81 := (11 − 1 − 1)1+1
Ï 86 := (1 + 1) × ((1 + 1) × (11 + 11) − 1)
:= (2/2 + 2)2+2
:= 2 × 2 × 22 − 2
:= 3 × 33
:= 3 + 3 + 3 × 33 − 3/3
4
:= 44 + 4 × (4 + 4)
:= (4 − 4/4)
:= 4 + (4 − 4/4)4 + 4/4
:= 5/5 + 5 × (5 + 5 + 5)
:= 5 × 5 + 55 + 5/5
:= 555/5 − 5 × 5
:= 6 + 6 + ((6 + 6)/6)6
:= 6 + 666/6 − 6 × 6
:= 6 + 6 + 6 + 66 + (6 + 6)/6
:= 77 − 7/7
:= 77 − 7 + 77/7
:= 7 + (7 + 7)/7 + 77
:= 8 × 8 + (88 + 8)/8
:= 8/8 + 88 − 8
:= 88 − (8 + 8)/8
:= 9 × 9 + (9 − 99)/(9 + 9)
:= 9 × 9
:= 9 × 9 + (9 × 9 + 9)/(9 + 9)
Ï 77 := 11 × (1 + (1 + 1) × (1 + 1 + 1))
:= 2 × 2 × 22 − 22/2
3
Ï 82 := 1 + (11 − 1 − 1)1+1
Ï 87 := 111 − (1 + 1) × (1 + 11)
:= 2 + 2 × 2 × (22 − 2)
:= 2 × 2 × 22 − 2/2
3
:= 3 + 3 × 33 + 3
:= 3 × 3 − 3/3 − 3
:= 3/3 + 3 × 3
:= (4 − 4/4)4 − 4
:= 4/4 + (4 − 4/4)4
:= 44 + 44 − 4/4
:= 55 + (55 + 55)/5
:= 5 × 5 + 55 + (5 + 5)/5
:= 55 + ((5 + 5)/5)5
:= 66 + 66/6
:= 6 + 6 + 6 + ((6 + 6)/6)6
:= 6 + 666/6 − 6 × 6 + 6
:= 77
:= 7 + 77 − (7 + 7)/7
:= 77 + (77 − 7)/7
:= 88 − 88/8
:= 88 − 8 + (8 + 8)/8
:= 88 − 8/8
:= 9 × 9 + (9 − 9 × 9)/(9 + 9)
:= 9/9 + 9 × 9
:= 99 − (99 + 9)/9
15
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 93 := ((1 + 1)11−1 − 1)/11
Ï 88 := 11 × (1 + 1)1+1+1
Ï 98 := 111 − 11 − 1 − 1
:= 2 + 2 × 2 × (22 + 2)
:= 2/2 + 2 × (2 × 22 + 2)
:= 2 × 2 × 22
3
:= 3 × 33 − 33/3
:= 3 × 33 − 3/3
:= 3 + 3 × (3 + 3)
:= 44 + 44
:= 4 + (444 − 4)/4 − 4 × 4
4
:= ((4 + 4) − 4)/44
5
:= 5 × 5 × 5 − ((5 + 5)/5)5
:= 5 × (5 × 5 − 5) − (5 + 5)/5
:= 66 + (66 + 66)/6
:= 666/6 − 6 − 6 − 6
:= (666 − 6)/6 − 6 − 6
:= 77 + 77/7
:= 7 + 77 + 7 + (7 + 7)/7
:= 7 × (7 + 7)
:= 88
:= 8 + 88 + 8 − 88/8
:= 8 + 88 + (8 + 8)/8
:= 99 − 99/9
:= 9 × 9 + (99 + 9)/9
:= 99 − 9/9
:= 5 × 5 + (5 /5 + 5)/(5 + 5)
Ï 89 := 111 − 11 − 11
Ï 94 := 1 + ((1 + 1)11−1 − 1)/11
Ï 99 := 11 × (11 − 1 − 1)
:= 2 + 2 × (2 × 22 + 2)
:= (22/2)2 − 22
:= 3 + 3 × (33 + 3) + 3/3
:= 3 × 33
:= 4 + 4 + (4 − 4/4)
:= (444 − 4)/4 − 4 × 4
:= 4 − 4 × 4 + 444/4
:= 5 × (5 × 5 − 5) − 55/5
:= 5 × (5 × 5 − 5) − 5 − 5/5
:= 5 × (5 × 5 − 5) − 5/5
:= 6 + 66 + 6 + 66/6
:= 6 × 6 + ((6 + 6)/6)6 − 6
:= 666/6 − 6 − 6
:= 77 + (77 + 7)/7
:= 7 + 77 + (77 − 7)/7
:= 7/7 + 7 × (7 + 7)
:= 8/8 + 88
:= 8 + 88 − (8 + 8)/8
:= 88 + 88/8
:= 9 + 9 × 9 − 9/9
:= 99 + (9 − 99)/(9 + 9)
:= 99
:= 2/2 + 2 × 2 × 22
:= 3 × (33 + 3) − 3/3
4
Ï 90 := (11 − 1) × (11 − 1 − 1)
Ï 100 := (11 − 1)1+1
Ï 95 := 111 − (1 + 1)1+1+1+1
:= (2 × (2 + 2) + 2)2
:= 2 + 2 × 2 × 22
:= 222/2 − 22+2
:= 3 × (33 + 3)
:= 3 × 33 − 3 − 3/3
:= 3/3 + 3 × 33
:= 44 + ((4 + 4)/4 + 44)
:= 444/4 − 4 × 4
:= 4 + 4 × (4 × 4 + 4 + 4)
:= 5 + (5 × 5 + 55 + 5)
:= 5 × (5 × 5 − 5) − 5
:= 5 × (5 × 5 − 5)
:= 6 + ((66 + 6 + 6) + 6)
:= 6 × 6 + 66 − 6 − 6/6
:= 6 × 6 + ((6 + 6)/6)6
:= 7 + (77 − 7/7 + 7)
:= 7 + 77 + 77/7
:= (7 + 7)/7 + 7 × (7 + 7)
:= 88 + (8 + 8)/8
:= 8 + 88 − 8/8
:= 88 + (88 + 8)/8
:= 9 + 9 × 9
:= 99 + (9 − 9 × 9)/(9 + 9)
:= 9/9 + 99
Ï 91 := 1 + (11 − 1) × (11 − 1 − 1)
:= 2 + 2 × 2 × 22 + 2/2
3
3
Ï 101 := 1 + (11 − 1)1+1
Ï 96 := (1 + 11) × (1 + 1)1+1+1
:= 2222/22
:= 2 × 2 × (22 + 2)
:= 3 × 33 − 3
:= 3 + 3 × 33 − 3/3
:= 4 + 44 + 44 − 4/4
:= 4 × (4 × 4 + 4 + 4)
:= 4444/44
:= 5 − 5 × 5 + 555/5
:= 5/5 + 5 × (5 × 5 − 5) − 5
:= 5/5 + 5 × (5 × 5 − 5)
:= 6 × 6 + 66 − 66/6
:= 6 × 6 + 66 − 6
:= 6 × 6 + 66 − 6/6
:= 7 + 77 + 7
:= 7 × (7 + 7) − (7 + 7)/7
:= 7777/77
:= 88 + (88/8 − 8)
:= 8 + 88
:= 8888/88
:= 9 + 9 × 9 + 9/9
:= 99 − (9 + 9 + 9)/9
:= 99 + (9 + 9)/9
:= 3 + (3/3 + 3)
Ï 92 := 11 + (11 − 1 − 1)1+1
:= 2 × (2 × 22 + 2)
3
:= 3 × 3 + 33/3
Ï 102 := 1 + 1 + (11 − 1)1+1
Ï 97 := 111 − 11 − 1 − 1 − 1
:= 2/2 + 2 × 2 × (22 + 2)
:= 2 + (2 × (2 + 2) + 2)2
:= 3/3 + 3 × 33 − 3
:= 3 + 3 × 33
4
:= 4 + 44 + 44
:= 4 × 4 + (4 − 4/4)
5
:= (444 − 4)/4 − 4 − 4
5
:= 5 + 5 + 55 + ((5 + 5)/5)
:= (5 + 5)/5 + 5 × (5 × 5 − 5)
:= 6 × 6 + 66 + (6 − 66)/6
:= 6 × 6 + 66 − 6 + 6/6
:= 6 × 6 + 66
:= 7 + 7 + 77 + 7/7
:= 7 × (7 + 7) − 7/7
:= 7 + 77 + 7 + 77/7
:= 88 + 8 × 8/(8 + 8)
:= 8 + 88 + 8/8
:= (888 − 8)/8 − 8
:= 9 × 9 + 99/9
:= 99 − (9 + 9)/9
:= 999/9 − 9
:= 5 + 55 + ((5 + 5)/5)
16
Inder J. Taneja
Ï 103 := 1 + 1 + 1 + (11 − 1)1+1
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 113 := 1 + 1 + 111
Ï 108 := 111 − 1 − 1 − 1
:= 2 + 2222/22
:= (222 − 2)/2 − 2
:= 2 + 222/2
:= 3 + 3 × 33 + 3/3
:= 3 × (33 + 3)
:= (333 + 3 + 3)/3
:= 444/4 − 4 − 4
:= 44 + 4 × 4 × 4
:= (444 + 4 + 4)/4
:= 5 + 5 × (5 × 5 − 5) − (5 + 5)/5
:= 55 + 55 − (5 + 5)/5
:= (555 + 5 + 5)/5
:= 6 × 6 + 66 + 6/6
:= 6 × (6 + 6 + 6)
:= (666 + 6 + 6)/6
:= (777 − 7)/7 − 7
:= 7 + 7777/77
:= (777 + 7 + 7)/7
:= 888/8 − 8
:= 8 + 88 + (88 + 8)/8
:= (888 + 8 + 8)/8
:= (999 + 9)/9 − 9
:= 9 + 99
:= (999 + 9 + 9)/9
Ï 104 := 1 + 1 + 1 + 1 + (11 − 1)1+1
Ï 114 := 1 + 1 + 1 + 111
Ï 109 := 111 − 1 − 1
:= 2 × 2 × (22 + 2 + 2)
:= 222/2 − 2
:= 3 + 3 + 3 × 33 − 3/3
:= 3/3 + 3 × (33 + 3)
:= 2 + (222 + 2)/2
:= 3 + 333/3
4
:= 44 + (4 + 4)/4
:= 4 + (444 − 4)/4
:= (5 − 5)/(5 × 5 + 5)
:= 55 + 55 − 5/5
:= 5 × 5 × 5 − 55/5
:= (666 − 6)/6 − 6
:= 6/6 + 6 × (6 + 6 + 6)
:= 6 + 6 × (6 + 6 + 6)
:= 777/7 − 7
:= 77/7 + 7 × (7 + 7)
:= ((7 + 7)/7)7 − 7 − 7
:= 8 + (88 + 8)
:= (888 − 8 − 8)/8
:= (888 + 88)/8 − 8
:= 99 + (9 × 9 + 9)/(9 + 9)
:= 9 + 99 + 9/9
:= (999 + 9 + 9 + 9)/9
:= 4 × 4 + 44 + 44
5
Ï 105 := 111 − (1 + 1) × (1 + 1 + 1)
Ï 115 := 1 + 1 + 1 + 1 + 111
Ï 110 := 111 − 1
:= 2 + 2 + 222/2
:= 222/2 − 2 − 2 − 2
:= (222 − 2)/2
:= 3 + 3 + 3 × 33
:= (333 − 3)/3
:= 3 + (333 + 3)/3
:= 4 + 4444/44
:= (444 − 4)/4
:= 4 + 444/4
:= 5 + 5 × (5 × 5 − 5)
:= 55 + 55
:= 5 + 55 + 55
:= 666/6 − 6
:= (666 − 6)/6
:= 6 + 6 × (6 + 6 + 6) + 6/6
:= 7 + 7 × (7 + 7)
:= (777 − 7)/7
:= 77 + 7 × 7 − 77/7
:= 8 + 8 + 88 + 8/8
:= (888 − 8)/8
:= 8 + 88 + 8 + 88/8
:= 9 + 99 − (9 + 9 + 9)/9
:= 99 + 99/9
:= 9 + 9 + 99 − (9 + 9)/9
Ï 106 := 111 − 1 − 1 − 1 − 1 − 1
Ï 116 := 1 + 1 + 1 + 1 + 1 + 111
Ï 111 := 111
:= 222/2
:= 2 + 2 + (222 + 2)/2
:= 3 + 3 + 3 × 33 + 3/3
:= 333/3
:= 3 + 3 + (333 − 3)/3
:= (444 − 4)/4 − 4
:= 444/4
:= 4 + 4 × (44 − 4 × 4)
:= 555/5 − 5
:= 555/5
:= 5 + 555/5
:= 2 + 2 × 2 × (22 + 2 + 2)
:= (666 + 6)/6 − 6
:= 666/6
:= 6 + (666 − 6)/6
:= 7 + 7 × (7 + 7) + 7/7
:= 777/7
:= 7 + 7 × (7 + 7) + 77/7
:= 8 + 8 + 88 + (8 + 8)/8
:= 888/8
:= 8 × (8 + 8) − (88 + 8)/8
:= 9 + 99 − (9 + 9)/9
:= 999/9
:= 9 + 9 + 99 − 9/9
Ï 107 := 111 − 1 − 1 − 1 − 1
Ï 117 := 111 + (1 + 1) × (1 + 1 + 1)
Ï 112 := 1 + 111
:= 222/2 − 2 − 2
:= (222 + 2)/2
:= (22/2)2 − 2 − 2
:= 3 × (33 + 3) − 3/3
:= (333 + 3)/3
:= 3 × (33 + 3 + 3)
:= 444/4 − 4
:= (444 + 4)/4
:= 4 + ((444 + 4 + 4)/4)
:= 5 + (555 + 5)/5
:= (555 + 5)/5 − 5
:= (555 + 5)/5
:= 6 × (6 + 6 + 6) − 6/6
:= (666 + 6)/6
:= 6 + 666/6
:= 7 + (7 + 7)/7 + 7 × (7 + 7)
:= (777 + 7)/7
:= 7 + (777 − 7)/7
:= 8 + (88/8 + 88)
:= (888 + 8)/8
:= 8 × (8 + 8) − 88/8
:= 9 + 99 − 9/9
:= (999 + 9)/9
:= 9 + (99 + 9)
17
Inder J. Taneja
Ï 118 := 111+1 − 1 − 1 − 1
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 128 := (1 + 1)1+(1+1)×(1+1+1)
Ï 123 := 1 + 1 + 11 × 11
:= 22 + 2 × 2 × (22 + 2)
:= 2 + (22/2)2
:= 2 × 22+2+2
:= 3 + 3 + (333 + 3)/3
:= 33 − 3 + 3 × 33
:= 3 + (3 − 3/3 + 3)3
:= 4 + 4 + (444 − 4)/4
:= 4 + 4 + 4 + 444/4
:= 4 × 4 × (4 + 4)
:= 5 + (555 + 5 + 5)/5
:= 5 × 5 × 5 − (5 + 5)/5
:= 5 + 5 × 5 × 5 − (5 + 5)/5
:= 6 + (666 + 6)/6
:= 6 + 6 + 666/6
:= ((6 + 6)/6)6+6/6
:= 7 + 777/7
:= (777 + 77 + 7)/7
:= 8 + (888 − 8)/8
:= ((7 + 7)/7)7
:= (888 + 88 + 8)/8
:= 9 + 99 + 9 + 9/9
:= 8 × (8 + 8)
:= (999 + 99 + 9)/9
:= 99 + 9 + 9 + 99/9
Ï 119 := 111+1 − 1 − 1
2
:= (22/2) − 2
Ï 124 := 1 + 1 + 1 + 11 × 11
Ï 129 := 11 × (1 + 11) − 1 − 1 − 1
:= 2 × (22+2+2 − 2)
:= (22×(2+2) + 2)/2
:= 3 × 3 + (333 − 3)/3
:= 3 + (33/3)
:= 4 + 4 + 444/4
:= 4 × 4 × (4 + 4) − 4
:= 4/4 + 4 × 4 × (4 + 4)
:= 5 × 5 × 5 − 5 − 5/5
:= 5 × 5 × 5 − 5/5
:= 5 + (5 × 5 × 5 − 5/5)
:= 6 + (666 + 6 + 6)/6
:= 6 + 6 + (666 + 6)/6
:= 6 + 6 + 6 + 666/6
:= 77 + 7 × 7 − 7
:= 7 + 7 + (777 − 7)/7
:= 8 + 888/8
:= 7/7 + ((7 + 7)/7)7
:= 8 × (8 + 8) − 8 × 8/(8 + 8)
:= 9 + (99/9 + 99)
:= 8/8 + 8 × (8 + 8)
:= 9 + 9 + 9 + 99 − (9 + 9)/9
:= 9 + 9 + 999/9
Ï 120 := 111+1 − 1
:= (2 + 2 + 2) × (22 − 2)
:= 3 + 3 × (33 + 3 + 3)
:= (4 + 4) × (44/4 + 4)
:= 5 × 5 × 5 − 5
:= 6 + 6 + 6 × (6 + 6 + 6)
:= 7 + (777 + 7 + 7)/7
:= 8 × (8 + 8) − 8
:= 9 + 999/9
Ï 121 := 11 × 11
3−3/3
:= 3 + (3 × 33 + 33 )
Ï 125 := 1 + 1 + 1 + 1 + 11 × 11
2
:= 2 + 2 × 22+2+2
:= 2 + (22/2) + 2
:= (3 − 3/3 + 3)
Ï 130 := (11 − 1) × (1 + 1 + 11)
3
:= 3 + 3 × 33 + 33 + 3/3
4
:= 44 + (4 − 4/4)
:= (44 + 4)/((4 + 4)/4)
:= 5 × 5 × 5
:= 5 + 5 × 5 × 5
:= 66 + 66 − 6 − 6/6
:= 66 + ((6 + 6)/6)6
:= 7 + 7 + 777/7
:= (7 + 7)/7 + ((7 + 7)/7)7
:= 8 + 8 × (8 + 8) − 88/8
:= 9 + 99 + 9 + 9 − 9/9
Ï 126 := 1 + 1 + 1 + 1 + 1 + 11 × 11
:= (22/2)2
:= 2 × 22+2+2 − 2
:= (33/3)3−3/3
:= 3 × (3 × 3 + 33)
:= 8 × (8 + 8) + (8 + 8)/8
:= 9 + 9 + (999 + 9)/9
Ï 131 := 11 × (1 + 11) − 1
:= 2 + (22×(2+2) + 2)/2
:= 3 + 3 + (3 + 3 − 3/3)3
:= (44/4)(4+4)/4
:= (4 − 4) × 4/(4 + 4)
:= 5 + 5 + 555/5
:= 4 + 444/4 + 4 × 4
:= 5/5 + 5 × 5 × 5
:= 66 × 66/(6 × 6)
:= 5 + 5 × 5 × 5 + 5/5
:= 66 + 66 − 6
:= ((7 + 7)/7)7 − 7
:= 66 + 66 − 6/6
:= 77 + 7 × 7
:= 7 + 7 + 7 + (777 − 7)/7
:= 8 × (8 + 8) − 8 + 8/8
:= 8 × (8 + 8) − (8 + 8)/8
:= 8 × (8 + 8) + 88/8 − 8
:= 9 + (999 + 9)/9
:= 9 + (99 + 9 + 9)
:= 9 + (999 + 99)/9
Ï 122 := 111 + 11
4
Ï 127 := 111 + (1 + 1)1+1+1+1
2×(2+2)
− 2)/2
Ï 132 := 11 × (1 + 11)
:= (222 + 22)/2
:= (2
:= (333 + 33)/3
3
:= 3 + (3 × 33 + 3/3)
:= 33 + 3 × 33
:= (444 + 44)/4
:= 4 × 4 + 444/4
:= 4 + 4 × 4 × (4 + 4)
:= (555 + 55)/5
:= 5 × 5 × 5 + (5 + 5)/5
:= 5 + 5 × 5 × 5 + (5 + 5)/5
:= (666 + 66)/6
:= 22 × (2 + 2 + 2)
:= 6 + 66 × 66/(6 × 6)
:= 66 + 66
:= (777 + 77)/7
:= 7/7 + (77 + 7 × 7)
:= 7 + 7 + 7 + 777/7
:= (888 + 88)/8
:= 8 × (8 + 8) − 8/8
:= 88 + 88 × 8/(8 + 8)
:= (999 + 99)/9
:= 99 + 9 + 9 + 9 + 9/9
:= 99 × (99 + 9)/(9 × 9)
18
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 133 := 1 + 11 × (1 + 11)
Ï 138 := 111 + (1 + 1 + 1)1+1+1
:= (2/2 + 2) × (2 × 22 + 2)
:= 22 + 222/2
:= 33 + 3 × 33 + 3/3
3
Ï 143 := 11 × (1 + 1 + 11)
:= 22 + (22/2)2
:= 33 + (333 − 3)/3
:= 3 + 333/3
:= 4 + 4 × 4 × (4 + 4) + 4/4
:= 4 + (4 + 4) × 4/(4 + 4) + 4
:= 4 × (4 + 4) + 444/4
4
:= 5 + 5 + 5 × 5 × 5 − (5 + 5)/5
:= (5 × 5 × 55 + 5)/(5 + 5)
:= 5 + (5 × 5 × 55 + 5)/(5 + 5)
:= 66 + 66 + 6/6
:= 6 + 66 + 66
:= (6 + 6) × (6 + 6) − 6/6
:= 7 + 77 + 7 × 7
:= 77 + 7 × 7 + (77 + 7)/7
:= 77 + 77 − 77/7
:= (8 − 8/8) × (88/8 + 8)
:= 8 + 8 × (8 + 8) + (8 + 8)/8
:= 8 + 8 + 8 × (8 + 8) − 8/8
:= 9 × (9 + 9) − 9 − 9 − 99/9
:= 9 + 9 + 9 + 999/9
:= 9 × (9 + 9) − 9 − 9 − 9/9
Ï 134 := 1 + 1 + 11 × (1 + 11)
Ï 139 := (11 − 1) × (1 + 1 + 1 + 11) − 1
Ï 144 := (1 + 11)1+1
:= 2 + 22 × (2 + 2 + 2)
:= (22×(2+2) + 22)/2
:= (2 × (2 + 2 + 2))2
:= 3 × 3 + (3 − 3/3 + 3)3
:= 33 + (333 + 3)/3
:= (3 + 3) × (33 − 3)
:= 4 + (44 + 4) × 4/(4 + 4)
:= 44/4 + 4 × 4 × (4 + 4)
:= 4 × (4 × (4 + 4) + 4)
:= 5 + 5 + 5 × 5 × 5 − 5/5
:= 5 × (5 × 5 + 5) − 55/5
:= (5/5 + 5) × (5 × 5 − 5/5)
:= 6 + ((6 + 6)/6)
:= 6 + 66 + 66 + 6/6
:= (6 + 6) × (6 + 6)
:= 7 + 77 + 7 × 7 + 7/7
:= 77/7 + ((7 + 7)/7)7
:= (7/7 + 7) × (77/7 + 7)
:= 8 + 8 × (8 + 8) − (8 + 8)/8
:= 8 × (8 + 8) + 88/8
:= 8 + 8 × (8 + 8) + 8
:= 9 + 9 + 9 + 9 + 99 − 9/9
:= 9 + 9 + 9 + (999 + 9)/9
:= (9 + 9) × (9 − 9/9)
6/6+6
Ï 135 := 1 + 1 + 1 + 11 × (1 + 11)
Ï 140 := (11 − 1) × (1 + 1 + 1 + 11)
Ï 145 := 1 + (1 + 11)1+1
:= 2 + 22 + 222/2
:= 2 × (2 × (22 + 2) + 22)
:= 2 + (22/2)2 + 22
:= 3 + 33 + 33 × 3
:= 3 + (333 − 3)/3 + 33
:= 3/3 + (3 + 3) × (33 − 3)
:= (4 − 4/4) × (44 + 4/4)
:= 4 × (4 × (4 + 4) + 4) − 4
:= 44 − 444/4
:= 5 + 5 + 5 × 5 × 5
:= 5 + 5 + 5 + 5 × 5 × 5
:= 5 × (5 × 5 + 5) − 5
:= 6 + 6 + 6 + 6 + 666/6
:= 6 + ((6 + 6)/6)6/6+6 + 6
:= 6/6 + (6 + 6) × (6 + 6)
:= 7 + ((7 + 7)/7)
:= 7 × (7 + 7 + 7) − 7
:= 7 × (7 + 7 + 7) − (7 + 7)/7
:= 8 + 8 × (8 + 8) − 8/8
:= 8 × (8 + 8) + (88 + 8)/8
:= 8 + 8 + 8 × (8 + 8) + 8/8
:= 9 + 9 + 9 + 9 + 99
:= 9 + 9 + (999 + 99)/9
:= 9/9 + (9 + 9) × (9 − 9/9)
7
Ï 136 := 1 + 1 + 1 + 1 + 11 × (1 + 11)
Ï 141 := (1 + 11)1+1 − 1 − 1 − 1
2
Ï 146 := 1 + 1 + (1 + 11)1+1
:= 2 + 2 + 22 × (2 + 2 + 2) + 2
:= 22 + (22/2) − 2
:= 2 + (2 × (2 + 2 + 2))2
:= (3/3 + 3) × (3/3 + 33)
:= 33 + 3 × (33 + 3)
:= 3 + (333 − 3)/3 + 33
:= 4 + 4 + 4 × 4 × (4 + 4)
:= 44 − 4 − 444/4
:= 44 + (4 − 444)/4
:= 5 × 5 + 555/5
:= 5 + 5 × 5 + 555/5
:= 5/5 + 5 × (5 × 5 + 5) − 5
:= 6 × 6 + 666/6 − 6
:= 6 × 6 + (666 − 6)/6
:= 7 + ((7 + 7)/7) + 7/7
:= 7/7 + 7 × (7 + 7 + 7) − 7
:= 7 × (7 + 7 + 7) − 7/7
:= 8 + 8 × (8 + 8)
:= 8 × 8 + 88 − 88/8
:= 8 + 8 + 8 × (8 + 8) + (8 + 8)/8
:= (9 − 9/9) × (9 + 9 − 9/9)
:= 9 + 99 × (99 + 9)/(9 × 9)
:= (9 + 9)/9 + (9 + 9) × (9 − 9/9)
6
:= 6 + ((6 + 6)/6) + 66
7
Ï 137 := 111 + (1 + 1) × (1 + 1 + 11)
:= 22+2 + (22/2)2
:= 33 + (333 − 3)/3
Ï 142 := (1 + 11)1+1 − 1 − 1
Ï 147 := 1 + 1 + 1 + (1 + 11)1+1
:= (2 × (2 + 2 + 2))2 − 2
:= 3 + (333 + 3)/3 + 3
3
4
:= (22/2 + 2)2 − 22
:= 3 + (3 + 3) × (33 − 3)
:= 4 + 4 × 4 × (4 + 4) + 4/4 + 4
:= 4 × 4 + (4 − 4) × 4/(4 + 4)
:= 4 + 444/4 + 4 × (4 + 4)
:= 5 × 5 + (555 + 5)/5
:= 5 + 5 × 5 + (555 + 5)/5
:= 5 × 5 + (555 + 55)/5
:= 6 + 66 + 66 − 6/6
:= (6 + 6) × (6 + 6) − (6 + 6)/6
:= 6 × 6 + 666/6
7
:= 77 + 77/7 + 7 × 7
:= 7 + 7 + ((7 + 7)/7)
:= 7 × (7 + 7 + 7)
:= 8 + 8 × (8 + 8) + 8/8
:= 8 + 8 + 8 × (8 + 8) − (8 + 8)/8
:= 8 + 8 × (8 + 8) + 88/8
:= 99 + 9 + 9 + 9 + 99/9
:= 9 × (9 + 9) − 9 − 99/9
:= 9 + 9 + 9 + 9 + 999/9
19
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 148 := 1 + 1 + 1 + 1 + (1 + 11)1+1
:= 2 + 2 + (2 × (2 + 2 + 2))2
:= (33 × 33 − 3)/(3 + 3)
Ï 153 := 11 × (1 + 1 + 1 + 11) − 1
:= 2 × ((2/2 + 2)2+2 − 2)
:= 2 × 22 + 222/2 − 2
3
Ï 158 := (1 + 1 + 11)1+1 − 11
3
:= 33 + (3 − 3/3 + 3)3
:= 3 × (3 − 3 + 3 )
4
:= 4 + 4 × (4 × (4 + 4) + 4)
:= 4 + 4 + 4 − 444/4
:= 4 × (44 − 4) − (4 + 4)/4
:= 5 × (5 × 5 + 5) − (5 + 5)/5
:= 5 + 5 × (5 × 5 + 5) − (5 + 5)/5
:= 5 × ((5 + 5)/5)5 − (5 + 5)/5
:= 6 × 6 + (666 + 6)/6
:= 6 + 666/6 + 6 × 6
:= 6 + 6 × 6 × 6 − ((6 + 6)/6)6
:= 7/7 + 7 × (7 + 7 + 7)
:= 77 + 77 − 7/7
:= 77/7 + 7 × (7 + 7 + 7)
:= 888/(8 − (8 + 8)/8)
:= 8/8 + 8 × 8 + 88
:= 8 + 88 − (8 + 8)/8 + 8 × 8
:= 99 + (9 × 99 − 9)/(9 + 9)
:= 9 × (9 + 9) − 9
:= 9 × (9 + 9) + (9 − 9 × 9)/(9 + 9)
Ï 149 := 1 + 1 + 1 + 1 + 1 + (1 + 11)1+1
2
:= 2 + (22/2 + 2) − 22
Ï 159 := 1 + (1 + 1 + 11)1+1 − 11
Ï 154 := 11 × (1 + 1 + 1 + 11)
:= 22 + 22 × (2 + 2 + 2)
:= 222/2 + 2 × (22 + 2)
:= (33 × 3 + 3)/(3 + 3)
:= 33/3 × (33/3 + 3)
:= (3 + 3) × 33 − 3
:= 4 + 44 − 444/4
:= 44 + (444 − 4)/4
:= 4 × (44 − 4) − 4/4
:= 5 × (5 × 5 + 5) − 5/5
:= 5 + 5 × (5 × 5 + 5) − 5/5
:= (55 + 55 )/(5 × 5 − 5)
:= 6 + (6 + 6) × (6 + 6) − 6/6
:= 6 + (666 + 6)/6 + 6 × 6
:= 6 × 6 + 6 + 6 + 666/6
:= 7 + ((7 + 7)/7) + 7 + 7
:= 77 + 77
:= 7 × 7 + (777 − 7)/7
:= 8 + 8 × 8 + 88 − 88/8
:= 8 × 8 + (8 + 8)/8 + 88
:= 8 + 8 × 8 + 88 − 8/8
:= 99 + (9 × 99 + 9)/(9 + 9)
:= 9/9 + 9 × (9 + 9) − 9
:= 9 × (9 + 9) − (9 + 9 + 9)/9
3
7
Ï 150 := (11 − 1) × (1 + 1 + 1 + 1 + 11)
:= 22 + 2 × 22+2+2
3
Ï 155 := 11 + (1 + 11)1+1
Ï 160 := (1 + 1) × ((11 − 1 − 1)1+1 − 1)
:= 2 × 22 + 222/2
:= 2 × 2 × 2 × (22 − 2)
3
3
:= 3/3 + (3 + 3) × 33 − 3
:= 3 + (3 + 3) × (3 − 3) + 3
:= 3 + (3 − 3/3 + 3) + 3
:= (44 + 44 )/((4 + 4)/4)
:= 44 + 444/4
:= 4 × (44 − 4)
:= 5 × (5 × 5 + 5)
:= 5 + 5 × (5 × 5 + 5)
:= 5 × ((5 + 5)/5)5
:= 6 + (6 + 6) × (6 + 6)
:= 66/6 + (6 + 6) × (6 + 6)
:= 6 + (666 + 6)/6 + 6 × 6 + 6
:= 7 + 77 + 77 − 77/7
:= 7/7 + 77 + 77
:= 7 × 7 + 777/7
:= 8 × 8 + 88 − (8 + 8)/8
:= 8 + 8 + 8 × (8 + 8) + 88/8
:= 8 + 8 × 8 + 88
:= 9 × (9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + 9 × (9 + 9) − 9
:= 9 × (9 + 9) − (9 + 9)/9
Ï 151 := 1 + (11 − 1) × (1 + 1 + 1 + 1 + 11)
:= 22 + (22×(2+2) + 2)/2
Ï 156 := (1 + 11) × (1 + 1 + 11)
:= 2 × (2 × 2 × (22 − 2) − 2)
3
Ï 161 := (1 + 1) × (11 − 1 − 1)1+1 − 1
:= ((22+2 + 2)2 − 2)/2
:= (3 + 3) × 33 − 33/3
:= (3 + 3) × (3 − 3/3)
:= (3 + 3) × 33 − 3/3
:= 44 + 444/4 − 4
:= 4 × (44 − 4) − 4
:= 4/4 + 4 × (44 − 4)
5
:= 5/5 + 5 × (5 × 5 + 5)
:= (5 − 5)/(5 × 5 − 5)
:= 5 + (55 − 5)/(5 × 5 − 5)
:= 6 + (6 + 6) × (6 + 6) + 6/6
:= 6 + (6 + 6) × (6 + 6) + 6
:= 6 + (6 + 6) × (6 + 6) + 66/6
:= 7 + (7/7 + 7) × (77/7 + 7)
:= 77 + 77 + (7 + 7)/7
:= 7 + 77 + 77
:= 8 × 8 + (88 − 8/8)
:= 8 + 888/(8 − (8 + 8)/8)
:= 8 + 8 × 8 + 88 + 8/8
:= 9 × (9 + 9) − 99/9
:= 9 × (9 + 9) + (9 + 9 + 9)/9 − 9
:= 9 × (9 + 9) − 9/9
Ï 157 := 1 + (1 + 11) × (1 + 1 + 11)
Ï 162 := (1 + 1) × (11 − 1 − 1)1+1
:= 2 × 2 × ((2 + 2 + 2)2 + 2)
:= 2 + 222/2 + 2 × 22
:= 2 × (2/2 + 2)2+2
:= 33 + (3 − 3/3 + 3)3
:= 3 + 33/3 × (33/3 + 3)
:= (3 + 3) × 33
:= 4 × (44 − 4) − 4 − 4
:= 4/4 + 4 × (44 − 4) − 4
Ï 152 := 11 × (1 + 1 + 1 + 11) − 1 − 1
:= (4 + 4)/4 + 4 × (44 − 4)
5
:= 5 × 5 × 5 + ((5 + 5)/5)
:= 5 + 5 × 5 × 5 + ((5 + 5)/5)5
:= 6 × 6 × 6 − ((6 + 6)/6)6
:= 6 + 6 + (6 + 6) × (6 + 6) + 6/6
:= 6 + (6 + 6) × (6 + 6) + 6 + 6
:= 77 + 77 − (7 + 7)/7
:= 77 + (7 + 7 + 7)/7 + 77
:= 7 + 77 + 77 + 7/7
:= 8 × 8 + 88
:= 88 + 88 − 8 − 88/8
:= 8 + 88 + 8 × 8 + (8 + 8)/8
:= 9 × (9 + 9) − 9/9 − 9
:= 9 × (9 + 9) + (9 − 99)/(9 + 9)
:= 9 × (9 + 9)
:= (5 + 5)/5 + 5 × (5 × 5 + 5)
20
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 163 := 1 + (1 + 1) × (11 − 1 − 1)1+1
:= ((22+2 + 2)2 + 2)/2
Ï 168 := (1 + 1 + 11)1+1 − 1
Ï 173 := 1 + 1 + 1 + 1 + (1 + 1 + 11)1+1
:= 2 × 2 × (2 × 22 − 2)
3
:= 2 + (22/2 + 2)2 + 2
:= 3/3 + (3 + 3) × 33
:= 3 + (3 + 3) × 3 + 3
:= (3 + 3) × 33 + 33/3
:= 4 + 4 × (44 − 4) − 4/4
:= 4 × 44 − 4 − 4
:= 4 × 44 − 4 + 4/4
:= 5 × 55 − (555 + 5)/5
:= 55 + (555 + 5 + 5)/5
:= 5 × (5 × 5 + 5 + 5) − (5 + 5)/5
:= 6 × (6 × 6 − 6) − 66/6 − 6
:= 6 × 6 + (66 + 66)
:= 6 × (6 × 6 − 6) − 6/6 − 6
:= 7 + 77 + 77 + (7 + 7)/7
:= 7 + ((77 + 77) + 7)
:= 77 + 7 × (7 + 7) − (7 + 7)/7
:= 8 × 8 + 88 + 88/8
:= 88 + (88 − 8)
:= 8 + 88 + 88 − 88/8
:= 9/9 + 9 × (9 + 9)
:= 9 + (9 × (9 + 9) − ((9 + 9 + 9)/9))
:= 99/9 + 9 × (9 + 9)
Ï 164 := (1 + 1) × (1 + (11 − 1 − 1)1+1 )
:= 2 + 2 × (2/2 + 2)2+2
3
Ï 169 := (1 + 1 + 11)1+1
Ï 174 := (1 + 1) × (111 − (1 + 1) × (1 + 11))
:= (22/2 + 2)2
:= 2 × 2 × 2 × 22 − 2
3
:= 3 + (3 + 3) × 3 − 3/3
:= 3 + 3 + (3 + 3) × 3 + 3/3
:= 3 + (3 + 3) × 33 + 3 × 3
:= 4 + 4 × (44 − 4)
:= 4 + 4 × 44 − 44/4
:= 4 × 44 − (4 + 4)/4
:= 5 + 5 × 55 − 555/5
:= 5 × (5 × 5 + 5 + 5) − 5/5
:= 5 × 55 − 555/5
:= (6/6 + 6 + 6)
(6+6)/6
:= 6 × (6 × 6 − 6) − 6
:= 77 + 77 + (77 − 7)/7
:= (7 − 7/7 + 7)
(7+7)/7
:= 77 + 7 × (7 + 7) − 7/7
:= 8 × 8 + 88 + (88 + 8)/8
:= 8/8 + 88 + 88 − 8
:= 8 × 8 + (888 − 8)/8
:= (9 + 9)/9 + 9 × (9 + 9)
:= 9 + 9 × (9 + 9) − (9 + 9)/9
:= 9 × (9 + 9) + (99 + 9)/9
6/6+6
:= 6 × 6 + ((6 + 6)/6)
Ï 165 := 11 × (1 + 1 + 1 + 1 + 11)
:= 2 × 22 + (22/2)2
Ï 175 := 11 × (1 + 1)1+1+1+1 − 1
Ï 170 := 1 + (1 + 1 + 11)1+1
:= 2 + 2 × (2 × (2 × 22 − 2))
:= 2 × 2 × 2 × 22 − 2/2
:= 3 + (3 × 333 + 3)/(3 + 3)
:= 333/3 + (3/3 + 3)3
:= 4 × 44 − 4 − (4 + 4)/4
:= 4 × 44 − 4/4
:= 5 + 5 × ((5 + 5)/5)
:= 5 × (5 × 5 + 5 + 5) − 5
:= 5 × (5 × 5 + 5 + 5)
:= 66 × (6 × 6 − 6)/(6 + 6)
:= 66 + (666 − 6)/6 − 6
:= 6/6 + 6 × (6 × 6 − 6) − 6
:= 3 + (3 + 3) × 3
3
:= 4 × 44 − 44/4
5
7
:= 77 + 7 × (7 + 7)
:= 77 + 77 + 77/7
:= 7 × 7 + ((7 + 7)/7) − 7
:= 88 + 88 − 88/8
:= 88 + (8 + 8)/8 − 8 + 88
:= 8 × 8 + 888/8
:= 9 + 9 × (9 + 9) − 9/9
:= 9 × (9 + 9) + (99 + 9 + 9)/9
:= 9 × (9 + 9) + (9 + 9 + 9)/9
Ï 166 := 1 + 11 × (1 + 1 + 1 + 1 + 11)
Ï 171 := 1 + 1 + (1 + 1 + 11)1+1
Ï 176 := 11 × (1 + 1)1+1+1+1
:= 2 × ((2/2 + 2)2+2 + 2)
:= 2 + (22/2 + 2)2
:= 2 × 2 × 2 × 22
:= 3 + (3 + 3) × 33 + 3/3
:= 3 × (33 + 33 + 3)
:= 3 + (3 + 3) × 33 + 33/3
:= 4 × 44 + (4 − 44)/4
:= 4 × 44 − 4 − 4/4
:= 4 × 44
:= 55 + 555/5
:= 5 + 55 + 555/5
:= 5/5 + 5 × (5 × 5 + 5 + 5)
:= 6 × 6 + 66 + ((6 + 6)/6)6
:= 66 + 666/6 − 6
:= 66 + (666 − 6)/6
:= 7 + (777 − 7)/7 + 7 × 7
:= (7 × 7 × 7 × 7 − 7)/(7 + 7)
:= 7/7 + 7 × (7 + 7) + 77
:= 8 × 8 − 8 + (888 − 8)/8
:= (8/8 + 8) × (88/8 + 8)
:= 88 + 88
:= 9 × (9 + 9) + (9 × 9 − 9)/(9 + 9)
:= 9 + 9 × (9 + 9)
:= (9 − 9/9) × (99 + 99)/9
Ï 177 := 1 + 11 × (1 + 1)1+1+1+1
Ï 167 := (1 + 1 + 11)1+1 − 1 − 1
Ï 172 := 1 + 1 + 1 + (1 + 1 + 11)1+1
:= (22/2 + 2)2 − 2
:= 2 × (2 × 2 × 22 − 2)
:= 2 × 2 × 2 × 22 + 2/2
:= (3 × 333 + 3)/(3 + 3)
:= 3 × 3 + (3 + 3) × 33 + 3/3
:= 3 × (33 + 33) − 3
:= 4 × 44 − 4 − 4 − 4/4
:= 4 × 44 − 4
:= 4 × 44 + 4/4
:= 55 + (555 + 5)/5
:= 5 + 55 + (555 + 5)/5
:= 55 + (555 + 55)/5
:= 6 × 6 + 66 + 66 − 6/6
:= 66 − 6 + (666 + 6)/6
:= 66 + 666/6
:= 7 + 7 × 7 + 777/7
:= (7 × 7 × 7 × 7 + 7)/(7 + 7)
:= 7 × 7 + ((7 + 7)/7)7
:= 8 × 8 − 8 + 888/8
:= 88 + 88 − 8 × 8/(8 + 8)
:= 88 + 88 + 8/8
:= 9 × (9 + 9) + (9 × 9 + 9)/(9 + 9)
:= 9 + 9 × (9 + 9) + 9/9
:= 99 + 9 × 9 − (9 + 9 + 9)/9
21
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 178 := (1 + 1) × (111 − 11 − 11)
Ï 183 := 1 + (1 + 1 + 11) × (1 + 1 + 1 + 11)
:= 2 + 2 × 2 × 2 × 22
:= 2 × 2 × (2 × 22 + 2) − 2/2
Ï 188 := 1 + 1 + ((1 + 1)11 − 1 − 1)/11
:= 2 × (2 × (2 × 22 + 2) + 2)
:= 3/3 + 3 × (3 + 33) − 3
:= (3 + 3) − 33
:= (3 + 3)3 − 33 − 3/3
:= 4 × 44 + (4 + 4)/4
:= 4 + 4 + 4 × 44 − 4/4
:= 444 − 44
:= 55 + 5 × 5 × 5 − (5 + 5)/5
:= (5 − (5 + 5)/5)5 − 55 − 5
:= (5 − (5 + 5)/5)5 − 55
:= 66 + (666 + 6)/6
3
3
:= 6 + 66 + 666/6
:= 6 + 6 × (6 × 6 − 6) + (6 + 6)/6
:= 7 + (7 × 7 × 7 × 7 − 7)/(7 + 7)
:= 7 + 77 + 7 × (7 + 7) + 7/7
:= 77 + 777/7
:= 88 + 88 + 8 + 8/8
:= 8 + 8 × 8 + 888/8
:= 88 + 88 + (88 + 8)/8
:= 99 + 9 × 9 − (9 + 9)/9
:= 9 × 9 − 9 + 999/9
:= 9 + 99 + 9 × 9 − 9/9
Ï 179 := 11 + (1 + 1 + 11)1+1 − 1
Ï 184 := (1 + 1) × (11 + (11 − 1 − 1)1+1 )
Ï 189 := (11 − 1 − 1) × (11 + 11 − 1)
:= 2 + (2 × 2 × 2 × 22) + 2/2
:= 2 × 2 × (2 × 22 + 2)
:= ((22 − 2)2 − 22)/2
:= 3 × (33 + 33) − 3/3
:= 3/3 + (3 + 3)3 − 33
:= (3 + 3)3 − 33
:= 4 + 4 × 44 − 4/4
:= 4 + 4 + 4 × 44
:= 4/4 + 444 − 44
:= 55 + 5 × 5 × 5 − 5/5
:= 5 + 55 + 5 × 5 × 5 − 5/5
:= 5 × (55 + 5) − 555/5
:= 6 × (6 × 6 − 6) − 6/6
:= 6 + 66 + (666 + 6)/6
:= 66 + 6 + 6 + 666/6
:= 7 + (7 × 7 × 7 × 7 + 7)/(7 + 7)
:= 7 + 7 × 7 + ((7 + 7)/7)
:= (7 + 7) × (7 + 7) − 7
:= 8 + (8/8 + 8) × (88/8 + 8)
:= 8 + 88 + 88
:= 8 + 8 × (8 + 8 + 8) − 88/8
:= 99 + 9 × 9 − 9/9
:= 9 × 9 − 9 + (999 + 9)/9
:= 9 + 99 + 9 × 9
Ï 180 := 11 + (1 + 1 + 11)1+1
7
Ï 185 := (1 + 1 + 1 + 11)1+1 − 11
Ï 190 := (11 − 1) × ((1 + 1) × (11 − 1) − 1)
:= 2 × (2 × 2 × 22 + 2)
:= 2 × 2 × (2 × 22 + 2) + 2/2
:= 222 − 2 × 22+2
:= 3 × (33 + 33)
:= 3 + (3 + 3)3 − 33 − 3/3
:= 3/3 + (3 + 3)3 − 33
:= 4 + 4 × 44
:= 4 + 4 + 4 × 44 + 4/4
:= 44 − (44 + 4 + 4)/4
:= 55 + 5 × 5 × 5
:= 5 + 55 + 5 × 5 × 5
:= 5 + 5 × 5 × 5 + 55 + 5
:= 6 × (6 × 6 − 6)
:= 6 + 6 × (6 × 6 − 6) − 6/6
:= (6 − 6/6) × ((6 + 6)/6 + 6 × 6)
:= 77 + (777 − 7)/7 − 7
:= (7 + 7) × (7 + 7) − 77/7
:= 7/7 + (7 + 7) × (7 + 7) − 7
:= 88 + 88 + 8 × 8/(8 + 8)
:= 8 + 88 + 88 + 8/8
:= 8 × (8 + 8 + 8) − (8 + 8)/8
:= 99 + 9 × 9
:= 9 + (9 − 9/9) × (99 + 99)/9
:= 9 + 99 + 9 × 9 + 9/9
Ï 181 := 1 + 11 + (1 + 1 + 11)1+1
Ï 186 := ((1 + 1)11 − 1 − 1)/11
Ï 191 := 1 + (11 − 1) × ((1 + 1) × (11 − 1) − 1)
:= 2/2 + 2 × (2 × 2 × 22 + 2)
:= 2 + 2 × 2 × (2 × 22 + 2)
:= 22 + (22/2 + 2)2
:= 3/3 + 3 × (33 + 33)
:= 3 + (3 + 3)3 − 33
:= 3 × (3/3 + 3)3 − 3/3
:= 4 + 4 × 44 + 4/4
:= 4 × 44 + (44 − 4)/4
:= 44 − (44 + 4)/4
:= 55 + 5 × 5 × 5 + 5/5
:= 5 + 5 × 5 × 5 + 55 + 5/5
:= 5 × 5 + 55 + 555/5
:= 6/6 + 6 × (6 × 6 − 6)
:= 6 + 6 × (6 × 6 − 6)
:= 6 × (6 × 6 − 6) + 66/6
:= 77 − 7 + 777/7
:= 77 + 7 × (7 + 7) + 77/7
:= 7 + ((7 + 7)/7)7 + 7 × 7 + 7
:= 8 × (8 + 8 + 8) + 88/8
:= 8 + 88 + 88 + (8 + 8)/8
:= 8 × (8 + 8 + 8) − 8/8
:= 9/9 + 99 + 9 × 9
:= 99 + 99 − (99 + 9)/9
:= 99 + 9 × 9 + 99/9
Ï 182 := (1 + 1 + 11) × (1 + 1 + 1 + 11)
:= 2 + 2 × (2 × 2 × 22 + 2)
Ï 187 := 1 + ((1 + 1)11 − 1 − 1)/11
Ï 192 := 111 + (11 − 1 − 1)1+1
:= 2 × 2 × 2 × 22 + 22/2
:= 2 × 2 × 2 × (22 + 2)
:= (3 + 3) − 33 − 3/3
:= 33 × (3 + 3) − 33/3
:= 3 × (3/3 + 3)3
:= 4 + 4 × 44 + (4 + 4)/4
:= 4 × 44 + 44/4
:= 4 × (44 + 4)
:= 55 + 5 × 5 × 5 + (5 + 5)/5
:= 55/5 × ((55 + 5)/5 + 5)
:= (5/5 + 5) × ((5 + 5)/5)5
3
:= 6 × (6 × 6 − 6) + (6 + 6)/6
:= 6 + 6 × (6 × 6 − 6) + 6/6
:= 6 + 6 × (6 × 6 − 6) + 6
:= 7 + 77 + 7 × (7 + 7)
:= 77 + (777 − 7)/7
:= 7 + (7 + 7) × (7 + 7) − 77/7
:= 8 + 8 × 8 + (888 − 8)/8
:= 88 + 88 + 88/8
:= 8 × (8 + 8 + 8)
:= 9 + 9 × (9 + 9) + 99/9
:= 99 + 99 − 99/9
:= 9 × 9 + 999/9
22
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 193 := 1 + 111 + (11 − 1 − 1)1+1
Ï 198 := (1 + 1) × 11 × (11 − 1 − 1)
Ï 203 := 1 + (1 + 1) × (1 + (11 − 1)1+1 )
:= 2 + (22/2 + 2)2 + 22
:= 2 + (22+2 − 2)2
:= 2 + ((22 − 2)2 + 2)/2
:= 3/3 + 3 × (3/3 + 3)3
:= 33 × (3 + 3)
:= 3 + 3 + 33 × (3 + 3) − 3/3
:= 4 × (44 + 4) + 4/4
:= 4 + 4 + (4 − 4 + 4)/4
:= 44 + (44 − 44 )/4
:= 5 + (5 − (5 + 5)/5)5 − 55
:= (5 + 5) × (5 × 5 − 5) − (5 + 5)/5
:= (55 − 5)/(5 + 5 + 5) − 5
:= 6 + 6 + 6 × (6 × 6 − 6) + 6/6
:= 6 × 66 × 6/(6 + 6)
:= 6 × 6 × 6 − 6 − 6 − 6/6
4
4
7
:= (7 + 7) × (7 + 7) − (7 + 7 + 7)/7
:= 77 + ((7 + 7)/7) − 7
:= 7 + (7 + 7) × (7 + 7)
:= 8/8 + 8 × (8 + 8 + 8)
:= 88 + (888 − 8)/8
:= 8 × (8 + 8 + 8) + 88/8
:= 9 × 9 + (999 + 9)/9
:= 99 + 99
:= 9 × 9 + (999 + 99)/9
Ï 194 := (1 + 1 + 1 + 11)1+1 − 1 − 1
Ï 199 := (1 + 1) × (11 − 1)1+1 − 1
Ï 204 := (1 + 1) × (1 + 1 + (11 − 1)1+1 )
:= (22+2 − 2)2 − 2
:= ((22 − 2)2 − 2)/2
:= 2 + 2 + 222 − 22
:= 33 × (3 + 3) − 3 − 3/3
:= 3/3 + 33 × (3 + 3)
:= 3 + 33 × (3 + 3) + 3
:= 4 + (4 − 4 + 4)/4
:= 4 + 4 + 44 − (44 + 4)/4
:= 44 + 4 × (44 − 4)
:= 5 × 5 × (5 + 5) − 55 − 5/5
:= (5 + 5) × (5 × 5 − 5) − 5/5
:= 66 + ((6 + 6)/6)6/6+6
:= 6 × 6 × 6 − 6 − 66/6
:= (7 + 7) × (7 + 7) − (7 + 7)/7
:= 77 + (777 + 77)/7
:= (8 + 8)/8 + 8 × (8 + 8 + 8)
:= 88 + 888/8
:= 9 × 9 + (999 + 9 + 9)/9
:= 99 + 99 + 9/9
4
4
Ï 195 := (1 + 1 + 1 + 11)1+1 − 1
:= 5 + (5 + 5) × (5 × 5 − 5) − 5/5
:= 6 × 6 × 6 − 6 − 6
:= 7 + (7 + 7) × (7 + 7) + 7/7
:= (88 + 8)/8 + 8 × (8 + 8 + 8)
:= (999/9 − 9) × (9 + 9)/9
Ï 200 := (1 + 1) × (11 − 1)1+1
Ï 205 := (1 + 1 + (1 + 1)11 )/(11 − 1)
:= 2 + 2 + ((22 − 2)2 + 2)/2
:= (22+2 − 2)2 − 2/2
:= 222 − 22
:= 33 × (3 + 3) − 3
:= 3 + 33 × (3 + 3) − 3/3
:= 4 + 44 − (44 + 4)/4
:= 4 + 4 + 4 × (44 + 4)
:= ((4 + 4)4 + 4)/(4 × 4 + 4)
:= 5 × 5 × (5 + 5) − 55
:= (5 + 5) × (5 × 5 − 5)
:= 5 + (5 + 5) × (5 × 5 − 5)
:= (6 × 66 − 6) × 6/(6 + 6)
:= 6 × 6 × 6 − 6 + (6 − 66)/6
:= 6 × 6 × 6 − 66/6
:= (7 + 7) × (7 + 7) − 7/7
:= (7 + 7) × (7 + 7) − 7 + 77/7
:= 77 + ((7 + 7)/7)7
:= 8 + 88 + 88 + 88/8
:= 8 + 8 × (8 + 8 + 8)
:= 88 + 8 × (8 + 8) − 88/8
:= 99 + 99 − (9 + 9 + 9)/9
:= 99 + 99 + (9 + 9)/9
:= 9 + 99 + 99 − (9 + 9)/9
:= (3 + 3)3 − 33/3
Ï 206 := 1 + (1 + 1 + (1 + 1)11 )/(11 − 1)
Ï 196 := (1 + 1 + 1 + 11)1+1
Ï 201 := 1 + (1 + 1) × (11 − 1)1+1
:= (22+2 − 2)2
:= ((22 − 2)2 + 2)/2
:= 222 − 22+2
:= 3/3 + 33 × (3 + 3) − 3
:= 3 + 33 × (3 + 3)
:= (3 + 3)3 + (3 − 33)/3
:= 4 + 4 × (44 + 4)
:= 44 − 44 − 44/4
:= 44 + 44 + 4 − (4 + 4)/4
:= 5/5 + (5 + 5) × (5 × 5 − 5)
:= 5 + (5 + 5) × (5 × 5 − 5) + 5/5
:= 66 + 66 + ((6 + 6)/6)
:= (6 × 66 + 6) × 6/(6 + 6)
:= 6 × 6 × 6 + (6 − 66)/6
:= (7 + 7) × (7 + 7)
:= 7 + (7 + 7) × (7 + 7) − (7 + 7)/7
:= 77 + 7/7 + ((7 + 7)/7)7
:= 8 × (8 + 8 + 8) + 8 × 8/(8 + 8)
:= 8 + 8 × (8 + 8 + 8) + 8/8
:= 8 + 88 + (888 − 8)/8
:= 99 + 99 − (9 + 9)/9
:= 9 + 9 × 9 + 999/9
:= 9 + 99 + 99 − 9/9
:= 5/5 + 5 × 5 × (5 + 5) − 55
6
Ï 197 := 1 + (1 + 1 + 1 + 11)1+1
Ï 202 := (1 + 1) × (1 + (11 − 1)1+1 )
Ï 207 := 11 + (1 + 1 + 1 + 11)1+1
:= 2/2 + (22+2 − 2)2
:= 2 + 222 − 22
:= 2/2 + 222 − 22+2
:= 33 × (3 + 3) − 3/3
:= 3 + 33 × (3 + 3) + 3/3
:= (3 + 3)3 − 3 × 3
:= 4 + 44 + (4 − 44 )/4
:= 44 + (4 − 44)/4 − 44
:= 44 − 4 − 44 − 4/4
:= 5 + (5/5 + 5) × ((5 + 5)/5)5
:= (5 + 5)/5 + (5 + 5) × (5 × 5 − 5)
:= (55 − 5 × 5 + 5)/(5 + 5 + 5)
:= (66 × (6 + 6 + 6) − 6)/6
:= 6 × 6 × 6 − 6 − 6 − (6 + 6)/6
:= 6 + (6 × 66 + 6) × 6/(6 + 6)
:= 7/7 + (7 + 7) × (7 + 7)
:= 7 + (7 + 7) × (7 + 7) − 7/7
:= (7 + 7) × (7 + 7) + 77/7
:= 88 + (888 − 8 − 8)/8
:= 8 + 8 × (8 + 8 + 8) + (8 + 8)/8
:= 8 + 88 + 888/8
:= 99 + 99 − 9/9
:= 9 + 9 × 9 + (999 + 9)/9
:= 9 + 99 + 99
23
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 208 := (1 + 1) × (111 − 1) − 1 − 11
:= 2 + 222 − 2
2+2
Ï 213 := (1 + 1) × (1 + 111) − 11
Ï 218 := (1 + 1) × (111 − 1 − 1)
:= 222 − 2 − 2
:= 2 + 222 − 22/2
:= 3 + (3 + 3)3 − 33/3
:= (3 + 3) − 3
:= 3 + (3 + 3)3 − 3/3
:= 4 × (44 + 4 + 4)
:= 44 − 44 + 4/4
:= 444 × 4/(4 + 4) − 4
:= (55 − 5)/(5 + 5 + 5)
:= 5 + (55 − 5)/(5 + 5 + 5)
:= (5 − (5 + 5)/5)5 − 5 × 5
:= 6 × 6 × 6 − 6 − (6 + 6)/6
:= 6 × 6 × 6 − 6 × 6/(6 + 6)
:= 6 × 6 × 6 + (6 + 6)/6
3
:= 7 × (7 + 7) + (777 − 7)/7
:= 7 + 77 + ((7 + 7)/7) + 7/7
:= 7/7 + 7 × (7 × 7 − 7) − 77
:= 8 + 8 + 8 × (8 + 8 + 8)
:= 8 + 88 + 8 × (8 + 8) − 88/8
:= 88 + 8 × (8 + 8) + (8 + 8)/8
:= 9 + 99 + 99 + 9/9
:= (9 + 9) × (9 + 9) − 999/9
:= 9 + 99 + 99 + 99/9
Ï 209 := 11 × ((1 + 1) × (11 − 1) − 1)
7
Ï 214 := 1 + (1 + 1) × (1 + 111) − 11
:= 222 − 2 × (2 + 2)
:= 222 − 22/2 − 2
Ï 219 := (1 + 1) × (111 − 1) − 1
:= 222 − 2/2 − 2
3
:= 33/3 + 33 × (3 + 3)
:= 3/3 + (3 + 3) − 3
:= 3 + (3 + 3)3
:= 4/4 + 4 × (44 + 4 + 4)
:= 44 − 44 + (4 + 4)/4
:= 44 + 4 × 44 − 4/4
5
:= (5 + 5 + 5)/(5 + 5 + 5)
:= 5 + (5 + 5 + 5)/(5 + 5 + 5)
:= 5 × 55 − 55 − 5/5
:= 6 × 6 × 6 − 6/6 − 6
:= 6 × 6 × 6 − (6 + 6)/6
:= 6 × 6 × 6 + 6 × 6/(6 + 6)
:= 7 × (7 + 7) + 777/7
:= 7 + (7 + 7) × (7 + 7) + 77/7
:= 77 + 7 + 7 + ((7 + 7)/7)7
:= 88/8 × (88/8 + 8)
:= 88 + 8 × (8 + 8) − (8 + 8)/8
:= 8 + 8 + 8 × (8 + 8 + 8) + 88/8
:= 99 + 99 + 99/9
:= (9 + 9)/9 × (99 + 9 − 9/9)
:= 9 + 99 + 999/9
Ï 210 := (11 − 1) × (11 + 11 − 1)
:= 222 − 2 × (2 + 2 + 2)
:= (3 + 3)3 − 3 − 3
5
Ï 215 := (1 + 1) × (1 + 1 + 111) − 11
:= 2 + 2 + 222 − 22/2
Ï 220 := (1 + 1) × (111 − 1)
:= 222 − 2
3
:= (3 + 3) − 3/3
:= 3 + (3 + 3)3 + 3/3
:= 4 − 44 − (4 + 4)/4
:= 4 + 44 − 44 − 4/4
:= 44 + 4 × 44
:= 5 + (5 + 5) × (5 × 5 − 5) + 5
:= 5 × 55 − 55 − 5
:= 55 × (5 − 5/5)
:= 6 × 6 × 6 − 6
:= 6 × 6 × 6 − 6/6
:= 6 + 6 × 6 × 6 − (6 + 6)/6
:= 7 + 7 + (7 + 7) × (7 + 7)
:= 7 × 7 × 7 − ((7 + 7)/7)7
:= 77/7 × (7 − 7/7 + 7 + 7)
:= 88 + (888 + 88)/8
:= 88 + 8 × (8 + 8) − 8/8
:= (8 + 8) × (888 − 8)/(8 × 8)
:= 99 + 999/9
:= ((9 + 9) × (99 + 9) − 9)/9
:= 99/9 × (99/9 + 9)
4
Ï 211 := (1 + 1) × 111 − 11
Ï 216 := (1 + 1) × (111 − (1 + 1 + 1))
Ï 221 := (1 + 1) × 111 − 1
:= 222 − 22/2
:= (2 + 2 + 2)2/2+2
:= 3/3 + (3 + 3)3 − 3 − 3
:= 222 − 2/2
:= (3 + 3)3
:= 3 + (3 + 3)3 − 3/3 + 3
4
4
:= 4 − 44 − 4/4
:= 4 + 4 − 44
:= 44 + 4 × 44 + 4/4
:= 55 + (5 − 5)/(5 × 5 − 5)
:= (5 − 5/5) × (55 − 5/5)
:= 55 × (5 − 5/5) + 5/5
:= 6 × 6 × 6 − 6 + 6/6
:= 6 × 6 × 6
:= 6 + (6 × 6 × 6 − 6/6)
5
:= 7 + 7 + (7 + 7) × (7 + 7) + 7/7
:= (7 − 7/7)
:= 8 + 8 × (8 + 8 + 8) + 88/8
:= 88 + 8 × (8 + 8)
:= (888 − 8 + 888)/8
:= 9 + 9 + 99 + 99
:= 99 + (999 + 99)/9
:= 99 + (999 + 9)/9
Ï 212 := 1 + (1 + 1) × 111 − 11
:= 222 + ((2 − 22)/2)
(7+7+7)/7
:= (777 + 777 − 7)/7
Ï 217 := (1 + 1) × (111 − 1 − 1) − 1
:= 222 − 2 − 2 − 2/2
3
Ï 222 := (1 + 1) × 111
:= 222
3
:= (3 + 3) − (3/3 + 3)
:= (3 + 3) + 3/3
:= 3 + (3 + 3)3 + 3
:= 44 − 44
:= 4 + 44 − 44 + 4/4
:= 444 × 4/(4 + 4)
5
5
:= (55 + 5 )/(5 + 5 + 5)
:= 5 + (55 + 5 )/(5 + 5 + 5)
:= (5 + 5) × 555/(5 × 5)
:= 6 × 6 × 6 − 6 + (6 + 6)/6
:= 6/6 + 6 × 6 × 6
:= 6 + 6 × 6 × 6
:= 77 + 7 + ((7 + 7)/7)
:= 7 × (7 × 7 − 7) − 77
:= ((7 + 7)/7) × 777/7
:= (8 × 8 × 8 − 88) × 8/(8 + 8)
:= 8/8 + 8 × (8 + 8) + 88
7
:= 99 + (999 + 9 + 9)/9
:= (8 + 8) × 888/(8 × 8)
9
:= 9 × 9 × 9 − ((9 + 9)/9)
24
:= (9 + 9) × 999/(9 × 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 228 := (1 + 1) × (1 + 1 + 1 + 111)
Ï 223 := 1 + (1 + 1) × 111
Ï 233 := 11 + (1 + 1) × 111
:= 222 + 22/2
:= 2 + 222 + 2 + 2
:= 2/2 + 222
:= 3 + (3 + 3) + 3/3 + 3
:= 3 + (3 + 3) + 3 × 3
:= 3 × (3 × 33 − 3) − 3/3
:= 44 + 44/4 − 44
:= 4 + 44 − 4 × (4 + 4)
:= 4 + 44 − 4 × 4 − 44/4
:= 5 + (5 − (5 + 5)/5)5 − 5 × 5
:= (5 − 5/5) × ((5 + 5)/5 + 55)
:= (5 − (5 + 5)/5)5 − 5 − 5
:= 6 + 6 × 6 × 6 + 6/6
:= 6 + 6 + 6 × 6 × 6
:= 6 + 6 × 6 × 6 + 66/6
:= 7 + (7 − 7/7)
:= (7 − 7/7) × (7 × 7 − 77/7)
:= 7 + 7 × (7 + 7) + ((7 + 7)/7)7
:= 8 + 88 + 8 × (8 + 8) − 8/8
:= (88/8 + 8) × (88 + 8)/8
:= 8 + (8 + 8 − 8/8)(8+8)/8
:= 9 × (9 + 9 + 9) − 99/9 − 9
:= 9 + 99 + 9 + 999/9
:= 9 × (9 + 9 + 9) − 9/9 − 9
3
3
(7+7+7)/7
Ï 229 := 1 + (1 + 1) × (1 + 1 + 1 + 111)
Ï 224 := (1 + 1) × (1 + 111)
Ï 234 := 1 + 11 + (1 + 1) × 111
:= (222 − 22)/2 − 2
:= 22×(2+2) − 22
:= 3 × 3 + ((3 + 3) − 3/3)
:= (33+3 − 33)/3 − 3
:= 3 × (3 × 33 − 3)
:= 44 − 4 × (4 + 4)
:= 44 − 44/4 − 4 × 4
:= 44 − 44 × 4/(4 + 4)
:= (5 − 5/5) × (55 + 5/5)
:= 5 + (5 − 5/5) × (55 + 5/5)
:= (5 − 5/5 + 5) × (5 × 5 + 5/5)
:= 6 + 6 × 6 × 6 + (6 + 6)/6
:= 6 + 6 × 6 × 6 + 6/6 + 6
:= 6 + 6 × 6 × 6 + 6 + 6
:= 77 + 7 × (7 + 7 + 7)
:= 7 + (7 + 7) × 777/(7 × 7)
:= (77 + 7/7) × (7 + 7 + 7)/7
:= 8 + 8 × (8 + 8) + 88
:= 8 + (888 + 888 − 8)/8
:= (8 × (8 + 8) − 88/8) × (8 + 8)/8
:= 9 + ((9 + 9) × (99 + 9) − 9)/9
:= 9 + 99/9 × (9 + 99/9)
:= 9 × (9 + 9 + 9) − 9
:= 2 + 222
3
Ï 225 := 1 + (1 + 1) × (1 + 111)
Ï 230 := (11 − 1) × (1 + 11 + 11)
:= 2 + 222 + 2/2
:= 222 + 2 × (2 + 2)
3
Ï 235 := 11 + (1 + 1) × (1 + 111)
:= 2 + 222 + 22/2
3
:= 3 + (3 + 3) + 33/3
:= 3 + (33+3 − 33)/3
:= 4/4 + 44 − 4 × (4 + 4)
:= 4 + 4 + 444 × 4/(4 + 4)
:= 44 − 4 − 4 × 4 − 4/4
:= 5 × (55 − 5 − 5)
:= 3 × 3 + (3 + 3)
:= 5 + 5 × (55 − 5 − 5)
:= 5 + 5 + 5 × (55 − 5 − 5)
:= 6 + 6 × 6/(6 + 6) + 6 × 6 × 6
:= 6 + 6 + 6 × 6 × 6 + (6 + 6)/6
:= 6 + 6 + 6 + 6 × 6 × 6 + 6/6
:= (7/7 + 7 + 7)(7+7)/7
:= (77 × (7 + 7 + 7) − 7)/7
:= 7 + (7 − 7/7) × (7 × 7 − 77/7)
:= 8 + (8 + 8) × 888/(8 × 8)
:= (8 × 8 × 88 + 8)/(8 + 8 + 8)
:= 9 + 99 + (999 + 99)/9
:= 9/9 + (9 × (9 + 9 + 9) − 9)
:= (8 − 8/8 + 8)
(8+8)/8
:= (9 + 9) × (9 + 9) − 99
Ï 236 := 1 + 11 + (1 + 1) × (1 + 111)
Ï 231 := 11 × (11 + 11 − 1)
Ï 226 := (1 + 1) × (1 + 1 + 111)
:= 2 + 222 + 2
:= (22 − 22)/2
:= 2 + 22×(2+2) − 22
:= 3 × 3 + (3 + 3)3 + 3/3
:= 33 + 33 × (3 + 3)
:= (33+3 − 3)/3 − 3 − 3
:= 4 + 444 × 4/(4 + 4)
:= 44 + 4 × 44 + 44/4
:= 44 − 4 × 4 − 4
:= 5/5 + 5 × (55 − 5 − 5)
:= 5 + 5 × (55 − 5 − 5) + 5/5
:= 5 × 5 × 5 + 555/5
:= 6 × 6 × 6 + (66 − 6)/6
:= 66 × (6 × 6 + 6)/(6 + 6)
:= 66 × 66/(6 + 6 + 6) − 6
:= 7 × (7 + 7) + ((7 + 7)/7)7
:= 77 + 77 + 77
:= (7 + 777/7) × (7 + 7)/7
:= 8 + 8 × (8 + 8) + (8 + 8)/8 + 88
:= 8 × (8 + 8) + 888/8 − 8
:= 8 + (88/8 + 8) × (88 + 8)/8
:= 9 + (9 + 9) × 999/(9 × 9)
:= 9 × (9 + 9 + 9) − 9 + (9 + 9)/9
2
9
:= 9 + 9 × 9 × 9 − ((9 + 9)/9)
Ï 227 := 1 + (1 + 1) × (1 + 1 + 111)
Ï 232 := 111 + 111+1
Ï 237 := 11 + (1 + 1) × (1 + 1 + 111)
:= 2 + 2 + 222 + 2/2
:= 2 + 222 + 2 × (2 + 2)
:= (222 − 2)/2 − 2 − 2
:= (3 + 3)3 + 33/3
:= (33+3 − 33)/3
:= 3 + 3 × (3 × 33 − 3)
:= 4 − 44 + 44 + 44/4
:= 44 − 4 × 4 − 4 − 4
:= 4/4 + 44 − (4 × 4 + 4)
:= 5 + (5 + 5) × 555/(5 × 5)
:= (5 + 5)/5 × (555/5 + 5)
:= 5 × 5 × 5 + (555 + 5)/5
:= 6 × 6 × 6 + 66/6
:= 6 + 6 × 6 × 6 + (66 − 6)/6
:= 6 + 66 × (6 × 6 + 6)/(6 + 6)
:= 7 + 77/7 × (7 − 7/7 + 7 + 7)
:= 7 × 7 × 7 − 777/7
:= 7 + (77 × (7 + 7 + 7) − 7)/7
:= 88 + 8 × (8 + 8) + 88/8
:= 88 + 8 + 8 + 8 × (8 + 8)
:= (8 + 8) × (8 + 8) − 88/8 − 8
:= ((9 + 9) × (99 + 9) + 99)/9
:= 9 × (9 + 9 + 9) − 99/9
:= 99 + 9 + 9 + 9 + 999/9
25
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 238 := (1 + 1) × (111+1 − 1 − 1)
Ï 243 := 1 + (1 + 1) × 111+1
:= 2 × ((22/2)2 − 2)
:= (222 + 2)/2
:= 2 + 222 /2 + 2 + 2
:= (33+3 + 3)/3 − 3 − 3
:= 3 × 3 × 33
:= 3 + (33+3 − 3)/3 + 3
:= 44 − 4 × 4 − (4 + 4)/4
:= (4 − 4/4)4/4+4
:= 44 − 4 − 4
:= (5 − (5 + 5)/5)5 − 5
:= (5 − (5 + 5)/5)5
:= 5 + (5 − (5 + 5)/5)5
:= 6 × 6 × 6 + (66 + 66)/6
:= (6 × 6/(6 + 6))6−6/6
Ï 248 := (1 + 1) × (1 + 1 + 1 + 111+1 )
:= 6 + 66 × 66/(6 + 6 + 6)
:= 777 − 7 × 77
:= ((7 + 7 + 7)/7)
:= 77 + (7 × 7 × 7 × 7 − 7)/(7 + 7)
:= 8 × (8 + 8) + (888 − 8)/8
:= (8 + 8/8) × (8 + 8 + 88/8)
:= (8 + 8) × (8 + 8) − 8
:= 9 × (9 + 9 + 9)
:= 9/9 + ((9 + 9)/9)9−9/9 − 9
Ï 244 := (1 + 1) × (1 + 111+1 )
Ï 249 := 1 + (1 + 1) × (1 + 1 + 1 + 111+1 )
7−(7+7)/7
9−9/9
:= ((9 + 9)/9)
−9−9
Ï 239 := (1 + 1) × (111+1 − 1) − 1
:= (222 − 2)/2 − 2
:= 2 + 222 /2
:= 2 + (222 + 2)/2 + 2 + 2
:= (33+3 − 3)/3 − 3
:= (33+3 + 3)/3
:= 33 + (3 + 3)3
:= 44 − 4 × 4 − 4/4
:= 4 + 44 − 4 × 4
:= 4 + 44 − 44/4
:= 5 × 5 × (5 + 5) − 55/5
:= 5/5 + (5 − (5 + 5)/5)5
:= 5 × 5 × (5 + 5) − 5/5
:= 6 + 6 + 6 × 6 × 6 + 66/6
:= ((6 + 6)/6) + 6 × (6 × 6 − 6)
:= 6 + (6 × 6/(6 + 6))6−6/6
:= 7 + 7 × 7 × 7 − 777/7
:= 7 × (7 × 7 − 7 − 7) − 7/7
:= ((7 + 7)/7)7/7+7 − 7
:= 8 × (8 + 8) + 888/8
:= (8 + 8) × (8 + 8) − (88 + 8)/8
:= 8/8 + (8 + 8) × (8 + 8) − 8
:= 9 × (9 + 9 + 9) + ((9 − 9 × 9)/(9 + 9))
:= 9 × (9 + 9 + 9) + 9/9
:= 9 + (999/9 + 9) × (9 + 9)/9
Ï 240 := (1 + 1) × (111+1 − 1)
2
6
Ï 245 := 1 + (1 + 1) × (1 + 111+1 )
2
Ï 250 := (11 − 1) × (1 + (1 + 1) × (1 + 11))
:= 22 /2 − 2
:= 2 + (22 + 2)/2
:= 2 × ((22/2)2 + 2 + 2)
:= 3 × 3 × 33 − 3
:= 3 + (33+3 − 3)/3
:= 3 + 3 + (33+3 + 3)/3
:= 44 − 4 × 4
:= 44 − 44/4
:= 44 − 4 − (4 + 4)/4
:= (5 + 5) × (5 × 5 − 5/5)
:= 5 × 5 × (5 + 5) − 5
:= 5 × 5 × (5 + 5)
:= 6 × (6 × 6 + 6) − 6 − 6
:= (6 + 6/6) × (6 × 6 − 6/6)
:= 6 × (6 × 6 + 6) − (6 + 6)/6
:= (7 − (7 + 7)/7) × (7 × 7 − 7/7)
:= 7 × (7 × 7 − 7 − 7)
:= (7 − (7 + 7)/7) × (7 × 7 + 7/7)
:= 8 × 8 + 88 + 88
:= (8 + 8) × (8 + 8) − 88/8
:= (8 + 8) × (8 + 8) + (8 + 8)/8 − 8
:= (999/9 + 9) × (9 + 9)/9
:= 9 × (9 + 9 + 9) + (9 + 9)/9
:= 9 + 9 × (9 + 9 + 9) − (9 + 9)/9
Ï 241 := (1 + 1) × 111+1 − 1
Ï 246 := (1 + 1) × (1 + 1 + 111+1 )
Ï 251 := 11 + (1 + 1) × (111+1 − 1)
:= (222 − 2)/2
:= 2 + 222 /2 + 2
:= (222 + 22)/2 − 2
:= (33+3 + 3)/3 − 3
:= 3 + 3 × 3 × 33
:= 3 × 3 + (33+3 − 3)/3
:= 4/4 + 44 − 4 × 4
:= 44 + (4 − 44)/4
:= 44 − 4/4 − 4
:= 5 + 555/5 + 5 × 5 × 5
:= 5/5 + 5 × 5 × (5 + 5) − 5
:= 5/5 + 5 × 5 × (5 + 5)
:= 6 × (6 × 6 + 6) − 6/6
:= 6 × (6 × 6 + 6) − 66/6
:= 6 × (6 × 6 + 6) − 6
:= 7 + (7/7 + 77) × (7 + 7 + 7)/7
:= 7/7 + 7 × (7 × 7 − 7 − 7)
:= 7 + 7 × (7 × 7 − 7 − 7) − 7/7
:= 8/8 + 8 × 8 + 88 + 88
:= (8 + 8) × (8 + 8) + (8 − 88)/8
:= 8 + (8/8 + 8) × (88/8 + 8 + 8)
:= 9 × (9 + 9 + 9) − (9 + 9)/9
:= 9 × (9 + 9 + 9) + (9 + 9 + 9)/9
:= 9 + 9 × (9 + 9 + 9) − 9/9
Ï 242 := (1 + 1) × 111+1
Ï 247 := 1 + (1 + 1) × (1 + 1 + 111+1 )
2
2
:= 22 /2
:= (33+3 − 3)/3
:= 2 + (22 + 2)/2 + 2
:= 22×(2+2) − 2 − 2
3+3
:= 3 × (3 × 33 + 3)
:= 3 + (3
:= 44 × 44/(4 + 4)
5
Ï 252 := (1 + 11) × (11 + 11 − 1)
+ 3)/3
4
:= 4 − 4/4 − 4 − 4
:= 44 − 4
5
:= (5 − (5 + 5)/5) − 5/5
:= 5 + (5 − (5 + 5)/5) − 5/5
:= (5 + 5)/5 + 5 × 5 × (5 + 5)
:= 66 × 66/(6 + 6 + 6)
:= 6/6 + 6 × (6 × 6 + 6) − 6
:= 6 × (6 × 6 + 6)
:= 77/7 × (7 + 7 + 7 + 7/7)
:= (7 + 7)/7 + 7 × (7 × 7 − 7 − 7)
:= 7 + 7 × (7 × 7 − 7 − 7)
:= 88 × (88 + 88)/(8 × 8)
:= (8 + 8) × (8 + 8) − 8 − 8/8
:= (8 × 8 × 8 − 8) × 8/(8 + 8)
:= 9 × (9 + 9 + 9) − 9/9
9−9/9
:= ((9 + 9)/9)
−9
26
:= 9 + 9 × (9 + 9 + 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 253 := 11 × (1 + 11 + 11)
2
Ï 258 := 1 + 1 + (1 + 1)(1+1)
1+1+1
Ï 263 := (1 + 1) × 11 × (1 + 11) − 1
:= 22 + (222 − 2)/2
2×(2+2)
:= (22 + 22)/2
:= 2 + 2
:= 3 × 3 + (33+3 + 3)/3
:= 3 + 3 × (3 × 33 + 3) + 3
:= (33 × (33 − 3) − 3)/3
:= 4/4 + 44 − 4
:= 44 + (4 + 4)/4
:= 4 + 44 − 4/4 + 4
:= 5 + 5 + (5 − (5 + 5)/5)5
:= (55 + 5)/(5 + 5) − 55
:= 5 × 55 − (55 + 5)/5
:= 6/6 + 6 × (6 × 6 + 6)
:= 6 + 6 × (6 × 6 + 6)
:= 66/6 + 6 × (6 × 6 + 6)
:= 7 + 7 × (7 × 7 − 7 − 7) + 7/7
:= (7 − 7/7) × ((7/7 − 7) + 7 × 7)
:= 7 + ((7 + 7)/7)7/7+7
:= 8 + (8 + 8) × (8 + 8) − 88/8
:= (8 + 8)/8 + (8 + 8) × (8 + 8)
:= 8 + (8 + 8) × (8 + 8) − 8/8
:= 9 + 9 × (9 + 9 + 9) + 9/9
:= 99 + 9 × (9 + 9) − (9 + 9 + 9)/9
:= 9 + 9 × (9 + 9 + 9) + 99/9
Ï 254 := 1 + 11 × (1 + 11 + 11)
:= 22×(2+2) − 2
:= (3
3+3
1+1+1
Ï 259 := 1 + 1 + 1 + (1 + 1)(1+1)
:= 2 + 22×(2+2) + 2/2
Ï 264 := (1 + 1) × 11 × (1 + 11)
:= 2 × 22 × (2 + 2 + 2)
3/3+3
+ 33)/3
:= 3 + (3/3 + 3)
:= 4 − (4 + 4)/4
:= 4 + 44 − 4/4
:= 4 + 44 + 4
:= 5 + 5 × 5 × (5 + 5) − 5/5
:= 5 × 55 − 55/5 − 5
:= 5 × 55 − 55/5
:= (6 + 6)/6 + 6 × (6 × 6 + 6)
:= 6 + 6 × (6 × 6 + 6) + 6/6
:= 6 + 6 × (6 × 6 + 6) + 6
:= 77 + ((7 + 7)/7) + 7 × 7
:= 7 × 7 × 7 − 77 − 7
:= 7 × 7 × 7 − (7 + 7)/7 − 77
:= (8 + 8) × (8 + 8) − (8 + 8)/8
:= 88/8 + (8 + 8) × (8 + 8) − 8
:= 8 + (8 + 8) × (8 + 8)
:= 99/9 + 9 × (9 + 9 + 9)
:= 99 + 9 × (9 + 9) − (9 + 9)/9
:= 9 × (9 + 9) + 999/9 − 9
4
7
Ï 255 := 111 + (1 + 11)1+1
2×(2+2)
:= 33 × (3 × 3 − 3/3)
Ï 260 := (1 + 1) × (11 − 1) × (1 + 1 + 11)
Ï 265 := 1 + (1 + 1) × 11 × (1 + 11)
:= 2 + 22×(2+2) + 2
:= 22 + (222 + 2)/2
:= 3 + 3 × (3 × 33 + 3)
:= 3 + (33+3 + 33)/3 + 3
:= (33 × (33 − 3) + 3)/3
:= 44 − 4/4
:= 4 + 44
:= 4 + 4/4 + 44 + 4
:= 5 + 5 × 5 × (5 + 5)
:= 5 + 5 × 5 × (5 + 5) + 5
:= 5 × 55 − 5 − 5
:= 6 × (6 × 6 + 6) + 6 × 6/(6 + 6)
:= 6 + 6 × (6 × 6 + 6) + (6 + 6)/6
:= 6 + 6 × (6 × 6 + 6) + 6/6 + 6
:= 7 × 7 × 7 − 77/7 − 77
:= 2
− 2/2
:= 7/7 + 7 × 7 × 7 − 77 − 7
:= 7 × 7 × 7 − 7/7 − 77
:= (8 + 8) × (8 + 8) − 8/8
:= (8 × 8 × 8 + 8) × 8/(8 + 8)
:= 8 + (8 + 8) × (8 + 8) + 8/8
:= 9 × (9 + 9 + 9) + (99 + 9)/9
:= 99 + 9 × (9 + 9) − 9/9
:= 9 + ((9 + 9)/9)9−9/9
Ï 256 := (1 + 1)(1+1)
1+1+1
Ï 261 := (1 + 1) × (11 × (1 + 11) − 1) − 1
:= 22×(2+2)
:= 2 + 2
:= (3/3 + 3)3/3+3
:= 4
2×(2+2)
+ 2/2 + 2
3
4
5−5/5
:= (5 − 5/5)
(6+6)/6+6
:= ((6 + 6)/6)
7/7+7
Ï 266 := (1 + 1) × (1 + 11 × (1 + 11))
:= 222 + 2 × 22
:= 3 × (3 × 3 + 3 + 3)
:= 3 + (33 × (33 − 3) − 3)/3
:= 4 + 4/4 + 44
:= 44 + (44 − 4)/4
:= 5 + (5 − 5/5)5−5/5
:= 5/5 + 5 × 55 − (5 + 5)
:= 6 + 6 × (6 × 6 + 6) + 6 × 6/(6 + 6)
:= (6/6 + 6) × ((6 + 6)/6 + 6 × 6)
7
:= ((7 + 7)/7)
:= 7 + ((7 + 7)/7) + 77 + 7 × 7
:= 7 × 7 × 7 − 77
:= (8 + 8) × (8 + 8)
:= (88/8 − 8) × (88 − 8/8)
:= 8 + (8 + 8) × (8 + 8) + (8 + 8)/8
:= ((9 + 9)/9)9−9/9
:= 99 + 9 × (9 + 9)
:= 9 + ((9 + 9)/9)9−9/9 + 9/9
1+1+1
Ï 257 := 1 + (1 + 1)(1+1)
2×(2+2)
Ï 262 := (1 + 1) × (11 × (1 + 11) − 1)
2
Ï 267 := 1 + (1 + 1) × (1 + 11 × (1 + 11))
:= 22 − 222
:= 22/2 + 22×(2+2)
:= 3 + (33+3 + 33)/3
:= 3 + (3/3 + 3)3/3+3 + 3
:= 3 × 3 × (33 + 3) − 3
:= 4/4 + 44
:= 4 + (4 + 4)/4 + 44
:= 44 + 44/4
:= (55 − 5)/(5 + 5) − 55
:= 5 + (55 − 5)/(5 + 5) − 55
:= 5 × 55 + ((5 + 5)/5 − 5 − 5)
:= 2/2 + 2
:= 6 + 6 × (6 × 6 + 6) − 6/6
(6+6)/6+6
:= 6 + ((6 + 6)/6)
:= 666 × 6/(6 + 6) − 66
:= 7/7 + ((7 + 7)/7)
:= (7 × 7 × 77 − 7)/(7 + 7) − 7
:= 7/7 + 7 × 7 × 7 − 77
:= 8/8 + (8 + 8) × (8 + 8)
:= 8 + (8 + 8) × (8 + 8) − (8 + 8)/8
:= 88/8 + (8 + 8) × (8 + 8)
:= 9/9 + ((9 + 9)/9)9−9/9
:= 9/9 + 9 × (9 + 9) + 99
:= 99/9 + ((9 + 9)/9)9−9/9
7/7+7
27
Inder J. Taneja
Ï 268 := (1 + 1) × (1 + 1 + 11 × (1 + 11))
:= 2 + 222 + 2 × 22
3
:= 3 + (33 × (3 − 3) + 3)/3
4
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 273 := (1 + 1 + 11) × (11 + 11 − 1)
:= (22/2 + 2) × (22 − 2/2)
3
Ï 278 := (1 + 1111)/(1 + 1 + 1 + 1)
:= 22 + 22×(2+2)
:= 3 + (33+3 − 3)/3 + 33
:= 3 + 3 × 3 × (3 + 3)
4
:= 44 + 44 × 4/(4 + 4)
:= 4 + 4 + 4 + 4
:= 4 × 4 + 4/4 + 4
:= 5 × 5 + (5 − (5 + 5)/5)5
:= 5 × 55 − (5 + 5)/5
:= 5 + 5 × 55 − (5 + 5)/5
:= (6 − (6 + 6)/6) × (66 + 6/6)
:= 6 + 666 × 6/(6 + 6) − 66
:= 6 × 6 + 66 × 66/(6 + 6 + 6)
:= 7 × 7 × 7 + (7 + 7)/7 − 77
:= 7 + 7 × 7 × 7 − 77
:= 7 × 7 × 7 + (77 + 7)/7 − 77
:= 8 + (8 × 8 × 8 + 8) × 8/(8 + 8)
:= 8 + (8 + 8) × (8 + 8) + 8/8 + 8
:= 88 + 8 × (8 + 8 + 8) − (8 + 8)/8
:= 9 + 9 × (9 + 9) − (9 + 9)/9 + 99
:= 9 × (9 + 9) + 999/9
:= 99 + 9 × 9 − 9/9 + 99
Ï 269 := 1 + (1 + 1) × (1 + 1 + 11 × (1 + 11))
:= 2 + 2
2×(2+2)
+ 22/2
Ï 274 := 11 × (1 + (1 + 1) × (1 + 11)) − 1
:= 2 + 2
2×(2+2)
+2
2+2
Ï 279 := 1 + (1 + 1111)/(1 + 1 + 1 + 1)
:= 2 + ((22 + 2)2 − 22)/2
:= 33 + (33+3 − 3)/3
:= 3 + 3 × 3 × (33 + 3) + 3/3
:= 3 × (3 × (33 + 3) + 3)
:= 4 + 4/4 + 44 + 4 + 4
:= 4 × 4 + ((4 + 4)/4 + 44 )
:= 4 + 44/4 + 44 + 4 + 4
:= 5 × 55 − (5/5 + 5)
:= 5 × 55 − 5/5
:= 5 + 5 × 55 − 5/5
:= 6 + 6 × (6 × 6 + 6) + 66/6
6
:= 6 × 6 × 6 + ((6 + 6)/6) − 6
:= 6 × 66 − 666/6 − 6
:= (7 × 7 × 77 − 7)/(7 + 7)
:= 7 + 7 × 7 × 7 − 77 + 7/7
:= 7 × (7 × 7 − 7) − 7/7 − 7 − 7
:= 8 + (88/8 − 8) × (88 − 8/8)
:= 8 + (8 + 8) × (8 + 8) + (8 + 8)/8 + 8
:= 88 + 8 × (8 + 8 + 8) − 8/8
:= 9 + 9 × (9 + 9) − 9/9 + 99
:= 9 + ((9 + 9)/9)9−9/9 + 9
:= 99 + 99 + 9 × 9
Ï 270 := (11 − 1) × (1 + 1 + 1)1+1+1
Ï 275 := 11 × (1 + (1 + 1) × (1 + 11))
Ï 280 := 111 + (1 + 1 + 11)1+1
:= 222 + 2 × (22 + 2)
:= 22 + (222 + 22)/2
:= 2 + 22×(2+2) + 22
:= 3 × 3 × (33 + 3)
:= 33 + (33+3 − 3)/3
:= (3 + 3)3 + (3/3 + 3)3
:= 4 + (44 − 4)/4 + 44
:= 4 + 44/4 + 44 + 4
:= 4 + 4 × 4 + 44 + 4
:= 5 × 55 − 5
:= 5 × 55
:= 5 + 5 × 55
:= 666 − 6 × 66
:= 66 + 6 × 6 × 6 − 6/6 − 6
:= 6 × 6 × 6 + ((6 + 6)/6)6
:= (7 × 7 × 77 + 7)/(7 + 7)
:= 77/7 × (77/7 + 7 + 7)
:= 7 × (7 × 7 − 7) − 7 − 7
:= 8 + 8 + (8 + 8) × (8 + 8) − (8 + 8)/8
:= 8 + (8 + 8) × (8 + 8) + 88/8
:= 88 + 8 × (8 + 8 + 8)
:= 9 + 9 × (9 + 9) + 99
:= 9 × (9 + 9) + (999 + 9 + 9)/9
:= 9/9 + 99 + 99 + 9 × 9
Ï 271 := 1 + (11 − 1) × (1 + 1 + 1)1+1+1
Ï 276 := (1 + 11) × (1 + 11 + 11)
:= 2 + 22×(2+2) + 22/2 + 2
:= 22 + 22×(2+2) − 2
:= 2 + ((22 + 2)2 − 22)/2 + 2
:= 33 + (33+3 + 3)/3
:= 33 + 3 × 3 × 33
:= 33 + (33+3 + 33)/3
:= 4 + 44/4 + 44
:= 4 + 4 × 4 + 44
:= 4 + 4 × 4 + 44 + 4/4 + 4
:= 5/5 + 5 × 55 − 5
:= 5 × 55 + 5/5
:= 5 + 5 × 55 + 5/5
:= 66 + 6 × 6 × 6 − 66/6
:= 66 + 6 × 6 × 6 − 6
:= 66 + 6 × 6 × 6 − 6/6
:= 7 + 7 × 7 × 7 − (7 + 7)/7 − 77
:= 7 + (7 × 7 × 77 − 7)/(7 + 7)
:= 7/7 + 7 × (7 × 7 − 7) − 7 − 7
:= 8 + (8 + 8) × (8 + 8) − 8/8 + 8
:= 8 + (8 × 8 × 8 + 8) × 8/(8 + 8) + 8
:= 8/8 + 8 × (8 + 8 + 8) + 88
:= (99/9 + 9 × 9) × (9 + 9 + 9)/9
:= 9 + 99/9 + 9 × (9 + 9) + 99
:= 9 + 9 × (9 + 9) + 99 + 9/9
Ï 272 := (1 + 1 + 11) × (11 + 11 − 1) − 1
Ï 277 := 1 + (1 + 11) × (1 + 11 + 11)
Ï 281 := 1 + 111 + (1 + 1 + 11)1+1
Ï 282 := (1 + 1) × ((1 + 11)1+1 − 1 − 1 − 1)
:= 22+2 + 22×(2+2)
:= ((22 + 2)2 − 22)/2
:= 2 + 22×(2+2) + 22 + 2
:= 3 + (33+3 − 3)/3 + 33
:= 33 + (33+3 + 3)/3
:= 3 + 3 × (3 × (33 + 3) + 3)
:= 4 × 4 + 44
:= 4 + 4 × 4 + 44 + 4/4
:= 4 + 44 × 4/(4 + 4) + 44
:= 5 × 55 + (5 + 5)/5 − 5
:= 5 × 55 + (5 + 5)/5
:= 5 + 5 × 55 + (5 + 5)/5
:= 6 + (6/6 + 6) × ((6 + 6)/6 + 6 × 6)
:= 66 + 6 × 6 × 6 − 6 + 6/6
:= 66 + 6 × 6 × 6
:= 7 + 7 × 7 × 7 − 7/7 − 77
:= 7 + (7 × 7 × 77 + 7)/(7 + 7)
:= 77 + ((7 + 7)/7)7 + 77
:= 8 + (8 + 8) × (8 + 8) + 8
:= (8 − 8/8 + 8) × (88/8 + 8) − 8
:= 88 + 8 × (8 + 8 + 8) + (8 + 8)/8
:= 99 + 99/9 + 9 × (9 + 9)
:= 99 + 99 − (9 + 9)/9 + 9 × 9
:= 9 + 999/9 + 9 × (9 + 9)
28
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 283 := (1 + 1) × ((1 + 11)1+1 − 1 − 1) − 1
Ï 288 := (1 + 1) × (1 + 11)1+1
Ï 293 := 1 + (1 + 1) × (1 + 1 + 1 + 11)1+1
:= ((22 + 2)2 − 2)/2 − 2 − 2
:= (22 + 2)2 /2
:= 2 + (22+2 + 2/2)2 + 2
:= 3 + (3/3 + 3)3 + (3 + 3)3
:= 3 × (3 × 33 − 3)
:= 3 × 3 × 33 − 3/3 − 3
:= 4 × 4 + 44/4 + 44
:= 4 + 4 × (4 + 4)
:= 4 + 4 × (4 + 4) + 44 + 4/4
:= 5 + 5 × 55 − (5 + 5)/5 + 5
:= (55 + 5)/(5 + 5) − 5 × 5
:= 5 + (55 + 5)/(5 + 5) − 5 × 5
:= 66 + 6 × 6 × 6 + 6/6
:= 6 × (6 × 6 + 6 + 6)
:= 6 + 6 × (6 × 6 + 6 + 6) − 6/6
:= 7 × (7 × 7 − 7) − 77/7
:= 7/7 + 7 × (7 × 7 − 7) − 7
:= 7 × (7 × 7 − 7) − 7/7
:= 8 + 8 + (8 + 8) × (8 + 8) + 88/8
:= 8 + 8 × (8 + 8 + 8) + 88
:= 8 + (8 − 8/8 + 8) × (88/8 + 8)
:= 9 + (99 + 99) + 9 × 9
:= ((9 + 9 + 9) × (99 − 9/9) − 9)/9
9−9/9
:= 9 + ((9 + 9)/9)
4
+9+9
Ï 284 := (1 + 1) × ((1 + 11)1+1 − 1 − 1)
Ï 289 := 1 + (1 + 1) × (1 + 11)1+1
Ï 294 := (1 + 1) × (1 + 1 + 1 + (1 + 11)1+1 )
:= 2 × ((2 × (2 + 2 + 2))2 − 2)
:= (22+2 + 2/2)2
:= 2 + (22 + 2)2 /2 + 2 + 2
:= (3/3 + 3) × ((3 + 3)3 − 3)/3
:= 3/3 + 3 × (3 × 33 − 3)
:= 3 × 3 × 33 − 3
4
:= 44 + 44 − 4 × 4
:= 44 + 4 − 44/4
:= 44 + 44 − (4 + 4)/4 − 4
:= 5 + 5 × 55 − 5/5 + 5
:= 5 × (55 + 5) − 55/5
:= 5 × (55 + 5) − 5/5 − 5
:= 66 + 6 × 6 × 6 + (6 + 6)/6
:= 6/6 + 6 × (6 × 6 + 6 + 6)
:= 6 + 6 × (6 × 6 + 6 + 6)
:= (7 − 77)/7 + 7 × (7 × 7 − 7)
:= (7 + 7)/7 + 7 × (7 × 7 − 7) − 7
:= 7 × (7 × 7 − 7)
:= (8 × (8 × 8 + 8) − 8) × 8/(8 + 8)
:= (8/8 + 8 + 8)(8+8)/8
:= 8 + 8 × 8 + (8 + 8) × 888/(8 × 8)
:= 9 × (9 + 9) + (999 + 99)/9
:= (9 + 9 − 9/9)(9+9)/9
:= ((9 + 9 + 9)/9) × (99 − 9/9)
Ï 285 := (1 + 1) × 11 × (1 + 1 + 11) − 1
Ï 290 := (1 + 1) × (1 + (1 + 11)1+1 )
Ï 295 := 11 × (1 + 1 + 1)1+1+1 − 1 − 1
:= 2 + (22+2 + 2/2)2 + 2 + 2
:= ((22 + 2)2 − 2)/2 − 2
:= 2 + (22 + 2)2 /2
:= 3 × (3 × 33 − 3) − 3
:= 3 + 3 × (3 × 33 − 3) − 3/3
4
:= 3/3 + 3 × 3 × 33 − 3
:= 44 + 44 − 4/4 − 4
:= 44 + 4/4 − 4 × 4 + 44
:= 44 + (4 − 44)/4 + 4
:= 5 + 5 × 55 + 5
:= 5 + (5 × 55 + 5) + 5
:= 5 × (55 + 5) − 5
:= 6 × 66 − 666/6
:= (6 + 6)/6 + 6 × (6 × 6 + 6 + 6)
:= 6 + 6 × (6 × 6 + 6 + 6) + 6/6
:= 7 × (7 × 7 − 7) − (7 + 7)/7 − 7
:= 7 + 7 × (7 × 7 − 7) − 77/7
:= 7/7 + 7 × (7 × 7 − 7)
:= (8 − 8/8 + 8) × (88/8 + 8)
:= 8/8 + (8/8 + 8 + 8)(8+8)/8
:= 8 + 88 + 88 + 888/8
:= (99 − 9/9) × (9 + 9 + 9)/9 − 9
:= (9/9 + 9) × (99/9 + 9 + 9)
:= (99 × (9 + 9 + 9) − (9 + 9))/9
Ï 286 := (1 + 1) × 11 × (1 + 1 + 11)
Ï 291 := 1 + (1 + 1) × (1 + (1 + 11)1+1 )
Ï 296 := 11 × (1 + 1 + 1)1+1+1 − 1
:= 22 × (22/2 + 2)
:= 2 + (22+2 + 2/2)2
:= 2 × ((2 × (2 + 2 + 2))2 + 2 + 2)
:= 33/3 × (33 − 3/3)
:= 3 + 3 × (3 × 33 − 3)
:= 3 × 3 × 33 − 3/3
:= 44 + 44 × 44/(4 + 4)
:= 4 + 4 × (4 + 4) − 4/4 + 44
:= 44 + 44 − 4
:= 5 × 55 + 55/5
:= 5 + 5 × 55 + 55/5
:= 5/5 + 5 × (55 + 5) − 5
:= 6 + ((6 + 6)/6) + 6 × 6 × 6
:= 6 + 6 × 66 − 666/6
:= ((6 + 6)/6 + 6) × (6 × 6 + 6/6)
:= 7 × (7 × 7 − 7) − 7/7 − 7
:= 7 × (7 × 7 − 7) − (7 + 7 + 7)/7
:= (7 + 7)/7 + 7 × (7 × 7 − 7)
:= 8 × 8 + (8 + 8) × 888/(8 × 8)
:= 88 + 8 × (8 + 8 + 8) + 88/8
:= 8 × 888/(8 + 8 + 8)
:= 99 × (9 − 9/9 + 9 + 9)/9
:= 99 + 9 × 9 + 999/9
:= (99 × (9 + 9 + 9) − 9)/9
6
Ï 287 := (1 + 1) × (1 + 11)1+1 − 1
Ï 292 := (1 + 1) × (1 + 1 + (1 + 11)1+1 )
Ï 297 := 11 × (1 + 1 + 1)1+1+1
:= ((22 + 2)2 − 2)/2
:= 2 + (22 + 2)2 /2 + 2
:= ((22 + 2)2 + 22)/2 − 2
:= 3 × (3 × 33 − 3) − 3/3
:= 3 + 3 × (3 × 33 − 3) + 3/3
:= 3 × 3 × 33
4
:= 44 + 44 + 4/4 − 4
4
:= 4 + 4 × (4 + 4) − 4/4
:= 4 + 4 × (4 + 4) + 4
:= 5 × 55 + (55 + 5)/5
:= 5 + (55 + 5)/5 + 5 × 55
6
:= 5 × 55 + (55 + 55)/5
:= 6 × (6 × 6 + 6 + 6) − 6/6
:= 6 + ((6 + 6)/6) + 6 × 6 × 6 + 6
:= 6 × 6 × 66/(6 + (6 + 6)/6)
:= 7 × (7 × 7 − 7) − 7
:= 7 × (7 × 7 − 7) − (7 + 7)/7
:= 7 × (7 × 7 − 7) + (7 + 7 + 7)/7
:= 88 + 88 + 888/8
:= (8 × (8 × 8 + 8) + 8) × 8/(8 + 8)
:= 8 + (8/8 + 8 + 8)(8+8)/8
:= (99 × (9 + 9 + 9) − 9)/9 − 9
:= 99 + 9 × 9 + (999 + 9)/9
:= 99 + 99 + 99
29
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 298 := 1 + 11 × (1 + 1 + 1)1+1+1
:= 2 × 22 + 2
2×(2+2)
−2
Ï 303 := (1 + 1 + 1) × (1 + (11 − 1)1+1 )
:= 222 + (2/2 + 2)
:= 3 + (3 × 3 × 33 + 3)
:= 3/3 + 3 × 3 × 33
Ï 308 := (1 + 1) × 11 × (1 + 1 + 1 + 11)
:= 22 × (22+2 − 2)
2+2
:= 33/3 + 3 × 3 × 33
:= 4 + 44 + 44 + 4
4
:= 44 + 44 − (4 + 4)/4
:= 4 + 44 − 4/4 + 4
:= 55 + (5 − (5 + 5)/5)5
:= (55 + 5)/(5 + 5) − 5 − 5
:= (55 + 5)/(5 + 5) − 5
:= (66 − 6)/6 + 6 × (6 × 6 + 6 + 6)
:= 6 + 6 × 6 × 66/(6 + (6 + 6)/6)
:= 66 + 66 × 66/(6 + 6 + 6)
:= 77/7 + 7 × (7 × 7 − 7) − 7
:= 7 + 7 × (7 × 7 − 7) + (7 + 7)/7
:= 7 + 7 × (7 × 7 − 7) + 7
:= (88 × (88/8 + 8 + 8) + 8)/8
:= 8 × (8 + 8 + 8) + 888/8
:= 88 × (8 × 8 − 8)/(8 + 8)
:= (99 × (9 + 9 + 9) + 9)/9
:= 9 + (99 − 9/9) × (9 + 9 + 9)/9
:= 99/9 × (9/9 + 9 + 9 + 9)
Ï 299 := 11 + (1 + 1) × (1 + 11)1+1
Ï 304 := 1 + (1 + 1 + 1) × (1 + (11 − 1)1+1 )
2+2
2
:= ((22 + 2)2 + 22)/2
:= 2 + (2
:= 3 + (3 × 3 × 33 − 3/3)
:= 3 + 3 × 3 × 33 + 3/3 + 3
4
+ 2) − 22
Ï 309 := 1 + (1 + 1) × 11 × (1 + 1 + 1 + 11)
:= 22 + ((22 + 2)2 − 2)/2
:= 3 + 3 × (3 × 33 + 3)
:= 44 + (44 − 44)/4
4
:= 44 + 4 − 4/4
:= 4 + 44 + 4
:= 5 × (55 + 5) − 5/5
:= 5 + 5 × (55 + 5) − 5/5
:= 5 + (5 × (55 + 5) − 5/5) + 5
:= 66/6 + 6 × (6 × 6 + 6 + 6)
:= ((6 + 6)/6 + 6) × ((6 + 6)/6 + 6 × 6)
:= 66 + (6 × 6/(6 + 6))6−6/6
:= 7 + (7 × (7 × 7 − 7) − (7 + 7)/7)
:= (7/7 + 7) × (7 × 7 − 77/7)
:= 7 + 7 × (7 × 7 − 7) + 7/7 + 7
:= (8 × (888 + 8) + 8)/(8 + 8 + 8)
:= (8 + 8) × (88/8 + 8)
:= 8/8 + 88 × (8 × 8 − 8)/(8 + 8)
:= (99 × (9 + 9 + 9) + 9 + 9)/9
:= (9 + 9) × (9 + 9) − 99/9 − 9
:= 99 + 999/9 + 99
Ï 310 := (1 + 1) × (11 + (1 + 11)1+1 )
Ï 300 := (1 + 1 + 1) × (11 − 1)1+1
Ï 305 := (11 × 111 − 1)/(1 + 1 + 1 + 1)
:= 2 × 22 + 22×(2+2)
:= 2 + (2/2 + 2)2+2 + 222
:= 2 + 22 × (22+2 − 2)
:= 3 + 3 × 3 × 33
:= 333 − 33 − 3/3
:= ((3/3 + 3) + 3)3 − 33
:= 44 + 44
:= 4 + 44 + 44 + 4/4
:= 44 + (44 − 4)/4 + 44
:= 5 × (55 + 5)
:= 5 + 5 × (55 + 5)
:= 5 + 5 × (55 + 5) + 5
:= (6 − 66) × (6/6 − 6)
:= (6 − 6/6) × (66 − 6 + 6/6)
:= (6 − 6/6) × ((6 + 6)/6 − 6 + 66)
:= 7 + 7 × (7 × 7 − 7) − 7/7
:= 77/7 + 7 × (7 × 7 − 7)
:= 7 + 7 × (7 × 7 − 7) + (7 + 7)/7 + 7
:= (8 × 8 × 8 + 88) × 8/(8 + 8)
:= 8/8 + (8 + 8) × (8 + 88/8)
:= 88 + (8 + 8) × 888/(8 × 8)
:= (9/9 + 99) × (9 + 9 + 9)/9
:= 9 + (99 × (9 + 9 + 9) − 9)/9
:= 99 + 99 + (999 + 9)/9
Ï 301 := 1 + (1 + 1 + 1) × (11 − 1)1+1
Ï 306 := (1 + 1) × (11 × (1 + 1 + 1 + 11) − 1)
Ï 311 := 1 + (1 + 1) × (11 + (1 + 11)1+1 )
:= 2 + ((22 + 2)2 + 22)/2
:= 22 × (22+2 − 2) − 2
:= 22 + (22+2 + 2/2)2
:= 3 + 3 × 3 × 33 + 3/3
:= 3 × (3 × 33 + 3)
:= 3 + 3 × 3 × 33 + 33/3
4
:= 4 + 44 + 4 + (4 + 4)/4
:= 44 + 44/4 + 44
:= 5/5 + 5 × (55 + 5)
:= 5 + 5 × (55 + 5) + 5/5
:= 55/5 + 5 × (55 + 5)
:= 6/6 + (6 − 66) × (6/6 − 6)
:= 6 + (6 − 66) × (6/6 − 6)
:= 6 + (6 − 6/6) × (66 − 6 + 6/6)
:= 7 + 7 × (7 × 7 − 7)
:= (7 − 7/7) × (7 × 7 + (7 + 7)/7)
:= 7 + 7 × (7 × 7 − 7) + (77 − 7)/7
:= 44 + 4/4 + 4
4
:= 8 + (8 − 8/8 + 8) × (88/8 + 8) + 8
:= (8 + 8)/8 + (8 + 8) × (8 + 88/8)
:= 8 + 888/8 + 8 × (8 + 8 + 8)
:= 9 × 9 + (99/9 + 9) × 99/9
:= (9 + 9) × (9 + 9 − 9/9)
:= (9 + 9) × (9 + 9) − (99 + 9 + 9)/9
Ï 302 := 1 + 1 + (1 + 1 + 1) × (11 − 1)1+1
Ï 307 := 111 + (1 + 1 + 1 + 11)1+1
2
:= (22+2 + 2)2 − 22
:= 3 + 3 × 3 × 33 − 3/3 + 3
Ï 312 := (1 + 1) × (1 + 11) × (1 + 1 + 11)
:= (22 + 2/2) − 222
:= (22 + 2) × (22/2 + 2)
:= 3/3 + 3 × (3 × 33 + 3)
:= 3 + 3 × (3 × 33 + 3) + 3
:= 4 + 44 − 4/4 + 4 + 4
:= 4 + 4 + 4 + 44 + 44
:= (5 + 5)/5 + 5 × (55 + 5)
:= (55 − 55)/(5 + 5)
:= (55 − 5)/(5 + 5)
:= 6 + ((6 + 6)/6 + 6) × (6 × 6 + 6/6)
:= 6 + (6 − 66) × (6/6 − 6) + 6/6
:= 6 × ((6 + 6)/6)6 − 6 − 6
:= 44 + (4 + 4)/4 + 4
4
4
:= 7 + 7 × (7 × 7 − 7) + 7/7
:= 7 + 7 × (7 × 7 − 7) − 7/7 + 7
:= 7 + 7 × (7 × 7 − 7) + 77/7
:= (88 − 8/8 + 8 × 8) × (8 + 8)/8
:= 88 × (8 × 8 − 8)/(8 + 8) − 8/8
:= 8 + (8 + 8) × (88/8 + 8)
:= (9 × (9 + 9) − 99/9) × (9 + 9)/9
:= 9 + (99 × (9 + 9 + 9) + 9)/9
:= (9 + 9) × (9 + 9) − (99 + 9)/9
30
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 313 := 1 + (1 + 1) × (1 + 11) × (1 + 1 + 11)
:= (2
2+2
2
+ 2) − 22/2
Ï 318 := (1 + 1 + 1) × (111 − 1) − 1 − 11
:= 2
2+2
Ï 323 := 1 + (1 + 1 + 1) × 111 − 11
:= ((22+2 + 2)2 ) − 2/2
× (22 − 2) − 2
:= 3 + (3/3 + 3 + 3)3 − 33
:= 3 + 3 × 33 + (3 + 3)3
:= 333 + ((3 − 33)/3)
:= 4 + (44 − 44)/4 + 44
:= 44 + (44 − 4 − 4)/4
:= 4 + (((44 − 4)/4) + 44 )
:= (55 + 5)/(5 + 5)
:= 5 + (55 + 5)/(5 + 5)
:= 5 + ((55 + 5)/(5 + 5) + 5)
:= 6 × (66 − 6 − 6) − 66/6
:= 66 + 6 × (6 × 6 + 6)
:= (6 × (66 − (6 + 6))) − 6/6
:= 7 + 7 × (7 × 7 − 7) + (77 + 7)/7
:= 7 × 7 × 7 − 77/7 − 7 − 7
:= 7/7 + (7 × 7 × 7 − (7 + 7 + 7))
:= 8 + (8 + 8) × (88/8 + 8) + 8/8
:= 8 × 8 + (8 + 8) × (8 + 8) − (8 + 8)/8
:= (8/8 + 8 + 8) × (88/8 + 8)
:= (9 + 9) × (9 + 9) − 99/9
:= 9 + 999/9 + 99 + 99
:= (9 + 9) × (9 + 9) − 9/9
Ï 314 := (1 + 1) × (1 + (1 + 11) × (1 + 1 + 11))
Ï 319 := 11 × ((11 − 1) × (1 + 1 + 1) − 1)
Ï 324 := ((1 + 1) × (11 − 1 − 1))1+1
:= 2 + (22 + 2) × (22/2 + 2)
:= 22+2 × (22 − 2) − 2/2
:= (22+2 + 2)2
:= 3 × 33 + (3 + 3)3 − 3/3
:= 333 − 33/3 − 3
:= 3 × (3 × (33 + 3))
4
4
:= 4 + (4 − 4 − 4)/4 − 4
:= 4 + (4 − 4)/4
:= 4 × (4 − 4/4)4
:= 5/5 + (55 + 5)/(5 + 5)
:= 5 + (55 + 5)/(5 + 5) + 5/5
:= (5/5 + 5) × (55 − 5/5)
:= (6 − 6/6) × ((6 + 6)/6) − 6
:= 66 + 6 × (6 × 6 + 6) + 6/6
:= 6 × (66 − (6 + 6))
:= 7 × (7 × 7 + 7) − (7/7 + 77)
:= 7 + 7 + 7 × (7 × 7 − 7) + 77/7
:= (77/7 + 7)(7+7)/7
:= 8 + (8 + 8) × (88/8 + 8) + (8 + 8)/8
:= 8 × 8 + (8 + 8) × (8 + 8) − 8/8
:= ((((8 + 8)/8) + 8) + 8)(8+8)/8
:= (9 + 9) × (9 + 9) − 9/9 − 9
:= 99/9 × (99/9 + 9 + 9)
:= (9 + 9) × (9 + 9)
4
6
Ï 315 := ((1 + 1)1+11 − 1)/(1 + 1 + 11)
4
Ï 320 := (11 − 1) × (11 × (1 + 1 + 1) − 1)
Ï 325 := 1 + ((1 + 1) × (11 − 1 − 1))1+1
:= 222 − (22/2 + 2)2
:= 22+2 × (22 − 2)
:= 2/2 + ((22+2 + 2)2 )
:= 3 × (3 × 33 + 3 + 3)
:= 333 + (3 − 33)/3 − 3
:= 3 + (333 − 33/3)
:= 4 + (4 − 4)/4 − 4
:= 4 × 4 × (4 × 4 + 4)
:= 4 + ((44 + 4)/4 + 44 )
:= (5 × 5 + 55 )/(5 + 5)
:= (5 + 5) × ((5 + 5)/5)5
4
4
:= 5 × (55 + 5 + 5)
6
:= (666 − 6 × 6) × 6/(6 + 6)
:= (6 − 6/6) × ((6 + 6)/6)
:= 6/6 + (6 × (66 − (6 + 6)))
:= 7 × (7 × 7 + 7) − 77
:= (7/7 + 7) × (7 × 7 − (7 + 7)/7 − 7)
:= 7 × 7 × 7 − (77/7 + 7)
:= 88/8 + (8 + 8) × (88/8 + 8)
:= 8 × (8 + 8 + 8 + 8 + 8)
:= (8/8 + 8 × 8) × ((8 − 88/8) + 8)
:= (9 + 9) × (9 + 9) − 9
:= (9 − 9/9) × (9 × 9 × 9 − 9)/(9 + 9)
:= 9/9 + (9 + 9) × (9 + 9)
Ï 316 := (1 + 1) × ((1 + 1 + 11)1+1 − 11)
:= 2 × (2 × ((2/2) + 2)2+2 − 2)
3
:= (3/3 + 3 + 3) − 3
Ï 321 := (1 + 1 + 1) × 111 − 1 − 11
:= 2/2 + (2
3
2+2
× (22 − 2))
Ï 326 := (1 + 1 + 1) × (111 − 1 − 1) − 1
:= 2 + ((22+2 + 2)2 )
:= 333 − (3 × 3 + 3)
:= 333 − ((3/3 + 3) + 3)
:= 44 + (44 + 4)/4
:= 4 + ((((44 + 4) + 4)/4) + 44 )
:= 5/5 + ((5 + 5) × ((5 + 5)/5)5 )
:= 5/5 + (5 × (55 + 5 + 5))
:= 6 × (6 × 6 + 6) + ((6 + 6)/6)
:= 6 × 6 × 6 + (666/6 − 6)
:= 6 + ((6 − 6/6) × ((6 + 6)/6)6 )
:= 7/7 + 7 × (7 × 7 + 7) − 77
:= 7 × 7 × 7 − (7/7 + 7 + 7 + 7)
:= 7 × 7 × 7 + (((7 − 77)/7) − 7)
:= 8 + 88 × (8 × 8 − 8)/(8 + 8)
:= 8/8 + (((8 + 8) × (8 + 8)) + 8 × 8)
:= 8 + ((((8 + 8) × (8 + 8)) − ((8 + 8)/8)) + 8 × 8)
:= 9/9 + (9 + 9) × (9 + 9) − 9
:= (9 + 9) × (9 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 + 9) × (9 + 9)
:= 4 × 4 + 44 + 4
4
:= 5 + 5 × (55 + 5) + 55/5
6
Ï 317 := 1 + (1 + 1) × ((1 + 1 + 11)1+1 − 11)
2
:= 2 + 22 − (22/2 + 2)
2
:= 333 + 33/3 − 33
4
Ï 322 := (1 + 1 + 1) × 111 − 11
:= ((2
2+2
+ 2) ) − 2
:= 333 − (3 + 3)
:= 333 − 33/3
4
Ï 327 := (1 + 1 + 1) × (111 − 1 − 1)
:= 2 + (((22+2 + 2)2 ) + 2/2)
2
:= 4 + (4 + 4)/4 − 4
:= 4 + (((4 + 4) + 4)/4)
:= 4 + ((((44 − 4)/4) + 44 ) + 4)
:= 5 + (55 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5)
:= 5 + (((55 − 5)/(5 + 5) + 5) + 5)
:= 66 + 6 × (6 × 6 + 6) − 6/6
:= (6 × (66 − (6 + 6))) − (6 + 6)/6
:= 6 × 6 × 6 + 666/6
:= 7 × 7 × 7 − (7 + 7 + 7)
:= 7 × 7 × 7 − (((7 + 7)/7 + 7) + 7)
:= (77/7 + 7)
(7+7)/7
−7
4
4
:= 8 + 88 × (8 × 8 − 8)/(8 + 8) + 8/8
:= 8 × 8 + (((8 + 8) × (8 + 8)) + ((8 + 8)/8))
:= 8 + ((((8 + 8) × (8 + 8)) − 8/8) + 8 × 8)
:= (9 + 9)/9 + (9 + 9) × (9 + 9) − 9
:= (9 + 9) × (9 + 9) − (9 + 9)/9
:= (9 + 9) × (9 + 9) + ((9 + 9 + 9)/9)
31
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 328 := 1 + (1 + 1 + 1) × (111 − 1 − 1)
Ï 338 := (1 + 1) × (1 + 1 + 11)1+1
Ï 333 := (1 + 1 + 1) × 111
:= 2 + (((22+2 + 2)2 ) + 2)
:= 222 + 222/2
:= 3/3 + (333 − (3 + 3))
:= 333
:= 2 × ((22/2 + 2)2 )
:= 3 + ((333 − 3/3) + 3)
4
4
:= 4 + (4 × (4 − 4/4) )
:= 4 + ((4 − 4/4) − 4)
:= 4/4 + ((4 − 4/4)4 + 44 )
:= 5 + (((55 + 5)/(5 + 5) + 5) + 5)
:= 5 × 5 + ((55 + 5)/(5 + 5) − 5)
:= 5 × 5 + (55 + 5)/(5 + 5)
:= 6 × 6 × 6 + (666 + 6)/6
:= 666 × 6/(6 + 6)
:= 6 + (6 × 66 − ((6 + 6)/6)6 )
:= 7 × 7 × 7 − (7/7 + 7 + 7)
:= 7 × 7 × 7 + (7 − 77)/7
:= 7 × 7 × 7 + ((7 + 7)/7 − 7)
:= 8 + (((8 + 8) × (8 + 8)) + 8 × 8)
:= (88/8 − 8) × 888/8
:= 8 × (8 × 8 − 8) + ((8 − 888)/8)
:= 9 + ((99/9) × (99/9 + 9 + 9))
:= 9 + (9 + 9) × (9 + 9)
:= 9 + (((9 × 9 + 9)/(9 + 9)) + (9 + 9) × (9 + 9))
4
Ï 329 := (1 + 1 + 1) × (111 − 1) − 1
Ï 334 := 1 + (1 + 1 + 1) × 111
Ï 339 := (1 + 1 + 1) × (1 + 1 + 111)
:= 2 × (((22/2 + 2)2 ) − 2)
:= (((22 + 2 + 2)2 ) + 2)/2
:= 3/3 + 333
:= 3 + (333 + 3)
:= 444 + ((4 − 444)/4)
:= (4 × ((4 − 4/4)4 + 4)) − 4/4
:= 55 + (5 × 55 − 5/5)
:= 5 + ((5 × 55 − 5/5) + 55)
:= ((55 − 55)/5) − 5 × 55
:= 6 × 66 − (66 + 6/6)
:= 6/6 + 666 × 6/(6 + 6)
:= 6 + 666 × 6/(6 + 6)
:= 7 × 7 × 7 − (7 + 7)
:= 7 × 7 × 7 − ((7 + 7)/7 + 7)
:= 7 + (7 × 7 × 7 − (77/7))
:= 8 + ((((8 + 8) × (8 + 8)) + 8/8) + 8 × 8)
:= (((8 + 8 + 8) × 888/8) + 8)/8
:= 8 + (((8/8 + 8 + 8) × (88/8 + 8)) + 8)
:= (9 + 9) × (9 + 9) + ((9 × 9 + 9)/(9 + 9))
:= 9 + ((9 + 9) × (9 + 9) + 9/9)
:= 9 + ((999 − 9)/((9 + 9 + 9)/9))
:= 2 + ((((2
2+2
2
+ 2) ) + 2/2) + 2)
:= 333 − (3/3 + 3)
4
4
:= 4 + (((4 + 4)/4 + 4 ) + 4)
Ï 335 := 1 + 1 + (1 + 1 + 1) × 111
Ï 340 := 1 + (1 + 1 + 1) × (1 + 1 + 111)
:= 22 × ((22/2 + 2) + 2)
:= 2 + (222/2 + 222)
:= 2 + (2 × ((22/2 + 2)2 ))
:= 333 − 3
:= 3 + (333 − 3/3)
:= (((3/3 + 3) + 3)3 ) − 3
Ï 330 := (1 + 1 + 1) × (111 − 1)
:= 4 + (((4 − 4) + 44)/4)
:= 4 + (((44 + 44 )/4) + 44 )
:= 55 + 5 × 55
:= 5 + (5 × 55 + 55)
:= 66 × (6 − 6/6)
:= (6 − 6/6) × (66 + 6/6)
:= 7/7 + (7 × 7 × 7 − (7 + 7))
:= 7 × 7 × 7 − (7/7 + 7)
:= 88/8 × ((88 + 88)/8 + 8)
:= ((8 − 8/8)88/8−8 ) − 8
:= ((8 × 88 − 8)/((8 + 8)/8)) − 8
:= (999 − 9)/((9 + 9 + 9)/9)
:= 99/9 + (9 + 9) × (9 + 9)
:= ((9 − 9/9) + 9) × (99/9 + 9)
4
4
Ï 331 := 1 + (1 + 1 + 1) × (111 − 1)
Ï 336 := (1 + 1 + 1) × (1 + 111)
:= 4 × ((4 − 4/4)4 + 4)
:= 5 + ((5 × 55 + 55) + 5)
:= (6 − 6/6) × (((6 + 6)/6) + 66)
:= 7 × 7 × 7 − (7 + 7 + 7)/7
Ï 341 := 11 + (1 + 1 + 1) × (111 − 1)
:= 222 + (222/2 − 2)
:= 2 × (2 × (2 × ((2 × 22) − 2)))
:= 2 + ((((22 + 2 + 2)2 ) + 2)/2)
:= 3/3 + (333 − 3)
:= 3 + 333
:= 3 × 3 + (333 − 3/3)
:= 4 × ((4 × (4 × 4 + 4)) + 4)
:= 4 + ((4 − 4/4)4 + 44 )
4
4
:= 4 + ((44 + 4 )/4)
:= 55 + (5 × 55 + 5/5)
:= (5/5 + 5) × (55 + 5/5)
:= 6/6 + 66 × (6 − 6/6)
:= 6 + 66 × (6 − 6/6)
:= 7 × 7 × 7 − (77 + 7)/7
:= 7 × 7 × 7 − 7
:= 6 + ((6 − 6/6) × (66 + 6/6))
:= 8 + ((8/8 + 8 + 8) × (88/8 + 8))
:= 88 + (((8 + 8) × (8 + 8)) − 8)
:= 9 + ((9 + 9) × (9 + 9) − ((9 + 9)/9))
:= (9 + 9) × (9 + 9) + (99 + 9)/9
Ï 332 := (1 + 1 + 1) × 111 − 1
:= 5 + ((5/5 + 5) × (55 + 5/5))
Ï 337 := 1 + (1 + 1 + 1) × (1 + 111)
:= 7 × 7 × 7 − (7 + 7)/7
:= 8 + ((88/8 − 8) × 888/8)
:= 9 + (((9 + 9) × (9 + 9) − 9/9) + 9)
Ï 342 := (1 + 1 + 1) × (1 + (1 + 1 + 111))
:= 2 + (22 × ((22/2 + 2) + 2))
:= (((22 + 2 + 2)2 ) − 2)/2
:= 2 × (((22/2 + 2)2 ) + 2)
:= 333 − 3/3
:= 3 + (333 + 3/3)
:= 3 × 3 + 333
:= 4 + ((4 × (4 − 4/4)4 ) + 4)
:= 5 × 5 + (55 − 55)/(5 + 5)
4
4
:= 4 + (4 − 4/4)
:= (4 − 44)/4 + ((4 + 4) × 44)
5
:= 5 × 5 + (5 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5 × 5)
:= 6 × 66 − ((6 + 6)/6)
:= 6 + (66 × (6 − 6/6) + 6/6)
:= 6 + (66 × (6 − 6/6) + 6)
:= 7 × 7 × 7 − 77/7
:= 7/7 + (7 × 7 × 7 − 7)
:= 7 × 7 × 7 − 7/7
:= 8 + (((((8 + 8)/8) + 8) + 8)(8+8)/8 )
:= 8 × (8 × 8 − 8) − 888/8
:= 88 + (((8 + 8) × (8 + 8)) − ((8 + 8)/8))
:= 9 + ((9 + 9) × (9 + 9) − 9/9)
:= 9 × 9 + (((9 + 9)/9)9−9/9 )
:= 9 + ((9 + 9) × (9 + 9) + 9)
6
32
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 343 := 11 + (1 + 1 + 1) × 111 − 1
Ï 348 := 1 + 11 + (1 + 1 + 1) × (1 + 111)
Ï 353 := 1 + 11 × (11 × (1 + 1 + 1) − 1)
:= 2 × ((2 × 2 × 2 × 22) − 2)
:= 2/2 + (22 × 22+2 )
:= ((3/3 + 3) + 3)3
:= 3 + ((333 + 3 × 3) + 3)
:= 3 × 3 + (333 + 33/3)
:= ((4 − 4/4) + 4)4−4/4
:= ((4 + 4) × 44) − 4
:= 4/4 + ((4 + 4) × 44)
:= 5 + ((55 + 5)/(5 + 5) + 5 × 5)
:= (5 − 5 − 5)/5 − 5 × 55
:= ((55 × ((5 + 5)/5)5 ) + 5)/5
:= 6 × (((6 + 6)/6)6 − 6)
:= 6 × (66 − 6) − 6/6 − 6
:= 222 + (22/2)
2
:= (6/6 + 6)6×6/(6+6)
:= 7 × 7 × 7
:= (8 − 8/8)88/8−8
:= 9 + (((9 + 9) × (9 + 9) + 9/9) + 9)
Ï 344 := 11 + (1 + 1 + 1) × 111
:= 2 × (2 × (2 × 2 × 22 − 2))
5
:= 7 + (7 × 7 × 7 − ((7 + 7)/7))
:= 7 × 7 × 7 + (77 − 7)/7
:= (8 × 88 − 8)/((8 + 8)/8)
:= 8/8 + (8 × (88/((8 + 8)/8)))
:= ((99 + 9)/9) × (99/9 + 9 + 9)
:= 9 + (((9 + 9) × (9 + 9) + (99/9)) + 9)
Ï 349 := 11 + (1 + 1) × (1 + 1 + 11)1+1
:= (22 × 2
2+2
Ï 354 := 1 + 1 + 11 × (11 × (1 + 1 + 1) − 1)
:= 2 + (22 × 22+2 )
) − 2/2 − 2
:= 333 + 33/3
:= 3 + ((((3/3 + 3) + 3)3 ) + 3)
:= 3 + (333 + (3 × (3 + 3)))
:= 44 + (44 + 44 )
:= 4/4 + (((4 + 4) × 44) − 4)
:= (4 + 4)/4 + ((4 + 4) × 44)
:= (55 − 5)/5 − (5 × 55 + 5)
:= (5 − 5)/5 − 5 × 55
:= 5 + ((55 − 5)/5 − 5 × 55)
:= 6 + ((6 × 66 − ((6 + 6)/6)6 ) + 6)
:= 6 × (66 − 6) − 66/6
:= 6 × (66 − 6) − 6
:= 7 + (7 × 7 × 7 − 7/7)
:= 7 × 7 × 7 + 77/7
:= 8/8 + ((8 × 88 − 8)/((8 + 8)/8))
:= (8 + 8)/8 + (8 × (88/((8 + 8)/8)))
:= 9 + (((9 − 9/9) + 9) × (99/9 + 9))
:= 9 × (9 + 9 + 9) + 999/9
:= 7/7 + 7 × 7 × 7
:= 88 + ((8 + 8) × (8 + 8))
:= 9 + ((9 + 9) × (9 + 9) + (99/9))
Ï 345 := 1 + 11 + (1 + 1 + 1) × 111
5
Ï 350 := 11 + (1 + 1 + 1) × (1 + 1 + 111)
2+2
Ï 355 := 11 + 11 + (1 + 1 + 1) × 111
:= 2 + ((22 × 22+2 ) + 2/2)
:= 2 + ((22/2)2 + 222)
:= (22 × 2
:= 3 + (333 + 3 × 3)
:= 3 + ((333 + 33/3) + 3)
:= 3 + ((((3/3 + 3) + 3)3 ) + 3 × 3)
:= 4 + (((4 − 4/4)4 + 44 ) + 4)
:= ((4 + 4) × 44) − (4 + 4)/4
:= 4 + (((4 + 4) × 44) − 4/4)
:= 5 × ((55 + 5 + 5) + 5)
:= 55 + (5 × (55 + 5))
:= 55 − (5 × 555 + 5)
)−2
6
:= 6 + (666 × 6/(6 + 6) + 6)
:= 7 × 7 × 7 + (7 + 7)/7
:= 8/8 + (((8 + 8) × (8 + 8)) + 88)
:= 9 + ((9 + 9) × (9 + 9) + ((99 + 9)/9))
Ï 346 := 1 + 1 + 11 + (1 + 1 + 1) × 111
:= (6 − 6/6) × (((6 + 6)/6) + 6)
:= 6/6 + (6 × (66 − 6) − 6)
:= 7 + 7 × 7 × 7
:= 7 × 7 × 7 + (77 + 7)/7
:= ((8 + 8)/8) × (888/8 + 8 × 8)
:= 88 + (((8 + 8) × (8 + 8)) + (88/8))
:= (((9 + 9)/9)9 ) − 9 × (9 + 9)
:= 99 + (((9 + 9)/9)9−9/9 )
Ï 351 := 11 × (11 × (1 + 1 + 1) − 1) − 1
Ï 356 := (1 + 1 + 1) × (111+1 − (1 + 1)) − 1
:= 22 + ((22+2 + 2)2 )
:= (22 × 22+2 ) − 2/2
:= 2 + ((22 × 22+2 ) + 2)
:= 3 + (((3/3 + 3) + 3)3 )
:= 3 × (3 × (33 + 3 + 3))
:= 33 + (333 − (3/3 + 3))
:= ((4 + 4) × 44) − 4/4
:= 4 + ((4 + 4) × 44)
:= ((4 + 4) × 44) − ((4 + 4)/4 + 4)
:= (55 + 5)/5 − (5 × 55 + 5)
:= 6 + ((6 − 6/6) × (((6 + 6)/6) + 66))
:= 7 × 7 × 7 + (7 + 7 + 7)/7
:= 88 + (((8 + 8) × (8 + 8)) + ((8 + 8)/8))
9−9/9
:= 9 + ((((9 + 9)/9)
) + 9 × 9)
Ï 347 := 11 + (1 + 1 + 1) × (1 + 111)
5
:= (5 + 5)/5 − 5 × 55
:= 5 + ((55 + 5)/5 − 5 × 55)
:= (666 + 6 × 6)/((6 + 6)/6)
:= (6 + 6)/6 + (6 × (66 − 6) − 6)
:= 7 + (7 × 7 × 7 + 7/7)
:= 7 + ((7 × 7 × 7 − 7/7) + 7)
:= 8 + ((8 − 8/8)88/8−8 )
:= (8 × 88 + 8)/((8 + 8)/8)
:= 9 + (((9 + 9) × (9 + 9) + 9) + 9)
:= (((9 × 9 × 9 × 9) + 9)/(9 + 9)) − 9
Ï 352 := 11 × (11 × (1 + 1 + 1) − 1)
Ï 357 := (1 + 1 + 1) × (111+1 − 1 − 1)
:= 2 + (((22/2)2 + 222) + 2)
:= 22 × 22+2
:= (2/2 + 2) × ((22/2)2 − 2)
:= 3 + (333 + 33/3)
:= 3 × 3 + (((3/3 + 3) + 3)3 )
:= 33 + (333 − 3)
:= (4 + 4) × 44
:= 4 + (((4 + 4) × 44) + 4/4)
:= ((4 + 4) × 44) − (4/4 + 4)
5
:= 55/5 × ((5 + 5)/5)
:= 5 + (55/5 × ((5 + 5)/5)5 )
:= 6 × (66 − 6) − (6/6 + 6 + 6)
:= 6 × (66 − 6) − ((6 + 6)/6 + 6)
:= 66 × 66/(6 + 6) − 6
:= 7 × 7 × 7 + (77/7 − 7)
:= 7 + (7 × 7 × 7 + ((7 + 7)/7))
:= 7 + (7 × 7 × 7 + 7)
:= ((8 × 88 − 8)/((8 + 8)/8)) − 8/8
:= 8 × (88/((8 + 8)/8))
:= 8/8 + ((8 × 88 + 8)/((8 + 8)/8))
:= 9 × 9 × (9 + 9) − 9999/9
:= ((9/9 + 9 + 9)(9+9)/9 ) − 9
:= ((9 − 9/9) + 9) × (((99 + 9)/9) + 9)
5
:= (55/5 × ((5 + 5)/5) ) − 5
33
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 358 := 1 + (1 + 1 + 1) × (111+1 − (1 + 1))
Ï 368 := 1 + 1 + (1 + 1 + 1) × (1 + 111+1 )
Ï 363 := 11 × 11 × (1 + 1 + 1)
:= 2 + (((22 × 22+2 ) + 2) + 2)
:= 2 + ((22 − (2/2 + 2))2 )
:= 2 × (2 × (2 × ((2 × 22) + 2)))
:= 33 + ((333 − 3) + 3/3)
:= 33 × 33/3
:= 3 + (((33 × 33 − 3)/3) + 3)
:= 4 + (((4 + 4) × 44) + (4 + 4)/4)
:= 4 + (444/4 − 4)
:= 4 × (44 + 44 + 4)
:= 5 + (((55 × ((5 + 5)/5)5 ) + 5)/5)
:= 55 + ((55 + 5)/(5 + 5) − 5)
:= 55 + (55 + 5)/(5 + 5)
:= 6 × (66 − 6) − (6 + 6)/6
:= 66 × 66/(6 + 6)
:= 6 + (6 × (66 − 6) + ((6 + 6)/6))
:= 7 + ((7 × 7 × 7 + 7/7) + 7)
:= 7 + (((7 × 7 × 7 − 7/7) + 7) + 7)
:= 7 + ((7 × 7 × 7 + (77/7)) + 7)
:= 8 × (8 × 8 − 8) − ((8 + 8)/8 + 88)
:= 88/8 + (8 × (88/((8 + 8)/8)))
:= 8 + (8 × (8 × 8 − 8) − 88)
:= ((9 + 9)/9) × ((9 × 9 − 9/9) + 99)
:= (99 × 99)/(9 + 9 + 9)
:= 9 + (((((9 + 9)/9)9 ) − 9 × (9 + 9)) + 9)
Ï 359 := (1 + 1 + 1) × (111+1 − 1) − 1
4
Ï 364 := 1 + 11 × 11 × (1 + 1 + 1)
Ï 369 := (1 + 1 + 1) × (1 + (1 + 111+1 ))
:= ((22 − (2/2 + 2))2 ) − 2
:= 2 × ((2 × (2 × 2 × 22 + 2)) + 2)
:= (2/2 + 2) × ((22/2)2 + 2)
:= 33 + (333 − 3/3)
:= (33 × 33 + 3)/3
:= 3 + (333 + 33)
:= 4 + ((((4 + 4) × 44) − 4/4) + 4)
:= 4 + ((((4 + 4) × 44) + 4) + 4)
:= (4/4 + 4)4 − 44
:= (5/5 + 5) × (55 + 5) − 5/5
:= (5 × (5 × (5 + 5 + 5))) − 55/5
:= (5 × (5 × (5 + 5 + 5))) − (5/5 + 5)
:= 6 × (66 − 6) − 6/6
:= 6 + (6 × (66 − 6) − ((6 + 6)/6))
:= 6 + 66 × 66/(6 + 6)
:= 7 + ((7 × 7 × 7 + ((7 + 7)/7)) + 7)
:= 7 + (7 × 7 × 7 + 7 + 7)
:= 7 + (((77 + 7)/7 + 7 × 7 × 7) + 7)
:= 8 × (8 × 8 − 8) − (8/8 + 88)
:= 8 + ((8 × 88 + 8)/((8 + 8)/8))
:= 8 + ((88/8 + 8)(8+8)/8 )
:= 9 + ((((9 + 9)/9)9 ) − 9 × (9 + 9))
:= ((9 × 9 × 9 × 9) − 9)/(9 + 9)
:= 9 + ((9 + 9) × (99/9 + 9))
Ï 360 := (1 + 1 + 1) × (111+1 − 1)
Ï 365 := 1 + 1 + 11 × 11 × (1 + 1 + 1)
2
:= 2 × (2 × (2 × 2 × 22 + 2))
Ï 370 := (1111 − 1)/(1 + 1 + 1)
:= 2 + (((22 − (2/2 + 2)) ) + 2)
:= 2 + (2 × (2 × (2 × ((2 × 22) + 2))))
:= 3 + 333
:= 3 + ((33 × 33 − 3)/3)
:= 33 + (((3/3 + 3) + 3)3 )
:= 4 + (((4 + 4) × 44) + 4)
:= (4/4 + 4)4 − (44 + 4)
:= 4 + ((444 − 4)/4 + 44 )
:= (5/5 + 5) × (55 + 5)
:= 5 + (5/5 + 5) × (55 + 5)
:= (5 × (5 × (5 + 5 + 5))) − 5
:= 6 × (66 − 6)
:= 6 + (6 × (66 − 6) − 6/6)
:= 6 × (66 − 6) + (66 − 6)/6
:= 7 + (((77 − 7)/7) + 7 × 7 × 7)
:= 7 + (((7 × 7 × 7 + 7/7) + 7) + 7)
:= 77 + (7 × (7 × 7 − 7) − 7/7)
:= 8 × (8 × 8 − 8) − 88
:= 888 − (8 × 8 × 8 + 88/8)
:= (8888 − 8)/(8 + 8 + 8)
:= (9 + 9) × (99/9 + 9)
:= ((9 × 9 × 9 × 9) + 9)/(9 + 9)
:= 9 + ((9/9 + 9 + 9)(9+9)/9 )
3
Ï 361 := (((1 + 1) × (11 − 1)) − 1)1+1
:= (22 − (2/2 + 2))2
3
:= 3 + (333 + 3/3)
Ï 366 := (1 + 1 + 1) × (1 + 111+1 )
Ï 371 := 1 + (1111 − 1)/(1 + 1 + 1)
:= 222 + ((2 × (2 + 2 + 2))2 )
:= 222 − (222/2 + 2)
:= 33 + 333
:= 33 + (333 + 33/3)
4
:= 4 + ((((4 + 4) × 44) + 4/4) + 4)
:= 4 + (444 − 4)/4
:= 5/5 + (5/5 + 5) × (55 + 5)
:= 4 + (444/4 + 44 )
:= (5/5 + 5) × ((55 + 5/5) + 5)
:= 6/6 + 6 × (66 − 6)
:= 5/5 + ((5 × (5 × (5 + 5 + 5))) − 5)
:= 6 + 6 × (66 − 6)
:= 7 + (7 × 7 × 7 + (77/7))
:= 66/6 + 6 × (66 − 6)
:= 7 + (((7 × 7 × 7 + ((7 + 7)/7)) + 7) + 7)
:= (88/8 + 8)(8+8)/8
:= 77 + 7 × (7 × 7 − 7)
:= ((8 + 8) × (8 + 8)) + ((888 − 8)/8)
:= (9/9 + 9 + 9)(9+9)/9
:= (8 − 8/8) × (8 × 8 − 88/8)
:= 9/9 + (((9 × 9 × 9 × 9) + 9)/(9 + 9))
:= 99/9 + ((9 + 9) × (99/9 + 9))
Ï 367 := 1 + (1 + 1 + 1) × (1 + 111+1 )
Ï 362 := 11 × 11 × (1 + 1 + 1) − 1
Ï 372 := 1 + 1 + (1111 − 1)/(1 + 1 + 1)
:= 2 + (2 × (2 × (2 × 2 × 22 + 2)))
:= 222/2 + 22×(2+2)
:= 2 × ((2 × (2 × ((2 × 22) + 2))) + 2)
:= (33 × 33 − 3)/3
:= 3 + ((33 × 33 + 3)/3)
:= 44 + ((444 − 4)/4 − 4)
:= 3 + ((333 + 33) + 3)
:= 44 + 444/4
:= 4 + (((4 + 4) × 44) + 4 × 4)
5
5
:= 55 + (5 − 55)/(5 + 5)
:= 55 + (5 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 55)
:= (6 + 6)/6 + 6 × (66 − 6)
:= 6 + (6 × (66 − 6) + 6/6)
:= 6 + (6 × (66 − 6) + 6)
:= 7 + ((77 + 7)/7 + 7 × 7 × 7)
:= 7 + ((((77 − 7)/7) + 7 × 7 × 7) + 7)
:= 7/7 + (7 × (7 × 7 − 7) + 77)
:= 8/8 + ((88/8 + 8)(8+8)/8 )
:= 888/8 + ((8 + 8) × (8 + 8))
:= 8 + (((8 × 88 + 8)/((8 + 8)/8)) + 8)
:= 9 + (((9 + 9)/9) × ((9 × 9 − 9/9) + 99))
:= 9 + ((99 × 99)/(9 + 9 + 9))
:= 9/9 + ((9/9 + 9 + 9)
(9+9)/9
)
34
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 378 := 1 + 11 + (1 + 1 + 1) × (1 + 111+1 )
Ï 373 := (11 + 11)1+1 − 111
2
Ï 383 := (1 + 11) × (11 × (1 + 1 + 1) − 1) − 1
:= 22 − 222/2
:= (22 − 2) − 22
:= 22 + ((22 − (2/2 + 2))2 )
:= 3 + ((((3/3 + 3) + 3)3 ) + 33 )
:= 3 × (3 × 33 + 33 )
:= ((3 + 3) × ((3/3 + 3)3 )) − 3/3
:= 4 + ((4/4 + 4)4 − 44 )
:= 44 + ((444 + 44)/4)
:= ((4 + 4) × (44 + 4)) − 4/4
:= 5 + ((55 + 5)/(5 + 5) + 55)
:= ((5 + 5)/5 + 5) × (55 − 5/5)
:= 5 + (((5 + 5)/5 + 5) × (55 − 5/5))
:= 6 + ((6 × (66 − 6) + 6/6) + 6)
:= 6 × 66 − 6 − 6 − 6
:= 6 × 66 − (6/6 + 6 + 6)
:= 77 + (7 × (7 × 7 − 7) + ((7 + 7)/7))
:= 7 × (7 × 7 + 7) − (7 + 7)
:= 7 × (7 × 7 + 7) − ((7 + 7)/7 + 7)
:= (8 × (8 × 8 − (8 + 8))) − 88/8
:= (8 − (8 + 8)/8) × (8 × 8 − 8/8)
:= (8 × (8 × 8 − (8 + 8))) − 8/8
:= 9 + (((9 × 9 × 9 × 9) − 9)/(9 + 9))
:= (9 + 9) × (((99 + 9)/9) + 9)
:= 9 + ((((9 × 9 × 9 × 9) + 9)/(9 + 9)) + 9)
2
Ï 374 := 11 × (1 + 11 × (1 + 1 + 1))
Ï 379 := (11 − 1) × (1 + 111/(1 + 1 + 1)) − 1
Ï 384 := (1 + 11) × (11 × (1 + 1 + 1) − 1)
:= 22 + (22 × 22+2 )
:= 2/2 + ((22 − 2) − 22)
:= 22+2 × (22 + 2)
:= 33/3 × (3/3 + 33)
:= 3 + ((((3/3 + 3) + 3)3 ) + 33)
:= (3 + 3) × ((3/3 + 3)3 )
:= (44 × (4 × 4 × 4 + 4))/(4 + 4)
:= 444 − (44 + 4)/4
:= (4 + 4) × (44 + 4)
:= (5 × (5 × (5 + 5 + 5))) − 5/5
:= 5 + ((5 × (5 × (5 + 5 + 5))) − 5/5)
:= ((55 + 5)/5) × ((5 + 5)/5)5
:= (66/6) × (6 × 6 − ((6 + 6)/6))
:= 6 × 66 − ((66/6) + 6)
:= 6 × ((6 + 6)/6)6
:= 7 × (7 × 7 + 7) − (77/7 + 7)
:= 7/7 + (7 × (7 × 7 + 7) − (7 + 7))
:= (7/7 + 7) × (7 × 7 − 7/7)
:= 888 − (8 × 8 × 8 + (8 + 8)/8)
:= 8 + ((8 − 8/8) × (8 × 8 − 88/8))
:= 8 × (8 × 8 − (8 + 8))
2
:= 9 + (((9 × 9 × 9 × 9) + 9)/(9 + 9))
Ï 375 := 1 + 11 × (1 + 11 × (1 + 1 + 1))
2
9
:= 9 × 99 − (((9 + 9)/9) )
Ï 380 := (11 − 1) × (1 + 111/(1 + 1 + 1))
:= ((9 + 9)/9) × (999/9 + 9 × 9)
Ï 385 := 11 × (1 + 1 + 11 × (1 + 1 + 1))
:= 2 + (22 − 222/2)
:= 2 + ((22 − 2) − 22)
:= 2/2 + (22+2 × (22 + 2))
:= 3 × ((3 − 3/3 + 3)3 )
:= 3 + ((33/3 × (3/3 + 33)) + 3)
:= 3/3 + ((3 + 3) × ((3/3 + 3)3 ))
:= 4 + ((444/4 + 44 ) + 4)
:= 444 − 4 × 4 × 4
:= 4/4 + ((4 + 4) × (44 + 4))
2
:= 5 × (5 × (5 + 5 + 5))
:= 5 + (5 × (5 × (5 + 5 + 5)))
:= 55 × ((5 + 5)/5 + 5)
:= 6 + (66 × 66/(6 + 6) + 6)
:= 6 × 66 + (((6 − 66)/6) − 6)
:= 6 × 66 − 66/6
:= 7 + (((7 × 7 × 7 + (77/7)) + 7) + 7)
:= 7 × (7 × 7 + 7) − (77 + 7)/7
:= 7 × (7 × 7 + 7) − 7
:= 888 − (8 × 8 × 8 + 8/8)
:= (8 × (88 + 8) − 8)/((8 + 8)/8)
:= 8/8 + (8 × (8 × 8 − (8 + 8)))
:= ((9 + 9) × (9 + 9 + 9)) − 999/9
:= (9/9 + 9 + 9) × (99/9 + 9)
:= 99/9 × ((((9 − 9/9) + 9) + 9) + 9)
Ï 376 := 1 + 1 + 11 × (1 + 11 × (1 + 1 + 1))
:= (22 − 2)2 − (22 + 2)
:= 33 + (((3/3 + 3) + 3)3 )
Ï 381 := 11 + (1111 − 1)/(1 + 1 + 1)
2
:= 2 + (((22 − 2) − 22) + 2/2)
3
:= 3 + (3 × (3 × 33 + 3 ))
4
Ï 386 := 1 + 11 × (1 + 1 + 11 × (1 + 1 + 1))
:= 2 + (22+2 × (22 + 2))
:= 33+3 − (((3/3 + 3) + 3)3 )
:= (4 + 4) × (44 − 4/4 + 4)
:= 444 + ((4 − 4 )/4)
:= 44 + ((44 + 4)/((4 + 4)/4))
:= 5/5 + (5 × (5 × (5 + 5 + 5)))
:= 5 + ((5 × (5 × (5 + 5 + 5))) + 5/5)
:= 5 × 55 + 555/5
:= 6 + (6 × (66 − 6) + ((66 − 6)/6))
:= 6 + ((66 × 66/(6 + 6) + 6) + 6)
:= 6 × 66 + (6 − 66)/6
:= (7/7 + 7) × 7 × 7 − (7 + 7)/7
:= 7 × (7 × 7 + 7) − 77/7
:= 7/7 + (7 × (7 × 7 + 7) − 7)
:= 888 − 8 × 8 × 8
:= 8 + ((8 × (8 × 8 − (8 + 8))) − (88/8))
:= (8 + 8)/8 + (8 × (8 × 8 − (8 + 8)))
:= 9 × 9 + ((99 × (9 + 9 + 9) − (9 + 9))/9)
:= 9 + (((99 × 99)/(9 + 9 + 9)) + 9)
:= (((9 + 9)/9)9 ) − ((99 + 9 + 9) + 9)
Ï 377 := 11 + (1 + 1 + 1) × (1 + 111+1 )
2
:= (22/2) + 2
:= (22+2 × (22 + 2)) − 2
2×(2+2)
:= 3 + (33/3 × (3/3 + 33))
4
Ï 382 := 1 + 11 + (1111 − 1)/(1 + 1 + 1)
4
:= 3333/3 − 3
4
3+3
Ï 387 := 1 + 1 + 11 × (1 + 1 + 11 × (1 + 1 + 1))
:= (22 − 2)2 − (22/2 + 2)
:= 3 + ((3 + 3) × ((3/3 + 3)3 ))
4
:= 4 + (((4/4 + 4) − 4 ) + 4)
:= 4 + ((4 − 4)/((4 + 4)/4))
:= 4 + (((4 + 4) × (44 + 4)) − 4/4)
:= (5 + 5)/5 + (5 × (5 × (5 + 5 + 5)))
:= 55 /5 − ((5 − (5 + 5)/5)5 )
:= 5 × 55 + (555 + 5)/5
:= 6 + (6 × (66 − 6) + (66/6))
:= 6 × 66 − ((6 + 6)/6 + 6 + 6)
:= 6 × 66 + (((6 − 66) + 6)/6)
:= 7 × (7 × 7 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 × (7 × 7 + 7)
:= ((7 + 7)/7) + (7 × (7 × 7 + 7) − 7)
:= 8/8 + (888 − 8 × 8 × 8)
:= (8 × (8 × 8 − (8 + 8))) − (8 + 8)/8
:= 888 + (88/8 − 8 × 8 × 8)
:= 9 × 9 + ((99 × (9 + 9 + 9) − 9)/9)
:= 9999/9 − 9 × 9 × 9
:= ((9 + 9) × (9 + 9 + 9)) − 99
35
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 388 := ((1 + 1) × (11 − 1))1+1 − 1 − 11
Ï 393 := (1 + 1 + 1) × (11 × (1 + 11) − 1)
Ï 398 := ((1 + 1) × (11 − 1))1+1 − 1 − 1
:= 2 × (((22+2 − 2)2 ) − 2)
:= 2/2 + (2 × ((22+2 − 2)2 ))
:= (22 − 2)2 − 2
:= 3 + (((3 + 3) × ((3/3 + 3)3 )) + 3/3)
:= (33 × (3 × 3 + 3)) − 3
:= 3 + ((33 × (3 × 3 + 3)) − 3/3)
:= 4 + ((4 + 4) × (44 + 4))
:= 4 + ((((4 + 4) × (44 + 4)) + 4/4) + 4)
:= 444 − ((4 + 4)/4 + 44)
:= 5 × 55 + (555 + 5 + 5)/5
:= 5 × 5 + ((5 + 5)/(5 + 5) + 55)
:= (5 − 5/5)5 − (55 + 5)/5
:= 6 × 66 − ((6 + 6)/6 + 6)
:= 6 × 66 − 6 × 6/(6 + 6)
:= (6 + 6)/6 + 6 × 66
5
:= ((7 × 777) − 7)/(7 + 7)
:= 7/7 + 7 × (7 × 7 + 7)
:= 7 + (7 × (7 × 7 + 7) − 7/7)
:= (8 × (88 + 8) + 8)/((8 + 8)/8)
:= 8 + ((8 × (8 × 8 − (8 + 8))) + 8/8)
:= 8 + ((8 − (8 + 8)/8) × (8/8 + 8 × 8))
:= 9 + (9 × 99 − (((9 + 9)/9)9 ))
:= 9 + (((9 + 9)/9) × (999/9 + 9 × 9))
:= ((9 + 9)/9) × ((9/9 + 99) + 99)
Ï 389 := ((1 + 1) × (11 − 1))1+1 − 11
Ï 394 := 1 + (1 + 1 + 1) × (11 × (1 + 11) − 1)
Ï 399 := ((1 + 1) × (11 − 1))1+1 − 1
:= (22 − 2)2 − 2/2
:= (22 − 2)2 − 22/2
:= 2 + (2 × ((22+2 − 2)2 ))
:= 33 + ((33 × 33 − 3)/3)
:= 3/3 + ((33 × (3 × 3 + 3)) − 3)
:= 3 + (33 × (3 × 3 + 3))
:= 4 + (((4 + 4) × (44 + 4)) + 4/4)
:= 4 + (((4 + 4)/4 + 4) × (4 + 4)/4)
:= 4 × 44 − (4/4 + 4)4
:= 5 × (5 × 5 + 55) − 55/5
:= (5 − 5/5)5 − (55 /5 + 5)
:= (5 − 5/5)5 − 55 /5
:= 6 × 66 − 6/6 − 6
:= 6 × 66 − (6 + 6)/6
:= 6 × 66 + (6 × 6/(6 + 6))
:= ((7 × 777) + 7)/(7 + 7)
4
:= ((7 + 7)/7) + 7 × (7 × 7 + 7)
:= 7 + 7 × (7 × 7 + 7)
:= (((8 − 8/8) × 888) + 8)/(8 + 8)
:= 8 + ((8 × (8 × 8 − (8 + 8))) + ((8 + 8)/8))
:= (8 − 8/8) × ((8/8 − 8) + 8 × 8)
:= 9 + ((9/9 + 9 + 9) × (99/9 + 9))
:= 9 × 9 + ((9 + 9) × (9 + 9) − (99/9))
:= (9/9 + 9 + 9) × (((99 + 9)/9) + 9)
Ï 390 := 1 + ((1 + 1) × (11 − 1))1+1 − 11
Ï 395 := 11 × (1 + 1 + 1) × (1 + 11) − 1
Ï 400 := ((1 + 1) × (11 − 1))1+1
:= (2 × ((22+2 − 2)2 )) − 2
:= (22 − 2)2 − (2/2 + 2 + 2)
:= (22 − 2)2
:= 33 + (33 × 33/3)
:= (33 × (3 × 3 + 3)) − 3/3
:= 3 + ((33 × (3 × 3 + 3)) + 3/3)
:= ((4 + 4)/4 + 4) × (4 + 4)/4
:= 44 + (((4 + 4) × 44) − 4/4)
:= 444 − 44
:= 5 + (55 × ((5 + 5)/5 + 5))
:= 5 × (5 × 5 + 55) − 5
:= 5 × (5 × 5 + 55)
:= 6 × 66 − 6
:= 6 × 66 − 6/6
:= 6 + (6 × 66 − ((6 + 6)/6))
:= 7 × (7 × 7 + 7) − (7 + 7)/7
:= 7 + (((7 × 777) − 7)/(7 + 7))
:= 7 + (7 × (7 × 7 + 7) + 7/7)
:= (8 − (8 + 8)/8) × (8/8 + 8 × 8)
:= 88/8 + (8 × (8 × 8 − (8 + 8)))
:= 8 + ((8 × (8 × 8 − (8 + 8))) + 8)
4
:= ((9 × 9 × 99) − 999)/(9 + 9)
Ï 391 := (1 + 1)11−1−1 − 111+1
9
:= (((9 + 9)/9) ) − (99 + 9 + 9)
Ï 396 := 11 × (1 + 1 + 1) × (1 + 11)
:= (99/9 + 9)(9+9)/9
Ï 401 := 1 + ((1 + 1) × (11 − 1))1+1
:= 2 + ((22 − 2)2 − 22/2)
:= 22 × (22+2 + 2)
:= 2/2 + (22 − 2)2
:= 33 + ((33 × 33 + 3)/3)
:= 33 × (3 × 3 + 3)
:= 3 + (((33 × (3 × 3 + 3)) − 3/3) + 3)
:= 444 + ((44 − 4 )/4)
:= 44 + ((4 + 4) × 44)
:= 4/4 + (444 − 44)
:= 5 + (555/5 + 5 × 55)
:= 5/5 + (5 × (5 × 5 + 55) − 5)
:= 5/5 + 5 × (5 × 5 + 55)
:= 6/6 + (6 × 66 − 6)
:= 6 × 66
:= 6 + (6 × 66 − 6/6)
:= 7 × (7 × 7 + 7) − 7/7
:= 7 + (((7 × 777) + 7)/(7 + 7))
:= 7 + (7 × (7 × 7 + 7) + ((7 + 7)/7))
:= (88 × (8 × 8 + 8))/(8 + 8)
:= 8 × 8 × 8 − 888/8
:= 9 × 9 + ((9 + 9) × (9 + 9) − 9)
:= (((9 + 9)/9)9 ) − 999/9
4
:= 8 + ((8 × (8 × 8 − (8 + 8))) − 8/8)
:= ((99/9 + 9)
(9+9)/9
)−9
Ï 392 := (1 + 1) × (1 + 1 + 1 + 11)1+1
Ï 397 := 1 + 11 × (1 + 1 + 1) × (1 + 11)
Ï 402 := 1 + 1 + ((1 + 1) × (11 − 1))1+1
:= 2 × ((22+2 − 2)2 )
:= (22 − 2)2 − 2/2 − 2
:= 2 + (22 − 2)2
:= (3/3 + 3) × (3 × 33 − 3/3)
:= 3/3 + (33 × (3 × 3 + 3))
:= 3 + ((33 × (3 × 3 + 3)) + 3)
:= 4 + (((4 + 4) × (44 + 4)) + 4)
:= 44 + (((4 + 4) × 44) + 4/4)
:= 444 + ((4 + 4)/4 − 44)
:= ((5 + 5)/5 + 5) × (55 + 5/5)
:= 5 + (((5 + 5)/5 + 5) × (55 + 5/5))
:= (5 + 5)/5 + 5 × (5 × 5 + 55)
:= (6 + 6)/6 + (6 × 66 − 6)
:= 6/6 + 6 × 66
:= 6 + 6 × 66
:= 7 × (7 × 7 + 7)
:= 7 + (7 × (7 × 7 + 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 × (7 × 7 + 7)
:= 8 + (8 × (8 × 8 − (8 + 8)))
:= 8/8 + ((88 × (8 × 8 + 8))/(8 + 8))
:= 8 × 8 × 8 + ((8 − 888)/8)
:= (9 − 9/9) × ((9 × 99 − 9)/(9 + 9))
:= (((9 + 9) × (99 + 99)) + 9)/9
:= (((9 + 9)/9)9 ) + ((9 − 999)/9)
36
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 403 := 1 + 1 + 1 + ((1 + 1) × (11 − 1))1+1
:= 2 + (22 − 2)2 + 2/2
Ï 413 := 1 + 1 + 11 + ((1 + 1) × (11 − 1))1+1
Ï 408 := 1 + 11 × 111/(1 + 1 + 1)
:= 2 × (2 + 2) + (22 − 2)2
:= 2 + (22 − 2)2 + 22/2
:= 3 + (33 × ((3 × 3 + 3) + 3))
:= 3 × 33 + (333 − 3/3)
:= 4 + (4 × 4 − (4/4 + 4) )
:= 4 + ((444 − 44) + 4)
:= 44 + ((4/4 + 4)4 − 44 )
:= 5 + ((5 − 5/5)5 − (55 + 5)/5)
:= (5 × 5 − 5/5) × (((55 + 5)/5) + 5)
:= 5 × (5 × 5 − 5) + (55 + 5)/(5 + 5)
:= 6 + (6 × 66 + 6/6)
:= 6 + (6 × 66 + 6)
:= 6 + (6 × 66 + (66/6))
:= 77/7 + 7 × (7 × 7 + 7)
:= (7/7 + 7) × ((7 + 7)/7 + 7 × 7)
:= 77 + (7 × 7 × 7 − 7)
:= 8 + ((8 × (8 × 8 − (8 + 8))) + (88/8))
:= 8 × 8 × 8 − (88 + 8 + 8)
:= 8 × 8 × 8 − (88/8 + 88)
:= 9 × 9 + ((9 + 9) × (9 + 9) − ((9 + 9)/9))
:= (99 × (9 + 9 + 9) + 999)/9
:= (((9 + 9)/9)9 ) − 99
:= 3 + (((33 × (3 × 3 + 3)) + 3/3) + 3)
4
4
Ï 404 := (1 + 1) × ((1 + 1) × (1 + (11 − 1)1+1 ))
2
2
:= 22/2 + (22 − 2) − 2
:= 2 + (22 − 2) + 2
3
:= 333 + (((3 + 3) − 3)/3)
:= 3 + ((3 × ((3 × 3 + 3) + 3)) + 3/3)
:= 4 + ((4/4 + 4) × (4 − 4/4) )
5
:= 5 + ((5 − 5/5) − 5 /5)
:= 6 + (((6 + 6)/6) + 6 × 66)
7
5
5
:= 5 + (((5 − 5/5) − 5 /5) + 5)
:= 6 + ((6 × 66 + 6/6) + 6)
:= 7 × 77 − (((7 + 7)/7) + 7)
:= 77 + (7 × 7 × 7 − (77/7))
:= 8 + ((88 × (8 × 8 + 8))/(8 + 8))
:= 8 + (8 × 8 × 8 − 888/8)
9
:= (((9 + 9)/9) ) − (99 + 9)
Ï 405 := 11 × 111/(1 + 1 + 1) − 1 − 1
Ï 414 := (1 + 1) × (11 + (1 + 1 + 1 + 11)1+1 )
:= 2 + 2 × (222 − 22+2 )
3
4
:= 4 + (444 − 44)
5
Ï 409 := 1 + 1 + 11 × 111/(1 + 1 + 1)
:= 9 + ((99/9 + 9)(9+9)/9 )
Ï 410 := 11 + ((1 + 1) × (11 − 1))1+1 − 1
2
:= 3 × 33 + 333
:= 4 + ((4 + 4)4 + 4)/((44 − 4)/4)
:= (5 − 5/5)5 − (555 + 55)
:= 6 + ((6 × 66 + 6) + 6)
:= 7/7 + ((7 × 7 × 7 − 7) + 77)
:= 8 × 8 × 8 + ((8 − 88)/8 − 88)
:= 9 + ((9 + 9) × (9 + 9) + 9 × 9)
Ï 415 := (1 + 1 + 11) × (11 × (1 + 1 + 1) − 1) − 1
:= 2 + (22 − 2) + 2/2 + 2
:= 2 + (22 − 2) + 2 × (2 + 2)
:= 2 + (22 − 2)2 + 22/2 + 2
:= 33 × ((3 × 3 + 3) + 3)
:= 3 + 33 × 333/33
:= 3/3 + (3 × 33 + 333)
2
4
:= (((4 + 4) ) + 4)/((44 − 4)/4)
:= 44 + (4 × (44 − 4) − 4/4)
:= 5 + 5 × (5 × 5 + 55)
:= 5 + (5 × (5 × 5 + 55) + 5)
:= 5 + ((5 × (5 × 5 + 55) + 5) + 5)
:= 6 + ((6 × 6/(6 + 6)) + 6 × 66)
:= 6 + ((((6 + 6)/6) + 6 × 66) + 6)
:= 6 + (((6 × 66 + 6/6) + 6) + 6)
:= 7 + ((7 × (7 × 7 + 7) − 7/7) + 7)
:= 7 + (7 × (7 × 7 + 7) + (77/7))
:= 7 + ((7/7 + 7) × ((7 + 7)/7 + 7 × 7))
:= 8 × 8 × 8 − ((88/8 + 88) + 8)
:= 8 + (((8 − 888)/8) + 8 × 8 × 8)
:= 8 × 8 × 8 − ((8/8 + 88) + 8)
:= 9 × (((9 + 9 + 9) + 9) + 9)
:= 9 + ((((9 + 9)/9)9 ) − 999/9)
:= 9 + (((9 + 9) × (9 + 9) + 9 × 9) + 9/9)
:= (4/4 + 4) × (4 − 4/4)4
Ï 406 := 11 × 111/(1 + 1 + 1) − 1
2
:= 2 + (22 − 2) + 2 + 2
:= 3/3 + (33 × ((3 × 3 + 3) + 3))
Ï 411 := 11 + ((1 + 1) × (11 − 1))1+1
:= 22/2 + (22 − 2)2
Ï 416 := (1 + 1 + 11) × (11 × (1 + 1 + 1) − 1)
:= 22+2 + (22 − 2)2
3
:= 3 + ((333 − 3/3) + 3 × 33 )
:= 3 × 3 + (333 − 3)
4
:= 4 + (((4 + 4)/4 − 44) + 444)
:= 44 + (444/4 + 4 )
:= 44 + (4 × (44 − 4))
:= 5 + (5 × (5 × 5 + 55) + 5/5)
:= 55/5 + 5 × (5 × 5 + 55)
:= (55/5 + 5) × (5 × 5 + 5/5)
:= 6 × 66 + (66 − 6)/6
:= 6 + (((6 × 6/(6 + 6)) + 6 × 66) + 6)
:= 6 + (((((6 + 6)/6) + 6 × 66) + 6) + 6)
7
:= 7 + (7 × (7 × 7 + 7) + 7)
:= 7 × 77 − ((7 + 7)/7)
:= ((77 × 77 − 7)/(7 + 7)) − 7
:= (8 − 8/8) × ((((8 + 8)/8) − 8) + 8 × 8)
:= 8 × 8 × 8 − (8888/88)
:= 8 × 8 × 8 − (88 + 8)
:= 9/9 + ((9 + 9) × (9 + 9) + 9 × 9)
Ï 407 := 11 × 111/(1 + 1 + 1)
2
9
:= (((9 + 9)/9) ) − ((9 + 9)/9 + 99)
Ï 412 := 1 + 11 + ((1 + 1) × (11 − 1))1+1
2+2
:= 9 × 9 + ((9 + 9) × (9 + 9) + (99/9))
Ï 417 := 11 × (1 + 111/(1 + 1 + 1)) − 1
:= (22 − 2/2)2 − 22 − 2
:= 2 + (22 − 2) + 2/2 + 2 + 2
:= 2 × (222 − 2
:= 33 × 333/33
:= 3/3 + ((3 × 33 − 3) + 333)
:= 3 + (3 × 33 + 333)
:= 4 + ((4 × 44 − (4/4 + 4)4 ) + 4)
:= 444 − 4 × (4 + 4)
:= 4/4 + (4 × (44 − 4) + 44 )
:= 55/5 × (((5 + 5)/5)5 + 5)
:= 5 + (55/5 × (((5 + 5)/5)5 + 5))
:= ((55 + 55 ) + 5)/(5 + 5 + 5)
:= 6 × 66 + 66/6
:= 6 + (((66 − 6)/6) + 6 × 66)
:= 6 × 66 + ((6 × 6 + 6)/((6 + 6)/6))
:= 7 + ((7 × (7 × 7 + 7) + 7/7) + 7)
:= 77 + (7 × 7 × 7 − (7/7 + 7))
:= ((77 × 77 + 7)/(7 + 7)) − 7
:= 8888/8 − 8 × 88
:= (888 − 8 × 8)/((8 + 8)/8)
:= 8/8 + (8 × 8 × 8 − (88 + 8))
:= 9 × 9 + ((9 + 9) × (9 + 9) + ((9 + 9)/9))
:= (((9 + 9)/9)9 ) − (9/9 + 99)
:= 9 + ((99 × (9 + 9 + 9) + 999)/9)
)
37
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 418 := 11 × (1 + 111/(1 + 1 + 1))
Ï 423 := 1 + (1 + 1) × ((1 + 1) × 111 − 11)
2
Ï 428 := (11 × ((1 + 1 + 1) × (1 + 1 + 11))) − 1
:= 22 + ((22 − 2) + 2/2)
:= 2 × (222 − 2 × (2 + 2))
:= 33/3 × (33/3 + 3 )
:= 3 × ((3 × (33 + 3)) + 33)
:= ((((3 + 3)3/3+3 ) − 3)/3) − 3
:= 44 + (4 × (44 − 4) + (4 + 4)/4)
:= 444 − ((4 × 4 + 4/4) + 4)
:= 444 − 4 × 4
:= 22 × (22 − 2/2 − 2)
3
5
:= 5 + ((55 + 5)/(5 + 5) + 5 × (5 × 5 − 5))
:= 55 + ((5 + 5)/(5 + 5) + 55)
:= (5 − 5/5) × ((555 + 5)/5 − 5)
:= 6 × 66 + ((66 + 66)/6)
:= 66 + (66 × 66/(6 + 6) − 6)
:= (6 + 6)/6 + ((6 × (66 + 6)) − 6)
:= 7 + (7 × 77 − ((7 + 7)/7) )
:= (77 × 77 − 7)/(7 + 7)
:= 7 × 77 − 777/7
:= 8 × 8 × 8 + (((8 + 8)/8) − (88 + 8))
:= 8 × 8 × 8 − (8/8 + 88)
:= 888/((8 + 8)/8) − 8 − 8
:= 9 + (((99/9 + 9)(9+9)/9 ) + 9)
:= 99 + (9 + 9) × (9 + 9)
:= (((9 × 9 × 99) + 9)/(9 + 9)) − (9 + 9)
7
Ï 419 := 1 + 11 × (1 + 111/(1 + 1 + 1))
:= (22 − 2/2)2 − 22
Ï 424 := (1 + 1) × (1 + ((1 + 1) × 111 − 11))
Ï 429 := 11 × ((1 + 1 + 1) × (1 + 1 + 11))
:= (2 × (222 − 2)) − 22/2
2
:= 2 + ((22 − 2) + 22)
:= ((33 + 3) × (33/3 + 3)) − 3/3
:= 3 × 3 + (((3/3 + 3) + 3) )
:= 33 + (33 × (3 × 3 + 3))
:= (44 × 44 − (44 + 4))/4
:= 444 − (4 × 4 + 4)
:= 4/4 + (444 − 4 × 4)
:= 555 − (555/5 + 5 × 5)
:= (5 − 5/5) × (555/5 − 5)
:= 555 − (5 × 5 × 5 + 5/5)
:= 6 + ((6 × 66 + (66/6)) + 6)
:= 6 × (66 − 6) + ((6 + 6)/6)6
:= 66 + 66 × 66/(6 + 6)
:= 77 + (7 × 7 × 7 − 7/7)
:= (77 × 77 + 7)/(7 + 7)
:= 8 + (8 × 8 × 8 − (8888/88))
:= 8 × 8 × 8 − 88
:= (((9 × 999) + 9)/(9 + 9)) − 9 × 9
:= 9/9 + ((9 + 9) × (9 + 9) + 99)
Ï 420 := (1 + 1) × (11 − 1) × (11 + 11 − 1)
3
3
Ï 425 := 1 + (1 + 1) × (1 + ((1 + 1) × 111 − 11))
:= 22 + (22 − 2)2 − 2
:= ((22 − 2/2)2 ) − 22+2
:= (33 + 3) × (33/3 + 3)
:= (((3 + 3) × ((3 + 3)3 − 3)) − 3)/3
:= 4 + (4 × (44 − 4) + 44 )
:= 4/4 + (444 − (4 × 4 + 4))
:= (55 + 5) × ((5 + 5)/5 + 5)
:= 5 × (5 × 5 + 55 + 5)
:= 6 × (((6 + 6)/6)6 + 6)
:= (6 × (66 + 6)) − 6/6 − 6
:= 77 + 7 × 7 × 7
:= 7 + ((7 × 77 − ((7 + 7)/7)7 ) + 7)
:= 88 × 88/(8 + 8) − 8 × 8
:= 8/8 + (8 × 8 × 8 − 88)
:= ((9 + 9)/9) × (999/9 + 99)
:= 99 + ((9 + 9) × (9 + 9) + ((9 + 9)/9))
Ï 421 := (1 + 1 + 1) × (1 + 11)1+1 − 11
Ï 426 := (1 + 1) × (1 + 1) × (1 + 111) − 11
:= 7 × 77 + ((7 − 777)/7)
:= 8 × (8 × 8 − 8) − (88/8 + 8)
:= (((9 + 9)/9)9 ) − (((9 + 9)/9) + 9 × 9)
Ï 430 := ((11 + 11 − 1)1+1 ) − 11
:= 2 + (2 × (222 − 2 × (2 + 2)))
:= (3 − 3/3) × ((3 + 3)3 − 3/3)
:= 44 + ((4 × 44) − (4 + 4)/4)
:= 555 − 5 × 5 × 5
:= (6 × (66 + 6)) − (6 + 6)/6
:= 7 + ((77 × 77 − 7)/(7 + 7))
:= 8 + (8 × 8 × 8 − ((8 + 8)/8 + 88))
:= (((9 + 9)/9)9 ) − (9/9 + 9 × 9)
Ï 431 := ((1 + 1 + 1) × (1 + 11)1+1 ) − 1
:= 2 × 222 − (22/2 + 2)
:= 2 + (((22 − 2/2)2 ) − 22)
:= (2 × (222 + 2)) − 22
:= (((3 + 3)3/3+3 ) − 33)/3
:= (3 + 3) × (((3 + 3)3 − 3)/3)
:= 44 + ((4 × 44) − 44/4)
:= 444 − ((4 + 4)/4 + 4 × 4)
:= 5 + ((55/5 + 5) × (5 × 5 + 5/5))
:= 5/5 + (5 × (5 × 5 + 55 + 5))
:= 5/5 + (555 − 5 × 5 × 5)
:= (6 × (66 + 6)) − 66/6
:= (6 × (66 + 6)) − 6
:= (6 × (66 + 6)) − 6/6
:= 7/7 + (7 × 7 × 7 + 77)
:= 7 + ((7 × 7 × 7 − 7/7) + 77)
:= 7 + ((77 × 77 + 7)/(7 + 7))
:= 8 + (8 × 8 × 8 − (88/8 + 88))
:= 8 × 8 × 8 + (((8 + 8)/8) − 88)
:= 8 + (8 × 8 × 8 − (8/8 + 88))
:= 9 + ((((9 + 9)/9)9 ) − (9/9 + 99))
:= (9 + 9) × (9 + 9) + (999/9 − 9)
:= (((9 + 9)/9)9 ) − 9 × 9
Ï 422 := (1 + 1) × (1 + 1) × 111 − 11
Ï 427 := 1 + (1 + 1) × ((1 + 1) × (1 + 111) − 11)
:= (((3 + 3)3/3+3 ) − 3)/3
:= 44 + ((4 × 44) − 4/4)
Ï 432 := (1 + 1 + 1) × (1 + 11)1+1
:= 22 + (22 − 2)2
:= 2 + (((22 − 2/2)2 ) − 22+2 )
:= 2 × ((2 + 2 + 2)2/2+2 )
:= 33 + ((33 × (3 × 3 + 3)) − 3/3)
:= (((3 + 3) × ((3 + 3)3 − 3)) + 3)/3
:= 3 × ((3 + 3) × (33 − 3))
:= 444 − (44/((4 + 4)/4))
:= 444 − (4 × 4 + 4/4)
:= 44 + (4 × 44)
:= 5 + (((55 + 55 ) + 5)/(5 + 5 + 5))
:= (5 + 5)/5 + (5 × (5 × 5 + 55 + 5))
:= 5 × 5 × 5 + (55 − 55)/(5 + 5)
:= ((6 − 66)/6) + (6 × (66 + 6))
:= 6/6 + ((6 × (66 + 6)) − 6)
:= 6 × (66 + 6)
:= 77 + (7 × 7 × 7 + ((7 + 7)/7))
:= 7 + (7 × 7 × 7 + 77)
:= ((7 + 7)/7 + 7) × (7 × 7 − 7/7)
:= 8 × 8 × 8 − ((8 + 8)/8 + 88)
:= 8 × 8 × 8 + (88/8 − (88 + 8))
:= 8 + (8 × 8 × 8 − 88)
:= (((9 × 9 × 99) − 9)/(9 + 9)) − (9 + 9)
:= 9 + ((9 + 9) × (9 + 9) + 99)
9
:= 9 + ((((9 + 9)/9) ) − 99)
38
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 433 := 1 + ((1 + 1 + 1) × (1 + 11)1+1 )
:= (((3 + 3)
3/3+3
Ï 438 := (1 + 1) × ((1 + 1) × (111 − 1) − 1)
) + 3)/3
:= 444 − 44/4
5
Ï 443 := ((1 + 1) × (1 + 1) × 111) − 1
:= 2 + ((22 − 2/2)2 )
:= (2 × (222 − 2)) − 2
:= 2 × 222 − 22/2
3
:= (3 + 3) × (((3 + 3) + 3)/3)
:= 333 + ((333 − 3)/3)
:= 444 − ((4 + 4)/4 + 4)
:= 444 − 4/4
5
:= 5 × 5 × 5 + ((5 + 5)/(5 + 5) − 5)
:= 5 × 5 × 5 + (5 + 5)/(5 + 5)
:= 555 − (555 + 5)/5
:= 6/6 + (6 × (66 + 6))
:= 6 + (6 × (66 + 6))
:= 66/6 + (6 × (66 + 6))
:= 777 − (7 × 7 × 7 + 7/7)
:= 7 + (((77 × 77 + 7)/(7 + 7)) + 7)
:= ((7 + 7)/7) + 7 × (7 × 7 + 7 + 7)
:= 8 + ((8 × 8 × 8 − 88) + 8/8)
:= (8 − 88)/8 + 8 × (8 × 8 − 8)
:= 888/((8 + 8)/8) − 8/8
:= 9 + (((9 + 9) × (9 + 9) + 99) + 9/9)
Ï 434 := 1 + (1 + ((1 + 1 + 1) × (1 + 11)1+1 ))
9
:= 9 + ((((9 + 9)/9) ) − (((9 + 9)/9) + 9 × 9))
Ï 439 := ((11 + 11 − 1)1+1 ) − 1 − 1
:= 9 + (((9 + 9) × (9 + 9) + (99/9)) + 99)
Ï 444 := (1 + 1) × (1 + 1) × 111
:= ((22 − 2/2)2 ) − 2
:= 2 × 222
:= (((3 + 3) × ((3 + 3)3 + 3)) + 3)/3
:= 333 + 333/3
:= 444 + (4 − 44)/4
:= 444 − (4/4 + 4)
:= 444
:= 5 + (555 − (5 × 5 × 5 + 5/5))
:= 555 − (555/5 + 5)
:= (5 − 5/5) × 555/5
:= (6 + 6)/6 + (6 × (66 + 6))
:= 6 + ((6 × (66 + 6)) + 6/6)
:= 6 + ((6 × (66 + 6)) + 6)
:= 777 − 7 × 7 × 7
:= 7 × (7 × 7 + 7 + 7) − (7 + 7)/7
:= (77/7 − 7) × 777/7
:= (8 − 8/8) × (8 × 8 − ((8 + 8)/8))
:= 8 × (8 × 8 − 8) − (8/8 + 8)
:= 888/((8 + 8)/8)
:= 99 + ((9 + 9) × (9 + 9) + (99/9))
:= 9 + ((((9 + 9)/9)9 ) − (9/9 + 9 × 9))
:= (999 × (9 − 9/9))/(9 + 9)
:= (2 × (222 − (2 + 2))) − 2
:= 3 + ((((3 + 3)
3/3+3
) − 3)/3)
Ï 435 := (1 + 1 + 1) × (1 + (1 + 11)1+1 )
Ï 440 := (1 + 1) × (1 + 1) × (111 − 1)
Ï 445 := 1 + ((1 + 1) × (1 + 1) × 111)
:= 2 + (2 × 222 − 22/2)
:= 2 × (222 − 2)
:= 2/2 + 2 × 222
:= 3 + (3 × 33 + 333)
:= (3/3 + 3) × ((333 − 3)/3)
:= 333 + ((333 + 3)/3)
:= 444 − ((4/4 + 4) + 4)
:= 444 − 4
:= 4/4 + 444
:= 5 + (555 − 5 × 5 × 5)
:= (5 + 5) × (55 − (55/5))
:= 555 − (55 + 55)
:= (6 × (66 + 6)) + (6 × 6/(6 + 6))
:= 6 + ((6 × (66 + 6)) + ((6 + 6)/6))
:= 6 + (((6 × (66 + 6)) + 6/6) + 6)
:= 7 + (7 × 77 − 777/7)
:= 7 × (7 × 7 + 7 + 7) − 7/7
:= 77/7 + (777 − 7 × 7 × 7)
:= 8 × 8 × 8 + (88/8 − 88)
:= 8 × (8 × 8 − 8) − 8
:= 8 + (8 × (8 × 8 − 8) − (88/8))
:= (9 + 9) × (9 + 9) + 999/9
:= 9 + ((((9 + 9)/9)9 ) − 9 × 9)
:= ((9 × 9 × 99) − 9)/(9 + 9)
Ï 436 := (1 + 1) × (1 + 1) × (111 − 1 − 1)
Ï 441 := (11 + 11 − 1)1+1
Ï 446 := (1 + 1) × (1 + (1 + 1) × 111)
:= (22 − 2/2)2
:= 2 + 2 × 222
:= 3 × (((3 + 3) × (33 − 3)) + 3)
:= ((3 × (33 × 33 )) + 3)/(3 + 3)
:= 444 − 4 − 4
:= 4/4 + (444 − 4)
:= 444 + (4 + 4)/4
:= 5 + ((555 − 5 × 5 × 5) + 5/5)
:= ((55/5 + 5) + 5)(5+5)/5
:= 5 + (((55/5 + 5) + 5)(5+5)/5 )
:= 6 + ((6 × (66 + 6)) − ((6 + 6)/6))
:= 2 × (222 − (2 + 2))
:= 3 + ((((3 + 3)
3/3+3
) + 3)/3)
:= 6 + ((6 × (66 + 6)) + (6 × 6/(6 + 6)))
:= 6 + (((6 × (66 + 6)) + ((6 + 6)/6)) + 6)
:= 7 + (((7 − 777)/7) + 7 × 77)
:= 7 × (7 × 7 + 7 + 7)
:= 7 + (7 × (7 × 7 + 7 + 7) − ((7 + 7)/7))
:= 888/((8 + 8)/8) − 8
:= 8/8 + (8 × (8 × 8 − 8) − 8)
:= 8 × (8 × 8 − 8) − (8 + 8)/8
:= (((9 × 9 × 99) − 9)/(9 + 9)) − 9
:= 9 × ((9 × 99 − 9)/(9 + 9))
:= ((9 × 9 × 99) + 9)/(9 + 9)
Ï 437 := 1 + ((1 + 1) × (1 + 1) × (111 − 1 − 1))
2
:= ((22 − 2/2) ) − 2 − 2
3
Ï 442 := 1 + ((11 + 11 − 1)1+1 )
Ï 447 := 1 + ((1 + 1) × (1 + (1 + 1) × 111))
:= 2 + (2 × 222 + 2/2)
:= 2 × 222 − 2
:= 3 × 33 + (((3/3 + 3) + 3) )
:= 3 + (333/3 + 333)
:= 444 − (4 + 4)/4
:= 4 + (444 − 4/4)
:= 5 × 5 × 5 + (5 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5 × 5 × 5)
:= ((55 − 5/5) + 5)/((5 + 5)/5 + 5)
:= 6 + ((6 × (66 + 6)) − 6/6)
:= ((66 − 6)/6) + (6 × (66 + 6))
:= ((6/6 + 6) × ((6 + 6)/6)6 ) − 6/6
:= 7 + (((77 × 77 − 7)/(7 + 7)) + 7)
:= 7/7 + 7 × (7 × 7 + 7 + 7)
:= 7 + (7 × (7 × 7 + 7 + 7) − 7/7)
:= 8 × (8 × 8 − 8) − 88/8
:= (8 + 8)/8 + (8 × (8 × 8 − 8) − 8)
:= 8 × (8 × 8 − 8) − 8/8
:= (((9 × 9 × 99) + 9)/(9 + 9)) − 9
:= 9/9 + (9 × ((9 × 99 − 9)/(9 + 9)))
:= 9/9 + (((9 × 9 × 99) + 9)/(9 + 9))
:= (((3 + 3) × ((3 + 3) + 3)) − 3)/3
:= 4 + (444 − 44/4)
5
3
39
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 448 := (1 + 1) × (1 + 1) × (1 + 111)
Ï 453 := 1 + (11 + ((11 + 11 − 1)1+1 ))
Ï 458 := 1 + (11 + ((1 + 1) × (1 + (1 + 1) × 111)))
:= 222 − (22 + 2 + 2)
:= 2/2 + (2 × ((222 + 2) + 2))
:= 2 × (222 + 2)
3
:= (3/3 + 3) × ((333 + 3)/3)
:= (3 × (3 + 3) × 3 ) − 33
:= (3 + 3)3 + ((33+3 − 3)/3)
:= 4 + 444
:= 4 + ((444 + 4/4) + 4)
:= 4 + ((44 − 4)/4 + 444)
:= (5 − 5/5) × (555 + 5)/5
:= 5 + ((5 − 5/5) × (555 + 5)/5)
6
6
:= (5 × 5 × 5 × 55 − 5)/(5 + 5 + 5)
:= (6/6 + 6) × ((6 + 6)/6)
:= 6 + (((6/6 + 6) × ((6 + 6)/6) ) − 6/6)
:= 66 + (((6 + 6)/6 − 6) + 6 × 66)
:= 7 + 7 × (7 × 7 + 7 + 7)
:= 7 × 7 × 7 + (777 − 7)/7
:= 7 + (77/7 × (7 × 7 − (7/7 + 7)))
:= 8 × (8 × 8 − 8)
:= 8 + ((8 × (8 × 8 − 8) − (88/8)) + 8)
:= 8 + (8 × (8 × 8 − 8) + ((8 + 8)/8))
:= (9 − 9/9) × ((999 + 9)/(9 + 9))
:= 9 + ((999 × (9 − 9/9))/(9 + 9))
:= 9 + ((((((9 + 9)/9)9 ) − 9 × 9) + 9) + 9)
Ï 449 := 1 + ((1 + 1) × (1 + 1) × (1 + 111))
:= 2/2 + (2 × (222 + 2))
Ï 454 := 11 + (((1 + 1) × (1 + 1) × 111) − 1)
:= 2 + (2 × ((222 + 2) + 2))
Ï 459 := 11 + ((1 + 1) × (1 + 1) × (1 + 111))
:= 22/2 + (2 × (222 + 2))
:= 3 + (((3 × (33 × 3 )) + 3)/(3 + 3))
:= 3/3 + ((3 × (3 + 3) × 3 ) − 33)
:= 3 × (3 × ((33 − 3) + 33 ))
:= 4 + (444 + 4/4)
:= 444 + (44 − 4)/4
:= 4 + (444 + 44/4)
:= 5 + ((5 − 5/5) × 555/5)
:= 5 + (((5 − 5/5) × 555/5) + 5)
:= (55 − 555)/5 − 55
:= 6 + ((6 × (66 + 6)) + (66/6))
:= 6 + ((6/6 + 6) × ((6 + 6)/6)6 )
:= 66 + (6 × 66 − (6 × 6/(6 + 6)))
:= 7 + (7 × (7 × 7 + 7 + 7) + 7/7)
:= 7 × 7 × 7 + 777/7
:= 7 + (7 × (7 × 7 + 7 + 7) + (77/7))
:= 8/8 + 8 × (8 × 8 − 8)
:= 8 + (8 × (8 × 8 − 8) − ((8 + 8)/8))
:= 88/8 + 8 × (8 × 8 − 8)
:= 9 + (((9 × 9 × 99) − 9)/(9 + 9))
:= (9 + 9 + 9) × ((9 − 9/9) + 9)
3
9
:= 9 + (((((9 + 9)/9) ) − 9 × 9) + 9)
Ï 450 := (1 + 1) × (1 + (1 + 1) × (1 + 111))
3
Ï 455 := 11 + ((1 + 1) × (1 + 1) × 111)
Ï 460 := (1 + 1) × ((11 − 1) × (1 + 11 + 11))
:= 2 + (2 × (222 + 2))
:= 22/2 + 2 × 222
:= 222 − (22 + 2)
:= (3 + 3) × ((3 × (33 − 3)) + 3)
:= (3 + 3)3 + (((33+3 − 3)/3) − 3)
:= (3 + 3)3 + ((33+3 + 3)/3)
:= 4 + (444 + (4 + 4)/4)
:= 444 + 44/4
:= 4 × 4 + 444
:= (5 + 5) × (55 − 5 − 5)
:= 5 + ((5 + 5) × (55 − 5 − 5))
:= 5 + (((5 + 5) × (55 − 5 − 5)) + 5)
:= 666 − 6 × 6 × 6
:= (6/6 + 6) × (66 − 6/6)
:= 6 × 66 + ((6 + 6)/6)6
:= (7 − 7/7) × (77 − (7 + 7)/7)
:= 7 × 77 − (77 + 7)
:= 7 × 77 − ((7 + 7)/7 + 77)
:= (8 + 8)/8 + 8 × (8 × 8 − 8)
:= 8 + (8 × (8 × 8 − 8) − 8/8)
:= 8 + (888/((8 + 8)/8) + 8)
:= 9 × ((9 × 99 + 9)/(9 + 9))
:= 9 + (((9 × 9 × 99) + 9)/(9 + 9))
:= (9 × (99 + 9)) − (((9 + 9)/9)9 )
Ï 451 := 1 + ((1 + 1) × (1 + (1 + 1) × (1 + 111)))
Ï 456 := 1 + (11 + ((1 + 1) × (1 + 1) × 111))
Ï 461 := ((1 + 1) × (11 × (11 + 11 − 1))) − 1
:= 2 + ((2 × (222 + 2)) + 2/2)
:= 2 × (((222 + 2) + 2) + 2)
:= 222 − (22 + 2/2)
:= 3 + ((3/3 + 3) × ((333 + 3)/3))
:= 3 + ((3 × (3 + 3) × 33 ) − 33)
:= (33 × (33/3 + 3)) − 3/3
:= 4 + ((444 − 4/4) + 4)
:= 4 + (444 + 4 + 4)
:= 4 × 4 + (444 + 4/4)
:= 5/5 + ((5 + 5) × (55 − 5 − 5))
:= 5 + (((5 + 5) × (55 − 5 − 5)) + 5/5)
:= 55/5 + ((5 + 5) × (55 − 5 − 5))
:= 66 + (6 × 66 − (66/6))
:= 66 + (6 × 66 − 6)
:= 66 + (6 × 66 − 6/6)
:= 77/7 × (7 × 7 − (7/7 + 7))
:= (7 − 7/7) × (77 − 7/7)
:= 7 × 77 − (7/7 + 77)
:= 88/8 + (8 × (8 × 8 − 8) − 8)
:= 8 + 8 × (8 × 8 − 8)
:= 8 × (8 × 8 − 8) + (88 + 8 + 8)/8
:= ((9 × 9 × 99) + 99)/(9 + 9)
:= (9 × (9 × 9 − (9 + 9))) − 999/9
:= 9/9 + ((9 × (99 + 9)) − (((9 + 9)/9)9 ))
Ï 452 := 11 + ((11 + 11 − 1)1+1 )
:= 2 × ((222 + 2) + 2)
Ï 457 := 11 + ((1 + 1) × (1 + (1 + 1) × 111))
:= 2 + (2 × 222 + 22/2)
3
3+3
:= (3/3 + 3) × (((333 − 3)/3) + 3)
:= (3 + 3) + (((3
:= 4 + (444 + 4)
:= 4 + (((444 + 4/4) + 4) + 4)
:= (5 + 5)/5 + ((5 + 5) × (55 − 5 − 5))
+ 3)/3) − 3)
5−5/5+5
:= (((5 + 5)/5)
) − 55
Ï 462 := (1 + 1) × (11 × (11 + 11 − 1))
:= 222 − 22
:= 33 × (33/3 + 3)
:= 4 × 4 + (444 + (4 + 4)/4)
:= 55/5 × ((((5 + 5)/5)5 + 5) + 5)
:= 66 + (((6 − 66)/6) + 6 × 66)
:= 66 + ((6 × 66 − 6) + 6/6)
:= 66 + 6 × 66
:= 77/7 + 7 × (7 × 7 + 7 + 7)
:= 7 + ((7 − 7/7) × (77 − (7 + 7)/7))
:= 77 × (7 − 7/7)
:= 8 + 888/((8 + 8)/8)
:= 8 + (8 × (8 × 8 − 8) + 8/8)
:= (8 − 8/8) × (((8 + 8)/8) + 8 × 8)
:= 9/9 + (((9 × 9 × 99) + 99)/(9 + 9))
:= ((9999 + 9)/(9 + 9)) − 99
:= 99 + ((99 × 99)/(9 + 9 + 9))
40
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 463 := 1 + ((1 + 1) × (11 × (11 + 11 − 1)))
Ï 468 := (1 + 1) × (1 + (11 + (1 + 1) × 111))
Ï 473 := (11 + 11)1+1 − 11
:= 22 + ((22 − 2/2)2 )
:= 222 − 22+2
:= 222 − 22/2
:= 3/3 + (33 × (33/3 + 3))
:= (3 + 3) × (3 × 33 − 3)
:= 33/3 + (33 × (33/3 + 3))
:= 4 + ((444 + 44/4) + 4)
:= 44 + (44 − 44)
:= 44/4 × (44 − 4/4)
:= 5 × (5 × 5 + 5) + (5 + 5)/(5 + 5)
:= (5 − 5/5) × ((555 + 5)/5 + 5)
:= 55/5 × (55 − ((55 + 5)/5))
:= 66 + (6 × 66 + 6/6)
:= 6 × (66 + 6 + 6)
:= 6 + ((6 × (66 + 6 + 6)) − 6/6)
5
:= 7/7 + (77 × (7 − 7/7))
:= (7 − 7/7) × (7/7 + 77)
:= 7 × 77 + (77/7 − 77)
:= 8 + ((8 × (8 × 8 − 8) − 8/8) + 8)
:= 8 × 8 × 8 − (88/((8 + 8)/8))
:= 8 + (((8 × (8 × 8 − 8) + 8/8) + 8) + 8)
:= 9 + ((((9 × 9 × 99) − 9)/(9 + 9)) + 9)
:= (9 + 9) × (((9 − 9/9) + 9) + 9)
:= 9 × 9 × 9 − (((9 + 9)/9)9−9/9 )
Ï 464 := (1 + 1) × (111 + 111+1 )
Ï 469 := 1 + ((1 + 1) × (1 + (11 + (1 + 1) × 111)))
Ï 474 := 1 + ((11 + 11)1+1 − 11)
:= 2 + (222 − 22)
:= 2/2 + (222 − 22+2 )
:= 222 + ((2 − 22)/2)
:= 3 + ((33 × (33/3 + 3)) − 3/3)
:= 3/3 + ((3 + 3) × (3 × 33 − 3))
:= (3 × ((3 + 3) × 33 − 3)) − 3
:= 4 + (444 + 4 × 4)
:= 4/4 + ((44 − 44) + 44 )
:= ((44 × (44 − 4/4)) + 4)/4
:= (5 − 5/5) × (555/5 + 5)
:= (5 − 5/5) − 555
:= 5 + ((5 − 5/5)5 − 555)
:= 66 + (((6 + 6)/6) + 6 × 66)
:= 6/6 + (6 × (66 + 6 + 6))
:= 6 + (6 × (66 + 6 + 6))
:= 7 × 77 + ((7 + 7)/7 − 77)
5
:= 7 + (77 × (7 − 7/7))
:= (7 − 7/7) × ((7 + 7)/7 + 77)
:= 8 + (8 × (8 × 8 − 8) + 8)
:= (8 − 8/8) × ((88/8 − 8) + 8 × 8)
:= (8 − (8 + 8)/8) × (88 − (8/8 + 8))
:= 9 + ((((9 × 9 × 99) + 9)/(9 + 9)) + 9)
:= 9 + ((9 × (99 + 9)) − (((9 + 9)/9)9 ))
:= ((9 + 9) × (9 + 9 + 9)) − (99 + 9)/9
Ï 465 := 1 + ((1 + 1) × (111 + 111+1 ))
Ï 470 := (1 + 1) × (11 + (1 + 1) × (1 + 111))
Ï 475 := 1 + (1 + ((11 + 11)1+1 − 11))
:= 2 + ((222 − 22) + 2/2)
:= 2 + (222 − 22+2 )
:= 2 + (222 − 22/2)
:= 3 + (33 × (33/3 + 3))
:= 3 + (((3 + 3) × (3 × 33 − 3)) − 3/3)
:= (3 × (3 + 3) × 33 ) − 33/3
:= 4 + ((444 + 4 × 4) + 4/4)
:= 4 + ((44/((4 + 4)/4)) + 444)
:= 444 + (4 × (4 + 4) − 4/4)
:= (5 × (5 × (5 × 5 − 5) − 5)) − 5
:= 5 × (5 × (5 × 5 − 5) − 5)
:= 5 × (5 × 5 × 5 − ((5 + 5)/5)5 )
:= 66 + ((6 × 6/(6 + 6)) + 6 × 66)
:= (6 + 6)/6 + (6 × (66 + 6 + 6))
:= 6 + ((6 × (66 + 6 + 6)) + 6/6)
:= 77 + (((7 × 777) − 7)/(7 + 7))
:= 7 + ((77 × (7 − 7/7)) + 7/7)
:= 7 + ((7 − 7/7) × (7/7 + 77))
:= 8 + ((8 × (8 × 8 − 8) + 8/8) + 8)
:= 8 + ((8 − 8/8) × (((8 + 8)/8) + 8 × 8))
:= 8 + ((8 × (8 × 8 − 8) + (88/8)) + 8)
:= 9 + ((9 × (9 × 9 − (9 + 9))) − 999/9)
:= (9 + 9)/9 + ((9 + 9) × (((9 − 9/9) + 9) + 9))
:= ((9 + 9) × (9 + 9 + 9)) − 99/9
Ï 466 := (1 + 1) × (11 + (1 + 1) × 111)
Ï 471 := (11 + 11)1+1 − (1 + 1 + 11)
2
:= 22 + 2 × 222
Ï 476 := (1 + 1) × ((1 + 1) × (111+1 − (1 + 1)))
:= 222 − 2 × (2 + 2)
:= 22 − (22/2 + 2)
:= 3 + ((33 × (33/3 + 3)) + 3/3)
:= 3 + ((3 + 3) × (3 × 3 − 3))
:= (33/3 + 3) × (3/3 + 33)
:= 444 + (44/((4 + 4)/4))
:= 4 × 4 + (444 + 44/4)
:= (55 + 5)/5 − (5 × ((5 + 5)/5)5 )
:= 444 + 4 × (4 + 4)
:= 5/5 + ((5 × (5 × (5 × 5 − 5) − 5)) − 5)
:= 5/5 + (5 × (5 × (5 × 5 − 5) − 5))
:= 6 × (66 − 6) + 666/6
:= (6/6 + 6) × (((6 + 6)/6) + 66)
6
:= 6 + (((6 + 6)/6) + 6 × 66)
3
7
:= 77 + (((7 × 777) + 7)/(7 + 7))
:= 7 × 7 × 7 + ((7 + 7)/7)
:= 7 × (77 − 7) − (7 + 7)
:= 8 + ((8 × (8 × 8 − 8) + ((8 + 8)/8)) + 8)
:= 8 + (((8 × (8 × 8 − 8) − 8/8) + 8) + 8)
:= 88 × 88/(8 + 8) − 8
:= ((9 + 9) × (9 + 9 + 9)) − (99/9 + 9)
:= 9 + (((99 × 99)/(9 + 9 + 9)) + 99)
:= (((9 + 9)/9)9 ) − ((9 + 9 + 9) + 9)
Ï 467 := 1 + ((1 + 1) × (11 + (1 + 1) × 111))
:= 22 + (2 × 222 + 2/2)
3
:= ((3 + 3) × (3 × 3 − 3)) − 3/3
4
4
:= 4 + (4 − (44 + 4/4))
5
:= 5 + ((5 − 5)/(5 + 5) + 5 × (5 × 5 + 5))
Ï 472 := (11 + 11)1+1 − 1 − 11
:= 222 − (2 × (2 + 2 + 2))
3
:= 3 + (((3 + 3) × (3 × 3 − 3)) + 3/3)
4
4
:= 4 + ((4 − 44) + 4 )
5
5
:= 55 + (((5 + 5 ) + 5)/(5 + 5 + 5))
Ï 477 := 1 + ((1 + 1) × ((1 + 1) × (111+1 − (1 + 1))))
:= 2 + ((222 − 22/2) + 2)
:= 3 × ((3 + 3) × 33 − 3)
:= 4 + (44/4 × (44 − 4/4))
:= (5 + 5)/5 + (5 × (5 × (5 × 5 − 5) − 5))
:= (6 × (66 + 6 + 6)) − 6/6
:= 6 + ((((6 + 6)/6) + 6 × 66) + 6)
:= 6 + (6 × (66 − 6) + 666/6)
:= 7 + (7 × 77 − ((7 + 7)/7 + 77))
:= 7 × (77 − 7) − (77/7 + 7)
:= 7/7 + (7 × (77 − 7) − (7 + 7))
:= 8 + (8 × (8 × 8 − 8) + (88/8))
:= 8 + ((8 × (8 × 8 − 8) + 8) + 8)
:= (8/8 + 8) × (8 × 8 − 88/8)
:= 9 + (((((9 × 9 × 99) − 9)/(9 + 9)) + 9) + 9)
:= ((9 + 9) × (9 + 9 + 9)) − 9
9
:= (((9 + 9)/9) ) − (((9 + 9 + 9) + 9) + 9)
6
41
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 478 := (1 + 1) × ((1 + 1) × (111+1 − 1) − 1)
Ï 483 := (11 + 11)1+1 − 1
Ï 488 := (1 + 1) × ((1 + 1) × (1 + 111+1 ))
:= 222 − (2 + 2 + 2)
:= 222 − 2/2
:= 2 + 222 + 2
:= 3/3 + (3 × ((3 + 3) × 33 − 3))
:= (3 × (3 + 3) × 33 ) − 3
:= 3 + (((3 − 3/3)3×3 ) − 33 )
:= 44 + (444/((4 + 4)/4))
:= (44 × 44 − 4)/4
:= 44 + 444
:= 5 + (55/5 × (55 − ((55 + 5)/5)))
:= 5 × 55 + ((55 − 5)/(5 + 5 + 5))
:= (5 − 5/5) × ((555 + 55)/5)
:= ((66 − 6)/6) + (6 × (66 + 6 + 6))
:= (6/6 + 6) × (6 × 6/(6 + 6) + 66)
:= ((6 + 6)/6 + 6) × ((66 − 6) + 6/6)
:= 7 + (((7 + 7)/7) + 7 × 7 × 7)
:= 7 × (77 − 7) − 7
:= 7 × (77 − 7) − (7 + 7)/7
:= (88 × 88 − (88 + 8))/(8 + 8)
:= 88 × 88/(8 + 8) − 8/8
:= 8 × 8 × 8 − 8 − 8 − 8
:= 9/9 + (((9 + 9) × (9 + 9 + 9)) − 9)
:= ((9 + 9 + 9)/9) × (9 × (9 + 9) − 9/9)
:= (9 + 9)/9 + ((9 + 9) × (9 + 9 + 9))
7
Ï 479 := ((1 + 1) × (1 + 1) × (111+1 − 1)) − 1
Ï 489 := 1 + ((1 + 1) × ((1 + 1) × (1 + 111+1 )))
Ï 484 := (11 + 11)1+1
:= 222 − (2/2 + 2 + 2)
:= 222
:= 2 + 222 + 2/2 + 2
:= ((3 − 3/3)3×3 ) − 33
:= 3/3 + ((3 × (3 + 3) × 33 ) − 3)
:= 3 + (3 × (3 + 3) × 33 )
:= ((44 × 44 − 4)/4) − 4
:= 44 × 44/4
:= 4 + ((44 × 44 + 4)/4)
5
:= 5 + (((5 − 5/5) − 555) + 5)
:= 55/5 × (55 − (55/5))
(6+6)/6
:= 555 − (55/5 + 55)
:= 666 − (666/6 + 66)
:= 66/6 + (6 × (66 + 6 + 6))
:= ((66 + 66)/6)
:= 7 × (77 − 7) − 77/7
:= 7/7 + (7 × (77 − 7) − 7)
:= 7 × (77 − 7) − 7/7
:= 8 × 8 × 8 − ((8/8 + 8 + 8 + 8) + 8)
:= 88 × 88/(8 + 8)
:= 8/8 + (8 × 8 × 8 − (8 + 8 + 8))
:= (9 + 9)/9 + (((9 + 9) × (9 + 9 + 9)) − 9)
:= ((99 + 99)/9)(9+9)/9
:= ((9 + 9 + 9)/9) + ((9 + 9) × (9 + 9 + 9))
Ï 480 := (1 + 1) × (1 + 1) × (111+1 − 1)
Ï 490 := (1 + 1) × (1 + ((1 + 1) × (1 + 111+1 )))
Ï 485 := 1 + (11 + 11)1+1
:= 222 − 2 − 2
:= 2/2 + 222
:= 2 + 222 + 2 + 2
:= 3 + (3 × ((3 + 3) × 33 − 3))
:= ((3 − 3/3)3×3 ) − 33
:= 3 + ((3 × (3 + 3) × 33 ) + 3/3)
:= (4 + 4) × (4 × 4 + 44)
:= (44 × 44 + 4)/4
:= 4 + (((44 × 44 + 4) + 4)/4)
:= 5 + (5 × (5 × (5 × 5 − 5) − 5))
:= 5 + ((5 × (5 × (5 × 5 − 5) − 5)) + 5)
:= (5 + 5) × (55 − (5/5 + 5))
:= 6 + ((6 × (66 + 6 + 6)) + 6)
:= 6 + ((6 × (66 + 6 + 6)) + (66/6))
:= (6/6 + 6) × (((6 + 6)/6)6 + 6)
:= (7/7 + 7) × (77/7 + 7 × 7)
:= ((7 + 7)/7) + (7 × (77 − 7) − 7)
:= 7 × (77 − 7)
:= 8 × (8 × 8 − (8/((8 + 8)/8)))
:= 8/8 + 88 × 88/(8 + 8)
:= (9/9 − 9 × 9) × (((9 + 9 + 9)/9) − 9)
Ï 481 := (11 + 11)1+1 − 1 − 1 − 1
9
:= (((9 + 9)/9) ) − (9 + 9 + 9)
Ï 486 := 1 + (1 + (11 + 11)1+1 )
:= 8 × 8 × 8 − (88 + 88)/8
:= (((9 × 999) − 9)/(9 + 9)) − 9
Ï 491 := 1 + ((1 + 1) × (1 + ((1 + 1) × (1 + 111+1 ))))
:= 222 − 2/2 − 2
:= 2 + 222
:= 2 + 222 + 2/2 + 2 + 2
:= 3 + ((3 × ((3 + 3) × 33 − 3)) + 3/3)
:= 3 × (3 + 3) × 33
:= 3 + ((((3 − 3/3)3×3 ) − 33 ) + 3)
:= ((44 × 44 + 4)/4) − 4
:= ((44 × 44 + 4) + 4)/4
:= 4 + (((44 × 44 − 4)/4) + 4)
:= (5555 − 5 )/5 − 5
:= (5555 − 55 )/5
:= 5 + (5555 − 55 )/5
:= (6/6 + 6 + 6) × (6 × 6 + 6/6)
:= 666 + (6 × (6 − 6 × 6))
:= 6 × 6 + ((6/6 + 6) × (66 − 6/6))
:= 7 × (77 − 7) − ((7 + 7)/7 + 7)
:= 7 + (7 × (77 − 7) − (77/7))
:= 7/7 + 7 × (77 − 7)
:= 8 × 8 × 8 + ((8 − 88)/8 − (8 + 8))
:= 8 + (88 × 88/(8 + 8) − 8/8)
:= (9 + 9) × (9 + 9 + 9)
:= (((9 × 999) + 9)/(9 + 9)) − 9
5
:= 8/8 + (8 × (8 × 8 − (8/((8 + 8)/8))))
:= 9 × 9 + ((99/9 + 9)
(9+9)/9
)
Ï 482 := (11 + 11)1+1 − 1 − 1
Ï 487 := 1 + (1 + (1 + (11 + 11)1+1 ))
Ï 492 := (1 + 1) × ((1 + 1) × (1 + (1 + 111+1 )))
:= 222 − 2
:= 2 + 222 + 2/2
:= 2 × (2 + 2) + 222
:= 3 + (((3 − 3/3)3×3 ) − 33)
:= 3/3 + (3 × (3 + 3) × 33 )
:= 3 + ((3 × (3 + 3) × 33 ) + 3)
:= (44 × 44 − (4 + 4))/4
:= 4 + ((44 × 44 − 4)/4)
:= 4 + (444 + 44)
:= ((5555 − 5 + 5)/5) − 5
:= (5555 − 5 + 5)/5
:= 5 + ((5555 − 55 + 5)/5)
:= 6 + ((6/6 + 6) × (((6 + 6)/6) + 66))
:= 6 + ((6/6 + 6 + 6) × (6 × 6 + 6/6))
:= 66 + ((6 × (66 + 6)) − 6)
:= 7 × (77 − 7) − (7/7 + 7)
:= 7 × (77 − 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + 7 × (77 − 7)
:= 8 × 8 × 8 − ((88 + 88)/8 + 8)
:= 8 × 8 × 8 − (8/8 + 8 + 8 + 8)
:= 8 + 88 × 88/(8 + 8)
:= ((9 × ((9 × (99 + 9)) − 9)) + 9)/(9 + 9)
:= 9/9 + ((9 + 9) × (9 + 9 + 9))
:= (((9 + 9)/9)9 ) − (99/9 + 9)
5
5
42
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 493 := 11 + ((11 + 11)1+1 − (1 + 1))
Ï 498 := (11 − 1)1+1+1 /(1 + 1) − 1 − 1
Ï 503 := 1 + 1 + (1 + 1)11−1−1 − 11
:= 22/2 + (222 − 2)
:= 22+2 + (222 − 2)
:= 22 + 222 − (2/2 + 2)
:= 3 + (((3 × (3 + 3) × 33 ) + 3/3) + 3)
:= 3 + (3 × ((3 + 3) × 33 + 3))
:= ((3 − 3/3)3×3 ) − 3 × 3
:= 4 + (((44 × 44 + 4)/4) + 4)
:= 44 + (44 × 44/(4 + 4))
:= (((4 + 4) × (44 − 4)) − 4)/4
:= 555 + ((5 − 55 /5)/(5 + 5))
:= 555 − ((5 + 5)/5 + 55)
:= (55 − (555 + 55))/5
:= ((6 + 6) × (6 × 6 + 6)) − 66/6
:= 66 + (6 × (66 + 6))
:= ((6 + 6) × (6 × 6 + 6)) − 6/6
:= 7 × (77 − 7) + (7 + 7 + 7)/7
:= 7 + (7 × (77 − 7) + 7/7)
:= 7 + ((7 × (77 − 7) − 7/7) + 7)
:= 8 × 8 × 8 − (88/8 + 8)
:= 8 × 8 × 8 + (((8 + 8)/8) − (8 + 8))
:= 8 × 8 × 8 − (8/8 + 8)
:= ((99 + 9)/9) + ((9 + 9) × (9 + 9 + 9))
:= (((9 + 9)/9)9 ) − 9
9
:= (((9 + 9)/9) ) − (9/9 + 9 + 9)
Ï 494 := 11 + ((11 + 11)1+1 − 1)
Ï 499 := (11 − 1)1+1+1 /(1 + 1) − 1
Ï 504 := (1 + 1) × ((1 + 11) × (11 + 11 − 1))
:= 2 + 22/2 + 222 + 2
:= 22 + 222 − 2
:= ((3 − 3/3)3×3 ) − (3 × (3 + 3))
:= ((3 × (3 × 333)) − 3)/(3 + 3)
:= (3 + 3) × (3 × 33 + 3)
:= ((44 × 44 − 4) + 44)/4
:= 4 + 44/4 × (44 + 4/4)
:= 44 + (44 − 4 − 4)
:= 555 − ((55 + 5/5) + 5)
:= 555 − (55 + 5/5)
:= 5 + (555 − (55 + 5/5))
:= ((6 − 66)/6) + ((6 + 6) × (6 × 6 + 6))
:= 66 + ((6 × (66 + 6)) + 6/6)
:= (6 + 6) × (6 × 6 + 6)
:= 77/7 + (7 × (77 − 7) − 7)
:= 7 + (7 × (77 − 7) + ((7 + 7)/7))
:= 7 + (7 × (77 − 7) + 7)
:= 8 × 8 × 8 + ((8 − 88)/8 − 8)
:= 8 × 8 × 8 − (88 + 8 + 8)/8
:= 8 × 8 × 8 − 8
:= ((9 × 999) − 9)/(9 + 9)
:= 9 + (((9 + 9) × (9 + 9 + 9)) + 9)
:= 2 + 2 × (2 + 2) + 22
2
9
:= (((9 + 9)/9) ) − (9 + 9)
Ï 495 := 11 + (11 + 11)1+1
Ï 500 := (11 − 1)1+1+1 /(1 + 1)
Ï 505 := (11111 − 1)/(11 + 11)
:= 22+2 + 222
:= 22 + 222 − 2/2
:= 3 × ((3 + 3) × 33 + 3)
:= (3/3 + 3) × ((3 − 3/3 + 3)3 )
:= 3/3 + ((3 + 3) × (3 × 33 + 3))
:= 44/4 × (44 + 4/4)
:= 4 + ((44 − 4 × 4) + 44 )
:= (((4 + 4) × (44 − 4)) + 4)/4
:= 55 × (5 − 5/5 + 5)
:= 5 × 5 × (5 × 5 − 5)
:= 5 + (5 × 5 × (5 × 5 − 5))
:= 6 × 66 + (666/6 − (6 + 6))
:= 66 + ((6 × (66 + 6)) + ((6 + 6)/6))
:= 6/6 + ((6 + 6) × (6 × 6 + 6))
:= 7 + (7 × (77 − 7) − ((7 + 7)/7))
:= 7 × (77 − 7) + (77 − 7)/7
:= 7 + ((7 × (77 − 7) + 7/7) + 7)
:= 8 × 8 × 8 − (8/8 + 8 + 8)
:= 8 × 8 × 8 − (88 + 8)/8
:= 8/8 + (8 × 8 × 8 − 8)
:= 9 + ((9 + 9) × (9 + 9 + 9))
:= ((9 × 999) + 9)/(9 + 9)
:= (9 + 9)/9 + ((((9 + 9)/9)9 ) − 9)
:= 22/2 + 22
2
Ï 496 := 1 + (11 + (11 + 11)1+1 )
:= 222 + (2 × (2 + 2 + 2))
:= 3/3 + (3 × ((3 + 3) × 33 + 3))
Ï 501 := (1 + 1)11−1−1 − 11
:= 2/2 + 2
2+2
Ï 506 := (1 + 1) × 11 × (1 + 11 + 11)
2
:= 22 + 222
3
:= ((3 − 3/3)3×3 ) − (3 + 3)
+ 22
:= ((3 + 3) × (3 × 3 + 3)) − 3
4
4
:= 4 × (4 × 4 × (4 + 4) − 4)
:= 4 + (4 − 44/4)
:= 44 + (44 − ((4 + 4)/4 + 4))
:= 5/5 + (55 × (5 − 5/5 + 5))
:= 5/5 + (5 × 5 × (5 × 5 − 5))
:= 5 + ((5 × 5 × (5 × 5 − 5)) + 5/5)
:= 6 × 66 + (666/6 − 6)
:= (6 + 6)/6 + ((6 + 6) × (6 × 6 + 6))
:= 7 + (7 × (77 − 7) − 7/7)
:= 77/7 + 7 × (77 − 7)
:= 7 + ((7 × (77 − 7) + ((7 + 7)/7)) + 7)
:= 8 × 8 × 8 − 8 − 8
:= 8 × 8 × 8 − 88/8
:= 8 × 8 × 8 + (((8 + 8)/8) − 8)
:= 9 + (((9 + 9) × (9 + 9 + 9)) + 9/9)
:= (((9 + 9)/9)9 ) − 99/9
:= 9 + (((9 + 9) × (9 + 9 + 9)) + (99/9))
6
:= ((6 + 6)/6) + (6 × (66 + 6))
Ï 497 := 1 + (1 + (11 + (11 + 11)1+1 ))
Ï 502 := 1 + (1 + 1)11−1−1 − 11
Ï 507 := (1 + 1 + 1) × (1 + 1 + 11)1+1
:= 2 + (22/2 + 222 )
:= 2 + 22+2 + 222
:= 22 + 222 + 2/2
:= 33/3 + (3 × (3 + 3) × 33 )
:= 3 + ((3 × 3 × 333 − 3)/(3 + 3))
:= 3 + ((3 + 3) × (3 × 33 + 3))
:= 4/4 + ((44 − 4 × 4) + 44 )
:= 44 + ((4 − 44)/4 + 44 )
:= 44 + (44 − (4/4 + 4))
:= 555 + ((5 + 5)/5 − (55 + 5))
:= 555 + ((5 + 5)/5 − 55)
:= (((5 + 5)/5)5−5/5+5 ) − 5
:= 66 + ((6 × (66 + 6)) − 6/6)
:= ((6 + 6) × (6 × 6 + 6)) − (6 + 6)/6
:= 6 × 66 + 666/6
:= 7 + 7 × (77 − 7)
:= 7 × (77 − 7) + (77 + 7)/7
:= 7 + (7 × (77 − 7) + ((77 − 7)/7))
:= 8/8 + (8 × 8 × 8 − (8 + 8))
:= 8 × 8 × 8 + (8 − 88)/8
:= 8 × 8 × 8 + (88/8 − (8 + 8))
:= 99/9 + ((9 + 9) × (9 + 9 + 9))
:= (((9 + 9)/9)9 ) − 9/9 − 9
:= (((9 + 9)/9)9 ) + ((9 − 99)/(9 + 9))
43
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 508 := 1 + (1 + 1 + 1) × (1 + 1 + 11)1+1
Ï 518 := (1 + 11 + 11)1+1 − 11
Ï 513 := 1 + (1 + 1)11−1−1
:= 2 + 222 + 22
:= 2/2 + 2
(2/2+2)2
:= 2 + (2 × (22×(2+2) + 2))
:= ((3 − 3/3)3×3 ) − (3/3 + 3)
:= 33+3 − (3 + 3)3
:= 3 + (((3 − 3/3)3×3 ) + 3)
:= 44 + (44 − 4)
:= 4/4 + (44 + 44 )
:= 4 + (((4 + 4)/4 + 44 ) + 44 )
:= ((55 − (555 + 5))/5) − 5
:= (55 − (555 + 5))/5
:= 5 + ((55 − (555 + 5))/5)
:= 6 × 66 + (666 + 6)/6
:= 6 + (666/6 + 6 × 66)
:= 6 + (((6 + 6)/6)6×6/(6+6)+6 )
:= 7 + (7 × (77 − 7) + (77/7))
:= 7/7 + ((7/7 + 7)(7+7+7)/7 )
:= 7 × 77 − (7 + 7 + 7)
:= 8 × 8 × 8 − (8/((8 + 8)/8))
:= 8/8 + 8 × 8 × 8
:= 9 + (((9 × 999) − 9)/(9 + 9))
Ï 509 := (1 + 1)11−1−1 − 1 − 1 − 1
:= 2 + 222 + 2/2 + 22
:= ((3 − 3/3)3×3 ) − 3
:= 4/4 + ((44 − 4) + 44 )
:= (55 − 555)/5 − 5
:= 6 + (((6 + 6) × (6 × 6 + 6)) − 6/6)
:= 7 + (7 × (77 − 7) + (77 + 7)/7)
:= 8 + (8 × 8 × 8 − 88/8)
:= 9 + (((9 × 999) + 9)/(9 + 9))
Ï 510 := (1 + 1)11−1−1 − 1 − 1
:= 2
:= 3
(2/2+2)2
3+3
−2
4
:= 4 + (4 − (4 + 4)/4)
:= 5 + ((5 × 5 × (5 × 5 − 5)) + 5)
:= 6 + ((6 + 6) × (6 × 6 + 6))
:= (7 − 7/7) × (7/7 + 77 + 7)
:= 8 × 8 × 8 − (8 + 8)/8
:= (((9 + 9)/9)9 ) − (9 + 9)/9
Ï 511 := (1 + 1)11−1−1 − 1
:= 2
(2/2+2)2
− 2/2
3×3
:= ((3 − 3/3)
4
) − 3/3
4
:= 4 + (4 − 4/4)
:= 555 + (55/5 − 55)
:= 6 + (((6 + 6) × (6 × 6 + 6)) + 6/6)
:= 7 × (77 + 7) − 77
:= 8 × 8 × 8 − 8/8
:= (((9 + 9)/9)9 ) − 9/9
Ï 512 := (1 + 1)11−1−1
:= 2
(2/2+2)2
3×3
:= (3 − 3/3)
4
:= 4 + 4
4
:= ((5 + 5)/5)5−5/5+5
:= ((6 + 6)/6)6×6/(6+6)+6
:= (7/7 + 7)
(7+7+7)/7
:= 8 × 8 × 8
9
:= ((9 + 9)/9)
:= 9/9 + (((9 + 9)/9) )
Ï 514 := 1 + 1 + (1 + 1)11−1−1
:= 2 + 2(2/2+2)
:= 9 + ((((9 × 999) + 9)/(9 + 9)) + 9)
Ï 519 := 1 + (1 + 11 + 11)1+1 − 11
:= 2 + ((2 × (22×(2+2) + 2)) + 2/2)
2
:= 3 + (((3 − 3/3)3×3 ) − 3/3)
:= 33 + (3 × (3 + 3) × 33 )
:= 44 + ((4 + 4)/4 + 44 )
:= (((4 + 4) × (44 + 4)) − 4)/4
:= (55 − 555)/5
:= 5 + (55 − 555)/5
:= 6 + ((666 + 6)/6 + 6 × 66)
:= 6 + ((666/6 + 6 × 66) + 6)
:= 7 × 77 − (77/7 + 7 + 7)
:= 7 + ((7/7 + 7)(7+7+7)/7 )
:= 8 × 8 × 8 + (8 + 8)/8
:= 8 + (8 × 8 × 8 − 8/8)
:= (9 + 9)/9 + (((9 + 9)/9)9 )
:= 9 + ((((9 + 9)/9)9 ) − ((9 + 9)/9))
Ï 515 := 1 + 1 + 1 + (1 + 1)11−1−1
Ï 520 := 1 + 1 + (1 + 11 + 11)1+1 − 11
:= 2 + ((2(2/2+2) ) + 2/2)
:= 2 × ((22×(2+2) + 2) + 2)
:= 3 + ((3 − 3/3)3×3 )
:= 3 × 3 + (((3 − 3/3)3×3 ) − 3/3)
:= 4 + ((44 − 4/4) + 44 )
:= 4 + ((44 + 44 ) + 4)
:= 55 /5 − (55 + 55)
:= (55 − 5)/(5/5 + 5)
:= 66/6 + ((6 + 6) × (6 × 6 + 6))
:= ((6 + 6)/6 + 6) × (66 − 6/6)
:= 7 + ((7 × (77 − 7) + (77/7)) + 7)
:= 7 × 77 − ((77 + 7)/7 + 7)
:= 8 × 8 × 8 + (88/8 − 8)
:= 8 + 8 × 8 × 8
2
3
− ((3 + 3) + 3)
4
:= 8 + (8 × 8 × 8 − ((8 + 8)/8))
9
9
:= (((9 + 9)/9) ) + ((9 + 9 + 9)/9)
Ï 516 := 1 + 1 + 1 + 1 + (1 + 1)11−1−1
:= 9 + ((((9 + 9)/9)9 ) − 9/9)
Ï 521 := 11 + (1 + 1)11−1−1 − (1 + 1)
:= 2 × (22×(2+2) + 2)
:= (22/2)2 + (22 − 2)2
:= 3 + (33+3 − (3 + 3)3 )
:= 3 × 3 + ((3 − 3/3)3×3 )
:= 4 + (44 + 44 )
:= (((4 + 4) × (44 + 4)) + 4)/4
:= (55 + 5)/5 − (55 + 55)
:= (55 + 5/5)/(5/5 + 5)
:= 6 + (((6 + 6) × (6 × 6 + 6)) + 6)
:= 6 + (((6 + 6) × (6 × 6 + 6)) + (66/6))
:= (7 − 7/7) × (((7 + 7)/7 + 77) + 7)
:= 7 × 77 − (77/7 + 7)
:= 8 × 8 × 8 + (8/((8 + 8)/8))
:= 8 + (8 × 8 × 8 + 8/8)
:= (((9 + 9)/9)9 ) + ((9 × 9 − 9)/(9 + 9))
:= 9 + (((9 + 9)/9)9 )
Ï 517 := 11 × (1 + (1 + 1) × (1 + 11 + 11))
Ï 522 := 11 + (1 + 1)11−1−1 − 1
:= 22 + (22/2 + 222 )
:= 2 + ((2 + 2 + 2)2 + 222 )
:= 3 + ((((3 − 3/3)3×3 ) − 3/3) + 3)
:= (3 + 3) × (3 × 33 + 3 + 3)
:= 4 + ((4/4 + 44 ) + 44 )
:= 44 + ((44 − 4)/4 + 44 )
:= 5 + ((5 + 5)/5)5−5/5+5
:= 5 + ((5 + 5)/5)5−5/5+5 + 5
:= (66/6) × (66/6 + 6 × 6)
:= 666 − (6 + 6) × (6 + 6)
:= 77/7 × 7 × 7 − (7 + 7)/7
:= 7 × 77 + (((7 − 77)/7) − 7)
:= 8 + ((8 × 8 × 8 − 88/8) + 8)
:= 8 + (8 × 8 × 8 + (8 + 8)/8)
:= 9 + ((((9 × 999) − 9)/(9 + 9)) + 9)
:= 9 + ((((9 + 9)/9)9 ) + 9/9)
44
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 523 := 11 + (1 + 1)11−1−1
:= 22/2 + (2
(2/2+2)2
)
Ï 533 := 1 + 1 + 1 + 1 + (1 + 11 + 11)1+1
Ï 528 := (1 + 11 + 11)1+1 − 1
:= 2 + (((22 + 2/2)2 ) + 2)
:= 22 × (22 + 2)
3
:= 3 + (((3 − 3/3)3×3 ) + (3 × (3 + 3)))
:= 33/3 + ((3 − 3/3)3×3 )
:= 33 × (3 − 33/3)
:= 44 + (44/4 + 44 )
:= 44 × ((4 + 4) + 4)
:= 555 − ((5 + 5)/5)5
:= 5 + (555 − ((5 + 5)/5) )
:= 555 − (55 + 55)/5
:= 6 + ((66/6) × (66/6 + 6 × 6))
:= 66 × ((6 + 6)/6 + 6)
:= 6 + ((66 × ((6 + 6)/6 + 6)) − 6/6)
:= 7 × 77 − (((7 + 7)/7 + 7) + 7)
:= 7 × 77 − 77/7
:= 7/7 + (7 × 77 − 7)
:= 8 × 8 × 8 + 88/8
:= 8 + (8 × 8 × 8 + 8)
:= 99/9 + (((9 + 9)/9)9 )
:= 9 + (((((9 + 9)/9) ) − ((9 + 9)/9)) + 9)
Ï 524 := 1 + 11 + (1 + 1)11−1−1
:= 4 + ((44 × ((4 + 4) + 4)) + 4/4)
5
:= 8 + ((88 + 8 + 8)/8 + 8 × 8 × 8)
9
:= 9 + ((((9 + 9)/9)9 ) + ((99 + 9)/9))
Ï 534 := 11 + 11 + (1 + 1)11−1−1
Ï 529 := (1 + 11 + 11)1+1
2
:= 2 × (222 − 222)
:= (22 + 2/2)2
:= 22 + (2(2/2+2) )
:= 3 + (((3 − 3/3)3×3 ) + 3 × 3)
:= (33 − 3/3 − 3)3−3/3
:= 3 + (33 × (3 + 3) + 333)
:= (44 × ((4 + 4) + 4)) − 4
:= 4/4 + (44 × ((4 + 4) + 4))
:= (4 + 4)/4 × (44/4 + 44 )
:= 5 + ((55 − 555)/5 + 5)
:= 555 − (5 × 5 + 5/5)
:= 5 + (555 − (5 × 5 + 5/5))
:= 6 × 66 + (((6 + 6)/6)6/6+6)
:= 6/6 + (66 × ((6 + 6)/6 + 6))
:= 6 + (66 × ((6 + 6)/6 + 6))
:= 7 × 77 − (7/7 + 7 + 7)
:= 8 × 8 × 8 + (88 + 8)/8
:= (((9 + 9)/9)9 ) + (99 + 9)/9
Ï 525 := 1 + 1 + 11 + (1 + 1)11−1−1
:= ((22 + 2/2)2 ) − 2 − 2
:= 3 + ((3 + 3) × (3 × 33 + 3 + 3))
:= 444 + (4 − 4/4)4
:= 5 × (5 × (5 × 5 − 5) + 5)
:= (6 − 6/6) × (666/6 − 6)
:= 7 × 77 − (7 + 7)
:= 8 × 8 × 8 + (88 + 8 + 8)/8
:= (((9 + 9)/9)9 ) + ((99 + 9 + 9)/9)
Ï 526 := (1 + 1) × ((1 + 1) × 11 × (1 + 11) − 1)
:= 7 × 77 + (7 − 77)/7
:= 7 × 77 + ((7 + 7)/7 − 7)
:= 8 + ((8 × 8 × 8 + 8/8) + 8)
:= 8 × 8 × 8 + (88 + 88)/8
:= 9 + (((((9 + 9)/9)9 ) − 9/9) + 9)
:= (((9 + 9)/9)9 ) + ((99 + 99)/9)
Ï 530 := 1 + (1 + 11 + 11)1+1
Ï 535 := 1 + 11 + 11 + (1 + 1)11−1−1
:= 2 + (22 × (22 + 2))
:= 2 + ((((22 + 2/2)2 ) + 2) + 2)
:= (3 × (3 + 3)) + ((3 − 3/3)3×3 )
:= (3333 − (3 × 3 + 3)3 )/3
:= (4 + 4)/4 + (44 × ((4 + 4) + 4))
:= (4/4 + 4) × (444/4 − 4)
:= 555 − 5 × 5
:= 5 + (555 − 5 × 5)
:= (6 + 6)/6 + (66 × ((6 + 6)/6 + 6))
:= 6 + ((66 × ((6 + 6)/6 + 6)) + 6/6)
:= 7 × 77 − ((7 + 7)/7 + 7)
:= 7 + (7 × 77 − (77/7))
:= 8 + ((8 × 8 × 8 + (8 + 8)/8) + 8)
:= 8 + (((8 × 8 × 8 − 8/8) + 8) + 8)
9
:= 9 + ((((9 + 9)/9) ) + 9)
:= (((99 × 99) − 9)/(9 + 9)) − 9
Ï 531 := 1 + 1 + (1 + 11 + 11)1+1
Ï 536 := (1 + 1) × (1 + 1) × (1 + 1 + 11 × (1 + 11))
2
2
:= (22 × (22 + 2)) − 2
:= 2 + ((22 + 2/2) )
:= 2 + ((2(2/2+2) ) + 22)
:= 3 + (((3 − 3/3)3×3 ) + 33/3)
:= 3 × ((3 × (33 + 33)) − 3)
:= 33 + (((3 − 3/3)3×3 ) − 3)
:= 4 + ((44 × ((4 + 4) + 4)) − 4/4)
:= 4 + ((44 × ((4 + 4) + 4)) + 4)
:= (44 × ((4 + 4) + 4)) − (4 + 4)/4
:= 5 + ((5 + 5/5)/(5/5 + 5))
:= 5/5 + (555 − 5 × 5)
:= 5 + ((555 − 5 × 5) + 5/5)
:= 6 + (((6 + 6)/6 + 6) × (66 − 6/6))
:= 6 + ((6 − 6/6) × (666/6 − 6))
:= ((6 + 6)/6 + 6) × (66 + 6/6)
:= 7/7 + (7 × 77 − (7 + 7))
:= 7 × 77 − (7/7 + 7)
:= 7 × 77 − (7 + 7 + 7)/7
:= 8 + ((8 × 8 × 8 − ((8 + 8)/8)) + 8)
:= 8 + (8 × 8 × 8 + 88/8)
:= 8 + (8 × 8 × 8 + 8 + 8)
:= 9 × 9 + (((9 × 9 × 99) − 9)/(9 + 9))
:= 9 × 9 × 9 − (99 + 99)
:= (((99 × 99) + 9)/(9 + 9)) − 9
5
Ï 527 := (1 + 11 + 11)1+1 − 1 − 1
Ï 532 := 1 + 1 + 1 + (1 + 11 + 11)1+1
Ï 537 := 1 + (1 + 1) × (1 + 1) × (1 + 1 + 11 × (1 + 11))
:= ((22 + 2/2)2 ) − 2
:= 2 + ((22 × (22 + 2)) + 2)
:= 2 × (2 + 2) + ((22 + 2/2)2 )
:= 3 + ((((3 − 3/3)3×3 ) + 3 × 3) + 3)
:= 3 + ((33 − 3/3 − 3)3−3/3 )
:= (3 × (3 × (33 + 33))) − 3
:= (44 × ((4 + 4) + 4)) − 4/4
:= 4 + (44 × ((4 + 4) + 4))
:= (4/4 + 4)4 − (44 + 44)
:= 555 + ((5 + 5)/5 − (5 × 5 + 5))
:= 555 + ((5 + 5)/5 − 5 × 5)
6
:= 5 + (((5 + 5)/5 − 5 × 5) + 555)
:= (66 × ((6 + 6)/6 + 6)) − 6/6
:= (6/6 + 6) × ((((6 + 6)/6) + 6) + 6)
:= 666/6 + ((6 × (66 + 6)) − 6)
:= 7 × 77 − (77 + 7)/7
:= 7 × 77 − 7
:= 7 × 77 − (7 + 7)/7
:= 8 + ((8 × 8 × 8 − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 8 × 8 × 8)
:= 8 + (((8 × 8 × 8 + 8/8) + 8) + 8)
:= 9 × 9 + (((9 × 9 × 99) + 9)/(9 + 9))
:= 9 + ((((9 + 9)/9)9 ) + (99/9))
:= 9 × (9 × 9 − 9) − 999/9
45
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 538 := 11 + ((1 + 11 + 11)1+1 − (1 + 1))
:= 22 + (2 × (2
2×(2+2)
+ 2))
Ï 543 := (1111 − 1)/(1 + 1) − 1 − 11
:= 2
2+2
2
+ (((22 + 2/2) ) − 2)
Ï 548 := (1111 − 11)/(1 + 1) − 1 − 1
:= 222 + 22+2+2
:= 33 + (((3 − 3/3)3×3 ) − 3/3)
:= 3 + (3 × (3 × (33 + 33)))
:= 3 + (((3 − 3/3)3×3 ) + 33)
:= 4 + ((4 + 4)/4 × (44/4 + 44 ))
:= ((4 + 4) × (4 × 4 × 4 + 4)) − 4/4
:= 4 + ((4 + 4) × (4 × 4 × 4 + 4))
:= 555 − (((55 + 5)/5) + 5)
:= 555 − (55 + 5)/5
:= 555 − ((5 + 5)/5 + 5)
:= 666 − (((6 + 6)/6)6/6+6)
:= 666/6 + (6 × (66 + 6))
:= 666 − ((666 + 6)/6 + 6)
:= 7 × 77 − 7/7
:= 7 × 77 + (77/7 − 7)
:= 7 + (7 × 77 + ((7 + 7)/7))
:= 8 + (((8 × 8 × 8 + (8 + 8)/8) + 8) + 8)
:= 8 + ((((8 × 8 × 8 − 8/8) + 8) + 8) + 8)
:= 8 × 8 + 88 × 88/(8 + 8)
:= 9 + ((((((9 + 9)/9)9 ) − 9/9) + 9) + 9)
:= (((99 × 99) − 9)/(9 + 9)) − 9/9
:= 9 + ((((((9 + 9)/9)9 ) + 9) + 9) + 9)
Ï 539 := 11 + ((1 + 11 + 11)1+1 − 1)
Ï 544 := (1111 − 1)/(1 + 1) − 11
Ï 549 := (1111 − 11)/(1 + 1) − 1
:= 2 × (22×(2+2) + 22+2 )
:= 22/2 + (22 × (22 + 2))
:= 22 + (((22 + 2/2)2 ) − 2)
3
:= 33 + ((3 − 3/3)3×3 )
:= (((3 × 3 + 3) + 3)/3) − 33
:= 3 × ((3 + 3)3 − 33)
:= 44/4 + (44 × ((4 + 4) + 4))
:= (4 + 4) × (4 × 4 × 4 + 4)
:= ((44 × 44 + 44 ) + 4)/4
:= 555 − (55/5 + 5)
:= 555 − 55/5
:= 555 − (5/5 + 5)
:= (6/6 + 6) × (66/6 + 66)
:= 666 − ((666 + 66)/6)
:= 666 − (666/6 + 6)
:= 7 × 77
:= 7 + (7 × 77 − ((7 + 7)/7))
:= 7 × 77 + (77 − 7)/7
:= 8 + ((8 × 8 × 8 + 88/8) + 8)
:= 8 + ((8 × 8 × 8 + 8 + 8) + 8)
:= 8 × 8 × 8 + 888/(8 + 8 + 8)
:= ((99 × 99) − 9)/(9 + 9)
:= 9 × (9 × 9 − 9) − 99
9
:= 9 + (((((9 + 9)/9) ) + 9) + 9)
Ï 540 := 11 + (1 + 11 + 11)1+1
Ï 545 := (1 + 1111)/(1 + 1) − 11
:= 2 × ((2 × (22 + 2)) + 222)
3×3
3
Ï 550 := (1111 − 11)/(1 + 1)
:= 22+2 + ((22 + 2/2)2 )
:= 22 + (22 × (22 + 2))
:= 3/3 + ((3 + 3)3 + 333)
:= 3 × (3 × (3 + 33))
:= 33 + ((3 − 3/3)
:= (4 − 4/4) × ((4 × 44) + 4)
:= 4/4 + ((4 + 4) × (4 × 4 × 4 + 4))
:= (4/4 + 4) × (444 − 4)/4
:= (5 + 5) × (55 − 5/5)
:= 555 − 5 − 5
:= (5 + 5) × 55
:= (6 + 6 + 6) × (6 × 6 − 6)
:= 6 + ((6/6 + 6) × (66/6 + 66))
:= (6 − 6/6) × ((666 − 6)/6)
:= 7/7 + 7 × 77
:= 7 + (7 × 77 − 7/7)
:= 7 × 77 + 77/7
:= 8 + ((((88 + 8)/8) + 8 × 8 × 8) + 8)
:= 8 + ((((8 × 8 × 8 + 8/8) + 8) + 8) + 8)
:= (8888 − 88)/(8 + 8)
:= (9 + 9 + 9) × (99/9 + 9)
:= ((99 × 99) + 9)/(9 + 9)
:= 9/9 + (9 × (9 × 9 − 9) − 99)
Ï 541 := 1 + (11 + (1 + 11 + 11)1+1 )
:= 2 + ((22 × (22 + 2)) + 22/2)
3
)
Ï 546 := 1 + ((1 + 1111)/(1 + 1) − 11)
:= 222 + ((2
2+2
2
Ï 551 := 1 + (1111 − 11)/(1 + 1)
:= 22 + ((22 + 2/2)2 )
+ 2) )
:= (3 + 3)3 + (333 − 3)
:= (3333 − 33 )/(3 + 3)
:= 4 + ((4/4 + 4) − (44 + 44))
:= (4 + 4)/4 + ((4 + 4) × (4 × 4 × 4 + 4))
:= 444 + (444/4 − 4)
:= 5/5 + ((5 + 5) × (55 − 5/5))
:= 5/5 + (555 − 5 − 5)
:= 5/5 + (5 + 5) × 55
:= 6/6 + ((6 + 6 + 6) × (6 × 6 − 6))
:= 6 + ((6 + 6 + 6) × (6 × 6 − 6))
:= 66/6 + ((6 + 6 + 6) × (6 × 6 − 6))
:= 7 × 77 + (7 + 7)/7
:= 7 + 7 × 77
:= 7 × 77 + (77 + 7)/7
:= 8 × (88 − 8) − (88/8 + 88)
:= (8 − 8/8) × ((8 − 88)/8 + 88)
:= 8 × (88 − 8) − (8/8 + 88)
:= 9 + (((((9 + 9)/9)9 ) + (99/9)) + 9)
:= ((9999 − 9)/(9 + 9)) − 9
:= (9999 − 9 × 9)/(9 + 9)
:= 3/3 + (3 × (3 × (3 + 33)))
4
Ï 542 := 1 + (1 + (11 + (1 + 11 + 11)1+1 ))
2
:= 2 + (((22 + 2/2) ) + 22/2)
Ï 547 := 1 + (1 + ((1 + 1111)/(1 + 1) − 11))
2
:= 2 + (((22 + 2/2) ) + 2
2+2
)
Ï 552 := 1 + (1 + (1111 − 11)/(1 + 1))
:= (22 + 2) × (22 + 2/2)
:= 3 + ((((3 × 3 + 3) + 3)/3) − 33)
:= 3 + ((3 + 3)3 + 333)
:= (4 + 4)/4 × ((44/4 + 44 ) + 4)
:= ((44 × 44 − 4) + 44 )/4
:= 44 + ((44 − 4) + 44 )
:= 555 − (55 + 5 + 5)/5
:= 555 + ((5 + 5)/5 − 5 − 5)
:= 555 + ((5 + 5)/5 − 5)
:= 6 + (((6 + 6)/6 + 6) × (66 + 6/6))
:= 6 + (((6 + 6 + 6) × (6 × 6 − 6)) + 6/6)
:= 6 + (((6 + 6 + 6) × (6 × 6 − 6)) + 6)
:= 7 × 77 + (7 + 7 + 7)/7
:= 7 + (7 × 77 + 7/7)
:= 7 + ((7 × 77 − 7/7) + 7)
:= 8 + ((88 + 88)/8 + 8 × 8 × 8)
:= (8888 − 8)/(8 + 8) − 8
:= 8 × (88 − 8) − 88
:= ((9999 − 9 × 9)/(9 + 9)) − 9
:= ((9999 + 9)/(9 + 9)) − 9
:= (9 − 9/9) × (9 × 9 − ((99 + 9)/9))
3×3
:= 3 + (((3 − 3/3)
3
)+3 )
3
46
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 553 := (1111 − 1)/(1 + 1) − 1 − 1
Ï 558 := 1 + (1 + (1 + 1111)/(1 + 1))
Ï 563 := 1 + (1 + ((11 + 1111)/(1 + 1)))
:= 2 + (((22 + 2/2) ) + 22)
:= 2 + ((((22 + 2) ) − 22) + 2)
:= ((22 + 2)2 ) − (22/2 + 2)
:= ((3333 + 3)/(3 + 3)) − 3
:= 3 × (((3 + 3)3 − 33) + 3)
:= ((3 × 3 + 3 + 3)3 + 3)/(3 + 3)
:= 4 + (((44 × 44 + 4 ) + 4)/4)
:= 4 + ((444 − 4)/4 + 444)
:= 4 + ((444/4 + 444) + 4)
:= 555 − (5 + 5)/5
:= 5 + (555 − (5 + 5)/5)
:= 5 + ((555 − (5 + 5)/5) + 5)
:= (6/6 + 6) × (66 + 6/6 + 6 + 6)
:= 666 − 6 × (6 + 6 + 6)
:= 666 − ((6 × 6 + 66) + 6/6)
2
4
2
:= 7 + (7 × 77 + 7)
:= 7 + ((77 + 7)/7 + 7 × 77)
:= 7 + ((7777 + 7)/(7 + 7))
:= 8/8 + (8 × (88 − 8) − 88)
:= (8/8 + 8) × (8 × 8 − ((8 + 8)/8))
:= 8 + (8888 − 8)/(8 + 8)
:= 9 + (((99 × 99) − 9)/(9 + 9))
:= (9 × (9 × 9 − (9 + 9))) − 9
:= 9 + ((((99 × 99) + 9)/(9 + 9)) + 9)
Ï 554 := (1111 − 1)/(1 + 1) − 1
Ï 559 := 1 + (1 + (1 + (1 + 1111)/(1 + 1)))
Ï 564 := ((1 + 1) × (1 + 11))1+1 − 1 − 11
:= ((22 + 2)2 ) − 22
:= 2 + (((2222 − 2)/(2 + 2)) + 2)
:= 222 + (2 × 2 × (22 − 2))
:= (3333 − 3 × 3)/(3 + 3)
:= 3 + ((3333 + 3)/(3 + 3))
:= (3 × ((3 + 3)3 − 33 )) − 3
:= 444 + (444 − 4)/4
:= 4 + (444/4 + 444)
:= (4 × (44 − 4)) − 444
:= 555 − 5/5
:= 5 + (555 − 5/5)
:= 5 + (555 − 5/5 + 5)
:= 666 − (666 + 6)/6
:= 6/6 + (666 − 6 × (6 + 6 + 6))
:= (6 + 6) × (66/6 + 6 × 6)
:= 7 + ((7 × 77 + 7/7) + 7)
:= 7 + (((7 × 77 − 7/7) + 7) + 7)
:= 7 + ((7 × 77 + (77/7)) + 7)
:= (8 + 8)/8 + (8 × (88 − 8) − 88)
:= 888/8 + 8 × (8 × 8 − 8)
:= 8 + ((8888 + 8)/(8 + 8))
:= 9 + (((99 × 99) + 9)/(9 + 9))
:= 9/9 + ((9 × (9 × 9 − (9 + 9))) − 9)
:= 9 + ((9999 − 9)/(9 + 9))
Ï 555 := (1111 − 1)/(1 + 1)
Ï 560 := (11 − 1) × (1 + 111)/(1 + 1)
Ï 565 := ((1 + 1) × (1 + 11))1+1 − 11
:= (2222 − 2)/(2 + 2)
:= ((22 + 2)2 ) − 22+2
:= ((22 + 2)2 ) − 22/2
:= (3333 − 3)/(3 + 3)
:= (3333 + 33 )/(3 + 3)
:= ((3 × 3 + 3)3 − 33)/3
:= 444 + 444/4
:= 4 + ((44 + 44 ) + 44 )
:= 4 + ((4/4 + 4)4 − 4 × 4 × 4)
:= 555
:= 5 + 555
:= 5 + 555 + 5
:= (6 − 6/6) × 666/6
:= (6 − 6/6) × (666 + 6)/6
:= 6/6 + ((6 + 6) × (66/6 + 6 × 6))
:= (7777 − 7)/(7 + 7)
:= 7 + ((7 × 77 + 7) + 7)
:= 7 + (((77 + 7)/7 + 7 × 77) + 7)
:= (8888 − 8)/(8 + 8)
:= (8 − 8/8) × (88 − 8)
:= 8 × (8 × 8 + 8) − 88/8
:= (9999 − 9)/(9 + 9)
:= (9 − 9/9) × (9 × 9 − 99/9)
:= 9 + ((9999 + 9)/(9 + 9))
Ï 556 := (1 + 1111)/(1 + 1)
Ï 566 := 11 + (1111 − 1)/(1 + 1)
Ï 561 := (11 + 1111)/(1 + 1)
:= 2 + (((22 + 2)2 ) − 22)
:= (2222 + 22)/(2 + 2)
:= ((2 − 22)/2) + ((22 + 2)2 )
:= (3333 + 3)/(3 + 3)
:= 33 × ((33/3 + 3) + 3)
:= (3 × ((3 + 3)3 − 33 )) − 3/3
4
4
4
:= 44 + (4 + 4 )
:= (4/4 + 4) − 4 × 4 × 4
:= 4 + (((4 − 44 )/4) + (4/4 + 4)4 )
:= 5/5 + 555
:= 5 + (555 + 5/5)
:= 555 + 55/5
:= (6666 + 6)/(6 + 6)
:= 6 + ((6 − 6/6) × 666/6)
:= 6 + ((6 − 6/6) × (666 + 6)/6)
:= (7777 + 7)/(7 + 7)
:= 7 + (((7 × 77 + 7/7) + 7) + 7)
:= 77 + (7 × (77 − 7) − 7/7)
:= (8888 + 8)/(8 + 8)
:= 8/8 + ((8 − 8/8) × (88 − 8))
:= (8 − 88)/8 + 8 × (8 × 8 + 8)
:= (9999 + 9)/(9 + 9)
:= (9999 + 99)/(9 + 9)
:= (9 × (9 × 9 − (9 + 9))) − 9/9
Ï 557 := 1 + (1 + 1111)/(1 + 1)
Ï 562 := 1 + ((11 + 1111)/(1 + 1))
2
2+2
)
Ï 567 := 11 + (1 + 1111)/(1 + 1)
:= 2 + (((22 + 2)2 ) − 22/2)
:= 2 + ((2222 − 2)/(2 + 2))
:= 2 + (((22 + 2) ) − 2
:= (3333 + 3 × 3)/(3 + 3)
:= ((3 × 3 + 3 + 3)3 − 3)/(3 + 3)
:= 3 × ((3 + 3)3 − 33 )
:= 44 + ((4/4 + 44 ) + 44 )
:= ((4 − 44 )/4) + (4/4 + 4)4
:= ((4/4 + 4) + 4) × ((44 − 4)/4)
:= 555 + (5 + 5)/5
:= 5 + (555 + (5 + 5)/5)
:= 555 + (55 + 5)/5
:= 6 + (((6 − 6/6) × 666/6) + 6)
:= 6/6 + (6666 + 6)/(6 + 6)
:= 6 + (6666 + 6)/(6 + 6)
:= 7 + (7 × 77 + (77/7))
:= 7 + ((7777 − 7)/(7 + 7))
:= 77 + 7 × (77 − 7)
:= 8 × (8 × 8 + 8) − (88/8 + 8)
:= (8 + 8)/8 + ((8 − 8/8) × (88 − 8))
:= (8/8 + 8) × (8 × 8 − 8/8)
:= (9 × (9 × 9 − (9 + 9))) − 9/9 − 9
:= 9 + ((((99 × 99) − 9)/(9 + 9)) + 9)
:= 9 × (9 × 9 − (9 + 9))
47
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 568 := 1 + (11 + (1 + 1111)/(1 + 1))
Ï 573 := ((1 + 1) × (1 + 11))1+1 − 1 − 1 − 1
Ï 578 := 1 + (1 + ((1 + 1) × (1 + 11))1+1 )
:= ((22 + 2)2 ) − 2 × (2 + 2)
:= ((22 + 2)2 ) − 2/2 − 2
:= 2 + ((22 + 2)2 )
:= 3 + (((3 × 3 + 3)3 − 33)/3)
:= ((3 × 3 + 3)3 /3) − 3
:= 3 + (((3 × 3 + 3)3 − 3)/3)
:= 4 + ((4 × (44 − 4)) − 444)
:= (4/4 + 4)4 − (44 + 4 + 4)
:= 4/4 + ((4/4 + 4)4 − (44 + 4))
:= (55 − 5 − 5)/5 − 55
:= 5 + ((55 − 5 − 5)/5 − 55)
:= 5 × 5 + (555 − (5 + 5)/5)
:= 6 + ((6666 + 6)/(6 + 6) + 6)
:= 66 + (666/6 + 6 × 66)
:= (6 + 6)/6 + (6 × ((6 × 6 − 6) + 66))
:= 7/7 + (7 × (77 − 7) + 77)
:= 7 × (77 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 × (77 + 7)
:= 8 × (8 × 8 + 8) − 8
:= 8 + (8 × (8 × 8 + 8) − 88/8)
:= (8 + 8)/8 + 8 × (8 × 8 + 8)
:= 9/9 + (9 × (9 × 9 − (9 + 9)))
:= 9 + (((9999 − 9)/(9 + 9)) + 9)
:= 99/9 + (9 × (9 × 9 − (9 + 9)))
Ï 569 := 1 + (1 + (11 + (1 + 1111)/(1 + 1)))
Ï 574 := ((1 + 1) × (1 + 11))1+1 − 1 − 1
Ï 579 := 1 + (1 + (1 + ((1 + 1) × (1 + 11))1+1 ))
:= 2 + ((((22 + 2)2 ) − 22/2) + 2)
:= ((22 + 2)2 ) − 2
:= 2 + (((22 + 2)2 ) + 2/2)
:= (((3 × 3 + 3)3 − 3)/3) − (3 + 3)
:= (((3 × 3 + 3)3 + 3)/3) − 3
:= 3 + ((3 × 3 + 3)3 /3)
:= 4 + (((4/4 + 4)4 − 4 × 4 × 4) + 4)
:= 44 + (((44 − 4 − 4)/4) + 44 )
:= 4 × 44 − (444 + 4/4)
:= (55 − 5)/5 − 55
:= 5 + ((55 − 5)/5 − 55)
:= 5 × 5 + (555 − 5/5)
:= 66 + (((6 + 6) × (6 × 6 + 6)) − 6/6)
:= (6 × ((6 × 6 − 6) + 66)) − (6 + 6)/6
:= 6 × 6 × 6 + 66 × 66/(6 + 6)
:= 7 + (((7777 − 7)/(7 + 7)) + 7)
:= 7 × (77 + 7) − (7 + 7)
:= 7 × (77 + 7) − ((7 + 7)/7 + 7)
:= 8/8 + (8 × (8 × 8 + 8) − 8)
:= 8 × (8 × 8 + 8) − (8 + 8)/8
:= 88/8 + (8 × (8 × 8 + 8) − 8)
:= (9 + 9)/9 + (9 × (9 × 9 − (9 + 9)))
:= (9 − ((9 + 9)/9)) × (9/9 + 9 × 9)
:= ((99 + 9)/9) + (9 × (9 × 9 − (9 + 9)))
Ï 570 := (11 − 1) × (1 + (1 + 111)/(1 + 1))
Ï 575 := ((1 + 1) × (1 + 11))1+1 − 1
Ï 580 := (1 + 1) × ((1 + 1) × (1 + (1 + 11)1+1 ))
:= ((22 + 2)2 ) − (2 + 2 + 2)
:= ((22 + 2)2 ) − 2/2
:= 2 + (((22 + 2)2 ) + 2)
:= 3 + (3 × ((3 + 3)3 − 33 ))
:= ((3 × 3 + 3)3 − 3)/3
:= 3 + (((3 × 3 + 3)3 + 3)/3)
:= 444 + ((44 − 4)/((4 + 4)/4))
:= 44 + (((44 − 4)/4) + 44 )
:= 4 × 44 − 444
:= 55 /5 − 55
:= 5 + (55 /5 − 55)
:= 5 × 5 + 555
:= 66 + ((6 + 6) × (6 × 6 + 6))
:= (6 × ((6 × 6 − 6) + 66)) − 6/6
:= (6 − 6/6) × (((666 − 6)/6) + 6)
:= 7 × (77 + 7) − (77/7 + 7)
:= 7/7 + (7 × (77 + 7) − (7 + 7))
:= 7 × (77 + 7) − (7/7 + 7)
:= (8 + 8)/8 + (8 × (8 × 8 + 8) − 8)
:= 8 × (8 × 8 + 8) − 8/8
:= 8 × (8 × 8 + 8) + (8/((8 + 8)/8))
:= 9 + ((9999 + 99)/(9 + 9))
:= 9 + ((9 × (9 × 9 − (9 + 9))) − 9/9)
:= 9 × 9 + (((9 × 999) − 9)/(9 + 9))
Ï 571 := 1 + (11 − 1) × (1 + (1 + 111)/(1 + 1))
Ï 576 := ((1 + 1) × (1 + 11))1+1
Ï 581 := 1 + ((1 + 1) × ((1 + 1) × (1 + (1 + 11)1+1 )))
:= ((22 + 2)2 ) − (2/2 + 2 + 2)
:= (22 + 2)2
:= 2 + ((((22 + 2)2 ) + 2/2) + 2)
:= (((3 × 3 + 3)3 + 3)/3) − (3 + 3)
:= (3 × 3 + 3)3 /3
:= 3 + ((((3 × 3 + 3)3 − 3)/3) + 3)
:= ((44 × (44 + 4 + 4)) − 4)/4
:= 4 × (4 × (4 × (4 + 4) + 4))
:= (4/4 + 4)4 − 44
:= (55 + 5)/5 − 55
:= 5 + ((55 + 5)/5 − 55)
:= 5 × 5 + (555 + 5/5)
:= 66 + (((6 + 6) × (6 × 6 + 6)) + 6/6)
:= 6 × ((6 × 6 − 6) + 66)
:= 6 + ((6 × ((6 × 6 − 6) + 66)) − 6/6)
:= 7 + (((7 × 77 + (77/7)) + 7) + 7)
:= (77 + 7)/7 × (7 × 7 − 7/7)
:= 7 × (77 + 7) − 7
:= 8 + ((8888 − 8)/(8 + 8) + 8)
:= 8 × (8 × 8 + 8)
:= 8 + ((8 × (8 × 8 + 8) − 88/8) + 8)
:= 9 + (((((99 × 99) − 9)/(9 + 9)) + 9) + 9)
:= 9 + (9 × (9 × 9 − (9 + 9)))
:= 9 × 9 + (((9 × 999) + 9)/(9 + 9))
Ï 572 := 11 + ((11 + 1111)/(1 + 1))
Ï 577 := 1 + ((1 + 1) × (1 + 11))1+1
Ï 582 := (1 + 1) × (1 + ((1 + 1) × (1 + (1 + 11)1+1 )))
:= 22 × (22 + 2 + 2)
:= 2/2 + ((22 + 2)2 )
:= 2 + ((((22 + 2)2 ) + 2) + 2)
:= (((3 × 3 + 3)3 − 3)/3) − 3
:= ((3 × 3 + 3)3 + 3)/3
:= 3 + (((3 × 3 + 3)3 /3) + 3)
:= 44 + (44 × ((4 + 4) + 4))
:= (4/4 + 4)4 − (44 + 4)
:= 4/4 + ((4/4 + 4)4 − 44)
:= (55 + 5 + 5)/5 − 55
:= 5 + ((55 + 5 + 5)/5 − 55)
:= 5 × 5 + (555 + (5 + 5)/5)
:= (66/6) × (((6 + 6)/6) − (6 + 6))
:= 6/6 + (6 × ((6 × 6 − 6) + 66))
:= 6 + (6 × ((6 × 6 − 6) + 66))
:= 7777/7 − 7 × 77
:= 7 × (77 + 7) − 77/7
:= 7/7 + (7 × (77 + 7) − 7)
:= 88 + 88 × 88/(8 + 8)
:= 8/8 + 8 × (8 × 8 + 8)
:= 8 + (8 × (8 × 8 + 8) − (8 + 8)/8)
:= 9 + (((((99 × 99) + 9)/(9 + 9)) + 9) + 9)
:= 9 + ((9 × (9 × 9 − (9 + 9))) + 9/9)
:= 9 × 9 + ((((9 + 9)/9)9 ) − (99/9))
6
48
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 583 := 11 × ((111 − 1)/(1 + 1) − 1 − 1)
2
Ï 588 := 1 + (11 + ((1 + 1) × (1 + 11))1+1 )
2+2
2
:= 2 + (((((22 + 2) ) + 2/2) + 2) + 2)
:= (2/2 + 2) × ((2
− 2) )
:= 3 + ((((3 × 3 + 3)3 + 3)/3) + 3)
:= 3 + (3 × (33 × (3 + 3) − 3))
:= 222 + (222/2 − 2)
:= 3 × 33 × (3 + 3) − 3/3
:= 44/4 × ((44 − 44)/4)
:= 4 + ((4 × 4 − 444) + 4)
:= (4/4 + 4)4 − 4 × (4 + 4)
:= 55/5 × (55 − (5 + 5)/5)
:= 5 × 55 + (55 + 5)/(5 + 5)
:= 55 /5 − ((5 + 5)/5)5
:= 6 + ((6 × ((6 × 6 − 6) + 66)) + 6/6)
:= 666 − (66 + 6 + 6)
:= 666 − (66 + 6/6 + 6)
:= ((7 + 7)/7) + (7 × (77 + 7) − 7)
:= 7 × (77 + 7)
:= 7 + (7 × (77 + 7) − ((7 + 7)/7))
:= 8 + (8 × (8 × 8 + 8) − 8/8)
:= ((88 + 8)/8) + 8 × (8 × 8 + 8)
:= 8 × 88 − 888/8
:= 9 + ((9 − ((9 + 9)/9)) × (9/9 + 9 × 9))
:= (99 − 9/9) × (9 − ((9 + 9 + 9)/9))
:= 9 × 9 + (((9 + 9)/9)9 )
Ï 584 := 1 + 11 × ((111 − 1)/(1 + 1) − 1 − 1)
:= 2 × (2 + 2) + ((22 + 2)2 )
3
:= 3 × 3 + (((3 × 3 + 3) − 3)/3)
4
4
Ï 593 := 11 × ((111 − 1)/(1 + 1) − 1) − 1
Ï 589 := 1 + (1 + (11 + ((1 + 1) × (1 + 11))1+1 ))
Ï 594 := 11 × ((111 − 1)/(1 + 1) − 1)
:= 2 + (((22 + 2)2 ) + 22/2)
:= 2 + (((22 + 2)2 ) + 22+2 )
:= 3 + ((3 × (33 × (3 + 3) − 3)) + 3/3)
:= 3 × 33 × (3 + 3)
4
:= 4 + (4 × 4 − 444)
:= 4 + (((4/4 + 4) − 44) + 4)
:= 4/4 + ((4/4 + 4)4 − 4 × (4 + 4))
:= 5 + ((555 − 5/5) + 5 × 5)
:= ((55 − 55)/5) − 5 × 5
:= 55/5 × (55 − 5/5)
:= ((6 + 6)/6 + 6) × (66 + 6/6 + 6)
:= 666 − (66/6 + 66)
:= 666 − (66 + 6)
:= 7 + (7 × (77 + 7) − (77/7))
:= 7/7 + 7 × (77 + 7)
:= 7 + (7 × (77 + 7) − 7/7)
:= 8 + 8 × (8 × 8 + 8)
:= 88 + (8 × 8 × 8 − 88/8)
:= 8 × 88 + ((8 − 888)/8)
:= 9 × 9 + ((((9 + 9)/9)9 ) − 9)
:= 9 + ((((9 × 999) − 9)/(9 + 9)) + 9 × 9)
:= 99 × (9 − ((9 + 9 + 9)/9))
Ï 585 := 11 + (((1 + 1) × (1 + 11))1+1 − (1 + 1))
Ï 590 := (11 − 1) × ((111+1 − 1)/(1 + 1) − 1)
Ï 595 := 111 + (11 + 11)1+1
:= 22/2 + (((22 + 2)2 ) − 2)
:= 22+2 + (((22 + 2)2 ) − 2)
:= 222 + 222/2
:= 3 × (33 × (3 + 3) − 3)
:= 3 + (((3 × 3 + 3)3 + 33)/3)
:= 3/3 + 3 × 33 × (3 + 3)
4
:= 4 + ((4/4 + 4) − 44)
:= (44 − 4)/4 × (((4 − 4)/4) − 4)
:= (44 × 44 + 444)/4
:= 5 + (555 + 5 × 5)
:= 5 + ((555 + 5 × 5) + 5)
:= 55 /5 − (5 × 5 + 5)
:= (6 − 6/6) × (666/6 + 6)
:= 666 + (((6 − 66)/6) − 66)
:= 6/6 + (666 − (66 + 6))
:= 7 × (77 + 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + 7 × (77 + 7)
:= 7 + 7 × (77 + 7)
:= 8 + (8 × (8 × 8 + 8) + 8/8)
:= 8 + ((8 × (8 × 8 + 8) − (8 + 8)/8) + 8)
:= 8 + (8 × (8 × 8 + 8) + (88/8))
:= 9 + ((9 × (9 × 9 − (9 + 9))) + 9)
:= 9 + ((((9 × 999) + 9)/(9 + 9)) + 9 × 9)
:= 9/9 + (99 × (9 − ((9 + 9 + 9)/9)))
Ï 586 := 11 + (((1 + 1) × (1 + 11))1+1 − 1)
4
Ï 591 := (1 + 1 + 1) × (1 + (1 + 1 + 1 + 11)1+1 )
Ï 596 := 1 + (111 + (11 + 11)1+1 )
:= 2 + (((22 + 2)2 ) + 2 × (2 + 2))
:= 2 + ((((22 + 2)2 ) + 22/2) + 2)
:= 22 + (((22 + 2)2 ) − 2)
:= 3 × 3 + (((3 × 3 + 3)3 + 3)/3)
:= 3 × 33 × (3 + 3) − 3
:= 3 + (3 × 33 × (3 + 3) − 3/3)
:= 4 + (((4/4 + 4) − 44) + 4/4)
:= 44/4 + (4 × 4 − 444)
:= (4 × (44 + 4)) − 444
:= 5 + (555 + 5 × 5 + 5/5)
:= 5 × 5 + (555 + (55/5))
:= (55 + 5)/5 − (5 × 5 + 5)
:= ((((6 + 6)/6)6+6 ) + 6)/(6/6 + 6)
:= 6 + ((6 − 6/6) × (666/6 + 6))
:= 666 − (((6 + 6)/6)6 + 6)
:= 7 × (77 + 7) − (7 + 7)/7
:= 7 × (77 + 7) + (7 + 7 + 7)/7
:= 7 + (7 × (77 + 7) + 7/7)
:= 8 + (8 × (8 × 8 + 8) + ((8 + 8)/8))
:= 8 + ((8 × (8 × 8 + 8) − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 8 × (8 × 8 + 8))
4
:= 9 + (((9 × (9 × 9 − (9 + 9))) + 9/9) + 9)
Ï 587 := 11 + ((1 + 1) × (1 + 11))1+1
4
9
:= 9 × 9 + ((((9 + 9)/9) ) − ((9 + 9)/9))
Ï 592 := (1 + 1) × (11 × (1 + 1 + 1)1+1+1 − 1)
:= (9999 + 9 × 9 × 9)/(9 + 9)
Ï 597 := 1 + (1 + (111 + (11 + 11)1+1 ))
:= 22/2 + ((22 + 2)2 )
:= 22+2 + ((22 + 2)2 )
:= 2 + (222/2 + 222 )
:= ((3 × 3 + 3)3 + 33)/3
:= 3/3 + (3 × 33 × (3 + 3) − 3)
:= 3 + 3 × 33 × (3 + 3)
:= 4 + (44/4 × ((4 − 44)/4))
:= 4 × ((4 × (4 × (4 + 4) + 4)) + 4)
:= 4 + ((4/4 + 4)4 − 4 × (4 + 4))
:= 555 + ((5 + 5)/5)5
:= 5 + (((5 + 5)/5)5 + 555)
:= (55 + 5 + 5)/5 − (5 × 5 + 5)
:= 66/6 + (6 × ((6 × 6 − 6) + 66))
:= 6 × 6 + (6666 + 6)/(6 + 6)
:= 666 − (6 × 6/(6 + 6) + 66)
:= 7 × (77 + 7) − 7/7
:= 77/7 + (7 × (77 + 7) − 7)
:= 7 + (7 × (77 + 7) + ((7 + 7)/7))
:= 88/8 + 8 × (8 × 8 + 8)
:= 8 + (8 × (8 × 8 + 8) + 8)
:= 8 + ((8 × 8 × 8 − 88/8) + 88)
:= 9 + ((9 × (9 × 9 − (9 + 9))) + (99/9))
:= 9 × 9 + ((((9 + 9)/9)9 ) − 9/9)
:= 999/9 + ((9 + 9) × (9 + 9 + 9))
4
49
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 598 := 11 + (11 + ((1 + 1) × (1 + 11))1+1 )
Ï 603 := 11 × (111 − 1)/(1 + 1) − 1 − 1
2
2
2
:= 22+2 × ((2 + 2 + 2)2 + 2)
:= 22 + ((22/2) − 2)
:= 22 + ((22 + 2) )
:= 3 + (3 × 33 × (3 + 3) + 3/3)
4
Ï 608 := (11 × 111 − 1)/(1 + 1) − 1 − 1
:= 33 + (((3 × 3 + 3)3 − 3)/3)
:= 3 × (33 × (3 + 3) + 3)
4
:= (4 + 4)/4 × ((44 − 4/4) + 4 )
:= 4 + ((((4 + 4) × (44 + 4 )) − 4)/4)
:= 44 + ((4 + 4) × 44)
:= (55 − 5 − 5)/5 − 5 × 5
:= (55 − (55 + 55))/5
:= ((55 − (55 + 5))/5) − 5
:= 666 − (((6 + 6)/6) + 66)
:= 6/6 + (666 − ((6 + 6)/6)6 )
:= 6 + (666 − ((6 + 6)/6)6 )
:= 7 × (77 + 7) + (77 − 7)/7
:= 7 + ((7 × (77 + 7) + 7/7) + 7)
:= (7/7 + 7) × (77 − 7/7)
:= 88 + (8 × 8 × 8 − ((8 + 8)/8))
:= 8 + ((8 × (8 × 8 + 8) + (88/8)) + 8)
:= 8 + (8 × 8 × 8 + 88)
:= 99 + (((9 × 999) − 9)/(9 + 9))
:= 9 + (99 × (9 − ((9 + 9 + 9)/9)))
:= 9 × 9 × 9 − (((999 + 9)/9) + 9)
Ï 599 := 1111 − (1 + 1)11−1−1
:= 22 + (((22 + 2)2 ) + 2/2)
3
:= ((33/3) ) − (3
3+3
Ï 604 := 11 × (111 − 1)/(1 + 1) − 1
:= 2 × (((2
2+2
2
+ 2) ) − 22)
Ï 609 := (11 × 111 − 1)/(1 + 1) − 1
:= 2 + (((22/2)2 + 222 ) + 2)
+ 3)
:= 33 + (((3 × 3 + 3)3 + 3)/3)
:= 33 + ((3 × 3 + 3)3 /3)
4
:= 44 + (((4 + 4) × 44) − 4)
:= (4/4 + 4)4 − 4 × 4
:= (5 − 5)/5 − 5 × 5
:= (55 × 55 − 5)/5
:= ((55 − 55)/5) − 5
:= 666 − (66 + 6/6)
:= 6 + (666 − (((6 + 6)/6) + 66))
:= 6 × (6 × 6 + 66) − 6 × 6/(6 + 6)
:= 77/7 + 7 × (77 + 7)
:= 7 + ((7 × (77 + 7) + ((7 + 7)/7)) + 7)
:= 77 + (7 × 77 − 7)
:= 88 + (8 × 8 × 8 − 8/8)
:= 88 + ((8/((8 + 8)/8)) + 8 × 8 × 8)
:= (8 − 8/8) × (88 − 8/8)
:= (((4 + 4) × (44 + 4 )) − 4)/4
5
:= 99 + (((9 × 999) + 9)/(9 + 9))
Ï 600 := (1 + 1) × (1 + 1 + 1) × (11 − 1)1+1
2
9
:= 9 × 9 + ((((9 + 9)/9) ) + (99/9))
Ï 605 := 11 × (111 − 1)/(1 + 1)
:= 9 × 9 × 9 − (999/9 + 9)
Ï 610 := (11 × 111 − 1)/(1 + 1)
:= 2 + (((22 + 2) ) + 22)
:= 222 + (22/2)2
:= 2 + ((2 × 22+2 ) + ((22 + 2)2 ))
:= 3 + (3 × 33 × (3 + 3) + 3)
:= 3 + (((33/3)3 ) − 33+3 )
:= 33 + (((3 × 3 + 3)3 + 3)/3)
:= (4/4 + 4)4 − (4 × 4 + 4)
:= 4/4 + ((4/4 + 4)4 − 4 × 4)
:= 55 × (55/5)
:= 55 + 555
:= 6 + (666 − 66 − 6/6)
:= 6 × (6 × 6 + 66) − (6 + 6)/6
:= 77 + (7 × 77 − (77/7))
:= 7/7 + (7 × 77 − 7 + 77)
:= 8 × 88 − (88/8 + 88)
:= 8 + ((8 × 8 × 8 + (8 + 8)/8) + 88)
:= 99/9 × ((999 − 9)/(9 + 9))
:= 99 + ((((9 + 9)/9)9 ) − 9/9)
4
:= (4 + 4) × ((44 + 4 )/4)
:= 5 × (5 × 5 × 5 − 5)
:= 666 − 66
:= (7/7 + 7) × (77 − (7 + 7)/7)
:= 88 + 8 × 8 × 8
:= 99 + ((((9 + 9)/9)9 ) − (99/9))
Ï 601 := 1 + (1 + 1) × (1 + 1 + 1) × (11 − 1)1+1
:= 2 + ((((22 + 2)2 ) + 22) + 2/2)
:= 3 + ((3 × 33 × (3 + 3) + 3/3) + 3)
4
:= (4/4 + 4) − ((4 × 4 + 4) + 4)
5
:= (5 + 5)/5 − 5 × 5
:= 6/6 + (666 − 66)
:= 7 + ((7 × (77 + 7) − 7/7) + 7)
:= 8/8 + (8 × 8 × 8 + 88)
:= 9 + (((((9 + 9)/9)9 ) − 9/9) + 9 × 9)
Ï 602 := (1 + 1) × (1 + (1 + 1 + 1) × (11 − 1)1+1 )
2
:= 2 + ((((22 + 2) ) + 22) + 2)
Ï 606 := 1 + 11 × (111 − 1)/(1 + 1)
:= 2 + (2 × (((2
2+2
2
+ 2) ) − 22))
Ï 611 := (1 + 11 × 111)/(1 + 1)
:= 222 + ((22×(2+2) − 2)/2)
:= 3 + (3 × (33 × (3 + 3) + 3))
:= 3 × 33 + ((3 − 3/3)3×3 )
:= 4/4 + ((4/4 + 4)4 − (4 × 4 + 4))
:= 4 + ((((4 + 4) × 44) − 4/4) + 44 )
:= (55 × 55 + 5)/5
:= 5 + ((55 × 55 + 5)/5)
:= 6 + (666 − 66)
:= 6 × (6 × 6 + 66) − 6/6
:= 7 + (7 × (77 + 7) + (77/7))
:= 77 + (((7 + 7)/7 − 7) + 7 × 77)
:= 8 + ((8 × 8 × 8 − ((8 + 8)/8)) + 88)
:= 88 + (8 × 8 × 8 + 88/8)
:= 9 + (((9 + 9) × (9 + 9 + 9)) + 999/9)
:= 99 + (((9 + 9)/9)9 )
Ï 607 := 1 + 1 + 11 × (111 − 1)/(1 + 1)
Ï 612 := 1 + (1 + 11 × 111)/(1 + 1)
:= 2 + ((22/2)2 + 222 )
:= 2 × ((22 × (22+2 − 2)) − 2)
:= 3 + ((3 × (33 × (3 + 3) + 3)) + 3/3)
:= (3 + 3) × (3 × 33 + 3)
:= (4 + 4)/4 × ((44 + 4 ) + 4/4)
:= 44 + (((4 + 4) × 44) − 4/4)
:= 4 + (((4 + 4) × 44) + 44 )
:= (55 + 5 + 5)/5 − 5 × 5
:= ((55 × 55 + 5) + 5)/5
:= (55 − (55 + 5 + 5))/5
:= 666 − ((6 + 6)/6)6
:= 6 + ((666 − 66) + 6/6)
:= 6 × (6 × 6 + 66)
3
:= ((33/3) ) − 3
3+3
4
:= 7 + (7 × (77 + 7) + 7)
:= 7 + (7 × (77 + 7) + (77 + 7)/7)
:= 7 + ((7 × 77 − (77/7)) + 77)
:= 88 + (8 × 8 × 8 + (8 + 8)/8)
:= 8 + ((8 × 8 × 8 − 8/8) + 88)
:= 88 + (((88 + 8)/8) + 8 × 8 × 8)
:= 9 + ((((9 + 9)/9)9 ) + 9 × 9)
:= 9 × 9 × 9 − (999 + 99)/9
:= 9 × 9 × 9 − (99 + 9 + 9)
50
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 613 := 1 + 1 + (1 + 11 × 111)/(1 + 1)
Ï 618 := 1 + 1 + 11 × (1 + 111)/(1 + 1)
Ï 623 := 111 + (1 + 1)11−1−1
:= 222 + ((22×(2+2) + 2)/2)
:= 2 + (2 × (22 × (22+2 − 2)))
:= ((2/2 + 2 + 2)2+2 ) − 2
:= 3/3 + ((3 + 3) × (3 × 33 + 3))
:= 3 + (3 × (3 + 3)3 − 33)
:= 3 + (3 × (3 + 3)3 − (33 + 3/3))
:= 4 + ((4/4 + 4)4 − 4 × 4)
:= 4 + ((4/4 + 4)4 − 44/4)
:= (4/4 + 4)4 − (4 + 4)/4
:= (55 − (55 + 5))/5
:= (55 − 5 − 5)/5 − 5
:= (55 − 5 − 5)/5
:= 6/6 + 6 × (6 × 6 + 66)
:= 6 + 6 × (6 × 6 + 66)
:= 66/6 + 6 × (6 × 6 + 66)
:= 7 + ((7 × (77 + 7) + (77/7)) + 7)
:= 77 + (7 × 77 + ((7 + 7)/7))
:= 7 + (7 × 77 + 77)
:= 8 + (8 × 88 − (88/8 + 88))
:= (8 + 8)/8 + 88 × (8 − 8/8)
:= 8 × 8 × 8 + 888/8
:= 9/9 + (9 × 9 × 9 − (99 + 9 + 9))
:= 9 × 9 × 9 − 999/9
:= 999/9 + (((9 + 9)/9)9 )
Ï 614 := (1 + (1 + 1) × (1 + 11))1+1 − 11
:= (2 × (22 × (22+2 − 2))) − 2
3
Ï 619 := 1 + 1 + 1 + 11 × (1 + 111)/(1 + 1)
:= (2 × 22) + (((22 + 2)2 ) − 2/2)
3+3
:= 3 × (3 + 3) − (3/3 + 33)
:= 3
:= (4/4 + 4)4 − 44/4
:= 444 + ((4 × 44) − 4/4)
+ ((3 − 333)/3)
Ï 624 := (1 + (1 + 1) × (1 + 11))1+1 − 1
:= (22 + 2) × (22 + 2 + 2)
:= 3 + (3 × ((3 + 3)3 − 3 × 3))
:= 4 × (4 × (44 − 4) − 4)
:= (5 − 55)/5
:= (5 − 5)/5 − 5
:= (55 − 5)/5
:= (6 + 6)/6 + 6 × (6 × 6 + 66)
:= 6 + (6 × (6 × 6 + 66) + 6/6)
:= 666 − (6 × 6 + 6)
5
5
:= 77 + (7 × 77 − ((7 + 7)/7))
:= 77 + (((7 + 7 + 7)/7) + 7 × 77)
:= (7/7 + 7) × (7/7 + 77)
:= 8 × 88 − ((8 + 8)/8 + 88)
:= 8 + ((8 × 8 × 8 + 88/8) + 88)
:= 8 + 88 × (8 − 8/8)
:= 999/9 + ((((9 + 9)/9)9 ) − 9)
:= 9 × 9 × 9 + ((9 − 999)/9)
:= (9 − 9/9) × (9 × 9 − ((9 + 9 + 9)/9))
Ï 615 := 11 × (1 + 111)/(1 + 1) − 1
Ï 620 := 11 + (11 × 111 − 1)/(1 + 1) − 1
Ï 625 := (1 + (1 + 1) × (1 + 11))1+1
:= (2/2 + 2 + 2) × ((22/2)2 + 2)
:= (2 × 22) + ((22 + 2)2 )
:= (2/2 + 2 + 2)2+2
:= 3 × (3 + 3)3 − 33
:= 3 × (3 + 3)3 − (33 + 3/3)
:= (3 − 3/3 + 3)3/3+3
:= (4 − 44)/4 + (4/4 + 4)4
:= 444 + (4 × 44)
:= (4/4 + 4)4
:= 55 /5 − 5 − 5
:= 55 /5 − 5
:= 55 /5
:= (66 × 66 − 666)/6
:= 6 + (6 × (6 × 6 + 66) + ((6 + 6)/6))
:= (6 − 6/6)6−(6+6)/6
:= 77 + (7 × 77 − 7/7)
:= 77 + ((7 × 77 − 7) + (77/7))
:= (7 − ((7 + 7)/7))77/7−7
:= 8 × 88 − (8/8 + 88)
:= 8 × 8 + ((8888 + 8)/(8 + 8))
:= 8 + (88 × (8 − 8/8) + 8/8)
:= 9 × 9 × 9 − (((999 + 9 + 9) + 9)/9)
:= 9 + ((((9 + 9)/9)9 ) + 99)
:= 9 × 9 + (((99 × 99) − 9)/(9 + 9))
Ï 616 := 11 × ((1 + 111)/(1 + 1))
:= 2 × (22 × (22+2 − 2))
3
Ï 621 := 11 + (11 × 111 − 1)/(1 + 1)
:= ((2/2 + 2 + 2)
2+2
)−2−2
3
Ï 626 := 1 + (1 + (1 + 1) × (1 + 11))1+1
:= 2 + ((22 + 2) × (22 + 2 + 2))
:= 3/3 + (3 × (3 + 3) − 33)
:= 3 × ((3 + 3) − 3 × 3)
:= 3/3 + ((3 − 3/3 + 3)3/3+3 )
:= (4 + 4) × ((4 − 4/4)4 − 4)
:= (4/4 + 4)4 − 4
:= 4/4 + (4/4 + 4)4
:= (55 + 5)/5 − 5 − 5
:= (55 + 5)/5 − 5
:= (55 + 5)/5
:= 6 + (6 × (6 × 6 + 66) − ((6 + 6)/6))
:= 6 + ((66 × 66 − 666)/6)
:= 6/6 + ((6 − 6/6)6−(6+6)/6 )
:= 77 + 7 × 77
:= 7 + ((7 × 77 − ((7 + 7)/7)) + 77)
:= (7 × (77 + 7 + 7)) − 77/7
:= 88 × (8 − 8/8)
:= 8 × (88 − 8) − (88/8 + 8)
:= 8 + ((((8 + 8)/8) − 88) + 8 × 88)
:= 99/9 × ((999 + 9)/(9 + 9))
:= 9 × 9 × 9 − (99 + 9)
:= 9 + (9 × 9 × 9 − ((999 + 9)/9))
Ï 617 := 1 + 11 × (1 + 111)/(1 + 1)
2
:= 22 + (222/2 + 22 )
:= 3
3+3
− ((333 + 3)/3)
4
Ï 622 := 11 + (1 + 11 × 111)/(1 + 1)
:= 222 + (22 − 2)2
Ï 627 := 11 × (1 + ((1 + 111)/(1 + 1)))
:= 2 + ((2/2 + 2 + 2)2+2 )
3
:= 3/3 + (3 × ((3 + 3) − 3 × 3))
4
:= 33 + 3 × 33 × (3 + 3)
:= (4/4 + 4) − 4 − 4
:= 4/4 + ((4/4 + 4) − 4)
:= (4 + 4)/4 + (4/4 + 4)4
:= (55 + 5 + 5)/5 − 5 − 5
:= (55 + 5 + 5)/5 − 5
:= (55 + 5 + 5)/5
:= 6 + (6 × (6 × 6 + 66) − 6/6)
:= 66 + (6666 + 6)/(6 + 6)
:= 666 − ((66 × 6/(6 + 6)) + 6)
:= 7/7 + (7 × 77 + 77)
:= 7 + ((7 × 77 − 7/7) + 77)
:= 77 + (7 × 77 + (77/7))
:= 8/8 + 88 × (8 − 8/8)
:= 8 × 8 × 8 + ((888 − 8)/8)
:= 88/8 + 88 × (8 − 8/8)
:= 9 × 9 × 9 − ((999 + 9)/9)
:= 9/9 + (9 × 9 × 9 − (99 + 9))
:= 9 + (9 × 9 × 9 − 999/9)
51
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 628 := 1 + (11 × (1 + ((1 + 111)/(1 + 1))))
2
2
Ï 633 := (1 + 1 + 1) × ((1 + 1) × 111 − 11)
2+2
:= (2 × (22+2 × (22 − 2))) − 2
:= 22 + ((2 × (2 + 2 + 2)) )
:= 2 × (2 + 2) + ((2/2 + 2 + 2)
:= 3 + ((3 − 3/3 + 3)3/3+3 )
:= 3 + (3 × ((3 + 3)3 − (3 + 3)))
:= (3 × ((3 + 3)3 − 3)) − 3/3
:= 4 + (4 × (4 × (44 − 4) − 4))
:= 4 + ((4/4 + 4)4 + 4)
:= (4 × (4 × (44 − 4))) − (4 + 4)/4
:= 5 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5)
:= ((55 + 55 + 5) + 5)/5
:= 666 − (((6 + 6)/6) + 6 × 6)
:= 666 − (66 × 6/(6 + 6))
:= (66/6) × (((6 + 6)/6)6 − 6)
:= 77 + ((77 + 7)/7 + 7 × 77)
:= 7 + ((7 × (77 + 7 + 7)) − (77/7))
:= 7/7 + (7 × (77 + 7 + 7))
:= 8 × (88 − 8) − (88 + 8)/8
:= 8/8 + (8 × (88 − 8) − 8)
:= 8 × (88 − 8) − (8 + 8)/8
:= 9 × 9 × 9 − ((9 + 9)/9 + 99)
:= 9 × 9 × 9 + (((9 + 9 + 9)/9) − 99)
:= 9 × (9 × 9 − 9) − 9/9 − 9
Ï 629 := 1 + (1 + (11 × (1 + ((1 + 111)/(1 + 1)))))
2+2
)
Ï 638 := 11 × (1 + 1 + (1 + 111)/(1 + 1))
Ï 634 := 1 + ((1 + 1 + 1) × ((1 + 1) × 111 − 11))
Ï 639 := 1 + 11 × (1 + 1 + (1 + 111)/(1 + 1))
:= 2 + (((22 + 2 + 2) ) − (2 × 22))
:= 2 + ((((2 + 2 + 2)2+2 ) − 22)/2)
:= 33+3 − (3 × 33 + 3/3)
:= 3 × (3 + 3)3 − (33/3 + 3)
:= 3 × ((3 + 3)3 − 3)
:= 4 + (4/4 + 4)4
:= 4 + (((4/4 + 4)4 + 4/4) + 4)
:= (4 × (4 × (44 − 4))) − 4/4
:= 5 + (5 − 5)/5
:= 5 + ((55 − 5)/5 + 5)
:= 5 + (((55 − 5)/5 + 5) + 5)
:= 666 − (6 × 6 + 6/6)
:= 6 + (666 − (((6 + 6)/6) + 6 × 6))
:= 6 + (666 − (66 × 6/(6 + 6)))
:= (7 × (77 + 7 + 7)) − (7/7 + 7)
:= 7 + ((7 × 77 + (77/7)) + 77)
:= ((7 + 7)/7) + (7 × (77 + 7 + 7))
:= 8 × (88 − 8) − 88/8
:= (8 + 8)/8 + (8 × (88 − 8) − 8)
:= 8 × (88 − 8) − 8/8
:= 9 × 9 × 9 − (9/9 + 99)
:= 9 + ((((99 × 99) − 9)/(9 + 9)) + 9 × 9)
:= 9 × (9 × 9 − 9) − 9
:= 2 + (((2/2 + 2 + 2)
) + 2)
5
Ï 630 := (11 − 1) × ((1 + 1 + 1) × (11 + (11 − 1)))
2
Ï 635 := 11 + ((1 + (1 + 1) × (1 + 11))1+1 − 1)
Ï 640 := 111 + (1 + 11 + 11)1+1
:= 2 + (((2 × (2 + 2 + 2))2 ) + 222 )
:= ((((2 + 2 + 2)2+2 ) − 22)/2) − 2
:= 2 × (22+2 × (22 − 2))
:= 3 × ((3 + 3)3 − (3 + 3))
:= (3 × ((3 + 3)3 − 3)) − (3/3 + 3)
:= 3/3 + (3 × ((3 + 3)3 − 3))
:= 4 + ((4/4 + 4)4 + 4/4)
:= (4/4 + 4)4 + (44 − 4)/4
:= 4 × (4 × (44 − 4))
:= 5 + 55 /5
:= 5 + (55 /5 + 5)
:= 5 + (55 /5 + 5 + 5)
:= 666 − 6 × 6
:= 6 + (666 − (6 × 6 + 6/6))
:= ((66 − 6)/6) × ((6 + 6)/6)6
:= (7 × (77 + 7 + 7)) − 7
:= (7 × (77 + 7 + 7)) − (7 + 7)/7
:= (7 − ((7 + 7)/7)) × ((7 + 7)/7)7
:= (8 − 8/8) × ((8 + 8)/8 + 88)
:= 8 + (88 × (8 − 8/8) + (88/8))
:= 8 × (88 − 8)
:= 9 × 9 × 9 − 99
:= 9 × 99 − (((9 + 9)/9)9−9/9 )
:= 9/9 + (9 × (9 × 9 − 9) − 9)
Ï 631 := 1 + ((11 − 1) × ((1 + 1 + 1) × (11 + (11 − 1))))
Ï 636 := 11 + (1 + (1 + 1) × (1 + 11))1+1
Ï 641 := 1 + 111 + (1 + 11 + 11)1+1
:= 2 + ((((2/2 + 2 + 2)2+2 ) + 2) + 2)
:= 2 × ((22+2 × (22 − 2)) − 2)
:= 2/2 + (2 × (22+2 × (22 − 2)))
:= 3/3 + (3 × ((3 + 3)3 − (3 + 3)))
:= (3 × ((3 + 3)3 − 3)) − 3
:= 3 + ((3 × ((3 + 3)3 − 3)) − 3/3)
:= 4 + ((4/4 + 4)4 + (4 + 4)/4)
:= (4 × (4 × (44 − 4))) − 4
:= 4 × 4 + (4/4 + 4)4
:= 5 + (55 + 5)/5
:= (55 + 55 )/5
:= 5 + ((55 + 55 )/5)
:= 6/6 + (666 − 6 × 6)
:= 6 + (666 − 6 × 6)
:= 666 + ((66/6) − 6 × 6)
:= 7 + ((7/7 + 7) × (7/7 + 77))
:= (7 × (77 + 7 + 7)) − 7/7
:= 77/7 + ((7 × (77 + 7 + 7)) − 7)
:= 8 × (88 − 8) − (8/8 + 8)
:= 8 × (88 − 8) − (8/((8 + 8)/8))
:= 8/8 + 8 × (88 − 8)
:= 9/9 + (9 × 9 × 9 − 99)
:= 9 × (9 × 9 − 9) − (99 + 9)/9
:= 9 × 9 × 9 + ((99/9) − 99)
Ï 632 := 111+1 + ((1 + 1)11−1−1 − 1)
2
Ï 637 := 1 + (11 + (1 + (1 + 1) × (1 + 11))1+1 )
2+2
) − 22)/2
Ï 642 := (1 + 1) × ((1 + 1 + 1) × 111 − 1 − 11)
:= 2 + (2 × (22+2 × (22 − 2)))
:= (22 + 2 + 2) − 2 × 22
:= (((2 + 2 + 2)
:= 3 + (33+3 − (3 × 33 + 3/3))
:= 3 × (3 + 3)3 − 33/3
:= 3 + (3 × ((3 + 3)3 − 3))
:= (4 × (4 × (44 − 4))) − 4 − 4
:= 4 + (((4/4 + 4)4 + 4) + 4)
:= 4 × 4 + ((4/4 + 4)4 + 4/4)
:= 5 + (55 + 5 + 5)/5
:= (55 + 55 + 5)/5
:= 5 + ((55 + 55 + 5)/5)
:= 666 + (((6 + 6)/6) − 6 × 6)
:= 6 + ((666 − 6 × 6) + 6/6)
:= (6 × 6 × (6 + 6 + 6)) − 6
:= (7/7 + 7) × ((7 + 7)/7 + 77)
:= 7 × (77 + 7 + 7)
:= 777 − (((7 + 7)/7)7 + 7)
:= 8 × (88 − 8) − 8
:= 8 + (8 × (88 − 8) − (88/8))
:= (8 + 8)/8 + 8 × (88 − 8)
:= 9 × 9 × 9 + (((9 + 9)/9) − 99)
:= 9 × (9 × 9 − 9) − 99/9
:= 9 × 9 × 9 + (((99 + 9)/9) − 99)
52
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 643 := (1 + 1) × ((1 + 1 + 1) × 111 − 11) − 1
Ï 648 := (1 + 1) × ((1 + 1) × (11 − 1 − 1))1+1
Ï 653 := (111+1+1 − 1)/(1 + 1) − 1 − 11
:= (22 − 2)2 + (222 + 2)/2
:= 2 × ((22+2 + 2)2 )
:= 222 + ((22/2 + 2)2 )
:= 3 + ((3 × ((3 + 3)3 − 3)) + 3/3)
:= 3 × (3 + 3)3
:= 3 + ((3 × (3 + 3)3 − 3/3) + 3)
:= 4 + ((4 × (4 × (44 − 4))) − 4/4)
:= (4 + 4) × (4 − 4/4)4
:= 44 + ((4/4 + 4)4 − 4 × 4)
:= 5 + (((55 + 55 + 5) + 5)/5)
:= 5 × 5 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5 × 5)
:= 6/6 + ((6 × 6 × (6 + 6 + 6)) − 6)
:= 6 × 6 × (6 + 6 + 6)
:= 666 − (6/6 + 6 + 6)
:= 7 + ((7 × (77 + 7 + 7)) − 7/7)
:= 77/7 + (7 × (77 + 7 + 7))
:= 7 × 77 + (((7 + 7)/7)7 − (7 + 7))
:= 88/8 + (8 × (88 − 8) − 8)
:= 8 + 8 × (88 − 8)
:= 88 + (8 × (8 × 8 + 8) − 88/8)
:= 99 + (((99 × 99) − 9)/(9 + 9))
:= 9 × (9 × 9 − 9)
:= 9 × (9 × 9 − 9) + ((9 × 9 + 9)/(9 + 9))
Ï 649 := 11 × ((111+1 − 1)/(1 + 1) − 1)
Ï 644 := (1 + 1) × ((1 + 1 + 1) × 111 − 11)
:= 2 × (((22+2 + 2)2 ) − 2)
:= (((2 + 2 + 2)2+2 ) + 2)/2
3
3
Ï 654 := (1 + 1) × ((1 + 1 + 1) × (111 − 1 − 1))
:= ((22 + 2 + 2)2 ) − 22
:= 3 × (3 + 3) − (3/3 + 3)
:= 3/3 + 3 × (3 + 3)
:= 4 + (4 × (4 × (44 − 4)))
:= 4 + (((4/4 + 4)4 + 4 × 4) + 4)
:= 4 + ((44 − 4)/4 × (44 + 4)/4)
:= 5 × 5 + ((5 − 5)/5 − 5)
:= 5 × 5 + (55 − 5)/5
:= 5 + ((55 − 5)/5 + 5 × 5)
:= 666 − ((66 + 66)/6)
:= 6/6 + (6 × 6 × (6 + 6 + 6))
:= 666 − 6 − 6
:= 7 + (7 × (77 + 7 + 7))
:= 777 − ((7 + 7)/7)7
:= 77 + (7 × (77 + 7) − (77/7))
:= 8 × (88 − 8) + (8/((8 + 8)/8))
:= 8 + (8 × (88 − 8) + 8/8)
:= 8 + ((8 × (88 − 8) − ((8 + 8)/8)) + 8)
:= 99 + (((99 × 99) + 9)/(9 + 9))
:= 9/9 + 9 × (9 × 9 − 9)
:= 9 + (9 × (9 × 9 − 9) − ((9 + 9 + 9)/9))
5
Ï 645 := 1 + (1 + 1) × ((1 + 1 + 1) × 111 − 11)
:= ((((2 + 2 + 2)
2+2
) − 2)/2) − 2
Ï 650 := 1 + 11 × ((111+1 − 1)/(1 + 1) − 1)
:= 2 + (2 × ((2
3
:= 3 × (3 + 3) − 3
:= 3 + (3 × (3 + 3)3 + 3)
2+2
2
+ 2) ))
Ï 655 := (1 + 1) × (1 + 1 + 1) × 111 − 11
:= 2 + (((22/2 + 2)2 ) + 222 )
:= 3 + (3 × (3 + 3)3 − 3/3)
:= 3 + ((3 × (3 + 3)3 + 3/3) + 3)
:= (44 − 4)/4 × (44 + 4)/4
:= (4 × (4 × (44 − 4) + 4)) − 4/4
:= 5 × 5 + (5 /5 − 5)
:= 5 × (5 × 5 × 5 + 5)
:= 5 + (55 /5 + 5 × 5)
:= (6 × 6 × 6 × 6 − 6)/((6 + 6)/6)
:= 666 + (((6 − 66)/6) − 6)
:= 666 − 66/6
:= 7 + ((7 × (77 + 7 + 7)) + 7/7)
:= 7 × 77 + 777/7
:= 7 + ((7 × (77 + 7 + 7)) + (77/7))
:= 8 + ((8 × (88 − 8) − (88/8)) + 8)
:= 8 + (8 × (88 − 8) + ((8 + 8)/8))
:= 8 + ((8 × (88 − 8) − 8/8) + 8)
:= 9 × (9 × 9 − 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 × (9 × 9 − 9)
:= 9 + (9 × (9 × 9 − 9) − ((9 + 9)/9))
4
:= 4 + ((4/4 + 4) + 4 × 4)
5
Ï 646 := (1 + 1) × (1 + (1 + 1 + 1) × 111 − 11)
:= (2 × ((22+2 + 2)2 )) − 2
Ï 651 := (1 + 1 + 1) × ((1 + 1) × (111 − 1 − 1) − 1)
:= 2 + ((((2 + 2 + 2)
3
:= 3/3 + (3 × (3 + 3) − 3)
4
:= 4 + (((4/4 + 4) + 4 × 4) + 4/4)
2+2
) + 2)/2)
Ï 656 := 1 + ((1 + 1) × (1 + 1 + 1) × 111 − 11)
:= 2 + (((22 + 2 + 2)2 ) − 22)
:= 3 + 3 × (3 + 3)3
:= (3 × ((3 + 3)3 + 3)) − 3/3
:= 44/4 + (4 × (4 × (44 − 4)))
:= 4 × (4 × (44 − 4) + 4)
:= 5 + (((55 + 55 )/5) + 5)
:= 5 × 5 + (5 + 5)/5
:= 5 + ((55 + 5)/5 + 5 × 5)
:= (6 × 6 × (6 + 6 + 6)) − (6 + 6)/6
:= (6 × 6 × 6 × 6 + 6)/((6 + 6)/6)
:= 666 + (6 − 66)/6
:= 7 + ((7 × (77 + 7 + 7)) + ((7 + 7)/7))
:= 7 + ((7 × (77 + 7 + 7)) + 7)
:= 7 + (777 − ((7 + 7)/7)7 )
:= 8 + (8 × (88 − 8) − ((8 + 8)/8))
:= 88/8 + 8 × (88 − 8)
:= 8 + (8 × (88 − 8) + 8)
:= 9 × (9 × 9 − 9) − (9 + 9)/9
:= 9 × (9 × 9 − 9) + ((9 + 9 + 9)/9)
:= 9 + (9 × (9 × 9 − 9) − 9/9)
5
Ï 647 := (1 + 1) × ((1 + 1) × (11 − 1 − 1))1+1 − 1
:= (((2 + 2 + 2)
2+2
) − 2)/2
Ï 652 := (1 + 1) × ((1 + 1 + 1) × (111 − 1 − 1) − 1)
:= 2 × (((2
3
2+2
2
+ 2) ) + 2)
3
Ï 657 := (1 + 1 + 1) × (((1 + 1) × (111 − 1)) − 1)
:= (2/2 + 2) × (222 − (2/2 + 2))
:= 3 × (3 + 3) − 3/3
:= 3 + (3 × (3 + 3) + 3/3)
:= 3 × ((3 + 3)3 + 3)
:= ((4 + 4) × (4 − 4/4)4 ) − 4/4
:= 4 + ((4 + 4) × (4 − 4/4)4 )
:= 4 × (4 + 4) + (4/4 + 4)4
:= ((55 + 55) + 55 )/5
:= 5 × 5 + (55 + 5 + 5)/5
:= 55 /5 + ((5 + 5)/5)5
:= 666 − ((6 + 6)/6 + 6 + 6)
:= 666 + (((6 − 66) + 6)/6)
:= 7 + ((7 − ((7 + 7)/7)) × ((7 + 7)/7) )
:= 7 + (((7 × (77 + 7 + 7)) + 7/7) + 7)
:= 7 + (777/7 + 7 × 77)
:= 8 + (8 × (88 − 8) − 8/8)
:= ((88 + 8)/8) + 8 × (88 − 8)
:= 8 + ((8 × (88 − 8) + 8/8) + 8)
:= 9 × (9 × 9 − 9) − 9/9
:= 9 × (9 × 9 − 9) + ((9 × 9 − 9)/(9 + 9))
:= 9 + 9 × (9 × 9 − 9)
:= (6 × 6 × (6 + 6 + 6)) − 6/6
7
53
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 658 := (1 + 1) × (((1 + 1 + 1) × (111 − 1)) − 1)
Ï 663 := (1 + 1 + 1) × ((1 + 1) × 111 − 1)
2
Ï 668 := (1 + 1) × (1 + (1 + 1 + 1) × 111)
:= 222 + ((22 − 2/2) )
:= 2 + ((2/2 + 2) × 222)
:= 3/3 + (3 × ((3 + 3) + 3))
:= 3 + ((3 × ((3 + 3)3 + 3)) + 3)
:= 3 + (33+3 − ((3/3 + 3)3 ))
:= 44 + ((4/4 + 4)4 − 44/4)
:= 4 + 4 × 4 × 44 − (44 + 4/4)
:= (4 × ((4 × 44) − (4 + 4))) − 4
:= (55 + 5)/5 + ((5 + 5)/5)5
:= 5 + ((55 + 5)/5 + ((5 + 5)/5)5 )
:= 55 + ((55 − (55 + 5))/5)
:= 666 − ((6 + 6)/6 + 6)
:= 666 − 6 × 6/(6 + 6)
:= 666 + (6 + 6)/6
:= (22 × (2 × (2 + 2) + 22)) − 2
3
:= 77 + (7 × (77 + 7) − 7)
:= 7 + ((777 − ((7 + 7)/7) ) + 7)
:= (7 × 7 × (7 + 7)) − (77/7 + 7)
:= 8 + ((8 × (88 − 8) + ((8 + 8)/8)) + 8)
:= 88 + (8 × (8 × 8 + 8) − 8/8)
:= 8 + (8 × 88 − (88/((8 + 8)/8)))
:= 9 + (9 × (9 × 9 − 9) + 9/9)
:= 9 + ((9 × (9 × 9 − 9) − ((9 + 9 + 9)/9)) + 9)
:= 9 + (9 × (9 × 9 − 9) + (99/9))
Ï 659 := ((1 + 1) × ((1 + 1 + 1) × (111 − 1))) − 1
2+2
7
Ï 664 := (1 + 1) × ((1 + 1 + 1) × 111 − 1)
Ï 669 := (1 + 1 + 1) × (1 + (1 + 1) × 111)
:= ((2/2 + 2) × 222) − 2
:= (2/2 + 2) × (222 + 2/2)
:= 33/3 + 3 × (3 + 3)3
:= (3 − 3/3) × (333 − 3/3)
:= 3 + (333 + 333)
:= 4 × 4 × 44 − (44 + 4/4)
:= 4 + (44 × (44/4 + 4))
:= 44 + (4/4 + 4)4
:= ((55 × (55 + 5)) − 5)/5
:= 5 + (((55 × (55 + 5)) − 5)/5)
:= 55 + ((55 − 55)/5)
:= 666 − 6/6 − 6
:= 666 − (6 + 6)/6
:= 666 + (6 × 6/(6 + 6))
:= 777 − (777/7 + 7)
:= 77 + (7 × (77 + 7) − 7/7)
:= ((7 − 77)/7) + ((7 × 7 × (7 + 7)) − 7)
:= 8 + (8 × (88 − 8) + (88/8))
:= 88 + 8 × (8 × 8 + 8)
:= 8 × 88 − ((88/8 + 8 + 8) + 8)
:= 99/9 + 9 × (9 × 9 − 9)
:= (9 − 9/9) × (((9 + 9)/9) + 9 × 9)
:= 9 + (9 × (9 × 9 − 9) + ((99 + 9)/9))
:= (((2 + 2 + 2)
) + 22)/2
Ï 660 := (1 + 1) × ((1 + 1 + 1) × (111 − 1))
:= 22 × (2 × (2 + 2) + 22)
3
:= 3 + (3 × ((3 + 3) + 3))
Ï 665 := (111+1+1 − 1)/(1 + 1)
2
:= ((22 + 2 + 2) ) − 22/2
:= 3
3+3
Ï 670 := 1 + ((1 + 1 + 1) × (1 + (1 + 1) × 111))
:= 2 + (((2/2 + 2) × 222) + 2)
:= 3 + ((((3 + 3) × 333) + 3)/3)
3
− ((3/3 + 3) )
:= 44 × (44/4 + 4)
:= 44 + ((4/4 + 4) − 4)
:= 44 + ((4/4 + 4)4 + 4/4)
:= 55 × ((55 + 5)/5)
:= 5 + (55 × ((55 + 5)/5))
:= 55 + (55 /5 − (5 + 5))
:= 666 − 6
:= 666 − 6/6
:= 6 + (666 − ((6 + 6)/6))
4
:= 7 × 77 + (((7 + 7)/7)7 − 7)
:= 77 + 7 × (77 + 7)
:= ((7 + 7)/7) × (7 × 7 × 7 − (7/7 + 7))
:= 8 × 88 − (88/((8 + 8)/8))
:= 8/8 + (8 × (8 × 8 + 8) + 88)
:= ((88 + 8) × (8 − 8/8)) − (8 + 8)/8
:= 9 × (9 × 9 − 9) + (99 + 9)/9
:= 9 + ((9 × (9 × 9 − 9) − 9/9) + 9)
:= 9 × (9 × 9 − 9) + ((99 + 99)/9)
Ï 661 := 1 + ((1 + 1) × ((1 + 1 + 1) × (111 − 1)))
:= 2 + ((((2 + 2 + 2)
2+2
) + 22)/2)
3
Ï 666 := (1 + 1) × (1 + 1 + 1) × 111
Ï 671 := 11 × ((1 + 111+1 )/(1 + 1))
:= 2 + ((2/2 + 2) × (222 + 2/2))
:= (2/2 + 2) × 222
:= 3 + ((3 × ((3 + 3) + 3)) + 3/3)
:= 3 × (((3 + 3) + 3) + 3)
:= 33/3 × (((3/3 + 3)3 ) − 3)
:= 4 + ((4/4 + 4)4 + 4 × (4 + 4))
:= ((4 + 4)/4 + 4) × 444/4
:= 44/4 × ((44 + 4)/4 − 4)
:= 5 × 5 + ((55 + 55 )/5)
:= 555 + 555/5
:= 5 + (555/5 + 555)
:= 6/6 + (666 − 6)
:= 666
:= 6 + (666 − 6/6)
:= 7 × 77 + (777 + 77)/7
:= (7 − 7/7) × 777/7
:= (7 × 7 × (7 + 7)) − (7/7 + 7 + 7)
:= ((88 + 8) × (8 − 8/8)) − 88/8
:= (8 − (8 + 8)/8) × 888/8
:= ((88 + 8) × (8 − 8/8)) − 8/8
:= 9 × (9 × 9 − 9) + ((99 + 9 + 9)/9)
:= 9 + (9 × (9 × 9 − 9) + 9)
:= 99/9 × (9 × 9 − (99/9 + 9))
Ï 662 := (1 + 1) × (1 + ((1 + 1 + 1) × (111 − 1)))
Ï 667 := 1 + (1 + 1) × (1 + 1 + 1) × 111
Ï 672 := (1 + 1) × ((1 + 1 + 1) × (1 + 111))
:= 2 + (22 × (2 × (2 + 2) + 22))
3
:= 2/2 + ((2/2 + 2) × 222)
:= (2/2 + 2) × (222 + 2)
:= 3 + (3 × (3 + 3) + 33/3)
:= (((3 + 3) × 333) + 3)/3
:= 33 + (3 × (3 + 3)3 − 3)
:= (4 + 4)/4 + (44 × (44/4 + 4))
:= 4444/4 − 444
:= 4 × ((4 × 44) − (4 + 4))
:= 5 + (((5 + 5)/5) + 5 /5)
:= 555 + (555 + 5)/5
:= (5/5 + 5) × (555 + 5)/5
:= 666 + ((6 + 6)/6 − 6)
:= 6/6 + 666
3
5
5
:= 6 + 666
7
:= 7 + (((7 × (77 + 7 + 7)) + (77/7)) + 7)
:= 7 × 77 + ((7 + 7)/7)
:= (7 + 7) × (7 × 7 − 7/7)
:= 88 + (8 × (8 × 8 + 8) − (8 + 8)/8)
:= 8 + ((8 × (88 − 8) + (88/8)) + 8)
:= (88 + 8) × (8 − 8/8)
:= 9 + (((9 × 9 + 9)/(9 + 9)) + 9 × (9 × 9 − 9))
:= 9 + ((9 × (9 × 9 − 9) + 9/9) + 9)
:= (9 − 9/9) × (((9 + 9 + 9)/9) + 9 × 9)
54
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 673 := 1 + ((1 + 1) × ((1 + 1 + 1) × (1 + 111)))
Ï 678 := 1 + (1 + (((1 + 1) × (1 + (1 + 11)))1+1 ))
Ï 683 := (1 + ((1 + 1)11 ))/(1 + 1 + 1)
:= ((22 + 2 + 2)2 ) − 2/2 − 2
:= 2 + ((22 + 2 + 2)2 )
:= 222 + (((22 − 2)2 − 2)/2)
:= ((33 − 3/3)3−3/3 ) − 3
:= 3 + (3 × (3 + 3)3 + 33 )
:= ((33/3)3 ) − 3 × (3 + 3)3
:= 4 + ((4/4 + 4)4 + 44)
:= (4 − 44)/4 + (4 × ((4 × 44) − 4))
:= (4 × ((4 × 44) − 4)) − (4/4 + 4)
:= 55 + ((55 − 5 − 5)/5 − 5)
:= 55 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 55)
:= 6 + (666 + 6/6)
:= 6 + (666 + 6)
:= 6 + (666 + (66/6))
:= 7 + ((7 − 7/7) × 777/7)
:= (7 × 7 × (7 + 7)) − (7/7 + 7)
:= (7 × 7 × (7 + 7)) − (7 + 7 + 7)/7
:= 8/8 + ((88 + 8) × (8 − 8/8))
:= 8 × 88 + ((8 − 88)/8 − (8 + 8))
:= 8 + ((8/8 + 8) × (88/8 + 8 × 8))
:= 9 + ((9 − 9/9) × (((9 + 9)/9) + 9 × 9))
:= 999/9 + (9 × (9 × 9 − (9 + 9)))
:= 9 + ((((9 + 9)/9)9 ) + 9 × (9 + 9))
Ï 674 := (1 + 1) × (1 + ((1 + 1 + 1) × (1 + 111)))
Ï 679 := (((1 + 1)11 ) − 11)/(1 + 1 + 1)
Ï 684 := 1 + ((1 + ((1 + 1)11 ))/(1 + 1 + 1))
:= ((22 + 2 + 2)2 ) − 2
:= 2 + (((22 + 2 + 2)2 ) + 2/2)
:= 2 × (2 × (((22/2 + 2)2 ) + 2))
:= 33 + (3 × (3 + 3)3 − 3/3)
:= 3 + ((33 − 3/3)3−3/3 )
:= 3 + (3 × (3 + 3)3 + 33)
:= 4 + (((4/4 + 4)4 + 44) + 4/4)
:= 4 + (44/4 + 4) × (44 + 4/4)
:= (4 × ((4 × 44) − 4)) − 4
5
:= 55 + ((55 − 5)/5 − 5)
:= 55 + (5 − 5)/5
:= 5 + ((55 − 5)/5 + 55)
:= 6 + (666 + ((6 + 6)/6))
:= 6 + (666 + 6/6 + 6)
:= 6 + ((666 + 6) + 6)
:= 7 + (((7 + 7)/7) + 7 × 77)
:= (7 × 7 × (7 + 7)) − 7
:= (7 × 7 × (7 + 7)) − (7 + 7)/7
:= 8 + ((8 − (8 + 8)/8) × 888/8)
:= 8 × 88 − (8/8 + 8 + 8 + 8)
:= 8 × 88 − (((88 + 8)/8) + 8)
:= 9 × (9 + 9) + (((9 + 9)/9)9 )
:= 9 × 9 × 9 − ((9 × 99 + 9)/(9 + 9))
:= (99 × (9 − ((9 + 9)/9))) − 9
7
Ï 675 := (((1 + 1) × (1 + (1 + 11)))1+1 ) − 1
Ï 680 := 1 + ((((1 + 1)11 ) − 11)/(1 + 1 + 1))
Ï 685 := 1 + (1 + ((1 + ((1 + 1)11 ))/(1 + 1 + 1)))
:= 2 + (((22 + 2 + 2)2 ) + 2)
:= 22/2 + (((22 + 2 + 2)2 ) − 2)
:= 33 + (3 × (3 + 3)3 − 3/3)
:= 33+3 − (33/3 + 33)
:= (44/4 + 4) × (44 + 4/4)
:= (4 + 4) × ((4 − 4/4)4 + 4)
:= 4 × 4 + ((4/4 + 4)4 + 44)
:= 5 × ((5 × 5 × 5 + 5) + 5)
:= 55 + 55 /5
:= 5 + (55 /5 + 55)
:= 6 + ((6 × 6/(6 + 6)) + 666)
:= 6 + ((666 + ((6 + 6)/6)) + 6)
:= 6 + ((666 + 6/6 + 6) + 6)
:= (7 × 7 × (7 + 7)) − 77/7
:= 7/7 + ((7 × 7 × (7 + 7)) − 7)
:= (7 × 7 × (7 + 7)) − 7/7
:= (8/8 + 8) × (88/8 + 8 × 8)
:= 8 × 88 − 8 − 8 − 8
:= 8 × 88 − (88/8 + 8)
:= 9 + ((9 × (9 × 9 − 9) + 9) + 9)
:= 9 × 9 × 9 + ((9 − 9 × 99)/(9 + 9))
:= 9 + ((((9 − 9/9) + 9) + 9)(9+9)/9 )
:= ((22 + 2 + 2)2 ) − 2/2
:= 3 × ((3 + 3)3 + 3 × 3)
Ï 676 := ((1 + 1) × (1 + (1 + 11)))1+1
Ï 681 := ((1 + ((1 + 1)11 ))/(1 + 1 + 1)) − 1 − 1
2
:= (22 + 2 + 2)2
:= 2 + ((((22 + 2 + 2) ) + 2/2) + 2)
3
:= (33 − 3/3)3−3/3
:= 33 + 3 × (3 + 3)
:= 4 + (4 × ((4 × 44) − (4 + 4)))
:= 4 + ((((4/4 + 4)4 + 44) + 4) + 4)
5
Ï 686 := (11 + (((1 + 1)11 ) − 1))/(1 + 1 + 1)
:= 2 × ((22/2)2 + 222)
:= (3 − 3/3) × (((3/3 + 3) + 3)3 )
5
:= (4 × ((4 × 44) − 4)) − (4 + 4)/4
:= 55 + ((5 + 5)/5 − 5)
:= 55 + (5 + 5)/5
:= 5 + ((55 + 5)/5 + 55)
:= 666 + (66 − 6)/6
:= 66 × (6 + 6) − 666/6
:= 6 + (((666 + ((6 + 6)/6)) + 6) + 6)
:= ((7 − 77)/7) + (7 × 7 × (7 + 7))
:= ((7 + 7)/7) + ((7 × 7 × (7 + 7)) − 7)
:= 7 × 7 × (7 + 7)
:= 8 × 88 + ((8 − 8 × 8)/((8 + 8)/8))
:= 8/8 + (8 × 88 − (8 + 8 + 8))
:= 8 × 88 + ((8 − 88)/8 − 8)
:= 9 × 99 − (999/9 + 99)
:= (99 − 9/9) × (9 − ((9 + 9)/9))
:= (((9 − 9/9) + 9) + 9)
(9+9)/9
Ï 677 := 1 + (((1 + 1) × (1 + (1 + 11)))1+1 )
Ï 682 := (((1 + 1)11 ) − (1 + 1))/(1 + 1 + 1)
Ï 687 := 11 + (((1 + 1) × (1 + (1 + 11)))1+1 )
:= 2/2 + ((22 + 2 + 2)2 )
:= 2 + ((((22 + 2 + 2)2 ) + 2) + 2)
:= 22/2 + ((22 + 2 + 2)2 )
:= (((3 + 3) × 333) + 33)/3
:= 3/3 + (3 × (3 + 3)3 + 33)
:= 33+3 − (3 × 3 + 33)
:= 4 + (((4/4 + 4)4 + 44) + 4)
:= 44/4 × ((44 − 4 − 4)/4)
:= (4 × ((4 × 44) − 4)) − 4/4
:= 55 + ((5 + 5 + 5)/5 − 5)
:= 55 + (5 + 5 + 5)/5
:= 5 + ((55 + 5 + 5)/5 + 55)
:= 666 + 66/6
:= 6 + ((66 − 6)/6 + 666)
:= ((6 × 6/(6 + 6))6 ) − (6 × 6 + 6)
:= (7 × 7 × (7 + 7)) − ((7 + 7)/7 + 7)
:= 7 + ((7 × 7 × (7 + 7)) − (77/7))
:= 7/7 + (7 × 7 × (7 + 7))
:= 8 × 88 − (88/8 + 8 + 8)
:= 8 × 88 − (88 + 88)/8
:= 8 × 88 − (8/8 + 8 + 8)
:= 9 + ((9 × (9 × 9 − 9) + (99/9)) + 9)
:= 99/9 × (9 × 9 − (9/9 + 9 + 9))
:= 9 + ((9 × (9 × 9 − (9 + 9))) + 999/9)
5
5
55
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 688 := (1 + 1) × (11 + (1 + 1 + 1) × 111)
:= 3 + (3
3+3
Ï 693 := 11 × ((1 + 1 + 1) × (11 + (11 − 1)))
− (33/3 + 33))
Ï 698 := (1 + 1 + 1) × (11 + (1 + 1) × 111) − 1
:= (2/2 + 2) × ((22 − 22)/2)
:= 22 + ((22 + 2 + 2)2 )
:= 33 × ((3 × (3 + 3)) + 3)
:= 3 + (33+3 − (3/3 + 33))
2
:= 2 × (2 × (2 × (2 × 2 × 22 − 2)))
4
:= 4 × ((4 × 44) − 4)
:= 44/4 × ((4 − 4)/4)
:= 44 + (444 − (4 + 4)/4)
:= (5 × 5 × 5 × 55 + 5)/(5 + 5)
:= 5 + ((5 × 5 × 5 × 55 + 5)/(5 + 5))
:= (5 × (5 + 5 + 5)) + (55 − 5 − 5)/5
:= 666 + ((66 + 66)/6)
:= ((6 × 6/(6 + 6)) ) − 6 × 6
:= ((7 + 7)/7) + (7 × 7 × (7 + 7))
:= ((66/6) × ((6 + 6)/6)6 ) − 6
:= 7 + (7 × 7 × (7 + 7))
:= 8 × 88 − 8 − 8
:= 777 − ((7 + 7)/7 + 77)
:= 8 × 88 − 88/8
:= 9 × 9 × 9 − ((9 × 9 × 9 + 9)/(9 + 9))
:= (8 + 8)/8 + (8 × 88 − 8)
:= 99 × (9 − ((9 + 9)/9))
:= 9 × 9 × 9 − (((99 + 99)/9) + 9)
6
Ï 694 := 11 + ((1 + ((1 + 1)11 ))/(1 + 1 + 1))
Ï 699 := (1 + 1 + 1) × (11 + (1 + 1) × 111)
:= 2 + (((22 + 2 + 2)2 ) + 22+2 )
:= 2 + ((22/2)2 + ((22 + 2)2 ))
:= 3 + ((3 − 3/3) × (((3/3 + 3) + 3) ))
:= 3/3 + (33 × ((3 × (3 + 3)) + 3))
:= 3 + (33+3 − 33)
:= 4/4 + (4 × ((4 × 44) − 4))
:= (4 − 44)/4 + 4 × 4 × 44
:= 44 + (444 − 4/4)
Ï 689 := 1 + ((1 + 1) × (11 + (1 + 1 + 1) × 111))
:= 2 + (((22 + 2 + 2)2 ) + 22/2)
3
5
:= 5 + (((5 − 5)/5 + 55) + 5)
5
:= 5 + ((((5 − 5)/5 + 55) + 5) + 5)
:= (55 − 5)/5 + (5 × (5 + 5 + 5))
:= 6 + ((666 + (66/6)) + 6)
:= 6 + (((66 + 66)/6) + 666)
:= 777 − (77/7 + 77)
:= 7 + ((7 × 7 × (7 + 7)) + 7/7)
:= 6 + (((6 × 6/(6 + 6))6 ) − 6 × 6)
:= 8/8 + (8 × 88 − (8 + 8))
:= 8 × 88 + (8 − 88)/8
:= 9 × 9 × 9 + ((9 − 9 × 9 × 9)/(9 + 9))
:= 9/9 + (99 × (9 − ((9 + 9)/9)))
Ï 690 := 11 + ((((1 + 1)11 ) − 11)/(1 + 1 + 1))
:= 2 + (((2/2 + 2) × 222) + 22)
:= 33 + (3 × ((3 + 3)3 + 3))
:= (4 + 4)/4 + (4 × ((4 × 44) − 4))
Ï 695 := 1 + 11 + (1 + (1 + 1)11 )/(1 + 1 + 1)
:= 222 + (222 − 22/2)
:= 3
3+3
− (3/3 + 33)
:= 777 − (7/7 + 77)
:= 88/8 + (8 × 88 − (8 + 8))
:= 9 × (9 × 9 + 9) − 999/9
Ï 700 := 1 + (1 + 1 + 1) × (11 + (1 + 1) × 111)
:= 2 × ((22 × 22+2 ) − 2)
:= 3 + ((33+3 − 33) + 3/3)
4
:= 4 + (444 − (4/4 + 4))
:= 44 + 444
:= 5 + ((55 /5 + 55) + 5)
:= 5 × 5 × (5 × 5 + 5) − 55
:= 6 + (((666 + 6) + 6) + 6)
:= 5 × (((5 × 5 × 5 + 5) + 5) + 5)
:= 6 × 6 + (666 − 6/6 − 6)
:= 77/7 + ((7 × 7 × (7 + 7)) − 7)
:= 6 × 6 + (666 − ((6 + 6)/6))
:= 7 + ((7 × 7 × (7 + 7)) + ((7 + 7)/7))
:= (8 + 8)/8 + (8 × 88 − (8 + 8))
:= 777 − 77
:= 8 × 88 − (8/8 + 8)
:= (9/9 + 9) × (9 × 9 − ((99 + 9)/9))
:= 8 × 88 − (8/((8 + 8)/8))
:= 9 + ((99 − 9/9) × (9 − ((9 + 9)/9)))
:= (9/9 + 9) × (9 × 9 − 99/9)
Ï 691 := ((1 + 1)11−1 ) − (1 + 1 + 1) × 111
:= 2 + ((((22 + 2 + 2)2 ) + 22/2) + 2)
:= 3
3+3
3
− (33/3 + 3 )
:= 4 + ((4 × ((4 × 44) − 4)) − 4/4)
5
Ï 696 := (1 + 1 + 1) × (111 + 111+1 )
:= 2 × (2 × ((2 × 2 × 2 × 22) − 2))
:= 3
3+3
− 33
Ï 701 := 1 + 1 + (1 + 1 + 1) × (11 + (1 + 1) × 111)
:= 2/2 + (2 × ((22 × 22+2 ) − 2))
:= 33+3 − (33 + 3/3)
4
:= 4 + (444 − 4)
:= 4/4 + (444 + 44 )
5
:= 55 + ((55 + 5 )/5)
:= 5 + (((55 + 5 )/5) + 55)
:= (55 + 5)/5 + (5 × (5 + 5 + 5))
:= 6 × 6 + (666 − 6/6)
:= 6 × 6 + (666 − 66/6)
:= 6 × 6 + (666 − 6)
:= 7 + ((7 × 7 × (7 + 7)) − ((7 + 7)/7))
:= (7 × 7 × (7 + 7)) + (77 − 7)/7
:= 7/7 + (777 − 77)
:= 8 × 88 − (88 + 8 + 8)/8
:= 8 × 88 − 8
:= 8 + (8 × 88 − (88/8))
:= 9 × 9 × 9 − ((99/9 + 9 + 9) + 9)
:= (9 − 9/9) × (99 − ((99 + 9)/9))
:= 9 × 9 × 9 − ((9/9 + 9 + 9) + 9)
Ï 692 := (11 × ((1 + 1 + 1) × (11 + (11 − 1)))) − 1
:= 2 × (((2
:= 3
3+3
2+2
2
+ 2) ) + 22)
− (3/3 + 33 + 3)
:= 4 + (4 × ((4 × 44) − 4))
5
Ï 697 := 1 + (1 + 1 + 1) × (111 + 111+1 )
2
2
:= (22/2) + ((22 + 2) )
:= 3/3 + (3
3+3
Ï 702 := (1 + 1) × ((11 × ((11 × (1 + 1 + 1)) − 1)) − 1)
:= (2 × (22 × 22+2 )) − 2
:= 33+3 − 33
− 33)
4
:= 4 + (44/4 × ((4 − 4)/4))
5
:= 4 × 4 × 44 − (4 + 4)/4
:= 55 + ((55 + 5 + 5)/5)
:= 5 + (((55 + 5 + 5)/5) + 55)
:= 55 + (((55 + 55) + 55 )/5)
:= 6 × 6 + (((6 − 66)/6) + 666)
:= 6 × 6 + ((666 − 6) + 6/6)
:= 6 × 6 + 666
:= 7 + ((7 × 7 × (7 + 7)) − 7/7)
:= 77/7 + (7 × 7 × (7 + 7))
:= ((7 + 7)/7) + (777 − 77)
:= 8 × 88 − (88 + 8)/8
:= 8/8 + (8 × 88 − 8)
:= 8 × 88 − (8 + 8)/8
:= 99 + ((((9 + 9)/9)9 ) + 9 × 9)
:= 9 + (9 × 9 × 9 − ((9 × 9 × 9 + 9)/(9 + 9)))
:= 9 × 9 × 9 − (9 + 9 + 9)
56
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 703 := (11 × ((1 + 1)(1+1)×(1+1+1) )) − 1
Ï 708 := (1 + 11) × ((111+1 − 1)/(1 + 1) − 1)
Ï 713 := (1 + (11 + 11)) × (1 + ((11 − 1) × (1 + 1 + 1)))
:= 2 + (222 + 22 )
:= ((2/2 + 2)2+2+2 ) − 22+2
:= 3/3 + (33+3 − 33 )
:= 3 + ((33+3 − 33 ) + 3)
:= 33+3 + (33/3 − 33 )
:= 4 × 4 × 44 − 4/4
:= 4 + 4 × 4 × 44
:= 4 + (4 × 4 × 44 + 4/4) + 4
:= 5 × 5 + ((55 − 5 − 5)/5 + 55)
:= ((55 + 5)/5) × (55 − 5/5 + 5)
:= (55 × (55 + 5 + 5) − (5 + 5))/5
:= 6 × 6 + (666 + 6/6)
:= 6 + (666 + 6 × 6)
:= 6 × 6 + (666 + (66/6))
:= (2 × (22 × 2
2+2
)) − 2/2
2
:= 7 + ((7 × 7 × (7 + 7)) + ((77 − 7)/7))
:= 7 + ((777 − 77) + 7/7)
:= 7 + ((777 − (7/7 + 77)) + 7)
:= 8 × 88 − 8/8
:= 8 × 88 + (8/((8 + 8)/8))
:= 8 + (8 × 88 + 8/8)
:= 9/9 + (9 × 9 × 9 − (9 + 9 + 9))
:= 9 × 9 × 9 − (((99 + 9)/9) + 9)
:= 9 × 9 × 9 + (((9 + 9)/9) − (9 + 9))
Ï 704 := 11 × ((1 + 1)(1+1)×(1+1+1) )
Ï 709 := 11 × 111 − (1 + 1)11−1−1
Ï 714 := (1 + 1) × ((1 + 1 + 1) × (111+1 − (1 + 1)))
:= 2 × (22 × 22+2 )
:= 2 + ((222 + 222 ) + 2/2)
:= (2 + 2 + 2) × ((22/2)2 − 2)
:= 33/3 × ((3/3 + 3)3 )
:= 33+3 − (33/3 + 3 × 3)
:= 3 + (33+3 − (3 × (3 + 3)))
:= 4 × (4 × 44)
:= 4 + 4 × 4 × 44 + 4/4
:= 4 × 4 × 44 + (44 − 4)/4
5
5
:= 5 + (((5 − 5)/5 + 55) + 5 × 5)
:= (55 × (55 + 5 + 5) − 5)/5
:= (66/6) × ((6 + 6)/6)
:= 6 + ((666 + 6/6) + 6 × 6)
:= (6 + 6) × (66 − 6) − 6
:= 7 + ((7 × 7 × (7 + 7)) + (77/7))
:= 7 + (((7 + 7)/7 − 77) + 777)
:= 7 + ((777 − 77) + 7)
:= 8 × 88
:= 8 + ((8 × 88 − (88/8)) + 8)
:= 8 + ((8 + 8)/8 + 8 × 88)
:= (9 − 9/9) × (99 − (99/9))
:= 9 × 9 × 9 − (99/9 + 9)
:= 9 × 9 × 9 + (((9 + 9 + 9)/9) − (9 + 9))
:= 5 × 5 + ((5 − 5)/5 + 55)
6
Ï 705 := 1 + 11 × ((1 + 1)(1+1)×(1+1+1) )
Ï 710 := (11 − 1) × ((1 + 11)1+1 /(1 + 1) − 1)
2
Ï 715 := 11 × (1 + ((1 + 1)(1+1)×(1+1+1) ))
:= 2/2 + (2 × (22 × 22+2 ))
:= 2 + ((222 + 22 ) + 2)
:= 22/2 + (2 × (22 × 22+2 ))
:= 3 + (33+3 − 33 )
:= 33+3 − ((3 × (3 + 3)) + 3/3)
:= 33+3 − (33/3 + 3)
:= 4/4 + 4 × 4 × 44
:= 4 + 4 × 4 × 44 + (4 + 4)/4
:= 44/4 + 4 × 4 × 44
:= 5 × 5 + (5 /5 + 55)
:= 5 + ((55 /5 + 55) + 5 × 5)
:= 55 × (55 + 5 + 5)/5
:= ((66 × ((6 + 6)/6)6 ) + 6)/6
:= 6 + ((66/6) × ((6 + 6)/6)6 )
:= (66/6) × (66 − 6/6)
:= 7 + ((7 × 7 × (7 + 7)) + (77 + 7)/7)
:= 777 + (((77 − 7)/7) − 77)
:= 7 + (((777 − 77) + 7/7) + 7)
:= 8/8 + 8 × 88
:= 8 + (8 × 88 − ((8 + 8)/8))
:= 88/8 + 8 × 88
:= 9 + ((9 − 9/9) × (99 − ((99 + 9)/9)))
:= 9 × 9 × 9 − (9/9 + 9 + 9)
:= 9 × 9 × 9 + (((9 − 99)/(9 + 9)) − 9)
5
Ï 706 := 1 + 1 + 11 × ((1 + 1)(1+1)×(1+1+1) )
Ï 711 := 1111 − ((1 + 1) × (11 − 1))1+1
Ï 716 := ((1 + 1)11 ) − (1 + 111+1+1 )
:= 222 + 222
:= (2222/2) − (22 − 2)2
:= 2 + ((2 + 2 + 2) × ((22/2)2 − 2))
:= 3 + ((33+3 − 33 ) + 3/3)
:= 33+3 − (3 × (3 + 3))
:= 33+3 + (((3 − 33)/3) − 3)
:= (4 + 4)/4 + 4 × 4 × 44
:= 4 + (4 × 4 × 44 − 4/4) + 4
:= (4 × ((4 × 44) + 4)) − 4
:= 5 × 5 + ((55 + 5)/5 + 55)
5
:= ((555 + 5 )/5) − 5 × 5
:= (55 × (55 + 5 + 5) + 5)/5
:= 6 + ((666 − ((6 + 6)/6)) + 6 × 6)
:= ((6 × 6/(6 + 6)) ) − 6 − 6 − 6
:= ((66 × (66 − 6/6)) + 6)/6
:= 7 + (777 − (7/7 + 77))
:= 77/7 + (777 − 77)
:= 7 × (77 + 7) + ((7 + 7)/7)7
:= (8 + 8)/8 + 8 × 88
:= 8 + (8 × 88 − 8/8)
:= 8 × 88 + (88 + 8)/8
:= 9 × 9 × 9 − ((99 + 99 + 9)/9)
:= 9 × 9 × 9 − (9 + 9)
:= 9 × 9 × 9 − ((99 + 9 + 9)/9)
Ï 707 := (11 − 1 − 1)1+1+1 − 11 − 11
6
Ï 712 := 1 + (1111 − (((1 + 1) × (11 − 1))1+1 ))
Ï 717 := ((1 + 1)11 ) − 111+1+1
:= 2/2 + (222 + 222 )
:= 2 × (((22 × 22+2 ) + 2) + 2)
:= 222 + (22/2 + 222 )
:= 3 + (33/3 × ((3/3 + 3)3 ))
:= 3/3 + (33+3 − (3 × (3 + 3)))
:= 33+3 − (3 × 3 + 3)
:= 4 + 4 × 4 × 44 − 4/4
:= 4 + 4 × 4 × 44 + 4
:= 4/4 + ((4 × ((4 × 44) + 4)) − 4)
5
5
5
:= 5 × 5 + ((5 + 5 + 5)/5 + 55)
:= 55 + (((5 + 5)/5) + 5 /5)
:= ((55 × (55 + 5 + 5) + 5) + 5)/5
:= 6 + ((666 − 6/6) + 6 × 6)
:= 6 × 6 + ((66 − 6)/6 + 666)
:= ((6 × 6/(6 + 6))6 ) − 6 − 6
:= 7 + (777 − 77)
:= 777 + ((77 + 7)/7 − 77)
:= 777 − (77/7 + 7 × 7)
:= 88/8 + (8 × 88 − 8)
:= 8 + 8 × 88
:= 8 × 88 + (88 + 8 + 8)/8
:= 9 × 9 × 9 − ((99 + 99)/9)
:= 9/9 + (9 × 9 × 9 − (9 + 9))
:= 9 × 9 × 9 − (99 + 9)/9
57
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 718 := (11 − 1 − 1)1+1+1 − 11
Ï 723 := (1 + 1 + 1) × (((1 + 1) × 111+1 ) − 1)
Ï 728 := (11 − 1 − 1)1+1+1 − 1
:= 2 × (((22 − (2/2 + 2))2 ) − 2)
:= (2/2 + 2) × (222 − 2)/2
:= 2 + (222 /2 + 222 )
:= 33+3 − 33/3
:= 33+3 − (3 + 3)
:= 33+3 − 3/3
:= (4 × ((4 × 44) + 4)) − (4 + 4)/4
:= 4 + ((4 × ((4 × 44) + 4)) − 4/4)
:= 4 + ((4 × ((4 × 44) + 4)) + 4)
:= 5 × 5 × (5 × 5 + 5) − ((5 + 5)/5)5
:= 5 × (5 × 5 − 5) + (55 − 5 − 5)/5
:= (55 + 5/5) × (55 + 5 + 5)/5
:= (6 + 6) × (66 − 6) − (6 + 6)/6
:= ((6 × 6/(6 + 6))6 ) − 6
:= ((6 × 6/(6 + 6))6 ) − 6/6
:= 7 + ((77/7 − 77) + 777)
:= 7 + (7 × (77 + 7) + ((7 + 7)/7)7 )
:= 777 − 7 × 7
:= 8 + ((8 × 88 − ((8 + 8)/8)) + 8)
:= 8 + (88/8 + 8 × 88)
:= 8 + (8 × 88 + 8 + 8)
:= 9 × 9 × 9 − 99/9
:= 9 × 9 × 9 + (((9 + 9 + 9)/9) − 9)
:= 9 × 9 × 9 − 9/9
Ï 719 := 1 + ((11 − 1 − 1)1+1+1 − 11)
Ï 724 := (1 + 1) × ((11 × (11 × (1 + 1 + 1))) − 1)
Ï 729 := (11 − 1 − 1)1+1+1
:= (2/2 + 2)2+2+2
:= ((22 − 2) × (2 + 2 + 2)2 ) − 2/2
:= 2 + (2 × ((22 − (2/2 + 2))2 ))
:= 33+3 + ((3 − 33)/3)
:= 3/3 + (33+3 − (3 + 3))
:= (4 × ((4 × 44) + 4)) − 4/4
:= 4 + (4 × ((4 × 44) + 4))
:= (4 − 4/4)4+(4+4)/4
:= (((55 + 5)(5+5)/5 ) − 5)/5
:= (55 − 5)/5 + 5 × (5 × 5 − 5)
:= (5 − (5 + 5)/5)5/5+5
:= (6 + 6) × (66 − 6) − 6/6
:= (66 × 66 − 6 − 6)/6
:= (6 × 6/(6 + 6))6
:= 777 − (((7 + 7)/7 + 7 × 7) + 7)
:= (7 × (7 × (7 + 7) + 7)) − 77/7
:= 7/7 + (777 − 7 × 7)
:= 8 + ((8 × 88 − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 8 × 88)
:= (8/8 + 8)88/8−8
:= 9 × 9 × 9 − 9/9 − 9
:= 9 × 9 × 9 + ((9 − 99)/(9 + 9))
:= 9 × 9 × 9
Ï 720 := (1 + 1) × ((1 + 1 + 1) × (111+1 − 1))
:= (22 − 2) × (2 + 2 + 2)2
Ï 725 := ((1 + 1) × (11 × (11 × (1 + 1 + 1)))) − 1
:= 33+3
Ï 730 := 1 + (11 − 1 − 1)1+1+1
:= 222 + (222 − 2)/2
:= 2/2 + ((2/2 + 2)2+2+2 )
:= 33+3 − (3/3 + 3)
:= 3/3 + 33+3
:= 4 × ((4 × 44) + 4)
:= 4 + ((4 × ((4 × 44) + 4)) + 4/4)
:= 4 + (44/4 × (((44 + 4) + 4)/4))
:= (5/5 + 5) × (5 × 5 × 5 − 5)
:= 5 × (5 × (5 × 5 + 5) − 5)
:= 5 + (5 × (5 × (5 × 5 + 5) − 5))
:= (6 + 6) × (66 − 6)
:= 3
3+3
−3×3
:= (66 × 66 − 6)/6
:= 6/6 + ((6 × 6/(6 + 6))6 )
:= 777 − ((7/7 + 7 × 7) + 7)
:= 777 − (((7 + 7 + 7)/7) + 7 × 7)
:= ((7 + 7)/7) + (777 − 7 × 7)
:= 8 + (8 × 88 + 8)
:= 8 + ((88 + 8 + 8)/8 + 8 × 88)
:= 8 + (((((8 + 8)/8) + 8 × 88) + 8) + 8)
:= 9 × 9 × 9 − 9
:= 9 × 9 × 9 + ((9 − 9 × 9)/(9 + 9))
:= 9/9 + 9 × 9 × 9
Ï 721 := (111 + 111+1+1 )/(1 + 1)
2
2
Ï 726 := (1 + 1) × (11 × (11 × (1 + 1 + 1)))
Ï 731 := 1 + (1 + (11 − 1 − 1)1+1+1 )
:= ((((2 + 2 + 2) + 2) ) − 2)/2
:= 22 × (22/2 + 22)
:= 2 + ((2/2 + 2)2+2+2 )
:= 3 + (33+3 − 33/3)
:= 33+3 − 3
:= 3 + (33+3 − 3/3)
:= 4/4 + (4 × ((4 × 44) + 4))
:= 44/4 × (((44 + 4) + 4)/4)
:= 44/4 + (4 × ((4 × 44) + 4))
:= 5/5 + (5 × (5 × (5 × 5 + 5) − 5))
:= ((555 + 55 )/5) − 5
:= 6/6 + (6 + 6) × (66 − 6)
:= 66 × (66/6)
:= 6 + ((66 × 66 − 6)/6)
:= 777 − (7 × 7 + 7)
:= (7 − 7/7) × (((7 + 7)/7)7 − 7)
:= 7 + ((7 × (7 × (7 + 7) + 7)) − (77/7))
:= 8 + ((8 × 88 + 8/8) + 8)
:= 8 × 88 + (88 + 88)/8
:= 8 + ((88/8 + 8 × 88) + 8)
:= 9/9 + (9 × 9 × 9 − 9)
:= 9 × 9 × 9 − (9 + 9 + 9)/9
:= 9 × 9 × 9 + ((9 + 9)/9)
:= (((55 + 5)
(5+5)/5
) + 5)/5
Ï 722 := (1 + 1) × ((((1 + 1) × (11 − 1)) − 1)1+1 )
:= 2 × ((22 − (2/2 + 2))2 )
Ï 727 := (11 − 1 − 1)1+1+1 − 1 − 1
Ï 732 := 1 + (1 + (1 + (11 − 1 − 1)1+1+1 ))
:= ((2/2 + 2)2+2+2 ) − 2
:= (2/2 + 2) × (222 /2 + 2)
:= 3/3 + (33+3 − 3)
:= 3 + 33+3
:= (4 + 4)/4 + (4 × ((4 × 44) + 4))
:= 4 + (((4 × ((4 × 44) + 4)) − 4/4) + 4)
:= 44 + (4 × ((4 × 44) − 4))
(5+5)/5
:= (5 + 5)/5 + (5 × (5 × (5 × 5 + 5) − 5))
:= (((555 + 55 ) + 5)/5) − 5
:= (66 × 66 + 6)/6
:= 66 + 666
:= 7/7 + (777 − (7 × 7 + 7))
:= 777 − (7/7 + 7 × 7)
:= 77/7 + (777 − (7 × 7 + 7))
:= 8 + ((((8 + 8)/8) + 8 × 88) + 8)
:= 8 + (((8 × 88 − 8/8) + 8) + 8)
:= 8 + ((((88 + 8)/8) + 8 × 88) + 8)
:= 9 × 9 × 9 + (((9 + 9)/9) − 9)
:= 9 × 9 × 9 − (9 + 9)/9
:= 9 × 9 × 9 + ((9 + 9 + 9)/9)
:= 3
3+3
− ((3/3 + 3) + 3)
:= ((((55 + 5)
) + 5) + 5)/5
:= (6 + 6)/6 + (6 + 6) × (66 − 6)
58
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 733 := 1 + (1 + (1 + (1 + (11 − 1 − 1)1+1+1 )))
Ï 738 := 1 + (11 × (1 + ((1 + 1) × (11 × (1 + 1 + 1)))))
Ï 743 := 1 + (1 + (1 + (11 + (11 − 1 − 1)1+1+1 )))
:= 2 + (((2/2 + 2)2+2+2 ) + 2)
:= 2 + (22+2 × ((2 × 22) + 2))
:= 2 + ((22×(2+2) + 222 ) + 2/2)
:= 3 + (33+3 + 3/3)
:= 3 × 3 + 33+3
:= 3 + (33+3 + 33/3)
:= 4 + ((4 − 4/4)4+(4+4)/4 )
:= 44 + ((44 × 44 − (4 + 4))/4)
:= 4 + (((44 × 44 − 4)/4) + 44 )
:= 55 + ((55 − 5 − 5)/5 + 55)
:= (((555 + 55 ) + 5) + 5)/5
:= 5 + ((((555 + 55 ) + 5) + 5)/5)
:= 6 + ((66 × 66 + 6)/6)
:= 6 + (666 + 66)
:= 6 + ((66/6) × (66 + 6/6))
:= (7 × (7 × (7 + 7) + 7)) − (7 + 7)/7
:= 777 + (((77 − 7)/7) − 7 × 7)
:= 7 + ((7 × (7 × (7 + 7) + 7)) + 7/7)
:= 8 + (((88 + 8 + 8)/8 + 8 × 88) + 8)
:= (8/8 + 8) × ((((8 + 8)/8) − 8) + 88)
:= 888/8 + (8 × (88 − 8) − 8)
:= 9 × 9 × 9 + ((9 × 9 − 9)/(9 + 9))
:= 9 + 9 × 9 × 9
:= 9 + (((9 × 9 + 9)/(9 + 9)) + 9 × 9 × 9)
Ï 734 := (1 + 1) × (1 + ((1 + 1 + 1) × (1 + 111+1 )))
2
Ï 739 := 11 + ((11 − 1 − 1)1+1+1 − 1)
Ï 744 := (1 + 11) × (1 + ((1 + 111+1 )/(1 + 1)))
:= 222 + (2(2/2+2) )
:= 222 + (22×(2+2) − 2/2)
:= (2 × 222 ) − (222 + 2)
:= 3 + ((33+3 − 3/3) + 3)
:= 3 × 3 + (33+3 + 3/3)
:= 3 + ((33+3 + 3 × 3) + 3)
:= (4 + 4)/4 × (444/4 + 44 )
:= 44 + ((44 × 44 − 4)/4)
:= 44 + (444 + 44 )
:= 5 + ((5 − (5 + 5)/5)5/5+5 )
:= 5 × 5 × 5 + ((55 − 55)/5)
:= 5 × 5 × 5 + ((55 − 5)/5 − 5)
:= 6 + (((6 × 6/(6 + 6))6 ) − 6/6)
:= 6 + (((66 × 66 + 6)/6) + 6)
:= 6 + ((666 + 66) + 6)
:= (7 × (7 × (7 + 7) + 7)) − 7/7
:= 77/7 + (777 − 7 × 7)
:= 7 + ((7 × (7 × (7 + 7) + 7)) + ((7 + 7)/7))
:= 8 + ((88 + 88)/8 + 8 × 88)
:= 8 + (((88/8 + 8 × 88) + 8) + 8)
:= 8 × (88 + 8) − 8 − 8 − 8
:= 9 × 9 × 9 + ((9 × 9 + 9)/(9 + 9))
:= 9 + (9 × 9 × 9 + 9/9)
:= 9 + ((9 × 9 × 9 − ((9 + 9 + 9)/9)) + 9)
Ï 735 := (1 + 1 + 1) × (1 + ((1 + 1) × (1 + 111+1 )))
Ï 740 := 11 + (11 − 1 − 1)1+1+1
Ï 745 := (1 + (1 + (1 + ((1 + 1)1+1+11 ))))/11
:= 2 + ((((2/2 + 2)2+2+2 ) + 2) + 2)
:= 222 + 22×(2+2)
:= 22+2 + ((2/2 + 2)2+2+2 )
:= 3 + (33+3 + 3)
:= 33+3 + 33/3
:= 33 + (33+3 − 33/3)
:= (4 − 4/4) × (44 − 44/4)
:= 4 + (4 × (((4 × 44) + 4) + 4))
:= 4 + (((44 × 44 + 4)/4) + 44 )
:= 55 + (55 /5 + 55)
:= 5 × 5 × (5 × 5 + 5) − 5 − 5
:= 5 × 5 × (5 × 5 + 5) − 5
:= 66/6 + ((6 × 6/(6 + 6))6 )
:= 6 + ((((66 × 66 + 6)/6) + 6) + 6)
:= 7 + ((7 × (7 × (7 + 7) + 7)) − ((7 + 7)/7))
:= 777 − ((77/7 + 7 + 7) + 7)
:= 8 × 88 + ((8 × 8 + 8)/((8 + 8)/8))
:= 8 + (((8/8 + 8)88/8−8 ) + 8)
:= 9 × 9 × 9 + (99/9)
:= 9 + ((9 × 9 × 9 − ((9 + 9)/9)) + 9)
:= 6 + ((6 × 6/(6 + 6))6 )
:= 7 × (7 × (7 + 7) + 7)
:= 8 + ((((8 × 88 − 8/8) + 8) + 8) + 8)
:= 9 + (9 × 9 × 9 − ((9 + 9 + 9)/9))
Ï 736 := 111 + (1 + (1 + 1) × (1 + 11))1+1
:= 22+2 × ((2 × 22) + 2)
:= 3 + ((33+3 + 3/3) + 3)
:= 4 × (((4 × 44) + 4) + 4)
Ï 741 := 1 + (11 + (11 − 1 − 1)1+1+1 )
:= 2/2 + (2
:= 3 + (3
2×(2+2)
3+3
2
+ 22 )
:= (2 × 222 ) − 222
:= 3 + ((33+3 + 33/3) + 3)
+ 3 × 3)
4
:= 4 + ((44 × 44 + 4)/4)
5
:= (555 + 55 )/5
Ï 746 := (1 + 1) × (((11 + 11)1+1 ) − 111)
:= 44 + 4 × 4 × 44 − (4 + 4)/4
:= 5 + (((555 + 55 )/5) + 5)
:= 5 + ((555 + 5 )/5)
6
:= 6 + (((6 × 6/(6 + 6))6 ) + 6/6)
:= 6 + (((6 × 6/(6 + 6)) ) + 6)
:= 6 + (((6 × 6/(6 + 6))6 ) + (66/6))
:= 7/7 + (7 × (7 × (7 + 7) + 7))
:= 7 + ((7 × (7 × (7 + 7) + 7)) − 7/7)
:= 77/7 + (7 × (7 × (7 + 7) + 7))
:= 8 + ((8 × 88 + 8 + 8) + 8)
:= 8 × 88 + 888/(8 + 8 + 8)
:= 8 × (88 + 8) − (88 + 88)/8
:= 9 + (9 × 9 × 9 − ((9 + 9)/9))
:= 9 × 9 × 9 + (99 + 9)/9
:= 9 + ((9 × 9 × 9 − 9/9) + 9)
Ï 737 := 11 × (1 + ((1 + 1) × (11 × (1 + 1 + 1))))
:= 2/2 + (2
2+2
× ((2 × 22) + 2))
Ï 742 := 1 + (1 + (11 + (11 − 1 − 1)1+1+1 ))
:= 2 + (2
2×(2+2)
2
+ 22 )
Ï 747 := ((1 + 1) × (11 × (1 + (11 × (1 + 1 + 1))))) − 1
:= 2/2 + ((2 × 222 ) − 222)
:= 3 × 3 + (33+3 − 3/3)
:= 3 + ((33+3 + 3/3) + 3 × 3)
:= 3 × ((3 + 3)3 + 33)
:= 4/4 + (4 × (((4 × 44) + 4) + 4))
:= 44 + (((44 × 44 + 4) + 4)/4)
:= 44 + 4 × 4 × 44 − 4/4
:= ((555 + 5 ) + 5)/5
:= 5 + (((555 + 5 ) + 5)/5)
:= ((555 + 55) + 55 )/5
:= (66/6) × (66 + 6/6)
:= 6 + ((((6 × 6/(6 + 6))6 ) + 6/6) + 6)
:= 6 + ((((6 × 6/(6 + 6))6 ) + 6) + 6)
:= ((7 + 7)/7) + (7 × (7 × (7 + 7) + 7))
:= 7 + (7 × (7 × (7 + 7) + 7))
:= (77 + 7)/7 + (7 × (7 × (7 + 7) + 7))
:= 8 + (((88 + 88)/8 + 8 × 88) + 8)
:= (8/8 + 8) × ((88/8 + 8 × 8) + 8)
:= 9 × 9 × 9 + ((99 + 9 + 9)/9)
:= 9 + (9 × 9 × 9 + 9)
5
5
:= 8 + ((8/8 + 8)
88/8−8
)
:= 9 + (9 × 9 × 9 − 9/9)
59
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 748 := (1 + 1) × (11 × (1 + (11 × (1 + 1 + 1))))
:= 22 × ((2 × 2
2+2
Ï 753 := (((1 + 11)1+1+1 )/(1 + 1)) − 111
:= 2 + (((2/2 + 2)
) + 2)
2+2+2
) + 22)
Ï 758 := (11 × ((1 + 1 + 1) × (1 + (11 + 11)))) − 1
:= 2 + (2 × ((22 − 2)2 − 22))
:= 3/3 + (33+3 + (3 × (3 + 3)))
:= 33 + (33+3 − 3)
:= 3 + ((33+3 − 3/3) + 33 )
:= 44 + 4 × 4 × 44
:= 4/4 + (4 × (444 − 44 ))
:= (4 + 4)/4 + ((4 − 4/4) × (44 − 4))
:= 5 × 5 × 5 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5 × 5 × 5)
:= 5 + (((55 − 5 − 5)/5 + 5 × 5 × 5) + 5)
:= (66/6) × (((6 + 6)/6) + 66)
:= 6 + (((((6 × 6/(6 + 6))6 ) + 6) + 6) + 6)
:= (6 + 6)/6 + (6 × ((66 − 6) + 66))
:= 7 + (((7 × (7 × (7 + 7) + 7)) − 7/7) + 7)
:= 7 + ((7 × (7 × (7 + 7) + 7)) + (77/7))
:= 777 − ((77 + 7)/7 + 7)
:= 8 × 88 + (88/((8 + 8)/8))
:= 8/8 + (8 × (88 + 8) − (8 + 8))
:= (8 − 88)/8 + 8 × (88 + 8)
:= 9 + ((9 × 9 × 9 + 9/9) + 9)
:= 9 + (((9 × 9 × 9 − ((9 + 9 + 9)/9)) + 9) + 9)
:= 9 + ((9 × 9 × 9 + (99/9)) + 9)
Ï 749 := 1 + ((1 + 1) × (11 × (1 + (11 × (1 + 1 + 1)))))
2+2+2
Ï 754 := 1 + ((((1 + 11)1+1+1 )/(1 + 1)) − 111)
Ï 759 := 11 × ((1 + 1 + 1) × (1 + (11 + 11)))
:= (2 × ((22 − 2)2 − 22)) − 2
:= (2/2 + 2) × ((222 + 22)/2)
:= 3 × 3 + (33+3 + 33/3)
:= 33 + ((33+3 − 3) + 3/3)
:= 3 + (33+3 + 33 )
:= 44 + 4 × 4 × 44 + 4/4
:= (4 + 4)/4 + (4 × (444 − 44 ))
:= (4 − 4/4) × ((4/4 − 4) + 44 )
:= 5 × 5 × 5 + (55 − 5)/5
:= 5 + ((55 − 5)/5 + 5 × 5 × 5)
:= 5 + (((55 − 5)/5 + 5 × 5 × 5) + 5)
:= 6 + (((66/6) × (66 + 6/6)) + 6)
:= 6 + ((66/6) × (((6 + 6)/6) + 66))
:= 6 × 6 + (((6 × 6/(6 + 6))6 ) − 6)
:= 22 + (((2/2 + 2)
) − 2)
:= 7 + ((7 × (7 × (7 + 7) + 7)) + 7)
:= 777 − ((((7 + 7)/7 + 7) + 7) + 7)
:= 777 − (77/7 + 7)
:= 8 × (88 + 8) − (88/8 + 8)
:= (8 + 8)/8 + (8 × (88 + 8) − (8 + 8))
:= 8 × (88 + 8) − (8/8 + 8)
:= 9 + (9 × 9 × 9 + (99/9))
:= 9 + (((9 × 9 × 9 − ((9 + 9)/9)) + 9) + 9)
:= 9 × (9 × 9 − 9) + 999/9
Ï 750 := (1 + 1) × (1 + (11 × (1 + (11 × (1 + 1 + 1)))))
2+2
Ï 755 := 111+1+1 − (((1 + 1) × (1 + 11))1+1 )
Ï 760 := 1 + (11 × ((1 + 1 + 1) × (1 + (11 + 11))))
:= (2 × ((22 − 2)2 − 22)) − 2/2
:= 2 × (((22 − 2)2 − 22) + 2)
:= 3 + (33+3 + (3 × (3 + 3)))
:= 33 + (33+3 − 3/3)
:= 3 + ((33+3 + 3/3) + 33 )
:= 44 + 4 × 4 × 44 + (4 + 4)/4
:= ((4 − 4/4) × (44 − 4)) − 4/4
:= 4 + ((4 − 4/4) × (44 − 4))
:= 5 × 5 × (5 × 5 + 5)
:= 5 + 5 × 5 × (5 × 5 + 5)
:= 5 + (5 × 5 × (5 × 5 + 5) + 5)
:= 66 × (6 + 6) − (6 × 6 + 6)
:= 66 × (6 + 6) − (6 × 6 + 6/6)
:= 6 × 6 + ((66 × 66 − 6 − 6)/6)
:= (7/7 + 7 + 7) × (7/7 + 7 × 7)
:= 2 + (22 × ((2 × 2
) + 2))
:= 777 − (7/7 + 7 + 7 + 7)
:= 777 + (((7 − 77)/7) − 7)
:= (8 − 88)/8 + (8 × (88 + 8) − 8)
:= 8 × (88 + 8) − (88 + 8 + 8)/8
:= 8 × (88 + 8) − 8
:= 9 + (((99 + 9)/9) + 9 × 9 × 9)
:= 9 + (((9 × 9 × 9 − 9/9) + 9) + 9)
:= 9 + (((99 + 99)/9) + 9 × 9 × 9)
Ï 751 := 11 + (11 + (11 − 1 − 1)1+1+1 )
2+2+2
Ï 756 := (1 + 1 + 1) × ((1 + 11) × (11 + (11 − 1)))
Ï 761 := 1 + (1 + (11 × ((1 + 1 + 1) × (1 + (11 + 11)))))
:= 2 × ((22 − 2)2 − 22)
:= 2 + ((2/2 + 2) × ((222 + 22)/2))
:= 33 + 33+3
:= 33 + (33+3 − 3/3)
:= (4 × (4 − 4)) − (4/4 + 4 )
:= (4 − 4/4) × (44 − 4)
:= ((4 − 4/4) × (44 − 4/4)) − 4
:= 5/5 + 5 × 5 × (5 × 5 + 5)
:= 5 + (5 × 5 × (5 × 5 + 5) + 5/5)
:= 5 × 5 + ((555 + 55 )/5)
:= 22 + ((2/2 + 2)
:= 33 + (3
3+3
)
− 33/3)
4
4
:= 6 × 6 + ((66/6) × (66 − 6/6))
:= 6 × ((66 − 6) + 66)
:= 6 × 6 + ((66 × 66 − 6)/6)
:= 777 − ((77 + 7)/7 + 7 + 7)
:= 777 − (7 + 7 + 7)
:= 777 − (((7 + 7)/7 + 7) + 7)
:= 888/8 + 8 × (88 − 8)
:= 8 × (88 + 8) − (88 + 8)/8
:= 8/8 + (8 × (88 + 8) − 8)
:= 9 × 9 × 9 + ((99 + 99)/9)
:= 9 + ((9 × 9 × 9 + 9) + 9)
:= ((99/9) × (9 × 9 − (99/9))) − 9
Ï 752 := 1 + (11 + (11 + (11 − 1 − 1)1+1+1 ))
2
Ï 757 := 1 + ((1 + 1 + 1) × ((1 + 11) × (11 + (11 − 1))))
Ï 762 := (1 + 1 + 1) × (1 + (11 × (1 + (11 + 11))))
:= 2 × ((22 − 2) − (22 + 2))
:= 2/2 + (2 × ((22 − 2) − 22))
:= (2/2 + 2) × (22×(2+2) − 2)
:= 33 + (33+3 − (3/3 + 3))
:= 33 + (33+3 + 3/3)
:= 33 + 33+3
:= 4 × (444 − 44 )
:= 4/4 + ((4 − 4/4) × (44 − 4))
:= (4 − 4/4) × (44 − (4 + 4)/4)
:= 5 × 5 × 5 + (55 + 5 + 5)/5
:= 5 + (5 × 5 × (5 × 5 + 5) + ((5 + 5)/5))
:= 5 × 5 + (((555 + 55 ) + 5)/5)
:= 6 + ((((6 × 6/(6 + 6))6 ) + (66/6)) + 6)
:= 6/6 + (6 × ((66 − 6) + 66))
:= 6 + (6 × ((66 − 6) + 66))
:= 777 − (77/7 + 7 + 7)
:= 7/7 + (777 − (7 + 7 + 7))
:= 777 − (7/7 + 7 + 7)
:= 8 × (88 + 8) − 8 − 8
:= 8 × (88 + 8) − 88/8
:= (8 + 8)/8 + (8 × (88 + 8) − 8)
:= 9 × 9 × 9 + ((99 + 99 + 9)/9)
:= 9 + (((9 × 9 × 9 + 9/9) + 9) + 9)
:= 9 × 9 × 9 + (99/((9 + 9 + 9)/9))
2
60
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 763 := (111 − 1 − 1) × (1 + ((1 + 1) × (1 + 1 + 1)))
Ï 768 := (1 + 1 + 1) × (1 + 1)(1+1)
1+1+1
Ï 773 := ((1 + ((1 + 1 + 1)1+1+1 ))1+1 ) − 11
:= 2/2 + ((2/2 + 2) × (22×(2+2) − 2))
:= 2 × (22+2 × (22 + 2))
:= (2 × 22) + ((2/2 + 2)2+2+2 )
:= 3/3 + (33+3 + 33)
:= 3 + ((33+3 + 33) + 3)
:= 33 + (33+3 + 33/3)
:= 4 × 44 − ((4/4 + 44 ) + 4)
:= 4 × (4 × (44 + 4))
:= 4 + ((4 × (4 × (44 + 4))) + 4/4)
:= 555 + ((55 − 5)/(5 + 5 + 5))
:= (5 × 5 − 5/5) × ((5 + 5)/5)5
:= 5 + ((5 × 5 − 5/5) × ((5 + 5)/5)5 )
:= 6 × 6 + ((66 × 66 + 6)/6)
:= (6 + 6) × ((6 + 6)/6)6
:= 6 + (((6 + 6) × ((6 + 6)/6)6 ) − 6/6)
:= 777 − (7 + 7)
:= (7 − 7/7) × ((7 + 7)/7)7
:= 7 + (777 − (77/7))
:= 88/8 + (8 × (88 + 8) − (8 + 8))
:= 8 × (88 + 8)
:= 8 + ((8 × (88 + 8) − (88/8)) + 8)
:= (9 − ((9 + 9)/9)) × (9/9 + 99 + 9)
:= 9 + (9 × (9 × 9 − 9) + 999/9)
:= 99 + ((((9 + 9)/9)9 ) + 9 × (9 + 9))
Ï 764 := 1 + ((111 − 1 − 1) × (1 + ((1 + 1) × (1 + 1 + 1))))
Ï 769 := 1 + (1 + 1 + 1) × (1 + 1)(1+1)
1+1+1
Ï 774 := 1 + (((1 + ((1 + 1 + 1)1+1+1 ))1+1 ) − 11)
:= 2 × ((2
2+2
× (22 + 2)) − 2)
:= 2/2 + (2 × (22+2 × (22 + 2)))
:= (2/2 + 2) × (22×(2+2) + 2)
:= 3 + ((3
3+3
− 3/3) + 33)
:= 3 + (((33+3 + 3/3) + 33) + 3)
:= 3 + ((33+3 + 33) + 3 × 3)
:= 4/4 + (4 × (4 × (44 + 4)))
:= (4 − 4/4) × ((4 + 4)/4 + 44 )
:= (5 × (5 − 55)) + ((5 − 5/5)5 − 5)
:= (5 × (5 − 55)) + (5 − 5/5)5
:= 6/6 + ((6 + 6) × ((6 + 6)/6)6 )
:= 6 + ((6 + 6) × ((6 + 6)/6)6 )
:= 7/7 + (777 − (7 + 7))
:= 777 − (7/7 + 7)
:= 777 − (7 + 7 + 7)/7
:= 8 × (88 + 8) − (8/((8 + 8)/8))
:= 8/8 + 8 × (88 + 8)
:= 8 + (8 × (88 + 8) − ((8 + 8)/8))
:= 9 + ((((9 × 9 × 9 − 9/9) + 9) + 9) + 9)
:= 9 × 9 × 9 + ((9 × 9 × 9 − 9)/(9 + 9))
:= 9 × 99 − (99 + 9 + 9)
4
4
:= 4 × 4 − (4 + 4)
5
:= 5 × (5 × 5 + 5) + ((5 − 55)/5)
6
:= 6 × 6 + (((6 × 6/(6 + 6)) ) − 6/6)
Ï 765 := (1 + 1 + 1) × (111 + ((1 + 11)1+1 ))
:= (2/2 + 2) × (2
:= 3 + (3
3+3
2×(2+2)
− 2/2)
Ï 770 := 11 × ((11 − 1 − 1)1+1 − 11)
:= 2 + (2 × (2
+ 33)
2+2
× (22 + 2)))
Ï 775 := 1111 − ((1 + 1 + 1) × (1 + 111))
:= 2 + (((22+2 + 2/2)2 ) + 222 )
:= 3 + ((33+3 + 33/3) + 33 )
:= 3333/3 − (333 + 3)
:= (4 + 4)/4 + (4 × (4 × (44 + 4)))
:= 4 + ((4 − 4/4) × (4/4 + 44 ))
:= 55 × ((5 − 5/5 + 5) + 5)
:= 5 × (5 × (5 × 5 + 5) + 5)
:= 6 × 6 + ((6 × 6/(6 + 6)) )
:= (6/6 + 6) × ((666 − 6)/6)
:= 66 × (6 + 6) − ((66/6) + 6)
:= 777 − (77 + 7)/7
:= 777 − 7
:= 777 − (7 + 7)/7
:= 8 + (8 × (88 + 8) − (88/8))
:= (8 + 8)/8 + 8 × (88 + 8)
:= 8 + (8 × (88 + 8) − 8/8)
:= 9 + (((9 × 9 × 9 + 9) + 9) + 9)
:= 99/9 × (9 × 9 − 99/9)
:= ((9 − 9/9) × (99 − 9/9)) − 9
4
:= (4 − 4/4) × (4 − 4/4)
:= 5 + ((5 × 5 × (5 × 5 + 5) + 5) + 5)
6
Ï 766 := (111 × (1 + ((1 + 1) × (1 + 1 + 1)))) − 11
:= (2 × (2
2+2
× (22 + 2))) − 2
Ï 771 := 1 + (11 × ((11 − 1 − 1)1+1 − 11))
2
2
:= 22 + ((((22 + 2) ) − 2)/2)
:= 3 + ((33+3 + 3/3) + 33)
:= 3 × 3 + (3
:= 4 × 44 − ((4 + 4)/4 + 44 )
3+3
:= 5 + (((555 + 5 )/5) + 5 × 5)
6
:= 6 × 6 + (((6 × 6/(6 + 6)) ) + 6/6)
:= 2 × (2 × (((22+2 − 2)2 ) − 2))
:= 3 + ((33+3 + 33/3) + 33)
+ 33)
4
5
Ï 776 := (111 × (1 + ((1 + 1) × (1 + 1 + 1)))) − 1
:= (4 − 4/4) × (4/4 + 4 )
:= 4 + ((4 × (4 × (44 + 4))) + 4)
:= 5/5 + (55 × ((5 − 5/5 + 5) + 5))
:= 5/5 + (5 × (5 × (5 × 5 + 5) + 5))
6
:= 6 + (((6 × 6/(6 + 6)) ) + 6 × 6)
:= 666 + (666 − 6)/6
:= 777 − 77/7
:= 7/7 + (777 − 7)
:= 777 − 7/7
:= 8 × (88 + 8) − (8 + 8)/8
:= 88/8 + (8 × (88 + 8) − 8)
:= 8 + 8 × (88 + 8)
:= 9 + ((((9 × 9 × 9 + 9/9) + 9) + 9) + 9)
:= 9 × 99 − (999/9 + 9)
:= (9 − 9/9) × (99 − ((9 + 9)/9))
Ï 767 := (1 + 1 + 1) × (1 + 1)(1+1)
:= (2 × 2 × 222) − (22/2)
3
3+3
1+1+1
−1
2
Ï 772 := 1 + (1 + (11 × ((11 − 1 − 1)1+1 − 11)))
:= 2 × ((2
2+2
× (22 + 2)) + 2)
Ï 777 := 111 × (1 + ((1 + 1) × (1 + 1 + 1)))
:= 222/2 × ((2/2 + 2 + 2) + 2)
:= 3 × 3 + ((33+3 + 3/3) + 33)
:= (33 × (33 + 3)) − 33
:= 4 + (4 × (4 × (44 + 4)))
:= (4 − 4/4) × ((44 − 4/4) + 4)
:= (((5 + 5)/5 + 5) × 555/5) − 5
:= ((5 + 5)/5 + 5) × 555/5
:= ((6 + 6) × ((6 + 6)/6) ) − 6/6
:= 666 + ((666 + 6)/6 − 6)
:= 666 + 666/6
:= 777 + (7 − 77)/7
:= ((7 + 7)/7) + (777 − 7)
:= 777
:= 8 × (88 + 8) − 8/8
:= 8 × (88 + 8) + (8/((8 + 8)/8))
:= 8 + (8 × (88 + 8) + 8/8)
:= 9 + (((9 × 9 × 9 + (99/9)) + 9) + 9)
:= 9 × 99 + (((9 − 999)/9) − 9)
:= (9 − ((9 + 9)/9)) × 999/9
:= 3 + (3
+ 33/3)
:= 4 × 44 − (4/4 + 44 )
5
:= 5 + ((((555 + 5 ) + 5)/5) + 5 × 5)
6
61
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 778 := 1 + (111 × (1 + ((1 + 1) × (1 + 1 + 1))))
2
Ï 783 := ((1 + ((1 + 1 + 1)1+1+1 ))1+1 ) − 1
2
Ï 788 := 11 + (111 × (1 + ((1 + 1) × (1 + 1 + 1))))
:= ((22 + 2 + 2 + 2) ) − 2/2
:= 2 + (((22 + 2 + 2 + 2)2 ) + 2)
:= 3 × (3 × (3 × 33 + 3 + 3))
:= (33 × (33 − 3)) − (3/3 + 3)
:= 4 + ((4 − 4/4) × ((4 + 4)/4 + 4 ))
:= (4 − 4/4) × ((4/4 + 44 ) + 4)
:= 4 + (4 × ((4 × (44 + 4)) + 4))
:= ((55 − 5)/(5 − 5/5)) − (5 + 5)/5
:= (((5 + 5)/5 + 55 ) + 5)/(5 − 5/5)
:= 5 + ((((5 + 5)/5 + 55 ) + 5)/(5 − 5/5))
:= 666 + (666 + 6)/6
:= 6 + (666/6 + 666)
:= (6 + 6)/6 + (66 × (6 + 6) − 6)
:= 7/7 + 777
:= 7 + (777 − 7/7)
:= 77/7 + 777
:= 8 + (8 × (88 + 8) + ((8 + 8)/8))
:= (8/8 + 8) × (88 − 8/8)
:= 8 + (((88 + 8)/8) + 8 × (88 + 8))
:= 9 × 9 × 9 + ((9 × 99 − 9)/(9 + 9))
:= 9 × 99 − (99 + 9)
:= 9 + (9 × 99 − ((999 + 9)/9))
:= (2 × (22 − 2) ) − 22
:= 3333/3 − 333
4
Ï 779 := 1 + (1 + (111 × (1 + ((1 + 1) × (1 + 1 + 1)))))
2
Ï 784 := (1 + ((1 + 1 + 1)1+1+1 ))1+1
Ï 789 := (1 + 1 + 1) × (((1 + 1) × (11 × (1 + 11))) − 1)
:= 2/2 + ((2 × (22 − 2) ) − 22)
:= (22 + 2 + 2 + 2)2
:= (2 × (22 − 2)2 ) − 22/2
:= (33 × 33 ) − ((333 + 3)/3)
:= (33 + 3/3)3−3/3
:= (33 × (33 − 3)) − 3
:= 44/4 + (4 × (4 × (44 + 4)))
:= 4 × ((4 × (44 + 4)) + 4)
:= 4 + ((4 × ((4 × (44 + 4)) + 4)) + 4/4)
:= ((5 − 5)/(5 − 5/5)) − 5/5
:= (55/5 + 55 )/(5 − 5/5)
:= 5 + ((55/5 + 55 )/(5 − 5/5))
:= 66 × (6 + 6) − (6/6 + 6 + 6)
:= (6/6 + 6) × (666 + 6)/6
:= 66 + (((6 × 6/(6 + 6))6 ) − 6)
:= ((7 + 7)/7) + 777
5
:= 7 + 777
:= 777 + (77 + 7)/7
:= 88/8 + 8 × (88 + 8)
:= 8 + (8 × (88 + 8) + 8)
:= 888 − (88/8 + 88)
:= 9 × 99 − ((999 + 9)/9)
:= (9 − 9/9) × (99 − 9/9)
:= 9 + (9 × 99 − 999/9)
Ï 780 := (11 − 1) × (111 − (11 × (1 + 1 + 1)))
2
Ï 785 := 1 + ((1 + ((1 + 1 + 1)1+1+1 ))1+1 )
Ï 790 := (1 + 1) × ((11 × ((1 + 1 + 1) × (1 + 11))) − 1)
:= 2 + ((2 × (22 − 2) ) − 22)
:= 2/2 + ((22 + 2 + 2 + 2) )
:= (22 × (2 + 2 + 2)2 ) − 2
:= (33 + 3) × (33 − 3/3)
:= 3/3 + ((33 + 3/3)3−3/3 )
:= 3/3 + ((33 × (33 − 3)) − 3)
:= (4 − 4/4) × (44 + 4)
:= 4/4 + (4 × ((4 × (44 + 4)) + 4))
:= 4 + ((4 × ((4 × (44 + 4)) + 4)) + (4 + 4)/4)
5
2
:= (5 − 5)/(5 − 5/5)
:= 5 + ((5 − 5)/(5 − 5/5))
:= 5 + (((55 − 5)/(5 − 5/5)) + 5)
5
:= (6 + 6) × (66 − 6/6)
:= 66 × (6 + 6) − 6/6 − 6
:= 66 × (6 + 6) − (6 + 6)/6
:= 777 + (7 + 7 + 7)/7
:= 7 + (777 + 7/7)
:= 7 + ((777 − 7/7) + 7)
:= ((88 + 8)/8) + 8 × (88 + 8)
:= 8 + ((8 × (88 + 8) + 8/8) + 8)
:= 88 + (8 × 88 − ((8 + 8)/8))
:= 9 × 99 − 999/9
:= 9 + ((9 − 9/9) × (99 − ((9 + 9)/9)))
:= 9 × 99 − ((9 + 9)/9 + 99)
Ï 781 := 11 × ((((1 + 11)1+1 )/(1 + 1)) − 1)
Ï 786 := 1 + (1 + ((1 + ((1 + 1 + 1)1+1+1 ))1+1 ))
Ï 791 := (11 × (((1 + 11)1+1 )/(1 + 1))) − 1
:= ((22 + 2 + 2 + 2)2 ) − 2/2 − 2
:= 2 + ((22 + 2 + 2 + 2)2 )
:= (22 × (2 + 2 + 2)2 ) − 2/2
:= 33/3 × (((3 + 3)3 − 3)/3)
:= (33 × (33 − 3)) − (3 + 3)
:= (33 × (33 − 3)) − 3/3
:= 4/4 + ((4 − 4/4) × (44 + 4))
:= (4 + 4)/4 + (4 × ((4 × (44 + 4)) + 4))
:= 44/4 + ((4 − 4/4) × (44 + 4))
:= (55 − 5/5)/(5 − 5/5)
:= 5 + ((55 − 5/5)/(5 − 5/5))
:= 55 + ((555 + 55 )/5)
:= 66 × (6 + 6) − 66/6
:= 66 × (6 + 6) − 6
:= 66 × (6 + 6) − 6/6
:= 77/7 + (777 − 7)
:= 7 + (((7 + 7)/7) + 777)
:= 7 + (777 + 7)
:= 88 + (8 × 88 − (88/8))
:= 8 + ((8 × (88 + 8) + ((8 + 8)/8)) + 8)
:= 88 + (8 × 88 − 8/8)
:= 9 × 99 + ((9 − 999)/9)
:= 9 + ((9 − ((9 + 9)/9)) × 999/9)
:= 9 × 99 − (9/9 + 99)
Ï 782 := 1 + (11 × ((((1 + 11)1+1 )/(1 + 1)) − 1))
Ï 787 := 111 + (((1 + 1) × (1 + (1 + 11)))1+1 )
Ï 792 := 11 × (((1 + 11)1+1 )/(1 + 1))
:= ((22 + 2 + 2 + 2)2 ) − 2
:= 2 + (((22 + 2 + 2 + 2)2 ) + 2/2)
:= 22 × (2 + 2 + 2)2
:= 33 + ((33+3 − 3/3) + 33 )
:= 3 + ((33 + 3/3)3−3/3 )
:= 33 × (33 − 3)
:= (4 + 4)/4 + ((4 − 4/4) × (44 + 4))
:= 4 + ((4 − 4/4) × ((4/4 + 44 ) + 4))
:= 44 × ((4 + 4)/4 + 4 × 4)
:= 5 + (((5 + 5)/5 + 5) × 555/5)
:= 5 + ((((5 + 5)/5 + 5) × 555/5) + 5)
:= 55 + (((555 + 55 ) + 5)/5)
:= ((6 − 66)/6) + 66 × (6 + 6)
:= 6/6 + (66 × (6 + 6) − 6)
:= 66 × (6 + 6)
:= 7 + (777 − ((7 + 7)/7))
:= 777 + (77 − 7)/7
:= 7 + ((777 + 7/7) + 7)
:= 8 + ((8 × (88 + 8) − ((8 + 8)/8)) + 8)
:= 8 + (8 × (88 + 8) + (88/8))
:= 88 + 8 × 88
:= 9 × 99 − (9/9 + 99 + 9)
:= 9999/9 − (9 + 9) × (9 + 9)
:= 99 × (9 − 9/9)
62
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 793 := 1 + (11 × (((1 + 11)1+1 )/(1 + 1)))
Ï 798 := (1 + 1) × ((((1 + 1) × (11 − 1))1+1 ) − 1)
Ï 803 := 11 × (1 + (((1 + 11)1+1 )/(1 + 1)))
:= 2/2 + (22 × (2 + 2 + 2)2 )
:= (2 × (22 − 2)2 ) − 2
:= 2 + ((2 × (22 − 2)2 ) + 2/2)
:= 3/3 + (33 × (33 − 3))
:= 3 + ((33 × (33 − 3)) + 3)
:= 33/3 + (33 × (33 − 3))
:= 4/4 + (44 × ((4 + 4)/4 + 4 × 4))
:= (4 − 4/4) × ((44 − 4)/4 + 44 )
:= 4 + (((4 × 4 + 4) × (44 − 4)) − 4/4)
:= 555 + (((5 − (5 + 5)/5)5 ) − 5)
:= 555 + ((5 − (5 + 5)/5)5 )
:= 5 + (((5 − (5 + 5)/5)5 ) + 555)
:= 6/6 + 66 × (6 + 6)
:= 6 + 66 × (6 + 6)
:= 66/6 + 66 × (6 + 6)
:= 7 + ((((7 + 7)/7) + 777) + 7)
:= 7 + (777 + 7 + 7)
:= 7 + (((77 + 7)/7 + 777) + 7)
:= 8/8 + (8 × 88 + 88)
:= 888 − ((8 + 8)/8 + 88)
:= 88 + (88/8 + 8 × 88)
:= 9/9 + (99 × (9 − 9/9))
:= 9 × (9 × 9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 × (9 × 9 + 9) − 9)
Ï 794 := 1 + (1 + (11 × (((1 + 11)1+1 )/(1 + 1))))
Ï 799 := ((1 + 1) × (((1 + 1) × (11 − 1))1+1 )) − 1
Ï 804 := 1 + (11 × (1 + (((1 + 11)1+1 )/(1 + 1))))
:= 2 + (22 × (2 + 2 + 2)2 )
:= (2 × (22 − 2)2 ) − 2/2
:= 2 × ((22 − 2)2 + 2)
:= 3 + ((33 × (33 − 3)) − 3/3)
:= (33 × (33 + 3)) − 33/3
:= (33 × (33 + 3)) − (3 + 3)
:= 444 + (((4 + 4) × 44) − (4 + 4)/4)
:= ((4 × 4 + 4) × (44 − 4)) − 4/4
:= 4 + ((4 × 4 + 4) × (44 − 4))
:= (5 × (5 × ((5 + 5)/5) )) − (5/5 + 5)
:= (5 × (5 × ((5 + 5)/5) )) − 5/5
:= 5 + ((5 × (5 × ((5 + 5)/5)5 )) − 5/5)
:= (6 + 6)/6 + 66 × (6 + 6)
:= 6 + (66 × (6 + 6) + 6/6)
:= 6 + (66 × (6 + 6) + 6)
:= 7 + (((77 − 7)/7) + 777)
:= 7 + (((777 + 7/7) + 7) + 7)
:= 7 + ((((777 − 7/7) + 7) + 7) + 7)
:= 88 + ((8 + 8)/8 + 8 × 88)
:= 888 − (8/8 + 88)
:= 88 + (((88 + 8)/8) + 8 × 88)
:= (9 + 9)/9 + (99 × (9 − 9/9))
:= 9 × (9 × 9 + 9) − 99/9
:= 9 × 99 + (((99 + 9)/9) − 99)
5
Ï 795 := 11 + ((1 + ((1 + 1 + 1)1+1+1 ))1+1 )
5
Ï 800 := (1 + 1) × (((1 + 1) × (11 − 1))1+1 )
Ï 805 := ((1 + 1)11 ) − (11 × (1 + (1 + 111)))
:= (2 × ((22 − 2)2 − 2)) − 2/2
:= 2 × (22 − 2)2
:= 2/2 + (2 × ((22 − 2)2 + 2))
:= 3 + (33 × (33 − 3))
:= 33+3 + (((3 + 3)3 − 3)/3)
:= 3 + ((((3 + 3)3 + 3)/3) + 33+3 )
:= 444 + (((4 + 4) × 44) − 4/4)
:= (4 × 4 + 4) × (44 − 4)
:= 4 + ((4/4 + 4)4 + (4 × 44))
:= (55 + 55 )/(5 − 5/5)
:= 5 × (5 × ((5 + 5)/5)5 )
:= 5 + (5 × (5 × ((5 + 5)/5)5 ))
:= 66 + ((6 × 6/(6 + 6))6 )
:= 6 + (66 × (6 + 6) + ((6 + 6)/6))
:= 6 + ((66 × (6 + 6) + 6/6) + 6)
:= 7 + (77/7 + 777)
:= 7 + (((((7 + 7)/7) + 777) + 7) + 7)
:= 7 + (777 + 7 + 7 + 7)
:= 8 + ((8 × (88 + 8) + (88/8)) + 8)
:= 888 − 88
:= 8 × 88 + (8888/88)
:= 9 × 99 + (((9 + 9 + 9)/9) − 99)
:= (9 − 9/9) × (9/9 + 99)
:= 9 × (9 × 9 + 9) + ((9 − 99)/(9 + 9))
Ï 796 := (1 + 1) × ((((1 + 1) × (11 − 1))1+1 ) − (1 + 1))
Ï 801 := 1 + ((1 + 1) × (((1 + 1) × (11 − 1))1+1 ))
Ï 806 := 1 + (((1 + 1)11 ) − (11 × (1 + (1 + 111))))
:= 2 × ((22 − 2)2 − 2)
:= 2/2 + (2 × (22 − 2)2 )
:= 2 + (2 × ((22 − 2)2 + 2))
:= 3 + ((33 × (33 − 3)) + 3/3)
:= 3 × ((3 × (3 × (33 + 3))) − 3)
:= (33 × (33 + 3)) − (3/3 + 3)
:= 444 + ((4 + 4) × 44)
:= (4 × 44) + (4/4 + 4)4
:= ((44 − 4)/4 × (4 − 4/4)4 ) − 4
:= 5 + (((555 + 55 )/5) + 55)
:= 5/5 + (5 × (5 × ((5 + 5)/5)5 ))
:= 5 + ((5 × (5 × ((5 + 5)/5)5 )) + 5/5)
:= 6 + (66 × (6 + 6) − ((6 + 6)/6))
:= 6 + (((6 × 6/(6 + 6))6 ) + 66)
:= 6 + ((66 × (6 + 6) + ((6 + 6)/6)) + 6)
:= 7 + ((77 + 7)/7 + 777)
:= 7 + ((((77 − 7)/7) + 777) + 7)
:= 7 + ((((777 + 7/7) + 7) + 7) + 7)
:= 88 + ((8/((8 + 8)/8)) + 8 × 88)
:= 8/8 + (888 − 88)
:= 8 + (888 − ((8 + 8)/8 + 88))
:= 9 × (9 × 9 + 9) − 9
:= 9 × (9 × 9 + 9) + ((9 − 9 × 9)/(9 + 9))
:= 9 + (9999/9 − (9 + 9) × (9 + 9))
Ï 797 := ((11 × (1 + ((1 + 11)1+1 ))) − 1)/(1 + 1)
Ï 802 := (1 + 1) × (1 + (((1 + 1) × (11 − 1))1+1 ))
Ï 807 := (1 + ((1 + 1) × (11 × (111 − 1))))/(1 + 1 + 1)
:= 2/2 + (2 × ((22 − 2) − 2))
:= 2 + (2 × (22 − 2) )
:= 2 + ((2 × ((22 − 2)2 + 2)) + 2/2)
:= 3 + (((33 × (33 − 3)) − 3/3) + 3)
:= 33+3 + (((3 + 3)3 + 3)/3)
:= (33 × (33 + 3)) − 3
:= (4 × 44) + ((4/4 + 4)4 − 4)
:= 4/4 + ((4/4 + 4)4 + (4 × 44))
2
2
:= 4 + ((((4 × 4 + 4) × (44 − 4)) − 4/4) + 4)
:= 5 + ((((555 + 5 ) + 5)/5) + 55)
:= (5 + 5)/5 + (5 × (5 × ((5 + 5)/5) ))
:= 5 + ((5 × (5 × ((5 + 5)/5)5 )) + ((5 + 5)/5))
:= 6 + (66 × (6 + 6) − 6/6)
:= 66 × (6 + 6) + (66 − 6)/6
:= 6 + ((((6 × 6/(6 + 6))6 ) + 66) + 6)
:= 7 + (((777 − 7/7) + 7) + 7)
:= 7 + ((77/7 + 777) + 7)
:= (7 × 7 × (7 + 7)) + (((7 + 7)/7)7 − 7)
:= 8 + (888 − (88/8 + 88))
:= 888 + (((8 + 8)/8) − 88)
:= 8 + (888 − (8/8 + 88))
:= 9 × (9 × 9 + 9) − ((99 + 9 + 9)/9)
:= 9/9 + (9 × (9 × 9 + 9) − 9)
:= 9 × (9 × 9 + 9) − (9 + 9 + 9)/9
5
5
63
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 808 := (1 + 1)1+1+1 × (1 + (11 − 1)1+1 )
Ï 813 := ((1 + 1) × (11 × (111/(1 + 1 + 1)))) − 1
Ï 818 := 1 + (1 + (((1 + 1)11 ) − (11 × (1 + 111))))
:= 2 × (((22 − 2)2 + 2) + 2)
:= 2 + ((2 × (22 − 2)2 ) + 22/2)
:= 22 + (2 × ((22 − 2)2 − 2))
:= 3/3 + ((33 × (33 + 3)) − 3)
:= 3 + (33 × (33 + 3))
:= 33 + ((33 × (33 − 3)) − 3/3)
:= 4 + (((4 × 4 + 4) × (44 − 4)) + 4)
:= 44 + ((4 × (4 × (44 + 4))) + 4/4)
:= 4 + (((44 − 4)/4 × (4 − 4/4)4 ) + 4)
:= 5 + ((((5 − (5 + 5)/5)5 ) + 555) + 5)
:= (55 × (5 + 5 + 5)) − (55 + 5)/5
:= (55 × (5 + 5 + 5)) − ((5 + 5)/5 + 5)
:= 6 + (66 × (6 + 6) + ((66 − 6)/6))
:= 6 × 6 + (666/6 + 666)
:= (6 − (((6 + 6)/6)6+6 ))/(6/6 − 6)
:= 7 × 7 + (777 − (77/7 + 7))
:= 7/7 + ((777 − (7 + 7)) + 7 × 7)
:= 7 × 7 + (777 − (7/7 + 7))
:= 8 + (888 − 88)
:= 888 − (88/8 + 8 × 8)
:= 8 + ((8/8 + 8) × ((8 + 8)/8 + 88))
:= 9 × (9 × 9 + 9) − (9 + 9)/9
:= 9 × (9 × 9 + 9) + ((9 + 9 + 9)/9)
:= 9 + (9 × (9 × 9 + 9) − 9/9)
Ï 809 := ((11 − 1) × (11 − 1 − 1)1+1 ) − 1
Ï 814 := (1 + 1) × (11 × (111/(1 + 1 + 1)))
:= 2/2 + (2 × (((22 − 2)2 + 2) + 2))
:= 22 + (22 × (2 + 2 + 2)2 )
:= (33 × (33 + 3)) − 3/3
:= 3 + ((33 × (33 + 3)) + 3/3)
4
Ï 819 := 111+1+1 − (1 + 1)11−1−1
:= ((22 + 2)2 ) + (222 + 2)/2
:= 3 × ((3 × (3 × (33 + 3))) + 3)
4
:= 4 + (((4/4 + 4) + (4 × 44)) + 4)
:= 4 + ((44 − 4)/4 × (4 − 4/4) )
:= (((4 + 4)4 ) − 4/4)/(4/4 + 4)
:= (55 × (5 + 5 + 5)) − (55/5 + 5)
:= (55 × (5 + 5 + 5)) − 55/5
:= (55 × (5 + 5 + 5)) − (5/5 + 5)
:= 6 + (66 × (6 + 6) + (66/6))
:= (66/6) × ((((6 + 6)/6) + 66) + 6)
:= 7 + (((77/7 + 777) + 7) + 7)
7
:= (6/6 + 6) × (666/6 + 6)
:= (7 × 7 × (7 + 7)) + ((7 + 7)/7)
:= 7 × 7 + (777 − 7)
:= 8 + ((888 − 88) + 8/8)
:= 8 × 88 + ((888 − 8)/8)
:= 8 + ((888 − 88) + (88/8))
:= 9 × (9 × 9 + 9) − 9/9
:= 99/9 × ((((9 + 9)/9) − 9) + 9 × 9)
:= 9 + 9 × (9 × 9 + 9)
Ï 810 := (11 − 1) × (11 − 1 − 1)1+1
Ï 815 := 1 + ((1 + 1) × (11 × (111/(1 + 1 + 1))))
Ï 820 := (11 − 1) × (1 + (11 − 1 − 1)1+1 )
:= 2 + (2 × (((22 − 2)2 + 2) + 2))
:= 22/2 + (2 × ((22 − 2)2 + 2))
:= 22 + ((2 × (22 − 2)2 ) − 2)
:= 33 × (33 + 3)
:= 3 + (((33 × (33 + 3)) − 3/3) + 3)
:= 33 + ((33 × (33 − 3)) + 3/3)
:= (44 − 4)/4 × (4 − 4/4)4
:= 4 × 4 × 44 + 444/4
:= (((4 + 4)4 ) + 4)/(4/4 + 4)
:= (5 + 5 + 5) × (55 − 5/5)
:= (55 × (5 + 5 + 5)) − 5 − 5
:= (55 × (5 + 5 + 5)) − 5
:= 6 + ((66 × (6 + 6) + 6) + 6)
:= 6 + ((66 × (6 + 6) + (66/6)) + 6)
:= 6 + (((66 + 66)/6) + 66 × (6 + 6))
:= (7 − 7/7) × (((7 + 7)/7) + 7)
:= 7 × 7 + (777 − (77/7))
:= 7/7 + (777 − 7 + 7 × 7)
:= (8/8 + 8) × ((8 + 8)/8 + 88)
:= 8 × 88 + 888/8
:= (((88/8 − 8)8 ) − 8/8)/8
:= 9 × (9 × 9 + 9)
:= 9 × (9 × 9 + 9) + ((9 × 9 + 9)/(9 + 9))
:= 9 + (9 × (9 × 9 + 9) + 9/9)
7
Ï 811 := 1 + (11 − 1) × (11 − 1 − 1)1+1
Ï 816 := ((1 + 1)11 ) − (11 × (1 + 111))
Ï 821 := 1 + ((11 − 1) × (1 + (11 − 1 − 1)1+1 ))
:= 22/2 + (2 × (22 − 2)2 )
:= 2 × ((22 − 2)2 + 2 × (2 + 2))
:= 22 + ((2 × (22 − 2)2 ) − 2/2)
:= 3/3 + (33 × (33 + 3))
:= 3 + ((33 × (33 + 3)) + 3)
:= 33/3 + (33 × (33 + 3))
:= 44 + (4 × 44 − (4/4 + 44 ))
:= 4 × (4 × (44 − 4) + 44)
:= 4 + ((4 × (44 + 4)) + (4/4 + 4)4 )
:= 5/5 + ((5 + 5 + 5) × (55 − 5/5))
:= 5/5 + ((55 × (5 + 5 + 5)) − (5 + 5))
:= 5/5 + ((55 × (5 + 5 + 5)) − 5)
:= 6 + (((66 × (6 + 6) + 6/6) + 6) + 6)
:= (6 + 6) × (((6 + 6)/6) + 66)
:= 6 × 6 + (66 × (6 + 6) − (6/6 + 6))
:= 77 + ((7 × (7 × (7 + 7) + 7)) − 7/7)
:= 7 × 7 + (((7 − 77)/7) + 777)
:= 7 + ((7 × 7 × (7 + 7)) + ((7 + 7)/7)7 )
:= 888 + (88/8 − 88)
:= 888 − (8 × 8 + 8)
:= 8 × (88 + 8 + 8) − 88/8
:= 9/9 + 9 × (9 × 9 + 9)
:= (9 − 9/9) × (999/9 − 9)
:= 99/9 + 9 × (9 × 9 + 9)
Ï 812 := 1 + 1 + (11 − 1) × (11 − 1 − 1)1+1
Ï 817 := 1 + (((1 + 1)11 ) − (11 × (1 + 111)))
:= 2 × ((((22 − 2)2 + 2) + 2) + 2)
:= ((22 + 2)2 ) + (222 − 2)/2
:= 3 + ((33 × (33 + 3)) − 3/3)
:= 3 + (((33 × (33 + 3)) + 3/3) + 3)
Ï 822 := (1 + 1) × (11 + (((1 + 1) × (11 − 1))1+1 ))
:= 22 + (2 × (22 − 2)2 )
4
:= 3 + ((33 × (33 − 3)) + 33 )
:= (4 + 4)/4 + ((((4 + 4)4 ) + 4)/(4/4 + 4))
:= 44 + (4 × (4 × (44 + 4)))
:= (4 × (44 + 4)) + (4/4 + 4)
:= ((5 + 5)/5 + 5) × (555/5 + 5)
:= 5 + (((5 + 5)/5 + 5) × (555/5 + 5))
:= (5 + 5)/5 + ((55 × (5 + 5 + 5)) − 5)
:= (6/6 + 6) × (((666 − 6)/6) + 6)
:= 6 × 6 + (66 × (6 + 6) − (66/6))
:= 6 × 6 + (66 × (6 + 6) − 6)
7
:= 77 + (7 × (7 × (7 + 7) + 7))
:= 7 + ((7 − 7/7) × (((7 + 7)/7) + 7))
:= 888 + ((88 + 8)/8 − 88)
:= 8/8 + (888 − (8 × 8 + 8))
:= 888 − (((8 + 8)/8) + 8 × 8)
:= (9 + 9)/9 + 9 × (9 × 9 + 9)
:= 9 + (9 × (9 × 9 + 9) − ((9 + 9)/9))
:= 9 × (9 × 9 + 9) + (99 + 9)/9
64
:= 7 + ((777 − (77/7)) + 7 × 7)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 823 := 1111 − ((1 + 1) × ((1 + 11)1+1 ))
Ï 828 := 1 + (((1 + 1)11 ) − 11 × 111)
Ï 833 := 1 + ((1 + (1 + 11)) × ((1 + 1)(1+1)×(1+1+1) ))
:= 22 + ((2 × (22 − 2)2 ) + 2/2)
:= (22+2 + 2) × ((2 × 22) + 2)
:= 22 + ((2 × (22 − 2)2 ) + 22/2)
:= 3 + (((33 × (33 − 3)) + 33 ) + 3/3)
:= 3 × 33 + 33+3
:= (3333 − 3/3)/(3/3 + 3)
:= 4 + ((((4 + 4)4 ) − 4/4)/(4/4 + 4))
:= (4 × (4 × (44 + 4 + 4))) − 4
:= 4/4 + (4 × (4 × (44 + 4 + 4)))
:= (55 × (5 + 5 + 5)) − (5 + 5)/5
:= 5 + ((55 × (5 + 5 + 5)) − ((5 + 5)/5))
:= 55 /5 + ((55 − 5)/(5 + 5 + 5))
:= 6 × 6 + ((66 × (6 + 6) − 6) + 6/6)
:= 6 × ((66 + 66) + 6)
:= 6 + ((66 × (6 + 6) − 6/6) + 6 × 6)
:= 7 × 7 + (777 − ((7 + 7 + 7)/7))
:= 7 × 7 + (((7 + 7)/7) + 777)
8
:= 7 + (777 + 7 × 7)
:= 888 − (8/8 + 8 × 8)
:= 8 + ((((88/8 − 8) ) − 8/8)/8)
:= 8/8 + 8 × (88 + 8 + 8)
:= 9 × (9 × 9 + 9) + ((99 + 9 + 9)/9)
:= 99 + 9 × 9 × 9
:= 9 + ((9 − 9/9) × (((999 + 9)/9) − 9))
Ï 824 := 1 + (1111 − ((1 + 1) × ((1 + 11)1+1 )))
2
Ï 829 := 1 + (1 + (((1 + 1)11 ) − 11 × 111))
2+2
:= (22 × ((2 + 2 + 2)2 + 2)) − 2
:= 2 + ((2 × (22 − 2) ) + 22)
:= 2/2 + ((2
:= 3 + ((33 × (33 + 3)) + 33/3)
:= 3/3 + (3 × 33 + 33+3 )
:= 3 + ((3 × 33 + 33+3 ) + 3)
:= 4 + ((((4 + 4)4 ) + 4)/(4/4 + 4))
:= 4 + (44/4 × ((44 + 44 )/4))
:= (4 + 4)/4 + (4 × (4 × (44 + 4 + 4)))
:= (55 × (5 + 5 + 5)) − 5/5
:= 5 + ((55 × (5 + 5 + 5)) − 5/5)
:= 5 + (((55 × (5 + 5 + 5)) − 5/5) + 5)
:= 6 × 6 + (66 × (6 + 6) + 6/6)
:= 6 + (66 × (6 + 6) + 6 × 6)
:= 7 × 7 + (777 − ((7 + 7)/7))
:= (77/7 × (77 − 7/7)) − 7
:= 7 + ((777 + 7 × 7) + 7/7)
:= 888 − 8 × 8
:= 8 + (8 × (88 + 8 + 8) − 88/8)
:= (8 + 8)/8 + 8 × (88 + 8 + 8)
:= (9 − 9/9) × (((999 + 9)/9) − 9)
:= 9/9 + (9 × 9 × 9 + 99)
:= (9999 + 9)/((99 + 9)/9)
6+6
:= 6 + ((6 − (((6 + 6)/6)
))/(6/6 − 6))
Ï 825 := 11 × (1 + ((1 + 1) × (111/(1 + 1 + 1))))
2
+ 2) × ((2 × 22) + 2))
Ï 834 := (1 + 1) × ((11 × (1 + (111/(1 + 1 + 1)))) − 1)
Ï 830 := (11 − 1) × (1 + (1 + (11 − 1 − 1)1+1 ))
2+2
:= (22 × ((2 + 2 + 2)2 + 2)) − 2/2
:= ((2 × 22) ) − (2222/2)
:= 2 + ((2
:= 33 + (33 × (33 − 3))
:= 3 + ((33+3 − 3/3) + 3 × 33)
:= 3 + ((33 − 3/3) × (33 − 3/3))
:= 44/4 × ((44 + 44 )/4)
:= (4 × (4 × (44 + 4 + 4))) − (4 + 4)/4
:= 4 + ((4 × (4 × (44 + 4 + 4))) − 4/4)
:= 55 × (5 + 5 + 5)
:= 5 + (55 × (5 + 5 + 5))
:= 5 + ((55 × (5 + 5 + 5)) + 5)
:= 6 + ((6/6 + 6) × (666/6 + 6))
:= 6 × 6 + (66 × (6 + 6) + ((6 + 6)/6))
:= 6 + ((66 × (6 + 6) + 6 × 6) + 6/6)
:= 7 × 7 + (777 − 7/7)
:= 7 × 7 + ((777 − 7) + (77/7))
:= (77 × 77 − (77 + 7))/7
:= 8/8 + (888 − 8 × 8)
:= 8 × (88 + 8 + 8) − (8 + 8)/8
:= 888 + (88/8 − 8 × 8)
:= 9 + ((9 − 9/9) × (999/9 − 9))
:= 9 + (9 × (9 × 9 + 9) + (99/9))
:= 9 + ((9 × 9 × 9 − ((9 + 9)/9)) + 99)
Ï 826 := ((1 + 1)11 ) − (1 + 11 × 111)
+ 2) × ((2 × 22) + 2))
Ï 835 := 1111 − ((1 + 11) × (1 + (11 + 11)))
Ï 831 := 1 + ((11 − 1) × (1 + (1 + (11 − 1 − 1)1+1 )))
Ï 836 := (1 + 1) × (11 × (1 + (111/(1 + 1 + 1))))
:= 22 + (2 × ((22 − 2)2 + 2))
:= (2/2 + 2) × ((((22 + 2) ) − 22)/2)
:= 22 × ((2 + 2 + 2)2 + 2)
:= 3/3 + ((33 × (33 − 3)) + 33)
:= 3 + (3 × 33 + 33+3 )
:= 33 + ((33 × (33 + 3)) − 3/3)
:= ((44 × ((44 + 44 )/4)) + 4)/4
:= (4 × (4 × (44 + 4 + 4))) − 4/4
:= 4 + (4 × (4 × (44 + 4 + 4)))
:= 5/5 + (55 × (5 + 5 + 5))
:= 5 + ((55 × (5 + 5 + 5)) + 5/5)
:= 55/5 + (55 × (5 + 5 + 5))
:= 6 × 6 + (66 × (6 + 6) − ((6 + 6)/6))
:= 6 × 6 + (((6 × 6/(6 + 6)) ) + 66)
:= (66/6) × ((((6 + 6)/6)6 + 6) + 6)
:= 7 × 7 + 777
:= 7 + ((777 − ((7 + 7)/7)) + 7 × 7)
:= 77/7 × (77 − 7/7)
:= 888 + (((8 + 8)/8) − 8 × 8)
:= 8 × (88 + 8 + 8) − 8/8
:= 888 + ((88 + 8)/8 − 8 × 8)
:= 99 + (9 × 9 × 9 − ((9 + 9)/9))
:= 9 × 9 × 9 + (999/9 − 9)
:= 9 + ((9 × 9 × 9 − 9/9) + 99)
Ï 827 := ((1 + 1)11 ) − 11 × 111
2
6
Ï 832 := (1 + (1 + 11)) × ((1 + 1)(1+1)×(1+1+1) )
Ï 837 := ((1 + 1)11 ) − (1 + (11 × (111 − 1)))
:= 2 + (((2 × 22)2 ) − (2222/2))
:= 2 × ((22 − 2)2 + 22+2 )
:= 2/2 + (22 × ((2 + 2 + 2)2 + 2))
:= 3 × 33 + (33+3 − 3/3)
:= (33 − 3/3) × (33 − 3/3)
:= 3 × (3 × ((3 × (33 + 3)) + 3))
:= (4 × (4 × (44 + 4 + 4))) − (4/4 + 4)
:= 4 × (4 × (44 + 4 + 4))
:= 44 + ((4/4 + 4)4 − 44)
:= (5 + 5)/5 + (55 × (5 + 5 + 5))
:= (5 × 5 + 5/5) × ((5 + 5)/5)5
:= 5 + ((5 × 5 + 5/5) × ((5 + 5)/5)5 )
:= 6 × 6 + (66 × (6 + 6) − 6/6)
:= (6/6 + 6 + 6) × ((6 + 6)/6)6
:= (66 × 66 + 666)/6
:= 7/7 + (777 + 7 × 7)
:= 7 + ((777 − 7/7) + 7 × 7)
:= 7 × 7 + (77/7 + 777)
:= 888 + (88/8 − (8 × 8 + 8))
:= 8 × (88 + 8 + 8)
:= 8 + ((8 × (88 + 8 + 8) − 88/8) + 8)
:= 99 + (9 × 9 × 9 − 9/9)
:= 9 × 9 × 9 + (((999 + 9)/9) − 9)
:= 9 + (9 × 9 × 9 + 99)
65
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 843 := 1 + (1 + ((((11 − 1) × (1 + 1 + 1)) − 1)1+1 ))
Ï 838 := ((1 + 1)11 ) − (11 × (111 − 1))
2
2
Ï 848 := 1 + (11 × (11 × (1 + ((1 + 1) × (1 + 1 + 1)))))
:= 2 + (22 × ((2 + 2 + 2) + 2))
:= 2 + (((((22 + 2/2) + 2) + 2) + 2) )
:= 2 × (((22 − 2)2 + 22) + 2)
:= 33 + ((33 × (33 + 3)) + 3/3)
:= 33 + (33 × (33 + 3))
:= 3 + (((3 − 3/3)3×3 ) + 333)
:= (4 − 44)/4 + 4 × (44 − 44)
:= 4 × (44 − 44) − (4/4 + 4)
:= 4 × (44 − 44)
:= 5 + (((55 − 5)/(5 + 5 + 5)) + 55 /5)
:= 55 /5 + (((5 − (5 + 5)/5)5 ) − 5 × 5)
:= (55/5 + 5) × (55 − (5 + 5)/5)
:= 6 + ((6/6 + 6 + 6) × ((6 + 6)/6) )
:= 6 + ((66 × 66 + 666)/6)
:= ((66 × (66/6 + 66)) + 6)/6
:= ((77 × 77 − (7 + 7))/7) − 7
:= 7 + (77/7 × (77 − 7/7))
:= (77 × 77 + 7)/7
:= 8 + (8 × (88 + 8 + 8) − ((8 + 8)/8))
:= 88/8 + 8 × (88 + 8 + 8)
:= 8 + (8 × (88 + 8 + 8) + 8)
:= 9 + ((9 × 9 × 9 + 99) + 9/9)
:= 9 + ((9999 + 9)/((99 + 9)/9))
:= 9 + ((9 × 9 × 9 + (99/9)) + 99)
6
Ï 839 := 1 + (((1 + 1)11 ) − (11 × (111 − 1)))
Ï 844 := (1 + 1) × ((1 + 1) × ((1 + 1) × 111 − 11))
Ï 849 := ((1 + 1)11 ) − (11 × (111 − 1 − 1))
:= ((2 − 22) × (2 − (2 × 22))) − 2/2
:= 2 × ((22 − 2)2 + 22)
:= 2 + ((2 × 222 ) − (22/2)2 )
:= 33+3 + ((333 − 3)/3)
:= 3 + (((333 + 3)/3) + 33+3 )
:= (33 × 33 ) − (3 × 3 + 33)
:= 4444/4 − (4 × 4 + 44 )
:= 4 × (44 − 44) − 4
:= 4/4 + 4 × (44 − 44)
:= ((5 + 5 + 5) × (55 + 5/5)) − 5/5
:= 5 × 55 + ((55 − 5)/5 − 55)
:= 5 × 5 + ((55 × (5 + 5 + 5)) − 5/5)
:= 6 × 6 + (66 × (6 + 6) + (66/6))
:= 6 + (((6/6 + 6 + 6) × ((6 + 6)/6)6 ) + 6)
:= 6 + (((66 × 66 + 666)/6) + 6)
:= ((77 × 77 − 7)/7) − 7
:= (77 × 77 − (7 + 7 + 7))/7
:= ((77 × 77 + 7) + 7)/7
:= 8 + (8 × (88 + 8 + 8) − 8/8)
:= 888 − (88/((8 + 8)/8))
:= 8 + ((8 × (88 + 8 + 8) + 8/8) + 8)
:= 99 + (9 × 9 × 9 + (99/9))
:= 9 + (((9 × 9 × 9 − ((9 + 9)/9)) + 99) + 9)
:= 9 + (999/9 + 9 × 9 × 9)
Ï 840 := 111 + (11 − 1 − 1)1+1+1
Ï 845 := (1 + (1 + (1 + 1 + 1))) × ((1 + (1 + 11))1+1 )
Ï 850 := 111+1 + (11 − 1 − 1)1+1+1
:= 2/2 + (2 × ((22 − 2)2 + 22))
:= 2 + (2 × (((22 − 2)2 + 22) + 2))
:= 333 + ((3 − 3/3)3×3 )
:= 33+3 + ((33/3)3−3/3 )
:= 4 × (4 − 44) − 4 − 4
:= 4/4 + (4 × (44 − 44) − 4)
:= (4 + 4)/4 + 4 × (44 − 44)
:= (5 + 5 + 5) × (55 + 5/5)
:= 5 + ((5 + 5 + 5) × (55 + 5/5))
:= 5 × ((5 × ((5 × 5 + 5) + 5)) − 5)
:= (6 + 6) × (((6 + 6)/6)6 + 6)
:= (6/6 + 6 + 6) × (66 − 6/6)
:= 66 × (6 + 6) + (((6 + 6)/6)6 − 6)
:= (77 × (77/7)) − 7
:= (77 × 77 − (7 + 7))/7
:= (((77 × 77 + 7) + 7) + 7)/7
:= 8 + 8 × (88 + 8 + 8)
:= 88 + (8 × (88 + 8) − (88/8))
:= 8 + ((8 × (88 + 8 + 8) + ((8 + 8)/8)) + 8)
:= 9 × 9 × 9 + 999/9
:= 9 + (((9 × 9 × 9 − 9/9) + 99) + 9)
:= 9 + (((999 + 9)/9) + 9 × 9 × 9)
:= (2 − 22) × (2 − (2 × 22))
:= 3
3+3
+ 333/3
4
Ï 841 := (((11 − 1) × (1 + 1 + 1)) − 1)1+1
:= ((((22 + 2/2) + 2) + 2) + 2)
:= 3
3+3
2
+ ((333 + 3)/3)
4
4
:= 4 + (((4/4 + 4) − 44) + 4 )
:= (5 × 5 − 5/5 + 5)
(5+5)/5
(6+6)/6
Ï 846 := (11 × (11 × (1 + ((1 + 1) × (1 + 1 + 1))))) − 1
2
Ï 851 := (1 + (11 + 11)) × (111/(1 + 1 + 1))
:= 2 + (2 × ((22 − 2) + 22))
:= (2 × (222 + 2)) − (22/2)2
:= 3 + ((33 × (33 + 3)) + 33)
:= 33+3 + ((33+3 + 3)/(3 + 3))
:= 4 × (44 − 44) − (4 + 4)/4
:= 4 + (4 × (44 − 44) − 4/4)
:= 5 + ((5 × 5 − 5/5 + 5)(5+5)/5 )
:= 5 × 5 + ((55 × (5 + 5 + 5)) + 5/5)
:= 666 + 6 × (6 × 6 − 6)
:= 6 + ((6/6 + 6 + 6) × (66 − 6/6))
:= ((77 × 77 + 7)/7) − 7
:= (77 × 77 − 7)/7
:= (77/7 × (7/7 + 77)) − 7
:= 8 + (8 × (88 + 8 + 8) + 8/8)
:= (8/8 + 8) × ((88 − ((8 + 8)/8)) + 8)
:= 8 + (8 × (88 + 8 + 8) + 88/8)
:= 9 × 9 × 9 + ((999 + 9)/9)
:= 9 + ((9 × 9 × 9 + 99) + 9)
:= 9 × 9 × 9 + (999 + 99)/9
:= (6 × 6 − (6/6 + 6))
Ï 842 := 1 + ((((11 − 1) × (1 + 1 + 1)) − 1)1+1 )
:= 2 + ((2 − 22) × (2 − (2 × 22)))
3+3
Ï 847 := 11 × (11 × (1 + ((1 + 1) × (1 + 1 + 1))))
:= (2 × 222 ) − (22/2)2
Ï 852 := 11 + ((((11 − 1) × (1 + 1 + 1)) − 1)1+1 )
:= 2 × ((2 × (222 + 2)) − 22)
:= 33/3 × (3 × 3 − (3/3 + 3))
:= (3/3 + 3) × ((3 + 3)3 − 3)
:= 4 × (44 − 44) − ((4 + 4)/4 + 4)
:= 4 × (44 − 44) − 4/4
:= 4 + 4 × (44 − 44)
:= 5 + (((5 × 5 + 5/5) × ((5 + 5)/5)5 ) + 5)
:= ((555 + 555) + 55 )/5
:= 5 + (((555 + 555) + 55 )/5)
:= 6 + ((66/6) × ((((6 + 6)/6)6 + 6) + 6))
:= (66/6) × (66/6 + 66)
:= 66 + (66 × (6 + 6) − 6)
:= (((77 × 77 + 7) + 7)/7) − 7
:= 77 × (77/7)
:= 7 + ((77 × 77 − (7 + 7))/7)
:= 3 + (((333 − 3)/3) + 3
)
3
:= 8 + (8 × (88 + 8 + 8) + ((8 + 8)/8))
:= 88/8 × (88 − 88/8)
:= 8 + (888 − (88/((8 + 8)/8)))
:= 9 × 9 × 9 + ((999 + 9 + 9)/9)
:= 9 + (((9 × 9 × 9 + 99) + 9/9) + 9)
:= ((99 + 9)/9) × (9 × 9 − (9/9 + 9))
66
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 853 := (((1 + 11)1+1+1 )/(1 + 1)) − 11
Ï 858 := 11 × (111 − (11 × (1 + 1 + 1)))
:= (2 × (222 − 2)) − 222/2
:= 22 × (2 × (22 − 2) − 2/2)
:= 3/3 + ((3/3 + 3) × ((3 + 3)3 − 3))
3
Ï 863 := (((1 + 11)1+1+1 )/(1 + 1)) − 1
:= ((22 + 2) × (2 + 2 + 2)2 ) − 2/2
:= (33 × 33 ) − (33 + 3/3)
:= 33 × (3 − 3/3)
4
:= 4 + (4 × (44 − 44) + 4/4)
:= 4 + ((4444 − 4)/4 − 4 )
:= (4 × (44 − 44 + 4)) − 4/4
:= 5 + ((55/5 + 5) × (55 − (5 + 5)/5))
:= 55 /5 + (((5 − (5 + 5)/5)5 ) − (5 + 5))
:= 55 /5 + (((5 − (5 + 5)/5)5 ) − 5)
:= (6 × (6 + 6) × (6 + 6)) − 66/6
:= 66 + 66 × (6 + 6)
:= (6 × (6 + 6) × (6 + 6)) − 6/6
:= 7 + ((77 × 77 − 7)/7)
:= 77/7 × (7/7 + 77)
:= 7 + ((((77 × 77 + 7) + 7)/7) + 7)
:= 888 − ((88/8 + 8 + 8) + 8)
:= 88 + (8 × (88 + 8) + ((8 + 8)/8))
:= 888 − (8/8 + 8 + 8 + 8)
:= 9 × 99 − ((99/9 + 9 + 9) + 9)
:= 9 + ((999/9 + 9 × 9 × 9) + 9)
:= 9 × 99 − ((9/9 + 9 + 9) + 9)
Ï 854 := 1 + ((((1 + 11)1+1+1 )/(1 + 1)) − 11)
Ï 859 := 1 + (11 × (111 − (11 × (1 + 1 + 1))))
:= 2 + (2 × ((2 × (222 + 2)) − 22))
:= 3
3+3
3
2
:= 2 + ((2 × 22 ) − 222/2)
3
+ ((3 − 3/3 + 3) )
:= 3/3 + (33 × (3 − 3/3))
Ï 864 := ((1 + 11)1+1+1 )/(1 + 1)
:= (22 + 2) × (2 + 2 + 2)2
:= 3 × 3 × (3 × 33 − 3)
:= (4444 − 4)/4 − 44
:= 4 + (4444/4 − 4 )
:= 4 × (44 − 44 + 4)
:= 555 + ((5 × (55 + 5)) − 5/5)
:= ((55/5 + 5) × (55 − 5/5)) − 5
:= (55/5 + 5) × (55 − 5/5)
:= (6/6 + 6) × ((666 + 66)/6)
:= 66 + (66 × (6 + 6) + 6/6)
:= 6 × (6 + 6) × (6 + 6)
:= 77 + 777
:= ((77 × 77 + 77) + 7)/7
:= (77/7 + 7) × (7 × 7 − 7/7)
4
:= 88 + (8 × (88 + 8) − ((8 + 8)/8))
:= 8 + ((8 × (88 + 8 + 8) + 88/8) + 8)
:= (88 + 8) × (8/8 + 8)
:= 9 × 99 − (((9/9 + 9 + 9) + 9) + 9)
:= 9 + ((((999 + 9)/9) + 9 × 9 × 9) + 9)
:= 9 × 99 − (9 + 9 + 9)
Ï 855 := 1111 − (1 + 1)(1+1)
1+1+1
Ï 860 := (1 + 1) × (((11 + (11 − 1))1+1 ) − 11)
Ï 865 := 1 + (((1 + 11)1+1+1 )/(1 + 1))
:= (2 × 222 ) − (222/2 + 2)
:= (2 − 22) × (2/2 − (2 × 22))
:= 2/2 + ((22 + 2) × (2 + 2 + 2)2 )
:= 3 × (3 × (3 × 33 − 3) − 3)
:= (3/3 + 3) × ((3 + 3)3 − 3/3)
:= 3/3 + (3 × 3 × (3 × 33 − 3))
:= (4/4 + 4) × ((4 × 44) − 4)
:= 4/4 + (4 × (44 − 44 + 4))
:= 555 + (5 × (55 + 5))
:= 5 + ((5 × (55 + 5)) + 555)
:= (5 × (5 × ((5 × 5 + 5) + 5))) − 5 − 5
:= 6 × 6 + ((6/6 + 6) × (666/6 + 6))
:= 66 + (66 × (6 + 6) + ((6 + 6)/6))
:= 6/6 + (6 × (6 + 6) × (6 + 6))
:= 7 + ((77 × 77 + 7)/7)
:= 7 + (((77 × 77 − 7)/7) + 7)
:= 7 + (77/7 × (7/7 + 77))
:= 88 + (8 × (88 + 8) − 8/8)
:= 888 + ((8 − 8 × 8)/((8 + 8)/8))
:= 8/8 + (88 + 8) × (8/8 + 8)
:= 9 × 99 − ((9 + 9 + 9) + 9)
:= 9 × 99 − (((99 + 99)/9) + 9)
:= 9/9 + (9 × 99 − (9 + 9 + 9))
:= 4444/4 − 4
4
Ï 856 := (11 − 1)1+1+1 − ((1 + 11)1+1 )
:= 2 × (2 × (222 − 2 × (2 + 2)))
:= (3 × 3
3+3
3
Ï 861 := (1 + 1 + 1) × (((1 + 1) × ((1 + 11)1+1 )) − 1)
2
:= (2/2 + 2) × ((22 + 2) − 2)/2
3
) − ((33/3) )
4
:= 4 + (4 × (4 − 44) + 4)
:= 5/5 + ((5 × (55 + 5)) + 555)
6
Ï 866 := 1 + (1 + (((1 + 11)1+1+1 )/(1 + 1)))
:= (2 × 2 × 222) − 22
:= 3 + (33 × (3 − 3/3))
:= 3 + ((33 × 33 ) − (33 + 3/3))
:= 4/4 + ((4/4 + 4) × ((4 × 44) − 4))
:= (4 + 4)/4 + (4 × (44 − 44 + 4))
5
:= 5 × 5 × 5 + ((555 + 5 )/5)
6
:= 5 + (((555 + 55 )/5) + 5 × 5 × 5)
:= 66 × (6 + 6) + ((6 + 6)/6)
:= 66 + (((6 × 6/(6 + 6)) ) + 66)
:= (6 + 6)/6 + (6 × (6 + 6) × (6 + 6))
:= 7 + (((77 × 77 + 7) + 7)/7)
:= 7 + (777 + 77)
:= 7 + (((77 × 77 + 77) + 7)/7)
:= 88 + 8 × (88 + 8)
:= 888 − (88/8 + 8 + 8)
:= 888 − (88 + 88)/8
:= (9 − 9/9) × ((99 − 9/9) + 9)
:= (9 × (99 + 9)) − 999/9
:= (9 + 9)/9 + (9 × 99 − (9 + 9 + 9))
Ï 857 := ((1 + 1) × ((11 + 11)1+1 )) − 111
2
:= (2 × 22 ) − 222/2
3
:= (33 × 3 ) − (3/3 + 33)
Ï 862 := (((1 + 11)1+1+1 )/(1 + 1)) − 1 − 1
:= 2 × (2 × 222 − 2) − 22
3
:= 3 + ((33 × (3 − 3/3)) + 3/3)
Ï 867 := 1 + (1 + (1 + (((1 + 11)1+1+1 )/(1 + 1))))
:= 2/2 + ((2 × 2 × 222) − 22)
:= 3 + (3 × 3 × (3 × 33 − 3))
:= 4 + ((4 × (4 − 44) + 4/4) + 4)
:= (4 × (4 − 44 + 4)) − (4 + 4)/4
:= 4 + ((4 × (44 − 44 + 4)) − 4/4)
:= ((5 + 5)/5)5 + (55 × (5 + 5 + 5))
:= 555 + (55 − 55)/(5 + 5)
:= 555 + (55 − 5)/(5 + 5)
:= 66 + (66 × (6 + 6) − 6/6)
:= (6 × (6 + 6) × (6 + 6)) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × (6 + 6) × (6 + 6))
4
4
:= ((77 × 77 − 7) + 77)/7
:= 7 + (((77 × 77 + 7)/7) + 7)
:= 7 + ((((77 × 77 − 7)/7) + 7) + 7)
:= 8/8 + (8 × (88 + 8) + 88)
:= 888 + ((8 − 88)/8 − (8 + 8))
:= 88 + (8 × (88 + 8) + (88/8))
:= 9 + (((9 × 9 × 9 + (99/9)) + 99) + 9)
:= 9 × 99 − (99/9 + 9 + 9)
:= 9 + (((999/9 + 9 × 9 × 9) + 9) + 9)
67
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 868 := 1111 − (1 + ((1 + 1) × 111+1 ))
:= 2 + ((2 × 2 × 222) − 22)
3
:= (3/3 + 3) × ((3 + 3) + 3/3)
4
Ï 873 := 1 + ((1 + 1)1+1+1 × (111 − 1 − 1))
Ï 878 := 1 + ((111 × (1 + 1)1+1+1 ) − 11)
:= 2 + (((((2 × 22) − 2)2 ) − 22)/2)
:= 2 × (((22 − 2/2)2 ) − 2)
:= 3 × (3 × (3 × 33 − 3) + 3)
:= (33 × 33 ) + (((3 − 33)/3) − 3)
4
4
:= 4 + (4 × (4 − 44 + 4))
:= 4 + ((4/4 + 4) − (4 + 4))
:= 44 × (4 × 4 + 4) − (4 + 4)/4
:= 55 /5 + ((5 − (5 + 5)/5)5 )
:= 5 + (((5 − (5 + 5)/5)5 ) + 55 /5)
:= (55 × (55/5 + 5)) − (5 + 5)/5
:= 6 + ((6 × (6 + 6) × (6 + 6)) − ((6 + 6)/6))
:= (6 + 6) × (6 + 6) + ((6 × 6/(6 + 6))6 )
:= 6 + (((6 × (6 + 6) × (6 + 6)) + ((6 + 6)/6)) + 6)
:= 7 + (777 + 77 + 7)
:= 777 + (7 × (7 + 7) − ((7 + 7)/7))
:= 7 + (7 × (77 + 7 × 7) − (77/7))
:= 888 − (((88 + 8)/8) + 8)
:= 8/8 + (888 − (8 + 8))
:= 888 + (8 − 88)/8
:= 9 × 99 − ((99 + 99 + 9)/9)
:= 9 × 99 − (9 + 9)
:= 9 × 99 − ((99 + 9 + 9)/9)
Ï 869 := 11 × ((11 − 1 − 1)1+1 − (1 + 1))
Ï 874 := 11 + ((((1 + 11)1+1+1 )/(1 + 1)) − 1)
Ï 879 := (11 − 1)1+1+1 − 111+1
:= (2222 − 222 )/2
:= 2 + (2 × (2 × (222 − (2 + 2))))
:= (2 × (2 × (222 − 2))) − 2/2
:= 33/3 + (33 × (33 − 3/3))
:= 3/3 + (3 × (3 × (3 × 33 − 3) + 3))
:= (33 × 33 ) − (3 × 3 + 3)
:= 44 × (4 × 4 + 4) − 44/4
:= 44 × (4 × 4 + 4) − ((4 + 4)/4 + 4)
:= 44 × (4 × 4 + 4) − 4/4
:= 5 + ((55/5 + 5) × (55 − 5/5))
:= (5 × 5 × (5 + 5)) + (5 − 5)/5
:= (55 × (55/5 + 5)) − 5/5
:= 6 + ((6 × (6 + 6) × (6 + 6)) − 6/6)
:= ((66 − 6)/6) + (6 × (6 + 6) × (6 + 6))
:= 6 + (((6 × 6/(6 + 6))6 ) + (6 + 6) × (6 + 6))
:= 77/7 × ((7 + 7)/7 + 77)
:= 777 + (7 × (7 + 7) − 7/7)
:= 7 × (77 + 7 × 7) − (7 + 7 + 7)/7
:= 888 − (88/8 + 8)
:= 888 + (((8 + 8)/8) − (8 + 8))
:= 888 − (8/8 + 8)
:= 9 × 99 − ((99 + 99)/9)
:= 9/9 + (9 × 99 − (9 + 9))
:= 9 × 99 − (99 + 9)/9
Ï 875 := 11 + (((1 + 11)1+1+1 )/(1 + 1))
Ï 880 := 11 × ((11 − 1 − 1)1+1 − 1)
Ï 870 := 1 + (11 × ((11 − 1 − 1)1+1 − (1 + 1)))
5
:= (2 × (2 × 222 + 2)) − 22
:= (2 × 2 × 222) − (22/2 + 2)
:= 2 × (2 × (222 − 2))
:= 3 + ((3 × 3 × (3 × 33 − 3)) + 3)
:= 33/3 + (3 × 3 × (3 × 33 − 3))
:= (33 × 33 ) − 33/3
:= 44 + ((4/4 + 4)4 − 44/4)
:= (4/4 + 4) × ((4 × 44) − 4/4)
:= 44 × (4 × 4 + 4)
:= (5 × (5 × ((5 × 5 + 5) + 5))) − 5
:= 5 × (5 × ((5 × 5 + 5) + 5))
:= 55 × (55/5 + 5)
:= 6 + (6 × (6 + 6) × (6 + 6))
:= 66/6 + (6 × (6 + 6) × (6 + 6))
:= 666 + (6 × 6 × 6 − (6 + 6)/6)
:= 7 × (77 + 7 × 7) − (77 + 7)/7
:= 777 + 7 × (7 + 7)
:= (7/7 + 7) × (777 − 7)/7
:= 888 + ((8 − 88)/8 − 8)
:= 888 − (88 + 8 + 8)/8
:= 888 − 8
:= 9 × 99 − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + (9 × 99 − (9 + 9))
:= 9 × 99 − 99/9
Ï 871 := 1111 − ((1 + 1) × (111+1 − 1))
Ï 876 := 1 + (11 + (((1 + 11)1+1+1 )/(1 + 1)))
Ï 881 := 1 + (11 × ((11 − 1 − 1)1+1 − 1))
:= ((((2 × 22) − 2)2 ) − 22)/2
:= 2 × ((2 × (222 − 2)) − 2)
:= ((((2 × 22) − 2)2 ) − 2)/2
:= 3 + ((3/3 + 3) × ((3 + 3)3 + 3/3))
:= (3/3 + 3) × ((3 + 3)3 + 3)
:= (33 × 33 ) + ((3 − 33)/3)
:= 4 × 4 + (4444/4 − 44 )
:= 44 × (4 × 4 + 4) − 4
:= 44 + (4/4 + 4)4
:= 5/5 + ((5 × (5 × ((5 × 5 + 5) + 5))) − 5)
:= 5/5 + (5 × (5 × ((5 × 5 + 5) + 5)))
:= 5/5 + (55 × (55/5 + 5))
:= 6 + ((6 × (6 + 6) × (6 + 6)) + 6/6)
:= 6 + ((6 × (6 + 6) × (6 + 6)) + 6)
:= 666 + (6 × 6 × 6 − 6/6)
:= 7 × (77 + 7 × 7) − 77/7
:= 7/7 + (777 + 7 × (7 + 7))
:= 7 × (77 + 7 × 7) − 7/7
:= 888 − (8/8 + 8 + 8)
:= 888 − (88 + 8)/8
:= 8/8 + 888 − 8
:= 9 × 99 − (99/9 + 9)
:= 9 × 99 + (((9 + 9 + 9)/9) − (9 + 9))
:= 9 × 99 − 9/9 − 9
Ï 872 := (1 + 1)1+1+1 × (111 − 1 − 1)
:= 2 × (2 × (222 − (2 + 2)))
3
:= (3/3 + 3) × (((3 + 3) − 3/3) + 3)
Ï 877 := (111 × (1 + 1)1+1+1 ) − 11
:= (2 × 2 × 222) − 22/2
3
:= (33 × 3 ) − (33/3 + 3)
4
4
Ï 882 := (1 + 1) × ((11 + (11 − 1))1+1 )
:= 2 × ((22 − 2/2)2 )
:= 3 × (3 × 3 × 33 − 3)
:= 44 × (4 × 4 + 4) − 4 − 4
:= 4 + ((4/4 + 4) − 4)
:= 4/4 + ((4/4 + 4)4 + 44 )
:= 5 + ((55 − 5)/(5 + 5) + 555)
:= (5 + 5)/5 + (5 × (5 × ((5 × 5 + 5) + 5)))
:= (5 + 5)/5 + (55 × (55/5 + 5))
:= 6 + ((6 × (6 + 6) × (6 + 6)) + ((6 + 6)/6))
:= 6 + (((6 × (6 + 6) × (6 + 6)) + 6/6) + 6)
:= 666 + 6 × 6 × 6
:= 7 + ((77/7 × (7/7 + 77)) + 7)
:= ((7 + 7)/7) + (777 + 7 × (7 + 7))
:= 7 × (77 + 7 × 7)
:= 888 − 8 − 8
:= 888 − 88/8
:= 888 + (((8 + 8)/8) − 8)
:= 9 × 99 − (9/9 + 9 + 9)
:= 9 × 99 + (((9 − 99)/(9 + 9)) − 9)
:= 9 × 99 − 9
68
Inder J. Taneja
Ï 883 := 1 + ((1 + 1) × ((11 + (11 − 1))1+1 ))
:= ((((2 × 22) − 2)2 ) + 2)/2
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 888 := 111 × (1 + 1)1+1+1
Ï 893 := 1 + (1 + (11 × (11 − 1 − 1)1+1 ))
:= 2 × 2 × 222
:= 2/2 + (2 × (2 × 222 + 2))
:= 3 + ((33 × 33 ) − 33/3)
:= (33 × 3 ) − 3
:= 3 + ((33 × 33 ) − 3/3)
:= 4 + (44 × (4 × 4 + 4) − 4/4)
:= (4 + 4) × 444/4
:= 4 + (((4 + 4) × 444 + 4)/4)
3
:= 5 + ((55 × (55/5 + 5)) − ((5 + 5)/5))
:= 5 × 55 + ((5 − (55 + 5))/5)
:= 5 × 55 + ((55 − 5 − 5)/5 − 5)
:= 6/6 + (666 + 6 × 6 × 6)
:= 6 + (666 + 6 × 6 × 6)
:= 666 + (6 × 6 × 6 + 66/6)
:= 7/7 + 7 × (77 + 7 × 7)
:= 777 + 777/7
:= 77/7 + 7 × (77 + 7 × 7)
:= 888 + (88/8 − (8 + 8))
:= 888
:= 8 + (888 − 88/8 + 8)
:= 9/9 + (9 × 99 − 9)
:= 9 × 99 − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 × 99
Ï 884 := (1 + 1) × (1 + ((11 + (11 − 1))1+1 ))
5
Ï 889 := 1 + (111 × (1 + 1)1+1+1 )
Ï 894 := 1 + (1 + (1 + (11 × (11 − 1 − 1)1+1 )))
:= 2 × (2 × 222 − 2)
:= 2/2 + (2 × 2 × 222)
:= 2 + (2 × (2 × 222 + 2))
:= (33 − 3/3) × (3/3 + 33)
:= 3/3 + ((33 × 33 ) − 3)
:= 3 + (33 × 33 )
:= 4 + 44 × (4 × 4 + 4)
:= ((4 + 4) × 444 + 4)/4
:= 4 + ((4 + 4)/4 × (444 + 4/4))
:= 5 + ((55 × (55/5 + 5)) − 5/5)
:= 5 × 55 + ((55 − 55)/5)
:= 5 × 55 + ((55 − 5)/5 − 5)
:= 666 + (6 × 6 × 6 + (6 + 6)/6)
:= 6 + ((666 + 6 × 6 × 6) + 6/6)
:= (6 × ((6 + 6) × (6 + 6) + 6)) − 6
:= ((7 + 7)/7) + 7 × (77 + 7 × 7)
:= 7 + 7 × (77 + 7 × 7)
:= (7 × ((7 + 7)/7)7 ) − (7 + 7)/7
:= 888 − (8/((8 + 8)/8))
:= 8/8 + 888
:= 8 + (888 − ((8 + 8)/8))
:= (9 + 9)/9 + (9 × 99 − 9)
:= 9 × 99 − (9 + 9)/9
:= 9 × 99 + ((9 + 9 + 9)/9)
Ï 890 := (11 × (11 − 1 − 1)1+1 ) − 1
Ï 895 := ((1 + 111) × (1 + 1)1+1+1 ) − 1
Ï 885 := 1 + ((1 + 1) × (1 + ((11 + (11 − 1))1+1 )))
:= 2 + (((((2 × 22) − 2)2 ) + 2)/2)
:= (2 × (2 × (222 + 2))) − 2/2
:= 2 + (2 × 2 × 222)
:= (33 × 33 ) − (3 + 3)
:= (33 × 3 ) − 3/3
:= 3 + ((33 × 33 ) + 3/3)
:= 4 + ((4/4 + 4)4 + 44 )
:= (4 + 4)/4 × (444 + 4/4)
:= 4 + (44/4 × (4 − 4/4)4 )
:= 5 + (55 × (55/5 + 5))
:= 5 + ((55 × (55/5 + 5)) + 5)
:= 55 /5 + (5 × 55 − 5)
:= (6 − 6/6) × (666/6 + 66)
:= 6 + ((666 + 6 × 6 × 6) + ((6 + 6)/6))
:= (6666/6) − 6 × 6 × 6
:= (7 × ((7 + 7)/7)7 ) − 77/7
:= 7 + (7 × (77 + 7 × 7) + 7/7)
:= (7 × ((7 + 7)/7)7 ) − 7/7
:= 8 + (888 − 88/8)
:= 9 × 99 + (((9 + 9 + 9)/9) − 9)
Ï 886 := (1 + 1) × (((1 + 1) × (1 + 1) × 111) − 1)
:= (2 × 2 × 222) − 2
:= 3/3 + ((33 × 33 ) − (3 + 3))
3
:= 888 + (8 + 8)/8
:= 8 + (888 − 8/8)
:= 9 × 99 − 9/9
:= 9 × 99 + ((9 × 9 − 9)/(9 + 9))
Ï 891 := 11 × (11 − 1 − 1)1+1
:= 2 + ((2 × 2 × 222) + 2/2)
:= 33 × 3
Ï 896 := (1 + 111) × (1 + 1)1+1+1
:= 2 × (2 × (222 + 2))
:= 3 + (((33 × 33 ) − 3/3) + 3)
3
:= 44/4 × (4 − 4/4)4
:= 4 × (44 − 4 × (4 + 4))
:= 5 + ((55 × (55/5 + 5)) + 5/5)
:= 5 × 55 + ((55 + 5)/5 − (5 + 5))
:= 5 × 55 + ((55 + 5)/5 − 5)
:= 6 + ((666 − ((6 + 6)/6)) + 6 × 6 × 6)
:= 6 + ((6 − 6/6) × (666/6 + 66))
:= (6/6 + 6) × (((6 + 6)/6)6/6+6)
:= 77/7 + (777 + 7 × (7 + 7))
:= 7 + (7 × (77 + 7 × 7) + ((7 + 7)/7))
:= 7 × ((7 + 7)/7)7
:= 888 − (8 + 8)/8
:= 888 + (88/8 − 8)
:= 8 + 888
:= 9 × 99 + ((9 − 99)/(9 + 9))
:= 9 × 99
:= 9 × 99 + ((9 × 9 + 9)/(9 + 9))
:= (4 + 4)/4 × (444 − 4/4)
Ï 887 := (111 × (1 + 1)1+1+1 ) − 1
:= (2 × 2 × 222) − 2/2
Ï 892 := 1 + (11 × (11 − 1 − 1)1+1 )
:= 2 × (2 × 222 + 2)
Ï 897 := 1 + ((1 + 111) × (1 + 1)1+1+1 )
:= 2/2 + (2 × (2 × (222 + 2)))
3
:= (33 × 33 ) − (3/3 + 3)
:= 3/3 + (33 × 3 )
:= 3 + ((33 × 33 ) + 3)
:= ((4 + 4) × 444 − 4)/4
:= 4 + (444 + 444)
:= 4 × 4 + ((4/4 + 4)4 + 44 )
5
:= 5 + ((55 × (55/5 + 5)) + ((5 + 5)/5))
:= 5 × 5 + ((5 − 5)/(5 + 5) + 555)
:= 5 × 55 + ((55 + 5 + 5)/5 − 5)
:= 6 + ((6 × 6 × 6 − 6/6) + 666)
:= 6 × 6 + (66 × (6 + 6) + ((6 + 6)/6)6 )
:= 666/6 + (66 × (6 + 6) − 6)
7
:= 7/7 + (7 × ((7 + 7)/7)7 )
:= 777 + (777 − 7)/7
:= 7 + ((7 × ((7 + 7)/7) ) − (77/7))
:= 888 − 8/8
:= 888 + (8/((8 + 8)/8))
:= 8 + (888 + 8/8)
:= 9 × 99 + ((9 − 9 × 9)/(9 + 9))
:= 9/9 + 9 × 99
:= 9 + (9 × 99 − ((9 + 9 + 9)/9))
69
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 898 := (((11 − 1) × (1 + 1 + 1))1+1 ) − 1 − 1
Ï 903 := (1 + 1)11−1 − 111+1
Ï 908 := ((11 − 1 − 1) × (1 + (11 − 1)1+1 )) − 1
:= 2 + (((2 × (2 + 2) + 22)2 ) + 2/2)
:= 2 × ((2 × ((222 + 2) + 2)) + 2)
:= 3 + (((33 × 3 ) + 3/3) + 3)
:= 3 + ((33 × 33 ) + 3 × 3)
:= 3 + (((33 × 33 ) + 33/3) + 3)
:= (4 + 4)/4 + (4 × (44 − 4 × (4 + 4)))
:= 4 + (((4 + 4) × 444 + 44)/4)
:= 44 + (4 × (44 − 44 + 4))
:= 5 × 55 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5 × 55)
:= (5 − 5/5)5 − (555/5 + 5)
:= 666/6 + 66 × (6 + 6)
:= 6 + ((6 × ((6 + 6) × (6 + 6) + 6)) + ((6 + 6)/6))
:= 2 + (2 × (2 × (222 + 2)))
3
:= (6 × ((6 + 6) × (6 + 6) + 6)) − (6 + 6)/6
7
7
:= ((7 + 7)/7) + (7 × ((7 + 7)/7) )
:= 7 + (7 × ((7 + 7)/7) )
:= (77 + 7)/7 + (7 × ((7 + 7)/7)7 )
:= 8 + (888 + ((8 + 8)/8))
:= 8 + (888 − 8/8 + 8)
:= 8 + (((88 + 8)/8) + 888)
:= 9 + (9 × 99 − ((9 + 9)/9))
:= 9 × 99 + (99 + 9)/9
:= 9 + ((9 × 99 − 9/9) + 9)
Ï 899 := (((11 − 1) × (1 + 1 + 1))1+1 ) − 1
Ï 904 := 1 + ((1 + 1)11−1 − 111+1 )
Ï 909 := (11 − 1 − 1) × (1 + (11 − 1)1+1 )
:= 22/2 + (2 × 2 × 222)
3
:= 3 × 3 + ((33 × 3 ) − 3/3)
:= 2 × (2 × ((222 + 2) + 2))
3
:= 3 + (((33 × 3 ) + 3 × 3) + 3/3)
:= 22 + ((2 × 2 × 222) − 2/2)
:= 3 × ((3 × 3 × 33 + 3) + 3)
:= ((4 + 4) × 444 + 44)/4
:= 4 + ((4 × (4 − 4 × (4 + 4))) + 4)
:= 4 × 44 − (444/4 + 4)
:= 5 × 55 + (55 − 5)/5
:= 5 + ((55 − 5)/5 + 5 × 55)
:= 5 + (((55 − 5)/5 + 5 × 55) + 5)
:= (6 × ((6 + 6) × (6 + 6) + 6)) − 6/6
:= 66 × (6 + 6) + (666 + 6)/6
:= 6 + (666/6 + 66 × (6 + 6))
4
:= 777 + (777 + 77)/7
:= 7 + ((7 × ((7 + 7)/7) ) + 7/7)
:= 7 + (((7 × ((7 + 7)/7)7 ) − 7/7) + 7)
:= 888 + 88/8
:= 8 + (888 + 8)
:= 8 + ((88 + 8 + 8)/8 + 888)
:= 9 + (9 × 99 − 9/9)
:= 9 × 99 + ((99 + 9 + 9)/9)
:= 9 + (9 × 99 + 9)
Ï 900 := ((11 − 1) × (1 + 1 + 1))1+1
Ï 905 := 1 + (1 + ((1 + 1)11−1 − 111+1 ))
7
Ï 910 := 1 + ((11 − 1 − 1) × (1 + (11 − 1)1+1 ))
:= (2 × (2 + 2) + 22)2
:= 22 + (((((2 × 22) − 2)2 ) + 2)/2)
:= 22 + (2 × 2 × 222)
:= 3 × (3 × 3 × 33 + 3)
:= 3 + ((33 × 33 ) + 33/3)
:= 3/3 + ((33 × 33 ) + (3 × (3 + 3)))
:= 4 + (4 × (44 − 4 × (4 + 4)))
:= 4 × 4 + (((4 + 4) × 444 + 4)/4)
:= 4 × 44 + (((4 − 444)/4) − 4)
:= 5 × (5 × 5 × 5 + 55)
:= 5 + (55 /5 + 5 × 55)
:= 5 + ((55 /5 + 5 × 55) + 5)
:= 6 × ((6 + 6) × (6 + 6) + 6)
:= 6 + ((6 × ((6 + 6) × (6 + 6) + 6)) − 6/6)
:= (6/6 + 6) × (((6 + 6)/6)6 + 66)
:= (77/7 + 7) × (7/7 + 7 × 7)
:= 777 + ((7 + 7)/7)7
:= 7 + ((7 × ((7 + 7)/7)7 ) + 7)
:= 888 + (88 + 8)/8
:= 8 + (888 + 8/8 + 8)
:= 888 + (88 + 88)/8
:= 9 + 9 × 99
:= 9 + (((9 × 9 + 9)/(9 + 9)) + 9 × 99)
:= 9 + ((9 × 99 + 9/9) + 9)
Ï 901 := 1 + (((11 − 1) × (1 + 1 + 1))1+1 )
Ï 906 := 1 + (1 + (1 + ((1 + 1)11−1 − 111+1 )))
Ï 911 := 11 + (((11 − 1) × (1 + 1 + 1))1+1 )
:= 2 + (2 × (2 × ((222 + 2) + 2)))
:= 22 + ((2 × 2 × 222) + 2/2)
:= 3 × 3 + ((33 × 3 ) + 3/3)
3
:= 3 + (((33 × 3 ) + 3 × 3) + 3)
:= 3 × 3 + ((33 × 33 ) + 33/3)
:= 4 + ((4 × (44 − 4 × (4 + 4))) + 4/4)
:= 4 + (4 × 44 − ((444 + 44)/4))
:= 4 × 44 − ((444 + 4 + 4)/4)
:= 5 × 55 + (55 + 5)/5
:= 5 + ((55 + 5)/5 + 5 × 55)
:= 5 × 55 + ((55 + 55 )/5)
:= 6 + (6 × ((6 + 6) × (6 + 6) + 6))
:= 66/6 + (6 × ((6 + 6) × (6 + 6) + 6))
:= 2/2 + ((2 × (2 + 2) + 22)2 )
3
:= 6/6 + (6 × ((6 + 6) × (6 + 6) + 6))
7
:= 7 + ((7 × ((7 + 7)/7) ) − ((7 + 7)/7))
:= 7/7 + (((7 + 7)/7) + 777)
:= 7 + (((7 × ((7 + 7)/7)7 ) + 7/7) + 7)
:= 888 + (88 + 8 + 8)/8
:= 8 + ((888 + ((8 + 8)/8)) + 8)
:= 8 + ((888 − 8/8 + 8) + 8)
:= 9 + (9 × 99 + 9/9)
:= 9 + ((9 × 99 − ((9 + 9 + 9)/9)) + 9)
:= 9 + ((99/9) + 9 × 99)
Ï 902 := 11 × (1 + (11 − 1 − 1)1+1 )
:= 2 + ((2 × (2 + 2) + 22)2 )
3
7
Ï 907 := 11 + ((1 + 111) × (1 + 1)1+1+1 )
:= 22/2 + (2 × (2 × (222 + 2)))
Ï 912 := (1 + 1)11−1 − (1 + 111)
:= 2 + ((2 × 2 × 222) + 22)
:= 33/3 + (33 × 3 )
:= 3 + ((33 × 3 ) − 33/3)
:= 3 + ((33 × 33 ) + (3 × (3 + 3)))
:= 4 × 44 − ((444 + 44)/4)
:= 4 × 4 + (44/4 × (4 − 4/4)4 )
:= 4 × ((44 − 4 × (4 + 4)) + 4)
:= 5 × 55 + (55 + 5 + 5)/5
:= 5 + ((55 + 5 + 5)/5 + 5 × 55)
:= 5 × 55 + ((55 + 55 + 5)/5)
:= (6 + 6)/6 + (6 × ((6 + 6) × (6 + 6) + 6))
:= 6 + ((6 × ((6 + 6) × (6 + 6) + 6)) + 6/6)
:= 6 + ((6 × ((6 + 6) × (6 + 6) + 6)) + 6)
3
7
3
7
:= 7 + (((7 + 7)/7)7 + 777)
:= 7 + ((7 × ((7 + 7)/7) ) − 7/7)
:= 77/7 + (7 × ((7 + 7)/7) )
:= 8 + ((888 − ((8 + 8)/8)) + 8)
:= 8 + (888 + 88/8)
:= 8 + (888 + 8 + 8)
:= 99/9 + 9 × 99
:= 9 + ((9 × 99 − ((9 + 9)/9)) + 9)
:= 9 + (((99 + 9)/9) + 9 × 99)
70
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 913 := (1 + 1)11−1 − 111
Ï 918 := (11 − 1 − 1) × (1 + (1 + (11 − 1)1+1 ))
:= (222/2 − 222)/2
:= 22 + (2 × (2 × (222 + 2)))
:= 33 + ((33 × 33 ) − 33/3)
:= 3 × (3 × (3 × 33 + 3))
:= (2 × (222 − 22)) − 2/2
:= 33 + ((33 × 33 ) − 3/3)
4
:= 4 + (((4 − 444)/4) + 4 × 4 )
:= 4 × 44 − 444/4
Ï 923 := 11 + ((1 + 1)11−1 − (1 + 111))
5
:= 44 + (44 × (4 × 4 + 4) − 4/4)
:= (5 − 5/5)5 − 555/5
:= 5 + ((5 − 5/5) − 555/5)
:= (5 × (55 + 5)) + (55 − 5 − 5)/5
:= (66/6) × (66/6 + 66 + 6)
:= 666 + 6 × (6 × 6 + 6)
:= (6/6 + 6 + 6) × ((66 − 6/6) + 6)
:= 77/7 × (77 − 7/7 + 7)
:= ((77 × (77 + 7) + 7)/7) − 7
:= (77 × (77 + 7) − 7)/7
:= 8 + ((888 + 8/8 + 8) + 8)
:= 8 + ((88 + 88)/8 + 888)
:= 8 + (((888 + 88/8) + 8) + 8)
:= 9 × 99 + ((99 + 99)/9)
:= 999 − 9 × 9
:= 9 + (((99 + 99 + 9)/9) + 9 × 99)
Ï 914 := 1 + ((1 + 1)11−1 − 111)
Ï 919 := 1 + ((11 − 1 − 1) × (1 + (1 + (11 − 1)1+1 )))
Ï 924 := 11 + ((1 + 1)11−1 − 111)
:= 22 + (2 × (2 × 222 + 2))
:= (2 × (222 − (22 + 2))) − 2/2
:= 2 × (222 − 22)
:= 33 + ((33 × 33 ) − (3/3 + 3))
:= 33 + ((33 × 33 ) + 3/3)
:= 33 + (33 × 33 )
:= 4 × 44 + ((4 − 444)/4)
:= 4444/4 − (4 × (44 + 4))
:= 44 + 44 × (4 × 4 + 4)
5
:= (5 − 5/5) − (55 + 55)
:= 5 + ((5 − 5/5) − (55 + 55))
:= (5 × (5 − 5 × 5)) + (5 − 5/5)5
:= 66 × (6 + 6) + ((666 + 66)/6)
:= 6/6 + (6 × (6 × 6 + 6) + 666)
:= 66 × ((6 + 6)/6 + 6 + 6)
:= 7 + ((7 × ((7 + 7)/7) ) + (77/7))
:= 7 + ((((7 + 7)/7)7 + 777) + 7)
:= 77 × (77 + 7)/7
:= 8 + (((888 + ((8 + 8)/8)) + 8) + 8)
:= 8 + (((888 − 8/8 + 8) + 8) + 8)
:= 888 + ((8 × 8 + 8)/((8 + 8)/8))
:= 9 × 99 + ((99 + 99 + 9)/9)
:= 9/9 + (999 − 9 × 9)
:= 9 × 99 + (99/((9 + 9 + 9)/9))
5
7
Ï 915 := 1 + (1 + ((1 + 1)11−1 − 111))
Ï 920 := (11 − 1) × (11 + (11 − 1 − 1)1+1 )
Ï 925 := (11111 − 11)/(1 + 11)
:= 2 + (222/2 − 222)/2
:= 2 × (222 − (22 + 2))
:= 2/2 + (2 × (222 − 22))
:= 33 + ((33 × 33 ) − 3)
:= 3 + (((33 × 33 ) − 3/3) + 33 )
:= 3/3 + ((33 × 33 ) + 33)
:= 4 × 44 + (((4 − 444) + 4)/4)
:= (4 + 4) × (444/4 + 4)
:= 44 + ((4/4 + 4)4 + 44 )
5
:= (5 + 5 + 5) × ((55 + 5/5) + 5)
:= 5 /5 + ((5 × (55 + 5)) − 5)
:= 5 × ((5 × 5 × 5 + 55) + 5)
:= 6 + ((666/6 + 66 × (6 + 6)) + 6)
:= 666 + (6 × (6 × 6 + 6) + ((6 + 6)/6))
:= 6/6 + (66 × ((6 + 6)/6 + 6 + 6))
:= 7777/7 − (7 + 7) × (7 + 7)
:= 7 + (77/7 × (77 − 7/7 + 7))
:= (77 × (77 + 7) + 7)/7
:= 8 + ((888 + 88/8) + 8)
:= 8 + ((888 + 8 + 8) + 8)
:= 888 + 888/(8 + 8 + 8)
:= 999 − (((9 + 9 + 9)/9) + 9 × 9)
:= 9 + (((99/9) + 9 × 99) + 9)
:= (((9 + 9)/9)9/9+9 ) − 99
Ï 916 := 1 + (1 + (1 + ((1 + 1)11−1 − 111)))
Ï 921 := 1 + ((11 − 1) × (11 + (11 − 1 − 1)1+1 ))
2
Ï 926 := (1 + 11111)/(1 + 11)
:= 2 × (22 − (22 + 2 + 2))
:= 2/2 + (2 × (22 − (22 + 2)))
:= 2 + (2 × (222 − 22))
:= 33 + (((33 × 33 ) − 3) + 3/3)
:= 3 + ((33 × 33 ) + 33 )
:= 3 + (((33 × 33 ) − 3/3) + 33)
:= (4 × (44 − 4 × 4)) − 44
:= 4 + ((4 × 44 − 444/4) + 4)
:= 44 + (((4/4 + 4)4 + 44 ) + 4/4)
:= 5 + (((55 + 55 )/5) + 5 × 55)
:= (5 × (55 + 5)) + ((55 + 5)/5 − 5)
:= (5 × (55 + 5)) + (55 + 5)/5
:= 6 + ((6/6 + 6) × (((6 + 6)/6)6 + 66))
:= 6 + (((666/6 + 66 × (6 + 6)) + 6) + 6)
:= (6 + 6)/6 + (66 × ((6 + 6)/6 + 6 + 6))
:= ((77 × (77 + 7) − 7)/7) − 7
:= (77 × (77 + 7) − (7 + 7 + 7))/7
:= ((77 × (77 + 7) + 7) + 7)/7
:= 8 + ((((88 + 8)/8) + 888) + 8)
:= 8 + (((888 + 8/8 + 8) + 8) + 8)
:= 8 + (((88 + 88)/8 + 888) + 8)
:= 999 − (((9 + 9)/9) + 9 × 9)
:= 9 × (9 × 9 + 9) + 999/9
:= 9 + (999 − (9/9 + 9 × 9))
2
Ï 917 := 1 + (1 + (1 + (1 + ((1 + 1)11−1 − 111))))
Ï 922 := 11 + (11 + (((11 − 1) × (1 + 1 + 1))1+1 ))
2
Ï 927 := 1 + ((1 + 11111)/(1 + 11))
:= (2 × (22 − 22)) − 2
:= 2 + ((2 × (222 − 22)) + 2/2)
:= 33 + ((33 × 33 ) − 3/3)
:= 3 + (((33 × 33 ) + 33 ) + 3/3)
:= 3 + ((33 × 33 ) + 33)
:= 4 + (4 × 44 − 444/4)
:= 4 + ((((4 − 444)/4) + 4 × 44 ) + 4)
:= (4 × (44 − 4)) − (4 − 4/4)4
:= 5 + (((55 + 55 + 5)/5) + 5 × 55)
:= 55 + ((55 − 5)/(5 + 5) + 555)
:= (5 × (55 + 5)) + (55 + 5 + 5)/5
:= 666 + (6 × (6 × 6 + 6) − 6/6)
:= 66 + (66 × (6 + 6) + ((6 + 6)/6)6 )
:= 66 + ((((6 × 6/(6 + 6))6 ) + 66) + 66)
:= (77 × (77 + 7)/7) − 7
:= (77 × (77 + 7) − (7 + 7))/7
:= (((77 × (77 + 7) + 7) + 7) + 7)/7
:= 2 + 2 + (2
22/2
− 222)/2
:= 8 + (((88 + 8 + 8)/8 + 888) + 8)
:= 8 + ((((888 + ((8 + 8)/8)) + 8) + 8) + 8)
:= (8/8 + 8) × (888/8 − 8)
:= 999 − (9/9 + 9 × 9)
:= 9 + (((99 + 99)/9) + 9 × 99)
:= 9 + (999 − 9 × 9)
71
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 928 := 1 + (1 + ((1 + 11111)/(1 + 11)))
:= 2 × ((222 − 22) + 2)
:= 3 + (((33 × 33 ) + 33) + 3/3)
Ï 933 := 1 + ((1 + 1) × ((1 + 1) × (11 + (1 + 1) × 111)))
:= 2/2 + (2 × (2 × 222 + 22))
3
:= 3 × 3 + ((33 × 3 ) + 33)
4
Ï 938 := 1 + (((1 + 1)11 ) − 1111)
:= 2 + (2 × (222 − 22+2 ))
:= 3 + ((3 × 333) − ((3/3 + 3)3 ))
:= 4 × (44 − ((4 × 4 + 4) + 4))
:= 4 + ((4 × (4 + 4)) − 444/4)
:= 4 + ((4444 − 4)/4 − (4 × 44))
:= ((5 + 5)/5)5 × (5 × 5 − 5/5 + 5)
:= 55 /5 + ((55 + 5)/(5 + 5) − 5)
:= 55 /5 + (55 + 5)/(5 + 5)
:= ((6 + 6)/6)6 + (6 × (6 + 6) × (6 + 6))
:= 6 × 6 × 6 + (((6 × 6/(6 + 6))6 ) − (6 + 6))
:= (6 + 6)/6 + ((6 + 6) × (66 + 6 + 6))
:= (77/7 × (7/7 + 77 + 7)) − 7
:= 7 + (((77 × (77 + 7) + 7) + 7)/7)
:= 7 + (7 × (77 + 7 × 7 + 7))
:= 8 × 8 × (8 + 8) − (88 + 8)
:= 8 × 8 + (888 − (88/8 + 8))
:= (8 + 8)/8 + (8 × 8 × (8 + 8) − 88)
:= 9 + ((999 − 9 × 9) + 9/9)
:= 9 + ((99/((9 + 9 + 9)/9)) + 9 × 99)
:= 9 + (((99/9) − 9 × 9) + 999)
Ï 929 := ((11 × ((1 + (1 + 11))1+1 )) − 1)/(1 + 1)
Ï 934 := ((1 + 1)11 ) − (1 + (1 + (1 + 1111)))
Ï 939 := 1 + (1 + (((1 + 1)11 ) − 1111))
:= 2/2 + (2 × 222 + 222 )
:= 2 + (2 × (2 × 222 + 22))
:= (((2/2 + 2)2 + 22)2 ) − 22
:= 33 + ((33 × 33 ) + 33/3)
:= 33+3 + ((3 + 3)3 − 33/3)
:= (33 × (33 + 3)) − 33
:= (4 × (44 + 4)) − 444/4
:= (4444 − 4)/4 − (4 × 44)
:= 4 × 44 − ((4 − 4/4)4 + 4)
:= 5 + ((5 × (5 − 5 × 5)) + (5 − 5/5)5 )
:= (55 × (((55 + 5)/5) + 5)) − 5/5
:= (55 + 5)/5 + (55 + 5)/(5 + 5)
:= ((6 + 6) × (66 + 6 + 6)) − (6 + 6)/6
:= 6 × 6 × 6 + (((6 × 6/(6 + 6))6 ) − 6)
:= 7 + ((77 × (77 + 7) − (7 + 7))/7)
:= (7 × (((7 + 7)/7)7 + 7)) − 77/7
:= 7 + (((77 × (77 + 7) + 7)/7) + 7)
:= 8/8 + (8 × 8 × (8 + 8) − (88 + 8))
:= 8 × 8 × (8 + 8) − ((8 + 8)/8 + 88)
:= 8 × 8 + 888 − (88 + 8 + 8)/8
:= 99/9 + (999 − 9 × 9)
:= 9 + ((((9 + 9)/9)9/9+9 ) − 99)
:= 99 + (999/9 + 9 × 9 × 9)
:= 66 + ((6 × (6 + 6) × (6 + 6)) − 6/6)
Ï 930 := (11 − 1) × (((1 + 1)11−1 − 1)/11)
Ï 935 := ((1 + 1)11 ) − (1 + (1 + 1111))
Ï 940 := 1 + (1 + (1 + (((1 + 1)11 ) − 1111)))
:= 2 + (2 × 222 + 222 )
:= 22/2 + (2 × (222 − 22))
:= 2 × ((222 − 22+2 ) + 2)
:= 3 + (((33 × 33 ) + 33) + 3)
:= (3 × 333) − ((3/3 + 3)3 )
:= 3/3 + ((33 × (33 + 3)) − 33)
:= (4 × (44 + 4)) + ((4 − 444)/4)
:= 44/4 × ((4 − 4/4)4 + 4)
:= (4 × (44 − (4 × 4 + 4))) − 4
:= 5 + ((5 × (55 + 5)) + 55 /5)
:= 55 × (((55 + 5)/5) + 5)
:= 5 + (55 × (((55 + 5)/5) + 5))
:= 66 + (6 × (6 + 6) × (6 + 6))
:= ((6 + 6) × (66 + 6 + 6)) − 6/6
:= 6 + (((6 + 6) × (66 + 6 + 6)) − ((6 + 6)/6))
:= 7 + ((77 × (77 + 7) − 7)/7)
:= 77/7 × (7/7 + 77 + 7)
:= 7 + ((((77 × (77 + 7) + 7) + 7)/7) + 7)
:= (8 − 8/8 + 8) × (8 × 8 − ((8 + 8)/8))
:= 8 × 8 × (8 + 8) − (8/8 + 88)
:= 8 + ((88/((8 + 8)/8)) + 888)
:= 9 + (9 × (9 × 9 + 9) + 999/9)
:= 9 + ((999 − (9/9 + 9 × 9)) + 9)
:= 9 × 99 + ((9 × 99 − 9)/(9 + 9))
Ï 941 := 1111 − (1 + ((1 + (1 + 11))1+1 ))
Ï 931 := (1 + ((11 − 1) × (1 + 1)11−1 ))/11
Ï 936 := ((1 + 1)11 ) − (1 + 1111)
:= (2 × (2 × 222 + 22)) − 2/2
:= 2 × (222 − 22+2 )
:= (2 × (222 − 2)) − (22 + 2/2)
:= 3 + ((((33 × 33 ) + 33) + 3/3) + 3)
:= (3 × 3 + 3) × (3 × 33 − 3)
:= 33+3 + ((3 + 3)3 − (3/3 + 3))
:= 4 + ((4 × (44 − 4)) − (4 − 4/4)4 )
:= 4 × 44 − (44 + 44)
:= 4/4 + ((4 × (44 − (4 × 4 + 4))) − 4)
:= 5 + ((5 × (55 + 5)) + (55 + 5)/5)
:= 5/5 + (55 × (((55 + 5)/5) + 5))
:= 5 + ((55 × (((55 + 5)/5) + 5)) + 5/5)
:= 66 + ((6 × (6 + 6) × (6 + 6)) + 6/6)
:= (6 + 6) × (66 + 6 + 6)
:= 6 + (((6 + 6) × (66 + 6 + 6)) − 6/6)
:= 7 × (77 + 7 × 7 + 7)
:= (77 + 7)/7 × (7/7 + 77)
:= 7 + ((7 × (((7 + 7)/7)7 + 7)) − (77/7))
:= 8 + ((((888 + 88/8) + 8) + 8) + 8)
:= 8 × 8 × (8 + 8) − 88
:= 8 × 8 + (888 − 88/8)
:= 9 × 99 + ((9 × 9 × 9 − 9)/(9 + 9))
:= 9 + ((999 − 9 × 9) + 9)
:= 9 × 99 + ((9 × 99 + 9)/(9 + 9))
Ï 932 := (1 + 1) × ((1 + 1) × (11 + (1 + 1) × 111))
Ï 937 := ((1 + 1)11 ) − 1111
Ï 942 := 1111 − ((1 + (1 + 11))1+1 )
:= 2 × (2 × 222 + 22)
:= 2/2 + (2 × (222 − 22+2 ))
:= (2 × (222 − 2)) − 22
:= (3 × 333) − (((3/3 + 3)3 ) + 3)
:= 3/3 + ((3 × 3 + 3) × (3 × 33 − 3))
:= 33+3 + ((3 + 3)3 − 3)
:= 4 + (4 × (44 − ((4 × 4 + 4) + 4)))
:= 4/4 + (4 × 44 − (44 + 44))
:= 4 × 44 − ((4 − 4/4)4 + 4/4)
:= 55 /5 + (55 − 55)/(5 + 5)
:= 55 /5 + (55 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 55 /5)
:= 66 + ((6 × (6 + 6) × (6 + 6)) + ((6 + 6)/6))
:= 6/6 + ((6 + 6) × (66 + 6 + 6))
:= 6 + ((6 + 6) × (66 + 6 + 6))
:= 7 + ((77 × (77 + 7) + 7)/7)
:= 7 + (((77 × (77 + 7) − 7)/7) + 7)
:= 7 + (77/7 × (7/7 + 77 + 7))
:= 888 + (88/((8 + 8)/8))
:= 8/8 + (8 × 8 × (8 + 8) − 88)
:= 8 × 8 + ((8 − 88)/8 + 888)
:= 9 × 99 + ((9 × 9 × 9 + 9)/(9 + 9))
:= 9 + (((999 − 9 × 9) + 9/9) + 9)
:= (9 × (99 + 9 + 9)) − 999/9
72
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 943 := 1 + (1111 − ((1 + (1 + 11))1+1 ))
Ï 948 := 11 + (((1 + 1)11 ) − 1111)
Ï 953 := 1 + ((1 + 1)1+1+1 × (111+1 − (1 + 1)))
:= 2/2 + ((2 × (222 − 2)) − 22)
:= 2 + ((2 × 222 ) − 22)
:= (2 × (222 − 2)) − 22/2
:= 3/3 + (((3 + 3)3 − 3) + 33+3 )
:= 3 + (33+3 + (3 + 3)3 )
:= ((3 + 3) × ((3 + 3) × 33 − 3)) − 3/3
:= 4 × 44 − (4 − 4/4)4
:= 4 + (4 × (44 − (4 × 4 + 4)))
:= 4 + (4 × 44 − ((44 + 44 )/4))
:= 5 + ((55 + 5)/(5 + 5) + 55 /5)
:= 5 + (((55 + 5)/(5 + 5) + 55 /5) + 5)
:= 55 + ((55 − 5 − 5)/5 + 5 × 55)
:= 6 + (((6 + 6) × (66 + 6 + 6)) + 6/6)
:= 6 + (((6 + 6) × (66 + 6 + 6)) + 6)
:= 6 + (((6 + 6) × (66 + 6 + 6)) + (66/6))
:= 7 + ((77 + 7)/7 × (7/7 + 77))
:= (77 + 7)/7 × ((7 + 7)/7 + 77)
:= 7 + ((7 × (((7 + 7)/7)7 + 7)) + 7/7)
:= 8 × 8 + (888 − (8/8 + 8))
:= 8 × 8 + (888 − (8/((8 + 8)/8)))
:= 8/8 + (888 + 8 × 8)
:= ((99 + 9)/9) × (9 × 9 − ((9 + 9)/9))
:= (9 × (99 + 9)) − (9/9 + 9 + 9)
9/9+9
:= (((9 + 9)/9)
)−9×9
Ï 944 := (1 + 1) × (((11 + 11)1+1 ) − (1 + 11))
Ï 949 := 1 + (11 + (((1 + 1)11 ) − 1111))
Ï 954 := (11 − 1 − 1) × (111 − (1 + (1 + (1 + 1 + 1))))
:= (2 × 22 ) − (22 + 2)
:= 2 + (((2 × 22 ) − 22) + 2/2)
:= 2 + (2 × (222 − 2 × (2 + 2)))
:= 33+3 + ((3 + 3)3 − 3/3)
:= 3 + ((33+3 + (3 + 3)3 ) + 3/3)
:= (3 + 3) × ((3 + 3) × 33 − 3)
:= 4 × (44 − (4 × 4 + 4))
:= 4 × 44 − ((44 + 44 )/4)
:= 4 × 44 − ((((44 + 4) + 4)/4) + 4)
:= (5 − 5/5)5 − (5 × 5 + 55)
:= (5 − 5/5)5 − (5 × (5 + 5 + 5))
:= 5 + ((5 − 5/5)5 − (5 × (5 + 5 + 5)))
:= 6 + (((6 + 6) × (66 + 6 + 6)) + ((6 + 6)/6))
:= (6/6 + 6 + 6) × (66 + 6/6 + 6)
:= 666 + 6 × (6 × 6 + 6 + 6)
:= (7/7 + 7) × (777/7 + 7)
:= 7 + ((77/7 × (7/7 + 77 + 7)) + 7)
:= 7 × 7 + (((7 + 7)/7)7 + 777)
:= 8 × 8 + 888 − 8
:= (8 × (8 × (8 + 8) − 8)) − 88/8
:= 8 × 8 + (888 + ((8 + 8)/8))
:= 999 + ((9 − 999)/(9 + 9))
:= 9999/9 − 9 × (9 + 9)
:= (9 × (99 + 9)) − (9 + 9)
2
2
Ï 945 := ((1 + 1) × (((11 + 11)1+1 ) − 11)) − 1
2
Ï 950 := ((1 + ((11 − 1) × (1 + 1 + 1)))1+1 ) − 11
Ï 955 := 1111 − ((1 + 11) × (1 + (1 + 11)))
:= (2 × 22 ) − (22 + 2/2)
:= (2 × (22 + 2)) − 22
:= (2 × 222 ) − (22/2 + 2)
:= 33+3 + (3 + 3)3
:= 3 + (((33+3 − 3/3) + (3 + 3)3 ) + 3)
:= (33 × 33 ) + ((3/3 + 3)3 )
:= 4/4 + (4 × (44 − (4 × 4 + 4)))
:= 4 + ((44 × ((4 × 44) − 4))/(4 + 4))
:= 4 × 44 − ((44 + 4)/4 + 4)
:= (5 × (5 × (55 + 5))) − 555
:= (5 + 5) × (5 × (5 × 5 − 5) − 5)
:= 5 + ((5 + 5) × (5 × (5 × 5 − 5) − 5))
:= (6 × 6 − (66/6)) × (((6 + 6)/6) + 6 × 6)
:= (((6 × 6 − 6) + 6/6)(6+6)/6 ) − 6
2
6
:= 6 × 6 × 6 + ((6 × 6/(6 + 6)) )
7
:= 7 × (((7 + 7)/7)7 + 7)
:= 77 × (7 + 7) − ((7 + 7)/7)
:= 7 + ((77 + 7)/7 × ((7 + 7)/7 + 77))
:= 8/8 + ((888 − 8) + 8 × 8)
:= 8 × 8 + (888 − ((8 + 8)/8))
:= 8 × 8 + (888 − 8 + 88/8)
:= (9 × (99 + 9)) − (9 + 9 + 9)
:= 999 + ((9 − 9 × 99)/(9 + 9))
:= 9/9 + ((9 × (99 + 9)) − (9 + 9))
Ï 946 := (1 + 1) × (((11 + 11)1+1 ) − 11)
Ï 951 := 1 + (((1 + ((11 − 1) × (1 + 1 + 1)))1+1 ) − 11)
Ï 956 := 1111 − (11 + ((1 + 11)1+1 ))
:= (2 × 222 ) − 22
:= 2/2 + ((2 × (222 + 2)) − 22)
:= 2 × (222 − (2 + 2 + 2))
:= 3/3 + (33+3 + (3 + 3)3 )
:= 3 + ((33+3 + (3 + 3)3 ) + 3)
:= 33+3 + ((3 + 3)3 + 33/3)
:= (44 × ((4 × 44) − 4))/(4 + 4)
:= 4 + ((4 × 44 − (4 − 4/4)4 ) + 4)
:= (4 × (44 − 4 × 4)) − 4
:= 55/5 × (555/5 − 5 × 5)
:= 5/5 + ((5 + 5) × (5 × (5 × 5 − 5) − 5))
:= 55 + ((55 + 5)/5 + 5 × 55)
:= 6/6 + (((6 × 6/(6 + 6))6 ) + 6 × 6 × 6)
:= 6 + (((6 × 6/(6 + 6))6 ) + 6 × 6 × 6)
:= 6 + ((6 × 6 − (66/6)) × (((6 + 6)/6) + 6 × 6))
:= 7/7 + (7 × (((7 + 7)/7) + 7))
:= 7 + ((7/7 + 7) × (777/7 + 7))
:= 77/7 + (7 × (((7 + 7)/7)7 + 7))
:= 88/8 × (88 − ((8 + 8)/8))
:= 8 × 8 + (888 − 8/8)
:= (88 × 88 − (88 + 8))/8
:= (9 × (99 + 9)) − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + ((9 × (99 + 9)) − (9 + 9))
7
9
:= 9 × 9 × (9 + 9) − (((9 + 9)/9) )
Ï 947 := 1 + ((1 + 1) × (((11 + 11)1+1 ) − 11))
Ï 952 := (1 + 1)1+1+1 × (111+1 − (1 + 1))
Ï 957 := 11 × (111 − ((1 + 1) × (1 + 11)))
:= 2/2 + ((2 × 22 ) − 22)
:= 2 × (22 − 2 × (2 + 2))
:= (2 × 222 ) − 22/2
:= 3 + ((33+3 − 3/3) + (3 + 3)3 )
:= (33 + 3/3) × (3/3 + 33)
:= 33 × ((33 − 3/3) + 3)
:= 4 + (4 × 44 − (4 − 4/4)4 )
:= (4 × (44 − 4 × 4)) − 4 − 4
:= 4/4 + ((4 × (44 − 4 × 4)) − 4)
:= 5 + (((55 − 5)/(5 + 5) + 55 /5) + 5)
:= (5 − 5/5) × (((5 − (5 + 5)/5)5 ) − 5)
:= 55/5 × (((5 + 5)/5)5 + 55)
:= 66/6 + ((6 + 6) × (66 + 6 + 6))
:= 6 × 66 + (6666 + 6)/(6 + 6)
:= 6 + ((((6 × 6/(6 + 6))6 ) + 6 × 6 × 6) + 6)
:= ((7 + 7)/7) + (7 × (((7 + 7)/7)7 + 7))
:= 7 + (7 × (((7 + 7)/7)7 + 7))
:= 7 + (77 × (7 + 7) − ((7 + 7)/7)7 )
:= 88/8 + (8 × 8 × (8 + 8) − 88)
:= 8 × 8 + 888
:= 88/8 × (88 − 8/8)
:= 9 × 99 + ((999 + 9)/(9 + 9))
:= (9 × (99 + 9)) − (99/9 + 9)
:= 99/9 × (99 − ((99 + 9)/9))
2
2
73
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 958 := 1 + (11 × (111 − ((1 + 1) × (1 + 11))))
Ï 963 := 1 + (1 + ((1 + ((11 − 1) × (1 + 1 + 1)))1+1 ))
Ï 968 := (1 + 1) × ((11 + 11)1+1 )
:= (2 × (222 − (2 + 2))) − 2
:= (2 × (222 − 2)) − 2/2
:= 2 × 222
:= 3 + ((33 × 33 ) + ((3/3 + 3)3 ))
:= 3 × (333 − (3 × 3 + 3))
:= (3/3 + 3) × ((33+3 − 3)/3)
4
4
:= 4 × 44 − (((44 + 4) + 4)/4)
:= 4 + (4 × 4 − (4 + 4)/4)
:= 4 + ((4 × (44 − 4 × 4)) + 4)
:= (5 − 5/5)5 − (55/5 + 55)
:= (5 − 5/5)5 − ((55 + 5/5) + 5)
:= (5 − 5/5)5 − (55 + 5/5)
:= (((6 + 6)/6)(66−6)/6) − 66
:= 6 × 6 × 6 × 6 − 666 × 6/(6 + 6)
:= (66/6) × (((66 + 66)/6) + 66)
:= (77 × 77 + 777)/7
:= 7 + ((7 × (((7 + 7)/7)7 + 7)) + (77/7))
:= (7/7 + 7) × (((7 + 7)/7)7 − 7)
:= (8 × (8 × (8 + 8) − 8)) − (8 + 8)/8
:= 8 × 8 + (888 + 88/8)
:= 88 × (88/8)
:= 9 + (9999/9 − 9 × (9 + 9))
:= (9 × (99 + 9)) − 9
:= 99/9 × (99 − (99/9))
Ï 959 := ((1 + 1)11 ) − ((11 × (1 + 1 + 1))1+1 )
2
Ï 964 := (1 + 1) × (((11 + 11)1+1 ) − (1 + 1))
2
:= 2 + ((2 × 22 ) − 22/2)
:= 2/2 + (2 × 222 )
:= 2 × (22 − 2)
3
Ï 969 := 1 + ((1 + 1) × ((11 + 11)1+1 ))
3
:= ((3 × 3 + 3/3) ) − (33 + 3)
:= (33 × (33 + 3)) − 3
4
:= 4 × 4 − (4 + 4)/4
:= 4 + (4 × (44 − 4 × 4))
:= 4 × 44 − (44/4 + 44)
:= (5 − 5/5)5 − (55 + 5 + 5)
:= (5 − 5/5)5 − (55 + 5)
:= (5 − 5/5)5 − 55
:= (6/6 + 6) × ((66 − 6/6 + 66) + 6)
:= 6 + ((((6 + 6)/6)(66−6)/6) − 66)
:= (6 × 6 × (6 × 6 − 6)) − 666/6
:= 77 + 7 × (77 + 7 × 7)
:= 7777/7 − 7 × (7 + 7 + 7)
:= (7 + 7) × (77 − 7) − 77/7
:= (8 × (8 × (8 + 8) − 8)) − 8/8
:= 8 + ((88 × 88 − (88 + 8))/8)
:= (88 × 88 + 8)/8
:= 999 + ((9 − 9 × 9 × 9)/(9 + 9))
:= 9/9 + ((9 × (99 + 9)) − 9)
:= (9 × (99 + 9)) − (9 + 9 + 9)/9
:= ((3 + 3) × (33 − 3/3)) − 3/3
4
Ï 960 := (1 + 11) × ((11 − 1 − 1)1+1 − 1)
:= 2 × (222 − (2 + 2))
:= (33 + 3) × (33 − 3/3)
:= 4 × (44 − 4 × 4)
Ï 965 := ((1 + 1) × (((11 + 11)1+1 ) − 1)) − 1
:= 2/2 + (2 × (222 − 2))
:= 2 + (2 × 222 )
:= (3 × 333) − (3/3 + 33)
:= 3 + (((3 × 3 + 3/3)3 ) − 33)
:= 4 + ((4 × (44 − 4 × 4)) + 4/4)
5
Ï 970 := (1 + 1) × (1 + ((11 + 11)1+1 ))
:= (44 × 44 + 4)/((4 + 4)/4)
:= (5 × 5 + 5) × ((5 + 5)/5)
:= 5 + ((5 × 5 + 5) × ((5 + 5)/5) )
:= 5/5 + ((5 − 5/5)5 − 55)
5
:= (6 − 66) × (((6 − 66)/6) − 6)
:= 66 + ((6 × ((6 + 6) × (6 + 6) + 6)) − 6/6)
:= ((6 − 666)/6) + (6 × 6 × (6 × 6 − 6))
:= 7/7 + (7 × (77 + 7 × 7) + 77)
:= 7 + ((77 × 77 + 777)/7)
:= ((7 − 77)/7) + (7 + 7) × (77 − 7)
:= 8 × (8 × (8 + 8) − 8)
:= 8 + (88/8 × (88 − 8/8))
:= ((88 × 88 + 8) + 8)/8
:= (9 − 9/9) × (999/9 + 9)
:= (9 + 9)/9 + ((9 × (99 + 9)) − 9)
:= (9 × (99 + 9)) − (9 + 9)/9
Ï 961 := (1 + ((11 − 1) × (1 + 1 + 1)))1+1
:= ((2/2 + 2)2 + 22)2
:= ((33 + 3/3) + 3)3−3/3
4
:= 4/4 + (4 × (4 − 4 × 4))
Ï 966 := (1 + 1) × (((11 + 11)1+1 ) − 1)
Ï 971 := 1 + ((1 + 1) × (1 + ((11 + 11)1+1 )))
:= (2 × 222 ) − 2
:= 2 + ((2 × 222 ) + 2/2)
:= (3 × 333) − 33
:= (33 × (33 + 3)) − 3/3
:= (44 × 44 − 4)/((4 + 4)/4)
:= 44/4 + (4 × (44 − 4 × 4))
(5+5)/5
:= 5 + (((5 × 5 + 5/5) + 5)(5+5)/5 )
:= ((5 − 5/5)5/5+5 ) − 55
(6+6)/6
:= 66 + (6 × ((6 + 6) × (6 + 6) + 6))
:= ((6 + 6 + 6) × (66 − (6 + 6))) − 6/6
:= (7 + 7) × (77 − (7/7 + 7))
:= (7 + 7) × (77 − 7) − ((7 + 7)/7 + 7)
:= 8/8 + (8 × (8 × (8 + 8) − 8))
:= (88 × 88 − (8 + 8))/8
:= 88/8 + (8 × (8 × (8 + 8) − 8))
:= (9 × (99 + 9)) − 99/9
:= 999 − (99/((9 + 9 + 9)/9))
:= (9 × (99 + 9)) − 9/9
:= ((5 × 5 + 5/5) + 5)
:= ((6 × 6 − 6) + 6/6)
:= (7 × 7 − (77/7 + 7))
(7+7)/7
Ï 962 := 1 + ((1 + ((11 − 1) × (1 + 1 + 1)))1+1 )
Ï 967 := 1111 − ((1 + 11)1+1 )
Ï 972 := (1 + 11) × (11 − 1 − 1)1+1
:= (2 × (222 − 2)) − 2
:= (2 × 222 ) − 2/2
:= 2 × (222 + 2)
:= (33 × 33 ) + (((3 + 3)3 − 3)/3)
:= ((3 × 3 + 3/3)3 ) − 33
:= 33 × (33 + 3)
:= (4 + 4)/4 + (4 × (44 − 4 × 4))
:= ((44 × (44 + 44)) − 4)/4
:= 4 × ((4 − 4/4)4+4/4 )
:= (5 × 5 + 5/5) × (((5 + 5)/5)5 + 5)
:= (5 − 5/5)5 − ((5 + 5)/5 + 55)
:= (5 − 5/5) × ((5 − (5 + 5)/5)5 )
:= 6/6 + (((6 × 6 − 6) + 6/6)(6+6)/6 )
:= 6 + (((6 × 6 − 6) + 6/6)(6+6)/6 )
:= (6 + 6 + 6) × (66 − (6 + 6))
:= (7 + 7) × (77 − 7) − (77/7 + 7)
:= 77 × (7 + 7) − 777/7
:= (7 + 7) × (77 − 7) − (7/7 + 7)
:= (8 + 8)/8 + (8 × (8 × (8 + 8) − 8))
:= (88 × 88 − 8)/8
:= 88 + (888 − (8/((8 + 8)/8)))
:= (9 × (99 + 9)) − 9/9 − 9
:= (9 × (99 + 9)) + ((9 − 99)/(9 + 9))
:= 9 × (99 + 9)
74
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 973 := 1 + ((1 + 11) × (11 − 1 − 1)1+1 )
Ï 978 := 1111 − (1 + 11 × (1 + 11))
Ï 983 := 1 + 1 + (11 − 1 − 1) × (111 − 1 − 1)
2
:= 2/2 + (2 × (22 + 2))
:= 2 + (2 × (22 + 2 + 2))
:= 22/2 + (2 × (222 + 2))
:= 3/3 + (33 × (33 + 3))
:= 3 + ((33 × (33 + 3)) + 3)
:= 33/3 + (33 × (33 + 3))
:= 4 + (4 × 44 − (44/4 + 44))
:= 4 × 44 − ((4 + 4)/4 + 44)
:= 4 + 4 × 44 − (44 + 4/4)
:= 5 + ((5 − 5/5)5 − (55 + 5/5))
:= 5 + (((5 − 5/5)5 − (55 + 5/5)) + 5)
:= (5 − 5/5)5 − ((55/5 + 5 × 5) + 5)
2
:= 6/6 + ((6 + 6 + 6) × (66 − (6 + 6)))
:= 6 + ((6 + 6 + 6) × (66 − (6 + 6)))
:= 66/6 + ((6 + 6 + 6) × (66 − (6 + 6)))
:= (7 + 7) × (77 − 7) − 7
:= (7 + 7) × (77 − 7) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 + 7) × (77 − 7)
:= 8 + ((88/8 × (88 − 8/8)) + 8)
:= 8 + (((88 × 88 + 8) + 8)/8)
:= 8 + (((88 × 88 − 8)/8) + 8)
:= 9/9 + (9 × (99 + 9))
:= 999 − (((99 + 9)/9) + 9)
:= 99/9 + (9 × (99 + 9))
Ï 974 := 1 + (1 + ((1 + 11) × (11 − 1 − 1)1+1 ))
:= 2 + (2 × (222 + 2))
3
:= 3 + ((3 × (33 + 3)) − 3/3)
Ï 984 := (1 + 11) × (1 + (11 − 1 − 1)1+1 )
Ï 979 := 11 × (111 − 11 − 11)
:= 22/2 + (2 × 222 )
:= 2 × (2 × (2 + 2) + 222 )
:= (3 × (333 − 3)) − 33/3
:= 3 + (3 × (333 − (3 + 3)))
4
:= 4 + ((44 × 44 + 4)/((4 + 4)/4))
:= 4 × 4 − (44 + 4/4)
:= 4 + (4 × 44 − 44)
:= 5 + ((5 − 5/5)5 − 55)
:= 5 + (((5 − 5/5)5 − 55) + 5)
:= 5 + ((((5 − 5/5)5 − 55) + 5) + 5)
:= (6 + 6)/6 + ((6 + 6 + 6) × (66 − (6 + 6)))
:= (6666/6) − (66 + 66)
:= 6 + (((6 + 6 + 6) × (66 − (6 + 6))) + 6)
:= 7/7 + ((7 + 7) × (77 − 7) − 7)
:= (7 + 7) × (77 − 7) − 7/7
:= 77/7 + ((7 + 7) × (77 − 7) − 7)
:= 8 + ((88 × 88 − (8 + 8))/8)
:= 88/8 × (8/8 + 88)
:= 8 + (888 + 88)
:= (9 + 9)/9 + (9 × (99 + 9))
:= 999 − (99/9 + 9)
:= (9 × (99 + 9)) + (99 + 9)/9
Ï 975 := 111 + (((1 + 11)1+1+1 )/(1 + 1))
Ï 980 := 1 + 11 × (111 − 11 − 11)
Ï 985 := 1 + (1 + 11) × (1 + (11 − 1 − 1)1+1 )
:= 2 + ((2 × (222 + 2)) + 2/2)
:= 2 × ((222 + 2 + 2) + 2)
:= 2 + ((2 × (222 + 2)) + 22/2)
:= 3 + (33 × (33 + 3))
:= (3 × (333 − (3 + 3))) − 3/3
:= (3 × 333) − (33/3 + 3)
4
4
:= 4 × 4 − 44
:= 4 + ((4 × 44 − 44) + 4/4)
:= 5 × ((5 × 5 × (5 + 5)) − 55)
:= 5 + (5 × ((5 × 5 × (5 + 5)) − 55))
:= 5 + ((5 × ((5 × 5 × (5 + 5)) − 55)) + 5)
:= 666/6 + (6 × (6 + 6) × (6 + 6))
:= (6 × 6 − 6/6) × (((66 + 66)/6) + 6)
:= 6 + ((6666/6) − (66 + 66))
:= 7 + ((7/7 + 7) × (((7 + 7)/7) − 7))
:= (7 + 7) × (77 − 7)
:= 7 + ((7 + 7) × (77 − 7) − ((7 + 7)/7))
:= 8 + ((88 × 88 − 8)/8)
:= ((88 × 88 + 88) + 8)/8
:= 8 + (((88 × 88 + 8)/8) + 8)
:= (9 × (99 + 9)) + ((9 + 9 + 9)/9)
:= 9 + ((9 × (99 + 9)) − 9/9)
:= 9 × 9 × 9 + (((9 + 9)/9)9−9/9 )
:= 4 × 4 − ((44 + 4/4) + 4)
7
Ï 976 := (1 + 1)1+1+1 × (1 + 111+1 )
Ï 986 := (11 − 1)1+1+1 − 1 − 1 − 1 − 11
Ï 981 := (11 − 1 − 1) × (111 − 1 − 1)
2
2
:= 2 + ((2 × 22 ) + 22/2)
:= 22 + (2 × (222 − 2))
3
:= 3 × (333 − (3 + 3))
:= (3 × (333 − 3)) − (3/3 + 3)
4
:= 4 × ((4 − 4 × 4) + 4)
:= 4/4 + (4 × 44 − 44)
:= 4 + (((4 + 4)/4 − 44) + 4 × 44 )
:= 5 + (((5 − 5/5)5/5+5 ) − 55 )
:= (5555/5) − (5 × 5 × 5 + 5)
:= 2 × (22 + 2 + 2)
:= 3 + ((3 × (33 + 3)) + 3/3)
:= (5555/5) − 5 × 5 × 5
6
:= ((6 + 6)/6 + 6) × ((666 + 66)/6)
:= 6 × (6 × 6 + 6) + ((6 × 6/(6 + 6)) )
:= ((66/6) + 6) × (((6 + 6)/6)6 − 6)
:= 7 + ((7 + 7) × (77 − 7) − (77/7))
:= 7/7 + (7 + 7) × (77 − 7)
:= 7 + ((7 + 7) × (77 − 7) − 7/7)
:= 88 + 888
:= (8/8 + 8) × ((888 − (8 + 8))/8)
:= 8 + ((((88 × 88 + 8) + 8)/8) + 8)
:= 999 − ((99 + 99 + 9)/9)
:= 9 + (9 × (99 + 9))
:= 999 − ((99 + 9 + 9)/9)
Ï 977 := 1 + (1 + 1)1+1+1 × (1 + 111+1 )
Ï 982 := 1 + (11 − 1 − 1) × (111 − 1 − 1)
Ï 987 := (11 − 1)1+1+1 − 1 − 1 − 11
:= 2/2 + (2 × (222 + 2 + 2))
:= 2 + (2 × ((222 + 2 + 2) + 2))
:= 22 + ((2 × (222 − 2)) + 2/2)
:= 3 + (((33 × (33 + 3)) − 3/3) + 3)
:= 3/3 + (3 × (333 − (3 + 3)))
:= (3 × (333 − 3)) − 3
4
:= 4/4 + (4 × ((4 − 4 × 4) + 4))
:= (4 + 4)/4 + (4 × 4 − 44)
:= (4 × (44 − 4 − 4)) − (4/4 + 4)
:= 5 + ((5 − 5/5) × ((5 − (5 + 5)/5)5 ))
:= 5 + (((5 − 5/5) × ((5 − (5 + 5)/5)5 )) + 5)
:= (5 − 5/5)5 − (((5 + 5)/5)5 + 5)
:= 6 + (((6 + 6 + 6) × (66 − (6 + 6))) − 6/6)
:= (((6 + 6)/6)(66−6)/6) − (6 × 6 + 6)
:= ((66 × (6 × 6 − 6)) − 6)/((6 + 6)/6)
:= (7 + 7) × (77 − 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + (7 + 7) × (77 − 7)
:= 7 + (7 + 7) × (77 − 7)
4
:= 8 + ((88 × 88 + 8)/8)
:= 8 + (((88 × 88 − (8 + 8))/8) + 8)
:= 8 + (88/8 × (8/8 + 88))
:= 999 − ((99 + 99)/9)
:= 9 + ((9 × (99 + 9)) + 9/9)
:= 999 − (99 + 9)/9
75
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 988 := (11 − 1)1+1+1 − 1 − 11
Ï 993 := 1 + 1 + 1 + (11 − 1 − 1) × (111 − 1)
Ï 998 := (11 − 1)1+1+1 − 1 − 1
:= 22 + ((2 × 222 ) − 2)
:= 2 + (((2 × 222 ) + 22) + 2/2)
:= 22 + (2 × (222 + 2 + 2))
:= (3 × 333) − 33/3
:= 3 + (3 × (333 − 3))
:= (3 × 333) − 3/3
4
:= (4 × (4 − 4 − 4)) − 4
:= 4/4 + (4 × (4 − 4 − 4))
:= (4 × (44 − 4)) + (4 − 44)/4
:= (5 − 5/5)5 − (55/5 + 5 × 5)
:= (5 − 5/5)5 − ((5 × 5 + 5/5) + 5)
:= (5 − 5/5)5 − (5 × 5 + 5/5)
:= (((6 + 6)/6)(66−6)/6) − 6 × 6
:= ((66 × (6 × 6 − 6)) + 6)/((6 + 6)/6)
:= 666 + (6 × 66 − ((6 + 6)/6)6 )
:= 7 + ((7 + 7) × (77 − 7) + 7/7)
:= 7 + (((7 + 7) × (77 − 7) − 7/7) + 7)
:= 7 + ((7 + 7) × (77 − 7) + (77/7))
:= 8 + (((88 × 88 + 88) + 8)/8)
:= 8 + ((((88 × 88 + 8)/8) + 8) + 8)
:= 888 + ((888 − 8)/8)
:= 999 − 99/9
:= 999 + (((9 + 9 + 9)/9) − 9)
:= 999 − 9/9
Ï 989 := (11 − 1)1+1+1 − 11
Ï 994 := (1 + 1)11−1 − (11 − 1) × (1 + 1 + 1)
Ï 999 := 111 × (11 − 1 − 1)
4
:= 22 + (2 × (222 + 2))
2
:= 22 + ((2 × 22 ) − 2/2)
:= (2/2 + 2)2 × 222/2
3
:= ((3 × 3 + 3/3) ) − (3 + 3)
:= (3 × (333 − 3)) − 3/3
:= 3 × 333
4
:= 4/4 + ((4 × (44 − 4 − 4)) − 4)
:= ((4/4 + 4) + 4) × 444/4
:= (4 + 4)/4 + (4 × (4 − 4 − 4))
:= (5 − 5/5)5 − ((5 × 5 + 5) + 5)
:= (5 − 5/5) − (5 × 5 + 5)
:= (5 − 5/5)5 − 5 × 5
:= 666 + ((6 × (66 − (6 + 6))) − 6/6)
:= (((66 − 6)/6)6×6/(6+6) ) − 6
:= (6 × 666)/(6 − ((6 + 6)/6))
:= 7 + ((7 + 7) × (77 − 7) + ((7 + 7)/7))
:= 7 + ((7 + 7) × (77 − 7) + 7)
:= ((7 + 7)/7 + 7) × 777/7
:= 888 + (8888/88)
:= 8 + (((((88 × 88 + 8) + 8)/8) + 8) + 8)
:= 888 + 888/8
:= 999 − 9/9 − 9
:= 999 + ((9 − 99)/(9 + 9))
:= 999
5
Ï 1000 := (11 − 1)1+1+1
Ï 995 := (11 − 1)1+1+1 − 1 − 1 − 1 − 1 − 1
Ï 990 := (11 − 1 − 1) × (111 − 1)
:= 22 + (2 × 222 )
:= 22 + ((2 × (222 + 2)) + 2/2)
:= 2 × (22+2 + 222 )
:= 3 × (333 − 3)
:= (3 × 333) − (3/3 + 3)
:= (3 × 3 + 3/3)3
:= (4 × (44 − 4 − 4)) − (4 + 4)/4
:= 4 + ((4 × (44 − 4 − 4)) − 4/4)
:= (4 × (44 − 4)) − 4 − 4
:= (5 + 5) × (5 × (5 × 5 − 5) − 5/5)
:= (5 × (5 + 5) × (5 × 5 − 5)) − 5
:= 5 × (5 + 5) × (5 × 5 − 5)
:= 6 × ((66 × (6 × 6 − 6))/(6 + 6))
:= 666 + (6 × 66 − (66 + 6/6))
:= ((66 − 6)/6)6×6/(6+6)
:= 77/7 × ((77 − 7/7 + 7) + 7)
:= 7 + (((7 + 7) × (77 − 7) + 7/7) + 7)
:= ((77 − 7)/7)(7+7+7)/7
:= 88/8 × ((8 + 8)/8 + 88)
:= 8 + ((88/8 × (8/8 + 88)) + 8)
:= 8 × 8 × (8 + 8) − 8 − 8 − 8
:= 999 − 9
:= 999 + ((9 − 9 × 9)/(9 + 9))
:= 9/9 + 999
Ï 996 := (1 + 1 + 1) × ((1 + 1 + 1) × 111 − 1)
Ï 991 := 1 + (11 − 1 − 1) × (111 − 1)
2
Ï 1001 := 1 + (11 − 1)1+1+1
:= 22 + ((2 × 222 ) + 2/2)
:= 2 + ((2 × (22 + 2)) + 22)
:= ((222/2 − 2)/2) − 22
:= 3/3 + (3 × (333 − 3))
:= (3 × 333) − 3
:= 3 + (3 × 333 − 3/3)
:= (4 × (4 − 4 − 4)) − 4/4
:= 4 + (4 × (44 − 4 − 4))
:= 5 + ((5555/5) − 5 × 5 × 5)
:= 5/5 + ((5 × (5 + 5) × (5 × 5 − 5)) − 5)
:= 5/5 + 5 × (5 + 5) × (5 × 5 − 5)
:= 6/6 + ((6 × (66 − (6 + 6))) + 666)
:= (6 + 6) × (66/6 + 66 + 6)
:= (6/6 + 6) × ((6 + 6) × (6 + 6) − 6/6)
:= 77/7 + (7 + 7) × (77 − 7)
:= (77 + 7)/7 × (77 − 7/7 + 7)
:= 77 × (7 − 7/7 + 7)
:= 888 + (888/8 − 8)
:= 8 × 8 × 8 + 88 × 88/(8 + 8)
:= 8/8 + (8 × 8 × (8 + 8) − (8 + 8 + 8))
:= 9/9 + (999 − 9)
:= 999 − (9 + 9 + 9)/9
:= (9 + 9)/9 + 999
4
Ï 992 := 1 + 1 + (11 − 1 − 1) × (111 − 1)
Ï 997 := (11 − 1)1+1+1 − (1 + 1 + 1)
:= 2 + ((2 × 22 ) + 22)
:= ((2/2 + 2)2 × 222/2) − 2
:= 3 + ((3 × (333 − 3)) − 3/3)
:= ((3 × 3 + 3/3)3 ) − 3
2
Ï 1002 := 1 + 1 + (11 − 1)1+1+1
:= 22×(2+2)+2 − 22
:= 3 + 3 × 333
4
4
:= (4 × (4 − 4)) − 44/4
:= 4 × (4 − 4 − 4)
5
:= 4 + (4 × (44 − 4) − 44/4)
5
5
:= 4 × 44 − (44/((4 + 4)/4))
5
:= 5 + ((5 − 5/5) − ((5 + 5)/5) )
:= (5 − 5/5) − ((5 + 5)/5)
6
:= 6 + (((66/6) + 6) × (((6 + 6)/6) − 6))
(6+6)/6
:= 6 × 6 + (((6 × 6 − 6) + 6/6)
(7+7)/7
:= (5 + 5)/5 + 5 × (5 + 5) × (5 × 5 − 5)
)
:= 66 + ((6 + 6) × (66 + 6 + 6))
:= (77 + 7)/7 + (7 + 7) × (77 − 7)
:= (((77 + 7)
:= 8 + ((888 + 88) + 8)
:= 888 + ((888 − (8 + 8))/8)
:= ((8 + 8)/8) × (8 × 8 × 8 − 88/8)
:= (9 + 9)/9 + (999 − 9)
:= 999 − (9 + 9)/9
:= 999 + ((9 + 9 + 9)/9)
) − 77)/7
76
:= 7/7 + 77 × (7 − 7/7 + 7)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1003 := 1 + 1 + 1 + (11 − 1)1+1+1
Ï 1008 := (11 − 1 − 1) × (1 + 111)
Ï 1013 := (1 + 1)11−1 − 11
:= (222/2 + 2)/2 − 22
:= 2 × (222 − 2 + 22)
:= (222/2 − 22)/2
:= 3 + ((3 × 3 + 3/3)3 )
:= 3 × (333 + 3)
:= 3 + (3 × 333 + 33/3)
4
:= 4 × (44 − 4) − (4/4 + 4)
:= 4 × (4 − 4)
:= 4 × 44 − 44/4
:= 5 + ((5 − 5/5)5 − (5 × 5 + 5/5))
:= (5 − 5/5)5 − (55/5 + 5)
:= (5 − 5/5)5 − 55/5
:= (66/6 + 6) × (66 − (6/6 + 6))
:= 6 × ((6 × 6 + 66) + 66)
:= (6 × 6 × (6 × 6 − 6)) − (66 + 6/6)
:= ((7 + 7)/7) + 77 × (7 − 7/7 + 7)
:= 7 + 77 × (7 − 7/7 + 7)
:= 7777/7 − 7 × (7 + 7)
:= 8 + ((88/8 × (8/8 + 88) + 8) + 8)
:= (8 + 8) × (8 × 8 − 8/8)
:= 8 × 8 × (8 + 8) − 88/8
:= 999 + ((9 × 9 − 9)/(9 + 9))
:= 9 + 999
:= (9999 + 9)/9 − 99
Ï 1004 := 1 + 1 + 1 + 1 + (11 − 1)1+1+1
Ï 1009 := 1 + (11 − 1 − 1) × (1 + 111)
Ï 1014 := 1 + (1 + 1)11−1 − 11
:= 2 + 22×(2+2)+2 − 22
:= 2/2 + 2 × (222 − 2 + 22)
:= 2 + 2 × (222 + 22)
:= 3 + ((3 × 333 − 3/3) + 3)
:= 3 × 3 + ((3 × 3 + 3/3)3 )
:= 3 + (3 × (333 + 3) + 3)
:= 4 × (4 − 4) − 4
:= 4/4 + 4 × (4 − 4)
:= 4 × 44 + (4 − 44)/4
:= 5 + ((5 − 5/5)5 − 5 × 5)
:= (5 − 5/5)5 − (5 + 5 + 5)
:= (5 − 5/5)5 − 5 − 5
:= (6 − ((6 + 6)/6)) × (6 × (6 × 6 + 6) − 6/6)
:= 6/6 + (66 × (6 + 6) + 6 × 6 × 6)
:= (6 × 6 × (6 × 6 − 6)) − 66
4
4
:= 7 + ((((77 + 7)
(7+7)/7
) − 77)/7)
:= (((77 + 7)
(7+7)/7
) + 7)/7
:= (7 − 7/7 + 7) × (7/7 + 77)
:= 8 × 8 × (8 + 8) − ((88 + 8)/8 + 8)
:= 8/8 + ((8 + 8) × (8 × 8 − 8/8))
:= (8 − 88)/8 + 8 × 8 × (8 + 8)
:= 999 + ((9 × 9 + 9)/(9 + 9))
:= 9 + (999 + 9/9)
:= 9 + ((999 − ((9 + 9 + 9)/9)) + 9)
Ï 1005 := 1 + 1 + 1 + 1 + 1 + (11 − 1)1+1+1
Ï 1015 := 1 + 1 + (1 + 1)11−1 − 11
Ï 1010 := (11111 − 1)/11
:= (2 + 222/2 + 2)/2 − 22
:= 2 × (222 + 22) − 2
:= 2 + (222/2 − 22)/2
:= 3 + (3 × 333 + 3)
:= 33/3 + 3 × 333
:= 3 + ((3 × (333 + 3) + 3/3) + 3)
:= 4/4 + (4 × (44 − 4) − 4)
:= (4 + 4)/4 + 4 × (44 − 4)
:= 5 + 5 × (5 + 5) × (5 × 5 − 5)
:= 5 + (5 × (5 + 5) × (5 × 5 − 5) + 5)
:= 5 × (((55 − 5)/(5 + 5 + 5)) − 5)
:= 6 + ((6 × 666)/(6 − ((6 + 6)/6)))
:= 6 × 6 × 6 + (66 × (6 + 6) + ((6 + 6)/6))
:= 6/6 + ((6 × 6 × (6 × 6 − 6)) − 66)
:= 7 + (((7 + 7) × (77 − 7) + (77/7)) + 7)
:= ((7 + 7)/7)7 + 7 × (77 + 7 × 7)
:= 7 × 7 × (7 + 7 + 7) − (7 + 7)
:= 8 × 8 × (8 + 8) − (88/8 + 8)
:= 888 + ((888 + 88)/8)
:= 8 × 8 × (8 + 8) − (8/8 + 8)
:= 9 + (999 − ((9 + 9 + 9)/9))
:= 99/9 + 999
:= (((9 + 9)/9)9/9+9 ) − 9
Ï 1006 := (11 − 1 − 1) × (1 + 111) − 1 − 1
Ï 1011 := 11 + (11 − 1)1+1+1
:= 4 × 44 − ((4/4 + 4) + 4)
Ï 1016 := 1 + 1 + 1 + (1 + 1)11−1 − 11
:= (2 × ((222 − 2) + 22)) − 2
:= 2 × (222 + 22) − 2/2
:= 2 × (222 + 22 + 2)
:= 3 + (((3 × 3 + 3/3)3 ) + 3)
:= 3 + (3 × (333 + 3))
:= (3 × ((333 + 3) + 3)) − 3/3
:= 4 × (4 − 4) − (4 + 4)/4
:= 4 + (4 × (4 − 4) − 4/4)
:= 4 × 44 − 4 − 4
:= 5 + (5 × (5 + 5) × (5 × 5 − 5) + 5/5)
:= 5 × 55 + (555 + 55 )/5
:= (5 + 5)/5 + ((5 − 5/5)5 − (5 + 5))
:= 6 + (((66 − 6)/6)6×6/(6+6) )
:= 6 + (((6 × 666)/(6 − ((6 + 6)/6))) + 6)
:= (6 × 6 × (6 × 6 − 6)) − ((6 + 6)/6)6
:= 7 + (((7 + 7)/7 + 7) × 777/7)
:= 7 × 7 × (7 + 7 + 7) − (77/7 + 7)
:= 7 + ((((77 + 7)(7+7)/7 ) + 7)/7)
:= 8 + ((888 − 8)/8 + 888)
:= 8 × 8 × (8 + 8) − (88 + 8 + 8)/8
:= 8 × 8 × (8 + 8) − 8
:= 9 + (999 − ((9 + 9)/9))
:= 999 + (99 + 9)/9
:= 9 + ((999 − 9/9) + 9)
4
Ï 1007 := (11 − 1 − 1) × (1 + 111) − 1
:= (2
22/2
− 2)/2 − 2
2+2
:= (3 × (333 + 3)) − 3/3
4
Ï 1012 := (1 + 1)11−1 − 1 − 11
Ï 1017 := (11 − 1 − 1) × (1 + 1 + 111)
:= 2 × (22 + 22)
:= 2 + (222/2 − 22)/2 + 2
:= 3 + (3 × (333 + 3) + 3/3)
:= 3 × ((333 + 3) + 3)
2
:= 4 × (4 − 4) − 4/4
:= 4 + 4 × (4 − 4)
:= 4 + (4 × 44 − 44/4)
:= (5 − 5/5)5 − (((55 + 5)/5) + 5)
:= (5 − 5/5)5 − (55 + 5)/5
:= (5 − 5/5)5 − ((5 + 5)/5 + 5)
:= 6 × 6 × 6 + (66 × (6 + 6) − 6/6)
:= (((6 + 6)/6)(66−6)/6) − 6 − 6
:= 6 × (6 × 6 + 6 + 6) + ((6 × 6/(6 + 6))6 )
:= (((77 + 7)(7+7)/7 ) − 7)/7
:= 77/7 + 77 × (7 − 7/7 + 7)
:= ((7/7 + 7) × ((7 + 7)/7)7 ) − 7
:= 8 + (888/8 + 888)
:= 8 × 8 × (8 + 8) − (88 + 8)/8
:= 8/8 + (8 × 8 × (8 + 8) − 8)
:= 9 + (999 − 9/9)
:= 9999/9 − 99
:= 9 + (999 + 9)
4
4
77
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1018 := ((1 + 1)11 − (1 + 11))/(1 + 1)
Ï 1023 := (1 + 1)11−1 − 1
Ï 1028 := 1 + (1 + 1 + 1 + (1 + 1)11−1 )
:= 2 + (2 × ((222 + 22) + 2))
:= (222/2 − 2)/2
:= 2 + ((22×(2+2)+2 ) + 2)
:= 3/3 + (3 × ((333 + 3) + 3))
:= 33 + (3 × 333 − 3)
:= (3 × (((3/3 + 3) + 3)3 )) − 3/3
:= 4 × 4 − ((4 + 4)/4 + 4)
:= 4 × 44 − 4/4
:= 4 + 4 × 44
:= (5 − 5/5)5 − (5/5 + 5)
:= (5 − 5/5)5 − 5/5
:= 5 + ((5 − 5/5)5 − 5/5)
:= (((6 + 6)/6)(66−6)/6) − 6
:= (((6 + 6)/6)(66−6)/6) − 6/6
:= 666 + (6 × (66 − 6) + ((6 + 6)/6))
:= 7 × 7 × (7 + 7 + 7) − 77/7
:= 7/7 + (7 × 7 × (7 + 7 + 7) − 7)
:= 7 × 7 × (7 + 7 + 7) − 7/7
:= (8 + 8)/8 + (8 × 8 × (8 + 8) − 8)
:= 8 × 8 × (8 + 8) − 8/8
:= 8 × 8 × (8 + 8) + (8/((8 + 8)/8))
:= 9 + ((999 + 9/9) + 9)
:= (((9 + 9)/9)9/9+9 ) − 9/9
:= 9 + ((99/9 + 999) + 9)
4
Ï 1019 := (1 + (1 + 1)11 − 11)/(1 + 1)
Ï 1029 := (11 + (1 + 1)11 − 1)/(1 + 1)
Ï 1024 := (1 + 1)11−1
:= ((222/2 − 2)/2) − 2 − 2
:= 22×(2+2)+2
:= 2 + (((222/2 + 2)/2) + 2)
:= 33/3 + (3 × (333 + 3))
:= (3 − 3/3)3×3+3/3
:= 3 × (((3/3 + 3) + 3)3 )
:= 4 × 44 − (4/4 + 4)
:= 4 × 44
:= 4 + (4 × 44 + 4/4)
:= (5 − 5/5)5 − 5
:= (5 − 5/5)5
:= 5 + (5 − 5/5)5
:= ((66 − 6) × (66/6 + 6)) − 6/6
:= ((6 + 6)/6)(66−6)/6
:= 666 + 66 × 66/(6 + 6)
:= ((7 − 77)/7) + 7 × 7 × (7 + 7 + 7)
:= (7/7 + 7) × ((7 + 7)/7)7
:= 7 × 7 × (7 + 7 + 7)
:= 88/8 + ((8 + 8) × (8 × 8 − 8/8))
:= 9 + (99/9 + 999)
Ï 1020 := (1 + 1)11−1 − 1 − 1 − 1 − 1
2
:= 8 × 8 × (8 + 8)
:= ((9999 − 9)/9) − 9 × 9
:= ((9 + 9)/9)9/9+9
Ï 1030 := (1 + 11 + (1 + 1)11 )/(1 + 1)
Ï 1025 := 1 + (1 + 1)11−1
:= 2 + (((22×(2+2)+2 ) + 2) + 2)
:= 2 × ((2(2/2+2) ) − 2)
:= (222/2 + 2)/2
:= 3 + (3 × ((333 + 3) + 3))
:= 33 + (3 × 333 − 3/3)
:= 4 × 44 − 4
:= 5/5 + ((5 − 5/5)5 − 5)
:= (66 − 6) × (66/6 + 6)
:= 7 + (7777/7 − 7 × (7 + 7))
:= 8 × 8 × (8 + 8) − (8/((8 + 8)/8))
:= 9 + (((99 + 9)/9) + 999)
Ï 1021 := (1 + 1)11−1 − 1 − 1 − 1
:= ((222/2 − 2)/2) − 2
:= 8 + ((8 × 8 × (8 + 8) − (88/8)) + 8)
:= 4/4 + 4 × 44
:= 3 + (((3 × 3 + 3/3)3 ) + 33 )
:= 4 + ((4 + 4)/4 + 4 × 44 )
:= 5 + ((5 − 5/5)5 + 5/5)
:= 5/5 + (5 − 5/5)5
:= 6/6 + (((6 + 6)/6)(66−6)/6)
:= 7 + (7 × 7 × (7 + 7 + 7) − (77/7))
:= 8/8 + 8 × 8 × (8 + 8)
:= 9 + (((999 − 9/9) + 9) + 9)
Ï 1026 := 1 + 1 + (1 + 1)11−1
:= 2 + (22×(2+2)+2 )
:= 6 + (((6 + 6)/6)(66−6)/6)
:= 7/7 + 7 × 7 × (7 + 7 + 7)
:= 8 + (8 × 8 × (8 + 8) − ((8 + 8)/8))
:= 9999/9 − 9 × 9
Ï 1031 := 1 + (1 + 11 + (1 + 1)11 )/(1 + 1)
:= 2 + ((((222/2 + 2)/2) + 2) + 2)
:= 33 + (3 × 333 − 3/3)
:= ((3 − 3/3)3×3+3/3 ) − 3
:= 3 × (333 + 3 × 3)
:= 4/4 + (4 × 44 − 4)
:= (4 + 4)/4 + 4 × 44
:= (5 + 5)/5 + ((5 − 5/5)5 − 5)
:= (5 + 5)/5 + (5 − 5/5)5
:= 6/6 + ((66 − 6) × (66/6 + 6))
:= 666 + 6 × (66 − 6)
:= 7 × 7 × (7 + 7 + 7) − (7/7 + 7)
:= ((7 + 7 + 7)/7) × (7 × 7 × 7 − 7/7)
:= 7 + ((7/7 + 7) × ((7 + 7)/7)7 )
:= 8 + (8 × 8 × (8 + 8) − (88/8))
:= (8 + 8)/8 + 8 × 8 × (8 + 8)
:= 8 + (8 × 8 × (8 + 8) − 8/8)
:= 9 + (9999/9 − 99)
:= 9 + (999 + 9 + 9)
:= (9999 + 9)/9 − 9 × 9
Ï 1022 := (1 + 1)11−1 − 1 − 1
:= 4 + ((4 × 44 − 4/4) + 4)
Ï 1027 := 1 + 1 + 1 + (1 + 1)11−1
:= 5 + ((5 − 5/5)5 + ((5 + 5)/5))
:= 6 + ((((6 + 6)/6)(66−6)/6) + 6/6)
Ï 1032 := 11 + (1 + 1)11−1 − 1 − 1 − 1
:= (22×(2+2)+2 ) − 2
:= 2 + ((222/2 + 2)/2)
:= 2 × (2 × (22×(2+2) + 2))
:= 3 + (3 × (333 + 3) + 33/3)
:= 33 + ((3 × 3 + 3/3)3 )
:= 33 + 3 × 333
:= 4 × 4 − (4 + 4)/4
:= 4 + (4 × 44 − 4/4)
:= 4 + (4 × 44 + 4)
:= (5 − 5/5)5 − (5 + 5)/5
:= 5 + ((5 − 5/5)5 − ((5 + 5)/5))
:= 5 + (5 − 5/5)5 − (5 + 5)/5 + 5
:= 6 + ((6 × 6 × (6 × 6 − 6)) − ((6 + 6)/6)6 )
:= 6/6 + (6 × (66 − 6) + 666)
:= 6 + (6 × (66 − 6) + 666)
:= 7 × 7 × (7 + 7 + 7) − 7
:= 7 × 7 × (7 + 7 + 7) − (7 + 7)/7
:= (7/7 + 7) × (((7 + 7)/7)7 + 7/7)
:= 8 × 8 × (8 + 8) − (8 + 8)/8
:= 88/8 + (8 × 8 × (8 + 8) − 8)
:= 8 + 8 × 8 × (8 + 8)
:= 9 + ((9999 + 9)/9 − 99)
:= 9 + (((999 + 9/9) + 9) + 9)
:= 9 + ((((9 + 9)/9)9/9+9 ) − 9/9)
4
78
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1033 := 11 + (1 + 1)11−1 − 1 − 1
Ï 1038 := 1 + 1 + 1 + 11 + (1 + 1)11−1
Ï 1043 := (1 + 1) × (11 − 1) + (1 + 1)11−1 − 1
:= ((222/2 + 22)/2) − 2
:= 2 + (2 × ((2 × (22×(2+2) + 2)) + 2))
:= 22 + (((222/2 − 2)/2) − 2)
:= 33 + ((3 × 3 + 3/3)3 )
:= 3 × ((((3/3 + 3) + 3)3 ) + 3)
:= 33 + (3 × 333 + 33/3)
:= 4 + ((4 × 44 + 4/4) + 4)
:= 4 × (44 + 4) − (4 + 4)/4
:= 4 + (4 × (44 + 4) − 4/4)
:= 5 + (((5 − 5/5)5 − 5/5) + 5)
:= 5 + ((((5 − 5/5)5 − 5/5) + 5) + 5)
:= 5 × 5 + ((5 − 5/5)5 − (5/5 + 5))
:= (6666/6) − (66 + 6 + 6)
:= (6 × (6 × (6 × 6 − 6) − 6)) − 6
:= (6 × (6 × (6 × 6 − 6) − 6)) − 6/6
:= (7777 − 7)/7 − 77
:= 7 + (((7/7 + 7) × ((7 + 7)/7)7 ) + 7)
:= 7 + (7 × 7 × (7 + 7 + 7) + 7)
:= 8 + (8 × 8 × (8 + 8) + 8/8)
:= 8 + ((8 × 8 × (8 + 8) − ((8 + 8)/8)) + 8)
:= 8 + (8 × 8 × (8 + 8) + (88/8))
:= 9 + (((9999 − 9)/9) − 9 × 9)
:= (9 × (99 + 9 + 9)) − 9/9 − 9
9/9+9
:= 9 + (((9 + 9)/9)
)
Ï 1034 := 11 + (1 + 1)11−1 − 1
Ï 1039 := 1 + 1 + 1 + 1 + 11 + (1 + 1)11−1
Ï 1044 := ((1 + 1) × (11 − 1)) + (1 + 1)11−1
:= 2 + (2 × (2 × (22×(2+2) + 2)))
:= 2 + (((222/2 + 22)/2) + 2)
:= 22 + ((22×(2+2)+2 ) − 2)
:= ((33/3)3 ) − 3 × 3 × 33
:= (3333 − (3 + 3)3 )/3
:= 3 + (3 × (333 + 3) + 33)
:= 4 × 44 + (44 − 4)/4
:= 4 × (44 + 4) − 4/4
:= 4 + 4 × (44 + 4)
:= 5 + ((5 − 5/5)5 + 5)
:= 5 + (((5 − 5/5)5 + 5) + 5)
:= 5 × 5 + ((5 − 5/5)5 − 5)
:= ((6666 − 66)/6) − 66
:= (6666/6) − (66 + 6)
:= 6 × (6 × (6 × 6 − 6) − 6)
:= 7777/7 − 77
:= ((77 − 7)/7) + 7 × 7 × (7 + 7 + 7)
:= 7 + ((7 × 7 × (7 + 7 + 7) + 7/7) + 7)
:= 8 + (8 × 8 × (8 + 8) + ((8 + 8)/8))
:= 8 + ((8 × 8 × (8 + 8) − 8/8) + 8)
:= ((88 + 8)/8) × (88 − 8/8)
:= 9 + ((((999 − 9/9) + 9) + 9) + 9)
:= 9 + (9999/9 − 9 × 9)
:= (9 × (99 + 9 + 9)) − 9
Ï 1035 := 11 + (1 + 1)11−1
Ï 1040 := 11 + ((11 + ((1 + 1)11 − 1))/(1 + 1))
Ï 1045 := 11 + (11 + ((1 + 1)11−1 − 1))
:= (222/2 + 22)/2
:= 2 × ((2 + 2 + 2)2 + 222 )
:= 22 + ((222/2 − 2)/2)
:= 3 + (3 × 333 + 33)
:= 33 + (3 × (333 + 3) − 3/3)
:= 33/3 × (3 × 33 − (3/3 + 3))
:= 4 × (4 + 4)
:= 4 + (4 × (44 + 4) + 4/4)
:= 55/5 + (5 − 5/5)5
:= 5 × ((55 − 5)/(5 + 5 + 5))
:= 55 × (5 × 5 − (5/5 + 5))
:= 6 + (66 × 66/(6 + 6) + 666)
:= (6/6 − 66) × (((6 − 66)/6) − 6)
:= (6666/6) − 66
:= 7 + (7 × 7 × (7 + 7 + 7) − 7/7)
:= 77/7 + 7 × 7 × (7 + 7 + 7)
:= 77/7 × ((77/7 + 77) + 7)
:= 88/8 + 8 × 8 × (8 + 8)
:= 8 + (8 × 8 × (8 + 8) + 8)
:= 88/8 × (88 − 8/8 + 8)
:= 9 + ((999 + 9 + 9) + 9)
:= 9 + ((9999 + 9)/9 − 9 × 9)
:= 9/9 + ((9 × (99 + 9 + 9)) − 9)
:= 44/4 + 4 × 4
4
Ï 1036 := 1 + 11 + (1 + 1)11−1
4
Ï 1041 := 11 + ((1 + (11 + (1 + 1)11 ))/(1 + 1))
Ï 1046 := 11 + 11 + (1 + 1)11−1
:= 2 × ((2 × (22×(2+2) + 2)) + 2)
:= 22+2 + ((222/2 + 2)/2)
:= 22 + (22×(2+2)+2 )
:= 3 + (((3 × 3 + 3/3)3 ) + 33)
:= 33 + (3 × (333 + 3))
:= 3 + ((3 × 333 + 33/3) + 33)
4
:= 4 + (4 × (44 + 4) + (4 + 4)/4)
:= 4 × (44 + 4) − 4
:= 4/4 + 4 × (4 + 4)
:= (5 − 5/5)5 + (55 + 5)/5
:= 5 + ((5 − 5/5)5 + ((55 + 5)/5))
:= 5/5 + (55 × (5 × 5 − (5/5 + 5)))
:= 6 + ((((6 + 6)/6)(66−6)/6) + 6)
:= (6666/6) − (((6 + 6)/6) + 6)
:= (6666 + 6)/6 − 66
:= 7 + 7 × 7 × (7 + 7 + 7)
:= 7 + (7777/7 − 77)
:= 7 + (7 × 7 × (7 + 7 + 7) + ((77 − 7)/7))
:= ((88 + 8)/8) + 8 × 8 × (8 + 8)
:= 8 + ((8 × 8 × (8 + 8) + 8/8) + 8)
:= (8888 − 8)/8 − 8 × 8
:= 9 + ((((999 + 9/9) + 9) + 9) + 9)
:= (9 × (99 + 9 + 9)) − (99 + 9)/9
:= (9 + 9)/9 + ((9 × (99 + 9 + 9)) − 9)
6
Ï 1042 := (1 + 1)11−1 + (1 + 1) × (11 − 1 − 1)
Ï 1047 := 1 + (11 + 11 + (1 + 1)11−1 )
:= 2 + ((222/2 + 22)/2)
:= 2 + ((22×(2+2)+2 ) + 22+2 )
:= 22 + ((222/2 + 2)/2)
:= 3 + (((33/3)3 ) − 3 × 3 × 33)
:= 3 + ((3333 − (3 + 3)3 )/3)
:= 3 × (((((3/3 + 3) + 3)3 ) + 3) + 3)
:= 4/4 + (4 × (44 + 4) − 4)
:= (4 + 4)/4 + 4 × (44 + 4)
:= (4444 − 44 )/4
:= (5 − 5/5)5 + (55 + 5 + 5)/5
:= (55 + 5/5)/(5 − (5 + 5)/5)
:= 5 + (55 + 5/5)/(5 − (5 + 5)/5)
:= (66/6 + 6) × ((66 − 6) + 6/6)
:= (6 × (6 × (6 × 6 − 6) − 6)) − (6 + 6)/6
:= (6666/6) − ((6 + 6)/6)6
:= 7 + (7 × 7 × (7 + 7 + 7) + 7/7)
:= 7 + ((7 × 7 × (7 + 7 + 7) − 7/7) + 7)
:= 7 + (7 × 7 × (7 + 7 + 7) + (77/7))
:= 8 × 8 × (8 + 8) + (88 + 8 + 8)/8
:= 8 + ((8 × 8 × (8 + 8) + ((8 + 8)/8)) + 8)
:= 8888/8 − 8 × 8
:= 9 + (((99/9 + 999) + 9) + 9)
:= 9 + ((((9 + 9)/9)9/9+9 ) + 9)
:= 9 + ((((9999 − 9)/9) − 9 × 9) + 9)
Ï 1037 := 1 + 1 + 11 + (1 + 1)11−1
79
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1048 := (1 + 1)11 − (11 − 1)1+1+1
:= 2 + ((22×(2+2)+2 ) + 22)
:= 3 + (33/3 × (3 × 33 − (3/3 + 3)))
Ï 1053 := (1 + 1 + 11) × (11 − 1 − 1)1+1
:= 2/2 + (2 × ((22 × (22 + 2)) − 2))
3
:= 3 × (33 + 3 + 3)
Ï 1058 := (1 + 1) × (1 + 11 + 11)1+1
:= 2 × ((22 + 2/2)2 )
:= 3 + (33 × 33 − (3/3 + 33))
4
:= 4 + (4 × (44 + 4) + 4)
:= 44 + (4 × (4 − 4) + 4/4)
:= (4 + 4)/4 + 4 × (44 + 4 + 4)
:= 5 × 5 + ((5 − 5/5)5 − 5/5)
:= 5 + (((5 − 5/5)5 − 5/5) + 5 × 5)
:= (5555 + 5 + 5)/5 − 55
:= 6 + ((6 × (6 × (6 × 6 − 6) − 6)) − ((6 + 6)/6))
:= 6 + ((6666/6) − ((6 + 6)/6)6 )
:= 6 + (((6666 + 6)/6 − 66) + 6)
:= 7 + ((7777/7 − 77) + 7)
:= 77 × (7 + 7) − (77/7 + 7 + 7)
:= 7/7 + ((7 + 7) × (77 − 7) + 77)
:= 8 + ((8 × 8 × (8 + 8) + 8) + 8)
:= 8 + (88/8 × (88 − 8/8 + 8))
:= ((88 × (88 + 8) + 8) + 8)/8
:= 9 + ((9999/9 − 9 × 9) + 9)
:= 9 × (99 + 9 + 9)
:= ((99/9) × (99 − ((9 + 9)/9))) − 9
Ï 1049 := 1 + ((1 + 1)11 − (11 − 1)1+1+1 )
Ï 1054 := 1 + ((1 + 1 + 11) × (11 − 1 − 1)1+1 )
Ï 1059 := 1 + ((1 + 1) × (1 + 11 + 11)1+1 )
:= 2 + (((222/2 + 2)/2) + 22)
:= 2 × (((22 + 2/2)2 ) − 2)
:= ((((2 × 22) + 2)2 ) + 2)/2
:= 33 × (33 + 3 + 3) − (3/3 + 3)
:= 3/3 + 33 × (33 + 3 + 3)
:= 3 + (33 × (33 − 3/3))
:= 4 + ((4 × (44 + 4) + 4/4) + 4)
:= 4 × (44 + 4 + 4) − (4 + 4)/4
:= 5 × 5 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 5 × 5)
:= 5 + (((5 − 5/5)5 + 5 × 5) + 5)
:= 6 + ((6 × (6 × (6 × 6 − 6) − 6)) − 6/6)
:= 6 × 6 + ((((6 + 6)/6)(66−6)/6) − 6)
:= 666 + (6 × 66 − (6 × 6/(6 + 6)))
:= ((7 + 7) × (77 − (7 + 7)/7)) − 7/7
:= (7777 − 7)/7 − (7 × 7 + 7)
:= ((88 × (88 + 8) + 8)/8) − 8
:= (88 × (88 + 8) − (8 + 8))/8
:= 999 + ((9 × 99 + 9)/(9 + 9))
:= 9/9 + (9 × (99 + 9 + 9))
Ï 1050 := (11 − 1) × (111 − ((1 + 1) × (1 + 1 + 1)))
Ï 1055 := 1111 − (1 + 111)/(1 + 1)
:= 4 + (4 × (44 + 4 + 4) − 4/4)
:= 77 × (7 + 7) − ((77 + 7)/7 + 7)
:= (((88 × (88 + 8) + 8) + 8) + 8)/8
:= 9 + ((9 × (99 + 9 + 9)) − ((9 + 9 + 9)/9))
Ï 1060 := (1 + 1) × (1 + (1 + 11 + 11)1+1 )
:= 2 + (2 × ((22 + 2/2)2 ))
:= 2 × (((22 + 2/2)2 ) − (2 + 2))
:= (2 × (22 × (22 + 2))) − 2/2
:= 33 × (33 + 3 + 3) − 3
:= 33 × 33 − 3/3 − 33
:= 4 × (44 + 4) + (44 − 4)/4
:= 4 × (44 + 4 + 4) − 4/4
:= (5 + 5) × (5 × (5 × 5 − 5) + 5)
:= 5 + ((5 + 5) × (5 × (5 × 5 − 5) + 5))
:= 6 + (6 × (6 × (6 × 6 − 6) − 6))
:= 666 + (6 × 66 − (6/6 + 6))
:= 6 × 6 + (((6 + 6)/6)(66−6)/6)
:= (7 + 7) × (77 − (7 + 7)/7)
:= 7777/7 − (7 × 7 + 7)
:= 77 × (7 + 7) − (77/7 + 7)
:= (((88 × (88 + 8) + 8) + 8)/8) − 8
:= (88 × (88 + 8) − 8)/8
:= ((88 × (8 + 8 + 8)) + 8)/((8 + 8)/8)
:= (9 × (99 + 9 + 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 × (99 + 9 + 9))
:= 9 + ((9 × (99 + 9 + 9)) − ((9 + 9)/9))
Ï 1051 := 1111 − (111+1 − 1)/(1 + 1)
Ï 1056 := (1 + 1) × ((1 + 11 + 11)1+1 − 1)
:= 3 + ((33 × (33 − 3/3)) + 3/3)
:= 4 + 4 × (44 + 4 + 4)
:= (5 + 5) × (555/5 − 5)
Ï 1061 := 1 + ((1 + 1) × (1 + (1 + 11 + 11)1+1 ))
:= (2 × ((22 × (22 + 2)) − 2)) − 2/2
:= 2 × (22 × (22 + 2))
:= 2 + (((((2 × 22) + 2)2 ) + 2)/2)
:= 33 + ((3 − 3/3)3×3+3/3 )
:= 33 × (33 − 3/3)
:= 33+3 + (333 − 3/3)
:= 44/4 + 4 × (44 + 4)
:= 4 × (44 + 4 + 4)
:= 4 + (4 × (44 + 4 + 4) + 4/4)
:= 5555/5 − (55 + 5)
:= 5555/5 − 55
:= 5 + (5555/5 − 55)
:= 6 + ((6666/6) − 66)
:= 66 × (((66 − 6)/6) + 6)
:= 666 + (6 × 66 − 6/6)
:= 7/7 + ((7 + 7) × (77 − (7 + 7)/7))
:= 77 + ((7 + 7) × (77 − 7) − 7/7)
:= (7777 − 7)/7 − 7 × 7
:= 8 + ((8 × 8 × (8 + 8) + (88/8)) + 8)
:= 88 × (88 + 8)/8
:= 8 + ((88/8 × (88 − 8/8 + 8)) + 8)
:= 99/9 × (99 − ((9 + 9 + 9)/9))
:= 9 + ((9 × (99 + 9 + 9)) − 9/9)
:= (9 × (99 + 9 + 9)) − (9 + 9)/9
Ï 1052 := 11 × 111 − (1 + 1 + 11)1+1
Ï 1057 := (11 × (1 + 1 + 1)) + (1 + 1)11−1
Ï 1062 := (1 + 1) × (1 + (1 + (1 + 11 + 11)1+1 ))
:= 2 × ((22 × (22 + 2)) − 2)
:= ((((2 × 22) + 2)2 ) − 2)/2
:= 2 × (((22 + 2/2)2 ) + 2)
:= 33 × (33 + 3 + 3) − 3/3
:= 3/3 + (33 × (33 − 3/3))
:= 33+3 + 333
:= 44 + 4 × (44 − 4)
:= 4/4 + 4 × (44 + 4 + 4)
:= 4 + (4 × (44 + 4 + 4) + (4 + 4)/4)
:= (5555 + 5)/5 − (55 + 5)
:= (5555 + 5)/5 − 55
:= 5 + ((5555 + 5)/5 − 55)
:= 6 + ((6666 + 6)/6 − 66)
:= 6 + (((6666/6) − 66) + 6)
:= 666 + 6 × 66
:= 7 + (77/7 × ((77/7 + 77) + 7))
:= 77 + (7 + 7) × (77 − 7)
:= 7777/7 − 7 × 7
:= 8 + (((88 + 8)/8) × (88 − 8/8))
:= (88 × (88 + 8) + 8)/8
:= 8 + ((88 × (88 + 8) − (8 + 8))/8)
:= (9 × (99 + 9 + 9)) − 9/9
:= 9 + (((9999/9 − 9 × 9) + 9) + 9)
:= 9 + (9 × (99 + 9 + 9))
80
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1063 := 1111 − ((1 + 1) × ((1 + 1) × (1 + 11)))
2
Ï 1068 := (1 + 11) × (111 − 11 − 11)
(2/2+2)2
Ï 1073 := 1111 − (1 + (111/(1 + 1 + 1)))
:= ((22/2 + 22)2 ) − 22+2
:= 2 + ((((((2 × 22) + 2) ) + 2)/2) + 2)
:= 2 × ((2
:= 3/3 + (33+3 + 333)
:= 3 + ((33+3 + 333) + 3)
:= 33+3 + (333 + 33/3)
:= 44 + (4 × 44 − (4/4 + 4))
:= 44 + 4 × 44
:= 4 + ((4 × 44 + 4/4) + 44)
:= 5 × 5 × 55 + ((5 − 55 )/(5 + 5))
:= 55 + ((5 − 5/5)5 − 55/5)
:= 55 + ((5 − 5/5)5 − (5/5 + 5))
:= 6/6 + (666 + 6 × 66)
) + 22)
:= 6 + (666 + 6 × 66)
:= (6 × 6 × (6 × 6 − 6)) − 6/6 − 6
:= 77 × (7 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 77 × (7 + 7)
:= ((7 + 7)/7) + (77 × (7 + 7) − 7)
:= 8 + ((88 × (88 + 8) − 8)/8)
:= (8/8 + 88) × ((88 + 8)/8)
:= 8 + (((88 × (88 + 8) + 8)/8) + 8)
:= 9 + ((9 × (99 + 9 + 9)) + 9/9)
:= 9 × 9 + (999 − ((99 + 9)/9))
:= 9 + ((9 × (99 + 9 + 9)) + (99/9))
Ï 1064 := (1 + 1)1+1+1 × (1 + (11 × (1 + 11)))
Ï 1069 := 1 + ((1 + 11) × (111 − 11 − 11))
Ï 1074 := 1111 − (111/(1 + 1 + 1))
:= 2 + (2 × (((22 + 2/2) ) + 2))
:= ((((2 × 22) + 2) ) + 22)/2
:= (2 + 2 + 2)2+2 − 222
:= 3 + ((33+3 − 3/3) + 333)
:= 33+3 + (3/3 + 3 + 3)3 − 3
:= 3 + (3 × ((333 − 3) + 33 ))
:= 44 + (4 × 44 − 4)
:= 44 + (4 × 44 + 4/4)
:= 4 + (((4 + 4)/4 + 4 × 44 ) + 44)
:= 5 + ((((5 − 5/5)5 + 5 × 5) + 5) + 5)
:= 55 + ((5 − 5/5)5 − (5 + 5))
:= 55 + ((5 − 5/5)5 − 5)
2
2
:= 666 + (((6 + 6)/6) + 6 × 66)
:= (6 × 6 × (6 × 6 − 6)) − 66/6
:= (6 × 6 × (6 × 6 − 6)) − 6
:= (7 + 7) × (77 − 7/7)
:= 7 + (7777/7 − 7 × 7)
:= 7 + (77 × (7 + 7) − (77/7))
:= 8 + (88 × ((88 + 8)/8))
:= (8 × (8 × (8 + 8) + 8)) − (88/8 + 8)
:= 8 + ((((88 × (88 + 8) + 8) + 8)/8) + 8)
:= 99/9 + (9 × (99 + 9 + 9))
:= 9 × 9 + (999 − (99/9))
:= (9 × (99 + 9)) + (999/9 − 9)
Ï 1065 := 1111 − ((1 + 1) × (1 + (11 + 11)))
:= (2222/2) − ((2 × 22) + 2)
:= 3 + (3
3+3
Ï 1070 := (11 − 1) × (111 − (1 + 1 + 1 + 1))
:= 2 + ((22×(2+2)+2 ) + 2 × 22)
3
:= 3 × 333 + (((3 + 3) − 3)/3)
+ 333)
4
4
Ï 1075 := 1111 − ((1 + 1 + 1) × (1 + 11))
:= 222/2 + (2 × (222 − 2))
:= 33 × 33 − (33/3 + 3)
:= 44 + ((4 × 4 − 4) + 4/4)
:= 44 + ((4 + 4)/4 + 4 × 4 )
:= ((44/4)4−4/4 ) − 44
:= 5 + ((5 + 5) × (555/5 − 5))
:= (5 + 5) × ((555 + 5)/5 − 5)
:= 5 × (5 × 55 − (55 + 5))
:= 66 + ((6 × 666)/(6 − ((6 + 6)/6)))
:= (6 − 6/6) × (6 × 6 × 6 − (6 + 6)/6)
:= (6666/6) − 6 × 6
:= 7/7 + ((7 + 7) × (77 − 7/7))
:= 77 × (7 + 7) − (7/7 + 7)
:= 77 × (7 + 7) − (7 + 7 + 7)/7
:= 8 + ((88 × (88 + 8) + 8)/8)
:= 8 + (((88 × (88 + 8) − (8 + 8))/8) + 8)
:= 8 + (88/8 × ((8/8 + 88) + 8))
:= ((99 + 9)/9) + (9 × (99 + 9 + 9))
:= 99 + ((9 × (99 + 9)) − 9/9)
:= 9999/9 − ((9 + 9 + 9) + 9)
Ï 1066 := (1 + 1 + 11) × (1 + (11 − 1 − 1)1+1 )
2
Ï 1071 := (11 − 1 − 1) × (111+1 − (1 + 1))
2
Ï 1076 := 1 + (1111 − ((1 + 1 + 1) × (1 + 11)))
:= 2 × ((((22 + 2/2) ) + 2) + 2)
:= 2 + (((((2 × 22) + 2) ) + 22)/2)
:= 2 + ((2 + 2 + 2)2+2 − 222)
:= 3 + ((33+3 + 333) + 3/3)
:= 3 × ((333 − 3) + 33 )
:= ((3 − 33)/3) + (33 × 33 − 3)
4
:= 44 + (4 × 4 − (4 + 4)/4)
:= 4 + ((4 × 4 − 4/4) + 44)
:= 4 + ((4 × 44 + 44) + 4)
:= 5 + (5555/5 − 55 + 5)
:= 5 + (5555/5 − 55 + 5 + 5)
:= 5/5 + (5 × (5 × 55 − (55 + 5)))
:= (6/6 + 6) × (666/6 + 6 × 6 + 6)
:= (6666 + 6)/6 − 6 × 6
:= 77 × (7 + 7) − 7
:= 77 × (7 + 7) − (7 + 7)/7
:= 8 + (((88 × (88 + 8) + 8) + 8)/8)
:= (8/8 + 8) × (888/8 + 8)
:= 8 + ((8/8 + 88) × ((88 + 8)/8))
:= (9/9 + 9 × 9) × ((99 + 9 + 9)/9)
:= 99 + (9 × (99 + 9))
:= 9 + ((99/9) × (99 − ((9 + 9)/9)))
4
(66−6)/6)
:= 6 + ((((6 + 6)/6)
+ 6 × 6)
:= 77 × (7 + 7) − (77 + 7)/7
Ï 1067 := 11 × (111 − (1 + (1 + 1 + 11)))
Ï 1072 := 1 + ((11 − 1 − 1) × (111+1 − (1 + 1)))
(2/2+2)2
:= (2222/2) − (2 × 22)
:= 2 × (((2
) + 22) + 2)
:= 33/3 + (33 × (33 − 3/3))
:= 33+3 + (((3/3 + 3) + 3)3 )
Ï 1077 := 1111 − (1 + 11 × (1 + 1 + 1))
:= 222/2 + ((2 × 222 ) − 2)
:= 33 × 33 − (3 × 3 + 3)
:= 44 + (4 × 44 − 4/4)
:= 4 + (4 × 4 + 44)
:= 4 × 44 + ((44 − 44)/4)
:= ((5555 + 55)/5) − 55
:= 55 − ((((5 + 5)/5)55/5 ) + 5)
:= 55 − (((5 + 5)/5)55/5 )
:= 6 + ((6 × 66 − 6/6) + 666)
:= (66 + 6/6) × (((66 − 6)/6) + 6)
:= (6 × 6 × (6 × 6 − 6)) − 6 × 6/(6 + 6)
:= 77 × (7 + 7) − 77/7
4
:= 7/7 + (77 × (7 + 7) − 7)
:= 77 × (7 + 7) − 7/7
:= 88/8 × ((8/8 + 88) + 8)
:= (8 × (8 × (8 + 8) + 8)) − 8 − 8
:= (8 × (8 × (8 + 8) + 8)) − 88/8
:= 99/9 × (99 − ((9 + 9)/9))
:= 9/9 + ((999 − 9) + 9 × 9)
:= ((99 × 99) − (99 + 9))/9
81
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1083 := (1 + 1 + 1) × ((1 + 1) × (11 − 1) − 1)1+1
Ï 1078 := 11 × (111 − 1 − 1 − 11)
Ï 1088 := (11 × (1 + 1 + 1))1+1 − 1
:= 22 + (2 × (22 × (22 + 2)))
:= (2/2 + 2) × ((22 − (2/2 + 2))2 )
:= ((22/2 + 22)2 ) − 2/2
:= 33 × 33 − 33/3
:= 33 × 33 − (3 + 3)
:= 33 × 33 − 3/3
4
:= 44 + ((44 − 4)/4 + 4 × 4 )
:= 44 + (4 × (4 + 4) − 4/4)
:= 4 × (4 × 4 + 44 )
:= 55 + ((5 − 5/5)5 − 5/5)
:= 5 + (((5 − 5/5)5 − 5/5) + 55)
:= (5 + 5)/5 × (555 − (55/5))
:= (6 × 6 × (6 × 6 − 6)) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × 6 × (6 × 6 − 6))
:= (66/6 + 6) × ((6 + 6)/6)6
:= 77 × (7 + 7)
:= 7 + (77 × (7 + 7) − ((7 + 7)/7))
:= 77 × (7 + 7) + (77 − 7)/7
:= 88/8 × (((8 + 8)/8 + 88) + 8)
:= (88/8 + 8) × ((8/8 − 8) + 8 × 8)
:= 8 × (8 × (8 + 8) + 8)
:= 99/9 × (99 − 9/9)
:= (9 × (99 + 9)) + 999/9
:= ((99 × 99) − 9)/9
Ï 1079 := 1 + (11 × (111 − 1 − 1 − 11))
:= 222/2 + (2 × 222 )
4
Ï 1084 := 1111 − ((1 + 1 + 1)1+1+1 )
Ï 1089 := (11 × (1 + 1 + 1))1+1
:= 22 + (2 × (((22 + 2/2)2 ) + 2))
:= (22/2 + 22)2
:= 3333/3 − 33
:= 33 × 33
:= 44 + (44/4 + 4 × 4 )
:= 44 + 4 × (44 + 4)
:= 4/4 + (4 × (4 × 4 + 44 ))
:= 55 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 55)
:= 5 + (((5 − 5/5)5 + 55) + 5)
:= (6 × 6 × (6 × 6 − 6)) − 6/6
:= 6 + ((6 × 6 × (6 × 6 − 6)) − ((6 + 6)/6))
:= 66 × 66/(6 − ((6 + 6)/6))
:= 7/7 + 77 × (7 + 7)
:= 7 + (77 × (7 + 7) − 7/7)
:= 77/7 + 77 × (7 + 7)
:= (8 × (8 × (8 + 8) + 8)) − (8/8 + 8)
:= (8 × (8 × (8 + 8) + 8)) − (8/((8 + 8)/8))
:= 8/8 + (8 × (8 × (8 + 8) + 8))
:= 9 × 9 + (999 − 9/9)
:= 9999/9 − (9 + 9 + 9)
:= 99 × (99/9)
:= ((3 − 33)/3) + 33 × 33
4
Ï 1080 := (11 − 1) × (111 − 1 − 1 − 1)
Ï 1085 := 1111 − ((1 + 1) × (1 + 1 + 11))
Ï 1090 := 1 + (11 × (1 + 1 + 1))1+1
:= 22 + (2 × ((22 + 2/2)2 ))
:= ((22/2 + 22)2 ) − 2 − 2
:= 2/2 + ((22/2 + 22)2 )
:= 3 × (333 + 33 )
:= 33 × 33 − (3/3 + 3)
:= 3/3 + 33 × 33
4
:= 44 + (4 × (4 + 4) − 4)
:= 44 + (4 × (4 + 4) + 4/4)
:= (4 + 4)/4 + (4 × (4 × 4 + 44 ))
:= (5 − 5/5) × (5 × 55 − 5)
:= 5 + ((5 − 5/5) × (5 × 55 − 5))
:= 55 × (5 × 5 − 5) − 5 − 5
:= 6 × 6 × (6 × 6 − 6)
:= 6 + ((6 × 6 × (6 × 6 − 6)) − 6/6)
:= 66 + (((6 + 6)/6)(66−6)/6)
:= ((7 + 7)/7) + 77 × (7 + 7)
:= 7 + 77 × (7 + 7)
:= 77 × (7 + 7) + (77 + 7)/7
:= (8 × (8 × (8 + 8) + 8)) − 8
:= 8 + ((8 × (8 × (8 + 8) + 8)) − (88/8))
:= (8 + 8)/8 + (8 × (8 × (8 + 8) + 8))
:= 9 × 9 + 999
:= (9999 + 9)/9 − (9 + 9 + 9)
:= ((99 × 99) + 9)/9
4
Ï 1081 := 1 + ((11 − 1) × (111 − 1 − 1 − 1))
Ï 1086 := 1111 − (1 + ((1 + 1) × (1 + 11)))
Ï 1091 := 1 + 1 + (11 × (1 + 1 + 1))1+1
:= 2 + (222/2 + (2 × 222 ))
:= ((22/2 + 22)2 ) − 2/2 − 2
:= 2 + ((22/2 + 22)2 )
:= 33 × 33 − 3
:= 3 + (33 × 33 − 3/3)
:= 3 + (33 × 33 − 33/3)
4
4
:= 4 + (((4 − 44)/4) + 4 × 4 )
4
4
:= 4 × 4 + (4 − 4 − 4)/4
:= 4 + (((44 − 4)/4) + 4 × 44 )
:= 5555/5 − (5 × 5 + 5)
:= 5555/5 − 5 × 5
:= 5 + (5555/5 − 5 × 5)
:= 6/6 + (6 × 6 × (6 × 6 − 6))
:= 6 + (6 × 6 × (6 × 6 − 6))
:= 66/6 + (6 × 6 × (6 × 6 − 6))
:= 77 × (7 + 7) + (7 + 7 + 7)/7
:= 7 + (77 × (7 + 7) + 7/7)
:= 7 + ((77 × (7 + 7) − 7/7) + 7)
:= 8/8 + ((8 × (8 × (8 + 8) + 8)) − 8)
:= (8 × (8 × (8 + 8) + 8)) − (8 + 8)/8
:= 88/8 + ((8 × (8 × (8 + 8) + 8)) − 8)
:= 9/9 + (999 + 9 × 9)
:= ((99 × 99) − (9 + 9 + 9))/9
:= (((99 × 99) + 9) + 9)/9
Ï 1082 := 1 + (1 + ((11 − 1) × (111 − 1 − 1 − 1)))
2
Ï 1087 := 1111 − (1 + 1) × (1 + 11)
Ï 1092 := 1 + 1 + 1 + (11 × (1 + 1 + 1))1+1
:= 2 + ((2 × ((22 + 2/2) )) + 22)
:= ((22/2 + 22)2 ) − 2
:= (22 + 2 + 2) × ((2 × 22) − 2)
:= 33 × 33 − ((3/3 + 3) + 3)
:= 3/3 + (33 × 33 − 3)
:= 3 + 33 × 33
:= 44 + (4 × (44 + 4) − (4 + 4)/4)
:= 4 × 44 + ((44 − 4)/4)
:= 4 + (4 × (4 × 4 + 44 ))
:= 5 + (55 − (((5 + 5)/5)55/5 ))
:= (5555 + 5)/5 − 5 × 5
:= 5 + ((5555 + 5)/5 − 5 × 5)
:= (6 + 6)/6 + (6 × 6 × (6 × 6 − 6))
:= 6 + ((6 × 6 × (6 × 6 − 6)) + 6/6)
:= 6 + ((6 × 6 × (6 × 6 − 6)) + 6)
:= 77/7 + (77 × (7 + 7) − 7)
:= 7 + (77 × (7 + 7) + ((7 + 7)/7))
:= 7 + (77 × (7 + 7) + 7)
:= (8 + 8)/8 + ((8 × (8 × (8 + 8) + 8)) − 8)
:= (8 × (8 × (8 + 8) + 8)) − 8/8
:= ((8888 − 88)/8) − 8
:= 9 × 9 + (((9 + 9)/9) + 999)
:= ((99 × 99) − (9 + 9))/9
:= ((9999 − 9)/9) − (9 + 9)
82
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1093 := 1111 − (1 + 1) × (11 − 1 − 1)
2
Ï 1103 := 1 + (1 + (1 + (1111 − 11)))
Ï 1098 := 1111 − 1 − 1 − 11
:= 2 + (((22/2 + 22) ) + 2)
:= (22 × ((2 × (22 + 2)) + 2)) − 2
:= (2222/2) − 2 × (2 + 2)
:= 3 + (33 × 33 + 3/3)
:= 3 × (333 + 33)
:= 3 + (33 × 33 + 33/3)
:= 4 + ((4444 − 4)/4 − 4 × 4)
:= 4444/4 − 4 − 4
4
:= 4 + ((4 × (4 × 4 + 4 )) + 4/4)
:= 55 × (5 × 5 − 5) − ((5 + 5)/5 + 5)
:= 55 × (5 × 5 − 5) − (5 + 5)/5
:= 5 + (55 × (5 × 5 − 5) − ((5 + 5)/5))
:= (6666/6) − 6 − 6 − 6
:= 666 + (6 × (66 + 6))
:= ((6666 − (6 + 6))/6) − 6
:= 7 + ((77 × (7 + 7) + 7/7) + 7)
:= (7777 + 7)/7 − (7 + 7)
:= (7777 − 7)/7 − 7
:= ((8888 − (8 + 8))/8) − 8 − 8
:= 8 + ((8 × (8 × (8 + 8) + 8)) + ((8 + 8)/8))
:= 8888/8 − 8
:= 9999/9 − (9 + 9)
:= 99 + 999
:= (9999 + 9)/9 − 9
Ï 1094 := 1 + (1111 − (1 + 1) × (11 − 1 − 1))
Ï 1104 := 1 + (1 + (1 + (1 + (1111 − 11))))
Ï 1099 := 1111 − 1 − 11
:= (2222 − (22 + 2))/2
:= (22 + 2) × ((2 × 22) + 2)
:= 3 + ((33 × 33 − 3/3) + 3)
:= 3 × 3 + (33 × 33 + 3/3)
:= 3 + (33 × 33 + 3 × 3 + 3)
:= (4444 − 4)/4 − 4 × 4
:= 4 + (4444/4 − 4 × 4)
:= 4 × ((4 × 4 + 44 ) + 4)
:= 55 × (5 × 5 − 5) − (5/5 + 5)
:= 55 × (5 × 5 − 5) − 5/5
:= 5 + (55 × (5 × 5 − 5) − 5/5)
:= ((6666 − 66)/6) − 6
:= (6666/6) − 6 − 6
:= ((6666 − 6)/6) − 6
:= 7 + ((77 × (7 + 7) + ((7 + 7)/7)) + 7)
:= 7 + ((77 × (7 + 7) + 7) + 7)
:= 7777/7 − 7
:= (8888 − 8)/8 − 8 − 8
:= 88/8 + (8 × (8 × (8 + 8) + 8))
:= 8 + ((8 × (8 × (8 + 8) + 8)) + 8)
:= (9999 + 9)/9 − (9 + 9)
:= 9 + (((99 × 99) + 9)/9)
:= (((9999 + 9) + 9)/9) − 9
:= 2 + ((22 + 2 + 2) × ((2 × 22) − 2))
Ï 1095 := 1111 − ((1 + 1)1+1+1+1 )
:= (2222/2) − 2
Ï 1100 := 1111 − 11
2+2
:= 3 + (33 × 33 + 3)
Ï 1105 := 1111 − ((1 + 1) × (1 + 1 + 1))
:= 22 × ((2 × (22 + 2)) + 2)
:= (2222/2) − (2 + 2 + 2)
:= 33/3 + 33 × 33
:= 3333/3 − (3 + 3)
4
:= 4444/4 − 4 × 4
:= 44 + 4 × (4 + 4 + 4)
:= 4 × 44 + (4 − 4/4)4
:= 55 × (5 × 5 − 5) − 5
:= 55 × (5 × 5 − 5)
:= 5 + 55 × (5 × 5 − 5)
:= 6 + (66 × 66/(6 − ((6 + 6)/6)))
:= (6666 − 66)/6
:= (6666/6) − 6
:= 7 + (77 × (7 + 7) + ((77 − 7)/7))
:= (7777 − 77)/7
:= (7777 + 7)/7 − 7
:= 8888/8 − 8 − 8
:= (8888 − 88)/8
:= (8/8 + 8 + 8) × (8/8 + 8 × 8)
:= 9 + (((99 × 99) − (9 + 9 + 9))/9)
:= 99/9 × (9/9 + 99)
:= 9 × 9 + (((9 + 9)/9)9/9+9 )
Ï 1096 := 1111 − (1 + 1 + 1 + 1 + 11)
Ï 1101 := 1 + (1111 − 11)
Ï 1106 := 1111 − 1 − 1 − 1 − 1 − 1
:= 2 × (22+2+2 + 222 )
:= ((2222 − 22) + 2)/2
:= 2 + ((22 + 2) × ((2 × 22) + 2))
:= 3 + ((33 × 33 + 3/3) + 3)
:= 3 + (33 × 33 + 3 × 3)
:= ((3333 + 3)/3) − (3 + 3)
:= 4 + ((4 × (4 × 4 + 44 )) + 4)
:= 4 × 44 + ((4 − 4/4)4 − 4)
:= (4444 − 4)/4 − 4
:= 5/5 + (55 × (5 × 5 − 5) − 5)
:= 5/5 + 55 × (5 × 5 − 5)
:= 5555/5 − 5
:= ((6666 − 66) + 6)/6
:= (6666 + 6)/6 − 6
(66−6)/6)
:= 6 + ((((6 + 6)/6)
+ 66)
:= 7 + (77 × (7 + 7) + (77/7))
:= (7777 − 77 + 7)/7
:= 77 + 7 × 7 × (7 + 7 + 7)
:= 8 + (8 × (8 × (8 + 8) + 8))
:= ((8888 − (8 + 8))/8) − 8
:= ((8888 + 88)/8) − 8 − 8
:= 9 + (((99 × 99) − (9 + 9))/9)
:= ((9999 − 9)/9) − 9
:= 9 + ((((99 × 99) − 9)/9) + 9)
Ï 1097 := 1111 − 1 − 1 − 1 − 11
Ï 1102 := 1 + (1 + (1111 − 11))
Ï 1107 := 1111 − 1 − 1 − 1 − 1
:= 2 + ((2222/2) − 22+2 )
:= 2 + (22 × ((2 × (22 + 2)) + 2))
:= (2222/2) − 2 − 2
:= 3 × 3 + (33 × 33 − 3/3)
:= 3333/3 − 3 × 3
:= 3 × ((333 + 33) + 3)
:= 4 + (((4 × (4 × 4 + 4 )) + 4/4) + 4)
:= (4444 − 4)/4 − 4 − 4
:= 4444/4 − 4
:= (5 + 5)/5 + (55 × (5 × 5 − 5) − 5)
:= (5 + 5)/5 + 55 × (5 × 5 − 5)
:= (5555 + 5)/5 − 5
4
:= 6 + ((6 × 6 × (6 × 6 − 6)) + (66/6))
:= ((6666 − (6 + 6 + 6))/6) − 6
:= (((6666 + 6) + 6)/6) − 6
:= 7777/7 − (7 + 7)
:= ((7777 − 7 − 7)/7) − 7
:= 7 + ((7777 − 77)/7)
:= 8 + ((8 × (8 × (8 + 8) + 8)) + 8/8)
:= (8888 − 8)/8 − 8
:= 8 + ((8 × (8 × (8 + 8) + 8)) + (88/8))
:= 9 + (((99 × 99) − 9)/9)
:= 9999/9 − 9
:= 9 + (999 + 99)
83
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1118 := (11 − 1) × (1 + 111) − 1 − 1
Ï 1113 := 1 + 1 + 1111
Ï 1108 := 1111 − 1 − 1 − 1
2
:= 2 × (((22 + 2) ) − 22)
:= 2 + (2222/2)
:= 2 + ((((2222 + 2)/2) + 2) + 2)
:= 3333/3 − 3
:= 3 + ((3333 − 3)/3)
:= 3 + ((3333 + 3)/3 + 3)
:= 4 + (4 × ((4 × 4 + 4 ) + 4))
:= ((4444 + 4) + 4)/4
:= 4 + ((4444 − 4)/4 + 4)
:= (5 + 5)/5 × (555 − 5/5)
:= ((5555 + 5) + 5)/5
:= 5 + (5555 + 5 + 5)/5
:= (6666 − (6 + 6 + 6))/6
:= ((6666 + 6) + 6)/6
:= 6 + (6666 + 6)/6
:= (7777 − 7 − 7 − 7)/7
:= 7 × 7 × 7 + (777 − 7)
:= 7 + 7777/7
:= 8 + ((8888 − 88)/8)
:= ((8888 + 8) + 8)/8
:= 8 + (8888 − 8)/8
:= 9 + ((((99 × 99) + 9)/9) + 9)
:= ((9999 + 9) + 9)/9
:= 9 + ((9999 − (9 + 9))/9)
4
Ï 1109 := 1111 − 1 − 1
Ï 1119 := (11 − 1) × (1 + 111) − 1
Ï 1114 := 1 + 1 + 1 + 1111
:= (2222/2) − 2
:= 2 + ((2222 + 2)/2)
:= 2 × (2 + 2) + (2222/2)
:= ((3333 + 3)/3) − 3
:= 3 + 3333/3
:= 3 + (33 × 33 + 33 )
:= (4444 − (4 + 4))/4
:= 4 + (4444 − 4)/4
:= 4 + (4444/4 + 4)
:= (5555 − (5 + 5))/5
:= 5 + ((5555 − (5 + 5))/5)
:= (((5 + 5) × (555 + 5)) − 5)/5
:= (6666 − (6 + 6))/6
:= (((6666 + 6) + 6) + 6)/6
:= 6 + (((6666 + 6) + 6)/6)
:= (7777 − 7 − 7)/7
:= ((7777 + 7 + 7) + 7)/7
:= 7 + (7777 + 7)/7
:= (8888 − (8 + 8))/8
:= (((8888 + 8) + 8) + 8)/8
:= 8 + 8888/8
:= (9999 − (9 + 9))/9
:= (((9999 + 9) + 9) + 9)/9
:= 9 + ((9999 − 9)/9)
Ï 1110 := 1111 − 1
Ï 1115 := 1 + 1 + 1 + 1 + 1111
Ï 1120 := (11 − 1) × (1 + 111)
:= (2222 − 2)/2
:= 2 + ((2222/2) + 2)
:= 2 × (((22 + 2)2 ) − 22+2 )
:= (3333 − 3)/3
:= 3 + ((3333 + 3)/3)
:= 3 × 3 + 3333/3
:= (4444 − 4)/4
:= 4 + 4444/4
:= 4 × (((4 × 4 + 44 ) + 4) + 4)
:= (5 + 5) × 555/5
:= 5 + (555 + 555)
:= (5 + 5) × (555 + 5)/5
:= (6666 − 6)/6
:= 6 + ((6666 − (6 + 6))/6)
:= 6 + ((((6666 + 6) + 6) + 6)/6)
:= (7777 − 7)/7
:= ((7777 + 77)/7) − 7
:= 7 × 7 × 7 + 777
:= (8888 − 8)/8
:= (8/((8 + 8)/8)) + 8888/8
:= 8 + (8 × 8 × (8 + 8) + 88)
:= (9999 − 9)/9
:= 9 + (((((99 × 99) − 9)/9) + 9) + 9)
:= 9 + 9999/9
Ï 1111 := 1111
:= 2222/2
Ï 1116 := 1 + 1 + 1 + 1 + 1 + 1111
:= 2 + (((2222 + 2)/2) + 2)
3
Ï 1121 := 11 + 1111 − 1
:= ((2222 − 2) + 22)/2
:= 3333/3
:= 3 + 33 × 33
:= 3 × 3 + ((3333 + 3)/3)
:= 4444/4
:= 4 + (4444 + 4)/4
:= 4 × (44 + 4) + (4 − 4/4)4
:= 5555/5
:= 5 + 5555/5
:= 5 + (5555/5 + 5)
:= 6666/6
:= 6 × (6 × (6 × 6 − 6) + 6)
:= (66 × (66/6 + 6)) − 6/6
:= 7777/7
:= 7 + ((7777 − 7 − 7)/7)
:= 7/7 + (7 × 7 × 7 + 777)
:= 8888/8
:= 8 + (((8888 − 88)/8) + 8)
:= 8 + (((8888 + 8) + 8)/8)
:= 9999/9
:= 9 + (999 + 99 + 9)
:= 9 + (9999 + 9)/9
Ï 1112 := 1 + 1111
Ï 1117 := 1111 + (1 + 1) × (1 + 1 + 1)
Ï 1122 := 11 + 1111
:= (2222 + 2)/2
:= 2 + (((2222/2) + 2) + 2)
:= (2222 + 22)/2
:= (3333 + 3)/3
:= 3 + 3333/3 + 3
:= 33 + 33 × 33
:= (4444 + 4)/4
:= 4 + (((4444 + 4) + 4)/4)
:= (4444 + 44)/4
:= (5555 + 5)/5
:= 5 + (5555 + 5)/5
:= (5555 + 55)/5
:= (6666 + 6)/6
:= 6 + (6666/6)
:= 66 × (66/6 + 6)
:= (7777 + 7)/7
:= 7 + (7777 − 7)/7
:= (7777 + 77)/7
:= 88 + 8 × 8 × (8 + 8)
:= 8 + ((8888 − (8 + 8))/8)
:= (8888 + 88)/8
:= (9999 + 9)/9
:= 9 + (((((99 × 99) + 9)/9) + 9) + 9)
:= (9999 + 99)/9
84
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1128 := (11 − 1) × (1 + 1 + 111) − 1 − 1
Ï 1123 := 1 + 11 + 1111
2
Ï 1133 := 11 + 11 + 1111
:= ((2222 + 22) + 2)/2
:= (2 × ((22 + 2) )) − (22 + 2)
:= 22 + (2222/2)
:= 3 + (3333/3 + 3 × 3)
:= 3 + ((3 × 3 + 3 + 3)3 /3)
:= ((33/3)3 ) − 33 × (3 + 3)
:= 4 + (4444/4 + 4 + 4)
:= 4 × 4 + (4444 + 4)/4
:= 44 + ((4 × (4 × 4 + 44 )) + 4/4)
:= ((5555 + 55) + 5)/5
:= 5 + (((5555 + 55) + 5)/5)
:= ((5555 + 55) + 55)/5
:= 6 + ((6666/6) + 6)
:= 6 + (66 × (66/6 + 6))
:= 66/6 + (66 × (66/6 + 6))
:= (7777 + 77 + 7)/7
:= 7/7 + (77 × (7 + 7) + 7 × 7)
:= 7 + (((7777 + 7)/7 + 7) + 7)
:= ((8888 + 88) + 8)/8
:= 8 × (8 × 8 + 88) − 88
:= 88/8 × (888/8 − 8)
:= 9 + (((9999 − 9)/9) + 9)
:= (9 × ((99 + 9 + 9) + 9)) − 9/9
9/9+9
:= 99 + (((9 + 9)/9)
)
Ï 1124 := 1 + 1 + 11 + 1111
Ï 1129 := (11 − 1) × (1 + 1 + 111) − 1
:= 2 + ((2222 + 22)/2)
:= 2 + ((2222/2) + 2
3
2+2
)
Ï 1134 := 1 + 11 + 11 + 1111
:= 22 + ((2222 + 2)/2)
:= ((3 × 3 + 3 + 3) − 3)/3
:= 3 + (((3 × 3 + 3 + 3) + 3)/3)
:= 33 × (3 × 3 + 33)
:= 4 + (4 × (((4 × 4 + 44 ) + 4) + 4))
:= 4 × 4 + (((4444 + 4) + 4)/4)
:= 4 × 44 + (444 − 4)/4
:= (5 − 5/5)5 + 5 × (5 × 5 − 5)
:= 5 + ((5 − 5/5)5 + 5 × (5 × 5 − 5))
:= 55 + ((5 − 5/5)5 + 55)
:= 6 + ((66 × (66/6 + 6)) + 6)
3
:= 6 + ((6666 + 6)/6 + 6)
:= 6 + (((6666/6) + 6) + 6)
:= 7 + ((7777 − 7)/7 + 7)
:= 7 + ((7777 + 77)/7)
:= 7 + (77 × (7 + 7) + 7 × 7)
:= (((8888 + 88) + 8) + 8)/8
:= 8 + ((((8888 + 8) + 8)/8) + 8)
:= 8 + (((8888 − 8)/8 + 8) + 8)
:= ((9 × 999) + 9/9)/(9 − 9/9)
:= 9 + (9999/9 + 9)
:= 9 × ((99 + 9 + 9) + 9)
Ï 1130 := (11 − 1) × (1 + 1 + 111)
Ï 1125 := 1 + 1 + 1 + 11 + 1111
:= 2 + (((2222 + 22) + 2)/2)
3
:= (2 × ((22 + 2)2 )) − 22
Ï 1135 := 111 + (1 + 1)11−1
:= 2 + ((2222/2) + 22)
:= (3 × 3 + 3 + 3) /3
:= 3 + ((((3 × 3 + 3 + 3) − 3)/3) + 3)
:= 33 + (3333/3 − 3)
:= 4 + (4 × (44 + 4) + (4 − 4/4)4 )
:= 4 + ((4444 + 44)/4 + 4)
:= 4 × 44 + 444/4
3
:= 5 + (5 × 5 × (55 − 5 − 5))
:= 5 + ((5 × 5 × (55 − 5 − 5)) + 5)
:= 6 × 66 + ((6 × 6/(6 + 6)) )
:= 6 + (((6666 + 6)/6 + 6) + 6)
:= 6 + ((((6666/6) + 6) + 6) + 6)
:= 7 + (7777/7 + 7)
:= 7 + ((7777 + 77 + 7)/7)
:= 7 + ((77 × (7 + 7) + 7 × 7) + 7/7)
:= 8 + (((8888 − (8 + 8))/8) + 8)
:= 8 + ((8888 + 88)/8)
:= 8 + ((8888/8 + 8) + 8)
:= (9 × ((99 + 9 + 9) + 9)) − 9
:= 9 + ((9999 + 9)/9 + 9)
:= 9/9 + (9 × ((99 + 9 + 9) + 9))
:= 5 × 5 × (55 − 5 − 5)
6
Ï 1126 := 1 + 1 + 1 + 1 + 11 + 1111
Ï 1131 := 1 + (11 − 1) × (1 + 1 + 111)
Ï 1136 := 1 + 111 + (1 + 1)11−1
:= (2 × (((22 + 2)2 ) − 2)) − 22
:= 22 + ((2222/2) − 2)
:= 2 × (((22 + 2)2 ) − 2 × (2 + 2))
:= ((3 × 3 + 3 + 3)3 + 3)/3
:= (33 × (3 × 3 + 33)) − 3
:= ((3 × 3 + 3 + 3)3 + 33)/3
:= 4 + (4444 + 44)/4
:= 4 + (4444/4 + 4 × 4)
:= 4 × ((44 − 4 × 4) + 44)
:= 5 + ((5555/5 + 5) + 5)
:= 5 × 5 + (5555/5 − 5)
:= 5 × 5 + 5555/5
6
:= 6 + (((((6666 + 6) + 6) + 6)/6) + 6)
:= 6 + (((6 × 6/(6 + 6)) ) + 6 × 66)
:= 6 × 6 + ((6666 − 66)/6)
:= 7 + ((7777 + 7)/7 + 7)
:= 7 + (((7777 − 7)/7 + 7) + 7)
:= 7 + (((7777 + 77)/7) + 7)
:= 8 + ((8888 − 8)/8 + 8)
:= 8 + (((8888 + 88) + 8)/8)
:= (8 + 8) × ((8 × 8 − 8/8) + 8)
:= 9/9 + ((9 × ((99 + 9 + 9) + 9)) − 9)
:= 9 + ((9999 + 99)/9)
:= (9 + 9)/9 + (9 × ((99 + 9 + 9) + 9))
Ï 1127 := 1111 + (1 + 1)1+1+1+1
:= 22+2 + (2222/2)
Ï 1137 := 1 + 1 + 111 + (1 + 1)11−1
Ï 1132 := 11 + 11 + 1111 − 1
:= 2 + ((2 × ((22 + 2)2 )) − 22)
3
:= 2 + (((2222/2) + 22) + 2)
:= 3 + (((3 × 3 + 3 + 3)3 − 3)/3)
:= 3 + ((3 × 3 + 3 + 3) + 3)/3 + 3
:= 3 + (33 × (3 × 3 + 33))
:= 4 × 4 + 4444/4
:= 44 + (4 × (4 × 4 + 44 ))
:= 44 + ((4/4 + 4)4 + 44 )
:= 5 + ((5555 + 55)/5)
:= 5 + (((5555 + 55)/5) + 5)
:= 5 × 5 + (5555 + 5)/5
:= 6 + ((66 × (66/6 + 6)) − 6/6)
:= ((66 − 6)/6) + (66 × (66/6 + 6))
:= 6 + ((((6 × 6/(6 + 6))6 ) + 6 × 66) + 6)
:= 7 × ((77 + 77) + 7)
:= 7 + ((7777/7 + 7) + 7)
:= (77/7 × (777/7 − 7)) − 7
:= 8 + (8888/8 + 8)
:= ((88 × (888/8 − 8)) − 8)/8
:= 8/8 + ((8 + 8) × ((8 × 8 − 8/8) + 8))
:= 9 + (((9999 − (9 + 9))/9) + 9)
9/9+9
:= 9 + ((((9 + 9)/9)
85
) + 99)
:= 9 + ((((9999 − 9)/9) + 9) + 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1138 := 1111 + (1 + 1 + 1)1+1+1
Ï 1143 := 1111 + 11 × (1 + 1 + 1) − 1
Ï 1148 := 1111 + (111/(1 + 1 + 1))
:= 2 + (2 × (((22 + 2) ) − 2 × (2 + 2)))
:= 2 + ((2 × ((22 + 2) )) − 22/2)
:= 2 × (((22 + 2)2 ) − 2)
:= 33 + 3333/3
:= 3 × ((3 × (3 × 33 + 33 )) + 3)
:= 3 + (((3333 + 3)/3) + 33)
:= 4 + ((444 − 4)/4 + 4 × 44 )
:= 4 × (4 + 4) + 4444/4
:= 444 + 4 × 4 × 44
:= 5 × 5 + (5555 + 5 + 5)/5
:= ((5 + 5)/5)5 + 5555/5
:= 5 × 5 × 5 + ((5 − 5/5)5 − 5/5)
:= ((6 + 6)/6)6 + ((6 × 6 × (6 × 6 − 6)) − 6)
:= 6 × 6 + ((((6666 + 6) + 6)/6) − 6)
:= 6 × 6 + (6666 + 6)/6
:= 7 × 7 + (77 × (7 + 7) + (77/7))
:= ((77 × (777/7 − 7)) − 7)/7
:= 77 + (77 × (7 + 7) − 7)
:= 8 + (((8888 + 88)/8) + 8)
:= (8/8 + 8) × (8 × (8 + 8) − 8/8)
:= 8 × 88 + 888/((8 + 8)/8)
:= 9 + ((9999/9 + 9) + 9)
:= 9 + (9 × ((99 + 9 + 9) + 9))
:= 9 + ((((9999 + 9)/9 + 9) + 9) + 9)
2
Ï 1139 := (11 − 1) × (1 + 1 + 1 + 111) − 1
2
Ï 1144 := 1111 + 11 × (1 + 1 + 1)
Ï 1149 := 1 + (1111 + (111/(1 + 1 + 1)))
:= (2 × ((22 + 2)2 )) − (22/2 + 2)
:= 2 × (22 × (22 + 2 + 2))
:= 2/2 + (2 × (((22 + 2)2 ) − 2))
:= 33 + ((3333 + 3)/3)
:= 33 + 3333/3
:= 33 + (33 × 33 + 33)
:= 4 + (444/4 + 4 × 44 )
:= 44 + (444 + 444)
:= 44 + ((4 − 4/4)4 + 4 × 44 )
:= ((55 − 555) + 55 )/5
:= 5 × 5 × 5 + ((5 − 5/5)5 − 5)
:= 5 × 5 × 5 + (5 − 5/5)5
:= (66/6 + 6) × (66 + 6/6)
:= 66/6 × (((666 − 6)/6) − 6)
:= 6 × 6 + (((6666 + 6) + 6)/6)
:= 7 + (((7777/7 + 7) + 7) + 7)
:= 77/7 × (777/7 − 7)
:= 7/7 + ((77 × (7 + 7) − 7) + 77)
:= 8 + ((((8888 + 88) + 8)/8) + 8)
:= (8 + 8) × (8 × 8 + 8) − 8
:= 8 + ((8 + 8) × (8 × 8 + 8) − (88/8))
:= 9 + (((9999 + 9)/9 + 9) + 9)
:= 9 + ((9 × ((99 + 9 + 9) + 9)) + 9/9)
:= 9 + ((((9999 + 99)/9) + 9) + 9)
Ï 1140 := (11 − 1) × (1 + 1 + 1 + 111)
Ï 1145 := 111+1 + (1 + 1)11−1
Ï 1150 := (1 + 1) × ((((1 + 1) × (1 + 11))1+1 ) − 1)
:= 2 × (((22 + 2)2 ) − (2 + 2 + 2))
:= 2/2 + (2 × (22 × (22 + 2 + 2)))
:= (2 × ((22 + 2)2 )) − 2
:= 3 + ((33 × (3 × 3 + 33)) + 3)
:= 33 + ((3333 + 3)/3)
:= 3 + ((3333/3 + 33) + 3)
:= 4 + (44 × (4 × 4 + 4) + 44 )
:= 4 + (((4 + 4) × 444 + 4)/4)
:= 44 + ((4444 − 4)/4 − 4)
:= (5 + 5) × (5 × 5 × 5 − (55/5))
:= (5 + 5) × (5 × 5 × 5 − 5) − 55
:= 5 × (5 × (55 − 5 − 5) + 5)
:= 66 + ((6 × 6 × (6 × 6 − 6)) − 6)
:= 6 + ((66/6 + 6) × (66 + 6/6))
:= ((6 × 6 − ((6 + 6)/6))(6+6)/6 ) − 6
:= (7/7 + 7 + 7) × (77 − 7/7)
:= (((7 + 7)/7 + 7) × ((7 + 7)/7)7 ) − 7
:= 7 + (((77 × (777/7 − 7)) − 7)/7)
:= ((88 + 8)/8) × (88 − 8/8 + 8)
:= 8/8 + ((8 + 8) × (8 × 8 + 8) − 8)
:= (8 + 8) × (8 × 8 + 8) − (8 + 8)/8
:= 9 + (((9999 + 99)/9) + 9)
:= 99/9 + (9 × ((99 + 9 + 9) + 9))
:= 999 + (9 × (9 + 9) − (99/9))
Ï 1141 := 1 + (11 − 1) × (1 + 1 + 1 + 111)
4
Ï 1146 := 1 + (111+1 + (1 + 1)11−1 )
Ï 1151 := ((1 + 1) × (((1 + 1) × (1 + 11))1+1 )) − 1
:= (2 × ((22 + 2)2 )) − 22/2
:= (2 × (((22 + 2)2 ) − 2)) − 2
:= (2 × ((22 + 2)2 )) − 2/2
:= 3 + (3333/3 + 33 )
:= 3333 − (3 × 33+3 )
:= 33 + (((3 × 3 + 3 + 3)3 − 3)/3)
:= 4 + (((4/4 + 4)4 + 44 ) + 44 )
:= 4 × 44 + ((444 + 44)/4)
:= 44 + (4444/4 − 4)
:= 5 + (5555/5 + 5 × 5)
:= 5 + ((5555/5 + 5 × 5) + 5)
:= 5/5 + (5 × (5 × (55 − 5 − 5) + 5))
:= 6 × 6 + ((6666/6) − 6)
:= 66 + (6 × 6 × (6 × 6 − 6))
:= (66 × (6 + 6 + 6)) − (6 × 6 + 6/6)
:= 7 + ((77 × (7 + 7) + 7 × 7) + 7)
:= 7 × 7 + (7777/7 − (7 + 7))
:= 7 + (77/7 × (777/7 − 7))
:= (8 + 8) × (8 × 8 + 8) − 88/8
:= (8 + 8)/8 + ((8 + 8) × (8 × 8 + 8) − 8)
:= (8 + 8) × (8 × 8 + 8) − 8/8
:= (9 − ((9 + 9)/9)) × (9 × (9 + 9) + 9/9)
:= 9 + (((((9999 − 9)/9) + 9) + 9) + 9)
:= 99 + ((9 × (99 + 9 + 9)) − 9/9)
Ï 1142 := 1 + 1 + (11 − 1) × (1 + 1 + 1 + 111)
Ï 1147 := 1111 + ((1 + 1 + 1) × (1 + 11))
Ï 1152 := (1 + 1) × (((1 + 1) × (1 + 11))1+1 )
:= 2 × ((22 + 2)2 )
:= (2 × (22 × (22 + 2 + 2))) − 2
:= (2 × (((22 + 2)2 ) − 2)) − 2/2
:= 3 + (((3333 + 3)/3) + 33 )
:= 3 + (3333/3 + 33)
:= 3 × ((3 + 3) × ((3/3 + 3)3 ))
:= 4 × (4 + 4) + (4444 − 4)/4
:= 4 + (4444/4 + 4 × (4 + 4))
:= 4 × (4 × (4 + 4) + 44 )
:= 5 + ((5555 + 5)/5 + 5 × 5)
:= 5 × 5 + ((5555 + 55)/5)
:= ((5 + 5)/5)5 × (55/5 + 5 × 5)
:= 6 × 6 + ((6666 + 6)/6 − 6)
:= 6 × 6 + (6666/6)
:= 6 × ((6 × (6 × 6 − 6) + 6) + 6)
:= 7 + (((77 × (7 + 7) + 7 × 7) + 7/7) + 7)
:= 7 + ((7/7 + 7 + 7) × (77 − 7/7))
:= ((7 + 7)/7 + 7) × ((7 + 7)/7)7
:= (8 − 88)/8 + (8 + 8) × (8 × 8 + 8)
:= 88/8 + ((8 + 8) × ((8 × 8 − 8/8) + 8))
:= (8 + 8) × (8 × 8 + 8)
:= 9 + ((9 × ((99 + 9 + 9) + 9)) − 9/9)
:= 9 + (((9999/9 + 9) + 9) + 9)
:= 99 + (9 × (99 + 9 + 9))
86
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1153 := 1 + ((1 + 1) × (((1 + 1) × (1 + 11))1+1 ))
Ï 1158 := 1 + (1 + ((1 + 11 × (1 + 1 + 1))1+1 ))
Ï 1163 := 11 + ((1 + 1) × (((1 + 1) × (1 + 11))1+1 ))
:= 2/2 + (2 × ((22 + 2)2 ))
:= 2 + (2 × (((22 + 2)2 ) + 2))
:= 22/2 + (2 × ((22 + 2)2 ))
:= ((3/3 + 3)3 ) + 33 × 33
:= 3 + (33 × ((33 − 3/3) + 3))
:= 3 + (3 − 3/3)3×3 + 3 × (3 + 3)3
:= 4/4 + (4 × (4 × (4 + 4) + 44 ))
:= 4 + ((4444 − 4)/4 + 44)
:= 4 + ((4444/4 + 44) + 4)
5
:= 5 + (((5 − 5/5)5 − 5/5) + 5 × 5 × 5)
:= 555 + ((5 − (55 + 55))/5)
:= 55 + ((5 + 5)/5 × (555 − 5/5))
:= 6 + ((6666/6) + 6 × 6)
:= 6 + (6 × ((6 × (6 × 6 − 6) + 6) + 6))
:= 66/6 + (6 × ((6 × (6 × 6 − 6) + 6) + 6))
:= 7 × 7 + (7777/7 − 7)
:= 7 × 7 + ((7777 − 7 − 7)/7)
:= 7 + ((77 × (7 + 7) + 77) + 7/7)
:= 8/8 + (8 + 8) × (8 × 8 + 8)
:= 8 + ((8 + 8) × (8 × 8 + 8) − ((8 + 8)/8))
:= 88/8 + (8 + 8) × (8 × 8 + 8)
:= 9/9 + ((9 × (99 + 9 + 9)) + 99)
:= 9 × 9 + (((99 × 99) − (99 + 9))/9)
:= (9 + 9)/9 + (999 + 9 × (9 + 9))
Ï 1154 := (1 + 1) × (1 + (((1 + 1) × (1 + 11))1+1 ))
Ï 1159 := 1 + (1 + (1 + ((1 + 11 × (1 + 1 + 1))1+1 )))
2
Ï 1164 := (1 + 11) × (111 − (1 + (1 + 1 + 11)))
:= 2 + (2 × ((22 + 2) ))
:= 2 + ((2 × (((22 + 2) ) + 2)) + 2/2)
:= 2 × (((((22 + 2)2 ) + 2) + 2) + 2)
:= (3 − 3/3) × (((3 × 3 + 3)3 + 3)/3)
:= 3 + ((3/3 + 33)3−3/3 )
:= 3 + ((33 × (33 + 3)) − 33 )
:= 44 + (4444 − 4)/4
:= 4 + (4444/4 + 44)
:= 444 + (4 × (4 × 44 + 4))
2
5
5
:= 5 + ((5 − 5/5) + 5 × 5 × 5)
:= 5 + (((5 − 5/5) + 5 × 5 × 5) + 5)
:= 55 + ((5555 − (5 + 5))/5)
:= 6 + ((6666 + 6)/6 + 6 × 6)
:= 6 + (((6666/6) + 6 × 6) + 6)
:= 6 × 6 × 6 × 6 − (66 + 66)
:= 77 + (77 × (7 + 7) − 7/7)
:= 7 × 7 + (7777 − 7)/7
:= (77 + 7)/7 × (7 × (7 + 7) − 7/7)
:= (8 + 8)/8 + (8 + 8) × (8 × 8 + 8)
:= 8 + ((8 + 8) × (8 × 8 + 8) − 8/8)
:= 8 × 8 + ((8888 − 88)/8)
:= 9 + ((9 × ((99 + 9 + 9) + 9)) + (99/9))
:= 9 × 9 + ((99/9) × (99 − 9/9))
:= 999/9 + (9 × (99 + 9 + 9))
Ï 1155 := ((1 + 11 × (1 + 1 + 1))1+1 ) − 1
Ï 1160 := (11 − 1) × (1 + 1 + 1 + 1 + 1 + 111)
Ï 1165 := 1111 + ((111 − 1)/(1 + 1) − 1)
:= 2 + ((2 × ((22 + 2) )) + 2/2)
:= 2 × ((((22 + 2) ) + 2) + 2)
:= 2 + ((2 × ((22 + 2)2 )) + 22/2)
:= 33 × ((33 − 3/3) + 3)
:= 33 × 33 + (((3 + 3)3 − 3)/3)
:= 33 + (3333/3 + 33 )
:= 44 + 4444/4
:= 4 + ((4 × (4 × (4 + 4) + 44 )) + 4)
:= 4 + ((44 − 4/4) × (44/4 + 4 × 4))
:= 55 + 55 × (5 × 5 − 5)
:= (5 + 5) × (555/5 + 5)
:= 5 + ((5 + 5) × (555/5 + 5))
2
2
:= 66/6 × (666/6 − 6)
:= 6 + (((6666 + 6)/6 + 6 × 6) + 6)
:= 66 + ((6666/6) − (6 + 6))
:= 77 + 77 × (7 + 7)
:= 7 × 7 + 7777/7
:= (7 + 7) × (77 + 7) − 77/7
:= 88/8 + ((8 + 8) × (8 × 8 + 8) − 8)
:= 8 + (8 + 8) × (8 × 8 + 8)
:= 8 + ((8/8 + 88) × (88 + 8 + 8)/8)
:= 99/9 × ((99 − ((9 + 9 + 9)/9)) + 9)
:= 999 + (9 × (9 + 9) − 9/9)
:= 9 + (((999/9 − 9)(9+9)/9 )/9)
Ï 1156 := (1 + 11 × (1 + 1 + 1))1+1
2
:= 2 × (((22 + 2) ) + 2)
Ï 1161 := 1111 + ((11 − 1)1+1 /(1 + 1))
Ï 1166 := 11 × (111 − (1 + (1 + 1 + 1 + 1)))
:= 2/2 + (2 × ((((22 + 2) ) + 2) + 2))
:= 2 + (2 × (((((22 + 2)2 ) + 2) + 2) + 2))
:= (33 × (33 + 3)) − 33
:= ((33/3)3 ) − ((3 + 3) × 33 + 3)
:= 4 + (4 × (4 × (4 + 4) + 4 ))
:= (44 − 4/4) × (44/4 + 4 × 4)
:= (44 × (44 − 44))/(4 + 4)
:= 55 + (55 × (5 × 5 − 5) + 5/5)
:= 55 + (5555/5 − 5)
:= 55 + 5555/5
2
:= (3/3 + 33)3−3/3
4
(6+6)/6
:= (6 × 6 − ((6 + 6)/6))
:= 6 + ((66/6) × (666/6 − 6))
:= 66 + ((6666 − 66)/6)
:= 7/7 + (77 × (7 + 7) + 77)
:= 7 × 7 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 7 × 7)
:= (8/((8 + 8)/8)) + (8 + 8) × (8 × 8 + 8)
:= 8 + ((8 + 8) × (8 × 8 + 8) + 8/8)
:= 8 × 8 + ((8888 − 8)/8 − 8)
:= ((999/9 − 9)(9+9)/9 )/9
:= 999 + 9 × (9 + 9)
:= 99/9 × ((99 − ((9 + 9)/9)) + 9)
Ï 1157 := 1 + ((1 + 11 × (1 + 1 + 1))1+1 )
Ï 1162 := 111+1+1 − (1 + 1 + 11)1+1
Ï 1167 := 11 + ((1 + 11 × (1 + 1 + 1))1+1 )
:= 2/2 + (2 × (((22 + 2)2 ) + 2))
:= 2 + (2 × ((((22 + 2)2 ) + 2) + 2))
:= 22/2 + (2 × (((22 + 2)2 ) + 2))
:= 3/3 + ((3/3 + 33)3−3/3 )
:= 3 + (((3/3 + 33)3−3/3 ) + 3)
:= 33 + (33 × (3 × 3 + 33))
:= 4 + ((4 × (4 × (4 + 4) + 44 )) + 4/4)
:= (4 + 4)/4 × ((4/4 + 4)4 − 44)
:= 444/4 + 4 × (44 + 4 + 4)
:= 5 + (((5 + 5)/5)5 × (55/5 + 5 × 5))
:= 55 + ((5555 + 5)/5 − 5)
:= 55 + (5555 + 5)/5
:= 6 + ((66 × (6 + 6 + 6)) − (6 × 6 + 6/6))
(6+6)/6
:= 6 + ((6 × 6 − ((6 + 6)/6))
)
:= ((6 × (6 × 66 − 6)) − 6)/((6 + 6)/6)
:= 77 + (77 × (7 + 7) + ((7 + 7)/7))
:= 7 + (77 × (7 + 7) + 77)
:= 7 + (7777/7 + 7 × 7)
:= (8/8 + 88) × (88 + 8 + 8)/8
:= 8 + ((8 + 8) × (8 × 8 + 8) + ((8 + 8)/8))
:= 8 × 8 + (8888/8 − 8)
:= (((9 + 9)/9)99/9 ) − 9 × 99
:= 9/9 + (999 + 9 × (9 + 9))
:= ((99 − 9/9) × ((99 + 9)/9)) − 9
87
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1168 := (1 + 11)1+1 + (1 + 1)11−1
2
Ï 1173 := 1 + (1111 + ((1 + 111+1 )/(1 + 1)))
Ï 1178 := 1 + (11 × (111 − (1 + 1 + 1 + 1)))
:= 2 × (((22 + 2) ) + 2 × (2 + 2))
:= 22 + ((2 × ((22 + 2) )) − 2/2)
:= 22 + (2 × (((22 + 2)2 ) + 2))
:= 3 + (3/3 + 33)3−3/3 + 3 × 3
:= 3 + (33 × 33 + 3 × 33 )
:= ((3 − 33)/3) + (33 × (33 + 3))
:= 4 × ((4 × (4 + 4) + 44 ) + 4)
:= (4 × 4 + 4/4) × ((44 + 4)/4 + 4)
:= 4 + (((4444 − 4) + 44 )/4)
:= 55 + (5555 + 5 + 5)/5
:= 555 + ((55 − 5 − 5)/5 − 5)
:= 555 + (55 − 5 − 5)/5
:= 6 + (((6 × 6 − ((6 + 6)/6))(6+6)/6 ) + 6)
:= ((6 × (6 × 66 − 6)) + 6)/((6 + 6)/6)
:= 66 + (6666 + 6)/6
:= (7 + 7) × (77 + 7) − (7/7 + 7)
:= (7 + 7) × (77 + 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + (7 + 7) × (77 + 7)
:= 8 + ((8 + 8) × (8 × 8 + 8) + 8)
:= 8 × 8 + ((8888 − (8 + 8))/8)
:= (88/8 + 8) × (8 × 8 − ((8 + 8)/8))
:= 9 × 9 + (((99 × 99) − (9 + 9))/9)
:= 9 + ((9 × (99 + 9 + 9)) + 999/9)
:= (((99 × (99 + 9)) − 9)/9) − 9
Ï 1169 := 1 + ((1 + 11)1+1 + (1 + 1)11−1 )
2
Ï 1174 := (1 + 1) × (11 + (((1 + 1) × (1 + 11))1+1 ))
Ï 1179 := (11 − 1 − 1) × ((11 × (1 + 11)) − 1)
:= ((((2 × 22) + 2)2 ) + 222)/2
:= 22 + (2 × ((22 + 2)2 ))
:= 2 + (22/2 × (222/2 − (2 + 2)))
:= ((33/3)3 ) − (3 + 3) × 33
:= (33 × (33 + 3)) − (33/3 + 3)
:= 3 × ((33 × (3 × 3 + 3)) − 3)
:= 4/4 + (4 × ((4 × (4 + 4) + 44 ) + 4))
:= ((4444 − 4) + 44 )/4
:= 4 + ((4444 + 44 )/4)
:= 555 + ((55 − 55)/5)
:= 555 + ((55 − 5)/5 − 5)
:= 555 + (55 − 5)/5
:= ((6 + 6 + 6) × (66 − 6/6)) − 6/6
:= 66 + ((6666 − (6 + 6 + 6))/6)
:= 66 + (((6666 + 6) + 6)/6)
:= (7 + 7) × (77 + 7) − 7
:= (7 + 7) × (77 + 7) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 + 7) × (77 + 7)
:= 8 + (((8 + 8) × (8 × 8 + 8) + 8/8) + 8)
:= 8 × 8 + (8888 − 8)/8
:= 8 + (((8 + 8) × (8 × 8 + 8) + (88/8)) + 8)
:= 9 × 9 + (((99 × 99) − 9)/9)
:= 9 × 9 + (9999/9 − (9 + 9))
:= 99 + (999 + 9 × 9)
Ï 1170 := (11 − 1) × (111 + ((1 + 1) × (1 + 1 + 1)))
Ï 1175 := 1111 + ((1 + 1)(1+1)×(1+1+1) )
Ï 1180 := (11 − 1) × (111+1 − (1 + 1 + 1))
:= 22 + (2 × (((22 + 2)2 ) − 2))
:= 22 + ((2 × ((22 + 2)2 )) + 2/2)
:= 2 + ((2 × (((22 + 2)2 ) + 2)) + 22)
:= (3 + 3) × (33 × (3 + 3) − 3)
:= ((3/3 + 3)3 ) + 3333/3
:= 3 + ((33 × (33 + 3)) − 33/3)
:= 4 + ((44 × (4 − 44))/(4 + 4))
:= (4444 + 44 )/4
:= (4 × ((44 − 4) + 44)) − 4
:= (5 + 5) × ((555 + 5)/5 + 5)
:= 55 /5 + (5 + 5) × 55
:= 55 /5 + 555
:= (6 + 6 + 6) × (66 − 6/6)
:= 66 + ((6666 − (6 + 6))/6)
:= (66 × (6 + 6 + 6)) − ((6 + 6)/6 + 6)
:= 7/7 + ((7 + 7) × (77 + 7) − 7)
:= (7 + 7) × (77 + 7) − 7/7
:= 77 + ((7777 − 7)/7 − 7)
:= (8/8 + 8) × (8 × (8 + 8) + ((8 + 8)/8))
:= 8 × 8 + 8888/8
:= 88 + (((8888 − 88)/8) − 8)
:= 9 + (999 + 9 × (9 + 9))
:= (((99 + 9) × (99 − 9/9)) − 9)/9
:= (((99 × (99 + 9)) + 9)/9) − 9
4
Ï 1171 := 1111 + (111+1 − 1)/(1 + 1)
Ï 1176 := (1 + 11) × (111 − 1 − 1 − 11)
Ï 1181 := 1 + (11 − 1) × (111+1 − 1 − 1 − 1)
:= 2 + (((((2 × 22) + 2)2 ) + 222)/2)
:= 2 + ((2 × ((22 + 2)2 )) + 22)
:= (((22/2 + 22) + 2)2 ) − (2 × 22)
:= 33 + (3333/3 + 33)
:= (3/3 + 3) × (3 × 3 × 33 − 3)
:= (33 × (33 + 3)) − ((3/3 + 3) + 3)
4
4
:= ((4444 + 4 )/4) − 4
:= (4 × ((4 − 4) + 44)) − 4 − 4
:= 5 + (5555/5 + 55)
:= 555 + ((55 + 5)/5 − 5)
:= 66 + ((6666/6) − 6)
:= (66 × (6 + 6 + 6)) − 6 − 6
:= 7 × 7 + ((7777 + 77)/7)
:= (7 + 7) × (77 + 7)
:= 8 + ((8 + 8) × (8 × 8 + 8) + (88/8))
:= 88 + (8 × (8 × (8 + 8) + 8))
:= 9 × 9 + (((99 × 99) + 9)/9)
Ï 1172 := 1111 + ((1 + 111+1 )/(1 + 1))
:= 4 + (44/4 × (444/4 − 4))
:= 555 + (55 + 5)/5
:= (66 × (6 + 6 + 6)) − 6/6 − 6
:= 77 + (7777/7 − 7)
:= (99 − 9/9) × ((99 + 9)/9)
Ï 1177 := 11 × (111 − (1 + 1 + 1 + 1))
:= 8 + (((8888 − (8 + 8))/8) + 8 × 8)
:= 9 × 9 + ((99/9) × (9/9 + 99))
Ï 1182 := 1 + 1 + (11 − 1) × (111+1 − 1 − 1 − 1)
:= 22 + ((2 × ((22 + 2)2 )) − 2)
:= 22/2 × (222/2 − (2 + 2))
:= 22 + (2 × ((((22 + 2)2 ) + 2) + 2))
:= 3 + (33/3)3 − (3 + 3) × 33
:= (33 × (33 + 3)) − 33/3
:= (33 × (33 + 3)) − (3 + 3)
:= 4 + (4 × ((4 × (4 + 4) + 44 ) + 4))
:= 44/4 × (444/4 − 4)
:= (4 × ((44 − 4) + 44)) − (4 + 4)/4
:= 5 + ((5555 + 5)/5 + 55)
:= 55 + ((5555 + 55)/5)
:= 555 + (55 + 5 + 5)/5
:= 66 + ((6666 + 6)/6 − 6)
:= 66 + (6666/6)
:= (66 × (6 + 6 + 6)) − 6
:= 7 + ((7 + 7) × (77 + 7) − (77/7))
:= 7/7 + (7 + 7) × (77 + 7)
:= 7 + ((7 + 7) × (77 + 7) − 7/7)
:= 8 + (((8888 − 88)/8) + 8 × 8)
:= 8/8 + ((8 × (8 × (8 + 8) + 8)) + 88)
:= 8 + ((8888 − 8)/8 + 8 × 8)
:= 9 × 9 + ((((99 × 99) + 9) + 9)/9)
:= 99/9 × ((99 − 9/9) + 9)
:= 9 × 9 + (((9999 − 9)/9) − 9)
88
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1183 := 1111 + (1 + 11)1+1 /(1 + 1)
Ï 1188 := 11 × (111 − 1 − 1 − 1)
Ï 1193 := 1 + 1111 + (11 − 1 − 1)1+1
:= 222 + (((2/2 + 2)2 + 22)2 )
:= 22 × (2 × (22 + 2 + 2) + 2)
:= 2 + (((2 × 222 ) + 222) + 2/2)
:= (3333 + (3 + 3)3 )/3
:= 33 × (33 + 3)
:= 3 + (((33 × (33 + 3)) − 3/3) + 3)
4
:= (4 × ((44 − 4) + 44)) − 4/4
:= 4 + (4 × ((4 − 4) + 44))
:= 4 + (4 × (44 + 44 ) − 44/4)
:= 5 + ((55 − 5 − 5)/5 + 555)
:= (55 − 5/5) × (55 + 55)/5
:= ((5 + 5)/5 × (55 − 5)/5) − 55
:= 6 + ((6666/6) + 66)
:= 66 × (6 + 6 + 6)
:= 6 + ((66 × (6 + 6 + 6)) − 6/6)
:= 7 + (7 + 7) × (77 + 7)
:= 77 + 7777/7
:= 7 + (((7777 − 7 − 7)/7) + 77)
:= 8 + (8888/8 + 8 × 8)
:= 88 + ((8888 − 88)/8)
:= 8/8 + ((8 + 8) × (88 − 8) − 88)
:= 9 × 9 + (9999/9 − 9)
:= 99 × ((99 + 9)/9)
:= 9 × 9 + (9999 + 9)/9
Ï 1189 := 1 + 11 × (111 − 1 − 1 − 1)
Ï 1194 := 11 × 111 − ((1 + 1 + 1)1+1+1 )
Ï 1184 := 11 × 111 − 111/(1 + 1 + 1)
:= 2 × (((22 + 2)2 ) + 22+2 )
:= 222 + ((2 × 222 ) − 2/2)
:= 222 + (2 × (222 + 2))
:= (33 × (33 + 3)) − (3/3 + 3)
:= 3/3 + (33 × (33 + 3))
:= 3 + ((33 × (33 + 3)) + 3)
4
:= 4 × ((4 − 4) + 44)
:= 4 × (44 + 4 ) − 44/4
:= 4 × (44 + 44 ) − ((4 + 4)/4 + 4)
:= 5 + ((55 − 5)/5 + 555)
:= 5 + (((55 − 5)/5 + 555) + 5)
:= ((55 − 5 + 55 )/5) − 55
:= 6 + ((6666 + 6)/6 + 66)
:= 6/6 + (66 × (6 + 6 + 6))
:= 6 + (66 × (6 + 6 + 6))
:= 7 + ((7 + 7) × (77 + 7) + 7/7)
:= 77 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 77)
:= 8 + ((8 × (8 × (8 + 8) + 8)) + 88)
:= ((88/8 + 8) × (8 × 8 − 8/8)) − 8
:= 8 + (((8888 + 88)/8) + 8 × 8)
:= 9 × 9 + ((9999 + 9)/9 − 9)
:= ((99 × (99 + 9)) + 9)/9
:= 9 × 9 + (((9999 + 9) + 9)/9)
4
Ï 1185 := 11 × 111 − (1 + 1 + 1) × (1 + 11)
Ï 1190 := (11 − 1) × (111+1 − 1 − 1)
2
Ï 1195 := 11 × 111 − ((1 + 1) × (1 + 1 + 11))
:= 222 + (2 × 22 )
:= (2 × (((22 + 2)2 ) + 22)) − 2/2
:= 3 + ((33 × (33 + 3)) − 3/3)
:= 3 + (((33 × (33 + 3)) + 3/3) + 3)
:= 4/4 + (4 × ((4 − 4) + 44))
:= (4 − 44)/4 + 4 × (44 + 44 )
:= 4 × (44 + 44 ) − (4/4 + 4)
:= 5 + (55 /5 + 555)
:= 5 × (((5 − (5 + 5)/5)5 ) − 5)
:= 5 × 5 × 5 × (5 + 5) − 55
:= 6 × 6 × 6 × 6 − 666/6
:= (6 + 6)/6 + (66 × (6 + 6 + 6))
:= 6 + ((66 × (6 + 6 + 6)) + 6/6)
:= 7 + ((7 + 7) × (77 + 7) + ((7 + 7)/7))
:= 7 + ((7 + 7) × (77 + 7) + 7)
:= 7 + (7777/7 + 77)
:= (8 − 8/8 + 8) × (88 − (8/8 + 8))
:= 88 + ((8888 − 8)/8 − 8)
:= 8 + ((((8888 + 88) + 8)/8) + 8 × 8)
:= 9 + ((99 − 9/9) × ((99 + 9)/9))
:= (((99 × (99 + 9)) + 9) + 9)/9
:= 9 + (((99 × (99 + 9)) − (9 + 9))/9)
:= (2 + 2 + 2)
2+2
− 222/2
:= (33 × (33 + 3)) − 3
4
Ï 1186 := 11 × (111 − 1 − 1 − 1) − 1 − 1
Ï 1191 := 1 + (11 − 1) × (111+1 − 1 − 1)
2
Ï 1196 := (11 × (111 − 1 − 1)) − 1 − 1 − 1
:= 222 + (2 × (222 − 2))
:= 2/2 + ((2 × 22 ) + 222)
:= 2 × (((22 + 2)2 ) + 22)
:= 3/3 + ((33 × (33 + 3)) − 3)
:= 3 + (33 × (33 + 3))
:= 3 × 3 + ((33 × (33 + 3)) − 3/3)
4
:= (4 + 4)/4 + (4 × ((44 − 4) + 44))
:= 4 × 4 + ((4444 + 4 )/4)
:= 4 × (44 + 44 ) − 4
:= 5 + ((55 + 5)/5 + 555)
:= 555 + ((55 + 55 )/5)
:= ((55 + 55 + 5)/5) − 55
:= (66 × (6 + 6 + 6)) − (6 + 6)/6
:= (6 × 6 × 66 + 6)/((6 + 6)/6)
:= 6 + ((66 × (6 + 6 + 6)) + ((6 + 6)/6))
:= 77 + ((7777 − 7 − 7)/7)
:= 7 + (((7 + 7) × (77 + 7) + 7/7) + 7)
:= 7 + ((7777 + 7)/7 + 77)
:= 8 × 8 + ((8888 + 88)/8)
:= 88 + (8888/8 − 8)
:= 8 + (((8888 − 88)/8) + 88)
:= ((99 × (99 + 9)) − (9 + 9))/9
:= 9 × 9 + ((9999 − 9)/9)
:= 9 + (((99 × (99 + 9)) − 9)/9)
Ï 1187 := 11 × (111 − 1 − 1 − 1) − 1
:= 2 + ((2 + 2 + 2)
2+2
− 222/2)
:= (33 × (33 + 3)) − 3/3
Ï 1192 := 1111 + (11 − 1 − 1)1+1
2
Ï 1197 := (11 × (111 − 1 − 1)) − 1 − 1
:= 2 + ((2 × 22 ) + 222)
:= 2/2 + (2 × (((22 + 2)2 ) + 22))
:= 3 + ((33 × (33 + 3)) + 3/3)
:= 3 × ((33 × (3 × 3 + 3)) + 3)
4
:= 4 + ((4 × ((4 − 4) + 44)) − 4/4)
:= 4 × (44 + 4 ) − 4 − 4
:= 4/4 + (4 × (44 + 44 ) − 4)
:= 5 + ((55 + 5 + 5)/5 + 555)
:= 555 + ((55 + 55 + 5)/5)
:= ((5 + 5)/5 × (55 + 5)/5) − 55
:= (66 × (6 + 6 + 6)) − 6/6
:= 6 + ((66 × (6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + ((6 × 6 × 66 + 6)/((6 + 6)/6))
4
:= 77 + (7777 − 7)/7
:= 7 + (((7 + 7) × (77 + 7) + (7 + 7)/7) + 7)
:= 7 + (((7 + 7) × (77 + 7) + 7) + 7)
:= 8 × 8 + (((8888 + 88) + 8)/8)
:= (8 + 8) × (88 − 8) − 88
:= (88/8 + 8) × (8 × 8 − 8/8)
:= ((99 × (99 + 9)) − 9)/9
:= 9 × 9 + 9999/9
:= 9 + (99 × ((99 + 9)/9))
89
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1198 := (11 × (111 − 1 − 1)) − 1
Ï 1203 := 1 + (1 + (1 + (1 + (11 × (111 − 1 − 1)))))
Ï 1208 := (11 × (111 − 1)) − 1 − 1
:= 2 + (2 × (((22 + 2) ) + 22))
:= (2/2 + 2) × ((22 − 2) + 2/2)
:= 2 + ((2/2 + 2) × ((22 − 2)2 + 2))
:= 3 × 3 + ((33 × (33 + 3)) + 3/3)
:= ((3 + 3) × (33 × (3 + 3) + 3)) − 3
:= 3 + (((33/3)3 ) − (3 × 33 + 33 ))
2
4
2
4
:= 4 × (44 + 4 ) − (4 + 4)/4
:= 4 + (4 × (44 + 4 ) − 4/4)
:= 4 + (4 × (44 + 44 ) + 4)
:= (5 + 5)/5 × ((55 − 5)/5 − 5 × 5)
:= 5 + ((5 + 5)/5 × ((55 − 5)/5 − 5 × 5))
:= (5 + 5)/5 × ((55 × 55 − 5)/5)
:= ((66 − 6)/6) + (66 × (6 + 6 + 6))
:= ((6 × (6 × 66 + 6)) − 6)/((6 + 6)/6)
:= (6 + 6)/6 + ((6 + 6 + 6) × (66 + 6/6))
:= 7 + ((((7 + 7) × (77 + 7) + 7/7) + 7) + 7)
:= (77/7 × (777 − 7)/7) − 7
:= 7 × (7 + 7) + (7777 − 7)/7
:= 88 + (8888 − 8)/8
:= 88/8 + ((8 + 8) × (88 − 8) − 88)
:= 8 × (8 × 8 + 88) − 8
:= 9 + (((99 × (99 + 9)) + 9)/9)
:= 9 × 9 + ((9999 + 99)/9)
:= 9 + ((99/9) × (9/9 + 99 + 9))
Ï 1204 := (11 × (111 − 1)) − ((1 + 1) × (1 + 1 + 1))
Ï 1199 := 11 × (111 − 1 − 1)
:= 2 + (((2/2 + 2) × (22 − 2)2 ) + 2)
:= 22/2 × (222/2 − 2)
:= 33/3 + (33 × (33 + 3))
3
Ï 1209 := (11 × (111 − 1)) − 1
:= (2 × 222 ) + (222 − 2)/2
:= (33 × ((3 + 3) + 3) − 3)/(3 + 3)
:= 3 + ((3 + 3) × (33 × (3 + 3) + 3))
:= 4 × (44 + 4 ) − 4/4
:= 4 + 4 × (44 + 44 )
:= 4 + ((4 × (44 + 44 ) + 4/4) + 4)
:= (5 + 5) × (5 × 5 × 5 − 5) − 5/5
:= 5 + ((5 + 5) × (5 × 5 × 5 − 5) − 5/5)
:= (55 × (55 + 55) − 5)/5
:= 66/6 + (66 × (6 + 6 + 6))
:= ((66/6) × ((666 − 6)/6)) − 6
:= ((6 × (6 × 66 + 6)) + 6)/((6 + 6)/6)
:= 77 + ((7777 + 77)/7)
:= 7 × ((7 × 7 × 7 × 7 + 7)/(7 + 7))
:= 7 × (7 + 7) + 7777/7
:= 88 + 8888/8
:= 8 × (8 × 8 + 88) − (88 + 8)/8
:= 8/8 + (8 × (8 × 8 + 88) − 8)
:= 99/9 × (9/9 + 99 + 9)
:= 9 + ((((99 × (99 + 9)) − (9 + 9))/9) + 9)
:= 99 + ((9999 − 9)/9)
4
Ï 1200 := 1 + (11 × (111 − 1 − 1))
:= (2/2 + 2) × (22 − 2)
2
:= 3 + ((33 × (33 + 3)) + 3 × 3)
4
Ï 1205 := 11 × 111 − ((1 + 1)1+1+1+1 )
2
Ï 1210 := 11 × (111 − 1)
:= (2/2 + 2 + 2) × (22 − 2)/2
:= 2 × ((22/2)2 + 222 )
:= ((33/3)3 ) − (3 × 33 + 33 )
:= 3 × 33 + 3333/3
4
:= 4 × (44 + 4 )
:= 4 + (4 × (44 + 4 ) + 4/4)
:= 44/4 × (444 − 4)/4
:= (5 + 5) × (5 × 5 × 5 − 5)
:= 5 + (5 + 5) × (5 × 5 × 5 − 5)
:= 55 × (55 + 55)/5
:= 6 + ((66 × (6 + 6 + 6)) + 6)
:= 6 + ((66 × (6 + 6 + 6)) + (66/6))
:= 66/6 × ((666 − 6)/6)
:= (7 × 7 − 7/7) × (77/7 + 7 + 7)
:= 7/7 + ((77 × (7 + 7) + 77) + 7 × 7)
:= 77/7 × (777 − 7)/7
:= (8 + 8) × (88/8 + 8 × 8)
:= 8 × (8 × 8 + 88) − 88/8
:= 88/8 × (888 − 8)/8
:= (9/9 + 9) × (999/9 + 9)
:= 9 + ((((99 × (99 + 9)) − 9)/9) + 9)
:= 99 + 9999/9
Ï 1201 := 1 + (1 + (11 × (111 − 1 − 1)))
Ï 1206 := (11 × (111 − 1)) − 1 − 1 − 1 − 1
Ï 1211 := 1 + (11 × (111 − 1))
:= 2 + (22/2 × (222/2 − 2))
:= (2/2 + 2) × ((22 − 2)2 + 2)
:= (2 × 222 ) + (222 + 2)/2
:= (3 × (33 + 3)) + 3333/3
:= (3 + 3) × (33 × (3 + 3) + 3)
:= 3 × 33 + ((3333 + 3)/3)
:= 4/4 + 4 × (44 + 44 )
:= 4 + (4 × (44 + 44 ) + (4 + 4)/4)
:= 44/4 + 4 × (44 + 44 )
:= 5/5 + (5 + 5) × (5 × 5 × 5 − 5)
:= 5 + ((5 + 5) × (5 × 5 × 5 − 5) + 5/5)
:= (55 × (55 + 55) + 5)/5
:= 6 + (((66 × (6 + 6 + 6)) + 6/6) + 6)
:= (6 + 6 + 6) × (66 + 6/6)
:= ((66 × ((666 − 6)/6)) + 6)/6
:= (7 × 7 × 7 × 7 + 7/7)/((7 + 7)/7)
:= 77 × (7 + 7) + ((7 + 7)/7)
:= (7 × (7 × (7 + 7) + 77)) − (7 + 7)
:= 8/8 + ((8 + 8) × (88/8 + 8 × 8))
:= 8 + ((8888 − 8)/8 + 88)
:= ((88 × (888 − 8)/8) + 8)/8
:= 9 + (9999/9 + 9 × 9)
:= 9 + ((99 × ((99 + 9)/9)) + 9)
:= 99 + (9999 + 9)/9
Ï 1202 := 1 + (1 + (1 + (11 × (111 − 1 − 1))))
7
Ï 1207 := (11 × (111 − 1)) − 1 − 1 − 1
Ï 1212 := 1 + (1 + (11 × (111 − 1)))
:= 2 + ((2/2 + 2) × (22 − 2)2 )
:= 2 + ((2/2 + 2 + 2) × (222 − 2)/2)
:= 2 + ((2 × 222 ) + 222 /2)
:= 3 + ((33 × (33 + 3)) + 33/3)
:= 3/3 + ((3 + 3) × (33 × (3 + 3) + 3))
:= 33 + ((33 × (33 + 3)) − 3)
:= (4 + 4)/4 + 4 × (44 + 44 )
:= 4 + ((4 × (44 + 44 ) − 4/4) + 4)
:= (4 × ((44 + 44 ) + 4)) − 4
:= (5 + 5)/5 + (5 + 5) × (5 × 5 × 5 − 5)
:= 5 + ((5 + 5) × (5 × 5 × 5 − 5) + ((5 + 5)/5))
:= (5 + 5)/5 × ((55 × 55 + 5)/5)
:= 6 + (((66 × (6 + 6 + 6)) + ((6 + 6)/6)) + 6)
:= 6/6 + ((6 + 6 + 6) × (66 + 6/6))
:= 6 + ((6 + 6 + 6) × (66 + 6/6))
:= 7 + ((7777/7 + 77) + 7)
:= 77/7 × 777/7 − (7 + 7)
:= 7/7 + ((7 × (7 × (7 + 7) + 77)) − (7 + 7))
:= (88/8 × (888 − 8)/8) − 8
:= 8 + (8888/8 + 88)
:= 8 × (8 × 8 + 88) − (8/((8 + 8)/8))
:= 9 + ((9999 + 9)/9 + 9 × 9)
:= 9 + ((((99 × (99 + 9)) + 9)/9) + 9)
:= ((99/9) × 999/9) − 9
90
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1213 := 1 + (1 + (1 + (11 × (111 − 1))))
2
2+2+2
Ï 1218 := 11 × 111 − 1 − 1 − 1
2
Ï 1223 := 1 + (1 + 11 × 111)
:= 22 + (2 × (((22 + 2) ) + 22))
:= 2 + (22/2 × 222/2)
:= 3 + (3333/3 + 3 × 33)
:= 3 + ((33 × (33 + 3)) + 33 )
:= ((33/3)3 ) − (3 × (33 + 3))
:= 4/4 + ((4 × ((44 + 44 ) + 4)) − 4)
:= (4 + 4)/4 + (4 × ((44 + 44 ) + 4))
:= (((44 × 444/4) + 4) + 4)/4
:= (5 × ((5 − (5 + 5)/5)5 )) − (5 + 5)/5
:= (5 + 5)/5 × (((55 − 55)/5) − 5)
:= (5 × (5 × 5 × (5 + 5) − 5)) − (5 + 5)/5
:= 6 × 6 + ((6666/6) + 66)
:= (6 × (6 × 6 × 6 − (6 + 6))) − 6
:= (6 × (6 × 6 × 6 − (6 + 6))) − 6/6
:= 7 + (77 × (7 + 7) + ((7 + 7)/7) )
:= (7 × (7 × (7 + 7) + 77)) − 7
:= (7 × (7 × (7 + 7) + 77)) − (7 + 7)/7
:= 88/8 × 888/8 − 8
:= (8 + 8)/8 + 8 × (8 × 8 + 88)
:= 8 + (8 × (8 × 8 + 88) − 8/8)
:= (((99 × 999/9) + 9)/9) − 9
:= 9 + (((9999 − 9)/9) + 99)
:= 22 + ((2/2 + 2)
)
7
Ï 1214 := 1 + (1 + (1 + (1 + (11 × (111 − 1)))))
:= 9 × 9 × 9 + ((((9 + 9)/9)9 ) − (9 + 9))
Ï 1224 := 1 + (1 + (1 + 11 × 111))
Ï 1219 := 11 × 111 − 1 − 1
:= 2 × (((22/2) + 22 ) + 2)
:= (22/2 × 222/2) − 2
:= (2 × 222 ) + 22×(2+2)
:= 33 + ((33 × (33 + 3)) − 3/3)
:= 3 + (((3 × 3 + 3/3)3 ) + (3 + 3)3 )
:= 333 + (33 × 33 )
:= 4 + (44/4 × (444 − 4)/4)
:= 4 + ((44/4 + 4) × (4 − 4/4)4 )
:= 4 + ((4 × ((44 + 44 ) + 4)) + 4)
:= (5 × ((5 − (5 + 5)/5)5 )) − 5/5
:= 5 + ((5 × ((5 − (5 + 5)/5)5 )) − 5/5)
:= (5 × (5 × 5 × (5 + 5) − 5)) − 5/5
:= (((66 × 666/6) − 6)/6) − 6
:= ((6 × 6 − 6/6)(6+6)/6 ) − 6
:= 6 × (6 × 6 × 6 − (6 + 6))
2
2
:= 77/7 × 777/7 − 7
:= 7/7 + ((7 × (7 × (7 + 7) + 77)) − 7)
:= 8 × (8 × 8 + 88) − (8 + 8)/8
:= 88/8 + (8 × (8 × 8 + 88) − 8)
:= 9 × 99 + ((9 + 9) × (9 + 9) − 9/9)
:= 9 + (9999/9 + 99)
Ï 1215 := 11 × 111 − ((1 + 1) × (1 + 1 + 1))
2
:= (2/2 + 2 + 2) × (22 + 2)/2
:= 3 × (33 × ((3 × 3 + 3) + 3))
:= (44/4 + 4) × (4 − 4/4)4
5
:= 5 × ((5 − (5 + 5)/5) )
:= ((7 + 7)/7 + 7) × (((7 + 7)/7)7 + 7)
:= ((22 × 222/2) − 2)/2
:= 4 + (4 × ((44 + 44 ) + 4))
:= 4 + (44/4 × 444/4)
:= ((66 × 666/6) − 6)/6
:= (77 × 777/7 − 7)/7
:= 9 × (((99 + 9 + 9) + 9) + 9)
:= ((99 × 999/9) − 9)/9
:= 2 × ((2 × 2
2
) + ((22 + 2) ))
:= (3 + 3)3 + ((3 × 3 + 3/3)3 )
:= 5 × (5 × 5 × (5 + 5) − 5)
:= (6 × 6 − 6/6)(6+6)/6
:= 7 × (7 × (7 + 7) + 77)
:= 8 + (8 × (8 × 8 + 88) + 8/8)
:= 9/9 + ((9 − 9/9) × (9 × (9 + 9) − 9))
Ï 1226 := 1 + (1 + (1 + (1 + (1 + 11 × 111))))
Ï 1221 := 11 × 111
:= 2 + ((2 × 222 ) + 22×(2+2) )
:= 22/2 × 222/2
:= 33 + (33 × (33 + 3))
:= 4 × ((44 + 44 ) + 4)
:= ((22/2 + 22) + 2)2
:= ((33 − 3/3) + 3)3−3/3
:= ((88 × 888/8) − 8)/8
2+2
:= (9 − 9/9) × (9 × (9 + 9) − 9)
:= ((33/3)3 ) − 333/3
:= 8 × (8 × 8 + 88) − 8/8
Ï 1216 := 11 × 111 − 1 − 1 − 1 − 1 − 1
:= 8 + 8 × (8 × 8 + 88)
Ï 1225 := 1 + (1 + (1 + (1 + 11 × 111)))
Ï 1220 := 11 × 111 − 1
:= 5 + (5 × ((5 − (5 + 5)/5)5 ))
:= ((66/6) × 666/6) − 6
:= (7 × (7 × (7 + 7) + 77)) − 7/7
:= 3 + (((33/3)3 ) − (3 × (33 + 3)))
:= 44/4 × 444/4
:= 4 + (((44 × 444/4) + 4)/4)
:= 5/5 + (5 × ((5 − (5 + 5)/5) ))
:= 55/5 × 555/5
:= 5 + 55/5 × 555/5
:= 6 + ((66/6) × ((666 − 6)/6))
:= 66/6 × 666/6
:= 6 + (((66 × 666/6) − 6)/6)
:= 7 + (7777/7 + 7 × (7 + 7))
:= 77/7 × 777/7
:= 7/7 + (7 × (7 × (7 + 7) + 77))
:= 8 × (8 × 8 + 88)
:= 88/8 × 888/8
:= 8 + (8 × (8 × 8 + 88) + ((8 + 8)/8))
:= 99/9 × 999/9
:= (9 + 9)/9 + ((9 − 9/9) × (9 × (9 + 9) − 9))
5
:= 9/9 + ((9 + 9) × (9 + 9) + 9 × 99)
Ï 1217 := 11 × 111 − 1 − 1 − 1 − 1
Ï 1227 := 11 × 111 + ((1 + 1) × (1 + 1 + 1))
Ï 1222 := 1 + 11 × 111
:= (22/2 × 222/2) − 2 − 2
:= ((22 × 222/2) + 2)/2
:= 2 + (((22/2 + 22) + 2)2 )
:= ((33/3)3 ) − (333/3 + 3)
:= ((33 × 333/3) + 3)/3
:= 3 + ((33 × 33 ) + 333)
:= 4/4 + (4 × ((44 + 44 ) + 4))
:= ((44 × 444/4) + 4)/4
:= 44/4 + (4 × ((44 + 44 ) + 4))
:= (5 + 5)/5 + (5 × ((5 − (5 + 5)/5)5 ))
:= ((55 × 555/5) + 5)/5
:= 5 + (((55 × 555/5) + 5)/5)
:= (6 × (6 × 6 × 6 − (6 + 6))) − 6/6 − 6
:= ((66 × 666/6) + 6)/6
:= 6 + ((66/6) × 666/6)
:= 7 + (77/7 × (777 − 7)/7)
:= (77 × 777/7 + 7)/7
:= 7 + ((77 × 777/7 − 7)/7)
:= 8/8 + 8 × (8 × 8 + 88)
:= ((88 × 888/8) + 8)/8
:= 88/8 + 8 × (8 × 8 + 88)
:= 9 + (((99/9) × (9/9 + 99 + 9)) + 9)
:= ((99 × 999/9) + 9)/9
:= 9 + ((((9999 − 9)/9) + 99) + 9)
91
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1228 := (11 × (1 + 111)) − 1 − 1 − 1 − 1
:= 2 × ((2 × (22 × (2
2+2
− 2))) − 2)
Ï 1238 := ((1 + 1) × (1 + 1 + 1)) + (11 × (1 + 111))
Ï 1233 := 1 + (11 × (1 + 111))
:= ((22 × (222 + 2)/2) + 2)/2
3
:= (2 × (((22 + 2)2 ) + 2 × 22)) − 2
:= 3 + (((33 − 3/3) + 3)3−3/3 )
:= 3 × ((3 × 3 − 3) + 333)
:= 3 + ((((33/3)3 ) − 3 × 33) + 3)
:= 44 × (44 − 4 × 4) − 4
:= 4/4 + 44 × (44 − 4 × 4)
:= ((4 + 4)/4 × ((4/4 + 4)4 − 4)) − 4
5
:= (5 + 5)/5 × ((55 − 55)/5)
:= 5 + ((5 + 5)/5 × ((5 − 55)/5))
:= (5 + 5)/5 × ((55 − 5)/5 − 5)
:= 6 + (((66 × 666/6) + 6)/6)
:= 6 + (((66/6) × 666/6) + 6)
:= 6 + ((66/6) × (666 + 6)/6)
:= 7 + 77/7 × 777/7
:= 7 + ((7 × (7 × (7 + 7) + 77)) + 7/7)
:= (7 × (((7 + 7)/7)7 + 7 × 7)) − 7/7
:= 8 + (((88 × 888/8) − 8)/8)
:= 8 + ((8 × (8 × 8 + 88) + 8/8) + 8)
:= 8 × (8 + 8) + (8888 − 8)/8
:= 9 + ((9999/9 + 99) + 9)
:= 9 + ((9 − 9/9) × (9 × (9 + 9) − 9))
:= 9 + ((((99 × 999/9) − 9)/9) + 9)
Ï 1229 := (11 × (1 + 111)) − 1 − 1 − 1
2
Ï 1234 := 1 + (1 + (11 × (1 + 111)))
2+2
− 2))))
Ï 1239 := 11 × 111 + (1 + 1) × (11 − 1 − 1)
:= (2222 + 22×(2+2) )/2
:= 2 + ((((22/2 + 22) + 2) ) + 2)
:= 2 + (2 × (2 × (22 × (2
:= ((33/3)3 ) − (3 × 33 + 3)
:= (33 × 33 ) + (((3/3 + 3) + 3)3 )
:= (3 × (3 × 33 + 333)) − 3
:= 4 + ((44/4 × 444/4) + 4)
:= (4 + 4)/4 + 44 × (44 − 4 × 4)
:= 4 × 4 × (4 + 4) + 4444/4
:= ((55 × 55 − 5) + 5 )/5
:= (((5 − 55) + 5 )/5) − 5
:= ((55 − 55) + 55 )/5
:= 6 × 6 × 6 × 6 − (66 + 6/6)
:= 6 + ((((66 × 666/6) + 6)/6) + 6)
:= (6/6 + 6) × (666/6 + 66)
:= 7 + ((77 × 777/7 + 7)/7)
:= 7 + (((77 × 777/7 − 7)/7) + 7)
:= 7 × (((7 + 7)/7)7 + 7 × 7)
:= 8 + 88/8 × 888/8
:= 8 + ((8 × (8 × 8 + 88) + ((8 + 8)/8)) + 8)
:= 8 × (8 + 8) + 8888/8
:= 9 + (((99 × 999/9) − 9)/9)
:= 9 + (((9 − 9/9) × (9 × (9 + 9) − 9)) + 9/9)
:= 9 + (((99/9) × 999/9) + 9)
5
5
Ï 1230 := (11 × (1 + 111)) − 1 − 1
5
Ï 1235 := 1 + (1 + (1 + (11 × (1 + 111))))
Ï 1240 := (11 − 1) × (1 + (1 + (1 + 111+1 )))
:= (2 × (2 + 2) + 2) × ((22/2)2 + 2)
:= 2 + (((22 × (222 + 2)/2) + 2)/2)
:= 2 × (((22 + 2)2 ) + 2 × 22)
:= (3 + 3) × ((3 + 3)3 − 33/3)
:= 3 + (((33/3)3 ) − 3 × 33)
:= 3/3 + ((3 × (3 × 33 + 333)) − 3)
:= (4/4 + 4) × ((4 − 44)/4 + 44 )
:= 4 + (44 × (44 − 4 × 4) − 4/4)
:= (4/4 + 4) × (44 − 4 − 4)
:= 5 + (5 × (5 × 5 × (5 + 5) − 5))
:= 5 + ((5 × (5 × 5 × (5 + 5) − 5)) + 5)
:= 5 × (((5 − (5 + 5)/5)5 ) + 5)
:= 6 × 6 × 6 × 6 − 66
:= 6 + (6 × 6 × 6 × 6 − (66 + 6/6))
:= 6 × 6 × 6 + (((6 + 6)/6)(66−6)/6)
:= (7 − 7/7) × (((7 + 7)/7)7 + 77)
:= 7 + (77/7 × 777/7 + 7)
:= 7/7 + (7 × (((7 + 7)/7)7 + 7 × 7))
:= 8 + (((88 × 888/8) + 8)/8)
:= (88/8 + 8) × (8/8 + 8 × 8)
:= 88 + (8 + 8) × (8 × 8 + 8)
:= 9 + ((99/9) × 999/9)
:= 99/9 + ((9 − 9/9) × (9 × (9 + 9) − 9))
:= 9 × 9 × 9 + ((((9 + 9)/9)9 ) − 9/9)
Ï 1236 := 1 + (1 + (1 + (1 + (11 × (1 + 111)))))
Ï 1231 := (11 × (1 + 111)) − 1
:= ((22 × (222 + 2)/2) − 2)/2
3
:= 2 × ((2 × (22 × (2
2+2
− 2))) + 2)
Ï 1241 := (11 × (1 + 1 + 111)) − 1 − 1
:= (22/2 × (222/2 + 2)) − 2
:= ((33/3) ) − (3 × 33 + 3/3)
:= 3 + (3 × ((3 × 33 − 3) + 333))
:= 33+3 + ((3 − 3/3)3×3 )
:= 44 × (44 − 4 × 4) − 4/4
:= 4 + 44 × (44 − 4 × 4)
:= 4/4 + ((4/4 + 4) × (44 − 4 − 4))
:= 5 × 5 × 5 + 5555/5
:= ((5/5 + 5)5−5/5 ) − 55
:= 6 + (6 × 6 × 6 × 6 − 66)
:= 66/6 + (6 × 6 × 6 × 6 − 66)
:= 7 + ((7 × (7 × (7 + 7) + 77)) − 7/7)
:= 77/7 + (7 × (7 × (7 + 7) + 77))
:= ((7 + 7)/7) + (7 × (((7 + 7)/7)7 + 7 × 7))
:= 8 + ((8 × (8 × 8 + 88) − 8/8) + 8)
:= ((88 + 8)/8) × (888/8 − 8)
:= 8/8 + ((8 + 8) × (8 × 8 + 8) + 88)
:= 9 + (((99 × 999/9) + 9)/9)
:= ((9 + 9)/9) × (9 × 9 × 9 − 999/9)
:= 9 × 9 × 9 + (((9 + 9)/9)9 )
5
:= ((55 × 55 + 5 ) + 5)/5
(6+6)/6
:= 6 + ((6 × 6 − 6/6)
)
Ï 1237 := 1 + (1 + (1 + (1 + (1 + (11 × (1 + 111))))))
Ï 1232 := 11 × (1 + 111)
:= 2 × (2 × (22 × (2
2+2
− 2)))
:= 2
2+2
Ï 1242 := (11 × (1 + 1 + 111)) − 1
:= 2 + (2 × (((22 + 2)2 ) + 2 × 22))
+ (22/2 × 222/2)
:= ((33/3)3 ) − 3 × 33
:= 3 + ((((3/3 + 3) + 3)3 ) + (33 × 33 ))
:= 3 × (3 × 33 + 333)
:= 44 × (44 − 4 × 4)
:= 4 + (44 × (44 − 4 × 4) + 4/4)
:= (4 + 4)/4 × ((4/4 + 4)4 − 4)
:= 55/5 × (555 + 5)/5
:= 5 + (55/5 × (555 + 5)/5)
:= (5 + 5)/5 × ((55 + 5)/5 − 5)
:= 66/6 × (666 + 6)/6
(6+6)/6
:= 6 + (((6 × 6 − 6/6)
) + 6)
:= 6 + ((6 × 6 × 6 × 6 − 66) + 6)
:= 7 + (7 × (7 × (7 + 7) + 77))
:= 77 + (7777/7 + 7 × 7)
:= (77/7 + 7) × (77 − (7/7 + 7))
:= 8 + (8 × (8 × 8 + 88) + 8)
:= 8 + (88/8 × 888/8 + 8)
:= 88 + ((8 + 8) × (8 × 8 + 8) + ((8 + 8)/8))
:= 9 × 9 × 9 + ((((9 + 9)/9)9 ) − 9)
:= 9 + (((9999/9 + 99) + 9) + 9)
:= 999 + 9 × (9 + 9 + 9)
92
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1248 := (1 + 1) × ((1 + (1 + 1) × (1 + 11))1+1 − 1)
Ï 1243 := 11 × (1 + 1 + 111)
:= 22/2 × (222/2 + 2)
3
:= 3/3 + (3 × (3 × 3 + 333))
:= 2 + (((((2 × (22 + 2)) + 2)2 ) + 2)/2)
:= (33 + 3 + 3) × (33 − 3/3)
:= 3 + (((33/3)3 ) − 3 × 33 )
4
4
4
:= 4 + (4 × (4 − 4 − 4))
:= 44 + (4 × (44 + 4 ) − 4/4)
Ï 1253 := (11 × (1 + (1 + 1 + 111))) − 1
:= 2 × ((22 + 2) × (22 + 2 + 2))
:= 4 + ((((4 + 4) × (4/4 + 4)4 ) − 4)/4)
5
:= ((5 + 5)/5 × (55 − 5)/5) − 5
:= (5 + 5)/5 × (5 − 5)/5
:= 66 + ((6666/6) + 66)
:= (6 × (6 × 6 × 6 − 6)) − 6 − 6
:= (6 × (6 × 6 × 6 − 6)) − 6/6 − 6
:= 77/7 × (777 + 7 + 7)/7
:= (77 + 7)/7 × (777/7 − 7)
:= 77 + (7 + 7) × (77 + 7)
:= 8 + ((88/8 + 8) × (8/8 + 8 × 8))
:= 8 + ((8 + 8) × (8 × 8 + 8) + 88)
:= (8 + 8) × (88 − 8) − (88/8 + 8 + 8)
:= 9/9 + (9 × (9 + 9 + 9) + 999)
:= 9 + ((((99/9) × 999/9) + 9) + 9)
:= 99/9 + (9 × (9 + 9 + 9) + 999)
:= 5 + ((5 + 5)/5 × (55 − 5)/5)
Ï 1249 := ((1 + 1) × (1 + (1 + 1) × (1 + 11))1+1 ) − 1
Ï 1244 := 1 + (11 × (1 + 1 + 111))
:= ((((2 × (22 + 2)) + 2)2 ) − 2)/2
2
:= 2 × ((22 − 2) + 222)
3
3
Ï 1254 := 11 × (1 + (1 + 1 + 111))
:= 2 × (((2/2 + 2 + 2)2+2 ) + 2)
:= 3 + (((3 − 3/3)3×3 ) + 33+3 )
:= ((33/3) ) − (3 × 3 + 3/3)
:= 33 × (33/3 + 33 )
:= 44 + 4 × (44 + 44 )
:= (((4 + 4) × (4/4 + 4)4 ) − 4)/4
:= 4 + (4 + 4)/4 × (4/4 + 4)4
:= ((55 − 5 + 55 )/5) − 5
:= (55 − 5 + 55 )/5
:= 5 + ((55 − 5 + 55 )/5)
:= 6 + (((66/6) × (666 + 6)/6) + 6)
:= (6 × (6 × 6 × 6 − 6)) − 66/6
:= (6 × (6 × 6 × 6 − 6)) − 6
:= (7 × (7 + 7) × (7 + 7)) − ((7 + 7)/7)
:= 7 + ((77/7 + 7) × (77 − (7/7 + 7)))
:= 7/7 + ((7 + 7) × (77 + 7) + 77)
:= 8 + (((88 + 8)/8) × (888/8 − 8))
:= (((88 + 8) × (88 + 8 + 8)) + 8)/8
:= (88/8 + 8) × (((8 + 8)/8) + 8 × 8)
:= (9 + 9)/9 + (9 × (9 + 9 + 9) + 999)
:= 9 + (((((9 + 9)/9)9 ) − 9/9) + 9 × 9 × 9)
:= 9 × 99 + ((99 × 99)/(9 + 9 + 9))
7
Ï 1245 := 1 + (1 + (11 × (1 + 1 + 111)))
:= 2 + (22/2 × (222/2 + 2))
Ï 1250 := (1 + 1) × (1 + (1 + 1) × (1 + 11))1+1
:= 2 × ((2/2 + 2 + 2)2+2 )
3
3
:= ((33/3) ) − 3 × 3
:= 3 + (3 × (3 × 3 + 333))
Ï 1255 := 1 + (11 × (1 + (1 + 1 + 111)))
:= 2 + ((((((2 × (22 + 2)) + 2)2 ) + 2)/2) + 2)
3
:= 3/3 + (33 × (33/3 + 33 ))
4
:= 44 + (4 × (44 + 44 ) + 4/4)
:= (4 + 4)/4 × (4/4 + 4)
:= 5 × 5 × 5 × (5 + 5) − 5
:= 5 × 5 × 5 × (5 + 5)
:= 6 + ((6/6 + 6) × (666/6 + 66))
:= ((6 − 66)/6) + (6 × (6 × 6 × 6 − 6))
:= 6/6 + ((6 × (6 × 6 × 6 − 6)) − 6)
:= (7/7 + 7 + 7) × (77 − 7/7 + 7)
:= (7/7 + 7 × 7) × (77/7 + 7 + 7)
:= 7 + ((77 + 7)/7 × (777/7 − 7))
:= 8 + ((88/8 × 888/8 + 8) + 8)
:= ((((8 + 8)/8) + 8)8×8/(8+8) )/8
:= 8 + ((8888/8 + 8 × (8 + 8)) + 8)
:= 999/9 + (9 × ((99 + 9 + 9) + 9))
Ï 1246 := 1 + (1 + (1 + (11 × (1 + 1 + 111))))
:= 2 × (((2/2 + 2 + 2)
2+2
) − 2)
3
:= 3 + ((3 × (3 × 3 + 333)) + 3/3)
4
:= (4 + 4)/4 × (4/4 + 4) − 4
5
5
:= (4/4 + 4) × (44 − 4/4 − 4)
:= 5 + 5 × 5 × 5 × (5 + 5)
9
:= 9 + ((((9 + 9)/9) ) + 9 × 9 × 9)
Ï 1251 := 1 + ((1 + 1) × (1 + (1 + 1) × (1 + 11))1+1 )
2
:= ((((2 × (22 + 2)) + 2) ) + 2)/2
3
Ï 1256 := 1 + (1 + (11 × (1 + (1 + 1 + 111))))
:= 2 + (2 × (((2/2 + 2 + 2)2+2 ) + 2))
:= 3 + ((((33/3)3 ) − 3 × 33 ) + 3)
:= 3 × ((3 × 3 + 333) + 3)
4
:= (((4 + 4) × (4/4 + 4) ) + 4)/4
5
:= 999 + (((9 + 9)/9)9−9/9 )
5
:= (4/4 + 4) × (44 − 4) − 4
:= ((5 + 5 + 5)/5) − 5
:= (5 + 5 + 5)/5
:= 6 × 6 + ((66/6) × ((666 − 6)/6))
:= 6 × 6 + (((66/6) × 666/6) − 6)
:= (6 + 6)/6 + ((6 × (6 × 6 × 6 − 6)) − 6)
:= 7 + (7 × (((7 + 7)/7) + 7 × 7))
:= 7 + ((7 × (7 + 7) × (7 + 7)) − ((7 + 7)/7)7 )
:= 777 + (7 × (77 − 7) − (77/7))
:= (8/8 + 88) × ((8 − (8 + 8)/8) + 8)
:= (8/8 + 8) × (8 × (8 + 8) + (88/8))
:= 88 × (8 + 8) − (8 × 8 + 88)
:= 9 + (9 × (9 + 9 + 9) + 999)
:= 9 × 99 + (((9 × 9 × 9 × 9) + 9)/(9 + 9))
7
9−9/9
:= 999 + ((((9 + 9)/9)
) − 9)
Ï 1247 := 1 + (1 + (1 + (1 + (11 × (1 + 1 + 111)))))
2
:= 5 + ((55 + 55 + 5)/5)
Ï 1252 := (11 × (1 + (1 + 1 + 111))) − 1 − 1
2+2
Ï 1257 := 1 + (1 + (1 + (11 × (1 + (1 + 1 + 111)))))
:= 222 + ((222/2 + 22)/2)
:= 22 + (((22/2 + 22) + 2) )
:= 2 + (2 × ((2/2 + 2 + 2)
:= ((33/3)3 ) − (3 × 33 + 3)
:= ((3/3 + 3)3 ) + (33 × (33 + 3))
:= 3 + (33 × (33/3 + 33 ))
:= 44 + ((4 × (44 − 4 − 4)) − 4/4)
:= ((4 + 4)/4 + 4)4 − 44
:= 4/4 + ((4/4 + 4) × (44 − 4) − 4)
:= ((5 + 5)/5 × (55 + 5)/5) − 5
:= (5 + 5)/5 × (55 + 5)/5
:= 5 + ((5 + 5)/5 × (55 + 5)/5)
:= (6 × (6 × 6 × 6 − 6)) − (6/6 + 6 + 6)
:= ((6 + 6)/6)6 + (66 × (6 + 6 + 6))
:= 6 × 6 + ((66/6) × 666/6)
:= 77 + ((7 + 7) × (77 + 7) − 7/7)
:= 7 + ((7/7 + 7 × 7) × (77/7 + 7 + 7))
:= 8 + (8888/8 + 8 × (8 + 8))
:= 8 × (88 + 8) + 88 × 88/(8 + 8)
:= 8 + ((((88 + 8) × (88 + 8 + 8)) + 8)/8)
:= 9 + (((((99 × 999/9) − 9)/9) + 9) + 9)
:= 9 × (9 + 9) + (((99 × 99) + 9)/9)
:= 9 × 9 + ((99 − 9/9) × ((99 + 9)/9))
77/7−7
:= (7 − 7/7)
−7×7
93
))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1258 := 11 × 111 + (111/(1 + 1 + 1))
2+2
Ï 1263 := (11 × (1 + (1 + (1 + 1 + 111)))) − 1 − 1
2
Ï 1268 := 1 + (1 + (1 + (11 × (1 + (1 + (1 + 1 + 111))))))
:= 222/2 + (2 × ((22 + 2) ))
:= 2 + ((222/2 + 222 )/2)
:= ((3 + 3)3/3+3 ) − 33
:= 3 + (((33/3)3 ) − (33 + 33))
:= (4 + 4)/4 × ((4/4 + 4) + 4)
:= 44 + (4 × (44 − 4) − 4/4)
:= 4 + (4 × (44 − 4) + 44 )
:= (5 + 5)/5 × ((55 − 5)/5 + 5)
:= 5 + ((5 + 5)/5 × ((55 − 5)/5 + 5))
:= (5 + 5)/5 × (((55 − 5)/5 + 5) + 5)
:= 2 × ((((2/2 + 2 + 2)
) + 2) + 2)
:= (3/3 + 33) × (3/3 + 33 + 3)
4
:= (6 × (6 × 6 × 6 − 6)) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × (6 × 6 × 6 − 6))
:= 6 + ((6 × (6 × 6 × 6 − 6)) + ((6 + 6)/6))
:= 7 × (7 + 7 + 7) + 7777/7
:= (7 + 7) × (77 + 7 + 7) − 77/7
:= 7/7 + (7 × (77 − 7) + 777)
:= 8 + (((((8 + 8)/8) + 8)8×8/(8+8) )/8)
:= 8 × 8 + (8888/8 + 88)
:= (8 + 8) × (88 − 8) − (88 + 8)/8
:= 9 × 9 + ((99/9) × ((99 − 9/9) + 9))
:= 9 × (9 + 9) + (((9999 − 9)/9) − 9)
:= 9 × 9 + (((99 × (99 + 9)) − 9)/9)
Ï 1259 := 1 + (11 × 111 + (111/(1 + 1 + 1)))
Ï 1264 := (11 × (1 + (1 + (1 + 1 + 111)))) − 1
Ï 1269 := 1111 + ((1 + 1 + 11)1+1 − 11)
:= 222/2 + (2 × (((22 + 2)2 ) − 2))
:= 2 × (((22 + 2 + 2)2 ) − (2 × 22))
:= 2 + (((222/2 + 222 ) + 2)/2)
:= (3 × (3 − 33 )) + ((33/3)3 )
:= 3/3 + (((3 + 3)3/3+3 ) − 33)
:= ((3 + 3)3/3+3 ) − 33
:= (4/4 + 4) × (44 − 4) − 4/4
:= 44 + 4 × (44 − 4)
:= 44 + (4 × 44 − 44/4)
:= 5 + (((55 − 5 + 55 )/5) + 5)
:= 5 × 5 × 55 − 555/5
:= 5 + (5 × 5 × 55 − 555/5)
:= (6 × (6 × 6 × 6 − 6)) − 6/6
:= 6 + ((6 × (6 × 6 × 6 − 6)) − ((6 + 6)/6))
:= 6 + ((6 × (6 × 6 × 6 − 6)) + (6 × 6/(6 + 6)))
:= ((77 − 7) × (77/7 + 7)) − 7/7
:= 77 + ((7777 − 7)/7 + 77)
:= 7 × 7 + ((77 × 777/7 − 7)/7)
:= 8 + ((8/8 + 8) × (8 × (8 + 8) + (88/8)))
:= (8 + 8) × (88 − (8/8 + 8))
:= (8 + 8) × (88 − 8) − 88/8
9
:= 9 × (9 + 9) + (9999/9 − 9)
:= 9 + ((9 + 9) × (9 × 9 − (99/9)))
:= 9 + (((((9 + 9)/9) ) + 9 × 9 × 9) + 9)
Ï 1260 := (1 + 11) × (111 − ((1 + 1) × (1 + 1 + 1)))
Ï 1265 := 11 × (1 + (1 + (1 + 1 + 111)))
Ï 1270 := (11 − 1) × (111 + ((1 + 1)1+1+1+1 ))
:= (2 − 22) × (2/2 − 22+2+2 )
:= 22/2 × ((222/2 + 2) + 2)
:= (2/2 + 2 + 2) × (22×(2+2) − 2)
:= (3 + 3) × ((3 + 3)3 − (3 + 3))
:= ((33/3)3 ) − (33 + 33)
:= 3 + (((33/3)3 ) − ((3/3 + 3)3 ))
:= (4/4 + 4) × (44 − 4)
:= 4/4 + (4 × (44 − 4) + 44 )
:= (4/4 + 4) × (44 − (4 + 4)/4)
:= 5 + (5 × 5 × 5 × (5 + 5) + 5)
:= 55 × (5 × 5 − ((5 + 5)/5))
:= (5 × (5 × 5 × (5 + 5) + 5)) − 5
:= 6 × (6 × 6 × 6 − 6)
:= 6 + ((6 × (6 × 6 × 6 − 6)) − 6/6)
:= ((66 − 6)/6) + (6 × (6 × 6 × 6 − 6))
:= (77 − 7) × (77/7 + 7)
:= 77 + (7777/7 + 77)
:= 7 × 7 + 77/7 × 777/7
:= (8 × 8 − 8/8) × ((88 + 8)/8 + 8)
:= 8/8 + ((8 + 8) × (88 − (8/8 + 8)))
:= (8 − 88)/8 + (8 + 8) × (88 − 8)
:= (9 + 9) × (9 × 9 − 99/9)
:= 9 × (9 + 9) + ((9999 + 9)/9 − 9)
:= (9 + 9) × 99 − (((9 + 9)/9)9 )
Ï 1261 := 1 + ((1 + 11) × (111 − ((1 + 1) × (1 + 1 + 1))))
Ï 1266 := 1 + (11 × (1 + (1 + (1 + 1 + 111))))
Ï 1271 := 11 × 111 + ((11 − 1)1+1 /(1 + 1))
:= ((((2 × (22 + 2)) + 2)2 ) + 22)/2
:= (222/2 + 222 )/2
:= 22 + (((((2 × (22 + 2)) + 2)2 ) − 2)/2)
:= 3/3 + ((3 + 3) × ((3 + 3)3 − (3 + 3)))
:= 3 + (((3 + 3)3/3+3 ) − 33)
:= ((33/3)3 ) − (33 + 33)
:= 4/4 + (4/4 + 4) × (44 − 4)
:= 4 × 4 + (4 + 4)/4 × (4/4 + 4)4
:= (4/4 + 4) × (44 − 4/4) − 4
:= ((55 + 55 ) + 55 )/5
:= 5 + (((55 + 55 ) + 55 )/5)
:= ((5/5 + 5)5−5/5 ) − 5 × 5
:= 6/6 + (6 × (6 × 6 × 6 − 6))
:= 6 + (6 × (6 × 6 × 6 − 6))
:= 66/6 + (6 × (6 × 6 × 6 − 6))
:= 7/7 + ((77 − 7) × (77/7 + 7))
:= 777 + (7 × (77 − 7) − 7/7)
:= (((7 + 7)/7)77/7 ) − 777
:= (8 + 8) × (88 − 8) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) × (88 − (8/8 + 8)))
:= (8 + 8) × (88 − 8) − (8/8 + 8)
:= 9/9 + ((9 + 9) × (9 × 9 − (99/9)))
:= 9 × (9 × (9 + 9) − 9) − 999/9
:= 9/9 + ((9 + 9) × 99 − (((9 + 9)/9)9 ))
Ï 1262 := (11 × (1 + 111)) + ((11 − 1) × (1 + 1 + 1))
:= 2 + ((2 − 22) × (2/2 − 2
2+2+2
))
Ï 1267 := 1 + (1 + (11 × (1 + (1 + (1 + 1 + 111)))))
:= ((2
22/2
2
+ 22 ) + 2)/2
Ï 1272 := (1 + 11) × (111 − (1 + (1 + 1 + 1 + 1)))
:= (2 + 2 + 2)2+2 − (22 + 2)
:= 3 + ((3 × (3 − 33 )) + ((33/3)3 ))
:= ((33/3)3 ) − ((3/3 + 3)3 )
:= 3 + (((3 + 3)3/3+3 ) − 33 )
:= 4 + ((4 + 4)/4 × ((4/4 + 4)4 + 4))
:= 4 + ((4 × (44 − 4) − 4/4) + 44 )
:= 44 + (4 × 44 − (4 + 4))
:= (5 + 5)/5 × ((55 + 5)/5 + 5)
:= 5 + ((5 + 5)/5 × ((55 + 5)/5 + 5))
:= (5 + 5)/5 × ((55 + 55 )/5)
:= (6 + 6)/6 + (6 × (6 × 6 × 6 − 6))
:= 6 + ((6 × (6 × 6 × 6 − 6)) + 6/6)
:= 6 + ((6 × (6 × 6 × 6 − 6)) + 6)
:= ((7 + 7)/7) + ((77 − 7) × (77/7 + 7))
:= 777 + 7 × (77 − 7)
:= (7 + 7) × (77 + 7 + 7) − (7 + 7)/7
88/8−8
:= 8 + ((88/8 + 8) × (((8 + 8)/8) + 8 × 8))
:= ((88/8)
:= (9 + 9)/9 + ((9 + 9) × (9 × 9 − (99/9)))
:= 9 × 9 + (((99 × (99 + 9)) − (9 + 9))/9)
)−8×8
94
:= (8 + 8) × (88 − 8) − 8
:= 9 × (9 + 9) + ((9999 − 9)/9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1273 := 1 + ((1 + 11) × (111 − (1 + (1 + 1 + 1 + 1))))
2
2
Ï 1278 := (11 − 1 − 1) × ((1 + 11)1+1 − (1 + 1))
2+2+2
Ï 1283 := ((1 + 11) × (111 − (1 + 1 + 1 + 1))) − 1
:= (2 + 2 + 2)2+2 − (22/2 + 2)
:= (22/2) + (2 × ((22 + 2) ))
:= ((22 − 2) × 2
:= (3 + 3) × 33 + 3333/3
:= (3 + 3) × ((3 + 3)3 − 3)
:= 33 + (((33/3)3 ) − 3 × 33 )
:= 4 + ((4 × 44 − 44/4) + 44 )
:= 44 + (4 × 44 − (4 + 4)/4)
:= 4 + ((4 × 44 − 4/4) + 44 )
:= 5 × 5 + ((5 + 5)/5 × (55 − 5)/5)
:= 5 + (((5 + 5)/5 × (55 − 5)/5) + 5 × 5)
:= 5 × 5 + ((5 + 5)/5 × ((55 − 5)/5 + 5))
)−2
:= 6 + (((6 × (6 × 6 × 6 − 6)) + 6/6) + 6)
:= 6 × 6 × 6 × 6 − 6 − 6 − 6
:= 6 × 6 × 6 × 6 − (6/6 + 6 + 6)
:= (7 + 7) × (77 + 7 + 7) − 7/7
:= (77/7 + 7) × ((7/7 − 7) + 77)
:= 7 + ((7 + 7) × (77 + 7 + 7) + ((7 + 7)/7))
:= 8/8 + ((8 + 8) × (88 − 8) − 8)
:= (8 + 8) × (88 − 8) − (8 + 8)/8
:= 88/8 + ((8 + 8) × (88 − 8) − 8)
:= 9 × (9 + 9) + 9999/9
:= (9 + 9) × (9 × 9 − (9/9 + 9))
:= 9 + ((9999 + 9)/9 + 9 × (9 + 9))
Ï 1274 := (1 + 1 + 11) × (111 − 1 − 1 − 11)
:= (2 + 2 + 2)
2+2
:= ((22 − 2) × 2
− 22
3
Ï 1279 := 1111 + ((1 + 1 + 11)1+1 − 1)
3
:= 3 + (((33/3) ) − (3 + 33))
2+2+2
Ï 1284 := (1 + 11) × (111 − (1 + 1 + 1 + 1))
:= 222 + 2 × (22 − 2)2
) − 2/2
3
:= 3/3 + ((3 + 3) × ((3 + 3) − 3))
4
:= (33 × (33 + 3 + 3)) − 3
4
:= 4 + ((4/4 + 4) × (44 − (4 + 4)/4))
:= 4 + (4 × 4 − 4/4)
:= 4 + (4 × 44 + 44 )
:= 5 × 5 + ((55 − 5 + 55 )/5)
:= 5 + (((55 − 5 + 55 )/5) + 5 × 5)
:= (((55 × (55 + 5)) − 5) + 55 )/5
:= 6 × 6 × 6 × 6 − ((66 + 66)/6)
:= 6 × 6 × 6 × 6 − (66/6 + 6)
:= 6 × 6 × 6 × 6 − 6 − 6
7
:= (7 + 7) × (77 + 7 + 7)
:= (((77 − 7) × ((7 + 7)/7) ) − 7)/7
:= ((77 − 7)/7) + (7 + 7) × (77 + 7 + 7)
:= (8 + 8)/8 + ((8 + 8) × (88 − 8) − 8)
:= (8 + 8) × (88 − 8) − 8/8
:= (8/((8 + 8)/8)) + (8 + 8) × (88 − 8)
:= 9 × (9 + 9) + (9999 + 9)/9
Ï 1275 := 111+1+1 − (1 + 111)/(1 + 1)
9
:= 9 + ((9 + 9) × 99 − (((9 + 9)/9) ))
:= ((99 + 9)/9) × ((99 − 9/9) + 9)
Ï 1285 := ((1 + 1 + 1) × (1 + 11))1+1 − 11
Ï 1280 := 1111 + (1 + 1 + 11)1+1
:= 2/2 + ((2 + 2 + 2)2+2 − 22)
:= (22 − 2) × 22+2+2
:= (2 + 2 + 2)2+2 − 22/2
:= ((3 + 3) × ((3 + 3)3 − 3)) − 3
:= 3 + (((33/3)3 ) − (33 + 33 ))
:= ((3 + 3)3/3+3 ) − 33/3
:= (4/4 + 4) × (44 − 4/4)
:= 44 + 4 × 44
:= (4/4 + 4) × (4/4 + 44 )
:= 5 × (5 × 5 × (5 + 5) + 5)
:= 5 + (5 × (5 × 5 × (5 + 5) + 5))
:= 5 + ((5 × (5 × 5 × (5 + 5) + 5)) + 5)
:= 6 × 6 × 6 × 6 + ((6 − (66 + 66))/6)
:= ((6 − 66)/6) + (6 × 6 × 6 × 6 − 6)
:= 6 × 6 × 6 × 6 − 66/6
7
:= 7/7 + (7 + 7) × (77 + 7 + 7)
:= ((77 − 7)/7) × ((7 + 7)/7)
:= 77/7 + (7 + 7) × (77 + 7 + 7)
:= (8/8 + 8 + 8) × (88/8 + 8 × 8)
:= (8 + 8) × (88 − 8)
:= 8 × 8 + 88/8 × 888/8
:= 9 + (9 × (9 × (9 + 9) − 9) − 999/9)
:= (9 − 9/9) × (9 × (9 + 9) − ((9 + 9)/9))
:= ((9 + 9) × (9 × 9 − 9)) − 99/9
Ï 1276 := 11 × (1 + 1 + 1 + 1 + 1 + 111)
:= 2 + ((2 + 2 + 2)
2+2
− 22)
:= 3/3 + (((3 + 3) × ((3 + 3)3 − 3)) − 3)
Ï 1281 := 1 + (1111 + (1 + 1 + 11)1+1 )
Ï 1286 := 1 + (((1 + 1 + 1) × (1 + 11))1+1 − 11)
:= 2/2 + ((22 − 2) × 22+2+2 )
:= 2 + 2 × (22 − 2)2 + 222
:= 3 + ((3 + 3) × ((3 + 3)3 − 3))
:= (33 × (33 + 3 + 3)) − 3/3
:= 4 + (4 × 4 − 4)
:= 4/4 + (4 × 4 + 4 )
:= 4 + (((4 + 4)/4 + 4 × 44 ) + 44 )
:= 5 × 5 + ((55 + 55 + 5)/5)
:= 5 + (((55 + 55 + 5)/5) + 5 × 5)
:= ((5/5 + 5)5−5/5 ) − 5 − 5
:= 66/6 × (((666 − 6)/6) + 6)
:= ((66/6) × (666/6 + 6)) − 6
:= ((6 − 66)/6) + 6 × 6 × 6 × 6
:= ((7 + 7)/7) + (7 + 7) × (77 + 7 + 7)
:= 7 + (7 + 7) × (77 + 7 + 7)
:= 7 + ((((77 − 7) × ((7 + 7)/7)7 ) − 7)/7)
:= (8 + 8) × (88 − 8) − (8/((8 + 8)/8))
:= 8/8 + (8 + 8) × (88 − 8)
:= 8 + ((8 + 8) × (88 − 8) − ((8 + 8)/8))
:= 99/9 × (((99 − 9/9) + 9) + 9)
:= 9 + (((9999 − 9)/9) + 9 × (9 + 9))
:= ((9 + 9) × (9 × 9 − 9)) − 9/9 − 9
4
4
4
Ï 1277 := 1 + 11 × (1 + 1 + 1 + 1 + 1 + 111)
:= 2 + (((2 + 2 + 2)2+2 − 22) + 2/2)
3
3
3
4
Ï 1282 := 1 + (1 + (1111 + (1 + 1 + 11)1+1 ))
:= 2 + ((22 − 2) × 22+2+2 )
3
:= ((33/3) ) − (3 + 3 )
Ï 1287 := 11 × (111 + ((1 + 1) × (1 + 1 + 1)))
:= 2 + ((2 + 2 + 2)2+2 − 22/2)
:= 3 + (((3 + 3) × ((3 + 3) − 3)) + 3/3)
:= 33 × (33 + 3 + 3)
:= 44 + ((4 + 4)/4 + 4 × 44 )
:= 4 × 44 + 4444/4
:= 5 + ((5 + 5)/5 × ((55 + 5 )/5))
:= (5 + 5)/5 × (((55 + 55 )/5) + 5)
:= 55/5 × ((555 + 5)/5 + 5)
:= 6 + ((6 × (6 × 6 × 6 − 6)) + (66/6))
:= 6 × 6 × 6 × 6 − ((6 + 6)/6 + 6 + 6)
:= 66/6 × (666/6 + 6)
:= 7 + (77/7 × 777/7 + 7 × 7)
:= 7 + ((7 + 7) × (77 + 7 + 7) + 7/7)
:= 7 + (((77 − 7)/7) × ((7 + 7)/7)7 )
:= 8 + ((8 + 8) × (88 − 8) − (88/8))
:= (8 + 8)/8 + (8 + 8) × (88 − 8)
:= 8 + ((8 + 8) × (88 − 8) − 8/8)
:= 9 + (9999/9 + 9 × (9 + 9))
:= ((9 + 9) × (9 × 9 − 9)) − 9
4
4
:= 4/4 + ((4 × 4 − 4) + 4 )
5
:= ((9 + 9) × (9 × 9 − 9)) − (9/9 + 9 + 9)
95
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1288 := 1 + 11 × (111 + (1 + 1) × (1 + 1 + 1))
Ï 1293 := ((1 + 1 + 1) × (1 + 11))1+1 − 1 − 1 − 1
Ï 1298 := 11 × (111+1 − (1 + 1 + 1))
:= 2 × (2 × (((22+2 + 2)2 ) − 2))
:= (2 + 2 + 2)2+2 − 2/2 − 2
:= 2 + (2 + 2 + 2)2+2
:= 3/3 + (33 × (33 + 3 + 3))
:= ((3 + 3)3/3+3 ) − 3
:= ((33/3)3 ) − 33
:= 4 + (4 × 4 + 4 + 4)
:= 4/4 + (((4 + 4)/4 + 4)4 − 4)
:= (4 + 4)/4 + ((4 + 4)/4 + 4)4
:= (55 + 5/5) × (5 × 5 − ((5 + 5)/5))
:= 5 + ((55 + 5/5) × (5 × 5 − ((5 + 5)/5)))
:= 5 × 55 + ((5 − 5/5)5 − 5/5)
:= 6 × 6 × 6 × 6 − ((6 + 6)/6 + 6)
:= 6 × 6 × 6 × 6 − 6 × 6/(6 + 6)
:= (6 + 6)/6 + 6 × 6 × 6 × 6
:= 7 + ((7 + 7) × (77 + 7 + 7) + 7)
:= (7 × (7 + 7) × (7 + 7)) − ((7 + 7)/7 + 77)
:= 77/7 × (777/7 + 7)
:= 8 + (8 + 8) × (88 − 8)
:= 8 + (88/8 × 888/8 + 8 × 8)
:= 88/8 × ((888 − 8)/8 + 8)
:= 9/9 + (((9 + 9) × (9 × 9 − 9)) − 9)
:= ((9 + 9) × (9 × 9 − 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + ((9 + 9) × (9 × 9 − 9))
4
4
Ï 1289 := (1 + 1 + 11) × (11 − 1)1+1 − 11
Ï 1294 := ((1 + 1 + 1) × (1 + 11))1+1 − 1 − 1
Ï 1299 := 1 + (11 × (111+1 − (1 + 1 + 1)))
:= 2 + (((2 + 2 + 2)2+2 − 22/2) + 2)
:= (2 + 2 + 2)2+2 − 2
:= 2 + ((2 + 2 + 2)2+2 + 2/2)
:= ((33/3)3 ) − (3 × 3 + 33)
:= 3/3 + (((3 + 3)3/3+3 ) − 3)
:= 3 + ((3 + 3)3/3+3 )
:= 4 + ((4/4 + 4) × (4/4 + 44 ))
:= ((4 + 4)/4 + 4)4 − (4 + 4)/4
:= 4 + (((4 + 4)/4 + 4)4 − 4/4)
5
5
:= 5 × 55 + ((5 − 5/5) − (5 + 5))
:= 5 × 55 + ((5 − 5/5) − 5)
:= 5 × 55 + (5 − 5/5)5
:= 6 × 6 × 6 × 6 − 6/6 − 6
:= 6 × 6 × 6 × 6 − (6 + 6)/6
:= 6 × 6 × 6 × 6 + (6 × 6/(6 + 6))
77/7−7
:= (7 × (7 + 7) × (7 + 7)) − (7/7 + 77)
:= ((77 × (777/7 + 7)) + 7)/7
:= 8 + ((8 + 8) × (88 − 8) + 8/8)
:= 8 + (((8 + 8) × (88 − 8) − ((8 + 8)/8)) + 8)
:= 8 + ((8 + 8) × (88 − 8) + (88/8))
:= (9 + 9)/9 + (((9 + 9) × (9 × 9 − 9)) − 9)
:= ((9 + 9) × (9 × 9 − 9)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + ((9 + 9) × (9 × 9 − 9))
:= (7 − 7/7)
−7
Ï 1290 := (11 − 1) × (11 × (1 + 11) − 1 − 1 − 1)
Ï 1295 := ((1 + 1 + 1) × (1 + 11))1+1 − 1
2+2
Ï 1300 := (1 + 1 + 11) × (11 − 1)1+1
:= 2 + ((2 + 2 + 2)2+2 + 2)
:= (2 + 2 + 2)2+2 − (2 + 2 + 2)
:= (2 + 2 + 2)
:= 3 + (33 × (33 + 3 + 3))
:= ((33/3)3 ) − (33 + 3)
:= 3 + (((3 + 3)3/3+3 ) + 3/3)
:= (4/4 + 4) × ((4 + 4)/4 + 44 )
:= ((4 + 4)/4 + 4)4 − 4/4
:= 4 + ((4 + 4)/4 + 4)4
:= 5 × ((55 + 5)/(5 + 5) − 55)
:= (5 × (5 × 55 − 5)) − 55
:= (5 + 5) × (5 × 5 × 5 + 5)
:= 6 × 6 × 6 × 6 − 6
:= 6 × 6 × 6 × 6 − 6/6
:= 6 + (6 × 6 × 6 × 6 − ((6 + 6)/6))
:= 7 × 7 × 7 × 7 − 7777/7
:= (7 × (7 + 7) × (7 + 7)) − 77
:= ((7 + 7)/7) × (777/7 + 7 × 77)
:= 8 + ((8 + 8) × (88 − 8) + ((8 + 8)/8))
:= 8 + (((8 + 8) × (88 − 8) − 8/8) + 8)
:= (8/8 + 8 × 8) × ((88 + 8)/8 + 8)
:= (9/9 + 9) × ((999/9 + 9) + 9)
:= ((9 + 9) × (9 × 9 − 9)) − 9/9
:= (9/9 + 99) × ((99 + 9 + 9)/9)
Ï 1291 := 1 + ((11 − 1) × (11 × (1 + 11) − 1 − 1 − 1))
− 2/2
Ï 1296 := ((1 + 1 + 1) × (1 + 11))1+1
Ï 1301 := 1 + (1 + 1 + 11) × (11 − 1)1+1
:= (2 + 2 + 2)2+2 − (2/2 + 2 + 2)
:= (2 + 2 + 2)2+2
:= 2 + (((2 + 2 + 2)2+2 + 2/2) + 2)
:= 3 + ((33 × (33 + 3 + 3)) + 3/3)
:= (3 + 3)3/3+3
:= 3 + (((33/3)3 ) − 33)
:= 44 + (44/4 + 4 × 44 )
:= ((4 + 4)/4 + 4)4
:= 4 + (((4 + 4)/4 + 4)4 + 4/4)
:= ((5/5 + 5)
5−5/5
)−5
:= 6/6 + (6 × 6 × 6 × 6 − 6)
:= (5/5 + 5)
5−5/5
:= 5 + ((5/5 + 5)5−5/5 )
:= 6 × 6 × 6 × 6
:= 6 + (6 × 6 × 6 × 6 − 6/6)
77/7−7
:= (7 − 7/7)
:= 7 + ((7 × (7 + 7) × (7 + 7)) − (7/7 + 77))
:= 88/8 + (8 + 8) × (88 − 8)
:= 8 + ((8 + 8) × (88 − 8) + 8)
:= (88/8 × (888/8 + 8)) − 8
:= 99 + (9999/9 + 9 × 9)
:= (9 + 9) × (9 × 9 − 9)
:= 9 × 9 + (((99 × 999/9) − 9)/9)
:= 77/7 × (777/7 + 7) − 7
Ï 1292 := 111+1+1 − ((1 + 1 + 1) × (1 + 1 + 11))
Ï 1297 := 1 + ((1 + 1 + 1) × (1 + 11))1+1
Ï 1302 := 1 + 1 + (1 + 1 + 11) × (11 − 1)1+1
:= (2 + 2 + 2)2+2 − 2 − 2
:= 2/2 + (2 + 2 + 2)2+2
:= 2 + (((2 + 2 + 2)2+2 + 2) + 2)
:= ((33/3)3 ) − (33 + 3 + 3)
:= 3/3 + ((3 + 3)3/3+3 )
:= 3 + (((3 + 3)3/3+3 ) + 3)
:= ((4 + 4)/4 + 4)4 − 4
:= 4/4 + ((4 + 4)/4 + 4)4
:= 4 + (((4 + 4)/4 + 4)4 + (4 + 4)/4)
:= 5/5 + (((5/5 + 5)5−5/5 ) − 5)
:= 5/5 + ((5/5 + 5)5−5/5 )
:= 5 + (((5/5 + 5)5−5/5 ) + 5/5)
:= (6 + 6)/6 + (6 × 6 × 6 × 6 − 6)
:= 6/6 + 6 × 6 × 6 × 6
:= 6 + 6 × 6 × 6 × 6
:= (7/7 − 77) × (((7 − 77)/7) − 7)
77/7−7
:= 7/7 + (7 − 7/7)
:= 7 + ((7 × (7 + 7) × (7 + 7)) − 77)
:= ((88 + 8)/8) + (8 + 8) × (88 − 8)
:= 88 × (8 + 8) − 888/8
:= 88 + (8 × (8 × 8 + 88) − ((8 + 8)/8))
:= 99 + ((9999 + 9)/9 + 9 × 9)
:= 9/9 + ((9 + 9) × (9 × 9 − 9))
:= 9 × 9 + ((99/9) × 999/9)
96
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1303 := 1 + 1 + 1 + (1 + 1 + 11) × (11 − 1)1+1
Ï 1308 := (1 + 11) × (111 − 1 − 1)
Ï 1313 := (1 + 1 + 11) × (1 + (11 − 1)1+1 )
:= 2 + ((((2 + 2 + 2)2+2 + 2/2) + 2) + 2)
:= 2 × (((22 + 2 + 2)2 ) − 22)
:= 2 + ((22/2 × ((22/2)2 − 2)) + 2)
:= ((33/3)3 ) − (33 + 3/3)
:= 3 + (((3 + 3)3/3+3 ) + 3 × 3)
:= ((33/3)3 ) − (3 × (3 + 3))
:= 4 + ((((4 + 4)/4 + 4)4 − 4/4) + 4)
:= 4 + ((((4 + 4)/4 + 4)4 + 4) + 4)
:= 4 × 4 + (((4 + 4)/4 + 4)4 + 4/4)
:= 55 + ((5 + 5)/5 × (55 − 5)/5)
:= 5 + (((5 + 5)/5 × (55 − 5)/5) + 55)
:= 55 + ((5 + 5)/5 × ((55 − 5)/5 + 5))
:= 6 + (6 × 6 × 6 × 6 + 6/6)
:= 6 + (6 × 6 × 6 × 6 + 6)
:= 6 + (6 × 6 × 6 × 6 + (66/6))
:= 7 + (7 − 7/7)
:= 7 × 77 + (777 − (7/7 + 7))
:= 7 × 77 + (777 − ((7 + 7 + 7)/7))
:= 88 + (8 × (8 × 8 + 88) − 8/8)
:= (88 × (888/8 + 8) − 8)/8
:= 8 + ((8 − 8/8 + 8) × (88 − 8/8))
:= 9 + (((9 + 9) × (9 × 9 − 9)) − ((9 + 9)/9))
:= ((99 + 9)/9) + ((9 + 9) × (9 × 9 − 9))
:= 9 + ((((9 + 9) × (9 × 9 − 9)) − 9/9) + 9)
77/7−7
Ï 1304 := 111+1+1 − (1 + 1 + 1)1+1+1
Ï 1309 := 11 × (111+1 − 1 − 1)
:= 2 × (2 × (((22+2 + 2)2 ) + 2))
:= 22/2 × ((22/2)2 − 2)
:= 2 + ((2 + 2 + 2)2+2 + 22+2 )
:= ((33/3)3 ) − 33
:= 33/3 + (((33/3)3 ) − 33)
:= (3 + 3) × ((3 + 3)3 + 3)
:= 4 + (((4 + 4)/4 + 4)4 + 4)
:= 4 + ((4/4 + 4) × ((4/4 + 44 ) + 4))
:= (((4 + 4) × (4/4 + 4)4 ) + 44 )/4
:= 5 + ((5 − 5/5)5 + 5 × 55)
:= 5 + (((5 − 5/5)5 + 5 × 55) + 5)
:= 5 × 5 × 55 − ((55 + 5/5) + 5)
:= 6 + (6 × 6 × 6 × 6 + ((6 + 6)/6))
:= 6 + ((6 × 6 × 6 × 6 + 6/6) + 6)
:= 6 + ((6 × 6 × 6 × 6 + 6) + 6)
:= ((7 + 7)/7)7 + (7 + 7) × (77 + 7)
:= 77 × ((77 − 7)/7 + 7)
:= 7 × 77 + (777 − ((7 + 7)/7))
:= 88 + 8 × (8 × 8 + 88)
:= 88/8 × (888/8 + 8)
:= (8 + 8)/8 + (88 × (8 + 8) − (88 + 8))
:= 9 + (((9 + 9) × (9 × 9 − 9)) − 9/9)
:= 99 + (9999/9 + 99)
:= 9 + (((9 + 9) × (9 × 9 − 9)) + 9)
Ï 1305 := (11 − 1 − 1) × (1 + (1 + 11)1+1 )
Ï 1310 := 1 + 11 × (111+1 − 1 − 1)
Ï 1314 := 1 + (1 + 1 + 11) × (1 + (11 − 1)1+1 )
Ï 1315 := 111+1+1 − (1 + 1)1+1+1+1
:= 22/2 + ((2 + 2 + 2)2+2 − 2)
:= ((2 + 2 + 2) × 222) − 22
:= (22/2)2/2+2 − 22+2
:= 3 × 3 + ((3 + 3)3/3+3 )
:= 3 + ((((33/3)3 ) − 33 ) + 3)
:= 3/3 + ((3 + 3) × ((3 + 3)3 + 3))
:= (4/4 + 4) × ((4/4 + 44 ) + 4)
:= (4/4 + 4) × (((4 + 4)/4 + 44 ) + 4)
:= ((44/4)4−4/4 ) − 4 × 4
:= 5 + ((5 + 5) × (5 × 5 × 5 + 5))
:= 5 + (((5 + 5) × (5 × 5 × 5 + 5)) + 5)
:= 5 × 5 × 55 − (55 + 5)
:= 6 + (6 × 6 × 6 × 6 + (6 × 6/(6 + 6)))
:= 6 + ((6 × 6 × 6 × 6 + ((6 + 6)/6)) + 6)
:= 6 + (((6 × 6 × 6 × 6 + 6/6) + 6) + 6)
:= 7 + 77/7 × (777/7 + 7)
:= 7 + ((7 − 7/7)77/7−7 + 7)
:= 7 × 77 + (777 − 7/7)
:= (8 − 8/8 + 8) × (88 − 8/8)
:= (88 × (888/8 + 8) + 8)/8
:= ((88/8)88/8−8 ) − 8 − 8
:= 9 + ((9 + 9) × (9 × 9 − 9))
:= 99 + ((9999 + 9)/9 + 99)
:= 9 + ((((9 + 9) × (9 × 9 − 9)) + 9/9) + 9)
Ï 1306 := (1 + 11) × (111 − 1 − 1) − 1 − 1
:= 2 + ((2 + 2 + 2)2+2 + 2 × (2 + 2))
:= 3 + (((33/3)3 ) − (33 + 3/3))
Ï 1311 := 1 + 1 + 11 × (111+1 − 1 − 1)
2
:= 2 + (22/2 × ((22/2) − 2))
3
:= ((3 + 3) × ((3 + 3) + 3)) − 3
4
Ï 1316 := 111+1+1 − 1 − 1 − 1 − 1 − 11
:= 22 + ((2 + 2 + 2)2+2 − 2)
:= 3 + (((33/3)3 ) − (3 × (3 + 3)))
:= (44 − 4)/4 + ((4 + 4)/4 + 4)4
:= 44 × 44 − (4/4 + 4)
:= 5 + (((5/5 + 5)5−5/5 ) + 5)
:= 5 + ((((5/5 + 5)5−5/5 ) + 5) + 5)
:= 5 × 5 + (((5/5 + 5)5−5/5 ) − 5)
:= ((66 − 6)/6) + 6 × 6 × 6 × 6
:= 6 + ((6 × 6 × 6 × 6 + (6 × 6/(6 + 6))) + 6)
:= 6 + (((6 × 6 × 6 × 6 + ((6 + 6)/6)) + 6) + 6)
:= 7 + (((77 × (777/7 + 7)) + 7)/7)
:= 7 + ((7 + 7) × (77 + 7) + ((7 + 7)/7)7 )
:= 7 × 77 + 777
:= 8 + (((8 − 888)/8) + 88 × (8 + 8))
:= 88 × (8 + 8) − ((8/8 + 88) + 8)
:= 8 + ((88 × (888/8 + 8) − 8)/8)
:= 9 + (((9 + 9) × (9 × 9 − 9)) + 9/9)
:= 99/9 × (999/9 + 9) − 9
:= 9 + (((9 + 9) × (9 × 9 − 9)) + (99/9))
Ï 1307 := (1 + 11) × (111 − 1 − 1) − 1
Ï 1312 := 1 + 1 + 1 + 11 × (111+1 − 1 − 1)
:= 4 + (4 × (44 + 4 + 4) + 44 )
Ï 1317 := 111+1+1 − 1 − 1 − 1 − 11
:= 22/2 + (2 + 2 + 2)2+2
:= 22+2 + (2 + 2 + 2)2+2
:= 22 + ((2 + 2 + 2)2+2 − 2/2)
:= 3 + (((33/3)3 ) − 33 )
:= 3/3 + (((3 + 3) × ((3 + 3)3 + 3)) − 3)
:= 3 + ((3 + 3) × ((3 + 3)3 + 3))
:= 44/4 + ((4 + 4)/4 + 4)
4
4
:= 4 × ((4 × (4 − 4/4) ) + 4)
:= (4 − 4/4) × (444 − (4/4 + 4))
:= 55 + ((5 + 5)/5 × (55 + 5)/5)
:= 5 + (((5 + 5)/5 × (5 + 5)/5) + 55)
:= 55 + ((5 + 5)/5 × ((55 + 5)/5 + 5))
:= 66/6 + 6 × 6 × 6 × 6
:= 6 + (((66 − 6)/6) + 6 × 6 × 6 × 6)
:= 6 × 6 × 6 × 6 + ((6 × 6 + 6)/((6 + 6)/6))
:= (7 + 7) × (7 + 7) + 7777/7
:= (7 × ((7 + 7) × (7 + 7) − 7)) − 77/7
:= 7/7 + (7 × 77 + 777)
5
:= 8 + (((8 + 8) × (88 − 8) + (88/8)) + 8)
:= 88 × (8 + 8) − (88 + 8)
:= 8 + (88/8 × (888/8 + 8))
:= 99/9 + ((9 + 9) × (9 × 9 − 9))
:= (9 − 9/9) × (((9 + 9)/9) + 9 × (9 + 9))
:= 9 + (((9 + 9) × (9 × 9 − 9)) + ((99 + 9)/9))
97
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1318 := 111+1+1 − 1 − 1 − 11
Ï 1323 := 1 + 1 + 1 + 11 × (111+1 − 1)
Ï 1328 := 111+1+1 − 1 − 1 − 1
:= 22 + (2 + 2 + 2)2+2
:= (2/2 + 2) × ((22 − 2/2)2 )
:= 2 × (((2/2 + 2) × 222) − 2)
:= 3 + (((3 + 3) × ((3 + 3)3 + 3)) + 3/3)
:= 33 + ((3 + 3)3/3+3 )
:= ((33/3)3 ) − 3
:= 4 + ((((4 + 4) × (4/4 + 4)4 ) + 44 )/4)
:= ((44/4)4−4/4 ) − 4 − 4
:= ((4 − 4/4) × 444) − 4
:= 5 × 5 × 55 − ((5 + 5)/5 + 55)
:= (5 × (5 × 55 − (5 + 5))) − (5 + 5)/5
:= 5 + ((5 × (5 × 55 − (5 + 5))) − ((5 + 5)/5))
:= ((6 + 6)/6) × (666 − 6/6 − 6)
:= 6 × 6 + ((66/6) × (666/6 + 6))
:= 6 × 6 × 6 + (6666 + 6)/6
:= 7 × 77 + (((7 + 7)/7) + 777)
:= 7 × ((7 + 7) × (7 + 7) − 7)
:= 7 + ((7 × ((7 + 7) × (7 + 7) − 7)) − ((7 + 7)/7))
:= 88 × (8 + 8) − ((8 + 8)/8 + 88)
:= ((88/8)88/8−8 ) − 8
:= 8 + 88 × (8 − 8/8 + 8)
:= 9 + ((9999/9 + 99) + 99)
:= 999 + (9 + 9) × (9 + 9)
Ï 1319 := 111+1+1 − 1 − 11
Ï 1324 := 1 + 1 + 1 + 1 + 11 × (111+1 − 1)
:= 9 + ((((9 + 9)/9)99/9 ) − 9 × 9 × 9)
Ï 1329 := 111+1+1 − 1 − 1
:= 22 + ((2 + 2 + 2)2+2 + 2/2)
:= 2 + (((2 + 2 + 2) × (222 − 2)) + 2)
:= (22/2)2/2+2 − 2
:= ((33/3)3 ) − (3 × 3 + 3)
:= ((33/3)3 ) − ((3/3 + 3) + 3)
:= 3/3 + (((33/3)3 ) − 3)
:= 4 + (((44/4)4−4/4 ) − 4 × 4)
:= 44 + (4 × 44 + 44 )
:= (4 − 4/4) × (444 − 4/4)
:= 5 × 5 × 55 − (55 + 5/5)
:= (5 × (55 + 5)) + (5 − 5/5)5
:= 5 + ((5 × (55 + 5)) + (5 − 5/5)5 )
:= (((6 + 6) × (666 − 6)) − 6)/6
:= ((6 + 6)/6)6 + (6 × (6 × 6 × 6 − 6))
:= 6 × 6 × 6 × 6 + (66 × 6/(6 + 6))
:= 7 + ((7 × ((7 + 7) × (7 + 7) − 7)) − (77/7))
:= 7/7 + (7 × ((7 + 7) × (7 + 7) − 7))
:= 88 × (8 + 8) − (8/8 + 88)
:= (((88 + 8)/8) × 888/8) − 8
:= (((9 + 9)/9)99/9 ) − 9 × 9 × 9
:= 9/9 + ((9 + 9) × (9 + 9) + 999)
Ï 1325 := 111+1+1 − (1 + 1) × (1 + 1 + 1)
Ï 1320 := 11 × (111+1 − 1)
:= (2 + 2 + 2) × (222 − 2)
:= 2 + ((2/2 + 2) × ((22 − 2/2)2 ))
:= (3/3 + 3) × (333 − 3)
:= ((33/3)3 ) − (3 + 3)
:= (4 − 4/4) × (444 − 4)
:= 44 + ((4 × 44 + 4/4) + 44 )
:= 55 × (5 × 5 − 5/5)
:= 5 × (5 × 55 − (5 + 5))
:= (6 + 6) × ((666 − 6)/6)
:= (6 × (6 × 6 × 6 + 6)) − 6/6 − 6
:= (77 + 7)/7 × (777 − 7)/7
:= ((7 + 7)/7) + (7 × ((7 + 7) × (7 + 7) − 7))
:= 88 × (8 − 8/8 + 8)
:= 8 + ((88/8 × (888/8 + 8)) + 8)
:= 99/9 × (999/9 + 9)
:= 9 + ((((9 + 9) × (9 × 9 − 9)) + (99/9)) + 9)
Ï 1321 := 1 + 11 × (111+1 − 1)
Ï 1326 := 111+1+1 − 1 − 1 − 1 − 1 − 1
:= 7 + ((7 × ((7 + 7) × (7 + 7) − 7)) − 7/7)
:= 8 + (88 × (8 − 8/8 + 8) + 8/8)
:= 9 + 99/9 × (999/9 + 9)
Ï 1330 := 111+1+1 − 1
:= ((2 + 2 + 2) × 222) − 2
:= ((33/3)3 ) − 3/3
:= ((44/4)4−4/4 ) − 4/4
:= 5 + (5 × (5 × 55 − (5 + 5)))
:= ((6 + 6)/6) × (666 − 6/6)
:= 7 + (7 × ((7 + 7) × (7 + 7) − 7))
:= ((88/8)88/8−8 ) − 8/8
:= (9/9 + 9 + 9) × (9 × 9 − 99/9)
Ï 1331 := 111+1+1
:= (22/2)2/2+2
:= 2/2 + ((2 + 2 + 2) × (222 − 2))
:= (2/2 + 2) × (2 × 222 − 2)
:= ((3 − 33)/3) + ((33/3)3 )
:= 3 + (((3 + 3)3/3+3 ) + 33 )
:= 4/4 + ((4 − 4/4) × (444 − 4))
:= (4 − 4/4) × (444 − (4 + 4)/4)
:= 5 × 5 + ((5/5 + 5)5−5/5 )
:= 5/5 + (5 × (5 × 55 − (5 + 5)))
:= (55/5)5−(5+5)/5
:= (6 × (6 × 6 × 6 + 6)) − 66/6
:= (6 × (6 × 6 × 6 + 6)) − 6
:= (66/6)6×6/(6+6)
:= (7 × ((7 + 7) × (7 + 7) − 7)) − (7 + 7)/7
:= (7/7 + 77) × ((77 − 7)/7 + 7)
:= (77/7)(7+7+7)/7
:= 8/8 + 88 × (8 − 8/8 + 8)
:= 8 + (88 × (8 + 8) − ((8 + 8)/8 + 88))
:= (88/8)88/8−8
:= 9 + ((9 − 9/9) × (((9 + 9)/9) + 9 × (9 + 9)))
:= ((9 − 9/9) + 9) × (9 × 9 − ((9 + 9 + 9)/9))
:= (99/9)(9+9+9)/9
Ï 1322 := 1 + 1 + 11 × (111+1 − 1)
Ï 1327 := 111+1+1 − 1 − 1 − 1 − 1
:= 2 + ((2 + 2 + 2) × (222 − 2))
:= (22/2)
3
2/2+2
−2−2
3
:= ((33/3) ) − 3 × 3
:= (33/3)3
:= (44/4)4−4/4
Ï 1332 := 1 + 111+1+1
:= (2 + 2 + 2) × 222
:= ((33/3) ) − (3/3 + 3)
:= 3/3 + ((33/3)3 )
:= 44 + ((4 × 4 − (4 + 4)/4) + 4 )
:= ((44/4)4−4/4 ) − 4
:= (4 − 4/4) × 444
:= (5 + 5)/5 + (55 × (5 × 5 − 5/5))
:= (5 + 5)/5 + (5 × (5 × 55 − (5 + 5)))
:= ((55 + 5)/5) × 555/5
:= ((6 + 6)/6) × ((666 − 6) + 6/6)
:= 6/6 + ((6 × (6 × 6 × 6 + 6)) − 6)
:= 6 × (6 × 6 × 6 + 6)
:= (7 × ((7 + 7) × (7 + 7) − 7)) − 7/7
4
4
:= 7 × 77 + (77/7 + 777)
:= (77 + 7)/7 × 777/7
:= (8 + 8)/8 + 88 × (8 − 8/8 + 8)
:= 8 + (88 × (8 + 8) − (8/8 + 88))
:= ((88 + 8)/8) × 888/8
:= 9 × (9 × 9 + 9) + (((9 + 9)/9)9 )
:= 9 + (((9999/9 + 99) + 99) + 9)
:= 9 + ((9 + 9) × (9 + 9) + 999)
98
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1338 := 11 + 111+1+1 − 1 − 1 − 1 − 1
Ï 1333 := 1 + 1 + 111+1+1
:= 2 + (22/2)2/2+2
:= (2/2 + 2) × (2 × 222 + 2)
Ï 1343 := 1 + 11 + 111+1+1
:= 22/2 + ((2 + 2 + 2) × 222)
:= 3 + (((33/3)3 ) − 3/3)
:= 3 + ((((33/3) ) + 3/3) + 3)
:= 3 + (((33/3)3 ) + 3 × 3)
:= 4/4 + ((4 − 4/4) × 444)
:= (4 + 4)/4 × ((4/4 + 4)4 + 44)
:= 4 + ((((44/4)4−4/4 ) + 4) + 4)
:= ((555 × ((55 + 5)/5)) + 5)/5
:= 5 × 5 × 55 − (((5 + 5)/5)5 + 5)
:= 5 × 5 × 55 − ((5 + 5)/5)5
:= 6/6 + (6 × (6 × 6 × 6 + 6))
:= 6 + (6 × (6 × 6 × 6 + 6))
:= 66/6 + (6 × (6 × 6 × 6 + 6))
3
:= ((777 × (77 + 7)/7) + 7)/7
:= 7 + ((77/7)
:= 88 × (8 + 8) − (88/8 + 8 × 8)
:= 8 + (((88/8)88/8−8 ) − 8/8)
:= 88 × (8 + 8) − (8/8 + 8 × 8)
:= 9 + (((9 + 9) × (9 + 9) + 999) + 9/9)
:= 9 × 9 × (9 + 9) − (999/9 + 9)
:= ((9 − 9/9) + 9) × (9 × 9 − ((9 + 9)/9))
Ï 1334 := 1 + 1 + 1 + 111+1+1
(7+7+7)/7
)
Ï 1339 := 11 + 111+1+1 − 1 − 1 − 1
:= 2 + ((2 + 2 + 2) × 222)
:= ((77 − 7)/7 + 7) × ((7 + 7)/7 + 77)
Ï 1344 := (1 + 11) × (1 + 111)
:= 2 × (2 + 2) + (22/2)2/2+2
:= (2 + 2 + 2) × (222 + 2)
:= 3 × 3 + (((33/3)3 ) − 3/3)
:= (3/3 + 3) × (333 + 3)
:= 4 + (((44/4)4−4/4 ) + 4)
:= 4 × ((4 × (4 × 4 + 4)) + 44 )
:= (5 × (5 × 55 − 5)) − (55/5 + 5)
:= (5 × (5 × 55 − 5)) − 55/5
:= (5 × 5 − 5/5) × (55 + 5/5)
:= (6 + 6)/6 + (6 × (6 × 6 × 6 + 6))
:= 6 + ((6 × (6 × 6 × 6 + 6)) + 6/6)
:= 6 + ((6 × (6 × 6 × 6 + 6)) + 6)
:= 77/7 + (7 × ((7 + 7) × (7 + 7) − 7))
:= 7 + ((77 + 7)/7 × 777/7)
:= (7 + 7) × (7 × (7 + 7) − ((7 + 7)/7))
:= 888 + (8 × (8 × 8 − 8) − ((8 + 8)/8))
:= 8 + ((88/8)88/8−8 )
:= 8 × (88 − 8 + 88)
3
:= 3 + ((33/3) )
:= 4 + (((44/4)
4−4/4
) − 4/4)
:= 99/9 + ((9 + 9) × (9 + 9) + 999)
Ï 1335 := 1 + 1 + 1 + 1 + 111+1+1
:= 2 + ((22/2)
2/2+2
+ 2)
:= 9 + ((9/9 + 9 + 9) × (9 × 9 − (99/9)))
Ï 1340 := 11 + 111+1+1 − 1 − 1
:= (2 × 22) + (2 + 2 + 2)2+2
:= ((99 + 9)/9) × ((999 + 9)/9)
Ï 1345 := 1 + (1 + 11) × (1 + 111)
:= 2/2 + ((2 + 2 + 2) × (222 + 2))
:= 3 + (((33/3)3 ) + 33/3)
:= 3 + (((33/3)3 ) + 3/3)
:= 3 × 3 + ((33/3)3 )
:= 4 + ((44/4)4−4/4 )
:= 44 + ((4 + 4)/4 + 4)4
:= 5 + ((5 × (5 × 55 − (5 + 5))) + 5)
:= (5 × (5 × 55 − 5)) − 5 − 5
:= (6 × 6/(6 + 6)) + (6 × (6 × 6 × 6 + 6))
:= 6 + ((6 × (6 × 6 × 6 + 6)) + ((6 + 6)/6))
:= (77 + 7)/7 + (7 × ((7 + 7) × (7 + 7) − 7))
:= 7 + (((777 × (77 + 7)/7) + 7)/7)
:= 7 + (((77/7)(7+7+7)/7 ) + 7)
:= (8 − 8/8 + 8) × (8/8 + 88)
:= 8 + (((88 + 8)/8) × 888/8)
:= 8/8 + 8 × (88 − 8 + 88)
:= (9 − (9 × 9 × 99))/(((9 + 9 + 9)/9) − 9)
:= 9 + (99/9)(9+9+9)/9
:= 9 × (9 + 9 + 9) + (9999/9 − 9)
Ï 1336 := 1 + 1 + 1 + 1 + 1 + 111+1+1
:= 2 + (((2 + 2 + 2) × 222) + 2)
:= 4/4 + ((4 × (4 × 4 + 44 )) + 44 )
:= (5 × (5 × 55 − 5)) − 5
:= (((6 + 6) × (666 + 6)) + 6)/6
Ï 1346 := 1 + 1 + (1 + 11) × (1 + 111)
Ï 1341 := 11 + 111+1+1 − 1
:= (2 × ((22 + 2 + 2)2 )) − 22/2
:= 2 + ((2 + 2 + 2) × (222 + 2))
:= 3 + ((((33/3) ) − 3/3) + 3)
:= 33 + ((3 + 3) × ((3 + 3)3 + 3))
:= 3 + ((((33/3)3 ) + 3 × 3) + 3)
:= 4 + ((4 − 4/4) × 444)
:= 44 + (((4 + 4)/4 + 4)4 + 4/4)
:= 4 + (((44/4)4−4/4 ) + 44/4)
:= 5 + (((55/5)5−(5+5)/5 ) + 5)
:= 5/5 + ((5 × (5 × 55 − 5)) − 5)
3
:= 5 + ((55/5)5−(5+5)/5 )
:= 6 + (((6 + 6)/6) × (666 − 6/6))
:= 7 + (((7 × ((7 + 7) × (7 + 7) − 7)) − 7/7) + 7)
:= 888 + 8 × (8 × 8 − 8)
:= 9 × 99 + (((9 × 9 × 99) − 9)/(9 + 9))
Ï 1337 := 1 + 1 + 1 + 1 + 1 + 1 + 111+1+1
:= 6 + ((6 × (6 × 6 × 6 + 6)) + (6 × 6/(6 + 6)))
:= 7 + ((7 × ((7 + 7) × (7 + 7) − 7)) + (77/7))
:= 8 + (88 × (8 + 8) − (88/8 + 8 × 8))
:= 9 × 9 × (9 + 9) − (99 + 9 + 9)
:= ((6 + 6)/6) × (666 + 6/6 + 6)
:= 7 + (((77 + 7)/7 × 777/7) + 7)
:= (8 + 8)/8 + 8 × (88 − 8 + 88)
:= 9 × 9 × (9 + 9) − ((999 + 9)/9)
Ï 1347 := 1 + 1 + 1 + (1 + 11) × (1 + 111)
Ï 1342 := 11 + 111+1+1
:= ((2 + 2 + 2) × (222 + 2)) − 2
:= 22+2 + (22/2)2/2+2
:= 3 + (((33/3)3 ) + 3)
:= 33/3 + ((33/3)3 )
:= 3 + ((3/3 + 3) × (333 + 3))
:= 4 + (((4 − 4/4) × 444) + 4/4)
:= 44/4 + ((44/4)4−4/4 )
:= 4 × 4 + ((44/4)4−4/4 )
:= 5 + (((55 + 5)/5) × 555/5)
:= 55/5 × ((555 + 55)/5)
:= (5 + 5)/5 + ((5 × (5 × 55 − 5)) − 5)
:= 2 + (((22/2)
:= 6 + ((66/6)
2/2+2
+ 2) + 2)
6×6/(6+6)
)
:= 66/6 × ((666 + 66)/6)
:= ((66/6) × ((666/6 + 6) + 6)) − 6
:= 7 + ((7 × ((7 + 7) × (7 + 7) − 7)) + 7)
:= 77/7 × (777 + 77)/7
:= 7 × 7 + 77/7 × (777/7 + 7)
:= 8/8 + (8 × (8 × 8 − 8) + 888)
:= 88/8 × ((888 + 88)/8)
:= 8 + (((88/8)88/8−8 ) + 8)
:= 9 × 99 + (((9 × 9 × 99) + 9)/(9 + 9))
:= 99/9 × (999 + 99)/9
:= 9 × 9 × (9 + 9) − 999/9
99
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1348 := 1 + 1 + 1 + 1 + (1 + 11) × (1 + 111)
2
Ï 1353 := 11 × (1 + 1 + 111+1 )
Ï 1358 := 1 + 1 + (1 + 11) × (1 + 1 + 111)
:= 2 + (2 × (((22 + 2 + 2)2 ) + 2))
2/2+2
:= 2 × (((22 + 2 + 2) ) − 2)
:= 22 + (22/2)
:= 3 + ((((33/3)3 ) + 33/3) + 3)
:= 3 + ((3 + 3) × ((3 + 3)3 + 3 × 3))
:= 33 + ((33/3)3 )
:= 4 × ((4 − 4/4)4 + 44 )
:= 4 + ((4 × ((4 − 4/4)4 + 44 )) + 4/4)
:= 4 + ((44 × (44 − 4) − 44 )/(4 + 4))
:= (5 × (5 × 55 − 5)) − (5 + 5)/5
:= 5 + ((5 × (5 × 55 − 5)) − ((5 + 5)/5))
:= (5 + 5)/5 × ((55 − 5)/5 + 55)
:= 6 + ((6 × (6 × 6 × 6 + 6)) + ((66 − 6)/6))
:= 66/6 × ((666/6 + 6) + 6)
:= 66 + ((6 × 6 × 6 × 6 − 6) + ((6 + 6)/6))
:= (77/7 − 7) × ((7 × 7 × 7 − 7) + 7/7)
:= 77/7 × ((777 + 77 + 7)/7)
:= (7 + 7) × (7 × (7 + 7) − 7/7)
:= 8 + ((((88 + 8)/8) × 888/8) + 8)
:= 8 + (8 × (88 − 8 + 88) + 8/8)
:= 8 + ((8 − 8/8 + 8) × ((8 + 8)/8 + 88))
:= 9 × 9 × (9 + 9) + ((9 − 999)/9)
:= 9 + (((99 + 9)/9) × ((999 + 9)/9))
:= 9 × 9 × (9 + 9) − (9/9 + 99)
Ï 1349 := 111+1+1 + (1 + 1) × (11 − 1 − 1)
2
Ï 1354 := 1 + 11 × (1 + 1 + 111+1 )
Ï 1359 := 1 + 1 + 1 + (1 + 11) × (1 + 1 + 111)
:= 2/2 + (2 × (((22 + 2 + 2) ) − 2))
:= 2 + (2 × ((22 + 2 + 2)2 ))
:= 2 + ((2 × (((22 + 2 + 2)2 ) + 2)) + 2/2)
:= (3 × (3 + 3)) + ((33/3)3 )
:= (3333 + 33+3 )/3
:= 3 × ((3 × (3 + 3) × 33 ) − 33)
:= 4/4 + (4 × ((4 − 4/4) + 4 ))
:= (44 × (44 − 4) − 44 )/(4 + 4)
:= 44 + (4444/4 − (4 + 4))
:= (5 × (5 × 55 − 5)) − 5/5
:= 5 + ((5 × (5 × 55 − 5)) − 5/5)
:= 5 × 5 × 55 − (55/5 + 5)
:= 6 + ((6 × (6 × 6 × 6 + 6)) + (66/6))
:= ((6 + 6)/6) × (666 + (66/6))
:= 6 + ((66/6) × ((666/6 + 6) + 6))
:= 7 + (77/7 × (777 + 77)/7)
:= (7 × (7 + 7) × (7 + 7)) − (77/7 + 7)
:= 7/7 + ((7 + 7) × (7 × (7 + 7) − 7/7))
4
4
:= (88/8 + 8) × ((8 × 8 − 8/8) + 8)
:= 8 + (8 × (88 − 8 + 88) + ((8 + 8)/8))
:= (8/8 + 8) × ((88 − 8/8) + 8 × 8)
:= 9 + ((99/9)(9+9+9)/9 + 9)
:= 9 × (9 + 9 + 9) + 9999/9
:= 9 × 9 × (9 + 9) − 99
Ï 1350 := 111+1+1 + (1 + 1) × (11 − 1) − 1
Ï 1355 := 11 + (1 + 11) × (1 + 111)
Ï 1360 := 1 + 1 + 1 + 1 + (1 + 11) × (1 + 1 + 111)
:= (2 × ((22 + 2 + 2) )) − 2
:= 2 + ((2 × ((22 + 2 + 2) )) + 2/2)
:= 2 × ((((22 + 2 + 2)2 ) + 2) + 2)
:= (3 + 3) × ((3 + 3)3 + 3 × 3)
:= 33 + (((33/3)3 ) − 3)
:= 3 + ((((33/3)3 ) − 3/3) + 33 )
:= 44 + ((4444 − 4)/4 − 4 × 4)
:= (4/4 + 4) × ((44/4 + 44 ) + 4)
:= 4 × (4 × ((4 − 4/4)4 + 4))
:= 5 × (5 × 55 − 5)
:= 5 + (5 × (5 × 55 − 5))
:= 5 + ((5 × (5 × 55 − 5)) + 5)
:= 6 + (((6 × (6 × 6 × 6 + 6)) + 6) + 6)
:= 66 + (6 × 6 × 6 × 6 − (6/6 + 6))
:= ((6 + 6)/6)6 + 6 × 6 × 6 × 6
:= ((77 − 7)/7) × (((7 + 7)/7)7 + 7)
:= ((7 − 77)/7) + ((7 × (7 + 7) × (7 + 7)) − 7)
:= (7 × (7 + 7) × (7 + 7)) − (77 + 7)/7
:= (8 − 8/8 + 8) × ((8 + 8)/8 + 88)
:= 88/8 + 8 × (88 − 8 + 88)
:= (88 − 8) × (8/8 + 8 + 8)
:= 9 × 9 × (9 + 9) − (99 + 9)
:= 9 + (9 × 9 × (9 + 9) − ((999 + 9)/9))
:= 9/9 + (9 × 9 × (9 + 9) − 99)
2
Ï 1351 := 111+1+1 + (1 + 1) × (11 − 1)
2
Ï 1356 := (1 + 11) × (1 + 1 + 111)
Ï 1361 := 111+1+1 + (11 − 1) × (1 + 1 + 1)
:= (2 × ((22 + 2 + 2)2 )) − 2/2
:= 2 × (((22 + 2 + 2)2 ) + 2)
:= 2/2 + (2 × ((((22 + 2 + 2)2 ) + 2) + 2))
:= 3 × 3 + (((33/3)3 ) + 33/3)
:= (3/3 + 3) × ((333 + 3) + 3)
:= 3 + (((33/3)3 ) + 33 )
:= 4 + (((44/4)4−4/4 ) + 4 × 4)
:= (4 − 4/4) × (444 + 4 + 4)
:= 4/4 + (4 × (4 × ((4 − 4/4)4 + 4)))
:= 5/5 + (5 × (5 × 55 − 5))
:= 5 + ((5 × (5 × 55 − 5)) + 5/5)
:= 55/5 + (5 × (5 × 55 − 5))
:= 6 + ((((6 + 6) × (666 + 6)) + 6)/6)
:= 66 + (6 × 6 × 6 × 6 − 6)
:= 66 + (6 × 6 × 6 × 6 − 6/6)
7
:= 77 + (7 + 7) × (77 + 7 + 7)
:= (7 − 7/7) × (((7 + 7)/7) + 7 × (7 + 7))
:= (7 × (7 + 7) × (7 + 7)) − 77/7
:= 8 + (88 × (8 + 8) − (8/8 + 8 × 8))
:= ((88 + 8)/8) + 8 × (88 − 8 + 88)
:= 8/8 + (88 − 8) × (8/8 + 8 + 8)
:= 9/9 + (9 × 9 × (9 + 9) − (99 + 9))
:= 9 + (9 × 9 × (9 + 9) − 999/9)
:= (9 + 9)/9 + (9 × 9 × (9 + 9) − 99)
Ï 1352 := 11 × (1 + (1 + 111+1 )) − 1
Ï 1357 := 1 + (1 + 11) × (1 + 1 + 111)
Ï 1362 := 11 × (1 + 1 + 1 + 111+1 ) − 1 − 1
:= 2 × ((22 + 2 + 2)2 )
:= 2/2 + (2 × (((22 + 2 + 2)2 ) + 2))
:= (22 × (22+2+2 − 2)) − 2
:= 3 + (((33/3)3 ) + (3 × (3 + 3)))
:= 33 + (((33/3)3 ) − 3/3)
:= (3 + 3) × ((3 + 3)3 + 33/3)
:= 4 + (4 × ((4 − 4/4)4 + 44 ))
:= 4/4 + ((4 − 4/4) × (444 + 4 + 4))
:= 44 + ((4444 − 4)/4 − 4)
:= (5 + 5)/5 + (5 × (5 × 55 − 5))
:= 5 + ((5 × (5 × 55 − 5)) + ((5 + 5)/5))
:= (5 + 5)/5 × ((55 + 5)/5 + 55)
:= 6 + (((6 + 6)/6) × (666 + 6/6 + 6))
:= 66 + ((6 × 6 × 6 × 6 − 6) + 6/6)
:= 66 + 6 × 6 × 6 × 6
:= (7 − 7/7 + 7) × (777/7 − 7)
:= (7 × (7 + 7) × (7 + 7)) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + (7 × (7 + 7) × (7 + 7))
:= 8 + 8 × (88 − 8 + 88)
:= 8 + ((88/8 + 8) × ((8 × 8 − 8/8) + 8))
:= (8 + 8)/8 + (88 − 8) × (8/8 + 8 + 8)
:= 9 + (((9 − 9/9) + 9) × (9 × 9 − ((9 + 9)/9)))
:= 9 × 9 × (9 + 9) − ((9 + 9)/9 + 99)
:= 999 + ((99 × 99)/(9 + 9 + 9))
100
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1363 := 11 × (1 + 1 + 1 + 111+1 ) − 1
Ï 1368 := (1 + 11) × (1 + 1 + 1 + 111)
Ï 1373 := 1 + (1 + 1 + 1 + 11)1+1+1 /(1 + 1)
:= 22/2 + (2 × ((22 + 2 + 2)2 ))
:= (2 × 222 ) + (22 − 2)2
:= 2 + ((((2 + 2 + 2)2 + 2/2)2 ) + 2)
:= 33 + (((33/3)3 ) − 3/3)
:= (3 + 3) × (((3 + 3)3 + 3 × 3) + 3)
:= 3 × 3 + (((33/3)3 ) + 33)
:= 44 + (4444/4 − 4)
:= 4 + (44 × (4 × (4 + 4) − 4/4))
:= 4/4 + (((4 + 4) × (4 × 44 − 4)) − 4)
:= 5 × 5 × 55 − (55 + 5)/5
:= 5 × 5 × 55 − ((5 + 5)/5 + 5)
:= 5 × 5 × 55 − (5 + 5)/5
:= 66 + (6 × 6 × 6 × 6 + 6/6)
:= 6 × (6 × 6 × 6 + 6 + 6)
:= 6 + ((6 × (6 × 6 × 6 + 6 + 6)) − 6/6)
:= (7 × (7 + 7) × (7 + 7)) − ((7 + 7)/7 + 7)
:= (77/7 + 7) × (77 − 7/7)
:= 7/7 + (7 × (7 + 7) × (7 + 7))
:= 8 + (8 × (88 − 8 + 88) + (88/8))
:= 88 + (8 + 8) × (88 − 8)
:= 8 × 8 + (88/8 × (888/8 + 8))
:= 9 + (9999/9 + 9 × (9 + 9 + 9))
:= 9 × (9 × (9 + 9) − 9) − 9
:= ((9 − 9 × 9)/(9 + 9)) + 9 × (9 × (9 + 9) − 9)
Ï 1364 := 11 × (1 + 1 + 1 + 111+1 )
Ï 1369 := (111/(1 + 1 + 1))1+1
Ï 1374 := 11 × (1 + 1 + 1 + 1 + 111+1 ) − 1
:= 22 × (22+2+2 − 2)
:= ((2 + 2 + 2)2 + 2/2)2
:= 22 + (2 × ((22 + 2 + 2)2 ))
:= 33 + ((33/3)3 )
:= (3/3 + 33 + 3)3−3/3
:= 33+3 + (3 × (3 + 3)3 − 3)
:= 44 × (4 × (4 + 4) − 4/4)
:= 4 + ((((4 + 4)4 ) − 4/4)/(4 − 4/4))
:= ((4 + 4) × (4 × 44 − 4)) − (4 + 4)/4
:= 5 × 5 × 55 − 55/5
:= 5 × 5 × 55 − (5/5 + 5)
:= 5 × 5 × 55 − 5/5
:= 66 + (6 × 6 × 6 × 6 + ((6 + 6)/6))
:= (6 × 6 + 6/6)
:= 6 + (6 × (6 × 6 × 6 + 6 + 6))
:= (7 × (7 + 7) × (7 + 7)) − (7/7 + 7)
:= (7 × (7 + 7) × (7 + 7)) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + (7 × (7 + 7) × (7 + 7))
:= 88 × (8 + 8) − (88/((8 + 8)/8))
:= 8/8 + ((8 + 8) × (88 − 8) + 88)
:= ((8 + 8)/8) × (8 × 88 − (8/8 + 8 + 8))
:= 999 + (((9 × 9 × 9 × 9) + 9)/(9 + 9))
:= 9/9 + (9 × (9 × (9 + 9) − 9) − 9)
:= 9 × (9 × (9 + 9) − 9) − (9 + 9 + 9)/9
Ï 1365 := 1 + 11 × (1 + 1 + 1 + 111+1 )
(6+6)/6
Ï 1375 := 11 × (1 + 1 + 1 + 1 + 111+1 )
Ï 1370 := 1 + (111/(1 + 1 + 1))1+1
:= 2/2 + (22 × (22+2+2 − 2))
:= 2 + ((2 × 222 ) + (22 − 2)2 )
:= (2 × 22) + (22/2)2/2+2
:= 3/3 + (((33/3)3 ) + 33)
:= 3 + ((((33/3)3 ) + 33) + 3)
:= 33/3 × ((3 − 3/3 + 3)3 )
:= (((4 + 4)4 ) − 4/4)/(4 − 4/4)
:= 4 + ((4444 − 4)/4 + 44 )
:= 44 + ((44/4)4−4/4 )
:= 5 × 5 × 55 − 5 − 5
:= 5 × 5 × 55 − 5
:= 5 × 5 × 55
:= 6 + ((6 × 6 + 6/6)(6+6)/6 )
:= (6/6 + 6 + 6) × (666/6 − 6)
:= 6/6 + ((6 × 6 + 6/6)
:= (7 × (7 + 7) × (7 + 7)) − 7
:= (7 × (7 + 7) × (7 + 7)) − (7 + 7)/7
:= 7 + ((77/7 + 7) × (77 − 7/7))
:= (8 − 8/8 + 8) × ((88/8 − 8) + 88)
:= 88 + ((8 + 8) × (88 − 8) + ((8 + 8)/8))
:= ((8 + 8) × (88 − ((8 + 8)/8))) − 8/8
:= 9 × (9 × (9 + 9) − 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 × (9 × (9 + 9) − 9) − 9)
:= 9 × (9 × (9 + 9) − 9) − (9 + 9)/9
Ï 1366 := 11 + (1 + 11) × (1 + 1 + 111) − 1
2+2+2
(6+6)/6
)
Ï 1371 := 1 + 1 + (111/(1 + 1 + 1))1+1
Ï 1376 := 1 + (11 × (1 + 1 + 1 + 1 + 111+1 ))
:= 2 + (((2 + 2 + 2)2 + 2/2)2 )
:= 22+2 × (2 × 2 × 22 − 2)
:= 3 + ((3 × ((3 + 3)3 − 3)) + 33+3 )
:= 3 + ((((33/3)3 ) + 33) + 3 × 3)
:= 4 + (4444 − 4)/4
:= 4 + (4444/4 + 44 )
:= (4 + 4) × (4 × 44 − 4)
:= 5/5 + (5 × 5 × 55 − (5 + 5))
:= 5/5 + (5 × 5 × 55 − 5)
:= 5/5 + 5 × 5 × 55
:= 6 + (((6 + 6)/6) + 6 × 6 × 6 × 6)
:= 666/6 + (6 × (6 × 6 × 6 − 6))
:= 6 + (((6 × 6 + 6/6)(6+6)/6 ) + 6/6)
:= 7/7 + ((7 × (7 + 7) × (7 + 7)) − 7)
:= (7 × (7 + 7) × (7 + 7)) − 7/7
:= 77/7 + ((7 × (7 + 7) × (7 + 7)) − 7)
:= 88 + ((8 + 8) × (88 − 8) − ((8 + 8)/8))
:= 88/8 + (88 − 8) × (8/8 + 8 + 8)
:= (8 + 8) × (88 − ((8 + 8)/8))
:= 9 × (9 × (9 + 9) − 9) − 99/9
:= 9 × 9 × (9 + 9) + (((99 + 9)/9) − 99)
:= 9 × (9 × (9 + 9) − 9) − 9/9
:= 2 + (22 × (2
− 2))
3
:= 3 + ((((33/3) ) − 3/3) + 33)
4
6
Ï 1367 := 11 + (1 + 11) × (1 + 1 + 111)
2
2
:= (((2 + 2 + 2) + 2/2) ) − 2
3
Ï 1372 := (1 + 1 + 1 + 11)1+1+1 /(1 + 1)
2
Ï 1377 := 1 + 1 + 11 × (1 + 1 + 1 + 1 + 111+1 )
:= 2 + ((22/2)2/2+2 + 2 × 22)
:= 22 + (2 × 2 × 222)
3
:= 3 + (((33/3) ) + 33)
:= (3/3 + 3) × (((3/3 + 3) + 3) )
:= 33 × ((33 − 3) + 33 )
:= 44 + 4444/4
:= ((4 + 4) × (4 × 44 − 4)) − 4
:= 4/4 + ((4 + 4) × (4 × 44 − 4))
:= (5 + 5)/5 + (5 × 5 × 55 − (5 + 5))
:= (5 + 5)/5 + (5 × 5 × 55 − 5)
:= (5 + 5)/5 + 5 × 5 × 55
6
:= (6 × (6 × 6 × 6 + 6 + 6)) − 6/6
:= 6 + ((((6 + 6)/6) + 6 × 6 × 6 × 6) + 6)
:= ((7 + 7)/7) + ((7 × (7 + 7) × (7 + 7)) − 7)
:= 6 + ((6 × (6 × 6 × 6 − 6)) + 666/6)
:= 7 × (7 + 7) × (7 + 7)
:= 7 + ((7 × (7 + 7) × (7 + 7)) − ((7 + 7)/7))
:= 88 + ((8 + 8) × (88 − 8) − 8/8)
:= 888 + 88 × 88/(8 + 8)
:= 8/8 + ((8 + 8) × (88 − ((8 + 8)/8)))
:= 9 × (9 × (9 + 9) − 9) − 9/9 − 9
:= 99 + (9999/9 + 9 × (9 + 9))
:= 9 × (9 × (9 + 9) − 9)
101
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1378 := 11 + 11 + (1 + 11) × (1 + 1 + 111)
Ï 1383 := 11 + ((1 + 1 + 1 + 11)1+1+1 /(1 + 1))
Ï 1388 := 1 + 111+1+1 + (1 + 111)/(1 + 1)
:= (2 × (22 − 2))2 − 222
:= (2/2 + 2) × (222 − 22 − 2/2)
:= 2 + (22 × (22+2+2 − 2/2))
:= 3 + (33/3 × ((3 − 3/3 + 3)3 ))
:= (33 × (3 × 3 + 33)) − 3
:= 3 + ((((33/3)3 ) + 33 ) + 33 )
4
:= 44 + (4444 + 44)/4
:= 4 × 4 + (4444/4 + 4 )
:= (4 × ((4 + 4) × 44 − 4)) − 4
:= 5 + (5 × 5 × 55 − ((5 + 5)/5))
:= 5 + ((5 × 5 × 55 − ((5 + 5)/5)) + 5)
:= 5 × 5 × 55 + (55 + 5 + 5)/5
:= ((66 − 6)/6) + (6 × (6 × 6 × 6 + 6 + 6))
:= ((66 × (6 × 6 + 6)) − 6)/((6 + 6)/6)
:= 6 + ((((6 + 6)/6)66/6 ) − 666)
:= 7 + ((7 × (7 + 7) × (7 + 7)) − 7/7)
:= 77/7 + (7 × (7 + 7) × (7 + 7))
:= ((7 + 7)/7) + (77 × (77/7 + 7))
:= (8 + 8)/8 + ((8 + 8) × (88 − ((8 + 8)/8)))
:= 88 × (8 + 8) − (8/8 + 8 + 8 + 8)
:= 88 × (8 + 8) − ((88 + 8)/8 + 8)
:= 9/9 + 9 × (9 × (9 + 9) − 9)
:= 9 + (9 × (9 × (9 + 9) − 9) − ((9 + 9 + 9)/9))
:= 99/9 + 9 × (9 × (9 + 9) − 9)
Ï 1379 := 11 + (1 + 11) × (1 + 1 + 1 + 111)
2
:= 2/2 + (2 × (22 − 2)) − 222
3
3
Ï 1384 := 1 + 11 + (1 + 1 + 1 + 11)1+1+1 /(1 + 1)
2
:= 2 × (((22 + 2 + 2) ) + 2
2+2
)
Ï 1389 := (1 + 11111)/(1 + 1)1+1+1
:= 22 + ((((2 + 2 + 2)2 + 2/2)2 ) − 2)
:= 3 × 3 + (((33/3) ) − 33)
:= 3/3 + ((33 × (3 × 3 + 33)) − 3)
:= 3 + (33 × (3 × 3 + 33))
:= 4 + (((44/4)4−4/4 ) + 44)
:= 4 + (((4 + 4) × (4 × 44 − 4)) + 4)
:= 4/4 + ((4 × ((4 + 4) × 44 − 4)) − 4)
:= 5 + (5 × 5 × 55 − 5/5)
:= 5 + ((5 × 5 × 55 − 5/5) + 5)
:= 5 × (5 × 55 + 5) − 55/5
:= 66/6 + (6 × (6 × 6 × 6 + 6 + 6))
:= ((66 × ((66 − 6) + 66)) − (6 + 6))/6
:= 666 + (((6 × 6/(6 + 6))6 ) − 6)
:= 7 + (7 × (7 + 7) × (7 + 7))
:= (77 + 7)/7 + (7 × (7 + 7) × (7 + 7))
:= 7 + ((7 × (7 + 7) × (7 + 7)) + ((77 − 7)/7))
:= 88 + ((8 + 8) × (88 − 8) + (88/8))
:= 88 × (8 + 8) − 8 − 8 − 8
:= 88 × (8 + 8) − (88/8 + 8)
:= (9 + 9)/9 + 9 × (9 × (9 + 9) − 9)
:= 9 + (9 × (9 × (9 + 9) − 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + 9 × (9 × (9 + 9) − 9)
Ï 1380 := 11 + ((111/(1 + 1 + 1))1+1 )
Ï 1385 := 111+1+1 + ((111 − 1)/(1 + 1) − 1)
Ï 1390 := 11 × 111 + (1 + 1 + 11)1+1
:= 2 + (2 × (22 − 2))2 − 222
:= 22+2 + (((2 + 2 + 2)2 + 2/2)2 )
:= (2 × (222 + 222 )) − 22
:= 3 + (3 × (3 + 3)3 + 33+3 )
:= 33 + (((33/3)3 ) + 33 )
:= 3 + ((33 × (3 × 3 + 33)) + 3/3)
4
:= (44 × (4 − 4)/(4 + 4)) − 4/4
:= 4 + (44 × (44 − 4)/(4 + 4))
:= 5 + 5 × 5 × 55
:= 5 + (5 × 5 × 55 + 5)
:= 5 + ((5 × 5 × 55 + 5) + 5)
:= 6 + ((6 × (6 × 6 × 6 + 6 + 6)) + 6)
:= ((66 × ((66 − 6) + 66)) − 6)/6
:= 666 + ((66 × 66 − 6 − 6)/6)
:= 7 + ((7 × (7 + 7) × (7 + 7)) + 7/7)
:= (77 × (77/7 + 7)) − 7/7
:= 7 + ((7 × (7 + 7) × (7 + 7)) + (77/7))
:= 8 + (88 × 88/(8 + 8) + 888)
:= 8/8 + (88 × (8 + 8) − (8 + 8 + 8))
:= ((8 + 8)/8) × (8 × 88 − (8/8 + 8))
:= ((9 + 9 + 9)/9) + 9 × (9 × (9 + 9) − 9)
:= 9 + (9 × (9 × (9 + 9) − 9) − 9/9)
:= 9 × 99 + (((9 × 999) − 9)/(9 + 9))
:= 4 + ((4 + 4) × (4 × 44 − 4))
Ï 1381 := 1 + 11 + ((111/(1 + 1 + 1))1+1 )
2
:= 2 + (2 × (22 − 2)) − 222 + 2/2
3
Ï 1386 := 11 × (1 + 1 + 1 + 1 + 1 + 111+1 )
Ï 1391 := (1 + 1 + 11) × (111 − 1 − 1 − 1 − 1)
2+2+2
:= 22 + (((2 + 2 + 2)2 + 2/2)2 )
:= 22 × (2
− 2/2)
:= 3 + ((33/3 × ((3 − 3/3 + 3) )) + 3)
:= 33 × (3 × 3 + 33)
:= 33 + (((33/3)3 ) + 33)
:= 4 + (((4 + 4) × (4 × 44 − 4)) + 4/4)
:= 44 × (44 − 4)/(4 + 4)
:= (4 × ((4 + 4) × 44 − 4)) − 4/4
:= 5 + (5 × 5 × 55 + 5/5)
:= 55/5 + 5 × 5 × 55
:= 5 + (5 × 5 × 55 + (55/5))
:= 6 + (((6 × 6 + 6/6)(6+6)/6 ) + 6)
:= 6 × ((66 × (6 × 6 + 6))/(6 + 6))
:= 666 + ((66 × 66 − 6)/6)
:= 7 + ((7 × (7 + 7) × (7 + 7)) + ((7 + 7)/7))
:= 77 × (77/7 + 7)
:= 7 + ((7 × (7 + 7) × (7 + 7)) + (77 + 7)/7)
:= 88 × (8 + 8) − (88/8 + 8 + 8)
:= ((8 + 8)/8) × (8 × 88 − (88/8))
:= 88 × (8 + 8) − (8/8 + 8 + 8)
:= 9 × (9 × (9 + 9) − 9) + ((9 × 9 − 9)/(9 + 9))
:= 9 + 9 × (9 × (9 + 9) − 9)
:= 9 × 99 + (((9 × 999) + 9)/(9 + 9))
Ï 1382 := 1 + 1 + 11 + ((111/(1 + 1 + 1))1+1 )
2
:= 2 + (2 × (22 − 2)) − 222 + 2
3
3
3
:= 3 + ((((33/3) ) − 3) + 3 )
4
Ï 1387 := 111+1+1 + (1 + 111)/(1 + 1)
Ï 1392 := (1 + 11) × (1 + 1 + 1 + 1 + 1 + 111)
:= 2/2 + (22 × (22+2+2 − 2/2))
:= 2 × ((((22 + 2 + 2)2 ) − 2) + 22)
:= 3/3 + (33 × (3 × 3 + 33))
:= 3 + ((33 × (3 × 3 + 33)) + 3)
4
:= (44 × (4 − 4)/(4 + 4)) − 4
:= 4/4 + (44 × (4 − 4)/(4 + 4))
:= 4 × ((4 + 4) × 44 − 4)
:= 5 + (5 × 5 × 55 + ((5 + 5)/5))
:= 5 × 5 × 55 + (55 + 5)/5
:= 5 + (5 × 5 × 55 + ((55 + 5)/5))
:= ((66 × ((66 − 6) + 66)) + 6)/6
:= 66 + ((6 × (6 × 6 × 6 + 6)) − 6)
:= ((77 − 7)/7) + (7 × (7 + 7) × (7 + 7))
:= 7/7 + (77 × (77/7 + 7))
:= 7 + ((77 × (77/7 + 7)) − 7/7)
:= (8 − 88)/8 + ((8 + 8) × (88 − 8/8))
:= (88/8 + 8) × ((8/8 + 8 × 8) + 8)
:= (8 + 8) × (88 − 8/8)
:= 9 × (9 × (9 + 9) − 9) + ((9 × 9 + 9)/(9 + 9))
:= 9 + (9 × (9 × (9 + 9) − 9) + 9/9)
:= 9 × 99 + ((((9 + 9)/9)9 ) − (99/9))
66/6
:= (((6 + 6)/6)
) − 666
102
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1393 := 1 + ((1 + 11) × (1 + 1 + 1 + 1 + 1 + 111))
Ï 1398 := 1 + (11 × (111 + (1 + 1)1+1+1+1 ))
Ï 1403 := (1 + 1 + 11) × (111 − 1 − 1 − 1) − 1
:= 2 + ((((2 + 2 + 2) + 2/2) ) + 22)
:= 2 + (2 × (((22 + 2 + 2) ) + 22))
:= 22 × 22+2+2 − (2/2 + 2 + 2)
:= (((3 + 3) × (33+3 − 33)) + 3)/3
:= 3 + ((3 × 3 + 3)3 − 333)
:= (3 × (33 − 3)) + ((33/3)3 )
:= 4/4 + (4 × ((4 + 4) × 44 − 4))
:= (4 − 44)/4 + 4 × (4 + 4) × 44
:= 4 × (4 + 4) × 44 − (4/4 + 4)
:= 5 × (5 × 55 + 5) − ((5 + 5)/5 + 5)
:= 5 × (5 × 55 + 5) − (5 + 5)/5
:= 5 + (5 × (5 × 55 + 5) − ((5 + 5)/5))
:= 666 + ((66 × 66 + 6)/6)
:= 66 + (6 × (6 × 6 × 6 + 6))
:= (6 × ((6 × 6 × 6 + 6 + 6) + 6)) − 6/6
:= 7 + (77 × (77/7 + 7))
:= ((7 + 7)/7) × (777 − (7/7 + 77))
:= (((7 + 7 + 7)/7)7 ) − (777 + 7)
:= 8/8 + ((8 + 8) × (88 − 8/8))
:= (8 − 88)/8 + 88 × (8 + 8)
:= 88/8 + ((8 + 8) × (88 − 8/8))
:= 9 + ((9 × (9 × (9 + 9) − 9) − ((9 + 9)/9)) + 9)
:= 9 + (9 × (9 × (9 + 9) − 9) + ((99 + 9)/9))
:= 9 × 99 + (((9 + 9)/9)9 )
2
2
2
Ï 1394 := (1 + 1 + (1 + 111)1+1 )/(11 − 1 − 1)
Ï 1399 := 1111 + ((1 + 1) × (1 + 11)1+1 )
:= 2 × ((22/2)2 + ((22 + 2)2 ))
:= (2222 + ((22 + 2)2 ))/2
3
3
:= 3 + ((((33/3) ) + 33) + 3 )
Ï 1404 := (1 + 1 + 11) × (111 − 1 − 1 − 1)
:= 2 × ((2 × (22 × 22+2 )) − 2)
3
:= 3 + (((3 × 3 + 3) − 333) + 3/3)
:= 3 × ((3 + 3) × (3 × 33 − 3))
:= 4 + ((44 × (4 − 4)/(4 + 4)) + 4)
:= 4 × (4 + 4) × 44 − ((4/4 + 4) + 4)
:= 4 × (4 + 4) × 44 − 4
:= 5 × (5 × 55 + 5) − (5/5 + 5)
:= 5 × (5 × 55 + 5) − 5/5
:= 5 + (5 × (5 × 55 + 5) − 5/5)
:= 666 + (((6 × 6/(6 + 6)) ) − 6/6)
:= 66 + ((6 × (6 × 6 × 6 + 6)) + 6/6)
:= 6 × ((6 × 6 × 6 + 6 + 6) + 6)
:= 7 + ((77 × (77/7 + 7)) + 7/7)
:= (((7 + 7) × (777 − 77)) − 7)/7
:= (77/7 + 7) × (7/7 + 77)
:= (8 + 8)/8 + ((8 + 8) × (88 − 8/8))
:= 88 × (8 + 8) − (8/8 + 8)
:= 88 × (8 + 8) − (8/((8 + 8)/8))
4
6
9
:= 9 × 99 + ((((9 + 9)/9) ) − 9)
Ï 1395 := (11 − 1 − 1) × (11 + (1 + 11)1+1 )
:= 2
2+2+2
+ (22/2)
:= 999 + ((99/9 + 9)
(9+9)/9
)
:= 9 + ((9 × (9 × (9 + 9) − 9) + 9) + 9)
Ï 1400 := (1 + (1 + 1 + 11)) × (11 − 1)1+1
Ï 1405 := 1 + ((1 + 1 + 11) × (111 − 1 − 1 − 1))
:= 2 × (2 × ((22 × 22+2 ) − 2))
2/2+2
:= 22 × 22+2+2 − 2/2 − 2
3
:= (3 × 3 + 3)3 − 333
:= 3 + ((((33/3) ) + 33) + 33)
:= 3/3 + (3 × ((3 + 3) × (3 × 33 − 3)))
:= 4 + ((4 × ((4 + 4) × 44 − 4)) − 4/4)
:= (4 + 4) × (4 × 44 − 4/4)
:= 4/4 + (4 × (4 + 4) × 44 − 4)
:= 5 × (5 × 55 + 5) − 5
:= 5 × (5 × 55 + 5)
:= 5 + 5 × (5 × 55 + 5)
:= 666 + ((6 × 6/(6 + 6)) )
:= 66 + ((6 × (6 × 6 × 6 + 6)) + ((6 + 6)/6))
:= 6 × 6 + ((6 × 6 + 6/6)(6+6)/6 )
:= 7 + ((77 × (77/7 + 7)) + ((7 + 7)/7))
:= 7 + ((77 × (77/7 + 7)) + 7)
:= 7 × (7 × 7 − 7) + 7777/7
:= 8 × 8 + ((88/8)88/8−8 )
:= 88 × (8 + 8) − 8
:= 8 + (88 × (8 + 8) − (88/8))
:= (99/9 + 9) × (9 × 9 − 99/9)
:= 9 + (((9 × (9 × (9 + 9) − 9) + 9/9) + 9) + 9)
6
:= 9 + (9 × (9 × (9 + 9) − 9) + 9)
Ï 1396 := 1 + ((11 − 1 − 1) × (11 + (1 + 11)1+1 ))
2
:= 2 × (((22 + 2 + 2) ) + 22)
:= 3/3 + ((3 × 3 + 3)3 − 333)
:= 4 + (4 × ((4 + 4) × 44 − 4))
:= 5/5 + (5 × (5 × 55 + 5) − 5)
:= ((6 + 6)/6)6 + (6 × (6 × 6 × 6 + 6))
Ï 1401 := 111+1+1 + (11 − 1 − 1)1+1 − 11
Ï 1406 := ((1 + 1 + 11) × (111 − 1 − 1)) − 11
:= 2 + ((2222 + ((22 + 2)2 ))/2)
:= 22 × 22+2+2 − 2
3
:= (3 × ((3 + 3) × (3 × 3 − 3))) − 3
:= 3 + ((3 × (33 − 3)) + ((33/3)3 ))
:= 4 + (4 × (4 + 4) × 44 − 44/4)
:= 4 × (4 + 4) × 44 − (4 + 4)/4
:= 5/5 + 5 × (5 × 55 + 5)
:= 5 + (5 × (5 × 55 + 5) + 5/5)
:= 6 + (((6 × 6/(6 + 6))6 ) + 666)
:= 6 × 6 × 6 × 6 + (666 − 6)/6
:= ((77 − 7)/7) + (77 × (77/7 + 7))
:= (77/7 × ((7 + 7)/7)7 ) − 7
:= ((77 × ((7 + 7)/7)7 ) − (7 + 7))/7
:= 88 × (8 + 8) − (88 + 8)/8
:= 8/8 + (88 × (8 + 8) − 8)
:= 88 × (8 + 8) − (8 + 8)/8
:= 9 + ((9 × (9 × (9 + 9) − 9) + 9/9) + 9)
:= 9 × 9 + 99/9 × (999/9 + 9)
:= 9 + ((9 × (9 × (9 + 9) − 9) + (99/9)) + 9)
Ï 1397 := 11 × (111 + (1 + 1)1+1+1+1 )
Ï 1402 := (1 + 1 + 11) × (111 − 1 − 1 − 1) − 1 − 1
2+2
:= 22/2 × ((22×(2+2) − 2)/2)
:= 2 + (2 × (2 × ((22 × 2
:= 33 + (((33/3)3 ) + 33)
:= ((33/3) ) + (((3 + 3) − 3)/3)
3
) − 2)))
:= 22 × 22+2+2 − 2/2
3
:= 4 × (4 + 4) × 44 − 44/4
:= 4 × 4 + (44 × (44 − 4)/(4 + 4))
:= (5 + 5)/5 + (5 × (5 × 55 + 5) − 5)
:= (5 + 5)/5 + 5 × (5 × 55 + 5)
Ï 1407 := 111 + ((1 + 1 + 1) × (1 + 11))1+1
:= 3 + (3 × ((3 + 3) × (3 × 33 − 3)))
:= 4 × (4 + 4) × 44 − 4/4
6
:= 5 × 5 × 55 + ((5 + 5)/5)5
:= 66 + ((66/6)6×6/(6+6) )
:= 6 + ((6 × (6 × 6 × 6 + 6)) + ((6 + 6)/6) )
:= 666/6 + 6 × 6 × 6 × 6
:= 77/7 + (77 × (77/7 + 7))
:= (((77 × ((7 + 7)/7)7 ) + 7)/7) − 7
:= (7 × ((7 + 7) × (7 + 7) + 7)) − (7 + 7)
:= 88 × (8 + 8) − 88/8
:= (8 + 8)/8 + (88 × (8 + 8) − 8)
:= 88 × (8 + 8) − 8/8
:= 9 + (9 × (9 × (9 + 9) − 9) + (99/9))
9
:= 9 × 99 + ((((9 + 9)/9) ) − 9/9)
103
:= 999/9 + ((9 + 9) × (9 × 9 − 9))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1408 := 11 × ((1 + 1)1+(1+1)×(1+1+1) )
Ï 1413 := 1 + (111+1+1 + (11 − 1 − 1)1+1 )
Ï 1418 := 1 + ((1 + 1 + 11) × (111 − 1 − 1))
:= 2/2 + (2 × (222 + 22 ))
:= 2 + (2 × ((222 + 222 ) + 2))
:= 33/3 × (((3 − 3/3 + 3)3 ) + 3)
:= 3 × (((3 + 3) × (3 × 33 − 3)) + 3)
:= 3 + ((((33/3)3 ) + 3 × 33 ) + 3)
:= 4 × (4 + 4) × 44
:= 4 + (4 × (4 + 4) × 44 + 4/4)
:= (44 − 4)/4 + 4 × (4 + 4) × 44
:= ((5 + 5)/5)5 × (55 − (55/5))
:= 5 + (((5 + 5)/5)5 × (55 − (55/5)))
:= 55 + (5 × 5 × 55 − ((55 + 5)/5))
:= 66/6 × (((6 + 6)/6)6/6+6)
:= 6 + (666/6 + 6 × 6 × 6 × 6)
:= 6 × 6 × 6 × 6 + ((666 + 66)/6)
:= 77/7 × ((7 + 7)/7)7
:= (7 × ((7 + 7) × (7 + 7) + 7)) − (7/7 + 7)
:= 7 + (((77/7 + 7) × (7/7 + 77)) + 7)
:= 88 × (8 + 8)
:= 8 + ((88 × (8 + 8) − (88/8)) + 8)
:= 8 + (88 × (8 + 8) + ((8 + 8)/8))
:= 99/9 × (((99/9 + 99) + 9) + 9)
:= ((9 + 9) × (9 × 9 − ((9 + 9)/9))) − 9
:= ((9 + 9)/9) × (9 × 9 × 9 − (99/9 + 9))
:= 22 × 2
2
2+2+2
Ï 1409 := 1 + 11 × ((1 + 1)1+(1+1)×(1+1+1) )
Ï 1414 := (1 + (1 + 1 + 11)) × (1 + (11 − 1)1+1 )
2
Ï 1419 := 11 × (11 × (1 + 11) − 1 − 1 − 1)
:= 2/2 + 22 × 22+2+2
:= 2 + (2 × (222 + 22 ))
:= 22/2 + 22 × 22+2+2
:= 3 × 33 + (((33/3)3 ) − 3)
:= 3 + ((((33/3)3 ) − 3/3) + 3 × 33 )
:= 33 + (33 × (3 × 3 + 33))
:= 4/4 + 4 × (4 + 4) × 44
:= 4 + (4 × (4 + 4) × 44 + (4 + 4)/4)
:= 44/4 + 4 × (4 + 4) × 44
:= 5 + ((5 × (5 × 55 + 5) − 5/5) + 5)
:= (5 × ((5 × 55 + 5) + 5)) − 55/5
:= 55 + (5 × 5 × 55 − (55/5))
:= 6 + ((6 × ((6 × 6 × 6 + 6 + 6) + 6)) − 6/6)
:= 6 + ((666 + 6)/6 + 6 × 6 × 6 × 6)
:= 6 + ((666/6 + 6 × 6 × 6 × 6) + 6)
:= ((77 × ((7 + 7)/7)7 ) + 7)/7
:= (7 × ((7 + 7) × (7 + 7) + 7)) − 7
:= 77/7 × (((7 + 7)/7)7 + 7/7)
:= 8/8 + 88 × (8 + 8)
:= 8 + (88 × (8 + 8) − ((8 + 8)/8))
:= 88/8 + 88 × (8 + 8)
:= 9 + ((99/9 + 9) × (9 × 9 − (99/9)))
:= 9/9 + (((9 + 9) × (9 × 9 − ((9 + 9)/9))) − 9)
:= 99/9 × ((999/9 + 9) + 9)
Ï 1410 := (11 − 1) × ((1 + 11)1+1 − 1 − 1 − 1)
Ï 1415 := ((1 + 1 + 11) × (111 − 1 − 1)) − 1 − 1
Ï 1420 := (11 − 1) × ((1 + 11)1+1 − (1 + 1))
:= 2 + 22 × 22+2+2
:= 2 + ((2 × (222 + 222 )) + 2/2)
:= 2 × (((222 + 222 ) + 2) + 2)
:= 3 + ((3 × ((3 + 3) × (3 × 33 − 3))) + 3)
:= 3 + (((33/3)3 ) + 3 × 33 )
:= 3/3 + ((33 × (3 × 3 + 33)) + 33)
:= (4 + 4)/4 + 4 × (4 + 4) × 44
:= 4 + ((4 × (4 + 4) × 44 − 4/4) + 4)
:= (4 × ((4 + 4) × 44 + 4)) − 4
:= 5 + (5 × (5 × 55 + 5) + 5)
:= 5 + ((5 × (5 × 55 + 5) + 5) + 5)
:= (5 × ((5 × 55 + 5) + 5)) − 5
:= 6 + (6 × ((6 × 6 × 6 + 6 + 6) + 6))
:= 66/6 + (6 × ((6 × 6 × 6 + 6 + 6) + 6))
:= 6 × 66 + (((6 + 6)/6)(66−6)/6)
:= (((7 + 7 + 7)/7)7 ) − 777
:= 7 + (77/7 × ((7 + 7)/7)7 )
:= (7 × ((7 + 7) × (7 + 7) + 7)) − 7/7
:= (8 + 8)/8 + 88 × (8 + 8)
:= 8 + (88 × (8 + 8) − 8/8)
:= ((88 + 8)/8) + 88 × (8 + 8)
:= ((99/9) × ((999/9 + 9) + 9)) − 9
:= 9 + (((9 × (9 × (9 + 9) − 9) + (99/9)) + 9) + 9)
:= (99/9 + 9) × (9 × 9 − (9/9 + 9))
Ï 1411 := 1 + (11 − 1) × ((1 + 11)1+1 − 1 − 1 − 1)
Ï 1416 := (1 + 11) × (111+1 − (1 + 1 + 1))
2
:= 2 + (22 × 22+2+2 + 2/2)
Ï 1421 := 1 + (11 − 1) × ((1 + 11)1+1 − 1 − 1)
:= 2 + (22 × 22+2+2 + 22/2)
:= 2 × ((222 + 22 ) + 2)
3
:= (3 × (33 + 3)) + ((33/3)3 )
:= 3 × 33 + (((33/3)3 ) − 3/3)
:= 3 + ((33 × (3 × 3 + 33)) + 3 )
:= 4 + (4 × (4 + 4) × 44 − 4/4)
:= 4 + (4 × (4 + 4) × 44 + 4)
:= 4/4 + ((4 × ((4 + 4) × 44 + 4)) − 4)
:= 55/5 + 5 × (5 × 55 + 5)
:= 5 + (5 × (5 × 55 + 5) + (55/5))
:= 5/5 + ((5 × ((5 × 55 + 5) + 5)) − 5)
:= 6 + ((6 × ((6 × 6 × 6 + 6 + 6) + 6)) + 6/6)
:= (6 + 6) × ((666 + 6)/6 + 6)
:= (6/6 + 6) × (6 × 6 × 6 − (6/6 + 6 + 6))
7
:= 7 + ((77/7 + 7) × (7/7 + 77))
:= 7 + (((77 × ((7 + 7)/7) ) + 7)/7)
:= 7 × ((7 + 7) × (7 + 7) + 7)
:= 88/8 + (88 × (8 + 8) − 8)
:= 8 + 88 × (8 + 8)
:= 88 × (8 + 8) + (88 + 8 + 8)/8
:= ((9 − 9/9) + 9) × (((9 + 9)/9) + 9 × 9)
:= 9 + (((9 + 9) × (9 × 9 − 9)) + 999/9)
:= 9 + (((((9 + 9)/9)9 ) + 9 × 99) + 9)
Ï 1412 := 111+1+1 + (11 − 1 − 1)1+1
Ï 1417 := (1 + 1 + 11) × (111 − 1 − 1)
2
2
:= (22/2) + (2 + 2 + 2)
:= 2 × (222 + 22 )
3
Ï 1422 := 1 + 1 + (11 − 1) × ((1 + 11)1+1 − (1 + 1))
2+2
:= ((2 + 2 + 2)2 + 2)2 − 22
3
3
:= 3 × 3 + ((33/3) )
:= 333 + (3333/3 − 3 )
:= 4 + 4 × (4 + 4) × 44
:= 4 + ((4 × (4 + 4) × 44 + 4/4) + 4)
5
:= 333 + 33 × 33
5
:= (4 × ((4 + 4) × 44 + 4)) − (4 + 4)/4
:= 5 + (5 × 5 × 55 + ((5 + 5)/5) )
:= 5 + ((5 × 5 × 55 + ((5 + 5)/5) ) + 5)
:= (5 + 5)/5 + ((5 × ((5 × 55 + 5) + 5)) − 5)
:= 6 + (((666 − 6)/6) + 6 × 6 × 6 × 6)
:= 6/6 + ((6 + 6) × ((666 + 6)/6 + 6))
:= 6 + ((6 + 6) × ((666 + 6)/6 + 6))
7
:= 7 + (7777/7 + 7 × (7 × 7 − 7))
:= 7 + ((((7 + 7 + 7)/7) ) − 777)
:= 7/7 + (7 × ((7 + 7) × (7 + 7) + 7))
:= 88 × (8 + 8) + (8/((8 + 8)/8))
:= 8 + (88 × (8 + 8) + 8/8)
:= 8 + ((88 × (8 + 8) − ((8 + 8)/8)) + 8)
:= 9 × 9 × (9 + 9) − ((9 × 9 × 9 + 9)/(9 + 9))
:= (9 + 9) × (9 × 9 − ((9 + 9)/9))
9
:= 9 + ((((9 + 9)/9) ) + 9 × 99)
104
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1423 := (1 + 1)11 − (1 + (1 + 1) × (1 + 11))1+1
Ï 1428 := (1 + 11) × (111+1 − 1 − 1)
Ï 1433 := 1 + 111 × (1 + 1 + 11) − 11
:= 2/2 + (((2 + 2 + 2) + 2) − 22)
:= 2 × ((2 + 2 + 2) × ((22/2) − 2))
:= ((2 + 2 + 2)2 + 2)2 − 22/2
:= 3/3 + (33 × 33 + 333)
:= (33/3 + 3) × (3 × 33 + 3)
:= 3 + (((33/3)3 ) + 3 × 33)
:= 4 + (4 × ((4 + 4) × 44 + 4))
:= 4 + (((4 × ((4 + 4) × 44 + 4)) + 4/4) + 4)
:= (((5 + 5)/5)55/5 ) − 55 /5
:= 55 + (5 × 5 × 55 − ((5 + 5)/5))
:= 5 + ((5 × 5 × 55 − ((5 + 5)/5)) + 55)
:= 6 + (((6 + 6) × ((666 + 6)/6 + 6)) + 6/6)
:= 66 + (6 × 6 × 6 × 6 + 66)
:= 66 + ((6 × (6 × 6 × 6 + 6 + 6)) − 6/6)
:= ((7 + 7)/7) + (7 × ((7 + 7) × (7 + 7) + 7))
:= 7 + (7 × ((7 + 7) × (7 + 7) + 7))
:= (77 + 7)/7 + (7 × ((7 + 7) × (7 + 7) + 7))
:= 8 + ((88 × (8 + 8) − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 88 × (8 + 8))
:= 8 + (((88 × (8 + 8) + 8/8) + 8) + 8)
:= 9/9 + ((9 + 9) × (9 × 9 − ((9 + 9)/9)))
:= (9 × (9 × (9 + 9) + 9)) − 999/9
:= (9 + 9)/9 + (9 × 9 × (9 + 9) − (9 + 9 + 9))
2
2
4
:= (4 × (4 + 4)) − (4/4 + 4)
4
Ï 1424 := (1 + 1)11−1 + ((1 + 1) × (11 − 1))1+1
2
Ï 1429 := ((1 + 1 + 11) × (111 − 1)) − 1
2+2+2
:= 22 + (2 × (222 + 222 ))
:= 22 + (22 × 2
:= 3 + ((3 × (33 + 3)) + ((33/3)3 ))
:= 3 × 33 + (((33/3)3 ) − 3/3)
:= 3 + (3 × (3 × ((3 + 3) × 33 − 3)))
:= 4 × ((4 + 4) × 44 + 4)
:= 4 + ((4 × ((4 + 4) × 44 + 4)) + 4/4)
:= 4 + ((44 × (44 − 4)/(4 + 4)) + 44)
:= (5 × ((5 × 55 + 5) + 5)) − 5/5
:= 55 + (5 × 5 × 55 − 5/5)
:= 5 + ((5 × 5 × 55 − 5/5) + 55)
:= 66 + ((6 × 6 × 6 × 6 + 66) + 6/6)
:= 66 + (6 × (6 × 6 × 6 + 6 + 6))
2
6/6+6)
:= 6 × 6 × 6 × 6 + (((6 + 6)/6)
:= 7 + (((((7 + 7 + 7)/7)7 ) − 777) + 7)
:= 8 + (88 × (8 + 8) + 8)
:= (9 + 9)/9 + ((9 + 9) × (9 × 9 − ((9 + 9)/9)))
Ï 1425 := ((1 + 11) × (111+1 − (1 + 1))) − 1 − 1 − 1
2
:= (2/2 + 2) × ((22 − 22/2) + 2)
:= 3 + (33 × 33 + 333)
− 2/2)
Ï 1434 := 1 + 1 + 111 × (1 + 1 + 11) − 11
:= 2 + (((2 + 2 + 2) + 2) − 22)
2
:= 7 + ((7 × ((7 + 7) × (7 + 7) + 7)) + 7/7)
:= (7 × (((7 + 7)/7)7 + 77)) − 7/7
:= 8 + ((88 + 8 + 8)/8 + 88 × (8 + 8))
:= 8 + (((88 × (8 + 8) + ((8 + 8)/8)) + 8) + 8)
:= 9 × 9 × (9 + 9) − (99/9 + 9 + 9)
:= ((9 + 9)/9) × (9 × 9 × 9 − ((99 + 9)/9))
Ï 1430 := (1 + 1 + 11) × (111 − 1)
Ï 1435 := 1 + 1 + 1 + (111 × (1 + 1 + 11) − 11)
:= 22 + 22 × 22+2+2
:= 2 + (((2 + 2 + 2)2 + 2)2 − 22/2)
3
:= 3 × 33 + ((33/3) )
:= 3 + ((3 × (3 × ((3 + 3) × 33 − 3))) + 3/3)
4
:= 4/4 + (4 × ((4 + 4) × 44 + 4))
:= 44 + (44 × (4 − 4)/(4 + 4))
:= 44/4 + (4 × ((4 + 4) × 44 + 4))
:= 5 × ((5 × 55 + 5) + 5)
:= 55 + 5 × 5 × 55
:= 5 + (5 × 5 × 55 + 55)
6
:= (6 − 6/6) × (6 × 66 − 666/6)
:= 66/6 × (((6 + 6)/6) + 66)
:= 66 + ((6 × 6 + 6/6)(6+6)/6 )
:= 77/7 + ((7 × ((7 + 7) × (7 + 7) + 7)) − 7)
:= (7 − 7/7 + 7) × (777 − 7)/7
:= 7 × (((7 + 7)/7)7 + 77)
:= 8 + ((88 × (8 + 8) + 8/8) + 8)
:= ((8 + 8)/8) × (88/8 + 8 × 88)
:= 8 + ((88 × (8 + 8) + 88/8) + 8)
:= 9 × 9 × (9 + 9) − (99/((9 + 9 + 9)/9))
Ï 1426 := ((1 + 11) × (111+1 − (1 + 1))) − 1 − 1
:= (22 + 2/2) × (22+2+2 − 2)
3+3
:= 99 + (99/9)
(9+9+9)/9
:= (9 + 9) × (9 + 9) + 9999/9
Ï 1431 := 1 + (1 + 1 + 11) × (111 − 1)
:= 22 + (22 × 22+2+2 + 2/2)
3
Ï 1436 := (1 + 1) × ((11 − 1 − 1)1+1+1 − 11)
:= 2 × (2 × (((22 − (2/2 + 2))2 ) − 2))
:= 3 × (3 × ((3 + 3) × 3 − 3))
:= 3 + ((((33/3)3 ) + 3 × 33) + 3)
:= (4 + 4)/4 + (4 × ((4 + 4) × 44 + 4))
:= 44 + ((4444 + 44 )/4)
:= (4 + 4) × (4 × 44 + 4) − 4
:= 5/5 + (5 × ((5 × 55 + 5) + 5))
:= 55 + (5 × 5 × 55 + 5/5)
:= ((((3 + 3) × 3
) + 3)/3) − 33
6
:= 5 + ((5 × 5 × 55 + 5/5) + 55)
:= 66 + (((6 + 6)/6) + 6 × 6 × 6 × 6)
:= 6 × 6 + (((6 × 6/(6 + 6)) ) + 666)
:= 6 + ((66/6) × (((6 + 6)/6)6 + 66))
:= 7 + (77/7 × (((7 + 7)/7)7 + 7/7))
:= ((77 − 7)/7) + (7 × ((7 + 7) × (7 + 7) + 7))
:= 7/7 + (7 × (((7 + 7)/7)7 + 77))
:= 8 + ((88 × (8 + 8) + ((8 + 8)/8)) + 8)
:= 8 + (((88 × (8 + 8) − 8/8) + 8) + 8)
:= 8 + ((((88 + 8)/8) + 88 × (8 + 8)) + 8)
:= (9 + 9) × (9 + 9) + (9999/9 − 9)
:= 9 × 9 × (9 + 9) − (9 + 9 + 9)
:= ((9 + 9)/9) × (9 × 9 × 9 − (99/9))
Ï 1427 := ((1 + 11) × (111+1 − (1 + 1))) − 1
6
Ï 1432 := 111 × (1 + 1 + 11) − 11
Ï 1437 := ((11 − 1) × (1 + 11)1+1 ) − 1 − 1 − 1
:= ((2/2 + 2) × (222 − 2/2)) − 22
:= 2 + (22 × 22+2+2 + 22)
:= (2/2 + 2) × (222 − (2/2 + 2 + 2))
:= 3 × 33 + (((33/3)3 ) − 3)
:= 3/3 + (3 × (3 × ((3 + 3) × 33 − 3)))
:= 3 × (((3 − 3/3)3×3 ) − 33)
:= 4 + ((44 × (4 + 4)) − (4/4 + 4)4 )
:= 4 + ((4 × ((4 + 4) × 44 + 4)) + 4)
:= 4/4 + ((4 + 4) × (4 × 44 + 4) − 4)
:= (5 + 5)/5 + (5 × ((5 × 55 + 5) + 5))
:= 55 + (5 × 5 × 55 + ((5 + 5)/5))
:= 5 + ((5 × 5 × 55 + ((5 + 5)/5)) + 55)
6
:= 66 + ((6 × 6 × 6 × 6 − 6/6) + 66)
:= ((6 + 6)/6) + (6 × (6 × 6 × 6 + 6 + 6))
:= ((6/6 + 6 + 6) × 666/6) − 6
:= 7 + ((7 × ((7 + 7) × (7 + 7) + 7)) − 7/7)
:= 77/7 + (7 × ((7 + 7) × (7 + 7) + 7))
:= ((7 × 7 − 77/7)(7+7)/7 ) − 7
:= 8 + (88 × (8 + 8) + 88/8)
:= 8 + ((88 × (8 + 8) + 8) + 8)
:= 8 × 8 × (8 + 8 + 8) − (88/8 + 88)
:= 9 × 9 × (9 + 9) − (((99 + 99)/9) + 9)
:= 9/9 + (9 × 9 × (9 + 9) − (9 + 9 + 9))
:= 9 × 9 × (9 + 9) − (((99 + 9)/9) + 9)
105
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1438 := ((11 − 1) × (1 + 11)1+1 ) − 1 − 1
2
2
:= ((2 + 2 + 2) + 2) − (2 + 2 + 2)
Ï 1448 := 11 × (11 × (1 + 11)) − 1 − 1 − 1 − 1
Ï 1443 := 111 × (1 + 1 + 11)
:= 2 + (((2 + 2 + 2)2 + 2)2 + 2)
:= (22/2 + 2) × 222/2
3
:= 3/3 + (3 × (((3 − 3/3)3×3 ) − 33))
:= 3 + ((3 + 3) × (3 × 3 × 3 − 3))
:= (3/3 + 3) × ((33 × 33 − 3)/3)
:= (4 + 4) × (4 × 44 + 4) − (4 + 4)/4
:= 4 + ((4 + 4) × (4 × 44 + 4) − 4/4)
:= 4 + ((4 + 4) × (4 × 44 + 4) + 4)
:= 5 × 5 × 55 + (55 /5 + 5)/(5 + 5)
:= 555/5 × (55 + 5 + 5)/5
:= 5 + (555/5 × (55 + 5 + 5)/5)
:= ((((6 + 6)/6) + 6 × 6)(6+6)/6 ) − 6
:= (6/6 + 6 + 6) × 666/6
:= 6 × 6 × (6 × 6 + 6) − ((6 + 6)/6)6
:= 77 + ((7 × (7 + 7) × (7 + 7)) − (77/7))
:= (7 − 7/7 + 7) × 777/7
:= 77 + ((7 × (7 + 7) × (7 + 7)) − 7/7)
:= 8 + ((88 + 88)/8 + 88 × (8 + 8))
:= 888/8 × (88 + 8 + 8)/8
:= 8 × 8 × (8 + 8 + 8) − 88
:= 9 × 9 × (9 + 9) − (99/9 + 9)
:= 999/9 × ((99 + 9 + 9)/9)
:= 9 × 9 × (9 + 9) − 9/9 − 9
Ï 1439 := ((11 − 1) × (1 + 11)1+1 ) − 1
Ï 1444 := 1 + 111 × (1 + 1 + 11)
Ï 1449 := (1 + 1 + 1) × ((11 + 11)1+1 − 1)
:= ((2 + 2 + 2)2 + 2)2 − (2/2 + 2 + 2)
:= ((2 + 2 + 2)2 + 2)2
:= (2/2 + 2) × (222 − 2/2)
:= (3 × (33 + 3)) + ((33/3)3 )
:= 333 + 3333/3
:= 3 × ((3 × (3 + 3) × 33 ) − 3)
:= 4 + (4 + 4) × (4 × 44 + 4)
:= (4 − 4/4) × ((44 × 44 − 4)/4)
:= (5 × (5 × (55 + 5))) − (55 + 5/5)
:= (55 − 5)/5 + (55 × (5 + 5 + 5))
:= (4 + 4) × (4 × 44 + 4) − 4/4
:= ((55 + 5) × (5 × 5 − 5/5)) − 5/5
(6+6)/6
:= (6 × (6 × (6 × 6 + 6) − (6 + 6))) − 6/6
:= (((6 + 6)/6) + 6 × 6)
:= 7 + ((7 × ((7 + 7) × (7 + 7) + 7)) + (77/7))
:= (7 × 7 − 77/7)(7+7)/7
:= 77 + (7 × (7 + 7) × (7 + 7))
:= ((8 + 8) × ((8 + 8)/8 + 88)) − 8/8
:= 888 + ((8888 + 8)/(8 + 8))
:= 8/8 + (8 × 8 × (8 + 8 + 8) − 88)
:= 9 × 9 × (9 + 9) − (9/9 + 9 + 9)
Ï 1440 := (11 − 1) × (1 + 11)1+1
:= ((99/9 + 9 + 9) + 9)
(9+9)/9
Ï 1445 := 1 + 1 + 111 × (1 + 1 + 11)
:= 6 + ((6/6 + 6 + 6) × 666/6)
:= 9 × 9 × (9 + 9) − 9
Ï 1450 := (11 − 1) × (1 + (1 + 11)1+1 )
:= 2 × ((22 − 2) × (2 + 2 + 2)2 )
:= 2/2 + ((2 + 2 + 2)2 + 2)2
:= ((2/2 + 2) × 222 ) − 2
:= (3 + 3) × (3 × 3 × 33 − 3)
:= 3 + (333/3 + ((33/3)3 ))
:= 3 + ((((3 + 3) × 33+3 ) − 33)/3)
:= (4 + 4) × (4 × 44 + 4)
:= 4 + ((4 + 4) × (4 × 44 + 4) + 4/4)
:= 44 + (4 × (4 + 4) × 44 − (4 + 4)/4)
:= (55 + 5) × (5 × 5 − 5/5)
:= (5 × (5 × (55 + 5))) − 55
:= 5 × (((5 × 55 + 5) + 5) + 5)
:= 6 × (6 × (6 × 6 + 6) − (6 + 6))
:= 6 × 6 × (6 × 6 + 6) − (66 + 6/6)
:= 6 + ((((6 + 6)/6) + 6 × 6)(6+6)/6 )
:= 7 × 7 × 7 + (7777/7 − (7 + 7))
:= 7/7 + ((7 × 7 − 77/7)(7+7)/7 )
:= 7 + ((7 − 7/7 + 7) × 777/7)
:= (8 + 8) × ((8 + 8)/8 + 88)
:= 88 × (8 + 8) + 888/(8 + 8 + 8)
:= (8 + 8)/8 + (8 × 8 × (8 + 8 + 8) − 88)
:= (9 + 9) × (9 × 9 − 9/9)
:= 9 × 9 × (9 + 9) − ((99 + 9 + 9)/9)
:= 9/9 + (9 × 9 × (9 + 9) − 9)
Ï 1441 := 11 × ((11 × (1 + 11)) − 1)
2
2
Ï 1446 := 1 + 1 + 1 + 111 × (1 + 1 + 11)
2
2
Ï 1451 := 11 × 11 × (1 + 11) − 1
:= ((2/2 + 2) × 222 ) − 2/2
:= ((2 + 2 + 2) + 2) − 2/2 − 2
:= 2 + ((2 + 2 + 2) + 2)
:= 3/3 + ((3 + 3) × (3 × 3 × 33 − 3))
:= (3 + 3) × (((33+3 + 3)/3) − 3)
:= (((3 + 3) × (33+3 − 3)) − 3)/3
:= 4/4 + (4 + 4) × (4 × 44 + 4)
:= 4 + ((4 + 4) × (4 × 44 + 4) + (4 + 4)/4)
:= 44 + (4 × (4 + 4) × 44 − 4/4)
:= 55 + (5 × 5 × 55 + (55/5))
:= 5/5 + ((5 × (5 × (55 + 5))) − 55)
:= 5/5 + (5 × (((5 × 55 + 5) + 5) + 5))
:= 6/6 + (6 × (6 × (6 × 6 + 6) − (6 + 6)))
:= 6 × 6 × (6 × 6 + 6) − 66
:= ((66 × (66 + 66)) − 6)/6
:= 7 + ((7 × (((7 + 7)/7) + 77)) − 7/7)
:= 7 × 7 × 7 + ((7777 − 7)/7 − 7)
:= 7 + ((7 × 7 − 77/7)(7+7)/7 )
:= 8/8 + ((8 + 8) × ((8 + 8)/8 + 88))
:= ((8 + 8)/8) × ((88/8 + 8 × 88) + 8)
:= 8 + (888/8 × (88 + 8 + 8)/8)
:= 9/9 + ((9 + 9) × (9 × 9 − 9/9))
:= 9 × 9 × (9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 × 9 × (9 + 9) − 9)
7
Ï 1442 := 111 + 111+1+1
Ï 1447 := 1 + 1 + 1 + 1 + 111 × (1 + 1 + 11)
:= ((2 + 2 + 2)2 + 2)2 − 2
2
2
Ï 1452 := 11 × 11 × (1 + 11)
:= 2 + (((2 + 2 + 2) + 2) + 2/2)
:= (2/2 + 2) × 222
:= 333/3 + ((33/3) )
:= (((3 + 3) × 33+3 ) − 33)/3
:= 33 × (33/3 + 33)
:= (4 + 4)/4 + (4 + 4) × (4 × 44 + 4)
:= 4 + (((4 + 4) × (4 × 44 + 4) − 4/4) + 4)
:= 44 + 4 × (4 + 4) × 44
:= 55 + (5 × 5 × 55 + ((55 + 5)/5))
:= (5 + 5)/5 + ((5 × (5 × (55 + 5))) − 55)
:= (5 + 5)/5 + (5 × (((5 × 55 + 5) + 5) + 5))
:= ((666 × (6/6 + 6 + 6)) − 6)/6
:= 6/6 + (6 × 6 × (6 × 6 + 6) − 66)
:= 66 × ((66 + 66)/6)
:= 7 + (7 × (((7 + 7)/7) + 77))
:= 7 × 7 × 7 + (7777/7 − 7)
:= (77 + 7)/7 × (((7 + 7)/7)7 − 7)
:= (8 + 8)/8 + ((8 + 8) × ((8 + 8)/8 + 88))
:= 8 × 8 × (8 + 8 + 8) − (8/8 + 88)
:= 88 × (8 + 8) + (88/((8 + 8)/8))
:= (9 + 9)/9 + ((9 + 9) × (9 × 9 − 9/9))
:= 9 × 9 × (9 + 9) − 99/9
:= 9 × 9 × (9 + 9) + (((9 + 9 + 9)/9) − 9)
3
7
106
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1453 := 1 + 11 × 11 × (1 + 11)
2
Ï 1458 := (1 + 1) × (11 − 1 − 1)1+1+1
2+2+2
Ï 1463 := 11 × (1 + 11 × (1 + 11))
:= 22/2 + ((2/2 + 2) × 222 )
:= 2/2 + ((2/2 + 2) × 22 )
:= 2 × ((2/2 + 2)
)
:= (((3 + 3) × (33+3 − 3)) + 3)/3
:= 3 × (3 × (3 + 3) × 33 )
:= (((3 + 3) × (33+3 + 3)) − 3)/3
:= 44 + (4 × (4 + 4) × 44 + 4/4)
:= (4 − 4/4)4 × ((4 + 4)/4 + 4 × 4)
:= 444 + (4 × 44 − (4/4 + 4))
:= ((5/5 + 5) × ((5 − (5 + 5)/5)5 )) − 5
:= (5/5 + 5) × ((5 − (5 + 5)/5)5 )
:= 5 + ((5/5 + 5) × ((5 − (5 + 5)/5)5 ))
:= ((66 × (66 + 66)) + 6)/6
:= 666 + 66 × (6 + 6)
:= (6/6 + 6) × (6 × 6 × 6 − (6/6 + 6))
:= 7 × 7 × 7 + (7777 − 7)/7
:= 7 + (((7 × 7 − 77/7)
:= 8 × 8 + (88 × (8 + 8) − (88/8 + 8))
:= ((8 + 8)/8) × ((8/8 + 8)88/8−8 )
:= 8 × 8 + (88 × (8 + 8) − (8/8 + 8))
:= 9 × 9 × (9 + 9)
:= 9 × 9 × (9 + 9) + ((9 × 9 + 9)/(9 + 9))
:= 9 × 9 × (9 + 9) + ((9 − 99)/(9 + 9))
Ï 1454 := 1 + 1 + 11 × 11 × (1 + 11)
(7+7)/7
) + 7)
Ï 1459 := 1 + (1 + 1) × (11 − 1 − 1)1+1+1
:= 77 × ((77 + 7)/7 + 7)
Ï 1464 := (1 + 11) × (1 + 111+1 )
:= 2 + ((2/2 + 2) × 222 )
:= 2/2 + (2 × ((2/2 + 2)2+2+2 ))
:= (2/2 + 2) × (222 + 2 + 2)
:= ((((3 + 3) × 33+3 ) − 3)/3) − 3
:= (((3 + 3) × 33+3 ) + 3)/3
:= (3 + 3) × ((33+3 + 3)/3)
:= 44 + (4 × (4 + 4) × 44 + (4 + 4)/4)
:= 4 + ((4 × (44 + 44 ) − 4/4) + 44 )
:= 444 + (4 × 44 − 4)
:= 55 + (5 × (5 × 55 + 5) − 5/5)
:= 5 + ((5 × (5 × 55 + 5) − 5/5) + 55)
:= (5 × 5 − 5/5) × ((55 + 5/5) + 5)
:= ((6 + 6)/6) × ((66 × 66 + 6)/6)
:= 6 + (((66 × (66 + 66)) + 6)/6)
:= 6 + (66 × (6 + 6) + 666)
:= 7 × 7 × 7 + 7777/7
:= ((7 + 7) × (7 × (7 + 7) + 7)) − 77/7
:= 7/7 + ((7 × 7 × (7 + 7)) + 777)
:= ((8/8 + 8 + 8) × (88 − ((8 + 8)/8))) − 8
:= 8 × (8 + 8) + ((88/8)88/8−8 )
:= 8 × 8 + (88 × (8 + 8) − 8)
:= 9/9 + 9 × 9 × (9 + 9)
:= 9 + (9 × 9 × (9 + 9) − ((9 + 9 + 9)/9))
:= ((9 + 9)/9) × (9 × 9 × 9 − ((9 + 9)/9))
Ï 1455 := 1 + 1 + 1 + 11 × 11 × (1 + 11)
Ï 1460 := (1 + 1) × (1 + (11 − 1 − 1)1+1+1 )
Ï 1465 := 1 + (1 + 11) × (1 + 111+1 )
:= (2/2 + 2) × (222 + 2/2)
:= 2 + (2 × ((2/2 + 2)2+2+2 ))
:= 2 + (((2/2 + 2) × 222 ) + 22/2)
:= 33+3 + (33+3 − 3)
:= 3 + ((((3 + 3) × 33+3 ) − 3)/3)
:= (((3 + 3) × (33+3 + 3)) + 3)/3
:= 44 + (4 × (44 + 44 ) − 4/4)
:= 4 + (4 × (44 + 44 ) + 44 )
:= 4/4 + ((4 × 44 − 4) + 444)
:= 55 + 5 × (5 × 55 + 5)
:= 5 + (5 × (5 × 55 + 5) + 55)
:= (5 × ((5 × (55 + 5)) − 5)) − 5 − 5
:= 6 + (((6/6 + 6 + 6) × 666/6) + 6)
:= 6 + (((6 + 6)/6) × ((66 × 66 + 6)/6))
:= (6 × (6 × (6 × 6 + 6) − 6)) − 66/6
:= 7 × 7 × 7 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 7 × 7 × 7)
:= 7 × 7 × 7 + ((7777 + 77)/7)
:= (8 − 8/8 + 8) × ((8/8 + 88) + 8)
:= 8 + ((88/((8 + 8)/8)) + 88 × (8 + 8))
:= 8/8 + ((88 × (8 + 8) − 8) + 8 × 8)
:= 9 × 9 × (9 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 × 9 × (9 + 9)
:= 9 + (9 × 9 × (9 + 9) − ((9 + 9)/9))
Ï 1456 := (1 + 1 + 11) × (1 + 111)
Ï 1461 := 11 × (1 + 11 × (1 + 11)) − 1 − 1
Ï 1466 := 11 + (1 + 1 + 11) × (1 + 111) − 1
:= 2 + (((2/2 + 2) × 222 ) + 2)
:= (2/2 + 2) × ((222 + 2/2) + 2)
:= 22 + ((2 + 2 + 2)2 + 2)2
:= (3 − 3/3) × (33+3 − 3/3)
:= 3 + (33+3 + 33+3 )
:= 3 + ((((3 + 3) × (33+3 + 3)) − 3)/3)
:= 44 + 4 × (44 + 44 )
:= 4 + ((4 × (44 + 44 ) + 44 ) + 4/4)
:= 444 + (4 × 44 − (4 + 4)/4)
:= (5 × 5 + 5/5) × (55 + 5/5)
:= 5 + ((5 × 5 + 5/5) × (55 + 5/5))
:= 5 + (((5 × 5 + 5/5) × (55 + 5/5)) + 5)
:= (6/6 + 6 + 6) × (666 + 6)/6
:= 66 + (((6 × 6/(6 + 6))6 ) + 666)
:= ((6 + 6)/6) × (((66 × 66 + 6)/6) + 6)
:= (7 + 7) × (777/7 − 7)
:= 7 + (7777/7 + 7 × 7 × 7)
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) − (77/7))
:= 8 + (8 × 8 × (8 + 8 + 8) − 88)
:= 8 × 8 + (88 × (8 + 8) − (88/8))
:= 8 × 8 + ((88 × (8 + 8) − 8) + ((8 + 8)/8))
:= 9 × 9 × (9 + 9) − (9 + 9)/9
:= 9 × 9 × (9 + 9) + ((9 + 9 + 9)/9)
:= 9 + (9 × 9 × (9 + 9) − 9/9)
Ï 1457 := 1 + (1 + 1 + 11) × (1 + 111)
2
Ï 1462 := 11 × (1 + 11 × (1 + 11)) − 1
2+2+2
) + 2)
Ï 1467 := 11 + (1 + 1 + 11) × (1 + 111)
:= 22 + (((2 + 2 + 2)2 + 2)2 + 2/2)
:= 2 + ((2/2 + 2) × (22 + 2/2))
:= 2 × (((2/2 + 2)
:= (((3 + 3) × 33+3 ) − 3)/3
:= 3 + ((((3 + 3) × 33+3 ) + 3)/3)
:= 3 × ((3 × (3 + 3) × 33 ) + 3)
:= 4/4 + (4 × (44 + 44 ) + 44 )
:= 4 + ((4 − 4/4)4 × ((4 + 4)/4 + 4 × 4))
:= 444 + (4 × 44 − 4/4)
:= 55 + (5 × (5 × 55 + 5) + ((5 + 5)/5))
:= 55 + (5 × 5 × 55 + ((5 + 5)/5)5 )
:= ((5 + 5)/5 × (555 + 55 )/5) − 5
:= 6 + (((66 × (66 + 66)) − 6)/6)
:= 6 + ((6/6 + 6 + 6) × (666 + 6)/6)
:= 6 + ((((6 × 6/(6 + 6))6 ) + 666) + 66)
:= 7/7 + ((7 + 7) × (777/7 − 7))
:= 777 + ((7 × 7 × (7 + 7)) − 7/7)
:= 77/7 + ((7 + 7) × (777/7 − 7))
:= 8 + ((8 × 8 × (8 + 8 + 8) − 88) + 8/8)
:= (8/8 + 8 + 8) × (88 − ((8 + 8)/8))
:= (8/8 + 8) × ((88/8 + 88) + 8 × 8)
:= 9 × 9 × (9 + 9) − 9/9
:= ((9 + 9)/9) × (9 × 9 × 9 + ((9 + 9)/9))
:= 9 + 9 × 9 × (9 + 9)
107
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1468 := (1 + 1 + 11) × (1 + 1 + 111) − 1
2
2
Ï 1473 := 11 × (1 + 1 + 11 × (1 + 11)) − 1
2
Ï 1478 := 11 + (1 + 1 + 11) × (1 + 1 + 111) − 1 − 1
:= 2 + (((2 + 2 + 2) + 2) + 22)
:= 22 + (((2/2 + 2) × 22 ) − 2/2)
:= (2 × (22×(2+2) + 222 )) − 2
:= 3 + ((((3 + 3) × (33+3 + 3)) + 3)/3)
:= ((3 + 3) × (3 × 3 × 33 + 3)) − 3
:= 3 + (((3 + 3) × (33 − 3)) + ((33/3)3 ))
:= 444 + 4 × 44
:= 4 + ((444 + 4 × 44 ) + 4/4)
:= 44 + (((44 × 444/4) + 4)/4)
:= (5 × (5 × (55 + 5))) − ((5 + 5)/5)5
:= (5 × ((5 × (55 + 5)) − 5)) − (5 + 5)/5
:= 5 + ((5 × ((5 × (55 + 5)) − 5)) − ((5 + 5)/5))
:= (6 × (6 × (6 × 6 + 6) − 6)) − ((6 + 6)/6 + 6)
:= 66 + (666/6 + 6 × 6 × 6 × 6)
:= (6 + 6)/6 + (6 × (6 × (6 × 6 + 6) − 6))
:= ((7 + 7) × (7 × (7 + 7) + 7)) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + ((7 + 7) × (7 × (7 + 7) + 7))
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) + 7/7)
:= 8 × 8 + (88 × (8 + 8) − (8/((8 + 8)/8)))
:= 8/8 + (88 × (8 + 8) + 8 × 8)
:= 8 + ((88 × (8 + 8) − ((8 + 8)/8)) + 8 × 8)
:= 9 + (9 × 9 × (9 + 9) + 9/9)
:= 9 + ((9 × 9 × (9 + 9) − ((9 + 9 + 9)/9)) + 9)
:= 9 + (9 × 9 × (9 + 9) + (99/9))
Ï 1469 := (1 + 1 + 11) × (1 + 1 + 111)
Ï 1474 := 11 × (1 + 1 + 11 × (1 + 11))
2
Ï 1479 := 11 + (1 + 1 + 11) × (1 + 1 + 111) − 1
:= (22/2 + 2) × (222/2 + 2)
:= 22 + ((2/2 + 2) × 22 )
:= (2 × (22 − 2))2 − (22/2)2
:= (((3 + 3) × 33+3 ) + 33)/3
:= (33 × 33 + 3333)/3
:= 3 + ((3 + 3) × (3 × 3 × 33 + 3))
:= 4/4 + (444 + 4 × 44 )
:= 4 + ((444 + 4 × 44 ) + (4 + 4)/4)
:= 444 + (44/4 + 4 × 44 )
:= (5 × ((5 × (55 + 5)) − 5)) − (5/5 + 5)
:= (5 × ((5 × (55 + 5)) − 5)) − 5/5
:= 5 + ((5 × ((5 × (55 + 5)) − 5)) − 5/5)
:= (6 × (6 × (6 × 6 + 6) − 6)) − 6/6 − 6
:= (66 + 6/6) × ((66 + 66)/6)
:= 6 × 6 + ((6/6 + 6 + 6) × 666/6)
:= ((7 + 7) × (7 × (7 + 7) + 7)) − 7/7
:= 77/7 + ((7 × 7 × (7 + 7)) + 777)
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) + ((7 + 7)/7))
:= 8 + ((88 × (8 + 8) − (88/8)) + 8 × 8)
:= 8 × 8 + (88 × (8 + 8) + ((8 + 8)/8))
:= (8/8 + 8 + 8) × (88 − 8/8)
:= 99/9 + 9 × 9 × (9 + 9)
:= 9 + ((9 × 9 × (9 + 9) − ((9 + 9)/9)) + 9)
:= 9 + (9 × 9 × (9 + 9) + ((99 + 9)/9))
Ï 1470 := 1 + (1 + 1 + 11) × (1 + 1 + 111)
2
Ï 1475 := 1 + 11 × (1 + 1 + 11 × (1 + 11))
2
Ï 1480 := 11 + (1 + 1 + 11) × (1 + 1 + 111)
:= (2/2 + 2) × ((22 + 2 + 2) + 2)
:= 22 + (((2/2 + 2) × 22 ) + 2/2)
:= 2 × (22×(2+2) + 222 )
:= 3 + (3 × ((3 × (3 + 3) × 33 ) + 3))
:= ((33/3)3 ) + ((3 + 3) × (33 − 3))
:= (3 − 3/3) × (33+3 + 33/3)
:= 444 + ((4 + 4)/4 + 4 × 44 )
:= (4 × ((4/4 + 4)4 − 44 )) − 4/4
:= 4 + (4 × ((4/4 + 4)4 − 44 ))
:= (5 × ((5 × (55 + 5)) − 5)) − 5
:= 5 × ((5 × (55 + 5)) − 5)
:= 5 + (5 × ((5 × (55 + 5)) − 5))
:= (6/6 + 6) × (6 × 6 × 6 − 6)
:= (6 × (6 × (6 × 6 + 6) − 6)) − 6/6
:= 6 + ((66 + 6/6) × ((66 + 66)/6))
:= (7 + 7) × (7 × (7 + 7) + 7)
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) − ((7 + 7)/7))
:= (7/7 + 7) × ((7 + 7) × (7 + 7) − (77/7))
:= 8 × 8 + (88 × (8 + 8) − ((8 + 8)/8))
:= 8 × 8 + ((88 × (8 + 8) − 8) + (88/8))
:= 8 + (88 × (8 + 8) + 8 × 8)
:= 9 × 9 × (9 + 9) + (99 + 9)/9
:= 9 + ((9 × 9 × (9 + 9) − 9/9) + 9)
:= ((9 + 9)/9) × (9 × 9 × 9 + (99/9))
Ï 1471 := 1 + 1 + (1 + 1 + 11) × (1 + 1 + 111)
:= 2 + ((22/2 + 2) × (222/2 + 2))
Ï 1476 := (1 + 11) × (1 + (1 + 111+1 ))
2
Ï 1481 := ((1 + 1 + 11) × (1 + 1 + 1 + 111)) − 1
:= 2 + (((2/2 + 2) × 22 ) + 22)
:= 2 + ((2 × (22 − 2))2 − (22/2)2 )
:= (3 + 3) × (3 × 3 × 33 + 3)
:= 33 + (((((3 + 3) × 33+3 ) − 3)/3) − 3)
:= 4 + ((4 × 4 − 4/4) + 444)
:= 4 × ((4/4 + 4)4 − 44 )
:= (4444 − 4/4)/(4 − 4/4)
:= 5/5 + ((5 × ((5 × (55 + 5)) − 5)) − 5)
:= 5/5 + (5 × ((5 × (55 + 5)) − 5))
:= 5 + ((5 × ((5 × (55 + 5)) − 5)) + 5/5)
:= 6/6 + ((6/6 + 6) × (6 × 6 × 6 − 6))
:= 6 × (6 × (6 × 6 + 6) − 6)
:= 6 + ((6 × (6 × (6 × 6 + 6) − 6)) − 6/6)
:= 7/7 + ((7 + 7) × (7 × (7 + 7) + 7))
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) − 7/7)
:= 77/7 + ((7 + 7) × (7 × (7 + 7) + 7))
:= (((3 + 3) × ((3
3+3
+ 3) + 3)) + 3)/3
4
:= 8 × 8 + (88 × (8 + 8) − 8/8)
:= 8 × 8 + (88 × (8 + 8) + (8/((8 + 8)/8)))
:= 8 + ((88 × (8 + 8) + 8/8) + 8 × 8)
:= 9 × 9 × (9 + 9) + ((99 + 9 + 9)/9)
:= 9 + (9 × 9 × (9 + 9) + 9)
:= 9 × 9 × (9 + 9) + ((99 + 99 + 9)/9)
Ï 1472 := 11 × (1 + 1 + 11 × (1 + 11)) − 1 − 1
:= 2 × (2
2+2
× ((2 × 22) + 2))
Ï 1477 := 1 + (1 + 11) × (1 + 1 + 111+1 )
2
Ï 1482 := (1 + 1 + 11) × (1 + 1 + 1 + 111)
:= 22 + ((2/2 + 2) × (22 + 2/2))
:= 2 + (2 × (22×(2+2) + 222 ))
:= 3 + ((((3 + 3) × 33+3 ) + 33)/3)
:= 3/3 + ((3 + 3) × (3 × 3 × 33 + 3))
:= (3 + 3) × (((33+3 + 3)/3) + 3)
:= 4 + (444 + 4 × 44 )
:= 4/4 + (4 × ((4/4 + 4)4 − 44 ))
:= 444 + (4 × (44 + 4) − (4 + 4)/4)
:= (5 + 5)/5 + (5 × ((5 × (55 + 5)) − 5))
:= (5 × 5 + 5/5) × ((5 + 5)/5 + 55)
:= 6/6 + (6 × (6 × (6 × 6 + 6) − 6))
:= 6 + (6 × (6 × (6 × 6 + 6) − 6))
:= ((7 + 7)/7) + ((7 + 7) × (7 × (7 + 7) + 7))
:= 7 + ((7 + 7) × (7 × (7 + 7) + 7))
:= (7/7 + 77) × ((77 + 7)/7 + 7)
:= 8 × ((88 + 88) + 8)
:= 88 + (88 × (8 + 8) − (88/8 + 8))
:= 8 + ((88 × (8 + 8) + ((8 + 8)/8)) + 8 × 8)
:= 9 + (((9 × 9 + 9)/(9 + 9)) + 9 × 9 × (9 + 9))
:= 9 + ((9 × 9 × (9 + 9) + 9/9) + 9)
:= ((9 + 9)/9) × (((99 + 9)/9) + 9 × 9 × 9)
5
:= (5 + 5)/5 × (555 + 5 )/5
6
:= ((6 + 6)/6) × ((66/6 + 6) + 6)
108
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1483 := 1 + ((1 + 1 + 11) × (1 + 1 + 1 + 111))
Ï 1488 := (1 + 11) × (1 + 1 + 1 + 111+1 )
Ï 1493 := (1 + 1)11 − (1111 − 1)/(1 + 1)
:= ((2/2 + 2) × (22/2 + 222 )) − 2
:= (22 + 2) × (22+2+2 − 2)
:= 2/2 + (2 × ((2 × 222 ) − 222))
:= 3 + ((3 − 3/3) × (33+3 + 33/3))
:= 3 + (3 × (3 × ((3 + 3) × 33 + 3)))
:= (3 + 3) × 33 + ((33/3)3 )
:= 444 + (4 × (44 + 4) − 4/4)
:= 4 + (4 × (44 + 4) + 444)
:= 4/4 + (44 × 44 − 444)
:= 5 × 5 + ((5/5 + 5) × ((5 − (5 + 5)/5)5 ))
:= (5/5 + 5) × (((5 − (5 + 5)/5)5 ) + 5)
:= (5 × (5 × (55 + 5))) − ((5 + 5)/5 + 5)
:= 6 + ((6 × (6 × (6 × 6 + 6) − 6)) + 6/6)
:= 6 + ((6 × (6 × (6 × 6 + 6) − 6)) + 6)
:= 6 + ((6 × (6 × (6 × 6 + 6) − 6)) + (66/6))
:= 777/7 + (7 × (7 + 7) × (7 + 7))
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) + (77/7))
:= 7 + (((77 × (((7 + 7)/7)7 + 7)) + 7)/7)
:= 8 × 8 + (88 × (8 + 8) + 88/8)
:= 88 + (88 × (8 + 8) − 8)
:= 8 + ((88 × (8 + 8) − (88/8)) + 88)
:= 9 + (((9 × 9 × (9 + 9) − ((9 + 9)/9)) + 9) + 9)
:= 999/9 + 9 × (9 × (9 + 9) − 9)
:= 9 + ((9 × (99 + 9)) + (((9 + 9)/9)9 ))
Ï 1484 := 11 × (1 + 1 + 1 + 11 × (1 + 11)) − 1
2×(2+2)
2
Ï 1489 := 1 + (1 + 11) × (1 + 1 + 1 + 111+1 )
Ï 1494 := 1 + (1 + 1)11 − (1111 − 1)/(1 + 1)
:= (2 × (22 − 2))2 − 222/2
:= 2 + (2 × ((2 × 222 ) − 222))
:= 3 + ((3 × (3 × ((3 + 3) × 33 + 3))) + 3/3)
:= (3 + 3) × ((3 + 3)3 + 33)
:= 444 + 4 × (4 + 4)
:= (4 − 4/4)4 + 4 × (4 + 4) × 44
:= (4 + 4)/4 + (44 × 44 − 444)
:= (5 × (5 × (55 + 5))) − (55/5 + 5)
:= (5 × (5 × (55 + 5))) − 55/5
:= (5 × (5 × (55 + 5))) − (5/5 + 5)
:= 6 + ((6 × (6 × (6 × 6 + 6) − 6)) + ((6 + 6)/6))
:= 6 + (((6 × (6 × (6 × 6 + 6) − 6)) + 6/6) + 6)
:= 2 × ((2
+ 22 ) + 2)
3
:= 3 + ((((3 + 3) × 3
3+3
) − 3)/3)
4
7
:= (6 × 6 × (66 − 6)) − 666
:= 7 + (((7 + 7) × (7 × (7 + 7) + 7)) + 7)
:= (77 × (7 + 7 + 7)) − ((7 + 7)/7)
:= (77/7 + 7) × (77 − 7/7 + 7)
:= 8 × 8 + (((88 + 8)/8) + 88 × (8 + 8))
:= 8/8 + ((88 × (8 + 8) − 8) + 88)
:= 88 + (88 × (8 + 8) − ((8 + 8)/8))
:= 9 + (((99 + 99)/9) + 9 × 9 × (9 + 9))
:= (9 + 9) × (((9 + 9)/9) + 9 × 9)
9
:= (9 × (99 + 9)) + (((9 + 9)/9) )
Ï 1490 := 1 + 1 + (1 + 11) × (1 + 1 + 1 + 111+1 )
Ï 1495 := (1 + 1 + 11) × (1 + 1 + 1 + 1 + 111)
:= (2/2 + 2) × (22/2 + 22 )
:= 2 + ((22 + 2) × (22+2+2 − 2))
:= ((2 × 22)2 ) − ((22 − 2/2)2 )
:= 3 × (3 × ((3 + 3) × 33 + 3))
:= 33 + ((((3 + 3) × 33+3 ) − 3)/3)
:= 3/3 + ((3 + 3) × ((3 + 3)3 + 33))
:= 4/4 + (4 × (44 + 4) + 444)
:= (4/4 + 4) × ((44 − (4 + 4)/4) + 44)
:= (4/4 + 4) × ((44 − 4/4) + 44 )
:= 55 × ((5 + 5)/5 + 5 × 5)
:= (5 × (5 × (55 + 5))) − 5 − 5
:= (5 × (5 × (55 + 5))) − 5
:= 6 + (((6/6 + 6 + 6) × 666/6) + 6 × 6)
:= 6 × 6 × (6 × 6 + 6) − ((66 + 66)/6)
:= (66 − 6/6) × ((66/6 + 6) + 6)
Ï 1485 := 11 × (1 + 1 + 1 + 11 × (1 + 11))
2
7
:= 7 + ((7 × (7 + 7) × (7 + 7)) + 777/7)
:= 7 + ((((7 + 7) × (7 × (7 + 7) + 7)) + (77/7)) + 7)
:= 88 + (88 × (8 + 8) − (88/8))
:= 88 + ((88 × (8 + 8) − 8) + ((8 + 8)/8))
:= 88 + (88 × (8 + 8) − 8/8)
:= 9 + ((9 × 9 × (9 + 9) + 9) + 9)
:= 9 + (((99 + 99 + 9)/9) + 9 × 9 × (9 + 9))
:= 9/9 + ((9 + 9) × (((9 + 9)/9) + 9 × 9))
:= 77/7 × (((7 + 7)/7) + 7)
Ï 1486 := 1 + 11 × (1 + 1 + 1 + 11 × (1 + 11))
2+2+2
Ï 1491 := (1 + 1)11 − 1 − (1 + 1111)/(1 + 1)
Ï 1496 := 11 × (1 + 1 + 1 + 1 + 11 × (1 + 11))
:= 2 + ((2 × (22 − 2)) − 222/2)
:= 2 × (22 × ((2 × 22+2 ) + 2))
:= 33 + ((((3 + 3) × 33+3 ) + 3)/3)
:= 33 + (33+3 + 33+3 )
:= 3 + ((3 + 3) × 33 + ((33/3)3 ))
:= 444 + (4 × (44 + 4) + (4 + 4)/4)
:= 44 × 44 − (444 + 4/4)
:= 4 + (44 × 44 − 444)
:= 5 × 5 × 55 + 555/5
:= 5 + (5 × 5 × 55 + 555/5)
:= 5/5 + ((5 × (5 × (55 + 5))) − 5)
:= ((66 − 6)/6) + (6 × (6 × (6 × 6 + 6) − 6))
:= (6/6 + 6) × (6 × 6 × 6 − (6 × 6/(6 + 6)))
:= 6 × 66 + ((6666 − 66)/6)
:= ((77 × (((7 + 7)/7) + 7)) + 7)/7
:= (7 + 7 + 7) × ((7/7 − 7) + 77)
:= 7 + ((77 × (7 + 7 + 7)) − ((7 + 7)/7)7 )
:= 88 + ((8 − 88)/8 + 88 × (8 + 8))
:= 8 + ((88 × (8 + 8) + 88/8) + 8 × 8)
:= 88 + 88 × (8 + 8)
:= 9 + (((9 × 9 × (9 + 9) + 9/9) + 9) + 9)
:= 9 × 9 × (9 + 9) + (99/((9 + 9 + 9)/9))
:= ((9 − 9/9) + 9) × (99 − (99/9))
:= ((22 + 2) × (2
2
− 2)) − 2
7
Ï 1487 := 11 + (1 + 11) × (1 + 1 + 111+1 )
Ï 1492 := (1 + 1)11 − (1 + 1111)/(1 + 1)
:= 2 + ((2/2 + 2) × (22/2 + 222 ))
:= 3 + (((((3 + 3) × 3
3+3
:= 2 × ((2 × 222 ) − 222)
3
) − 3)/3) + 3 )
4
:= 33 + ((((3 + 3) × 3
3+3
Ï 1497 := 1 + 11 × (1 + 1 + 1 + 1 + 11 × (1 + 11))
:= (2 × 222 ) + ((22 + 2/2)2 )
) + 3)/3)
:= 3 + ((3 + 3) × ((3 + 3)3 + 33))
:= 44 × 44 − 444
:= 4 + ((44 × 44 − 444) + 4/4)
:= 5 × 5 × 55 + (555 + 5)/5
:= 5 + ((555 + 5)/5 + 5 × 5 × 55)
:= (5 + 5)/5 + ((5 × (5 × (55 + 5))) − 5)
:= 66/6 + (6 × (6 × (6 × 6 + 6) − 6))
:= ((6/6 + 6) × (6 × 6 × 6 − (6 + 6)/6)) − 6
:= 6 + ((6/6 + 6) × (6 × 6 × 6 − (6 × 6/(6 + 6))))
:= (44 × (44 + 4)) − (4/4 + 4)
7
7
:= 77 + ((((7 + 7 + 7)/7) ) − 777)
:= 7 + (77/7 × (((7 + 7)/7) + 7))
:= 77 + ((7 × ((7 + 7) × (7 + 7) + 7)) − 7/7)
:= 8 + ((8/8 + 8 + 8) × (88 − 8/8))
:= 88 + (88 × (8 + 8) − (8/((8 + 8)/8)))
:= 8/8 + (88 × (8 + 8) + 88)
:= 9 + ((9 × 9 × (9 + 9) + (99/9)) + 9)
:= ((9 + 9)/9) × (((9 × 9 × 9 − 9/9) + 9) + 9)
:= 9 + (9 × (9 × (9 + 9) − 9) + 999/9)
109
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1498 := (1 + 1)11 − (1111 − 11)/(1 + 1)
Ï 1508 := (1 + 1)11−1 + (11 + 11)1+1
Ï 1503 := (1 + 1 + 1) × ((1 + 1)11−1−1 − 11)
:= 2 + (2 × (22 × ((2 × 22+2 ) + 2)))
:= (2/2 + 2)2 × (((22/2 + 2)2 ) − 2)
:= 222 + (22×(2+2)+2 )
:= ((33 × 333) − 3)/(3 + 3)
:= 3 × (((3 + 3) × (3 × 33 + 3)) − 3)
:= 3 + (((3/3 + 3) + 3) × ((3 + 3)3 − 3/3))
:= 44 + ((4 + 4)/4 × ((4/4 + 4)4 − 4))
:= (4 × (444 − 4)) − (4/4 + 44 )
:= 4 + ((4 + 4) × (444 − 44 ))
:= (5 × (5 × (55 + 5))) − (5 + 5)/5
:= 5 + ((5 × (5 × (55 + 5))) − ((5 + 5)/5))
:= ((55 + 5/5)/((5 + 5)/5)) − 55
:= (6/6 + 6) × (6 × 6 × 6 − (6 + 6)/6)
:= (((6 − 66) + 6)/6) + 6 × 6 × (6 × 6 + 6)
:= (6 + 6)/6 + (6 × 6 × (6 × 6 + 6) − 6)
:= 77 + (7 × ((7 + 7) × (7 + 7) + 7))
:= 7 × (7 × 7 + 7) + 7777/7
:= 7 + ((((7 + 7 + 7)/7)7 ) − (7 × 7 × (7 + 7)))
:= 88 + (88 × (8 + 8) + ((8 + 8)/8))
:= 8 + ((88 × (8 + 8) − 8/8) + 88)
:= 88 + (((88 + 8)/8) + 88 × (8 + 8))
:= 999 + (((9 × 999) − 9)/(9 + 9))
:= 9 + ((9 + 9) × (((9 + 9)/9) + 9 × 9))
:= 9 × 9 × (9 + 9) + ((9 × 99 + 9)/(9 + 9))
Ï 1499 := 11 + (1 + 11) × (1 + 1 + 1 + 111+1 )
:= ((2 × (22 − 2) − 2/2)2 ) − 22
3
Ï 1504 := 1 + (1 + 1 + 1) × ((1 + 1)11−1−1 − 11)
Ï 1509 := 1 + (1 + 1)11−1 + (11 + 11)1+1
:= 2 × (2 × ((22 − 2)2 − (22 + 2)))
:= 222 + ((222/2 + 2)/2)
3
:= ((3 × 333) + 3)/(3 + 3)
:= 3 × (((3 − 3/3)3×3 ) − 3 × 3)
:= ((3 × 333) + 33)/(3 + 3)
4
4
:= ((4/4 + 4) × (44 + 4 )) − 4/4
:= (4 + 4) × (444 − 4 )
:= (5 × (5 × (55 + 5))) − 5/5
:= 5 + ((5 × (5 × (55 + 5))) − 5/5)
:= 5 + (((5 × (5 × (55 + 5))) − 5/5) + 5)
:= 6 × 6 × (6 × 6 + 6) − (6/6 + 6 + 6)
:= 6 × 6 × (6 × 6 + 6) − ((6 + 6)/6 + 6)
:= 6 × 6 × (6 × 6 + 6) − 6 × 6/(6 + 6)
:= 7 + ((77/7 × (((7 + 7)/7)7 + 7)) + 7)
:= (7/7 + 7) × (777/7 + 77)
:= (((7 + 7)/7)77/7 ) − 7 × 77
:= 88 + ((88 × (8 + 8) − 8) + (88/8))
:= 8 + (88 × (8 + 8) + 88)
:= 8 + ((88/8 + 8) × (88 − (8/8 + 8)))
:= 999 + (((9 × 999) + 9)/(9 + 9))
:= 9 + (((9 + 9) × (((9 + 9)/9) + 9 × 9)) + 9/9)
:= ((9 + 9 + 9)/9) × ((((9 + 9)/9)9 ) − 9)
Ï 1500 := (1 + 1 + 1) × (11 − 1)1+1+1 /(1 + 1)
:= 2 × ((22 × ((2 × 22+2 ) + 2)) + 2)
3
:= (3 × 3 + 3) × ((3 − 3/3 + 3) )
4
:= 4 × 44 + ((44 × 44 + 4)/4)
Ï 1505 := (1 + 11)1+1+1 − 1 − (1 + 1) × 111
Ï 1510 := ((1 + 1 + 1) × (1 + 1 + 11))1+1 − 11
:= 222 + (((222/2 − 2)/2) − 2)
:= 2 + ((22×(2+2)+2 ) + 222 )
3
:= ((3/3 + 3) + 3) × ((3 + 3) − 3/3)
:= 3/3 + (3 × (((3 − 3/3)3×3 ) − 3 × 3))
4
:= (4/4 + 4) × (44 + 4 )
:= 4/4 + ((4 + 4) × (444 − 4 ))
:= 4 + ((4 + 4)/4 × (4/4 + 4)4 + 44 )
:= 5 × (5 × (55 + 5))
:= 5 + (5 × (5 × (55 + 5)))
:= 5 + ((5 × (5 × (55 + 5))) + 5)
:= 6 × 6 × (6 × 6 + 6) − 6 − 6
:= (6/6 + 6) × (6 × 6 × 6 − 6/6)
:= ((7 + 7)/7)7 + (7 × (7 + 7) × (7 + 7))
:= 7 × (7 × 7 × 7 − ((7 + 7)/7)7 )
:= 6 × 6 × (6 × 6 + 6) − (6 + 6)/6
:= 7 + (7777/7 + 7 × (7 × 7 + 7))
:= 88 + (88 × (8 + 8) + (8/((8 + 8)/8)))
:= 8 + ((88 × (8 + 8) + 88) + 8/8)
:= 88 × (8 + 8) + ((888 − 8)/8 − 8)
:= 999 + ((((9 + 9)/9)9 ) − (99/9))
:= 9 + (((9 − 9/9) + 9) × (99 − (99/9)))
:= 999 + ((((9 + 9)/9)9 ) − 9/9)
Ï 1501 := 1 + (1 + 1 + 1) × ((11 − 1)1+1+1 /(1 + 1))
Ï 1506 := (1 + 11)1+1+1 − (1 + 1) × 111
Ï 1511 := 1111 + ((1 + 1) × (11 − 1))1+1
:= 2 + (((2 × (22 − 2) − 2/2)2 ) − 22)
:= 222 + ((22×(2+2)+2 ) − 2)
:= (22 − 2)2 + 2222/2
:= 3 + (((33 × 333) − 3)/(3 + 3))
:= 3 + (3 × (((3 + 3) × (3 × 33 + 3)) − 3))
:= 3 × 333 + ((3 − 3/3)3×3 )
:= 4/4 + ((4/4 + 4) × (44 + 44 ))
:= 44 + (4 + 4)/4 × (4/4 + 4)4
:= ((44 − 4) × ((4 + 4)/4 + 4)) − 4/4
:= 5/5 + (5 × (5 × (55 + 5)))
:= 5 + ((5 × (5 × (55 + 5))) + 5/5)
:= 55/5 + (5 × (5 × (55 + 5)))
:= 6 × 6 × (6 × 6 + 6) − 66/6
:= 6 × 6 × (6 × 6 + 6) − 6
:= 6 × 6 × (6 × 6 + 6) − 6/6
:= (((7 + 7 + 7)/7)7 ) − (7 × 7 × (7 + 7))
:= (77 × (7 + 7 + 7)) − 777/7
:= ((77 + 7) × (77/7 + 7)) − 7/7
:= (88/8 + 8) × (88 − (8/8 + 8))
:= 8 + ((88 × (8 + 8) + ((8 + 8)/8)) + 88)
:= 888/8 + (88 × (8 + 8) − 8)
:= (9/9 + 9 + 9) × (9 × 9 − ((9 + 9)/9))
:= 9 + ((9 × (9 × (9 + 9) − 9) + 999/9) + 9)
:= 999 + (((9 + 9)/9)9 )
Ï 1502 := (1 + 1 + 1) × ((1 + 1)11−1−1 − 11) − 1
:= 2 × (((2/2 + 2)
2+2+2
) + 22)
3
:= 3 + (((3 × 333) + 3)/(3 + 3))
4
4
Ï 1507 := 11 × (111 + ((1 + 1) × (1 + 1 + 11)))
2
:= 22 + ((2
22/2
:= 2 × (2 × ((22 − 2)2 − 22))
− 2)/2)
3
:= 3 × ((3 + 3) × (3 × 33 + 3))
:= 3 + (((3 × 333) + 33)/(3 + 3))
4−4/4
:= 4 + ((4 + 4)/4 × (4/4 + 4) − 4)
:= 4 × 44 + ((44/4)
:= (5 + 5)/5 + (5 × (5 × (55 + 5)))
:= 5 + ((5 × (5 × (55 + 5))) + ((5 + 5)/5))
:= ((6 − 66)/6) + 6 × 6 × (6 × 6 + 6)
:= 6/6 + (6 × 6 × (6 × 6 + 6) − 6)
:= 7 × (7 × 7 + 7) + (7777 − 7)/7
Ï 1512 := (11 − 1 − 1) × ((1 + 1 + 11)1+1 − 1)
:= (44 − 4) × ((4 + 4)/4 + 4)
)
:= ((55 + 5)/5) + (5 × (5 × (55 + 5)))
:= 6 × 6 × (6 × 6 + 6)
7
:= 7 + ((7 × (7 + 7) × (7 + 7)) + ((7 + 7)/7) )
:= (77 + 7) × (77/7 + 7)
:= 8 + ((88 × (8 + 8) − ((8 + 8)/8)) + 88)
:= 88 + (88 × (8 + 8) + 88/8)
:= (8 + 8 + 8) × (8 × 8 − 8/8)
:= 999 + ((((9 + 9)/9)9 ) − 9)
:= 9 × 9 × (9 + 9) + ((9 × 99 − 9)/(9 + 9))
:= (9 + 9) × (((9 + 9 + 9)/9) + 9 × 9)
110
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1513 := (1 + ((111 − 1)/(1 + 1))1+1 )/(1 + 1)
Ï 1518 := 11 × (111 + (1 + 1 + 1)1+1+1 )
Ï 1523 := 1 + 1 + ((1 + 1 + 1) × (1 + 1 + 11))1+1
:= 2 + 2222/2 + (22 − 2)2
:= 222 + (2 + 2 + 2)2+2
:= 2 + ((2 × (22 − 2) − 2/2)2 )
:= 3/3 + (3 × ((3 + 3) × (3 × 33 + 3)))
:= 3 × (((3 − 3/3)3×3 ) − (3 + 3))
:= ((33/3)3 ) + (3 × ((3/3 + 3)3 ))
:= 4/4 + ((44 − 4) × ((4 + 4)/4 + 4))
:= ((4 + 4)/4 + 4) × ((4/4 − 4) + 44 )
:= 4 + (4 × 444 − (4/4 + 44 ))
:= (55 × 55 + 5/5)/((5 + 5)/5)
:= 5 + ((55 × 55 + 5/5)/((5 + 5)/5))
:= (5 × ((5 × (55 + 5)) + 5)) − (5 + 5)/5
:= 6/6 + 6 × 6 × (6 × 6 + 6)
:= 6 + 6 × 6 × (6 × 6 + 6)
:= 66/6 + 6 × 6 × (6 × 6 + 6)
:= 7/7 + ((77 + 7) × (77/7 + 7))
:= (7 × (7 × (7 × 7 − 7) − 77)) − 7/7
:= 77/7 + ((77 + 7) × (77/7 + 7))
:= (8/8 + 8 + 8) × (8/8 + 88)
:= 88 × (8 + 8) + (888 − 8)/8
:= ((9 − 9/9) + 9) × ((9 × 9 − 9/9) + 9)
Ï 1514 := 1 + (1 + ((111 − 1)/(1 + 1))1+1 )/(1 + 1)
:= 88/8 + ((8 + 8 + 8) × (8 × 8 − 8/8))
9
:= 9 + (((9 + 9 + 9)/9) × ((((9 + 9)/9) ) − 9))
:= (9 + 9)/9 + ((9 × (99 + 9 × 9)) − 99)
Ï 1524 := 1 + 1 + 1 + ((1 + 1 + 1) × (1 + 1 + 11))1+1
Ï 1519 := (1 + 1)11 − (1 + 11 + 11)1+1
:= 2 + 2 × 2 × ((22 − 2)2 − 22)
:= ((2 × (22 − 2) − 2/2)2 ) − 2
:= (2 + 2 + 2) × (22×(2+2) − 2)
:= (3 + 3)3 + (((33/3)3 ) − 33)
:= ((3/3 + 3) + 3) × ((3 + 3)3 + 3/3)
:= 3 + ((33 + 3 + 3)3−3/3 )
:= 44 + ((4 + 4)/4 × ((4/4 + 4)4 + 4))
:= 4 × 444 − (4/4 + 44 )
:= 4 + (4 × 444 − 44 )
:= (5 × ((5 × (55 + 5)) + 5)) − 55/5
:= (5 × ((5 × (55 + 5)) + 5)) − (5/5 + 5)
:= (5 × ((5 × (55 + 5)) + 5)) − 5/5
:= (6 + 6)/6 + 6 × 6 × (6 × 6 + 6)
:= 6 + (6 × 6 × (6 × 6 + 6) + 6/6)
:= 6 + (6 × 6 × (6 × 6 + 6) + 6)
:= ((7 + 7)/7) + ((77 + 7) × (77/7 + 7))
:= 7 × (7 × (7 × 7 − 7) − 77)
:= ((7 + 7)/7) × (777 − (7/7 + 7 + 7))
:= 8/8 + ((8/8 + 8 + 8) × (8/8 + 88))
:= 888/8 + 88 × (8 + 8)
:= ((88 + 8)/8) × (8 × (8 + 8) − 8/8)
:= 9 × 9 × (9 + 9) + ((999 + 9)/(9 + 9))
:= (9 × (9 × (9 + 9) + 9)) − (99/9 + 9)
:= 9 × 9 + (999/9 × ((99 + 9 + 9)/9))
Ï 1515 := (1 + 1 + 1 + 1 + 11) × (1 + (11 − 1)1+1 )
Ï 1520 := ((1 + 1 + 1) × (1 + 1 + 11))1+1 − 1
Ï 1525 := (1 + 1 + 1) × (1 + 1)11−1−1 − 11
:= (2/2 + 2) × (222 − 2/2 + 22)
:= 2 + ((2 + 2 + 2)2+2 + 222)
:= 2 + (((2 × (22 − 2) − 2/2)2 ) + 2)
:= 3 + (3 × ((3 + 3) × (3 × 33 + 3)))
:= (3 + 3)3 + (((33/3)3 ) − 33 )
:= (3 × ((3 − 3/3)3×3 )) − 33/3
:= 4 × 444 − ((4/4 + 44 ) + 4)
:= 4 × 444 − 44
:= 4 + 4 × 444 − 44 + 4/4
:= 5 + (((5 × (5 × (55 + 5))) + 5) + 5)
:= (5 × ((5 × (55 + 5)) + 5)) − 5
:= 5 × ((5 × (55 + 5)) + 5)
:= (6 × 6/(6 + 6)) + 6 × 6 × (6 × 6 + 6)
:= 6 + (6 × 6 × (6 × 6 + 6) + ((6 + 6)/6))
:= 6 + ((6 × 6 × (6 × 6 + 6) + 6/6) + 6)
:= (7/7 + 7 + 7) × (7777/77)
:= 7/7 + (7 × (7 × (7 × 7 − 7) − 77))
:= (((7 + 7) × (777 − (7 + 7))) − 7)/7
:= (8 + 8) × (88 − 8/8 + 8)
:= 8 × 8 × (8 + 8 + 8) − 88/8
:= 8 + ((88 × (8 + 8) + 88/8) + 88)
9
:= 9 × 9 × (9 + 9) + (((((9 + 9)/9) ) + 9/9)/9)
Ï 1516 := 1 + ((1 + 1 + 1 + 1 + 11) × (1 + (11 − 1)1+1 ))
9
:= 9 × 9 + (((99/9 + 9 + 9) + 9)(9+9)/9 )
:= 9 + ((((9 + 9)/9) ) + 999)
Ï 1521 := ((1 + 1 + 1) × (1 + 1 + 11))1+1
2
Ï 1526 := (1 + (1 + 1 + 11)) × (111 − 1 − 1)
:= 2 × ((2 × ((22 − 2)2 − 22)) + 2)
:= (2 × (22 − 2) − 2/2)
:= 2 + ((2 + 2 + 2) × (22×(2+2) − 2))
:= 3 + ((3 × ((3 + 3) × (3 × 33 + 3))) + 3/3)
:= (33 + 3 + 3)3−3/3
:= 33 × (3 + 3) + (((33/3)3 ) − 3)
:= 4 × 444 − (44 + 4)
:= 4/4 + (4 × 444 − 44 )
:= (((4 + 4)/4 + 4) × (44 − 4/4)) − 4
:= 5 + ((5 × (5 × (55 + 5))) + (55/5))
:= 5/5 + ((5 × ((5 × (55 + 5)) + 5)) − 5)
:= 6 + (6 × 6 × (6 × 6 + 6) − ((6 + 6)/6))
77/7
:= 7 + ((((7 + 7)/7)
) − 7 × 77)
6
:= 66 × (6 + 6) + ((6 × 6/(6 + 6)) )
:= (((7 − 77)/7) + 7 × 7)
(7+7)/7
:= 5/5 + (5 × ((5 × (55 + 5)) + 5))
:= (6/6 + 6) × (6 × 6 × 6 + (6 + 6)/6)
:= 7 + (7 × (7 × (7 × 7 − 7) − 77))
:= 8 × 8 × (8 + 8 + 8) − ((88 + 8)/8 + 8)
:= 8 + ((8/8 + 8 + 8) × (8/8 + 88))
:= (8 − 88)/8 + 8 × 8 × (8 + 8 + 8)
:= 9 + (((9 × 99 − 9)/(9 + 9)) + 9 × 9 × (9 + 9))
:= (9 × (99 + 9 × 9)) − 99
:= (9 + 9) × 99 − (((9 + 9)/9)9−9/9 )
Ï 1517 := (11 × (111 + (1 + 1 + 1)1+1+1 )) − 1
Ï 1522 := 1 + ((1 + 1 + 1) × (1 + 1 + 11))1+1
2
Ï 1527 := 1 + (1 + 1 + 1 + 11) × (111 − 1 − 1)
:= ((2 × (22 − 2) − 2/2) ) − 2 − 2
:= 2/2 + ((2 × (22 − 2) − 2/2) )
:= 2 + ((((2 × (22 − 2) − 2/2)2 ) + 2) + 2)
:= 3 + ((((33/3)3 ) − 33) + (3 + 3)3 )
:= 3/3 + ((33 + 3 + 3)3−3/3 )
:= 3 × (((3 − 3/3)3×3 ) − 3)
:= 4/4 + (4 × 444 − (44 + 4))
:= (4 + 4)/4 + (4 × 444 − 44 )
:= 4 + ((4 × 444 − (4/4 + 44 )) + 4)
:= 5 + ((5 × (5 × (55 + 5))) + ((55 + 5)/5))
:= (5 + 5)/5 + ((5 × ((5 × (55 + 5)) + 5)) − 5)
:= (5 + 5)/5 + (5 × ((5 × (55 + 5)) + 5))
:= 6 + (6 × 6 × (6 × 6 + 6) − 6/6)
:= ((66 − 6)/6) + 6 × 6 × (6 × 6 + 6)
:= 6 + (((6 × 6/(6 + 6))6 ) + 66 × (6 + 6))
:= (7 × (7 × (7 × 7 − 7) − 77)) − (7 + 7)/7
:= ((77 + 7)/7 × ((7 + 7)/7)7 ) − (7 + 7)
:= (((7 + 7) × (777 − (7 + 7))) + 7)/7
2
:= 8 × 8 × (8 + 8 + 8) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) × (88 − 8/8 + 8))
:= 8 × 8 × (8 + 8 + 8) − (8/8 + 8)
:= (9 × (9 × (9 + 9) + 9)) − ((99 + 99)/9)
:= 9/9 + ((9 × (99 + 9 × 9)) − 99)
:= (9 × (9 × (9 + 9) + 9)) − (99 + 9)/9
111
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1528 := 1 + (1 + (1 + 1 + 1 + 11) × (111 − 1 − 1))
Ï 1533 := (1 + 1 + 1) × ((1 + 1)11−1−1 − 1)
Ï 1538 := 1 + (1 + ((1 + 1 + 1) × (1 + 1)11−1−1 ))
:= 2 × (2 × ((22+2 × (22 + 2)) − 2))
:= (2/2 + 2) × ((2
:= 3/3 + (3 × (((3 − 3/3)3×3 ) − 3))
:= (3 × ((3 − 3/3)3×3 )) − 3
:= 3 + ((3 × ((3 − 3/3)3×3 )) − 3/3)
:= 4 + ((4 × 444 − 44 ) + 4)
:= 4/4 + ((4 × (4 + 4) × (44 + 4)) − 4)
:= (4 + 4)/4 + (4 × (4 + 4) × (44 + 4))
(2/2+2)2
) − 2/2)
:= 2 + ((2 + 2 + 2) × 22×(2+2) )
:= 5 + ((5 × ((5 × (55 + 5)) + 5)) − ((5 + 5)/5))
:= ((5/5 + 5) − 555/5)/5
:= ((55 + 5/5)/((5 + 5)/5)) − 5 × 5
:= 6 + (6 × 6 × (6 × 6 + 6) + ((66 − 6)/6))
:= (6/6 + 6) × ((6 × 6/(6 + 6)) + 6 × 6 × 6)
:= ((6 − 66)/6) + 6 × 6 × (6 × 6 + 6 + 6)
:= (77 × (7 + 7 + 7)) − (77 + 7)
:= ((7 + 7)/7) × (777 − (7/7 + 7))
:= 8 × 8 × (8 + 8 + 8) − 8
:= 8 + (8 × 8 × (8 + 8 + 8) − (88/8))
:= (8 + 8)/8 + 8 × 8 × (8 + 8 + 8)
:= (9 × (9 × (9 + 9) + 9)) − 99/9
:= ((9 + 9 + 9)/9) + ((9 × (9 × (9 + 9) + 9)) − 9)
:= (9 × (9 × (9 + 9) + 9)) − 9/9
:= 7 + ((((7 − 77)/7) + 7 × 7)
(7+7)/7
)
Ï 1529 := 11 × (((11 − 1) × (1 + (1 + 1 + 11))) − 1)
5
Ï 1534 := 1 + ((1 + 1 + 1) × ((1 + 1)11−1−1 − 1))
Ï 1539 := (1 + 1 + 1) × (1 + (1 + 1)11−1−1 )
:= 2 × (2 + 2) + ((2 × (22 − 2) − 2/2)2 )
:= ((2 + 2 + 2) × 22×(2+2) ) − 2
:= 2 + (((2 + 2 + 2) × 22×(2+2) ) + 2/2)
:= 33 × (3 + 3) + ((33/3)3 )
:= 3/3 + ((3 × ((3 − 3/3)3×3 )) − 3)
:= 3 + (3 × ((3 − 3/3)3×3 ))
4
:= 4 + (((4 × 444 − 44 ) + 4/4) + 4)
:= 4 + (((4 + 4)/4 + 4) × (4 − 4/4))
:= 4 + ((4 × (4 + 4) × (44 + 4)) − 4/4)
:= 5 + ((5 × ((5 × (55 + 5)) + 5)) − 5/5)
:= (5 × 5 + 5/5) × (55 − 5/5 + 5)
:= (55 × (5 × 5 + 5)) − 555/5
:= 6 + (6 × 6 × (6 × 6 + 6) + (66/6))
:= (6/6 + 6 + 6) × ((666 + 6)/6 + 6)
:= 6 + ((6/6 + 6) × ((6 × 6/(6 + 6)) + 6 × 6 × 6))
:= ((77 + 7)/7 × ((7 + 7)/7) ) − 7
:= (7 − 7/7 + 7) × (777/7 + 7)
:= (((7 + 7) × (777 − 7)) − 7)/7
:= 8/8 + (8 × 8 × (8 + 8 + 8) − 8)
:= 8 × 8 × (8 + 8 + 8) − (8 + 8)/8
:= 88/8 + (8 × 8 × (8 + 8 + 8) − 8)
:= (9 × (9 × (9 + 9) + 9)) − 9/9 − 9
:= ((9 − 99)/(9 + 9)) + (9 × (9 × (9 + 9) + 9))
:= 9 × (9 × (9 + 9) + 9)
7
Ï 1530 := (11 − 1) × ((11 × (1 + (1 + 1 + 11))) − 1)
:= (2/2 + 2) × ((2
(2/2+2)2
) − 2)
Ï 1535 := ((1 + 1 + 1) × (1 + 1)11−1−1 ) − 1
:= ((2 + 2 + 2) × 22×(2+2) ) − 2/2
3×3
:= (3 × ((3 − 3/3)
:= ((4 + 4)/4 + 4) × (44 − 4/4)
:= (4 × (4 + 4) × (44 + 4)) − 4/4
:= 6 + ((6 × 6 × (6 × 6 + 6) + 6) + 6)
:= 22 × ((2 × (22 + 2)) + 22)
:= 3 + ((3 × ((3 − 3/3)3×3 )) + 3/3)
:= 3 + (3 × (((3 − 3/3)3×3 ) − 3))
:= 5 + (5 × ((5 × (55 + 5)) + 5))
Ï 1540 := (111 − 1) × (1 + (1 + 1 + 11))
)) − 3/3
5
:= 4 + (4 × (4 + 4) × (44 + 4))
:= 5 × ((55 + 5)/(5 + 5) − 5)
:= 5 × (5 − 55)/(5 + 5)
:= 6 + ((6 × 6 × (6 × 6 + 6) + (66/6)) + 6)
7
:= (6/6 + 6) × ((6 × 6 × 6 − (6 + 6)/6) + 6)
:= 77 × ((7 − 7/7 + 7) + 7)
:= (77/7 + 7) × (7/7 + 77 + 7)
:= (((77 + 7) × ((7 + 7)/7) ) − 7)/7
:= (8 + 8)/8 + (8 × 8 × (8 + 8 + 8) − 8)
:= 8 × 8 × (8 + 8 + 8) − 8/8
:= (8/((8 + 8)/8)) + 8 × 8 × (8 + 8 + 8)
:= ((9 − 9 × 9)/(9 + 9)) + (9 × (9 × (9 + 9) + 9))
:= 9/9 + (9 × (9 × (9 + 9) + 9))
:= (9 × (9 × (9 + 9) + 9)) − 9
Ï 1531 := 1 + ((11 − 1) × ((11 × (1 + (1 + 1 + 11))) − 1))
:= 2 + (((2 × (22 − 2) − 2/2)2 ) + 2 × (2 + 2))
Ï 1536 := (1 + 1 + 1) × (1 + 1)11−1−1
:= (2 + 2 + 2) × 2
Ï 1541 := 1 + ((111 − 1) × (1 + (1 + 1 + 11)))
2×(2+2)
:= ((2 × 22 + 2/2)2 ) − 222
:= 3 + ((3 × (((3 − 3/3)3×3 ) − 3)) + 3/3)
:= 3 × ((3 − 3/3)3×3 )
:= (3 + 3)3 + (((33/3)3 ) − (3 + 3))
:= 44 + (4/4 + 4) × (44 − 4/4)
:= 4 × (4 + 4) × (44 + 4)
:= 4 + ((4 × (4 + 4) × (44 + 4)) + 4/4)
:= 5 + ((5 × ((5 × (55 + 5)) + 5)) + 5/5)
:= 5/5 + (5 × (5 − 55)/(5 + 5))
:= 5/5 + (5 × ((55 + 5)/(5 + 5) − 5))
:= 6 × (((6 + 6)/6)6+(6+6)/6 )
:= (66 + 6/6) × ((66/6 + 6) + 6)
:= 6 + (((6 × 6 × (6 × 6 + 6) + 6/6) + 6) + 6)
5
7
:= (77 + 7)/7 × ((7 + 7)/7)
:= (((7 + 7) × (777 − 7)) + 7)/7
:= 88/8 + ((8 + 8) × (88 − 8/8 + 8))
:= 8 × 8 × (8 + 8 + 8)
:= 8 + ((8 × 8 × (8 + 8 + 8) − (88/8)) + 8)
:= 9/9 + ((9 × (9 × (9 + 9) + 9)) − 9)
:= ((9 + 9 + 9)/9) × (((9 + 9)/9)9 )
:= (9 + 9)/9 + (9 × (9 × (9 + 9) + 9))
:= 77 + (7777/7 + 7 × 7 × 7)
Ï 1532 := 11 + ((1 + 1 + 1) × (1 + 1 + 11))1+1
:= 2 × ((2 × (2
2+2
× (22 + 2))) − 2)
Ï 1537 := 1 + ((1 + 1 + 1) × (1 + 1)11−1−1 )
:= 2/2 + ((2 + 2 + 2) × 2
2×(2+2)
)
Ï 1542 := 1 + (1 + ((111 − 1) × (1 + (1 + 1 + 11))))
:= (2 × 22 − 2)2 − 222
:= 3 + (33 × (3 + 3) + ((33/3)3 ))
:= 3/3 + (3 × ((3 − 3/3)3×3 ))
:= 3 + ((3 × ((3 − 3/3)3×3 )) + 3)
:= (4 × (4 + 4) × (44 + 4)) − 4
:= 4/4 + (4 × (4 + 4) × (44 + 4))
:= ((4 + 4)/4 + 4) × (4/4 + 44 )
:= ((5 + 5)/5)5 + (5 × (5 × (55 + 5)))
:= ((55 − 5/5)/((5 + 5)/5)) − 5 × 5
:= 5 + (((55 − 5/5)/((5 + 5)/5)) − 5 × 5)
:= 6 + ((6/6 + 6) × (6 × 6 × 6 + (6 + 6)/6))
:= 6 × 6 × (6 × 6 + 6 + 6) − 66/6
:= 6 × 6 × (6 × 6 + 6 + 6) − 6
7
:= ((7 + 7)/7) × (777 − (77/7))
:= (((77 + 7) × ((7 + 7)/7) ) + 7)/7
:= ((7 + 7)/7) × ((777 − 7) + 7/7)
:= 8 × 8 × (8 + 8 + 8) − (8/((8 + 8)/8))
:= 8/8 + 8 × 8 × (8 + 8 + 8)
:= 8 + (8 × 8 × (8 + 8 + 8) − ((8 + 8)/8))
:= (9 + 9)/9 + ((9 × (9 × (9 + 9) + 9)) − 9)
:= (9 × (9 × (9 + 9) + 9)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 × (9 × (9 + 9) + 9))
112
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1548 := 1 + ((1 + 1 + 11) × (111+1 − (1 + 1)))
Ï 1543 := (111 × (1 + (1 + 1 + 11))) − 11
2
2×(2+2)
Ï 1553 := (111 × (1 + (1 + 1 + 11))) − 1
:= 222 + (22/2)2/2+2
:= 22 + ((2 × (22 − 2) − 2/2) )
:= (2 + 2 + 2) × (2
:= ((((3 × (3 + 3)) + 3)3 ) − 3)/(3 + 3)
:= 3 + (3 × (((3 − 3/3)3×3 ) + 3))
:= 3 + (33/3)3 + (3 + 3)3 + 3
:= 4 + (((4 × (4 + 4) × (44 + 4)) − 4/4) + 4)
:= 44 + (((4 + 4)/4 + 4)4 − 4)
:= 4/4 + (((4 + 4)/4 + 4)4 + 44 )
:= 5 + (((55 + 5/5)/((5 + 5)/5)) − 5 × 5)
:= (((5/5 + 5)5 − (55/5))/5) − 5
:= ((5/5 + 5)5 − (55/5))/5
:= 6/6 + (6 × 6 × (6 × 6 + 6 + 6) − 6)
:= 6 × (6 × (6 × 6 + 6) + 6)
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) − 6/6
:= (((7 + 7) × 777) − 77)/7
:= ((((7 + 7) × 777) + 7)/7) − 7
:= (((7 + 7) × 777) − 7)/7
:= 8 + (8 × 8 × (8 + 8 + 8) − 8/8)
:= ((88 + 8)/8) + 8 × 8 × (8 + 8 + 8)
:= 8 + ((8 × 8 × (8 + 8 + 8) + 8/8) + 8)
:= 999 + (((99 × 99) − 9)/(9 + 9))
:= 9 + (9 × (9 × (9 + 9) + 9))
:= 9 + ((((99 × 99) + 9)/(9 + 9)) + 999)
Ï 1544 := 1 + ((111 × (1 + (1 + 1 + 11))) − 11)
2
2
3
3
+ 2)
Ï 1549 := ((1 + 1 + 11) × (111+1 − 1)) − 11
2
:= 2 × 22 + ((22 + 2) )
Ï 1554 := 111 × (1 + (1 + 1 + 11))
:= 2 + ((22/2 + 2) × ((22/2) − 2))
:= 222 × ((2/2 + 2 + 2) + 2)
:= (3 + 3) + (((33/3) ) − 3)
:= ((((3 × (3 + 3)) + 3)3 ) + 33)/(3 + 3)
:= 3 × ((((3 − 3/3)3×3 ) + 3) + 3)
:= 4 + ((4 × (4 + 4) × (44 + 4)) + 4)
:= 4/4 + ((((4 + 4)/4 + 4)4 − 4) + 44 )
:= 444 + (4444 − 4)/4
:= ((5/5 + 5) − (55 + 5/5))/5
:= (5 × (55 − 5)/(5 + 5)) − 55/5
:= ((5/5 + 5)5 − (5/5 + 5))/5
:= (6 + 6)/6 + 6 × (6 × (6 × 6 + 6) + 6) − 6
:= 6/6 + 6 × (6 × (6 × 6 + 6) + 6)
:= 6 + 6 × (6 × (6 × 6 + 6) + 6)
5
:= ((((7 + 7) × 777) − 77) + 7)/7
:= (((7 + 7)/7) × (777 + 7/7)) − 7
:= (7 + 7) × 777/7
:= 8 + 8 × 8 × (8 + 8 + 8)
:= 8 × 8 × (8 + 8 + 8) + (88 + 8 + 8)/8
:= 888/8 × ((8 − (8 + 8)/8) + 8)
:= 999 + (((99 × 99) + 9)/(9 + 9))
:= 9 + ((9 × (9 × (9 + 9) + 9)) + 9/9)
:= 9 + ((((9 + 9 + 9)/9) × (((9 + 9)/9)9 )) + 9)
Ï 1545 := 1 + (1 + ((111 × (1 + (1 + 1 + 11))) − 11))
2
:= 2 + (((2 × (22 − 2) − 2/2) ) + 22)
Ï 1550 := (11 − 1) × (11 + (1 + 11)1+1 )
Ï 1555 := 1 + (111 × (1 + (1 + 1 + 11)))
:= 2 + ((2 + 2 + 2) × (22×(2+2) + 2))
:= 2 × 222 + 2222/2
:= 3 + (((33/3)3 ) + (3 + 3)3 )
:= ((333 × (33/3 + 3)) + 3)/3
:= 4 + ((((4 + 4)/4 + 4) × (4/4 + 44 )) + 4)
:= 444 + 4444/4
:= 5 + (5 × ((5 + 5)/(5 + 5) − 5))
:= 5 × (((5 × (55 + 5)) + 5) + 5)
:= 55 + (5 × (5 × (55 + 5)))
:= 6 × 6 × (6 × 6 + 6) + (66 × 6/(6 + 6))
:= (6 + 6)/6 + 6 × (6 × (6 × 6 + 6) + 6)
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) + 6/6
:= (((7 + 7)/7) × (777 − 7/7)) − 7
:= 7 + ((((7 + 7) × 777) − 77)/7)
:= (((7 + 7) × 777) + 7)/7
3×3
:= 3 × (((3 − 3/3)
) + 3)
:= 4 + (((4 × (4 + 4) × (44 + 4)) + 4/4) + 4)
5
:= 8 + (8 × 8 × (8 + 8 + 8) + 8/8)
9
:= 9 + (((9 + 9 + 9)/9) × (((9 + 9)/9) ))
Ï 1546 := ((1 + 1 + 11) × (111+1 − (1 + 1))) − 1
:= 8 + ((8 × 8 × (8 + 8 + 8) − ((8 + 8)/8)) + 8)
:= 8 + (8 × 8 × (8 + 8 + 8) + (88/8))
:= 99/9 + (9 × (9 × (9 + 9) + 9))
:= 99 + (9 × 9 × (9 + 9) − ((9 + 9)/9))
Ï 1551 := 11 × ((1 + 11)1+1 − (1 + 1 + 1))
Ï 1556 := 1 + (1 + (111 × (1 + (1 + 1 + 11))))
:= 2 + (2 × 222 + ((22 + 2)2 ))
:= 222 + ((22/2)2/2+2 − 2)
:= 2 × ((2 × (22 − 2))2 − 22)
:= 3 + (((((3 × (3 + 3)) + 3)3 ) − 3)/(3 + 3))
:= 33 × (33/3 + 33 + 3)
:= 3 × 3 + (((33/3)3 ) + (3 + 3)3 )
:= 4 + (((4 + 4)/4 + 4) × (4/4 + 44 ))
:= 44 + (((4 + 4)/4 + 4)4 − 4/4)
:= 4 + (((4 + 4)/4 + 4)4 + 44 )
:= 55 − ((5 − 5/5)5 + 555)
:= 5/5 + (5 × (((5 × (55 + 5)) + 5) + 5))
:= (((5/5 + 5)5 − 5/5) + 5)/5
:= 6 × (6 × (6 × 6 + 6) + 6) − (6 + 6)/6
:= 66/6 × ((666/6 − 6) + 6 × 6)
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) + (6 + 6)/6
:= ((((7 + 7) × 777) − 7)/7) − 7
:= (((7 + 7) × (777 − 7)) + 77)/7
:= ((7 + 7)/7) × (777 + 7/7)
:= 8 + (8 × 8 × (8 + 8 + 8) + ((8 + 8)/8))
:= 8 + ((8 × 8 × (8 + 8 + 8) − 8/8) + 8)
:= 8 + (8 × 8 × (8 + 8 + 8) + ((88 + 8)/8))
:= ((99 + 9)/9) + (9 × (9 × (9 + 9) + 9))
:= 99 + (9 × 9 × (9 + 9) − 9/9)
:= 9 + ((9 × (9 × (9 + 9) + 9)) − ((9 + 9)/9))
Ï 1547 := (1 + 1 + 11) × (111+1 − 1 − 1)
2
:= (22/2 + 2) × ((22/2) − 2)
Ï 1552 := (111 × (1 + (1 + 1 + 11))) − 1 − 1
:= 2 × (2 × (2 × (((2
2+2
2
Ï 1557 := 1 + (1 + (1 + (111 × (1 + (1 + 1 + 11)))))
− 2) ) − 2)))
:= 2 + 2222/2 + 2 × 222
3
:= (3 + 3)3 + ((33/3)3 )
:= 3 + ((3 × (3 + 3) + 3) + 33)/(3 + 3)
:= 3 × ((3 × (3 + 3) × 33 ) + 33)
:= 44/4 + (4 × (4 + 4) × (44 + 4))
:= 4 × ((4 + 4) × (44 + 4) + 4)
:= 4 + ((((4 + 4)/4 + 4)4 + 44 ) + 4/4)
:= ((55 − 5/5)/((5 + 5)/5)) − (5 + 5 + 5)
:= ((55 − 5/5)/((5 + 5)/5)) − 5 − 5
:= (55 − (55/5))/((5 + 5)/5)
:= 6 × (6 × (6 × 6 + 6) + 6) − 6/6
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) − (6 + 6)/6
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) + 6 × 6/(6 + 6)
:= 777 + (777 − 7)
:= ((7 + 7)/7) × (777 − 7/7)
:= (((((7 + 7) × 777) + 7) + 7) + 7)/7
:= 88/8 + 8 × 8 × (8 + 8 + 8)
:= 8 + (8 × 8 × (8 + 8 + 8) + 8)
:= (88/8 − 8) × ((8 × 8 × 8 − 8/8) + 8)
:= 9 + ((9 × (9 × (9 + 9) + 9)) − 9/9)
:= (9 × (9 × (9 + 9) + 9)) + ((99 + 9 + 9)/9)
:= 99 + 9 × 9 × (9 + 9)
113
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1558 := ((1 + 1 + 11) × (111+1 − 1)) − 1 − 1
2
Ï 1563 := 1 + (11 × ((1 + 11)1+1 − (1 + 1)))
22/2
2
Ï 1568 := (1 + 111) × (1 + (1 + 1 + 11))
:= 2 × ((22 + 2 + 2 + 2)2 )
:= 2 + 2 × (2 × (22 − 2) − 22)
:= 2
:= 3 + (((333 × (33/3 + 3)) + 3)/3)
:= 3 × (((3 − 3/3)3×3 ) + 3 × 3)
:= (33/3 + 3) × ((333 + 3)/3)
:= 4 + ((4444 − 4)/4 + 444)
:= 4 + ((4444/4 + 444) + 4)
:= (4 + 4) × ((4 × (44 + 4)) + 4)
:= ((5 + 5/5)/((5 + 5)/5)) − 5
:= (55 + 5/5)/((5 + 5)/5)
:= 5 + ((55 + 5/5)/((5 + 5)/5))
:= 6 × (6 × (6 × 6 + 6) + 6) + (66 − 6)/6
:= ((66 × (66 + 66)) + 666)/6
:= (6/6 + 6) × ((6 × 6 × 6 + (6 + 6)/6) + 6)
5
− (22 + 2/2)
:= ((7 + 7)/7) × (((7 + 7)/7) + 777)
:= 7 + (((7 + 7)/7) × (777 + 7/7))
:= 7 × (7 × (7 + 7 + 7) + 77)
:= ((8 + 8)/8) × (8 × (88 + 8) + (88/8))
:= 8 + ((8 × 8 × (8 + 8 + 8) + (88/8)) + 8)
:= 8 + ((8 + 8 + 8) × (8/8 + 8 × 8))
:= 9/9 + (9 × 9 × (9 + 9) + 99)
:= ((9 + 9 + 9)/9) × ((((9 + 9)/9)9 ) + 9)
:= 99 + (9 × 9 × (9 + 9) + (99/9))
Ï 1559 := ((1 + 1 + 11) × (111+1 − 1)) − 1
Ï 1564 := (1 + 1)11 − ((11 + 11)1+1 )
Ï 1569 := 1 + ((1 + 111) × (1 + (1 + 1 + 11)))
:= 2/2 + (2 × ((22 + 2 + 2 + 2)2 ))
:= (2 × (222 + 2)) + 2222/2
:= 2
:= 3 + ((((33/3)3 ) + (3 + 3)3 ) + 3 × 3)
:= 33 + ((3 × ((3 − 3/3)3×3 )) + 3/3)
:= 33 + (3 × ((3 − 3/3)3×3 ))
:= 4 + (4444/4 + 444)
:= 4 + ((44 + 4) × ((4 + 4)/4 + 4))
:= 4/4 + ((4 + 4) × ((4 × (44 + 4)) + 4))
:= (5 × (55 − 5)/(5 + 5)) − 5/5
:= 6 × (6 × (6 × 6 + 6) + 6) + 66/6
:= 7 + (((7 + 7)/7) × (777 − 7/7))
:= 8 × (8 × 8 − 8) + 8888/8
:= 9 + ((9 × (9 × (9 + 9) + 9)) + (99/9))
Ï 1560 := (1 + 1 + 11) × (111+1 − 1)
2
22/2
− 22
2
:= (5 × (55 + 5)/(5 + 5)) − 5/5
:= 5 + ((5 × (55 + 5)/(5 + 5)) − 5/5)
:= (6 − ((6 + 6)/6)) × ((6 × 66 − 6) + 6/6)
:= 6 + (((66 × (66 + 66)) + 666)/6)
:= ((((7 + 7) × 777) − 7) + 77)/7
:= (((7 + 7) × (777 + 7)) + 7)/7
:= (8/8 + 8 + 8) × ((8/((8 + 8)/8)) + 88)
:= 8 + (((8 + 8 + 8) × (8/8 + 8 × 8)) + 8/8)
:= ((9 − 9/9) + 9) × ((99/9) + 9 × 9)
:= 9 × 9 × (9 + 9) + 999/9
Ï 1565 := 11 + (111 × (1 + (1 + 1 + 11)))
Ï 1570 := 1 + (1 + ((1 + 111) × (1 + (1 + 1 + 11))))
:= 2/2 + (222/2 − 222 )
:= 2 + (2 × ((22 + 2 + 2 + 2)2 ))
:= ((33/3)3 ) + (3 × (3 × 33 − 3))
:= 33 + ((3 × (3 + 3) + 3)3 − 3)/(3 + 3)
:= 4 + (((4 + 4)/4 + 4) × ((4/4 + 44 ) + 4))
:= 5 × (55 − 5)/(5 + 5)
:= 4 + (((44 + 4) × ((4 + 4)/4 + 4)) + 4/4)
:= 5 × (55 + 5)/(5 + 5)
:= 5 + (5 × (55 + 5)/(5 + 5))
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) + 6
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) + 66/6
:= ((6 + 6)/6) × (66 × (6 + 6) − (6/6 + 6))
:= 7 + ((((7 + 7) × 777) − 7)/7)
:= (((7 + 7) × 777) + 77)/7
:= ((7 + 7)/7) × ((777 + 7/7) + 7)
:= (8 + 8 + 8) × (8/8 + 8 × 8)
:= ((88/8 + 8) × (88 − 8/8)) − 88
:= 8 + ((888/8 × ((8 − (8 + 8)/8) + 8)) + 8)
:= 9 × 9 × (9 + 9) + (999/9 − 9)
:= 9 + ((9 × 9 × (9 + 9) − 9/9) + 99)
:= 9 × 9 × (9 + 9) + ((999 + 9)/9)
:= 2 × (2 × (22 − 2) − 22 + 2)
:= (33 − 3/3) × (33 + 33)
:= (44 + 4) × ((4 + 4)/4 + 4)
Ï 1561 := 1 + ((1 + 1 + 11) × (111+1 − 1))
22/2
2
Ï 1566 := 1 + (11 + (111 × (1 + (1 + 1 + 11))))
Ï 1571 := (11 × (11 × (1 + 1 + 11))) − 1 − 1
:= 2 + (222/2 − 222 )
:= ((22/2 + 2) × (22/2)2 ) − 2
:= 3/3 + ((33 − 3/3) × (33 + 33))
:= 3 × ((3 + 3) × (3 × 33 + 3 + 3))
:= 3 × 3 × 33 + (33/3)3 − 3
:= 4/4 + ((44 + 4) × ((4 + 4)/4 + 4))
:= ((4 + 4)/4 + 4) × ((4/4 + 44 ) + 4)
:= 4 + (((4 + 4) × ((4 × (44 + 4)) + 4)) − 4/4)
:= 2
− ((22 + 2/2) + 2)
:= 5/5 + (5 × (5 − 5)/(5 + 5))
:= 5/5 + (5 × (55 + 5)/(5 + 5))
:= 5 + ((5 × (55 + 5)/(5 + 5)) + 5/5)
:= (6/6 + 6) × (6 × 6 × 6 + 6/6 + 6)
:= 6 + 6 × (6 × (6 × 6 + 6) + 6) + 6 + 6
:= (((6 + 6) × (66 × (6 + 6) − 6)) − 6)/6
:= 7 + (777 + 777)
:= ((7 + 7)/7) × ((777 − 7/7) + 7)
:= 7 + (((((7 + 7) × 777) − 7) + 77)/7)
:= 8/8 + ((8 + 8 + 8) × (8/8 + 8 × 8))
:= (8/8 − 88) × ((8 − 88)/8 − 8)
:= 88/8 + ((8 + 8 + 8) × (8/8 + 8 × 8))
:= 9 × 9 × (9 + 9) + (((999 + 9)/9) − 9)
:= 9 + (9 × 9 × (9 + 9) + 99)
:= 9 × 9 × (9 + 9) + ((999 + 9 + 9)/9)
5
Ï 1562 := 11 × ((1 + 11)1+1 − (1 + 1))
Ï 1567 := ((1 + 111) × (1 + (1 + 1 + 11))) − 1
Ï 1572 := (1 + 11) × ((11 × (1 + 11)) − 1)
:= (2 × ((22 + 2 + 2 + 2)2 )) − 2/2
:= 2 × (((22 + 2 + 2 + 2)2 ) + 2)
:= 33 + ((3 × ((3 − 3/3)3×3 )) − 3/3)
:= 3/3 + (3 × ((3 + 3) × (3 × 33 + 3 + 3)))
:= 3 + ((3 × ((3 − 3/3)3×3 )) + 33)
:= 4 × 44 + (44 × (44 − 4)/(4 + 4))
:= ((4 + 4) × ((4 × (44 + 4)) + 4)) − 4/4
:= 4 + ((4 + 4) × ((4 × (44 + 4)) + 4))
:= 2
22/2
2
− (22 + 2)
:= (5 − 5/5)/((5 + 5)/5)
:= 5 + ((5 − 5/5)/((5 + 5)/5))
:= 5 + (((55 − 5/5)/((5 + 5)/5)) + 5)
:= 66/6 × ((6 + 6) × (6 + 6) − ((6 + 6)/6))
:= 6 + ((6/6 + 6) × (6 × 6 × 6 + 6/6 + 6))
:= 66 + (6 × 6 × (6 × 6 + 6) − 6)
:= 7 + ((((7 + 7) × 777) + 7)/7)
:= (((7 + 7) × (777 + 7)) − 7)/7
:= 7 + ((((7 + 7) × 777) + 77)/7)
:= 8 + (888/8 × ((8 − (8 + 8)/8) + 8))
:= 8 + (8888/8 + 8 × (8 × 8 − 8))
:= (((8 × 8 − 8)(8+8)/8 ) + 8)/((8 + 8)/8)
:= 99/9 × (9 × (9 + 9) − (99/9 + 9))
:= 9 + ((9 × 9 × (9 + 9) + 99) + 9/9)
:= 9 + (((9 + 9 + 9)/9) × ((((9 + 9)/9)9 ) + 9))
5
5
114
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1573 := 11 × (11 × (1 + 1 + 11))
Ï 1578 := 11 + (((1 + 111) × (1 + (1 + 1 + 11))) − 1)
Ï 1583 := (11 × (1 + 11)1+1 ) − 1
:= (22/2 + 2) × (22/2)2
:= (2 × (22 − 2))2 − 22
:= (2 × (22 × (2 + 2 + 2)2 )) − 2/2
:= ((33/3)3 ) + ((33+3 − 3)/3)
:= 33 + (3 × (((3 − 3/3)3×3 ) + 3))
:= ((33/3)3 ) + (3 × (3 × 33 + 3))
:= (44 × (4 × (4 + 4) + 4)) − 44/4
:= ((4 + 4)/4 + 4) × (((44 − 4/4) + 4) + 4)
:= (44 × (4 × (4 + 4) + 4)) − 4/4
:= 5 + (((55 + 5/5)/((5 + 5)/5)) + 5)
:= 555 + ((5 − 5/5)5 − 5/5)
:= 5 + (((5 − 5/5)5 − 5/5) + 555)
:= 66/6 × ((6 + 6) × (6 + 6) − 6/6)
:= 66 + 6 × 6 × (6 × 6 + 6)
:= (66 × ((6 + 6 + 6) + 6)) − 6/6
:= (7 − 7/7 + 7) × (((7 + 7)/7) − 7)
:= (7 − 7/7) × ((((7 + 7)/7)7+7/7 ) + 7)
:= (((7 + 7) × (777 + 7 + 7)) + 7)/7
:= 888 + (8 × 88 − (88/8 + 8))
:= 888 + ((((8 + 8)/8) − (8 + 8)) + 8 × 88)
:= 888 + (8 × 88 − (8/8 + 8))
:= 9 + (((9 − 9/9) + 9) × ((99/9) + 9 × 9))
:= 9 + (9 × 9 × (9 + 9) + 999/9)
:= ((9 + 9) × (99 − (99/9))) − 9/9
7
Ï 1574 := 1 + (11 × (11 × (1 + 1 + 11)))
Ï 1579 := 11 + ((1 + 111) × (1 + (1 + 1 + 11)))
Ï 1584 := 11 × (1 + 11)1+1
:= 2 + (2 × (((22 + 2 + 2 + 2)2 ) + 2))
:= 2/2 + ((2 × (22 − 2))2 − 22)
:= 2 × (22 × (2 + 2 + 2)2 )
:= 3 × 3 × 33 + ((33/3)3 )
:= (((3 + 3) × ((33 × (33 − 3)) − 3)) + 3)/3
:= 33 × (3 × 33 − 33)
:= 44 + (((4 + 4)/4 + 4) × (44 − 4/4))
:= (44 × (4 × (4 + 4) + 4)) − (4/4 + 4)
:= 44 × (4 × (4 + 4) + 4)
:= 555 + ((5 − 5/5)5 − 5)
:= 555 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 555)
:= ((6 + 6)/6) × ((66 × (6 + 6) − 6) + 6/6)
:= 66 + (6 × 6 × (6 × 6 + 6) + 6/6)
:= 66 × ((6 + 6 + 6) + 6)
:= 7 + ((((7 + 7) × (777 + 7)) − 7)/7)
:= (((7 + 7) × (777 + 7)) + 77)/7
:= (77/7 + 7) × (77/7 + 77)
:= 8 + ((8/8 − 88) × ((8 − 88)/8 − 8))
:= 8 + (((8 + 8 + 8) × (8/8 + 8 × 8)) + (88/8))
:= 88 × ((((8 + 8)/8) + 8) + 8)
:= 9 + (((9 × 9 × (9 + 9) − 9/9) + 99) + 9)
:= 9 + (((999 + 9)/9) + 9 × 9 × (9 + 9))
:= (9 + 9) × (99 − (99/9))
Ï 1585 := 1 + (11 × (1 + 11)1+1 )
Ï 1575 := 1 + (1 + (11 × (11 × (1 + 1 + 11))))
Ï 1580 := (11 − 1) × ((1 + 1 + 11)1+1 − 11)
:= 2 + ((22/2 + 2) × (22/2)2 )
:= 2 + ((2 × (22 − 2))2 − 22)
:= 2/2 + (2 × (22 × (2 + 2 + 2)2 ))
:= 3 × (((3 + 3) × (3 × 33 + 3 + 3)) + 3)
:= 33 + (((33/3)3 ) + (3 + 3)3 )
:= 3/3 + (33 × (3 × 33 − 33))
:= ((4/4 + 4) + 4) × (4 × 44 − 4/4)
:= (44 × (4 × (4 + 4) + 4)) − 4
:= 4/4 + (44 × (4 × (4 + 4) + 4))
:= 5 × ((5 × 5 + 55 )/(5 + 5))
:= 5 + (5 × ((5 × 5 + 55 )/(5 + 5)))
:= 5 × ((55 − 5)/(5 + 5) + 5)
:= (6 × 6 − 6/6) × (666/6 − 66)
:= (6 − ((6 + 6)/6)) × (6 × 66 − 6/6)
:= 6/6 + (66 × ((6 + 6 + 6) + 6))
:= 7 + (7 × (7 × (7 + 7 + 7) + 77))
:= 7 + ((7 − 7/7 + 7) × (((7 + 7)/7)7 − 7))
:= 7 × 7 + ((77 + 7)/7 × ((7 + 7)/7)7 )
:= (8/8 + 8) × (888/8 + 8 × 8)
:= 888 + (8 × 88 − ((88 + 8)/8))
:= 8/8 + ((888 − 8) + 8 × 88)
:= 9 + ((9 × 9 × (9 + 9) + 99) + 9)
:= (99/9 + 9) × (9 × 9 − ((9 + 9)/9))
:= 9/9 + ((9 + 9) × (99 − (99/9)))
Ï 1576 := 1 + (1 + (1 + (11 × (11 × (1 + 1 + 11)))))
Ï 1581 := (11 × (1 + 11)1+1 ) − 1 − 1 − 1
Ï 1586 := 1 + (1 + (11 × (1 + 11)1+1 ))
:= (2 × (22 − 2))2 − (22 + 2)
:= (2/2 + 2) × (((22 + 2/2)2 ) − 2)
:= 2 + (2 × (22 × (2 + 2 + 2)2 ))
:= 3 + (((33+3 − 3)/3) + ((33/3)3 ))
:= (33 × (3 × 33 − 33)) − 3
:= (33 − 3/3) × (((3/3 + 3)3 ) − 3)
:= (44 × (4 × (4 + 4) + 4)) − 4 − 4
:= 4/4 + ((44 × (4 × (4 + 4) + 4)) − 4)
:= (4 + 4)/4 + (44 × (4 × (4 + 4) + 4))
:= 5/5 + (5 × ((5 × 5 + 55 )/(5 + 5)))
:= 5 + ((5 × ((5 × 5 + 55 )/(5 + 5))) + 5/5)
:= 5/5 + (5 × ((55 − 5)/(5 + 5) + 5))
:= ((6 + 6)/6)6 + 6 × 6 × (6 × 6 + 6)
:= (66 × ((6 + 6 + 6) + 6)) − 6 × 6/(6 + 6)
:= (6 + 6)/6 + (66 × ((6 + 6 + 6) + 6))
:= 7 + ((((7 + 7) × (777 + 7)) + 7)/7)
:= (((7 + 7) × (777 + 7 + 7)) − 7)/7
:= (7 × 7 × 77) − (((7 + 7 + 7)/7)7 )
:= 888 + (8 × 88 − (8 + 8))
:= 888 + (8 × 88 − (88/8))
:= 888 + ((((8 + 8)/8) − 8) + 8 × 88)
:= ((9 − 9/9) + 9) × (((99 + 9)/9) + 9 × 9)
:= 9 + ((9/9 + 9 + 9) × (((9 + 9)/9) + 9 × 9))
:= (9 − 9/9) × ((99 − 9/9) + 99)
Ï 1577 := 1 + (1 + (1 + (1 + (11 × (11 × (1 + 1 + 11))))))
Ï 1582 := (11 × (1 + 11)1+1 ) − 1 − 1
Ï 1587 := 1 + (1 + (1 + (11 × (1 + 11)1+1 )))
:= (2 × (22 − 2))2 − (22 + 2/2)
:= (2 × (22 × (2 + 2 + 2)2 )) − 2
:= (2/2 + 2) × ((22 + 2/2)2 )
:= 3 + (3 × 3 × 33 + ((33/3)3 ))
:= 3/3 + ((33 × (3 × 33 − 33)) − 3)
:= 3 + (33 × (3 × 33 − 33))
:= 4 + ((44 × (4 × (4 + 4) + 4)) − 44/4)
:= (44 × (4 × (4 + 4) + 4)) − (4 + 4)/4
:= 44 + ((44/4)4−4/4 )
:= 5 + ((((55 − 5/5)/((5 + 5)/5)) + 5) + 5)
:= 5 × 5 + ((55 − (55/5))/((5 + 5)/5))
:= 5 × 5 + ((55 − 5/5)/((5 + 5)/5))
:= 66 + (6 × 6 × (6 × 6 + 6) − 6/6)
:= ((6 + 6)/6) × (66 × (6 + 6) − 6/6)
:= 666/6 + (6 × (6 × (6 × 6 + 6) − 6))
7
:= (7 × (7 × (7 × 7 − (7 + 7)))) − ((7 + 7)/7)7
:= 7 + (((7 + 7)/7) × ((777 + 7/7) + 7))
:= 7 × (((7 + 7)/7) + 7 × (7 + 7))
:= 8/8 + ((888 − (8 + 8)) + 8 × 88)
:= 888 + ((8 − 88)/8 + 8 × 88)
:= 888 + ((88/8 − (8 + 8)) + 8 × 88)
:= (9/9 + 9 + 9) × (((9 + 9)/9) + 9 × 9)
:= ((9 + 9)/9) × (9 × 99 − (9/9 + 99))
:= 9 + ((9 × 9 × (9 + 9) + 999/9) + 9)
115
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1588 := 1 + (1 + (1 + (1 + (11 × (1 + 11)1+1 ))))
Ï 1593 := (11 × (1 + (1 + 11)1+1 )) − 1 − 1
Ï 1598 := 1 + (1 + (1 + (11 × (1 + (1 + 11)1+1 ))))
:= 2 × ((22 × (2 + 2 + 2)2 ) + 2)
:= 2/2 + (2 × (2 × ((22 − 2)2 − 2)))
:= (2 × (22 − 2))2 − 2
:= 3 + ((33 × (3 × 33 − 33)) + 3/3)
:= 3 × (33 × (3 + 3) + 333)
:= 33 × 33 + (((3 − 3/3)3×3 ) − 3)
:= 4 + (44 × (4 × (4 + 4) + 4))
:= ((4/4 + 4) + 4) × (4 × 44 + 4/4)
:= (4 × (444 − 44)) − (4 + 4)/4
:= 5 × 5 + ((55 + 5/5)/((5 + 5)/5))
:= (6 − ((6 + 6)/6)) × (6 × 66 + 6/6)
5
:= (5 × ((5 + 5) × ((5 + 5)/5)5 )) − (5 + 5)/5
:= 5 + (((5 + 5/5)/((5 + 5)/5)) + 5 × 5)
6
:= (6 × (6 + 6) × (6 + 6)) + ((6 × 6/(6 + 6)) )
7
:= ((6 + 6)/6) × ((66 × (6 + 6) + 6/6) + 6)
:= 7 + ((((7 + 7) × (777 + 7 + 7)) − 7)/7)
:= 7 + ((7 × 7 × 77) − (((7 + 7 + 7)/7) ))
:= ((7 + 7)/7) + ((7 + 7 + 7) × (77 − 7/7))
:= 888 + (8 × 88 − (8/((8 + 8)/8)))
:= 8/8 + (888 + 8 × 88)
:= 8 × (8 × (8 + 8 + 8) + 8) − (8 + 8)/8
:= 9 + ((((999 + 9)/9) + 9 × 9 × (9 + 9)) + 9)
:= 9 + ((9 + 9) × (99 − (99/9)))
:= ((9 + 9)/9) × (9 × (9 × 9 + 9) − (99/9))
Ï 1589 := (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 ) − 11
Ï 1594 := (11 × (1 + (1 + 11)1+1 )) − 1
Ï 1599 := (1 + 1 + 11) × (1 + (1 + 111+1 ))
:= (2 × (22 − 2))2 − 22/2
:= 2 + (2 × (2 × ((22 − 2)2 − 2)))
:= (2 × (22 − 2))2 − 2/2
:= 3 + ((33 − 3/3) × (((3/3 + 3)3 ) − 3))
:= 3/3 + (3 × (33 × (3 + 3) + 333))
:= ((3 + 3) × ((3 × (3 × (33 + 3))) − 3)) − 3
:= 4 + ((44 × (4 × (4 + 4) + 4)) + 4/4)
:= (44 − 4)/4 + (44 × (4 × (4 + 4) + 4))
:= (4 × (444 − 44)) − 4/4
:= 5 + (((5 − 5/5) + 555) + 5)
:= 55 /5 + ((5 − 5/5)5 − 55)
:= (5 × ((5 + 5) × ((5 + 5)/5)5 )) − 5/5
:= 6 + ((66 × ((6 + 6 + 6) + 6)) − 6/6)
:= 6 + ((6 − ((6 + 6)/6)) × (6 × 66 + 6/6))
:= (6/6 + 6 + 6) × ((666/6 + 6) + 6)
:= ((7 + 7 + 7) × (77 − 7/7)) − 7
:= 7 × (77 − 7) + (7777/7 − 7)
:= (77 × (7 + 7 + 7)) − (77/7 + 7)
:= 8 × (8 × (8 + 8 + 8) + 8) − 88/8
:= 888 + ((8 + 8)/8 + 8 × 88)
:= 8 × (8 × (8 + 8 + 8) + 8) − 8/8
:= 9 + ((99/9 + 9) × (9 × 9 − ((9 + 9)/9)))
:= 9 + (((9 + 9) × (99 − (99/9))) + 9/9)
:= (9 × (99 + 9 × 9)) − (((99 + 9)/9) + 9)
5
Ï 1590 := (1 + 1 + 1) × (1 + (1 + 11 + 11)1+1 )
Ï 1595 := 11 × (1 + (1 + 11)1+1 )
Ï 1600 := ((1 + 1) × ((1 + 1) × (11 − 1)))1+1
:= (2 × (22 − 2))2
:= (2 × (2 × ((22 − 2)2 − 2))) − 2
:= 222 + 2222/2
:= 3 + ((33 × (3 × 33 − 33)) + 3)
:= 33/3 + (33 × (3 × 33 − 33))
:= ((3/3 + 3)3 ) × ((3/3 − 3) + 33 )
:= 4 + ((44 × (4 × (4 + 4) + 4)) + (4 + 4)/4)
:= 44/4 + (44 × (4 × (4 + 4) + 4))
:= 4 × (444 − 44)
:= 5 × ((55 + 5)/(5 + 5) + 5)
:= 55 × (5 × 5 − 5/5 + 5)
:= 5 × ((5 + 5) × ((5 + 5)/5)5 )
:= 6 + (66 × ((6 + 6 + 6) + 6))
:= 66/6 + (66 × ((6 + 6 + 6) + 6))
:= ((6 + 6)/6)6 × (6 × 6 − 66/6)
:= 7 + ((((7 + 7) × (777 + 7 + 7)) + 7)/7)
:= ((7 + 7 + 7) × (77 − 7/7)) − 7/7
:= (7 × 7 − ((7 + 7)/7 + 7))(7+7)/7
:= 888 + (8 × 88 − ((8 + 8)/8))
:= 888 + ((8 × 88 − 8) + (88/8))
:= 8 × (8 × (8 + 8 + 8) + 8)
:= (9/9 + 9) × (9 × (9 + 9) − ((9 + 9 + 9)/9))
:= 99/9 + ((9 + 9) × (99 − (99/9)))
:= (99/9 + 9) × (9 × 9 − 9/9)
Ï 1591 := 1 + ((1 + 1 + 1) × (1 + (1 + 11 + 11)1+1 ))
2
Ï 1596 := 1 + (11 × (1 + (1 + 11)1+1 ))
Ï 1601 := 1 + (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 )
:= 2 × ((2 × (22 − 2))2 − 2)
:= 2/2 + (2 × (22 − 2))2
:= 3 + (3 × (33 × (3 + 3) + 333))
:= 33 × 33 + ((3 − 3/3)3×3 )
:= (4 × (444 − 44)) − 4
:= 4/4 + (4 × (444 − 44))
:= 5/5 + (5 × ((5 + 5)/(5 + 5) + 5))
:= 5/5 + (55 × (5 × 5 − 5/5 + 5))
:= 5/5 + (5 × ((5 + 5) × ((5 + 5)/5)5 ))
:= 6 + ((66 × ((6 + 6 + 6) + 6)) + 6/6)
:= 6 + ((66 × ((6 + 6 + 6) + 6)) + 6)
:= 6 + ((66 × ((6 + 6 + 6) + 6)) + (66/6))
:= 7 + ((77/7 + 7) × (77/7 + 77))
:= (7 + 7 + 7) × (77 − 7/7)
:= 7 × (77 − 7) + 7777/7
:= 888 + (8 × 88 − 8/8)
:= 888 + ((8/((8 + 8)/8)) + 8 × 88)
:= 8/8 + 8 × (8 × (8 + 8 + 8) + 8)
:= ((99/9 + 9) × (9 × 9 − 9/9)) − 9
:= (9/9 + 9 + 9) × (((9 + 9 + 9)/9) + 9 × 9)
:= (((9 + 9)/9)9 ) + (99 × (99/9))
:= 2 + ((2 × (22 − 2)) − 22/2)
3
:= (((3 + 3) × ((33 × (3 − 3)) + 3)) + 3)/3
:= 4 + (((44/4)
4−4/4
4
)+4 )
5
Ï 1592 := (11 × (1 + (1 + 11)1+1 )) − 1 − 1 − 1
Ï 1597 := 1 + (1 + (11 × (1 + (1 + 11)1+1 )))
Ï 1602 := 1 + (1 + (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 ))
:= 2 × (2 × ((22 − 2)2 − 2))
:= (2 × (22 − 2))2 − 2/2 − 2
:= 2 + (2 × (22 − 2))2
:= ((33/3)3 ) + (3 × (3 × 33 + 3 + 3))
:= (3 × (3 + 3) × 33 ) + 3333/3
:= (3 + 3) × ((3 × (3 × (33 + 3))) − 3)
:= 4 + ((44 × (4 × (4 + 4) + 4)) + 4)
:= 4/4 + ((4 × (444 − 44)) − 4)
:= (4 + 4)/4 + (4 × (444 − 44))
:= 5 + (((5 − 5/5)/((5 + 5)/5)) + 5 × 5)
:= (5 + 5)/5 + (55 × (5 × 5 − 5/5 + 5))
:= (5 + 5)/5 + (5 × ((5 + 5) × ((5 + 5)/5)5 ))
:= 6 + ((66 × ((6 + 6 + 6) + 6)) + ((6 + 6)/6))
5
:= (((6 + 6) × (66 × (6 + 6) + 6)) + 6)/6
:= 6 + (((66 × ((6 + 6 + 6) + 6)) + 6) + 6)
:= 7 × 7 + ((((7 + 7) × 777) − 77)/7)
:= 7/7 + ((7 + 7 + 7) × (77 − 7/7))
:= 7 × 7 + ((((7 + 7) × 777) − 7)/7)
:= 888 + 8 × 88
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) − (88/8))
:= (8 + 8)/8 + 8 × (8 × (8 + 8 + 8) + 8)
:= 9 × 9 + ((((9 + 9)/9)9 ) + 999)
:= ((9 + 9) × (9 + 9 + 9)) + 9999/9
:= (9 + 9) × ((9 × 9 − 9/9) + 9)
116
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1603 := 1 + (1 + (1 + (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 )))
Ï 1608 := (1 + 11) × (1 + (1 + (11 × (1 + 11))))
Ï 1613 := 1 + ((1 + 1 + 11) × (1 + (1 + (1 + 111+1 ))))
:= 2 + ((2 × (22 − 2))2 + 2/2)
:= 2 × (2 × ((22 − 2)2 + 2))
:= 2 + ((2 × (22 − 2))2 + 22/2)
:= (3 × 3 + 3)3 − ((3 − 3/3 + 3)3 )
:= (33 − 3) × (((3/3 + 3)3 ) + 3)
:= 3 + ((((3 × (3 + 3))3 − 3)/3) − 333)
:= 4 + ((4 × (444 − 44)) − 4/4)
:= 4 + ((4 × (444 − 44)) + 4)
:= 4/4 + ((4 × ((444 − 44) + 4)) − 4)
:= 5 + ((5 × ((5 + 5) × ((5 + 5)/5)5 )) − ((5 + 5)/5))
:= 55 + (((5/5 + 5)5 − (55/5))/5)
:= 55 + (((55 + 5/5)/((5 + 5)/5)) − 5)
:= 6 + ((((6 + 6) × (66 × (6 + 6) + 6)) + 6)/6)
:= (6 − ((6 + 6)/6)) × (6 × 66 + 6)
:= 66 + 6 × (6 × (6 × 6 + 6) + 6) − 6/6
:= (77 × (7 + 7 + 7)) − (7 + 7)
:= (77 × (7 + 7 + 7)) − ((7 + 7)/7 + 7)
:= 7 + ((77 × (7 + 7 + 7)) − (77/7))
:= 888 + (88/8 + 8 × 88)
:= 8 + 8 × (8 × (8 + 8 + 8) + 8)
:= 88 + (8 × 8 × (8 + 8 + 8) − (88/8))
:= 9/9 + ((9 + 9) × ((9 × 9 − 9/9) + 9))
:= (9 × (99 + 9 × 9)) − (99 + 9)/9
:= (9 + 9)/9 + ((9 × (99 + 9 × 9)) − 9)
Ï 1604 := (1 + 1)11 − ((1 + 1) × (1 + 1) × 111)
Ï 1609 := 1 + ((1 + 11) × (1 + (1 + (11 × (1 + 11)))))
Ï 1614 := ((1 + 11)1+1+1 ) − (1 + 1 + 1 + 111)
:= 2 + ((2 × (22 − 2))2 + 2)
:= 2/2 + (2 × (2 × ((22 − 2)2 + 2)))
:= 2 + (2 × ((2 × ((22 − 2)2 + 2)) + 2))
:= 3 + ((3 × (3 × (33 + 3))) + ((33/3)3 ))
:= (33 × (33 + 33)) − 33/3
:= (33 × (33 + 33)) − (3 + 3)
:= 4 + (4 × (444 − 44))
:= 4 + (((4 × (444 − 44)) + 4/4) + 4)
:= (4 × ((444 − 44) + 4)) − (4 + 4)/4
5
5
:= 5 × 5 + ((5 − 5/5) + 555)
:= 5 + (((5 − 5/5) + 555) + 5 × 5)
:= (5/5 + 5) × (5 × 55 − (5/5 + 5))
:= (6 − ((6 + 6)/6)) × ((6 × 66 − 6/6) + 6)
:= 6/6 + ((6 − ((6 + 6)/6)) × (6 × 66 + 6))
:= 66 + 6 × (6 × (6 × 6 + 6) + 6)
:= 7/7 + ((77 × (7 + 7 + 7)) − (7 + 7))
:= (77 × (7 + 7 + 7)) − (7/7 + 7)
:= ((7 + 7 + 7)/7) × (7 × 77 − 7/7)
:= 888 + (((88 + 8)/8) + 8 × 88)
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) + 8/8)
:= 8 × 8 × 8 + ((8888 − 8)/8 − 8)
:= (9 + 9)/9 + ((9 + 9) × ((9 × 9 − 9/9) + 9))
:= (9 × (99 + 9 × 9)) − 99/9
:= (((9 + 9)/9)9 ) + (9999/9 − 9)
Ï 1605 := (11 × (1 + (1 + (1 + 11)1+1 ))) − 1
Ï 1610 := 1 + (1 + ((1 + 11) × (1 + (1 + (11 × (1 + 11))))))
Ï 1615 := ((1 + 11)1+1+1 ) − (1 + 1 + 111)
:= 2 + (((2 × (22 − 2))2 + 2/2) + 2)
:= 2 + (2 × (2 × ((22 − 2)2 + 2)))
:= 2 + (((2 × (22 − 2))2 + 22/2) + 2)
:= 3333 − (3 × 3 + 3)3
:= (((3 × (3 + 3))3 − 3)/3) − 333
:= 3 + ((((3 × (3 + 3))3 + 3)/3) − 333)
:= 4 + ((4 × (444 − 44)) + 4/4)
:= (4/4 + 4) × (((44 + 4 + 4)/4) + 44 )
:= (4 × ((444 − 44) + 4)) − 4/4
5
:= 5 × (((5 − 5)/(5 + 5) + 5) + 5)
:= 5 × (((55 + 5)/(5 + 5) + 5) + 5)
:= 6 + ((6/6 + 6 + 6) × ((666/6 + 6) + 6))
:= (6 × 6 − 6/6) × (((66 − 6)/6) + 6 × 6)
:= 66 + 6 × (6 × (6 × 6 + 6) + 6) + 6/6
:= (77 × (7 + 7 + 7)) − (77 + 7)/7
:= (77 × (7 + 7 + 7)) − 7
:= (77 × (7 + 7 + 7)) − (7 + 7)/7
:= (8 − 8/8 + 8) × ((88/8 + 88) + 8)
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) + ((8 + 8)/8))
:= 8 × 8 × 8 + (8888/8 − 8)
:= 9 + ((9/9 + 9 + 9) × (((9 + 9 + 9)/9) + 9 × 9))
:= (9/9 + 9) × (9 × (9 + 9) − 9/9)
:= ((9 − 99)/(9 + 9)) + (9 × (99 + 9 × 9))
5
:= 5 + (5 × ((5 + 5) × ((5 + 5)/5) ))
Ï 1606 := 11 × (1 + (1 + (1 + 11)1+1 ))
Ï 1611 := 11 + (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 )
2
Ï 1616 := ((1 + 11)1+1+1 ) − (1 + 111)
:= 22/2 + (2 × (22 − 2))2
:= 2 × (2 × (((22 − 2)2 + 2) + 2))
:= 3 + ((3 × 3 + 3) − ((3 − 3/3 + 3) ))
:= 3 × ((3 × (3 × (33 + 33))) − 3)
:= (3 × 3 + 3)3 − ((333 + 3)/3)
:= 4 + ((4 × (444 − 44)) + (4 + 4)/4)
:= 44/4 + (4 × (444 − 44))
:= 4 × ((444 − 44) + 4)
:= 2 + (((2 × (22 − 2)) + 2) + 2)
3
3
5
:= 5 + ((5 × ((5 + 5) × ((5 + 5)/5) )) + 5/5)
:= 555/5 + (5 × (5 × (55 + 5)))
:= 5 + ((5 × (5 × (55 + 5))) + 555/5)
6
:= 6 + (((6 + 6)/6) × (6 × 6 − 66/6))
:= (6 × (6 × 66 − 6)) − ((6 × 6/(6 + 6)) )
:= (((6 + 6)/6)66/6 ) − (6 × (66 + 6))
:= (77 × (7 + 7 + 7)) − 77/7
:= 7/7 + ((77 × (7 + 7 + 7)) − 7)
:= (77 × (7 + 7 + 7)) − 7/7
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) − ((8 + 8)/8))
:= 88/8 + 8 × (8 × (8 + 8 + 8) + 8)
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) + 8)
:= ((9/9 − 9) + 9 × 9) × ((99 + 99)/9)
:= (9 × (99 + 9 × 9)) − 9
:= ((9 + 9)/9) × (9 × (9 × 9 + 9) − ((9 + 9)/9))
6
Ï 1607 := 1 + (11 × (1 + (1 + (1 + 11)1+1 )))
Ï 1612 := (1 + 1 + 11) × (1 + (1 + (1 + 111+1 )))
Ï 1617 := ((1 + 11)1+1+1 ) − 111
:= 222/2 − ((22 − 2/2)2 )
:= 2 × ((2 × ((22 − 2)2 + 2)) + 2)
:= 22 + ((2222/2) + 222 )
:= 33 + (3 × 3 × 33 + ((33/3)3 ))
:= (((3 × (3 + 3))3 + 3)/3) − 333
:= (33 × (33 + 33)) − 3
:= (4 × ((444 − 44) + 4)) − 4
:= 4/4 + (4 × ((444 − 44) + 4))
:= 4 + (((4 × (444 − 44)) − 4/4) + 4)
5
:= 5 + ((5 × ((5 + 5) × ((5 + 5)/5) )) + ((5 + 5)/5))
:= 55 + ((5 − (55/5))/((5 + 5)/5))
:= 55 + ((55 − 5/5)/((5 + 5)/5))
:= ((6 − ((6 + 6)/6)) × (6 × 66 + 6)) − 6/6
:= ((6 + 6)/6)6 + 6 × (6 × (6 × 6 + 6) + 6)
:= 66/6 × (666/6 + 6 × 6)
:= ((7 − 77)/7) + (77 × (7 + 7 + 7))
:= ((7 + 7)/7) + ((77 × (7 + 7 + 7)) − 7)
:= 77 × (7 + 7 + 7)
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) − 8/8)
:= 8 × 8 × 8 + ((8888 − 88)/8)
:= 8 + ((8 × (8 × (8 + 8 + 8) + 8) + 8/8) + 8)
:= (9 × (99 + 9 × 9)) − ((99 + 9 + 9)/9)
:= 9/9 + ((9 × (99 + 9 × 9)) − 9)
:= (9 × (99 + 9 × 9)) − (9 + 9 + 9)/9
5
117
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1618 := 1 + (((1 + 11)1+1+1 ) − 111)
Ï 1623 := 1111 + (1 + 1)11−1−1
Ï 1628 := 11 + (((1 + 11)1+1+1 ) − 111)
:= 2 + ((2 × (22 − 2))2 + 22+2 )
:= 22 + ((2 × (22 − 2))2 + 2/2)
:= 22 × (2 × (2 + 2 + 2)2 + 2)
:= 3/3 + ((33 × (33 + 33)) − 3)
:= 3 + (33 × (33 + 33))
:= ((33/3)3 ) + 3 × 3 × 33
:= (4 + 4)/4 + (4 × ((444 − 44) + 4))
:= 44 + (4444/4 + 44 )
:= 44 + (44 × (4 × (4 + 4) + 4))
:= 55 + ((55 + 5/5)/((5 + 5)/5))
:= (5 × (5 × (55 + 5 + 5))) − (5 + 5)/5
:= 5 + ((5 × (5 × (55 + 5 + 5))) − ((5 + 5)/5))
:= 6 × (666 − 6 × 66) − (6 + 6)/6
:= 666/6 + 6 × 6 × (6 × 6 + 6)
:= 66/6 × ((666 + 6)/6 + 6 × 6)
:= 7/7 + (77 × (7 + 7 + 7))
:= 7 + ((77 × (7 + 7 + 7)) − 7/7)
:= 77/7 + (77 × (7 + 7 + 7))
:= 8 + ((8 × (8 × (8 + 8 + 8) + 8) + ((8 + 8)/8)) + 8)
:= 8 × 8 × 8 + 8888/8
:= 88/8 × (888/(8 − (8 + 8)/8))
:= (9 × (99 + 9 × 9)) − (9 + 9)/9
:= (((9 + 9)/9)9 ) + 9999/9
:= 9 + ((9 × (99 + 9 × 9)) − 9/9)
Ï 1619 := 1 + (1 + (((1 + 11)1+1+1 ) − 111))
Ï 1624 := 1 + (1111 + (1 + 1)11−1−1 )
Ï 1629 := 1 + (11 + (((1 + 11)1+1+1 ) − 111))
:= 22 + ((2 × (22 − 2))2 − (2/2 + 2))
:= 2 + ((2 × (22 − 2))2 + 22)
:= 2 + ((((2 × 22) − 2/2)2 ) − 222)
:= (33 × (33 + 33)) − 3/3
:= 3 + ((33 × (33 + 33)) + 3/3)
:= (3 × 3 + 3)3 − 3 × 33
:= ((4 × 4 + 4) × (4 − 4/4)4 ) − 4/4
:= 4 + ((4 × 4 + 4) × (4 − 4/4)4 )
:= 4 × (44 − 4) + ((4/4 + 4)4 − 4)
:= (5 × (5 × (55 + 5 + 5))) − (5/5 + 5)
:= (5 × (5 × (55 + 5 + 5))) − 5/5
:= 5 + ((5 × (5 × (55 + 5 + 5))) − 5/5)
:= 6 × (666 − 6 × 66) − 6/6
:= 6 × 6 × (6 × 6 + 6) + (666 + 6)/6
:= 6 + (6 × 6 × (6 × 6 + 6) + 666/6)
:= ((7 + 7)/7) + (77 × (7 + 7 + 7))
:= 7 + (77 × (7 + 7 + 7))
:= (77 + 7)/7 + (77 × (7 + 7 + 7))
:= 8 + (8 × (8 × (8 + 8 + 8) + 8) + (88/8))
:= 88 + 8 × 8 × (8 + 8 + 8)
:= (8/8 + 8) × (8 × (8 + 8 + 8) − (88/8))
:= (9 × (99 + 9 × 9)) − 9/9
:= ((9 + 9)/9) × (9 × (9 × 9 + 9) + ((9 + 9)/9))
:= 9 + (9 × (99 + 9 × 9))
Ï 1620 := (1 + 1) × ((11 − 1) × (11 − 1 − 1)1+1 )
Ï 1625 := 1 + (1 + (1111 + (1 + 1)11−1−1 ))
Ï 1630 := (11 − 1) × (1 + ((1 + 1) × (11 − 1 − 1)1+1 ))
:= 22 + ((2 × (22 − 2))2 − 2)
:= 2 + (((2 × (22 − 2))2 + 22) + 2/2)
:= 2 + (22 × (2 × (2 + 2 + 2)2 + 2))
:= 33 × (33 + 33)
:= ((33/3)3 ) + (3 × 3 × 33 − 3)
:= 3/3 + ((3 × 3 + 3)3 − 3 × 33)
:= (4 × 4 + 4) × (4 − 4/4)4
:= 4 + (((4 × 4 + 4) × (4 − 4/4)4 ) + 4/4)
:= 4 + (((4 + 4)/4 + 4) × ((44/4 + 44 ) + 4))
:= (5/5 + 5) × (5 × 55 − 5)
:= 5 × (5 × (55 + 5 + 5))
:= 5 + (5 × (5 × (55 + 5 + 5)))
:= 6 × (666 − 6 × 66)
:= (6/6 − 66) × ((66/6) − 6 × 6)
:= 6 + (6 × 6 × (6 × 6 + 6) + (666 + 6)/6)
:= ((7 + 7 + 7)/7) + (77 × (7 + 7 + 7))
:= 7 + ((77 × (7 + 7 + 7)) + 7/7)
:= 7 + (((77 × (7 + 7 + 7)) − 7/7) + 7)
:= ((88 + 8)/8) × ((8 × (8 + 8) − 8/8) + 8)
:= 8/8 + (8 × 8 × (8 + 8 + 8) + 88)
:= 8 + ((8888 − 8)/8 + 8 × 8 × 8)
:= 9 × (99 + 9 × 9)
:= (9 × (99 + 9 × 9)) + ((9 × 9 + 9)/(9 + 9))
:= 9 + ((9 × (99 + 9 × 9)) + 9/9)
Ï 1621 := 1 + ((1 + 1) × ((11 − 1) × (11 − 1 − 1)1+1 ))
Ï 1626 := 1 + (1 + (1 + (1111 + (1 + 1)11−1−1 )))
Ï 1631 := 1 + (11 − 1) × (1 + (1 + 1) × (11 − 1 − 1)1+1 )
:= 22 + ((2 × (22 − 2))2 − 2/2)
:= 2 + (((2 × (22 − 2))2 + 22) + 2)
:= (((2 × 22) + 2)2 ) − (222 + 2/2)
:= 3/3 + (33 × (33 + 33))
:= 3 + ((33 × (33 + 33)) + 3)
:= 3 + (((33/3)3 ) + 3 × 3 × 33)
:= 4/4 + ((4 × 4 + 4) × (4 − 4/4)4 )
:= ((4 + 4)/4 + 4) × ((44/4 + 44 ) + 4)
:= 44 + (((44/4)4−4/4 ) + 44 )
:= 5/5 + ((5/5 + 5) × (5 × 55 − 5))
:= 5/5 + (5 × (5 × (55 + 5 + 5)))
:= 5 + ((5 × (5 × (55 + 5 + 5))) + 5/5)
:= 6/6 + 6 × (666 − 6 × 66)
:= 6 + 6 × (666 − 6 × 66)
:= 6 + ((6/6 − 66) × ((66/6) − 6 × 6))
:= 77/7 + ((77 × (7 + 7 + 7)) − 7)
:= 7 + ((77 × (7 + 7 + 7)) + ((7 + 7)/7))
:= 7 + ((77 × (7 + 7 + 7)) + 7)
:= 8 × 8 × 8 + ((8888 − (8 + 8))/8)
:= 88 + (8 × 8 × (8 + 8 + 8) + ((8 + 8)/8))
:= 8 + (8888/8 + 8 × 8 × 8)
:= 9 + ((9 × (99 + 9 × 9)) − ((9 + 9 + 9)/9))
:= 99/9 + (9 × (99 + 9 × 9))
:= 9/9 + (9 × (99 + 9 × 9))
Ï 1622 := 1111 + ((1 + 1)11−1−1 − 1)
Ï 1627 := 11 + (((1 + 11)1+1+1 ) − (1 + 111))
:= 22 + (2 × (22 − 2))2
:= (((2 × 22) − 2/2)2 ) − 222
:= (((2 × 22) + 2)2 ) − 222
:= 3 + ((33 × (33 + 33)) − 3/3)
:= 3 + (((33 × (33 + 33)) + 3/3) + 3)
:= 3 + ((3 × 3 + 3)3 − 3 × 33)
:= 44 + ((4444 − 4)/4 + 44 )
:= 4 + ((4444/4 + 44 ) + 44 )
:= 4 × (((444 − 44) + 4) + 4)
:= 5 + (((5 − 5/5)/((5 + 5)/5)) + 55)
:= (5 + 5)/5 + (5 × (5 × (55 + 5 + 5)))
:= ((5 + 5)/5)5 × ((5/5 − 5) + 55)
:= (6 + 6)/6 + 6 × (666 − 6 × 66)
:= 6 + (6 × (666 − 6 × 66) + 6/6)
:= 6 + (6 × (666 − 6 × 66) + 6)
:= 7 + ((77 × (7 + 7 + 7)) − ((7 + 7)/7))
:= ((77 − 7)/7) + (77 × (7 + 7 + 7))
:= 7 + (((77 × (7 + 7 + 7)) + 7/7) + 7)
:= 8 × 8 × 8 + (8888 − 8)/8
:= ((88/8 + 8) × (8/8 + 88)) − 8 × 8
:= (88 + 8) × (8/8 + 8 + 8)
:= (9 + 9)/9 + (9 × (99 + 9 × 9))
:= 9 + ((9 × (99 + 9 × 9)) − ((9 + 9)/9))
:= ((99 + 9)/9) + (9 × (99 + 9 × 9))
5
118
Ï 1632 := (1 + 1) × ((1 + 1)11 − (11 × (1 + 111)))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1633 := (1 + 1 + 1 + 11)1+1+1 − 1111
Ï 1638 := ((1 + 1) × 111+1+1 ) − (1 + 1)11−1
Ï 1643 := ((1 + 1)1+11 ) − (11 × (1 + (1 + 1) × 111))
:= 22 + ((2 × (22 − 2))2 + 22/2)
:= 2 + ((2 × ((22 + 2)2 )) + 222 )
:= (2 × 22 − 2)2 − (22/2)2
:= ((3 × (3 × 33 × 33)) − 3)/(3 + 3)
:= 3 × (((3 + 3)3 − 3) + 333)
:= (3 × 3 + 3)3 − ((3 × 33 + 3/3) + 3)
:= 44 + ((44 × (44 − 4)/(4 + 4)) − 4)
:= (((4 − 4/4)4+4 ) + 44/4)/4
:= (5 − 5/5)5 + (((55 − 55)/5) − 5)
:= (5/5 + 5) × (5 × 55 − ((5 + 5)/5))
:= (5 − 5/5)5 + ((55 − 5)/5 − 5)
:= ((66/6 + 6) + 6) × ((66 − 6/6) + 6)
:= (6/6 + 6) × ((6 × 6 × 6 + 6 + 6) + 6)
:= (66 × (6 × 6 − 66/6)) − 6/6 − 6
:= 7 + (((77 × (7 + 7 + 7)) + ((7 + 7)/7)) + 7)
:= (7 + 7 + 7) × (7/7 + 77)
:= 7 × 77 + (7777/7 − 7)
:= 8/8 + ((88 + 8) × (8/8 + 8 + 8))
:= (8/8 − 8 × 8) × ((8 − 88)/8 − (8 + 8))
:= 88/8 + ((88 + 8) × (8/8 + 8 + 8))
:= (9 × (99 + 9 × 9)) + ((99 + 9 + 9)/9)
:= 9 + ((9 × (99 + 9 × 9)) + 9)
:= (9 × (99 + 9 × 9)) + ((99 + 99 + 9)/9)
4
:= 4 × (4 − 4) + (4/4 + 4)
4
Ï 1634 := 11 + (1111 + (1 + 1)11−1−1 )
2
Ï 1639 := 11 × (1 + (1 + (1 + (1 + (1 + (1 + 11)1+1 )))))
2
Ï 1644 := (1 + 11) × (111 + ((1 + 1) × (1 + 1 + 11)))
:= 2 + ((((2 × 22) + 2) ) − 22 )
:= (22 × (22 + 2)) + 2222/2
:= 2 × (2 × (22 − 2)2 + 22)
:= ((3 × (3 × 33 × 33)) + 3)/(3 + 3)
:= 3/3 + ((3 × 3 + 3)3 + (3 × (3 − 33)))
:= (3 × 3 + 3)3 − (3 × 33 + 3)
:= 4/4 + (4 × (44 − 4) + (4/4 + 4)4 )
:= (((4 − 4/4)4+4 ) − (4/4 + 4))/4
:= 44 + (4 × (444 − 44))
:= 55 + ((5 − 5/5) + 555)
:= (55 × (5 × 5 + 5)) − 55/5
:= 55 /5 + ((5 − 5/5)5 − 5)
:= (((6 + 6)/6) + 6 × 6) × ((6 × 6 + 6/6) + 6)
:= 66/6 × (((6 + 6) × (6 + 6) − 6/6) + 6)
:= (66 × (6 × 6 − 66/6)) − 6
:= 7 + ((77 × (7 + 7 + 7)) + ((77 − 7)/7))
:= 7/7 + ((7 + 7 + 7) × (7/7 + 77))
:= 7 + (((7 + 7 + 7) × (7/7 + 77)) − 7/7)
:= (88/8 + 8) × (88 − ((8 + 8)/8))
:= 8 + ((8888/8 + 8 × 8 × 8) + 8)
:= ((88 + 8)/8) × ((8 × (8 + 8) + 8/8) + 8)
:= 9 + ((9 × (99 + 9 × 9)) + ((9 × 9 + 9)/(9 + 9)))
:= 9 + (((9 × (99 + 9 × 9)) + 9/9) + 9)
:= 9 + ((((9 + 9 + 9)/9) × (((9 + 9)/9)9 )) + 99)
5
Ï 1635 := (111 − 1 − 1) × (1 + 1 + 1 + 1 + 11)
2
Ï 1640 := 1111 + (1 + 11 + 11)1+1
Ï 1645 := 1 + ((1 + 11) × (111 + ((1 + 1) × (1 + 1 + 11))))
:= 2 × (2 × (22 − 2)2 − 2 + 22)
2
:= 22 + ((2 × ((22 + 2) )) − 2/2)
3
:= 2 + ((2 × 22 − 2)2 − (22/2)2 )
3
:= 3 × (((3 − 3/3)3×3 ) + 33)
:= 3 + ((((33/3) ) − 3 ) + 333)
:= 3/3 + ((3 × 3 + 3)3 − (3 × 33 + 3))
:= (44/4 + 4) × ((44 + 4)/4 + 44)
:= (44 − 4) × ((4/4 − 4) + 44)
:= 4 × 44 + ((4/4 + 4)4 − 4)
:= 5 + ((5 × (5 × (55 + 5 + 5))) + 5)
:= (55 × (5 × 5 + 5)) − 5 − 5
:= (55 × (5 × 5 + 5)) − 5
:= (6 − 6/6) × (666/6 + 6 × 6 × 6)
:= ((6 + 6)/6 + 6) × (6 × 6 × 6 − (66/6))
:= (7/7 + 7) × (((7 + 7)/7)7 + 77)
:= (6 × 6 − 6/6) × (66/6 + 6 × 6)
:= 7 + ((77 × (7 + 7 + 7)) + (77/7))
:= 88 + (8 × 8 × (8 + 8 + 8) + (88/8))
:= 8 + ((88 + 8) × (8/8 + 8 + 8))
:= ((88/8 + 8) × (88 − 8/8)) − 8
:= (99/9 + 9) × (9/9 + 9 × 9)
:= 999 + (9 × (9 × 9 − 9) − ((9 + 9)/9))
9
:= 99 + (((9 + 9 + 9)/9) × (((9 + 9)/9) ))
Ï 1636 := 1 + ((111 − 1 − 1) × (1 + 1 + 1 + 1 + 11))
:= 7 + ((7 + 7 + 7) × (7/7 + 77))
Ï 1641 := 1 + (1111 + (1 + 11 + 11)1+1 )
2
Ï 1646 := 11 + ((111 − 1 − 1) × (1 + 1 + 1 + 1 + 11))
2
:= 222 + (2 × ((22 + 2)2 ))
:= (2 × 22 − 2) − ((22/2) + 2)
:= 2222 − ((22 + 2)2 )
:= 3 + (((3 × (3 × 33 × 33)) − 3)/(3 + 3))
:= 3 + ((3 × 3 + 3)3 + (3 × (3 − 33)))
:= (3 × 3 + 3)3 − (3 × 33 + 3/3)
:= 44 × 44 − (44 + 44 )
:= 4 × 44 + ((4/4 + 4)4 − (4 + 4))
:= 4 + ((44 × (44 − 4)/(4 + 4)) + 44 )
:= 55/5 + (5 × (5 × (55 + 5 + 5)))
:= 5/5 + ((55 × (5 × 5 + 5)) − (5 + 5))
:= 5/5 + ((55 × (5 × 5 + 5)) − 5)
6
6
:= 6 × 6 + (((6 + 6)/6) × (6 × 6 − 66/6))
:= 6 × 6 × 66 − (((6 × 6/(6 + 6)) ) + 6)
:= (((6 + 6)/6)66/6 ) − (6 × 66 + 6)
:= 7 + ((77 × (7 + 7 + 7)) + (77 + 7)/7)
:= (((7 + 7 + 7)/7)7 ) − (7 × 77 + 7)
:= 7 + (((7 + 7 + 7) × (7/7 + 77)) + 7/7)
:= 8 + (88/8 × (888/(8 − (8 + 8)/8)))
:= 8 + (((88 + 8) × (8/8 + 8 + 8)) + 8/8)
:= ((8 + 8)/8) × (888 − (8/8 + 8 × 8))
:= 9 + (((9 × (99 + 9 × 9)) − ((9 + 9)/9)) + 9)
:= 9 + ((9 × (99 + 9 × 9)) + ((99 + 9)/9))
:= 999 + (9 × (9 × 9 − 9) − 9/9)
Ï 1637 := 1 + (1 + ((111 − 1 − 1) × (1 + 1 + 1 + 1 + 11)))
Ï 1642 := 1 + (1 + (1111 + (1 + 11 + 11)1+1 ))
Ï 1647 := 111 + ((1 + 1 + 1) × (1 + 1)11−1−1 )
:= 2/2 + ((2 × ((22 + 2)2 )) + 222 )
:= (2 × 22) + ((2 × (22 − 2))2 − 2)
:= 2/2 + 2222 − (22 + 2)2
:= 333 + (((33/3)3 ) − 33 )
:= 3 + (((3 × 3 + 3)3 + (3 × (3 − 33))) + 3/3)
:= 3 × ((3 + 3)3 + 333)
:= 4 + (4 × (44 − 4) + (4/4 + 4)4 )
:= 44 + (44 × (44 − 4)/(4 + 4))
:= (4 × (444 − 4 × (4 + 4))) − 4/4
5
:= 5 + (((5 + 5)/5) × ((5/5 − 5) + 55))
:= (5 + 5)/5 + ((55 × (5 × 5 + 5)) − (5 + 5))
:= (5 + 5)/5 + ((55 × (5 × 5 + 5)) − 5)
:= 6 + (((6/6 − 66) × ((66/6) − 6 × 6)) + 6)
:= 66 + (6 × 6 × (6 × 6 + 6) + ((6 + 6)/6) )
:= 6 × 6 × 66 − ((6 × 6/(6 + 6))6 )
:= ((7 + 7 + 7) × (7/7 + 77)) − 7/7
:= 7 + (((77 × (7 + 7 + 7)) + (77/7)) + 7)
:= 7 + ((7/7 + 7) × (((7 + 7)/7)7 + 77))
6
:= ((8 + 8) × (888/8 − 8)) − 88/8
:= 8 + ((88/8 + 8) × (88 − ((8 + 8)/8)))
:= 888/8 + 8 × 8 × (8 + 8 + 8)
:= 9 + (((9 × (99 + 9 × 9)) − 9/9) + 9)
:= ((9 + 9)/9) × (9 × (9 × 9 + 9) + (99/9))
:= 999 + 9 × (9 × 9 − 9)
119
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1648 := (1 + 1)11 − ((1 + 1) × (11 − 1))1+1
Ï 1653 := (1 + 1 + 1) × (1 + ((1111 − 11)/(1 + 1)))
Ï 1658 := ((1 + 1 + 111)1+1 ) − 11111
:= 222/2 − (22 − 2)2
:= (2 × 22 − 2)2 − 222/2
:= 2 + (((2 × 22) + 2) × (2 + 2 + 2)2 )
:= 3/3 + (3 × ((3 + 3)3 + 333))
:= 33 + (33 × (33 + 33))
:= 333 + (((33/3)3 ) − (3 + 3))
:= 4 × (444 − 4 × (4 + 4))
:= 4 + ((4/4 + 4)4 + 4 × 44 )
:= 4 + ((((4/4 + 4)4 + 4 × 44 ) + 4/4) + 4)
:= (5 − 5/5)5 + (55 − 5)/5
:= 5 + ((5 − 5/5)5 + (55 − 5)/5)
:= 5 + (((5 − 5/5)5 + (55 − 5)/5) + 5)
:= ((6 + 6)/6)6 + (66 × ((6 + 6 + 6) + 6))
:= 6 + (6 × 6 × 66 − ((6 × 6/(6 + 6))6 ))
:= 6 + ((((6 + 6)/6)66/6 ) − 6 × 66)
:= (((7 + 7 + 7)/7)7 ) − 7 × 77
:= 7/7 + ((7 + 7) × (777/7 + 7))
:= ((7/7 + 7 + 7) × 777/7) − 7
:= (8 + 8) × (888/8 − 8)
:= (88/8 + 8) × (88 − 8/8)
:= 8 + ((8 − 8/8 + 8) × (888 − 8)/8)
:= 9/9 + (9 × (9 × 9 − 9) + 999)
:= (9/9 + 9 + 9) × (99 − ((99 + 9)/9))
:= 9 + (((99/9 + 9) × (9/9 + 9 × 9)) + 9)
Ï 1649 := ((111 − 1) × (1 + 1 + 1 + 1 + 11)) − 1
Ï 1654 := (1 + 1) × ((1 + 1)11 − 11 × 111)
Ï 1659 := (1 + 1 + 1) × (((1111 − 1)/(1 + 1)) − (1 + 1))
:= (((2 × 22) + 2) × (2 + 2 + 2) ) − 2
:= ((2 × (22 − 2) + 2/2)2 ) − 22
:= 3 + ((3 × 3 + 3)3 − (3 × 33 + 3/3))
:= 3/3 + ((33 × (33 + 33)) + 33)
:= 3 + ((3 × (3 − 33 )) + (3 × 3 + 3)3 )
:= 4 × 44 + (4/4 + 4)4
:= 4 + (((4/4 + 4)4 + 4 × 44 ) + 4/4)
:= 44 + (4 × (4 + 4) × 44 − (4/4 + 4))
:= 55 /5 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 55 /5)
:= 5 + (((5 − 5/5)5 + 55 /5) + 5)
:= (66 × (6 × 6 − 66/6)) − 6/6
:= (6 × (6 × 6 × 6 − 6 + 66)) − (6 + 6)/6
:= 666/6 + 6 × 6 × (6 × 6 + 6 + 6)
:= 2/2 + (2
22/2
2
− (22 − 2) )
2
:= 7 × 77 + (7777 − 7)/7
:= ((7 + 7)/7) + ((7 + 7) × (777/7 + 7))
:= 7 + ((7 + 7) × (777/7 + 7))
:= 8/8 + ((8 + 8) × (888/8 − 8))
:= 8/8 + ((88/8 + 8) × (88 − 8/8))
:= 88/8 + ((8 + 8) × (888/8 − 8))
:= 9 + ((99/9 + 9) × (9/9 + 9 × 9))
:= ((9 + 9)/9) × ((9 × 9 × 9 − 9/9) + 99)
:= 9 + ((9 × (9 × (9 + 9) + 9)) + 999/9)
Ï 1650 := (111 − 1) × (1 + 1 + 1 + 1 + 11)
Ï 1655 := 1 + ((1 + 1) × ((1 + 1)11 − 11 × 111))
:= 2 + (222/2 − (22 − 2)2 )
:= 2 + ((2 × 22 − 2)2 − 222/2)
:= (22 − 2) × (((2/2 + 2)2+2 ) + 2)
:= 3 + (3 × ((3 + 3)3 + 333))
:= 333 + (((33/3)3 ) − 3 × 3)
:= (3 − 3/3 + 3) × (333 − 3/3)
:= 4/4 + ((4/4 + 4)4 + 4 × 44 )
:= (4/4 + 4) × (((44 + 44 )/4) + 44 )
:= 44 + (4 × (4 + 4) × 44 − 4)
:= 55 × (5 × 5 + 5)
:= 5 + (55 × (5 × 5 + 5))
:= 5 + ((55 × (5 × 5 + 5)) + 5)
:= 66 × (6 × 6 − 66/6)
:= (6 × (6 × 6 × 6 − 6 + 66)) − 6/6
:= (6 − 6/6) × (6 × 66 − ((6 + 6)/6)6 )
:= 7 × 77 + 7777/7
:= 7 + ((((7 + 7 + 7)/7)7 ) − 7 × 77)
:= 7 + (((7 + 7) × (777/7 + 7)) + 7/7)
:= (8 − 8/8 + 8) × (888 − 8)/8
:= 888 + (8 × (88 + 8) − 8/8)
:= ((8 + 8) × (88 + 8 + 8)) − (8/((8 + 8)/8))
:= 999/9 + (9 × (9 × (9 + 9) + 9))
:= ((9 + 9) × ((99/9) + 9 × 9)) − 9/9
:= (99/9 + 9) × (((9 + 9)/9) + 9 × 9)
Ï 1651 := 1 + ((111 − 1) × (1 + 1 + 1 + 1 + 11))
:= (22/2 + 2) × ((2
2×(2+2)
− 2)/2)
Ï 1656 := (1 + 1) × (1 + ((1 + 1)11 − 11 × 111))
:= ((2 × 22) + 2) × (2 + 2 + 2)
Ï 1660 := (11 − 1) × (1 + (11 × (1 + 1 + 1 + 1 + 11)))
Ï 1661 := (((1 + 1 + 1) × 1111) − 11)/(1 + 1)
:= 2 + (((2 × (22 − 2) + 2/2)2 ) − 22)
2
:= 3 + ((3 × ((3 + 3)3 + 333)) + 3/3)
:= 3 × (((3 + 3)3 + 333) + 3)
:= 333 + (((33/3)3 ) − 3)
:= 4 + ((4 × (444 − 4 × (4 + 4))) − 4/4)
:= 4 + ((4 × (444 − 4 × (4 + 4))) + 4)
:= 4 × (44 + 4) + ((4/4 + 4)4 − 4)
:= 5/5 + (55 × (5 × 5 + 5))
:= 5 + ((55 × (5 × 5 + 5)) + 5/5)
:= 55/5 + (55 × (5 × 5 + 5))
:= 6/6 + (66 × (6 × 6 − 66/6))
:= 6 × (6 × 6 × 6 − 6 + 66)
:= 6 + ((6 × (6 × 6 × 6 − 6 + 66)) − 6/6)
:= 7 × 77 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 7 × 77)
:= 7 × 77 + ((7777 + 77)/7)
:= (88 + 8 + 8)/8 × (8 × (8 + 8) − 8/8)
:= 888 + 8 × (88 + 8)
:= 8 + ((88/8 + 8) × (88 − 8/8))
:= (9 × (9 × (9 + 9) + 9)) + ((999 + 9)/9)
:= (9 + 9) × ((99/9) + 9 × 9)
:= 99/9 × (9 × (9 + 9) − (99/9))
Ï 1652 := 1 + (1 + ((111 − 1) × (1 + 1 + 1 + 1 + 11)))
Ï 1657 := ((1 + 1 + 1) × ((1 + 1111)/(1 + 1))) − 11
Ï 1662 := (1 + 1 + 1) × (((1111 − 1)/(1 + 1)) − 1)
:= 2 × ((2 × ((22 − 2) + 2)) + 22)
:= 2/2 + (((2 × 22) + 2) × (2 + 2 + 2) )
:= (2/2 + 2) × (((22 + 2)2 ) − 22)
:= 33 + ((33 × (33 + 33)) − 3/3)
:= (3 × 3 + 3)3 + ((3 − (3 + 3)3 )/3)
:= (3 × 3 + 3)3 − (33 + 33)
:= 4 + (4 × (444 − 4 × (4 + 4)))
:= 4 + (((4/4 + 4)4 + 4 × 44 ) + 4)
:= 44 + (4 × (4 + 4) × 44 − (4 + 4)/4)
:= (5 + 5)/5 + (55 × (5 × 5 + 5))
:= 5 + ((55 × (5 × 5 + 5)) + ((5 + 5)/5))
:= (5/5 + 5) × (5 × 55 + ((5 + 5)/5))
:= 6/6 + (6 × (6 × 6 × 6 − 6 + 66))
:= 6 + (6 × (6 × 6 × 6 − 6 + 66))
:= 7 + (7777/7 + 7 × 77)
:= 7 + (((((7 + 7 + 7)/7)7 ) − 7 × 77) + 7)
:= ((8 + 8) × (88 + 8 + 8)) − (88 + 8)/8
:= 8/8 + (8 × (88 + 8) + 888)
:= ((8 + 8) × (88 + 8 + 8)) − (8 + 8)/8
:= ((99/9) × (9 × (9 + 9) − (99/9))) − 9
:= 9/9 + ((9 + 9) × ((99/9) + 9 × 9))
:= (9 + 9) × 99 − (999/9 + 9)
2
66/6
:= (((6 + 6)/6)
) − 6 × 66
:= (7 + 7) × (777/7 + 7)
2
120
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1663 := (111 × (1 + 1 + 1 + 1 + 11)) − 1 − 1
:= ((22 + 2 + 2) × 2
2+2+2
Ï 1673 := 1 + (11 × ((11 × (1 + (1 + 1 + 11))) − (1 + 1)))
Ï 1668 := (1 + 1 + 1) × ((1 + 1111)/(1 + 1))
:= 2 × ((22 × ((2 + 2 + 2) + 2)) − 2)
:= 2/2 + (2 × (22 × ((2 + 2 + 2)2 + 2)))
:= 3 + (333 × (3 − 3/3 + 3))
:= 3 × 3 + (((33/3)3 ) + 333)
2
) − 2/2
:= 333 + (((33/3)3 ) − 3/3)
4
:= 4 + ((4 × (44 + 4) + (4/4 + 4)4 ) + 4)
:= 44 + (4 × (4 + 4) × 44 − 4/4)
:= 4 + (4 × (4 + 4) × 44 + 4 )
:= (((5 + 5 + 5) × 555) − (5 + 5))/5
:= (5 − (5 + 5)/5) × (555 + 5/5)
:= 5 + ((5 − (5 + 5)/5) × (555 + 5/5))
:= 6 + ((6 × (6 × 6 × 6 − 6 + 66)) + 6/6)
:= 6 + ((6 × (6 × 6 × 6 − 6 + 66)) + 6)
:= (6 × (6 × 66 − 6)) − (666 + 6/6)
:= 77/7 + ((7 + 7) × (777/7 + 7))
:= 7 × 7 + ((77 × (7 + 7 + 7)) + ((7 + 7)/7))
:= 7 + ((77 × (7 + 7 + 7)) + 7 × 7)
:= ((8 + 8) × (88 + 8 + 8)) − 8/8
:= (8/((8 + 8)/8)) + ((8 + 8) × (88 + 8 + 8))
:= 8/8 + 88 × (88/8 + 8)
:= (9 + 9) × 99 + (((9 − 999)/9) − 9)
:= 99 + (9 × 9 × (9 + 9) + 999/9)
:= (9 + 9) × 99 − (9/9 + 99 + 9)
Ï 1669 := 1 + ((1 + 1 + 1) × ((1 + 1111)/(1 + 1)))
Ï 1664 := (111 × (1 + 1 + 1 + 1 + 11)) − 1
Ï 1674 := (1 + 1 + 1) × (1 + (1 + ((1 + 1111)/(1 + 1))))
:= 2 + ((222/2 × ((22/2 + 2) + 2)) + 2)
:= 2 + (2 × (22 × ((2 + 2 + 2)2 + 2)))
:= 3 + (((3 × 3333) − 3)/(3 + 3))
:= 3 × (3 × (((3 + 3)3 − 33) + 3))
:= 4 × (4 × (44 − 4) + 44 )
:= 4 + (4 × (44 + 4) + (4/4 + 4)4 )
:= 44 × 44 − (((4 + 4)/4 + 44 ) + 4)
:= (((5 + 5 + 5) × 555) − 5)/5
:= 5 + ((((5 + 5 + 5) × 555) − 5)/5)
:= 5 × 5 + ((5 − 5/5)5 + 55 /5)
:= (6/6 + 6 + 6) × (((6 + 6)/6)6/6+6)
:= (((6 × 6 − 6/6) + 6)
:= (7 − 7/7 + 7) × ((7 + 7)/7)7
:= 7 + ((((((7 + 7 + 7)/7) ) − 7 × 77) + 7) + 7)
:= ((7 × 7 − (7/7 + 7))(7+7)/7 ) − 7
:= (8 + 8) × (88 + 8 + 8)
:= 8 + (((88/8 + 8) × (88 − 8/8)) + 8)
:= (8 + 8)/8 + 88 × (88/8 + 8)
:= 9 + (((9 + 9) × ((99/9) + 9 × 9)) − 9/9)
:= 9 + ((99/9 + 9) × (((9 + 9)/9) + 9 × 9))
:= (9 + 9) × 99 − (99 + 9)
:= (22 + 2 + 2) × 2
2+2+2
:= 333 + ((33/3)3 )
(6+6)/6
)−6−6
:= (6 × (6 × 66 − 6)) − 666
7
Ï 1670 := (11 − 1) × ((1 + 1 + 11)1+1 − (1 + 1))
Ï 1665 := 111 × (1 + 1 + 1 + 1 + 11)
Ï 1675 := 11 + ((111 × (1 + 1 + 1 + 1 + 11)) − 1)
2
:= 222/2 × ((22/2 + 2) + 2)
:= (2 × (22 × ((2 + 2 + 2) + 2))) − 2
:= (((2 × 22) + 2)2 ) − ((22 − 2/2)2 )
:= 333 × (3 − 3/3 + 3)
:= 3 + ((((33/3)3 ) + 333) + 3)
:= (3 × 33+3 ) − ((3 − 3/3)3×3 )
:= 4 × (44 + 4) + (4/4 + 4)4
:= 4 + (((4 − 444)/4) + 4 × 444)
:= 44 × 44 − ((4/4 + 44 ) + 4)
:= (5 + 5 + 5) × 555/5
:= 5 + ((5 + 5 + 5) × 555/5)
:= 5 × ((5 × 55 + 55) + 5)
6/6+6)
:= (6 − 6/6) × 666 × 6/(6 + 6)
:= 6 + ((6/6 + 6 + 6) × (((6 + 6)/6)
:= (7/7 + 7 + 7) × 777/7
:= 7 + (((7 + 7) × (777/7 + 7)) + (77/7))
)
:= (7 − ((7 + 7)/7)) × (7 × 7 × 7 − (7/7 + 7))
:= 8/8 + ((8 + 8) × (88 + 8 + 8))
:= 88 × (88/8 + 8) − (8 + 8)/8
:= 88/8 + ((8 + 8) × (88 + 8 + 8))
:= 9 + ((9 + 9) × ((99/9) + 9 × 9))
:= (9 + 9) × 99 − ((999 + 9)/9)
:= 9 + ((99 − 9/9) × ((9 − 9/9) + 9))
Ï 1671 := (1 + 1 + 1) × (1 + ((1 + 1111)/(1 + 1)))
Ï 1666 := 1 + (111 × (1 + 1 + 1 + 1 + 11))
2
:= (66 + 6/6) × (6 × 6 − 66/6)
Ï 1676 := 11 + (111 × (1 + 1 + 1 + 1 + 11))
:= (2 × (22 × ((2 + 2 + 2) + 2))) − 2/2
:= 2 × ((22 × ((2 + 2 + 2)2 + 2)) + 2)
:= ((3 × 3333) − 3)/(3 + 3)
:= 3 + ((333 × (3 − 3/3 + 3)) + 3)
:= 3 + ((((33/3)3 ) + 333) + 3 × 3)
:= 4 × 444 + ((4 − 444)/4)
:= 44 × 44 − (((4/4 + 44 ) + 4) + 4)
:= 44 × 44 − (44 + 4)
:= 5 + (5555/5 + 555)
:= 5 × 5 + ((55 × (5 × 5 + 5)) + 5/5)
:= 6 + ((6 − 6/6) × (6 × 66 − ((6 + 6)/6) ))
:= 6 + ((6 − 6/6) × 666 × 6/(6 + 6))
:= 6/6 + ((66 + 6/6) × (6 × 6 − 66/6))
:= 7 × (777 − 7 × 77)
:= 7 + ((7 − 7/7 + 7) × ((7 + 7)/7)7 )
:= (77/7 × ((77 − 7/7) + 77)) − 7
:= (8 + 8)/8 + ((8 + 8) × (88 + 8 + 8))
:= 88 × (88/8 + 8) − 8/8
:= ((88 + 8)/8) + ((8 + 8) × (88 + 8 + 8))
:= (99 − 9/9) × ((9 − 9/9) + 9)
:= (9 + 9) × 99 − 999/9
:= (9 + 9)/9 + ((9 + 9) × 99 − (99 + 9))
:= 2 + ((22 + 2 + 2) × 2
2+2+2
)
:= 555 + 5555/5
6
Ï 1667 := (1 + ((1 + 1 + 1) × 1111))/(1 + 1)
Ï 1672 := 11 × ((11 × (1 + (1 + 1 + 11))) − (1 + 1))
Ï 1677 := 1 + (11 + (111 × (1 + 1 + 1 + 1 + 11)))
:= 2 × (22 × ((2 + 2 + 2)2 + 2))
:= ((2 × (22 − 2) + 2/2)2 ) − 2 − 2
:= ((3 × 3333) + 33)/(3 + 3)
:= 3 + (3 × (3 × (((3 + 3)3 − 33) + 3)))
:= 4 + ((4 × (4 + 4) × 44 − 4/4) + 4 )
:= 44 × (44 − ((4 + 4)/4 + 4))
:= 4/4 + (44 × 44 − (44 + 4))
:= 555 + (5555 + 5)/5
:= 5 + ((5555 + 5)/5 + 555)
:= 2 + (222/2 × ((22/2 + 2) + 2))
:= 3 + (((33/3)3 ) + 333)
4
:= 66/6 + (6 × (6 × 6 × 6 − 6 + 66))
:= 5 × 5 + ((55 × (5 × 5 + 5)) + ((5 + 5)/5))
6
:= 66/6 × (6 × 6 × 6 − ((6 + 6)/6) )
:= 6 + (((6 − 6/6) × 666 × 6/(6 + 6)) + 6)
:= 7/7 + ((77 × (7 + 7 + 7)) + 7 × 7)
:= 7 + ((7/7 + 7 + 7) × 777/7)
:= 7 × 7 + ((77 × (7 + 7 + 7)) + (77/7))
:= 888 + (8 × (88 + 8) + (88/8))
:= 88 × (88/8 + 8)
:= (88 + 8 + 8)/8 × (8 × (8 + 8) + 8/8)
:= 9/9 + ((99 − 9/9) × ((9 − 9/9) + 9))
:= 99/9 × (9 × (9 + 9) − (9/9 + 9))
:= 9 + ((9 × 9 × (9 + 9) + 999/9) + 99)
121
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1678 := 11 + ((1 + ((1 + 1 + 1) × 1111))/(1 + 1))
Ï 1683 := 11 × ((11 × (1 + (1 + 1 + 11))) − 1)
Ï 1688 := ((11 − 1) × (1 + 1 + 11)1+1 ) − 1 − 1
:= (2 × 22)2 − (22×(2+2) + 2)
:= 2 + ((2 × (22 − 2) + 2/2)2 )
:= 2 × (2 × ((22 − 2)2 + 22))
:= 3 + 3 × 33+3 − (3 − 3/3)3×3
:= 33 × ((33 − 3) + 33 )
:= ((3 × (3 × 3 + 3 + 3)3 ) + 3)/(3 + 3)
:= 44 × 44 − ((4 + 4)/4 + 44 )
:= 4 + (44 × 44 − (4/4 + 44 ))
:= 4 + ((44 × 44 − 44 ) + 4)
:= ((5/5 + 5) × (5 × 55 + 5)) − (5 + 5)/5
:= 5 × 5 × 55 + ((55 + 5)/(5 + 5) − 5)
:= 5 × 5 × 55 + (55 + 5)/(5 + 5)
:= 6 + ((66/6) × (6 × 6 × 6 − ((6 + 6)/6)6 ))
:= 66/6 × (666/6 + 6 × 6 + 6)
:= (6 + 6)/6 + ((6 × (6 × 6 × 6 + 66)) − 6)
:= 7 + (((7 − 7/7 + 7) × ((7 + 7)/7) ) + 7)
:= 77/7 × ((77 − 7/7) + 77)
:= 7 + ((7 × 7 − (7/7 + 7))(7+7)/7 )
:= 8 + (88 × (88/8 + 8) − ((8 + 8)/8))
:= 88/8 + 88 × (88/8 + 8)
:= 8 + (88 × (88/8 + 8) + 8)
:= (9 × (9 × 9 − (9 + 9))) + 9999/9
:= 99 × ((9 − 9/9) + 9)
:= (9 − 9/9) × (((999 + 9)/9) + 99)
7
Ï 1679 := ((11 − 1) × (1 + 1 + 11)1+1 ) − 11
Ï 1684 := 1 + (11 × ((11 × (1 + (1 + 1 + 11))) − 1))
2
2
2
:= 2 + ((((2 + 2 + 2) + 22) )/2)
:= ((2 × (22 − 2) + 2/2) ) − 2
3
3
Ï 1689 := ((11 − 1) × (1 + 1 + 11)1+1 ) − 1
:= 2 + ((2222/2) + ((22 + 2)2 ))
:= (33 − 3/3 − 3) × (((3 + 3)3 + 3)/3)
:= 3/3 + (33 × ((3 − 3) + 3 ))
:= (3 × 3 + 3)3 − (33 + 3 + 3)
:= 44 × 44 − (4/4 + 44 )
:= 4 + (44 × 44 − 44 )
:= 4 + (((44 × 44 − 44 ) + 4/4) + 4)
:= ((5/5 + 5) × (5 × 55 + 5)) − 5/5
:= 5 + (((5/5 + 5) × (5 × 55 + 5)) − 5/5)
:= 5 × 5 + ((((5 + 5 + 5) × 555) − 5)/5)
:= ((66/6 + 6) + 6) × (66 + 6/6 + 6)
:= (((6 + 6)/6)6+6 ) − (6 × (6 × 66 + 6))
:= (6 − 6 × 6) × (6 − 66) − 666/6
:= 7 + (((7/7 + 7 + 7) × 777/7) + 7)
:= 7 + (((77 × (7 + 7 + 7)) + (77/7)) + 7 × 7)
:= (((7 + 7)/7) × ((77 × 77 + 7)/7)) − 7
:= 8 + (88 × (88/8 + 8) − 8/8)
:= ((88 + 8)/8) + 88 × (88/8 + 8)
:= 8 + ((88 × (88/8 + 8) + 8/8) + 8)
:= 9 + ((9 + 9) × 99 − ((999 + 9)/9))
:= 9/9 + (99 × ((9 − 9/9) + 9))
:= 9 + (((9 + 9) × 99 − 999/9) + 9)
Ï 1680 := (11 − 1) × ((1 + 1 + 11)1+1 − 1)
Ï 1685 := (1 + 1)11 − (11 × (11 × (1 + 1 + 1)))
:= 2 × ((2 − 22) × (2 − (2 × 22)))
Ï 1690 := (11 − 1) × (1 + 1 + 11)1+1
:= 2 + (((2 × (22 − 2) + 2/2)2 ) + 2)
:= 2 + (2 × (2 × ((22 − 2)2 + 22)))
:= 3 + ((33 × ((33 − 3) + 33 )) − 3/3)
:= (3 × 3 + 3)3 − (33/3 + 33 )
:= 4 + ((44 × 44 − 44 ) + 4/4)
:= 44 × 44 + ((44 − 4)/4 − 44 )
:= 5 + ((5/5 + 5) × (5 × 55 + 5))
:= 5 × ((55 + 5)/(5 + 5) + 5 × 5)
:= 6 × (((6 + 6)/6) + 6 × 6 × 6)
:= (6 × (6 × 6 × 6 + 66)) − 6/6 − 6
:= (6 × (6 × 6 × 6 + 66)) − (6 + 6)/6
:= (77 + 7) × ((7 − 7/7 + 7) + 7)
:= (7 − ((7 + 7)/7)) × ((7 × 7 × 7 − 7) + 7/7)
:= 7 + (77/7 × ((77 − 7/7) + 77))
:= 8 + 88 × (88/8 + 8)
:= 8 + ((88 + 8 + 8)/8 × (8 × (8 + 8) + 8/8))
:= (8/8 + 8 × 8) × (((((8 + 8)/8) + 8) + 8) + 8)
:= 9 + ((9 + 9) × 99 − 999/9)
:= (9 + 9)/9 + (99 × ((9 − 9/9) + 9))
:= (9 + 9) × 99 − ((99/9) + 9 × 9)
3
3
:= (3 + 3/3) × (3 + 33)
:= 44 × 44 − 4
4
:= (5/5 + 5) × (5 × 55 + 5)
6
Ï 1681 := (1 + ((1 + 1) × ((1 + 1) × (11 − 1))))1+1
2
:= (2 × (22 − 2) + 2/2)
3
:= ((33/3 + 3 ) + 3)
Ï 1686 := 1 + ((1 + 1)11 − (11 × (11 × (1 + 1 + 1))))
2
:= (2 × (2 × ((22 − 2) + 22))) − 2
3−3/3
3
3
Ï 1691 := 1 + ((11 − 1) × (1 + 1 + 11)1+1 )
:= 2222 − (((22 + 2/2)2 ) + 2)
:= 3 + (33 × ((3 − 3) + 3 ))
:= 33 + (((33/3)3 ) + 333)
:= 4/4 + (44 × 44 − 4 )
:= 4 + ((44 × 44 − 44 ) + (4 + 4)/4)
:= 44/4 + (44 × 44 − 44 )
:= 5/5 + ((5/5 + 5) × (5 × 55 + 5))
:= (5/5 + 5) × ((5 × 55 + 5/5) + 5)
:= 5 + ((5/5 + 5) × ((5 × 55 + 5/5) + 5))
:= (6 × (6 × 6 × 6 + 66)) − 6
:= (6 × (6 × 6 × 6 + 66)) − 6/6
:= 77 + ((77 × (7 + 7 + 7)) − (7/7 + 7))
:= 77 + (((7 + 7 + 7)/7) × (7 × 77 − 7/7))
:= 8 + (88 × (88/8 + 8) + 8/8)
:= 8 × (8 × 8 + 8) + (8888 − 8)/8
:= (88/8 + 8) × (8/8 + 88)
:= (9 + 9) × 99 − ((9 + 9)/9 + 99)
:= (9 + 9) × 99 + (((9 + 9 + 9)/9) − 99)
:= 9 + ((9 + 9) × 99 − (9/9 + 99))
4
:= ((6 × 6 − 6/6) + 6)
(6+6)/6
:= (7 × 7 − (7/7 + 7))
(7+7)/7
Ï 1682 := 1 + ((1 + ((1 + 1) × ((1 + 1) × (11 − 1))))1+1 )
2
2
Ï 1687 := 1111 + (((1 + 1) × (1 + 11))1+1 )
Ï 1692 := 1 + (1 + ((11 − 1) × (1 + 1 + 11)1+1 ))
:= (((2 + 2 + 2) + 22) )/2
:= ((22 + 2)2 ) + 2222/2
:= 2 × ((2 × ((22 − 2)2 + 22)) + 2)
:= (33 × ((33 − 3) + 33 )) − 3/3
:= ((3 × (3 × 3 + 3 + 3)3 ) − 3)/(3 + 3)
:= (3 × 3 + 3)3 − (33 + 3)
:= (4 + 4)/4 + (44 × 44 − 44 )
:= 4 + ((44 × 44 − (4/4 + 44 )) + 4)
:= ((4 + 4) × (44 − 44)) − 4
:= ((5 + 5)/5)5 + (55 × (5 × 5 + 5))
:= 5 × 5 × 55 + (55 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5 × 5 × 55)
:= 6/6 + (((6 × 6 − 6/6) + 6)(6+6)/6 )
:= 6 + (((6 × 6 − 6/6) + 6)(6+6)/6 )
:= 6 × (6 × 6 × 6 + 66)
:= 77 + ((77 × (7 + 7 + 7)) − 7)
:= ((7 + 7)/7) × ((77 × 77 − 7)/7)
:= 8 + (88 × (88/8 + 8) + ((8 + 8)/8))
:= 8 × (8 × 8 + 8) + 8888/8
:= 8/8 + ((88/8 + 8) × (8/8 + 88))
:= (9 + 9) × 99 − (9/9 + 99)
:= 9 + ((9 × (9 × 9 − (9 + 9))) + 9999/9)
:= 9 + (99 × ((9 − 9/9) + 9))
:= 7/7 + ((7 × 7 − (7/7 + 7))
(7+7)/7
)
122
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1693 := (11 × (11 × (1 + (1 + 1 + 11)))) − 1
2
Ï 1698 := (1 + 1 + 1) × (11 + ((1111 − 1)/(1 + 1)))
2+2
2
:= 22 + ((2 × (22 − 2) + 2/2)2 )
:= 2222 − ((22 + 2/2) )
:= 2 + (((2
:= 3/3 + ((3 × 3 + 3)3 − (33 + 3))
:= 3 + ((3 × 3 + 3)3 − 33)
:= 3 + ((3 × 3 + 3)3 − (33 + 3/3))
:= 44 + ((4/4 + 4)4 + 4 × 44 )
:= (4 + 4)/4 + ((4 + 4) × (44 − 44))
:= 4 + ((((4 + 4)4 ) − (((4 − 4/4) + 4)4 )) + 4)
:= 5 + ((55 + 5)/(5 + 5) + 5 × 5 × 55)
:= (5 − (5 + 5)/5) × (555 + (55/5))
:= 55 + ((5 − 5/5)5 + (55 − 5)/5)
:= 6/6 + (6 × (6 × 6 × 6 + 66))
:= 6 + (6 × (6 × 6 × 6 + 66))
:= 66/6 + (6 × (6 × 6 × 6 + 66))
:= 77 + ((77 × (7 + 7 + 7)) − 7/7)
:= (7 − 7/7) × (7 × (7 × 7 − 7) − (77/7))
:= 7 + (((7 + 7)/7) × ((77 × 77 + 7)/7))
:= ((8 × 8 − 8/8) × (88/8 + 8 + 8)) − 8
:= 8 + ((8/8 + 8 × 8) × (((((8 + 8)/8) + 8) + 8) + 8))
:= 888 + (888/8 + 8 × 88)
:= 9 + ((99 × ((9 − 9/9) + 9)) + 9/9)
:= (9 + 9) × 99 − (((9 + 9 + 9)/9) + 9 × 9)
:= (9 + 9)/9 + (9 × ((99 + 9 × 9) + 9))
Ï 1694 := 11 × (11 × (1 + (1 + 1 + 11)))
− 2) × (22/2) ) + 2)
Ï 1703 := (1 + 1 + 11) × ((11 × (1 + 11)) − 1)
Ï 1699 := ((11 − 1) × (1 + (1 + 1 + 11)1+1 )) − 1
:= ((22 − 2/2) × ((2/2 + 2)2+2 )) − 2
:= (22+2 − 2) × (22/2)2
3
Ï 1704 := (1 + 11) × ((1 + 11)1+1 − (1 + 1))
:= 2 × (2 × ((2 × (222 + 2)) − 22))
:= (3 × 3 + 3) − 3/3 − 33
:= 3 + (((3 × 3 + 3) − 33) + 3/3)
:= 3 + ((3 × 3 + 3)3 − 33 )
:= ((4 + 4) × (44 − 44)) − (4 + 4)/4
:= 4 + (((4 + 4)4 ) − (((4 − 4/4) + 4)4 ))
:= 4 + (((4 + 4) × (44 − 44)) + 4)
:= 55 + ((55 × (5 × 5 + 5)) − (55/5))
:= 55 + (((5 − 5/5)5 − 5) + 55 /5)
:= 55 + ((5 − 5/5)5 + 55 /5)
:= (6 + 6)/6 + (6 × (6 × 6 × 6 + 66))
:= 6 + ((6 × (6 × 6 × 6 + 66)) + 6/6)
:= 6 + 6 × (6 × 6 × 6 + 66) + 6
:= 77 + (77 × (7 + 7 + 7))
:= 7 × (77 + 7) + 7777/7
:= (7 × (7 × (7 × 7 − (7 + 7)))) − 77/7
3
:= 88/8 × (((8 + 8)/8 + 88) + 8 × 8)
:= 8 + ((88/8 + 8) × (8/8 + 88))
:= (8 + 8 + 8) × ((8 × 8 − 8/8) + 8)
:= 99/9 + (99 × ((9 − 9/9) + 9))
:= (9 + 9) × 99 − (((9 + 9)/9) + 9 × 9)
:= (9 + 9) × 99 + (((9 + 9 + 9)/9) − 9 × 9)
Ï 1695 := 1 + (11 × (11 × (1 + (1 + 1 + 11))))
Ï 1700 := (11 − 1) × (1 + (1 + 1 + 11)1+1 )
:= (2 × 22)2 + ((2 − 222 )/2)
:= (2 × 22 − 2)2 − 22+2+2
:= 2 + (((2 × (22 − 2) + 2/2)2 ) + 22)
:= (3 × 3 + 3)3 − 33
:= (3 × 3 + 3)3 − (33 + 3/3)
:= 3 + (((3 × 3 + 3)3 − 33 ) + 3/3)
:= ((4 + 4)4 ) − (((4 − 4/4) + 4)4 )
:= 4 + ((4 + 4) × (44 − 44))
:= 44/4 × (444/4 + 44)
:= 5 − (5 × 5 × 55 + 55)
:= 5 × (((5 × 55 + 55) + 5) + 5)
:= 55 + (55 × (5 × 5 + 5))
:= (6 − 6/6) × (666 × 6/(6 + 6) + 6)
:= 6 + ((6 × (6 × 6 × 6 + 66)) + ((6 + 6)/6))
:= 6 + 6 × (6 × 6 × 6 + 66) + 6/6 + 6
:= 7/7 + ((77 × (7 + 7 + 7)) + 77)
:= 7 + (((77 × (7 + 7 + 7)) − 7/7) + 77)
:= 77/7 × (7/7 + 77 + 77)
:= 8 + (8888/8 + 8 × (8 × 8 + 8))
:= (8/8 + 8 + 8) × (((88 + 8)/8) + 88)
:= 8/8 + ((8 + 8 + 8) × ((8 × 8 − 8/8) + 8))
:= (9 + 9) × 99 + (((99 + 9)/9) − 99)
:= ((9 − 9/9) + 9) × (9/9 + 99)
:= 99/9 × ((((9 + 9)/9) − 9) + 9 × (9 + 9))
5
Ï 1696 := 1 + (1 + (11 × (11 × (1 + (1 + 1 + 11)))))
Ï 1701 := 1 + ((11 − 1) × (1 + (1 + 1 + 11)1+1 ))
Ï 1705 := 11 × (11 + (1 + 11)1+1 )
Ï 1706 := 1 + (11 × (11 + (1 + 11)1+1 ))
:= 2 + ((22+2 − 2) × (22/2)2 )
:= (22 − 2/2) × ((2/2 + 2)2+2 )
:= (2 × (2 × (222 − 2))) − 222
:= 3/3 + ((3 × 3 + 3)3 − 33)
:= (3 × 3 + 3)3 − 33
:= 33/3 + ((3 × 3 + 3)3 − 33)
:= (4 + 4) × (44 − 44)
:= 4 + (((4 + 4) × (44 − 44)) + 4/4)
:= (44 − 4)/4 + ((4 + 4) × (44 − 44))
:= (55/5 + 5) × (555/5 − 5)
:= ((5 + 5)/5 + 5) × ((5 − (5 + 5)/5)5 )
:= 55 + ((55 × (5 × 5 + 5)) + 5/5)
:= 6 + ((6 × (6 × 6 × 6 + 66)) − ((6 + 6)/6))
:= (6/6 + 6) × ((6 × 6/(6 + 6))6−6/6 )
:= 6 + 6 × (6 × 6 × 6 + 66) + (6 + 6)/6 + 6
:= ((7 + 7)/7) × ((77 × 77 + 7)/7)
:= 7 + ((77 × (7 + 7 + 7)) + 77)
:= 7 + (7777/7 + 7 × (77 + 7))
:= 8 + ((88 × (88/8 + 8) + 8) + 8)
:= (8 × 8 − 8/8) × (88/8 + 8 + 8)
:= (8 + 8)/8 + ((8 + 8 + 8) × ((8 × 8 − 8/8) + 8))
:= (9 − 9/9) × (((999 + 9 + 9)/9) + 99)
:= 9 × ((99 + 9 × 9) + 9)
:= 9 × 9 × 9 + (999 − ((99 + 99)/9))
Ï 1697 := 1 + (1 + (1 + (11 × (11 × (1 + (1 + 1 + 11))))))
2
2
Ï 1702 := ((1 + 1 + 11) × ((11 × (1 + 11)) − 1)) − 1
Ï 1707 := 1 + (1 + (11 × (11 + (1 + 11)1+1 )))
:= 2 + ((2 × 22 − 2)2 − 22+2+2 )
:= 2 + ((((2 × (22 − 2) + 2/2)2 ) + 22) + 2)
:= 3/3 + ((3 × 3 + 3)3 − 33 )
:= 3 + (((3 × 3 + 3)3 − 33 ) + 3)
:= 44/4 + ((4 + 4) × (44 − 44))
:= 5 + (((55 − 5)/(5 + 5) + 5 × 5 × 55) + 5)
:= 4 + (((4 + 4) × (44 − 44)) + (4 + 4)/4)
:= 55 + (((55 × (5 × 5 + 5)) − 5) + ((5 + 5)/5))
:= 55 + ((55 × (5 × 5 + 5)) + ((5 + 5)/5))
:= 6 + ((6 × (6 × 6 × 6 + 66)) − 6/6)
:= ((66 − 6)/6) + (6 × (6 × 6 × 6 + 66))
:= 6 + (6/6 + 6) × (6 × 6/(6 + 6))6−6/6
:= (7 × (7 − 77)) + (((7 + 7 + 7)/7)7 )
:= 7 + (((77 × (7 + 7 + 7)) + 77) + 7/7)
:= (7 × (7 × (7 × 7 − (7 + 7)))) − (7/7 + 7)
:= 2 + (((2 − 22 )/2) + (2 × 22) )
:= 3 + ((3 × 3 + 3)3 − (3/3 + 33))
:= 4/4 + ((4 + 4) × (44 − 44))
8×8/(8+8)
:= ((8 − 8/8)
) − 8 × 88
:= 9 + ((9 − 9/9) × (((999 + 9)/9) + 99))
:= ((88/8 + 8) × ((8 + 8)/8 + 88)) − 8
:= 8 + (((88/8 + 8) × (8/8 + 88)) + 8)
:= 9/9 + (9 × ((99 + 9 × 9) + 9))
:= 9 + ((9 + 9) × 99 − (((9 + 9 + 9)/9) + 9 × 9))
123
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1708 := (1 + (1 + 1 + 11)) × (1 + 111+1 )
Ï 1713 := 1 + ((1 + 1)11 − ((1 + 1 + 1) × (1 + 111)))
Ï 1718 := 1 + (((1 + 11)1+1+1 ) − 11)
:= 2 + ((2 × (2 × (222 − 2))) − 222)
:= (2 × 22)2 − (222 + 2/2)
:= 2 + (((2 × 22)2 − 222) + 2)
:= ((3/3 + 3) + 3) × ((33+3 + 3)/3)
:= 3 + ((3 × 3 + 3)3 − (3 × (3 + 3)))
:= ((3 − 33)/3) + (3 × 3 + 3)3
:= (4 × (444 − 4 × 4)) − 4
:= 4/4 + (4 × (444 − 4 × 4))
:= 4 + ((4 × (444 − 4 × 4)) + (4 + 4)/4)
:= 5 + (((5 − 5/5)5 + (55 − 5)/5) + 55)
:= (5 − (5 + 5)/5) × ((55 + 5)/5 − 55)
:= 55 − (5 × 5 × 55 + ((5 + 5)/5)5 )
:= 6 × 6 × 66 − (666 + (6 + 6)/6)
:= 66 + (6 × 6 × 66 − ((6 × 6/(6 + 6))6 ))
:= ((6 − 66)/6) + (6 × 6 × (6 × 6 + 6 + 6))
:= (7 × (7 × (7 × 7 − (7 + 7)))) − 7
:= (7 × (7 × (7 × 7 − (7 + 7)))) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 × (7 × (7 × 7 − (7 + 7))))
:= 8 + ((8/8 + 8 + 8) × (((88 + 8)/8) + 88))
:= 8/8 + ((888 − 8 × 8) + 888)
:= 8 + ((88/8 + 8) × ((8 + 8)/8 + 88))
:= 9 + ((9 + 9) × 99 − (((9 + 9)/9) + 9 × 9))
:= (9 + 9) × 99 + (((99 + 9)/9) − 9 × 9)
:= 99 + ((9 × (99 + 9 × 9)) − 9/9)
Ï 1709 := 1 + (1 + 1 + 1 + 11) × (1 + 111+1 )
2
2
Ï 1714 := (1 + 1)11 − (1 + (1 + 1 + 1) × 111)
Ï 1719 := 1 + (1 + (((1 + 11)1+1+1 ) − 11))
:= (2 × 22)2 − 222
:= (2/2 + 2) × (((22 + 2)2 ) − (2/2 + 2))
:= (3 × 3 + 3) − ((3 × (3 + 3)) + 3/3)
:= (3 × 3 + 3)3 − (33/3 + 3)
:= (3 × 3 + 3)3 − 3 × 3
:= 4/4 + ((4 × (444 − 4 × 4)) − 4)
:= (4 + 4)/4 + (4 × (444 − 4 × 4))
:= ((44 − 4) × (44 − 4/4)) − 4/4
:= 22 + (((22/2 + 22) + 2) )
3
5
5
5
:= 5 + (((5 − 5/5) + 5 /5) + 55)
:= 5 − (5 × (5 × 55 + 5) + (55/5))
:= 55 + ((((5 + 5 + 5) × 555) − 5)/5)
:= 6 × 6 × 66 − (666 + 6/6)
:= (((6 + 6)/6)6+6 ) − (6 × 6 × 66 + 6)
:= (((6 − 66) + 6)/6) + (6 × 6 × (6 × 6 + 6 + 6))
:= 7/7 + ((7 × (7 × (7 × 7 − (7 + 7)))) − 7)
:= (7 × (7 × (7 × 7 − (7 + 7)))) − 7/7
:= 77/7 + ((7 × (7 × (7 × 7 − (7 + 7)))) − 7)
:= 8 + ((8 × 8 − 8/8) × (88/8 + 8 + 8))
:= (((8 + 8)/8) × (888 + 8/8)) − 8 × 8
:= (8/8 + 8) × (8 × (8 + 8 + 8) − 8/8)
:= 9 + (((9 − 9/9) + 9) × (9/9 + 99))
:= 9 × 9 × (9 + 9) + (((9 + 9)/9)9−9/9 )
:= 99 + (9 × (99 + 9 × 9))
Ï 1710 := (11 − 1) × (1 + (1 + (1 + 1 + 11)1+1 ))
2
:= 2222 − (2(2/2+2) )
Ï 1715 := (1 + 1)11 − (1 + 1 + 1) × 111
:= 2/2 + ((2 × 22)2 − 222)
Ï 1720 := 1 + (1 + (1 + (((1 + 11)1+1+1 ) − 11)))
:= (2 − 22) × (2 − 2 × 2 × 22)
:= (3 × 3 + 3) − (3 × (3 + 3))
:= ((3 − 33)/3) + ((3 × 3 + 3)3 − 3)
:= 3 + ((3 × 3 + 3)3 − 33/3)
:= 4 × 444 − ((44 + 4 + 4)/4)
:= 4 + (4 × 444 − (44 + 4)/4)
:= (44 − 4) × (44 − 4/4)
:= 5 + ((55 × (5 × 5 + 5)) + 55)
:= 5 + (((55 × (5 × 5 + 5)) + 55) + 5)
:= 6 × 6 × 66 − 666
:= 6 + (6 × 6 × 66 − (666 + 6/6))
:= ((6 + 6)/6 + 6) × (6 × 6 × 6 − 6/6)
3
:= 55 − (5 × (5 × 55 + 5) + 5)
:= (7 − ((7 + 7)/7)) × (7 × 7 × 7 − 7/7)
:= 7 × (7 × (7 × 7 − (7 + 7)))
:= (7 − ((7 + 7)/7)) × (7 × 7 × 7 + 7/7)
:= (88/8 + 8) × ((8 + 8)/8 + 88)
:= (8 − 8/8) × ((8 + 8) × (8 + 8) − (88/8))
:= 8 × (8 × (8 + 8) + 88) − 8
:= 9 + (9 × ((99 + 9 × 9) + 9))
:= 9 × 9 × 9 + (999 − ((99 + 9 + 9)/9))
:= 9/9 + ((9 × (99 + 9 × 9)) + 99)
Ï 1711 := 1 + ((11 − 1) × (1 + (1 + (1 + 1 + 11)1+1 )))
Ï 1716 := 11 × ((1 + 11) × (1 + 1 + 11))
Ï 1721 := 1 + (1 + (1 + (1 + (((1 + 11)1+1+1 ) − 11))))
:= 222/2 + (2 × (22 − 2))2
:= 2 + ((2 × 22)2 − 222)
:= (22/2)2 + (2 × (22 − 2))2
:= 3/3 + ((3 × 3 + 3)3 − (3 × (3 + 3)))
:= (3 × 3 + 3)3 − (3 × 3 + 3)
:= (3 × 3 + 3)3 − ((3/3 + 3) + 3)
:= 4 × 444 − (44 + 4)/4
:= 4 + (4 × (444 − 4 × 4))
:= 4/4 + ((44 − 4) × (44 − 4/4))
:= (5555 + 5 )/5 − 5 × 5
:= (5/5 + 5) × (5 × 55 + (55/5))
:= 5 + ((5/5 + 5) × (5 × 55 + (55/5)))
:= 6/6 + (6 × 6 × 66 − 666)
:= 6 + (6 × 6 × 66 − 666)
:= (6 × 6 × (6 × 6 + 6 + 6)) − 6/6 − 6
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) − (77/7))
:= 7/7 + (7 × (7 × (7 × 7 − (7 + 7))))
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) − 7/7)
:= 88 + (8888/8 + 8 × 8 × 8)
:= 8 × (8 × (8 + 8) + 88) − (88 + 8)/8
:= 8/8 + (8 × (8 × (8 + 8) + 88) − 8)
:= 9 + ((9 × ((99 + 9 × 9) + 9)) + 9/9)
:= 9 × 9 × 9 + (999 − ((99 + 9)/9))
:= 9 + ((9 × ((99 + 9 × 9) + 9)) + (99/9))
5
Ï 1712 := (1 + 1)11 − ((1 + 1 + 1) × (1 + 111))
Ï 1717 := ((1 + 11)1+1+1 ) − 11
Ï 1722 := (1 + (1 + 1 + 11)) × (1 + (1 + 111+1 ))
:= (2 × 22)2 − (222 + 2)
:= 2 + (((2 × 22)2 − 222) + 2/2)
:= (2/2 + 2) × (((22 + 2)2 ) − 2)
:= 33/3 + ((3 × 3 + 3)3 − 33 )
:= (3 × 3 + 3)3 − 33/3
:= (3 × 3 + 3)3 − (3 + 3)
:= 4 × (444 − 4 × 4)
:= 4 + ((4 × (444 − 4 × 4)) + 4/4)
:= (4 + 4)/4 + ((44 − 4) × (44 − 4/4))
:= (55/5 + 5) × ((555 + 5)/5 − 5)
:= ((55 × 55 + 5555) + 5)/5
:= (5/5 + 5) × (((55 + 5)/5) + 5 × 55)
:= (6 + 6)/6 + (6 × 6 × 66 − 666)
:= (6 × 6 × (6 × 6 + 6 + 6)) − 66/6
:= (6 × 6 × (6 × 6 + 6 + 6)) − 6
:= 7 + (77/7 × (7/7 + 77 + 77))
:= ((7 + 7)/7) + (7 × (7 × (7 × 7 − (7 + 7))))
:= 7 + (7 × (7 × (7 × 7 − (7 + 7))))
:= (8 + 8) × ((88/8 + 88) + 8)
:= 8 × (8 × (8 + 8) + 88) − 88/8
:= (8 + 8)/8 + (8 × (8 × (8 + 8) + 88) − 8)
:= 99/9 + (9 × ((99 + 9 × 9) + 9))
:= 9 × 9 × 9 + (999 − (99/9))
:= (9/9 + 9 × 9) × (((99 + 9)/9) + 9)
124
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1723 := ((1 + 11)1+1+1 ) − 1 − 1 − 1 − 1 − 1
Ï 1728 := (1 + 11)1+1+1
Ï 1733 := 1 + (1 + (1 + (1 + (1 + ((1 + 11)1+1+1 )))))
:= 2 + ((2 × (22 − 2))2 + (22/2)2 )
:= (2/2 + 2) × ((22 + 2)2 )
:= 2 + ((2/2 + 2) × (((22 + 2)2 ) + 2/2))
:= 3/3 + ((3 × 3 + 3)3 − (3 + 3))
:= (3 × 3 + 3)3
:= 3 + (((3 × 3 + 3)3 − 3/3) + 3)
:= 44/4 + (4 × (444 − 4 × 4))
:= 4 × (4 × 44 + 44 )
:= 4/4 + (4 × 444 − 44)
:= (((55 + 5)/5)5−(5+5)/5 ) − 5
:= ((55 + 5)/5)5−(5+5)/5
:= 5 + (((55 + 5)/5)5−(5+5)/5 )
:= 6/6 + ((6 × 6 × (6 × 6 + 6 + 6)) − 6)
:= 6 × 6 × (6 × 6 + 6 + 6)
:= 6 + ((6 × 6 × (6 × 6 + 6 + 6)) − 6/6)
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) + 7/7)
:= ((77 + 7)/7)(7+7+7)/7
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) + (77/7))
:= 8 + ((8 − 8/8) × ((8 + 8) × (8 + 8) − (88/8)))
:= 8 × (8 × (8 + 8) + 88)
:= 8 × 8 × 8 + 88/8 × 888/8
:= 9 × 9 × 9 + 999
:= (9 + 9) × 99 + ((9 − 9 × 99)/(9 + 9))
9−9/9
:= 9 + ((((9 + 9)/9)
) + 9 × 9 × (9 + 9))
Ï 1724 := ((1 + 11)1+1+1 ) − 1 − 1 − 1 − 1
Ï 1734 := ((1 + 1) × (1 + 1 + 1)) + ((1 + 11)1+1+1 )
Ï 1729 := 1 + ((1 + 11)1+1+1 )
:= 2 + ((2/2 + 2) × (((22 + 2)2 ) − 2))
:= 2/2 + ((2/2 + 2) × ((22 + 2)2 ))
:= (2/2 + 2) × (((22 + 2)2 ) + 2)
:= (3 × 3 + 3)3 − (3/3 + 3)
:= 3/3 + (3 × 3 + 3)3
:= 3 + ((3 × 3 + 3)3 + 3)
:= (4 × (4 × 44 + 44 )) − 4
:= 4/4 + (4 × (4 × 44 + 44 ))
:= (4 + 4)/4 + (4 × 444 − 44)
:= 5 − (5 × (5 × 55 + 5) + 5/5)
:= 5 + (55 × (5 × 5 − 5) + (55 − 5)/5)
:= ((5555 − (5 + 5)) + 55 )/5
:= (6 + 6)/6 + ((6 × 6 × (6 × 6 + 6 + 6)) − 6)
:= 6/6 + (6 × 6 × (6 × 6 + 6 + 6))
:= 6 + (6 × 6 × (6 × 6 + 6 + 6))
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) + ((7 + 7)/7))
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) + 7)
:= (7 × (7 × 7 × 7 − 77)) − ((7 + 7)/7)7
:= 8 × (8 × (8 + 8) + 88) − (8/((8 + 8)/8))
:= 8/8 + 8 × (8 × (8 + 8) + 88)
:= 8 + (8 × (8 × (8 + 8) + 88) − ((8 + 8)/8))
:= 9/9 + (9 × 9 × 9 + 999)
:= ((9 − 9/9) + 9) × (999/9 − 9)
5
99/9
:= (((9 + 9)/9)
) − (9 + 9) × (9 + 9)
Ï 1725 := ((1 + 11)1+1+1 ) − 1 − 1 − 1
Ï 1730 := 1 + (1 + ((1 + 11)1+1+1 ))
Ï 1735 := 1 + (((1 + 1) × (1 + 1 + 1)) + ((1 + 11)1+1+1 ))
:= (2/2 + 2) × (((22 + 2)2 ) − 2/2)
:= 2 + ((2/2 + 2) × ((22 + 2)2 ))
:= 2/2 + ((2/2 + 2) × (((22 + 2)2 ) + 2))
:= (3 × 3 + 3)3 − 3
:= 3 + ((3 × 3 + 3)3 − 3/3)
:= 3 + (((3 × 3 + 3)3 + 3/3) + 3)
:= 4/4 + ((4 × (4 × 44 + 44 )) − 4)
:= (4 + 4)/4 + (4 × (4 × 44 + 44 ))
:= 4 + 4 × 444 − (44 + 4/4)
:= 5 + (55 − 5 × (5 × 55 + 5))
:= (55 × ((5 + 5)/5)5 ) − 5 × 5
:= (6 + 6)/6 + (6 × 6 × (6 × 6 + 6 + 6))
:= 6 + ((6 × 6 × (6 × 6 + 6 + 6)) + 6/6)
:= 77 + ((((7 + 7 + 7)/7) ) − 7 × 77)
:= 7 + (((7 × (7 × (7 × 7 − (7 + 7)))) + 7/7) + 7)
:= 7 + (((77 + 7)/7)(7+7+7)/7 )
:= 8 + (8 × (8 × (8 + 8) + 88) − (88/8))
:= (8 + 8)/8 + 8 × (8 × (8 + 8) + 88)
:= 8 + (8 × (8 × (8 + 8) + 88) − 8/8)
:= 9 × 9 × 9 + (999 − ((9 + 9 + 9)/9))
:= 9 × 9 × 9 + (((9 + 9)/9) + 999)
:= 9 + ((9 × 9 × 9 − ((9 + 9)/9)) + 999)
Ï 1731 := 1 + (1 + (1 + ((1 + 11)1+1+1 )))
Ï 1736 := 11 + (((1 + 11)1+1+1 ) − (1 + 1 + 1))
5
:= 5 − 5 × (5 × 55 + 5)
:= (6 × 6 × (6 × 6 + 6 + 6)) − 6 × 6/(6 + 6)
7
Ï 1726 := ((1 + 11)1+1+1 ) − 1 − 1
:= ((2/2 + 2) × ((22 + 2)2 )) − 2
:= (2/2 + 2) × (((22 + 2)2 ) + 2/2)
:= 2222 − (222 + 2)
:= 3/3 + ((3 × 3 + 3)3 − 3)
:= 3 + (3 × 3 + 3)3
:= 3 × 3 + ((3 × 3 + 3)3 − 3/3)
:= (4 × (4 × 44 + 44 )) − (4 + 4)/4
:= 4 × 444 − (44 + 4/4)
:= 4 + (4 × 444 − 44)
5
:= (5555 + 55 )/5
5
:= 5 + (5/5 − 5 × (5 × 55 + 5))
:= (5555 + 5 )/5 − 5
:= (6 × 6 × (6 × 6 + 6 + 6)) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × 6 × (6 × 6 + 6 + 6))
:= ((6 + 6)/6 + 6) × (6 × 6 × 6 + 6/6)
:= 77/7 + (7 × (7 × (7 × 7 − (7 + 7))))
:= 7 + (((7 × (7 × (7 × 7 − (7 + 7)))) + ((7 + 7)/7)) + 7)
:= 7 + (((7 × (7 × (7 × 7 − (7 + 7)))) + 7) + 7)
:= 8 × (8 × (8 + 8) + 88) − (8 + 8)/8
:= 88/8 + (8 × (8 × (8 + 8) + 88) − 8)
:= 8 + 8 × (8 × (8 + 8) + 88)
:= 9 × 9 × 9 + (999 − ((9 + 9)/9))
:= 999/9 + (9 × (99 + 9 × 9))
:= 9 + ((999 − 9/9) + 9 × 9 × 9)
Ï 1727 := ((1 + 11)1+1+1 ) − 1
Ï 1732 := 1 + (1 + (1 + (1 + ((1 + 11)1+1+1 ))))
Ï 1737 := 11 + (((1 + 11)1+1+1 ) − (1 + 1))
:= ((2/2 + 2) × ((22 + 2)2 )) − 2/2
:= 2 × ((2 × 2 × 222) − 22)
:= 2222 − (222 + 2/2)
:= (3 × 3 + 3)3 − 3/3
:= 3 + ((3 × 3 + 3)3 + 3/3)
:= 3 × 3 + (3 × 3 + 3)3
:= 4 × 444 − 44
:= 4 + ((4 × 444 − 44) + 4/4)
4
:= (4 × (4 × 44 + 4 )) − 4/4
5
:= 55 + ((5 + 5)/5 − 5 × (5 × 55 + 5))
:= (((5555 + 5 ) + 5)/5) − 5
:= ((5555 + 55 ) + 5)/5
:= (6 × 6 × (6 × 6 + 6 + 6)) − 6/6
:= 6 + ((6 × 6 × (6 × 6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + ((6 × 6 × (6 × 6 + 6 + 6)) + (6 × 6/(6 + 6)))
7
:= 7 + (((((7 + 7 + 7)/7) ) − 7 × 77) + 77)
:= 7 + ((((7 × (7 × (7 × 7 − (7 + 7)))) + 7/7) + 7) + 7)
:= 8 × (8 × (8 + 8) + 88) − 8/8
:= (8/((8 + 8)/8)) + 8 × (8 × (8 + 8) + 88)
:= 8 + (8 × (8 × (8 + 8) + 88) + 8/8)
:= 9 × 9 × 9 + (999 − 9/9)
:= (9 + 9) × 99 − ((9 × 99 + 9)/(9 + 9))
:= 9 + (9 × 9 × 9 + 999)
:= 7 + ((7 − ((7 + 7)/7)) × (7 × 7 × 7 + 7/7))
125
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1743 := 1 + (1 + (1 + (1 + (11 + ((1 + 11)1+1+1 )))))
Ï 1738 := 11 + (((1 + 11)1+1+1 ) − 1)
Ï 1748 := ((1 + 1) × (11 − 1)) + ((1 + 11)1+1+1 )
:= 2222 − 222
:= 2/2 + ((2 × 22 − 2)2 − 22)
:= (2 × 22 − 2)2 − 22+2
:= 3 × 3 + ((3 × 3 + 3)3 + 3/3)
:= 3 + (((3 × 3 + 3)3 + 3 × 3) + 3)
:= 3 × 3 + ((3 × 3 + 3)3 + 33/3)
:= 4 + ((4 × 444 − 44) + (4 + 4)/4)
:= (4 × (444 − (4 + 4))) − 4/4
:= 4 + (4 × (444 − (4 + 4)))
:= (((5555 + 55 ) + 5) + 5)/5
:= 5 + ((((5555 + 55 ) + 5) + 5)/5)
:= 55 − (5 × 5 × 55 + ((5 + 5)/5))
:= ((66 − 6)/6) + (6 × 6 × (6 × 6 + 6 + 6))
:= (6/6 + 6) × (((6 × 6/(6 + 6))6−6/6 ) + 6)
:= 6 + ((((6 + 6)/6 + 6) × (6 × 6 × 6 + 6/6)) + 6)
:= 77/7 × (7 × (7 + 7 + 7) + (77/7))
:= 7 × ((((7 + 7)/7)7+7/7 ) − 7)
:= (77 − 7/7) × ((((7 + 7)/7 + 7) + 7) + 7)
:= 8 + (8 × (8 × (8 + 8) + 88) + ((8 + 8)/8))
:= 8 + ((8 × (8 × (8 + 8) + 88) − 8/8) + 8)
:= 8 + ((88 − 8/8) × ((88 + 8)/8 + 8))
:= 9 + ((9 × 9 × 9 + 999) + 9/9)
:= 9 + (((9 − 9/9) + 9) × (999/9 − 9))
:= (9/9 + 9 + 9) × ((99/9) + 9 × 9)
Ï 1739 := 11 + ((1 + 11)1+1+1 )
Ï 1744 := (111 − 1 − 1) × (1 + 1)1+1+1+1
Ï 1749 := 11 + (11 + (((1 + 11)1+1+1 ) − 1))
:= 2/2 + (2222 − 222 )
:= 2 + ((2 × 22 − 2)2 − 22)
:= 2/2 + ((2 × 22 − 2)2 − 22+2 )
:= 33/3 + (3 × 3 + 3)3
:= 33 + ((3 × 3 + 3)3 − 33/3)
:= 3 + ((3 × 3 + 3)3 + (3 × (3 + 3)))
:= 44/4 + (4 × (4 × 44 + 44 ))
:= 4 × (444 − (4 + 4))
:= (4 × (444 − 4)) − 44/4
:= 5 − (5 × 5 × 55 + (55/5))
:= 55 − ((5 × 5 × 55 + 5/5) + 5)
:= 55 − (5 × 5 × 55 + 5/5)
:= 66/6 + (6 × 6 × (6 × 6 + 6 + 6))
:= ((6 + 6)/6 + 6) × (6 × 6 × 6 + (6 + 6)/6)
:= 66/6 × ((666/6 + 6 × 6 + 6) + 6)
5
7+7/7
:= 77/7 × ((777 − 7)/7 + 7 × 7)
:= ((77/7 + 7) × (7 × (7 + 7) − 7/7)) − 7
:= 7/7 + (7 × ((((7 + 7)/7)
:= 88/8 + 8 × (8 × (8 + 8) + 88)
:= 8 + (8 × (8 × (8 + 8) + 88) + 8)
:= 88/8 × (((88 − 8/8) + 8 × 8) + 8)
:= 9 × 9 × 9 + (99/9 + 999)
:= 9 × 9 × 9 + ((((9 + 9)/9)9/9+9 ) − 9)
:= 99/9 × (9 × (9 + 9) − ((9 + 9 + 9)/9))
Ï 1740 := 1 + (11 + ((1 + 11)1+1+1 ))
) − 7))
Ï 1745 := 1 + ((111 − 1 − 1) × (1 + 1)1+1+1+1 )
Ï 1750 := 11 + (11 + ((1 + 11)1+1+1 ))
:= 2 + (2222 − 222 )
:= 2 + (((2 × 22 − 2)2 − 22) + 2/2)
:= 2 + ((2 × 22 − 2)2 − 22+2 )
:= 3 + ((3 × 3 + 3)3 + 3 × 3)
:= 3 + (((3 × 3 + 3)3 + 33/3) + 3)
:= 33 + ((3 × 3 + 3)3 − 33/3)
:= (4 × (444 − (4 + 4))) − 4
:= 4/4 + (4 × (444 − (4 + 4)))
:= (4 − 44)/4 + (4 × (444 − 4))
5
:= 5 − ((5 × 5 × 55 + 5) + 5)
:= 5 − (5 × 5 × 55 + 5)
:= 6 + ((6 × 6 × (6 × 6 + 6 + 6)) + 6)
:= 6 + ((6 × 6 × (6 × 6 + 6 + 6)) + (66/6))
5
:= 55 − 5 × 5 × 55
:= (6/6 + 6) × (6 × (6 × 6 + 6) − ((6 + 6)/6))
:= 7 + (((7 × (7 × (7 × 7 − (7 + 7)))) + (77/7)) + 7)
:= ((7 + 7)/7) + (77 × (7 + 7 + 7))
:= (7 + 7) × (777/7 + 7 + 7)
:= (88 − 8/8) × ((88 + 8)/8 + 8)
:= 8 + ((8 × (8 × (8 + 8) + 88) + 8/8) + 8)
:= 8 × (88 − 8) + (8888 − 8)/8
:= 9 + ((9 × (99 + 9 × 9)) + 999/9)
:= ((9 + 9) × (99 − ((9 + 9)/9))) − 9/9
:= 9 × (9 × 9 − 9) + (9999/9 − 9)
Ï 1741 := 1 + (1 + (11 + ((1 + 11)1+1+1 )))
7
Ï 1746 := ((1 + 11)1+1+1 ) + (1 + 1) × (11 − 1 − 1)
2
Ï 1751 := 1 + (11 + (11 + ((1 + 11)1+1+1 )))
:= 2 + (((2 × 22 − 2)2 − 22) + 2)
:= (2 × 22 − 2)2 − (22/2 + 2)
:= 3 + (((3 × 3 + 3) + 3/3) + 3 × 3)
:= (3 × (3 + 3)) + (3 × 3 + 3)3
:= 33 + ((3 × 3 + 3)3 − (3/3 + 3))
:= 4/4 + ((4 × (444 − (4 + 4))) − 4)
:= (4 + 4)/4 + (4 × (444 − (4 + 4)))
:= (4 × (444 − 4)) − ((4/4 + 4) + 4)
:= (2 × 22 − 2) − (22 + 2/2)
3
:= 5 + (5555 + 55 )/5
:= 5 + ((5555 + 5 )/5 + 5)
:= 55 + (5/5 − 5 × 5 × 55)
:= 666 + (6 × 6 × (6 × 6 − 6))
:= (66/6 + 6) × ((6 × 6 + 66) + 6/6)
:= 77 + ((7 − 7/7 + 7) × ((7 + 7)/7) )
:= (77/7 + 7) × (7 × (7 + 7) − 7/7)
:= 7/7 + ((7 + 7) × (777/7 + 7 + 7))
:= 88 + ((88/8 + 8) × (88 − 8/8))
:= ((8 + 8)/8) × ((888 − (8 + 8)) + 8/8)
:= 8 × (88 − 8) + 8888/8
:= 9 × 9 × 9 + (9999/9 − 99)
:= (9 + 9) × (99 − ((9 + 9)/9))
:= ((9 − 9/9) + 9) × (((999 + 9)/9) − 9)
5
:= 6 + (((6 × 6 × (6 × 6 + 6 + 6)) + 6/6) + 6)
7
Ï 1742 := 1 + (1 + (1 + (11 + ((1 + 11)1+1+1 ))))
:= (2 × 22 − 2)2 − 22
Ï 1747 := ((1 + 1) × (11 − 1)) + (((1 + 11)1+1+1 ) − 1)
:= (2 × 22 − 2)2 − (22+2 + 2/2)
Ï 1752 := (1 + 11) × (1 + (1 + (1 + 11)1+1 ))
:= 2 × (2 × ((2 × (222 − 2)) − 2))
:= 3 + ((3 × 3 + 3) + 33/3)
:= 3/3 + ((3 × 3 + 3) + (3 × (3 + 3)))
:= 33 + ((3 × 3 + 3)3 − 3)
:= (4 × (444 − (4 + 4))) − (4 + 4)/4
:= 4 + ((4 × (444 − (4 + 4))) − 4/4)
:= (4 × (444 − 4)) − 4 − 4
3
3
:= 5 + (((5555 + 5 ) + 5)/5)
:= ((5555 + 55) + 5 )/5
:= 55 + ((5 + 5)/5 − 5 × 5 × 55)
:= 6 + (((6 + 6)/6 + 6) × (6 × 6 × 6 + 6/6))
:= 66 + (((6 × 6 − 6/6) + 6)(6+6)/6 )
:= (6 × 6 + 6) × (6 × 6 + 6) − 6 − 6
:= 7/7 + ((77/7 + 7) × (7 × (7 + 7) − 7/7))
:= 7 + ((77 × (7 + 7 + 7)) + ((7 + 7)/7)7 )
:= ((8 + 8)/8) × (888 − (8/8 + 8 + 8))
:= 8 + (8 × (8 × (8 + 8) + 88) + (88/8))
:= 88 + ((8 + 8) × (88 + 8 + 8))
:= ((9 + 9)/9) × (9 × 99 − (99/9 + 9))
:= 9/9 + ((9 + 9) × (99 − ((9 + 9)/9)))
:= 9 × 9 + ((9 + 9) × 99 − 999/9)
5
(7+7+7)/7
:= 7 + ((((77 + 7)/7)
5
) + 7)
126
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1753 := 1 + ((1 + 11) × (1 + (1 + (1 + 11)1+1 )))
Ï 1758 := (1 + 1) × ((11 − 1)1+1+1 − 111+1 )
Ï 1763 := (((1 + 1) × (11 + (11 − 1)))1+1 ) − 1
:= (2 × 22 − 2)2 − 22/2
:= (2 × 22 − 2)2 − (2 + 2 + 2)
:= (2 × 22 − 2)2 − 2/2
:= 33 + (((3 × 3 + 3)3 − 3) + 3/3)
:= 3 + ((3 × 3 + 3)3 + 33 )
:= 3 + (((3 × 3 + 3)3 − 3/3) + 33)
:= 4 + ((4 × (444 − 4)) − 44/4)
:= (4 × (444 − 4)) − (4 + 4)/4
:= 4 + ((4 × (444 − 4)) − 4/4)
:= 5 + (55 − (5 × 5 × 55 + ((5 + 5)/5)))
:= (55 × ((5 + 5)/5)5 ) − (5 + 5)/5
:= 5 + ((55 × ((5 + 5)/5)5 ) − ((5 + 5)/5))
:= (6 × 6 + 6) × (6 × 6 + 6) − 66/6
:= (6 × 6 + 6) × (6 × 6 + 6) − 6
:= (6 × 6 + 6) × (6 × 6 + 6) − 6/6
:= ((7 + 7) × (77 + 7 × 7)) − 77/7
:= 7/7 + (((7 + 7) × (77 + 7 × 7)) − 7)
:= ((7 + 7) × (77 + 7 × 7)) − 7/7
:= 8/8 + (((8 + 8) × (88 + 8 + 8)) + 88)
:= ((8 + 8)/8) × (888 − 8/8 − 8)
:= 8 + ((8/8 + 8 × 8) × (88/8 + 8 + 8))
:= ((9 + 9)/9) × (9 × 99 − ((99 + 9)/9))
:= (9 + 9) × 99 − (9/9 + 9 + 9)
9/9+9
:= 9 × 9 × 9 + (((9 + 9)/9)
)
Ï 1754 := 1 + (1 + ((1 + 11) × (1 + (1 + (1 + 11)1+1 ))))
Ï 1759 := ((111 − 1) × (1 + 1)1+1+1+1 ) − 1
Ï 1764 := ((1 + 1) × (11 + (11 − 1)))1+1
:= (2 × (2 × 2 × 222)) − 22
:= (2 × 22 − 2)2 − (2/2 + 2 + 2)
:= ((2 × 22) − 2)2
:= 33 + ((3 × 3 + 3)3 − 3/3)
:= 3 + (((3 × 3 + 3)3 + 33 ) + 3/3)
:= 3 + ((3 × 3 + 3)3 + 33)
:= 4 × 444 − (44/((4 + 4)/4))
:= (4 × (444 − 4)) − 4/4
:= 4 + (4 × (444 − 4))
5
:= 5 + (5 − (5 × 5 × 55 + 5/5))
:= (55 × ((5 + 5)/5) ) − 5/5
:= 5 + ((55 × ((5 + 5)/5)5 ) − 5/5)
:= ((6 − 66)/6) + (6 × 6 + 6) × (6 × 6 + 6)
:= 6/6 + ((6 × 6 + 6) × (6 × 6 + 6) − 6)
:= (6 × 6 + 6) × (6 × 6 + 6)
:= ((7 − 77)/7) + ((7 + 7) × (77 + 7 × 7))
:= ((7 + 7)/7) + (((7 + 7) × (77 + 7 × 7)) − 7)
:= (7 + 7) × (77 + 7 × 7)
:= ((8 + 8)/8) × (888 − 88/8)
:= (88 × ((88 + 8)/8 + 8)) − 8/8
:= 888 + (888 − ((88 + 8)/8))
:= (9 + 9) × 99 − ((9/9 + 9 + 9) + 9)
:= 9 × (9 × 9 − 9) + 9999/9
:= (9 + 9) × (99 − 9/9)
5
Ï 1755 := (11 − 1 − 1) × ((1 + 1 + 1 + 11)1+1 − 1)
Ï 1765 := 1 + (((1 + 1) × (11 + (11 − 1)))1+1 )
Ï 1760 := (111 − 1) × (1 + 1)1+1+1+1
:= 2 + ((2 × 22 − 2)2 − 22/2)
:= 2 × (2 × (2 × (222 − 2)))
:= 2/2 + (2 × 22 − 2)2
:= 33 + (3 × 3 + 3)3
:= 33 + ((3 × 3 + 3)3 − 3/3)
:= 3 + (((3 × 3 + 3)3 + 33) + 3/3)
:= (4 × (444 − 4)) − (4/4 + 4)
:= 4 × (444 − 4)
:= 4 × 444 − 44/4
5
5
:= 55 × ((5 + 5)/5)
:= 5 + (55 × ((5 + 5)/5)5 )
:= (6 + 6)/6 + ((6 × 6 + 6) × (6 × 6 + 6) − 6)
:= 6/6 + (6 × 6 + 6) × (6 × 6 + 6)
:= (7 − 7/7 + 7) × (((7 + 7)/7) + 7)
:= 77/7 × (777/7 + 7 × 7)
:= 7/7 + ((7 + 7) × (77 + 7 × 7))
:= (8/8 + 8 × 8) × (88/8 + 8 + 8)
:= 88 × ((88 + 8)/8 + 8)
:= 888 + (888 − 88/8)
:= (9 + 9) × 99 − (9 + 9 + 9)
:= 99/9 × (9 × (9 + 9) − ((9 + 9)/9))
:= 9/9 + ((9 + 9) × (99 − 9/9))
:= 5 + (5 − 5 × 5 × 55)
:= (6/6 − 66) × (6 − (66 × 6/(6 + 6)))
7
Ï 1756 := 1 + (11 − 1 − 1) × ((1 + 1 + 1 + 11)1+1 − 1)
Ï 1761 := 1 + ((111 − 1) × (1 + 1)1+1+1+1 )
Ï 1766 := 1 + (1 + (((1 + 1) × (11 + (11 − 1)))1+1 ))
:= 2 × (2 × (((22 − 2/2)2 ) − 2))
:= (2 × 22 − 2)2 − 2/2 − 2
:= 2 + (2 × 22 − 2)2
:= 33 + ((3 × 3 + 3)3 + 3/3)
:= 33 + (3 × 3 + 3)3
:= 33 + ((3 × 3 + 3)3 + 33/3)
:= (4 × (444 − 4)) − 4
:= 4/4 + (4 × (444 − 4))
:= (4 − 44)/4 + 4 × 444
:= 5 + ((5 − 5 × 5 × 55) + 5/5)
:= 5/5 + (55 × ((5 + 5)/5)5 )
:= 5 + ((55 × ((5 + 5)/5)5 ) + 5/5)
:= ((6 + 6)/6)6 + (6 × (6 × 6 × 6 + 66))
:= (6 × 6 + 6) × (6 × 6 + 6) − 6 × 6/(6 + 6)
:= (6 + 6)/6 + (6 × 6 + 6) × (6 × 6 + 6)
:= ((7 + 7) × (77 + 7 × 7)) − (7/7 + 7)
:= ((7 + 7 + 7)/7) × (7 × (77 + 7) − 7/7)
:= ((7 + 7)/7) + ((7 + 7) × (77 + 7 × 7))
:= ((8 + 8)/8) × ((8 − 88)/8 + 888)
:= 8/8 + (88 × ((88 + 8)/8 + 8))
:= (((8 + 8)/8) × (888 − 8/8)) − 8
:= 9/9 + ((9 + 9) × 99 − (9 + 9 + 9))
:= (9 + 9) × 99 − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + ((9 + 9) × (99 − 9/9))
5
Ï 1757 := (1 + 1)11 − (1 + ((1 + 1) × (1 + (1 + 11)1+1 )))
Ï 1762 := (((1 + 1) × (11 + (11 − 1)))1+1 ) − 1 − 1
Ï 1767 := 1 + (1 + (1 + (((1 + 1) × (11 + (11 − 1)))1+1 )))
:= 2 + (((2 × 22 − 2)2 − 22/2) + 2)
:= (2 × 22 − 2)2 − 2
:= 2 + ((2 × 22 − 2)2 + 2/2)
:= 3 + (((3 × 3 + 3)3 − 3/3) + 33 )
:= 3/3 + ((3 × 3 + 3)3 + 33)
:= 3 + (((3 × 3 + 3)3 + 33) + 3)
:= (4 + 4)/4 + (4 × (444 − 4))
:= 4/4 + ((4 × (444 − 4)) − 4)
:= 4 × 444 − ((4/4 + 4) + 4)
5
:= 5 + (((5 + 5)/5 − 5 × 5 × 55) + 5 )
:= (5 + 5)/5 + (55 × ((5 + 5)/5) )
:= 5 + ((55 × ((5 + 5)/5)5 ) + ((5 + 5)/5))
:= (6/6 + 6) × (6 × (6 × 6 + 6) − 6/6)
:= (6 × 6 + 6) × (6 × 6 + 6) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × 6 + 6) × (6 × 6 + 6)
5
:= ((7 + 7) × (77 + 7 × 7)) − 7
:= ((7 + 7) × (77 + 7 × 7)) − (7 + 7)/7
:= 7 + (77/7 × (777/7 + 7 × 7))
:= 888 + (888 − (88/8 + 8))
:= ((8 + 8)/8) × ((888 − 8) + 8/8)
:= 888 + (888 − 8/8 − 8)
:= 9 + ((9/9 + 9 + 9) × ((99/9) + 9 × 9))
:= (9 + 9) × 99 − (99/9 + 9)
:= (9 + 9) × 99 + (((9 + 9 + 9)/9) − (9 + 9))
127
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1768 := (1 + 1)1+1+1 × ((1 + 1) × 111 − 1)
Ï 1773 := (1 + 1)11 − (11 × (1 + ((1 + 1) × (1 + 11))))
Ï 1778 := 1 + (1 + (111 × (1 + 1)1+1+1+1 ))
:= 2 + ((2 × 22 − 2)2 + 2)
:= 2/2 + (2 × ((2 × 2 × 222) − 2))
:= 2 + (2 × (2 × 2 × 222))
:= 3 + ((((3 × 3 + 3)3 + 33) + 3/3) + 3)
:= 3 × (3 × 33 × (3 + 3) − 3)
:= (3 × 3 × 33 × (3 + 3)) − (3/3 + 3)
:= 4 × 444 − 4 − 4
:= 4/4 + (4 × 444 − 4)
:= (4 + 4)/4 + 4 × 444
55/5
:= (((5 + 5)/5)
:= (((5 + 5)/5)55/5 ) − (5 × 55 + 5)
) − 5 × 55
:= 5 + ((((5 + 5)/5)55/5 ) − 5 × 55)
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) − ((6 + 6)/6))
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) + (6 × 6/(6 + 6)))
:= 666 + (6666 + 6)/6
:= 77/7 + (((7 + 7) × (77 + 7 × 7)) − 7)
:= 7 + (((7 + 7) × (77 + 7 × 7)) + ((7 + 7)/7))
:= 7 + (((7 + 7) × (77 + 7 × 7)) + 7)
:= 888 + 888 − 8
:= 8 + ((888 − 88/8) + 888)
:= ((8 + 8)/8) × (888 + 8/8)
:= 9 + (9999/9 + 9 × (9 × 9 − 9))
:= (9 + 9) × 99 − 9
:= ((9 + 9)/9) × (9 × 99 − ((9 + 9)/9))
Ï 1769 := 1 + ((1 + 1)1+1+1 × ((1 + 1) × 111 − 1))
Ï 1774 := (1 + 1) × ((111 × (1 + 1)1+1+1 ) − 1)
Ï 1779 := 1 + (1 + (1 + (111 × (1 + 1)1+1+1+1 )))
:= 2 + (((2 × 22 − 2)2 + 2/2) + 2)
:= (2 × (2 × 2 × 222)) − 2
:= 2 + ((2 × (2 × 2 × 222)) + 2/2)
:= 3 + (((3 × 3 + 3)3 + 33/3) + 33 )
:= 3/3 + (3 × (3 × 33 × (3 + 3) − 3))
:= (3 × 3 × 33 × (3 + 3)) − 3
:= 4 + (4 × 444 − 44/4)
:= 4 × 444 − (4 + 4)/4
:= 4 + (4 × 444 − 4/4)
:= 5 + (((55 × ((5 + 5)/5) ) − 5/5) + 5)
:= 5 + ((5 × (5 − 5 × 55)) − 5/5)
:= 5 + (((5 × (5 − 5 × 55)) − 5/5) + 55 )
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) − 6/6)
:= ((66 − 6)/6) + (6 × 6 + 6) × (6 × 6 + 6)
:= 666 + (((6666 + 6) + 6)/6)
:= 7 + (((7 + 7) × (77 + 7 × 7)) − ((7 + 7)/7))
:= ((77 − 7)/7) + ((7 + 7) × (77 + 7 × 7))
:= 7 + ((((7 + 7) × (77 + 7 × 7)) + 7/7) + 7)
:= 8/8 + ((888 − 8) + 888)
:= ((8 + 8)/8) × (888 − 8/8)
:= 8/8 + (((8 + 8)/8) × (888 + 8/8))
:= (9 + 9) × 99 − ((99 + 9 + 9)/9)
:= 9/9 + ((9 + 9) × 99 − 9)
:= (9 + 9) × 99 − (9 + 9 + 9)/9
5
Ï 1770 := (11 − 1) × (1 + (11 × (1 + 1)1+1+1+1 ))
5
Ï 1775 := (111 × (1 + 1)1+1+1+1 ) − 1
:= 22/2 + (2 × 22 − 2)2
:= 2 + (((2 × 22 − 2)2 + 2) + 2)
Ï 1780 := (1 + 1) × ((11 × (11 − 1 − 1)1+1 ) − 1)
:= 2 × ((2 × 2 × 222) + 2)
:= 3 × 3 + ((3 × 3 + 3) + 33)
:= (((3 + 3) × ((33 × 3 ) − 3)) − 3)/3
:= 3/3 + ((3 × 3 × 33 × (3 + 3)) − 3)
:= 4 × 444 − ((4 + 4)/4 + 4)
:= 4 × 444 − 4/4
:= 4 + 4 × 444
:= 5 + ((55 × ((5 + 5)/5)5 ) + 5)
:= 5 × ((5 × (55 + 5)) + 55)
:= 5 + ((5 × (5 − 5 × 55)) + 55 )
:= 6 + (6 × 6 + 6) × (6 × 6 + 6)
:= 66/6 + (6 × 6 + 6) × (6 × 6 + 6)
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) + ((66 − 6)/6))
:= 7 + (((7 + 7) × (77 + 7 × 7)) − 7/7)
:= 77/7 + ((7 + 7) × (77 + 7 × 7))
:= ((7 + 7) × ((7 + 7)/7)7 ) − (77 + 7)/7
:= (((8 + 8)/8) × (888 + 8/8)) − 8
:= 888 + (888 − 8/8)
:= ((8 + 8)/8) × (888 + ((8 + 8)/8))
:= (9 + 9) × 99 − (99 + 9)/9
:= (9 + 9)/9 + ((9 + 9) × 99 − 9)
:= (9 + 9) × 99 − (9 + 9)/9
3
3
Ï 1771 := 11 × (((1 + 1) × (11 − 1 − 1)1+1 ) − 1)
Ï 1776 := 111 × (1 + 1)1+1+1+1
Ï 1781 := ((1 + 1) × (11 × (11 − 1 − 1)1+1 )) − 1
:= 2 × (2 × 2 × 222)
:= 2/2 + (2 × ((2 × 2 × 222) + 2))
:= 33/3 × ((3 + 3) × 3 − 3/3)
:= 3 + (3 × (3 × 33 × (3 + 3) − 3))
:= (3 × 3 × 33 × (3 + 3)) − 3/3
:= 4 × 444 − (4/4 + 4)
:= 4 × 444
:= 4 + (4 × 444 + 4/4)
:= 55/5 + (55 × ((5 + 5)/5)5 )
:= (55/5 + 5) × 555/5
:= 5 + ((55/5 + 5) × 555/5)
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) + 6/6)
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) + 6)
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) + (66/6))
:= 7 + ((7 + 7) × (77 + 7 × 7))
:= 777/7 × (((7 + 7)/7 + 7) + 7)
:= ((7 + 7) × ((7 + 7)/7)7 ) − 77/7
:= 88/8 + (88 × ((88 + 8)/8 + 8))
:= (8 + 8) × 888/8
:= (8 × ((8 × (8 + 8) + 88) + 8)) − 88/8
:= (9 + 9) × 99 − 99/9
:= (9 + 9) × 99 + (((9 + 9 + 9)/9) − 9)
:= (9 + 9) × 99 − 9/9
:= (2 × ((2 × 2 × 222) − 2)) − 2/2
3
Ï 1772 := 1 + (11 × (((1 + 1) × (11 − 1 − 1)1+1 ) − 1))
:= 2 × ((2 × 2 × 222) − 2)
Ï 1777 := 1 + (111 × (1 + 1)1+1+1+1 )
:= 2/2 + (2 × (2 × 2 × 222))
3
3
:= 33 + ((3 × 3 + 3) + 33/3)
Ï 1782 := (1 + 1) × (11 × (11 − 1 − 1)1+1 )
:= 22 × ((2/2 + 2)2+2 )
:= (((3 + 3) × ((33 × 3 ) − 3)) + 3)/3
:= 3 × 3 × 33 × (3 + 3)
:= 4/4 + 4 × 444
:= 4 + (4 × 444 + (4 + 4)/4)
:= ((55 + 5)/5) + (55 × ((5 + 5)/5) )
:= (555 × (55/5 + 5) + 5)/5
:= 5 + ((555 × (55/5 + 5) + 5)/5)
:= 6 + ((6 × 6 + 6) × (6 × 6 + 6) + ((6 + 6)/6))
:= 666 + (6666/6)
:= 66 × ((66 × 6/(6 + 6)) − 6)
:= 7 + (((7 + 7) × (77 + 7 × 7)) + 7/7)
:= 7 + ((((7 + 7) × (77 + 7 × 7)) − 7/7) + 7)
:= (77/7 + 7) × (7 × (7 + 7) + 7/7)
:= ((8 + 8)/8) × (888 − ((8 + 8)/8))
:= 8/8 + (888 + 888)
:= 8 + (((8 + 8)/8) × (888 − 8/8))
:= (9 + 9) × 99 − 9/9 − 9
:= (9 + 9) × 99 + ((9 − 99)/(9 + 9))
:= (9 + 9) × 99
:= 4 × 444 − 4
5
128
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1783 := 1 + ((1 + 1) × (11 × (11 − 1 − 1)1+1 ))
Ï 1788 := 1 + (11 + (111 × (1 + 1)1+1+1+1 ))
Ï 1793 := 1 + ((1 + 111) × (1 + 1)1+1+1+1 )
:= 2/2 + (22 × ((2/2 + 2)2+2 ))
:= 2 + ((2 × 22 − 2)2 + 22)
:= 2/2 + (2 × (2 × (2 × (222 + 2))))
:= 3/3 + (3 × 3 × 33 × (3 + 3))
:= 3 + ((3 × 3 × 33 × (3 + 3)) + 3)
:= 33/3 + (3 × 3 × 33 × (3 + 3))
:= 4 + ((4 × 444 − 4/4) + 4)
:= (4 × (444 + 4)) − 4
:= 4/4 + (4 × (444 + 4))
5−(5+5)/5
:= 55 + (((55 + 5)/5)
:= (5/5 + 5) × (((5 − (5 + 5)/5) ) + 55)
:= 5/5 + (((5 + 5)/5)5 × (55 + 5/5))
:= (6 − 6 × 6) × (6 − 66) − 6 − 6
:= (6 − 6 × 6) × (6 − 66) − 6/6 − 6
5
)
:= 6 + ((6666/6) + 666)
:= 7 + (777/7 × (((7 + 7)/7 + 7) + 7))
:= 7 + (((7 + 7) × ((7 + 7)/7) ) − (77/7))
:= 7/7 + ((7 + 7) × ((7 + 7)/7)7 )
:= 8 + ((888 − 8/8) + 888)
:= ((8 × 8 × (8 × 8 − 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 × ((8 × (8 + 8) + 88) + 8))
:= 9/9 + (9 + 9) × 99
:= 9 + ((9 + 9) × 99 − ((9 + 9 + 9)/9))
:= 99/9 + (9 + 9) × 99
7
Ï 1784 := (1 + 1) × (1 + (11 × (11 − 1 − 1)1+1 ))
Ï 1789 := ((1 + 1) × (((11 − 1) × (1 + 1 + 1))1+1 )) − 11
Ï 1794 := 1 + (1 + ((1 + 111) × (1 + 1)1+1+1+1 ))
:= 2 × (2 × (2 × 222 + 2))
:= 2 + (((2 × 22 − 2)2 + 22) + 2/2)
:= 2 + (2 × (2 × (2 × (222 + 2))))
:= 3 + ((3 × 3 × 33 × (3 + 3)) − 3/3)
:= ((3/3 + 3)3 ) + ((3 × 3 + 3)3 − 3)
:= 3 + (3 × (3 × 33 × (3 + 3) + 3))
:= 4 + (4 × 444 + 4)
:= 4/4 + ((4 × (444 + 4)) − 4)
:= (4 + 4)/4 + (4 × (444 + 4))
:= (5 × 5 + 5) × (55 + 5) − 55/5
:= 55 − ((55/5)5−(5+5)/5 )
:= (6 − 6 × 6) × (6 − 66) − 66/6
:= (6 − 6 × 6) × (6 − 66) − 6
:= ((7 + 7) × ((7 + 7)/7) ) − (7/7 + 7)
:= 7 + ((77/7 + 7) × (7 × (7 + 7) + 7/7))
:= ((7 + 7)/7) + ((7 + 7) × ((7 + 7)/7)7 )
:= 8 + (888 + 888)
:= 8 + ((8 × ((8 × (8 + 8) + 88) + 8)) − (88/8))
:= ((8 + 8)/8) × (888 + 8/8 + 8)
:= (9 + 9)/9 + (9 + 9) × 99
:= 9 + ((9 + 9) × 99 − ((9 + 9)/9))
:= (9 + 9) × 99 + (99 + 9)/9
:= 5 × 5 + ((55 × ((5 + 5)/5)5 ) − 5/5)
:= 6 + ((6666 + 6)/6 + 666)
7
Ï 1785 := 1 + ((1 + 1) × (1 + (11 × (11 − 1 − 1)1+1 )))
Ï 1790 := (1 + 1) × (((1 + 111) × (1 + 1)1+1+1 ) − 1)
Ï 1795 := (1 + 1)11 − (11 × (1 + (11 + 11)))
:= 22 + ((2 × 22 − 2)2 − 2/2)
:= (2 × (2 × (2 × (222 + 2)))) − 2
:= 2 + ((2 × (2 × (2 × (222 + 2)))) + 2/2)
:= 3 + (3 × 3 × 33 × (3 + 3))
:= (3 × (3 × 33 × (3 + 3) + 3)) − 3/3
:= 3 + (((3/3 + 3)3 ) + (3 × 3 + 3)3 )
:= 4 + ((4 × 444 + 4/4) + 4)
:= (4 × (444 + 4)) − (4 + 4)/4
:= 4 + ((4 × (444 + 4)) − 4/4)
:= 5 × 5 + (55 × ((5 + 5)/5) )
:= (5 × 5 + 5) × (55 + 5) − 5 − 5
:= (5 × 5 + 5) × (55 + 5) − 5
:= (66/6 + 6) × (666/6 − 6)
:= (6 − 6/6) × (6 × (66 − 6) − ((6 + 6)/6))
:= 6/6 + ((6 − 6 × 6) × (6 − 66) − 6)
5
7
7
:= ((7 + 7) × ((7 + 7)/7) ) − 7
:= ((7 + 7) × ((7 + 7)/7) ) − (7 + 7)/7
:= (((7 + 7 + 7)/7)7 ) − 7 × (7 × 7 + 7)
:= 8 + ((888 + 888) + 8/8)
:= ((8 + 8)/8) × (888 − 8/8 + 8)
:= 8 + ((888 + 888) + (88/8))
:= (9 + 9) × 99 + ((9 + 9 + 9)/9)
:= 9 + ((9 + 9) × 99 − 9/9)
:= (9 + 9) × 99 + ((99 + 9 + 9)/9)
Ï 1791 := ((1 + 111) × (1 + 1)1+1+1+1 ) − 1
Ï 1796 := 1 + ((1 + 1)11 − (11 × (1 + (11 + 11))))
Ï 1786 := 11 + ((111 × (1 + 1)1+1+1+1 ) − 1)
2
:= (2 × (2 × (2 × (222 + 2)))) − 2/2
:= 2 × ((2 × (2 × (222 + 2))) + 2)
:= 3 + ((3 × 3 × 33 × (3 + 3)) + 3/3)
:= 3 × (3 × 33 × (3 + 3) + 3)
:= 3 + ((3 × 3 × 33 × (3 + 3)) + 33/3)
:= 4 × 444 + (44 − 4)/4
:= (4 × (444 + 4)) − 4/4
:= 4 + (4 × (444 + 4))
:= 22 + (2 × 22 − 2)
5
:= 5 + (((55/5 + 5) × 555/5) + 5)
:= 55 + (5555 + 5 )/5
:= 5/5 + ((5 × 5 + 5) × (55 + 5) − 5)
:= (((6 + 6)/6) + 6 × 6) × (66/6 + 6 × 6)
:= 6 + ((66/6 + 6) × (666/6 − 6))
:= (6 + 6)/6 + ((6 − 6 × 6) × (6 − 66) − 6)
:= 7/7 + (((7 + 7) × ((7 + 7)/7) ) − 7)
:= ((7 + 7) × ((7 + 7)/7)7 ) − 7/7
:= 77/7 + (((7 + 7) × ((7 + 7)/7)7 ) − 7)
:= 8 + (((8 + 8)/8) × (888 + 8/8))
:= (8/8 + 8) × (888/8 + 88)
:= ((8 × 8 × (8 × 8 − 8)) + 8)/((8 + 8)/8)
:= ((9 + 9)/9) × (((9 + 9)/9) + 9 × 99)
:= 9 + (9 + 9) × 99
:= 9 + (((9 × 9 + 9)/(9 + 9)) + (9 + 9) × 99)
7
Ï 1792 := (1 + 111) × (1 + 1)1+1+1+1
Ï 1787 := 11 + (111 × (1 + 1)1+1+1+1 )
Ï 1797 := 11 × 111 + (((1 + 1) × (1 + 11))1+1 )
:= 22 + ((2 × 22 − 2)2 + 2/2)
:= 2 × (2 × (2 × (222 + 2)))
:= (((3 + 3) × ((33 × 33 ) + 3)) − 3)/3
:= ((3/3 + 3)3 ) + (3 × 3 + 3)3
:= 44/4 + 4 × 444
:= 4 × (444 + 4)
:= 4 + ((4 × (444 + 4)) + 4/4)
:= (((5 + 5)/5) × (55 + 5/5)) − 5
:= ((5 + 5)/5)5 × (55 + 5/5)
:= 5 + (((5 + 5)/5)5 × (55 + 5/5))
:= (6 − 6 × 6) × (6 − 66) − (6/6 + 6 + 6)
:= (6/6 + 6) × (((6 + 6)/6)6+(6+6)/6 )
:= (6 − 6 × 6) × (6 − 66) − 6 × 6/(6 + 6)
5
7
7
:= 22 + ((2 × 22 − 2)2 + 22/2)
:= ((3 + 3) × (3 × 3 × 33 + 3)) − 3
:= ((7 + 7)/7) + (((7 + 7) × ((7 + 7)/7) ) − 7)
:= (7 + 7) × ((7 + 7)/7)
:= (7 × 7 × (7 + 7)) + 7777/7
:= 888 + (888 + 88/8)
:= 8 × ((8 × (8 + 8) + 88) + 8)
:= ((88/8 + 8) × (88 − 8/8 + 8)) − 8
:= (9 + 9) × 99 + ((9 × 9 + 9)/(9 + 9))
:= 9 + ((9 + 9) × 99 + 9/9)
:= 9 + (((9 + 9) × 99 − ((9 + 9 + 9)/9)) + 9)
129
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1798 := (1 + 1) × ((((11 − 1) × (1 + 1 + 1))1+1 ) − 1)
Ï 1803 := 1 + ((1 + 1) × (1 + (((11 − 1) × (1 + 1 + 1))1+1 )))
Ï 1808 := (1 + 1)11 − ((1 + 1) × (111+1 − 1))
:= ((2 × 22 + 2/2)2 ) − 222
:= 2 × (2 × (2 × ((222 + 2) + 2)))
:= 3 + ((((3/3 + 3) ) + (3 × 3 + 3) ) + 3)
:= 3 + ((3 + 3) × (3 × 3 × 33 + 3))
:= 3 × 33 + ((3 × 3 + 3)3 − 3/3)
:= 4 + ((4 × (444 + 4)) + (4 + 4)/4)
:= 44/4 + (4 × (444 + 4))
:= 4 × (444 + 4 + 4)
:= (5 × 5 + 5) × (55 + 5) − (5 + 5)/5
:= 5 + ((5 × 5 + 5) × (55 + 5) − ((5 + 5)/5))
:= (55/5 + 5) × (555 + 5 + 5)/5
:= (6 − 6 × 6) × (6 − 66) − (6 + 6)/6
:= 666/6 + (6 × (6 × 6 × 6 + 66))
:= 6 + ((6 − 6 × 6) × (6 − 66) + ((6 + 6)/6))
:= 22 + (2 × (2 × 2 × 222))
3
3
:= 7 + (((7 + 7) × ((7 + 7)/7) ) − 7/7)
:= 77/7 + ((7 + 7) × ((7 + 7)/7) )
:= (7/7 + 7) × (((7 + 7)/7)7 + 7 × (7 + 7))
:= ((8 + 8)/8) × (888 + 88/8)
:= 88/8 + (8 × ((8 × (8 + 8) + 88) + 8))
:= (8 + 8) × ((888 + 8 + 8)/8)
:= 9 + (((9 + 9) × 99 − ((9 + 9)/9)) + 9)
:= 9 + ((9 + 9) × 99 + ((99 + 9)/9))
:= 9 + ((((9 + 9) × 99 − 9/9) + 9) + 9)
7
Ï 1799 := ((1 + 1) × (((11 − 1) × (1 + 1 + 1))1+1 )) − 1
7
Ï 1804 := (1 + 1) × (11 × (1 + (11 − 1 − 1)1+1 ))
Ï 1809 := 1 + ((1 + 1)11 − ((1 + 1) × (111+1 − 1)))
:= (2 × ((2 × (2 + 2) + 22)2 )) − 2/2
:= 2 × (((2 × (2 + 2) + 22)2 ) + 2)
:= 2 + (((2 − 222 )/2) + 222/2 )
:= (3 × 3 + 3)3 + (((3 + 3)3 − 3)/3)
:= 3 + (((3 + 3) × (3 × 3 × 33 + 3)) + 3/3)
:= 3 × (3 × (33 × (3 + 3) + 3))
:= ((4 − 4/4) + 4) × (4/4 + 4 )
:= 44 + (4 × (444 − 4))
:= 4/4 + (4 × (444 + 4 + 4))
:= (5 × 5 + 5) × (55 + 5) − 5/5
:= 5 + ((5 × 5 + 5) × (55 + 5) − 5/5)
:= 5 + (((5 × 5 + 5) × (55 + 5) − 5/5) + 5)
:= (6 − 6 × 6) × (6 − 66) − 6/6
:= 6 + ((6 − 6 × 6) × (6 − 66) − ((6 + 6)/6))
:= (66 + 6/6) × ((66 × 6/(6 + 6)) − 6)
:= 7 + ((7 + 7) × ((7 + 7)/7)7 )
:= 7 + (7777/7 + (7 × 7 × (7 + 7)))
:= 7 + (((((7 + 7 + 7)/7)7 ) − 7 × (7 × 7 + 7)) + 7)
:= (8 − 8/8) × ((8 + 8) × (8 + 8) + 8/8)
:= 8 + (((8 × 8 × (8 × 8 − 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8) × ((888 + 8 + 8)/8))
:= 9 + (((9 + 9) × 99 − 9/9) + 9)
:= 99/9 × (((9 + 9)/9) + 9 × (9 + 9))
:= 9 + ((9 + 9) × 99 + 9 + 9)
4
Ï 1800 := (1 + 1) × (((11 − 1) × (1 + 1 + 1))1+1 )
Ï 1805 := (1 + 1)11 − (1 + ((1 + 1) × 111+1 ))
22/2
2
− (22 + 2)/2
Ï 1810 := (11 − 1) × (1 + (11 + (1 + 1 + 11)1+1 ))
:= 2 × ((2 × (2 + 2) + 22)2 )
:= 2
:= (3 + 3) × (3 × 3 × 33 + 3)
:= 3 × 3 + ((3 × 3 + 3) − (3/3 + 3))
:= 3/3 + ((3 × 3 + 3)3 + 3 × 33 )
:= 4 + ((4 × (444 + 4)) + 4)
:= 44 + ((4 × (444 − 4)) + 4/4)
:= (4 + 4)/4 + (4 × (444 + 4 + 4))
:= (5 × 5 + 5) × (55 + 5)
:= 5 + (5 × 5 + 5) × (55 + 5)
:= 5 + ((5 × 5 + 5) × (55 + 5) + 5)
:= (6 − 6 × 6) × (6 − 66)
:= 6 + ((6 − 6 × 6) × (6 − 66) − 6/6)
3
:= 2 + ((2 × 22 − 2)2 + 2 × 22)
3
7
:= (6 − 6/6) × (6 × (66 − 6) + ((6 + 6)/6))
:= 7 + (((7 + 7) × ((7 + 7)/7) ) + 7/7)
:= 7 + ((((7 + 7) × ((7 + 7)/7) ) − 7/7) + 7)
:= 7 + (((7 + 7) × ((7 + 7)/7)7 ) + (77/7))
:= 8 + (8 × ((8 × (8 + 8) + 88) + 8))
:= (88/8 + 8) × (88 − 8/8 + 8)
:= ((8 + 8)/8) × ((888 + 8/8 + 8) + 8)
:= 9 + ((9 + 9) × 99 + 9)
:= (9 + 9) × 99 + ((99 + 99 + 9)/9)
:= 9 + ((((9 + 9) × 99 + 9/9) + 9) + 9)
7
Ï 1801 := 1 + ((1 + 1) × (((11 − 1) × (1 + 1 + 1))1+1 ))
:= 2/2 + (2 × ((2 × (2 + 2) + 22)2 ))
:= 3/3 + ((3 + 3) × (3 × 3 × 33 + 3))
Ï 1806 := (1 + 1)11 − ((1 + 1) × 111+1 )
:= 2
22/2
Ï 1811 := 11 + ((1 + 1) × (((11 − 1) × (1 + 1 + 1))1+1 ))
2
− 22 /2
3
:= 3 × (((33/3) ) − 3
3+3
:= 22/2 + (2 × ((2 × (2 + 2) + 22)2 ))
)
4
:= 3 + (((3 × 3 + 3)3 − 3/3) + 3 × 33 )
:= 4 + (((4 × (444 + 4)) + 4/4) + 4)
:= ((4 − 4/4) + 4) × ((4 + 4)/4 + 4 )
:= 4 + ((4 × (444 + 4 + 4)) − 4/4)
:= 5/5 + (5 × 5 + 5) × (55 + 5)
:= 5 + ((5 × 5 + 5) × (55 + 5) + 5/5)
:= 55/5 + (5 × 5 + 5) × (55 + 5)
:= 6/6 + (6 − 6 × 6) × (6 − 66)
:= 6 + (6 − 6 × 6) × (6 − 66)
:= 66/6 + (6 − 6 × 6) × (6 − 66)
7
7
:= 7 + (((7 + 7) × ((7 + 7)/7) ) + ((7 + 7)/7))
:= 7 + (((7 + 7) × ((7 + 7)/7) ) + 7)
:= 777 + (7777/7 − 77)
:= 8 + ((8 × ((8 × (8 + 8) + 88) + 8)) + 8/8)
:= 8 + (((8 + 8)/8) × (888 + 88/8))
:= 8 + ((8 × ((8 × (8 + 8) + 88) + 8)) + (88/8))
:= 9 + (((9 + 9) × 99 + 9/9) + 9)
:= ((9 + 9)/9) × (((99 + 9)/9) + 9 × 99)
:= 9 + (((9 + 9) × 99 + (99/9)) + 9)
Ï 1802 := (1 + 1) × (1 + (((11 − 1) × (1 + 1 + 1))1+1 ))
Ï 1807 := 1 + ((1 + 1)11 − ((1 + 1) × 111+1 ))
Ï 1812 := (1 + 1 + 1) × (11 × (111 − 1)/(1 + 1) − 1)
:= 2 + (2 × ((2 × (2 + 2) + 22)2 ))
:= 222/2 + ((2 − 222 )/2)
:= 2 × ((2 × (2 × ((222 + 2) + 2))) + 2)
:= 3 + ((((3 + 3)3 − 3)/3) + (3 × 3 + 3)3 )
:= 3/3 + (((3 × 3 + 3)3 − 3) + 3 × 33 )
:= 3 + ((3 × 3 + 3)3 + 3 × 33 )
:= (44 − 4)/4 + (4 × (444 + 4))
:= (4 × (444 + 4 + 4)) − 4/4
:= 4 + (4 × (444 + 4 + 4))
:= (5 + 5)/5 + (5 × 5 + 5) × (55 + 5)
:= 5 + ((5 × 5 + 5) × (55 + 5) + ((5 + 5)/5))
:= (5/5 + 5) × ((5 × (55 + 5)) + ((5 + 5)/5))
:= (6 + 6)/6 + (6 − 6 × 6) × (6 − 66)
:= 6 + ((6 − 6 × 6) × (6 − 66) + 6/6)
:= 6 + ((6 − 6 × 6) × (6 − 66) + 6)
7
7
:= 7 + ((((7 + 7 + 7)/7) ) − 7 × (7 × 7 + 7))
:= 7 + ((((7 + 7) × ((7 + 7)/7) ) + 7/7) + 7)
:= (7 × (7 × 7 × 7 − 77 − 7)) − 7/7
:= 8 + (((8 + 8)/8) × (888 + 8/8 + 8))
:= 8 × 88 + (8888/8 − 8)
:= ((88 + 8)/8) × ((88 − 8/8) + 8 × 8)
:= 9 + ((9 + 9) × 99 + (99/9))
:= 9 + ((((9 + 9) × 99 − ((9 + 9)/9)) + 9) + 9)
:= 9 + (((9 + 9) × 99 + ((99 + 9)/9)) + 9)
130
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1813 := (1 + 1)11 − (11 + ((1 + 1) × (1 + 111)))
Ï 1818 := 1 + ((1 + 1)11 − (11 × (11 + (11 − 1))))
Ï 1823 := (1 + 1)11 − (1 + ((1 + 1) × (1 + 111)))
:= (2 × 22)2 − ((22/2)2 + 2)
:= ((2 × (22 + 2))2 ) − (222 + 2)
:= (2 × 22)2 − (222/2 + 2)
:= 3 + (((3 × 3 + 3)3 + 3 × 33 ) + 3/3)
:= 33+3 + 33 × 33
:= 3 × 33 + ((3 × 3 + 3)3 − (3/3 + 3))
:= 4 + ((4 × (444 + 4 + 4)) + 4/4)
:= 44 + (4 × 444 − (4 + 4)/4)
:= 4 + ((4 × 444 − 4/4) + 44)
:= (5 × (5 × (55 + 5))) + (55 + 5)/(5 + 5)
:= (5/5 + 5) × ((55 + 5)/(5 + 5) − (5 + 5))
:= ((5/5 + 5) × (55 + 5)/(5 + 5)) − 55
:= 6 + (((6 − 6 × 6) × (6 − 66) + 6/6) + 6)
:= 6 + (((6 − 6 × 6) × (6 − 66) + 6) + 6)
:= 6 + (((6 − 6 × 6) × (6 − 66) + (66/6)) + 6)
:= 7 × (7 × 7 × 7 − 77 − 7)
:= (77/7 + 7) × (7777/77)
:= 77 + ((77/7 + 7) × (7 × (7 + 7) − 7/7))
:= 8 + ((88/8 + 8) × (88 − 8/8 + 8))
:= 8 + (((8 + 8)/8) × ((888 + 8/8 + 8) + 8))
:= 8 + (8888/8 + 8 × 88)
:= 9 + (((99 + 99)/9) + (9 + 9) × 99)
:= (9 + 9) × ((9 + 9)/9 + 99)
:= (9 + 9) × 99 + ((9 × 9 × 9 + 9)/(9 + 9))
Ï 1814 := (11 × (11 × (1 + 1 + 1 + 1 + 11))) − 1
Ï 1819 := 1 + (1 + ((1 + 1)11 − (11 × (11 + (11 − 1)))))
Ï 1824 := (1 + 1)11 − ((1 + 1) × (1 + 111))
:= 22 + (2 × (2 × (2 × (222 + 2))))
:= 2 + (((2 × 22)2 − (22/2)2 ) + 2)
:= 222/2 − (222 + 2)
:= ((33 × ((3 + 3) × 33 + 3)) − 3)/3
:= 3/3 + (33 × 33 + 33+3 )
:= 3 × 33 + ((3 × 3 + 3)3 − 3)
:= 4 + ((4 × (444 + 4 + 4)) + (4 + 4)/4)
:= 44 + (4 × 444 − 4/4)
:= 4 + (4 × 444 + 44)
5
5
5−5/5
:= 55 + ((55 × ((5 + 5)/5) ) − 5/5)
:= 5 − ((((5/5 + 5)
:= 6 + (((6 − 6 × 6) × (6 − 66) + ((6 + 6)/6)) + 6)
:= (66/6 + 6) × (6 × (6 + 6 + 6) − 6/6)
:= 7/7 + (7 × (7 × 7 × 7 − 77 − 7))
) + 5) + 5)
:= 55 − (((5/5 + 5)5−5/5 ) + 5)
:= 66 + ((6 × 6 + 6) × (6 × 6 + 6) − 6)
:= 7 + ((7 × (7 × 7 × 7 − 77 − 7)) − 7/7)
:= (7 × 7 − 7/7) × (7 × 7 − 77/7)
:= 8 × 88 + (8888 − 8)/8
:= (8/8 + 8 + 8) × ((88/8 + 88) + 8)
:= (88 + 8) × (88/8 + 8)
:= 9 + (((99 + 99 + 9)/9) + (9 + 9) × 99)
:= 9/9 + ((9 + 9) × ((9 + 9)/9 + 99))
:= ((99 + 9)/9) × (9 × (9 + 9) − (9/9 + 9))
Ï 1815 := 11 × (11 × (1 + 1 + 1 + 1 + 11))
Ï 1820 := (11 − 1) × ((1 + 1 + 11) × (1 + (1 + 1 + 11)))
Ï 1825 := (1 + 1)11 − (1 + (1 + 1) × 111)
:= (2 × 22)2 − (22/2)2
:= 2 × ((2 × 2 × 222) + 22)
:= (2 × 22)2 − 222/2
:= 33 + (3 × 3 × 33 × (3 + 3))
:= 3 + ((33 × 33 − 3/3) + 33+3 )
:= 3/3 + (((3 × 3 + 3)3 − 3) + 3 × 33)
:= 44 + 4 × 444
:= 44 × 44 − 444/4
:= 44 + (4 × 444 − (4/4 + 4))
5
:= (5 × (5 × (5 × (5 + 5 + 5)))) − 55
:= 55 + (55 × ((5 + 5)/5) )
6
:= 5 × ((5/5 + 5) × (55 + 5) + 5)
:= (6 × 6 − 6/6) × (((6 + 6)/6) − (6 + 6))
:= (6 − 6/6) × ((6 × (66 − 6) − 6/6) + 6)
:= (7/7 + 7 + 7) × (((7 + 7)/7) − 7)
:= 7 + (7 × (7 × 7 × 7 − 77 − 7))
:= 7 + ((77/7 + 7) × (7777/77))
:= 8 × 88 + 8888/8
:= (8 − 8/8) × ((8 × 8 × 8 + 8)/((8 + 8)/8))
:= 8/8 + ((88 + 8) × (88/8 + 8))
:= (99/9 + 9) × ((9/9 + 9 × 9) + 9)
:= ((9 + 9) × (999/9 − 9)) − 99/9
:= (6 − 6/6) × 66 × 66/(6 + 6)
7
:= 99/9 × (((9 + 9 + 9)/9) + 9 × (9 + 9))
Ï 1826 := (1 + 1)11 − (1 + 1) × 111
Ï 1816 := 1 + (11 × (11 × (1 + 1 + 1 + 1 + 11)))
Ï 1821 := (1 + 1)11 − (1 + ((1 + 1) × (1 + 1 + 111)))
:= 2 × (((2 × 2 × 222) − 2) + 22)
:= 2/2 + (2 × ((2 × 2 × 222) + 22))
:= 222/2 − 222
:= ((33 × ((3 + 3) × 33 + 3)) + 3)/3
:= 3 + (33 × 33 + 33+3 )
:= 3 × 33 + ((3 × 3 + 3)3 − 3/3)
:= 44 + (4 × 444 − 4)
:= 44 + (4 × 444 + 4/4)
:= 44 × 44 + ((4 − 444)/4)
:= 5 + ((5 × 5 + 5) × (55 + 5) + (55/5))
:= 5 + (((5 − (5 + 5 ))/5) − 55)
:= 55 − ((5 − 5/5)5 + 5 × 55)
:= ((6 + 6)/6 + 6) × (6 × 6 × 6 + 66/6)
:= 6 + ((6 − 6/6) × 66 × 66/(6 + 6))
:= (((6 + 6)/6)66/6 ) − (6 × 6 × 6 + 6)
:= 7/7 + ((7/7 + 7 + 7) × (((7 + 7)/7)7 − 7))
:= 7 + ((7 × (7 × 7 × 7 − 77 − 7)) + 7/7)
:= 777/7 + (7 × (7 × (7 × 7 − (7 + 7))))
:= 88 + 8 × (8 × (8 + 8) + 88)
:= 8 + (((88/8 + 8) × (88 − 8/8 + 8)) + 8)
:= (8 + 8)/8 + ((88 + 8) × (88/8 + 8))
:= 9 × 9 × 9 + (((99 × 99) − (9 + 9))/9)
:= 9 + ((((9 + 9) × 99 + ((99 + 9)/9)) + 9) + 9)
:= 9 + ((((99 × 99) − 9)/9) + 9 × 9 × 9)
Ï 1817 := (1 + 1)11 − (11 × (11 + (11 − 1)))
5
5
5
Ï 1822 := (1 + 1)11 − ((1 + 1) × (1 + 1 + 111))
Ï 1827 := 1 + ((1 + 1)11 − (1 + 1) × 111)
:= 2 + ((2 × 22)2 − (22/2)2 )
:= 222 + (2 × (22 − 2))2
:= (((2 × 22) − 2/2)2 ) − 22
:= 33+3 + (33 × 33 − 3/3)
:= 3 + ((33 × 33 + 33+3 ) + 3/3)
:= 3 × 33 + (3 × 3 + 3)3
:= 44 + ((4 × 444 − 4) + 4/4)
:= 44 + (4 × 444 + (4 + 4)/4)
:= (4 − 4/4) × ((4/4 + 4)4 − 4 × 4)
:= 5 × 5 + (((5 + 5)/5)5 × (55 + 5/5))
:= 5 + ((((5 + 5)/5)5 × (55 + 5/5)) + 5 × 5)
:= (5 − (5 + 5)/5) × (((55 − 55)/5) − 5)
:= 6 + ((6 − 6 × 6) × (6 − 66) + (66/6))
:= 6 + (((6 + 6)/6 + 6) × (6 × 6 × 6 + 66/6))
:= 6 + (((6 − 6/6) × 66 × 66/(6 + 6)) + 6)
7
:= ((7 × 7 − 7/7) × (7 × 7 − 77/7)) − 7
:= 7 + ((7/7 + 7 + 7) × (((7 + 7)/7) − 7))
:= 8/8 + (8 × (8 × (8 + 8) + 88) + 88)
:= 8 + ((8888 − 8)/8 + 8 × 88)
:= (8/8 + 8) × (8 × (8 + 8 + 8) + (88/8))
:= 9 × 9 × 9 + (((99 × 99) − 9)/9)
:= 9 × 9 × 9 + (9999/9 − (9 + 9))
:= 9 + ((9 + 9) × ((9 + 9)/9 + 99))
131
:= 7 + ((7 × (7 × 7 × 7 − 77 − 7)) + 7)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1828 := (1 + 1)11 − ((1 + 1) × (111 − 1))
Ï 1833 := (1 + 1 + 1) × (1 + 11 × 111)/(1 + 1)
Ï 1838 := 111 + (((1 + 11)1+1+1 ) − 1)
:= 2 + (222/2 − 222)
:= (((2 × 22) − 2/2)2 ) − 22+2
:= (((2 × 22) − 2/2)2 ) − 22/2
:= 3/3 + ((3 × 3 + 3)3 + 3 × 33)
:= (3 × ((3 + 3) × (3 × 33 + 3))) − 3
:= (3 × 3 + 3)3 + ((333 − 3)/3)
:= 4 + ((4 × 444 + 44) + 4)
:= ((44 − 4/4)(4+4)/4 ) − 4 × 4
:= 4 × 444 + (44 − 4 − 4)/4
:= 55 − (((5/5 + 5)5−5/5 ) + 5/5)
:= 5 + (55 − (((5/5 + 5)5−5/5 ) + 5/5))
:= 5 × 55 + ((55 + 5/5)/((5 + 5)/5))
:= ((6 + 6)/6)6 + (6 × 6 + 6) × (6 × 6 + 6)
:= 6 × 6 × (66 − 6 − 6) − 666/6
:= 6 + ((((6 + 6)/6)66/6 ) − 6 × 6 × 6)
:= 7 + (((7 × (7 × 7 × 7 − 77 − 7)) + 7/7) + 7)
:= 7 + ((7 × (7 × (7 × 7 − (7 + 7)))) + 777/7)
:= 7 + (((7 × 7 − 7/7) × (7 × 7 − 77/7)) + 7)
:= ((8 × (8 × (8 × 8 − 8) + 8)) + 8)/((8 + 8)/8)
:= 8 + (((88 + 8) × (88/8 + 8)) + 8/8)
:= 8 × 8 + (((8 + 8)/8) × (888 − 8/8))
:= 9 + (((9 + 9) × ((9 + 9)/9 + 99)) + 9/9)
:= (9 + 9) × (99 + 9) − 999/9
:= 9 × 9 × 9 + ((9999 − (9 + 9))/9)
Ï 1829 := 1 + ((1 + 1)11 − ((1 + 1) × (111 − 1)))
Ï 1834 := (1 + (1 + 1 + 11)) × ((11 × (1 + 11)) − 1)
Ï 1839 := 111 + ((1 + 11)1+1+1 )
:= 2 + ((((2 × 22) − 2/2)2 ) − 22)
:= 2 + (2 × (2 × (222 − (22 + 2 + 2))))
:= 2 + (22/2 × (((22/2 + 2)2 ) − 2))
:= ((33 × 333 + 3)/(3 + 3)) − 3
:= 3 + ((33 × 333 − 3)/(3 + 3))
:= 3 + (3 × ((3 + 3) × (3 × 33 + 3)))
:= 4 + (44 × 44 − 444/4)
:= 44 + ((4 × (444 + 4)) − (4 + 4)/4)
:= 4 × 444 + ((44 − 4)/4)
:= 55 − ((5/5 + 5)5−5/5 )
:= 5 + (55 − ((5/5 + 5)5−5/5 ))
:= 5 + ((55 − ((5/5 + 5)5−5/5 )) + 5)
:= (6 + 6 + 6) × (6 × 6 + 66) − (6 + 6)/6
:= 666/6 + (6 × 6 × (6 × 6 + 6 + 6))
:= 7 + (((7/7 + 7 + 7) × (((7 + 7)/7) − 7)) + 7)
:= 7 × ((((7 × 7 × 77) − 7)/(7 + 7)) − 7)
:= 777 + (7777/7 − 7 × 7)
:= 8 × 8 + ((888 − 88/8) + 888)
:= 8 + (((88 + 8) × (88/8 + 8)) + ((8 + 8)/8))
:= 8 × 8 + ((888 − 8/8) + 888)
:= 9 + ((99/9 + 9) × ((9/9 + 9 × 9) + 9))
:= ((9 + 9)/9) × (999 − (9/9 + 9 × 9))
:= 9 × 9 × 9 + ((9999 − 9)/9)
:= 66 + ((6 × 6 + 6) × (6 × 6 + 6) − 6/6)
7
Ï 1830 := (1 + 1 + 1) × (11 × 111 − 1)/(1 + 1)
Ï 1835 := 1 + (1 + 1 + 1 + 11) × (11 × (1 + 11) − 1)
Ï 1840 := 1 + (111 + ((1 + 11)1+1+1 ))
:= 2 + ((222/2 − 222) + 2)
:= 2 + ((((2 × 22) − 2/2)2 ) − 22+2 )
:= 2 × (2 × (222 − (22 + 2)))
:= 3 + ((3 × 3 + 3)3 + 3 × 33)
:= 3 + ((33 × 333 + 3)/(3 + 3))
:= 33+3 + 3333/3
:= 4 + (((4 − 444)/4) + 44 × 44)
:= 44 + ((4 × (444 + 4)) − 4/4)
:= 4 × (444 + 4 × 4)
5
:= (5/5 + 5) × ((5 × (55 + 5)) + 5)
:= 5 × ((5 − 5)/(5 + 5) + 55)
:= 5 × ((55 + 5)/(5 + 5) + 55)
:= 66 + (6 × 6 + 6) × (6 × 6 + 6)
:= (6 + 6 + 6) × (6 × 6 + 66) − 6/6
:= ((6 × 6/(6 + 6))6 ) + (6666/6)
:= (7 − 7/7) × (7 × (7 × 7 − 7) + (77/7))
:= 7 × 77 + (7 − 7/7)77/7−7
:= 77 + (((7 + 7) × (77 + 7 × 7)) − 7/7)
:= (8 − 8/8 + 8) × ((888 + 88)/8)
:= 88/8 + ((88 + 8) × (88/8 + 8))
:= 8 × 8 + (888 + 888)
:= 9 × 9 × 9 + (((9999 − 9)/9) − 9)
:= ((9 + 9) × (999/9 − 9)) − 9/9
:= 9 × 9 × 9 + 9999/9
Ï 1831 := ((11 × (1 + 1 + 1) × 111) − 1)/(1 + 1)
Ï 1836 := (1 + 11) × ((11 × (1 + (1 + 1 + 11))) − 1)
Ï 1841 := 1 + (1 + (111 + ((1 + 11)1+1+1 )))
:= 2 + (((((2 × 22) − 2/2)2 ) − 22) + 2)
:= 2 × ((2 × (2 × (222 + 2))) + 22)
:= (((2 × 22) − 2/2)2 ) − 2 × (2 + 2)
:= (33 × 333 − 3)/(3 + 3)
:= 3 × ((3 + 3) × (3 × 33 + 3))
:= 33+3 + ((3333 + 3)/3)
:= 44 + (4 × 444 + 44/4)
:= 44 + (4 × (444 + 4))
:= 4/4 + (4 × (444 + 4 × 4))
5
5
5
:= 5 + (5 − ((5 − 5/5) + 5 × 55))
:= 5/5 + (5 × ((5 − 5)/(5 + 5) + 55))
:= 5/5 + (5 × ((55 + 5)/(5 + 5) + 55))
:= 66 + ((6 × 6 + 6) × (6 × 6 + 6) + 6/6)
:= (6 + 6 + 6) × (6 × 6 + 66)
:= 6 + ((6 + 6 + 6) × (6 × 6 + 66) − 6/6)
:= 7 + ((7 × 7 − 7/7) × (7 × 7 − 77/7))
:= (77 + 7)/7 × ((77 − 7/7) + 77)
:= 7 × ((((7 + 7)/7)7+7/7 ) + 7)
:= 8 + ((8888/8 + 8 × 88) + 8)
:= ((88 + 8)/8) × ((8 × 8 + 88) + 8/8)
:= 8/8 + ((888 + 888) + 8 × 8)
:= 9 × 9 × 9 + (9999/9 − 9)
:= (9 + 9) × (999/9 − 9)
:= 9 × 9 × 9 + (9999 + 9)/9
Ï 1832 := (1 + (11 × (1 + 1 + 1) × 111))/(1 + 1)
Ï 1837 := 11 × ((1 + 1 + 11)1+1 − (1 + 1))
Ï 1842 := 1 + (1 + (1 + (111 + ((1 + 11)1+1+1 ))))
:= 2 × (2 × (222 − (22 + 2 + 2)))
:= 22/2 × (((22/2 + 2)2 ) − 2)
:= 2 + (2 × (2 × (222 − (22 + 2))))
:= (33 × 333 + 3)/(3 + 3)
:= 3/3 + (3 × ((3 + 3) × (3 × 33 + 3)))
:= (3 × (3 × (3 + 3)3 − 33)) − 3
:= 44 + ((4 × (444 + 4)) − 4)
:= 44 + ((4 × (444 + 4)) + 4/4)
:= (4 + 4)/4 + (4 × (444 + 4 × 4))
5
5
:= ((5 + 5)/5) + (5 × 5 + 5) × (55 + 5)
:= 5 × 55 + ((5 − 5/5)/((5 + 5)/5))
:= (5/5 + 5) × (55 − 55)/(5 + 5)
:= (((6 + 6)/6)66/6 ) − 6 × 6 × 6
:= 6/6 + (6 + 6 + 6) × (6 × 6 + 66)
:= 6 + (6 + 6 + 6) × (6 × 6 + 66)
7
7
:= (((7/7 − 7) + 7 × 7)(7+7)/7 ) − 7
:= 77 + ((7 − 7/7 + 7) × (((7 + 7)/7) + 7))
:= (((7 + 7 + 7)/7) ) − (7 × 7 × 7 + 7)
:= 8 + ((88 + 8) × (88/8 + 8))
:= 88/8 × ((888/8 − 8) + 8 × 8)
:= 8 × 8 + (((8 + 8)/8) × (888 + 8/8))
:= 9 × 9 × 9 + ((9999 + 9)/9 − 9)
:= 9/9 + ((9 + 9) × (999/9 − 9))
:= 9 + ((9 + 9) × (99 + 9) − 999/9)
132
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1843 := 111+1+1 + (1 + 1)11−1−1
2
Ï 1848 := 11 × ((1 + 1 + 11)1+1 − 1)
Ï 1853 := 1 + ((1 + 1) × ((1 + 11111)/(1 + 11)))
:= (((2 × 22) − 2/2) ) − (2 + 2 + 2)
:= 2 × (2 × (22 − 22))
:= 2 + ((((2 × 22) − 2/2)2 ) + 2)
:= 3 + (3333/3 + 33+3 )
:= 3 + (3 × (3 × (3 + 3)3 − 33))
:= (3 × 3 + 3)3 + ((3 − 3/3 + 3)3 )
:= 4 + (4 × 444 + ((44 − 4)/4))
:= 44 × (44 − (4 + 4)/4)
:= 4 + ((44 − 4/4)(4+4)/4 )
:= (5 × (5 × (5 × (5 + 5 + 5)))) − ((5 + 5)/5)5
:= (5/5 + 5) × ((55 + 5)/(5 + 5) − 5)
:= 5 + ((5/5 + 5) × ((55 + 5)/(5 + 5) − 5))
:= (((6 × 6 + 6/6) + 6)(6+6)/6 ) − 6
:= 66 × (((66 + 66)/6) + 6)
:= 666 + ((66 × (6 + 6 + 6)) − 6/6)
:= ((7 + 7)/7) + (7 × (7 × (7 × 7 − (7 + 7))))
:= 77 × (((77 − 7)/7 + 7) + 7)
:= 777 + (77 × (7 + 7) − ((7 + 7)/7))
:= (88/8 + 8) × ((8/8 + 88) + 8)
:= 88 × ((88 + 8 + 8)/8 + 8)
:= (8/8 + 8 + 8) × ((888 − (8 + 8))/8)
:= (9/9 + 9 + 9) × (99 − ((9 + 9)/9))
:= 9 + (((9999 − 9)/9) + 9 × 9 × 9)
:= 9 × 9 + ((9 + 9) × 99 − (9/9 + 9))
7
Ï 1844 := 1 + (111+1+1 + (1 + 1)11−1−1 )
2
Ï 1849 := (((1 + 1) × (11 + 11)) − 1)1+1
2
Ï 1854 := (1 + 1) × (1 + ((1 + 11111)/(1 + 11)))
:= 2 × ((2 × (222 − 22)) − 2)
:= ((2 × 22) − 2/2)
:= 2 + (2 × ((2 × (222 − 22)) + 2))
:= (3 × 33+3 ) − (((3/3 + 3) + 3)3 )
:= ((3 × 3 + 33) + 3/3)3−3/3
:= (3 × 33+3 ) − 333
:= 4 + (4 × (444 + 4 × 4))
:= (44 − 4/4)(4+4)/4
:= 4 + (((44 − 4/4)(4+4)/4 ) + 4/4)
:= (5 − 5/5)5 + ((55 × (5 + 5 + 5)) − 5)
5
:= (5 − 5/5) + (55 × (5 + 5 + 5))
(6+6)/6
:= 5 + ((55 × (5 + 5 + 5)) + (5 − 5/5)5 )
:= 6 + (((((6 + 6)/6)66/6 ) − 6 × 6 × 6) + 6)
:= ((6 × 6 + 6/6) + 6)
:= (((7 + 7 + 7)/7)7 ) − 7 × 7 × 7
:= ((7/7 − 7) + 7 × 7)(7+7)/7
:= 777 + (77 × (7 + 7) − 7/7)
:= 8 + (((88 + 8)/8) × ((8 × 8 + 88) + 8/8))
:= 8/8 + ((8 × (8 × (8 + 8) − 8)) + 888)
:= 888 + ((88 × 88 − (8 + 8))/8)
:= (9 + 9) × (99 + 9) − (9/9 + 99)
:= 9 + (9999/9 + 9 × 9 × 9)
:= 9 × 9 + ((9 + 9) × 99 − 9)
Ï 1845 := (1 + 1 + 1) × (11 × (1 + 111)/(1 + 1) − 1)
Ï 1850 := 1 + ((((1 + 1) × (11 + 11)) − 1)1+1 )
:= 666 + (66 × (6 + 6 + 6))
Ï 1855 := (11 × (1 + 1 + 11)1+1 ) − 1 − 1 − 1 − 1
:= (((2 × 22) − 2/2)2 ) − 2 − 2
:= 2 + (2 × (2 × (222 − 22)))
:= 2 + (((((2 × 22) − 2/2)2 ) + 2) + 2)
:= 3 × (3 × (3 + 3)3 − 33)
:= (3 × 3 + 3)3 + ((33+3 + 3)/(3 + 3))
:= 3/3 + ((3 × 33+3 ) − 333)
:= ((44 − 4/4)(4+4)/4 ) − 4
:= 4/4 + ((44 − 4/4)(4+4)/4 )
:= 44 × 44 − (4 − 4/4)4
:= (5 × ((5 × (5 × (5 + 5 + 5))) − 5)) − 5
:= 5 × ((5 × (5 × (5 + 5 + 5))) − 5)
:= 5 + (5 × ((5 × (5 × (5 + 5 + 5))) − 5))
:= (6 − 6/6) × (66 × 66/(6 + 6) + 6)
:= 6/6 + (((6 × 6 + 6/6) + 6)(6+6)/6 )
:= 6 + (((6 × 6 + 6/6) + 6)(6+6)/6 )
:= 7/7 + ((((7 + 7 + 7)/7)7 ) − 7 × 7 × 7)
:= 7/7 + (((7/7 − 7) + 7 × 7)(7+7)/7 )
:= 777 + 77 × (7 + 7)
:= 888 + (88/8 × (88 − 8/8))
:= 8 + ((((8 + 8)/8) × (888 + 8/8)) + 8 × 8)
:= 888 + ((88 × 88 − 8)/8)
:= (9 + 9) × (99 + 9) − 99
:= 9 + ((9999 + 9)/9 + 9 × 9 × 9)
:= 9/9 + (((9 + 9) × 99 − 9) + 9 × 9)
Ï 1846 := (1 + 1 + 11) × ((1 + 11)1+1 − (1 + 1))
Ï 1851 := 1 + (1 + ((((1 + 1) × (11 + 11)) − 1)1+1 ))
Ï 1856 := (11 × (1 + 1 + 11)1+1 ) − 1 − 1 − 1
:= 2 × (2 × 222 + 222 )
:= (2 × (2 × (222 − 22))) − 2
:= 2 + (((2 × 22) − 2/2)2 )
:= 3/3 + (3 × (3 × (3 + 3)3 − 33))
:= (3 × 33+3 ) − (333 + 3)
:= ((3/3 + 3)3 ) × ((33 − 3/3) + 3)
:= 4 + ((4 × (444 + 4 × 4)) + (4 + 4)/4)
:= 44/4 + (4 × (444 + 4 × 4))
:= 4 × ((444 + 4 × 4) + 4)
:= ((55 × 555/5) + 55 )/5
:= 5/5 + (5 × ((5 × (5 × (5 + 5 + 5))) − 5))
:= (55/5 + 5) × (555/5 + 5)
:= 6 + (((6 × 6/(6 + 6))6 ) + (6666/6))
:= ((66/6 + 6) × 666/6) − 6 × 6
:= ((6 + 6)/6)6 × (6 × 6 − (6/6 + 6))
:= ((7 + 7)/7) × ((77 × (77 + 7) − 7)/7)
:= 7 + ((((7 + 7 + 7)/7)7 ) − 7 × 7 × 7)
:= 7 + (((7/7 − 7) + 7 × 7)(7+7)/7 )
:= 8 + ((((8 + 8)/8) × (888 − 8/8)) + 8 × 8)
:= 8 + ((88/8 + 8) × ((8/8 + 88) + 8))
:= 8 × (((8 × (8 + 8) + 88) + 8) + 8)
:= 9/9 + ((9 + 9) × (99 + 9) − 99)
:= 9 × 9 + ((9 + 9) × 99 − ((99 + 9)/9))
:= 9 × 9 + (((9 + 9) × 99 − 9) + ((9 + 9)/9))
Ï 1857 := (11 × (1 + 1 + 11)1+1 ) − 1 − 1
Ï 1847 := (11 × ((1 + 1 + 11)1+1 − 1)) − 1
Ï 1852 := (1 + 1) × ((1 + 11111)/(1 + 11))
:= (((2 × 22) − 2/2)2 ) − 2
:= 2 × ((2 × (222 − 22)) + 2)
:= 2 × (2 + 2) + (((2 × 22) − 2/2)2 )
:= 3 + ((3 × 33+3 ) − (((3/3 + 3) + 3)3 ))
:= 3 + (((3 × 3 + 33) + 3/3)3−3/3 )
:= 3 + ((3 × 33+3 ) − 333)
:= (44 × (44 − (4 + 4)/4)) − 4/4
:= 4 + (44 × (44 − (4 + 4)/4))
:= 4 × 444 + (4 − 4/4)4
:= 5 + ((5/5 + 5) × (55 − 55)/(5 + 5))
:= (5 + 5)/5 + (5 × ((5 × (5 × (5 + 5 + 5))) − 5))
:= (5 − (5 + 5)/5) × ((55 − 5)/5 − 5)
:= 66/6 + (6 + 6 + 6) × (6 × 6 + 66)
:= 666 + ((66 × (6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + (((66/6 + 6) × 666/6) − 6 × 6)
:= 777 + (77 × (7 + 7) − (7/7 + 7))
:= ((7 − 77)/7) + (7 × (7 × 7 × 7 − 77))
:= ((7 + 7)/7) + (77 × (7 + 7) + 777)
:= 888 + ((8 × (8 × (8 + 8) − 8)) − 8/8)
:= 88 × (8 + 8) + 888/((8 + 8)/8)
:= 888 + ((88 × 88 + 8)/8)
:= (9 + 9)/9 + ((9 + 9) × (99 + 9) − 99)
:= 9 × 9 + ((9 + 9) × 99 − (99/9))
:= 9 + ((((9999 − 9)/9) + 9 × 9 × 9) + 9)
133
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1858 := (11 × (1 + 1 + 11)1+1 ) − 1
Ï 1863 := (1 + (11 + 11)) × (11 − 1 − 1)1+1
Ï 1868 := (11 × (1 + (1 + 1 + 11)1+1 )) − 1 − 1
:= 2 + (2 × (2 × 222 + 222 ))
:= (22 + 2/2) × ((2/2 + 2)2+2 )
:= 2 × ((2 × (2 × 222 + 22)) + 2)
:= 3 + (((3 × 33+3 ) − 333) + 3/3)
:= 3 × (3 × ((3 + 3)3 − 3 × 3))
:= ((33 − 3) × (3 × 33 − 3)) − (3/3 + 3)
:= 4/4 + (4 × 444 + (4 − 4/4)4 )
:= (4 − 4/4) × ((4/4 + 4)4 − 4)
:= (44 × (4 + 4)) − (4 × 44 + 4)
:= ((5 − (5 + 5)/5) × ((55 + 5)/5 − 5)) − 5
:= (5 − (5 + 5)/5) × ((55 + 5)/5 − 5)
:= 5 + ((5 − (5 + 5)/5) × ((55 + 5)/5 − 5))
:= ((6 + 6)/6)6 + ((6 − 6 × 6) × (6 − 66) − 6)
:= (6 × (6 × (66 + 6))) − ((6 × 6/(6 + 6))6 )
:= (6 × (6 − 6 × 6)) + (((6 + 6)/6)66/6 )
:= 7 + (((((7 + 7 + 7)/7)7 ) − 7 × 7 × 7) + 7)
:= 7/7 + (7 × (7 × 7 × 7 − 77))
:= 7 + ((7 × (7 × 7 × 7 − 77)) − 7/7)
:= 888 + (88 × 88 + 8 + 8)/8
:= 8 + (((88 × 88 − 8)/8) + 888)
:= 88 + (((8 + 8)/8) × (888 + ((8 + 8)/8)))
:= 9 + ((9999/9 + 9 × 9 × 9) + 9)
:= 9 × (99 + 99 + 9)
:= (((9 + 9)/9)99/9 ) − (99 + 9 × 9)
Ï 1859 := 11 × (1 + 1 + 11)1+1
Ï 1864 := 1 + ((1 + (11 + 11)) × (11 − 1 − 1)1+1 )
:= 22/2 × ((22/2 + 2)2 )
Ï 1869 := (11 × (1 + (1 + 1 + 11)1+1 )) − 1
:= 22 + ((((2 × 22) − 2/2)2 ) − 2)
:= 2 × (2 × (2 × 222 + 22))
3
:= 3 + (((3/3 + 3)3 ) × ((33 − 3/3) + 3))
:= 3/3 + (3 × (3 × ((3 + 3) − 3 × 3)))
:= ((33 − 3) × (3 × 33 − 3)) − 3
:= 4 + (44 × 44 − (4 − 4/4)4 )
:= 44 + (4 × 444 + 44)
:= ((4 − 44 )/4) + (44 × 44 − 4)
:= (((5 + 5 + 5) × (55 /5 − 5)) − 5)/5
:= 66/6 × ((6/6 + 6 + 6)(6+6)/6 )
5
5
5
:= 5 − (((55 + 5 ) + 5 )/5)
:= 55 − (((55 + 55 + 5)/5) + 5)
6
:= ((6 + 6)/6) + (6 − 6 × 6) × (6 − 66)
:= 66 × (6 × 6 − 6) − 666/6
:= ((7 + 7)/7) + (7 × (7 × 7 × 7 − 77))
:= 7 + (7 × (7 × 7 × 7 − 77))
:= 88/8 × ((88 − 8 + 88) + 8/8)
:= 88 + (888 + 888)
:= (8 − 8/8) × ((8 + 8) × (8 + 8) + (88/8))
:= 99/9 × ((9 × (9 + 9) − ((9 + 9)/9)) + 9)
:= 9/9 + ((9 + 9) × 99 + 9 × 9)
:= 99 + ((9 + 9) × 99 − ((99 + 9)/9))
:= 77/7 × ((7 − 7/7 + 7)
(7+7)/7
)
Ï 1860 := 1 + (11 × (1 + 1 + 11)1+1 )
Ï 1865 := 1 + (1 + ((1 + (11 + 11)) × (11 − 1 − 1)1+1 ))
Ï 1870 := 11 × (1 + (1 + 1 + 11)1+1 )
:= 2 × ((2 × 222 + 222 ) + 2)
:= 22+2 + (((2 × 22) − 2/2)2 )
:= 22 + (2 × (2 × (222 − 22)))
:= (3 × (3 × ((3 + 3)3 − 3 × 3))) − 3
:= 3 + ((((3 × (3 + 3))3 − 3)/3) − 3 × 33 )
:= 3/3 + (((33 − 3) × (3 × 33 − 3)) − 3)
(4+4)/4
)
:= (4 × 4 + 4/4) × (444 − 4)/4
:= 4 + (4 × ((444 + 4 × 4) + 4))
:= 4 × 4 + ((44 − 4/4)
:= (5 + 5 + 5) × (5 × 5 × 5 − 5/5)
:= (5 × (5 × (5 × (5 + 5 + 5)))) − 5 − 5
:= (5 × (5 × (5 × (5 + 5 + 5)))) − 5
:= (6 − 66) × (6 − (6 × 6 + 6/6))
:= 66 + ((6 − 6 × 6) × (6 − 66) − 6/6)
:= (66/6 + 6) × ((666 − 6)/6)
:= (7 × (7 × 7 × 7 − 77)) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 × (7 × 7 × 7 − 77))
:= 7 + ((7 × (7 × 7 × 7 − 77)) + 7/7)
:= 8 + (888/((8 + 8)/8) + 88 × (8 + 8))
:= 8 + (((88 × 88 + 8)/8) + 888)
:= (8/8 + 8 + 8) × (888 − 8)/8
:= 9 × 9 + ((9 + 9) × 99 − ((9 + 9 + 9)/9))
:= 9 × 9 + ((9 + 9) × 99 + ((9 + 9)/9))
:= 99 + ((9 + 9) × 99 − (99/9))
Ï 1861 := 1 + (1 + (11 × (1 + 1 + 11)1+1 ))
Ï 1866 := (1 + 1 + 1) × (11 + (1 + 11 × 111)/(1 + 1))
Ï 1871 := 1 + (11 × (1 + (1 + 1 + 11)1+1 ))
:= 2 + (22/2 × ((22/2 + 2)2 ))
:= 2 + (2 × (2 × (2 × 222 + 22)))
:= 22 + (((2 × 22) − 2/2)2 )
:= 3/3 + ((3 × (3 × ((3 + 3)3 − 3 × 3))) − 3)
:= 3 + (3 × (3 × ((3 + 3)3 − 3 × 3)))
:= ((33 − 3) × (3 × 33 − 3)) − 3/3
:= 4 + (4 × 444 + (4 − 4/4)4 )
:= (4 − 4/4) × (((4/4 + 4)4 − 4) + 4/4)
:= 44 × 44 − (44 + 4)/4
:= (((5 + 5 + 5) × (55 /5 − 5)) + 5)/5
:= 5 + ((((5 + 5 + 5) × (55 /5 − 5)) + 5)/5)
:= 55 + (((5 − (55 + 55 ))/5) − 5)
:= 6 + ((((6 × 6 + 6/6) + 6)(6+6)/6 ) + 6)
:= 66 + (6 − 6 × 6) × (6 − 66)
:= (6 × ((6 × (6 × 6 + 6) − 6) + 66)) − 6/6
:= (7 × (7 × 7 × 7 − 77)) − 7/7
:= 77/7 + (77 × (7 + 7) + 777)
:= 7 + ((7 × (7 × 7 × 7 − 77)) + ((7 + 7)/7))
:= 8 + ((8/8 + 8 + 8) × ((888 − (8 + 8))/8))
:= 88 + (((8 + 8)/8) × (888 + 8/8))
:= 8 × (88 + 8) + (8888/8 − 8)
:= 9 × 9 + ((9 + 9) × 99 − ((9 + 9)/9))
:= 9 × 9 + ((9 + 9) × 99 + ((9 + 9 + 9)/9))
:= 9 + (((9 + 9) × 99 − 9/9) + 9 × 9)
Ï 1862 := 1 + (1 + (1 + (11 × (1 + 1 + 11)1+1 )))
Ï 1867 := (11 × (1 + (1 + 1 + 11)1+1 )) − 1 − 1 − 1
:= (2 × (2 × (2 × 222 + 22))) − 2
:= 2 + ((((2 × 22) − 2/2)2 ) + 22+2 )
:= 2 × (2 × (222 − 22+2 ))
:= (((3 × (3 + 3))3 − 3)/3) − 3 × 33
:= 3 + ((3 × (3 × ((3 + 3)3 − 3 × 3))) + 3/3)
:= (33 − 3) × (3 × 33 − 3)
:= 44 × 44 + ((4 − (44 + 44 ))/4)
:= 4 + ((4 − 4/4) × ((4/4 + 4)4 − 4))
:= 4 × ((44 − 44) + 44 )
:= 5 + ((5 − (5 + 5)/5) × ((55 − 5)/5 − 5))
:= ((5/5 + 5) × (55 − 5)/(5 + 5)) − 5
:= (5/5 + 5) × (55 − 5)/(5 + 5)
:= 6 + (((6 + 6)/6)6 × (6 × 6 − (6/6 + 6)))
:= 66 + ((6 − 6 × 6) × (6 − 66) + 6/6)
:= 6 × ((6 × (6 × 6 + 6) − 6) + 66)
:= 7 × (7 × 7 × 7 − 77)
:= 7 + ((7 × (7 × 7 × 7 − 77)) − ((7 + 7)/7))
:= (77/7 + 7) × (777/7 − 7)
:= 88 + (((8 + 8)/8) × (888 − 8/8))
:= 888 + 88/8 × (8/8 + 88)
:= (8 + 8) × (8 × (8 + 8) − (88/8))
:= 9 × 9 + ((9 + 9) × 99 − 9/9)
:= 9 + (((9999/9 + 9 × 9 × 9) + 9) + 9)
:= 9 + ((9 + 9) × 99 + 9 × 9)
134
Ï 1872 := (1 + 1 + 11) × (1 + 11)1+1
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1873 := 1 + ((1 + 1 + 11) × (1 + 11)1+1 )
Ï 1878 := (1 + 1)11 − (1 + (1 + 1 + 11)1+1 )
Ï 1883 := 11 + ((1 + 1 + 11) × (1 + 11)1+1 )
:= 2 + ((((2 × 22) − 2/2)2 ) + 22)
:= 2 + (2 × ((2 × (222 − 22+2 )) + 2))
:= 2 + (22/2 × (((22/2 + 2)2 ) + 2))
:= 3/3 + ((33 − 3) × (3 × 33 − 3))
:= (33 × ((33 + 33 ) + 3)) − 3
:= 3 + ((3 × 3 × (3 + 3)3 ) − ((3/3 + 3)3 ))
:= ((4 − 44 )/4) + 44 × 44
:= (4 − 4/4) × ((4/4 + 4)4 + 4/4)
:= 4 + (((4 − 4/4) × (4/4 + 4)4 ) + 4)
:= 55 − ((5 + 5)/5 × (55 + 5)/5)
:= (5/5 + 5) × (55 + 5)/(5 + 5)
:= 5 + ((5/5 + 5) × (55 + 5)/(5 + 5))
:= 6/6 + (6 × ((6 × (6 × 6 + 6) − 6) + 66))
:= 6 × 6 × (66 − 6 − 6) − 66
:= 6 + (6 × 6 × (66 − 6 − 6) − (66 + 6/6))
:= 77/7 + (7 × (7 × 7 × 7 − 77))
:= 7 + (((7 × (7 × 7 × 7 − 77)) + ((7 + 7)/7)) + 7)
:= 7 × (((7 × 7 × 77) − 7)/(7 + 7))
:= 8/8 + ((8 + 8) × (8 × (8 + 8) − (88/8)))
:= 8 + ((8/8 + 8 + 8) × (888 − 8)/8)
:= 8 + ((8/8 + 8 + 8 + 8) × (88/8 + 8 × 8))
:= 9 + (((9 + 9) × 99 + 9 × 9) + 9/9)
:= (((9 − 9/9) + 9) × 999/9) − 9
:= 99 + ((9 + 9) × 99 + ((9 + 9)/9))
Ï 1874 := (1 + 1) × ((1 + 1)11 − 1111)
2
2+2
Ï 1879 := (1 + 1)11 − (1 + 1 + 11)1+1
Ï 1884 := (1 + 11) × (1 + ((1 + 11) × (1 + 1 + 11)))
:= 222/2 − ((22/2 + 2)2 )
:= 2 × ((2 × (222 − 2)) − 22)
:= 3 + ((33 + 3/3) × (((3/3 + 3)3 ) + 3))
:= 3 + (33 × ((33 + 33 ) + 3))
:= 4 + ((4 − 4/4) × (4/4 + 4)4 )
:= 44 + (4 × (444 + 4 × 4))
:= 5 − ((5 + 5 + 5)/5)
:= 5 + (55 − ((55 + 55 + 5)/5))
:= 5 + ((55 − ((55 + 55 + 5)/5)) + 5)
:= 6 + ((((6 + 6)/6)66/6 ) + (6 × (6 − 6 × 6)))
:= 6/6 + (6 × 6 × (66 − 6 − 6) − 66)
:= 6 + (6 × 6 × (66 − 6 − 6) − 66)
:= 777 + (7777/7 − (7 + 7))
:= 7 + ((77/7 + 7) × (777/7 − 7))
:= 7/7 + (7 × (((7 × 7 × 77) − 7)/(7 + 7)))
:= 8 + ((((8 + 8)/8) × (888 + 8/8)) + 88)
:= 8 × (88 + 8) + 8888/8
:= 88 + (((8 × 8 × (8 × 8 − 8)) + 8)/((8 + 8)/8))
:= 9 × 9 + ((9 + 9) × 99 + (99/9))
:= 99 + ((9 + 9) × 99 − ((9 + 9)/9))
:= (9 + 9) × 99 + (999/9 − 9)
:= 2 + (2 × (2 × (22 − 2
)))
:= 3 + (((33 − 3) × (3 × 33 − 3)) − 3/3)
4
:= 44 × 44 + (((4 − 4 ) + 4)/4)
5
5
5
Ï 1875 := 1 + ((1 + 1) × ((1 + 1)11 − 1111))
Ï 1885 := (1 + 1 + 11) × (1 + (1 + 11)1+1 )
Ï 1880 := 1 + ((1 + 1)11 − (1 + 1 + 11)1+1 )
:= (2/2 + 2) × ((2/2 + 2 + 2)2+2 )
:= 2 × (2 × ((222 − 22+2 ) + 2))
:= (22/2)2 + (2 × 22 − 2)2
:= 3 + ((33 − 3) × (3 × 33 − 3))
:= (3 × 3 × (3 + 3)3 ) − ((3/3 + 3)3 )
:= 3 + ((33 × ((33 + 33 ) + 3)) + 3/3)
:= (4 − 4/4) × (4/4 + 4)4
:= (44 − 4) × (44 − 4/4 + 4)
:= 4 + (44/4 × (4 × 44 − (4/4 + 4)))
:= 5 × (5 × (5 × (5 + 5 + 5)))
:= 5 + (5 × (5 × (5 × (5 + 5 + 5))))
:= 5 + ((5 × (5 × (5 × (5 + 5 + 5)))) + 5)
:= 6 + (66 × (6 × 6 − 6) − 666/6)
:= 6 × 6 × (66 − 6 − 6) − ((6 + 6)/6)
:= (6/6 − 66) × ((6/6 − 6 × 6) + 6)
:= 7 + (((7 × (7 × 7 × 7 − 77)) − 7/7) + 7)
:= 7 + ((7 × (7 × 7 × 7 − 77)) + (77/7))
:= ((7 + 7)/7) + (7 × (((7 × 7 × 77) − 7)/(7 + 7)))
:= (8/8 + 8 + 8 + 8) × (88/8 + 8 × 8)
:= 8 + ((8 + 8) × (8 × (8 + 8) − (88/8)))
:= 8 + ((8/8 − (8 × 8 × 88))/(8 − 88/8))
:= 9 × 9 + ((9 + 9) × 99 + ((99 + 9)/9))
:= 99 + ((9 + 9) × 99 − 9/9)
:= (9 + 9) × 99 + (((999 + 9)/9) − 9)
Ï 1876 := (1 + 1) × (1 + ((1 + 1)11 − 1111))
6
Ï 1881 := 11 × (1 + (1 + (1 + 1 + 11)1+1 ))
Ï 1886 := 1 + ((1 + 1 + 11) × (1 + (1 + 11)1+1 ))
:= 2 × ((2 × (222 − 22+2 )) + 2)
:= 22/2 × (((22/2 + 2)2 ) + 2)
:= 2 + (2 × ((2 × (222 − 2)) − 22))
:= (33 + 3/3) × (((3/3 + 3)3 ) + 3)
:= 33 × ((33 + 33 ) + 3)
:= (33 − 3/3 − 3) × (3 × 33 + 3/3)
:= 4 + (4 × ((44 − 44) + 44 ))
:= 44/4 × (4 × 44 − (4/4 + 4))
:= 4 × 444 + (444 − 4)/4
:= 55 + ((5 − (55 + 55 ))/5)
:= 5 + ((5 × (5 × (5 × (5 + 5 + 5)))) + 5/5)
:= 55 + ((55 − (55 + 55 ))/5)
:= 6 + ((66/6 + 6) × ((666 − 6)/6))
:= ((66/6 + 6) × 666/6) − 6
:= ((66/6 + 6) × 666/6) − 6/6
:= 7 + ((7 × (7 × 7 × 7 − 77)) + 7)
:= 777 + (7777/7 − 7)
:= ((7 × 7 × 77) − 7/7)/((7 + 7)/7)
:= 88 + (((8 × 8 × (8 × 8 − 8)) − 8)/((8 + 8)/8))
:= (88/8 + 8) × (88/8 + 88)
:= ((8/8 + 8 + 8) × 888/8) − 8/8
:= 99 + (((9 − 99)/(9 + 9)) + (9 + 9) × 99)
:= 99 + (9 + 9) × 99
:= (((9 + 9)/9)99/9 ) − 9 × (9 + 9)
Ï 1877 := 1 + ((1 + 1) × (1 + ((1 + 1)11 − 1111)))
Ï 1882 := 1 + (11 × (1 + (1 + (1 + 1 + 11)1+1 )))
Ï 1887 := 111 × (1 + (1 + 1)1+1+1+1 )
:= 2 + ((2 × 22 − 2)2 + 222/2)
:= (2 × ((2 × (222 − 2)) − 22)) − 2
:= 222/2 × (22+2 + 2/2)
:= (3 × 3 × (3 + 3)3 ) − (((3/3 + 3)3 ) + 3)
:= 3/3 + (33 × ((33 + 33 ) + 3))
:= (3 × (3 × ((3 + 3)3 − (3 + 3)))) − 3
:= 4 + (((4 − 44 )/4) + 44 × 44)
:= 4 + ((4 − 4/4) × ((4/4 + 4)4 + 4/4))
:= (4 − 4/4) × ((4/4 + 4)4 + 4)
:= 5 + ((5/5 + 5) × (55 − 5)/(5 + 5))
:= 5 + (((5/5 + 5) × (55 − 5)/(5 + 5)) + 5)
:= (5 − (5 + 5)/5) × ((55 − 5)/5 + 5)
:= 6 × 6 × (66 − 6 − 6) − (66 + 6/6)
:= 6 + (((66/6 + 6) × ((666 − 6)/6)) + 6)
:= (66/6 + 6) × 666/6
:= 7 + (((7 × (7 × 7 × 7 − 77)) + 7/7) + 7)
:= 777 + ((7777 + 7)/7 − 7)
:= 777 + (7777 − 7)/7
:= (8/8 − (8 × 8 × 88))/(8 − 88/8)
:= 8/8 + ((88/8 + 8) × (88/8 + 88))
:= (8/8 + 8 + 8) × 888/8
:= (((9 + 9)/9)99/9 ) − (9 × (9 + 9) + 9)
:= 9/9 + ((9 + 9) × 99 + 99)
:= ((9 − 9/9) + 9) × 999/9
135
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1888 := 1 + (111 × (1 + (1 + 1)1+1+1+1 ))
Ï 1898 := (1 + 1 + 11) × (1 + (1 + (1 + 11)1+1 ))
Ï 1893 := (1 + 1)11 − (11 + (1 + 11)1+1 )
:= 2 × (2 × 222 − (22 + 2))
:= 2/2 + (2 × (2 × 222 − 22))
:= 2 + ((2 × 22)2 + (2 × (2 − 22)))
:= 3/3 + ((3 × (3 × ((3 + 3)3 − (3 + 3)))) − 3)
:= 3 + (3 × (3 × ((3 + 3)3 − (3 + 3))))
:= (33 − 3/3) × (((3 + 3)3 + 3)/3)
:= (4 + 4) × (44 − (4 × 4 + 4))
:= 4/4 + (44 × (44 − 4/4))
:= 4 + ((44 × (44 − 4/4)) + (4 + 4)/4)
:= ((5 + 5)/5)5 × (55 − 5/5 + 5)
:= (5 − (5 + 5)/5) × ((55 + 5)/5 + 5)
:= 5 + ((5 − (5 + 5)/5) × ((55 + 5)/5 + 5))
:= 6/6 + ((66/6 + 6) × 666/6)
:= 6 + ((66/6 + 6) × 666/6)
:= ((66/6 + 6) × (666 + 6)/6) − 6
7
:= 777 + 7777/7
:= (7 × (7 − 7 × 7)) + (((7 + 7 + 7)/7) )
:= 7 + (((7 + 7) × (((7 + 7)/7)7 + 7)) + 7/7)
:= (8 + 8) × ((888 − 8)/8 + 8)
:= ((8 + 8) × (888/8 + 8)) − 88/8
:= 8 + ((8 − 8/8 + 8) × (8 × (8 + 8) − ((8 + 8)/8)))
:= 999 + (9 × 99 − ((9 + 9)/9))
:= (9 + 9) × 99 + 999/9
:= 9 + ((9 × 99 − 9/9) + 999)
Ï 1889 := ((1 + 1) × (11 − 1)1+1+1 ) − 111
Ï 1894 := (1 + 1)11 − (11 × (1 + (1 + 1 + 11)))
:= 2 + (222/2 × (22+2 + 2/2))
2
Ï 1899 := (11 − 1 − 1) × ((1 + 1) × 111 − 11)
:= 2 + (2 × (2 × 22 − 22))
:= (2/2 + 2)2 × (222 − 22/2)
:= (3 × (3 × ((3 + 3) − (3 + 3)))) − 3/3
:= 3 + ((3 × (3 × ((3 + 3)3 − (3 + 3)))) + 3/3)
:= 3 × ((3 × ((3 + 3)3 − (3 + 3))) + 3)
:= 4/4 + ((4 + 4) × (44 − (4 × 4 + 4)))
:= (4 + 4)/4 + (44 × (44 − 4/4))
:= (4 − 4/4) × (((4/4 + 4)4 + 4) + 4)
3
5
:= ((5/5 + 5)5 /(5 − 5/5)) − 55
:= 5 + (((5/5 + 5) /(5 − 5/5)) − 55)
:= (5 × ((5 × (5 × (5 + 5 + 5))) + 5)) − 5/5
:= ((6 + 6 + 6) × (666/6 − 6)) − 6/6
:= 6 + (((66/6 + 6) × 666/6) + 6/6)
:= 6 + (((66/6 + 6) × 666/6) + 6)
:= 777 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 777)
:= 777 + ((7777 + 77)/7)
:= 8 + ((88/8 + 8) × (88/8 + 88))
:= 8 + (((8/8 + 8 + 8) × 888/8) − 8/8)
:= 88/8 + ((8 + 8) × ((888 − 8)/8 + 8))
:= 999 + (9 × 99 − 9/9)
:= (9 + 9) × 99 + ((999 + 9)/9)
:= 9 + (999 + 9 × 99)
Ï 1890 := 1 + (((1 + 1) × (11 − 1)1+1+1 ) − 111)
Ï 1895 := 1 + ((1 + 1)11 − (11 × (1 + (1 + 1 + 11))))
Ï 1900 := (11 − 1)1+1 × (((1 + 1) × (11 − 1)) − 1)
:= (2 × 22)2 − ((2 × 22) + 2)
:= 2 + ((2 × (2 × 222 − 22)) + 2/2)
:= 2 × ((2 × (222 + 2)) − 22)
:= 3 × (3 × ((3 + 3)3 − (3 + 3)))
:= ((33 − 3/3) × (((3 + 3)3 + 3)/3)) − 3
:= (((3 × (3 + 3))3 − 33)/3) − 33
:= (44 + 4/4) × (44 − (4 + 4)/4)
:= 4 + 44 × 44 − (44 + 4/4)
:= 4 + ((44 × (44 − 4/4)) + 4)
:= (5 + 5 + 5) × (5 × 5 × 5 + 5/5)
:= (5 × ((5 × (5 × (5 + 5 + 5))) + 5)) − 5
:= 5 × ((5 × (5 × (5 + 5 + 5))) + 5)
:= (6 + 6 + 6) × (666/6 − 6)
:= (6 − 6/6) × (6 × 66 − (66/6 + 6))
:= (6 − 6/6) × ((((6 − 66)/6) − 6) + 6 × 66)
:= (7 + 7) × (((7 + 7)/7)7 + 7)
:= 7 + (7777/7 + 777)
:= (7/7 + 7 × 7) × (7 × 7 − 77/7)
:= (8 − 8/8 + 8) × (8 × (8 + 8) − ((8 + 8)/8))
:= 8 + ((8/8 + 8 + 8) × 888/8)
:= (88/8 + 8) × (((88 + 8)/8) + 88)
99/9
:= 9 + ((((9 + 9)/9)
:= 999 + 9 × 99
Ï 1891 := (1 + 1)11 − (1 + ((1 + 11) × (1 + 1 + 11)))
2
) − 9 × (9 + 9))
Ï 1896 := (1 + 11) × ((1 + 1 + 11)1+1 − 11)
:= (9/9 + 9 + 9) × (9/9 + 99)
Ï 1901 := 1 + ((11 − 1)1+1 × (((1 + 1) × (11 − 1)) − 1))
:= (2 × 22) − (2 × 22 + 2/2)
:= 2 × ((2 × 222 − 22) + 2)
:= 2/2 + (2 × ((2 × (222 + 2)) − 22))
:= 3/3 + (3 × (3 × ((3 + 3)3 − (3 + 3))))
:= 3 + ((3 × (3 × ((3 + 3)3 − (3 + 3)))) + 3)
:= 3 + ((33 − 3/3) × (((3 + 3)3 + 3)/3))
:= 44 × 44 − (44 + 4/4)
:= 4 + 44 × (44 − 4/4)
5
:= 4 + (((44 × (44 − 4/4)) + 4/4) + 4)
5
:= (((5 + 5 + 5) × (5 /5 + 5)) + 5)/5
:= 5 + ((((5 + 5 + 5) × (5 /5 + 5)) + 5)/5)
:= 5/5 + (5 × ((5 × (5 × (5 + 5 + 5))) + 5))
:= 6 + ((6/6 − 66) × ((6/6 − 6 × 6) + 6))
:= 6 + ((6 + 6 + 6) × (666/6 − 6))
:= (6 × (6 × (6 × 6 + 6) + 66)) − 6/6 − 6
:= 7 + ((7777 + 7)/7 + 777)
:= 77/7 + ((7 + 7) × (((7 + 7)/7)7 + 7))
:= 8 + (((8/8 + 8 + 8 + 8) × (88/8 + 8 × 8)) + 8)
:= (8 + 8 + 8) × (88 − (8/8 + 8))
:= 888 + (8 × 8 × (8 + 8) − (88/8))
:= 9/9 + (999 + 9 × 99)
:= 9 + (((9 − 9/9) + 9) × 999/9)
:= 99/9 + (999 + 9 × 99)
7
:= 7/7 + ((7 + 7) × (((7 + 7)/7) + 7))
Ï 1892 := 11 × (1 + (1 + (1 + (1 + 1 + 11)1+1 )))
Ï 1897 := 1 + ((1 + 11) × ((1 + 1 + 11)1+1 − 11))
Ï 1902 := (1 + 1)11 − (1 + (1 + (1 + 11)1+1 ))
:= 2 × (2 × 222 − 22)
:= 2/2 + ((2 × 22)2 + (2 × (2 − 22)))
:= 2 + (2 × ((2 × (222 + 2)) − 22))
:= 3 + ((3 × (3 × ((3 + 3)3 − (3 + 3)))) − 3/3)
:= (((3 × (3 + 3))3 − 33)/3) − (33 + 3)
:= (3 × (3 × (3 + 3)3 − 3)) − 33
:= 44 × (44 − 4/4)
:= 4 + ((44 × (44 − 4/4)) + 4/4)
:= 44 × 44 + ((44 − 4)/4 − 44)
5
5
:= 5 + ((5 − (5 + 5)/5) × ((5 − 5)/5 + 5))
:= 5 × 5 + ((5/5 + 5) × (5 − 5)/(5 + 5))
:= (5/5 + 5) × ((55 − 5)/(5 + 5) + 5)
:= 6 + (((66/6 + 6) × 666/6) − 6/6)
:= 66 × (6 + 6) + ((6666/6) − 6)
:= (6 × (6 × (6 × 6 + 6) + 66)) − 6
7
7
:= 7 + ((7 + 7) × (((7 + 7)/7) + 7))
:= 7 + ((7777/7 + 777) + 7)
:= 88 × (8 + 8) + 88 × 88/(8 + 8)
:= 8/8 + ((8 + 8 + 8) × (88 − (8/8 + 8)))
:= ((8 + 8)/8) × ((888 − 8/8) + 8 × 8)
:= 99 + ((9 + 9) × 99 + (99/9))
:= 9 + ((999 − ((9 + 9)/9)) + 9 × 99)
:= 9 + ((9 + 9) × 99 + 999/9)
:= ((7 + 7)/7) + ((7 + 7) × (((7 + 7)/7) + 7))
136
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1908 := (1 + 11) × (1 + ((1 + 1 + 11)1+1 − 11))
Ï 1903 := (1 + 1)11 − (1 + (1 + 11)1+1 )
Ï 1913 := (1 + 1)11 − (1 + (1 + (1 + (11 × (1 + 11)))))
:= (2 × 22)2 − (22/2 + 22)
:= 2 + ((2 × (2 × (222 − 2))) − 22)
:= (2 × 22)2 − (22 + 2/2)
:= 3 + ((((3 × (3 + 3))3 − 33)/3) − 33)
:= 3 × ((3 × ((3 + 3)3 − 3)) − 3)
:= 3 + ((((3 × (3 + 3))3 − 3)/3) − 33)
:= 44/4 + (44 × (44 − 4/4))
:= 4 + (4 × (444 + 4 × (4 + 4)))
:= 4 + (44 × 44 − (44/4 + 4 × 4))
:= 5 × 5 + ((5/5 + 5) × (55 + 5)/(5 + 5))
:= (5/5 + 5) × ((55 + 5)/(5 + 5) + 5)
:= 5 + ((5/5 + 5) × ((55 + 5)/(5 + 5) + 5))
:= 66 × (6 + 6) + (6666/6)
:= 6 × (6 × (6 × 6 + 6) + 66)
:= 66 × (6 × 6 − 6) − (66 + 6/6)
:= 777/7 + ((7 + 7) × ((7 + 7)/7) )
:= (77/7 + 7) × ((7 × (7 + 7) + 7/7) + 7)
:= ((7/7 + 7 + 7) × ((7 + 7)/7)7 ) − 7
:= ((8 + 8) × (888/8 + 8)) − 8/8
:= ((8 + 8) × (8 × (8 + 8) − 8)) − (88 + 8)/8
:= 8/8 + (8 × 8 × (8 + 8) + 888)
:= 99/9 × (99/9 + 9 × (9 + 9))
:= 9 + ((999 + 9 × 99) + 9)
:= 9 + (((9 − 9/9) + 9) × ((999 + 9)/9))
7
Ï 1909 := 1 + ((1 + 11) × (1 + ((1 + 1 + 11)1+1 − 11)))
Ï 1904 := (1 + 1)11 − (1 + 11)1+1
Ï 1914 := (1 + 1)11 − (1 + (1 + (11 × (1 + 11))))
:= 2 × (2 × (222 − 2 × (2 + 2)))
:= (22 + 2/2) × (((2/2 + 2)2+2 ) + 2)
:= (2 × 22)2 − 22
:= ((33/3)3 ) + (((3 × 3 + 3)3 /3) − 3)
:= 3/3 + (3 × ((3 × ((3 + 3)3 − 3)) − 3))
:= (3 × (3 × ((3 + 3)3 − 3))) − 3
:= 4 × (444 + 4 × (4 + 4))
:= 44 × 44 − (44/4 + 4 × 4)
:= 44/4 × (4 × 44 − (4 + 4)/4)
5
5
:= 5 + (5 − 55/5 × 555/5)
:= 55 × 55 − 5555/5
:= (66/6 + 6) × (666 + 6)/6
:= 6/6 + (6 × (6 × (6 × 6 + 6) + 66))
:= 66 × (6 × 6 − (6/6 + 6))
:= (7 × ((7 × 7 × 7 − 77) + 7)) − 7
:= (7 × ((7 × 7 × 7 − 77) + 7)) − (7 + 7)/7
:= 77/7 × ((7 × (7 + 7) − 7/7) + 77)
:= (8 + 8) × (888/8 + 8)
:= ((8 + 8) × (8 × (8 + 8) − 8)) − 88/8
:= (88 − 8/8) × (88 + 88)/8
:= ((9 − 9/9) + 9) × ((999 + 9)/9)
:= 9 + ((9/9 + 9 + 9) × (9/9 + 99))
:= 99/9 × (((99 + 9)/9) + 9 × (9 + 9))
:= 5 − 55/5 × 555/5
Ï 1905 := 1 + ((1 + 1)11 − (1 + 11)1+1 )
Ï 1910 := 11 + ((11 − 1 − 1) × ((1 + 1) × 111 − 11))
Ï 1915 := (1 + 1)11 − (1 + (11 × (1 + 11)))
:= 2 + ((2 × 22)2 − (22/2 + 22))
:= (2 × 22)2 − (22 + 2 + 2)
:= 2/2 + ((2 × 22)2 − 22)
:= (3 × ((3 × ((3 + 3)3 − 3)) − 3)) − 3
:= (((3 × (3 + 3))3 − 3)/3) − 33
:= 3 + ((((3 × (3 + 3))3 + 3)/3) − 33)
:= 4/4 + (4 × (444 + 4 × (4 + 4)))
:= (4 − 44)/4 + ((4 + 4) × (44 − 4 × 4))
:= 44 × 44 − ((4 × 4 + 4/4) + 4)
:= 5 + (5 × ((5 × (5 × (5 + 5 + 5))) + 5))
:= 55 − (5 × ((5 − (5 + 5)/5)5 ))
:= (55 × ((5 × 5 + 5) + 5)) − 5 − 5
:= 6 + ((((66/6 + 6) × 666/6) + 6) + 6)
:= 6 + ((66/6 + 6) × (666 + 6)/6)
:= 6/6 + (66 × (6 × 6 − (6/6 + 6)))
:= 7/7 + ((7 × ((7 × 7 × 7 − 77) + 7)) − 7)
:= (7 × ((7 × 7 × 7 − 77) + 7)) − 7/7
:= 7 + ((77/7 + 7) × ((7 × (7 + 7) + 7/7) + 7))
:= 8/8 + ((8 + 8) × (888/8 + 8))
:= 888 + (8 × 8 × (8 + 8) − ((8 + 8)/8))
:= 88/8 + ((8 + 8) × (888/8 + 8))
:= 9 + ((((9 − 9/9) + 9) × 999/9) + 9)
:= 9 + ((999 + 9 × 99) + (99/9))
:= 9 × 99 + (((9 + 9)/9)9/9+9 )
Ï 1906 := 1 + (1 + ((1 + 1)11 − (1 + 11)1+1 ))
Ï 1911 := (1 + 1 + 11) × (1 + (1 + (1 + (1 + 11)1+1 )))
Ï 1916 := (1 + 1)11 − (11 × (1 + 11))
:= (2 × (2 × (222 − 2))) − 22
:= (2 × 22)2 − ((22 + 2/2) + 2)
:= 2 + ((2 × 22)2 − 22)
:= (3 × (3 × ((3 + 3)3 − 3))) − 33/3
:= (3 × 3 × (3 + 3)3 ) − 33
:= (((3 × (3 + 3))3 − 3)/3) − 33
:= (4 + 4)/4 + (4 × (444 + 4 × (4 + 4)))
:= 4 × 44 + (((4 + 4) × 444 − 4)/4)
:= 44 × 44 − (4 × 4 + 4)
:= 5 + ((5 × ((5 × (5 × (5 + 5 + 5))) + 5)) + 5/5)
:= 55 + (5/5 − (5 × ((5 − (5 + 5)/5)5 )))
:= 55 + ((5 − (55 × (55 + 55)))/5)
:= (6 × (6 × (6 × 6 + 6) + 66)) − (6 + 6)/6
:= 666/6 + (6 − 6 × 6) × (6 − 66)
:= 66 × (6 × 6 − 6) − ((6 + 6)/6)6
:= ((7/7 + 7 + 7) × ((7 + 7)/7)7 ) − (7 + 7)
:= 7 × ((7 × 7 × 7 − 77) + 7)
:= 7 + ((7 × ((7 × 7 × 7 − 77) + 7)) − ((7 + 7)/7))
:= (8 + 8)/8 + ((8 + 8) × (888/8 + 8))
:= 888 + (8 × 8 × (8 + 8) − 8/8)
:= ((8 + 8) × (8 × (8 + 8) − 8)) − (8/((8 + 8)/8))
:= 9 + (((9 + 9) × 99 + 999/9) + 9)
:= 9 × 9 × 9 + (((99 × (99 + 9)) − 9)/9)
9/9+9
:= 9 × 99 + ((((9 + 9)/9)
) − 9)
Ï 1907 := 1 + (1 + (1 + ((1 + 1)11 − (1 + 11)1+1 )))
Ï 1912 := 1 + ((1 + 1 + 11) × (1 + (1 + (1 + (1 + 11)1+1 ))))
Ï 1917 := 1 + ((1 + 1)11 − (11 × (1 + 11)))
:= 2/2 + ((2 × (2 × (222 − 2))) − 22)
:= (2 × 22)2 − (22 + 2)
:= 2 + (((2 × 22)2 − 22) + 2/2)
:= ((33/3)3 ) + ((3 × 3 + 3)3 /3)
:= (((3 × (3 + 3))3 + 3)/3) − 33
:= 3 × (3 × ((3 + 3)3 − 3))
:= 4 + ((44 × (44 − 4/4)) + 44/4)
:= (4 + 4) × (44 − (4 × 4 + 4/4))
:= 4/4 + (44 × 44 − (4 × 4 + 4))
:= 5 + ((5/5 + 5) × ((5 − 5)/(5 + 5) + 5))
:= 5 + (((5/5 + 5) × ((5 − 5)/(5 + 5) + 5)) + 5)
:= 55 + ((5 + 5)/5 × ((5 − 55 × 55)/5))
:= (6 × (6 × (6 × 6 + 6) + 66)) − 6/6
:= 66 × (6 × 6 − 6) − (((6 + 6)/6) + 66)
:= ((6 × 6/(6 + 6))6 ) + (66 × (6 + 6 + 6))
:= 7 + ((7/7 + 7 × 7) × (7 × 7 − 77/7))
:= 7/7 + (7 × ((7 × 7 × 7 − 77) + 7))
:= 7 + ((7 × ((7 × 7 × 7 − 77) + 7)) − 7/7)
:= 8 × 8 + ((88/8 + 8) × ((8/8 + 88) + 8))
:= 888 + 8 × 8 × (8 + 8)
:= 8 + (((8 + 8) × (8 × (8 + 8) − 8)) − (88/8))
:= 9 + (((9 × 99 − 9/9) + 999) + 9)
:= 9 + ((99/9) × (99/9 + 9 × (9 + 9)))
:= 999 + (999 − 9 × 9)
5
5
137
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1918 := 1 + (1 + ((1 + 1)11 − (11 × (1 + 11))))
Ï 1923 := 1 + ((1 + 1) × ((1 + ((11 − 1) × (1 + 1 + 1)))1+1 ))
Ï 1928 := 1 + ((1 + 1)11 − 111+1 )
:= 2 + (((2 × 22)2 − 22) + 2)
:= (2 × 22)2 − (22/2 + 2)
:= 2 × (2 × (222 − 2))
:= 3/3 + (3 × (3 × ((3 + 3)3 − 3)))
:= 3 + ((3 × (3 × ((3 + 3)3 − 3))) + 3)
:= 33/3 + (3 × (3 × ((3 + 3)3 − 3)))
:= 44 × 44 − ((4 + 4)/4 + 4 × 4)
:= 4 + (44 × 44 − (4 × 4 + 4/4))
:= 44 × 44 − 4 − 4
:= ((5 + 5)/5 + 5) × (5 × 55 − 5/5)
:= (((5 + 5)/5)55/5 ) − 5 × 5 × 5
:= 5 + ((((5 + 5)/5)55/5 ) − 5 × 5 × 5)
6
:= ((6 × 6 − 6) × ((6 + 6)/6) ) − (6 + 6)/6
:= 6 + ((66 × (6 × 6 − 6) − ((6 + 6)/6)6 ) + 6)
:= 6 × 6 + ((66/6 + 6) × 666/6)
7
:= 7 + (7 × ((7 × 7 × 7 − 77) + 7))
:= (7 × 7 × (7 × 7 − 7)) − (((7 + 7)/7) + 7)
:= 7 + (((7/7 + 7 + 7) × ((7 + 7)/7)7 ) + 7/7)
:= ((8 + 8) × (8 × (8 + 8) − 8)) − (8 + 8)/8
:= 888 + (8 × 8 × (8 + 8) + (88/8))
:= 8 + ((8 + 8) × (8 × (8 + 8) − 8))
:= 9 + (((9/9 + 9 + 9) × (9/9 + 99)) + 9)
:= (9 + 9) × (99 + 9) − (((99 + 9)/9) + 9)
:= 9 + ((9/9 + 9 + 9) × ((9 + 9)/9 + 99))
Ï 1919 := 1 + (1 + (1 + ((1 + 1)11 − (11 × (1 + 11)))))
2
:= (2 × 22) − (2
2+2
+ 2/2)
Ï 1924 := (1 + 1)11 − (1 + (1 + (1 + 111+1 )))
Ï 1929 := 1 + (1 + ((1 + 1)11 − 111+1 ))
:= 2 × ((2 × (222 − 2)) − 2)
:= 2/2 + (2 × (2 × (222 − 2)))
:= 3 + ((((3 × (3 + 3)) − 3)/3) − 3 )
:= (3 × (3 × (3 + 3)3 − 3)) − 33/3
:= 44 × 44 − (4 × 4 + 4/4)
:= 3 + (3 × ((3 × ((3 + 3)3 − 3)) + 3))
:= 4 + ((4 + 4) × (44 − 4 × 4))
:= 4 + (44 × 44 − 44/4)
3
3
:= (55 × ((5 × 5 + 5) + 5)) − (5/5 + 5)
:= ((6 × 6 − 6) × ((6 + 6)/6)6 ) − 6/6
:= (55 × ((5 × 5 + 5) + 5)) − 5/5
:= 5 + ((55 × ((5 × 5 + 5) + 5)) − 5/5)
6
:= 7 + ((7 × ((7 × 7 × 7 − 77) + 7)) + 7/7)
:= ((8 + 8) × (8 × (8 + 8) − 8)) − 8/8
:= (9/9 + 9 + 9) × ((9 + 9)/9 + 99)
Ï 1920 := (111+1 − 1) × (1 + 1)1+1+1+1
:= (6 × 6 + 6/6) × (((6 + 6)/6) − (6 + 6))
:= (((6 × 666) − 6)/((6 + 6)/6)) − 66
:= (77 × (77/7 + 7 + 7)) − 7/7
:= (7 × 7 × (77 + 7)) − (((7 + 7 + 7)/7)7 )
:= (8/((8 + 8)/8)) + ((8 + 8) × (8 × (8 + 8) − 8))
:= 8 + (((8 + 8) × (8 × (8 + 8) − 8)) + 8/8)
:= (9 + 9) × (99 + 9) − (99/9 + 9)
:= 9 + ((9/9 + 9) × (999/9 + 9 × 9))
Ï 1925 := (1 + 1)11 − (1 + (1 + 111+1 ))
2
:= 2 × (2 × (22 − (2 + 2)))
Ï 1930 := 1 + (1 + (1 + ((1 + 1)11 − 111+1 )))
2
:= (2 × 22) − 22/2
:= 2 + (2 × (2 × (222 − 2)))
:= ((33/3)3 ) + 3 × 33 × (3 + 3)
:= (((3 × (3 + 3))3 − 33)/3) − 3
:= 44 × 44 − 44/4
:= 44 × 44 − ((4 + 4)/4 + 4)
:= (6 × 6 − 6) × ((6 + 6)/6)6
:= 55 × ((5 × 5 + 5) + 5)
:= 5 + (55 × ((5 × 5 + 5) + 5))
:= (6 − 6/6) × (6 × 66 − (66/6))
:= (6 − 6/6) × (((6 − 66)/6) + 6 × 66)
:= (7/7 + 7 + 7) × ((7 + 7)/7)7
:= 77 × (77/7 + 7 + 7)
:= (7 × 7 × (7 × 7 − 7)) − ((7 + 7)/7)7
:= (8 + 8) × (8 × (8 + 8) − 8)
:= 88/8 × (888/8 + 8 × 8)
:= 8 + (((8 + 8) × (8 × (8 + 8) − 8)) + ((8 + 8)/8))
:= (9/9 + 9) × (999/9 + 9 × 9)
:= (9 + 9) × (99 + 9) − (9/9 + 9 + 9)
:= 9 + (9999/9 + 9 × (9 × 9 + 9))
:= 3 + (3 × (3 × ((3 + 3)3 − 3)))
:= (4 + 4) × (44 − 4 × 4)
:= (55 + 5) × ((5 + 5)/5)5
Ï 1921 := 1 + ((111+1 − 1) × (1 + 1)1+1+1+1 )
:= (((2
2+2+2
2
− 2) ) − 2)/2
Ï 1926 := (1 + 1)11 − (1 + 111+1 )
Ï 1931 := 1 + (1 + (1 + (1 + ((1 + 1)11 − 111+1 ))))
:= (2 × (2 × (222 − 2))) − 2
:= (2 × 22)2 − (2/2 + 2 + 2)
:= 3 × ((3 × ((3 + 3)3 − 3)) + 3)
:= (((3 × (3 + 3))3 − 3)/3) − (3 × 3 + 3)
:= (4 − 44)/4 + 44 × 44
:= 44 × 44 − (4/4 + 4)
:= 5/5 + ((55 + 5) × ((5 + 5)/5) )
:= 5/5 + (55 × ((5 × 5 + 5) + 5))
:= 5 + ((55 × ((5 × 5 + 5) + 5)) + 5/5)
:= 6/6 + ((6 × 6 − 6) × ((6 + 6)/6)6 )
:= 6 + ((6 × 6 − 6) × ((6 + 6)/6)6 )
:= 6 + ((6 − 6/6) × (6 × 66 − (66/6)))
:= 7/7 + ((7/7 + 7 + 7) × ((7 + 7)/7)7 )
:= 7/7 + (77 × (77/7 + 7 + 7))
:= 7 + ((77 × (77/7 + 7 + 7)) − 7/7)
:= 8/8 + ((8 + 8) × (8 × (8 + 8) − 8))
:= 8 + (((8 + 8) × (8 × (8 + 8) − 8)) − ((8 + 8)/8))
:= 88/8 + ((8 + 8) × (8 × (8 + 8) − 8))
:= 9 × (9 × 9 + 9) + 9999/9
:= (9 + 9) × ((99 − 9/9) + 9)
:= (9 + 9) × (99 + 9) − ((99 + 9 + 9)/9)
3
:= 3 + ((3 × (3 × ((3 + 3) − 3))) + 3/3)
4
:= 4/4 + ((4 + 4) × (4 − 4 × 4))
5
Ï 1922 := (1 + 1) × ((1 + ((11 − 1) × (1 + 1 + 1)))1+1 )
:= ((2
2+2+2
2
Ï 1927 := (1 + 1)11 − 111+1
Ï 1932 := (1 + 1) × ((1 + 1) × (((11 + 11)1+1 ) − 1))
2
− 2) )/2
:= 2 + ((2 × 22) − 22/2)
3
:= 2 × (2 × 222 − 2)
3
:= (((3 × (3 + 3)) + 33)/3) − 33
4
:= (4 + 4)/4 + ((4 + 4) × (4 − 4 × 4))
5
:= (5 + 5)/5 + ((55 + 5) × ((5 + 5)/5) )
6
:= 6 + (66 × (6 × 6 − 6) − ((6 + 6)/6) )
:= 3/3 + (3 × ((3 × ((3 + 3) − 3)) + 3))
:= (3 × (3 × (3 + 3)3 − 3)) − 3
:= 44 × 44 − ((4/4 + 4) + 4)
:= 44 × 44 − 4
:= (5 + 5)/5 + (55 × ((5 × 5 + 5) + 5))
6
:= 6 + (((6 × 6 − 6) × ((6 + 6)/6) ) + 6/6)
7
:= ((5 + 5)/5 + 5) × (5 × 55 + 5/5)
:= 6 × 6 × (66 − 6 − 6) − 6 − 6
:= 7 + ((7/7 + 7 + 7) × ((7 + 7)/7) )
:= 7 + (77 × (77/7 + 7 + 7))
:= (8 + 8)/8 + ((8 + 8) × (8 × (8 + 8) − 8))
:= 8 + (((8 + 8) × (8 × (8 + 8) − 8)) − 8/8)
:= ((8 + 8)/8) × ((88 × 88 − (8 + 8))/8)
:= ((9 + 9)/9) × ((9 × (99 + 9)) − (99/9))
:= 9/9 + ((9 + 9) × ((99 − 9/9) + 9))
:= ((99 + 9)/9) × (9 × (9 + 9) − 9/9)
:= 77/7 + (7 × ((7 × 7 × 7 − 77) + 7))
138
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1938 := 1 + ((1 + 1)11 − 111)
Ï 1933 := (1 + 1)11 − (1 + (1 + 1 + 1 + 111))
Ï 1943 := (1 + 1)11 − (111 − ((1 + 1) × (1 + 1 + 1)))
:= (2 × 22)2 − 2/2 − 2
:= 2 + (2 × 22)2
:= (2 × (2 × (222 + 2))) − 2/2
:= ((3 × (3 + 3))3 − 33)/3
:= 3 + (3 × (3 × (3 + 3)3 − 3))
:= ((3 × (3 + 3))3 − 3)/3
:= 4/4 + (44 × 44 − 4)
:= (4 + 4)/4 + 44 × 44
:= 4 + ((44 × 44 − 4/4) + 4)
:= 55 + ((5/5 + 5) × (55 + 5)/(5 + 5))
:= (5 + 5)/5 × ((5 − 5/5)5 − 55)
:= ((5/5 + 5)5 /(5 − 5/5)) − 5/5
:= 6 × 6 × (66 − 6 − 6) − 66/6
:= 6 × 6 × (66 − 6 − 6) − 6
:= 6 × 6 × (66 − 6 − 6) − 6/6
:= 7 + ((77 × (77/7 + 7 + 7)) + 7/7)
:= 77 + ((7 × (7 × 7 × 7 − 77)) − 7/7)
:= 7 + ((((7 + 7)/7 − 7) + 7 × 7)(7+7)/7 )
:= 8 + (88/8 × (888/8 + 8 × 8))
:= ((8 + 8)/8) × ((88 × 88 + 8)/8)
:= 8 + ((8 − 8/8 + 8) × (8 × (8 + 8) + 8/8))
:= (9 + 9) × (99 + 9) − 99/9
:= (9/9 + 9 + 9) × (999/9 − 9)
:= (9 + 9) × (99 + 9) − 9/9
Ï 1934 := (1 + 1)11 − (1 + 1 + 1 + 111)
Ï 1939 := 1 + (1 + ((1 + 1)11 − 111))
Ï 1944 := (1 + 1) × ((1 + 11) × (11 − 1 − 1)1+1 )
:= (2 × 22)2 − 2
:= 2 + ((2 × 22)2 + 2/2)
:= 2 × (2 × (222 + 2))
:= (((3 × (3 + 3))3 − 3)/3) − 3 × 3
:= (((3 × (3 + 3))3 + 3)/3) − (3 + 3)
:= 3 × 3 × (3 + 3)3
:= 44 × 44 − (4 + 4)/4
:= 4 + (44 × 44 − 4/4)
:= 4 + (44 × 44 + 4)
5
5
:= ((5/5 + 5) /(5 − 5/5)) − 5
:= (5/5 + 5)5 /(5 − 5/5)
:= 6/6 + (6 × 6 × (66 − 6 − 6) − 6)
:= 6 × (6 × (66 − 6 − 6))
:= 7 + (((7/7 + 7 + 7) × ((7 + 7)/7) ) + 7)
:= 77 + (7 × (7 × 7 × 7 − 77))
:= (7 − 7/7) × ((77/7 + 7)(7+7)/7 )
:= ((8 + 8)/8) × ((88 × 88 − 8)/8)
:= 8 + (((8 + 8) × (8 × (8 + 8) − 8)) + (88/8))
:= 8 + (88 × (88 + 88)/8)
:= (9 + 9) × (99 + 9) − 9/9 − 9
:= 99 + (9999/9 + 9 × 9 × 9)
:= (9 + 9) × (99 + 9)
:= ((5/5 + 5) /(5 − 5/5)) − 5 − 5
:= ((6 − 66)/6) + 6 × 6 × (66 − 6 − 6)
7
Ï 1940 := 1 + (1 + (1 + ((1 + 1)11 − 111)))
Ï 1935 := (1 + 1)11 − (1 + 1 + 111)
Ï 1945 := 1 + ((1 + 1) × ((1 + 11) × (11 − 1 − 1)1+1 ))
:= (2 × 22)2 − 2/2
:= 2 + ((2 × 22)2 + 2)
:= 2/2 + (2 × (2 × (222 + 2)))
:= 3 × (3 × (3 + 3)3 − 3)
:= (((3 × (3 + 3))3 − 3)/3) − 3
:= ((3 × (3 + 3))3 + 3)/3
:= 44 × 44 − 4/4
:= 4 + 44 × 44
:= 4 + ((44 × 44 + 4/4) + 4)
:= 5 + ((55 × ((5 × 5 + 5) + 5)) + 5)
:= 5 + (((55 × ((5 × 5 + 5) + 5)) + 5) + 5)
:= 55 − (55 /5 + 555)
:= (6 × 6/(6 + 6) + 6) × (6 × 6 × 6 − 6/6)
:= (6 + 6)/6 + (6 × 6 × (66 − 6 − 6) − 6)
:= 6/6 + 6 × 6 × (66 − 6 − 6)
7
:= (7/7 + 7 + 7) × (((7 + 7)/7) + 7/7)
:= 7/7 + ((7 × (7 × 7 × 7 − 77)) + 77)
:= 7 + (((7 × (7 × 7 × 7 − 77)) − 7/7) + 77)
:= (8 − 8/8 + 8) × (8 × (8 + 8) + 8/8)
:= ((8 + 8)/8) × (88 × 88 + 8 + 8)/8
:= 8 + ((88 × (88 + 88)/8) + 8/8)
:= (9 + 9) × (99 + 9) − 9
:= (99/9 + 9) × (99 − ((9 + 9)/9))
:= 9/9 + (9 + 9) × (99 + 9)
Ï 1941 := 1 + (1 + (1 + (1 + ((1 + 1)11 − 111))))
Ï 1936 := (1 + 1)11 − (1 + 111)
Ï 1946 := 11 + ((1 + 1)11 − (1 + 1 + 111))
2
2
:= 2 + (((2 × 22) + 2/2) + 2)
:= (2 × 22)
:= 2 + (2 × (2 × (222 + 2)))
3
3
:= 3 + (((3 × (3 + 3)) − 33)/3)
:= (3 × 3 × (3 + 3) ) − 3
:= 3 + (((3 × (3 + 3))3 − 3)/3)
:= 44 × 44
:= 4 + (44 × 44 + 4/4)
:= 44 × 44 + (44 − 4)/4
:= 5 + ((55 − (55/5))(5+5)/5 )
:= 55 + (((5 − 55 )/5) − 555)
:= (55 − (55/5))
(5+5)/5
:= ((((6 + 6)/6) + 6 × 6) + 6)
(6+6)/6
:= ((6 − (6
6−6/6
)) + 6)/((6 + 6)/6 − 6)
:= (6 + 6)/6 + 6 × 6 × (66 − 6 − 6)
7
(7+7)/7
:= 7 + ((((7/7 + 7 + 7) × ((7 + 7)/7) ) + 7) + 7)
:= 7 + ((7 × (7 × 7 × 7 − 77)) + 77)
:= 88 × (88 + 88)/8
:= 8 + ((88/8 × (888/8 + 8 × 8)) + 8)
:= 8 + (((8 + 8)/8) × ((88 × 88 + 8)/8))
:= 9/9 + ((9 + 9) × (99 + 9) − 9)
:= (9 + 9) × (99 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 + 9) × (99 + 9)
:= (((7 + 7)/7 − 7) + 7 × 7)
Ï 1942 := 1 + (1 + (1 + (1 + (1 + ((1 + 1)11 − 111)))))
Ï 1937 := (1 + 1)11 − 111
Ï 1947 := 11 + ((1 + 1)11 − (1 + 111))
:= 2/2 + (2 × 22)2
:= 2 + (((2 × 22)2 + 2) + 2)
:= 22/2 + (2 × 22)2
:= (((3 × (3 + 3))3 − 3)/3) − (3 + 3)
:= ((3 × (3 + 3))3 − (3 + 3))/3
:= 3 + (3 × 3 × (3 + 3)3 )
:= 4/4 + 44 × 44
:= 4 + (44 × 44 + (4 + 4)/4)
:= 44/4 + 44 × 44
5
:= 55 + ((((5 − 55 ) + 5)/5) − 555)
:= ((5/5 + 5) /(5 − 5/5)) − (5 + 5)/5
:= 5 + (((5 + 5)/5 + 5) × (5 × 55 + 5/5))
:= 6 × 6 × (66 − 6 − 6) − (6 + 6)/6
:= 6 × 6 × (66 − 6 − 6) − 6/6 − 6
:= 66/6 × (666/6 + 66)
7
:= (7 × ((7 − 7 × 7) + 7)) + (((7 + 7 + 7)/7) )
:= 77/7 × (((7 + 7)/7)7 + 7 × 7)
:= 8/8 + (88 × (88 + 88)/8)
:= 8 + (((8 + 8)/8) × ((88 × 88 − 8)/8))
:= 88/8 × ((88 + 88) + 8/8)
:= (9 + 9)/9 + ((9 + 9) × (99 + 9) − 9)
:= (9 + 9) × (99 + 9) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 + 9) × (99 + 9)
7
:= 7 + ((7 × 7 × (7 × 7 − 7)) − ((7 + 7)/7) )
139
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1953 := 1 + ((1 + 111+1 ) × (1 + 1)1+1+1+1 )
Ï 1948 := 11 + ((1 + 1)11 − 111)
2
2
2+2
:= 22 + (2 × 22)2
:= 2 × ((2 × (22 + 2)) + 2)
:= 2/2 + ((2 × 22) + 2
:= 3 + (((3 × (3 + 3))3 + 3)/3)
:= 3 × (3 × (3 + 3)3 + 3)
:= 3 + (((3 × (3 + 3))3 + 33)/3)
:= 4 + ((44 × 44 + 4) + 4)
:= 4 × 4 + (44 × 44 + 4/4)
:= 44/4 × (4 × 44 + (4 + 4)/4)
:= 5 + (((5/5 + 5)5 /(5 − 5/5)) − 5/5)
:= (5 − (5 + 5)/5) × ((55 + 5)/5 + 5 × 5)
:= 55 − ((5555 + 5)/5 + 55)
:= 6 + (6 × 6 × (66 − 6 − 6) − ((6 + 6)/6))
:= 6 + ((66/6) × (666/6 + 66))
:= 66/6 × ((666 + 6)/6 + 66)
:= ((7 − 777)/7) + (7 × 7 × (7 × 7 − 7))
:= 777 + (7 + 7) × (77 + 7)
:= ((7 + 7)/7) × ((7 + 7) × (77 − 7) − 7/7)
:= 8 + (((8 + 8)/8) × (88 × 88 + 8 + 8)/8)
:= 8/8 + ((8 + 8) × ((888 + 88)/8))
:= (8/8 + 88) × (88 + 88)/8
:= ((9 + 9)/9) × ((9 × (99 + 9)) + ((9 + 9)/9))
:= 9 + (9 + 9) × (99 + 9)
:= 9 + ((((9 + 9)/9)99/9 ) − 99)
Ï 1949 := 1 + (11 + ((1 + 1)11 − 111))
)
Ï 1958 := (1 + 1) × (11 × (111 − 11 − 11))
Ï 1954 := 1 + (1 + ((1 + 111+1 ) × (1 + 1)1+1+1+1 ))
Ï 1959 := 11 + (11 + ((1 + 1)11 − 111))
:= 2 + ((2 × 22)2 + 22/2)
:= 2 + ((2 × 22)2 + 22+2 )
:= 22 + ((2 × 22)2 + 2/2)
:= 3 + ((((3 × (3 + 3))3 − 3)/3) + 3)
:= 3 × 3 + (((3 × (3 + 3))3 + 3)/3)
:= 3 + ((3 × (3 × (3 + 3)3 + 3)) + 3)
:= 4 + (((44 × 44 + 4/4) + 4) + 4)
:= 4 × 4 + (44 × 44 + (4 + 4)/4)
:= ((4 + 4) × (44 − 44/4)) − 4/4
5
5
:= 5 + ((5/5 + 5) /(5 − 5/5))
:= 5 + (((5/5 + 5) /(5 − 5/5)) + 5)
:= 55 − (5555/5 + 55)
:= 6 + (6 × 6 × (66 − 6 − 6) − 6/6)
:= ((66 − 6)/6) + 6 × 6 × (66 − 6 − 6)
:= ((6 × (666 − (6 + 6))) − 6)/((6 + 6)/6)
7
:= (7 × (7 × (7 × 7 − 7) − (7 + 7))) − 77/7
:= 7 + (77/7 × (((7 + 7)/7) + 7 × 7))
:= (7 × (7 × (7 × 7 − 7) − (7 + 7))) − 7/7
:= (8 × (8 + 8) × (8 + 8)) − (88/8 + 88)
:= ((8 + 8)/8) × (((88 × 88 + 8)/8) + 8)
:= (8 × (8 + 8) × (8 + 8)) − (8/8 + 88)
:= 9 + ((9 + 9) × (99 + 9) + 9/9)
:= ((99 − 9/9) × (99/9 + 9)) − 9/9
99/9
:= (((9 + 9)/9)
) − 99
Ï 1955 := (1 + 1)11 − (((1 + 1)11−1 − 1)/11)
Ï 1950 := 1 + (1 + (11 + ((1 + 1)11 − 111)))
Ï 1960 := (11 − 1) × (1 + 1 + 1 + 11)1+1
:= 2 + (2 × ((2 × (222 + 2)) + 2))
:= 22 + ((2 × 22)2 − (2/2 + 2))
:= 2 + ((2 × 22)2 + 22)
:= 3 + ((3 × 3 × (3 + 3)3 ) + 3)
:= ((3 × (3 + 3))3 + 33)/3
:= 33 + (((3 × (3 + 3))3 − 33)/3)
:= 4 + (44 × 44 + (44 − 4)/4)
:= 4 + ((44 × 44 + 44/4) + 4)
:= (4 + 4) × (44 − 44/4)
:= (5 + 5 + 5) × (5 × 5 × 5 + 5)
:= 5 + ((5 + 5 + 5) × (5 × 5 × 5 + 5))
:= ((5 + 5)/5 + 5) × (5 × 55 + 5)
:= 666 + (6 × 6 × 6 × 6 − ((6 + 6)/6))
:= 6 + 6 × 6 × (66 − 6 − 6)
:= 66/6 + 6 × 6 × (66 − 6 − 6)
:= (7/7 + 77) × (77/7 + 7 + 7)
:= (7 − ((7 + 7)/7)) × (7 × (7 × 7 + 7) − 7/7)
:= 7 × (7 × (7 × 7 − 7) − (7 + 7))
:= ((8 + 8)/8) × (((88 × 88 − 8)/8) + 8)
:= 8 + (88/8 × ((88 + 88) + 8/8))
:= (8 × (8 + 8) × (8 + 8)) − 88
:= 9 + ((9 + 9) × (99 + 9) − ((9 + 9 + 9)/9))
:= 99/9 + (9 + 9) × (99 + 9)
:= (99 − 9/9) × (99/9 + 9)
Ï 1951 := 1 + (1 + (1 + (11 + ((1 + 1)11 − 111))))
Ï 1956 := (1 + 1) × (1111 − (1 + (11 × (1 + 11))))
Ï 1961 := 1 + (11 − 1) × (1 + 1 + 1 + 11)1+1
:= 2 + (((2 × 22)2 + 22/2) + 2)
:= 22 + ((2 × 22)2 − 2)
:= 2 + (((2 × 22)2 + 22) + 2/2)
:= 3 + ((((3 × (3 + 3))3 + 3)/3) + 3)
:= 3 + (3 × (3 × (3 + 3)3 + 3))
:= 3 + ((((3 × (3 + 3))3 + 33)/3) + 3)
:= 4 + (44 × 44 + 44/4)
:= 4 + (44 × 44 + 4 × 4)
:= 4 + ((((4 − 4/4) + 4)4 ) − 444)
:= 5/5 + ((5 + 5 + 5) × (5 × 5 × 5 + 5))
:= 5 + (((5 + 5 + 5) × (5 × 5 × 5 + 5)) + 5/5)
:= 5 × 5 + ((55 − (55/5))(5+5)/5 )
:= 6 + (6 × 6 × (66 − 6 − 6) + 6/6)
:= 6 + (6 × 6 × (66 − 6 − 6) + 6)
:= 666 + (6 × 6 × 6 × 6 − 6/6)
:= (77/7 − 7) × (7 × (77 − 7) − 7/7)
:= 7/7 + (7 × (7 × (7 × 7 − 7) − (7 + 7)))
:= 8 × 8 + ((8/8 + 8 + 8) × 888/8)
:= (((8 × 8 × 8 × 8) − 8)/((8 + 8)/8)) − 88
:= 8/8 + ((8 × (8 + 8) × (8 + 8)) − 88)
:= 9 + ((9 + 9) × (99 + 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + (9 + 9) × (99 + 9)
:= 9 + (((9 + 9) × (99 + 9) − 9/9) + 9)
:= 7 + ((7 − 7/7) × ((77/7 + 7)
(7+7)/7
))
Ï 1952 := (1 + 111+1 ) × (1 + 1)1+1+1+1
2
Ï 1957 := (1 + 1)11 − (1 + (11 − 1) × (11 − 1 − 1))
2
Ï 1962 := (1 + 1) × ((11 − 1 − 1) × (111 − 1 − 1))
:= 2 × (2 × (22 + 2 + 2))
:= 22 + ((2 × 22) − 2/2)
:= 2 + (((2 × 22)2 + 22) + 2)
:= 3 × 3 + (((3 × (3 + 3))3 − 3)/3)
:= 3 + ((((3 × (3 + 3))3 + 3)/3) + 3 × 3)
:= 3 × ((3 × (3 + 3)3 + 3) + 3)
:= 4 × (444 + 44)
5
4
:= 4 + ((44/((4 + 4)/4)) + 44 × 44)
5
:= (((4 − 4/4) + 4) ) − 444
:= ((5 + 5)/5) × ((55 + 5/5) + 5)
:= 5 + (((5 + 5)/5) × ((55 + 5/5) + 5))
:= (55 − 5)/(5 + 5) + (55 × (5 × 5 + 5))
:= 6 + (6 × 6 × (66 − 6 − 6) + ((6 + 6)/6))
:= 6 + ((6 × 6 × (66 − 6 − 6) + 6/6) + 6)
:= 666 + 6 × 6 × 6 × 6
:= 777 + ((7 + 7) × (77 + 7) − 7/7)
:= 7 + ((7/7 + 77) × (77/7 + 7 + 7))
:= ((7 + 7)/7) + (7 × (7 × (7 × 7 − 7) − (7 + 7)))
:= (8 + 8) × ((888 + 88)/8)
:= (88/8 + 8) × (888/8 − 8)
:= (8 + 8)/8 + ((8 × (8 + 8) × (8 + 8)) − 88)
:= 9 + ((9 + 9) × (99 + 9) − 9/9)
:= (9/9 + 9 + 9) × (((999 + 9)/9) − 9)
:= 9 + ((9 + 9) × (99 + 9) + 9)
140
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1963 := 1 + ((1 + 1) × ((11 − 1 − 1) × (111 − 1 − 1)))
2
Ï 1968 := 1 + ((1 + 1)11 − (11 − 1 − 1)1+1 )
2
2+2
Ï 1973 := 1 + (1 + ((11 − 1 − 1) × (((1 + 1) × (111 − 1)) − 1)))
:= 2/2 + ((2 × 22)2 + (2 + 2 + 2)2 )
:= 2 + ((((2 × 22) + 22) + 2/2) + 2)
:= 2 × (2 × 22 + 2
:= 3/3 + (3 × ((3 × (3 + 3)3 + 3) + 3))
:= (3 × (3 × ((3 + 3)3 + 3))) − 3
:= 3 + ((((3 × (3 + 3))3 − 3)/3) + 33 )
:= 4 × 4 + (44 × 44 + 44/4)
:= 4 × ((444 + 44) + 4)
:= 4/4 + (((4 + 4) × (44 − 4)) − 44)
:= (55 + 5)/(5 + 5) + (55 × (5 × 5 + 5))
:= (55/5 + 5) × (5 × 5 × 5 − ((5 + 5)/5))
:= (5 × (5 × 55 − 5)) + (55 − 5 − 5)/5
:= 6/6 + (6 × 6 × 6 × 6 + 666)
:= 66 × (6 × 6 − 6) − 6 − 6
:= 66 × (6 × 6 − 6) − 6/6 − 6
:= 7 + ((77/7 − 7) × (7 × (77 − 7) − 7/7))
:= (7 × 7 − 7/7) × (7 × 7 − (7/7 + 7))
:= 777/7 + (7 × (7 × 7 × 7 − 77))
:= 8 + ((88/8 × ((88 + 88) + 8/8)) + 8)
:= 8 + ((8 × (8 + 8) × (8 + 8)) − 88)
:= (8 × ((8 + 8) × (8 + 8) − 8)) − 88/8
:= 9 + (((9 + 9) × (99 + 9) + 9/9) + 9)
Ï 1964 := (1 + 1) × (1 + ((11 − 1 − 1) × (111 − 1 − 1)))
2
:= 2 + ((((2 × 22) + 22) + 2) + 2)
3
)
99/9
:= 9/9 + ((((9 + 9)/9)
) − 9 × 9)
:= 9 + (((9 + 9) × (99 + 9) + (99/9)) + 9)
Ï 1969 := 11 × (11 + ((1 + 1 + 11)1+1 − 1))
Ï 1974 := (1 + 1) × ((11 − 1)1+1+1 − (1 + 1 + 11))
2
:= 22 + ((2 × 22) + 22/2)
3
:= 2 + ((2 × 22)2 + (2 + 2 + 2)2 )
3
:= 3 + (((3 × (3 + 3)) − (3 + 3))/3)
:= 3 + (3 × (3 × ((3 + 3)3 + 3)))
:= 44 + (44 × 44 − 44/4)
:= 44 + (44 × 44 − ((4 + 4)/4 + 4))
:= 5 + (5 − (5555/5 + 55))
:= 5 × 5 + ((5/5 + 5)5 /(5 − 5/5))
:= (55 − 5)/5 + (5 × (5 × 55 − 5))
:= 666 + (6 × 6 × 6 × 6 + ((6 + 6)/6))
:= 66 × (6 × 6 − 6) − 66/6
:= 3 × 3 + (((3 × (3 + 3)) + 33)/3)
4
:= 4 + ((4 + 4) × (4 − 44/4))
5
:= 66 × (6 × 6 − 6) − 6
7
:= 7 × 7 + ((7/7 + 7 + 7) × ((7 + 7)/7) )
:= (7 − 7/7) × (7 × 7 × 7 − (7 + 7))
:= (((8 × 8 × 8 × 8) + 8)/((8 + 8)/8)) − 88
:= 8 + (((8 × (8 + 8) × (8 + 8)) − 88) + 8/8)
:= (8 − 88)/8 + (8 × ((8 + 8) × (8 + 8) − 8))
:= 9 + ((9 + 9) × (99 + 9) + (99/9))
:= 9 + ((99 − 9/9) × (99/9 + 9))
:= 9 × 9 + ((9 + 9) × 99 + 999/9)
77/7
:= (((7 + 7)/7)
) − (77 + 7)
Ï 1965 := (1 + 1)11 − (1 + (1 + (11 − 1 − 1)1+1 ))
Ï 1970 := 111 + (11 × (1 + 1 + 11)1+1 )
2
2+2
:= 22 + ((2 × (2 × (222 + 2))) − 2/2)
:= 2 + ((2 × 22) + 2 × 2
:= ((3 + 3) × 333) − 33
:= 3 + (((3 × (3 + 3)) − 3)/3)
4
:= 4 + (((((4 − 4/4) + 4) ) − 444) + 4)
3
Ï 1975 := 1111 + (((1 + 11)1+1+1 )/(1 + 1))
)
:= (2 × 22)2 + (2 × (22 − 2) − 2/2)
3
:= 3 + ((3 × (3 × ((3 + 3)3 + 3))) + 3/3)
4
:= 4 + ((4444/((4 + 4)/4)) − 4 )
5
:= 44 + (44 × 44 − (4/4 + 4))
5
:= 5 + (((5 + 5)/5 + 5) × (5 × 55 + 5))
:= (5 × ((5 − 5/5) − 5)) − 5
:= ((6 × 666) − 66)/((6 + 6)/6)
:= (6 − 6/6) × (6 × 66 − ((6 + 6)/6))
:= 77 + (7777/7 + 777)
:= ((77 − 7)/7) × ((7 + 7) × (7 + 7) + 7/7)
:= 7 + ((7 × 7 − 7/7) × (7 × 7 − (7/7 + 7)))
:= 8 + ((88/8 + 8) × (888/8 − 8))
:= 8 + (((8 × (8 + 8) × (8 + 8)) − 88) + ((8 + 8)/8))
:= (8 × ((8 + 8) × (8 + 8) − 8)) − (8/8 + 8)
:= 9 + ((9 + 9) × (99 + 9) + ((99 + 9)/9))
:= 9 × 9 × (9 + 9) + (((9 + 9)/9)9 )
:= 9 + ((9 + 9) × (99 + 9) + ((99 + 99)/9))
Ï 1966 := (1 + 1)11 − (1 + (11 − 1 − 1)1+1 )
:= 22 + (2 × (2 × (222 + 2)))
:= 3/3 + (((3 + 3) × 333) − 33)
Ï 1971 := (11 − 1 − 1) × (((1 + 1) × (111 − 1)) − 1)
2
:= 6/6 + (66 × (6 × 6 − 6) − 6)
Ï 1976 := (1 + 1) × ((11 − 1)1+1+1 − (1 + 11))
:= 2 × ((2 × 222 − 2) + 22)
:= 2 + (((2 × 22) + 22/2) + 22)
3
:= 33 + (((3 × (3 + 3))3 − 3)/3)
:= 3 × (3 × ((3 + 3) + 3))
4
:= (4444/((4 + 4)/4)) − 44
:= 5 × (5 × (5 × 5 + 55) − 5)
4
:= 4 + ((4 × (4 + 4)) − (4 − 4/4) )
5
:= 44 + (44 × 44 − 4)
5
:= 5 + ((((5 + 5)/5 + 5) × (5 × 55 + 5)) + 5/5)
:= 5/5 + ((5 × ((5 − 5/5) − 5)) − 5 )
:= 5/5 + (5 × (5 × (5 × 5 + 55) − 5))
:= 66 × (6 × 6 − 6) − ((6 + 6)/6 + 6 + 6)
:= 6 + (((6 × 666) − 66)/((6 + 6)/6))
:= (6 + 6)/6 + (66 × (6 × 6 − 6) − 6)
:= 7 + ((7 × (7 × (7 × 7 − 7) − (7 + 7))) − 7/7)
:= (((7 + 7)/7)
:= 8 + ((8/8 + 88) × (88 + 88)/8)
:= 88/8 + ((8 × (8 + 8) × (8 + 8)) − 88)
:= (8 × ((8 + 8) × (8 + 8) − 8)) − 8
:= ((9 + 9)/9) × ((9 × (99 + 9)) + (99/9))
:= 999 + (9 × (99 + 9))
:= 9 + ((((9 + 9)/9)99/9 ) − 9 × 9)
Ï 1967 := (1 + 1)11 − (11 − 1 − 1)1+1
:= 2
22/2
− ((2/2 + 2)
3
2+2
:= (77 − 7/7) × ((77 + 7)/7 + 7 + 7)
) − 77
Ï 1972 := 1 + ((11 − 1 − 1) × (((1 + 1) × (111 − 1)) − 1))
2
)
:= 2 × ((2 × (22 − 2)) + 22)
3
:= 3 + ((((3 × (3 + 3)) − 3)/3) − 3)
4
77/7
4
3
3
:= 3 + (((3 × (3 + 3)) + 3)/3)
4
Ï 1977 := ((1 + 1) × ((11 − 1)1+1+1 − 11)) − 1
:= 2/2 + ((2 × 22)2 + 2 × (22 − 2))
:= 33 + (3 × 3 × (3 + 3)3 )
:= (4 × (4 + 4)) − (4 − 4/4)
:= ((4 + 4) × (4 − 4)) − 44
:= 44 + ((44 × 44 − 4) + 4/4)
:= ((5 + 5)/5 + 5) × ((5 × 55 + 5/5) + 5)
:= (((55 + 5)/5) + 5) × (555/5 + 5)
:= (5 + 5)/5 + (5 × (5 × (5 × 5 + 55) − 5))
:= 66 × (6 × 6 − 6) − (6/6 + 6 + 6)
:= 66 × (6 × 6 − 6) − ((6 + 6)/6 + 6)
:= 66 × (6 × 6 − 6) − 6 × 6/(6 + 6)
77/7
:= 7 + (7 × (7 × (7 × 7 − 7) − (7 + 7)))
:= 7/7 + ((((7 + 7)/7)
:= 8 + ((8 × (8 + 8) × (8 + 8)) − (8/8 + 88))
:= (8 × ((8 + 8) × (8 + 8) − 8)) − (88 + 8)/8
:= 8/8 + ((8 × ((8 + 8) × (8 + 8) − 8)) − 8)
:= 9/9 + ((9 × (99 + 9)) + 999)
:= (99 × (99/9 + 9)) − (9 + 9 + 9)/9
99/9
:= (((9 + 9)/9)
)−9×9
141
) − 77)
:= 7 + (((77 − 7)/7) × ((7 + 7) × (7 + 7) + 7/7))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1978 := (1 + 1) × ((11 − 1)1+1+1 − 11)
Ï 1983 := 1 + ((1 + 1) × (1 + ((11 − 1 − 1) × (111 − 1))))
Ï 1988 := (1 + 1)11 − (111+1 − 1)/(1 + 1)
:= (2 × 22) + ((2 × 22)2 − 2)
:= 2 + (((2 × 22)2 + 2 × 22) + 2/2)
:= 2 × ((2 × (222 + 2)) + 22)
:= 33 + (((3 × (3 + 3))3 + 3)/3)
:= 3 + (33 × (33 + 33))
:= 33 + (((3 × (3 + 3))3 + 33)/3)
:= 44 + (44 × 44 − (4 + 4)/4)
:= (44 × (4 + 4)) − (44 + 4)/4
:= 4 + ((4 + 4) × (44 − 4 − 4))
:= (55 × (55/5 + 5 × 5)) − (5 + 5)/5
:= (((5 + 5)/5)55/5 ) − (55 + 5 + 5)
:= (((5 + 5)/5)55/5 ) − (55 + 5)
:= 66 × (6 × 6 − 6) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + 66 × (6 × 6 − 6)
:= 6 + (66 × (6 × 6 − 6) + ((6 + 6)/6))
:= 7 + ((((7 + 7)/7)77/7 ) − 77)
:= ((7 + 7)/7) + ((7 × 7 × (7 × 7 − 7)) − 77)
:= 7 + ((7 × 7 × (7 × 7 − 7)) − 77)
:= (8 + 8)/8 + ((8 × ((8 + 8) × (8 + 8) − 8)) − 8)
:= (8 × ((8 + 8) × (8 + 8) − 8)) − 8/8
:= 888 + ((8888 − 88)/8)
:= ((9 + 9)/9) × (999 − (9/9 + 9))
:= ((9 + 9 + 9)/9) + (99 × (99/9 + 9))
:= 9 + ((99 × (99/9 + 9)) − 9/9)
Ï 1979 := 1 + ((1 + 1) × ((11 − 1)1+1+1 − 11))
:= (2 × 22) + ((2 × 22)2 − 2/2)
Ï 1984 := (1 + 1)11 − ((1 + 1)(1+1)×(1+1+1) )
2
:= 2 × ((2 × 22 + 22) + 2)
3
Ï 1989 := (11 − 1 − 1) × ((1 + 1) × 111 − 1)
:= (2/2 + 2)2 × (222 − 2/2)
:= ((33/3)3 ) + 3 × (3 + 3)3
:= 3 + ((33 × (3 + 33)) + 3/3)
:= ((3 + 3) × 333) − 3 × 3
:= 44 + (44 × 44 − 4/4)
:= (4 + 4) × (44 − 4 − 4)
:= 4 + (((4 + 4) × (44 − 4 − 4)) + 4/4)
:= (55 × (55/5 + 5 × 5)) − 5/5
:= (55/5 + 5) × (5 × 5 × 5 − 5/5)
:= (((5 + 5)5−5/5 ) − 55)/5
:= 66 × (6 × 6 − 6) − 6/6
:= 6 + (66 × (6 × 6 − 6) − ((6 + 6)/6))
:= (66/6 + 6) × (666/6 + 6)
7
:= (7 × 7 × (7 × 7 − 7)) − ((7 + 7)/7 + 77)
:= (((7 + 7 + 7)/7) ) − ((7 + 7) × (7 + 7) + 7)
:= 8 + (((8 × (8 + 8) × (8 + 8)) − 88) + (88/8))
:= 8 × ((8 + 8) × (8 + 8) − 8)
:= (8/8 + 8 + 8) × (8 × (8 + 8) − (88/8))
:= 9 × 99 + (9999/9 − (9 + 9))
:= 9 + (99 × (99/9 + 9))
:= (99 × (99/9 + 9)) − 9/9
Ï 1980 := (1 + 1) × ((11 − 1 − 1) × (111 − 1))
Ï 1985 := 1 + ((1 + 1)11 − ((1 + 1)(1+1)×(1+1+1) ))
2
:= 7 + (((7 × 7 × (7 × 7 − 7)) − 77) + 7/7)
Ï 1990 := 1 + ((11 − 1 − 1) × ((1 + 1) × 111 − 1))
:= 2 × (2 × 22 + 22)
:= 2/2 + ((2 × 22) + 2 × (22 + 2))
:= 2 + (2 × ((2 × (222 + 2)) + 22))
:= 33 × (33 + 33)
:= 3 + ((((33/3)3 ) + 3 × (3 + 3)3 ) + 3)
:= 3 + (((3 + 3) × 333) − 33/3)
2
:= 44 + 44 × 44
:= 4/4 + ((4 + 4) × (4 − 4 − 4))
:= 4 + (((4 + 4) × (44 − 4 − 4)) + (4 + 4)/4)
:= 55 × (55/5 + 5 × 5)
:= 5 + (55 × (55/5 + 5 × 5))
:= (5 × (5 − 5/5)5 ) − (55 + 5)
:= 66 × (6 × 6 − 6)
:= 6 + (66 × (6 × 6 − 6) − 6/6)
:= (6 − 6/6) × (((6 + 6)/6) + 6 × 66)
4
77/7
:= (77/7 + 7) × (777 − 7)/7
:= 7 + (((((7 + 7)/7)
:= ((((8 + 8)/8) + 8) + 8) × (888 − 8)/8
:= 8/8 + (8 × ((8 + 8) × (8 + 8) − 8))
:= 99 × (99/9 + 9)
Ï 1981 := 1 + ((1 + 1) × ((11 − 1 − 1) × (111 − 1)))
99/9
:= 9 + (((((9 + 9)/9)
) − 77) + 7)
) − 9 × 9) + 9)
Ï 1986 := (1 + 1)11 − (1 + ((1 + 111+1 )/(1 + 1)))
:= ((7 + 7)/7)7 + (7 × (7 × 7 × 7 − 77))
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) − ((8 + 8)/8))
:= 9 + ((99 × (99/9 + 9)) + 9/9)
Ï 1991 := 11 × (1 + (11 + (1 + 1 + 11)1+1 ))
:= 2/2 + ((2 × 22)2 + 2 × 22)
:= 222 + (2 × 22 − 2)2
:= 2 + ((2/2 + 2)2 × (222 − 2/2))
:= 3/3 + (33 × (33 + 33))
:= 3 + ((33 × (33 + 33)) + 3)
:= 33/3 + (33 × (33 + 33))
:= 44 + (44 × 44 + 4/4)
:= (4 + 4)/4 + ((4 + 4) × (44 − 4 − 4))
:= 44 + (44 × 44 + 44/4)
:= 5/5 + (55 × (55/5 + 5 × 5))
:= 5 + ((55 × (55/5 + 5 × 5)) + 5/5)
:= ((((5 + 5)5−5/5 ) + 5)/5) − 5 − 5
:= 6/6 + 66 × (6 × 6 − 6)
:= 6 + 66 × (6 × 6 − 6)
:= 66/6 + 66 × (6 × 6 − 6)
:= (7 × 7 × (7 × 7 − 7)) − 77
:= (7 − 7/7) × (7 × 7 × 7 − (77 + 7)/7)
:= (((7 + 7 + 7)/7)7 ) − (7 + 7) × (7 + 7)
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) − (88/8))
:= (8 + 8)/8 + (8 × ((8 + 8) × (8 + 8) − 8))
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) − 8/8)
:= 9/9 + (99 × (99/9 + 9))
:= 99 + (((9 − 9/9) + 9) × 999/9)
:= 99/9 + (99 × (99/9 + 9))
Ï 1982 := (1 + 1) × (1 + ((11 − 1 − 1) × (111 − 1)))
Ï 1987 := (1 + 1)11 − ((1 + 111+1 )/(1 + 1))
2
Ï 1992 := (1 + 1)11 − (1 + 111)/(1 + 1)
:= 2 + ((2 × 22)2 + 2 × 22)
:= 2/2 + ((2 × 22 − 2) + 222)
:= 2 × (((2 × (222 + 2)) + 22) + 2)
:= 3 + (((33/3)3 ) + 3 × (3 + 3)3 )
:= ((3 + 3) × 333) − 33/3
:= (3 + 3) × (333 − 3/3)
:= 44 + (44 × 44 + (4 + 4)/4)
:= 4 + ((44 × (4 + 4)) − (44 + 4)/4)
:= 4 + (((4 + 4) × (44 − 4 − 4)) + 4)
5−5/5
:= (5 + 5)/5 + (55 × (55/5 + 5 × 5))
:= (((5 + 5)
:= (6 + 6)/6 + 66 × (6 × 6 − 6)
:= 6 + (66 × (6 × 6 − 6) + 6/6)
:= 6 + (66 × (6 × 6 − 6) + 6)
:= 7/7 + ((7 × 7 × (7 × 7 − 7)) − 77)
) − (55 + 5 + 5))/5
:= ((55 + 5)/5) × (555/5 + 55)
:= 7 + ((77/7 + 7) × (777 − 7)/7)
:= (7 − 7/7) × (7 × 7 × 7 − (77/7))
:= (8 × ((8 + 8) × (8 + 8) − 8)) − (8 + 8)/8
:= 88/8 + ((8 × ((8 + 8) × (8 + 8) − 8)) − 8)
:= 8 + (8 × ((8 + 8) × (8 + 8) − 8))
:= (9 + 9)/9 + (99 × (99/9 + 9))
:= 999 + (999 − (99/9))
:= 99 + ((9 + 9) × 99 + 999/9)
142
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 1993 := (1 + 1)11 − (111 − 1)/(1 + 1)
Ï 1998 := (1 + 1) × (111 × (11 − 1 − 1))
Ï 2003 := 1 + ((1 + 1) × (1 + (11 − 1)1+1+1 ))
:= 2 × 222 + ((222/2 + 2)/2)
:= 222 × (2/2 + 2)2
:= (2 × 22 + 2/2)2 − 22
:= 3/3 + ((3 + 3) × (333 − 3/3))
:= (3 + 3) × 333
:= 3 + (((3 + 3) × 333 − 3/3) + 3)
:= (44 × (4 + 4)) − (44/4 + 44)
:= 44 × 44 + (4 − 4 − 4)/4
:= 44 × (4 + 4) − (44 + 4/4)
:= (((5 + 5)/5)55/5 ) − 55
:= (((5 + 5)5−5/5 ) − (5 + 5))/5
:= 5 + ((((5 + 5)5−5/5 ) − (5 + 5))/5)
:= 6 + ((66 × (6 × 6 − 6) + 6/6) + 6)
:= 6 × 666 × 6/(6 + 6)
:= 6 + ((66 × (6 × 6 − 6) + (66/6)) + 6)
:= 7777/7 + 7 × (77 + 7 × 7)
:= (77/7 + 7) × 777/7
:= 7/7 + (7 × (7 × (7 × 7 − 7) − 7) − 7)
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) + 8/8)
:= 888 + (8888 − 8)/8
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) + (88/8))
:= 9 × 99 + (9999/9 − 9)
:= (9 + 9) × 999/9
:= 9 × 99 + (9999 + 9)/9
4
Ï 1994 := 1 + (1 + 1)11 − (111 − 1)/(1 + 1)
Ï 1999 := ((1 + 1) × (11 − 1)1+1+1 ) − 1
Ï 2004 := (1 + 1) × (1 + 1 + (11 − 1)1+1+1 )
:= (222 × (2/2 + 2)2 ) − 2 − 2
:= 2/2 + (222 × (2/2 + 2)2 )
:= 222/2 − (2 × 22)
:= ((3 + 3) × 333) − (3/3 + 3)
:= 3/3 + ((3 + 3) × 333)
:= 3 + ((3 + 3) × 333 + 3)
:= (4 + 4)/4 × (4 × (4 − 4) − 44/4)
:= 44 × 44 + ((4 − 4)/4)
:= 44 × (4 + 4) − 44
:= ((((5 + 5)5−5/5 ) − 5)/5) − 5
:= (((5 + 5)5−5/5 ) − 5)/5
:= 5 + ((((5 + 5)5−5/5 ) − 5)/5)
:= 6 + ((66 × (6 × 6 − 6) + ((6 + 6)/6)) + 6)
:= 6/6 + (6 × 666 × 6/(6 + 6))
:= 6 + (6 × 666 × 6/(6 + 6))
4
4
77/7
:= (7 + 7)/7 + (7 × (7 × (7 × 7 − 7) − 7) − 7)
:= (7 × (7 × (7 × 7 − 7) − 7)) − (7/7 + 7 + 7)
:= (((7 + 7)/7)
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) + ((8 + 8)/8))
:= 888 + 8888/8
:= ((8 × 8 × 8 × 8) − 88)/((8 + 8)/8)
:= ((9 + 9)/9) × (999 − ((9 + 9)/9))
:= 9/9 + (999 + 999)
:= 9 × 99 + (((9999 + 9) + 9)/9)
Ï 1995 := (1 + 1 + 1) × ((111+1+1 − 1)/(1 + 1))
)−7×7
Ï 2000 := (1 + 1) × (11 − 1)1+1+1
Ï 2005 := 1 + ((1 + 1) × (1 + 1 + (11 − 1)1+1+1 ))
:= (((2 × 22) + 2)2 ) − (22/2)2
:= 2222 − 222
:= 2 + ((2 × 22 + 2/2)2 − 22)
:= ((3 + 3) × 333) − 3
:= 3 + (((3 + 3) × 333) − 3/3)
:= 3 + (((3 + 3) × 333 + 3/3) + 3)
4
:= 44/4 + ((4 + 4) × (4 − 4 − 4))
:= 4 × 4 × 4 + 44 × 44
:= 4/4 + (44 × (4 + 4) − 44)
:= (5 × (5 − 5/5)5 ) − 55
:= 5 × 5 × (5 × 5 + 55)
:= 5 + (5 × 5 × (5 × 5 + 55))
:= ((6 × 666) − 6)/((6 + 6)/6)
:= (6 + 6)/6 + (6 × 666 × 6/(6 + 6))
:= (6 − 6/6) × ((6 × 66 − 6/6) + 6)
:= (7 × (7 × (7 × 7 − 7) − 7)) − (7 + 7)
:= 7/7 + ((((7 + 7)/7)77/7 ) − 7 × 7)
:= 7 + (77/7 + 7) × 777/7
:= 88/8 + (8 × ((8 + 8) × (8 + 8) − 8))
:= 8 + ((8 × ((8 + 8) × (8 + 8) − 8)) + 8)
:= 8 + (((8888 − (8 + 8))/8) + 888)
:= ((9 + 9)/9) × (999 + 9/9)
:= 9 + (((9 + 9)/9) × (999 − 9/9))
:= 999 + (999 − ((9 + 9 + 9)/9))
Ï 1996 := (1 + 1) × ((11 − 1)1+1+1 − (1 + 1))
Ï 2001 := 1 + ((1 + 1) × (11 − 1)1+1+1 )
Ï 2006 := (1 + 1) × (1 + 1 + 1 + (11 − 1)1+1+1 )
:= (222 × (2/2 + 2)2 ) − 2
:= 2/2 + 2222 − 222
:= 2 + (222/2 − (2 × 22))
:= 3/3 + (((3 + 3) × 333) − 3)
:= 3 + (3 + 3) × 333
:= 3 × 3 + ((3 + 3) × 333 − 3/3)
:= 4 × 4 + (44 × 44 + 44)
:= 44 × 44 + (4 + 4)/4
:= (4 + 4)/4 + (44 × (4 + 4) − 44)
:= ((((5 + 5)5−5/5 ) + 5)/5) − 5
:= (((5 + 5)5−5/5 ) + 5)/5
:= 5 + ((((5 + 5)5−5/5 ) + 5)/5)
:= 6 + (66 × (6 × 6 − 6) + ((66 − 6)/6))
:= ((6 × 666) + 6)/((6 + 6)/6)
:= (((6 + 6)/6)66/6 ) − (6 × 6 + 6)
:= 7/7 + ((7 × (7 × (7 × 7 − 7) − 7)) − (7 + 7))
:= 7 × (7 × (7 × 7 − 7) − 7) − (7/7 + 7)
:= 7 + ((((7 + 7)/7)77/7 ) − 7 × 7)
:= ((88 + 8)/8) + (8 × ((8 + 8) × (8 + 8) − 8))
:= (8/8 − 88) × (8/8 − (8 + 8 + 8))
:= 8 + ((8888 − 8)/8 + 888)
:= ((9 + 9)/9) × (999 − 9/9)
:= 9 × 99 + ((9999 − 9)/9)
:= 9 + ((999 − 9/9) + 999)
4
Ï 1997 := ((1 + 1) × (111 × (11 − 1 − 1))) − 1
2
Ï 2002 := (1 + 1) × (1 + (11 − 1)1+1+1 )
Ï 2007 := (11 − 1 − 1) × (1 + (1 + 1) × 111)
:= (222 × (2/2 + 2) ) − 2/2
:= 2 + 2222 − 222
:= (2/2 + 2)2 × (222 + 2/2)
:= ((3 + 3) × 333) − 3/3
:= 3 + ((3 + 3) × 333 + 3/3)
:= 3 × 3 + (3 + 3) × 333
4
4
:= 44 × 44 + ((4 + 4)/4 − 4)
:= 44 × 44 + ((4 + 4 + 4)/4)
:= 4 + 44 × (4 + 4) − (44 + 4/4)
:= (((((5 + 5)5−5/5 ) + 5) + 5)/5) − 5
:= ((((5 + 5)5−5/5 ) + 5) + 5)/5
:= 5 + (((((5 + 5)5−5/5 ) + 5) + 5)/5)
:= 6 + (66 × (6 × 6 − 6) + (66/6))
:= 6/6 + (((6 × 666) + 6)/((6 + 6)/6))
:= 6 + (((6 × 666) + 6)/((6 + 6)/6))
:= 77 + ((7/7 + 7 + 7) × ((7 + 7)/7) )
:= 7 × (7 × (7 × 7 − 7) − 7) − 7
:= 7 × (7 × (7 × 7 − 7) − 7) − (7 + 7)/7
:= 888 + ((8888 − (8 + 8))/8)
:= ((8 − 888)/8) + (88 × (8 + 8 + 8))
:= 8 + (8888/8 + 888)
:= 999 + (999 − 9/9)
:= 9 × 99 + 9999/9
:= 9 + (999 + 999)
7
143
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2008 := 1 + ((11 − 1 − 1) × (1 + (1 + 1) × 111))
Ï 2013 := 1 + ((1 + 1)11 − ((1 + 1 + 1) × (1 + 11)))
Ï 2018 := (1 + 1)11 − (11 − 1) × (1 + 1 + 1)
:= 222/2 + (2 × (2 − 22))
:= 2/2 + (222/2 − (2 + 2 + 2)2 )
:= 2 + (2 × (2 × ((222 − 2) + 22)))
:= 3 × 3 + ((3 + 3) × 333 + 3/3)
:= 33 × (((3/3 + 3)3 ) − 3)
:= 3 + (((33/3 + 3)3 ) − 33+3 )
:= 4 + (44 × (4 + 4) − 44)
:= 4/4 + ((4 + 4) × (44 − 4) − 4)
:= (4 + 4)/4 + (4 + 4) × (44 − 4)
:= 55 − ((5555 + 5)/5 + 5)
:= 55 − (5555 + 5)/5
:= 5 + (55 − (5555 + 5)/5)
:= ((6 + 6)/6)6 + 6 × 6 × (66 − 6 − 6)
:= (6 × (666 + 6) − 6)/((6 + 6)/6)
:= 6 + ((((6 + 6)/6)66/6 ) − 6 × 6)
:= 7 × (7 × (7 × 7 − 7) − 7) − 7/7
:= 7 + (((((7 + 7)/7)77/7 ) − 7 × 7) + 7)
:= 7 + (7 × (7 × (7 × 7 − 7) − 7) + ((7 + 7)/7))
:= 88 + ((8 + 8) × (8 × (8 + 8) − 8))
:= 88/8 × ((888/8 + 8 × 8) + 8)
:= 8 + (((8888 + 88)/8) + 888)
:= 9 + ((999 + 999) + 9/9)
:= 9 × 99 + ((9999 + 99)/9)
:= ((9 + 9)/9) × ((999 + 9/9) + 9)
Ï 2014 := (1 + 1)11 − (1 + 11 × (1 + 1 + 1))
Ï 2019 := 1 + ((1 + 1)11 − ((11 − 1) × (1 + 1 + 1)))
Ï 2009 := 11 + ((1 + 1) × (111 × (11 − 1 − 1)))
:= (2 × 22 + 2/2)2 − 22+2
:= 2 + (222/2 − (2 + 2 + 2)2 )
:= (2 × 22 + 2/2)2 − (2 + 2 + 2)
:= 33/3 + (3 + 3) × 333
:= 3/3 + (33 × (((3/3 + 3)3 ) − 3))
:= 3 + (3 + 3) × (333 + 3)
:= (4 + 4) × (44 − 4) − (4 + 4)/4
:= 4 + ((4 + 4) × (44 − 4) − 4/4)
:= 5 − (5555/5 + 5)
:= 55 − 5555/5
:= 5 + (55 − 5555/5)
:= ((6 − 6/6) × (6 × 66 + 6)) − 6/6
:= 6 × 6 + (66 × (6 × 6 − 6) − ((6 + 6)/6))
:= (6 × (666 + 6) + 6)/((6 + 6)/6)
:= 7 × (7 × (7 × 7 − 7) − 7)
:= 7 + (7 × (7 × (7 × 7 − 7) − 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 × (7 × (7 × 7 − 7) − 7)
:= 8 + ((8/8 − 88) × (8/8 − (8 + 8 + 8)))
:= 8 + (((8888 − 8)/8 + 888) + 8)
:= 88 + (((8 + 8) × (8 × (8 + 8) − 8)) + (88/8))
:= 9 + (((9 + 9)/9) × (999 + 9/9))
:= ((9 + 9)/9) × ((999 − 9/9) + 9)
:= 9 + ((((9999 − 9)/9) + 9 × 99) + 9)
4
:= 4 + ((4 × (4 + 4) − 44) + 4/4)
5
Ï 2010 := 11 + (((1 + 1) × (11 − 1)1+1+1 ) − 1)
Ï 2015 := (1 + 1)11 − (11 × (1 + 1 + 1))
Ï 2020 := (1 + 1) × ((11111 − 1)/11)
:= 222/2 − (22/2 + 22)
:= 2 × (2 × (222 + 22) − 2)
:= ((33/3 + 3)3 ) − 33+3
:= 3 + ((3 + 3) × (333 + 3) + 3/3)
:= (4 + 4) × (4 − 4) − ((4 + 4)/4 + 4)
:= (4 + 4) × (44 − 4) − 4/4
:= 4 + (4 + 4) × (44 − 4)
:= 5 + ((5 × 5 × (5 × 5 + 55)) + 5)
:= 55 + ((5 − 5555)/5)
:= 55 + ((55 × (5 − 5 × 5)) − 5)
:= (6 − 6/6) × (6 × 66 + 6)
:= 6 × 6 + (66 × (6 × 6 − 6) − 6/6)
:= (6 − 6/6) × ((((6 + 6)/6) + 6 × 66) + 6)
:= 7/7 + 7 × (7 × (7 × 7 − 7) − 7)
:= 7 + (7 × (7 × (7 × 7 − 7) − 7) − 7/7)
:= 77/7 + 7 × (7 × (7 × 7 − 7) − 7)
:= 888 + ((8888 + 88)/8)
:= 8 + ((8888/8 + 888) + 8)
:= ((8 × (8 × 8 × 8 − 8)) + 8)/((8 + 8)/8)
:= 9 + (((9999 − 9)/9) + 9 × 99)
:= ((9 + 9) × ((999 + 9)/9)) − 9/9
:= ((9 + 9)/9) × (99/9 + 999)
:= 2 + (2
22/2
+ (2 × (2 − 22)))
:= 3 + ((3 + 3) × 333 + 3 × 3)
4
Ï 2011 := 11 + ((1 + 1) × (11 − 1)1+1+1 )
Ï 2016 := 1 + ((1 + 1)11 − (11 × (1 + 1 + 1)))
Ï 2021 := (1 + 1)11 − (1 + 1 + 1)1+1+1
:= 2 + ((2 × 22 + 2/2)2 − 22+2 )
:= 2 × (2 × ((222 − 2) + 22))
:= (2 × 22 + 2/2)2 − 2 − 2
:= 3 + (((3 + 3) × 333 + 3 × 3) + 3/3)
:= (3 + 3) × (333 + 3)
:= (3 − 3/3)33/3 − 33
:= (4 + 4) × (44 − 4) − (4/4 + 4)
:= (4 + 4) × (44 − 4)
:= 4 + ((4 + 4) × (44 − 4) + 4/4)
:= 5 + ((((5 + 5)5−5/5 ) + 55)/5)
:= 5 + ((5 + 5)5−5/5 + 55)/5 + 5
:= 6/6 + ((6 − 6/6) × (6 × 66 + 6))
:= 6 × (66 × (6 − 6/6) + 6)
:= 6 + ((66 × (6 × 6 − 6) − 6/6) + 6 × 6)
:= (7 + 7)/7 + 7 × (7 × (7 × 7 − 7) − 7)
:= 7 + 7 × (7 × (7 × 7 − 7) − 7)
:= (77 + 7)/7 + 7 × (7 × (7 × 7 − 7) − 7)
:= 8 + (((8 × ((8 + 8) × (8 + 8) − 8)) + (88/8)) + 8)
:= 8 × ((8 × 8 × 8 − 8)/((8 + 8)/8))
:= (8 × (8 + 8) × (8 + 8)) − (88/8 + 8 + 8)
:= 9 + (9999/9 + 9 × 99)
:= (9 + 9) × ((999 + 9)/9)
:= (((9 + 9)/9)99/9 ) − (9 + 9 + 9)
:= (((5 + 5)
5−5/5
) + 55)/5
Ï 2012 := (1 + 1)11 − ((1 + 1 + 1) × (1 + 11))
Ï 2017 := 1 + (1 + ((1 + 1)11 − (11 × (1 + 1 + 1))))
Ï 2022 := (1 + 1) × (11 + (11 − 1)1+1+1 )
:= 222/2 − (2 + 2 + 2)2
:= (2 × 22)2 + (2/2 + 2)2+2
:= 222/2 − 22 − 2 − 2
:= 3 + ((3 + 3) × 333 + 33/3)
:= 3/3 + (3 + 3) × (333 + 3)
:= 3 + ((3 + 3) × (333 + 3) + 3)
4
4
:= (4 + 4) × (4 − 4) − 4
:= 4/4 + (4 + 4) × (4 − 4)
:= 4 + ((4 + 4) × (44 − 4) + (4 + 4)/4)
:= 55 − (5555 + 5 + 5)/5
:= 5 + (55 − (5555 + 5 + 5)/5)
:= (5/5 + 5) × ((55 − 5)/(5 + 5) + 5 × 5)
:= (((6 + 6)/6)66/6 ) − 6 × 6
:= 6 × 6 + (66 × (6 × 6 − 6) + 6/6)
:= 6 + (66 × (6 × 6 − 6) + 6 × 6)
:= 7 + ((77/7 + 7) × 777/7 + 7)
:= 7 + (7 × (7 × (7 × 7 − 7) − 7) + 7/7)
:= (7 − 7/7) × ((7 × 7 × 7 − 7) + 7/7)
:= ((8 × (8 × 8 × 8 − 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 × ((8 × 8 × 8 − 8)/((8 + 8)/8)))
:= (88 × (8 + 8 + 8)) − ((8 + 8)/8 + 88)
:= 9 + ((9999 + 9)/9 + 9 × 99)
:= 9/9 + ((9 + 9) × ((999 + 9)/9))
:= ((9 + 9)/9) × (((99 + 9)/9) + 999)
144
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2023 := (1 + 1)11 − (1 + ((1 + 1) × (1 + 11)))
Ï 2028 := (1 + 11) × (1 + 1 + 11)1+1
Ï 2033 := (1 + 1)11 − 1 − 1 − 1 − 1 − 11
:= (2 × 22 + 2/2)2 − 2
:= 2 + (222/2 − 22)
:= 2/2 + (222/2 − 22+2 )
:= 3 + (((3 + 3) × (333 + 3) + 3/3) + 3)
:= 3 + ((3 + 3) × 333 + 33 )
:= 33+3 + ((33/3)3 − 33 )
:= 4 + (((4 + 4) × (4 − 4) − 4/4) + 4)
:= 44 + (4 × 444 − 4)
:= 4/4 + (4 × 444 + 44 )
:= (((5 + 5)/5)55/5 ) − 5 × 5
:= 5 + ((((5 + 5)/5)55/5 ) − 5 × 5)
:= (((5 + 5)/5)55/5 ) − (5 + 5 + 5)
:= 6 + ((66 × (6 × 6 − 6) + 6 × 6) + 6/6)
:= 6 + ((66 × (6 × 6 − 6) + 6 × 6) + 6)
:= (6 × (666 × 6/(6 + 6) + 6)) − 6/6
:= 7 + (7 × (7 × (7 × 7 − 7) − 7) + 7)
:= (7 − 7/7) × ((7 × 7 × 7 − 7) + ((7 + 7)/7))
:= ((7 + 7 + 7)/7)7 − 77 − 77
:= (8/8 + 8 + 8) × (888/8 + 8)
:= 8 + (((8 × (8 × 8 × 8 − 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8) × (8 × (8 + 8) − 8/8))
:= (((9 + 9)/9)99/9 ) − (99/9 + 9)
:= (9/9 + 9 + 9) × ((99 − 9/9) + 9)
4
9/9+9
:= 999 + (((9 + 9)/9)
)
Ï 2024 := (1 + 1)11 − ((1 + 1) × (1 + 11))
Ï 2029 := 1 + ((1 + 11) × (1 + 1 + 11)1+1 )
Ï 2034 := (1 + 1)11 − 1 − 1 − 1 − 11
:= 2 × (2 × (222 + 22))
:= 2 + ((2 × 22 + 2/2)2 + 2)
:= 2 + (222/2 − 22+2 )
:= 3 + ((3 − 3/3)33/3 − 33 )
:= (((3 × 33 − 3)3−3/3 ) + 3)/3
:= (3 + 3) × (333 + 3 + 3)
:= 4 + ((4 + 4) × (4 − 4) + 4)
:= 4 + ((44 + 4/4)(4+4)/4 )
:= 44 + (4 × 444 + (4 + 4)/4)
:= 5 + (55 − 5555/5 + 5)
:= 5 + ((55 − 5555/5 + 5) + 5)
:= (55 × (((5 + 5)/5)5 + 5)) − 5/5
:= 6 + (((((6 + 6)/6)66/6 ) − 6 × 6) + 6)
:= ((6 − 6/6) × (6 × 66 + (66/6))) − 6
:= 6 × (666 × 6/(6 + 6) + 6)
:= 7 + ((7 × (7 × (7 × 7 − 7) − 7) + 7/7) + 7)
:= 7 + ((7 − 7/7) × ((7 × 7 × 7 − 7) + 7/7))
:= (((7 + 7)/7)77/7 ) − (7 + 7)
:= 88 × ((8 − 8/8 + 8) + 8)
:= (8 × (8 + 8) × (8 + 8)) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) × (8 × (8 + 8) − 8/8))
:= 9 × 9 + ((9 + 9) × (99 + 9) − 9/9)
:= 9 + (((9 + 9)/9) × (99/9 + 999))
:= 9 + ((9 + 9) × (99 + 9) + 9 × 9)
4
Ï 2025 := (1 + 1)11 − (1 + (11 + 11))
Ï 2030 := (1 + 1)11 − (1 + 1) × (11 − 1 − 1)
Ï 2035 := (1 + 1)11 − 1 − 1 − 11
:= (2 × 22 + 2/2)2
:= 2 + ((222/2 − 22) + 2)
:= 222/2 − (22/2 + 2)
:= 3 × (3 × (3 + 3)3 + 33 )
:= 33 + ((3 + 3) × 333 − 3/3)
:= 3/3 + ((3 + 3) × (333 + 3 + 3))
:= (44 + 4/4)
:= 4 + ((4 + 4) − 44)/((4 + 4)/4)
:= 4 + ((4 × 444 − 4/4) + 44 )
:= 5 × (5 × (5 × 5 + 55) + 5)
:= 5 + ((55 × (5 − 5 × 5)) + 55 )
:= 55 × (((5 + 5)/5)5 + 5)
:= 6 × 6 × 6 × 6 + ((6 × 6/(6 + 6))6 )
:= (6 × 6 − 6/6) × (((6 + 6)/6)6 − 6)
:= (6 − 6/6) × (6 × 66 + (66/6))
:= (7/7 + 7 + 7) × (((7 + 7)/7)7 + 7)
:= 7 × (7 × (7 × 7 − 7) + 7) − 77
:= 7/7 + ((((7 + 7)/7)77/7 ) − (7 + 7))
:= 8/8 + (88 × ((8 − 8/8 + 8) + 8))
:= ((8 + 8)/8) × (8 × 8 × (8 + 8) − (8/8 + 8))
:= 88/8 + (88 × ((8 − 8/8 + 8) + 8))
:= 9 × ((9 + 9) × (9 + 9) − 99)
:= (((9 + 9)/9)99/9 ) − (9 + 9)
:= 9 + (((9 + 9) × (99 + 9) + 9 × 9) + 9/9)
4
(4+4)/4
Ï 2036 := (1 + 1)11 − 1 − 11
Ï 2026 := (1 + 1)11 − 11 − 11
Ï 2031 := 1 + ((1 + 1)11 − (1 + 1) × (11 − 1 − 1))
:= 222/2 − 22
:= 2 + (((2 × 22 + 2/2)2 + 2) + 2)
:= 222/2 − (2 × (2 + 2 + 2))
:= 33 + ((3 + 3) × 333 + 3/3)
:= 33 + (3 + 3) × 333
:= (3/3 + 3) × ((3 − 3/3)3×3 − 3)
:= (((4 + 4)4 ) − 44)/((4 + 4)/4)
:= 44 + (4 × 444 − 4/4)
:= 4 + (4 × 444 + 44 )
:= 5 × 5 + ((((5 + 5)5−5/5 ) + 5)/5)
:= 5 + (((55 × (5 − 5 × 5)) + 5/5) + 55 )
:= 5/5 + (55 × (((5 + 5)/5)5 + 5))
:= 6 + ((6 − 6/6) × ((((6 + 6)/6) + 6 × 66) + 6))
:= ((6 × 666) + 66)/((6 + 6)/6)
:= (((6 + 6)/6)66/6 ) − 6 − 6
:= ((7 + 7 + 7)/7) − ((77 + 77) + 7)
:= 7/7 + (7 × (7 × (7 × 7 − 7) + 7) − 77)
:= 7 × 7 × (7 × 7 − 7) − (7/7 + 7 + 7 + 7)
:= (8 + 8)/8 + (88 × ((8 − 8/8 + 8) + 8))
:= (8 × (8 + 8) × (8 + 8)) − (8/8 + 8 + 8)
:= (8 × (8 + 8) × (8 + 8)) − (88 + 8)/8
7
:= 9/9 + ((9 + 9) × (99 + 9) + 9 × 9)
Ï 2027 := 1 + ((1 + 1)11 − (11 + 11))
99/9
:= 9/9 + ((((9 + 9)/9)
) − (9 + 9))
Ï 2032 := (1 + 1)11 − (1 + 1)1+1+1+1
:= 9 × 9 + ((9 + 9) × (99 + 9) + (99/9))
Ï 2037 := (1 + 1)11 − 11
:= 222/2 − 22/2
:= 2 + (2 × 22 + 2/2)2
:= 222/2 − 22+2
:= 33+3 + ((33/3)3 − 33)
:= 3/3 + ((3 + 3) × 333 + 33)
:= 3 + (3 + 3) × (333 + 3 + 3)
:= 44/4 + (4 + 4) × (4 − 4)
:= 4 × ((4 − 4) + 4 )
:= 44 × (4 + 4) − 44/4
:= 5 × 5 + (((((5 + 5)5−5/5 ) + 5) + 5)/5)
:= ((5 + 5)/5)5 + (5 × 5 × (5 × 5 + 55))
:= (((5 + 5)/5)55/5 ) − 55/5
:= 6 × 6 + (66 × (6 × 6 − 6) + (66/6))
:= 6/6 + (((6 × 666) + 66)/((6 + 6)/6))
:= (((6 + 6)/6)66/6 ) − 66/6
:= (((7 + 7)/7)77/7 ) − (7 + 7 + 7)
:= 7 + ((7/7 + 7 + 7) × (((7 + 7)/7)7 + 7))
:= (7 + 7 + 7) × (7 × (7 + 7) − 7/7)
:= 88/8 + (8 × ((8 × 8 × 8 − 8)/((8 + 8)/8)))
:= (8 + 8) × (8 × (8 + 8) − 8/8)
:= (8 × (8 + 8) × (8 + 8)) − 88/8
:= 9 + (((9 + 9)/9) × ((999 + 9/9) + 9))
:= 9 + ((((9 + 9)/9)9/9+9 ) + 999)
:= (((9 + 9)/9)99/9 ) − 99/9
4
4
4
145
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2038 := 1 + ((1 + 1)11 − 11)
Ï 2043 := (1 + 1)11 − 1 − 1 − 1 − 1 − 1
Ï 2048 := (1 + 1)11
:= ((2 − 22)/2) + 222/2
:= 222/2 − (2/2 + 2 + 2)
:= 222/2
:= 3 + (3 + 3) × (333 + 3 + 3) + 3/3
:= 3 × (3 × (3 + 3)3 + 33)
:= (3 − 3/3)33/3
:= (4 − 44)/4 + 44 × (4 + 4)
:= 44 × (4 + 4) − (4/4 + 4)
:= 44 × (4 + 4)
:= (5 + 5)/5 × ((5 − 5/5)5 − 5)
:= (((5 + 5)/5)55/5 ) − 5
:= ((5 + 5)/5)55/5
:= (6 − 66)/6 + (((6 + 6)/6)66/6 )
:= 6/6 + ((((6 + 6)/6)66/6 ) − 6)
:= ((6 + 6)/6)66/6
:= 7/7 + ((7 + 7 + 7) × (7 × (7 + 7) − 7/7))
:= 7 × 7 × (7 × 7 − 7) − (7/7 + 7 + 7)
:= ((7 + 7)/7)77/7
:= (8 − 88)/8 + (8 × (8 + 8) × (8 + 8))
:= 88/8 + ((8 + 8) × (8 × (8 + 8) − 8/8))
:= 8 × (8 + 8) × (8 + 8)
:= (((9 + 9)/9)99/9 ) − 9/9 − 9
:= 99 + (9 + 9) × (99 + 9)
:= ((9 + 9)/9)99/9
Ï 2039 := 1 + (1 + (1 + 1)11 − 11)
:= 2 + (2
22/2
Ï 2044 := (1 + 1)11 − 1 − 1 − 1 − 1
− 22/2)
33/3
:= (3 − 3/3)
:= 2
22/2
:= 2/2 + 222/2
−2−2
3
−3×3
:= 3/3 + (3 × (3 × (3 + 3) + 33))
4
4
:= 4 × (4 + 4) − (4/4 + 4 + 4)
Ï 2049 := 1 + (1 + 1)11
:= 3/3 + (3 − 3/3)33/3
:= 4 × (4 + 4) − 4
:= 4/4 + 44 × (4 + 4)
:= 5 × 5 + (5 − 5555/5)
:= 5/5 + ((((5 + 5)/5)55/5 ) − 5)
:= 5/5 + (((5 + 5)/5)55/5 )
:= 66 + (66 × (6 × 6 − 6) − (6/6 + 6))
:= ((6 + 6)/6)6 + 66 × (6 × 6 − 6)
:= 7 × 7 × (7 × 7 − 7) − ((77 + 7)/7 + 7)
:= 6/6 + (((6 + 6)/6)66/6 )
:= 7 × 7 × (7 × 7 − 7) − (7 + 7)
:= (8 × (8 + 8) × (8 + 8)) − (8/8 + 8)
:= 7/7 + (((7 + 7)/7)77/7 )
:= ((8 × 8 × 8 × 8) − 8)/((8 + 8)/8)
:= (((9 + 9)/9)99/9 ) − 9
:= 8/8 + (8 × (8 + 8) × (8 + 8))
:= 9/9 + ((9 + 9) × (99 + 9) + 99)
:= 9/9 + (((9 + 9)/9)99/9 )
5
Ï 2040 := 1 + (1 + (1 + (1 + 1)11 − 11))
:= 2
22/2
Ï 2045 := (1 + 1)11 − 1 − 1 − 1
− 2 × (2 + 2)
:= 2
3
:= (3 + 3) × ((((3/3 + 3) + 3) ) − 3)
4
22/2
Ï 2050 := 1 + (1 + (1 + 1)11 )
− 2/2 − 2
33/3
:= (3 − 3/3)
:= 2 + 222/2
−3
:= 3 + ((3 − 3/3)33/3 − 3/3)
4
:= (4 + 4) × (4 − 4/4)
:= 4/4 + (4 × (4 + 4) − 4)
5
5
:= (4 + 4)/4 + 44 × (4 + 4)
:= 5 + (55 × (((5 + 5)/5) + 5))
:= 5 + ((55 × (((5 + 5)/5) + 5)) + 5)
:= 5 × ((5 × (5 × 5 + 55) + 5) + 5)
:= 66 + (66 × (6 × 6 − 6) − 6)
:= 66 + (66 × (6 × 6 − 6) − 6/6)
:= (6 + 6)/6 + (((6 + 6)/6)66/6 )
:= 7 × 7 × (7 × 7 − 7) − (77/7 + 7)
:= 7/7 + (7 × 7 × (7 × 7 − 7) − (7 + 7))
:= (8 × (8 + 8) × (8 + 8)) − 8
:= 8 + ((8 × (8 + 8) × (8 + 8)) − (88/8))
:= 9/9 + ((((9 + 9)/9)99/9 ) − 9)
:= 99 + ((9 + 9) × (99 + 9) + ((9 + 9)/9))
Ï 2041 := 1 + (1 + (1 + (1 + (1 + 1)11 − 11)))
:= 2 + ((2
22/2
− 22/2) + 2)
Ï 2046 := (1 + 1)11 − 1 − 1
:= 2
3
:= 3/3 + (3 + 3) × ((3/3 + 3 + 3) − 3)
4
22/2
−2
3
:= 3 + (3 × (3 × (3 + 3) + 33))
:= 44 × (4 + 4) − (4 + 4)/4
:= 55 − (((5 − 5/5)5 + 55) + 5)
:= 55 − ((5 − 5/5)5 + 55)
:= (((6 + 6)/6)66/6 ) − 6/6 − 6
:= 66 + 66 × (6 × 6 − 6)
:= (((7 + 7)/7)77/7 ) − 7
:= (7 − 7/7) × (7 × 7 × 7 − ((7 + 7)/7))
:= 8/8 + ((8 × (8 + 8) × (8 + 8)) − 8)
:= (8 × (8 + 8) × (8 + 8)) − (8 + 8)/8
:= 99 + ((9 + 9) × (99 + 9) − ((9 + 9)/9))
:= (((9 + 9)/9)99/9 ) − (9 + 9)/9
:= 2
22/2
:= 2
− (2 + 2 + 2)
33/3
:= (3 − 3/3)
:= (3 − 3/3)
−3−3
:= 4 × (4 + 4) − ((4 + 4)/4 + 4)
:= (((5 + 5)/5)
66/6
:= (((6 + 6)/6)
) − (5/5 + 5)
77/7
:= 7/7 + ((((7 + 7)/7)
) − 7)
:= (8 + 8)/8 + ((8 × (8 + 8) × (8 + 8)) − 8)
:= 99 + ((9 + 9) × (99 + 9) − 9/9)
:= 2 + (222/2 + 2/2)
:= 3 + (3 − 3/3)33/3
:= 4 + (44 × (4 + 4) − 4/4)
:= 5 + (55 − ((5 − 5/5)5 + 55))
:= ((((6 + 6)/6)6+6 ) + 6)/((6 + 6)/6)
:= 7 × 7 × (7 × 7 − 7) − 7
:= 88/8 + ((8 × (8 + 8) × (8 + 8)) − 8)
:= (((9 + 9)/9)9 ) + (9 × (9 × (9 + 9) + 9))
:= 2 + 222/2 + 2
− 3/3
4
:= 4 × (4 + 4) − 4/4
:= (3 + 3) × (333 + 3 × 3)
:= 4 + 44 × (4 + 4)
55/5
) − 5/5
:= 5 + ((((5 + 5)/5)55/5 ) − 5/5)
66/6
) − 6/6
:= 6 + (66 × (6 × 6 − 6) + 66)
:= (((5 + 5)/5)
:= (((6 + 6)/6)
)−6
:= (9 + 9)/9 + (((9 + 9)/9)99/9 )
Ï 2052 := 1 + (1 + (1 + (1 + (1 + 1)11 )))
− 2/2
33/3
4
55/5
Ï 2047 := (1 + 1)11 − 1
22/2
:= (8 + 8)/8 + (8 × (8 + 8) × (8 + 8))
Ï 2051 := 1 + (1 + (1 + (1 + 1)11 ))
:= 4 + (4 × (4 + 4) − 44/4)
Ï 2042 := (1 + 1)11 − ((1 + 1) × (1 + 1 + 1))
:= 7 × 7 × (7 × 7 − 7) − (7/7 + 7)
:= 7 × 7 × (7 × 7 − 7) − 77/7
:= 7/7 + (7 × 7 × (7 × 7 − 7) − 7)
:= (8 × (8 + 8) × (8 + 8)) − 8/8
:= ((8 × 8 × 8 × 8) + 8)/((8 + 8)/8)
99/9
:= (((9 + 9)/9)
) − 9/9
146
:= (99 + 9) × (9/9 + 9 + 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2053 := 1 + (1 + (1 + (1 + (1 + (1 + 1)11 ))))
Ï 2058 := 11 + ((1 + 1)11 − 1)
Ï 2063 := 1 + (1 + (1 + (1 + (11 + (1 + 1)11 ))))
:= 2 + ((222/2 + 2/2) + 2)
:= 2 + (222/2 + 2 × (2 + 2))
:= 2 + ((222/2 + 22/2) + 2)
:= 3/3 + ((3 + 3) × (333 + 3 × 3))
:= (3 + 3) × (((3/3 + 3) + 3)3 )
:= 3 + ((33/3)3 + 33+3 )
:= 4 + (44 × (4 + 4) + 4/4)
:= 44 × (4 + 4) + (44 − 4)/4
:= 4 + (44 × (4 + 4) + 44/4)
:= 5 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5)
:= 5 + (((((5 + 5)/5)55/5 ) + 5) + 5)
:= 6 + ((((6 + 6)/6)66/6 ) − 6/6)
:= 6 × ((6/6 + 6)6×6/(6+6) )
:= 6 + (((6 + 6)/6)6+6 + 6)/((6 + 6)/6) + 6
:= (7 + 7)/7 + (7 × 7 × (7 × 7 − 7) − 7)
:= 7 × 7 × (7 × 7 − 7)
:= 7 + (7 × 7 × (7 × 7 − 7) − ((7 + 7)/7))
:= 8 + (((8 × (8 + 8) × (8 + 8)) − (88/8)) + 8)
:= 8 + ((8 × (8 + 8) × (8 + 8)) + ((8 + 8)/8))
:= 8 + (((8 × (8 + 8) × (8 + 8)) − 8/8) + 8)
:= 9/9 + ((99 + 9) × (9/9 + 9 + 9))
Ï 2054 := (1 + 1)11 + ((1 + 1) × (1 + 1 + 1))
99/9
:= 9 + ((((9 + 9)/9)
) + 9/9)
:= 99/9 + ((99 + 9) × (9/9 + 9 + 9))
Ï 2064 := (1 + 1)11 + (1 + 1)1+1+1+1
Ï 2059 := 11 + (1 + 1)11
:= 2 + ((222/2 + 2) + 2)
:= 22/2 + 222/2
:= 22+2 + 222/2
:= 3 + ((3 − 3/3)33/3 + 3)
:= 3/3 + (3 + 3) × (3/3 + 3 + 3)3
:= 3 + ((3 × 3 + 3)3 + 333)
:= 4 + (44 × (4 + 4) + (4 + 4)/4)
:= 44/4 + 44 × (4 + 4)
:= 4 × ((44 + 44 ) + 4)
:= 5 + ((((5 + 5)/5)55/5 ) + 5/5)
:= 55/5 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + (55/5))
:= 6 + (((6 + 6)/6)66/6 )
:= 66/6 + (((6 + 6)/6)66/6 )
:= 6 + (6 × ((6/6 + 6)6×6/(6+6) ))
:= 7 + (7 × 7 × (7 × 7 − 7) − (77/7))
:= 7/7 + 7 × 7 × (7 × 7 − 7)
:= 7 + 7 × 7 × (7 × 7 − 7) − 7/7
:= 8 + ((8 × (8 + 8) × (8 + 8)) − ((8 + 8)/8))
:= 88/8 + (8 × (8 + 8) × (8 + 8))
:= 8 + ((8 × (8 + 8) × (8 + 8)) + 8)
:= 99 + ((9 + 9) × (99 + 9) + (99/9))
:= 99/9 + (((9 + 9)/9)99/9 )
:= 9 + ((9 + 9) × (99 + 9) + 999/9)
Ï 2055 := 1 + ((1 + 1)11 + ((1 + 1) × (1 + 1 + 1)))
Ï 2065 := 1 + ((1 + 1)11 + (1 + 1)1+1+1+1 )
Ï 2060 := 1 + (11 + (1 + 1)11 )
:= 2 + (((222/2 + 2/2) + 2) + 2)
:= 222/2 + (2 × (2 + 2 + 2))
:= 2/2 + (222/2 + 22+2 )
:= 3 + ((3 + 3) × (333 + 3 × 3))
:= 33+3 + (33/3)3
:= 3 + (((3 × 3 + 3)3 + 333) + 3/3)
:= 4 + ((4 × (4 + 4) − 4/4) + 4)
:= 4 + ((44 × (4 + 4) + 4) + 4)
:= 4 × 4 + (44 × (4 + 4) + 4/4)
:= 55 + (5 × 5 × (5 × 5 + 55))
:= 5 + ((5 × 5 × (5 × 5 + 55)) + 55)
:= 55 + ((5 + 5) × (5 − 555/5))
:= 6 + ((6 + 6)/6)66/6 + 6
:= (6 × 6 − 6/6) × (66 − (6/6 + 6))
:= (7 + 7)/7 + 7 × 7 × (7 × 7 − 7)
:= 7 + 7 × 7 × (7 × 7 − 7)
:= 8 + ((8 × (8 + 8) × (8 + 8)) − 8/8)
:= 8 + (((8 × 8 × 8 × 8) + 8)/((8 + 8)/8))
:= 8 + (((8 × (8 + 8) × (8 + 8)) + 8/8) + 8)
:= 999/9 + (9 + 9) × (99 + 9)
:= 9 + ((9 × (9 × (9 + 9) + 9)) + (((9 + 9)/9)9 ))
:= 9 + (((((9 + 9)/9)99/9 ) − 9/9) + 9)
4
66/6
:= 6 + ((((6 + 6)/6)
77/7
:= 7 + (((7 + 7)/7)
) + 6/6)
)
Ï 2056 := 11 + ((1 + 1)11 − (1 + 1 + 1))
Ï 2061 := 1 + (1 + (11 + (1 + 1)11 ))
Ï 2066 := (1 + 1)11 + (1 + 1) × (11 − 1 − 1)
:= 2 × (2 + 2) + 222/2
:= 2 + (222/2 + 22/2)
:= 2 + (222/2 + 22+2 )
:= (((3 × (3 + 3))3 + 333) + 3)/3
:= 333 + (3 × 3 + 3)3
:= 3 + (((33/3)3 + 33+3 ) + 3)
:= 4 + (44 × (4 + 4) + 4)
:= 4 + (((44 × (4 + 4) + 4/4) + 4) + 4)
:= 4 × 4 + (44 × (4 + 4) + (4 + 4)/4)
:= 55 + ((((5 + 5)5−5/5 ) + 5)/5)
:= 5 + (((((5 + 5)5−5/5 ) + 5)/5) + 55)
:= 55 + ((((5 + 5)5−5/5 ) + 55)/5)
:= 6 + ((((6 + 6)/6)66/6 ) + ((6 + 6)/6))
:= 6 + ((6 + 6)/6)66/6 + 6 + 6/6
:= 6 + (((((6 + 6)/6)66/6 ) + 6) + 6)
:= 7 × 7 × (7 × 7 − 7) − (7 + 7)/7
:= (7 + 7 + 7)/7 + 7 × 7 × (7 × 7 − 7)
:= 7 + (7 × 7 × (7 × 7 − 7) + 7/7)
:= 8 + (8 × (8 + 8) × (8 + 8))
:= (8 × (8 + 8) × (8 + 8)) + (88 + 8 + 8)/8
:= 8 + (((8 × (8 + 8) × (8 + 8)) + ((8 + 8)/8)) + 8)
:= 9 + ((99 + 9) × (9/9 + 9 + 9))
:= 9 + ((((9 + 9)/9)99/9 ) + 9)
99/9
:= 9 + ((((9 + 9)/9)
) − 9/9)
Ï 2057 := 11 + ((1 + 1)11 − (1 + 1))
Ï 2062 := 1 + (1 + (1 + (11 + (1 + 1)11 )))
Ï 2067 := (1 + 1)11 + (((1 + 1) × (11 − 1)) − 1)
:= 22/2 + 222/2 − 2
:= 22+2 + 222/2 − 2
:= 22 + (222/2 − (2/2 + 2))
:= 3 × 3 + (3 − 3/3)33/3
:= 3/3 + ((3 × 3 + 3)3 + 333)
:= 3 + (((3 × 3 + 3)3 + 333) + 3)
:= 4 + ((44 × (4 + 4) + 4/4) + 4)
:= 4 + (44 × (4 + 4) + (44 − 4)/4)
:= 4 + ((44 × (4 + 4) + 44/4) + 4)
:= (((5 + 5) × ((5 − 5/5)5 + 5)) − 5)/5
:= 5 + ((((5 + 5) × ((5 − 5/5)5 + 5)) − 5)/5)
:= ((55 + 5/5) × (((5 + 5)/5)5 + 5)) − 5
:= 6 + (((((6 + 6)/6)6+6 ) + 6)/((6 + 6)/6))
:= 6 + ((6 + 6)/6)66/6 + 6 + (6 + 6)/6
:= 66 + (((6 × 666) + 6)/((6 + 6)/6))
77/7
:= 7 × 7 × (7 × 7 − 7) − 7/7
:= 7 + ((((7 + 7)/7)
:= 8 + ((8 × (8 + 8) × (8 + 8)) + 8/8)
:= 8 + (((8 × (8 + 8) × (8 + 8)) − ((8 + 8)/8)) + 8)
:= 8 + ((8 × (8 + 8) × (8 + 8)) + (88/8))
:= 9 + (((99 + 9) × (9/9 + 9 + 9)) + 9/9)
:= 9 + (((((9 + 9)/9)99/9 ) + 9/9) + 9)
99/9
:= 9 + (((9 + 9)/9)
)
) + 7)
147
:= 7 + (7 × 7 × (7 × 7 − 7) + ((7 + 7)/7))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2068 := (1 + 1)11 + ((1 + 1) × (11 − 1))
Ï 2073 := 1 + ((1 + 1)11 + ((1 + 1) × (1 + 11)))
Ï 2078 := (1 + 1)11 + ((11 − 1) × (1 + 1 + 1))
:= 22 + 222/2 − 2
:= 2 + 222/2 + 22 + 2/2
:= 22 + 222/2 + 2 × (2 + 2)
:= 33/3 × ((3 + 3)3 − (33 + 3/3))
:= (3 × (33+3 − 33 )) − 33
:= 3 + ((3 − 3/3)33/3 + 33 )
:= 4 + (44 × (4 + 4) + 4 × 4)
:= 4 + ((4 + 4) × (44 + 4) − 44/4)
:= (4 + 4) × (44 + 4) − (4 + 4)/4
:= 5 × 5 + ((((5 + 5)/5)55/5 ) − 5)
:= 5 × 5 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 × 5)
:= 66/6 × ((6 × (6 × 6 − 6) + ((6 + 6)/6)) + 6)
:= 6 + ((((6 × 666) + 6)/((6 + 6)/6)) + 66)
:= 6 × 6 + ((((6 + 6)/6)66/6 ) − 6)
:= 77/7 × (777/7 + 77)
:= 7 + ((7 × 7 × (7 × 7 − 7) + 7/7) + 7)
:= 7 + ((7 × 7 × (7 × 7 − 7) − 7/7 + 7) + 7)
:= 8 + ((((8 × 8 × 8 × 8) + 8)/((8 + 8)/8)) + 8)
:= 8 + ((((8 × (8 + 8) × (8 + 8)) + 8/8) + 8) + 8)
:= 8 + ((88 + 88)/8 + (8 × (8 + 8) × (8 + 8)))
:= 9 + ((((9 + 9)/9)99/9 ) + (99/9))
:= 9 + (((9 + 9) × (99 + 9) + 999/9) + 9)
:= (99 × (((99 + 9)/9) + 9)) − 9/9
Ï 2074 := (1 + 1)11 + ((1 + 1) × (1 + 1 + 11))
Ï 2069 := 11 + (11 + ((1 + 1)11 − 1))
:= 22 + (222/2 − 2/2)
:= 2 + 2
22/2
Ï 2079 := 1 + ((1 + 1)11 + ((11 − 1) × (1 + 1 + 1)))
:= 2 × 222 + 2222/2
+ 22 + 2
3
:= 3 × 3 + ((33/3)3 + 33+3 )
:= (3/3 + 33) × (((3/3 + 3) ) − 3)
:= 3 × (33 × ((3 × (3 + 3)) + 3))
:= (4 + 4) × (44 + 4) − 44/4
:= 4 + ((4 + 4)4 + 44)/((4 + 4)/4)
:= (4 + 4) × (44 + 4) − 4/4
:= 55 + (55 − 5555/5)
:= 5 + ((55 − 5555/5) + 55 )
:= ((5 + 5) × ((55 − 5)/(5 + 5 + 5))) − 5/5
:= 66/6 + (6 × ((6/6 + 6)6×6/(6+6) ))
:= (66/6 + 6) × ((666 + 66)/6)
:= 6 × 6 + (((((6 + 6)/6)66/6 ) − 6) + 6/6)
:= 77/7 + 7 × 7 × (7 × 7 − 7)
:= 7 + ((7 × 7 × (7 × 7 − 7) + ((7 + 7)/7)) + 7)
:= 7 + ((7 × 7 × (7 × 7 − 7) + 7) + 7)
:= (8/8 + 8 + 8) × ((888 + 88)/8)
:= (88/8 − 8) × (8 × 88 − (88/8))
:= (9 × (99 + 9)) + (9999/9 − 9)
:= 99 × (((99 + 9)/9) + 9)
:= 8 + ((8 × (8 + 8) × (8 + 8)) + (88 + 8 + 8)/8)
9
:= 99 + (9 × 9 × (9 + 9) + (((9 + 9)/9) ))
Ï 2070 := 11 + (11 + (1 + 1)11 )
Ï 2075 := (1 + 1)11 + (1 + 1 + 1)1+1+1
Ï 2080 := (1 + 1)11 + ((11 × (1 + 1 + 1)) − 1)
:= 22 + 222/2
:= 2 + 222/2 + 22 + 2/2 + 2
:= 222/2 + 2 × 22+2
:= 3 × ((3 × ((3 + 3)3 + 3)) + 33)
:= 33 + (3 − 3/3)33/3
:= 3/3 + (3 × (33 × ((3 × (3 + 3)) + 3)))
:= (((4 + 4)4 ) + 44)/((4 + 4)/4)
:= (4 + 4) × (4 + 4) − (4/4 + 4)
:= (4 + 4) × (44 + 4)
:= (5 × (5 × (5 × 5 + 55 + 5))) − 55
:= 5 × (((5 × (5 × 5 + 55) + 5) + 5) + 5)
:= (5 + 5) × ((55 − 5)/(5 + 5 + 5))
:= 6 × ((666 × 6/(6 + 6) + 6) + 6)
:= 66 + (((6 − 6/6) × (6 × 66 + 6)) − 6/6)
:= (6/6 − 66) × (6 − ((6 + 6)/6 + 6 × 6))
:= (7 − 7/7) × (7 × 7 × 7 + ((7 + 7)/7))
:= 7 + (7 × 7 × (7 × 7 − 7) + ((77 − 7)/7))
:= 7 + (((7 × 7 × (7 × 7 − 7) + 7/7) + 7) + 7)
:= ((8 + 8)/8) × (8 × 8 × (8 + 8) + (88/8))
:= 8 + (((8 × (8 + 8) × (8 + 8)) + (88/8)) + 8)
:= 8 × ((8 × 8 × 8 + 8)/((8 + 8)/8))
99/9
:= 9/9 + (99 × (((99 + 9)/9) + 9))
:= (9/9 + 9) × (99 + 99 + 9)
Ï 2071 := 1 + (11 + (11 + (1 + 1)11 ))
4
:= 9 + (((((9 + 9)/9)
) + 9) + 9)
Ï 2076 := 1 + ((1 + 1)11 + (1 + 1 + 1)1+1+1 )
Ï 2081 := (1 + 1)11 + (11 × (1 + 1 + 1))
:= 22 + 222/2 + 2/2
:= 2 + 222/2 + 22 + 2 + 2
:= 22 + (222/2 + 22/2)
:= (3 × 3 + 3)3 + (((3/3 + 3) + 3)3 )
:= (3 + 3) × ((((3/3 + 3) + 3)3 ) + 3)
:= 33 + (3 − 3/3)33/3
:= (4 + 4) × (44 + 4) − (4/4 + 4 + 4)
:= (4 + 4) × (44 + 4) − 4
:= 4/4 + (4 + 4) × (44 + 4)
:= 55 − (((5 − 5/5)5 + 5 × 5) + 5)
:= 55 − ((5 − 5/5)5 + 5 × 5)
:= 5 + (55 − ((5 − 5/5)5 + 5 × 5))
:= 6 + ((6 × 6 − 6/6) × (66 − (6/6 + 6)))
:= 66 + ((6 − 6/6) × (6 × 66 + 6))
:= (6 × (6 × (((6 + 6)/6)6 − 6))) − 6/6 − 6
:= 7 + (7 × 7 × (7 × 7 − 7) − 7/7 + 7)
:= 7 + 7 × 7 × (7 × 7 − 7) + 77/7
:= 7 + (((7 × 7 × (7 × 7 − 7) + ((7 + 7)/7)) + 7) + 7)
:= 8 + ((((8 × (8 + 8) × (8 + 8)) − 8/8) + 8) + 8)
:= ((8 × (8 × 8 × 8 + 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 × ((8 × 8 × 8 + 8)/((8 + 8)/8)))
:= (9/9 + 9 + 9) × (9/9 + 99 + 9)
:= 9 × 9 × (9 + 9 + 9) − 999/9
:= 9 × 9 + (((9 + 9)/9) × (999 + 9/9))
Ï 2072 := (1 + 1)11 + ((1 + 1) × (1 + 11))
Ï 2077 := (1 + 1)11 + (((11 − 1) × (1 + 1 + 1)) − 1)
Ï 2082 := 1 + ((1 + 1)11 + (11 × (1 + 1 + 1)))
:= 2 + 222/2 + 22
:= 2 × 222 + 2222/2 − 2
:= ((2 × (22 + 2))2 ) − 222
:= 33 + ((3 − 3/3)33/3 − 3)
:= 3 × 33+3 + ((3 − 333)/3)
:= 3 + (3 × (33 × ((3 × (3 + 3)) + 3)))
:= (4 + 4) × ((4 − 4/4) + 4)
:= 4/4 + ((4 + 4) × (4 + 4) − 4)
:= (4 + 4)/4 + (4 + 4) × (44 + 4)
:= (55 + 5/5) × (((5 + 5)/5)5 + 5)
:= 5 + ((55 + 5/5) × (((5 + 5)/5)5 + 5))
:= 5 + (((55 + 5/5) × (((5 + 5)/5)5 + 5)) + 5)
:= 6 + ((((((6 + 6)/6)66/6 ) + 6) + 6) + 6)
:= (66 + 6/6) × ((6 × 6 − 6) + 6/6)
:= (6 × (6 × (((6 + 6)/6)6 − 6))) − 6
:= 7 + (7 × 7 × (7 × 7 − 7) + 7)
:= 7 + 7 × 7 × (7 × 7 − 7) + (77 + 7)/7
:= ((7 + 7 + 7)/7)7 − (7 × (7 + 7) + 7)
:= 8 + (((8 × (8 + 8) × (8 + 8)) + 8) + 8)
:= ((8 + 8 + 8) × (88 − 8/8)) − 88/8
:= ((88/8 + 8) × (888 − 8)/8) − 8
:= 9 + (((99 + 9) × (9/9 + 9 + 9)) + (99/9))
:= 9 × 9 + (((9 + 9)/9) × (999 − 9/9))
:= (9 × (99 + 9)) + ((9999 − 9)/9)
4
4
148
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2083 := 1 + (1 + ((1 + 1)11 + (11 × (1 + 1 + 1))))
Ï 2088 := (1 + 1)11 + ((1 + 1) × ((1 + 1) × (11 − 1)))
Ï 2093 := 1 + ((1 + 1)11 + ((1 + 1) × (11 + 11)))
:= 2 + ((222/2 + 22/2) + 22)
:= 222/2 + 2 × (22 − 2)
:= 2/2 + (222/2 + 2 × 22)
:= 3 + 3 × 33 × (3 × (3 + 3) + 3) + 3/3
:= 3 × (33+3 − 33)
:= 33 + ((33/3)3 + 33+3 )
:= 4 + ((4 + 4) × (44 + 4) − 4/4)
:= 4 + ((4 + 4) × (44 + 4) + 4)
:= 44 + (44 × (4 + 4) + 4/4)
:= 5 + (((((5 + 5)/5)55/5 ) + 5 × 5) + 5)
:= (5 + 5)/5 × (((5 − 5/5)5 − 5) + 5 × 5)
:= 55 + ((5 + 5)/5 × ((5 − 5/5)5 − 5))
:= 6 + ((66 + 6/6) × ((6 × 6 − 6) + 6/6))
:= 6 × (6 × (((6 + 6)/6)6 − 6))
:= 6 × 6 × (66 − 6) − (66 + 6/6)
:= 7 + (7 × 7 × (7 × 7 − 7) + 77/7 + 7)
:= (7 − 7/7) × ((7 × 7 × 7 − ((7 + 7)/7)) + 7)
:= 7 × (7 × (7 × 7 − 7) + 7) − (7 + 7)
:= 8 + ((((8 × (8 + 8) × (8 + 8)) + (88/8)) + 8) + 8)
:= (8 + 8 + 8) × (88 − 8/8)
:= (88 × (8 + 8 + 8)) − (88/8 + 8)
:= (9 × (99 + 9)) + 9999/9
:= 9 × 9 × (9 + 9 + 9) − 99
:= ((99 + 9 + 9)/9) × (9 × (9 + 9) − 9/9)
Ï 2084 := (1 + 1)11 + ((1 + 1 + 1) × (1 + 11))
Ï 2089 := ((1 + 1) × (1111 − 11)) − 111
Ï 2094 := (1 + 1)11 + ((1 + 1) × (1 + (11 + 11)))
:= 222 + (2 × (22 − 2))2
:= 2/2 + (222/2 + 2 × (22 − 2))
:= (2 × 22 + 2)2 − 22
:= 3 + ((3 − 3/3)33/3 + 33)
:= 3/3 + (3 × (33+3 − 33))
:= 3 + ((3 × (33+3 − 33)) + 3)
:= 4 + (4 + 4) × (44 + 4)
:= 4 + (((4 + 4) × (44 + 4) + 4/4) + 4)
:= 44 + (44 × (4 + 4) + (4 + 4)/4)
:= (5 − 5/5) × ((55 + 5/5)/(5/5 + 5))
:= 55 − ((5 − 5/5)5 + ((55 + 5)/5))
:= 55 − (((5 − 5/5)5 + ((5 + 5)/5)) + 5)
:= 6 × 6 + (((6 + 6)/6)66/6 )
:= 6/6 + (6 × (6 × (((6 + 6)/6)6 − 6)))
:= 6 × 6 × (66 − 6) − 66
:= 7 + (7 × 7 × (7 × 7 − 7) + (77 + 7)/7 + 7)
:= ((7 + 7 + 7)/7)7 − 7 × (7 + 7)
:= (7 − 7/7) × ((7 × 7 × 7 − 7/7) + 7)
:= ((8 × (8 × 8 × 8 + 8)) + 8)/((8 + 8)/8)
:= 8/8 + ((8 + 8 + 8) × (88 − 8/8))
:= ((8 + 8)/8) × (8888/8 − 8 × 8)
:= 9/9 + (9 × 9 × (9 + 9 + 9) − 99)
:= (9 × (9 × (9 + 9 + 9) − 9)) − (99 + 9)/9
99/9
:= 9 + ((((((9 + 9)/9)
) + 9) + 9) + 9)
Ï 2085 := (1 + 1)11 + (111/(1 + 1 + 1))
Ï 2090 := (111 − 1) × (((1 + 1) × (11 − 1)) − 1)
Ï 2095 := 1 + ((1 + 1)11 + ((1 + 1) × (1 + (11 + 11))))
:= 2/2 + ((2 × (22 − 2))2 + 222 )
:= 2 × 22 + 222/2 − 2
:= 2/2 + (2 × 22 + 2)2 − 22
:= (3 × (33+3 − 33)) − 3
:= 3 + ((3 × (33+3 − 33)) − 3/3)
:= (3 × (33+3 − 33 )) − 33/3
:= 4 + ((4 + 4) × (44 + 4) + 4/4)
:= 44 + (44 × (4 + 4) − (4 + 4)/4)
:= 4 + ((44 × (4 + 4) − 4/4) + 44)
:= 5 + ((5 + 5) × ((55 − 5)/(5 + 5 + 5)))
:= 55 + (55 × (((5 + 5)/5)5 + 5))
:= ((55 + 5) × ((5 × 5 + 5) + 5)) − 5
:= 6 + ((((6 + 6)/6)66/6 ) + 6 × 6)
:= 6/6 + (6 × 6 × (66 − 6) − 66)
66/6
:= 6 × 6 + ((((6 + 6)/6)
) + 6/6)
:= 77 + (7 × (7 × (7 × 7 − 7) − 7) − 7/7)
:= 7 × 7 + ((((7 + 7)/7)77/7 ) − 7)
:= (88/8 − 8) × (8 × 88 − (8/8 + 8))
:= (88/8 + 8) × (888 − 8)/8
:= 9 + (9 × 9 × (9 + 9 + 9) − 999/9)
Ï 2086 := ((1 + 1 + 11)1+1+1 ) − 111
:= (9/9 + 9 + 9) × (99/9 + 99)
Ï 2091 := 1 + ((111 − 1) × (((1 + 1) × (11 − 1)) − 1))
:= 7 × (7 × (7 × 7 − 7) + 7) − (77 + 7)/7
:= (88 × (8 + 8 + 8)) − (8/8 + 8 + 8)
:= (9 × (9 × (9 + 9 + 9) − 9)) − 99/9
Ï 2096 := (1 + 1)11 + ((1 + 1) × ((1 + 1) × (1 + 11)))
:= 2 + (2 × 22 + 2)2 − 22
:= 2 + ((2 × (22 − 2))2 + 222 )
:= 2 × 22 + (222/2 − 2/2)
:= 3/3 + 3 × (33+3 − 33) − 3
:= 3 + (3 × (33+3 − 33))
:= 4 + ((4 + 4) × (44 + 4) + (4 + 4)/4)
:= 44 + (44 × (4 + 4) − 4/4)
:= 55 − ((((5 − 5/5)5 + 5) + 5) + 5)
:= 55 − (((5 − 5/5)5 + 5) + 5)
:= (6 × (6 × (((6 + 6)/6)6 − 6))) − (6 + 6)/6
:= 666/6 + 66 × (6 × 6 − 6)
:= 77 + 7 × (7 × (7 × 7 − 7) − 7)
:= (7 + 7) × (77 − 7) + 7777/7
:= 7 + (((7 + 7 + 7)/7)7 − 7 × (7 + 7))
:= ((8 + 8 + 8) × (88 − 8/8)) − (8 + 8)/8
:= (88/8 − 8) × ((8 × 88 − 8) + 8/8)
:= (88 × (8 + 8 + 8)) − 8 − 8
:= 999 + (((99 × 99) − (9 + 9))/9)
:= 999/9 + (99 × (99/9 + 9))
:= (9 × (9 × (9 + 9 + 9) − 9)) − 9/9 − 9
Ï 2087 := 1 + (((1 + 1 + 11)1+1+1 ) − 111)
:= 2
22/2
+ (2 × (22 − 2) − 2/2)
Ï 2092 := (1 + 1)11 + ((1 + 1) × (11 + 11))
:= 2 × 22 + 2
:= ((33/3 + 3)3 ) − 3 × (3 + 3)3
:= 4 + (44 × (4 + 4) + 44)
:= 55 − ((5 − 5/5)5 + 5)
:= 6 × 6 × (66 − 6) − ((6 + 6)/6)6
Ï 2097 := (11 − 1 − 1) × (11 + (1 + 1) × 111)
:= 2 + (2 × 22 + 2)2 − 22 + 2/2
22/2
:= (3 × (33+3 − 33)) − 3/3
:= 3 + ((3 × (33+3 − 33)) + 3/3)
:= 3 × ((33+3 − 33) + 3)
:= 4 + (((4 + 4) × (44 + 4) − 4/4) + 4)
:= 44 + 44 × (4 + 4)
:= 4 + ((44 × (4 + 4) + 44) + 4/4)
:= 55 + (5/5 − ((((5 − 5/5)5 + 5) + 5) + 5))
:= 55 + (5/5 − (((5 − 5/5)5 + 5) + 5))
:= 55 + (5/5 − ((5 − 5/5)5 + 5))
:= (6 × (6 × (((6 + 6)/6)6 − 6))) − 6/6
:= 66 × (6 × 6 − 6) + (666 + 6)/6
:= 6 + (66 × (6 × 6 − 6) + 666/6)
:= 7/7 + (7 × (7 × (7 × 7 − 7) − 7) + 77)
:= 7 × 77 + ((((7 + 7) × 777) − 7)/7)
:= 7 × 7 + (((7 + 7)/7)77/7 )
:= ((8 + 8 + 8) × (88 − 8/8)) − 8/8
:= ((8 × 8 × 8 × 8) + 88)/((8 + 8)/8)
:= 8/8 + ((88 × (8 + 8 + 8)) − (8 + 8))
:= 999 + (((99 × 99) − 9)/9)
:= 9 + (9999/9 + (9 × (99 + 9)))
:= (9 × (9 × (9 + 9 + 9) − 9)) − 9
149
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2098 := (1 + 1)11 + ((11 − 1)1+1 /(1 + 1))
2
Ï 2103 := (1 + 1)11 + (111 − 1)/(1 + 1)
2
Ï 2108 := (111 × (((1 + 1) × (11 − 1)) − 1)) − 1
:= 2 × (2 × (((22 + 2/2)2 ) − 2))
:= 2 + (2 × 22 + 2) − 22 + 2
:= 2 + 2222 − (22/2)
:= 3/3 + (3 × ((33+3 − 33) + 3))
:= (3 × (33+3 − 33 )) − 3
:= 3 + ((3 × (33+3 − 33 )) − 3/3)
:= 4 + ((44 × (4 + 4) + (4 + 4)/4) + 44)
:= 44 + (44 × (4 + 4) + 44/4)
:= 44 × (44 + 4) − 4
:= 55 + ((((5 + 5)/5)55/5 ) − 5)
:= 55 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 55)
:= ((6 − 66) × (6/6 − 6 × 6)) − (6 + 6)/6
:= ((6 × (666 + 6 × 6)) − 6)/((6 + 6)/6)
:= 66 + ((((6 + 6)/6)66/6 ) − 6)
:= 7/7 + ((((7 + 7)/7)77/7 ) + 7 × 7)
:= ((7 + 7 + 7)/7)7 − 77 − 7
:= 7/7 + 7 × (7 × (7 × 7 − 7) + 7)
:= 8 + ((88/8 + 8) × (888 − 8)/8)
:= (88 × (8 + 8 + 8)) − (8/8 + 8)
:= (88 × (8 + 8 + 8)) − (8/((8 + 8)/8))
:= 9/9 + ((9 × (9 × (9 + 9 + 9) − 9)) − 9)
:= (9 × (9 × (9 + 9 + 9) − 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 × (9 × (9 + 9 + 9) − 9))
Ï 2099 := ((1 + 1) × (1111 − 1)) − 111+1
2
:= 2 × (2 × (22 × (22 + 2) − 2))
:= 2222 − (22/2) − 2
:= 33/3 + (3 × (3
3+3
Ï 2104 := (1 + 1)11 + (1 + 111)/(1 + 1)
− 33))
:= 3/3 + ((3 × (3
3+3
3
− 3 )) − 3)
Ï 2109 := 111 × (((1 + 1) × (11 − 1)) − 1)
:= 2222 − (222/2 + 2)
:= 3 + (3 × (33+3 − 33 ))
:= 4 + (((44 × (4 + 4) − 4/4) + 44) + 4)
:= 44 × (44 + 4) − 4 − 4
:= 4/4 + (44 × (44 + 4) − 4)
:= 55 − ((5 − 5/5)5 + ((5 + 5)/5))
:= 5 + (55 − ((5 − 5/5)5 + ((5 + 5)/5)))
:= 555/5 × (5 × 5 − (5/5 + 5))
:= ((6 − 66) × (6/6 − 6 × 6)) − 6/6
:= 6 + (((6 − 66) × (6/6 − 6 × 6)) − ((6 + 6)/6))
:= 666/6 × ((6/6 + 6 + 6) + 6)
77/7
:= 7 + ((((7 + 7)/7)
:= ((88/8 − 8)8−8/8 ) − 88
:= (88 × (8 + 8 + 8)) − 8
:= (88/8 + 8) × 888/8
:= 9 + ((9/9 + 9 + 9) × (99/9 + 99))
:= (9 × (9 × (9 + 9 + 9) − 9)) − (9 + 9)/9
:= (9/9 + 9 + 9) × 999/9
Ï 2100 := (11 + (11 − 1)) × (11 − 1)1+1
) + 7 × 7)
:= (7 + 7)/7 + 7 × (7 × (7 × 7 − 7) + 7)
:= 7 × (7 × (7 × 7 − 7) + 7) − (7/7 + 7)
Ï 2105 := 1 + ((1 + 1)11 + (1 + 111)/(1 + 1))
Ï 2110 := 1 + (111 × (((1 + 1) × (11 − 1)) − 1))
:= (2 × 22 + 2)2 − 22+2
:= (2 × 22 + 2)2 − 22/2
:= (2 × 22 + 2)2 − (2 + 2 + 2)
:= 3 + (3 × ((33+3 − 33) + 3))
:= (3 × (33+3 − 33 )) − 3/3
:= 3 + ((3 × (33+3 − 33 )) + 3/3)
:= 4 + ((44 × (4 + 4) + 44) + 4)
:= 4 + (44 × (44 + 4) − 44/4)
:= 44 × (44 + 4) − (4 + 4)/4
:= (55 + 5) × ((5 × 5 + 5) + 5)
:= 5 + ((55 + 5) × ((5 × 5 + 5) + 5))
:= 5 + (((55 + 5) × ((5 × 5 + 5) + 5)) + 5)
:= (6 − 66) × (6/6 − 6 × 6)
:= 6 + (((6 − 66) × (6/6 − 6 × 6)) − 6/6)
:= ((66 − 6)/6) × (6 × 6 × 6 − 6 + 6/6)
:= (7 − 7/7) × (7 × 7 × 7 + 7)
:= 7 × (7 × (7 × 7 − 7) + 7) − (7 + 7)/7
:= ((7 + 7 + 7)/7)7 − 77
:= (88 × (8 + 8 + 8)) − (88 + 8)/8
:= 8/8 + ((88 × (8 + 8 + 8)) − 8)
:= (88 × (8 + 8 + 8)) − (8 + 8)/8
:= (9/9 + 9) × (999/9 + 99)
:= (9 × (9 × (9 + 9 + 9) − 9)) − 9/9
:= 999 + 9999/9
Ï 2106 := 1 + (1 + ((1 + 1)11 + (1 + 111)/(1 + 1)))
Ï 2111 := ((1 + 1) × 1111) − 111
Ï 2101 := ((1 + 1) × 1111) − 111+1
:= 2222 − (22/2)
2
:= 2 + (2 × (2 × ((22 × (22 + 2)) − 2)))
3
3+3
3
:= 2222 − 222/2
:= (33 × ((3/3 + 3)3 )) − 3/3
:= (33 × ((3/3 + 3) )) − 33/3
:= 3 × (3
−3 )
:= 44 × (44 + 4) − 44/4
:= 44 × (44 + 4) − ((4 + 4)/4 + 4)
5
5
:= 44 × (44 + 4) − 4/4
:= 55 − (5 − 5/5)5
:= 5 + (5 − (5 − 5/5) )
:= 5 + ((55 − (5 − 5/5)5 ) + 5)
:= 6/6 + ((6 − 66) × (6/6 − 6 × 6))
:= 6 + ((6 − 66) × (6/6 − 6 × 6))
:= 66/6 + ((6 − 66) × (6/6 − 6 × 6))
:= 7/7 + ((7 − 7/7) × (7 × 7 × 7 + 7))
:= 7 × (7 × (7 × 7 − 7) + 7) − 7/7
:= 7/7 + (((7 + 7 + 7)/7)7 − 77)
:= (88 × (8 + 8 + 8)) − 88/8
:= (8 + 8)/8 + ((88 × (8 + 8 + 8)) − 8)
:= (88 × (8 + 8 + 8)) − 8/8
:= 999 + (9999/9 − 9)
:= 9 × (9 × (9 + 9 + 9) − 9)
:= 999 + (9999 + 9)/9
Ï 2102 := 1 + (((1 + 1) × 1111) − 111+1 )
2
:= 2 + (2 × 22 + 2) − 2
:= (3 × (3
3+3
2+2
3
− 3 )) − (3/3 + 3)
:= (4 − 44)/4 + 44 × (44 + 4)
5
5
Ï 2107 := ((1 + 1) × (1111 − (1 + 1))) − 111
:= 2 + ((2 × 22 + 2)2 − 22/2)
:= 3/3 + (3 × (3
3+3
:= 2 × (2 × (22 × (22 + 2)))
3
:= 33 × ((3/3 + 3)3 )
− 3 ))
:= 44 × (44 + 4) − (4/4 + 4)
5
Ï 2112 := 1 + (((1 + 1) × 1111) − 111)
:= 44 × (44 + 4)
5
:= 5 + ((5/5 − (5 − 5/5) ) + 5 )
:= 55 + (55/5 − (5 − 5/5)5 )
:= 6 + (6 × 6 × (66 − 6) − ((6 + 6)/6)6 )
:= 6 + (((6 − 66) × (6/6 − 6 × 6)) + 6/6)
:= 66 × (((6 + 6)/6 − 6) + 6 × 6)
:= (7 + 7)/7 + ((7 − 7/7) × (7 × 7 × 7 + 7))
:= 7 × (7 × (7 × 7 − 7) + 7)
:= 5 + (5/5 − (5 − 5/5) )
8−8/8
:= (8 − 88)/8 + (88 × (8 + 8 + 8))
:= 8 + (((88/8 − 8)
:= 999 + ((9999 + 9)/9 − 9)
:= 9/9 + (9 × (9 × (9 + 9 + 9) − 9))
) − 88)
150
:= 7 + (7 × (7 × (7 × 7 − 7) + 7) − ((7 + 7)/7))
:= 88 × (8 + 8 + 8)
:= 99/9 × (999/9 + 9 × 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2113 := ((1 + 1) × (1 + 1111)) − 111
2
Ï 2118 := 1 + (1 + (((1 + 1) × (1 + (11 + 11)))1+1 ))
2
Ï 2123 := 1 + (11 + (((1 + 1) × 1111) − 111))
:= 2 + (2 × 22 + 2)2 + 2/2 + 2 + 2
:= (2 × 22 + 2) − 2/2 − 2
:= 2 + (2 × 22 + 2)
:= 3/3 + (33 × ((3/3 + 3)3 ))
:= 3 + ((33 × ((3/3 + 3)3 )) + 3)
:= 3 × 33+3 − ((3/3 + 3)3 )
:= 4/4 + 44 × (44 + 4)
:= 4 + (44 × (44 + 4) + (4 + 4)/4)
:= 44/4 + 44 × (44 + 4)
:= 5 + (((((5 + 5)/5)55/5 ) + 55) + 5)
:= 555 + ((55 + 5/5)/((5 + 5)/5))
:= (5 × (5 + 5 + 5)) + (((5 + 5)/5)55/5 )
:= (6 × (6 × (66 − 6) − 6)) − 66/6
:= (6 × (6 × (66 − 6) − 6)) − 6
:= (6 × (6 × (66 − 6) − 6)) − 6/6
77/7
:= 7 + ((7 × 7 − ((7 + 7 + 7)/7))(7+7)/7 )
:= 7 + (7 × (7 × (7 × 7 − 7) + 7) − 7/7)
:= 77 + ((((7 + 7)/7)
:= 8/8 + (88 × (8 + 8 + 8))
:= 8 + ((88 × (8 + 8 + 8)) − ((8 + 8)/8))
:= 88/8 + (88 × (8 + 8 + 8))
:= 9 + ((9 × (9 × (9 + 9 + 9) − 9)) − ((9 + 9)/9))
:= 9 + ((9/9 + 9 + 9) × 999/9)
:= 9 + (((9 × (9 × (9 + 9 + 9) − 9)) − 9/9) + 9)
Ï 2114 := 1 + (((1 + 1) × (1 + 1111)) − 111)
) − 7)
Ï 2119 := 1 + (1 + (1 + (((1 + 1) × (1 + (11 + 11)))1+1 )))
Ï 2124 := 1111 + (1 + 1)11−1 − 11
:= (2 × 22 + 2)2 − 2
:= 2 + (2 × 22 + 2)2 + 2/2
:= 2 × 2 × ((22 + 2/2)2 + 2)
:= 3 + ((33 × ((3/3 + 3)3 )) − 3/3)
:= 3 + (((33 × ((3/3 + 3)3 )) + 3/3) + 3)
:= 3 × (((33+3 − 33 ) + 3) + 3)
:= (4 + 4)/4 + 44 × (44 + 4)
:= 4 + ((44 × (44 + 4) − 4/4) + 4)
:= 44 + (4 + 4) × (44 + 4)
:= 5 + (555/5 × (5 × 5 − (5/5 + 5)))
:= (5 − 5/5)5 + (55 × (5 × 5 − 5) − 5)
:= (5 − 5/5)5 + 55 × (5 × 5 − 5)
:= 6/6 + ((6 × (6 × (66 − 6) − 6)) − 6)
:= 6 × (6 × (66 − 6) − 6)
:= 7 + 7 × (7 × (7 × 7 − 7) + 7)
:= (77 + 7)/7 + 7 × (7 × (7 × 7 − 7) + 7)
:= 7 + ((((7 + 7 + 7)/7)7 − 77) + 7)
:= (8 + 8)/8 + (88 × (8 + 8 + 8))
:= 8 + ((88 × (8 + 8 + 8)) − 8/8)
:= ((88 + 8)/8) + (88 × (8 + 8 + 8))
:= 9 + ((9 × (9 × (9 + 9 + 9) − 9)) − 9/9)
:= 9 + (9999/9 + 999)
:= 9 + ((9 × (9 × (9 + 9 + 9) − 9)) + 9)
66/6
:= 66 + (((6 + 6)/6)
)
Ï 2115 := (((1 + 1) × (1 + (11 + 11)))1+1 ) − 1
Ï 2120 := 11 + (111 × (((1 + 1) × (11 − 1)) − 1))
Ï 2125 := 1 + (1111 + (1 + 1)11−1 − 11)
:= (2 × 22 + 2)2 − 2/2
:= 2 + (2 × 22 + 2)2 + 2
:= 22/2 + (2 × 22 + 2)2 − 2
:= 3 + (33 × ((3/3 + 3)3 ))
:= 3 × 33+3 − (((3/3 + 3)3 ) + 3)
:= 3/3 + (3 × (((33+3 − 33 ) + 3) + 3))
:= 4 + (44 × (44 + 4) − 4/4)
:= 4 + (44 × (44 + 4) + 4)
:= 44 + ((4 + 4) × (44 + 4) + 4/4)
:= (5 − 5/5) × (555 − 5 × 5)
:= 5 × (5 × (5 × 5 + 55 + 5))
5
:= 555 + (5 × (5 − 5)/(5 + 5))
66/6
:= 6 + (666/6 × ((6/6 + 6 + 6) + 6))
:= 6 + ((((6 + 6)/6)
:= 7 + (7 × (7 × (7 × 7 − 7) + 7) + 7/7)
:= 7 + ((7 × (7 × (7 × 7 − 7) + 7) − 7/7) + 7)
:= 88/8 + ((88 × (8 + 8 + 8)) − 8)
:= 8 + (88 × (8 + 8 + 8))
:= 8 + (((88/8 + 8) × 888/8) + 8)
:= 9 + (9 × (9 × (9 + 9 + 9) − 9))
:= 9 × 9 + ((((9 + 9)/9)99/9 ) − 9)
:= 9 + (((9 × (9 × (9 + 9 + 9) − 9)) + 9/9) + 9)
Ï 2116 := ((1 + 1) × (1 + (11 + 11)))1+1
:= (2 × 22 + 2)
2
) + 66)
:= 6/6 + (6 × (6 × (66 − 6) − 6))
Ï 2121 := (11 + (11 − 1)) × (1 + (11 − 1)1+1 )
:= 77 + (((7 + 7)/7)77/7 )
Ï 2126 := 11 + ((((1 + 1) × (1 + (11 + 11)))1+1 ) − 1)
:= 2 + (2 × 22 + 2)2 + 2/2 + 2
:= 2 + ((2 × 22 + 2)2 + 2 × (2 + 2))
:= 3 + ((33 × ((3/3 + 3) )) + 3/3)
:= 3 × 3 + (33 × ((3/3 + 3)3 ))
:= 3 + (3 × 33+3 − ((3/3 + 3)3 ))
:= 4 + 44 × (44 + 4)
:= 4 + ((44 × (44 + 4) + 4/4) + 4)
:= 4 + (44 × (44 + 4) + (44 − 4)/4)
:= 5 × 5 + (55 − ((5 − 5/5)5 + 5))
:= 5 × 5 + (55 − (5 − 5/5)5 )
:= ((6 × 6/(6 + 6))6/6+6) − 66
:= (6 + 6)/6 + (6 × (6 × (66 − 6) − 6))
:= (7 × 7 − ((7 + 7 + 7)/7))
:= 7 + (7 × (7 × (7 × 7 − 7) + 7) + 7)
:= 7/7 + ((((7 + 7)/7)77/7 ) + 77)
:= (8/((8 + 8)/8)) + (88 × (8 + 8 + 8))
:= 8 + ((88 × (8 + 8 + 8)) + 8/8)
:= 8 + (((88 × (8 + 8 + 8)) − ((8 + 8)/8)) + 8)
:= 9 + ((9 × (9 × (9 + 9 + 9) − 9)) + 9/9)
:= 9 + ((99/9) × (999/9 + 9 × 9))
:= 9 + ((9 × (9 × (9 + 9 + 9) − 9)) + (99/9))
3
5
5
:= 5 + (((5 − (5 − 5/5) ) + 5) + 5)
:= (((66 − 6)/6) + 6 × 6)
(6+6)/6
(7+7)/7
Ï 2117 := 1 + (((1 + 1) × (1 + (11 + 11)))1+1 )
Ï 2122 := 11 + (((1 + 1) × 1111) − 111)
Ï 2127 := 11 + (((1 + 1) × (1 + (11 + 11)))1+1 )
:= 2/2 + (2 × 22 + 2)2
:= 2 + (2 × 22 + 2)2 + 2 + 2
:= 22/2 + (2 × 22 + 2)2
:= 33/3 + (3 × (33+3 − 33 ))
:= 3 × 3 + ((33 × ((3/3 + 3)3 )) + 3/3)
:= 3 × 33+3 − (33 + 33)
:= (44 − 4)/4 + 44 × (44 + 4)
:= 4 + (44 × (44 + 4) + 44/4)
:= 4 + (44 × (44 + 4) + 4/4)
5
5
:= 5 + ((55/5 − (5 − 5/5) ) + 5 )
:= 5 + (((55/5 − (5 − 5/5) ) + 5 ) + 5)
:= 5 × 5 + ((5/5 − (5 − 5/5)5 ) + 55 )
:= (6 × (6 × (66 − 6) − 6)) − 6/6 − 6
:= (6 × (6 × (66 − 6) − 6)) − (6 + 6)/6
:= 6 + (((6 × 6/(6 + 6))6/6+6) − 66)
:= 7 + (((7 + 7 + 7)/7)7 − 77)
:= 7 + ((7 × (7 × (7 × 7 − 7) + 7) + 7/7) + 7)
:= 77 + (7 × 7 × (7 × 7 − 7) − (7/7 + 7))
:= 8 + ((88/8 + 8) × 888/8)
:= 8 + ((88 × (8 + 8 + 8)) + ((8 + 8)/8))
:= 8 + (((88 × (8 + 8 + 8)) − 8/8) + 8)
:= 99/9 + (9 × (9 × (9 + 9 + 9) − 9))
:= 9 + (((9 × (9 × (9 + 9 + 9) − 9)) − ((9 + 9)/9)) + 9)
:= 9 + (((9/9 + 9 + 9) × 999/9) + 9)
5
151
5
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2128 := (1 + 111) × (((1 + 1) × (11 − 1)) − 1)
:= 2 × ((2 × (((22 + 2/2)2 ) + 2)) + 2)
:= 3/3 + (3 × 33+3 − (33 + 33))
Ï 2133 := 1111 + ((1 + 1)11−1 − (1 + 1))
:= 22 + (2 × 22 + 2)2
:= 22 + (2222 − 222/2)
:= 3 × (3
3+3
:= 33 + ((33 × ((3/3 + 3)3 )) − 3/3)
− (3 × (3 + 3)))
4
Ï 2138 := (1 + 1)11 + (11 − 1) × (11 − 1 − 1)
4
:= 4 × ((44 × ((4 + 4) + 4)) + 4)
:= 4 + ((4 − 4/4) + 4 × (4 + 4))
:= 4 + ((4444 − 4)/4 + 4 × 44 )
:= 5 × 5 + ((((5 + 5)/5)55/5 ) + 55)
:= 55 + (((5 + 5)/5)5 − (5 − 5/5)5 )
:= 5 × 555 − ((55 + 55 + 5)/5)
:= 6 + ((6 × (6 × (66 − 6) − 6)) − ((6 + 6)/6))
:= 6 + ((((6 × 6/(6 + 6))6/6+6) − 66) + 6)
:= 6 × 6 × (66 − 6) − ((66 + 66)/6)
:= 77 + (7 × 7 × (7 × 7 − 7) − 7)
:= 77 + (7 × 7 × (7 × 7 − 7) − ((7 + 7)/7))
:= ((7 + 7 + 7)/7)7 − 7 × 7
:= 8 + ((88 × (8 + 8 + 8)) + 8)
:= (88/8 − 8) × ((8 × 88 − 8/8) + 8)
:= 88 + ((8 × (8 + 8) × (8 + 8)) + ((8 + 8)/8))
:= (9/9 + 9 + 9) × ((999 + 9)/9)
:= (9 + 9 + 9) × (9 × 9 − ((9 + 9)/9))
:= 9 + ((((9 + 9)/9)99/9 ) + 9 × 9)
Ï 2129 := (1 + 1)11 + (11 − 1 − 1)1+1
Ï 2134 := 1111 + ((1 + 1)11−1 − 1)
2
:= 2 + ((2 × 22 + 2) + 22/2)
3
:= 2222 − 2 × 2 × 22
33/3
:= 3 × 3 + (3 − 3/3)
4
Ï 2139 := 1 + ((1 + 1)11 + (11 − 1) × (11 − 1 − 1))
:= 3/3 + (3 × (3
4
3+3
:= 22 + ((2 × 22 + 2)2 + 2/2)
− (3 × (3 + 3))))
4
:= (4 − 4/4) + 4 × (4 + 4)
:= 4 × 4 + (4444 − 4)/4
5
:= 33 + (33 × ((3/3 + 3)3 ))
:= 4 + (4444/4 + 4 × 44 )
5
:= 5 + ((5 × (5 × (55 + 5))) + (5 − 5)/5)
:= 555 + ((5 − 5/5) + 555)
:= 5 × 555 − ((55 + 55 )/5)
:= 6 + ((6 × (6 × (66 − 6) − 6)) − 6/6)
:= ((66 − 6)/6) + (6 × (6 × (66 − 6) − 6))
:= ((66 × 66 − 6)/((6 + 6)/6)) − 6 × 6
:= 7/7 + ((7 × 7 × (7 × 7 − 7) − 7) + 77)
:= 77 + 7 × 7 × (7 × 7 − 7) − 7/7
:= 7/7 + (((7 + 7 + 7)/7)7 − 7 × 7)
:= 88 + ((8 × (8 + 8) × (8 + 8)) − ((8 + 8)/8))
:= 8 + (((88 × (8 + 8 + 8)) + (88/8)) + 8)
:= 9/9 + ((9 + 9 + 9) × (9 × 9 − ((9 + 9)/9)))
:= ((99/9 + 9) × 999/9) − 9 × 9
:= 8 + (((88 × (8 + 8 + 8)) + 8/8) + 8)
99/9
:= 9 × 9 + (((9 + 9)/9)
)
Ï 2130 := 1 + ((1 + 1)11 + (11 − 1 − 1)1+1 )
Ï 2135 := 1111 + (1 + 1)11−1
Ï 2140 := 11 + ((1 + 1)11 + (11 − 1 − 1)1+1 )
:= 22+2 + ((2 × 22 + 2)2 − 2)
:= (222/2 + 2222)/2
:= 2 + ((2 × 22 + 2)2 + 22)
:= (33 + 3) × (((3 + 3)3 − 3)/3)
:= 3 + ((33 − 3/3) × (3 × 33 + 3/3))
:= 33 + ((33 × ((3/3 + 3)3 )) + 3/3)
:= 4 × 4 + (44 × (44 + 4) + (4 + 4)/4)
:= 4 × 44 + 4444/4
:= 4 + ((4 + 4) × (44/4 + 44 ))
:= 5 + (5 × (5 × (5 × 5 + 55 + 5)))
:= 5 + ((5 × (5 × (5 × 5 + 55 + 5))) + 5)
:= (5 − 5/5) × ((555 − 5 × 5) + 5)
:= 6 + (6 × (6 × (66 − 6) − 6))
:= 66/6 + (6 × (6 × (66 − 6) − 6))
:= ((66 − 6)/6) × (6 × 6 × 6 − (6 + 6)/6)
:= (7 − 7/7) × ((77 + 7)/7 + 7 × 7 × 7)
:= 77 + 7 × 7 × (7 × 7 − 7)
:= 8 + (((88 × (8 + 8 + 8)) + ((8 + 8)/8)) + 8)
:= (7 + 7)/7 + (((7 + 7 + 7)/7)7 − 7 × 7)
:= 88 + ((8 × (8 + 8) × (8 + 8)) − 8/8)
:= 88 + (((8 × 8 × 8 × 8) + 8)/((8 + 8)/8))
:= 9 + (((9 × (9 × (9 + 9 + 9) − 9)) + (99/9)) + 9)
:= (99/9 + 9) × ((99 − 9/9) + 9)
99/9
:= 9/9 + ((((9 + 9)/9)
) + 9 × 9)
Ï 2131 := 1 + (1 + ((1 + 1)11 + (11 − 1 − 1)1+1 ))
2
:= 2 + (((2 × 22 + 2) + 22/2) + 2)
3
3
:= (((3 + 3) × ((3 + 3) − 3)) + 3)/3
4
4
:= 4 + ((4 − 4/4) × (4/4 + 4) )
5
5
Ï 2136 := 1 + (1111 + (1 + 1)11−1 )
Ï 2141 := (1 + 1)11 + (((1 + 1)11−1 − 1)/11)
2
:= 2222 − (2/2 + 2)2+2
:= 22 + ((2 × 22 + 2) − 2)
:= 3 + (3 × (3
3+3
− (3 × (3 + 3))))
4
:= (33/3)3 + (33 × (33 + 3))
:= (4 − 4/4 + 4)4 − 44 − 4
:= (4 + 4) × (44/4 + 4 )
5
5
:= 5 + ((5 × 5 − (5 − 5/5) ) + 5 )
:= 5 + (((5 × 5 − (5 − 5/5) ) + 5 ) + 5)
:= 6 + ((6 × (6 × (66 − 6) − 6)) + 6/6)
:= 6 + ((6 × (6 × (66 − 6) − 6)) + 6)
:= 6 + ((6 × (6 × (66 − 6) − 6)) + (66/6))
:= 7/7 + (7 × 7 × (7 × 7 − 7) + 77)
:= 7 + (7 × 7 × (7 × 7 − 7) − 7/7 + 77)
:= 8 + ((88 × (8 + 8 + 8)) + (88/8))
:= 88 + (8 × (8 + 8) × (8 + 8))
:= 8 + ((88/8 − 8) × ((8 × 88 − 8/8) + 8))
:= ((99/9 + 9) × ((99 − 9/9) + 9)) − 9
:= 9 + ((((9/9 + 9 + 9) × 999/9) + 9) + 9)
:= 9 + ((9 × (99 + 9 × 9)) + (((9 + 9)/9)9 ))
7
:= ((7 + 7 + 7)/7) − (7 × 7 + 7)
Ï 2132 := 1111 + ((1 + 1)11−1 − 1 − 1 − 1)
Ï 2137 := 1 + (1 + (1111 + (1 + 1)11−1 ))
:= ((5 − 55 )/5) + (5 × 555 − (5 + 5))
Ï 2142 := (1 + 1) × ((11 − 1 − 1) × (111+1 − (1 + 1)))
:= 22+2 + (2 × 22 + 2)2
:= 22 + ((2 × 22 + 2)2 − 2/2)
:= 2 + (((2 × 22 + 2)2 + 22) + 2)
:= (33 − 3/3) × (3 × 33 + 3/3)
:= 3 + ((3 × (33+3 − (3 × (3 + 3)))) + 3/3)
:= 3 × ((33+3 − (3 × (3 + 3))) + 3)
:= 4 + (44 × (44 + 4) + 4 × 4)
:= 4/4 + ((4 + 4) × (44/4 + 44 ))
:= 4/4 + (4 − 4/4 + 4)4 − 44 − 4
(5+5)/5+5
:= ((5 − (5 + 5)/5)
) − 55
:= 6 + ((6 × (6 × (66 − 6) − 6)) + ((6 + 6)/6))
77/7
:= 7 + ((((7 + 7)/7)
) + 77)
:= 8 + ((88 × (8 + 8 + 8)) + ((88 + 8)/8))
9
:= (((9 + 9)/9) ) + (9 × (99 + 9 × 9))
(5+5)/5+5
:= 5 + (((5 − (5 + 5)/5)
) − 55)
:= 6 + (((6 × (6 × (66 − 6) − 6)) + 6/6) + 6)
7
:= (5 − 5/5 + 5) × (((5 − (5 + 5)/5)5 ) − 5)
:= 6 × (66 × 66/(6 + 6) − 6)
:= ((7 + 7 + 7)/7) − (7/7 + 7 × 7)
:= 7 + (7 × 7 × (7 × 7 − 7) + 77)
:= 8/8 + ((8 × (8 + 8) × (8 + 8)) + 88)
:= (8/8 + 8 + 8) × (8 × (8 + 8) − ((8 + 8)/8))
:= 9 + ((9/9 + 9 + 9) × ((999 + 9)/9))
:= (9 + 9) × ((99/9 + 99) + 9)
152
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2143 := 11 × ((1 + 1 + 1 + 11)1+1 − 1) − 1 − 1
Ï 2148 := 111 + ((1 + 1)11 − 11)
Ï 2153 := 111 + ((1 + 1)11 − ((1 + 1) × (1 + 1 + 1)))
:= 2 + (2222 − (2/2 + 2)2+2 )
:= 2 × ((2 + 2 + 2)2+2 − 222)
:= ((22/2 + 2)2/2+2 ) − (2 × 22)
:= 3 × 33+3 − (33/3 + 33)
:= 3 + ((3 × (33+3 − 3)) − 33)
:= 3 × 33+3 − 3/3 − 33
:= (44 − 4)/4 + (4 + 4) × (44 + 4)
:= 4 + ((4 + 4) × ((44 + 4 + 4) + 4))
:= 4 + (((((4 − 4/4) + 4)4 ) − 44 ) + 4)
:= 5 × 555 − ((55 + 5 + 5)/5 + 5)
:= 5 × 555 − (55 + 5 + 5)/5
:= 5 + (5 × 555 − (55 + 5 + 5)/5)
:= 6 × 6 × (66 − 6) − (66/6 + 6)
:= 6 × 6 × (66 − 6) − 6 − 6
:= 6 × 6 × (66 − 6) − 6/6 − 6
:= 7 + ((7 × 7 × (7 × 7 − 7) + 77) + 7/7)
:= 77 × (7 + 7 + 7 + 7) − (7/7 + 7)
:= 7 + ((((7 + 7)/7)77/7 ) + 7 × (7 + 7))
:= 8 + (((8 × (8 + 8) × (8 + 8)) − 8/8) + 88)
:= 8 + ((((8 × 8 × 8 × 8) + 8)/((8 + 8)/8)) + 88)
:= 8 + ((((8 × (8 + 8) × (8 + 8)) + 88) + 8/8) + 8)
:= 9/9 + ((9 + 9) × ((99/9 + 99) + 9))
:= 9/9 + ((((9 + 9)/9)99/9 ) + 99)
:= (9 + 9)/9 + (((9 + 9) × (999/9 + 9)) − 9)
Ï 2144 := 11 × ((1 + 1 + 1 + 11)1+1 − 1) − 1
:= 2
2+2
× (22 × (2 + 2 + 2) + 2)
3
:= (33 − 3/3) × (((3/3 + 3) ) + 3)
4
:= (4 + 4) × ((4 + 4 + 4) + 4)
5
:= 5 × 555 − ((5 + 5)/5 + 5)
Ï 2149 := 1 + (111 + ((1 + 1)11 − 11))
Ï 2154 := (11 × (1 + 1 + 1 + 11)1+1 ) − 1 − 1
:= 22 + ((2 × 22 + 2)2 + 22/2)
:= 222 + (2 × (2 × 222 − 2))
:= 3 × 33+3 − (33/3 + 33 )
:= 3 × 33+3 − 33
:= 4 + ((((4 − 4/4) + 4)4 ) − 44 )
:= 44 + (44 × (44 + 4) − (4 + 4)/4)
:= 5 × 555 − (5 + 5)/5
:= 5 + (5 × 555 − (55 + 5)/5)
:= (6 − 66)/6 + (6 × 6 × (66 − 6) − 6)
:= 6 × 6 × (66 − 6) − 66/6
:= 6 × 6 × (66 − 6) − 6
:= 7 + (((7 + 7 + 7)/7)7 − (7/7 + 7 × 7))
:= 77 × (7 + 7 + 7 + 7) − 7
:= ((7 + 7)/7) × (77 × (7 + 7) − 7/7)
:= 8 + ((8 × (8 + 8) × (8 + 8)) + 88)
:= (88 × (8 + 8 + 8)) + 888/(8 + 8 + 8)
:= 8 × 8 + ((88/8 + 8) × (888 − 8)/8)
:= (9 + 9)/9 + ((9 + 9) × ((99/9 + 99) + 9))
:= 9 + ((99/9 + 9) × ((99 − 9/9) + 9))
:= ((9 + 9 + 9)/9) × (9 × 9 × 9 − (99/9))
5
Ï 2145 := 11 × ((1 + 1 + 1 + 11)1+1 − 1)
:= 22/2 × (((2
2+2
2
− 2) ) − 2/2)
Ï 2150 := 1 + (1 + (111 + ((1 + 1)11 − 11)))
Ï 2155 := (11 × (1 + 1 + 1 + 11)1+1 ) − 1
:= 222 + (2 × (2 × (222 − 2)))
:= ((22 × ((22+2 − 2)2 )) − 2)/2
:= (3 × 3 + 3/3) × ((3 + 3)3 − 3/3)
:= 3/3 + (3 × 33+3 − 33)
:= 4 + (4 − 4/4 + 4)4 − 44 + 4/4
:= 44 + (44 × (44 + 4) − 4/4)
:= 55 × (55 − (55/5 + 5))
:= 5 × (555 − 5 × 5 × 5)
:= 5 + 5 × (555 − 5 × 5 × 5)
:= (66 − 6/6) × (66 × 6/(6 + 6))
:= (6 − 66)/6 + 6 × 6 × (66 − 6)
:= 6/6 + (6 × 6 × (66 − 6) − 6)
:= 7 + (((7 + 7 + 7)/7)7 − 7 × 7)
:= 7/7 + (77 × (7 + 7 + 7 + 7) − 7)
:= 77 × (7 + 7 + 7 + 7) − 7/7
:= 8 + (((8 × (8 + 8) × (8 + 8)) + 88) + 8/8)
:= (8/8 + 8 + 8 + 8) × (88 − ((8 + 8)/8))
:= 8 + (((8 × (8 + 8) × (8 + 8)) + (88/8)) + 88)
:= (9/9 + 9) × (((9 + 9) × (99 + 9) − 9)/9)
:= 9 + (((((9 + 9)/9)99/9 ) − 9/9) + 99)
:= (3 × (3
3+3
− 3)) − 33
4
:= (((4 − 4/4) + 4) ) − 4
4
99/9
:= 99 + ((((9 + 9)/9)
) − ((9 + 9)/9))
Ï 2146 := 1 + 11 × ((1 + 1 + 1 + 11)1+1 − 1)
:= 2 + (2 × 2 × (22 + 2) + 2
:= 3/3 + ((3 × (3
3+3
22/2
)
− 3)) − 33)
4
4
:= 4/4 + ((((4 − 4/4) + 4) ) − 4 )
Ï 2151 := 1 + (1 + (1 + (111 + ((1 + 1)11 − 11))))
2
2
Ï 2156 := 11 × (1 + 1 + 1 + 11)1+1
:= (2/2 + 2) × ((22 − 2)/2 − 2)
:= 22 × (2 × 2 × (22 + 2) + 2)
:= 3 × (33+3 − (3 × 3 + 3))
:= 3 + (3 × 33+3 − (3/3 + 33))
4
:= 4 × (4 + 4) + 4444/4
:= 44 + 44 × (44 + 4)
:= ((5 − 5 )/5) + (5 × 555 − 5)
:= ((5 − 55 )/5) + 5 × 555
:= 55 + (55 − (5 − 5/5)5 )
:= (6 × 6 + 6/6) × (((6 + 6)/6)6 − 6)
:= ((6 × 6/(6 + 6))6/6+6) − 6 × 6
:= (6 + 6)/6 + (6 × 6 × (66 − 6) − 6)
:= (7 + 7)/7 + (77 × (7 + 7 + 7 + 7) − 7)
:= 77 × ((7 + 7 + 7) + 7)
:= (8/8 + 8) × (888/8 + 8 × (8 + 8))
:= (8 − 8/8) × ((88 × (8 × 8 − 8))/(8 + 8))
:= ((9 + 9) × (999/9 + 9)) − 9
:= 9 + ((((9 + 9)/9)99/9 ) + 99)
5
77/7
:= 7 × (7 + 7) + (((7 + 7)/7)
)
:= 8 + (((8 × (8 + 8) × (8 + 8)) + ((8 + 8)/8)) + 88)
99/9
:= 99 + ((((9 + 9)/9)
) − 9/9)
Ï 2147 := 111 + ((1 + 1)11 − (1 + 11))
Ï 2152 := 1 + (1 + (1 + (1 + (111 + ((1 + 1)11 − 11)))))
Ï 2157 := 111 + (1 + 1)11 − (1 + 1)
:= 222 + ((2 × 22)2 − 22/2)
:= (2 + 2 + 2)2 + (2 × 22 + 2)2
:= 222 + ((2 × 22)2 − 2/2)
:= 3 × 33 + (3 − 3/3)33/3
:= 3/3 + (3 × (33+3 − (3 × 3 + 3)))
:= 3 + (3 × 33+3 − 33)
:= 44 + (44 × (44 + 4) − 4)
:= 44 + (44 × (44 + 4) + 4/4)
4
4
:= 4 + (4 + 4) × (4 + 4) + (4 − 4)/4
5
5
:= 5 × 555 + ((((5 − 5 ) + 5)/5) − 5)
:= 5 × 555 + (((5 − 5 ) + 5)/5)
:= 5 + ((((5 − 55 ) + 5)/5) + 5 × 555)
:= 6 × 6 × (66 − 6) − (6/6 + 6 + 6)
:= 6 × 6 × (66 − 6) − ((6 + 6)/6 + 6)
:= 6 × 6 × (66 − 6) − 6 × 6/(6 + 6)
:= 77 × (7 + 7 + 7 + 7) − ((7 + 7)/7 + 7)
:= 88 + ((8 × (8 + 8) × (8 + 8)) + (88/8))
99/9
:= 99 + (((9 + 9)/9)
)
7
:= 7 + ((((7 + 7 + 7)/7) − 7 × 7) + 7)
:= 7/7 + 77 × (7 + 7 + 7 + 7)
:= 8 + (((8 × (8 + 8) × (8 + 8)) + 88) + 8)
:= ((8 + 8) × (8 × (8 + 8) + 8)) − (88/8 + 8)
:= 9/9 + (((9 + 9) × (999/9 + 9)) − 9)
:= ((9 + 9 + 9)/9) × (9 × 9 × 9 − (9/9 + 9))
153
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2158 := 111 + (1 + 1)11 − 1
Ï 2163 := 1 + (1 + (1 + (1 + (111 + (1 + 1)11 ))))
Ï 2168 := 111+1 + ((1 + 1)11 − 1)
:= 222 + (2 × 22)2
:= 2 + ((222/2 + 222/2 ) + 2)
:= 2 + ((2 × (2 × (222 + 2))) + 222)
:= 3 + ((3 × 33+3 − 33) + 3/3)
:= 3 + (3 × (33+3 − 3 × 3))
:= (3 × (33+3 − (3 + 3))) − 3/3
:= 44 × (4 + 4) + (444 − 4)/4
:= 4 + (444/4 + 44 × (4 + 4))
:= (4 + 4) × ((44/4 + 44 ) + 4)
:= (5 + 5)/5 × ((5 − 5/5)5 + 55)
:= 5 + ((5 + 5)/5 × ((5 − 5/5)5 + 55))
:= 5 × 5 × 5 + ((((5 + 5)/5)55/5 ) − 5)
:= 6 × 6 × (66 − 6) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + 6 × 6 × (66 − 6)
:= 6 + (6 × 6 × (66 − 6) + ((6 + 6)/6))
:= (7 + 7)/7 + 77 × (7 + 7 + 7 + 7)
:= 7 + 77 × (7 + 7 + 7 + 7)
:= ((7 + 7)/7) × ((77 × (7 + 7) − 7/7) + 7)
:= (888 − 8)/8 + (8 × (8 + 8) × (8 + 8))
:= ((88/8 − 8)8−8/8 ) − 8 − 8 − 8
:= ((8 + 8) × (8 × (8 + 8) + 8)) − 8
:= ((9 + 9)/9) × ((999 − 9/9) + 9 × 9)
:= ((9 + 9 + 9)/9) × ((9 × 9 × 9 − 9) + 9/9)
:= 9 × 9 × (9 + 9 + 9) − (9/9 + 9 + 9)
Ï 2159 := 111 + (1 + 1)11
:= 222/2 + 2
:= 3 × 3
3+3
Ï 2164 := 1 + (1 + (1 + (1 + (1 + (111 + (1 + 1)11 )))))
22/2
3
− (3 + 3/3)
4
:= (22 − 2)2 + (2 × 22 − 2)2
:= (22/2)2 + 222/2
:= (((3 × 3 + 3/3) + 3)3 ) − 33
:= 3 × (33+3 − (3 + 3))
:= 4 + (((4 + 4)4 ) − 44 × 44)
:= 444/4 + 4 × (4 + 4)
5
Ï 2169 := 111+1 + (1 + 1)11
5
(5+5)/5
:= 4 + ((4/4 + 4) × (444 − 44/4))
:= 5 + ((5 × 555 − (5 + 5)/5) + 5)
:= 5 − (((5 × 5 + 5/5) + 5)
:= 6 × 6 × (66 − 6) − 6/6
:= 6 + (6 × 6 × (66 − 6) − ((6 + 6)/6))
)
:= 5 × 555 − ((55 × 55 + 5)/5)
:= ((66 × 66 − 6)/((6 + 6)/6)) − 6
:= 7 + (77 × (7 + 7 + 7 + 7) + 7/7)
:= 777/7 + 7 × 7 × (7 × 7 − 7)
:= 888/8 + (8 × (8 + 8) × (8 + 8))
:= ((8 + 8) × (8 × (8 + 8) + 8)) − (88 + 8)/8
:= 8/8 + (((8 + 8) × (8 × (8 + 8) + 8)) − 8)
:= ((9 + 9) × (999/9 + 9)) − 9/9
:= 9999/9 + (9 × (99 + 9 + 9))
:= 9 × 9 × (9 + 9 + 9) − (9 + 9)
7
:= 7 × 7 + (((7 + 7 + 7)/7) − 77)
Ï 2160 := 1 + (111 + (1 + 1)11 )
Ï 2165 := 111 + ((1 + 1)11 + ((1 + 1) × (1 + 1 + 1)))
Ï 2170 := 1 + (111+1 + (1 + 1)11 )
:= 2 + ((2 × 22)2 + 222)
:= 222 + ((2 × (22 − 2) + 2/2)2 )
:= 2 × ((22/2 + 22)2 − (2 + 2))
:= 3 × (33+3 − 3 × 3)
:= 33/3 + (3 × 33+3 − 33)
:= 3/3 + (3 × (33+3 − (3 + 3)))
:= ((4 + 4)4 ) − 44 × 44
:= (4/4 + 4) × (444 − 44/4)
:= (4/4 + 4) × ((4 − 44)/4 + 444)
:= (55 + 5) × (55/5 + 5 × 5)
:= ((5 − 5/5) × 555) − 55
:= 5 + (((5 − 5/5) × 555) − 55)
:= 6 × 6 × (66 − 6)
:= 6 + (6 × 6 × (66 − 6) − 6/6)
:= ((66 − 6)/6) + 6 × 6 × (66 − 6)
:= (77/7 − 7) × (7 × 77 + 7/7)
:= 7 + (77 × (7 + 7 + 7 + 7) + ((7 + 7)/7))
:= 7 + (77 × (7 + 7 + 7 + 7) + 7)
:= (8 + 8) × ((8 × (8 + 8) − 8/8) + 8)
:= ((8 + 8) × (8 × (8 + 8) + 8)) − 88/8
:= (8 + 8)/8 + (((8 + 8) × (8 × (8 + 8) + 8)) − 8)
:= (9 + 9) × (999/9 + 9)
Ï 2161 := 1 + (1 + (111 + (1 + 1)11 ))
99/9
:= 9 + (((((9 + 9)/9)
) + 99) + 9)
Ï 2166 := 111+1 + ((1 + 1)11 − (1 + 1 + 1))
:= 9/9 + (9 × 9 × (9 + 9 + 9) − (9 + 9))
Ï 2171 := 1 + (1 + (111+1 + (1 + 1)11 ))
:= 2 + (222/2 + 222/2 )
:= 222 + (2 × (2 × (222 + 2)))
:= 2 + ((22/2)2 + 222/2 )
:= 3/3 + (3 × (33+3 − 3 × 3))
:= (3 × (33+3 − (3 + 3))) − 3
:= 3 + ((3 × (33+3 − (3 + 3))) − 3/3)
:= 4 × 4 + ((((4 − 4/4) + 4)4 ) − 44 )
:= (4 − 44)/4 + (4 + 4) × (4 × 4 + 44 )
:= (4 + 4) × (4 × 4 + 44 ) − (4/4 + 4)
:= 5 + ((55 − (5 − 5/5)5 ) + 55 )
:= 5 + (((55 − (5 − 5/5)5 ) + 55 ) + 5)
:= ((5 − 55 )/5) + ((5 × (555 + 5)) − 5)
:= 6/6 + 6 × 6 × (66 − 6)
:= 6 + 6 × 6 × (66 − 6)
:= 66/6 + 6 × 6 × (66 − 6)
:= 7 + (((7 + 7)/7) × (77 × (7 + 7) − 7/7))
:= ((7 + 7 + 7)/7)7 − (7 + 7 + 7)
:= 7 + ((77 × (7 + 7 + 7 + 7) + 7/7) + 7)
:= 8/8 + ((8 + 8) × ((8 × (8 + 8) − 8/8) + 8))
:= (8 − 88)/8 + ((8 + 8) × (8 × (8 + 8) + 8))
:= ((88/8 − 8)8−8/8 ) − 8 − 8
:= 9/9 + ((9 + 9) × (999/9 + 9))
:= 9 × 9 × (9 + 9 + 9) − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + (9 × 9 × (9 + 9 + 9) − (9 + 9))
Ï 2162 := 1 + (1 + (1 + (111 + (1 + 1)11 )))
:= 2 + (((2 × 22)2 + 222) + 2)
Ï 2167 := 11 × (1 + (1 + 1 + 1 + 11)1+1 )
Ï 2172 := 1 + (1 + (1 + (111+1 + (1 + 1)11 )))
:= (22/2)2 + 222/2 − 2
:= 2222 − ((2 × (22 + 2)) + 2)
:= (3 × (33+3 − 3)) − 33/3
:= 3 + (3 × (33+3 − (3 + 3)))
:= 4 + ((444 − 4)/4 + 44 × (4 + 4))
:= 4 + ((444/4 + 44 × (4 + 4)) + 4)
:= (4 + 4) × (4 × 4 + 44 ) − 4
:= 5 × 555 + (((55 − 55 ) + 5)/5)
:= ((5 + 5)/5 × 5555/5) − 55
:= (5 + 5)/5 × (5555/5 − 5 × 5)
:= (6 + 6)/6 + 6 × 6 × (66 − 6)
:= 6 + (6 × 6 × (66 − 6) + 6/6)
:= 6 + (6 × 6 × (66 − 6) + 6)
:= 7 + (77 × (7 + 7 + 7 + 7) − 7/7)
:= 77/7 + 77 × (7 + 7 + 7 + 7)
:= ((7 + 7 + 7)/7)7 − (7/7 + 7 + 7)
:= (8 + 8)/8 + ((8 + 8) × ((8 × (8 + 8) − 8/8) + 8))
:= ((8 + 8) × (8 × (8 + 8) + 8)) − (8/8 + 8)
:= ((8 + 8) × (8 × (8 + 8) + 8)) − (8/((8 + 8)/8))
:= (9 + 9)/9 + ((9 + 9) × (999/9 + 9))
:= 99/9 × ((99 − 9/9) + 99)
:= ((99 + 9)/9) × ((9/9 + 99) + 9 × 9)
:= 3 + (3 × 3
3+3
3
− (3 + 3/3))
154
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2173 := 1 + (1 + (1 + (1 + (111+1 + (1 + 1)11 ))))
Ï 2178 := (1 + 1) × (11 × (1 + 1 + 1))1+1
Ï 2183 := 1 + (1 + (1 + ((1 + 1)11 + (11 × (1 + 11)))))
:= 2 + (((22/2)2 + 222/2 ) + 2)
:= 2 × (22/2 + 22)2
:= ((2 × (22 + 2))2 ) − (22/2)2
:= 3 × 33+3 − (33/3 + 3)
:= 3 × (33+3 − 3)
:= 3 × 33+3 − (3/3 + 3)
:= 4/4 + ((4 + 4) × (4 × 4 + 44 ) − 4)
:= (4 + 4)/4 + (4 + 4) × (4 × 4 + 44 )
:= ((4 − 4/4)4+4−4/4 ) − 4
:= 5 × 5 × 5 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 × 5 × 5)
:= (((5 + 5)/5)5 + 5) × (55 − 5/5 + 5)
:= 6 + ((6 × 6 × (66 − 6) + 6/6) + 6)
:= 6 × 66 × 66/(6 + 6)
:= (6 × 6 + 6/6) × (66 − (6/6 + 6))
7
:= (77/7 + 7) × (((7 + 7)/7) − 7)
:= 7 + (((7 + 7 + 7)/7)7 − (77/7))
:= 8 + (((8 + 8) × (8 × (8 + 8) + 8)) − (88/8))
:= (8 + 8)/8 + ((8 + 8) × (8 × (8 + 8) + 8))
:= 8 + (((8 + 8) × (8 × (8 + 8) + 8)) − 8/8)
:= 9 + (9999/9 + (9 × (99 + 9 + 9)))
:= 9 × 9 × (9 + 9 + 9) − 9
:= ((9 − 9 × 9)/(9 + 9)) + 9 × 9 × (9 + 9 + 9)
7
:= ((7 + 7 + 7)/7) − (7 + 7)
Ï 2174 := (1 + 1) × (1111 − ((1 + 1) × (1 + 11)))
:= 2 × ((22/2 + 22)2 − 2)
Ï 2179 := 1 + ((1 + 1) × (11 × (1 + 1 + 1))1+1 )
:= (((22+2+2 + 2)2 ) + 2)/2
3+3
Ï 2184 := (1 + 1 + 1) × ((11 − 1 − 1)1+1+1 − 1)
:= 2 + (2222 + (2 × (2 − 22)))
:= 3 × 33+3 − 3
:= (3 × (33+3 − 3)) − (3/3 + 3)
:= 3/3 + (3 × (3
:= ((((4 + 4)4 ) − 4) + 44 )/((4 + 4)/4)
:= 4 + ((4 + 4) × (4 × 4 + 44 ) − 4/4)
:= 4 + ((4 + 4) × (4 × 4 + 44 ) + 4)
:= (5 × (555 + 5)) − (55 + 5)/5
:= 5 + ((5 × (555 + 5)) − (55 + 5)/5)
:= ((5 + 5)/5 + 5) × (55 − 5)/(5 + 5)
:= 6 + ((6 × 6 × (66 − 6) + ((6 + 6)/6)) + 6)
:= 6/6 + (6 × 66 × 66/(6 + 6))
:= 6 + (6 × 66 × 66/(6 + 6))
− 3))
7
:= 7/7 + (((7 + 7 + 7)/7)7 − (7 + 7))
:= ((7 + 7 + 7)/7) − (7/7 + 7)
:= 77 + 7 × (7 × (7 × 7 − 7) + 7)
:= ((8 + 8) × (8 × (8 + 8) + 8)) − (8 + 8)/8
:= ((88/8 − 8)8−8/8 ) − 8
:= 8 + ((8 + 8) × (8 × (8 + 8) + 8))
:= ((9 + 9)/9) × (((99 × 99) − (9 + 9))/9)
:= 9/9 + (9 × 9 × (9 + 9 + 9) − 9)
:= ((9 + 9 + 9)/9) × (9 × 9 × 9 − 9/9)
Ï 2175 := ((1 + 1 + 11)1+1+1 ) − 11 − 11
Ï 2180 := (1 + 1)11 + (11 × (1 + 11))
Ï 2185 := ((1 + 1 + 11)1+1+1 ) − 1 − 11
:= ((22/2 + 2)2/2+2 ) − 22
:= 2 + (2 × (22/2 + 22)2 )
:= ((2/2 + 2)2/2+2+2+2 ) − 2
:= (3 × (33+3 − 3)) − 3
:= 3 + ((3 × (33+3 − 3)) − 3/3)
:= 3/3 + (3 × 33+3 − 3)
:= (4 + 4) × (4 × 4 + 44 ) − 4/4
:= 4 + (4 + 4) × (4 × 4 + 44 )
:= 4 + (((4 + 4) × (4 × 4 + 44 ) + 4/4) + 4)
:= 5 × ((555 − 5 × 5 × 5) + 5)
:= (5 − 5/5) × (555 − 5 − 5)
:= 5 + ((5 − 5/5) × (555 − 5 − 5))
:= (66 × 66 − 6)/((6 + 6)/6)
:= 66 + ((((6 + 6)/6)66/6 ) + 66)
:= 6 × (6 × (66 − 6) + 6) − 66/6
:= ((7 + 7 + 7)/7)7 − (77 + 7)/7
:= ((7 + 7 + 7)/7)7 − 7
:= ((8 + 8) × (8 × (8 + 8) + 8)) − 8/8
:= 8/8 + (((88/8 − 8)8−8/8 ) − 8)
:= 9 × 9 × (9 + 9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 × 9 × (9 + 9 + 9) − 9)
Ï 2176 := (1 + 1) × ((11 × (1 + 1 + 1))1+1 − 1)
Ï 2181 := 1 + ((1 + 1)11 + (11 × (1 + 11)))
:= ((7 + 7 + 7)/7)7 − (7 + 7)/7
:= 8 + (((8 + 8) × (8 × (8 + 8) + 8)) + 8/8)
:= 9 × 9 × (9 + 9 + 9) − (9 + 9)/9
Ï 2186 := ((1 + 1 + 11)1+1+1 ) − 11
:= 2222 − (2 + 2 + 2)2
:= 2222 − ((2 × 22) + 2)
:= 2 + ((((22+2+2 + 2)2 ) + 2)/2)
:= 3 × 33+3 − 33/3
:= 3 + (3 × (33+3 − 3))
:= (4 + 4) × (4 × 4 + 44 )
:= 4 + ((4 + 4) × (4 × 4 + 44 ) + 4/4)
:= (5 − 5/5) × (555 − (55/5))
:= 5 + ((5 × (555 + 5)) + ((5 − 55 )/5))
:= ((6 + 6)/6)6 × (6 × 6 − ((6 + 6)/6))
:= (66 × 66 + 6)/((6 + 6)/6)
:= ((7 + 7 + 7)/7)7 − 77/7
:= 7/7 + (((7 + 7 + 7)/7)7 − 7)
:= (8 + 8) × (8 × (8 + 8) + 8)
:= (88/8 × (888/8 + 88)) − 8
:= ((88/8 − 8)8−8/8 ) − 8/8
:= 9 × 9 × (9 + 9 + 9) − 99/9
:= ((9 + 9 + 9)/9) + (9 × 9 × (9 + 9 + 9) − 9)
:= 9 × 9 × (9 + 9 + 9) − 9/9
Ï 2177 := ((1 + 1) × (11 × (1 + 1 + 1))1+1 ) − 1
Ï 2182 := 1 + (1 + ((1 + 1)11 + (11 × (1 + 11))))
:= 3 × 33+3 − 3/3
:= ((4 − 4/4)4+4−4/4 ) − 4/4
:= (55 + 5)/5 + (5 × (55 − 5)/(5 + 5))
:= ((6 × 6/(6 + 6))6/6+6) − 6/6
:= ((7 + 7 + 7)/7)7 − 7/7
Ï 2187 := (1 + 1 + 1)1+(1+1)×(1+1+1)
:= (((22+2+2 + 2)2 ) − 2)/2
:= 2 × ((22/2 + 22)2 + 2)
:= (2/2 + 2)2/2+2+2+2
:= (3 × (33+3 − 3)) − 3/3
:= 3 + ((3 × (33+3 − 3)) + 3/3)
:= 3 × 33+3
:= 4/4 + (4 + 4) × (4 × 4 + 44 )
:= 4 + ((4 + 4) × (4 × 4 + 44 ) + (4 + 4)/4)
:= (4 − 4/4)4+4−4/4
:= (((5 − 55 ) + 5)/5) + (5 × (555 + 5))
:= ((5 − (5 + 5)/5)(5+5)/5+5 ) − 5
:= (5 − (5 + 5)/5)(5+5)/5+5
:= 6 + (6 × 6 × (66 − 6) + (66/6))
:= 6 + (((6 + 6)/6)6 × (6 × 6 − ((6 + 6)/6)))
:= (6 × 6/(6 + 6))6+6/6
:= 7 + ((77 × (7 + 7 + 7 + 7) + 7) + 7)
:= (7 + 7)/7 + (((7 + 7 + 7)/7)7 − 7)
:= ((7 + 7 + 7)/7)7
:= 8/8 + ((8 + 8) × (8 × (8 + 8) + 8))
:= 8 + (((8 + 8) × (8 × (8 + 8) + 8)) − ((8 + 8)/8))
:= (88/8 − 8)8−8/8
:= 9 × 9 × (9 + 9 + 9) − 9/9 − 9
:= ((9 + 9)/9) × ((((99 × 99) + 9) + 9)/9)
:= 9 × 9 × (9 + 9 + 9)
155
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2188 := 1 + ((1 + 1 + 1)1+(1+1)×(1+1+1) )
Ï 2193 := 1 + ((1 + 1)11 + (1 + 11)1+1 )
Ï 2198 := 1 + ((1 + 1 + 11)1+1+1 )
:= 2 + (2222 − (2 + 2 + 2)2 )
:= ((2 × (22 + 2))2 ) − 222/2
:= 2222 − (22 + 2)
:= 3/3 + 3 × 33+3
:= 3 + (3 × 33+3 + 3)
:= 33/3 + 3 × 33+3
:= 44 + (44 × 44 − 4)
:= 4/4 + (44 × 44 + 44 )
:= 4 + ((44 × 44 + (4 + 4)/4) + 44 )
:= 55 + (((5 − 55 )/(5 + 5)) − 55 /5)
:= 5 + ((((5 − 55 )/(5 + 5)) − 55 /5) + 55 )
:= (5 + 5)/5 × (55 × (5 × 5 − 5) − 5/5)
:= 6/6 + ((6 × 6/(6 + 6))6/6+6)
:= 6 + ((6 × 6/(6 + 6))6/6+6)
:= 6 × (6 × (66 − 6) + 6) + (6 + 6)/6
:= 7/7 + ((7 + 7 + 7)/7)7
:= 7 + (((7 + 7 + 7)/7)7 − 7/7)
:= 77/7 + ((7 + 7 + 7)/7)7
:= 8/8 + ((88/8 − 8)8−8/8 )
:= (8/8 + 8 + 8) × (8 × (8 + 8) + 8/8)
:= 88 + ((88 × (8 + 8 + 8)) − ((8 + 8)/8))
:= 9/9 + 9 × 9 × (9 + 9 + 9)
:= 9 + (((9 + 9 + 9)/9) × (9 × 9 × 9 − 9/9))
:= 99/9 + 9 × 9 × (9 + 9 + 9)
Ï 2189 := 11 × (((1 + 1) × (11 − 1)1+1 ) − 1)
Ï 2194 := ((1 + 1 + 11)1+1+1 ) − 1 − 1 − 1
Ï 2199 := 1 + (1 + ((1 + 1 + 11)1+1+1 ))
:= 2 + ((2/2 + 2)2/2+2+2+2 )
:= 2 + ((2 × 22)2 + 22×(2+2) )
:= 2 + ((22/2 + 2)2/2+2 )
:= 3 + (3 × 33+3 − 3/3)
:= (((3 × 3 + 3/3) + 3)3 ) − 3
:= 3 + (3 × (33+3 + 3))
:= 44 + ((((4 − 4/4) + 4)4 ) − 44 )
:= 44 + (44 × 44 + (4 + 4)/4)
:= ((4/4 + 4) × (444 − 4)) − 4/4
:= 5 + (((5 + 5)/5 + 5) × (55 − 5)/(5 + 5))
:= (5 × (5 + 5) × 55) − (555 + 5/5)
:= ((5 + 5) × (55 × (5 − 5/5))) − 5/5
:= 6 × 6 × (66 − 6 + 6) − 6/6 − 6
:= 6 × (6 × (66 − 6) + 6) − (6 + 6)/6
:= 6 + (((6 × 6/(6 + 6))6/6+6) + 6)
:= (7 + 7)/7 + ((7 + 7 + 7)/7)7
:= 7 + ((7 + 7 + 7)/7)7
:= (77 + 7)/7 + ((7 + 7 + 7)/7)7
:= 88/8 × (888/8 + 88)
:= 8 + (((88/8 − 8)8−8/8 ) − 8/8)
:= 88 + ((88 × (8 + 8 + 8)) − 8/8)
:= 9 + (9 × 9 × (9 + 9 + 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + 9 × 9 × (9 + 9 + 9)
:= (9 + 9)/9 + 9 × 9 × (9 + 9 + 9)
Ï 2190 := (11 − 1) × (((1 + 1) × (111 − 1)) − 1)
:= 2222 − (2 × 2
:= 3 + 3 × 3
2+2
Ï 2195 := ((1 + 1 + 11)1+1+1 ) − 1 − 1
:= ((22/2 + 2)
)
3+3
:= (3 × (3
3+3
2/2+2
)−2
+ 3)) − 3/3
Ï 2200 := (1 + 1) × (1111 − 11)
:= 2222 − 22
:= 3 + (((3 × 3 + 3/3) + 3)3 )
:= 4 + (((4 − 4/4)4+4−4/4 ) + 4)
:= (4/4 + 4) × (444 − 4)
:= 5 × ((5 + 5)/(5 + 5) + 5 × 5 × 5)
:= (5 × (5 + 5) × 55) − 555
:= (5 + 5) × (55 × (5 − 5/5))
:= 6 × (6 × (66 − 6) + 6) − 6
:= 6 × (6 × (66 − 6) + 6) − 6/6
:= 6 + 6 × (6 × (66 − 6) + 6) − (6 + 6)/6
:= (7 + 7 + 7)/7 + ((7 + 7 + 7)/7)7
:= 7 + (((7 + 7 + 7)/7)7 + 7/7)
:= 7 + ((((7 + 7 + 7)/7)7 − 7/7) + 7)
:= ((8 + 8)/8) × (8888/8 − (8 + 8))
:= 8 + ((88/8 − 8)8−8/8 )
:= 88 + (88 × (8 + 8 + 8))
:= ((9 + 9 + 9)/9) + 9 × 9 × (9 + 9 + 9)
:= 9 + (9 × 9 × (9 + 9 + 9) − 9/9)
:= (99/9 + 9) × (99/9 + 99)
:= 44 + (44 × 44 − (4 + 4)/4)
5
Ï 2191 := (1 + 1)11 + (11 × (1 + 1 + 11))
:= 2 + (((2/2 + 2)2/2+2+2+2 ) + 2)
Ï 2196 := ((1 + 1 + 11)1+1+1 ) − 1
:= 2/2 + (2222 − 22)
:= 2222 − 22 − 2 − 2
3+3
Ï 2201 := 1 + ((1 + 1) × (1111 − 11))
:= 3 + (3 × 33+3 + 33/3)
:= 3 + (3 × 33+3 + 3/3)
:= 3 × (3
:= 4 + ((4 − 4/4)4+4−4/4 )
:= 4 + (44 × 44 + 44 )
:= 4 + (((4/4 + 4 + 4) + 4)4−4/4 )
:= ((5 + 5)/5 + 5) × (55 + 5)/(5 + 5)
:= (5 − 5/5) × (555 − (5/5 + 5))
:= 5/5 + ((5 + 5) × (55 × (5 − 5/5)))
:= 6 × (6 × (66 − 6) + 6) − 6 + 6/6
:= 6 × (6 × (66 − 6) + 6)
:= 6 + 6 × (6 × (66 − 6) + 6) − 6/6
:= 7 + (7 × (7 × (7 × 7 − 7) + 7) + 77)
:= 7 + (((7 + 7 + 7)/7)7 + ((7 + 7)/7))
:= 7 + (((7 + 7 + 7)/7)7 + 7)
:= 8 + ((((8 + 8) × (8 × (8 + 8) + 8)) − 8/8) + 8)
:= 8 + (((88/8 − 8)8−8/8 ) + 8/8)
:= 8/8 + ((88 × (8 + 8 + 8)) + 88)
:= 9 × 9 + (9999/9 + 999)
:= 9 + 9 × 9 × (9 + 9 + 9)
:= 9 × (9 + 9) + ((((9 + 9)/9)99/9 ) − 9)
Ï 2192 := (1 + 1)11 + (1 + 11)1+1
2
:= (2 × 22) + 2
:= 3 + ((3 × 3
2×(2+2)
3+3
− 3/3) + 3)
+ 3)
Ï 2197 := (1 + 1 + 11)1+1+1
:= (22/2 + 2)
Ï 2202 := (1 + 1) × (1 + (1111 − 11))
2/2+2
:= 2 + (2222 − 22)
:= ((3 × 3 + 3/3) + 3)
3
:= 3 + ((3 × (33+3 + 3)) + 3)
:= 44 + 44 × 44
:= ((4/4 + 4 + 4) + 4)4−4/4
:= 44 + (44 × 44 + (44 − 4)/4)
:= 5 + ((5 − (5 + 5)/5)(5+5)/5+5 )
:= ((55 + 5 + 5)/5)5−(5+5)/5
:= 5 + ((55 + 5 + 5)/5)5−(5+5)/5
:= 6 + (((6 × 6/(6 + 6))6/6+6) − 6/6)
:= 6 × (6 × (66 − 6) + 6) + 6/6
:= 6 + 6 × (6 × (66 − 6) + 6)
7
(7+7+7)/7
:= 7 + ((((7 + 7 + 7)/7)7 + 7/7) + 7)
:= 7 + (((7 + 7 + 7)/7) − ((7 + 7)/7))
:= (7 − 7/7 + 7)
:= 8 + (((8 + 8) × (8 × (8 + 8) + 8)) + 8)
:= ((88 + 8 + 8)/8)88/8−8
:= 88 + ((88 × (8 + 8 + 8)) + ((8 + 8)/8))
:= 9 + (9 × 9 × (9 + 9 + 9) + 9/9)
:= ((9 + 9)/9) × (((9999 − 9)/9) − 9)
:= 9 × 9 × (9 + 9 + 9) + ((9 × 9 + 9)/(9 + 9))
156
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2203 := 1 + ((1 + 1) × (1 + (1111 − 11)))
:= 2 + ((2222 − 22) + 2/2)
Ï 2208 := 11 + ((1 + 1 + 11)1+1+1 )
:= 2 + (2222 − 2
3
2+2
Ï 2213 := ((1 + 1) × (1 + 1111)) − 11
:= 2 + (2222 − 22/2)
)
:= 3 + ((((3 × 3 + 3/3) + 3) ) + 3)
:= 3 + (3 × ((33+3 + 3) + 3))
:= 33 + (3 × 33+3 − 3/3)
:= 4 × 4 + ((4 − 4/4)4+4−4/4 )
:= (4 + 4) × ((4 × 4 + 44 ) + 4)
:= 4 + ((44 − 4/4 + 4)(4+4)/4 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 × (5 × 5 + 5))
:= (5 − 5/5) × (((5 + 5)/5 − 5) + 555)
:= ((5 − 5/5) × 555) − ((5 + 5)/5 + 5)
:= 6 + 6 × (6 × (66 − 6) + 6) + 6/6
:= 6 × (6 × (66 − 6) + 6) + 6 + 6
:= 6 + (6 × 6 × (66 − 6 + 6) + (66/6))
:= 7 + ((((7 + 7 + 7)/7) + ((7 + 7)/7)) + 7)
:= 7 + ((7 + 7 + 7)/7) + 7 + 7
:= 7 + ((7 × (7 × (7 × 7 + 7) − 77)) + 7/7)
:= 8 + (((88/8 − 8)8−8/8 ) + 8)
:= 8 + ((88 × (8 + 8 + 8)) + 88)
:= ((88 + 8 + 8)/8)88/8−8 + 8 + 8
:= 9 + ((9 × 9 × (9 + 9 + 9) − ((9 + 9)/9)) + 9)
:= 9 + (9 × 9 × (9 + 9 + 9) + ((99 + 9)/9))
:= (9 + 9 + 9) × (9/9 + 9 × 9) − 9/9
7
Ï 2204 := (1 + 1) × (1 + (1 + (1111 − 11)))
7
Ï 2209 := ((1 + 1) × (1111 − 1)) − 11
:= ((2 × 22 + 2/2) + 2)
:= 2 + ((2222 − 22) + 2)
3+3
2
:= (3 × ((33+3 + 3) + 3)) − 3/3
:= 3 + ((3 × ((3
:= 4 + ((4/4 + 4) × (444 − 4))
:= (44 − 4/4 + 4)(4+4)/4
:= (5 − 5/5) × ((5 + 5) × 55 + 5/5)
:= ((5 − 5/5) × 555) − 55/5
:= 6 + (6 × 6 × (66 − 6 + 6) + ((6 + 6)/6))
:= (66/6 + 6 × 6)
+ 3) + 3)) + 3/3)
(6+6)/6
Ï 2214 := 1 + (((1 + 1) × (1 + 1111)) − 11)
:= 2222 − 2 × (2 + 2)
:= 3 × (33+3 + 3 × 3)
:= (4 + 4)/4 × (4444/4 − 4)
:= ((5 − 5/5) × 555) − (5/5 + 5)
:= 6 × (66 × 66/(6 + 6) + 6)
:= (7 × 7 − (7 + 7)/7)(7+7)/7
:= 7 + ((((((7 + 7 + 7)/7)7 − 7/7) + 7) + 7) + 7)
:= ((8 + 8)/8) × ((8888 − 8)/8 − 8)
:= (8 × 8 − (8/8 + 8 + 8))(8+8)/8
:= (((8 + 8)/8) × 8888/8) − 8
:= 9 + ((9 × 9 × (9 + 9 + 9) − 9/9) + 9)
:= 9 + ((99/9 + 9) × (99/9 + 99))
:= 7 + ((7 − 7/7 + 7)
(7+7+7)/7
)
Ï 2205 := 1 + ((1 + 1) × (1 + (1 + (1111 − 11))))
2
:= (2/2 + 2 + 2) × ((22 − 2/2) )
Ï 2210 := (11 − 1) × ((1 + 1) × 111 − 1)
:= 2222 − (2 × (2 + 2 + 2))
:= (9 + 9 + 9) × (9/9 + 9 × 9)
Ï 2215 := ((1 + 1) × (1 + (1 + 1111))) − 11
:= 2 + ((2222 − 22/2) + 2)
:= 3 + ((3 × (33+3 + 3)) + 33/3)
:= 33 + (3 × 33+3 + 3/3)
:= (4/4 + 4) × ((444 − 4) + 4/4)
:= (4/4 + 4) × (444 − (4 + 4)/4)
:= (4/4 + 4) × (444 − 4/4)
:= 5 + ((5 + 5) × (55 × (5 − 5/5)))
:= ((5 − 5/5) × 555) − 5 − 5
:= ((5 − 5/5) × 555) − 5
:= 6 + ((((6 × 6/(6 + 6))6/6+6 ) + 6) + 6)
:= ((6 + 6)/6) × ((6666/6) − 6)
:= 6 + ((66/6 + 6 × 6)(6+6)/6 )
:= 7 × (7 × (7 × 7 + 7) − 77)
:= 7/7 + (7 × 7 − (7 + 7)/7)(7+7)/7
:= 7 + (((((7 + 7 + 7)/7)7 + 7) + 7) + 7)
:= 8 + ((88 + 8 + 8)/8)88/8−8
:= (8/8 + 8 + 8) × (8 × (8 + 8) + ((8 + 8)/8))
:= 888/8 + ((88 × (8 + 8 + 8)) − 8)
:= 9 + (9 × 9 × (9 + 9 + 9) + 9)
:= 9 × (9 + 9) + (((9 + 9)/9)99/9 )
:= 9/9 + (9 + 9 + 9) × (9/9 + 9 × 9)
:= 3 × ((3
3+3
+ 3) + 3)
Ï 2206 := 11 + (((1 + 1 + 11)1+1+1 ) − (1 + 1))
Ï 2211 := ((1 + 1) × 1111) − 11
Ï 2216 := (1 + 1) × (1111 − (1 + 1 + 1))
:= 2222 − 22/2
:= 2222 − (2 + 2 + 2)
:= 3 × 3 + (((3 × 3 + 3/3) + 3)3 )
:= 33 + (3 × 33+3 − 3)
:= 3 + ((3 × 33+3 − 3/3) + 33 )
:= 4 × 4444/(4 + 4) − 4 × 4
:= ((4/4 + 4) × (444 − 4/4)) − 4
:= (4/4 + 4) × 444 − 4
:= 5 + (((5 + 5) × (55 × (5 − 5/5))) + 5/5)
:= ((5 − 5/5) × (555 − 5/5)) − 5
:= (5 − 5/5) × (555 − 5/5)
:= 6 × (6 × (66 − 6) + 6) + (66 − 6)/6
:= (66 + 6/6) × (66 × 6/(6 + 6))
:= (((6 + 6)/6) × (6666/6)) − 6
:= 7/7 + (7 × (7 × (7 × 7 + 7) − 77))
:= 7 + (((7 − 7/7 + 7)(7+7+7)/7 ) + 7)
:= 7 + ((7 × 7 − (7 + 7)/7)(7+7)/7 )
:= ((8 + 8)/8) × (8888/8 − 8)
:= 8 + ((((88/8 − 8)8−8/8 ) + 8) + 8)
:= ((88 + 8) × (8 + 8 + 8)) − 88
:= 9 + ((9 × 9 × (9 + 9 + 9) + 9/9) + 9)
:= ((99/9 + 9) × 999/9) − 9
:= 9 + ((9 × 9 × (9 + 9 + 9) + (99/9)) + 9)
:= 2222 − 2
2+2
Ï 2207 := 11 + (((1 + 1 + 11)1+1+1 ) − 1)
2
Ï 2212 := 1 + (((1 + 1) × 1111) − 11)
Ï 2217 := ((1 + 1) × (1111 − (1 + 1))) − 1
:= ((2 − 22)/2) + 2222
:= 2222 − (2/2 + 2 + 2)
:= 33 + ((3 × 33+3 − 3) + 3/3)
:= 3 + (3 × 33+3 + 33 )
:= ((4 + 4) × ((4 × 4 + 44 ) + 4)) − 4/4
:= 4 + ((4 + 4) × ((4 × 4 + 44 ) + 4))
:= 4/4 + ((4/4 + 4) × 444 − 4)
:= ((5 + 5)/5 × (5555/5 − 5)) − 5
:= (5 + 5)/5 × (5555/5 − 5)
:= ((5 + 5)/5 × 5555/5) − 5
:= 6 × (6 × (66 − 6) + 6) + 66/6
:= ((6 + 6)/6) × ((6666 + 6)/6 − 6)
:= 6 + ((66 + 6/6) × (66 × 6/(6 + 6)))
:= 7 + (((((7 + 7 + 7)/7)7 − 7/7) + 7) + 7)
:= 7 + (7 × (7 × (7 × 7 + 7) − 77))
:= 7 × (7 × 7 × 7 − 7) − (((7 + 7)/7)7 + 7)
:= 8 + (((88 × (8 + 8 + 8)) − 8/8) + 88)
:= (((8 + 8)/8) × (8888 − 8)/8) − 8
:= 8 + ((8 × 8 − (8/8 + 8 + 8))(8+8)/8 )
:= 9 + (9 × 9 × (9 + 9 + 9) + (99/9))
:= ((9/9 + 9 + 9) + 9) × (9 × 9 − ((9 + 9)/9))
:= 999/9 + (9 × (9 × (9 + 9 + 9) − 9))
:= (((2 × 22 + 2/2) + 2) ) − 2
:= 33/3 + (3 × (3
3+3
+ 3))
157
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2218 := (1 + 1) × (1111 − (1 + 1))
:= 3 + ((3 × 3
Ï 2228 := (1 + 1) × (1 + (1 + (1 + 1111)))
Ï 2223 := 1 + ((1 + 1) × 1111)
:= 2 + ((2222 + 2) + 2)
:= 2/2 + 2222
:= 2222 − 2 − 2
3+3
3
+ 3 ) + 3/3)
:= 3 + (3 × 3
3+3
:= 33 + ((3 × (33+3 + 3)) − 3/3)
+ 33)
:= 4 × 4444/(4 + 4) − 4
:= 4 + ((4/4 + 4) × 444 − 4/4)
:= 4 + ((4/4 + 4) × 444 + 4)
:= ((5 − 5/5) × 555) − (5 + 5)/5
:= 5 + (((5 − 5/5) × 555) − ((5 + 5)/5))
:= (5 − 5/5) × (555 + (5 + 5)/5)
:= ((6 + 6)/6) × ((6666 − (6 + 6))/6)
:= 6 × 6 + ((6 × 6/(6 + 6))
6/6+6)
:= 6 + (((6 + 6)/6) × (6666/6))
:= 7 × 7 + (((7 + 7 + 7)/7)7 − (7/7 + 7))
:= (7 + 7)/7 × (7777 − 7 − 7)/7
:= 7 + (((7 × 7 − (7 + 7)/7)
:= ((8 + 8)/8) × ((8888 − (8 + 8))/8)
:= 888/8 + (88 × (8 + 8 + 8))
:= 8 + (((8 + 8)/8) × (8888 − 8)/8)
:= ((9 + 9)/9) × ((9999 − (9 + 9))/9)
:= 9 + (9 + 9 + 9) × (9/9 + 9 × 9)
:= 99 + ((((9 + 9)/9)99/9 ) + 9 × 9)
Ï 2219 := ((1 + 1) × (1111 − 1)) − 1
:= 2222 − 2/2 − 2
(7+7)/7
) + 7)
Ï 2229 := 1 + ((1 + 1) × (1 + (1 + (1 + 1111))))
Ï 2224 := (1 + 1) × (1 + 1111)
:= 2 + 2222
:= 2 + (((2222 + 2/2) + 2) + 2)
:= 33 + (((3 × 3 + 3/3) + 3)3 )
:= 33 + (3 × (33+3 + 3))
:= (4/4 + 4) × 444 − 4/4
:= 4 + (4/4 + 4) × 444
:= 4 + ((4/4 + 4) × (444 + 4/4))
:= 33 + (3 × 3
3+3
− 3/3)
:= ((5 − 5/5) × 555) − 5/5
:= (5 − 5/5) × (555 + 5/5)
:= 5 + ((5 − 5/5) × (555 + 5/5))
:= ((66 − 6) × (6 × 6 + 6/6)) − 6/6
:= ((6 + 6)/6) × (6666 + 6)/6
:= 6 × (6 × 66 − 6) − 666/6
:= 7 + ((7 × (7 × (7 × 7 + 7) − 77)) + 7)
:= ((7 + 7)/7) × (7777 + 7)/7
:= 7 × 7 + (((7 + 7 + 7)/7)7 − 7)
)
:= (8 + 8) × (8 × (8 + 8) + (88/8))
:= (8 × (8 × (8 + 8 + 8) + 88)) − 88/8
) + 9 × (9 + 9))
:= ((9 + 9)/9) × (9999 + 9)/9
:= 9 + ((99/9 + 9) × 999/9)
:= 888 + ((88/8)
88/8−8
99/9
:= 9 + ((((9 + 9)/9)
Ï 2220 := (1 + 1) × (1111 − 1)
Ï 2230 := (11 − 1) × (1 + (1 + 1) × 111)
:= 2 + (2222 + 2/2)
:= 2222 − 2
:= 33 + 3 × 3
Ï 2225 := 1 + ((1 + 1) × (1 + 1111))
3
:= 2 × (2 + 2) + 2222
3
:= 33 + (((3 × 3 + 3/3) + 3)3 )
:= 3 + ((33 × 3 ) + (33/3) )
3+3
:= (4/4 + 4) × 444
:= (4/4 + 4) × (444 + 4/4)
:= 4 + (4 × 4444/(4 + 4) + 4)
:= (5 − 5/5) × 555
:= 5 + ((5 − 5/5) × 555)
:= 5 + (((5 − 5/5) × 555) + 5)
:= (66 − 6) × (6 × 6 + 6/6)
:= 66 + (6 × 6 × (66 − 6) − 6/6)
:= 6 + (6 × 6 × (66 − 6) + ((6 + 6)/6)6 )
:= ((7 + 7)/7) × (7777 − 7)/7
:= 7 × 7 + (((7 + 7 + 7)/7)7 − (77/7))
:= 7/7 + ((((7 + 7 + 7)/7)7 − 7) + 7 × 7)
:= ((8 + 8)/8) × (8888 − 8)/8
:= (8/8 + 88) × (8/8 + 8 + 8 + 8)
:= 8 + (((8 + 8)/8) × 8888/8)
:= (99/9 + 9) × 999/9
:= 99/9 + (9 + 9 + 9) × (9/9 + 9 × 9)
:= 9 + (((99/9 + 9) × 999/9) + 9/9)
Ï 2221 := ((1 + 1) × 1111) − 1
Ï 2231 := 11 + ((1 + 1) × (1111 − 1))
:= 2 + (2222 + 2)
:= 2222 − 2/2
:= 3/3 + (3 × 3
Ï 2226 := (1 + 1) × (1 + (1 + 1111))
3+3
+ 33)
:= 3 + ((3 × 3
3+3
:= 22/2 + (2222 − 2)
:= 33 + (3 × 33+3 + 33/3)
+ 33) + 3)
:= 4/4 + (4/4 + 4) × 444
:= 4 + 4 × 4444/(4 + 4)
:= 44 + ((4 − 4/4)4+4−4/4 )
:= 5/5 + ((5 − 5/5) × 555)
:= 5 + (((5 − 5/5) × 555) + 5/5)
:= 55/5 + ((5 − 5/5) × 555)
:= 6/6 + ((66 − 6) × (6 × 6 + 6/6))
:= 66 + 6 × 6 × (66 − 6)
:= (6 × ((6 × (66 − 6) + 6) + 6)) − 6/6
:= (((7 + 7)/7) × 7777/7) − 7/7
:= 7 × (7 × 7 × 7 − 7 − 7) − 77
:= 7 + (((7 + 7)/7) × (7777 + 7)/7)
:= (((8 + 8)/8) × 8888/8) − 8/8
:= ((8 + 8)/8) × (((8888 + 8) + 8)/8)
:= 8 + ((88 × (8 + 8 + 8)) + 888/8)
:= 9/9 + ((99/9 + 9) × 999/9)
:= ((9 + 9)/9) × (((9999 + 9) + 9)/9)
:= 9 + (((9 + 9)/9) × 9999/9)
Ï 2222 := (1 + 1) × 1111
Ï 2227 := 1 + ((1 + 1) × (1 + (1 + 1111)))
:= 2222
:= 2 + ((2222 + 2/2) + 2)
Ï 2232 := 11 + (((1 + 1) × 1111) − 1)
:= 2 + (2222 + 2 × (2 + 2))
3
3
:= (3 − 3/3) × 3333/3
:= 3 + ((((3 × 3 + 3/3) + 3) ) + 3 )
:= 3 + ((3 × (33+3 + 3)) + 33)
:= 4 × 4444/(4 + 4)
:= 4 + (((4/4 + 4) × 444 − 4/4) + 4)
:= (4/4 + 4 + 4) × (44 − 4 − 4)
:= (5 + 5)/5 × 5555/5
:= 5 + ((5 + 5)/5 × 5555/5)
:= (5 + 5)/5 × (5555/5 + 5)
:= ((6 + 6)/6) × (6666/6)
:= 66 + (6 × 6 × (66 − 6) + 6/6)
:= 6 × ((6 × (66 − 6) + 6) + 6)
:= ((7 + 7)/7) × 7777/7
:= ((7 × 7 − 7/7)(7+7)/7 ) − 77
:= 77 + (77 × (7 + 7 + 7 + 7) − 7/7)
:= ((8 + 8)/8) × 8888/8
:= 8 + (((88/8)88/8−8 ) + 888)
:= (8/8 + 8) × ((8 + 8) × (8 + 8) − 8)
:= ((9 + 9)/9) × 9999/9
:= 9 + (((9 + 9)/9) × ((9999 − (9 + 9))/9))
:= 9 + ((9 + 9 + 9) × (9/9 + 9 × 9) + 9)
158
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2238 := (1 + 1) × (((11 − 1) × (1 + 111)) − 1)
Ï 2233 := 11 + ((1 + 1) × 1111)
:= 22/2 + 2222
:= 2
2+2
Ï 2243 := ((1 + 1) × (11 + 1111)) − 1
:= 22 + (2222 − 2/2)
+ 2222
:= 3 + ((((3 × 3 + 3/3) + 3) ) + 33)
:= 33 + (3 × ((33+3 + 3) + 3))
:= 3 + (((3 − 3/3)3×3 ) + (3 × 3 + 3)3 )
:= 4 + (((4/4 + 4) × (444 + 4/4)) + 4)
:= 4 × 4 + 4 × 4444/(4 + 4)
:= (4 × (4/4 + 4)4 ) − (4/4 + 44 )
:= 5 + ((5 − 5/5) × (555 + (5 + 5)/5))
:= ((5 − 5/5) × (555 + 5)) − (5 + 5)/5
:= 5 + (((5 − 5/5) × (555 + 5)) − ((5 + 5)/5))
:= 6/6 + (6 × ((6 × (66 − 6) + 6) + 6))
:= 6 + (6 × ((6 × (66 − 6) + 6) + 6))
:= (66 × (6 × 6 − ((6 + 6)/6))) − 6/6
3
:= 77 + 77 × (7 + 7 + 7 + 7)
:= 7 × 7 + (((7 + 7 + 7)/7) + ((7 + 7)/7))
:= 7 + (((7 + 7 + 7)/7)7 + 7 × 7)
:= 8 + ((8/8 + 88) × (8/8 + 8 + 8 + 8))
:= ((8 + 8)/8) × (8888/8 + 8)
:= 8 × 8 + (((88/8 − 8)8−8/8 ) − 8)
:= 9 + (((9 + 9)/9) × (9999 + 9)/9)
:= 9 + (((99/9 + 9) × 999/9) + 9)
:= (9 + 9)/9 + ((9 + 9 + 9) × (((9 + 9)/9) + 9 × 9))
7
Ï 2234 := 1 + (11 + ((1 + 1) × 1111))
:= 2222 + (2 × (2 + 2 + 2))
Ï 2239 := ((1 + 1) × ((11 − 1) × (1 + 111))) − 1
Ï 2244 := (1 + 1) × (11 + 1111)
:= 2/2 + (2222 + 22+2 )
:= 22 + 2222
:= 3 × 3 + ((((3 × 3 + 3/3) + 3)3 ) + 33)
:= 3 + (3 × (33+3 + (3 × (3 + 3))))
:= 4 + ((4 × 4444/(4 + 4) + 4) + 4)
:= ((4/4 + 4) × (444 + 4)) − 4/4
:= (4 × (4/4 + 4)4 ) − 44
:= 5 + (((5 − 5/5) × (555 + 5/5)) + 5)
:= ((5 − 5/5) × (555 + 5)) − 5/5
:= (5 − 5/5) × ((555 + 5/5) + 5)
:= ((6 + 6)/6) × ((6666/6) + 6)
:= 6 + ((6 × ((6 × (66 − 6) + 6) + 6)) + 6/6)
:= 66 × (6 × 6 − ((6 + 6)/6))
:= ((7 + 7) × (777/7 + 7 × 7)) − 7/7
:= 7 + ((((7 + 7 + 7)/7)7 + 7 × 7) + 7/7)
:= 8 + (((8 + 8)/8) × (((8888 + 8) + 8)/8))
:= (8 × (8 × (8 + 8 + 8) + 88)) − 8/8
:= ((8 + 8)/8) × ((8888 + 88)/8)
:= 9 + ((9 + 9 + 9) × (9/9 + 9 × 9) + (99/9))
:= 99 + ((99/9 + 9) × ((99 − 9/9) + 9))
:= ((9 + 9)/9) × ((9999 + 99)/9)
:= 3 + ((3 × 3
3+3
+ 33/3) + 33)
:= 7 + (((7 × 7 − 7/7)(7+7)/7 ) − 77)
Ï 2235 := 11 + ((1 + 1) × (1 + 1111))
:= 2 + (2222 + 22/2)
Ï 2240 := (1 + 1) × ((11 − 1) × (1 + 111))
:= 2 + (2222 + 2
2+2
:= 22 + (2222 + 2/2)
)
:= (3 × (33+3 + 33 )) − 33
:= (3 × 3 + 3) + ((3 − 3/3)
:= (4/4 + 4) × ((444 − 4/4) + 4)
:= (4/4 + 4) × (444 + 4)
:= ((5 − 5/5) × (555 + 5)) − 5
:= (5 − 5/5) × (555 + 5)
3
:= 6 + (6 × (6 × 66 − 6) − 666/6)
Ï 2245 := 1 + ((1 + 1) × (11 + 1111))
3×3
:= 3 + ((3 × (33+3 + (3 × (3 + 3)))) + 3/3)
)
:= 4/4 + ((4 × (4/4 + 4)4 ) − 44 )
:= 5 + ((5 − 5/5) × (555 + 5))
6
:= (6 × 6 − 6/6) × ((6 + 6)/6)
:= 6/6 + (66 × (6 × 6 − ((6 + 6)/6)))
:= 7 + ((((7 + 7 + 7)/7)7 + ((7 + 7)/7)) + 7 × 7)
:= 8 × 8 + (((88/8 − 8)8−8/8 ) − (8 + 8))
:= (7 + 7) × (777/7 + 7 × 7)
:= 8 × (8 × (8 + 8 + 8) + 88)
:= 8 × 8 × (8 + 8) + 88/8 × 888/8
:= 9 + (((9 + 9)/9) × (((9999 + 9) + 9)/9))
:= ((9 + 9)/9) × (9999/9 + 9)
:= 9999/9 + (9 × ((99 + 9 + 9) + 9))
:= 7 × 7 + (((7 + 7 + 7)/7)7 − 7/7)
Ï 2236 := 1 + (11 + ((1 + 1) × (1 + 1111)))
2+2
Ï 2241 := 1 + ((1 + 1) × ((11 − 1) × (1 + 111)))
Ï 2246 := (1 + 1) × (1 + (11 + 1111))
:= 22 + (2222 − (2/2 + 2))
:= 2 + (2222 + 22)
:= 3 + (((((3 × 3 + 3/3) + 3) ) + 33) + 3)
:= 3 × (33+3 + (3 × (3 + 3)))
:= 33 + ((3 × 33+3 − 3/3) + 33)
:= 4 × 4 + (4/4 + 4) × 444
:= 4/4 + ((4/4 + 4) × (444 + 4))
:= (4 + 4)/4 + ((4 × (4/4 + 4)4 ) − 44 )
:= (5 − 5/5) × (555 − 5/5 + 5)
:= 5/5 + ((5 − 5/5) × (555 + 5))
:= 5 + (((5 − 5/5) × (555 + 5)) + 5/5)
:= ((6 + 6)/6) × ((6666 + 6)/6 + 6)
:= 66 + ((66 × 66 − 6)/((6 + 6)/6))
:= 6 + ((6 × 6 − 6/6) × ((6 + 6)/6)6 )
:= 7 × 7 + ((7 + 7 + 7)/7)7
:= 7 × (7 × 7 × 7 − 7) − 777/7
:= 7 × 7 + ((7 − 7/7 + 7)(7+7+7)/7 )
:= ((8 + 8)/8) × ((8888 − 8)/8 + 8)
:= 8/8 + (8 × (8 × (8 + 8 + 8) + 88))
:= 8 + (((8 + 8)/8) × (8888/8 + 8))
:= ((9 + 9)/9) × (((9999 − (9 + 9))/9) + 9)
:= (9 + 9 + 9) × (((9 + 9)/9) + 9 × 9)
:= 99 + ((((9 + 9)/9)99/9 ) + 99)
:= 2
+ (2222 − 2)
3
Ï 2237 := 11 + ((1 + 1) × (1 + (1 + 1111)))
:= 2 + ((2222 + 22/2) + 2)
:= 33 + ((3 × ((3
3+3
+ 3) + 3)) − 3/3)
:= 4 × 4 + ((4/4 + 4) × 444 + 4/4)
Ï 2242 := (1 + 1) × (11 + (1111 − 1))
Ï 2247 := 1 + ((1 + 1) × (1 + (11 + 1111)))
:= 22 + (2222 − 2)
:= 3/3 + (3 × (3
3+3
:= 2 + ((2222 + 22) + 2/2)
+ (3 × (3 + 3))))
:= 4 + (4 × 4444/(4 + 4) + 4 × 4)
(5+5)/5+5
:= 5 + ((5 + 5)/5 × (5555/5 + 5))
:= 55 + ((5 − (5 + 5)/5)
:= 6 + ((6 × ((6 × (66 − 6) + 6) + 6)) − 6/6)
:= ((6 + 6)/6 + 6 × 6) × (66 − (6/6 + 6))
)
:= 33 + (3 × 33+3 + 33)
:= 4 + ((4 × (4/4 + 4)4 ) − (4/4 + 44 ))
:= 5 × 5 + ((5 + 5)/5 × 5555/5)
:= 66 + ((66 × 66 + 6)/((6 + 6)/6))
:= 7/7 + (((7 + 7 + 7)/7)7 + 7 × 7)
:= 7 + ((((7 + 7 + 7)/7) − 7/7) + 7 × 7)
:= 8 + ((8 × (8 × (8 + 8 + 8) + 88)) − (88/8))
:= (8 + 8)/8 + (8 × (8 × (8 + 8 + 8) + 88))
:= 8 + ((8 × (8 × (8 + 8 + 8) + 88)) − 8/8)
:= 9 + (((((9 + 9)/9)99/9 ) + 99) + 9 × 9)
:= 9/9 + ((9 + 9 + 9) × (((9 + 9)/9) + 9 × 9))
:= (((99 + 9)/9) + 9) × ((99 − 9/9) + 9)
7
159
:= 7 × 7 × 7 × 7 − 77 − 77
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2248 := (1 + 1) × (1 + (1 + (11 + 1111)))
3
Ï 2253 := 11 + ((1 + 1) × (11 + (1111 − 1)))
3+3
3+3
Ï 2258 := (1 + 1) × (((11 − 1) × (1 + 1 + 111)) − 1)
:= (2 + 2 + 2)2 + 2222
:= 22 + ((2222 − 2) + 22/2)
:= 2 + ((2222 + 22) + 2)
:= 3 × 33+3 + (((3 + 3)3 − 3)/3)
− 3)
:= 33 + (3 × 3
4
:= 4 + ((4 × (4/4 + 4) ) − 4 )
:= 44 + ((44 − 4/4 + 4)(4+4)/4 )
:= 4 + (4 × 4444/(4 + 4) + 4 × (4 + 4))
:= (5 − 5/5) × ((555 + (5 + 5)/5) + 5)
:= 5 + ((5 − 5/5) × ((555 + (5 + 5)/5) + 5))
:= 55 + (((5 − 55 )/(5 + 5)) − 555)
:= ((3/3 + 3) ) + (3 × 3
4
6/6+6)
+ 33)
6/6+6)
:= 6 × 6 + (((6 + 6)/6) × (6666/6))
:= 6 × 6 × 66 − (((6 + 6)/6)
:= 66 + ((6 × 6/(6 + 6))
:= 7 + (7 × (7 × 7 × 7 − 7) − 777/7)
:= (7 × (7 × 7 × 7 − 7 − 7 − 7)) − 7/7
:= 7/7 + ((((7 + 7 + 7)/7)7 − 7) + 77)
:= 8 + (8 × (8 × (8 + 8 + 8) + 88))
:= ((88/8 + 8) × (888/8 + 8)) − 8
:= 8 + ((8/8 + 8 + 8 + 8) × ((8 + 8)/8 + 88))
:= (9 × (9 × (9 + 9 + 9) + 9)) − (99/9 + 9)
:= 9 + (((9 + 9)/9) × ((9999 + 99)/9))
:= (9 × (9 × (9 + 9 + 9) + 9)) − 9/9 − 9
Ï 2249 := 1 + ((1 + 1) × (1 + (1 + (11 + 1111))))
Ï 2254 := 11 + (((1 + 1) × (11 + 1111)) − 1)
Ï 2259 := (1 + 1)11 + ((1 + 1) × 111 − 11)
:= 2 + (((2222 + 22) + 2/2) + 2)
:= 2222 + 2 × 22+2
:= 2/2 + ((2 + 2 + 2)2 + 2222)
:= 33 + ((33 × 33 ) + (33/3)3 )
:= 3 + (((3/3 + 3)3 ) + 3 × 33+3 )
:= 3 × ((33+3 − 3) + 33 )
:= 4 + (((4 × (4/4 + 4)4 ) − 44 ) + 4/4)
:= 4 × (4 + 4) + 4 × 4444/(4 + 4)
:= (4/4 + 4 + 4) × (44 − 4/4 − 4)
:= (5 × ((5 + 5) × (55 − 5 − 5))) − 5/5
:= 5 + ((5 × ((5 + 5) × (55 − 5 − 5))) − 5/5)
:= (5 − 5/5 + 5) × (5 × 5 × (5 + 5) + 5/5)
:= 6 + ((66 × (6 × 6 − ((6 + 6)/6))) − 6/6)
:= 6 × 6 × 66 − ((666 + 66)/6)
:= 6 × 6 × 66 − (666/6 + 6)
:= (7 × 7 − 7/7) × (7 × 7 − (7 + 7)/7) − 7
:= 7 × (7 × 7 × 7 − 7 − 7 − 7)
:= 7 × 7 × 7 × 7 − ((((7 + 7)/7)7 + 7) + 7)
:= 8 + ((8 × (8 × (8 + 8 + 8) + 88)) + 8/8)
:= ((8 + 8)/8) × ((8888/8 + 8) + 8)
:= 8 + (((88/8 − 8)8−8/8 ) + 8 × 8)
:= 9 + (((9 + 9)/9) × (9999/9 + 9))
:= (99 − 9/9) × ((99 + 99 + 9)/9)
:= (9 × (9 × (9 + 9 + 9) + 9)) − 9
Ï 2250 := (11 − 1) × (1 + ((1 + 1) × (1 + 111)))
Ï 2255 := 11 + ((1 + 1) × (11 + 1111))
Ï 2260 := (1 + 1) × ((11 − 1) × (1 + 1 + 111))
:= 2 + (((2222 + 22) + 2) + 2)
:= 22 + (2222 + 22/2)
:= 3 × ((33+3 + (3 × (3 + 3))) + 3)
:= 33/3 × ((3 + 3)3 − 33/3)
:= 2 × ((2 × ((22 + 2)2 )) − 22)
4
:= 3/3 + (3 × ((33+3 − 3) + 33 ))
4
:= (4/4 + 4) × ((444 + (4 + 4)/4) + 4)
:= 4 + (44 × 44 + ((4 − 4)/4))
:= 5 × ((5 + 5) × (55 − 5 − 5))
:= 5 + (5 × ((5 + 5) × (55 − 5 − 5)))
:= 6 + (66 × (6 × 6 − ((6 + 6)/6)))
:= 66/6 × (6 × 6 × 6 − (66/6))
:= 7 + ((((7 + 7 + 7)/7)7 + 7 × 7) + 7)
:= 7/7 + (7 × (7 × 7 × 7 − 7 − 7 − 7))
:= (8/8 + 8 + 8 + 8) × ((8 + 8)/8 + 88)
:= 8 + (((8 × (8 × (8 + 8 + 8) + 88)) − 8/8) + 8)
:= (9 × (9 × (9 + 9 + 9) + 9)) − (9 + 9)
:= 9 + (((((9 + 9)/9)99/9 ) + 99) + 99)
Ï 2251 := 1 + ((11 − 1) × (1 + ((1 + 1) × (1 + 111))))
:= (((22 − 2) × (222 + 2)) + 22)/2
3
:= ((3/3 + 3) ) + 3 × 3
3+3
Ï 2256 := 1 + (11 + ((1 + 1) × (11 + 1111)))
:= 4 × 4 × 4 + ((4 − 4/4)
)
:= (5 − 5/5) × (555 + 5 + 5)
:= 6 × 6 × 66 + (((6 − 666)/6) − 6)
:= 7 + ((7 × (7 × 7 × 7 − 7 − 7 − 7)) − 7/7)
:= ((8 × 8 × (8 × 8 + 8)) − 88)/((8 + 8)/8)
:= 9/9 + ((9 × (9 × (9 + 9 + 9) + 9)) − 9)
Ï 2261 := 1 + ((1 + 1) × ((11 − 1) × (1 + 1 + 111)))
:= 2 + (2222 + 2 × 22+2 )
:= 2/2 + (2 × ((2 × ((22 + 2)2 )) − 22))
3+3
:= 3 + ((((3 + 3)3 − 3)/3) + 3 × 33+3 )
:= (3 × ((3
4+4−4/4
:= 44 + (44 × (4 + 4) − 44)
3
− 3) + 3 )) − 3
:= 4 × (4 × (44 − 4) − 444)
:= 4/4 + ((44 × (4 + 4) − 44) + 44 )
:= 5/5 + (5 × ((5 + 5) × (55 − 5 − 5)))
:= 5 + ((5 × ((5 + 5) × (55 − 5 − 5))) + 5/5)
:= 6 + ((66 × (6 × 6 − ((6 + 6)/6))) + 6/6)
:= 55 + ((55/5 + 5) × (5/5 − 55))
:= 6 + ((66 × (6 × 6 − ((6 + 6)/6))) + 6)
:= 7 + (((((7 + 7 + 7)/7)7 + 7 × 7) + 7/7) + 7)
:= 6 + ((66/6) × (6 × 6 × 6 − (66/6)))
:= (7 × 7 − 7/7) × (7 × 7 − (7 + 7)/7)
:= 8 × 8 + ((88/8 − 8)8−8/8 )
:= 7 + (7 × (7 × 7 × 7 − 7 − 7 − 7))
:= 8 + ((8 × (8 × (8 + 8 + 8) + 88)) + 8)
:= (88/8 + 8) × (888/8 + 8)
:= 9/9 + ((9 × (9 × (9 + 9 + 9) + 9)) − (9 + 9))
:= (9 × (9 × (9 + 9 + 9) + 9)) − (99 + 9)/9
:= (9 + 9)/9 + ((9 × (9 × (9 + 9 + 9) + 9)) − 9)
Ï 2252 := 1 + (1 + ((11 − 1) × (1 + ((1 + 1) × (1 + 111)))))
2
:= 2 × ((2 × (((22 + 2) ) − 2)) − 22)
3
:= 3/3 + (((3/3 + 3) ) + 3 × 3
4
3+3
)
4
:= 4 + (((4 × (4/4 + 4) ) − 4 ) + 4)
Ï 2257 := 11 + ((1 + 1) × (1 + (11 + 1111)))
:= 2 + ((2222 + 22/2) + 22)
:= (3 × (3
3+3
:= 2222 + 2 × (22 − 2)
3
+ 3 )) − 33/3
:= 3 + (3 × ((33+3 − 3) + 33 ))
4
:= 4/4 + ((4 + 4) × (4 + 4) + 4 × 44)
5
5
Ï 2262 := (1 + 1) × (1 + ((11 − 1) × (1 + 1 + 111)))
5
:= 44 + (4 × 4444/(4 + 4) − 4)
:= (5 + 5)/5 + (5 × ((5 + 5) × (55 − 5 − 5)))
:= 5 − (((5 − (5 + 5)/5) ) + 5 /5)
:= (5 × 5 + 5/5) × (((5 + 5)/5)5 + 55)
:= 6 + (((6 × 6 − 6/6) × ((6 + 6)/6)6 ) + 6)
:= (6 × 6 + 6/6) × (66 − 6 + 6/6)
:= 66 + 6 × 6 × (66 − 6 + 6)
7
:= 77 + (((7 + 7 + 7)/7) − 7)
:= 7 + ((7 × (7 × 7 × 7 − 7 − 7 − 7)) + 7/7)
:= 8 + (((8 + 8)/8) × ((8888 + 88)/8))
:= 8 + (((8 × (8 × (8 + 8 + 8) + 88)) + 8/8) + 8)
:= (8/8 − 88) × ((8 − 88)/8 − (8 + 8))
:= (9 + 9)/9 + ((9 × (9 × (9 + 9 + 9) + 9)) − (9 + 9))
:= (9 × (9 × (9 + 9 + 9) + 9)) − 99/9
:= ((9 + 9)/9) × (((9999 + 99)/9) + 9)
:= (7 × (7 × 7 × 7 − 7 − 7 − 7)) − (7 + 7)/7
160
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2263 := 1 + ((1 + 1) × (1 + ((11 − 1) × (1 + 1 + 111))))
:= 2/2 + (2222 + 2 × (22 − 2))
:= 3 + ((3 × ((3
3+3
Ï 2268 := (1 + 1)11 + ((1 + 1) × (111 − 1))
:= 2 + ((222/2 + 222) + 2/2)
:= 2 + (2222 + 2 × 22)
3
− 3) + 3 )) + 3/3)
4
:= 3 × (3
3+3
Ï 2273 := 1 + ((1 + 1)11 + ((1 + 1) × (1 + 111)))
3
:= 3 + (((3 × (33+3 + 33 )) − 3/3) + 3)
+3 )
4
:= 4 + ((4/4 + 4 + 4) × (4 − 4/4 − 4))
:= (4 − 4) × (4/4 + 4 + 4)
:= 4/4 + ((4 + 4) × (44 − 4) + 44 )
:= 5 + ((((5 − 55 )/(5 + 5)) − 555) + 55 )
:= (5 − 5/5) × (((55 + 5)/5) + 555)
:= (5 − 5/5)5 + ((55 − 5 + 55 )/5)
:= 6 + ((6 × 6 + 6/6) × (66 − 6 + 6/6))
:= 6 × (6 × 66 − (6 + 6 + 6))
:= 6 × (6 × 66 − 6) − (66 + 6/6)
:= 77 + (((7 + 7 + 7)/7) − 7/7)
:= 7 × ((77/7 + 7)(7+7)/7 )
:= 7 × 7 × 7 × 7 − ((7 + 7)/7)7
:= 88 + (((8 + 8) × (8 × (8 + 8) + 8)) − 8/8)
:= (8/8 + 8) × ((8 × 8 × 8 − 8)/((8 + 8)/8))
:= ((8 − 8/8)8×8/(8+8) ) − 8 × (8 + 8)
:= 9 + ((99 − 9/9) × ((99 + 99 + 9)/9))
:= 9 × (9 × (9 + 9 + 9) + 9)
:= ((9 × 9 + 9)/(9 + 9)) + (9 × (9 × (9 + 9 + 9) + 9))
7
Ï 2269 := (1 + 1)11 + ((1 + 1) × 111 − 1)
Ï 2264 := (1 + 1) × (11 + (11 + (1111 − 1)))
:= 222 + (2
:= 2 × 22 + (2222 − 2)
3
33/3
22/2
− 2/2)
:= 3/3 + (3 × (3
:= (3 + 3) + (3 − 3/3)
3+3
3
+ 3 ))
4
:= 44 + (4/4 + 4) × 444
:= (5 − 5/5) × (555 + (55/5))
66/6
:= 6 × 6 × 6 + (((6 + 6)/6)
:= 2 + ((222/2 + 222) + 2)
:= 3 + ((3 × (33+3 + 33 )) + 3)
:= 4/4 + (4 − 4) × (4/4 + 4 + 4)
:= 4 + ((444 + ((4 + 4)4 ))/((4 + 4)/4))
:= 5 + ((5 − 5/5) × (555 + (55/5)))
:= (5 − 5/5)5 + 5 × 5 × 5 × (5 + 5)
:= 6/6 + (6 × (6 × 66 − (6 + 6 + 6)))
)
Ï 2274 := (1 + 1)11 + ((1 + 1) × (1 + 1 + 111))
:= 6 × (6 × 66 − 6) − 66
:= 77 + ((7 + 7 + 7)/7)
:= 7/7 + (7 × ((77/7 + 7)(7+7)/7 ))
:= 88 + ((8 + 8) × (8 × (8 + 8) + 8))
:= 8 + ((88/8 + 8) × (888/8 + 8))
:= 8 + ((888/8 − 8) × (88 + 88)/8)
:= ((9 − 9 × 9)/(9 + 9)) + (9 × (9 × (9 + 9 + 9) + 9))
:= 9/9 + (9 × (9 × (9 + 9 + 9) + 9))
:= 9 + ((9 × (9 × (9 + 9 + 9) + 9)) − ((9 + 9 + 9)/9))
7
Ï 2270 := (1 + 1)11 + (1 + 1) × 111
Ï 2265 := ((1 + 1) × (11 + (11 + 1111))) − 1
:= 222 + 2
:= 2 × 22 + (2222 − 2/2)
:= (3 × (3
3+3
3
22/2
:= 3 + ((3 × (3
+ 3 )) − 3
3+3
:= 7 × (7 × 7 × 7 − 7) − 7/7 − 77
Ï 2275 := 1 + ((1 + 1)11 + ((1 + 1) × (1 + 1 + 111)))
:= 2 + (((222/2 + 222) + 2/2) + 2)
3
+ 3 )) − 3/3)
4
:= (3 × (33+3 + 33)) − 33/3
:= 44 + ((4/4 + 4) × 444 + 4/4)
:= (444 + ((4 + 4) ))/((4 + 4)/4)
:= (4/4 + 4) × (444 + 44/4)
:= 5 + ((5 − 5/5) × (555 + 5 + 5))
:= 55 + (((5 − 5/5) × 555) − 5)
:= 5 × (((5 + 5) × (55 − 5 − 5)) + 5)
:= 6 × 6 × 66 − 666/6
:= 6 + ((((6 + 6)/6)66/6 ) + 6 × 6 × 6)
:= (6 × 6 − 6/6) × (66 − 6/6)
7
7
:= 7/7 + (((7 + 7 + 7)/7) + 77)
:= 7 + ((((7 + 7 + 7)/7) − 7/7) + 77)
:= 7 × (7 × 7 × 7 − 7) − 77
:= 8/8 + (((8 + 8) × (8 × (8 + 8) + 8)) + 88)
:= 8 + ((8/8 − 88) × ((8 − 88)/8 − (8 + 8)))
:= 88 + ((88/8 − 8)8−8/8 )
:= (9 × (9 × (9 + 9 + 9) + 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 × (9 × (9 + 9 + 9) + 9))
:= 9 + ((9 × (9 × (9 + 9 + 9) + 9)) − ((9 + 9)/9))
Ï 2271 := 1 + ((1 + 1)11 + (1 + 1) × 111)
Ï 2266 := (1 + 1) × (11 + (11 + 1111))
Ï 2276 := (1 + 1) × (1111 + (1 + 1 + 1)1+1+1 )
:= 2/2 + (222/2 + 222)
:= 2 + (((222/2 + 222) + 2) + 2)
:= 3 + (3 × (33+3 + 33 ))
:= 333 + (((3 × (3 + 3))3 − 3)/3)
:= 44 + ((4 + 4) × (44 − 4) − 4/4)
:= 4 + ((4 + 4) × (44 − 4) + 44 )
:= 5 + (((55/5 + 5) × (5/5 − 55)) + 5 )
:= 55 + ((5 − 5/5) × (555 − 5/5))
:= 6 × 6 × 66 + ((6 − 666)/6)
:= (5 − 5/5) × ((55 − 5)/5 − 55)
:= 6 + (6 × 6 × 66 − 666/6)
:= 6 × (6 × 66 − 6) − ((6 + 6)/6)6
:= 2 × 22 + 2222
:= 3/3 + ((3 × (3
3+3
3
+ 3 )) − 3)
:= 44 + 4 × 4444/(4 + 4)
5
7
7
:= 7 × 7 × 7 × 7 − (((7 + 7)/7) + 7)
:= 7 + (((7 + 7 + 7)/7) + 77)
:= (888/8 − 8) × (88 + 88)/8
:= 7/7 + (7 × (7 × 7 × 7 − 7) − 77)
:= 8 + ((((8 + 8) × (8 × (8 + 8) + 8)) − 8/8) + 88)
:= (9 × (9 × (9 + 9 + 9) + 9)) − (9 + 9)/9
:= ((8 × (8 × (8 × 8 + 8) − 8)) + 8)/((8 + 8)/8)
:= ((9 + 9 + 9)/9) + (9 × (9 × (9 + 9 + 9) + 9))
:= 9 + ((9 × (9 × (9 + 9 + 9) + 9)) − 9/9)
Ï 2267 := 1 + ((1 + 1) × (11 + (11 + 1111)))
:= 2/2 + (2222 + 2 × 22)
:= (3 × (3
3+3
Ï 2272 := (1 + 1)11 + ((1 + 1) × (1 + 111))
:= 2 + (2
3
22/2
+ 222)
:= 3 + ((3 × (33+3 + 33 )) + 3/3)
+ 3 )) − 3/3
4
4
:= (4 − 4) × (4/4 + 4 + 4) − 4/4
4
:= 4 + (4 + 4) × (4 − 4)
5
Ï 2277 := 11 × (11 + (1 + 1 + 1 + 11)1+1 )
:= 22/2 + (2222 + 2 × 22)
:= 3 × ((33+3 + 33 ) + 3)
:= (4/4 + 4 + 4) × ((4/4 − 4) + 44 )
:= 5 + ((5 × 5 + 5/5) × (((5 + 5)/5) + 55))
:= (5 + 5)/5 × (5555/5 + 5 × 5)
:= 55 + ((5 + 5)/5 × 5555/5)
:= (6 × (6 × 66 − (6 + 6 + 6))) − 6/6
:= 6 + (((6 − 666)/6) + 6 × 6 × 66)
:= 6 + ((6 × 6 × 66 − 666/6) + 6)
:= 777/7 + 77 × (7 + 7 + 7 + 7)
:= 7 + ((((7 + 7 + 7)/7)7 + 77) + 7/7)
:= (7 + 7)/7 + (7 × (7 × 7 × 7 − 7) − 77)
:= 8 + (((8 + 8) × (8 × (8 + 8) + 8)) + 88)
:= 88/8 × ((888/8 + 88) + 8)
:= 999 + (9999/9 + 9 × (9 + 9))
:= 9 + (9 × (9 × (9 + 9 + 9) + 9))
:= 88 + (((88/8 − 8)
8−8/8
) − 8)
:= (9 × (9 × (9 + 9 + 9) + 9)) − 9/9
161
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2278 := 1 + (11 × (11 + (1 + 1 + 1 + 11)1+1 ))
2
Ï 2283 := 11 + ((1 + 1)11 + ((1 + 1) × (1 + 111)))
2
Ï 2288 := (1 + 1) × (1111 + (11 × (1 + 1 + 1)))
:= ((2 × (22 + 2)) ) − 22 − 2 − 2
:= 2/2 + (((2 × (22 + 2)) ) − 22)
:= 2 × (2 × (22 × (22 + 2 + 2)))
:= 3/3 + (3 × ((33+3 + 33 ) + 3))
:= (3 × (33+3 + 33)) − 3
:= 3 + ((3 × (33+3 + 33)) − 3/3)
:= (4 − 44)/4 + 44 × (44 + 4 + 4)
:= 44 × (44 + 4 + 4) − (4/4 + 4)
:= 44 × (44 + 4 + 4)
:= 5 + (((55 − 5 + 55 )/5) + (5 − 5/5)5 )
:= (55/5 × ((55 − 5)/(5 + 5 + 5))) − 5
:= 55/5 × ((55 − 5)/(5 + 5 + 5))
:= (66 + 6/6) × (6 × 6 − ((6 + 6)/6))
:= 6 + (((6 × 6 × 66 − 666/6) + 6) + 6)
:= 66 + (((6 + 6)/6) × (6666/6))
:= 7 × 7 × 7 × 7 − (777/7 + 7)
:= (7/7 + 7) × (7 × (7 × 7 − 7) − (7/7 + 7))
7
:= 7 + ((((7 + 7 + 7)/7) + 77) + 7)
8−8/8
:= 8 × 8 + ((((8 + 8)/8) × 8888/8) − 8)
:= 8 + (((88/8 − 8)
:= 9 + ((9 × (9 × (9 + 9 + 9) + 9)) + 9/9)
:= 99 + (((9 + 9 + 9)/9) × (9 × 9 × 9 − 9/9))
Ï 2279 := (1 + 1)11 + (11 × (11 + (11 − 1)))
) + 88)
Ï 2284 := 1 + (11 + ((1 + 1)11 + ((1 + 1) × (1 + 111))))
:= 88 × (((((8 + 8)/8) + 8) + 8) + 8)
:= 9 + ((9 × (9 × (9 + 9 + 9) + 9)) + (99/9))
Ï 2289 := (11 + (11 − 1)) × (111 − 1 − 1)
:= 2 + (((2 × (22 + 2)) ) − 22)
:= 222/2 + (222 − 2)/2
:= 33/3 + (3 × (33+3 + 33 ))
:= 3/3 + ((3 × (33+3 + 33)) − 3)
:= 3 + (3 × (33+3 + 33))
:= (44 − 4/4) × (44 − 44)/4
:= 44 × (44 + 4 + 4) − 4
:= 4/4 + 44 × (44 + 4 + 4)
:= 5 + (5 × 5 × 5 × (5 + 5) + (5 − 5/5)5 )
:= (5 − 5/5) × ((55 + 5)/5 − 55)
:= 5 + ((5 − 5/5) × ((55 + 5)/5 − 55))
:= 6 + (6 × (6 × 66 − 6) − (66 + 6/6))
:= 6 + ((66 + 6/6) × (6 × 6 − ((6 + 6)/6)))
:= (6/6 + 6) × (666/6 + 6 × 6 × 6)
:= 7 × 7 × 7 × 7 − (777 + 77)/7
:= 7 × (7 + 7) + (((7 + 7 + 7)/7)7 − 7/7)
:= 7 × (7 × 7 × 7 − 7 − 7) − (7 + 7)
:= ((8 + 8 + 8) × (88 − 8/8 + 8)) − 8/8
:= 8 + (((8 × (8 × (8 × 8 + 8) − 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((88 × (8 + 8) − 8) + 888)
:= 99/9 + (9 × (9 × (9 + 9 + 9) + 9))
:= 99 + (9 × 9 × (9 + 9 + 9) − ((9 + 9)/9))
:= 9 + ((9/9 + 9 + 9) × (999/9 + 9))
:= 2
22/2
2
+ ((22 − 22)/2)
Ï 2280 := (1 + 1) × ((11 − 1) × (1 + 1 + 1 + 111))
2
Ï 2285 := 11 + ((1 + 1)11 + ((1 + 1) × (1 + 1 + 111)))
Ï 2290 := (1 + 1)11 + ((1 + 1) × 111+1 )
:= ((2 × (22 + 2))2 ) − (22 + 2)
:= 2 + ((((2 × (22 + 2))2 ) − 22) + 2/2)
:= 222/2 + 222 /2
:= 3 + (3 × ((33+3 + 33 ) + 3))
:= (3 × (33+3 + 33)) − 3/3
:= 3 + ((3 × (33+3 + 33)) + 3/3)
:= 44 × (44 + 4 + 4) − 4 − 4
:= 4/4 + (44 × (44 + 4 + 4) − 4)
:= (4 + 4)/4 + 44 × (44 + 4 + 4)
:= (5 − 5/5) × (55 /5 − 55)
:= 5 + ((5 − 5/5) × (55 /5 − 55))
:= 55 − (((55 × (5 + 5 + 5)) + 5) + 5)
:= 6 + (6 × (6 × 66 − 6) − 66)
:= 66/6 + (6 × (6 × 66 − 6) − 66)
:= 6 + (((66 + 6/6) × (6 × 6 − ((6 + 6)/6))) + 6)
:= 7 + (7 × 7 × 7 × 7 − ((7 + 7)/7)7 )
:= 7 × (7 + 7) + ((7 + 7 + 7)/7)7
:= 7 × 7 × 7 × 7 − 777/7
:= (8 + 8 + 8) × (88 − 8/8 + 8)
:= 888 + (88 × (8 + 8) − (88/8))
:= (9/9 + 9 + 9) × (999/9 + 9)
:= 99 + (9 × 9 × (9 + 9 + 9) − 9/9)
Ï 2281 := 11 + ((1 + 1)11 + (1 + 1) × 111)
Ï 2286 := (1 + 1) × (1111 + ((11 × (1 + 1 + 1)) − 1))
:= ((2 × (22 + 2))2 ) − (22 + 2/2)
:= 22+2+2 + 2222
:= 3 + ((3 × ((33+3 + 33 ) + 3)) + 3/3)
:= 3 × (33+3 + 33)
:= 44 + ((44 + 4/4)(4+4)/4 )
:= (4/4 + 4 + 4) × (44 − (4 + 4)/4)
:= 5 + ((55 × (5 × 5 + 5)) + (55 + 5)/5)
:= ((55 + 55 )/5) + (55 × (5 × 5 + 5))
:= 6 + ((6 × 6 − 6/6) × (66 − 6/6))
:= 6 + ((6 × (6 × 66 − 6) − 66) + 6)
:= 7 + (7 × (7 × 7 × 7 − 7) − (7/7 + 77))
:= 7/7 + (((7 + 7 + 7)/7)7 + 7 × (7 + 7))
:= 8/8 + ((8 + 8 + 8) × (88 − 8/8 + 8))
:= 8 × 8 + (((8 + 8)/8) × 8888/8)
:= 9/9 + ((9/9 + 9 + 9) × (999/9 + 9))
:= 99 + 9 × 9 × (9 + 9 + 9)
Ï 2282 := 1 + (11 + ((1 + 1)11 + (1 + 1) × 111))
:= 888 + ((88 × (8 + 8) − 8) + ((8 + 8)/8))
:= (9/9 + 9) × (((99/9) × (99/9 + 9)) + 9)
Ï 2291 := 1 + ((1 + 1)11 + ((1 + 1) × 111+1 ))
:= 222/2 + (222 + 2)/2
:= 3 + (((3 × (33+3 + 33)) − 3/3) + 3)
Ï 2287 := (1 + 1)11 + (((1 + 1) × (111+1 − 1)) − 1)
:= 4 + (44 × (44 + 4 + 4) − 4/4)
:= 555 + (5555 + 55 )/5
:= (6 × (6 × ((6 + 6)/6)6 )) − (6/6 + 6 + 6)
:= 7 × 7 × 7 × 7 + ((7 − 777)/7)
:= 8 + ((((88/8 − 8)8−8/8 ) + 88) + 8)
:= 9 × (9 + 9 + 9) + (((9 + 9)/9)99/9 )
Ï 2292 := (1 + 1)11 + ((1 + 1) × (1 + 111+1 ))
:= ((2 × (22 + 2))2 ) − 22
:= 2/2 + (22+2+2 + 2222)
:= 2 × ((2 × (((22 + 2)2 ) − 2)) − 2)
:= (3 × (33+3 + 33)) − (3/3 + 3)
:= 3/3 + (3 × (33+3 + 33))
:= 3 + ((3 × (33+3 + 33)) + 3)
:= 44 + ((((4 + 4)4 ) − 44)/((4 + 4)/4))
:= 44 × (44 + 4 + 4) − 4/4
:= 4 + 44 × (44 + 4 + 4)
:= 5 + (((5 + 5)/5 × 5555/5) + 55)
:= 55 + ((5 + 5)/5 × (5555/5 + 5))
:= 55 + (((5 − 55 )/(5 + 5 + 5)) − 55 /5)
:= 6 + (6 × (6 × 66 − 6) − ((6 + 6)/6)6 )
:= 6 + (((6 × 6 − 6/6) × (66 − 6/6)) + 6)
:= (6 × (6 × ((6 + 6)/6)6 )) − 6 − 6
:= 7 + (7 × (7 × 7 × 7 − 7) − 77)
:= 7 + ((7 × 7 × 7 × 7 − ((7 + 7)/7)7 ) + 7)
:= 7 + (((7 + 7 + 7)/7)7 + 7 × (7 + 7))
:= (8 + 8)/8 + ((8 + 8 + 8) × (88 − 8/8 + 8))
:= 888 + (88 × (8 + 8) − (8/8 + 8))
:= ((88 + 8)/8) × (8 × (8 + 8 + 8) − 8/8)
:= (9 + 9) × 99 + (((9 × 999) + 9)/(9 + 9))
:= 9/9 + (9 × 9 × (9 + 9 + 9) + 99)
:= 9 × 9 + (((99/9 + 9) × 999/9) − 9)
162
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2293 := (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ) − 11
Ï 2298 := (11 × (11 × (((1 + 1) × (11 − 1)) − 1))) − 1
Ï 2303 := (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ) − 1
:= ((2 × (22 + 2))2 ) − 22/2
:= 2 + (2 × (2 × (((22 + 2)2 ) − 2)))
:= ((2 × (22 + 2))2 ) − 2/2
:= 3 + (((3 × (33+3 + 33)) + 3/3) + 3)
:= 3 + (3 × ((33+3 + 33) + 3))
:= (33/3)3 + (33 × (33 + 3))
:= 4 + (44 × (44 + 4 + 4) + 4/4)
:= 44 + (44 × (4 + 4) − ((4 + 4)/4 + 4))
:= 44 + (44 × (4 + 4) − 4/4)
:= 5 + (55/5 × ((55 − 5)/(5 + 5 + 5)))
:= 55 − ((55 × (5 + 5 + 5)) + ((5 + 5)/5))
:= 5 + ((((5 + 5)/5)55/5 ) + 5 × 5 × (5 + 5))
:= (6 × (6 × ((6 + 6)/6)6 )) − 66/6
:= (6 × (6 × ((6 + 6)/6)6 )) − 6
:= (6 × (6 × ((6 + 6)/6)6 )) − 6/6
:= ((7 − 77)/7) + 7 × (7 × 7 × 7 − 7 − 7)
:= 777/7 + ((7 + 7 + 7)/7)7
:= 7 × (7 × 7 × 7 − 7 − 7)
:= ((88 + 8) × (8 + 8 + 8)) − 88/8
:= 888 + (88 × (8 + 8) + ((8 + 8)/8))
:= ((88 + 8) × (8 + 8 + 8)) − 8/8
:= (9 + 9) × 99 + ((((9 + 9)/9)9 ) − 9/9)
:= 999/9 + 9 × 9 × (9 + 9 + 9)
:= 9 + ((9 + 9) × 99 + (((9 + 9)/9)9 ))
Ï 2294 := 1 + ((((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ) − 11)
Ï 2299 := 11 × (11 × (((1 + 1) × (11 − 1)) − 1))
Ï 2304 := ((1 + 1) × ((1 + 1) × (1 + 11)))1+1
:= (2 × (2 × (((22 + 2)2 ) − 2))) − 2
:= (22/2)2 × (22 − (2/2 + 2))
:= (2 × (22 + 2))2
:= (3 × ((33+3 + 33) + 3)) − 3/3
:= 3 × 33+3 + ((333 + 3)/3)
:= (33 − 3) × (3 × 33 − 3)
:= 4 + (44 × (44 + 4 + 4) + (4 + 4)/4)
:= 4 + ((4/4 + 4 + 4) × (44 − 4/4))
:= 44 + 44 × (4 + 4)
:= 55 − (((55 × (5 + 5 + 5)) + 5/5) + 5)
:= 55 − ((55 × (5 + 5 + 5)) + 5/5)
:= (55 − ((5 + 5)/5 + 5))(5+5)/5
:= ((6 + 6)/6) × ((6666/6) + 6 × 6)
:= 6/6 + ((6 × (6 × ((6 + 6)/6)6 )) − 6)
:= 6 × (6 × ((6 + 6)/6)6 )
:= 7 × (7 × 7 × 7 − 7 − 7) − ((7 + 7)/7 + 7)
:= 7 + ((((7 + 7 + 7)/7)7 + 7 × (7 + 7)) + 7)
:= (7 × 7 − 7/7)(7+7)/7
:= 888 + (88 × (8 + 8) − ((8 + 8)/8))
:= 88/8 × (88/8 × (88/8 + 8))
:= (88 + 8) × (8 + 8 + 8)
:= 99/9 × ((99/9 + 99) + 99)
:= 9 × (((9 + 9)/9)9−9/9 )
9
:= (9 + 9) × 99 + (((9 + 9)/9) )
Ï 2295 := (11 − 1 − 1) × (111 + (1 + 11)1+1 )
Ï 2305 := 1 + (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 )
Ï 2300 := (1 + (11 + 11)) × (11 − 1)1+1
:= 2 + (((2 × (22 + 2))2 ) − 22/2)
:= 2 × ((2 × ((22 + 2)2 )) − 2)
:= 2/2 + ((2 × (22 + 2))2 )
:= 3 × ((33+3 + 33) + 3)
:= (3/3 + 3) × (((3 × 3 + 3)3 − 3)/3)
:= 3/3 + ((33 − 3) × (3 × 33 − 3))
:= (4/4 + 4 + 4) × (44 − 4/4)
:= 44 + (44 × (4 + 4) − 4)
:= 4/4 + (44 × (4 + 4) + 44 )
:= 55 − ((55 × (5 + 5 + 5)) + 5)
:= 55 − (55 × (5 + 5 + 5))
:= 5 + (55 − (55 × (5 + 5 + 5)))
:= 6 × (6 × 66 + 6) − (666/6 + 6)
:= (6 + 6)/6 + ((6 × (6 × ((6 + 6)/6)6 )) − 6)
:= 6/6 + (6 × (6 × ((6 + 6)/6)6 ))
:= 7 × (7 × 7 × 7 − 7 − 7) − (7/7 + 7)
:= (7/7 + 7 × 7) × (7 × 7 − ((7 + 7 + 7)/7))
:= 7/7 + ((7 × 7 − 7/7)(7+7)/7 )
:= 888 + (88 × (8 + 8) − 8/8)
:= ((8 × 8 × (8 × 8 + 8)) − 8)/((8 + 8)/8)
:= 8/8 + ((88 + 8) × (8 + 8 + 8))
:= 9 + (9 × 9 × (9 + 9 + 9) + 99)
:= (9/9 + 99) × ((99 + 99 + 9)/9)
:= 9/9 + (9 × (((9 + 9)/9)9−9/9 ))
Ï 2296 := (1 + 1) × (1111 + (111/(1 + 1 + 1)))
Ï 2301 := 1 + ((1 + (11 + 11)) × (11 − 1)1+1 )
Ï 2306 := 1 + (1 + (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ))
2
:= ((2 × (22 + 2))2 ) − 2/2 − 2
:= 2 + ((2 × (22 + 2))2 )
+ 33) + 3))
:= 33 + (3 × (33+3 + 33 ))
:= 3 + ((33 × (33 + 3)) + (33/3)3 )
:= 4/4 + ((44 × (4 + 4) − 4) + 44 )
:= 44 + (44 × (4 + 4) + (4 + 4)/4)
:= 55 + (5/5 − (55 × (5 + 5 + 5)))
:= 5 + ((5/5 − (55 × (5 + 5 + 5))) + 55 )
:= (6 × (6 × ((6 + 6)/6) )) − ((6 + 6)/6 + 6)
:= 6 × (6 × 66 + 6) − 666/6
:= (6 + 6)/6 + (6 × (6 × ((6 + 6)/6)6 ))
:= 7 × (7 × 7 × 7 − 7 − 7) − 7
:= 7 × (7 × 7 × 7 − 7 − 7) − (7 + 7)/7
:= 77 + ((((7 + 7 + 7)/7)7 − 7) + 7 × 7)
:= 888 + 88 × (8 + 8)
:= 8 + (((88 + 8) × (8 + 8 + 8)) − (88/8))
:= (8 + 8)/8 + ((88 + 8) × (8 + 8 + 8))
:= (9/9 + 9 × 9) × ((9/9 + 9 + 9) + 9)
:= 9 × 9 + ((99/9 + 9) × 999/9)
:= (9 + 9)/9 + (9 × (((9 + 9)/9)9−9/9 ))
:= 2 × (2 × (((22 + 2) ) − 2))
:= 3/3 + (3 × ((3
3+3
:= 4 + (44 × (44 + 4 + 4) + 4)
5
:= 5 + (5/5 − ((55 × (5 + 5 + 5)) + 5))
6
Ï 2297 := 111 + (((1 + 1 + 11)1+1+1 ) − 11)
2
:= 2/2 + (2 × (2 × (((22 + 2) ) − 2)))
:= 33/3 + (3 × (3
3+3
+ 33))
:= 4 + ((44 × (44 + 4 + 4) + 4/4) + 4)
5
:= 5 + ((5 + 5)/5 − ((55 × (5 + 5 + 5)) + 5))
6
:= (6 × (6 × ((6 + 6)/6) )) − 6/6 − 6
(7+7)/7
:= ((7 × 7 − 7/7)
)−7
:= 8/8 + (88 × (8 + 8) + 888)
:= 99 + (9 × 9 × (9 + 9 + 9) + (99/9))
Ï 2302 := (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ) − 1 − 1
2
:= ((2 × (22 + 2)) ) − 2
:= 2 + (((2 × (22 + 2))2 ) + 2/2)
:= 3 + (((333 + 3)/3) + 3 × 3
4
Ï 2307 := 111 + (((1 + 1 + 11)1+1+1 ) − 1)
3+3
)
4
:= 4 + (4 × (4 + 4) − (4 + 4)/4)
5
:= 5 + ((5 + 5)/5 − (55 × (5 + 5 + 5)))
6
:= 3 + ((33 − 3) × (3 × 33 − 3))
:= 4 + ((44 × (4 + 4) − 4/4) + 44 )
:= 5 + (((5 + 5)/5 − (55 × (5 + 5 + 5))) + 55 )
:= (6 × (6 × ((6 + 6)/6) )) − (6 + 6)/6
:= 6 + (6 × (6 × 66 + 6) − 666/6)
:= 7 × (7 × 7 × 7 − 7 − 7) − 7/7
:= 77/7 + (7 × (7 × 7 × 7 − 7 − 7) − 7)
:= ((88 + 8) × (8 + 8 + 8)) − (8 + 8)/8
:= 888 + (88 × (8 + 8) + 88/8)
9−9/9
:= (9 × (((9 + 9)/9)
)) − (9 + 9)/9
163
:= 9 + (9 × 9 × (9 + 9 + 9) + 999/9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2308 := 111 + ((1 + 1 + 11)1+1+1 )
Ï 2313 := 1 + ((1 + 1) × ((1 + 11 × (1 + 1 + 1))1+1 ))
Ï 2318 := 111+1 + ((1 + 1 + 11)1+1+1 )
:= 2 + (((2 × (22 + 2))2 ) + 2)
:= 2/2 + (2 × (2 × (((22 + 2)2 ) + 2)))
:= 2 + (2 × ((2 × (((22 + 2)2 ) + 2)) + 2))
:= (3/3 + 3) × (((3 × 3 + 3)3 + 3)/3)
:= 3 × ((33+3 + 33) + 3 × 3)
:= 33 + ((3 × (33+3 + 33)) − 3/3)
:= 4 + (44 × (4 + 4) + 44 )
:= (4/4 + 4 + 4) × (4/4 + 44 )
:= ((4 + 4)4 ) − (4 × 444 + (4 + 4)/4)
:= (5 + 5)/5 × (((5 − 5/5)5 + 5 × 5 × 5) + 5)
:= 5 × 5 + (55/5 × ((55 − 5)/(5 + 5 + 5)))
:= 5 × 55 + ((((5 + 5)/5)55/5 ) − 5)
:= 6 × 6 × 66 − (((6 + 6)/6) + 66)
:= 6/6 + (6 × 6 × 66 − ((6 + 6)/6)6 )
:= 6 + (6 × 6 × 66 − ((6 + 6)/6)6 )
:= 7 + (7 × (7 × 7 × 7 − 7 − 7) − ((7 + 7)/7))
:= 77 + (((7 + 7 + 7)/7)7 + 7 × 7)
:= 7 + (((7 × 7 − 7/7)(7+7)/7 ) + 7)
:= ((8 × 8 × (8 × 8 + 8)) + 8)/((8 + 8)/8)
:= 8 + (((88 + 8) × (8 + 8 + 8)) + 8/8)
:= (88/8 + 8) × ((888 + 88)/8)
:= 9 + (9 × 9 × (9 + 9 + 9) + ((999 + 9)/9))
Ï 2309 := ((111 − 1) × (11 + (11 − 1))) − 1
:= 2 + ((((2 × (22 + 2)) ) + 2/2) + 2)
3
:= ((33 × ((3 + 3) − (3 + 3))) − 3)/3
4
:= 4 + ((4 × (4 + 4) + 4 ) + 4/4)
:= 5 + ((55 − ((5 + 5)/5 + 5))
(5+5)/5
)
:= 6 × 6 × 66 − (66 + 6/6)
:= 7 + (7 × (7 × 7 × 7 − 7 − 7) − 7/7)
:= 8 + ((((88 + 8) × (8 + 8 + 8)) − (88/8)) + 8)
:= 9 + ((9/9 + 99) × ((99 + 99 + 9)/9))
Ï 2310 := (111 − 1) × (11 + (11 − 1))
:= (9/9 + 9 + 9) × (999 + 99)/9
))
:= 33 × ((((3/3 + 3)3 ) + 3) + 3)
:= 4 + ((44 × (4 + 4) + (4 + 4)/4) + 44 )
5
:= 55 × ((((5 + 5)/5) + 5) + 5)
:= 66 × (6 × 6 − 6/6)
:= 7 + 7 × (7 × 7 × 7 − 7 − 7)
:= 2 + ((((2 × (22 + 2))2 ) + 22/2) + 2)
:= 3 + (((33 × ((3 + 3)3 − (3 + 3))) + 3)/3)
:= 33 + (3 × (33+3 + 33))
:= 4/4 + ((4/4 + 4 + 4) × (4/4 + 44 ))
:= ((4 + 4)4 ) − (4 × 444 + 4/4)
:= 5 + (55 − (5 + 5)/5 + 5)(5+5)/5 + 5
:= 5555 − (555/5 + 55 )
:= 6 + (6 × 6 × 66 − (((6 + 6)/6) + 66))
:= 6 + ((6 × 6 × 66 − ((6 + 6)/6)6 ) + 6/6)
:= 77/7 + 7 × (7 × 7 × 7 − 7 − 7)
:= (7 + 7)/7 + (7 × 7 × 7 × 7 − (77 + 7))
:= 8 + (((88 + 8) × (8 + 8 + 8)) + ((8 + 8)/8))
:= 8 + ((((88 + 8) × (8 + 8 + 8)) − 8/8) + 8)
9−9/9
:= 9 + ((9 × (((9 + 9)/9)
)) + 9/9)
:= 99/9 × (999/9 + 99)
:= 3 + (((3 − 3/3 + 3)3 ) + 3 × 33+3 )
:= 3/3 + ((3 × (33+3 + 33)) + 33)
:= 44 + (44 × (4 + 4) + 44/4)
:= 4 × (4 × 44 − 444)
:= 5 + (55 × ((((5 + 5)/5)5 + 5) + 5))
:= (5 − 5/5) × (555 + 5 × 5)
:= 6 + (6 × 6 × 66 − (66 + 6/6))
:= 6 × 6 × 66 + (((66 − 6)/6) − 66)
7
:= ((7 + 7)/7) + ((7 + 7 + 7)/7)
:= 88/8 + ((88 + 8) × (8 + 8 + 8))
:= 99/9 + (9 × (((9 + 9)/9)
:= (2 × (2 × (((22 + 2)2 ) + 2))) − 2/2
:= ((33 × ((3 + 3)3 − (3 + 3))) + 3)/3
4
4
:= 4 + (((4 × (4 + 4) − 4/4) + 4 ) + 4)
5
:= 5 + (55/5 − (55 × (5 + 5 + 5)))
:= 6/6 + (66 × (6 × 6 − 6/6))
:= 7 + ((7 × 7 − 7/7)(7+7)/7 )
:= 8 + (((88 + 8) × (8 + 8 + 8)) − 8/8)
:= ((9 × 9 − 9/9) × (99/9 + 9 + 9)) − 9
Ï 2312 := (1 + 1) × ((1 + 11 × (1 + 1 + 1))1+1 )
:= 2 × (2 × (((22 + 2)2 ) + 2))
:= 3 × 33+3 + ((3 − 3/3 + 3)3 )
:= 4 + ((44 × (4 + 4) + 44 ) + 4)
5
:= 5 + (((55 + 5)/5) − (55 × (5 + 5 + 5)))
6
:= 6 × 6 × 66 − ((6 + 6)/6)
(7+7)/7
) + 7/7)
:= 8 + ((88 + 8) × (8 + 8 + 8))
9
:= 9 + (((9 + 9) × 99 + (((9 + 9)/9) )) + 9)
Ï 2320 := (11 − 1) × (111 + 111+1 )
:= 2 × (2 × ((((22 + 2)2 ) + 2) + 2))
9−9/9
Ï 2311 := 1 + ((111 − 1) × (11 + (11 − 1)))
:= 9 + ((99/9) × (999/9 + 99))
:= 22/2 + ((2 × (22 + 2))2 )
7
:= 8 + (((88 + 8) × (8 + 8 + 8)) − ((8 + 8)/8))
Ï 2319 := (111 × (11 + (11 − 1))) − 1 − 11
:= 2 + (2 × (2 × (((22 + 2) ) + 2)))
2
Ï 2315 := 11 + (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 )
:= 2 + ((((2 × (22 + 2))2 ) + 2) + 2)
:= 7 + (((7 × 7 − 7/7)
:= 9 + (9 × (((9 + 9)/9)
Ï 2314 := (1 + 1) × (1 + ((1 + 11 × (1 + 1 + 1))1+1 ))
2
4
9−9/9
))
Ï 2316 := 1 + (11 + (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ))
:= 7 + ((((7 + 7 + 7)/7)7 + 77) + 7 × 7)
:= 8 + (((88 + 8) × (8 + 8 + 8)) + 8)
:= (9 × 9 − 9/9) × (99/9 + 9 + 9)
Ï 2321 := 11 × ((1 + 1) × 111 − 11)
:= 2 × ((2 × (((22 + 2) ) + 2)) + 2)
:= 22/2 × (222 − 22/2)
:= 3 + ((3 × (33+3 + 33)) + 33 )
:= (33/3)3 + (3 × (333 − 3))
:= ((4 + 4)4 ) − (4 × 444 + 4)
:= 4/4 + (4 × (4 × 44 − 444))
:= (5 − 5/5) × ((555 − 5/5) + 5 × 5)
:= 5/5 + ((5 − 5/5) × (555 + 5 × 5))
:= 6 + (66 × (6 × 6 − 6/6))
:= 66/6 + (66 × (6 × 6 − 6/6))
2
:= 7 × 7 × 7 × 7 − (7/7 + 77 + 7)
:= 7 + (7 × (7 × 7 × 7 − 7 − 7) + (77/7))
:= 8 + (((8 × 8 × (8 × 8 + 8)) + 8)/((8 + 8)/8))
:= 8 + ((((88 + 8) × (8 + 8 + 8)) + 8/8) + 8)
:= 9 + ((9 × 9 × (9 + 9 + 9) + 999/9) + 9)
:= 99 + (((9 + 9)/9) × 9999/9)
Ï 2317 := 111+1 + (((1 + 1 + 11)1+1+1 ) − 1)
Ï 2322 := 1 + (11 × ((1 + 1) × 111 − 11))
:= 2 + (((2 × (22 + 2))2 ) + 22/2)
:= 2 + (((2 × (22 + 2))2 ) + 22+2 )
:= ((3/3 + 3) + 3) × ((333 − 3) + 3/3)
:= 3 + ((3 × (33+3 + 33)) + 33)
:= 4 + ((4/4 + 4 + 4) × (4/4 + 44 ))
:= (4/4 + 4 + 4) × ((4 + 4)/4 + 44 )
:= ((5 + 5)/5 + 5) × ((5 × 55 + 55) + 5/5)
:= (55 − 5/5) × (55 − ((55 + 5)/5))
:= 6 + ((66 × (6 × 6 − 6/6)) + 6/6)
:= 6 + ((66 × (6 × 6 − 6/6)) + 6)
:= 7 × 7 × 7 × 7 − 77 − 7
:= 7 × 7 × 7 × 7 − ((7 + 7)/7 + 77)
:= (88 + 8 + 8)/8 + ((88 + 8) × (8 + 8 + 8))
:= (8/8 + 8) × ((8 + 8) × (8 + 8) + ((8 + 8)/8))
:= 9 + ((9 × 9 × (9 + 9 + 9) + ((999 + 9)/9)) + 9)
:= (9 + 9) × ((999/9 + 9) + 9)
164
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2323 := 1 + (1 + (11 × ((1 + 1) × 111 − 11)))
Ï 2328 := (111 × (11 + (11 − 1))) − 1 − 1 − 1
2
Ï 2333 := 111 + ((1 + 1) × 1111)
:= 2 + (((2 × (22 + 2)) ) + 22)
:= 222/2 + 2222
:= (333 × ((3/3 + 3) + 3)) − 3
:= 3 + (3 × 333 + (33/3)3 )
:= (4 × ((4/4 + 4)4 − 44)) − 4/4
:= 4 + (4 × ((4/4 + 4)4 − 44))
:= ((44 × (44 − 44)) + 4)/4
:= 5 × 55 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 × 55)
:= 5 + (((((5 + 5)/5)55/5 ) + 5 × 55) + 5)
:= 6 × (6 × 66 − 6) − (66/6 + 6)
:= 6 × (6 × 66 − 6) − 6 − 6
:= 6 × (6 × 66 − 6) − 6/6 − 6
:= 7 × 7 × 7 × 7 − 7/7 − 77
:= ((7 + 7 + 7)/7) × (777 − 7/7)
:= 7 + ((7 × 7 × 7 × 7 − 77) + ((7 + 7)/7))
:= 8 + (((88 + 8) × (8 + 8 + 8)) + (88/8))
:= (8 + 8 + 8) × ((8/8 + 88) + 8)
:= ((8 + 8) × (8 × 8 + 88)) − (88/8 + 88)
:= 9/9 + ((9 + 9) × ((999/9 + 9) + 9))
:= 9 + (((99/9) × (999/9 + 99)) + 9)
:= 9 + (((9/9 + 9 + 9) + 9) × (((9 + 9)/9) + 9 × 9))
:= 2 + (22/2 × (222 − 22/2))
:= 3 + (((3 × (3
3+3
+ 33)) + 33) + 3/3)
Ï 2324 := 1 + (1 + (1 + (11 × ((1 + 1) × 111 − 11))))
2
Ï 2329 := (111 × (11 + (11 − 1))) − 1 − 1
Ï 2334 := 1 + (111 + ((1 + 1) × 1111))
:= 22 + (((2 × (22 + 2)) ) − 2)
:= (222/2 × (22 − 2/2)) − 2
:= 22 + (2 × (2 × (((22 + 2)2 ) + 2)))
:= ((3/3 + 3) + 3) × (333 − 3/3)
:= 3/3 + ((333 × ((3/3 + 3) + 3)) − 3)
:= 3 + (333 × ((3/3 + 3) + 3))
4
4
:= 4 × ((4/4 + 4) − 44)
:= (((44 × (4 − 44)) + 4)/4) − 4
:= (((44 × (44 − 44)) + 4) + 4)/4
:= 55 − ((5 × (5 × ((5 + 5)/5)5 )) + 5/5)
:= 5 + (((5 + 5) × (5 × 5 × 5 + 5)) + (5 − 5/5)5 )
:= 55 − ((555 + 55 )/5 + 55)
:= 6 + ((6 × 6 × 66 − ((6 + 6)/6)6 ) + 6)
:= 6 × (6 × 66 − 6) − 66/6
:= 6 × (6 × 66 − 6) − 6
:= 7 × 7 × 7 × 7 − 77
:= 7 + (7 × 7 × 7 × 7 − ((7 + 7)/7 + 77))
:= 7 × (7 + 7 + 7) + ((7 + 7 + 7)/7)7
:= 8 + ((((8 × 8 × (8 × 8 + 8)) + 8)/((8 + 8)/8)) + 8)
:= 8/8 + ((8 + 8 + 8) × ((8/8 + 88) + 8))
:= ((8 + 8)/8) × ((8888/8 − 8) + 8 × 8)
:= ((9/9 + 9 + 9) + 9) × (((9 + 9)/9) + 9 × 9)
:= 9 + ((9 × 9 − 9/9) × (99/9 + 9 + 9))
:= (9999/((9 + 9 + 9)/9)) − 999
Ï 2325 := 1 + (1 + (1 + (1 + (11 × ((1 + 1) × 111 − 11)))))
2
Ï 2330 := (111 × (11 + (11 − 1))) − 1
2
:= 22 + (((2 × (22 + 2)) ) − 2/2)
:= 2 + ((((2 × (22 + 2)) ) + 22) + 2)
:= 3333 − (3 × (333 + 3))
:= 3 × 333 + (33/3)
4
3
Ï 2335 := 111 + ((1 + 1) × (1 + 1111))
:= 2 + (222/2 + 2222)
:= 3 + ((333 × ((3/3 + 3) + 3)) + 3/3)
:= 4/4 + (4 × ((4/4 + 4) − 44))
:= ((44 × (4 − 44)) − (4 + 4))/4
:= 4 + (((44 × (44 − 44)) − 4)/4)
:= 55 − (5 × (5 × ((5 + 5)/5)5 ))
:= 5 + (55 − (5 × (5 × ((5 + 5)/5)5 )))
:= 5 + ((55 − (5 × (5 × ((5 + 5)/5)5 ))) + 5)
:= ((6/6 + 6) × 666 × 6/(6 + 6)) − 6
:= (6 − 66)/6 + 6 × (6 × 66 − 6)
:= 6/6 + (6 × (6 × 66 − 6) − 6)
:= 7/7 + (7 × 7 × 7 × 7 − 77)
:= 7 + (7 × 7 × 7 × 7 − (7/7 + 77))
:= 77/7 + (7 × 7 × 7 × 7 − 77)
:= 8 × 8 + ((88/8 + 8) × (888/8 + 8))
:= 8 + ((8/8 + 8) × ((8 + 8) × (8 + 8) + ((8 + 8)/8)))
:= 8 + (8888 − ((88/8 − 8)8 ))
:= 999/9 + (9 + 9 + 9) × (9/9 + 9 × 9)
:= 999 + (99/9)(9+9+9)/9
:= 9 + ((((99 × 99) − 9)/(9 + 9)) + (9 + 9) × 99)
4
Ï 2326 := 11 + (11 + (((1 + 1) × ((1 + 1) × (1 + 11)))1+1 ))
:= 22 + ((2 × (22 + 2))2 )
3
:= 3 + (((333 + 3)/3) + 3 × 3
3+3
)
4
:= 4 + ((4/4 + 4 + 4) × ((4 + 4)/4 + 4 ))
5
5
:= 5 + (5/5 − (5 × (5 × ((5 + 5)/5) )))
Ï 2331 := 111 × (11 + (11 − 1))
Ï 2336 := 1 + (111 + ((1 + 1) × (1 + 1111)))
:= 222/2 × (22 − 2/2)
:= (2 × 22)2 + (22 − 2)2
:= 333 × ((3/3 + 3) + 3)
:= (33 − 3/3) × (((3 + 3)3 + 3)/3)
:= ((44 × (44 − 44)) − 4)/4
:= 44 + (4 + 4) × (44 + 4)
:= 555/5 × ((55/5 + 5) + 5)
:= 5 + (555/5 × ((55/5 + 5) + 5))
:= 6 × (6 × 66 − 6) − ((6 + 6)/6 + 6 + 6)
:= (6/6 + 6) × 666 × 6/(6 + 6)
:= (6 + 6)/6 + (6 × (6 × 66 − 6) − 6)
:= (7 + 7)/7 + (7 × 7 × 7 × 7 − 77)
:= 7 + (7 × 7 × 7 × 7 − 77)
:= (7/7 + 7) × (7 × (7 × 7 − 7) − ((7 + 7)/7))
:= 8 + ((88/8 + 8) × ((888 + 88)/8))
:= (8 × 8 − 8/8) × 888/(8 + 8 + 8)
:= 8 + ((8 + 8 + 8) × ((8/8 + 88) + 8))
:= (9 + 9) × 99 + (((99 × 99) − 9)/(9 + 9))
:= 9 × (9 × (9 + 9) + 99) − (9 + 9)
:= 9 × (9 × (9 + 9) + 99) − ((99 + 9 + 9)/9)
Ï 2327 := 111 + ((1 + 1) × (1111 − (1 + 1 + 1)))
2
:= 22 + (((2 × (22 + 2)) ) + 2/2)
Ï 2332 := 1 + (111 × (11 + (11 − 1)))
Ï 2337 := 111 + ((1 + 1) × (1 + (1 + 1111)))
:= 22 × ((2 × 2 × (22 + 2 + 2)) + 2)
:= 2 + 222/2 + 2222 + 2
:= 3 × 333 + ((33/3) − 3)
:= 3/3 + (333 × ((3/3 + 3) + 3))
:= ((33 + 3) × (3 × 33 − 3)) − 3
:= (((44 × (44 − 44)) − 4)/4) − 4
:= 44 × (44 − 44)/4
:= (((4 − 4/4) + 4)4 ) − 4 × 4 × 4
:= 55 − (((5 − (5 + 5)/5)5 ) + 555)
:= (5 + 5)/5 × (5555/5 + 55)
:= 5 + ((5 + 5)/5 × (5555/5 + 55))
:= 6 × (6 × 66 − 6) − (6/6 + 6 + 6)
:= 6 × (6 × 66 − 6) − ((6 + 6)/6 + 6)
:= 6 × (6 × 66 − 6) − 6 × 6/(6 + 6)
:= 7 × 7 × 7 × 7 + (((7 + 7 + 7)/7) − 77)
:= 7 + (7 × 7 × 7 × 7 − 77 + 7/7)
:= 7 × (7 × 7 × 7 − 7) − (7/7 + 7 + 7)
:= 8888 − ((88/8 − 8)8 )
:= (88/((8 + 8)/8)) × (8 × 8 − 88/8)
:= ((8 − 8/8)8×8/(8+8) ) − 8 × 8
:= 9 + ((9/9 + 9 + 9) × (999 + 99)/9)
:= 9/9 + (9 × (9 × (9 + 9) + 99) − (9 + 9))
:= 9 × (9 × (9 + 9) + 99) − (99 + 9)/9
3
165
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2338 := 1 + (111 + ((1 + 1) × (1 + (1 + 1111))))
:= 222 + (2 × 22 + 2)2
Ï 2343 := 11 × (((1 + 1) × (1 + 111)) − 11)
Ï 2348 := (1 + 1)11 + ((1 + 1 + 1) × (11 − 1)1+1 )
:= (22/2)2 + 2222
:= 2 × ((2 × ((22 + 2)2 )) + 22)
:= 33 × (((3 + 3)3 − 3)/3)
:= (33 × (3 × 33 + 3 + 3)) − 3/3
:= ((4 − 4 )/4) + (((4 − 4/4) + 4) )
:= 44/4 × ((4/4 − 44) + 44 )
:= 44 + (44 × (4 + 4) + 44 )
:= (5 + 5)/5 × (((55 − 55)/5) + 555)
:= 55/5 × (((55 − 5)/(5 + 5 + 5)) + 5)
:= 55 − (((5 + 5)/5 + 5) × 555/5)
:= 6 × (6 × 66 − 6) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + 6 × (6 × 66 − 6)
:= 6 + (6 × (6 × 66 − 6) + ((6 + 6)/6))
:= 7 × (7 × 7 × 7 − 7) − (7 + 7)
:= 7 × (7 × 7 × 7 − 7) − ((7 + 7)/7 + 7)
:= 7 + (7 × (7 × 7 × 7 − 7) − (77/7))
:= ((8 + 8) × (8 × 8 + 88)) − (8/8 + 88)
:= ((8 × 8 × (8 × 8 + 8)) + 88)/((8 + 8)/8)
:= 99/9 × ((9 + 9) × (9 + 9) − 999/9)
:= 9 × (9 × (9 + 9) + 99) − 9/9
:= ((3/3 + 3) + 3) × (333 + 3/3)
4
4
8×8/(8+8)
:= 8/8 + (((8 − 8/8)
) − 8 × 8)
:= 9 × (9 × (9 + 9) + 99) − 99/9
Ï 2339 := 111 + ((1 + 1) × (1 + (1 + (1 + 1111))))
Ï 2344 := 1 + (11 × (((1 + 1) × (1 + 111)) − 11))
2
2
Ï 2349 := ((1 + 111) × (11 + (11 − 1))) − 1 − 1 − 1
:= 2/2 + ((2 × 22 + 2) + 222)
:= 2 × (((2 × ((22 + 2) )) − 2) + 22)
:= 2/2 + (((2 × (22 + 2))2 ) + 2 × 22)
:= (33/3)3 + (3 × (333 + 3))
:= ((33 × ((3 + 3)3 − 3)) + 3)/3
:= 33 × (3 × 33 + 3 + 3)
:= (44 + 4) × (4/4 + 4 + 4) − 4/4
:= 4 + (44 + 4) × (4/4 + 4 + 4)
:= (4/4 + 4 + 4) × ((4/4 + 44 ) + 4)
5
5
:= 55 + (((55 − 5/5)/(5/5 − 5)) − 5)
:= 5 + ((5 − 5/5)/(5/5 − 5))
:= 5 + (((55 − 5/5)/(5/5 − 5)) + 55 )
:= 6 × (6 × 66 − 6) − 6/6
:= 6 + (6 × (6 × 66 − 6) − ((6 + 6)/6))
:= 6 + (6 × (6 × 66 − 6) + (6 × 6/(6 + 6)))
:= 7/7 + (7 × (7 × 7 × 7 − 7) − (7 + 7))
:= 7 × (7 × 7 × 7 − 7) − (7/7 + 7)
:= 7 × (7 × 7 × 7 − 7) − (7 + 7 + 7)/7
:= 8 + ((8 × 8 − 8/8) × 888/(8 + 8 + 8))
:= ((8 + 8) × (8 × 8 + 88)) − 88
:= (88 − 8/8) × (88/8 + 8 + 8)
:= 9 × (9 × (9 + 9) + 99) − 9/9 − 9
:= ((9 − 99)/(9 + 9)) + 9 × (9 × (9 + 9) + 99)
:= 9 × (9 × (9 + 9) + 99)
Ï 2340 := (11 − 1) × (1 + (11 + (1 + 1) × 111))
Ï 2345 := 111+1 + ((1 + 1) × (1 + 1111))
2
2
Ï 2350 := (11 − 1) × (11 + ((1 + 1) × (1 + 111)))
:= 2 + ((2 × 22 + 2) + 222)
:= 2 + ((22/2) + 2222)
:= 2 + (((2 × (22 + 2))2 ) + 2 × 22)
:= (33 + 3) × (3 × 33 − 3)
:= 3 + (((33 × ((3 + 3)3 − 3)) − 3)/3)
:= 3/3 + (33 × (3 × 33 + 3 + 3))
:= (44 + 4) × (4/4 + 4 + 4)
:= 4 + (((((4 − 4/4) + 4)4 ) − 4 × 4 × 4) + 4)
:= (4444 + 44 )/((4 + 4)/4)
:= 55 + (((55 − 5)/(5/5 − 5)) − 5)
:= 5 × ((5 − 5/5)5 − 555)
:= 5 × ((5 × (5 × (5 × 5 − 5) − 5)) − 5)
:= 6 × (6 × 66 − 6)
:= 6 + (6 × (6 × 66 − 6) − 6/6)
:= ((66 − 6)/6) + 6 × (6 × 66 − 6)
:= 7 × (7 × 7 × 7 − 7) − (77 + 7)/7
:= 7 × (7 × 7 × 7 − 7) − 7
:= 7 × (7 × 7 × 7 − 7) − (7 + 7)/7
8×8/(8+8)
:= (8/8 + 8) × ((8 × 8 × 8 + 8)/((8 + 8)/8))
:= 8 + (((8 − 8/8)
:= 9 × (9 × (9 + 9) + 99) − 9
:= (9 − ((9 + 9)/9)) × ((9 + 9) × (9 + 9) + (99/9))
Ï 2341 := 11 + ((111 × (11 + (11 − 1))) − 1)
) − 8 × 8)
Ï 2346 := 1 + (111+1 + ((1 + 1) × (1 + 1111)))
:= ((8 + 8)/8) × (8888/8 + 8 × 8)
:= 9/9 + 9 × (9 × (9 + 9) + 99)
Ï 2351 := ((1 + 111) × (11 + (11 − 1))) − 1
:= (22/2)2 + (2222 − 2)
:= 2 × 22 + (((2 × (22 + 2))2 ) − 2)
:= 2222 + ((22×(2+2) + 2)/2)
:= 3/3 + ((33 + 3) × (3 × 33 − 3))
:= 3 + (33 × (((3 + 3)3 − 3)/3))
:= (((3 × 33 + 3)3−3/3 ) − 3)/3
:= 4 + ((((4 − 4/4) + 4)4 ) − 4 × 4 × 4)
:= (((4 − 4/4) + 4)4 ) − 44/4 − 44
:= 44/4 + (44 + 4) × (4/4 + 4 + 4)
:= 55 + ((55/5 + 55 )/(5/5 − 5))
:= (((5 + 5)/5 + 5)5−5/5 ) − 55
:= 5 + ((((5 + 5)/5 + 5)5−5/5 ) − 55)
:= 6/6 + 6 × (6 × 66 − 6)
:= 6 + 6 × (6 × 66 − 6)
:= 66/6 + 6 × (6 × 66 − 6)
:= 7 × (7 × 7 × 7 − 7) − 77/7
:= 7/7 + (7 × (7 × 7 × 7 − 7) − 7)
:= 7 × (7 × 7 × 7 − 7) − 7/7
:= ((88 − 8/8) × (88/8 + 8 + 8)) − 8
:= (8 + 8)/8 + (((8 + 8) × (8 × 8 + 88)) − 88)
:= 8 + (((8 + 8) × (8 × 8 + 88)) − (8/8 + 88))
:= 9/9 + (9 × (9 × (9 + 9) + 99) − 9)
:= 9 × (9 × (9 + 9) + 99) − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 × (9 × (9 + 9) + 99)
Ï 2347 := 111+1 + ((1 + 1) × (1 + (1 + 1111)))
Ï 2342 := 11 + (111 × (11 + (11 − 1)))
2
2
:= 2 × (((2 × ((22 + 2)2 )) + 22) + 2)
:= 2 + (((2 × 22 + 2) + 222) + 2)
:= 2 + (((22/2) + 2222) + 2)
:= ((33 × ((3 + 3)3 − 3)) − 3)/3
:= 3 + (((33 × ((3 + 3)3 − 3)) + 3)/3)
4
4
Ï 2352 := (1 + 111) × (11 + (11 − 1))
4
:= 3 + (33 × (3 × 33 + 3 + 3))
:= 4 + (44/4 × ((4/4 − 44) + 4 ))
:= ((4 + 4) + 4) × ((4 × (44 + 4)) + 4)
:= 5 + ((((5 + 5)/5 + 5 ) + 5)/(5/5 − 5))
:= 55 + (((55 − 5)/(5/5 − 5)) + ((5 + 5)/5))
:= (55 + 5/5) × ((((5 + 5)/5)5 + 5) + 5)
:= (6 + 6)/6 + 6 × (6 × 66 − 6)
:= 6 + (6 × (6 × 66 − 6) + 6/6)
:= 6 + (6 × (6 × 66 − 6) + 6)
:= ((7 − 77)/7) + 7 × (7 × 7 × 7 − 7)
:= (7 + 7)/7 + (7 × (7 × 7 × 7 − 7) − 7)
:= 7 × (7 × 7 × 7 − 7)
:= 4 + ((((4 − 4/4) + 4) ) + ((4 − 4 )/4))
5
5
:= ((8 + 8) × (8 × 8 + 88)) − ((8 + 8)/8 + 88)
:= ((8 × (8 − 888)) − 8/8)/(8 − 88/8)
:= 8 + (((8 + 8) × (8 × 8 + 88)) − 88)
:= (9 + 9)/9 + (9 × (9 × (9 + 9) + 99) − 9)
:= 9 × (9 × (9 + 9) + 99) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + 9 × (9 × (9 + 9) + 99)
166
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2353 := 1 + ((1 + 111) × (11 + (11 − 1)))
:= 22 + (222/2 × (22 − 2/2))
3
3−3/3
Ï 2358 := (1 + 1) × ((11 − 1 − 1) × ((11 × (1 + 11)) − 1))
2
:= 22 /2 + (2 × 22 + 2)
2
Ï 2363 := 11 + ((1 + 111) × (11 + (11 − 1)))
:= 22 + ((2222 − 2) + (22/2)2 )
:= 3 × ((33 × (33 − 3)) − (3 + 3))
:= 33 + (3 × 333 + (33/3)3 )
:= (((4 − 4/4) + 4)4 ) − (44 + 4)
:= 4/4 + ((((4 − 4/4) + 4)4 ) − 44)
:= 44 + (44 × (44 + 4) − (4/4 + 4))
:= 5 + ((((5 + 5)/5)55/5 ) + (5 × (55 + 5)))
:= (5 + 5)/5 × ((55 − 5)/5 + 555)
:= 5 + ((5 + 5)/5 × ((55 − 5)/5 + 555))
:= 6 + ((6 × (6 × 66 − 6) + 6/6) + 6)
:= 6 × 6 × 66 − 6 − 6 − 6
:= 6 × 6 × 66 − (6/6 + 6 + 6)
:= 7/7 + 7 × (7 × 7 × 7 − 7)
:= 7 + (7 × (7 × 7 × 7 − 7) − 7/7)
:= 77/7 + 7 × (7 × 7 × 7 − 7)
:= 8 + (((8 + 8)/8) × (8888/8 + 8 × 8))
:= (8/8 + 8 + 8) × (8 × (8 + 8) + (88/8))
:= 9 + 9 × (9 × (9 + 9) + 99)
:= 9 + (((99 − 9/9) + 9) × ((99 + 99)/9))
:= (((3 × 3 + 3)
) + 3)/3
8×8/(8+8)
:= 8 + ((((8 − 8/8)
) − 8 × 8) + 8)
:= ((9 × 9 − 9)/(9 + 9)) + 9 × (9 × (9 + 9) + 99)
Ï 2354 := 1 + (1 + ((1 + 111) × (11 + (11 − 1))))
Ï 2359 := 1 + ((1 + 1) × ((11 − 1 − 1) × ((11 × (1 + 11)) − 1))) Ï 2364 := (1 + 11) × (1 + (1 + 1 + 1 + 11)1+1 )
:= 22 × (222/2 − (2 + 2))
:= (222 + 2)/2 + (2 × 22 + 2)2
:= 2 × ((2 × ((((22 + 2)2 ) + 2) + 2)) + 22)
:= 33/3 × (((3 + 3)3 − 3) + 3/3)
:= 3/3 + (3 × ((33 × (33 − 3)) − (3 + 3)))
:= (3 × ((33 × (33 − 3)) − 3)) − 3
:= 4 + ((4444 + 44 )/((4 + 4)/4))
:= 4 + (((44/4)4−4/4 ) + 4 × 44 )
:= 44 + (44 × (44 + 4) − 4)
:= 5 + ((((55 − 5/5)/(5/5 − 5)) + 55 ) + 5)
:= 55 + ((55 − ((5 + 5)/5 + 5))(5+5)/5 )
:= 55 − ((555 + 55 )/5 + 5 × 5)
:= 66/6 × (6 × 6 × 6 − (6 + 6)/6)
:= 6 × 6 × 66 − (66/6 + 6)
:= 6 × 6 × 66 − 6 − 6
:= (7 + 7)/7 + 7 × (7 × 7 × 7 − 7)
:= 7 + 7 × (7 × 7 × 7 − 7)
:= (77 + 7)/7 + 7 × (7 × 7 × 7 − 7)
:= 88/8 × ((8 × (8 + 8) − ((8 + 8)/8)) + 88)
:= 8 × 8 + ((88 × (8 + 8) − 8/8) + 888)
:= 8 × 8 + (((8 × 8 × (8 × 8 + 8)) − 8)/((8 + 8)/8))
:= ((99 − 9/9) + 9) × ((99 + 99)/9)
:= 9 + (9 × (9 × (9 + 9) + 99) + 9/9)
:= ((99 + 9)/9) × ((99 − 9/9) + 99)
Ï 2360 := (1 + 1) × ((11 − 1) × (111+1 − (1 + 1 + 1)))
Ï 2355 := 111+1+1 + (1 + 1)11−1
2
2
Ï 2365 := 11 × (((1 + 1) × (1 + 1 + 111)) − 11)
:= 22 + (222/2 + 2222)
:= 2 + ((2 × 22 + 2) + 22 /2)
:= 22 + ((22/2)2 + 2222)
:= 3 + ((33 × (3 × 33 + 3 + 3)) + 3)
:= 33/3 + (33 × (3 × 33 + 3 + 3))
:= 33/3 × ((3 + 3)3 − 3/3)
:= 4 × 44 + ((44/4)4−4/4 )
:= (44 − 4) × (((44 − 4)/4) − 4)
:= 4 + (((((4 − 4/4) + 4)4 ) − 44) + 4)
:= 5 + (5 × ((5 × (5 × (5 × 5 − 5) − 5)) − 5))
:= (5 + 5) × (555/5 + 5 × 5 × 5)
:= 55 × (55 − ((55 + 5)/5))
:= 6 × 6 × 66 + ((6 − (66 + 66))/6)
:= (6 − 66)/6 + (6 × 6 × 66 − 6)
:= 6 × 6 × 66 − 66/6
:= (7 + 7 + 7)/7 + 7 × (7 × 7 × 7 − 7)
:= 7 + (7 × (7 × 7 × 7 − 7) + 7/7)
:= 7 + ((7 × (7 × 7 × 7 − 7) − 7/7) + 7)
:= 88/8 + (((8 + 8) × (8 × 8 + 88)) − 88)
:= 8 × 8 + (88 × (8 + 8) + 888)
:= 88/8 × ((8 × (8 + 8) − 8/8) + 88)
:= 9 + (9 × (9 × (9 + 9) + 99) − ((9 + 9 + 9)/9))
:= 99/9 + 9 × (9 × (9 + 9) + 99)
:= 99/9 × (((9 + 9) × (99 + 9) − 9)/9)
Ï 2356 := 1 + (111+1+1 + (1 + 1)11−1 )
:= 2 + (22 × (222/2 − (2 + 2)))
3
:= 3 + ((((3 × 3 + 3)
3−3/3
) + 3)/3)
4
:= ((4 + 4) × (44 + 4 )) − 44
5
Ï 2361 := 1 + ((1 + 1) × ((11 − 1) × (111+1 − (1 + 1 + 1))))
2
2
Ï 2366 := (1 + 11)1+1 + ((1 + 1) × 1111)
:= 2 + ((22 + 2)/2 + (2 × 22 + 2) )
:= 2222 + ((2 × (2 + 2 + 2))2 )
:= 3 + (3 × ((33 × (33 − 3)) − (3 + 3)))
:= (3 × ((33 × (33 − 3)) − 3)) − 3/3
:= 4 + ((((4 − 4/4) + 4)4 ) − 44)
:= 44 + (44 × (44 + 4) − (4 + 4)/4)
:= 55 /5 + (5555 + 55 )/5
:= 5/5 + (55 × (55 − ((55 + 5)/5)))
:= 6 + (6 × (6 × 66 − 6) + ((66 − 6)/6))
:= 6 × 6 × 66 + (6 − 6 × 6)/((6 + 6)/6)
:= (6 − 66)/6 + 6 × 6 × 66
:= 77/7 + (7 × (7 × 7 × 7 − 7) − 7)
:= 7 + (7 × (7 × 7 × 7 − 7) + ((7 + 7)/7))
:= 7 + 7 × (7 × 7 × 7 − 7) + 7
:= 8 + (((8 × 8 × (8 × 8 + 8)) + 88)/((8 + 8)/8))
:= 8/8 + ((88 × (8 + 8) + 888) + 8 × 8)
:= 8 × 8 + (((88 + 8) × (8 + 8 + 8)) − ((8 + 8)/8))
:= 9 + (9 × (9 × (9 + 9) + 99) − ((9 + 9)/9))
:= ((99 + 9)/9) + 9 × (9 × (9 + 9) + 99)
:= 9 + ((9 × (9 × (9 + 9) + 99) − 9/9) + 9)
:= (5 − 5/5) × (((5 − 55)/5) − 5 × 5)
Ï 2357 := 1 + (1 + (111+1+1 + (1 + 1)11−1 ))
Ï 2362 := 11 + (((1 + 111) × (11 + (11 − 1))) − 1)
Ï 2367 := 1 + ((1 + 11)1+1 + ((1 + 1) × 1111))
:= (222 − 2)/2 + (2 × 22 + 2)2
:= 2 + (2 × 22 + 2)2 + 222 /2 + 2
:= 2 + (((22/2)2 + 2222) + 22)
:= 3 + (33/3 × (((3 + 3)3 − 3) + 3/3))
:= (33/3 × ((3 + 3)3 − 3/3)) − 3
:= 3 × ((33 × (33 − 3)) − 3)
:= (((4 − 4/4) + 4)4 ) − 44
:= 4 + (((((4 − 4/4) + 4)4 ) − 44) + 4/4)
:= 44 + (44 × (44 + 4) − 4/4)
:= 55 + (((5 + 5)/5)5 × (5/5 − 5 × 5))
:= (5 + 5)/5 × ((55 + 5)/5 + 555)
:= 5 + ((5 + 5)/5 × ((55 + 5)/5 + 555))
:= 6 + (6 × (6 × 66 − 6) + (66/6))
:= 6 × 6 × 66 − ((6 + 6)/6 + 6 + 6)
:= 6 × 6 × 66 + (((6 − 66) + 6)/6)
:= 7 + (7 × (7 × 7 × 7 − 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 × (7 × 7 × 7 − 7)
:= 7 + ((7 × (7 × 7 × 7 − 7) + 7/7) + 7)
:= 8 + ((88 − 8/8) × (88/8 + 8 + 8))
:= 8 + (88/8 × ((8 × (8 + 8) − ((8 + 8)/8)) + 88))
:= 8 × 8 + (((88 + 8) × (8 + 8 + 8)) − 8/8)
:= 9 + (9 × (9 × (9 + 9) + 99) − 9/9)
:= ((99 + 9 + 9)/9) + 9 × (9 × (9 + 9) + 99)
:= 9 + (9 × (9 × (9 + 9) + 99) + 9)
167
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2368 := ((1 + 1)1+11 ) − ((1 + 11)1+1+1 )
2
:= 2 × (2 × (((22 + 2) ) + 2
2+2
))
Ï 2373 := (11 + (11 − 1)) × (1 + 1 + 111)
:= (22 − 2/2) × (222/2 + 2)
:= 2 + 22 × ((222 − 2)/2 − 2)
:= 3 + (33/3 × ((3 + 3)3 − 3/3))
:= (3 × (33 × (3 − 3))) − 3
:= 3 + (((33 × (3 + 3)3 ) − 3)/3)
:= 44 + 44 × (44 + 4)
:= 4 + ((((4 − 4/4) + 4)4 ) − 4 × (4 + 4))
:= 444 + (44 × 44 − (4 + 4)/4)
:= ((5 + 5)/5)5 × ((5 × (5 + 5 + 5)) − 5/5)
:= 55 − (5 × 5 × (5 × 5 + 5) + ((5 + 5)/5))
:= 55 − (((555 + 55) + 55 )/5)
:= (6 × 6 + 6/6) × ((6 + 6)/6)6
:= 6 × 6 × 66 − 6 × 6/(6 + 6)
:= (6 + 6)/6 + 6 × 6 × 66
:= 7 + ((7 × (7 × 7 × 7 − 7) + ((7 + 7)/7)) + 7)
:= 7 × (7 × 7 × 7 + 7) − 77
:= 7 × 7 × 7 × 7 − ((((7 + 7)/7 + 7) + 7) + 7)
:= 8 × (8 × 888/(8 + 8 + 8))
:= 8 + (88/8 × ((8 × (8 + 8) − 8/8) + 88))
:= (8 + 8)/8 + (88 × (88/8 + 8 + 8))
:= 9 + ((9 × (9 × (9 + 9) + 99) + 9/9) + 9)
:= ((99 × (9 × 9 − 9)) − 9)/((9 + 9 + 9)/9)
:= (9/9 + 9 × 9) × (99/9 + 9 + 9)
Ï 2369 := 1 + (((1 + 1)1+11 ) − ((1 + 11)1+1+1 ))
3
Ï 2378 := (1 + 1) × (1 + (11 × (111 − 1 − 1 − 1)))
Ï 2374 := 1 + ((11 + (11 − 1)) × (1 + 1 + 111))
Ï 2379 := 1 + ((1 + 1) × (1 + (11 × (111 − 1 − 1 − 1))))
:= 2/2 + (((2 × (22 + 2))2 ) + 22+2+2 )
:= 22 × ((222 − 2)/2 − 2) − 2
:= (((2/2 + 2 + 2) + 2)2+2 ) − 22
:= (((33 × (3 + 3)3 ) − 3)/3) − 3 − 3
:= (((33 × (3 + 3)3 ) + 3)/3) − 3
:= 3 + (3 × (33 × (33 − 3)))
:= (((4 − 4/4) + 4)4 ) − 4 × (4 + 4)
:= (((4 − 4/4) + 4)4 ) − (44/4 + 4 × 4)
:= 444 + (44 × 44 − 4/4)
:= 55 − ((5 × 5 × (5 × 5 + 5) + 5/5) + 5)
:= 5 − (5 × 5 × (5 × 5 + 5) + 5/5)
:= 5 + ((5 × (5 × 55 − 5)) + (5 − 5/5)5 )
:= 6 × 6 × 66 − 6/6 − 6
:= 6 × 6 × 66 − (6 + 6)/6
:= 6 × 6 × 66 + (6 × 6/(6 + 6))
:= 7 + (7 × (7 × 7 × 7 − 7) + ((77 − 7)/7))
:= 7/7 + (7 × (7 × 7 × 7 + 7) − 77)
:= 7 × 7 × 7 × 7 − (7/7 + 7 + 7 + 7)
:= 8/8 + (((88 + 8) × (8 + 8 + 8)) + 8 × 8)
:= (88 × (88/8 + 8 + 8)) − (8 + 8)/8
:= 8 × 8 + (((88 + 8) × (8 + 8 + 8)) + (88/8))
:= 9 + (9 × (9 × (9 + 9) + 99) + (99/9))
:= ((9 + 9)/9) × (((99 × (99 + 9)) − 9)/9)
:= 999/9 + (9 × (9 × (9 + 9 + 9) + 9))
Ï 2370 := (11 − 1) × (11 + ((1 + 1) × (1 + 1 + 111)))
5
Ï 2375 := ((1 + 1) × (11 × (111 − 1 − 1 − 1))) − 1
Ï 2380 := (1 + 1) × ((11 − 1) × (111+1 − (1 + 1)))
:= (2 × (2 + 2 + 2)2+2 ) − 222
:= 2 + ((22 − 2/2) × (222/2 + 2))
:= 2 × (2 × 222 + 222)
:= 3 + (3 × ((33 × (33 − 3)) − 3))
:= ((33 × (3 + 3)3 ) − 3)/3
:= 3 + (((33 × (3 + 3)3 ) + 3)/3)
:= 4/4 + ((((4 − 4/4) + 4)4 ) − 4 × (4 + 4))
:= ((44 × (44 − 44 + 4)) − 4)/4
:= 444 + 44 × 44
:= 55 − (5 × 5 × (5 × 5 + 5) + 5)
:= 5 × (5 × (5 × (5 × 5 − 5) − 5))
:= 5 + (5 × (5 × (5 × (5 × 5 − 5) − 5)))
:= 6 × 6 × 66 − 6
:= 6 × 6 × 66 − 6/6
:= 6 + (6 × 6 × 66 − ((6 + 6)/6))
:= 7 + (7 × (7 × 7 × 7 − 7) + (77/7))
:= (7 + 7)/7 + (7 × (7 × 7 × 7 + 7) − 77)
:= 7 × 7 × 7 × 7 − (7 + 7 + 7)
:= 8 × 8 + (((88 + 8) × (8 + 8 + 8)) + ((8 + 8)/8))
:= (88 × (88/8 + 8 + 8)) − 8/8
:= ((88 + 8)/8 + 8) × (888/8 + 8)
:= 9 + (9 × (9 × (9 + 9) + 99) + ((99 + 9)/9))
:= 9 × 9 + ((9 + 9) × 99 + (((9 + 9)/9)9 ))
:= (99/9 + 9) × ((99/9 + 99) + 9)
Ï 2371 := ((11 + (11 − 1)) × (1 + 1 + 111)) − 1 − 1
Ï 2376 := (1 + 1) × (11 × (111 − 1 − 1 − 1))
Ï 2381 := (1 + 1)11 + (1 + 1 + 1) × 111
:= ((22 − 2/2) × (222/2 + 2)) − 2
:= 22 × ((222 − 2)/2 − 2)
:= 2 + ((((2/2 + 2 + 2) + 2)2+2 ) − 22)
:= (((33 × (3 + 3)3 ) + 3)/3) − 3 − 3
:= 3 × (33 × (33 − 3))
:= 3 + ((((33 × (3 + 3)3 ) − 3)/3) + 3)
:= 4 + ((44 × (44 + 4) − 4/4) + 44 )
:= 44 × ((44 − 4)/4 + 44)
:= (((4 − 4/4) + 4)4 ) − 4 × 4 − 4
:= 55 + (5/5 − (5 × 5 × (5 × 5 + 5) + 5))
:= 55 + (5/5 − 5 × 5 × (5 × 5 + 5))
:= 5 + ((5/5 − 5 × 5 × (5 × 5 + 5)) + 55 )
:= 6/6 + (6 × 6 × 66 − 6)
:= 6 × 6 × 66
:= 6 + (6 × 6 × 66 − 6/6)
:= 7 + (7 × (7 × 7 × 7 − 7) + (77 + 7)/7)
:= 7 × 7 × 7 × 7 − (77/7 + 7 + 7)
:= 7/7 + (7 × 7 × 7 × 7 − (7 + 7 + 7))
:= 8 + ((8/8 + 8 + 8) × (8 × (8 + 8) + (88/8)))
:= 88 × (88/8 + 8 + 8)
:= 88 + (((88 + 8) × (8 + 8 + 8)) − (88/8))
:= ((99 + 99)/9) + 9 × (9 × (9 + 9) + 99)
:= 999 + 9 × (9 × (9 + 9) − 9)
:= 9 + ((((9 + 9)/9)99/9 ) + (9 + 9) × (9 + 9))
Ï 2372 := ((11 + (11 − 1)) × (1 + 1 + 111)) − 1
Ï 2377 := 1 + ((1 + 1) × (11 × (111 − 1 − 1 − 1)))
Ï 2382 := 1 + ((1 + 1)11 + (1 + 1 + 1) × 111)
:= 2 × ((2 × (222 − 2)) + 222)
:= (2 × 22)2 + ((22 − 2/2)2 )
:= 2 + ((2 × 22)2 + 2 × 222)
:= (((33 × (3 + 3)3 ) − 3)/3) − 3
:= ((33 × (3 + 3)3 ) + 3)/3
:= 3 + ((3 × (33 × (33 − 3))) + 3)
:= 4 + (44 × (44 + 4) + 44 )
:= (((4 − 4/4) + 4)4 ) − ((4 × 4 + 4) + 4)
:= 4/4 + ((((4 − 4/4) + 4)4 ) − (4 × 4 + 4))
:= (5 − 5/5) × (55 /5 − ((5 + 5)/5)5 )
:= 55 + ((5 + 5)/5 − 5 × 5 × (5 × 5 + 5))
:= 5 + (((5 + 5)/5 − 5 × 5 × (5 × 5 + 5)) + 55 )
:= (6 + 6)/6 + (6 × 6 × 66 − 6)
:= 6/6 + 6 × 6 × 66
:= 6 + 6 × 6 × 66
:= 7 × 7 × 7 × 7 − ((77 + 7)/7 + 7)
:= 7 × (7 × 7 × 7 + 7) − 7/7 − 77
:= 7 + ((7 × (7 × 7 × 7 − 7) + (77/7)) + 7)
:= ((8 × 8 × 88) − 888)/((8 + 8)/8)
:= 8/8 + (88 × (88/8 + 8 + 8))
:= 8 + ((88 × (88/8 + 8 + 8)) − ((8 + 8)/8))
:= (9 + 9) × (9 + 9) + (((9 + 9)/9)99/9 )
:= 9/9 + (9 × (9 × (9 + 9) − 9) + 999)
:= 9 + (((99 × (9 × 9 − 9)) − 9)/((9 + 9 + 9)/9))
168
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2383 := 1 + (1 + ((1 + 1)11 + (1 + 1 + 1) × 111))
Ï 2388 := 1 + (11 × ((1 + 1) × (111 − 1 − 1) − 1))
Ï 2393 := ((1 + 1) × ((11 × (111 − 1 − 1)) − (1 + 1))) − 1
:= (2 + 2 + 2) × ((22 − 2) − 2)
:= 2 + (((22/2 + 2)2 ) + 2222)
:= 3 + ((((33 × (3 + 3)3 ) + 3)/3) + 3)
:= 3 + (3 × ((33 × (33 − 3)) + 3))
:= 3 + ((33/3 × ((3 + 3)3 + 3/3)) + 3)
:= 4 + ((44 × 44 − 4/4) + 444)
:= 4 + ((44 × 44 + 444) + 4)
:= (((4 − 4/4) + 4)4 ) − 4 − 4
:= 55 − ((((555 + 55 ) + 5)/5) + 5)
:= 55 − (((555 + 55 ) + 5)/5)
:= 5 + (55 − (((555 + 55 ) + 5)/5))
:= 6 + (6 × 6 × 66 + 6/6)
:= 6 + (6 × 6 × 66 + 6)
:= 6 + (6 × 6 × 66 + (66/6))
:= 7 × 7 × 7 × 7 − (77/7 + 7)
:= 7/7 + (7 × 7 × 7 × 7 − (7 + 7))
:= 7 × 7 × 7 × 7 − (7/7 + 7)
:= 8 + ((88 × (88/8 + 8 + 8)) − 8/8)
:= ((88 + 8)/8) × (888/8 + 88)
:= ((8 − 8/8)8×8/(8+8) ) − 8
:= 9 + (((9 + 9)/9) × (((99 × (99 + 9)) − 9)/9))
:= ((99 + 9)/9) × ((9/9 + 99) + 99)
:= 99 + ((9 + 9) × 99 + (((9 + 9)/9)9 ))
:= 2 + (((((2/2 + 2 + 2) + 2)
2+2
) − 22) + 2)
2
Ï 2384 := (1 + 1)11 + ((1 + 1 + 1) × (1 + 111))
Ï 2389 := 1 + (1 + (11 × ((1 + 1) × (111 − 1 − 1) − 1)))
2
:= 2/2 + ((2 + 2 + 2) × ((22 − 2)2 − 2))
:= 2 × ((2 × 22 + 222) + 2)
3
:= 3 × 3 + (((33 × (3 + 3) ) − 3)/3)
:= 3 + ((3 × ((33 × (3 − 3)) + 3)) + 3/3)
4
:= 4 + (44 × 44 + 444)
5
3
:= 4 + ((((4 − 4/4) + 4) ) − 4 × 4)
5
:= 5 − ((555 + 5 )/5 + 5)
5
5
Ï 2394 := (1 + 1) × ((11 × (111 − 1 − 1)) − (1 + 1))
:= 2 + (2 × (2 × (((22 + 2)2 ) + 22)))
:= 3 × (((33 × (33 − 3)) + 3) + 3)
:= 4 + ((((4 − 4/4) + 4)4 ) − 44/4)
:= 5 − (555 + 5 )/5
:= 5 + (55 − (555 + 55 )/5)
:= 6 + (6 × 6 × 66 + ((6 + 6)/6))
:= 6 + (6 × 6 × 66 + 6/6 + 6)
:= 6 + ((6 × 6 × 66 + 6) + 6)
:= ((7 − 77)/7) + (7 × 7 × 7 × 7 − 7)
:= 7 × 7 × 7 × 7 − (77 + 7)/7
:= 7 × 7 × 7 × 7 − 7
8×8/(8+8)
:= 8 + (88 × (88/8 + 8 + 8))
:= ((8 − 8/8)
:= (9 − 9/9) × ((99 × (9 + 9 + 9) + 9)/9)
:= 9 + ((99/9 + 9) × ((99/9 + 99) + 9))
Ï 2385 := 1 + ((1 + 1)11 + ((1 + 1 + 1) × (1 + 111)))
) − (88 + 8)/8
Ï 2390 := (11 − 1) × (((1 + 1) × (111+1 − 1)) − 1)
:= (2/2 + 2)2+2 + ((2 × (22 + 2))2 )
:= 2 + ((2 + 2 + 2) × ((22 − 2)2 − 2))
:= 3 × ((33 × (33 − 3)) + 3)
:= 3 + (33/3 × ((3 + 3)3 + 3/3))
:= (((4 − 4/4) + 4)4 ) − 4 × 4
:= 5 + ((5 × (5 × (5 × (5 × 5 − 5) − 5))) + 5)
4
:= (((4 − 4/4) + 4) ) − 44/4
5
5
:= 8/8 + (((8 − 8/8)8×8/(8+8) ) − 8)
:= 9 + (9 × ((((9 + 9)/9)9−9/9 ) + 9))
Ï 2395 := ((1 + 1) × ((11 × (111 − 1 − 1)) − 1)) − 1
:= 22 × (222/2 − 2) − 2/2 − 2
:= (((3/3 + 3) + 3)3/3+3 ) − 3 − 3
:= (((4 − 4/4) + 4)4 ) − ((4 + 4)/4 + 4)
:= 5 + ((5 − (555 + 5 ))/5)
:= 6 + (6 × 6 × 66 + (6 × 6/(6 + 6)))
:= 55 × 55 − (55 /5 + 5)
:= 6 + ((6 × 6 × 66 + ((6 + 6)/6)) + 6)
:= 7 × 7 × 7 × 7 − (((7 + 7)/7 + 7) + 7)
:= 7 × 7 × 7 × 7 − 77/7
:= ((6/6 + 6)6−(6+6)/6 ) − 6
:= ((8 − 8/8)8×8/(8+8) ) − 8 − 8
:= ((8 − 8/8)8×8/(8+8) ) − 88/8
:= 9 × ((((9 + 9)/9)9−9/9 ) + 9)
:= ((8/8 + 88) × (88/8 + 8 + 8)) − 8
:= 9 + (((((9 + 9)/9)99/9 ) + (9 + 9) × (9 + 9)) + 9)
:= 9 × (9 + 9) × (9 + 9) − ((((9 + 9)/9)9 ) + 9)
Ï 2386 := (11 × ((1 + 1) × (111 − 1 − 1) − 1)) − 1
2
:= ((2 + 2 + 2) × ((22 − 2) − 2)) − 2
3
Ï 2391 := 1 + ((11 − 1) × (((1 + 1) × (111+1 − 1)) − 1))
:= 2222 + ((22/2 + 2)2 )
3
:= 7/7 + (7 × 7 × 7 × 7 − 7)
Ï 2396 := (1 + 1) × ((11 × (111 − 1 − 1)) − 1)
:= 22 × (222/2 − 2) − 2
:= 3 × 3 + (((33 × (3 + 3) ) + 3)/3)
:= 3 + ((3 × ((33 × (3 − 3)) + 3)) + 3)
:= 3 × 3 + (33/3 × ((3 + 3)3 + 3/3))
:= 4/4 + ((((4 − 4/4) + 4)4 ) − 4 × 4)
:= (4 − 44)/4 + (((4 − 4/4) + 4)4 )
:= ((4 + 4) × (44 + 44 )) − 4
:= 55 + (55/5 − 5 × 5 × (5 × 5 + 5))
:= (((5 + 5)/5 + 5)5−5/5 ) − 5 − 5
:= (((5 + 5)/5 + 5)5−5/5 ) − 5
:= 6 × 6 × 66 + (66 − 6)/6
:= 6 + ((6 × 6 × 66 + (6 × 6/(6 + 6))) + 6)
:= 6 + (((6 × 6 × 66 + ((6 + 6)/6)) + 6) + 6)
:= 7 × 7 × 7 × 7 − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 × 7 × 7 × 7
:= (7 + 7)/7 + (7 × 7 × 7 × 7 − 7)
:= 8 + ((88 × (88/8 + 8 + 8)) + ((8 + 8)/8))
:= 88 + (((88 + 8) × (8 + 8 + 8)) − 8/8)
:= ((8 × (8 × 8 × 8 + 88)) − 8)/((8 + 8)/8)
:= 9/9 + (9 × ((((9 + 9)/9)9−9/9 ) + 9))
:= ((99/9 + 9) × (999/9 + 9)) − 9
:= 9 + (((9/9 + 9 × 9) × (99/9 + 9 + 9)) + 9)
Ï 2387 := 11 × ((1 + 1) × (111 − 1 − 1) − 1)
:= 22/2 × (222 − (2/2 + 2 + 2))
3
Ï 2392 := 11 + ((1 + 1)11 + (1 + 1 + 1) × 111)
:= 2 × (2 × (((22 + 2)2 ) + 22))
3/3+3
Ï 2397 := ((1 + 1) × (11 × (111 − 1 − 1))) − 1
:= 22 × (222/2 − 2) − 2/2
:= (3 × (33 × (33 + 3))) − 33
:= 33/3 × ((3 + 3) + 3/3)
:= (((3/3 + 3) + 3)
:= (4 + 4)/4 + ((((4 − 4/4) + 4)4 ) − 4 × 4)
:= (4 + 4) × ((44 − 4/4) + 44 )
:= (((4 − 4/4) + 4)4 ) − 4
:= 55 − ((((555 + 55 ) + 5) + 5)/5)
:= (5 − 5/5) × ((55 − 5 − 5)/5 − 5 × 5)
:= 5/5 + ((((5 + 5)/5 + 5)5−5/5 ) − 5)
:= 66/6 + 6 × 6 × 66
:= 6 + (6 × 6 × 66 + ((66 − 6)/6))
:= 6 × 6 × 66 + ((6 × 6 + 6)/((6 + 6)/6))
:= 7 × 7 × 7 × 7 − (7 + 7)
:= 7 × 7 × 7 × 7 − ((7 + 7)/7 + 7)
:= 7 + (7 × 7 × 7 × 7 − (77/7))
:= 88/8 + (88 × (88/8 + 8 + 8))
:= 88 + ((88 + 8) × (8 + 8 + 8))
:= (88/8 − 8) × (888 − (8/8 + 88))
:= 9 + ((9/9 + 9 × 9) × (99/9 + 9 + 9))
:= (((9 − 9/9) + 9) + 9) × ((99/9) + 9 × 9)
:= 9 + (((99 + 9)/9) × ((9/9 + 99) + 99))
)−3×3
169
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2398 := (1 + 1) × (11 × (111 − 1 − 1))
:= 22 × (222/2 − 2)
Ï 2403 := 1 + (1 + ((1 + ((1 + 1) × (1 + 1 + 1)))1+1+1+1 ))
:= 2 + (((2/2 + 2 + 2) + 2)
3/3+3
2+2
Ï 2408 := (11 × (((1 + 1) × (111 − 1)) − 1)) − 1
:= ((2/2 + 2 + 2) × (222 − 2)) − 2
)
:= (3 + 3)3 + 3 × 33+3
:= ((33 × ((3 + 3)3 + 3)) − 3)/3
:= 4/4 + ((((4 − 4/4) + 4)4 ) − 4)
:= (4 + 4)/4 + (((4 − 4/4) + 4)4 )
:= 4 + (((4 + 4) × (44 + 44 )) + 4)
:= 55 × 55 − (55 + 5 + 5)/5
:= 5 + (55 × 55 − (55 + 5 + 5)/5)
:= (5 − 5/5) × ((55 + 5 + 5)/5 − 5 × 5)
:= 66/6 × (6 × 6 × 6 + (6 + 6)/6)
:= 6 × 6 × 6 + ((6 × 6/(6 + 6))6/6+6)
:= (6 + 6)/6 + (6 × (6 × 66 + 6) − 6)
:= 7 × 7 × 7 × 7 − (7 + 7 + 7)/7
:= (7 + 7)/7 + 7 × 7 × 7 × 7
:= 7 + 7 × 7 × 7 × 7
:= ((8 + 8)/8) × (8888/8 + 88)
:= (8/8 + 88) × (88/8 + 8 + 8)
:= 8 + ((88 + 8) × (8/8 + 8 + 8 + 8))
:= (9/9 + 99 + 9) × ((99 + 99)/9)
:= (9 + 9 + 9) × ((9 × 9 − 9/9) + 9)
:= ((9 + 9) × (9 × 9 − 9)) + (9999 + 9)/9
:= (((3/3 + 3) + 3)
)−3
Ï 2399 := 1 + ((1 + 1) × (11 × (111 − 1 − 1)))
:= (((2/2 + 2 + 2) + 2)2+2 ) − 2
3
3
Ï 2404 := (1 + 1) × (1 + (1 + (1 + (11 × (111 − 1 − 1)))))
2
:= 2 + (((2 + 2 + 2) × (22 − 2) ) + 2)
Ï 2409 := 11 × (((1 + 1) × (111 − 1)) − 1)
:= 22/2 × (222 − (2/2 + 2))
:= 33 × (((3 + 3)3 + 3)/3)
:= 3 + ((((33 × (3 + 3) ) − 3)/3) − 3)
:= 3 + (((3/3 + 3) + 3)
:= (((4 − 4/4) + 4)4 ) − (4 + 4)/4
:= 4 + ((4 + 4) × (44 + 44 ))
:= 4 + ((((4 − 4/4) + 4)4 ) + 4)
:= 5 × 5 × 55 + (5 − 5/5)5
:= 5 + (5 × 5 × 55 + (5 − 5/5)5 )
:= 5 + ((5 × 5 × 55 + (5 − 5/5)5 ) + 5)
:= 6 + ((6 × 6 × 66 + (66/6)) + 6)
:= ((6 + 6)/6)6 + 6 × (6 × 66 − 6)
:= 6 × 6 × 66 + (66 × 6/(6 + 6))
:= 7 × 7 × 7 × 7 − (7 + 7)/7
3/3+3
)
:= 7 × 7 × 7 × 7 + (7 + 7 + 7)/7
:= 7 + (7 × 7 × 7 × 7 + 7/7)
:= ((8 − 8/8)8×8/(8+8) ) − (8 + 8)/8
:= ((8 × (8 × 8 × 8 + 88)) + 8)/((8 + 8)/8)
:= 8 + ((8 − 8/8)8×8/(8+8) )
:= ((99/9 + 9) × (999/9 + 9)) − 9/9
:= 9 × (9 + 9) × (9 + 9) − (((9 + 9)/9)9 )
:= 9 + ((99/9 + 9) × (999/9 + 9))
Ï 2400 := (1 + 1) × (1 + (11 × (111 − 1 − 1)))
:= (2 + 2 + 2) × (22 − 2)2
3
:= (3 + 3) + (3 × 3
3+3
Ï 2405 := ((1 + 1) × ((11 × (111 − 1)) − (1 + 1))) − 11
:= 2 + ((((2/2 + 2 + 2) + 2)
− 3)
2+2
) + 2)
Ï 2410 := 1 + (11 × (((1 + 1) × (111 − 1)) − 1))
:= (2/2 + 2 + 2) × (222 − 2)
:= (((33 × ((3 + 3)3 + 3)) − 3)/3) − 3
:= ((33 × ((3 + 3)3 + 3)) + 3)/3
:= (4 + 4) × (44 + 4 )
:= 4 + (((4 − 4/4) + 4)4 )
:= 4 + (((((4 − 4/4) + 4)4 ) + 4/4) + 4)
:= (5 × 5 − 5) × (5 × 5 × 5 − 5)
4
:= 5 + ((5 × 5 − 5) × (5 × 5 × 5 − 5))
:= 5 + (((5 × 5 − 5) × (5 × 5 × 5 − 5)) + 5)
:= 6 × (6 × 66 + 6) − 6 − 6
:= (6 × 6 + 6/6) × (66 − 6/6)
:= 6 × (6 × 66 + 6) − (6 + 6)/6
:= 7 × 7 × 7 × 7 − 7/7
:= 77/7 + (7 × 7 × 7 × 7 − 7)
:= 7 + (7 × 7 × 7 × 7 + ((7 + 7)/7))
:= (88 + 8) × (8/8 + 8 + 8 + 8)
:= (8/8 + 8 × 8) × 888/(8 + 8 + 8)
:= 8 + (((8 − 8/8)8×8/(8+8) ) + 8/8)
:= (99/9 + 9) × (999/9 + 9)
:= 9/9 + (9 × (9 + 9) × (9 + 9) − (((9 + 9)/9)9 ))
:= 9 + ((9 − ((9 + 9)/9))(9×9−9)/(9+9) )
Ï 2401 := (1 + ((1 + 1) × (1 + 1 + 1)))1+1+1+1
:= ((2/2 + 2 + 2) + 2)2+2
:= (2/2 + 2) × (2 × (22 − 2)2 + 2)
:= ((3/3 + 3) + 3)3/3+3
:= ((4 − 4/4) + 4)
4
:= ((5 + 5)/5 + 5)
5−5/5
Ï 2406 := (11 × (((1 + 1) × (111 − 1)) − 1)) − 1 − 1 − 1
:= 3 + (3 × 3
3+3
Ï 2411 := 1 + (1 + (11 × (((1 + 1) × (111 − 1)) − 1)))
:= 2 + (22/2 × (222 − (2/2 + 2)))
:= ((33/3 + 3)3 ) − 333
3
+ (3 + 3) )
:= 4 + ((((4 − 4/4) + 4)4 ) + 4/4)
:= 44/4 + ((4 + 4) × (44 + 44 ))
:= (6/6 + 6)6−(6+6)/6
:= 5 + (((5 + 5)/5 + 5)5−5/5 )
:= 5 + ((((5 + 5)/5 + 5)5−5/5 ) + 5)
:= 6 × (6 × 66 + 6) − 6
:= 6 × (6 × 66 + 6) − 6/6
:= 7 × 7 × 7 × 7
:= (8 − 8/8)8×8/(8+8)
:= 7 + (7 × 7 × 7 × 7 − ((7 + 7)/7))
:= 7 × 7 × 7 × 7 + (77 − 7)/7
:= 8 + (((8 + 8)/8) × (8888/8 + 88))
:= 8 + ((8/8 + 88) × (88/8 + 8 + 8))
:= (9 − ((9 + 9)/9))(9×9−9)/(9+9)
:= ((9 + 9 + 9)/9) × ((9 × (9 × 9 + 9) − 9) + 9/9)
:= (9 × 9 + 9) × (9 + 9 + 9) − (9/9 + 9 + 9)
Ï 2402 := 1 + ((1 + ((1 + 1) × (1 + 1 + 1)))1+1+1+1 )
2
:= 2 + ((2 + 2 + 2) × (22 − 2) )
3
3
Ï 2407 := (11 × (((1 + 1) × (111 − 1)) − 1)) − 1 − 1
:= 2 + (((((2/2 + 2 + 2) + 2)
2+2
) + 2) + 2)
Ï 2412 := (1 + 11) × (1 + ((1 + 1) × (11 − 1)1+1 ))
:= (2 + 2 + 2) × ((22 − 2)2 + 2)
:= 3 + ((((3/3 + 3) + 3)3/3+3 ) + 3)
:= 3 + (33 × (((3 + 3)3 + 3)/3))
:= 4/4 + (((4 − 4/4) + 4) )
:= 4 + ((((4 − 4/4) + 4)4 ) + (4 + 4)/4)
:= 44/4 + (((4 − 4/4) + 4)4 )
:= 5/5 + (((5 + 5)/5 + 5)5−5/5 )
:= 5 + ((((5 + 5)/5 + 5)5−5/5 ) + 5/5)
:= 55/5 + (((5 + 5)/5 + 5)5−5/5 )
:= 6/6 + ((6/6 + 6)6−(6+6)/6 )
:= 6 + ((6/6 + 6)6−(6+6)/6 )
:= 6 × (6 × 66 + 6)
:= 7/7 + 7 × 7 × 7 × 7
:= 7 + (7 × 7 × 7 × 7 − 7/7)
:= 77/7 + 7 × 7 × 7 × 7
:= 3 + (((33 × (3 + 3) ) − 3)/3)
4
8×8/(8+8)
:= 8/8 + ((8 − 8/8)
)
:= 999 + ((((9 + 9)/9)9 ) + 9 × 99)
8×8/(8+8)
:= 8 + (((8 − 8/8)
) − ((8 + 8)/8))
:= ((9 + 9) × (9 × 9 − 9)) + 9999/9
170
:= 88/8 + ((8 − 8/8)8×8/(8+8) )
:= (9 × 9 + 9) × (9 + 9 + 9) − (9 + 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2413 := 1 + ((1 + 11) × (1 + ((1 + 1) × (11 − 1)1+1 )))
2
Ï 2418 := (1 + 1) × ((11 × (111 − 1)) − 1)
2
2
Ï 2423 := 1 + ((1 + 1) × (1 + (11 × (111 − 1))))
:= 2/2 + ((2 + 2 + 2) × ((22 − 2) + 2))
:= 22 + ((2 × 22) − 2)
:= 2 + (((22 × (222 − 2)) + 2)/2)
:= 3 + (((33 × ((3 + 3)3 + 3)) + 3)/3)
:= (3 × ((33 × (33 + 3)) − 3)) − 3
:= 3 + (33 × 33 + (33/3)3 )
:= 4 + (((((4 − 4/4) + 4)4 ) + 4) + 4)
:= 4 × 4 + ((((4 − 4/4) + 4)4 ) + 4/4)
:= 4 + ((4 × (4/4 + 4)4 ) − (4 − 4/4)4 )
:= 5 + ((5 − 5/5) × ((55 + 5 + 5)/5 − 5 × 5))
:= 5555 − (((55 + 5)/5) + 55 )
:= 5555 − (((5 + 5)/5 + 55 ) + 5)
:= 6/6 + 6 × (6 × 66 + 6)
:= 6 + 6 × (6 × 66 + 6)
:= 66/6 + 6 × (6 × 66 + 6)
:= 7 × 7 × 7 × 7 + (77 + 7)/7
:= 7 + (7 × 7 × 7 × 7 + ((77 − 7)/7))
:= 7 + (((7 × 7 × 7 × 7 + 7/7) + 7) + 7)
:= 8 + ((((8 − 8/8)8×8/(8+8) ) + 8/8) + 8)
:= ((8 + 8) × (8 × 8 + 88)) − (8/8 + 8)
:= (9 × 9 + 9) × (9 + 9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + ((9 × 9 + 9) × (9 + 9 + 9) − 9)
:= (88/8 + 8) × (8 × (8 + 8) − 8/8)
9
:= 9 + (9 × (9 + 9) × (9 + 9) − (((9 + 9)/9) ))
Ï 2414 := (1 + 1) × ((11 × (111 − 1)) − (1 + 1 + 1))
Ï 2419 := ((1 + 1) × (11 × (111 − 1))) − 1
Ï 2424 := (1 + 1) × (1 + (1 + (11 × (111 − 1))))
:= 2 + ((2 + 2 + 2) × ((22 − 2)2 + 2))
:= ((22 × (222 − 2)) − 2)/2
:= 2 + (((2 × 22)2 + 222 ) + 2)
:= 3 + (((33/3 + 3)3 ) − 333)
:= (3 × (33 × (33 + 3))) − 33/3
:= 3 + (3 × ((33 × (33 + 3)) − 3))
:= 4 + ((((((4 − 4/4) + 4)4 ) + 4/4) + 4) + 4)
:= (4 × (4/4 + 4)4 ) − (4 − 4/4)4
:= 4 + 44 × (44/4 + 44)
:= 5 × 5 + (55 − (555 + 55 )/5)
:= 5555 − (55/5 + 55 )
:= (5 − 5/5) × ((55 × 55 + 5)/5)
:= (6 + 6)/6 + 6 × (6 × 66 + 6)
:= 6 + (6 × (6 × 66 + 6) + 6/6)
:= 7 + ((7 × 7 × 7 × 7 − 7/7) + 7)
:= 7 + (7 × 7 × 7 × 7 + (77/7))
:= 8/8 + ((88/8 + 8) × (8 × (8 + 8) − 8/8))
:= 8 + (((8/8 + 88) × (88/8 + 8 + 8)) + 8)
:= (9 + 9)/9 + ((9 × 9 + 9) × (9 + 9 + 9) − (9 + 9))
:= (9 × 9 + 9) × (9 + 9 + 9) − 99/9
Ï 2415 := ((1 + 1) × ((11 × (111 − 1)) − (1 + 1))) − 1
Ï 2420 := (1 + 1) × (11 × (111 − 1))
:= 6 + (6 × (6 × 66 + 6) + 6)
:= 7 + (((7 × 7 × 7 × 7 + ((7 + 7)/7)) + 7) + 7)
:= ((8 + 8) × (8 × 8 + 88)) − 8
:= ((9 + 9)/9) × (((99/9) × 999/9) − 9)
Ï 2425 := 1 + ((1 + 1) × (1 + (1 + (11 × (111 − 1)))))
:= (22/2)2 + ((2 × (22 + 2))2 )
:= 222/2 + ((2 × (22 + 2))2 )
:= 22 × (222 − 2)/2
:= 3 + ((33 × (((3 + 3)3 + 3)/3)) + 3)
:= 33 × 33 + (33/3)3
:= (4/4 + 4) × ((44 × 44 − 4)/4)
:= 44 × (44/4 + 44)
:= 4 + (((((4 − 4/4) + 4)4 ) + 4 × 4) + 4)
:= (55 × (55 − (55/5))) − 5
:= 55 × (55 − (55/5))
:= 5555 − (55 + 5)
:= (6 × 6/(6 + 6)) + 6 × (6 × 66 + 6)
:= 6 + (6 × (6 × 66 + 6) + ((6 + 6)/6))
:= 6 + ((6 × (6 × 66 + 6) + 6/6) + 6)
:= 7 + (7 × 7 × 7 × 7 + 7)
:= 7 + (7 × 7 × 7 × 7 + (77 + 7)/7)
:= 7 + ((7 × 7 × 7 × 7 + ((77 − 7)/7)) + 7)
:= (88/((8 + 8)/8)) × (8 × 8 − (8/8 + 8))
:= 8/8 + (((8 + 8) × (8 × 8 + 88)) − 8)
:= (9/9 + 9) × (9 × (9 + 9 + 9) − 9/9)
:= ((9 − 99)/(9 + 9)) + (9 × 9 + 9) × (9 + 9 + 9)
:= 888/8 + ((88 + 8) × (8 + 8 + 8))
9−9/9
:= 999/9 + (9 × (((9 + 9)/9)
))
Ï 2416 := (1 + 1) × ((11 × (111 − 1)) − (1 + 1))
:= 3 + ((3 × ((33 × (33 + 3)) − 3)) + 3/3)
Ï 2421 := 1 + ((1 + 1) × (11 × (111 − 1)))
Ï 2426 := (1 + 1) × (1 + (1 + (1 + (11 × (111 − 1)))))
2
:= 22 + (2 × (2 × 22 − 2))
:= ((22 × (222 − 2)) + 2)/2
:= (22 × 222/2) − 22+2
:= 3 + ((((33 × ((3 + 3)3 + 3)) + 3)/3) + 3)
:= 3 × ((33 × (33 + 3)) − 3)
:= (3 × (33 × (33 + 3))) − (3/3 + 3)
:= 4 × (((4 + 4) × 44 − 4) + 44 )
:= 4 + ((((4 − 4/4) + 4)4 ) + 4 × 4)
:= (((4/4 + 4) × (44 × 44 + 4)) + 4)/4
:= (5 − 5/5) × ((55 × 55 − 5)/5)
:= 5/5 + (55 × (55 − (55/5)))
:= 5 × 5 + (((5 + 5)/5 + 5)5−5/5 )
:= 6 + (6 × (6 × 66 + 6) − ((6 + 6)/6))
:= 6 + (6 × (6 × 66 + 6) + (6 × 6/(6 + 6)))
:= 6 + ((6 × (6 × 66 + 6) + ((6 + 6)/6)) + 6)
:= 7 + ((7 × 7 × 7 × 7 + 7/7) + 7)
:= 7 + (((7 × 7 × 7 × 7 − 7/7) + 7) + 7)
:= 7 + ((7 × 7 × 7 × 7 + (77/7)) + 7)
:= (8 + 8) × ((88 − 8/8) + 8 × 8)
:= ((8 + 8) × (8 × 8 + 88)) − 88/8
:= (8 + 8)/8 + (((8 + 8) × (8 × 8 + 88)) − 8)
:= 9 + (((9 + 9) × (9 × 9 − 9)) + 9999/9)
:= (9 × 9 + 9) × (9 + 9 + 9) − 9
:= ((9 − 9 × 9)/(9 + 9)) + (9 × 9 + 9) × (9 + 9 + 9)
2
Ï 2417 := ((1 + 1) × ((11 × (111 − 1)) − 1)) − 1
:= (((22 × (222 − 2)) − 2)/2) − 2
Ï 2422 := (1 + 1) × (1 + (11 × (111 − 1)))
Ï 2427 := ((1 + 1) × (11 × 111 − (1 + 1))) − 11
:= 2 + ((2 × 22) + 22 )
:= 2 + (((2 × (22 + 2))2 ) + (22/2)2 )
:= 3/3 + (3 × ((33 × (33 + 3)) − 3))
:= (3 × (33 × (33 + 3))) − 3
:= 4 × 4 + (((4 − 4/4) + 4) )
:= 4 + (((((4 − 4/4) + 4)4 ) + 4 × 4) + 4/4)
:= (4 × ((4 + 4) × 44 + 44 )) − (4/4 + 4)
:= 5 + ((((5 + 5)/5 + 5)5−5/5 ) + (55/5))
:= (5 + 5)/5 + (55 × (55 − (55/5)))
:= (5 + 5)/5 + (5555 − (55 + 5))
:= 6 + (6 × (6 × 66 + 6) − 6/6)
3
:= 33 × 33 + ((33/3) − 3)
4
2
2
:= ((66 − 6)/6) + 6 × (6 × 66 + 6)
:= 6 + ((6 × (6 × 66 + 6) + (6 × 6/(6 + 6))) + 6)
:= 7 + ((7 × 7 × 7 × 7 + ((7 + 7)/7)) + 7)
:= 7 + ((7 × 7 × 7 × 7 + 7) + 7)
:= 7 + ((7 × 7 × 7 × 7 + (77 + 7)/7) + 7)
:= 8 + (((8 − 8/8)8×8/(8+8) ) + 8)
:= (8 − 88)/8 + ((8 + 8) × (8 × 8 + 88))
:= 88/8 + ((8 + 8) × ((88 − 8/8) + 8 × 8))
:= 99 × (9 + 9 + 9) − (((9 + 9)/9)9−9/9 )
:= 9/9 + ((9 × 9 + 9) × (9 + 9 + 9) − 9)
:= ((9 + 9 + 9)/9) × (9 × (9 × 9 + 9) − 9/9)
171
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2428 := (11 × ((1 + 1) × 111 − 1)) − 1 − 1 − 1
2
2
Ï 2433 := 1 + (1 + (11 × ((1 + 1) × 111 − 1)))
Ï 2438 := (1 + 1) × (11 × 111 − (1 + 1))
:= (22 × 222/2) − 2 − 2
:= 22 + (2 × (2 × (22 + 2)))
:= 2 + (22/2 × (222 − 2/2))
:= 33 + (((3/3 + 3) + 3)3/3+3 )
3
:= 3 + (3 × (3 × (3 + 3)))
:= (3 × ((33 × (33 + 3)) + 3)) − 3/3
:= (4 × ((4 + 4) × 44 + 44 )) − 4
:= 4 × (4 + 4) + (((4 − 4/4) + 4)4 )
:= (4 + 4)/4 + (4 × ((4/4 + 4)4 − 4 × 4))
:= 5555 − ((5 + 5)/5 + 55 )
:= 5 + (5555 − ((5 + 5)/5 + 55 ))
:= 55 + ((5 − 5 × 5 × 5 × 55)/(5 + 5))
:= 6 + (6 × (6 × 66 + 6) + ((66 − 6)/6))
:= 66 + ((((6 − 66) + 6)/6) + 6 × 6 × 66)
:= 6 + (((6 + 6)/6)6 × ((6 + 6)/6 + 6 × 6))
:= 77 + (7 × (7 × 7 × 7 − 7) − 7/7)
:= 7 + (((7 × 7 × 7 × 7 + (77/7)) + 7) + 7)
:= 7 × (7 × 7 × 7 + 7) − (77 + 7)/7
:= ((8 + 8) × (8 × 8 + 88)) − (8/((8 + 8)/8))
:= 8/8 + ((8 + 8) × (8 × 8 + 88))
:= 8 + (((8 + 8) × (8 × 8 + 88)) − ((8 + 8)/8))
:= (9 × 9 + 9) × (9 + 9 + 9) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 × 9 + 9) × (9 + 9 + 9)
:= 9 + ((9 × 9 + 9) × (9 + 9 + 9) − 9/9)
Ï 2429 := (11 × ((1 + 1) × 111 − 1)) − 1 − 1
:= (22/2 × (222 − 2/2)) − 2
3
3
3
Ï 2434 := 1 + (1 + (1 + (11 × ((1 + 1) × 111 − 1))))
Ï 2439 := ((1 + 1) × (11 × 111 − 1)) − 1
:= (22 × 222/2) − 2 × (2 + 2)
:= (((22 × 222) − 2)/2) − 2
3
3
:= (3 × (3 × (3 + 3))) − 3/3
:= 3 + ((3 × (3 × (3 + 3))) + 3/3)
:= 3 × ((33 × (33 + 3)) + 3)
:= 44 + ((((4 − 4/4) + 4)4 ) − 4 × 4)
:= 44 + ((((4 − 4/4) + 4)4 ) − 44/4)
:= 4 + ((4 × (4/4 + 4)4 ) − (44 + 4)/4)
:= 5555 − (55 + 5/5)
:= 5 + (5555 − (55 + 5/5))
:= 55 − (((55 + 5)/5 + 55) + 5)
6
:= 6 + (6 × (6 × 66 + 6) + (66/6))
:= 6 × 6 × 66 + (((6 + 6)/6) − 6)
:= 66 + (6 × 6 × 66 − (6 × 6/(6 + 6)))
:= 77 + 7 × (7 × 7 × 7 − 7)
:= 7 × (7 × 7 × 7 + 7) − (((7 + 7)/7 + 7) + 7)
:= 7 × (7 × 7 × 7 + 7) − 77/7
:= 8 + (((8 + 8) × (8 × 8 + 88)) − (88/8))
:= (8 + 8)/8 + ((8 + 8) × (8 × 8 + 88))
:= 8 + (((8 + 8) × (8 × 8 + 88)) − 8/8)
:= (9 × 9 + 9) × (9 + 9 + 9) − 9/9
:= ((9 × 9 − 9)/(9 + 9)) + (9 × 9 + 9) × (9 + 9 + 9)
:= 9 + (9 × 9 + 9) × (9 + 9 + 9)
Ï 2430 := (11 × ((1 + 1) × 111 − 1)) − 1
2
:= (2/2 + 2 + 2) × (22 + 2)
3
Ï 2435 := ((1 + 1) × (1 + (1 + 11 × 111))) − 11
:= 2 + ((22/2 × (222 − 2/2)) + 2)
3
3
3
Ï 2440 := (1 + 1) × (11 × 111 − 1)
:= (22 × 222/2) − 2
:= 3 × (3 × (3 + 3))
:= 3 + (((3 × (3 × (3 + 3))) − 3/3) + 3)
:= 3/3 + (3 × ((33 × (33 + 3)) + 3))
:= (4/4 + 4) × (((44 × 44 + 4) + 4)/4)
:= (4 × (4/4 + 4)4 ) − (44 + 4)/4
:= 4 + (4 × ((4/4 + 4)4 − 4 × 4))
:= 5555 − 55
:= 5 + (5555 − 55 )
:= 5 + (5555 − 55 + 5)
:= 6 + ((6 × (6 × 66 + 6) + 6) + 6)
:= 66 + (6 × 6 × 66 − (6/6 + 6))
:= 6 × 6 × 66 + ((6 + 6)/6)6
:= 7/7 + (7 × (7 × 7 × 7 − 7) + 77)
:= 7 × (7 × 7 × 7 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 × (7 × 7 × 7 + 7)
:= ((8 + 8) × (8 × 8 + 88)) − (8 + 8)/8
:= 88/8 + (((8 + 8) × (8 × 8 + 88)) − 8)
:= 8 + ((8 + 8) × (8 × 8 + 88))
:= (9 × 9 + 9) × (9 + 9 + 9)
:= ((9 × 9 + 9)/(9 + 9)) + (9 × 9 + 9) × (9 + 9 + 9)
:= 9 + ((9 × 9 + 9) × (9 + 9 + 9) + 9/9)
Ï 2431 := 11 × ((1 + 1) × 111 − 1)
Ï 2436 := (1 + 1) × (11 × 111 − (1 + 1 + 1))
Ï 2441 := ((1 + 1) × 11 × 111) − 1
:= 22/2 × (222 − 2/2)
:= (22 × 222/2) − (2 + 2 + 2)
:= 3/3 + (3 × (33 × (33 + 3)))
:= ((22 × 222) − 2)/2
:= 3 + ((3 × (33 × (33 + 3))) + 3)
:= 33/3 + (3 × (33 × (33 + 3)))
4
:= (4 × ((4 + 4) × 44 + 4 )) − 4/4
5
4
:= 4 × ((4/4 + 4) − 4 × 4)
:= 44 + ((((4 − 4/4) + 4)4 ) − 4)
5
:= 5/5 + (5555 − 5 )
:= 5 + ((5555 − 5 ) + 5/5)
:= 66 + (6 × 6 × 66 − (66/6))
:= 55 + (((5 − 55 )/5) − (55 + 5))
:= 66 + (6 × 6 × 66 − 6)
:= 77 + (7 × (7 × 7 × 7 − 7) + ((7 + 7)/7))
:= 66 + (6 × 6 × 66 − 6/6)
:= 7 × (7 × 7 × 7 + 7) − (7 + 7)
:= ((8 + 8) × (8 × 8 + 88)) − 8/8
:= 7 × (7 × 7 × 7 + 7) − ((7 + 7)/7 + 7)
:= (8 − 8/8) × ((8 × 88 − 8)/((8 + 8)/8))
:= 9/9 + (9 × 9 + 9) × (9 + 9 + 9)
:= 8 + (((8 + 8) × (8 × 8 + 88)) + 8/8)
:= 9 + (((9 + 9 + 9)/9) × (9 × (9 × 9 + 9) − 9/9))
:= 99/9 + (9 × 9 + 9) × (9 + 9 + 9)
Ï 2432 := 1 + (11 × ((1 + 1) × 111 − 1))
Ï 2437 := ((1 + 1) × (11 × 111 − (1 + 1))) − 1
:= 2 + ((2/2 + 2 + 2) × (222 + 2))
:= (((22 × 222) − 2)/2) − 2 − 2
:= 3 + ((3 × (33 × (33 + 3))) − 3/3)
:= 3 + (((3 × (33 × (33 + 3))) + 3/3) + 3)
4
:= 4 × ((4 + 4) × 44 + 4 )
5
:= (5 + 5)/5 + (5555 − 5 )
6
:= ((6 + 6)/6) × ((6 + 6)/6 + 6 × 6)
7
4
Ï 2442 := (1 + 1) × 11 × 111
:= 22 × 222/2
:= 3 + (3 × ((33 × (33 + 3)) + 3))
:= 4 + ((((4 − 4/4) + 4) ) + 4 × (4 + 4))
:= 44/4 × (444/((4 + 4)/4))
:= 5 + ((5555 − 55 ) + ((5 + 5)/5))
:= 555/5 × (55 + 55)/5
:= 6 × 6 + ((6/6 + 6)
6−(6+6)/6
)
:= 66 + 6 × 6 × 66
:= 7/7 + (7 × (7 × 7 × 7 + 7) − (7 + 7))
:= 7 × (7 × 7 × 7 + 7) − (7/7 + 7)
:= (8 + 8) × (8 × 8 + 88)
:= 8 + ((((8 + 8) × (8 × 8 + 88)) − (88/8)) + 8)
:= 888/8 × (88 + 88)/8
:= (9 + 9)/9 + (9 × 9 + 9) × (9 + 9 + 9)
:= 9 + ((9 × 9 + 9) × (9 + 9 + 9) − ((9 + 9)/9))
:= 999/9 × ((99 + 99)/9)
:= ((7 + 7)/7) × ((77 + 7)/7 + 7)
172
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2443 := 1 + ((1 + 1) × 11 × 111)
:= ((22 × 222) + 2)/2
Ï 2448 := (1 + 1) × (1 + (1 + (1 + 11 × 111)))
:= 2
22/2
+ (22 − 2)
2
Ï 2453 := 11 × (1 + (1 + 1) × 111)
:= 22/2 × (222 + 2/2)
:= ((33 × (((3 + 3) + 3) + 3)) + 3)/3
:= (33 − 3) × (3 × 33 + 3)
:= 33 + (33 × 33 + (33/3)3 )
:= 44 + ((4 − 4/4)4+4−4/4 )
:= 4 × (((4 + 4) × 44 + 44 ) + 4)
:= 4 + (((((4 − 4/4) + 4)4 ) + 44) + 4)
:= 55 − ((55 + 5 + 5)/5 + 55)
:= 5 + (55 − ((55 + 5 + 5)/5 + 55))
:= 55 − ((5/5 + 5) × (555 + 5)/5)
:= 66 + (6 × 6 × 66 + 6/6)
:= 6 × (6 × 66 + 6 + 6)
:= 6 + ((6 × (6 × 66 + 6 + 6)) − 6/6)
:= 7 × (7 × 7 × 7 + 7) − 7
:= 7 × (7 × 7 × 7 + 7) − (7 + 7)/7
:= (7 + 7 + 7)/7 + 7 × (7 × 7 × 7 + 7)
:= 88/8 + ((8 + 8) × (8 × 8 + 88))
:= 8 + (((8 + 8) × (8 × 8 + 88)) + 8)
:= 88/8 × (((8 × (8 + 8) − 8/8) + 88) + 8)
:= 9/9 + (999/9 × ((99 + 99)/9))
:= 9 + ((9 × 9 + 9) × (9 + 9 + 9) + 9)
:= 99/9 × (9 × (9 + 9 + 9) − (99/9 + 9))
3
Ï 2444 := (1 + 1) × (1 + 11 × 111)
Ï 2449 := 1 + ((1 + 1) × (1 + (1 + (1 + 11 × 111))))
Ï 2454 := 1 + (11 × (1 + (1 + 1) × 111))
:= 2 + (22 × 222/2)
:= ((((2 × (22 + 2)) + 22) ) − 2)/2
:= 2 × ((((22/2 + 22) + 2)2 ) + 2)
:= 3 + ((3 × (33 × (33 + 3))) + 33/3)
:= 3/3 + ((33 − 3) × (3 × 33 + 3))
:= 33+3 + ((3 × 3 + 3)3 − 3)
:= 44 + ((4 + 4) × (44 + 44 ))
:= 4 + ((((4 − 4/4) + 4)4 ) + 44)
:= 4 + ((44 − 4)/4 × (44 − 44/4))
:= 55 − ((55 + 5)/5 + 55)
:= 5 + (55 − ((55 + 5)/5 + 55))
:= 5 + ((55 − ((55 + 5)/5 + 55)) + 5)
:= 66 + (6 × 6 × 66 + ((6 + 6)/6))
:= 6/6 + (6 × (6 × 66 + 6 + 6))
:= 6 + (6 × (6 × 66 + 6 + 6))
2
:= 7/7 + (7 × (7 × 7 × 7 + 7) − 7)
:= 7 × (7 × 7 × 7 + 7) − 7/7
:= 77/7 + (7 × (7 × 7 × 7 + 7) − 7)
:= ((88 + 8)/8) + ((8 + 8) × (8 × 8 + 88))
:= 8 + ((((8 + 8) × (8 × 8 + 88)) + 8/8) + 8)
:= ((8 + 8)/8) × (8 × (8 × 8 + 88) + (88/8))
:= ((9 + 9)/9) × (((99 × 999/9) + 9)/9)
:= 9 + (((9 × 9 + 9) × (9 + 9 + 9) + 9/9) + 9)
:= 999 + (9 × 9 × (9 + 9) − ((9 + 9 + 9)/9))
Ï 2445 := 1 + ((1 + 1) × (1 + 11 × 111))
:= 2 + (((22 × 222) + 2)/2)
3
Ï 2450 := 11 + (((1 + 1) × (11 × 111 − 1)) − 1)
2
Ï 2455 := 1 + (1 + (11 × (1 + (1 + 1) × 111)))
:= 2 + (22/2 × (222 + 2/2))
:= 2 × (((22/2 + 22) + 2) )
:= ((3 − 3) × (3 × 33 + 3)) − 3
:= 3 + (((33/3 + 3) ) − 3 × 3 × 33)
:= 3/3 + (((3 × 3 + 3)3 − 3) + 33+3 )
:= 44 + (((4 − 4/4) + 4)4 )
:= (44 − 4)/4 × (44 − 44/4)
:= (4 × (4/4 + 4)4 ) − (44 + 4/4)
:= 55 − (55 /5 + 55)
:= (5 + 5) × (5 × 5 × (5 + 5) − 5)
:= 5 + ((5 + 5) × (5 × 5 × (5 + 5) − 5))
:= 66 + (6 × 6 × 66 + (6 × 6/(6 + 6)))
:= (6 + 6)/6 + (6 × (6 × 66 + 6 + 6))
:= 6 + ((6 × (6 × 66 + 6 + 6)) + 6/6)
:= (7 + 7)/7 + (7 × (7 × 7 × 7 + 7) − 7)
:= 7 × (7 × 7 × 7 + 7)
:= 7 + (7 × (7 × 7 × 7 + 7) − ((7 + 7)/7))
:= (88/8 − 8) × (888/8 + 8 × 88)
:= 8 + (888/8 × (88 + 88)/8)
:= 8 + (((((8 + 8) × (8 × 8 + 88)) − 8/8) + 8) + 8)
:= 999 + (9 × 9 × (9 + 9) − ((99 + 9)/9))
:= 9 + ((9 × 9 + 9) × (9 + 9 + 9) + (99/9))
:= 999 + (9 × 9 × (9 + 9) − ((9 + 9)/9))
Ï 2446 := (1 + 1) × (1 + (1 + 11 × 111))
3
Ï 2451 := 11 + ((1 + 1) × (11 × 111 − 1))
Ï 2456 := 1 + (1 + (1 + (11 × (1 + (1 + 1) × 111))))
:= 2 + ((22 × 222/2) + 2)
:= (22/2 × (222 + 2/2)) − 2
:= (2 × (22 + 2) + 2)2 − (2 × 22)
:= 3 + (((33 × (((3 + 3)3 + 3) + 3)) + 3)/3)
:= 3 + ((33 − 3) × (3 × 33 + 3))
:= 33+3 + ((3 × 3 + 3)3 − 3/3)
:= 44 + ((((4 − 4/4) + 4)4 ) + 4/4)
:= 4 + ((44 × ((4 + 4) + 4)) − (4/4 + 4)4 )
:= (4 × (4/4 + 4)4 ) − 44
:= 55 + (((5 − 55 )/5) − 55)
:= 5 + ((((5 − 55 )/5) − 55) + 55 )
:= (5 − 5/5) × ((55 − 55)/5)
:= 6 + (6 × 6 × 66 + ((6 + 6)/6)6 )
:= 666/6 + 6 × (6 × 66 − 6)
:= 6 + ((6 × (6 × 66 + 6 + 6)) + ((6 + 6)/6))
:= 7 + (7 × (7 × 7 × 7 + 7) − (77/7))
:= 7/7 + 7 × (7 × 7 × 7 + 7)
:= 7 + (7 × (7 × 7 × 7 + 7) − 7/7)
:= 8 + ((((8 + 8) × (8 × 8 + 88)) − ((8 + 8)/8)) + 8)
:= (88/8 + 8) × (8 × (8 + 8) + 8/8)
:= 8 + ((((8 + 8) × (8 × 8 + 88)) + 8) + 8)
:= 999 + (9 × 9 × (9 + 9) − (99/9))
:= 9 + (999/9 × ((99 + 99)/9))
:= 999 + (9 × 9 × (9 + 9) − 9/9)
Ï 2447 := 1 + ((1 + 1) × (1 + (1 + 11 × 111)))
:= 2 + ((((22 × 222) + 2)/2) + 2)
Ï 2452 := (11 × (1 + (1 + 1) × 111)) − 1
2
:= 2 + (2 × (((22/2 + 22) + 2) ))
Ï 2457 := 11 + ((1 + 1) × (1 + (1 + 11 × 111)))
:= 2 + ((22/2 × (222 + 2/2)) + 2)
:= ((33/3 + 3) ) − 3 × 3 × 33
:= 3 + (((3 − 3) × (3 × 33 + 3)) + 3/3)
:= 33+3 + (3 × 3 + 3)3
:= (44 × ((4 + 4) + 4)) − (4/4 + 4)4
:= 4 × (((4/4 + 4)4 − 4 × 4) + 4)
:= 4/4 + ((4 × (4/4 + 4)4 ) − 44)
:= 55 + ((((5 − 55 ) + 5)/5) − 55)
:= (5 − 5/5) × ((55 − (55 + 5))/5)
:= 5 + ((5 − 5/5) × ((55 − (55 + 5))/5))
:= (6 × (6 × 66 + 6 + 6)) − 6/6
:= 6 + ((6 × 6 × 66 + ((6 + 6)/6)6 ) + 6)
:= 6 + (6 × (6 × 66 − 6) + 666/6)
:= 7 × (7 × 7 × 7 + 7) − (7 + 7 + 7)/7
:= (7 + 7)/7 + 7 × (7 × 7 × 7 + 7)
:= 7 + 7 × (7 × 7 × 7 + 7)
:= 8 + ((((8 + 8) × (8 × 8 + 88)) − 8/8) + 8)
:= (((88 × (8 × 8 − 8)) − 8)/((8 + 8)/8)) − 8
:= 8 × 8 + (((8 − 8/8)8×8/(8+8) ) − 8)
:= 9 + (((9 × 9 + 9) × (9 + 9 + 9) − 9/9) + 9)
:= ((99 + 99)/9) + (9 × 9 + 9) × (9 + 9 + 9)
:= 999 + 9 × 9 × (9 + 9)
3
3
173
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2458 := (1 + 1) × ((11 × (1 + 111)) − (1 + 1 + 1))
:= 2
2+2
Ï 2463 := ((1 + 1) × (11 × (1 + 111))) − 1
+ (22 × 222/2)
Ï 2468 := (1 + 1) × (1 + (1 + (11 × (1 + 111))))
:= 2 + 22 × (222 + 2)/2 + 2
:= ((22 × (222 + 2)) − 2)/2
:= 3/3 + ((3 × 3 + 3)3 + 33+3 )
:= 33 + (3 × (3 × (3 + 3)))
:= 33+3 + ((3 × 3 + 3)3 + 33/3)
:= (4 + 4)/4 + ((4 × (4/4 + 4)4 ) − 44)
:= (44 × (44 + 4 + 4 + 4)) − 4/4
:= 4 × ((4/4 + 4)4 − (4 + 4))
:= 55 − ((555 + 5)/5 + 555)
:= 5 × 555 + ((5 − 55 )/(5 + 5))
:= 55 − (((5 + 5)/5)5 + 55 /5)
:= ((66 − 6)/6) + (6 × (6 × 66 + 6 + 6))
:= 6 + ((6 × (6 × 66 − 6) + 666/6) + 6)
:= 6 × 6 + (((6 + 6)/6)6 × ((6 + 6)/6 + 6 × 6))
:= 7 + (7 × (7 × 7 × 7 + 7) + 7/7)
:= 7 + ((7 × (7 × 7 × 7 + 7) − 7/7) + 7)
:= 7 + (7 × (7 × 7 × 7 + 7) + (77/7))
:= 8 + ((888/8 × (88 + 88)/8) + 8)
:= (8 × ((88 × (8 × 8 − 8))/(8 + 8))) − 8/8
:= ((88 × (8 × 8 − 8)) + 8)/((8 + 8)/8)
:= 9/9 + (9 × 9 × (9 + 9) + 999)
:= ((9 + 9 + 9)/9) × (9 × (9 × 9 + 9) + (99/9))
:= 99/9 + (9 × 9 × (9 + 9) + 999)
Ï 2459 := ((1 + 1) × ((11 × (1 + 111)) − (1 + 1))) − 1
:= 2
2+2
+ (((22 × 222) + 2)/2)
3
3
Ï 2464 := (1 + 1) × (11 × (1 + 111))
Ï 2469 := 1 + ((1 + 1) × (1 + (1 + (11 × (1 + 111)))))
:= 22 × (222 + 2)/2
:= 2 + ((22 × (222 + 2) + 2)/2 + 2)
3
3
:= 3 + (((3 × 3 + 3)3 − 3/3) + 33+3 )
:= 3/3 + ((3 × (3 × (3 + 3))) + 33)
:= (33 × ((3 × (33 − 3)) + 3)) − 3 − 3
:= 4 + (4 × (4/4 + 4)4 ) − (44 + 4/4)
:= 44 × (44 + 4 + 4 + 4)
:= 4 + ((((4 − 4/4) + 4)4 ) + 4 × 4 × 4)
:= 55 − (555/5 + 555)
:= 5 − (((55 + 5 )/5) + 5 × 5)
:= 55 − (((55 + 5)/5 + 5 × 5) + 5)
:= 66/6 × ((6 × 6 × 6 + (6 + 6)/6) + 6)
:= 666/6 + (6 × 6 × 66 − (6 + 6 + 6))
:= 7 + 7 × (7 × 7 × 7 + 7) + 7
:= 7 + (7 × (7 × 7 × 7 + 7) + (77 + 7)/7)
:= 66/6 + (6 × (6 × 66 + 6 + 6))
:= 7 + (7 × (7 × 7 × 7 + 7) + ((7 + 7)/7))
:= 8 + ((88/8 + 8) × (8 × (8 + 8) + 8/8))
:= (9 + 9)/9 + (9 × 9 × (9 + 9) + 999)
Ï 2460 := (1 + 1) × ((11 × (1 + 111)) − (1 + 1))
2
:= (22 − 2) × ((22/2) + 2)
3
:= 3 + ((3 × 3 + 3) + 3
3+3
)
5
5
:= 8 × ((88 × (8 × 8 − 8))/(8 + 8))
:= 8/8 + (((88 × (8 × 8 − 8)) + 8)/((8 + 8)/8))
:= ((9/9 + 9 + 9) + 9) × (99 − (99/9))
:= 9 + (9 × (9 × (9 + 9) + 99) + 999/9)
Ï 2465 := 1 + ((1 + 1) × (11 × (1 + 111)))
:= ((22 × (222 + 2)) + 2)/2
Ï 2470 := (1 + 1) × (1 + (1 + (1 + (11 × (1 + 111)))))
:= 2 + (22 × (222 + 2)/2 + 2 + 2)
:= (33/3)3 + (33 × (3 × 3 + 33))
4
:= (3 × 3 + 3/3) × (((33+3 + 3)/3) + 3)
:= 4 + ((4 × (4/4 + 4)4 ) − 44)
:= 4 × 4 × 4 + (((4 − 4/4) + 4) )
:= (5 − 5/5) × (55 /5 − (5 + 5))
:= 5 − (55 × ((55 + 5)/5))
:= 55 × 55 − 555
:= 6 + ((6 × (6 × 66 + 6 + 6)) + (66/6))
:= (66 − 6/6) × ((6 + 6)/6 + 6 × 6)
:= 7 + ((7 × (7 × 7 × 7 + 7) + 7/7) + 7)
:= 77 + (7 × 7 × 7 × 7 − (7/7 + 7))
:= 6 + ((6 × (6 × 66 + 6 + 6)) + 6)
:= ((77 − 7)/7) + 7 × (7 × 7 × 7 + 7)
5
8×8/(8+8)
:= ((88 × (8 × 8 − 8)) − 8)/((8 + 8)/8)
:= 8 × 8 + ((8 − 8/8)
:= 999/9 + 9 × (9 × (9 + 9) + 99)
:= 9 + ((9 × 9 × (9 + 9) − 9/9) + 999)
Ï 2461 := ((1 + 1) × ((11 × (1 + 111)) − 1)) − 1
)
Ï 2466 := (1 + 1) × (1 + (11 × (1 + 111)))
:= 4 + ((((4 − 4/4) + 4)4 ) + (44 + 4)/4)
:= (88/8 + 8) × (8 × (8 + 8) + ((8 + 8)/8))
:= (9/9 + 9) × ((((9 + 9)/9)9−9/9 ) − 9)
Ï 2471 := 1 + ((1 + 1) × (1 + (1 + (1 + (11 × (1 + 111))))))
:= 2 + (22 × (222 + 2)/2)
:= (2 × (2 + 2 + 2)2+2 ) − (22/2)2
:= 3 + ((3 × (33 × (33 + 3))) + 33)
:= (33 × ((3 × (33 − 3)) + 3)) − (3/3 + 3)
:= 4 × 4 + ((((4 − 4/4) + 4)4 ) + 44)
:= (44 + 4)/4 + (((4 − 4/4) + 4)4 )
:= 4 + ((4 × ((4/4 + 4)4 − (4 + 4))) − 4/4)
:= 5 + ((5 − 5/5) × ((55 − 55)/5))
:= 5 + ((5 − (55 × (55 + 5)))/5)
:= 55 + (((5 − 55 )/5) − (5 × 5 + 5))
:= 6 + (((6 × (6 × 66 + 6 + 6)) + 6/6) + 6)
:= 6 + (((6 × (6 × 66 + 6 + 6)) + 6) + 6)
:= 66 + ((6 × 6 + 6/6) × (66 − 6/6))
:= 77/7 + 7 × (7 × 7 × 7 + 7)
:= 7 + ((7 × (7 × 7 × 7 + 7) + ((7 + 7)/7)) + 7)
:= 77 + (7 × 7 × 7 × 7 − 7)
:= (((22 × (222 + 2)) − 2)/2) − 2
:= 3 + (((3 × 3 + 3)3 + 33+3 ) + 3/3)
5
8×8/(8+8)
:= 8/8 + (((88 × (8 × 8 − 8)) − 8)/((8 + 8)/8))
:= 8/8 + (((8 − 8/8)
:= ((999 + 9)/9) + 9 × (9 × (9 + 9) + 99)
:= 9 + (9 × 9 × (9 + 9) + 999)
Ï 2462 := (1 + 1) × ((11 × (1 + 111)) − 1)
) + 8 × 8)
Ï 2467 := 1 + ((1 + 1) × (1 + (11 × (1 + 111))))
:= ((8 + 8 + 8) × (888/8 − 8)) − 8/8
:= ((9 × 9 − 9/9) × (((99 + 99)/9) + 9)) − 9
Ï 2472 := 11 + (((1 + 1) × ((11 × (1 + 111)) − 1)) − 1)
:= (22 × (222 + 2)/2) − 2
:= 2 + (22 × (222 + 2) + 2)/2
:= 2 × (2 × ((2 × (22 × (22+2 − 2))) + 2))
:= 33 + ((3 × (33 × (33 + 3))) − 3/3)
:= 3 + (((3 × (33 × (33 + 3))) + 33) + 3/3)
:= (33 × ((3 × (33 − 3)) + 3)) − 3
:= (44 + 4)/4 + ((((4 − 4/4) + 4)4 ) − 4)
:= (4 × ((4/4 + 4)4 − (4 + 4))) − 4/4
:= 4 + (4 × ((4/4 + 4)4 − (4 + 4)))
:= 5 × 555 − (55 + 5)/(5 + 5)
:= 5 + (5 × 555 − (55 + 5)/(5 + 5))
:= (5 − 5/5) × ((55 − 5 − 5)/5 − 5)
:= 6 + (((6 × (6 × 66 + 6 + 6)) + ((6 + 6)/6)) + 6)
:= 66 + ((6/6 + 6)6−(6+6)/6 )
:= 66 + (6 × (6 × 66 + 6) − 6)
:= (77 + 7)/7 + 7 × (7 × 7 × 7 + 7)
:= 77 + (7 × 7 × 7 × 7 − (77/7))
:= 7/7 + ((7 × 7 × 7 × 7 − 7) + 77)
:= (8 × ((88 × (8 × 8 − 8))/(8 + 8))) − (8 + 8)/8
:= 8 + (((88/8 + 8) × (8 × (8 + 8) + 8/8)) + 8)
:= (8 + 8 + 8) × (888/8 − 8)
:= 9 + ((99/9) × (9 × (9 + 9 + 9) − (99/9 + 9)))
:= 9 + ((9 × 9 × (9 + 9) + 999) + 9/9)
:= (9 − 9/9) × ((999/9 + 99) + 99)
174
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2473 := 11 + ((1 + 1) × ((11 × (1 + 111)) − 1))
2
2
Ï 2478 := 1 + (1 + (1 + (11 × (1 + ((1 + 1) × (1 + 111))))))
2×(2+2)
Ï 2483 := ((1 + 1) × ((11 × (1 + 1 + 111)) − 1)) − 1
:= 22 × (222/2 + 2) − 2/2 − 2
:= ((22/2 + 2) ) + ((2 × (22 + 2)) )
:= 2222 + 2
:= 3/3 + ((33 × ((3 × (33 − 3)) + 3)) − 3)
:= 3 + (33 × ((3 × (33 − 3)) + 3))
:= (3 × (3 × 33 + 33+3 )) − 3/3
:= (4 × ((4/4 + 4)4 − 4)) − 44/4
:= 44 + 4 × 4444/(4 + 4)
:= (4 × ((4/4 + 4)4 − 4)) − 4/4
:= 55 − ((55 + 5 + 5)/5 + 5 × 5)
:= 55 − (((55 + 55) + 55 )/5)
:= 55 − (((55 + 55 + 5)/5) + 5)
:= ((6 × 6 + 6/6) × (66 + 6/6)) − 6
:= 66 + 6 × (6 × 66 + 6)
:= (6 × ((6 × 66 + 6 + 6) + 6)) − 6/6
:= 77 + ((7 × 7 × 7 × 7 − 7) + ((7 + 7)/7))
:= 77 + 7 × 7 × 7 × 7
:= 7 + ((7 × 7 × 7 × 7 − ((7 + 7)/7)) + 77)
:= 8 + (((8 − 8/8)8×8/(8+8) ) + 8 × 8)
:= 8 + ((88/8 + 8) × (8 × (8 + 8) + ((8 + 8)/8)))
:= 8 + (88/8 × ((8 − 8/8 + 8)(8+8)/8 ))
:= 9 + (((9/9 + 9 + 9) + 9) × (99 − (99/9)))
:= (9 − ((9 + 9 + 9)/9)) × ((((9 + 9)/9)9 ) − 99)
:= ((9 + 9 + 9) × ((99/9) + 9 × 9)) − 9/9
Ï 2474 := (11 × (1 + ((1 + 1) × (1 + 111)))) − 1
Ï 2479 := 11 + ((1 + 1) × (1 + (1 + (11 × (1 + 111)))))
Ï 2484 := (1 + 1) × ((11 × (1 + 1 + 111)) − 1)
:= 2/2 + (2222 + 22×(2+2) )
:= 22 × (222/2 + 2) − 2
:= (33 × ((3 × (33 − 3)) + 3)) − 3/3
:= 3 + ((33 × ((3 × (33 − 3)) + 3)) + 3/3)
:= 3 × (3 × 33 + 33+3 )
:= 44 + (4 × 4444/(4 + 4) − 4)
:= (4 × ((4/4 + 4)4 − 4)) − (4/4 + 4)
:= 4 × ((4/4 + 4)4 − 4)
:= 55 − ((55 + 5)/5 + 5 × 5)
:= 5 + (55 − ((55 + 5)/5 + 5 × 5))
:= 55 − (((55 + 55 )/5) + 5)
:= 66 + ((6 × (6 × 66 + 6) − 6) + ((6 + 6)/6))
:= (6 × 6 + 6/6) × (66 + 6/6)
:= 6 × ((6 × 66 + 6 + 6) + 6)
:= 7 + ((7 × 7 × 7 × 7 − (77/7)) + 77)
:= 7/7 + (7 × 7 × 7 × 7 + 77)
:= 7 + ((7 × 7 × 7 × 7 − 7/7) + 77)
:= (8 + 8)/8 + ((8 + 8 + 8) × (888/8 − 8))
:= 8 + (((8 + 8 + 8) × (888/8 − 8)) − 8/8)
:= 2222 + (2
2×(2+2)
− (2 + 2))
:= (99 × (((9 − ((9 + 9)/9)) + 9) + 9)) − 9/9
Ï 2475 := 11 × (1 + ((1 + 1) × (1 + 111)))
:= 22/2 × ((222 + 2/2) + 2)
9−9/9
:= 9 + ((9/9 + 9) × ((((9 + 9)/9)
) − 9))
Ï 2480 := (1 + 1) × ((11 − 1) × (1 + (1 + (1 + 111+1 ))))
:= 2 + (2222 + 2
2×(2+2)
:= 8 + ((((88 × (8 × 8 − 8)) + 8)/((8 + 8)/8)) + 8)
:= (9 + 9 + 9) × ((99/9) + 9 × 9)
Ï 2485 := ((1 + 1) × (11 × (1 + 1 + 111))) − 1
:= 22 × (222/2 + 2) − 2/2
)
:= 33 × ((3 × (3 − 3)) + 3)
:= ((3 + 3/3) + 3) × (3 × 3 − 3/3)
:= 3/3 + (3 × (3 × 33 + 33+3 ))
:= (44 + 4/4) × (44/4 + 44)
:= 4 × (444 + 4 × 44)
:= 4/4 + (4 × ((4/4 + 4)4 − 4))
:= 55 × (55 − 5 − 5)
3
3
3
:= 5 + (55 × (55 − 5 − 5))
:= 5 + ((55 × (55 − 5 − 5)) + 5)
:= 666/6 + (6 × 6 × 66 − (6 + 6))
:= 66 + (6 × (6 × 66 + 6) + ((6 + 6)/6))
:= 6 + ((6 × 6 + 6/6) × (66 + 6/6))
:= 7 + ((7 × (7 × 7 × 7 + 7) + (77/7)) + 7)
:= 77 + (7 × 7 × 7 × 7 + ((7 + 7)/7))
:= 7 + (7 × 7 × 7 × 7 + 77)
:= 8 + ((8 + 8 + 8) × (888/8 − 8))
:= ((8 + 8 + 8) × (88 + 8 + 8)) − 88/8
:= (9 × 9 − 9/9) × (((99 + 99)/9) + 9)
:= 9/9 + ((9 + 9 + 9) × ((99/9) + 9 × 9))
:= 88/8 × ((8 − 8/8 + 8)
(8+8)/8
)
:= 99 × (((9 − ((9 + 9)/9)) + 9) + 9)
Ï 2476 := 1 + (11 × (1 + ((1 + 1) × (1 + 111))))
:= 2222 + (2
2×(2+2)
− 2)
Ï 2481 := (1 + 1 + 1) × ((1 + 1)11 − 11 × 111)
:= (2 × (2 + 2 + 2)2+2 ) − 222/2
3+3
Ï 2486 := (1 + 1) × (11 × (1 + 1 + 111))
:= 22 × (222/2 + 2)
:= 3 + ((3 × (3 × 33 + 33+3 )) − 3/3)
:= 3/3 + (33 × ((3 × (33 − 3)) + 3))
:= (3 × (3 × 33 + 3
)) − 3
:= 44 + (4/4 + 4) × 444
:= 4/4 + (4 × (444 + 4 × 44))
:= 55 + (((5 − 55 )/5) − 5 × 5)
:= 5 + ((((5 − 5 )/5) − 5 × 5) + 5 )
:= 55 + (((5 − 55 )/5) − (5 + 5 + 5))
:= ((6 + 6)/6)6 + 6 × (6 × 66 + 6)
:= 666/6 + (6 × 6 × 66 − 6)
:= 6 × 6 × 66 + (666 − 6)/6
:= 77 + (7 × 7 × 7 × 7 − ((7 + 7)/7))
:= 7 × (7 × 7 − 7) + ((7 + 7 + 7)/7)
5
:= (4 + 4)/4 + (4 × ((4/4 + 4)4 − 4))
5
7
8×8/(8+8)
:= 8 + (((88 × (8 × 8 − 8)) + 8)/((8 + 8)/8))
:= 88 + (((8 − 8/8)
:= 9/9 + (99 × (((9 − ((9 + 9)/9)) + 9) + 9))
:= 9 × 9 + ((99/9 + 9) × (999/9 + 9))
Ï 2477 := 1 + (1 + (11 × (1 + ((1 + 1) × (1 + 111)))))
:= 2 + (22/2 × ((222 + 2/2) + 2))
) − 8)
Ï 2482 := (1 + 1) × ((11 × (1 + 1 + 111)) − (1 + 1))
2
:= (2 × (22 + 2 + 2)) − 222
:= 7 + ((7 × 7 × 7 × 7 + 77) + 7/7)
:= (8 − 88)/8 + ((8 + 8 + 8) × (88 + 8 + 8))
:= (9 + 9)/9 + ((9 + 9 + 9) × ((99/9) + 9 × 9))
Ï 2487 := 1 + ((1 + 1) × (11 × (1 + 1 + 111)))
:= 2/2 + 22 × (222/2 + 2)
3
:= 3 + ((33 × ((3 × (3 − 3)) + 3)) − 3/3)
:= (3/3 + 33) × (((3 + 3) + 3)/3)
:= 3 + (3 × (3 × 33 + 33+3 ))
:= 4 + ((4 × ((4/4 + 4)4 − 4)) − 44/4)
:= (4 − 4/4)4 + (((4 − 4/4) + 4)4 )
:= 4 + ((4 × ((4/4 + 4)4 − 4)) − 4/4)
:= 55 + ((((5 − 55 ) + 5)/5) − 5 × 5)
:= 5 + (((((5 − 55 ) + 5)/5) − 5 × 5) + 55 )
:= 55 − (((55 + 55 + 5) + 5)/5)
:= 66 + (6 × (6 × 66 + 6) − 6/6)
:= 6 + (6 × (6 × 66 + 6) + ((6 + 6)/6)6 )
:= 666/6 + 6 × 6 × 66
:= 77 + (7 × 7 × 7 × 7 − 7/7)
:= 77 + ((7 × 7 × 7 × 7 − 7) + (77/7))
:= 7 + ((7 × 7 × 7 × 7 + ((7 + 7)/7)) + 77)
:= 8 × 8 + ((88/8 + 8) × (8 × (8 + 8) − 8/8))
:= 8 × 88 + (((8 + 8)/8) × (888 + 8/8))
3
:= 9 + ((9 × 9 × (9 + 9) + 999) + (99/9))
9
:= ((9 + 9)/9) × ((((9 + 9)/9) ) + 9 × 9 × 9)
175
:= ((8 + 8 + 8) × (88 + 8 + 8)) − (8/8 + 8)
:= ((9 + 9 + 9)/9) × ((9 × 9 × 9 + 99) + 9/9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2488 := (1 + 1) × (1 + (11 × (1 + 1 + 111)))
:= 2 + 22 × (222/2 + 2)
Ï 2493 := 1 + ((1 + 1)11 + ((1 + 1) × (1 + 1) × 111))
:= 2/2 + (2
3+3
22/2
+ 2 × 222)
Ï 2498 := 1 + (11 × (1 + ((1 + 1) × (1 + 1 + 111))))
:= (2 × (22 + 2) + 2)2 − 2
:= 3 × ((3 × 33 + 33+3 ) + 3)
:= (3 × (3 + 3))3 − (3333 + 3/3)
:= 4 + (4 × ((4/4 + 4) − 4))
:= 4 + ((4 × (4/4 + 4)4 ) − 44/4)
:= (4 × (4/4 + 4)4 ) − (4 + 4)/4
:= 55 − ((55 + 55 + 5)/5)
:= 55 − ((55 + 5 + 5)/5 + 5)
:= 55 − (55 + 5 + 5)/5
:= 6 × 6 × 66 + (666 + 6)/6
:= 6 + (666/6 + 6 × 6 × 66)
:= 6 × 6 × 66 + ((666 + 66)/6)
:= 3 + ((3 × (3 × 33 + 3
)) + 3/3)
4
:= (7 × (7 × 7 × 7 + 7 + 7)) − 77/7
:= ((7/7 + 7 × 7)
:= ((8 + 8 + 8) × (88 + 8 + 8)) − 8
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) − (88/8))
:= (8 + 8)/8 + ((8 + 8 + 8) × (88 + 8 + 8))
:= 9 × (9 × (9 + 9) − 9) + 9999/9
:= 9 + ((9 + 9 + 9) × ((99/9) + 9 × 9))
:= 9 + (((9 × 9 − 9/9) × (((99 + 99)/9) + 9)) + 9)
Ï 2489 := 1 + ((1 + 1) × (1 + (11 × (1 + 1 + 111))))
(7+7)/7
)−7
Ï 2494 := (1 + 1)11 + ((1 + 1) × (1 + (1 + 1) × 111))
:= (7 × (7 × 7 × 7 + 7 + 7)) − 7/7
Ï 2499 := (((11 − 1)1+1 /(1 + 1))1+1 ) − 1
:= 222/2 + ((22 − 2/2)2 )
:= 2 + (222/2 + 2 × 222)
:= (2 × (22 + 2) + 2)2 − 2/2
:= (((33 + 3) × ((3 + 3)3 + 33)) − 3)/3
:= 3/3 + (3 × ((3 × 33 + 33+3 ) + 3))
:= (3 × (3 + 3))3 − 3333
:= (4 × (4/4 + 4)4 ) − 44/4
:= (4 × (4/4 + 4)4 ) − ((4 + 4)/4 + 4)
:= (4 × (4/4 + 4)4 ) − 4/4
:= 55 − ((55 + 55 )/5)
:= 55 − ((55 + 5)/5 + 5)
:= 55 − (55 + 5)/5
:= 6 + ((6 × ((6 × 66 + 6 + 6) + 6)) − 6/6)
:= 6 + ((666 + 6)/6 + 6 × 6 × 66)
:= 6 + ((666/6 + 6 × 6 × 66) + 6)
:= 7/7 + (((7/7 + 7 × 7)(7+7)/7 ) − 7)
:= 7 × (7 × 7 × 7 + 7 + 7)
:= 77 + (7 × 7 × 7 × 7 + (77/7))
8×8/(8+8)
:= 88 + ((8 − 8/8)
)
:= ((8 + 8 + 8) × (88 + 8 + 8)) − (8 + 8)/8
:= 9 + ((9 × 9 − 9/9) × (((99 + 99)/9) + 9))
Ï 2490 := (1 + 1) × (1 + (1 + (11 × (1 + 1 + 111))))
:= 2 + (22 × (222/2 + 2) + 2)
:= 3 + ((3 × (3 × 33 + 33+3 )) + 3)
:= (4 − 44)/4 + (4 × (4/4 + 4)4 )
:= 9 + (((9 + 9 + 9) × ((99/9) + 9 × 9)) + 9/9)
Ï 2495 := 11 + ((1 + 1) × ((11 × (1 + 1 + 111)) − 1))
:= (2 × (22 + 2) + 2)2 − (2/2 + 2 + 2)
:= 33/3 + (3 × (3 × 33 + 33+3 ))
:= (4 × (4/4 + 4)4 ) − (4/4 + 4)
:= 55 − (55 /5 + 5 + 5)
:= 6 + (6 × ((6 × 66 + 6 + 6) + 6))
:= 55 − (55 /5 + 5)
:= 66/6 + (6 × ((6 × 66 + 6 + 6) + 6))
:= 88/8 + (((8 + 8 + 8) × (88 + 8 + 8)) − 8)
:= 99 + ((99/9 + 9) × (999/9 + 9))
Ï 2500 := ((11 − 1)1+1 /(1 + 1))1+1
:= (2 × (22 + 2) + 2)2
:= 3 + (33/3 × ((3 + 3)3 + 33/3))
:= 4 × (4/4 + 4)4
:= 55 − 55 /5
:= ((6 + 6)/6 + 6 × 6 + 6 + 6)(6+6)/6
:= (7/7 + 7 × 7)(7+7)/7
:= 8/8 + (((8 − 8/8)8×8/(8+8) ) + 88)
:= 7 + ((7 × (7 × 7 × 7 + 7 + 7)) − (77/7))
:= ((8 + 8 + 8) × (88 + 8 + 8)) − 8/8
:= ((((8 + 8)/8) − (8 + 8)) + 8 × 8)(8+8)/8
:= (((9 + 9)/9) + 9 × 9) × ((((99 + 9)/9) + 9) + 9)
:= 99/9 + ((9 + 9 + 9) × ((99/9) + 9 × 9))
:= ((9 × 99 + 9)/(9 + 9))(9+9)/9
:= 77 + (7 × 7 × 7 × 7 + (77 + 7)/7)
Ï 2491 := 1 + ((1 + 1) × (1 + (1 + (11 × (1 + 1 + 111)))))
:= 2 + (((22 − 2/2)2 ) + 222/2 )
Ï 2496 := (11 × (1 + ((1 + 1) × (1 + 1 + 111)))) − 1
Ï 2501 := 1 + (((11 − 1)1+1 /(1 + 1))1+1 )
:= (2 × (22 + 2) + 2)2 − 2 − 2
:= 2/2 + (2 × (22 + 2) + 2)2
:= (33 − 3/3) × (3 × 33 − 3)
:= ((33/3 + 3)3 ) − 3 × 3 × 33
:= (4 × (4/4 + 4)4 ) − 4
:= 4/4 + (4 × (4/4 + 4)4 )
:= 5 + (((5 − 5 )/5) − (5 + 5))
:= 55 + (((5 − 55 )/5) − 5)
:= 55 + ((5 − 55 )/5)
:= 6 + (((6 × 6 + 6/6) × (66 + 6/6)) + 6)
:= 6 + ((6 × ((6 × 66 + 6 + 6) + 6)) + 6)
:= (6 × 6 − 6/6 + 6) × (66 − 6 + 6/6)
:= (7 × (7 × 7 × 7 + 7 + 7)) − (7/7 + 7)
:= 7 + ((7 × 7 × 7 × 7 + (77/7)) + 77)
:= 7/7 + ((7/7 + 7 × 7)(7+7)/7 )
:= 88 + ((8/8 + 88) × (88/8 + 8 + 8))
:= (8 + 8 + 8) × (88 + 8 + 8)
:= 88 + ((88/8 + 8) × (8 × (8 + 8) − 8/8))
:= (9 − 9/9) × ((9 + 9) × (9 + 9) − ((99 + 9)/9))
:= 9 × 9 × 9 + ((9 + 9) × 99 − (9/9 + 9))
3
3
:= (((3 + 3) × ((3 + 3) + 33)) + 3)/3
4
:= 444 + (4 × (4 + 4) − 4/4)
5
5
:= (((9 × 99 + 9)/(9 + 9))
(9+9)/9
)−9
Ï 2492 := (1 + 1)11 + ((1 + 1) × (1 + 1) × 111)
Ï 2497 := 11 × (1 + ((1 + 1) × (1 + 1 + 111)))
Ï 2502 := 1 + (1 + (((11 − 1)1+1 /(1 + 1))1+1 ))
:= 222/2 + 2 × 222
:= (2 × (22 + 2) + 2)2 − 2/2 − 2
:= 2 + (2 × (22 + 2) + 2)2
:= (33 + 3/3) × ((3 × (33 + 3)) − 3/3)
:= 33/3 × ((3 + 3)3 + 33/3)
:= 3 + ((3 × (3 + 3))3 − 3333)
:= 444 + 44 × (4 + 4)
:= 4/4 + ((4 × (4/4 + 4)4 ) − 4)
:= (4 + 4)/4 + (4 × (4/4 + 4)4 )
:= (5 − 5/5) × (55 − 5 − 5)/5
:= 55 + ((((5 − 55 ) + 5)/5) − 5)
:= 55 + (((5 − 55 ) + 5)/5)
:= 6 + (((666 − 6)/6) + 6 × 6 × 66)
:= 66/6 × (6 × 6 × 6 + 66/6)
:= (66 × ((6 + 6)/6 + 6 × 6)) − 6
:= (7 × (7 × 7 × 7 + 7 + 7)) − 7
:= (7 × (7 × 7 × 7 + 7 + 7)) − (7 + 7)/7
:= (7 + 7)/7 + ((7/7 + 7 × 7)(7+7)/7 )
:= (8 − 8/8) × ((8 × 88 + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8 + 8) × (88 + 8 + 8))
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) − ((8 + 8)/8))
:= 9 + (9 × (9 × (9 + 9) − 9) + 9999/9)
:= 9 × 9 × 9 + ((9 + 9) × 99 − 9)
9
:= (((9 + 9)/9) ) + (99 × (99/9 + 9))
176
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2503 := ((11 × (1 + 1)11 ) − 1)/(11 − 1 − 1)
Ï 2508 := (1 + 1) × (11 × (1 + 1 + 1 + 111))
:= 2 + (2 × (22 + 2) + 2)2 + 2/2
:= 2 + ((2 × (22 + 2) + 2)2 + 22/2)
:= 22 × ((222 + 2)/2 + 2)
3
:= 3 + ((33/3 × ((3 + 3)3 + 33/3)) + 3)
Ï 2513 := 1 + (1 + (11 + (((11 − 1)1+1 /(1 + 1))1+1 )))
:= 3 + ((33 × ((3 × (33 + 3)) + 3)) − 3/3)
:= 33 × ((((3 + 3) + 3)/3) + 3)
4
:= 4 + ((4 × (4/4 + 4)4 ) − 4/4)
:= 4 + ((4 × (4/4 + 4) ) + 4)
:= 4/4 + ((4 × ((4/4 + 4)4 + 4)) − 4)
:= 5 + (55 − (55 + 5 + 5)/5)
:= (5 − 5/5) × (55 + 5 + 5)/5
:= 5 + ((5 − 5/5) × (55 + 5 + 5)/5)
:= 6 + 66/6 × (6 × 6 × 6 + 66/6)
:= 66 × ((6 + 6)/6 + 6 × 6)
:= (6/6 + 6) × (6 × (66 − 6) − 6/6)
:= 77/7 + ((7 × (7 × 7 × 7 + 7 + 7)) − 7)
:= 7 + (((7/7 + 7 × 7)(7+7)/7 ) + 7/7)
:= 7 + ((7 × (7 × 7 × 7 + 7 + 7)) + 7)
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) − 8/8)
:= (88/((8 + 8)/8)) × ((8/8 − 8) + 8 × 8)
:= 8 + ((((8 + 8 + 8) × (88 + 8 + 8)) + 8/8) + 8)
:= 9/9 + (((9 + 9) × 99 − 9) + 9 × 9 × 9)
:= 9 × 9 × 9 + ((9 + 9) × 99 − ((9 + 9 + 9)/9))
:= 9 × 9 × 9 + ((9 + 9) × 99 + ((9 + 9)/9))
Ï 2504 := 1 + (((11 × (1 + 1)11 ) − 1)/(11 − 1 − 1))
Ï 2509 := 1 + ((1 + 1) × (11 × (1 + 1 + 1 + 111)))
Ï 2514 := 11 + (((11 × (1 + 1)11 ) − 1)/(11 − 1 − 1))
:= 2 + (2 × (22 + 2) + 2)2 + 2
:= 222 + (2 × 22 + 2/2)2
:= 2 + ((2 × 22)2 + ((22 + 2)2 ))
:= 3 + (((33/3 + 3)3 ) − 3 × 3 × 33 )
:= 3/3 + (33 × ((((3 + 3)3 + 3)/3) + 3))
:= 3 + (33 × ((3 × (33 + 3)) + 3))
:= 4 + (4 × (4/4 + 4)4 )
:= 4 + (((4 × (4/4 + 4)4 ) + 4/4) + 4)
:= (4 × ((4/4 + 4)4 + 4)) − (4 + 4)/4
:= 5 + (55 − (55 + 5)/5)
:= 5 + ((55 − (55 + 5)/5) + 5)
:= 5 + (((55 − (55 + 5)/5) + 5) + 5)
:= 6 × 6 × 66 + (((6 + 6)/6)6/6+6)
:= 6/6 + (66 × ((6 + 6)/6 + 6 × 6))
:= ((6 + 6) × (6 × 6 × 6 − 6)) − 6
:= 7 + ((7 × (7 × 7 × 7 + 7 + 7)) − ((7 + 7)/7))
:= ((77 − 7)/7) + (7 × (7 × 7 × 7 + 7 + 7))
:= 7 + (((7/7 + 7 × 7)(7+7)/7 ) + 7)
:= 8 + ((8 + 8 + 8) × (88 + 8 + 8))
:= 88 + (((8 + 8) × (8 × 8 + 88)) − (88/8))
:= 8 + ((((8 + 8 + 8) × (88 + 8 + 8)) + ((8 + 8)/8)) + 8)
:= (9 − 9/9) × ((9 + 9) × (9 + 9) − (99/9))
Ï 2505 := ((1 + (11 + 11)) × (111 − 1 − 1)) − 1 − 1
:= 9 + (((9 × 99 + 9)/(9 + 9))
(9+9)/9
)
Ï 2510 := (1 + 1) × (1 + (11 × (1 + 1 + 1 + 111)))
:= 9 × 9 × 9 + ((9 + 9) × 99 + ((9 + 9 + 9)/9))
Ï 2515 := 1 + (11 + (((11 × (1 + 1)11 ) − 1)/(11 − 1 − 1)))
:= 2 + (2 × (22 + 2) + 2)2 + 2/2 + 2
:= 2 + 22 × ((222 + 2)/2 + 2)
:= 2 + (((2 × (22 + 2) + 2)2 + 22/2) + 2)
:= (33 × ((3 × (33 + 3)) + 3)) − 3 − 3
:= (33 × ((3 × (33 + 3)) + 3)) − 3/3
:= 3 + ((33 × ((3 × (33 + 3)) + 3)) + 3/3)
:= 4 + ((4 × (4/4 + 4)4 ) + 4/4)
:= (44 − 4)/4 + (4 × (4/4 + 4)4 )
:= (4 × ((4/4 + 4)4 + 4)) − 4/4
:= 5 + (55 − 55 /5)
:= 5 + ((55 − 55 /5) + 5)
:= 55 − (555 + 55)
:= 6 + (((666/6 + 6 × 6 × 66) + 6) + 6)
:= (6 + 6)/6 + (66 × ((6 + 6)/6 + 6 × 6))
:= 6/6 + (((6 + 6) × (6 × 6 × 6 − 6)) − 6)
:= 7 + ((7 × (7 × 7 × 7 + 7 + 7)) − 7/7)
:= 77/7 + (7 × (7 × 7 × 7 + 7 + 7))
:= 7 × 7 × 7 × 7 + (((7 + 7)/7)7 − (7 + 7))
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) + 8/8)
:= 88 × (8 + 8) + ((8888 − 8)/8 − 8)
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) + (88/8))
:= 9 + ((9 − 9/9) × ((9 + 9) × (9 + 9) − ((99 + 9)/9)))
:= 9 × 9 × 9 + ((9 + 9) × 99 − 9/9)
:= ((9 × 9 + 9)/(9 + 9)) × ((((9 + 9)/9)9 ) − 9)
Ï 2506 := (1 + 1) × ((11 × (1 + 1 + 1 + 111)) − 1)
:= 2 + (2 × (22 + 2) + 2)2 + 2 + 2
Ï 2511 := 11 + (((11 − 1)1+1 /(1 + 1))1+1 )
:= 22/2 + (2 × (22 + 2) + 2)2
3
3
:= 3 × 3 + (33/3 × ((3 + 3) + 33/3))
3
Ï 2516 := (1 + 1) × (11 × 111 + (111/(1 + 1 + 1)))
:= 22+2 + (2 × (22 + 2) + 2)2
:= 3 × ((3 × (3 + 3)) + 3)
:= 333 + (3 × 33+3 − (3/3 + 3))
:= 44/4 + (4 × (4/4 + 4)4 )
:= 4 × ((4/4 + 4)4 + 4)
:= 5 + (((5 − 5 )/5) + 5 )
:= 55 + ((55 − 55 )/5)
:= 5 + (((55 − 55 )/5) + 55 )
:= 66 + (6 × 6 × 66 + ((6 + 6)/6)6 )
:= (66 × 66 + 666)/((6 + 6)/6)
:= (6 × 6 + 6/6) × (((6 + 6)/6) + 66)
:= 7 + (7 × (7 × 7 × 7 + 7 + 7))
:= 7 × 7 × 7 × 7 + (777 − 7)/7
:= 77 + (7 × (7 × 7 × 7 + 7) − (77/7))
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) + ((8 + 8)/8))
:= 88 × (8 + 8) + (8888/8 − 8)
:= ((8 × 8 × (88 − 8)) − 88)/((8 + 8)/8)
:= 9 × ((99 + 99) + 9 × 9)
:= 9 + ((9/9 + 99 + 9) × ((99 + 99 + 9)/9))
4
:= 4 + ((4 × (4/4 + 4) ) + (4 + 4)/4)
5
5
9
:= (((9 × 9 + 9)/(9 + 9)) × ((((9 + 9)/9) ) − 9)) − 9
Ï 2507 := (1 + (11 + 11)) × (111 − 1 − 1)
Ï 2512 := 1 + (11 + (((11 − 1)1+1 /(1 + 1))1+1 ))
2
2
Ï 2517 := 11 + ((1 + 1) × ((11 × (1 + 1 + 1 + 111)) − 1))
:= (22 + 2/2) × (222/2 − 2)
:= (2 × 22) + ((22 + 2) )
:= 2/2 + ((2 × (22 + 2) + 2)2 + 22+2 )
:= (33 × ((3 × (33 + 3)) + 3)) − (3/3 + 3)
:= 3/3 + (33 × ((3 × (33 + 3)) + 3))
:= 333 + (3 × 33+3 − 3)
:= 4 + (((4 × (4/4 + 4)4 ) − 4/4) + 4)
:= (4 × ((4/4 + 4)4 + 4)) − 4
:= 4/4 + (4 × ((4/4 + 4)4 + 4))
:= 5 + ((((5 − 55 ) + 5)/5) + 55 )
:= 55 + (((55 − 55 ) + 5)/5)
:= 5 + ((((55 − 55 ) + 5)/5) + 55 )
:= (66 × ((6 + 6)/6 + 6 × 6)) − 6/6
6
:= ((6 + 6)/6) + (6 × (6 × 66 + 6 + 6))
:= 666/6 + (6 × (6 × 66 + 6) − 6)
:= 777/7 + 7 × 7 × 7 × 7
:= 7 + ((7 × (7 × 7 × 7 + 7 + 7)) + (77/7))
:= 88/8 + ((8 + 8 + 8) × (88 + 8 + 8))
:= 8 + (((8 + 8 + 8) × (88 + 8 + 8)) + 8)
:= 88 × (8 + 8) + ((8888 − (8 + 8))/8)
:= (9/9 + 99 + 9) × ((99 + 99 + 9)/9)
:= 9/9 + ((9 + 9) × 99 + 9 × 9 × 9)
:= 9 + (((9 + 9) × 99 − ((9 + 9 + 9)/9)) + 9 × 9 × 9)
:= 7 + ((7/7 + 7 × 7)
(7+7)/7
)
177
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2518 := 11 + ((1 + (11 + 11)) × (111 − 1 − 1))
2
:= 2 + (((2 × (22 + 2))2 ) + 222)
:= 3/3 + ((3 × 33+3 − 3) + 333)
:= 3 + (3 × 33+3 + 333)
:= ((33/3 + 3)3 ) − (3 + 3)3
:= (4 + 4)/4 + (4 × ((4/4 + 4)4 + 4))
:= 4 + (((44 − 4) × (44 − 4) − 4)/4)
:= 4 × ((4 × (4 × (44 − 4))) − (4 + 4))
:= 55 − (((55 × 55 + 5) + 5)/5)
:= 5 × 5 + (55 − (55 + 5 + 5)/5)
:= (5 − 5/5) × ((55 + 5 + 5)/5 + 5)
:= ((6 + 6) × (6 × 6 × 6 − 6)) − (6 + 6)/6
:= 666/6 + 6 × (6 × 66 + 6)
:= (6 × (6 × (66 + 6))) − ((6 + 6)/6)6
:= 7 + ((777 − 7)/7 + 7 × 7 × 7 × 7)
:= 7 × 7 × 7 + (((7 + 7 + 7)/7)7 − 7)
:= 7/7 + (7 × (7 × 7 × 7 + 7) + 77)
:= 88 × (8 + 8) + (8888 − 8)/8
:= (8/8 − 88) × (8 − 888/(8 + 8 + 8))
:= 8 + (((8 + 8) × (8 × 8 + 88)) + 88)
:= 9 × 9 + (999/9 × ((99 + 99)/9))
:= 99 + ((9 × 9 + 9) × (9 + 9 + 9) − 9/9)
:= 9 + ((((9 × 99 + 9)/(9 + 9))
2
Ï 2528 := ((111 − 1) × (1 + (11 + 11))) − 1 − 1
:= 22 + ((2 × (22 + 2) + 2) + 2/2)
:= 2 + ((2 × (22 + 2) + 2) + 2
2+2
Ï 2523 := (1 + 1 + 1) × ((((11 − 1) × (1 + 1 + 1)) − 1)1+1 )
)
(9+9)/9
) + 9)
Ï 2519 := 11 × (1 + ((1 + 1) × (1 + 1 + 1 + 111)))
Ï 2524 := 1 + ((1 + 1 + 1) × ((((11 − 1) × (1 + 1 + 1)) − 1)1+1 )) Ï 2529 := ((111 − 1) × (1 + (11 + 11))) − 1
:= 22/2 × (((222 − 22)/2) − 2)
:= 2 + ((2 × (22 + 2) + 2)2 + 22)
:= 222 + 222/2 − 2/2 − 2
:= 333 + (3 × 33+3 − 3/3)
:= 3 + ((3 × 33+3 + 333) + 3/3)
:= 3 × ((33 × (33 + 3)) + 33)
:= ((44 − 4) × (44 − 4) − 4)/4
:= 4 + ((4 × ((4/4 + 4)4 + 4)) + 4)
:= 4 × 4 × (4 + 4) + (((4 − 4/4) + 4)4 )
:= 55 − ((55 × 55 + 5)/5)
:= 5 × 5 + (55 − (55 + 5)/5)
:= 5 + ((5 × 5 − (55 + 5)/5) + 55 )
:= ((6 + 6) × (6 × 6 × 6 − 6)) − 6/6
:= 6 + (((6 + 6) × (6 × 6 × 6 − 6)) − ((6 + 6)/6))
:= 6 + (6 × (6 × 66 + 6) + 666/6)
:= 7 + (777/7 + 7 × 7 × 7 × 7)
:= (((7 + 7)/7 + 7 × 7)(7+7)/7 ) − 77
:= 7 × 7 × 7 × 7 + ((7 + 7)/7)7
:= 88 × (8 + 8) + 8888/8
:= ((8 × (8 × (88 − 8) − 8)) − 8)/((8 + 8)/8)
:= 8 × (8 + 8) + ((8 − 8/8)8×8/(8+8) )
:= 9 + (((9 + 9) × 99 − 9/9) + 9 × 9 × 9)
:= 9 + (((9 × 9 + 9)/(9 + 9)) × ((((9 + 9)/9)9 ) − 9))
:= 99 + (9 × 9 + 9) × (9 + 9 + 9)
Ï 2520 := (11 + (11 − 1)) × (111+1 − 1)
2
:= 22 + ((2 × (22 + 2) + 2) − 2)
Ï 2525 := (1 + ((1 + 1) × (1 + 11))) × (1 + (11 − 1)1+1 )
Ï 2530 := (111 − 1) × (1 + (11 + 11))
:= 222 + (((2 × (22 + 2))2 ) − 2/2)
:= 22 × (222/2 + 2 + 2)
:= ((33/3 + 3)3 ) − ((3 + 3)3 + 3)
:= 333 + (((3 × 3 + 3/3) + 3)3 )
:= 4 + (4 × ((4/4 + 4)4 + 4))
:= 4 + (((44 − 4) × (44 − 4) + 4)/4)
:= (44 − 4)/4 × ((4/4 − 4) + 44 )
:= 55 − (55 × (55/5))
:= 5 × ((5 × 5 × (5 × 5 − 5)) + 5)
:= 5 + ((5 × (5 − 5 × 5 × 5)) + 55 )
:= (6 + 6) × (6 × 6 × 6 − 6)
:= 6 + (((6 + 6) × (6 × 6 × 6 − 6)) − 6/6)
:= ((66 − 6)/6) + ((6 + 6) × (6 × 6 × 6 − 6))
:= 77 + (7 × (7 × 7 × 7 + 7) − 7)
:= 77 + (7 × (7 × 7 × 7 + 7) − ((7 + 7)/7))
:= 7 × 7 × 7 + ((7 + 7 + 7)/7)7
:= 88 + ((8 + 8) × (8 × 8 + 88))
:= ((8 − 88/8) + 8) × ((8 × 8 × 8 − 8) + 8/8)
:= ((8 − 8/8 + 8) + 8) × (888 − 8)/8
:= ((9 + 9)/9 + 99) × (((9 − ((9 + 9)/9)) + 9) + 9)
:= 9/9 + ((9 × 9 + 9) × (9 + 9 + 9) + 99)
:= 333 + 3 × 3
3+3
:= 9 + ((9 + 9) × 99 + 9 × 9 × 9)
Ï 2521 := 1 + ((11 + (11 − 1)) × (111+1 − 1))
2
Ï 2526 := 1 + ((1 + ((1 + 1) × (1 + 11))) × (1 + (11 − 1)1+1 ))
Ï 2531 := 1 + ((111 − 1) × (1 + (11 + 11)))
:= 222 + ((2 × (22 + 2))2 )
:= 222 + (222/2 − 2/2)
:= 3 + ((3 × 33+3 + 333) + 3)
:= 3 + (((33/3 + 3)3 ) − (3 + 3)3 )
:= 4 + ((((44 − 4) × (44 − 4) + 4) + 4)/4)
:= (4 × (((4/4 + 4)4 + 4) + 4)) − 4/4
:= 5 + ((5 − 55 × 55)/5)
:= 5 × 5 + (((5 − 55 )/5) + 55 )
:= 5 + ((((5 − 55 )/5) + 5 × 5) + 55 )
:= 6/6 + ((6 + 6) × (6 × 6 × 6 − 6))
:= 6 + ((6 + 6) × (6 × 6 × 6 − 6))
:= 66/6 + ((6 + 6) × (6 × 6 × 6 − 6))
:= 7 + ((((7/7 + 7 × 7)(7+7)/7 ) + 7) + 7)
:= 77 + (7 × (7 × 7 × 7 + 7) − 7/7)
:= 7/7 + (((7 + 7 + 7)/7)7 + 7 × 7 × 7)
:= 8/8 + (((8 + 8) × (8 × 8 + 88)) + 88)
:= 8 + ((8888 − 8)/8 + 88 × (8 + 8))
:= 88 + (((8 + 8) × (8 × 8 + 88)) + (88/8))
:= 9 + (((9 + 9) × 99 + 9 × 9 × 9) + 9/9)
:= 99 + (((9 + 9 + 9)/9) × (9 × (9 × 9 + 9) − 9/9))
:= 9 + (((9 + 9) × 99 + (99/9)) + 9 × 9 × 9)
:= 22 + ((2 × (22 + 2) + 2) − 2/2)
:= 3/3 + (3 × 3
3+3
+ 333)
4
:= ((44 − 4) × (4 − 4) + 4)/4
5
Ï 2522 := 1 + (1 + ((11 + (11 − 1)) × (111+1 − 1)))
2
Ï 2527 := (1 + (11 × (1 + 11))) × (((1 + 1) × (11 − 1)) − 1)
2
Ï 2532 := (1 + 11) × ((1 + 1) × 111 − 11)
:= 2/2 + (((2 × (22 + 2)) ) + 222)
:= 222 + 222/2
:= 3 + ((3 × 33+3 − 3/3) + 333)
:= 333 + ((((3 × 3 + 3/3) + 3)3 ) − 3)
:= 3 + ((3 × (33+3 + 3)) + 333)
:= (((44 − 4) × (44 − 4) + 4) + 4)/4
:= 44/4 + (4 × ((4/4 + 4)4 + 4))
:= 4 × (((4/4 + 4)4 + 4) + 4)
:= 5 + (((5 − 55 × 55) + 5)/5)
:= 5 × 5 + ((((5 − 55 ) + 5)/5) + 55 )
:= 55 + (((5 + 5)/5)5 − 55 /5)
:= (6 + 6)/6 + ((6 + 6) × (6 × 6 × 6 − 6))
:= 6 + (((6 + 6) × (6 × 6 × 6 − 6)) + 6/6)
:= 6 + (((6 + 6) × (6 × 6 × 6 − 6)) + 6)
:= 7 × 7 × 7 × 7 + (((7 + 7)/7) − 7)
:= 77 + 7 × (7 × 7 × 7 + 7)
:= 7 + ((7 × (7 × 7 × 7 + 7) − ((7 + 7)/7)) + 77)
:= 88 + (((8 + 8) × (8 × 8 + 88)) + ((8 + 8)/8))
:= 8 + (8888/8 + 88 × (8 + 8))
:= ((8 × (8 × (88 − 8) − 8)) + 8)/((8 + 8)/8)
:= 9 × 9 × 9 + ((9 + 9) × 99 + (99/9))
:= 99 + ((9 × 9 + 9) × (9 + 9 + 9) − ((9 + 9)/9))
:= 999/9 + ((9 × 9 + 9) × (9 + 9 + 9) − 9)
:= 22 + (2 × (22 + 2) + 2)
5
7
178
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2533 := 1 + ((1 + 11) × ((1 + 1) × 111 − 11))
:= 2/2 + (2
22/2
Ï 2538 := (11 × (11 × (11 + (11 − 1)))) − 1 − 1 − 1
2
+ 22 )
:= 2 + (((2
22/2
Ï 2543 := 1 + (1 + (11 × (11 × (11 + (11 − 1)))))
:= 2 + ((22/2)2 × (22 − 2/2))
2
+ 22 ) + 2) + 2)
:= 3 + ((((3 × 3 + 3/3) + 3)3 ) + 333)
:= (3 × 33 × 33) − 33+3
:= ((33/3 + 3)3 ) − (33 × (3 + 3) + 3)
:= 4/4 + (4 × (((4/4 + 4)4 + 4) + 4))
:= ((44 × (4 × 4 + 4)) − 44)/((4 + 4)/4)
:= 44 + ((4 × (4/4 + 4)4 ) − 4/4)
:= 5 + ((5 − 5/5) × ((55 + 5 + 5)/5 + 5))
:= 55 − (((5 + 5)/5)5 + 555)
:= 5 + (55 − (((5 + 5)/5)5 + 555))
:= 6 + ((((6 + 6) × (6 × 6 × 6 − 6)) + 6/6) + 6)
:= 6 + ((((6 + 6) × (6 × 6 × 6 − 6)) + 6) + 6)
:= (6 × ((6 × (66 + 6)) − 6)) − (6/6 + 6 + 6)
:= 7 + ((7 × (7 × 7 × 7 + 7) − 7/7) + 77)
:= 77 + (7 × (7 × 7 × 7 + 7) + (77/7))
:= 7 + ((7 × 7 × 7 × 7 + ((7 + 7)/7)7 ) + 7)
:= 8 + (((8 − 88/8) + 8) × ((8 × 8 × 8 − 8) + 8/8))
:= 8 + (((8 − 8/8 + 8) + 8) × (888 − 8)/8)
:= 888/8 + ((8 + 8) × (8 × 8 + 88))
:= ((9/9 + 9 × 9) × (((99 + 99)/9) + 9)) − 9
:= 999 + (9 × (9 × (9 + 9) + 9))
:= ((99/9) × (9 × (9 + 9 + 9) − (99/9))) − 9
Ï 2534 := 1 + (1 + ((1 + 11) × ((1 + 1) × 111 − 11)))
:= 2 + (2
22/2
2
Ï 2539 := (11 × (11 × (11 + (11 − 1)))) − 1 − 1
2
+ 22 )
:= 2 × ((2 + 2 + 2)2+2 − (22 + 2))
:= ((22/2) × (22 − 2/2)) − 2
3
3
:= 3 + ((((33/3 + 3) ) − (3 + 3) ) + 3)
:= 3/3 + ((3 × 33 × 33) − 3
4
:= 3 + (33 × (3 × 33 − (3/3 + 3)))
)
4
:= 4 + ((44 − 4)/4 × ((4/4 − 4) + 4 ))
5
3+3
:= 4 + ((4 + 4)/4 × (44 − (4/4 + 4)))
5
Ï 2544 := (1 + 11) × (1 + ((1 + 1) × 111 − 11))
5
:= 4 × ((4 × (4 × (44 − 4))) − 4)
:= 5 + (((5 × 5 − (5 + 5)/5) + 5 ) + 5)
:= ((5 − 5/5) × ((55 + 5 )/5)) − 5
:= (5 − 5/5) × ((55 + 55 )/5)
:= 6 + ((6 × (6 × (66 + 6))) − ((6 + 6)/6)6 )
:= (6 × ((6 × (66 + 6)) − 6)) − (66/6 + 6)
:= (6 × ((6 × (66 + 6)) − 6)) − 6 − 6
:= 7 + (7 × (7 × 7 × 7 + 7) + 77)
:= 77 + (7 × (7 × 7 × 7 + 7) + (77 + 7)/7)
:= 7 + ((((7 + 7 + 7)/7)7 + 7 × 7 × 7) + 7)
:= 88 + ((88/8 + 8) × (8 × (8 + 8) + 8/8))
:= (8 + 8) × (((88 − 8/8) + 8 × 8) + 8)
:= 9/9 + ((9 × (9 × (9 + 9) + 9)) + 999)
:= 99 × (9 + 9 + 9) − ((999/9 + 9) + 9)
:= 8 + (((8888 − 8)/8 + 88 × (8 + 8)) + 8)
99/9
:= ((9 + 9) × (9 + 9 + 9)) + (((9 + 9)/9)
)
Ï 2535 := (1 + 1 + 11) × ((1 + 1 + 1 + 11)1+1 − 1)
:= 2 + ((2
22/2
2
Ï 2540 := (11 × (11 × (11 + (11 − 1)))) − 1
:= (2 − 22) × ((2 − 2
+ 22 ) + 2/2)
2×(2+2)
Ï 2545 := 1 + ((1 + 11) × (1 + ((1 + 1) × 111 − 11)))
:= 2 + (((22/2)2 × (22 − 2/2)) + 2)
)/2)
:= 3 × (((3 − 3/3)3×3 ) + 333)
:= (3 × 3 + 3/3) × ((33+3 + 33)/3)
:= (3 − 3/3 + 3) × (((3 − 3/3)3×3 ) − 3)
:= (44 + 4)/4 × (44 − (4/4 + 4))
:= 44 + ((4 × (4/4 + 4)4 ) − 4)
:= 44 + ((4 × (4/4 + 4)4 ) + 4/4)
:= 5 + (((5 × (5 − 5 × 5 × 5)) + 55 ) + 5)
:= 55 − ((555 + 5 × 5) + 5)
:= 55 − (555 + 5 × 5)
:= (6 − 6/6) × (666/6 + 6 × 66)
:= ((66 + 6/6) × ((6 + 6)/6 + 6 × 6)) − 6
:= (6 × ((6 × (66 + 6)) − 6)) − 66/6
:= 7 + ((7 × (7 × 7 × 7 + 7) + (77/7)) + 77)
:= 7 + ((7 × (7 × 7 × 7 + 7) + 77) + 7/7)
:= (7 × ((7 × 7 × 7 + 7 + 7) + 7)) − (7/7 + 7)
:= 8 + ((8888/8 + 88 × (8 + 8)) + 8)
:= ((88 + 8)/8 + 8) × (8 × (8 + 8) − 8/8)
:= (888/8 × ((8 − 8/8 + 8) + 8)) − 8
:= 999 + (((9 + 9 + 9)/9) × (((9 + 9)/9)9 ))
:= (9/9 + 9) × (9 × (9 + 9 + 9) + (99/9))
:= 9 × 9 × (9 + 9) + (((99 × 99) − (9 + 9))/9)
Ï 2536 := 1 + (1 + 1 + 11) × ((1 + 1 + 1 + 11)1+1 − 1)
:= 2 + ((2
22/2
Ï 2541 := 11 × (11 × (11 + (11 − 1)))
Ï 2546 := 1 + (1 + ((1 + 11) × (1 + ((1 + 1) × 111 − 11))))
:= (22/2)2 × (22 − 2/2)
2
+ 22 ) + 2)
3
:= 3 + (((((3 × 3 + 3/3) + 3) ) + 333) + 3)
:= 2222 + ((22+2 + 2)2 )
3
:= 33 × (3 × 3 − (3/3 + 3))
:= ((33/3 + 3)3 ) − 33 × (3 + 3)
4
4
:= 4 + ((44 − 4/4) × (((4 − 4)/4) − 4))
:= 4 + (4 × (((4/4 + 4) + 4) + 4))
5
:= 5 × 5 + (((55 − 55 )/5) + 55 )
5
:= 5 + ((((55 − 5 )/5) + 5 × 5) + 5 )
:= ((44 − 4)/4 × (44 − 4/4)) − 4
:= 55 + (5/5 − (555 + 5 × 5))
:= (6/6 + 6) × 66 × 66/(6 + 6)
:= (66 + 6/6) × ((6 + 6)/6 + 6 × 6)
:= (7 + 7 + 7) × (((7 + 7)/7)7 − 7)
:= (7 × ((7 × 7 × 7 + 7 + 7) + 7)) − (7 + 7)/7
:= 8 + ((((8 + 8) × (8 × 8 + 88)) + 88) + 8)
:= 888 + ((88/8 + 8) × (88 − 8/8))
:= ((8 + 8)/8) × (((8 + 8) × (88 − 8) − 8) + 8/8)
:= 999 + ((9 × (9 × (9 + 9) + 9)) − ((9 + 9)/9))
:= 999/9 + (9 × 9 + 9) × (9 + 9 + 9)
:= 9 × 9 × (9 + 9) + (((99 × 99) − 9)/9)
Ï 2537 := (11 × (11 × (11 + (11 − 1)))) − 1 − 1 − 1 − 1
Ï 2542 := 1 + (11 × (11 × (11 + (11 − 1))))
:= 6 + (((6 + 6) × (6 × 6 × 6 − 6)) + ((66 − 6)/6))
7
:= 7 + (7 × 7 × 7 × 7 + ((7 + 7)/7) )
Ï 2547 := (1 + 1) × (111+1+1 − 1 − 1) − 111
:= ((22/2)2 × (22 − 2/2)) − 2 − 2
:= 2 + ((2 − 22) × ((2 − 22×(2+2) )/2))
:= (222 + 2)/2 + ((2 × (22 + 2))2 )
:= 3 × 3 + (((33/3 + 3)3 ) − (3 + 3)3 )
:= 3/3 + (33 × (3 × 33 − (3/3 + 3)))
:= (33 × (3 × 33 − 3)) − 33
:= (44 − 4/4) × (((44 − 4)/4) − 4)
:= 44 + ((4 × (4/4 + 4)4 ) − (4 + 4)/4)
:= 4 + (((4 × (4/4 + 4)4 ) − 4/4) + 44)
:= 5 + ((((5 + 5)/5)5 − 55 /5) + 55 )
:= 55 + (55/5 × ((5 + 5)/5 − 55))
:= 55 + ((5 − 5/5) × (55 − 5 − 5)/5)
:= 6 + (((6 + 6) × (6 × 6 × 6 − 6)) + (66/6))
:= 66 + (6 × (6 × 66 + 6) + ((6 + 6)/6)6 )
7
:= 7 + (((7 + 7 + 7)/7) + 7 × 7 × 7)
8×8/(8+8)
:= 8 + (((8 − 8/8)
) + 8 × (8 + 8))
:= 999 + ((9 × (9 × (9 + 9) + 9)) − 9/9)
7
:= 6 + ((6/6 + 6) × 66 × 66/(6 + 6))
:= 7/7 + ((7 + 7 + 7) × (((7 + 7)/7) − 7))
:= (7 × ((7 × 7 × 7 + 7 + 7) + 7)) − 7/7
:= ((8 + 8)/8) × ((8 + 8) × (88 − 8) − (8/8 + 8))
:= (((8 − 88/8) + 8) × (8 × 8 × 8 − 8/8)) − 8
:= (9/9 + 9 × 9) × (((99 + 99)/9) + 9)
:= 9 + ((9 × (9 × (9 + 9) + 9)) + 999)
179
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2548 := (1 + 1 + 11) × (1 + 1 + 1 + 11)1+1
:= 2 × ((2 + 2 + 2)2+2 − 22)
Ï 2558 := (1 + 1)11 + ((1 + 1)11−1−1 − (1 + 1))
Ï 2553 := 111 × (1 + (11 + 11))
:= (2 × ((22 − 2) × 22+2+2 )) − 2
:= (22 + 2/2) × 222/2
3
:= (33 − 3/3) × (3 × 33 − 3/3)
:= ((3 × 3 + 3) × ((3 + 3) − 3)) − 3
4
:= 3 + (((3 × 3 + 3) × ((3 + 3)3 − 3)) − 3/3)
:= 4 + ((4 × (4/4 + 4)4 ) + 44)
:= 4 + ((4 × (44 − 4) − 44)/4)
:= (44 × (44 − 4) − (4 + 4))/4
:= (5 − 5/5) × ((55 + 55 + 5)/5)
:= 55 + (55 − (55 + 5 + 5)/5)
:= 55 − (((55 + 5)/5) + 555)
:= (6 × ((6 × (66 + 6)) − 6)) − ((6 + 6)/6 + 6)
:= 666/6 × ((66/6 + 6) + 6)
:= (6 + 6)/6 + (6 × ((6 × (66 + 6)) − 6))
:= 7 × ((7 × 7 × 7 + 7 + 7) + 7)
:= 777/7 × ((((7 + 7)/7 + 7) + 7) + 7)
:= 77 + (((7 + 7 + 7)/7)7 + 7 × (7 × 7 − 7))
:= (((8 × 8 × (88 − 8)) − 8)/((8 + 8)/8)) − 8
:= 888/8 × ((8 − 8/8 + 8) + 8)
:= ((8 + 8)/8) × ((8 + 8) × (88 − 8) − 8/8)
:= (99 − 9/9) × (((9 − 9/9) + 9) + 9)
:= 999/9 × ((99 + 99 + 9)/9)
:= 9 × 9 × (9 + 9) + ((99/9) × (9/9 + 99))
Ï 2549 := ((1 + 1) × (111+1+1 − 1)) − 111
Ï 2554 := 1 + (111 × (1 + (11 + 11)))
:= 2/2 + (2 × ((2 + 2 + 2)2+2 − 22))
:= 22 + (222/2 + 222 )
:= 3 + (((33/3 + 3)3 ) − 33 × (3 + 3))
:= 3/3 + (((3 × 3 + 3) × ((3 + 3)3 − 3)) − 3)
4
Ï 2559 := (1 + 1)11 + ((1 + 1)11−1−1 − 1)
:= (((22 − 2) × 22×(2+2) ) − 2)/2
4
:= 3 + ((3 × 3 + 3) × ((3 + 3)3 − 3))
:= (4 × (44 − 4) − 44)/4
:= 4 + ((44 − 4)/4 × (4 − 4/4))
:= (44 × (44 − 4) − 4)/4
:= 5 + ((5 − 5/5) × ((55 + 55 )/5))
:= 55 + (55 − (55 + 5)/5)
:= 55 − (555 + (55/5))
:= (6 × ((6 × (66 + 6)) − 6)) − 6/6 − 6
:= (6 × ((6 × (66 + 6)) − 6)) − (6 + 6)/6
:= 6 + (666/6 × ((66/6 + 6) + 6))
:= 7/7 + (7 × ((7 × 7 × 7 + 7 + 7) + 7))
:= 7 + ((7 × ((7 × 7 × 7 + 7 + 7) + 7)) − 7/7)
:= 77/7 + (7 × ((7 × 7 × 7 + 7 + 7) + 7))
:= (8 × ((8 + 8) × (8 + 8) + 8 × 8)) − 88/8
:= 8/8 + (888/8 × ((8 − 8/8 + 8) + 8))
:= (8 × ((8 + 8) × (8 + 8) + 8 × 8)) − 8/8
:= 9 + ((9/9 + 9) × (9 × (9 + 9 + 9) + (99/9)))
:= ((9 − 999)/9) + (99 × (9 + 9 + 9) − 9)
:= 9 × 9 × (9 + 9) + (((9999 − 9)/9) − 9)
Ï 2550 := (11 − 1) × (111 + (1 + 11)1+1 )
Ï 2555 := 1 + (1 + (111 × (1 + (11 + 11))))
Ï 2560 := (11 − 1) × (1 + 1)(1+1)×(1+1+1)
:= 2 + (2 × ((2 + 2 + 2)2+2 − 22))
:= 2 + ((22 + 2/2) × 222/2)
:= 2 × ((22 − 2) × 22+2+2 )
:= 3 + ((33 × (3 × 33 − 3)) − 33 )
:= ((3 × 3 + 3) × ((3 + 3)3 − 3)) − 3/3
:= (3 − 3/3 + 3) × ((3 − 3/3)3×3 )
:= (44 − 4)/4 × (44 − 4/4)
:= ((44 × (44 − 4) − 4)/4) − 4
:= 4 × (4 × (4 × (44 − 4)))
:= (5 + 5) × (5 × 5 × (5 + 5) + 5)
:= 55 + (5 − 5 /5)
:= 55 − (555 + 5 + 5)
5
5
:= (6 × ((6 × (66 + 6)) − 6)) − 6
:= (6 × ((6 × (66 + 6)) − 6)) − 6/6
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) − ((6 + 6)/6))
:= (7/7 + 7 × 7) × ((7 + 7)/7 + 7 × 7)
:= 7 + (7 × ((7 × 7 × 7 + 7 + 7) + 7))
:= ((7 + 7)/7)7 × ((7 − 7/7 + 7) + 7)
:= (((8 + 8)/8) + 8) × ((8 + 8) × (8 + 8) − 8/8)
:= ((8 − 88/8) + 8) × (8 × 8 × 8 − 8/8)
:= 8 × ((8 + 8) × (8 + 8) + 8 × 8)
:= 9 + ((9 × 9 + 9) × (9 + 9 + 9) + 999/9)
:= 9 + ((((99 × 99) − 9)/9) + 9 × 9 × (9 + 9))
:= (9/9 + 9) × (((9 + 9)/9)9−9/9 )
Ï 2551 := ((1 + 1) × 111+1+1 ) − 111
Ï 2556 := 1 + (1 + (1 + (111 × (1 + (11 + 11)))))
:= ((22 + 2/2) × 222/2) − 2
Ï 2561 := 1 + (11 − 1) × (1 + 1)(1+1)
1+1+1
:= 2 × (((22 − 2) × 22+2+2 ) − 2)
:= (((22 − 2) × 22×(2+2) ) + 2)/2
:= (3 × 3 + 3) × ((3 + 3)3 − 3)
:= 33 + (((33/3 + 3)3 ) − (3 + 3)3 )
:= (4 × (4 × (4 × (44 − 4)))) − 4
:= (44 × (44 − 4) + 4)/4
:= 55 + ((((5 − 5 )/5) − 5) + 5 )
:= 55 + (((5 − 55 )/5) + 55 )
:= 5 + ((((5 − 55 )/5) + 55) + 55 )
:= 6/6 + ((6 × ((6 × (66 + 6)) − 6)) − 6)
:= 6 × ((6 × (66 + 6)) − 6)
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) − 6/6)
:= 7 + (((((7 + 7 + 7)/7) + 7 × 7 × 7) + 7) + 7)
:= 7 + ((7 × ((7 × 7 × 7 + 7 + 7) + 7)) + 7/7)
:= 777/7 + 7 × (7 × 7 × 7 + 7)
:= (8 × ((8 + 8) × (8 + 8) + 8 × 8)) − (8/8 + 8)
:= ((8 × 8 × (88 − 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 × ((8 + 8) × (8 + 8) + 8 × 8))
:= 99 + (9 × 9 × (9 + 9) + 999)
:= 99 × (9 + 9 + 9) − ((999 + 9)/9)
3
:= 3 + ((3 − 3/3) × (3 × 33 − 3/3))
4
:= ((4 × (44 − 4) − 4)/4) − 4 − 4
5
5
7
9−9/9
:= ((9/9 + 9) × (((9 + 9)/9)
)) − 9
Ï 2552 := 11 × (111 + 111+1 )
Ï 2557 := 1 + (1 + (1 + (1 + (111 × (1 + (11 + 11))))))
Ï 2562 := (11 + (11 − 1)) × (1 + 111+1 )
:= 2 × (((2 + 2 + 2)2+2 − 22) + 2)
:= 2 + (((22 + 2/2) × 222/2) + 2)
:= 2 + (2 × ((22 − 2) × 22+2+2 ))
:= 33/3 × ((33+3 − 33)/3)
:= 3/3 + ((3 × 3 + 3) × ((3 + 3)3 − 3))
:= 3 + (((3 × 3 + 3) × ((3 + 3)3 − 3)) + 3)
:= (4 + 4) × (((44 − 4)/4) + 44 )
:= ((44 × (44 − 4) + 4)/4) − 4
:= ((44 × (44 − 4) + 4) + 4)/4
:= 55 + (((((5 − 55 ) + 5)/5) − 5) + 55 )
:= 55 + ((((5 − 55 ) + 5)/5) + 55 )
:= 5 + (((((5 − 55 ) + 5)/5) + 55) + 55 )
:= 6 + ((66 + 6/6) × ((6 + 6)/6 + 6 × 6))
:= 6/6 + (6 × ((6 × (66 + 6)) − 6))
:= 6 + (6 × ((6 × (66 + 6)) − 6))
:= 77/7 × (7 × 7 × 7 − 777/7)
:= 7 + ((7/7 + 7 × 7) × ((7 + 7)/7 + 7 × 7))
:= 7 + ((7 × ((7 × 7 × 7 + 7 + 7) + 7)) + 7)
:= (8 × ((8 + 8) × (8 + 8) + 8 × 8)) − 8
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 8 × 8)) − (88/8))
:= (8 + 8)/8 + (8 × ((8 + 8) × (8 + 8) + 8 × 8))
:= 99/9 × (9 × (9 + 9 + 9) − (99/9))
:= 9 + ((99 − 9/9) × (((9 − 9/9) + 9) + 9))
:= 99 × (9 + 9 + 9) − 999/9
180
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2563 := 11 × (11 + (1 + 1) × 111)
Ï 2568 := (1 + 1) × ((1 + 11) × (111 − (1 + 1 + 1 + 1)))
Ï 2573 := (11 × (1 + (11 + (1 + 1) × 111))) − 1
:= 2 × (2 × (22 − 2) + 22 )
:= (22 × ((22/2)2 − (2 + 2))) − 2/2
:= (33 × (3 × 33 − 3)) − 3 − 3
:= (33 × (3 × 33 − 3)) − 3/3
:= 4 + ((4 × (44 − 4) − 4)/4)
:= 4 + ((4 × (4 × (4 × (44 − 4)))) + 4)
:= 4 × 44 + ((((4 − 4/4) + 4)4 ) − 4)
:= 55 − ((555 + (5 + 5)/5) + 5)
:= 55 − (555 + (5 + 5)/5)
:= 5 + (55 − (555 + (5 + 5)/5))
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) + 6/6)
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) + 6)
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) + (66/6))
:= 7 + (((7 × ((7 × 7 × 7 + 7 + 7) + 7)) + 7/7) + 7)
:= 7 + (7 × (7 × 7 × 7 + 7) + 777/7)
:= 7 × 77 + ((((7 + 7)/7)77/7 ) − (7 + 7))
:= 8 + (((8 − 88/8) + 8) × (8 × 8 × 8 − 8/8))
:= 8 + (8 × ((8 + 8) × (8 + 8) + 8 × 8))
:= 8 + ((88/8 + 8) × ((8 × (8 + 8) − 8/8) + 8))
:= 99/9 × (9 × (9 + 9 + 9) − (9/9 + 9))
:= 9 × 9 × (9 + 9) + ((9999 − 9)/9)
:= 99 × (9 + 9 + 9) − (9/9 + 99)
2
:= 22/2 × (222 + 22/2)
3
:= (33 × (3 × 3 − 3)) − 33/3
4
Ï 2564 := 1 + (11 × (11 + (1 + 1) × 111))
:= 2 × (((22 − 2) × 2
2+2+2
Ï 2569 := (11 − 1) × (1 + (1 + 1)(1+1)
:= (2 × (2 + 2 + 2)
) + 2)
:= ((3 − 33)/3) + (33 × (3 × 33 − 3))
2
2+2
1+1+1
)−1
) − (22 + 2/2)
:= 3/3 + ((33 × (3 × 33 − 3)) − (3 + 3))
4
:= 4 + (((4 × (44 − 4) + 4)/4) + 4)
:= 4 + (4 × (4 × (4 × (44 − 4))))
:= 22 × ((22/2)2 − (2 + 2))
:= 33 × (3 × 33 − 3)
:= 4 + ((44 − 4)/4 × (4/4 + 44 ))
5
5
Ï 2574 := 11 × (1 + (11 + (1 + 1) × 111))
:= 5 − ((555 + 5/5) + 5)
:= 5 − (555 + 5/5)
:= 5 + (55 − (555 + 5/5))
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) + ((6 + 6)/6))
:= 6 + (((6 × ((6 × (66 + 6)) − 6)) + 6/6) + 6)
:= 66 × ((66 × 6/(6 + 6)) + 6)
:= 7 + (((7/7 + 7 × 7) × ((7 + 7)/7 + 7 × 7)) + 7)
:= 77 + ((7 × (7 × 7 × 7 + 7 + 7)) − 7)
:= 7 + ((((7 + 7)/7)7 × ((7 − 7/7 + 7) + 7)) + 7)
:= ((8 × 8 × (88 − 8)) + 8)/((8 + 8)/8)
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 8 × 8)) + 8/8)
:= ((8 + 8)/8) × (((8 + 8) × (88 − 8) − 8/8) + 8)
:= 99 × (9 + 9 + 9) − (9/9 + 99 + 9)
:= 9 × 9 × (9 + 9) + 9999/9
:= 99 × (((9 − 9/9) + 9) + 9)
Ï 2565 := 1 + (1 + (11 × (11 + (1 + 1) × 111)))
Ï 2570 := (11 − 1) × (1 + (1 + 1)(1+1)
2+2
1+1+1
:= 2 + (22/2 × (222 + 22/2))
:= (2 × (2 + 2 + 2)
:= 3 × 3 × (3 × (3 × 33 − 3) − 3)
:= (33 × (3 × 33 − 3)) − (3/3 + 3)
4
)
Ï 2575 := ((1 + 111) × (1 + (11 + 11))) − 1
) − 22
:= 22 + ((22 + 2/2) × 222/2)
:= 3/3 + (33 × (3 × 33 − 3))
4
:= 4 + ((4 × (44 − 4) + 4)/4)
:= (44 − 4)/4 × (4/4 + 4 )
5
:= 4 + ((44 × (44 − 4) + 44)/4)
5
:= 5 − (555 + 5)
:= 5 − 555
:= 6 + ((666/6 × ((66/6 + 6) + 6)) + 6)
7
:= 5 + (55 − 555)
:= 6666 − (((6 + 6)/6)6+6 )
:= (6 × (6 × (66 + 6))) − (66/6 + 6)
:= ((77 + 7)/7 + 7) × (((7 + 7)/7) + 7)
:= 77 + (((7/7 + 7 × 7)
:= (88/8 + 8) × ((8 × (8 + 8) − 8/8) + 8)
:= (((8 + 8)/8) + 8) × ((8 + 8) × (8 + 8) + 8/8)
:= (8/8 + 8 + 8 + 8) × (888/8 − 8)
:= 99 × (9 + 9 + 9) − (99 + 9)
:= 9 × 9 × (9 + 9) + (9999 + 9)/9
:= 9/9 + (99 × (((9 − 9/9) + 9) + 9))
Ï 2566 := 1 + (1 + (1 + (11 × (11 + (1 + 1) × 111))))
:= (2 × ((2 + 2 + 2)
2+2
− 2)) − 22
(7+7)/7
Ï 2571 := (11 − 1) × (1 + (1 + 1)(1+1)
:= 2/2 + ((2 × (2 + 2 + 2)
2+2
:= 77 + ((7 × (7 × 7 × 7 + 7 + 7)) − 7/7)
) − 7)
1+1+1
)+1
) − 22)
Ï 2576 := (1 + 111) × (1 + (11 + 11))
:= 2 + (22 × ((22/2)2 − (2 + 2)))
:= 3 + ((33 × (3 × 33 − 3)) − 33/3)
:= (33 × (3 × 33 − 3)) − 3
:= 3 + ((33 × (3 × 33 − 3)) − 3/3)
:= 4 + (((44 × (44 − 4) + 4) + 4)/4)
:= (44 × (44 − 4) + 44)/4
:= 4 × ((4 × (4 × (44 − 4))) + 4)
:= 5 + (5/5 − (555 + 5))
:= 55 + (5/5 − 555)
:= 5 + ((55 − 555) + 5/5)
:= ((66 − 6)/6) + (6 × ((6 × (66 + 6)) − 6))
:= ((66 × (66 + 6 + 6)) − 6)/((6 + 6)/6)
:= 6 + (6666 − (((6 + 6)/6)6+6 ))
:= 7 + ((7 × ((7 × 7 × 7 + 7 + 7) + 7)) + (77/7))
:= ((7 + 7)/7)7 + (7 × (7 × 7 × 7 + 7) − 7)
:= 77 + (7 × (7 × 7 × 7 + 7 + 7))
:= 8 + (((8 + 8)/8) × ((8 + 8) × (88 − 8) − 8/8))
:= 88/8 + (8 × ((8 + 8) × (8 + 8) + 8 × 8))
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 8 × 8)) + 8)
:= 9/9 + (99 × (9 + 9 + 9) − (99 + 9))
:= 9 + (99 × (9 + 9 + 9) − 999/9)
:= (9 + 9)/9 + (99 × (((9 − 9/9) + 9) + 9))
5
Ï 2567 := 1111 + ((1 + 1 + 11) × (1 + 111))
Ï 2572 := (11 × (1 + (11 + (1 + 1) × 111))) − 1 − 1
2+2
Ï 2577 := 1 + ((1 + 111) × (1 + (11 + 11)))
:= 222/2 + ((22 + 2/2)2 )
:= 2 + ((22/2 × (222 + 22/2)) + 2)
:= 2 + ((2 × (2 + 2 + 2)
:= 33/3 + ((3 × 3 + 3) × ((3 + 3)3 − 3))
:= 3/3 + ((33 × (3 × 33 − 3)) − 3)
:= 3 + (33 × (3 × 33 − 3))
:= 4 + (((44 × (44 − 4) − 4)/4) + 4)
:= (4 × ((4 × (4 × (44 − 4))) + 4)) − 4
:= 4 × 44 + (((4 − 4/4) + 4)4 )
:= 55 + ((5 + 5)/5 − (555 + 5))
:= 55 + ((5 + 5)/5 − 555)
:= 5 + (((5 + 5)/5 − 555) + 55 )
:= 66/6 + (6 × ((6 × (66 + 6)) − 6))
:= 6 + ((6 × ((6 × (66 + 6)) − 6)) + ((66 − 6)/6))
7
) − 22)
7
:= ((66 × (66 + 6 + 6)) + 6)/((6 + 6)/6)
:= 7 + (((7 + 7)/7) × ((7 − 7/7 + 7) + 7))
:= 7 × (7 × 7 + 7) + (((7 + 7 + 7)/7) − 7)
:= 77 + ((7/7 + 7 × 7)(7+7)/7 )
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 8 × 8)) − 8/8)
:= 8 + (((8 × 8 × (88 − 8)) + 8)/((8 + 8)/8))
:= 8 + (((8 × ((8 + 8) × (8 + 8) + 8 × 8)) + 8/8) + 8)
:= ((9 − 9/9) + 9) × (9 × (9 + 9) − (99/9))
:= 99 × (9 + 9 + 9) − ((9 + 9)/9 + 99)
:= 9 + (((9999 − 9)/9) + 9 × 9 × (9 + 9))
181
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2578 := 1 + (1 + ((1 + 111) × (1 + (11 + 11))))
Ï 2583 := (11 + (11 − 1)) × (1 + (1 + 111+1 ))
Ï 2588 := 1 + (11 + ((1 + 111) × (1 + (11 + 11))))
2
2
:= 2 + ((22 × ((22/2) − (2 + 2))) + 2)
:= (22 − 2/2) × ((22/2) + 2)
:= 2 × ((2 + 2 + 2)2+2 − 2)
:= 3 + ((33 × (3 × 33 − 3)) + 3/3)
:= 3 × ((33 × (33 − 3/3)) + 3)
:= (3/3 + 3) × (3 × (3 + 3)3 − 3/3)
:= 4 + (((44 − 4)/4 × (4/4 + 44 )) + 4)
:= ((44 − 4)/4) × ((4/4 − 4) + 44)
:= (4 × ((4 + 4) × (4 − 4/4)4 )) − 4
:= 5 + ((55 − (555 + (5 + 5)/5)) + 5)
:= 55 + ((55 + 5 + 5)/5 − 555)
:= (5 − 5/5) × (((55 + 55) + 55 )/5)
:= ((6 + 6)/6) × (6 × 6 × 6 × 6 − (6/6 + 6))
:= 6 × 66 + ((6 × 6/(6 + 6))6/6+6)
:= (6 + 6)/6 + ((6 × (6 × (66 + 6))) − 6)
:= 7 + ((7 × (7 × 7 × 7 + 7 + 7)) + 77)
:= 7/7 + ((((7 + 7)/7)77/7 ) + 7 × 77)
:= 8 + ((((8 + 8)/8) + 8) × ((8 + 8) × (8 + 8) + 8/8))
:= ((8/8 + 8 + 8) × (8 × 8 + 88)) − 8/8
:= ((8 × (8 × (88 − 8) + 8)) − 8)/((8 + 8)/8)
:= 9 + (9999/9 + 9 × 9 × (9 + 9))
:= 9 + (99 × (((9 − 9/9) + 9) + 9))
:= ((9 + 9)/9) × (((9 + 9) × (9 × 9 − 9)) − ((9 + 9)/9))
7
:= ((7 + 7)/7) + 7 × (7 × 7 × 7 + 7)
Ï 2579 := 1 + (1 + (1 + ((1 + 111) × (1 + (11 + 11)))))
Ï 2584 := (11 × (11 + ((1 + 1) × (1 + 111)))) − 1
:= 2 × ((2 + 2 + 2)2+2 − (2 + 2))
:= 2 + (((22 + 2/2)2 ) + 222/2 )
Ï 2589 := (1 + 1 + 1) × ((((1 + 11)1+1+1 )/(1 + 1)) − 1)
:= 2/2 + (2 × ((2 + 2 + 2)2+2 − 2))
3
:= 3 + (((33 × (3 × 33 − 3)) − 3/3) + 3)
:= 3 + ((3 × (3 × 33 − 3)) − 33/3)
:= (33 × (3 × 33 − 3)) − 3
:= 4 + (((44 × (44 − 4) + 44)/4) + 4)
:= 4 + ((4 × ((4 × (4 × (44 − 4))) + 4)) + 4)
:= ((44 + 4) × (44 − 4) − 44)/4
5
:= 5 + ((55 − (555 + 5/5)) + 5)
:= 5 + (((5 − (555 + 5/5)) + 5) + 5)
:= 5 × 5 + (55 − ((555 + 5/5) + 5))
:= (6 × (6 × (66 + 6))) − (6/6 + 6 + 6)
:= (6 × (6 × (66 + 6))) − ((6 + 6)/6 + 6)
:= (6 × (6 × (66 + 6))) − 6 × 6/(6 + 6)
:= 7 × (7 × 7 + 7) + ((7 + 7 + 7)/7)
:= 7 + (((7/7 + 7 × 7)(7+7)/7 ) + 77)
:= 77 + (777/7 + 7 × 7 × 7 × 7)
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 8 × 8)) + (88/8))
:= (8/8 + 8 + 8) × (8 × 8 + 88)
:= 8 + ((8/8 − 88 × 88)/(8 − 88/8))
:= 9 + ((9999 + 9)/9 + 9 × 9 × (9 + 9))
:= (9 − 9/9) × ((9 + 9) × (9 + 9) − 9/9)
:= 9 + ((99/9 + 9) × ((999/9 + 9) + 9))
7
Ï 2580 := (11 − 1) × (1 + 1 + (1 + 1)(1+1)
1+1+1
)
Ï 2590 := (1 + 1) × (((1 + 1 + 1) × (1 + 11))1+1 − 1)
Ï 2585 := 11 × (11 + ((1 + 1) × (1 + 111)))
:= 2 × ((2 + 2 + 2)2+2 − (2 + 2 + 2))
:= 2 + ((22 − 2/2) × ((22/2)2 + 2))
:= (2 × (2 + 2 + 2)2+2 ) − 2
:= 3 + ((33 × (3 × 33 − 3)) + 3)
:= 33/3 + (33 × (3 × 33 − 3))
:= 3/3 + ((33 × (3 × 33 − 3)) − 3)
:= 4 + (4 × ((4 × (4 × (44 − 4))) + 4))
:= 4 + (((((4 − 4/4) + 4)4 ) + 4 × 44) + 4)
:= (44 − 4)/4 × ((44 − 4/4) + 4)
:= 5 + ((55 − 555) + 5)
:= 5 + (((55 − 555) + 5) + 5)
:= 5 × 5 + (55 − (555 + 5))
:= (6 + 6) × (6 × 6 × 6 − 6/6)
77/7
:= (6 × (6 × (66 + 6))) − 6/6 − 6
:= (6 × (6 × (66 + 6))) − (6 + 6)/6
7
:= 7 + (7 × (7 × 7 × 7 + 7) + ((7 + 7)/7) )
:= (7 × (7 × (7 × 7 − 7) + 77)) − 7
:= ((88 + 8)/8 + 8) × (8 × (8 + 8) + 8/8)
:= 8/8 + ((8/8 + 8 + 8) × (8 × 8 + 88))
:= (8 − 8/8) × ((8888 − 8)/(8 + 8 + 8))
:= (99/9 + 9) × ((999/9 + 9) + 9)
:= 99/9 + (99 × (((9 − 9/9) + 9) + 9))
:= ((9 + 9)/9) × (((9 + 9) × (9 × 9 − 9)) − 9/9)
:= 7 × 77 + ((((7 + 7)/7)
) − 7)
Ï 2581 := ((1 + 1) × ((1 + 1 + 1) × (1 + 11))1+1 ) − 11
Ï 2586 := 1 + (11 × (11 + ((1 + 1) × (1 + 111))))
Ï 2591 := ((1 + 1) × ((1 + 1 + 1) × (1 + 11))1+1 ) − 1
:= (2 × (2 + 2 + 2)2+2 ) − 22/2
:= (2 × ((2 + 2 + 2)2+2 − 2)) − 2
:= (2 × (2 + 2 + 2)2+2 ) − 2/2
:= (33 × (3 × 33 − 3)) − 33/3
:= (33 × (3 × 33 − 3)) − 3 − 3
:= (33 × (3 × 33 − 3)) − 3/3
:= 4 + ((((4 − 4/4) + 4)4 ) + 4 × 44)
:= 4 × 4 + ((44 − 4)/4 × (4/4 + 44 ))
:= (4 × ((4 + 4) × (4 − 4/4)4 )) − 4/4
:= 55 + (55/5 − 555)
:= 5 + ((55/5 − 555) + 55 )
:= 5 + (((55/5 − 555) + 55 ) + 5)
:= (6 × (6 × (66 + 6))) − 66/6
:= (6 × (6 × (66 + 6))) − 6
:= (6 × (6 × (66 + 6))) − 6/6
7
:= 7777/7 + ((7 + 7) × (7 × (7 + 7) + 7))
:= 7 + (((7 + 7 + 7)/7) + 7 × (7 × 7 + 7))
:= 7/7 + ((7 × (7 × (7 × 7 − 7) + 77)) − 7)
:= (8/8 − 88 × 88)/(8 − 88/8)
:= (8 + 8)/8 + ((8/8 + 8 + 8) × (8 × 8 + 88))
:= ((88 + 8) × (88/8 + 8 + 8)) − 8/8
:= (99/9 + 9 + 9) × ((9 × 9 − 9/9) + 9)
:= ((99 + 9)/9) + (99 × (((9 − 9/9) + 9) + 9))
:= 99 × (9 + 9 + 9) − (9/9 + 9 × 9)
Ï 2582 := (11 + 11 − 1) × (1 + 1 + 111+1 ) − 1
Ï 2587 := 11 + ((1 + 111) × (1 + (11 + 11)))
Ï 2592 := (1 + 1) × ((1 + 1 + 1) × (1 + 11))1+1
:= ((2 × 2 × 22)2 + 2)/(2/2 + 2)
:= (2 × ((2 + 2 + 2)2+2 − 2)) − 2/2
:= 2 × (2 + 2 + 2)2+2
:= ((33/3 + 3)3 ) − (3 + 3) × 33
:= 3/3 + ((33 × (3 × 33 − 3)) − (3 + 3))
:= 33 × (3 × 33 − 3)
:= ((44 × (4 × 4 + 4)) + 44)/((4 + 4)/4)
:= 4 + (((44 − 4)/4) × ((4/4 − 4) + 44))
:= 4 × ((4 + 4) × (4 − 4/4)4 )
:= 55 + (((55 + 5)/5) − 555)
:= ((5/5 + 5)5 /(5 − (5 + 5)/5)) − 5
:= (5/5 + 5)5 /(5 − (5 + 5)/5)
:= (6 − 66)/6 + (6 × (6 × (66 + 6)))
:= 6/6 + ((6 × (6 × (66 + 6))) − 6)
:= 6 × (6 × (66 + 6))
77/7
:= (7/7 + 7) × ((77/7 + 7)(7+7)/7 )
:= 7 + (((7 × (7 × 7 × 7 + 7 + 7)) − 7/7) + 77)
:= 7 × 77 + (((7 + 7)/7)
:= ((8 + 8)/8) × ((8 + 8) × (88 − 8) + (88/8))
:= (88 + 8 + 8)/8 × (888/8 + 88)
:= (88 + 8) × (88/8 + 8 + 8)
:= 9 + (99 × (9 + 9 + 9) − (9/9 + 99))
:= 9 + ((9999/9 + 9 × 9 × (9 + 9)) + 9)
:= (9 + 9) × ((9 + 9) × (9 − 9/9))
182
)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2593 := 1 + ((1 + 1) × ((1 + 1 + 1) × (1 + 11))1+1 )
Ï 2598 := ((1 + (11 + 11)) × (1 + 1 + 111)) − 1
Ï 2603 := (1 + 1)11 + ((1111 − 1)/(1 + 1))
:= 2/2 + (2 × (2 + 2 + 2)2+2 )
:= 2 + (2 × ((2 + 2 + 2)2+2 + 2))
:= 2 + (((2 × (22 + 2) + 2/2) + 2)2 )
:= 3/3 + (33 × (3 × 33 − 3))
:= 3 + ((33 × (3 × 33 − 3)) + 3)
:= 33/3 + (33 × (3 × 33 − 3))
:= 4/4 + (4 × ((4 + 4) × (4 − 4/4)4 ))
:= ((44 + 4) × (44 − 4) − (4 + 4))/4
:= 4 + (((44 + 4) × (44 − 4) − 4)/4)
:= 5 × 5 + (55 − (555 + (5 + 5)/5))
:= 555 + ((((5 + 5)/5)55/5 ) − 5)
:= 555 + (((5 + 5)/5)55/5 )
:= 6/6 + (6 × (6 × (66 + 6)))
:= 6 + (6 × (6 × (66 + 6)))
:= 66/6 + (6 × (6 × (66 + 6)))
:= 7 + ((((7 + 7 + 7)/7) + 7 × (7 × 7 + 7)) + 7)
:= 7/7 + (7 × (7 × (7 × 7 − 7) + 77))
:= 7 + ((7 × (7 × (7 × 7 − 7) + 77)) − 7/7)
:= 8/8 + ((88 + 8) × (88/8 + 8 + 8))
:= 8 + ((8 − 8/8) × ((8888 − 8)/(8 + 8 + 8)))
:= (88/8 + 8) × ((8 × (8 + 8) + 8/8) + 8)
:= 9/9 + ((9 + 9) × ((9 + 9) × (9 − 9/9)))
:= 9 + (((99/9 + 9) × ((999/9 + 9) + 9)) + 9)
:= 99/9 + ((9 + 9) × ((9 + 9) × (9 − 9/9)))
7
Ï 2594 := (1 + 1) × (1 + ((1 + 1 + 1) × (1 + 11))1+1 )
Ï 2604 := (1 + 11) × ((1 + 1) × (111 − 1 − 1) − 1)
Ï 2599 := (1 + (11 + 11)) × (1 + 1 + 111)
:= (22 + 2/2) × (222/2 + 2)
:= 2 × ((((2 + 2 + 2)2+2 + 2) + 2) + 2)
:= 3 + ((33 × (3 × 33 − 3)) − 3/3)
:= 3 + (((33 × (3 × 33 − 3)) + 3/3) + 3)
:= 3333 − 33+3
:= 4 + ((44 − 4)/4 × ((44 − 4/4) + 4))
:= ((44 + 4) × (44 − 4) − 4)/4
:= 4 + ((44 + 4) × (44 − 4)/4)
:= 5 × 5 + (55 − (555 + 5/5))
:= (5 × ((55 − 5)/(5/5 + 5))) − 5/5
:= 55 − ((55 + 5/5)/(5/5 + 5))
:= (6 + 6)/6 + (6 × (6 × (66 + 6)))
:= 6 + ((6 × (6 × (66 + 6))) + 6/6)
:= 6 + ((6 × (6 × (66 + 6))) + 6)
:= (((7 + 7)/7 + 7 × 7)(7+7)/7 ) − 7
:= (7 + 7)/7 + (7 × (7 × (7 × 7 − 7) + 77))
:= 7 + (7 × (7 × (7 × 7 − 7) + 77))
:= (8 + 8)/8 + ((88 + 8) × (88/8 + 8 + 8))
:= 8 + (((88 + 8) × (88/8 + 8 + 8)) − 8/8)
:= ((8 × 8 × (88 − 8)) + 88)/((8 + 8)/8)
:= (9 + 9)/9 + ((9 + 9) × ((9 + 9) × (9 − 9/9)))
:= 9 + (((9 + 9)/9) × (((9 + 9) × (9 × 9 − 9)) − 9/9))
:= ((99 + 9)/9) + ((9 + 9) × ((9 + 9) × (9 − 9/9)))
:= 2 + (2 × (2 + 2 + 2)
2+2
)
Ï 2595 := 1 + ((1 + 1) × (1 + ((1 + 1 + 1) × (1 + 11))1+1 ))
Ï 2600 := 1 + ((1 + (11 + 11)) × (1 + 1 + 111))
:= 2 × (((2 + 2 + 2)
:= 2 + ((2 × (2 + 2 + 2)2+2 ) + 2/2)
2+2
Ï 2605 := 1 + ((1 + 11) × ((1 + 1) × (111 − 1 − 1) − 1))
:= 2 + ((((2 × (22 + 2) + 2/2) + 2)2 ) + 2)
+ 2) + 2)
:= (33 − 3/3) × (3 × 33 + 3/3)
:= 3/3 + (3333 − 33+3 )
:= (44 + 4) × (44 − 4)/4
:= 4 + (((44 + 4) × (44 − 4) + 4)/4)
:= 5 × 5 + (55 − 555)
:= 5 × ((55 − 5)/(5/5 + 5))
:= 5 + (5 × ((55 − 5)/(5/5 + 5)))
:= (6 × 6/(6 + 6)) + (6 × (6 × (66 + 6)))
:= 6 + ((6 × (6 × (66 + 6))) + ((6 + 6)/6))
:= 6 + (((6 × (6 × (66 + 6))) + 6/6) + 6)
:= (7 × (7 × (7 × 7 − 7) + 77)) − (7 + 7)/7
:= (7/7 + 7) × (7 × 7 × 7 − (77/7 + 7))
:= 7 + ((7 × (7 × (7 × 7 − 7) + 77)) + 7/7)
:= 88/8 + ((8/8 + 8 + 8) × (8 × 8 + 88))
:= 8 + ((88 + 8) × (88/8 + 8 + 8))
:= ((8 − 88/8) + 8) × ((8 × 8 × 8 + 8/8) + 8)
:= ((9 + 9 + 9)/9) + ((9 + 9) × ((9 + 9) × (9 − 9/9)))
:= (9 − 9/9) × ((9 + 9) × (9 + 9) + 9/9)
:= ((9 × 9 + 9)/(9 + 9)) × ((((9 + 9)/9)9 ) + 9)
:= 3 + (33 × (3 × 33 − 3))
:= (44 × (((44 − 4)/4) − 4)) − 4/4
Ï 2596 := (1 + 1) × (11 × (111+1 − (1 + 1 + 1)))
Ï 2601 := (1 + ((11 − 1)1+1 /(1 + 1)))1+1
Ï 2606 := (11 × (11 + ((1 + 1) × (1 + 1 + 111)))) − 1
2
:= 2 + (2 × ((((2 + 2 + 2)2+2 + 2) + 2) + 2))
:= 2 × ((2 + 2 + 2)2+2 + 2)
:= ((2 × (22 + 2) + 2/2) + 2)
:= 3 + ((33 × (3 × 33 − 3)) + 3/3)
:= 3 × ((3 × 3 × (3 × 33 − 3)) + 3)
:= 3 + ((33 × (3 × 33 − 3)) + 33/3)
:= ((44 + 4) × (44 − 4) + 4)/4
:= 4 + ((((44 + 4) × (44 − 4) + 4) + 4)/4)
:= ((5/5 − 5) + 55)(5+5)/5
:= 5 + (((5/5 − 5) + 55)(5+5)/5 )
:= 6 + ((6 × (6 × (66 + 6))) + (6 × 6/(6 + 6)))
:= 6 + (((6 × (6 × (66 + 6))) + ((6 + 6)/6)) + 6)
:= 44 × (((44 − 4)/4) − 4)
:= 5 × 5 + ((55 − 555) + 5/5)
:= 6 + ((6 × (6 × (66 + 6))) − ((6 + 6)/6))
(7+7)/7
:= 77 + (7 × 7 × 7 × 7 + ((7 + 7)/7)7 )
:= (7 × (7 × (7 × 7 − 7) + 77)) − 7/7
:= ((7 + 7)/7 + 7 × 7)
:= ((8 × (8 × (88 − 8) + 8)) + 8)/((8 + 8)/8)
:= (8/8 + 8) × ((8/8 + 8 + 8)(8+8)/8 )
:= 88 + ((8888 − 8)/8 + 88 × (8 + 8))
:= 99/9 × ((9 × (9 + 9 + 9) − 9) + ((9 + 9)/9))
:= 9 + ((9 + 9) × ((9 + 9) × (9 − 9/9)))
:= 9/9 + (((9 × 9 + 9)/(9 + 9)) × ((((9 + 9)/9)9 ) + 9))
Ï 2597 := 1 + ((1 + 1) × (11 × (111+1 − (1 + 1 + 1))))
Ï 2602 := 1 + ((1 + ((11 − 1)1+1 /(1 + 1)))1+1 )
2
2
Ï 2607 := 11 × (11 + ((1 + 1) × (1 + 1 + 111)))
:= 2 + ((2 × 22 + 2) + 22 )
:= 22/2 + (2 × ((2 + 2 + 2)2+2 + 2))
:= 3 + (((33 × (3 × 33 − 3)) − 3/3) + 3)
:= 33 + ((33 × (3 × 33 − 3)) + 3/3)
:= 3 + (3333 − 33+3 )
:= 4/4 + (44 × (((44 − 4)/4) − 4))
:= (((44 + 4) × (44 − 4) + 4) + 4)/4
:= 4 + ((((44 + 4) × (44 − 4) − 4)/4) + 4)
:= 5 + ((5/5 + 5)5 /(5 − (5 + 5)/5))
:= 55 + (((5 + 5)/5)5 − 555)
:= 5 + ((((5 + 5)/5)5 − 555) + 55 )
:= 6 + ((6 × (6 × (66 + 6))) − 6/6)
:= ((66 − 6)/6) + (6 × (6 × (66 + 6)))
:= 2/2 + (2 × ((2 + 2 + 2)
2+2
+ 2))
(7+7)/7
:= 77 + (((7 + 7 + 7)/7)7 + 7 × 7 × 7)
:= 7 × (7 × (7 × 7 − 7) + 77)
:= 7/7 + (((7 + 7)/7 + 7 × 7)
:= (8 − 8/8) × ((8 − 8/8) × (8 × 8 − 88/8))
:= ((88 − 8/8) × ((88 + 88)/8 + 8)) − 8
:= 88 + (8888/8 + 88 × (8 + 8))
:= 9 + (((9 + 9)/9) × (((9 + 9) × (9 × 9 − 9)) − ((9 + 9)/9)))
:= 9 + (((9 + 9) × ((9 + 9) × (9 − 9/9))) + 9/9)
:= (9 × 9 − ((9 + 9)/9)) × (99/((9 + 9 + 9)/9))
183
)
:= 6 + (((6 × (6 × (66 + 6))) + (6 × 6/(6 + 6))) + 6)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2608 := 1 + (11 × (11 + ((1 + 1) × (1 + 1 + 111))))
Ï 2613 := (1 + 1 + 11) × (1 + ((1 + 1) × (11 − 1)1+1 ))
Ï 2618 := (1 + 1) × 11 × (111+1 − 1 − 1)
:= 2 × ((2 + 2 + 2)2+2 + 2 × (2 + 2))
:= 22 + ((2 × (2 + 2 + 2)2+2 ) − 2/2)
:= 22 × ((22/2)2 − 2)
:= (3 − 3/3) × ((33/3)3 − 33 )
:= 3 + ((33 + 3) × (3 × 33 + 3 + 3))
:= 33 + ((33 × (3 × 33 − 3)) − 3/3)
:= 4 × (((4 + 4) × (4 − 4/4)4 ) + 4)
:= 44 + ((((4 − 4/4) + 4)4 ) − 44)
:= 44/4 × (44 − ((4 + 4)/4 + 4 × 4))
:= 5 + ((((5 + 5)/5)55/5 ) + 555)
:= 55 − (((5 + 5)/5)5−5/5+5 )
:= 55/5 × (((5 − (5 + 5)/5)5 ) − 5)
:= 6 + ((6 × (6 × (66 + 6))) + ((66 − 6)/6))
:= (66 + 6/6) × ((66 × 6/(6 + 6)) + 6)
:= (6 − 66)/6 + (6 × ((6 × (66 + 6)) + 6))
:= (7 × (7 × 77 − 7)) − 7777/7
:= 77 × (7 × 7 − (7/7 + 7 + 7))
:= (8 + 8) × ((88/8 + 88) + 8 × 8)
:= 8 × 8 × 8 + ((88 × (8 + 8 + 8)) − (88/8))
:= (888/8 + 8) × (88 + 88)/8
:= (9 − 9/9) × ((9 + 9) × (9 + 9) + ((9 + 9)/9))
:= ((9 + 9 + 9)/9) × (9 × 99 − (99/9 + 9))
:= 999 + ((9 × (99 + 9 × 9)) − 9/9)
:= 7 + (((7 + 7)/7 + 7 × 7)
(7+7)/7
)
Ï 2609 := (1 + 1)11 + ((11 + 1111)/(1 + 1))
Ï 2614 := (1 + 1) × (((1 + 11) × (111 − 1 − 1)) − 1)
:= 2 + ((2 × ((2 + 2 + 2)2+2 + 2)) + 22/2)
:= 22 + (2 × (2 + 2 + 2)2+2 )
:= (3 × (33 × 33 )) − ((3/3 + 3)3 )
:= (3 − 3/3) × (((33/3)3 − 33 ) + 3)
:= 4 + ((((44 + 4) × (44 − 4) + 4)/4) + 4)
Ï 2619 := 1 + ((1 + 1) × 11 × (111+1 − 1 − 1))
:= 2/2 + (22 × ((22/2)2 − 2))
4
:= 4 + ((44 − 4)/4 × ((4/4 + 4 ) + 4))
5
:= 5 + (55 − ((55 + 5/5)/(5/5 + 5)))
:= 55 + (5 − (555 + (55/5)))
:= 6 + ((6 × (6 × (66 + 6))) + (66/6))
:= ((6 + 6)/6) × (6 × 6 × 6 × 6 + (66/6))
7
:= 3 × (3 × (3 × (3 × 33 − 3) + 3))
:= (4 × (4 × (4 × (44 − 4) + 4))) − (4/4 + 4)
:= 55 + (55 − ((555 + 5/5) + 5))
:= 6 + ((66 + 6/6) × ((66 × 6/(6 + 6)) + 6))
:= 7 × 7 × (7 × 7 + 7) − (((7 + 7)/7)7 + 7)
:= 7 + ((((7 + 7 + 7)/7) + 7 × 7 × 7) + 77)
:= 7/7 + (77 × (7 × 7 − (7/7 + 7 + 7)))
:= 8 + ((8/8 + 8) × ((8/8 + 8 + 8)(8+8)/8 ))
:= 8 × 8 + ((((8 + 8)/8) + 8) × ((8 + 8) × (8 + 8) − 8/8))
:= (88/8 + 8 + 8) × ((8/8 + 88) + 8)
:= 9 + ((9 − 9/9) × ((9 + 9) × (9 + 9) + 9/9))
:= 9 + (((9 × 9 + 9)/(9 + 9)) × ((((9 + 9)/9)9 ) + 9))
:= 999 + (9 × (99 + 9 × 9))
Ï 2620 := (1 + 1) × (1 + 11 × (111+1 − 1 − 1))
Ï 2610 := 11 + ((1 + (11 + 11)) × (1 + 1 + 111))
Ï 2615 := ((1 + 1) × ((1 + 11) × (111 − 1 − 1))) − 1
:= 22 + (2 × ((2 + 2 + 2)2+2 − 2))
:= 22 + ((2 × (2 + 2 + 2)2+2 ) + 2/2)
:= 2 + (22 × ((22/2)2 − 2))
:= (33 + 3) × (3 × 33 + 3 + 3)
:= 33/3 + (3333 − 33+3 )
:= 33 + ((33 × (3 × 33 − 3)) + 3/3)
:= (44 − 4)/4 × ((4/4 + 44 ) + 4)
:= 4 + ((4 × (4/4 + 4)4 ) + 444/4)
:= (4 × (4 × (4 × (44 − 4) + 4))) − 4
:= 5 + ((5 × ((55 − 5)/(5/5 + 5))) + 5)
:= 5 × (555 − ((5 + 5)/5)5 )
:= 55 + (55 − (555 + 5))
:= 6 + (((6 × (6 × (66 + 6))) + 6) + 6)
:= 6 + (((6 × (6 × (66 + 6))) + (66/6)) + 6)
:= ((6 + 6)/6)6 + (6 × ((6 × (66 + 6)) − 6))
:= 777/7 + (7 × (7 × 7 × 7 + 7 + 7))
:= 7 + ((((7 + 7)/7 + 7 × 7)(7+7)/7 ) + 7)
:= 7 + ((7 × (7 × 77 − 7)) − 7777/7)
:= (88 − 8/8) × ((88 + 88)/8 + 8)
:= 888 + (8 × (8 × (8 + 8) + 88) − 8/8)
:= 8 × 8 + (((8 × 8 × (88 − 8)) − 8)/((8 + 8)/8))
:= (9/9 + 9) × (9 × (9 + 9) + 99)
:= (9 × (9 × 9 − (9 + 9))) + (((9 + 9)/9)99/9 )
:= 9/9 + ((9 × (99 + 9 × 9)) + 999)
Ï 2611 := 111 + (((11 − 1)1+1 /(1 + 1))1+1 )
Ï 2616 := (1 + 1) × ((1 + 11) × (111 − 1 − 1))
Ï 2621 := 1 + (1 + 1) × (1 + 11 × (111+1 − 1 − 1))
:= 222/2 + (2 × (22 + 2) + 2)2
:= (22 + 2) × (222/2 − 2)
:= 2 + ((22 × ((22/2)2 − 2)) + 2/2)
:= 3 + ((3 − 3/3) × ((33/3)3 − 33 ))
:= 33 + ((33 × (3 × 33 − 3)) − 3)
:= ((33 − 3/3) × (3 × 33 + 3/3)) − 3
:= 444/4 + (4 × (4/4 + 4)4 )
:= 4 × 4 + ((44 + 4) × (44 − 4)/4)
:= 44 + ((((4 − 4/4) + 4)4 ) + 4 × 44)
:= 55 + ((555 − 55 )/5)
:= 5 + (((555 − 55 )/5) + 55 )
:= 5 + ((((555 − 55 )/5) + 55 ) + 5)
:= 6 + ((((6 × (6 × (66 + 6))) + 6/6) + 6) + 6)
:= (6 + 6) × (6 × 6 × 6 + (6 + 6)/6)
:= (6 × ((6 × (66 + 6)) + 6)) − 6/6 − 6
:= 7 + ((7 × (7 × (7 × 7 − 7) + 77)) + 7)
:= 7 × 7 × (7 × 7 + 7) − ((7 + 7)/7)7
:= 777 + (((7 + 7 + 7)/7)7 − 7 × 7 × 7)
:= 8 + ((88/8 + 8) × ((8 × (8 + 8) + 8/8) + 8))
:= 888 + 8 × (8 × (8 + 8) + 88)
:= 8 × 8 × 8 + ((88/8 + 8) × 888/8)
:= 9/9 + ((9/9 + 9) × (9 × (9 + 9) + 99))
:= 9 × 9 × 9 + (((9 − 9/9) + 9) × 999/9)
:= (9 + 9)/9 + ((9 × (99 + 9 × 9)) + 999)
Ï 2612 := 11 + ((1 + ((11 − 1)1+1 /(1 + 1)))1+1 )
2+2
Ï 2617 := 1 + ((1 + 1) × ((1 + 11) × (111 − 1 − 1)))
2
Ï 2622 := (1 + (11 + 11)) × (1 + 1 + 1 + 111)
:= (22 × ((22/2) − 2)) − 2/2
:= (22 − 2)2 + 2222
:= 3 + ((3 × (33 × 33 )) − ((3/3 + 3)3 ))
:= (3 + 3)3 + (((3/3 + 3) + 3)3/3+3 )
:= 3 + ((33 × (3 × 33 − 3)) + 33 )
:= 4 + (4 × (((4 + 4) × (4 − 4/4)4 ) + 4))
:= 4 + (((((4 − 4/4) + 4)4 ) − 44) + 44 )
:= (4 × (4 × (4 × (44 − 4) + 4))) − (4 + 4)/4
:= 55 + (((555 − 55 ) + 5)/5)
:= 5 + ((((555 − 55 ) + 5)/5) + 55 )
:= 55 + (((555 − 55 ) + 55)/5)
:= 6 + ((((6 × (6 × (66 + 6))) + ((6 + 6)/6)) + 6) + 6)
:= (6 × ((6 × (66 + 6)) + 6)) − 66/6
:= (6 × ((6 × (66 + 6)) + 6)) − 6
:= 7 + (((7 × (7 × (7 × 7 − 7) + 77)) + 7/7) + 7)
:= (7 × (7 × (7 × 7 + 7) − 7)) − 7/7 − 77
:= (7 × 7 − 77/7) × (77 − (7/7 + 7))
:= 8 + (((8 × 8 × (88 − 8)) + 88)/((8 + 8)/8))
:= 8/8 + (8 × (8 × (8 + 8) + 88) + 888)
:= 8 × 8 × 8 + ((88 × (8 + 8 + 8)) − ((8 + 8)/8))
:= 9 + ((9 − 9/9) × ((9 + 9) × (9 + 9) + ((9 + 9)/9)))
:= 9 × 9 × 9 + ((9 + 9) × 99 + 999/9)
:= 22 + ((2 × (2 + 2 + 2)
) − 2)
:= 9 + (((9 + 9) × ((9 + 9) × (9 − 9/9))) + (99/9))
184
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2623 := 1 + ((1 + (11 + 11)) × (1 + 1 + 1 + 111))
:= 22 + (((2 × (22 + 2) + 2/2) + 2)2 )
Ï 2628 := (1 + 11) × (((1 + 1) × (111 − 1)) − 1)
:= (2 + 2 + 2) × ((2 × (222 − 2)) − 2)
3
Ï 2633 := (1 + 1 + 1 + 11)1+1+1 − 111
:= 2 + ((22/2 × ((222 − 2)/2 − 2)) + 2)
:= ((33/3 + 3)3 ) − 333/3
:= ((3 − 3/3) × ((33/3)3 − 3)) − 33
:= (3 × 3 + 3) × ((3 + 3) + 3)
:= (44 − 4/4) × ((44 + 4)/4 − 4)
:= 4 + (4 × (4 × (4 × (44 − 4) + 4)))
5
:= 4 + (44/4 × (44 − (4 × 4 + 4/4)))
5
:= 5 + (55/5 × (((5 − (5 + 5)/5)5 ) − 5))
:= (5 − 5/5) × (((5 + 5)/5) + 5 /5)
:= 5 + ((5 − 5/5) × (((5 + 5)/5)5 + 55 /5))
:= 6/6 + ((6 × ((6 × (66 + 6)) + 6)) − 6)
:= 6 × ((6 × (66 + 6)) + 6)
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) − 6/6)
:= 7 + (7 × 7 × (7 × 7 + 7) − ((7 + 7)/7) )
:= (77/7 + 7) × (7 × (7 + 7 + 7) − 7/7)
:= 7 × 7 × (7 × 7 + 7) − 777/7
:= 8 × 8 × 8 + ((88 × (8 + 8 + 8)) − 8/8)
:= 8 × 8 + (((8 × 8 × (88 − 8)) + 8)/((8 + 8)/8))
:= 8 + (((88 × (8 + 8 + 8)) + 8 × 8 × 8) + 8/8)
:= 99 × (9 + 9 + 9) − ((9 × 99 + 9)/(9 + 9))
:= 9 + ((9 × (99 + 9 × 9)) + 999)
:= 99 × (9 + 9 + 9) + ((9 − 9 × 9 × 9)/(9 + 9))
7
Ï 2624 := (1 + 1)11 + (((1 + 1) × (1 + 11))1+1 )
Ï 2629 := 11 × (((1 + 1) × (111+1 − 1)) − 1)
Ï 2634 := 1 + ((1 + 1 + 1 + 11)1+1+1 − 111)
:= 222/2 + ((22 + 2)2 )
:= 22/2 × ((222 − 2)/2 − 2)
:= (2 + 2 + 2) × (((22 − 2/2)2 ) − 2)
:= (33 − 3/3) × (3 × 33 + 3/3)
:= 3/3 + ((3 × 3 + 3) × ((3 + 3)3 + 3))
:= 3 + (((3 × 3 + 3) × ((3 + 3)3 + 3)) + 3)
4
:= 44/4 × (4 − (4 × 4 + 4/4))
:= 4 × (4 × (4 × (44 − 4) + 4))
:= ((4 + 4)/4 + 4) × (444 − (4/4 + 4))
5
:= 55 + (55 − (555 + 5/5))
:= 5 + ((55 − (555 + 5/5)) + 5 )
:= 55 + (((55 − 5555)/5) − 5)
:= ((6 + 6)/6)6 × (6 × 6 − 6/6 + 6)
:= 6/6 + (6 × ((6 × (66 + 6)) + 6))
:= 6 + (6 × ((6 × (66 + 6)) + 6))
:= (7/7 + 7) × (7 × 7 × 7 − (7/7 + 7 + 7))
:= 7 + ((7 × 7 − 77/7) × (77 − (7/7 + 7)))
:= ((7 − 777)/7) + 7 × 7 × (7 × 7 + 7)
:= 8 × (((8 + 8) × (8 + 8) + 8 × 8) + 8)
:= 88/8 × (888/8 + 8 × (8 + 8))
:= (88/8 − 8) × ((8 − 88)/8 + 888)
:= (9/9 + 9 × 9) × (((99 + 99 + 9)/9) + 9)
:= 9 + (((9 × (99 + 9 × 9)) + 999) + 9/9)
:= (9 + 9) × (9 × (9 + 9) − 9) − (999/9 + 9)
Ï 2625 := 1 + ((1 + 1)11 + (((1 + 1) × (1 + 11))1+1 ))
Ï 2630 := 1 + (11 × (((1 + 1) × (111+1 − 1)) − 1))
Ï 2635 := ((1 + 1) × (111+1+1 − (1 + 1 + 11))) − 1
:= 2/2 + (222/2 + ((22 + 2)2 ))
:= 2 + ((2 + 2 + 2) × ((2 × (222 − 2)) − 2))
:= (2/2 + 2 + 2) × ((22 + 2/2)2 − 2)
:= 33 + (33 × (3 × 33 − 3))
:= 3 + (((3 + 3)3/3+3 ) + (33/3)3 )
:= ((3 − 3/3) × (33/3)3 ) − 33
:= 4/4 + (4 × (4 × (4 × (44 − 4) + 4)))
:= (4 − 44)/4 + (44 × (4 × 4 + 44))
:= (44 × (4 × 4 + 44)) − (4/4 + 4)
:= 5 × (5 × (5 × (5 × 5 − 5) + 5))
:= 5 + ((55 − 555) + 55)
:= 5 + (((55 − 555) + 55) + 5)
:= (6 × 6 − 6/6) × (666/6 − 6 × 6)
:= (6 + 6)/6 + (6 × ((6 × (66 + 6)) + 6))
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) + 6/6)
:= 7 + (77 × (7 × 7 − (7/7 + 7 + 7)))
:= 7 + ((7 × 7 × (7 × 7 + 7) − ((7 + 7)/7)7 ) + 7)
:= (7 × (7 × (7 × 7 + 7) − (7 + 7))) − 77/7
:= 8/8 + ((88 × (8 + 8 + 8)) + 8 × 8 × 8)
:= (((8 + 8)/8) + 8) × (((8 + 8) × (8 + 8) − 8/8) + 8)
:= 8 × 8 × 8 + ((88 × (8 + 8 + 8)) + (88/8))
:= 9 + ((((9 − 9/9) + 9) × 999/9) + 9 × 9 × 9)
:= 99/9 + ((9 × (99 + 9 × 9)) + 999)
:= (9 + 9 + 9) × (99 − 9/9) − 99/9
Ï 2626 := (1 + 1) × (1 + 1 + 11) × (1 + (11 − 1)1+1 )
Ï 2631 := 1 + (1 + (11 × (((1 + 1) × (111+1 − 1)) − 1)))
:= 2 + (22/2 × ((222 − 2)/2 − 2))
:= 2 + (222/2 + ((22 + 2)2 ))
Ï 2636 := (1 + 1) × (111+1+1 − (1 + 1 + 11))
:= 2 × ((2 + 2 + 2)2+2 + 22)
3
:= 3/3 + ((33 × (3 × 33 − 3)) + 33)
:= 3 + ((3 × 3 + 3) × ((3 + 3) + 3))
:= ((33/3 + 3)3 ) − (3 × (33 + 3))
:= (4 + 4)/4 + (4 × (4 × (4 × (44 − 4) + 4)))
:= 444 + ((4 − 4/4)4+4−4/4 )
:= (44 × (4 × 4 + 44)) − 4
:= 55 + ((5 − 555) + 5/5)
:= 5 + (((5/5 − 555) + 55) + 55 )
:= 5 × 5 + (((555 − 55 )/5) + 55 )
:= (6 × ((6 × (66 + 6)) + 6)) − (6 + 6)/6
:= 666/6 + ((6 + 6) × (6 × 6 × 6 − 6))
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) + ((6 + 6)/6))
:= 7 × 7 × (7 × 7 + 7) − (777/7 + 7)
:= ((7 + 7) × (777/7 + 77)) − 7/7
:= 7 × (77 + 7) + (((7 + 7)/7)77/7 )
:= 8 + ((888/8 + 8) × (88 + 88)/8)
:= (88/8 − 8) × (888 − 88/8)
:= ((8 × (8 × (88 − 8) + 8)) + 88)/((8 + 8)/8)
:= (((9 − 9/9) + 9) + 9) × ((9 + 9)/9 + 99)
:= (((9 + 9 + 9)/9) × (9 × 99 − (99/9))) − 9
:= (9 + 9 + 9) × (99 − 9/9) − 9/9 − 9
5
Ï 2627 := ((1 + 11) × (((1 + 1) × (111 − 1)) − 1)) − 1
Ï 2632 := (1 + 1 + 1 + 11)1+1+1 − (1 + 111)
Ï 2637 := ((1 + 1) × (111+1+1 − (1 + 11))) − 1
:= (22/2 × ((222 − 2)/2 − 2)) − 2
:= 2 × (((2 + 2 + 2)2+2 − 2) + 22)
:= 2/2 + (2 × ((2 + 2 + 2)2+2 + 22))
:= (33/3)3 + ((3 + 3)3/3+3 )
:= 3 + (((3 × 3 + 3) × ((3 + 3)3 + 3)) + 3/3)
:= 3 × ((33 × 33 ) − (3 × 3 + 3))
:= (44 × (4 × 4 + 44)) − 4 − 4
:= 4/4 + (44 × (4 × 4 + 44) − 4)
4
:= 4 + ((44 − 4/4) × ((4 + 4)/4 − 4))
5
5
:= 55 + (((5 + 5)/5 − 555) + 5 )
:= 5 + ((((5 + 5)/5 − 555) + 55) + 5 )
:= 5 × 5 + ((((555 − 55 ) + 5)/5) + 55 )
:= (6 × ((6 × (66 + 6)) + 6)) − 6/6
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) − ((6 + 6)/6))
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) + (6 × 6/(6 + 6)))
:= (7 + 7) × (777/7 + 77)
:= ((7 + 7)/7 + 7) × (7 × (7 × 7 − 7) − 7/7)
:= 8 + ((88/8 + 8 + 8) × ((8/8 + 88) + 8))
:= 8 + ((88 × (8 + 8 + 8)) + 8 × 8 × 8)
:= (88/8 − 8) × (888 − 8/8 − 8)
:= 9 + (((9 × (99 + 9 × 9)) − 9/9) + 999)
:= 99 × (9 + 9 + 9) − ((9 × 9 × 9 + 9)/(9 + 9))
:= (9 + 9 + 9) × (99 − 9/9) − 9
7
:= ((7 + 7)/7) + (7 × (7 × 7 × 7 + 7 + 7))
185
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2638 := (1 + 1) × (111+1+1 − (1 + 11))
Ï 2643 := 1 + (1 + 1) × (1 + 11 × (111+1 − 1))
Ï 2648 := (1 + 1) × (111+1+1 − 1) − 1 − 11
:= 2 + (2 × ((2 + 2 + 2)2+2 + 22))
:= 2 + (((22 × (222 /2 − 2)) + 2)/2)
:= 2 + ((2 + 2 + 2) × ((22 − 2/2)2 ))
:= 3 + (((3 − 3/3) × (33/3)3 ) − 33 )
:= 3 + (33 × (3 × 33 − 3/3))
:= 3 + (((33/3 + 3)3 ) − 3 × 33)
:= (44 × (4 × 4 + 44)) − (4 + 4)/4
:= 4 + (44 × (4 × 4 + 44) − 4/4)
:= 4 + (44 × (4 × 4 + 44) + 4)
:= 55 + ((55 − (5555 + 5))/5)
:= (5 × (555 − 5 × 5)) − ((5 + 5)/5 + 5)
:= (5 × (555 − 5 × 5)) − (5 + 5)/5
:= ((66 − 6)/6) + (6 × ((6 × (66 + 6)) + 6))
:= ((6 × (666 + 6 × 6 × 6)) − 6)/((6 + 6)/6)
:= 66 + ((6 × (6 × (66 + 6))) + ((6 − 66)/6))
77/7
:= (7 + 7)/7 + (7 × (7 × (7 × 7 + 7) − (7 + 7)))
:= (7 × (7 × (7 × 7 + 7) − (7 + 7))) − (7/7 + 7)
:= 7 + ((((7 + 7)/7)
:= ((8 + 8)/8) × (88 × (8 + 8) − (8/8 + 88))
:= (88/8 − 8) × ((888 − 8) + 8/8)
:= 8 + (88 × ((88 + 88)/8 + 8))
:= 9/9 + ((9 + 9 + 9) × (99 − 9/9) − 9)
:= ((9 + 9 + 9)/9) × (9 × 99 − (9/9 + 9))
:= (9 + 9)/9 + (9 + 9 + 9) × (99 − 9/9)
Ï 2639 := ((1 + 1) × 11 × (111+1 − 1)) − 1
) + 7 × (77 + 7))
Ï 2644 := (1 + 1) × (1 + 1 + 11 × (111+1 − 1))
Ï 2649 := (1 + 1) × (111+1+1 − 1) − 11
:= ((22 × (222 /2 − 2)) − 2)/2
:= 2 × (((2 + 2 + 2) × (222 − 2)) + 2)
:= (22/2 × (222 − 2)/2) − 2
:= (3 × (33 × 33 )) − 3/3 − 33
:= (3 − 3/3) × ((33/3)3 − 3 × 3)
:= 3 + (3 × (3 × (3 × 3 × 33 − 3)))
:= (44 × (4 × 4 + 44)) − 4/4
:= 4 + (44 × (4 × 4 + 44))
:= 44 + ((((4 − 4/4) + 4)4 ) − (4 + 4))
:= 55 + ((55 − 5555)/5)
:= (5 × (555 − 5 × 5)) − (5/5 + 5)
:= (5 × (555 − 5 × 5)) − 5/5
:= 66/6 + (6 × ((6 × (66 + 6)) + 6))
:= (6 − ((6 + 6)/6)) × ((666 − 6) + 6/6)
:= (66 × (6 × 6 + 6)) − ((666/6 + 6) + 6)
:= (7 × (7 × (7 × 7 + 7) − (7 + 7))) − 7
:= (7 × (7 × (7 × 7 + 7) − (7 + 7))) − (7 + 7)/7
:= 7 × 77 + (((7 + 7 + 7)/7)7 − 77)
:= (88 × ((88 + 88)/8 + 8)) − 8/8
:= 88 + (((8 × 8 × (88 − 8)) − 8)/((8 + 8)/8))
:= 8 + ((88 × ((88 + 88)/8 + 8)) + 8/8)
:= (99/9 + 9 + 9) × ((9/9 + 9 × 9) + 9)
:= 99 × (9 + 9 + 9) − (99/9 + 9 + 9)
:= ((9 + 9 + 9)/9) × ((9 × 99 − 9) + 9/9)
Ï 2640 := (1 + 1) × 11 × (111+1 − 1)
Ï 2645 := 1 + (1 + 1) × (1 + 1 + 11 × (111+1 − 1))
Ï 2650 := (1 + 1) × 111+1+1 − 1 − 11
:= 2 × ((2 + 2 + 2) × (222 − 2))
:= (2/2 + 2 + 2) × ((22 + 2/2)2 )
:= ((22 × (222 − 2)/2) − 2)/2
:= 33 × (3 × 33 − 3/3)
:= ((33/3 + 3)3 ) − 3 × 33
:= (3 − 3/3) × ((33/3)3 − (3 + 3))
:= 44 × (4 × 4 + 44)
:= 4 + (44 × (4 × 4 + 44) + 4/4)
:= 4 + (((44 − 4)/4) × (44 − (4 + 4)/4))
:= (5 + 5) × (5 × 55 − (55/5))
:= (5 × (555 − 5 × 5)) − 5
:= 5 × (555 − 5 × 5)
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) + 6)
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) + (66/6))
:= ((6 + 6)/6)6 + ((6 × (6 × (66 + 6))) − 6)
:= (7 × 7 − 7/7) × ((7 × 7 − 7/7) + 7)
:= (7 × (7 × (7 × 7 + 7) − (7 + 7))) − 7/7
:= 7 × 7 + (((7 + 7)/7 + 7 × 7)(7+7)/7 )
:= 88 × ((88 + 88)/8 + 8)
:= 8 + ((88/8 − 8) × (888 − 8/8 − 8))
:= (((8 + 8)/8) + 8) × (((8 + 8) × (8 + 8) + 8/8) + 8)
:= ((9 + 9 + 9)/9) × (9 × 99 − (99/9))
:= (9 + 9 + 9) × (99 − 9/9) − 9/9
:= (9/9 + 9) × ((((9 + 9)/9)9−9/9 ) + 9)
Ï 2641 := 1 + ((1 + 1) × 11 × (111+1 − 1))
:= ((22 × (222 /2 − 2)) + 2)/2
3
Ï 2646 := (1 + 1) × (1 + 1 + 1 + 11 × (111+1 − 1))
2
Ï 2651 := 11 × (((1 + 1) × 111+1 ) − 1)
:= (2 + 2 + 2) × ((22 − 2/2) )
:= 22/2 × (222 − 2)/2
:= 33/3 × (((33+3 + 3)/3) − 3)
:= 3/3 + (33 × (3 × 3 − 3/3))
:= 3 × (3 × (3 × 3 × 33 − 3))
:= 4/4 + (44 × (4 × 4 + 44))
:= ((44 − 4)/4) × (44 − (4 + 4)/4)
:= 55 + (55/5 × (55/5 − 55))
:= 5/5 + ((5 × (555 − 5 × 5)) − 5)
:= 5/5 + (5 × (555 − 5 × 5))
:= 6 + (((6 × ((6 × (66 + 6)) + 6)) + 6/6) + 6)
:= 666 + 66 × (6 × 6 − 6)
:= 66 + ((6 × (6 × (66 + 6))) − (6/6 + 6))
:= 7 + (((7 − 777)/7) + 7 × 7 × (7 × 7 + 7))
:= 7 × (7 × (7 × 7 + 7) − (7 + 7))
:= 8/8 + (88 × ((88 + 88)/8 + 8))
:= ((8 + 8)/8) × (((88/8)88/8−8 ) − 8)
:= 8 + ((88/8 − 8) × ((888 − 8) + 8/8))
:= 9 × 9 + ((9/9 + 9) × (((9 + 9)/9)9−9/9 ))
:= (9 + 9 + 9) × (99 − 9/9)
:= 99/9 × (9 × (9 + 9 + 9) − ((9 + 9)/9))
Ï 2642 := (1 + 1) × (1 + 11 × (111+1 − 1))
:= 44/4 + (44 × (4 × 4 + 44))
Ï 2647 := (1 + 1) × (111+1+1 − 1 − 1) − 11
2
:= 7 + ((7 × (7 × (7 × 7 + 7) − (7 + 7))) − ((7 + 7)/7))
Ï 2652 := (1 + 11) × ((1 + 1) × 111 − 1)
:= 2 + (2 × ((2 + 2 + 2) × (222 − 2)))
:= 2 + ((2/2 + 2 + 2) × ((22 + 2/2) ))
:= (2 + 2 + 2) × (2 × 222 − 2)
:= ((33/3 + 3)3 ) − (3 × 33 + 3)
:= 3/3 + (3 × (3 × (3 × 3 × 33 − 3)))
:= (33 − 3/3) × (3 × 33 + 3)
:= (4 + 4)/4 + (44 × (4 × 4 + 44))
:= 4 + ((44 × (4 × 4 + 44) − 4/4) + 4)
:= 4 + ((44 × (4 × 4 + 44) + 4) + 4)
:= 55 + (((5/5 + 5) /(5 − (5 + 5)/5)) − 5)
:= 55 + ((5/5 + 5)5 /(5 − (5 + 5)/5))
:= (5 + 5)/5 + (5 × (555 − 5 × 5))
:= 6 + (((6 × ((6 × (66 + 6)) + 6)) + ((6 + 6)/6)) + 6)
:= 66 + ((6 × (6 × (66 + 6))) − (66/6))
:= 66 + ((6 × (6 × (66 + 6))) − 6)
5
:= 7 + ((7 × (7 × (7 × 7 + 7) − (7 + 7))) − (77/7))
:= 7/7 + (7 × (7 × (7 × 7 + 7) − (7 + 7)))
:= 7 + ((7 × (7 × (7 × 7 + 7) − (7 + 7))) − 7/7)
:= (8 + 8)/8 + (88 × ((88 + 88)/8 + 8))
:= 8888/8 + 8 × 8 × (8 + 8 + 8)
:= 88 + (((8 × 8 × (88 − 8)) + 8)/((8 + 8)/8))
:= 99 × (9 + 9 + 9) − (((99 + 99)/9) + 9)
:= 9/9 + (9 + 9 + 9) × (99 − 9/9)
:= (((9 − 9/9) + 9) + 9) × (999/9 − 9)
186
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2653 := 1 + ((1 + 11) × ((1 + 1) × 111 − 1))
Ï 2658 := (1 + 1) × (111+1+1 − 1 − 1)
Ï 2663 := 1 + ((1 + 1) × 111+1+1 )
:= 2 + (22/2 × (222 − 2)/2)
:= 2 × ((22/2)2/2+2 − 2)
:= 2/2 + (22 × (22/2)2 )
:= (3 × ((33 × 33 ) − 3)) − 33/3
:= 3 + (3 × ((33 × 33 ) − (3 + 3)))
:= ((33/3 + 3)3 ) − 3 × 33
:= 44 + ((((4 − 4/4) + 4)4 ) − 4)
:= 4/4 + ((((4 − 4/4) + 4)4 ) + 44 )
:= (444 × ((4 + 4)/4 + 4)) − 4/4
:= 5 + ((5 × (555 − 5 × 5)) − ((5 + 5)/5))
:= 5 × 555 − ((555 + 5)/5 + 5)
:= 5 × 555 − (555 + 5)/5
:= ((6 + 6) × (6 × 6 × 6 + 6)) − 66/6
:= 66 + (6 × (6 × (66 + 6)))
:= ((6 + 6) × (6 × 6 × 6 + 6)) − 6/6
:= 7 + (7 × (7 × (7 × 7 + 7) − (7 + 7)))
:= (7 − 7/7) × (7 × (7 × 7 + 7 + 7) + ((7 + 7)/7))
:= 7 + ((7/7 + 7) × (7 × 7 × 7 − (77/7)))
:= ((8 + 8 + 8) × 888/8) − 88/8
:= (88/8 − 8) × (888 − ((8 + 8)/8))
:= ((8 + 8 + 8) × 888/8) − 8/8
:= 99 × (9 + 9 + 9) − (99/9 + 9)
:= 9 + (((9 + 9 + 9)/9) × ((9 × 99 − 9) + 9/9))
:= 99 × (9 + 9 + 9) − 9/9 − 9
Ï 2654 := 1 + (1 + ((1 + 11) × ((1 + 1) × 111 − 1)))
Ï 2659 := (1 + 1) × (111+1+1 − 1) − 1
Ï 2664 := (1 + 1) × (1 + 111+1+1 )
:= 2 + ((2 + 2 + 2) × (2 × 222 − 2))
:= (22 × (22/2)2 ) − 2/2 − 2
:= 2 + (22 × (22/2)2 )
:= (3 × (3 − 33)) + ((33/3 + 3)3 )
:= ((3 − 3/3) × (33/3)3 ) − 3
:= 3 × ((33 × 33 ) − 3)
:= 4/4 + (((((4 − 4/4) + 4)4 ) − 4) + 44 )
:= 4 + ((44/4 + 4) × (4 × 44 + 4/4))
:= 444 × ((4 + 4)/4 + 4)
:= 5 + ((5 × (555 − 5 × 5)) − 5/5)
:= 5 × 555 − (555/5 + 5)
:= (5 × 5 − 5/5) × 555/5
:= (6 − 66)/6 + ((6 + 6) × (6 × 6 × 6 + 6))
:= 66 + ((6 × (6 × (66 + 6))) + 6/6)
:= (6 + 6) × (6 × 6 × 6 + 6)
:= 7 + ((7 × (7 × (7 × 7 + 7) − (7 + 7))) + 7/7)
:= 7 × 7 × (7 × 7 + 7) − (7/7 + 77 + 7)
:= (7/7 + 7) × (((7 − 77)/7) + 7 × 7 × 7)
:= (8 − 88)/8 + ((8 + 8 + 8) × 888/8)
:= ((88/8 − 8) × (888 + 8/8)) − 8
:= (8 + 8 + 8) × 888/8
:= 99 × (9 + 9 + 9) − (9/9 + 9 + 9)
:= 9 + (9999/9 + (9 × (9 × (9 + 9) + 9)))
:= 99 × (9 + 9 + 9) − 9
Ï 2655 := (1 + 1) × (1 + 1 + 111+1+1 ) − 11
:= 2 + ((22/2 × (222 − 2)/2) + 2)
Ï 2660 := (1 + 1) × (111+1+1 − 1)
Ï 2665 := 1 + (1 + 1) × (1 + 111+1+1 )
2
:= (22 × (22/2) ) − 2
:= ((222 × (22 + 2)) + 2)/2
3
:= 3 × ((33 × 3 ) − (3 + 3))
:= (3 − 3/3) × ((33/3) − 3/3)
:= 3 + ((3 − 3/3) × (33/3)3 )
:= (44/4 + 4) × (4 × 44 + 4/4)
:= 4 × (((4/4 + 4)4 − 4) + 44)
:= 4 + (((((4 − 4/4) + 4)4 ) + 44 ) + 4)
:= 5 + (5 × (555 − 5 × 5))
:= 5 + ((5 × (555 − 5 × 5)) + 5)
:= 5 × 555 − (55 + 55)
:= (66 × (6 × 6 + 6)) − (666/6 + 6)
:= (6 − ((6 + 6)/6)) × (666 − 6/6)
:= 6/6 + ((6 + 6) × (6 × 6 × 6 + 6))
:= (7 − ((7 + 7)/7)) × (7 × 77 − (7/7 + 7))
:= (77 − 7) × (7 × 7 − 77/7)
:= (7 − ((7 + 7)/7)) × ((7 × 77 − 7) + 7/7)
:= (8 − 8/8 + 8) × ((88 + 88) + 8/8)
:= (8 − 8/8) × ((8 × (88 + 8) − 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8 + 8) × 888/8)
:= 99 × (9 + 9 + 9) − (9 + 9)
:= 99 × (9 + 9 + 9) − ((99 + 9 + 9)/9)
:= 9/9 + (99 × (9 + 9 + 9) − 9)
3
Ï 2656 := (1 + 1) × (111+1+1 − 1 − 1 − 1)
:= 2 × (2 × (((2/2 + 2) × 222) − 2))
3
:= (3 − 3/3) × ((33/3) − 3)
Ï 2661 := (1 + 1) × 111+1+1 − 1
Ï 2666 := (1 + 1) × (1 + 1 + 111+1+1 )
2
:= (22 × (22/2) ) − 2/2
:= 2 + ((22 × (22/2)2 ) + 2)
3
:= (3 × ((33 × 3 ) − 3)) − 3
:= 3 + (((33/3 + 3)3 ) − 3 × 33 )
4
4
:= 4 × ((44 × (44/4 + 4)) + 4)
:= 4 + ((((4 − 4/4) + 4) ) + 4 )
:= 5 + ((5 × (555 − 5 × 5)) + 5/5)
:= 55/5 + (5 × (555 − 5 × 5))
:= 5 + ((5 × (555 − 5 × 5)) + (55/5))
:= ((6 + 6)/6) + (6 × (6 × (66 + 6)))
:= (66 × (6 × 6 + 6)) − 666/6
:= (6 + 6)/6 + ((6 + 6) × (6 × 6 × 6 + 6))
:= (7/7 + 7) × (7 × 7 × 7 − (77/7))
:= 7/7 + ((77 − 7) × (7 × 7 − 77/7))
:= 7 × 7 × (7 × 7 + 7) − 7/7 − 77
:= ((8 + 8 + 8) × 888/8) − 8
:= (88/8 − 8) × (888 − 8/8)
:= (8 + 8)/8 + ((8 + 8 + 8) × 888/8)
:= 9/9 + (99 × (9 + 9 + 9) − (9 + 9))
:= 99 × (9 + 9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (99 × (9 + 9 + 9) − 9)
6
Ï 2657 := (1 + 1) × (111+1+1 − 1 − 1) − 1
2
:= (22 × (22/2) ) − (2/2 + 2 + 2)
3
:= (((3 + 3) × ((33/3) − 3)) + 3)/3
4
4
:= 4 + (((4 − 4/4) + 4) )
:= 5 + ((5 × (555 − 5 × 5)) + ((5 + 5)/5))
:= (44 − 4/4) × (44 − 4 − 4)/4
Ï 2662 := (1 + 1) × 111+1+1
:= 22 × (22/2)
Ï 2667 := 1 + (1 + 1) × (1 + 1 + 111+1+1 )
2
:= 2 + (((222 × (22 + 2)) + 2)/2)
:= (3 − 3/3) × (33/3)3
:= 3 + (3 × ((33 × 33 ) − 3))
:= (4 + 4)/4 × ((44/4)
4−4/4
:= (5 + 5)/5 × ((55/5)
5−(5+5)/5
)
:= 4 + ((444 × ((4 + 4)/4 + 4)) − 4/4)
)
:= 66 + ((6 × (6 × (66 + 6))) − 6/6)
:= ((6 + 6) × (6 × 6 × 6 + 6)) − (6 + 6)/6
:= 77/7 + (7 × (7 × (7 × 7 + 7) − (7 + 7)))
:= 7 × 7 × 77 − 7777/7
:= 5 + ((5 × (555 − 5 × 5)) + ((55 + 5)/5))
:= 6 + ((66 × (6 × 6 + 6)) − 666/6)
:= 7 × 7 × (7 × 7 + 7) − 77
88/8−8
:= 8/8 + (((8 + 8 + 8) × 888/8) − 8)
:= ((8 + 8)/8) × ((88/8)
:= (9 + 9)/9 + (99 × (9 + 9 + 9) − (9 + 9))
:= 99 × (9 + 9 + 9) − 99/9
187
)
:= (88/8 − 8) × (888 + 8/8)
:= ((9 + 9 + 9)/9) + (99 × (9 + 9 + 9) − 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2668 := (1 + 1) × (1 + 1 + 1 + 111+1+1 )
Ï 2673 := 11 + ((1 + 1) × 111+1+1 )
2
Ï 2678 := 1 + (1 + ((1 + 11) × (1 + (1 + 1) × 111)))
:= 22/2 × (22 + 2)/2
:= 2 + (2 + 2 + 2) × (2 × 222 + 2)
:= (3 − 3/3) × ((33/3) + 3)
:= 3 × (33 × 33 )
:= 3 + (((3 × (33 × 33 )) − 3/3) + 3)
:= 4 + (444 × ((4 + 4)/4 + 4))
:= (4/4 + 4)4 + 44 × (4 + 4)
:= (4 + 4)/4 + (4 × ((4/4 + 4)4 + 44))
:= (55/5 × ((5 − (5 + 5)/5)5 )) − 5
:= 55/5 × ((5 − (5 + 5)/5)5 )
:= 5 + (55/5 × ((5 − (5 + 5)/5)5 ))
:= (6 − ((6 + 6)/6)) × (666 + 6/6)
:= 66/6 × ((6 × 6/(6 + 6))6−6/6 )
:= 6 + ((((6 + 6) × (6 × 6 × 6 + 6)) + ((6 + 6)/6)) + 6)
:= 7/7 + (7 × 7 × (7 × 7 + 7) − 77)
:= 7 + (7 × 7 × (7 × 7 + 7) − (7/7 + 77))
:= 77 + (((7 + 7)/7 + 7 × 7)(7+7)/7 )
:= 8/8 + ((88/8 − 8) × (888 + 8/8))
:= 8 + (((8 + 8 + 8) × 888/8) + 8/8)
:= 8 + ((88/8 − 8) × (888 + ((8 + 8)/8)))
:= ((9 − 99)/(9 + 9)) + 99 × (9 + 9 + 9)
:= 99 × (9 + 9 + 9)
:= 99 × (9 + 9 + 9) + ((9 × 9 + 9)/(9 + 9))
:= 2 × (((2 + 2 + 2) × 222) + 2)
3
Ï 2669 := 1 + (1 + 1) × (1 + 1 + 1 + 111+1+1 )
Ï 2674 := 1 + (11 + ((1 + 1) × 111+1+1 ))
:= 2 + ((((222 × (22 + 2)) + 2)/2) + 2)
:= (2 + 2 + 2) × (2 × 222 + 2) − 2
:= 2 + (((22/2 × (222 + 2)/2) + 2) + 2)
:= (3 × (33 × 33 )) − (3/3 + 3)
:= 3/3 + (3 × (33 × 33 ))
:= 3 + ((3 × (33 × 33 )) + 3)
:= (4/4 + 4)4 + (44 × (4 + 4) − 4)
:= 4 + (((4 + 4)/4 + 4) × (444 + 4/4))
:= 4 + ((4 × ((4/4 + 4)4 + 44)) − 4/4)
:= 5 + ((5 × 5 − 5/5) × 555/5)
:= (5 − 5/5)5 + (55 × (5 × 5 + 5))
:= 5 + ((55 × (5 × 5 + 5)) + (5 − 5/5)5 )
:= 6 + (((6 + 6) × (6 × 6 × 6 + 6)) − 6/6)
:= 6 + ((6 − ((6 + 6)/6)) × (666 + 6/6))
:= 6 + ((66/6) × ((6 × 6/(6 + 6))6−6/6 ))
:= 7 + (7 × 7 × 77 − 7777/7)
Ï 2679 := 1 + (1 + (1 + ((1 + 11) × (1 + (1 + 1) × 111))))
:= 7 + (7 × 7 × (7 × 7 + 7) − 77)
:= (7 × 7 − (7 + 7)/7) × (7/7 + 7 × 7 + 7)
:= 8 + ((88/8 − 8) × (888 − 8/8))
:= 8 + (((8 + 8 + 8) × 888/8) + ((8 + 8)/8))
:= ((8 + 8) × (88 − 8 + 88)) − (8/8 + 8)
:= ((9 − 9 × 9)/(9 + 9)) + 99 × (9 + 9 + 9)
:= 9/9 + 99 × (9 + 9 + 9)
:= 9 + (((9 + 9 + 9)/9) × (9 × 99 − 9/9))
Ï 2670 := 11 + (((1 + 1) × (111+1+1 − 1)) − 1)
Ï 2675 := 11 + (1 + 1) × (1 + 111+1+1 )
Ï 2680 := (1 + 1) × (11 + 111+1+1 − 1 − 1)
:= 2 + (2 × (((2 + 2 + 2) × 222) + 2))
:= 2 + (22/2 × (222 + 2)/2)
:= 2 × ((2 + 2 + 2)2+2 + 2 × 22)
:= (3 × (33 × 33 )) − 3
:= 3 + ((3 × (33 × 33 )) − 3/3)
:= 3 + (((3 × (33 × 33 )) + 3/3) + 3)
:= ((4 + 4)/4 + 4) × (444 + 4/4)
:= (4 × ((4/4 + 4)4 + 44)) − 4/4
:= 4 + (4 × ((4/4 + 4)4 + 44))
:= (5 × (555 − 5 − 5)) − 55
:= 5 × ((555 − 5 × 5) + 5)
:= 5 + (5 × ((555 − 5 × 5) + 5))
:= 6 + ((6 + 6) × (6 × 6 × 6 + 6))
:= 66/6 + ((6 + 6) × (6 × 6 × 6 + 6))
:= 7 × (77 − 7) + (((7 + 7 + 7)/7)7 − 7)
:= (66 + 6/6) × ((6 × 6 − ((6 + 6)/6)) + 6)
:= 7 + ((7 × 7 × (7 × 7 + 7) − 77) + 7/7)
:= (7/7 + 7) × (7 × 7 × 7 − (7/7 + 7))
:= (88/8 − 8) × (888 + ((8 + 8)/8))
:= 8 + ((88/8 − 8) × (888 + 8/8))
:= ((8 + 8) × (88 − 8 + 88)) − 8
:= ((9 + 9 + 9)/9) × (9 × 99 − 9/9)
:= (9 + 9)/9 + 99 × (9 + 9 + 9)
:= 9 + (99 × (9 + 9 + 9) − ((9 + 9)/9))
Ï 2671 := 11 + ((1 + 1) × (111+1+1 − 1))
:= (22/2 × (222 + 2)/2) − 2
Ï 2676 := (1 + 11) × (1 + (1 + 1) × 111)
:= (2 + 2 + 2) × (2 × 222 + 2)
3
:= 3/3 + ((3 × (33 × 33 )) − 3)
:= 3 + (3 × (33 × 3 ))
4
Ï 2681 := ((1 + 1) × (11 + (111+1+1 − 1))) − 1
:= 222 + ((22/2 + 2)2/2+2 )
:= (3 × ((33 × 33 ) + 3)) − 3/3
:= (44 × (44 − 4) + 444)/4
:= 4 × ((4/4 + 4) + 44)
:= ((4/4 + 4)4+4/4 ) − 444
:= 5/5 + ((5 × (555 − 5 − 5)) − 55)
:= 5/5 + (5 × ((555 − 5 × 5) + 5))
:= 55 + ((5/5 − 5) × 555/5)
:= 6 + (((6 + 6) × (6 × 6 × 6 + 6)) + 6/6)
:= 6 + (((6 + 6) × (6 × 6 × 6 + 6)) + 6)
:= 6 + (((6 + 6) × (6 × 6 × 6 + 6)) + (66/6))
:= 7 + ((7/7 + 7) × (((7 − 77)/7) + 7 × 7 × 7))
:= 7 + ((7 × 7 × 77 − 7777/7) + 7)
:= (7 × (7 × (7 × 7 + 7) − 7)) − (7 + 7)
:= 8 + (((8 + 8 + 8) × 888/8) − 8/8)
:= ((88 + 8)/8) + ((8 + 8 + 8) × 888/8)
:= 8/8 + (((8 + 8) × (88 − 8 + 88)) − 8)
:= 99 × (9 + 9 + 9) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + 99 × (9 + 9 + 9)
:= 9 + (99 × (9 + 9 + 9) − 9/9)
Ï 2672 := 11 + (((1 + 1) × 111+1+1 ) − 1)
Ï 2677 := 1 + ((1 + 11) × (1 + (1 + 1) × 111))
Ï 2682 := (1 + 1) × (11 + (111+1+1 − 1))
:= 2 × ((((2 + 2 + 2) × 222) + 2) + 2)
:= 2 + ((22/2 × (222 + 2)/2) + 2)
:= (2 × (22 + 2 + 2))2 − 22
:= (3 × (33 × 33 )) − 3/3
:= 3 + ((3 × (33 × 33 )) + 3/3)
:= 3 × ((33 × 33 ) + 3)
:= 4 × ((4 × (4 × 44 − (4 + 4))) − 4)
:= 4 + ((4/4 + 4)4 + 44 × (4 + 4))
:= ((4 + 4)/4 + 4) × ((444 − 4/4) + 4)
5
55/5
5
:= 5 + ((5/5 − 5) × (555 + 5)/5)
:= 55 + ((555 + 5)/5 − 555)
:= 6 + (((6 + 6) × (6 × 6 × 6 + 6)) + ((6 + 6)/6))
:= 6 + ((((6 + 6) × (6 × 6 × 6 + 6)) + 6/6) + 6)
:= ((6 × 6 + 6) × ((6 + 6)/6)6 ) − 6
:= (7/7 + 7) × (7 × 7 × 7 − ((7 + 7)/7 + 7))
:= 7 × (77 − 7) + ((7 + 7 + 7)/7)7
:= 7/7 + ((7 × (7 × (7 × 7 + 7) − 7)) − (7 + 7))
:= 8 + ((8 + 8 + 8) × 888/8)
:= ((8 + 8) × (88 − 8 + 88)) − 88/8
:= (8 − (8 + 8)/8) × (8 × (8 × 8 − 8) − 8/8)
:= 99 × (9 + 9 + 9) − 9/9
:= 99 × (9 + 9 + 9) + ((9 × 9 − 9)/(9 + 9))
:= 9 + 99 × (9 + 9 + 9)
:= (5 − 5)/5 + (((5 + 5)/5)
)
188
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2683 := ((1 + 1) × (11 + 111+1+1 )) − 1
2
Ï 2693 := 1 + ((1 + 1) × (1 + (1 + ((1 + 11) × (1 + 111)))))
Ï 2688 := (1 + 1) × ((1 + 11) × (1 + 111))
:= (2 × (22 + 2 + 2))2 − 22/2
:= 2 × ((2 + 2 + 2) × (222 + 2))
:= (22 × (22 /2 + 2) − 2)/2
3
:= 3/3 + (3 × ((33 × 33 ) + 3))
:= 3 + ((3 × ((33 × 3 ) + 3)) + 3)
:= 33/3 + (3 × ((33 × 33 ) + 3))
:= (44 × ((44 + 4)/4 − 4)) − 4/4
:= 4 × (4 × (4 × 44 − (4 + 4)))
:= 4 + ((4 × (4 × (4 × 44 − (4 + 4)))) + 4/4)
:= 5 + ((55/5 × ((5 − (5 + 5)/5)5 )) + 5)
:= (5 × 5 − 5/5) × (555 + 5)/5
:= 5 + ((5 × 5 − 5/5) × (555 + 5)/5)
:= 6/6 + (((6 × 6 + 6) × ((6 + 6)/6)6 ) − 6)
:= (6 × 6 + 6) × ((6 + 6)/6)
:= 6 + (((6 × 6 + 6) × ((6 + 6)/6)6 ) − 6/6)
:= (7 × (7 × (7 × 7 + 7) − 7)) − (77 + 7)/7
:= (7/7 + 7) × (7 × 7 × 7 − 7)
:= (7 × (7 × (7 × 7 + 7) − 7)) − (7 + 7)/7
:= 8 + (((88/8 − 8) × (888 + 8/8)) + 8)
:= (8 + 8) × (88 − 8 + 88)
:= 8 + ((88/8 − 8) × (888 − 8/8 + 8))
:= 9 + (99 × (9 + 9 + 9) + 9/9)
:= 9 + ((((9 + 9 + 9)/9) × (9 × 99 − 9/9)) + 9)
:= 9 + (99 × (9 + 9 + 9) + (99/9))
Ï 2684 := (1 + 1) × (11 + 111+1+1 )
6
Ï 2689 := 1 + ((1 + 1) × ((1 + 11) × (1 + 111)))
Ï 2694 := (11 × (1 + ((1 + 1) × (1 + 111+1 )))) − 1
:= 22 + (22 × (22/2)2 )
:= (((22 + 2) × (222 + 2)) + 2)/2
:= (2 × (2 × (((22 + 2 + 2)2 ) − 2))) − 2
:= 33/3 + (3 × (33 × 33 ))
:= 33 + ((3 − 3/3) × (33/3)3 )
:= 3 + (3 × (((33 × 33 ) + 3) + 3))
:= 44 × ((44 + 4)/4 − 4)
:= 4/4 + (4 × (4 × (4 × 44 − (4 + 4))))
:= ((44 × (44 − 44/4)) − 4)/4
:= 55 − (((55/5 + 5) + 5)(5+5)/5 )
:= (5 × (555 + 5)) − 555/5
:= (5 × (5 + 5) × 55) − (55 + 5/5)
:= (6 − ((6 + 6)/6)) × ((666 − 6/6) + 6)
:= 6/6 + ((6 × 6 + 6) × ((6 + 6)/6) )
:= 6 + ((6 × 6 + 6) × ((6 + 6)/6)6 )
:= (7 × (7 × (7 × 7 + 7) − 7)) − 77/7
:= 7/7 + ((7/7 + 7) × (7 × 7 × 7 − 7))
:= (7 × (7 × (7 × 7 + 7) − 7)) − 7/7
:= (((8 × 8 − 8) × (88 + 8)) − 8)/((8 + 8)/8)
:= 8/8 + ((8 + 8) × (88 − 8 + 88))
:= (8 − (8 + 8)/8) × (8 × (8 × 8 − 8) + 8/8)
:= 99/9 + 99 × (9 + 9 + 9)
:= 9 + ((99 × (9 + 9 + 9) − ((9 + 9)/9)) + 9)
:= 9 + (99 × (9 + 9 + 9) + ((99 + 9)/9))
Ï 2685 := 1 + ((1 + 1) × (11 + 111+1+1 ))
6
Ï 2690 := (1 + 1) × (1 + ((1 + 11) × (1 + 111)))
Ï 2695 := 11 × (1 + ((1 + 1) × (1 + 111+1 )))
:= ((22 × (222 /2 + 2)) + 2)/2
:= 2 + (2 × ((2 + 2 + 2) × (222 + 2)))
:= 22/2 × ((222 + 2)/2 + 2)
:= 3 + (3 × ((33 × 33 ) + 3))
:= ((33/3 + 3)3 ) − (33 + 33 )
:= 33 + ((3 − 3/3) × (33/3)3 )
:= 4/4 + (44 × ((44 + 4)/4 − 4))
:= (4 + 4)/4 + (4 × (4 × (4 × 44 − (4 + 4))))
:= 44/4 × (44 − 44/4)
:= 5 + ((5 × ((555 − 5 × 5) + 5)) + 5)
:= (5 + 5) × (5 × 55 − (5/5 + 5))
:= 55 × (55 − (5/5 + 5))
:= ((6 × 6 + 6) × ((6 + 6)/6) ) − 6 × 6/(6 + 6)
:= (6 + 6)/6 + ((6 × 6 + 6) × ((6 + 6)/6)6 )
:= (6/6 + 6) × (6 × 66 − (66/6))
:= ((7 − 77)/7) + (7 × (7 × (7 × 7 + 7) − 7))
:= (7 − ((7 + 7)/7)) × (7 × 77 − 7/7)
:= 7 × (7 × (7 × 7 + 7) − 7)
:= (88/8 − 8) × (888 − 8/8 + 8)
:= (8 + 8)/8 + ((8 + 8) × (88 − 8 + 88))
:= 8 + (((8 + 8) × (88 − 8 + 88)) − 8/8)
:= ((99 + 9)/9) + 99 × (9 + 9 + 9)
:= 9 + ((99 × (9 + 9 + 9) − 9/9) + 9)
:= 99/9 × (9 × (9 + 9 + 9) + ((9 + 9)/9))
6
Ï 2686 := (1 + 1) × (1 + (11 + 111+1+1 ))
Ï 2691 := 1 + ((1 + 1) × (1 + ((1 + 11) × (1 + 111))))
Ï 2696 := 1 + (11 × (1 + ((1 + 1) × (1 + 111+1 ))))
:= 2 + ((22 × (22/2)2 ) + 22)
:= 2 + ((((22 + 2) × (222 + 2)) + 2)/2)
:= 2 × (2 × (((22 + 2 + 2)2 ) − 2))
:= 3 + ((3 × ((33 × 33 ) + 3)) + 3/3)
:= 3 × (((33 × 33 ) + 3) + 3)
:= 3 + ((3 × ((33 × 33 ) + 3)) + 33/3)
:= (4 + 4)/4 + (44 × ((44 + 4)/4 − 4))
:= (44/4 × (44 − 44/4)) − 4
:= (4 + 4) × ((4 − 4/4)4 + 44 )
:= 5 + (((5/5 − 5) × 555/5) + 55 )
:= 5/5 + ((5 + 5) × (5 × 55 − (5/5 + 5)))
:= 5/5 + (55 × (55 − (5/5 + 5)))
:= ((6 × 6 + 6) × ((6 + 6)/6)6 ) − (6 + 6)/6
:= ((66/6 + 6) + 6) × (666/6 + 6)
:= 6 + (((6 × 6 + 6) × ((6 + 6)/6)6 ) + ((6 + 6)/6))
:= (7 × (7 × (7 × 7 + 7) − 7)) − ((7 + 7)/7 + 7)
:= ((8 + 8) × (88 − 8 + 88)) − (8 + 8)/8
:= 99 × (9 + 9 + 9) + ((99 + 9 + 9)/9)
Ï 2687 := ((1 + 1) × ((1 + 11) × (1 + 111))) − 1
:= 7 + ((7 × (7 × (7 × 7 + 7) − 7)) − (77/7))
:= 7/7 + (7 × (7 × (7 × 7 + 7) − 7))
:= (88/8 − 8) × (888 + 8/8 + 8)
:= 8 + ((8 + 8) × (88 − 8 + 88))
:= 9 + (99 × (9 + 9 + 9) + 9)
:= 9 × (9 × 9 − 9) + (((9 + 9)/9)99/9 )
Ï 2692 := (1 + 1) × (1 + (1 + ((1 + 11) × (1 + 111))))
Ï 2697 := 1 + (1 + (11 × (1 + ((1 + 1) × (1 + 111+1 )))))
:= (((22 + 2) × (222 + 2)) − 2)/2
:= 2 × (((2 + 2 + 2) × (222 + 2)) + 2)
:= 2 + (22/2 × ((222 + 2)/2 + 2))
:= 3 + ((3 × (33 × 33 )) + 33/3)
:= 3/3 + (3 × (((33 × 33 ) + 3) + 3))
:= 33 + ((3 × (33 × 33 )) − 3)
:= (4 × (4 × (4 × 44 − (4 + 4)))) − 4/4
:= 4 + (4 × (4 × (4 × 44 − (4 + 4))))
:= 44 + (((((4 − 4/4) + 4)4 ) − 4) + 44 )
5
5
:= 5 − ((5 + 5)/(5 + 5) + 5 × 5 × 5)
6
5
5
:= 5 + (5 − ((5 + 5)/(5 + 5) + 5 × 5 × 5))
6
:= (5 + 5)/5 + (55 × (55 − (5/5 + 5)))
:= ((6 + 6)/6) + (6 × ((6 × (66 + 6)) + 6))
:= 666/6 + ((6 × (6 × (66 + 6))) − 6)
:= (7 × (7 × (7 × 7 + 7) − 7)) − (7/7 + 7)
:= (7 × (7 × (7 × 7 + 7) − 7)) − (7 + 7 + 7)/7
:= (7 + 7)/7 + (7 × (7 × (7 × 7 + 7) − 7))
:= ((8 + 8) × (88 − 8 + 88)) − 8/8
:= (((8 × 8 − 8) × (88 + 8)) + 8)/((8 + 8)/8)
:= 8 + (((8 + 8) × (88 − 8 + 88)) + 8/8)
:= 9 + (99 × (9 + 9 + 9) + ((9 × 9 + 9)/(9 + 9)))
:= 9 + ((99 × (9 + 9 + 9) + 9/9) + 9)
:= ((9 + 9 + 9)/9) × ((9 × 99 − 9/9) + 9)
:= ((6 × 6 + 6) × ((6 + 6)/6) ) − 6/6
189
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2698 := 11 + (((1 + 1) × ((1 + 11) × (1 + 111))) − 1)
Ï 2703 := (((1 + 1) × ((1 + 1) × (1 + 1 + 11)))1+1 ) − 1
Ï 2708 := (1 + 1) × (1 + (11 × (1 + (1 + 111+1 ))))
:= 2 + (2 × (2 × (((22 + 2 + 2)2 ) − 2)))
:= (2 × (22 + 2 + 2))2 − 2/2
:= 2 + ((2 × (22 + 2 + 2))2 + 2)
:= 3 + (((3 − 3/3) × (33/3)3 ) + 33)
:= 3 + ((3 × (33 × 33 )) + 33 )
:= ((33/3 + 3)3 ) − (33 + 3)
:= 4 + (((44 × (44 − 44/4)) − 4)/4)
:= ((44 + 4 + 4)(4+4)/4 ) − 4/4
:= 4 + ((44 + 4 + 4)(4+4)/4 )
:= ((5 + 5) × (5 × 55 − 5)) − (5 + 5)/5
:= 5 + (((5 + 5) × (5 × 55 − 5)) − ((5 + 5)/5))
:= 55 − ((55 + 55 + 5)/(5 + 5 + 5))
:= ((6 + 6)/6 + 6 × 6) × ((66 − 6/6) + 6)
:= 666/6 + (6 × (6 × (66 + 6)))
:= (66 × (6 × 6 + 6)) − ((6 + 6)/6)6
:= (7 + 7 + 7)/7 + (7 × (7 × (7 × 7 + 7) − 7))
:= 7 + ((7 × (7 × (7 × 7 + 7) − 7)) + 7/7)
:= 7 + (((7 × (7 × (7 × 7 + 7) − 7)) − 7/7) + 7)
:= 8 + (((8 + 8) × (88 − 8 + 88)) + ((8 + 8)/8))
:= 8 + ((((8 + 8) × (88 − 8 + 88)) − 8/8) + 8)
:= ((8 × (8 × 88 − (8 + 8))) − 88)/((8 + 8)/8)
:= ((9 + 9 + 9) × (9/9 + 99)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) × ((9 × 99 + 9/9) + 9)
:= 9 + (9 × 9 × (9 + 9 + 9) + (((9 + 9)/9)9 ))
Ï 2699 := 11 + ((1 + 1) × ((1 + 11) × (1 + 111)))
2
:= (2 × (22 + 2 + 2)) − (2/2 + 2 + 2)
3
Ï 2704 := ((1 + 1) × ((1 + 1) × (1 + 1 + 11)))1+1
:= (2 × (22 + 2 + 2))
2
3
3
:= 3 + ((3 × (33 × 3 )) − 3/3)
4
Ï 2709 := 1 + ((1 + 1) × (1 + (11 × (1 + (1 + 111+1 )))))
:= 2 + (((2 × (22 + 2 + 2))2 + 2/2) + 2)
3
:= 3 + (((3 × (33 × 3 )) + 3 ) + 3/3)
:= 3 + ((3 × (33 × 33 )) + 33)
(4+4)/4
:= ((44 − 4)/4) × (44 − 4/4)
:= 4 + (44/4 × (4 − 44/4))
:= (44 + 4 + 4)
:= ((5 + 5) × (5 × 55 − 5)) − 5/5
:= 5 + (((5 + 5) × (5 × 55 − 5)) − 5/5)
:= 5 × 555 − (55/5 + 55)
:= (6 × (666 − 6 × 6 × 6)) − 6/6
:= (((6 + 6)/6)6 − (6 + 6))(6+6)/6
:= 6 + ((6 × (6 × (66 + 6))) + 666/6)
:= 77/7 + ((7/7 + 7) × (7 × 7 × 7 − 7))
:= (((7 + 7 + 7)/7) + 7 × 7)(7+7)/7
:= 7 + ((7 × (7 × (7 × 7 + 7) − 7)) + 7)
:= 8 + (((8 + 8) × (88 − 8 + 88)) + 8)
:= (88/8 − 8) × ((888 − 8/8 + 8) + 8)
:= 8 × 8 × 8 + ((88/8 − 8)
8−8/8
)
9
:= (((9 + 9)/9) ) + 9 × 9 × (9 + 9 + 9)
Ï 2700 := (1 + 11) × (1 + ((1 + 1) × (1 + 111)))
:= (9 × 9 − (99/9 + 9 + 9))
(9+9)/9
Ï 2705 := 1 + (((1 + 1) × ((1 + 1) × (1 + 1 + 11)))1+1 )
:= 9 + ((9 + 9 + 9) × (9/9 + 99))
Ï 2710 := (1 + 1) × (11 + ((1 + 11) × (1 + 111)))
:= 2 × ((2 × ((22 + 2 + 2) )) − 2)
:= 2/2 + (2 × (22 + 2 + 2))2
:= 2 + (((2 × (22 + 2 + 2))2 + 2) + 2)
:= 3 × ((33 × 33 ) + 3 × 3)
:= 33 + ((3 × (33 × 33 )) − 3/3)
:= ((33/3 + 3)3 ) − 3/3 − 33
:= (4/4 + 4 + 4) × (44 + 44 )
:= 4/4 + ((44 + 4 + 4)(4+4)/4 )
:= (((44 − 4) × (44 − 4/4)) + 4)/4
:= (5 + 5) × (5 × 55 − 5)
:= 5 + ((5 + 5) × (5 × 55 − 5))
:= 5 + (((5 + 5) × (5 × 55 − 5)) + 5)
:= 6 × (666 − 6 × 6 × 6)
:= (66 × (6 × 6 + 6)) − (66 + 6/6)
:= 6 + ((((6 + 6)/6)6 − (6 + 6))(6+6)/6 )
:= (7 − ((7 + 7)/7)) × (7 × 77 + 7/7)
:= ((77 − 7)/7) + (7 × (7 × (7 × 7 + 7) − 7))
:= 7 + (((7 × (7 × (7 × 7 + 7) − 7)) + 7/7) + 7)
:= 8 + ((((8 × 8 − 8) × (88 + 8)) + 8)/((8 + 8)/8))
:= (88 × ((8 + 8 + 8) + 8)) − 888/8
:= 8 + (((8 − (8 + 8)/8) × (8 × (8 × 8 − 8) + 8/8)) + 8)
:= (9 + 9 + 9) × (9/9 + 99)
:= 9 + ((((9 + 9)/9)99/9 ) + 9 × (9 × 9 − 9))
:= 9 + (((9 + 9 + 9) × (9/9 + 99)) + 9/9)
2
Ï 2701 := 1 + ((1 + 11) × (1 + ((1 + 1) × (1 + 111))))
Ï 2706 := (1 + 1) × (11 × (1 + (1 + 111+1 )))
Ï 2711 := ((1 + 1) × ((1 + 11) × (1 + 1 + 111))) − 1
:= (2 × (22 + 2 + 2))2 − 2/2 − 2
:= 2 + (2 × (22 + 2 + 2))2
:= (2 × (22 − 2))2 + 2222/2
:= 33 + ((3 × (33 × 33 )) + 3/3)
:= 33 + (3 × (33 × 33 ))
:= ((33/3 + 3)3 ) − 33
:= 44 + ((((4 − 4/4) + 4)4 ) + 44 )
:= 44/4 × ((4 − 44)/4 + 44 )
:= 4 + ((44 × (44 − 4) − (44 + 4))/4)
:= 5/5 + ((5 + 5) × (5 × 55 − 5))
:= 5 + (((5 + 5) × (5 × 55 − 5)) + 5/5)
:= 55/5 + ((5 + 5) × (5 × 55 − 5))
:= 6/6 + (6 × (666 − 6 × 6 × 6))
:= 66 × (6 × 6 − 6/6 + 6)
:= 6 + ((66 × (6 × 6 + 6)) − (66 + 6/6))
:= 7 + ((7 × (7 × (7 × 7 + 7) − 7)) − 7/7)
:= 77/7 + (7 × (7 × (7 × 7 + 7) − 7))
:= 7 + ((((7 + 7 + 7)/7) + 7 × 7)(7+7)/7 )
:= ((8/8 + 8 × 8) + 8) × 888/(8 + 8 + 8)
:= 88/8 × ((8 − 88)/8 + (8 + 8) × (8 + 8))
:= 8888/8 + 8 × (8 × (8 + 8 + 8) + 8)
:= 9/9 + ((9 + 9 + 9) × (9/9 + 99))
:= ((9 + 9 + 9)/9) × ((99/9) + 9 × 99)
:= 99/9 + ((9 + 9 + 9) × (9/9 + 99))
Ï 2712 := (1 + 1) × ((1 + 11) × (1 + 1 + 111))
Ï 2702 := 1 + (1 + ((1 + 11) × (1 + ((1 + 1) × (1 + 111)))))
Ï 2707 := 1 + ((1 + 1) × (11 × (1 + (1 + 111+1 ))))
2
:= (2 × (22 + 2 + 2)) − 2
:= 2 + ((2 × (22 + 2 + 2))2 + 2/2)
:= 2 × (2 × (((22 + 2 + 2)2 ) + 2))
:= ((33/3 + 3)3 ) − (3 × 3 + 33)
:= 3/3 + ((3 × (33 × 33 )) + 33)
:= 3 + (((3 × (33 × 33 )) + 33) + 3)
:= ((44 + 4 + 4)(4+4)/4 ) − (4 + 4)/4
:= (44 × (44 − 4) − (44 + 4))/4
:= 4 + (((44 + 4 + 4)(4+4)/4 ) + 4)
:= (5 + 5)/5 + ((5 + 5) × (5 × 55 − 5))
:= 5 + (((5 + 5) × (5 × 55 − 5)) + ((5 + 5)/5))
:= 55 × 55 − (55 + 5)/(5 + 5)
:= (6 + 6)/6 + (6 × (666 − 6 × 6 × 6))
:= 6/6 + (66 × (6 × 6 − 6/6 + 6))
:= 6 + (66 × (6 × 6 − 6/6 + 6))
:= 7 + (7 × (7 × (7 × 7 + 7) − 7))
:= 7 + ((7 − ((7 + 7)/7)) × (7 × 77 + 7/7))
:= 7 × 77 + (((7 + 7 + 7)/7)7 − (7 + 7))
:= 8 + ((8 − (8 + 8)/8) × (8 × (8 × 8 − 8) + 8/8))
:= 8 + (((8 + 8) × (88 − 8 + 88)) + (88/8))
:= (8 + 8 + 8) × ((888 + 8 + 8)/8)
:= 9 + ((99 × (9 + 9 + 9) + (99/9)) + 9)
:= 9 + (((9 + 9 + 9) × (9/9 + 99)) − ((9 + 9)/9))
:= 9 + (((9 + 9 + 9)/9) × ((9 × 99 + 9/9) + 9))
190
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2713 := 1 + ((1 + 1) × ((1 + 11) × (1 + 1 + 111)))
Ï 2718 := 1 + (11 × (1 + ((1 + 1) × (1 + (1 + 111+1 )))))
Ï 2723 := 11 + ((1 + 1) × ((1 + 11) × (1 + 1 + 111)))
:= 2/2 + (2 × (2 × (((22 + 2 + 2) ) + 2)))
:= 2 + (2 × ((2 × (((22 + 2 + 2) ) + 2)) + 2))
:= 22 + ((2 × (22 + 2 + 2))2 − (2/2 + 2))
:= 3 + (((33/3 + 3)3 ) − (3/3 + 33))
:= 3 + ((3 × ((33 × 33 ) + 3)) + 33)
:= 3 + ((((33/3 + 3)3 ) − 33 ) + 3)
:= 4 + (((44 − 4)/4) × (44 − 4/4))
:= ((44 − 4) × (4 × 4 × 4 + 4)) − (4 + 4)/4
:= (((44 × (44 − 4 − 4)) − 4)/4) − 4
:= 55 × 55 + ((5 − 55 )/(5 + 5))
:= 5 × 555 − ((5 + 5)/5 + 55)
:= (5 × (555 − 5 − 5)) − (5 + 5)/5
:= 6 + ((66 × (6 × 6 − 6/6 + 6)) + 6/6)
:= 6 + ((66 × (6 × 6 − 6/6 + 6)) + 6)
:= (6/6 + 6) × (6 × 66 − (6/6 + 6))
:= 7 + ((7 × (7 × (7 × 7 + 7) − 7)) + (77/7))
:= 7 + (((((7 + 7 + 7)/7) + 7 × 7)(7+7)/7 ) + 7)
:= 7 × (((7 × 777) + 7)/(7 + 7))
:= 8 + ((88 × ((8 + 8 + 8) + 8)) − 888/8)
:= ((((8 + 8)/8) + 8) + 8) × ((88 − 8/8) + 8 × 8)
:= 8 + ((88/8 − 8) × ((888 + 8/8 + 8) + 8))
:= (9 + 9) × (9 × (9 + 9) − (99/9))
:= 99 × (9 + 9 + 9) + ((9 × 99 + 9)/(9 + 9))
2
:= 9 + ((9 × 9 − (99/9 + 9 + 9))
(9+9)/9
)
Ï 2714 := (1 + 1) × (1 + ((1 + 11) × (1 + 1 + 111)))
2
Ï 2719 := 1 + (1 + (11 × (1 + ((1 + 1) × (1 + (1 + 111+1 ))))))
Ï 2724 := (1 + 11) × (1 + ((1 + 1) × (1 + 1 + 111)))
:= 2 + (2 × (2 × (((22 + 2 + 2)2 ) + 2)))
:= 2 + (((2 × (22 + 2 + 2)) + 22/2) + 2)
:= 2 × ((22 × (22+2+2 − 2)) − 2)
:= 3 + (((33/3 + 3)3 ) − 33)
:= 3 + ((3 − 3/3) × ((33/3)3 + 33 ))
:= 3 + ((33 × (3 × 33 + 3)) − 33)
:= 4 + ((((44 − 4) × (44 − 4/4)) + 4)/4)
:= ((44 − 4) × (4 × 4 × 4 + 4)) − 4/4
:= 4 + ((44 − 4) × (4 × 4 × 4 + 4))
:= 5 × 555 − ((55 + 5/5) + 5)
:= 5 × 555 − (55 + 5/5)
:= (5 × (555 − 5 − 5)) − 5/5
:= ((6/6 + 6) × (6 × 66 − 6)) − 66/6
:= (6 + 6) × (6 × 6 × 6 + 66/6)
:= ((7 × 777) − (77/7))/((7 + 7)/7)
:= 7 × 77 + (((7 + 7 + 7)/7)7 − 7)
:= 7/7 + (7 × (((7 × 777) + 7)/(7 + 7)))
:= ((8 − 8/8 + 8) + 8) × ((888 − 8)/8 + 8)
:= (88 × ((8 + 8 + 8) + 8)) − ((8/8 + 88) + 8)
:= (((8 × 8 × 88) − 8)/((8 + 8)/8)) − 88
:= 99 × (9 + 9 + 9) + ((9 × 9 × 9 + 9)/(9 + 9))
:= 9/9 + ((9 + 9) × (9 × (9 + 9) − (99/9)))
:= (9 × ((9 + 9) × (9 + 9) − 9)) − 999/9
:= 666 + (((6 + 6)/6)66/6 )
Ï 2715 := 1 + ((1 + 1) × (1 + ((1 + 11) × (1 + 1 + 111))))
2
Ï 2720 := ((1 + ((1 + 1)1+1+11 ))/(1 + 1 + 1)) − 11
Ï 2725 := (111 − 1 − 1) × (1 + ((1 + 1) × (1 + 11)))
:= 22/2 + (2 × (22 + 2 + 2))2
:= 2 × (2 × ((((22 + 2 + 2)2 ) + 2) + 2))
:= 22 + ((2 × (22 + 2 + 2))2 − 2/2)
:= 33 + (3 × ((33 × 33 ) + 3))
:= 3 + (((33/3 + 3)3 ) − 33 )
:= (3 × 3 + 3)3 + (((3 × 3 + 3/3)3 ) − 3)
:= 44/4 + ((44 + 4 + 4)(4+4)/4 )
:= (44 − 4) × (4 × 4 × 4 + 4)
:= (((44 × (44 − 4 − 4)) + 4)/4) − 4
:= 5 × 555 − (55 + 5)
:= 5 × 555 − 55
:= 5 × (555 − 5 − 5)
:= 6 + (((6 × (6 × (66 + 6))) + 666/6) + 6)
:= 6 + ((((6 + 6)/6)66/6 ) + 666)
:= 6/6 + ((6 + 6) × (6 × 6 × 6 + 66/6))
:= 7 × (7 × (7 × 7 + 7) + 7) − 7/7 − 77
:= ((7 × 777) + 7/7)/((7 + 7)/7)
:= 7 × 77 + (((7 + 7 + 7)/7)7 − 7/7)
:= (88/8 − 8) × ((888 + 8/8 + 8) + 8)
:= 8 × (((8 × 88 − 8)/((8 + 8)/8)) − 8)
:= 8 + (88/8 × ((8 + 8) × (8 + 8) − (8/8 + 8)))
:= 9 + (((9 + 9 + 9)/9) × ((99/9) + 9 × 99))
:= ((9 − 9/9) + 9) × (9 × (9 + 9) − ((9 + 9)/9))
:= 9 + (((9/9 + 9 + 9) + 9) × (99 − ((9 + 9)/9)))
Ï 2716 := (1 + 1) × (1 + (1 + ((1 + 11) × (1 + 1 + 111))))
Ï 2721 := 1 + (((1 + ((1 + 1)1+1+11 ))/(1 + 1 + 1)) − 11)
Ï 2726 := 1 + ((111 − 1 − 1) × (1 + ((1 + 1) × (1 + 11))))
:= 2 × ((2 × (((22 + 2 + 2)2 ) + 2)) + 2)
:= 2/2 + ((2 × (22 + 2 + 2))2 + 22+2 )
:= 22 + (2 × (22 + 2 + 2))2
:= (3 − 3/3) × ((33/3)3 + 33 )
:= (33 × (3 × 33 + 3)) − 33
:= ((33/3 + 3)3 ) − (3 × (3 + 3))
:= ((44 − 4) × (4 × 4 × 4 + 4)) − 4
:= 4/4 + ((44 − 4) × (4 × 4 × 4 + 4))
:= ((44 × (44 − 4 − 4)) − (4 + 4))/4
:= 5/5 + (5 × 555 − (55 + 5))
:= 5/5 + (5 × 555 − 55)
:= 5/5 + (5 × (555 − 5 − 5))
:= (6/6 + 6) × (6 × 66 − ((6 + 6)/6 + 6))
:= 6 + ((((6 × (6 × (66 + 6))) + 666/6) + 6) + 6)
:= (66/6 + 6 × 6) × (((6 + 6)/6)6 − 6)
:= 7 × (((7 × 777) − 7)/(7 + 7))
:= ((7/7 + 7) × (7 × 7 × 7 − ((7 + 7)/7))) − 7
:= 7 × 77 + ((7 + 7 + 7)/7)7
:= (8 − 8/8) × ((8 × (88 + 8) + 8)/((8 + 8)/8))
:= ((8 × 8 − 88/8)(8+8)/8 ) − 88
:= (88 × ((8 + 8 + 8) + 8)) − ((8 + 8)/8 + 88)
:= ((9/9 + 9 + 9) + 9) × (99 − ((9 + 9)/9))
:= ((9/9 + 9) × (999/9 + 9 × (9 + 9))) − 9
:= (9 + 9 + 9) × ((9 + 9)/9 + 99) − 9/9
Ï 2717 := 11 × (1 + ((1 + 1) × (1 + (1 + 111+1 ))))
Ï 2722 := (1 + 1 + 1 + 11)1+1+1 − 11 − 11
:= 2 + ((2 × (22 + 2 + 2))2 + 22/2)
:= 222 + (2 × (22 + 2) + 2)2
:= 22 + ((2 × (22 + 2 + 2))2 + 2/2)
:= ((33/3 + 3)3 ) − 33
:= 3/3 + ((33 × (3 × 33 + 3)) − 33)
:= 3 × ((33 × 33 ) + (3 × (3 + 3)))
:= (4 + 4)/4 + ((44 − 4) × (4 × 4 × 4 + 4))
:= ((44 × (44 − 4 − 4)) − 4)/4
:= 5 + (55 × 55 − (5 + 5)/(5 + 5))
:= (5 + 5)/5 + (5 × 555 − 55)
:= (5 + 5)/5 + (5 × (555 − 5 − 5))
:= 66/6 + (66 × (6 × 6 − 6/6 + 6))
:= 6 + ((6/6 + 6) × (6 × 66 − ((6 + 6)/6 + 6)))
:= 66 + ((66 × (6 × 6 + 6)) − 666/6)
4
:= 44/4 × (4 − (4/4 + 4 + 4))
5
Ï 2727 := (((1 + 1)1+1+11 ) − 11)/(1 + 1 + 1)
:= 7/7 + (7 × (((7 × 777) − 7)/(7 + 7)))
:= 7 × 7 × (7 × 7 + 7) − (7/7 + 7 + 7 + 7)
:= 7 + (((7 × 777) + 7/7)/((7 + 7)/7))
:= 88/8 × ((8 + 8) × (8 + 8) − (8/8 + 8))
:= 8 + (((8 − 8/8 + 8) + 8) × ((888 − 8)/8 + 8))
:= (88 × ((8 + 8 + 8) + 8)) − (8/8 + 88)
:= 99/9 × ((((9 + 9)/9)9−9/9 ) − 9)
:= 9999/9 + ((9 × (99 + 9 × 9)) − 9)
:= (9 + 9 + 9) × ((9 + 9)/9 + 99)
191
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2728 := (1 + 1) × (11 × (1 + (1 + (1 + 111+1 ))))
Ï 2733 := (1 + 1 + 1 + 11)1+1+1 − 11
Ï 2738 := (1 + 1) × ((111/(1 + 1 + 1))1+1 )
:= 2 × (22 × (22+2+2 − 2))
:= (22+2 − 2)2/2+2 − 22/2
:= 2 × (((2 + 2 + 2)2 + 2/2)2 )
:= (3 − 3/3) × ((33/3)3 + 33)
:= ((33/3 + 3)3 ) − 33/3
:= ((33/3 + 3)3 ) − 3 − 3
:= 44 × (44 − 4 − 4)/4
:= 4 + (((44 × (44 − 4 − 4)) + 4)/4)
:= (4 + 4)/4 + (4 × ((4 × (4 × 44 − 4)) − 4))
:= 55/5 × (((5 − (5 + 5)/5)5 ) + 5)
:= 5 + (55/5 × (((5 − (5 + 5)/5)5 ) + 5))
:= 5 × 555 − (((5 + 5)/5)5 + 5)
:= ((6 + 6)/6)6 + ((6 + 6) × (6 × 6 × 6 + 6))
:= 666/6 + ((6 × ((6 × (66 + 6)) + 6)) − 6)
:= (6 + 6)/6 + (6 × ((6 × 66 − 6) + 66))
:= (7/7 + 7) × (7 × 7 × 7 − ((7 + 7)/7))
:= 7 × 7 × (7 × 7 + 7) − 77/7
:= 7/7 + 7 × (7 × (7 × 7 + 7)) − 7
:= 88 × (((8 − 8/8 + 8) + 8) + 8)
:= (8 × ((8 + 8) × (8 + 8) + 88)) − (88/8 + 8)
:= 8 + ((88 × (((8 − 8/8 + 8) + 8) + 8)) + ((8 + 8)/8))
:= 9/9 + (9 + 9 + 9) × ((9 + 9)/9 + 99)
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) − 999/9)
:= 9/9 + (((9 − 9/9) + 9) × (9 × (9 + 9) − 9/9))
Ï 2729 := 1 + ((1 + 1) × (11 × (1 + (1 + (1 + 111+1 )))))
Ï 2734 := 1 + ((1 + 1 + 1 + 11)1+1+1 − 11)
Ï 2739 := 1 + ((1 + 1) × ((111/(1 + 1 + 1))1+1 ))
:= 2/2 + (2 × (22 × (22+2+2 − 2)))
:= 2 × ((((2 + 2 + 2)2 + 2/2)2 ) − 2)
:= (((2 × (2 + 2 + 2)2 + 2)2 ) + 2)/2
:= 3 + (((33/3 + 3)3 ) − (3 × (3 + 3)))
:= ((3 − 33)/3) + ((33/3 + 3)3 )
:= 33 × ((3 × 33 − 3/3) + 3)
:= ((44 × (44 − 4 − 4)) + 4)/4
:= 4 + ((44 + 4)/4 × (44 − (4 + 4)/4))
:= 44/4 × ((44 − 44/4) + 4)
:= 5 + ((5 × (555 − 5 − 5)) − 5/5)
:= (5 × (5 + 5) × 55) − (55/5 + 5)
:= (5 × (5 + 5) × 55) − 55/5
:= ((6 − 6/6)6−6/6 ) − 6 × 66
:= (66 × (6 × 6 + 6)) − ((6 + 6)/6 + 6 × 6)
:= 666/6 + (6 × ((6 × (66 + 6)) + 6))
:= 7 × 7 × (7 × 7 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 × 7 × (7 × 7 + 7)
:= (7 + 7)/7 + 7 × (7 × (7 × 7 + 7)) − 7
:= 8/8 + (88 × (((8 − 8/8 + 8) + 8) + 8))
:= 8 × 8 × 8 + (((8 + 8)/8) × 8888/8)
:= 88/8 × (((8 + 8) × (8 + 8) − 8) + 8/8)
:= 9 + (((9 − 9/9) + 9) × (9 × (9 + 9) − ((9 + 9)/9)))
:= (9 + 9) × (9 × (9 + 9) − 9) − (99/9 + 9)
:= 9 + ((9/9 + 9) × (999/9 + 9 × (9 + 9)))
Ï 2730 := (1 + 1) × ((((1 + 1)1+11 ) − 1)/(1 + 1 + 1))
Ï 2735 := 1 + (1 + ((1 + 1 + 1 + 11)1+1+1 − 11))
Ï 2740 := (1 + 1) × (1 + ((111/(1 + 1 + 1))1+1 ))
:= 2 + (2 × (22 × (22+2+2 − 2)))
:= ((22 + 2/2) × ((22/2)2 − 2)) − 2
:= 2 + (2 × (((2 + 2 + 2)2 + 2/2)2 ))
:= 3 + ((3 + 3) × 333 + 33+3 )
:= ((33/3 + 3)3 ) − 3 × 3
:= ((33/3 + 3)3 ) − (3/3 + 3)
:= (44 + 4)/4 × (44 − (4 + 4)/4)
:= (4 × 4 × 4 × 44) − (4 − 4/4)4
:= 4 + (4 × ((4 × (4 × 44 − 4)) − 4))
:= 5 + (5 × (555 − 5 − 5))
:= 5 + ((5 × (555 − 5 − 5)) + 5)
:= (5 + 5) × (5 × 55 − 5/5)
:= (6/6 + 6) × (6 × 66 − 6)
:= (66 × (6 × 6 + 6)) − (6 × 6 + 6/6)
:= 6 + ((66 × (6 × 6 + 6)) − ((6 + 6)/6 + 6 × 6))
:= 7 × 7 × (7 × 7 + 7) − (7 + 7)
:= 7 × 7 × (7 × 7 + 7) − ((7 + 7)/7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7) − (77/7))
:= (8 + 8)/8 + (88 × (((8 − 8/8 + 8) + 8) + 8))
:= 8 + ((88 × ((8 + 8 + 8) + 8)) − (8/8 + 88))
:= ((8 × (8 × 88 − 8)) − 88)/((8 + 8)/8)
:= (9/9 + 9) × (999/9 + 9 × (9 + 9))
:= (9 + 9) × (9 × (9 + 9) − 9) − (9/9 + 9 + 9)
:= 9 + (9999/9 + (9 × (99 + 9 × 9)))
Ï 2731 := (1 + ((1 + 1)1+1+11 ))/(1 + 1 + 1)
Ï 2736 := (1 + 1) × ((1 + 11) × (1 + 1 + 1 + 111))
Ï 2741 := (1 + 1 + 1 + 11)1+1+1 − 1 − 1 − 1
:= ((222/2+2 ) + 2/2)/(2/2 + 2)
:= 2 × (2 × 222 + (22 − 2)2 )
:= 2 + ((((2 × (2 + 2 + 2)2 + 2)2 ) + 2)/2)
:= 3 + (((3 × 3 + 3/3)3 ) + (3 × 3 + 3)3 )
:= (33 − 3) × (333/3 + 3)
:= ((33/3 + 3)3 ) − 3
:= 4 + (((44 × (44 − 4 − 4)) − 4)/4)
:= 4 × ((4 × (4 × 44 − 4)) − 4)
:= (4 × (4 × (4 × 44 − 4))) − 44/4
:= 5 + ((5 × (555 − 5 − 5)) + 5/5)
:= 55/5 + (5 × (555 − 5 − 5))
:= 5/5 + ((5 + 5) × (5 × 55 − 5/5))
:= 6/6 + ((6/6 + 6) × (6 × 66 − 6))
:= 6 × ((6 × 66 − 6) + 66)
:= 6 + ((66 × (6 × 6 + 6)) − (6 × 6 + 6/6))
:= 7/7 + (7 × 7 × (7 × 7 + 7) − (7 + 7))
:= (7/7 + 7) × (7 × 7 × 7 − 7/7)
:= 7 × 7 × (7 × 7 + 7) − (7 + 7 + 7)/7
:= 8 × 8 + ((88/8 − 8) × (888 + 8/8))
:= 8 + (88 × (((8 − 8/8 + 8) + 8) + 8))
:= (8 × ((8 + 8) × (8 + 8) + 88)) − 88/8
:= 9999/9 + (9 × (99 + 9 × 9))
:= (9 + 9) × (9 × (9 + 9) − (9/9 + 9))
:= (9 + 9) × (9 × (9 + 9) − 9) − ((99 + 9 + 9)/9)
Ï 2732 := 1 + ((1 + ((1 + 1)1+1+11 ))/(1 + 1 + 1))
Ï 2737 := (1 + (11 + 11)) × (111+1 − 1 − 1)
Ï 2742 := (1 + 1 + 1 + 11)1+1+1 − 1 − 1
:= 2 × ((22 × (22+2+2 − 2)) + 2)
:= (22 + 2/2) × ((22/2)2 − 2)
:= (22+2 − 2)2/2+2 − 2
:= ((33/3 + 3)3 ) − (3 × 3 + 3)
:= ((3/3 + 3)3 ) + (3 × (33 × 33 ))
:= 3/3 + (((33/3 + 3)3 ) − 3)
:= 4 + (44 × (44 − 4 − 4)/4)
:= 4/4 + (4 × ((4 × (4 × 44 − 4)) − 4))
:= (4 − 44)/4 + (4 × (4 × (4 × 44 − 4)))
:= 5 + ((5 × (555 − 5 − 5)) + ((5 + 5)/5))
:= ((55 + 5)/5) + (5 × (555 − 5 − 5))
:= (5 + 5)/5 + ((5 + 5) × (5 × 55 − 5/5))
:= (6 + 6)/6 + ((6/6 + 6) × (6 × 66 − 6))
:= 6/6 + (6 × ((6 × 66 − 6) + 66))
:= 6 + (6 × ((6 × 66 − 6) + 66))
:= 7 × 7 × (7 × 7 + 7) − (77 + 7)/7
:= 7 × 7 × (7 × 7 + 7) − 7
:= 7 × 7 × (7 × 7 + 7) − (7 + 7)/7
:= (((8 × 8 × 88) + 8)/((8 + 8)/8)) − 88
:= ((8 − 8/8 + 8) + 8) × (888/8 + 8)
:= (8 − 88)/8 + (8 × ((8 + 8) × (8 + 8) + 88))
:= ((9 + 9)/9) × (9 × (9 × (9 + 9) − 9) − (99/9))
:= ((9 − 9/9) + 9) × (9 × (9 + 9) − 9/9)
:= (9 + 9) × (9 × (9 + 9) − 9) − (99 + 9)/9
192
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2743 := (1 + 1 + 1 + 11)1+1+1 − 1
Ï 2748 := 1 + (1 + (1 + (1 + (1 + 1 + 1 + 11)1+1+1 )))
Ï 2753 := 1 + ((1 + 1)11−1 + ((1 + 11)1+1+1 ))
:= (22+2 − 2)2/2+2 − 2/2
:= 2 + ((22+2 − 2)2/2+2 + 2)
:= 2 + (((22 + 2/2)2 ) + 2222)
:= ((33/3 + 3)3 ) − 3/3
:= 3 + (((33/3 + 3)3 ) + 3/3)
:= 3 × 3 + ((33/3 + 3)3 )
:= 4 + (44/4 × ((44 − 44/4) + 4))
:= (4 × (4 × (4 × 44 − 4))) − 4
:= 4/4 + (4 × (4 × (4 × 44 − 4)))
:= 5 × 555 − ((5 + 5)/5)5
:= (5 × (5 + 5) × 55) − (5 + 5)/5
:= 5 + ((5 × (5 + 5) × 55) − ((5 + 5)/5))
:= 6 + ((6 × ((6 × 66 − 6) + 66)) + 6/6)
:= 6 + ((6 × ((6 × 66 − 6) + 66)) + 6)
:= 6 + ((66 + 6/6) × (6 × 6 − 6/6 + 6))
:= 7 × 7 × (7 × 7 + 7) − 7/7
:= 77/7 + 7 × (7 × (7 × 7 + 7)) − 7
:= 7 + (7 × 7 × (7 × 7 + 7) + ((7 + 7)/7))
:= (8 × ((8 + 8) × (8 + 8) + 88)) − (8/8 + 8)
:= ((8 × (8 × 88 − (8 + 8))) − 8)/((8 + 8)/8)
:= 8/8 + (8 × ((8 + 8) × (8 + 8) + 88))
:= (9 + 9) × (9 × (9 + 9) − 9) − 99/9
:= ((9 + 9 + 9)/9) + ((9 + 9) × (9 × (9 + 9) − 9) − 9)
:= (9 + 9) × (9 × (9 + 9) − 9) − 9/9
Ï 2744 := (1 + (1 + 1 + 11))1+1+1
Ï 2749 := ((111 − 1) × (1 + ((1 + 1) × (1 + 11)))) − 1
Ï 2754 := 11 + ((1 + 1 + 1 + 11)1+1+1 − 1)
:= (22+2 − 2)2/2+2
:= 2222 + (22 + 2/2)2 − 2
:= 2 + ((22 × (((22/2)2 + 2) + 2)) + 2)
:= (33/3 + 3)3
:= 3 + ((((33/3 + 3)3 ) − 3/3) + 3)
:= 33 × (3 × 33 + 3)
:= ((44 − 4)/4 + 4)4−4/4
:= 4/4 + ((4 × (4 × (4 × 44 − 4))) − 4)
:= (4 + 4)/4 + (4 × (4 × (4 × 44 − 4)))
:= (5 × (5 + 5) × 55) − 5/5
:= 5 + ((5 × (5 + 5) × 55) − 5/5)
:= (5 × (5 + 5) × 55) − (5/5 + 5)
:= (66 × (6 × 6 + 6)) − ((66/6 + 6) + 6)
:= (66 × (6 × 6 + 6)) − 6 − 6 − 6
:= 7 × 7 × (7 × 7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 × 7 × (7 × 7 + 7)
:= 8 × ((8 − 8/8)88/8−8 )
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 88)) − (88/8))
:= (8 + 8)/8 + (8 × ((8 + 8) × (8 + 8) + 88))
:= ((9 − 99)/(9 + 9)) + (9 + 9) × (9 × (9 + 9) − 9)
:= (9 + 9) × (9 × (9 + 9) − 9)
:= ((6 + 6)/6 + 6 + 6)
6×6/(6+6)
:= (99 − 9/9) × ((9/9 + 9 + 9) + 9)
Ï 2745 := 1 + (1 + 1 + 1 + 11)1+1+1
Ï 2750 := (111 − 1) × (1 + ((1 + 1) × (1 + 11)))
Ï 2755 := 11 + (1 + 1 + 1 + 11)1+1+1
:= 2/2 + (22+2 − 2)2/2+2
:= 22 × (((22/2)2 + 2) + 2)
:= 22/2 + (22+2 − 2)2/2+2
:= 3/3 + ((33/3 + 3)3 )
:= 3 + (((33/3 + 3)3 ) + 3)
:= 3/3 + (33 × (3 × 33 + 3))
:= (44 + 4/4) × ((44 + 4)/4 − 4)
:= 44/4 × (44 − ((4 + 4)/4 + 4))
:= 4 + ((4 × (4 × (4 × 44 − 4))) − 4/4)
:= (5 × (5 + 5) × 55) − 5
:= 5 × (5 + 5) × 55
:= 5 + (5 × (5 + 5) × 55)
6×6/(6+6)
:= 6 + (((6 + 6)/6 + 6 + 6)
:= 7/7 + 7 × 7 × (7 × 7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7) − 7/7)
:= 77/7 + 7 × 7 × (7 × 7 + 7)
:= 8/8 + (8 × ((8 − 8/8)88/8−8 ))
:= (8/8 + 8 + 8 + 8) × (888 − 8)/8
:= 88 + ((88/8 − 8) × (888 + 8/8))
:= (9 + 9) × (9 × (9 + 9) − 9) − 9
:= 99/9 × ((9 × (9 + 9 + 9) − ((9 + 9)/9)) + 9)
:= 9/9 + (9 + 9) × (9 × (9 + 9) − 9)
Ï 2751 := 1 + ((111 − 1) × (1 + ((1 + 1) × (1 + 11))))
Ï 2756 := 1 + (11 + (1 + 1 + 1 + 11)1+1+1 )
Ï 2746 := 1 + (1 + (1 + 1 + 1 + 11)1+1+1 )
)
:= (66 × (6 × 6 + 6)) − (66/6 + 6)
:= 6 + ((6 × ((6 × (66 + 6)) + 6)) + 666/6)
:= 2 + (22+2 − 2)2/2+2
:= 2222 + ((22 + 2/2)2 )
:= 2 × ((2 × (22 − 2))2 − 222)
:= 3 + (((33/3 + 3)3 ) − 3/3)
:= (33 × (3 × 33 + 3)) − 3
:= 3 + (((33/3 + 3)3 ) + 3 × 3)
:= (4 × (4 × (4 × 44 − 4))) − ((4 + 4)/4 + 4)
:= (4 × (4 × (4 × 44 − 4))) − 4/4
:= 4 + (4 × (4 × (4 × 44 − 4)))
:= 5/5 + ((5 × (5 + 5) × 55) − 5)
:= 5/5 + (5 × (5 + 5) × 55)
:= 5 + ((5 × (5 + 5) × 55) + 5/5)
:= (((6 + 6)/6)6 × ((6 × 6 + 6/6) + 6)) − 6
:= (6/6 + 6) × (6 × 66 − (6 × 6/(6 + 6)))
:= (6 − 66)/6 + ((66 × (6 × 6 + 6)) − 6)
:= (7 + 7)/7 + 7 × 7 × (7 × 7 + 7)
:= 7 + 7 × 7 × (7 × 7 + 7)
:= (77 + 7)/7 + 7 × 7 × (7 × 7 + 7)
:= (8 × ((8 + 8) × (8 + 8) + 88)) − 8/8
:= ((8 × (8 × 88 − (8 + 8))) + 8)/((8 + 8)/8)
:= (9 + 9) × (9 × (9 + 9) − 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 + 9) × (9 × (9 + 9) − 9)
:= (8 + 8)/8 + (8 × ((8 − 8/8)88/8−8 ))
:= 9/9 + ((9 + 9) × (9 × (9 + 9) − 9) − 9)
Ï 2747 := 1 + (1 + (1 + (1 + 1 + 1 + 11)1+1+1 ))
:= 2 + ((22+2 − 2)2/2+2 + 2/2)
Ï 2752 := (1 + 1)11−1 + ((1 + 11)1+1+1 )
:= 2 + (22 × (((22/2)2 + 2) + 2))
3
Ï 2757 := 1 + (1 + (11 + (1 + 1 + 1 + 11)1+1+1 ))
:= 2 + ((22+2 − 2)2/2+2 + 22/2)
:= 3 + ((33/3 + 3)3 )
:= 3 × 3 + (((33/3 + 3) ) − 3/3)
:= 3 + (33 × (3 × 33 + 3))
:= (4 × (4 × (4 × 44 − 4))) − (4/4 + 4)
:= 4 × (4 × (4 × 44 − 4))
:= 4 + ((4 × (4 × (4 × 44 − 4))) + 4/4)
:= (5 + 5)/5 + ((5 × (5 + 5) × 55) − 5)
:= (5 + 5)/5 + (5 × (5 + 5) × 55)
:= 5 + ((5 × (5 + 5) × 55) + ((5 + 5)/5))
6
:= ((6 + 6)/6) × ((6 × 6 + 6/6) + 6)
:= 6 + ((6/6 + 6) × (6 × 66 − (6 × 6/(6 + 6))))
:= (7 + 7 + 7)/7 + 7 × 7 × (7 × 7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7) + 7/7)
:= 7 + ((7 × 7 × (7 × 7 + 7) − 7/7) + 7)
:= 8 + (88/8 × (((8 + 8) × (8 + 8) − 8) + 8/8))
:= 8 × ((8 + 8) × (8 + 8) + 88)
:= 8 + (((8 × ((8 + 8) × (8 + 8) + 88)) − (88/8)) + 8)
:= (9 + 9)/9 + ((9 + 9) × (9 × (9 + 9) − 9) − 9)
:= (9 + 9) × (9 × (9 + 9) − 9) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 + 9) × (9 × (9 + 9) − 9)
:= (66 + 6/6) × (6 × 6 − 6/6 + 6)
193
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2758 := ((1 + (11 + 11)) × (111+1 − 1)) − 1 − 1
Ï 2763 := 111 + ((1 + 11) × ((1 + 1) × 111 − 1))
Ï 2768 := 1 + (1 + (1 + (((1 + 1)1+11 ) − 111+1+1 )))
:= 2 + (2 × ((2 × (22 − 2))2 − 222))
:= 2 + (22/2 × (((222 + 22)/2) − 2))
:= 2 + ((22+2 − 2)2/2+2 + 22)
:= 3 + ((33 × (3 × 33 + 3)) + 3/3)
:= 3 × (((33 × 33 ) + 33 ) + 3)
:= 33 + (((33/3 + 3)3 ) − 3)
:= 4 + ((4 × (4 × (4 × 44 − 4))) + (4 + 4)/4)
:= 44/4 + (4 × (4 × (4 × 44 − 4)))
:= 4 × ((4 × (4 × 44 − 4)) + 4)
:= 5 + (((5 − 5 )/(5 + 5)) − 55)
:= 5 × 555 − (55 + 5)/5
:= 5 × 555 − ((5 + 5)/5 + 5)
:= (6/6 + 6) × (6 × 66 − ((6 + 6)/6))
:= (((6 − 66) + 6)/6) + (66 × (6 × 6 + 6))
:= (6 + 6)/6 + ((66 × (6 × 6 + 6)) − 6)
:= 7 + 7 × (7 × (7 × 7 + 7) + 7)
:= 7 + (7 × 7 × (7 × 7 + 7) + (77 + 7)/7)
:= (7/7 + 7) × (((7 + 7 + 7)/7) + 7 × 7 × 7)
:= 8 + ((8/8 + 8 + 8 + 8) × (888 − 8)/8)
:= 88/8 + (8 × ((8 + 8) × (8 + 8) + 88))
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 88)) + 8)
:= ((9 + 9)/9) × (9 × (9 × (9 + 9) − 9) + ((9 + 9)/9))
:= 9 + (9 + 9) × (9 × (9 + 9) − 9)
:= 9 × 9 × 9 + ((((9 + 9)/9)99/9 ) − 9)
5
5
Ï 2759 := ((1 + (11 + 11)) × (111+1 − 1)) − 1
Ï 2764 := (1 + 1)1+11 − 1 − 111+1+1
Ï 2769 := 111 + (1 + 1) × (111+1+1 − 1 − 1)
:= 22 + ((22 + 2/2) × ((22/2)2 − 2))
:= 22 + ((22+2 − 2)2/2+2 − 2)
:= (2/2 + 2) × ((2 × (222 − 22)) − 2/2)
:= 3 + ((((33/3 + 3)3 ) + 3 × 3) + 3)
:= 3 + ((33 × (3 × 33 + 3)) − 33/3)
:= (33 × (3 × 33 + 3)) − 3
:= 4 + (((4 × (4 × (4 × 44 − 4))) − 4/4) + 4)
:= (44 × ((44 − 4)/4)) − 4 − 4
:= ((44 × (44 − 4) + 4)/4) − 4
:= 5 × 555 − (55/5 + 5)
:= 5 × 555 − 55/5
:= 5 × 555 − (5/5 + 5)
:= (66 × (6 × 6 + 6)) − (6/6 + 6 + 6)
:= (66 × (6 × 6 + 6)) − ((6 + 6)/6 + 6)
:= (66 × (6 × 6 + 6)) − 6 × 6/(6 + 6)
:= 7 + ((7 × 7 × (7 × 7 + 7) + 7/7) + 7)
:= 7 + (((7 × 7 × (7 × 7 + 7) − 7/7) + 7) + 7)
:= 7 + ((7 × 7 × (7 × 7 + 7) + (77/7)) + 7)
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 88)) − 8/8)
:= ((8 × 8 − 8/8) × (88/((8 + 8)/8))) − 8
:= 8 + (((8 × ((8 + 8) × (8 + 8) + 88)) + 8/8) + 8)
:= 9 × 9 × 9 + ((((9 + 9)/9)99/9 ) − (9 + 9))
:= 9 + ((9 + 9) × (9 × (9 + 9) − 9) + 9/9)
:= 99 + (((9 + 9 + 9)/9) × (9 × 99 − 9/9))
Ï 2760 := (1 + (11 + 11)) × (111+1 − 1)
Ï 2765 := ((1 + 1)1+11 ) − 111+1+1
Ï 2770 := 111 + (((1 + 1) × (111+1+1 − 1)) − 1)
:= (2 + 2 + 2) × (222 − (22 + 2))
:= 22 + ((22+2 − 2)2/2+2 − 2/2)
:= (22 × ((2 × 22+2+2 ) − 2)) − 2
:= 3 + ((33 × (3 × 33 + 3)) + 3)
:= 3 + (((33/3 + 3)3 ) + (3 × (3 + 3)))
:= 33 + (((33/3 + 3)3 ) − 3/3)
:= 4 + ((4 × (4 × (4 × 44 − 4))) + 4)
:= 4 + (44/4 × (44 − 4/4 − 4))
:= (44 × (44 − 4) − (4 + 4))/4
:= 5 + ((5 × (5 + 5) × 55) + 5)
:= 5 × 555 − 5 − 5
:= 5 × 555 − 5
:= (66 × (6 × 6 + 6)) − 6 − 6
:= (6/6 + 6) × (6 × 66 − 6/6)
:= (66 × (6 × 6 + 6)) − (6 + 6)/6
:= (7/7 + 7) × (7 × 7 × 7 + ((7 + 7)/7))
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) + 7)
:= 7 + ((7 × 7 × (7 × 7 + 7) + (77 + 7)/7) + 7)
:= 8 + (8 × ((8 + 8) × (8 + 8) + 88))
:= (8 − 8/8) × ((8 × (8 × 8 − (8 + 8))) + (88/8))
:= 8 + (((8 × ((8 + 8) × (8 + 8) + 88)) + ((8 + 8)/8)) + 8)
:= 9 + ((9 + 9) × (9 × (9 + 9) − 9) − ((9 + 9 + 9)/9))
:= 99/9 + (9 + 9) × (9 × (9 + 9) − 9)
:= 99 + (99 × (9 + 9 + 9) − ((9 + 9)/9))
Ï 2761 := 1 + ((1 + (11 + 11)) × (111+1 − 1))
Ï 2766 := 1 + (((1 + 1)1+11 ) − 111+1+1 )
Ï 2771 := 111 + ((1 + 1) × (111+1+1 − 1))
:= 22/2 × (((222 + 22)/2) − 2)
:= 22 + (22+2 − 2)2/2+2
:= (22 × ((2 × 22+2+2 ) − 2)) − 2/2
:= (33 × (3 × 33 + 3)) − 33/3
:= (33 × (3 × 33 + 3)) − 3 − 3
:= 33 + ((33/3 + 3)3 )
:= 44/4 × (44 − 4/4 − 4)
:= ((44 × (44 − 4) − (4 + 4))/4) − 4
:= (44 × (44 − 4) − 4)/4
:= 55/5 + (5 × (5 + 5) × 55)
:= 5/5 + (5 × 555 − (5 + 5))
:= 5/5 + (5 × 555 − 5)
:= (66 × (6 × 6 + 6)) − 66/6
:= (66 × (6 × 6 + 6)) − 6
:= (66 × (6 × 6 + 6)) − 6/6
:= 7 + (7 × 7 × (7 × 7 + 7) + ((77 − 7)/7))
:= 7 + (((7 × 7 × (7 × 7 + 7) + 7/7) + 7) + 7)
:= 77 + ((7 × (7 × (7 × 7 + 7) − 7)) − 7/7)
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 88)) + 8/8)
:= 8 + (((8/8 + 8 + 8 + 8) × (888 − 8)/8) + 8)
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 88)) + (88/8))
:= 9 + ((9 + 9) × (9 × (9 + 9) − 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + (9 + 9) × (9 × (9 + 9) − 9)
:= 99 + (99 × (9 + 9 + 9) − 9/9)
Ï 2762 := 1 + (1 + ((1 + (11 + 11)) × (111+1 − 1)))
2
Ï 2767 := 1 + (1 + (((1 + 1)1+11 ) − 111+1+1 ))
2+2
2/2+2
Ï 2772 := 11 × ((1 + 11) × (11 + (11 − 1)))
:= 22 × ((2 × 22+2+2 ) − 2)
:= 2 + ((2 + 2 + 2) × (22 − (22 + 2)))
:= 22 + ((2
:= (3 × (3 + 3)) + ((33/3 + 3)3 )
:= 33 + (((33/3 + 3)3 ) − (3/3 + 3))
:= 33 × (3 × 33 + 3)
:= ((44 × (44 − 4/4 − 4)) + 4)/4
:= ((44 × (44 − 4) − 4)/4) − 4
:= 44 × ((44 − 4)/4)
:= ((55 + 5)/5) + (5 × (5 + 5) × 55)
:= (5 + 5)/5 + (5 × 555 − (5 + 5))
:= (5 + 5)/5 + (5 × 555 − 5)
:= (6 − 66)/6 + (66 × (6 × 6 + 6))
:= 6/6 + ((66 × (6 × 6 + 6)) − 6)
:= 66 × (6 × 6 + 6)
− 2)
+ 2/2)
:= 7 + (7 × 7 × (7 × 7 + 7) + (77/7))
:= 7 + ((7/7 + 7) × (7 × 7 × 7 + ((7 + 7)/7)))
:= 77 + (7 × (7 × (7 × 7 + 7) − 7))
:= 8 + ((8 × ((8 + 8) × (8 + 8) + 88)) + ((8 + 8)/8))
:= (888/8 × (8/8 + 8 + 8 + 8)) − 8
:= (8 × 8 − 8/8) × (88/((8 + 8)/8))
:= 9 + ((9 + 9) × (9 × (9 + 9) − 9) − 9/9)
:= ((99 + 9 + 9)/9) + (9 + 9) × (9 × (9 + 9) − 9)
:= 99 + 99 × (9 + 9 + 9)
194
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2773 := 111 + ((1 + 1) × 111+1+1 )
Ï 2778 := (1 + 11111)/(1 + 1 + 1 + 1)
:= (22 + 2/2) × (22/2)2
:= 3/3 + (33 × (3 × 33 + 3))
:= 3 + ((33 × (3 × 33 + 3)) + 3)
:= 3 + ((((33/3 + 3)3 ) + 33) + 3)
:= (44 × (44 − 4) + 4)/4
:= 4 + (((44 × (44 − 4) + 4) + 4)/4)
:= 44/4 × ((4/4 − 4) + 44 )
) − 2))
2
Ï 2783 := 11 × (11 × (1 + (11 + 11)))
:= (2/2 + 2) × ((2 × (22 − 22)) + 2)
:= 2/2 + (22 × ((2 × 2
2+2+2
:= 5 × 555 − (5 + 5)/5
:= 5 + (5 × 555 − ((5 + 5)/5))
:= 5 + ((5 × 555 − ((5 + 5)/5)) + 5)
:= 6/6 + (66 × (6 × 6 + 6))
:= 6 + (66 × (6 × 6 + 6))
:= 66/6 + (66 × (6 × 6 + 6))
:= 7/7 + ((7 × (7 × (7 × 7 + 7) − 7)) + 77)
:= (7 − 7/7) × ((77 × (7 − 7/7)) + 7/7)
:= ((7 − 77)/7) + 7 × (7 × (7 × 7 + 7) + 7)
:= 8/8 + ((8 × 8 − 8/8) × (88/((8 + 8)/8)))
:= ((8 + 8)/8) × (88 × (8 + 8) − (88/8 + 8))
:= 8 + (888/8 × (8/8 + 8 + 8 + 8))
99/9
:= 9/9 + (99 × (9 + 9 + 9) + 99)
:= 9/9 + ((((9 + 9)/9)
Ï 2774 := (111 × (1 + ((1 + 1) × (1 + 11)))) − 1
2+2+2
) + 9 × 9 × 9)
Ï 2779 := 1 + ((1 + 11111)/(1 + 1 + 1 + 1))
:= 99 + (99 × (9 + 9 + 9) + (99/9))
Ï 2784 := 1 + (11 × (11 × (1 + (11 + 11))))
:= ((22 + 2/2) × (22/2)2 ) − 2 − 2
:= (2 + 2 + 2) × ((222 − 22) + 2)
:= 3 + (((33/3 + 3)3 ) + 33 )
:= 3 + (((33 × (3 × 33 + 3)) + 3/3) + 3)
:= (3/3 + 3) × (33+3 − 33)
:= ((44 × (44 − 4) + 4) + 4)/4
:= 4 + (((44 × (44 − 4) − 4)/4) + 4)
:= 4 × ((444 − 4) + 44 )
:= 5 × 555 − 5/5
:= 5 + (5 × 555 − 5/5)
:= 5 + ((5 × 555 − 5/5) + 5)
:= (6 + 6)/6 + (66 × (6 × 6 + 6))
:= 6 + ((66 × (6 × 6 + 6)) + 6/6)
:= 6 + ((66 × (6 × 6 + 6)) + 6)
:= 2 + (22 × ((2 × 2
) − 2))
:= 7 + (((7/7 + 7) × (7 × 7 × 7 + ((7 + 7)/7))) + 7)
:= 7 × (7 × (7 × 7 + 7) + 7) − 7 − 7
:= (7/7 + 7) × ((7 × 7 × 7 − ((7 + 7)/7)) + 7)
:= (888/8 × (8/8 + 8 + 8 + 8)) − 8/8
:= (((8 × (8 × 88 − 8)) − 8)/((8 + 8)/8)) − 8/8
:= 8 × ((8 × 88 − 8)/((8 + 8)/8))
:= 9 + ((9 + 9) × (9 × (9 + 9) − 9) + (99/9))
:= 999 + ((9 + 9) × 99 − ((9 + 9)/9))
:= 999/9 + 99 × (9 + 9 + 9)
Ï 2775 := 111 × (1 + ((1 + 1) × (1 + 11)))
Ï 2780 := 1 + (1 + ((1 + 11111)/(1 + 1 + 1 + 1)))
Ï 2785 := 1 + (1 + (11 × (11 × (1 + (11 + 11)))))
:= 222/2 × ((22 + 2/2) + 2)
:= 2 × ((2 × (222 + 22 )) − 22)
:= 2 + ((22 + 2/2) × (22/2)2 )
:= 3 + (33 × (3 × 33 + 3))
:= 3 + (((33/3 + 3)3 ) + 33)
:= 3/3 + ((3/3 + 3) × (33+3 − 33))
:= 4 + ((44 × (44 − 4) − 4)/4)
:= 4 + ((44 × ((44 − 4)/4)) + 4)
:= 4/4 + (4 × ((444 − 4) + 44 ))
:= 5 × 555
:= 5 + 5 × 555
:= 5 + (5 × 555 + 5)
:= 6 + ((66 × (6 × 6 + 6)) + ((6 + 6)/6))
:= 6 + (((66 × (6 × 6 + 6)) + 6/6) + 6)
:= 7 × (77 + 7) + ((7 + 7 + 7)/7)
:= 7/7 + (7 × (7 × (7 × 7 + 7) + 7) − (7 + 7))
:= 7 × (7 × (7 × 7 + 7) + 7) − (7/7 + 7)
:= 888/8 × (8/8 + 8 + 8 + 8)
:= ((8 × (8 × 88 − 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 × ((8 × 88 − 8)/((8 + 8)/8)))
:= 999/9 + (99 × (9 + 9 + 9) − 9)
:= 999 + ((9 + 9) × 99 − 9/9)
:= 99 × (9 + 9 + 9) + ((999 + 9)/9)
2
:= (6 × 6/(6 + 6)) + (66 × (6 × 6 + 6))
7
Ï 2776 := 1 + (111 × (1 + ((1 + 1) × (1 + 11))))
Ï 2781 := (11 × (11 × (1 + (11 + 11)))) − 1 − 1
Ï 2786 := 11 + (111 × (1 + ((1 + 1) × (1 + 11))))
:= ((22 + 2/2) × (22/2)2 ) − 2
:= 222 + (((2 × (22 + 2))2 ) − 2)
:= 3 + ((33 × (3 × 33 + 3)) + 3/3)
:= 3 × (((33 × 33 ) + 33) + 3)
:= 3 × 3 + (((33/3 + 3)3 ) + 33)
:= 4 + (44 × ((44 − 4)/4))
:= 4 + (((44 × (44 − 4) + 4)/4) + 4)
:= 4 × 4 + ((44 × (44 − 4) − (4 + 4))/4)
:= 5/5 + 5 × 555
:= 5 + (5 × 555 + 5/5)
:= 55/5 + 5 × 555
:= 6 + ((66 × (6 × 6 + 6)) − ((6 + 6)/6))
:= 6 + ((66 × (6 × 6 + 6)) + (6 × 6/(6 + 6)))
:= (6/6 + 6) × (((6 + 6)/6) + 6 × 66)
:= (7/7 + 7) × ((7 × 7 × 7 − 7) + (77/7))
:= 7 × (7 × (7 × 7 + 7) + 7) − (77 + 7)/7
:= 7 × (7 × (7 × 7 + 7) + 7) − 7
:= 88 + ((8 + 8) × (88 − 8 + 88))
:= (88/8 + 8 + 8) × (888/8 − 8)
:= (8 + 8)/8 + (8 × ((8 × 88 − 8)/((8 + 8)/8)))
:= 999 + (9 + 9) × 99
:= 9 + ((((9 + 9)/9)99/9 ) + 9 × 9 × 9)
:= 2 + ((22 × ((2 × 2
2+2+2
) − 2)) + 2)
99/9
:= 9 × 9 × 9 + ((((9 + 9)/9)
) − 9/9)
Ï 2777 := 1 + (1 + (111 × (1 + ((1 + 1) × (1 + 11)))))
Ï 2782 := (11 × (11 × (1 + (11 + 11)))) − 1
2
2
Ï 2787 := 1 + (11 + (111 × (1 + ((1 + 1) × (1 + 11)))))
:= 2 + (222/2 × ((22 + 2/2) + 2))
:= 2 × ((((2 + 2 + 2) + 2/2) ) + 22)
:= 2 + (((22 + 2/2) × (22/2)2 ) + 2)
:= 33 + ((33/3 + 3)3 )
:= 33 + ((33 × (3 × 33 + 3)) + 3/3)
:= 33 + (33 × (3 × 33 + 3))
:= 4 + ((44 × (44 − 4) + 4)/4)
:= 4 + ((((44 × (44 − 4) + 4) + 4)/4) + 4)
:= 4 + (44/4 × ((4/4 − 4) + 44 ))
:= (5 + 5)/5 + 5 × 555
:= 5 + (5 × 555 + ((5 + 5)/5))
:= 5 × 555 + (55 + 5)/5
:= 6 + ((66 × (6 × 6 + 6)) − 6/6)
:= ((66 − 6)/6) + (66 × (6 × 6 + 6))
:= 6 + (((66 × (6 × 6 + 6)) + (6 × 6/(6 + 6))) + 6)
:= 77 + ((7 − ((7 + 7)/7)) × (7 × 77 + 7/7))
:= 7 × (7 × (7 × 7 + 7) + 7) − 77/7
:= 7/7 + (7 × (7 × (7 × 7 + 7) + 7) − 7)
:= 8/8 + (((8 + 8) × (88 − 8 + 88)) + 88)
:= ((8 + 8)/8) × (88 × (8 + 8) − (8/8 + 8 + 8))
:= (((8 × (8 × 88 − 8)) + 8)/((8 + 8)/8)) − 8/8
:= 9 × 9 × 9 + (((9 + 9)/9)99/9 )
:= 9/9 + ((9 + 9) × 99 + 999)
:= 9 × (9 + 9) × (9 + 9) − ((999/9 + 9) + 9)
195
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2788 := 1 + (1 + (11 + (111 × (1 + ((1 + 1) × (1 + 11))))))
2
2
Ï 2793 := (11 + (11 − 1)) × (1 + (11 × (1 + 11)))
2×(2+2)
Ï 2798 := 1 + (1 + ((1 + 11) × (11 + (1 + 1) × 111)))
:= 2222 + ((22 + 2)2 )
:= 22 + ((2 × (22 + 2)) )
:= ((22 × (2
:= (3/3 + 33) × (3 × 33 + 3/3)
:= 3 + ((33 + 3) × ((3 × (33 + 3)) + 3))
:= 33 + (((33/3 + 3)3 ) + 33 )
:= 4 + (4 × ((444 − 4) + 44 ))
:= 4 + (((44 × (44 − 4) + 4)/4) + 4 × 4)
:= 4 + (44/4 × (44 − (4 + 4)/4))
:= 55 + (((5 − 55 )/(5 + 5)) − 5 × 5)
:= (5 × (555 + 5)) − ((5 + 5)/5 + 5)
:= (5 × (555 + 5)) − (5 + 5)/5
:= 6 + ((66 × (6 × 6 + 6)) + ((66 − 6)/6))
:= (6/6 + 6) × ((6 × 6/(6 + 6)) + 6 × 66)
:= (6 − 66)/6 + (6 × (6 × (66 + 6 + 6)))
:= (7 + 7)/7 + (7 × (7 × (7 × 7 + 7) + 7) − 7)
:= 7 × (7 × (7 × 7 + 7) + 7)
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) − ((7 + 7)/7))
− 2)) − 2)/2
(8+8)/8
:= ((8 × (8 × 88 − 8)) + 8)/((8 + 8)/8)
:= ((8 × 8 − 88/8)
:= ((9 − 9/9) + 9) × (((9 + 9)/9) + 9 × (9 + 9))
:= 9 + (99 × (9 + 9 + 9) + 999/9)
Ï 2789 := 11 + ((1 + 11111)/(1 + 1 + 1 + 1))
)−8−8
Ï 2794 := 11 × (1 + (11 × (1 + (11 + 11))))
:= ((8 + 8)/8) × (88 × (8 + 8) − (8/8 + 8))
:= 9 + ((((9 + 9) × 99 − 9/9) + 999) + 9)
Ï 2799 := ((1 + 111) × (1 + ((1 + 1) × (1 + 11)))) − 1
:= 2/2 + (((2 × (22 + 2))2 ) + 222 )
:= 22 × ((22×(2+2) − 2)/2)
:= 2/2 + (2222 + ((22 + 2)2 ))
:= 3 + ((((33/3 + 3)3 ) + 33) + 3 × 3)
:= 33/3 × ((33+3 + 33)/3)
:= 33 + (33 × (3 × 33 + 3))
:= 4 × 4 + ((44 × (44 − 4) + 4)/4)
:= 44/4 × (44 − (4 + 4)/4)
:= (4 × (444 + 44 )) − 4/4
:= (5 × (555 + 5)) − 55/5
:= (5 × (555 + 5)) − (5/5 + 5)
:= (5 × (555 + 5)) − 5/5
:= 6 + ((66 × (6 × 6 + 6)) + (66/6))
:= 66/6 × (6 × (6 × 6 + 6) + ((6 + 6)/6))
:= 6 + ((6/6 + 6) × ((6 × 6/(6 + 6)) + 6 × 66))
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) − (77/7))
:= 7/7 + 7 × (7 × (7 × 7 + 7) + 7)
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) − 7/7)
:= 8 + ((88/8 + 8 + 8) × (888/8 − 8))
:= ((8 + 8)/8) × (88 × (8 + 8) − (88/8))
:= (88 × ((8 + 8 + 8) + 8)) − (8/8 + 8 + 8)
:= 9 + (((9 + 9) × 99 − 9/9) + 999)
:= 99/9 × (9 × (9 + 9 + 9) + (99/9))
:= 9 + (((9 + 9) × 99 + 999) + 9)
Ï 2790 := 1 + (11 + ((1 + 11111)/(1 + 1 + 1 + 1)))
2
2
Ï 2795 := 1 + (11 × (1 + (11 × (1 + (11 + 11)))))
2×(2+2)
:= 2 + (2222 + ((22 + 2)2 ))
:= 2 + (((2 × (22 + 2)) ) + 22 )
:= ((22 × (2
:= (33 + 3) × ((3 × (33 + 3)) + 3)
:= 33 + ((((33/3 + 3)3 ) − 3) + 33 )
:= (33 + 3/3) × (3 × 33 + 3/3)
:= (44 + 4/4) × (44 − 4 − 4)/4
:= (44 − 4/4) × (44 + 4)/4
:= 4 × (444 + 44 )
:= 5 + ((5 × 555 + 5) + 5)
:= (5 × (555 + 5)) − 5
:= 5 × (555 + 5)
:= 6 + (((66 × (6 × 6 + 6)) + 6) + 6)
:= (6/6 + 6 + 6) × (6 × 6 × 6 − 6/6)
:= (((6 + 6)/6)6+6 ) − 6 × 6 × 6 × 6
:= (77/7 + 7) × (7/7 + 77 + 77)
:= (7 + 7)/7 + 7 × (7 × (7 × 7 + 7) + 7)
:= 7 + 7 × (7 × (7 × 7 + 7) + 7)
:= (8 × 8 − ((8 + 8)/8)) × (8 × 8 − (88/8 + 8))
:= 88/8 + (8 × ((8 × 88 − 8)/((8 + 8)/8)))
:= (8 + 8) × (888/8 + 8 × 8)
:= 9 + ((9 + 9) × 99 + 999)
Ï 2791 := (11 × (1 + (11 × (1 + (11 + 11))))) − 1 − 1 − 1
− 2)) + 2)/2
Ï 2800 := (1 + 111) × (1 + ((1 + 1) × (1 + 11)))
99/9
:= 9 + (((((9 + 9)/9)
) + 9 × 9 × 9) + 9)
Ï 2796 := (1 + 11) × (11 + (1 + 1) × 111)
:= (9/9 + 99) × ((9/9 + 9 + 9) + 9)
Ï 2801 := 1 + ((1 + 111) × (1 + ((1 + 1) × (1 + 11))))
:= (((22 × (22×(2+2) − 2)) − 2)/2) − 2
:= 2 + (22 × ((22×(2+2) − 2)/2))
:= (((22 − 2)2 × (22+2 − 2)) + 2)/2
:= 3 + ((3/3 + 33) × (3 × 33 + 3/3))
:= 33 + ((33 × (3 × 33 + 3)) − 3)
:= 3 + ((((33/3 + 3)3 ) + 33 ) + 33 )
:= ((44 − 4/4) × (44 + 4)/4) − 4
:= (4 × (444 + 44 )) − 4
:= 4/4 + (4 × (444 + 44 ))
:= 5 + (5 × 555 + (55/5))
:= 5/5 + ((5 × (555 + 5)) − 5)
:= 5/5 + (5 × (555 + 5))
:= 6 + ((((66 × (6 × 6 + 6)) + 6/6) + 6) + 6)
:= (6 × (6 × (66 + 6 + 6))) − 6 − 6
:= (6 × (6 × (66 + 6 + 6))) − 6/6 − 6
:= 7 × (7 × (7 × 7 + 7) + 7) − (7 + 7)/7
:= (7 + 7 + 7)/7 + 7 × (7 × (7 × 7 + 7) + 7)
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) + 7/7)
:= 8 + ((888/8 × (8/8 + 8 + 8 + 8)) + 8)
:= 8 + (((8 × (8 × 88 − 8)) + 8)/((8 + 8)/8))
:= ((8 × 8 − 88/8)(8+8)/8 ) − 8
:= 9 + (((9 + 9) × 99 + 999) + 9/9)
:= 9 × (9 + 9) × (9 + 9) − (999/9 + 9)
:= 9 + (((9 + 9) × 99 + 999) + (99/9))
Ï 2792 := (11 × (1 + (11 × (1 + (11 + 11))))) − 1 − 1
2
Ï 2797 := 1 + ((1 + 11) × (11 + (1 + 1) × 111))
2×(2+2)
Ï 2802 := 1 + (1 + ((1 + 111) × (1 + ((1 + 1) × (1 + 11)))))
:= 2 + ((2222 + ((22 + 2)2 )) + 2)
:= 2 × (2 × (((22 + 2 + 2) ) + 22))
:= 2 + (((22 × (2
:= 3 × 33 + (((33/3 + 3)3 ) − 33)
:= 3 + (33/3 × ((33+3 + 33)/3))
:= 3 + ((33 × (3 × 33 + 3)) + 33 )
:= (4 × (444 + 44 )) − 4 − 4
:= 4/4 + ((4 × (444 + 44 )) − 4)
:= (4 + 4)/4 + (4 × (444 + 44 ))
:= 5 + (5 × 555 + ((55 + 5)/5))
:= (5 + 5)/5 + ((5 × (555 + 5)) − 5)
:= (5 + 5)/5 + (5 × (555 + 5))
:= 6 + ((6/6 + 6) × (((6 + 6)/6) + 6 × 66))
:= (6 × (6 × (66 + 6 + 6))) − 66/6
:= (6 × (6 × (66 + 6 + 6))) − 6
:= 7 × (7 × (7 × 7 + 7) + 7) − 7/7
:= 77/7 + (7 × (7 × (7 × 7 + 7) + 7) − 7)
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) + ((7 + 7)/7))
:= 8 + (8 × ((8 × 88 − 8)/((8 + 8)/8)))
:= (88 × ((8 + 8 + 8) + 8)) − (88/8 + 8)
:= ((8 + 8)/8) × ((88 × (8 + 8) − 8) + 8/8)
:= 99/9 + ((9 + 9) × 99 + 999)
:= 9 + (((9 − 9/9) + 9) × (((9 + 9)/9) + 9 × (9 + 9)))
:= 9 + ((99 × (9 + 9 + 9) + 999/9) + 9)
− 2)) + 2)/2)
196
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2803 := (11 × (111 + (1 + 11)1+1 )) − 1 − 1
2
Ï 2808 := (1 + 11) × (1 + (11 + (1 + 1) × 111))
:= 2 × (2 × ((2 × (22 × 2
:= ((22/2 + 2 × 22) ) − 222
3
2+2
)) − 2))
3
Ï 2813 := 11 × (1 + 1)(1+1)
:= (((22 × 2
2×(2+2)
1+1+1
−1−1−1
) − 2)/2) − 2
:= 3 + ((3 + 3/3) × (3 × 33 + 3/3))
:= (33 + 3) × (3 × 3 − 3)
:= 3 + ((((33/3 + 3)3 ) + 33) + 33)
:= 4 + ((4 × (444 + 44 )) − 4/4)
:= (4 × 4 × 4 × 44) − 4 − 4
:= 4/4 + ((4 × 4 × 4 × 44) − 4)
5
5
:= 5 + ((5 × (555 + 5)) − ((5 + 5)/5))
:= 5 + (((5 − 5 )/(5 + 5)) − 5)
:= 55 + ((5 − 55 )/(5 + 5))
:= 6/6 + ((6 × (6 × (66 + 6 + 6))) − 6)
:= 6 × (6 × (66 + 6 + 6))
:= 6 + ((6 × (6 × (66 + 6 + 6))) − 6/6)
:= 77 + (((7 + 7 + 7)/7) + 7 × 77)
:= (7/7 + 7) × ((7 × 7 × 7 + 7/7) + 7)
:= 77 + ((7/7 + 7) × (7 × 7 × 7 − 7/7))
:= (88 × ((8 + 8 + 8) + 8)) − (88 + 8 + 8)/8
:= (88 × ((8 + 8 + 8) + 8)) − 8
:= 8 + (88/8 × ((8 + 8) × (8 + 8) − 8/8))
:= 9 + ((99/9) × (9 × (9 + 9 + 9) + (99/9)))
:= (99 + 9) × (((9 − 9/9) + 9) + 9)
:= (99/9 + 9 + 9) × (99 − ((9 + 9)/9))
7
Ï 2804 := (11 × (111 + (1 + 11)1+1 )) − 1
:= 2 × ((2 × (2 × ((22 × 2
2+2
) − 2))) + 2)
Ï 2809 := ((111 − 1)/(1 + 1) − (1 + 1))1+1
1+1+1
Ï 2814 := 11 × (1 + 1)(1+1)
2
:= (2 × 22 × 2
3
:= (2 × (22 + 2 + 2) + 2/2)
2+2+2
−1−1
)−2
:= 33 + (((33/3 + 3)3 ) + 33)
:= 3/3 + ((33 + 3) × (3 × 3 − 3))
:= 3 + (((33 + 3) × (3 × 33 − 3)) + 3)
:= 4 + (4 × (444 + 44 ))
:= 4 + (44/4 × (44 − 4/4))
:= (4 × 4 × 4 × 44) − (4 + 4)/4
(5+5)/5
:= 5 + ((5 × (555 + 5)) − 5/5)
:= (55 − (5 + 5)/5)
:= (6 + 6)/6 + ((6 × (6 × (66 + 6 + 6))) − 6)
:= 6/6 + (6 × (6 × (66 + 6 + 6)))
:= 77/7 + 7 × (7 × (7 × 7 + 7) + 7)
:= ((77/7 − 7) + 7 × 7)
(7+7)/7
:= (((8 × 8 × 88) − 8)/((8 + 8)/8)) − 8
:= (8 × 8 − 88/8)
:= 9 × (9 + 9) × (9 + 9) − ((999 + 9)/9)
:= 9 + ((9/9 + 99) × ((9/9 + 9 + 9) + 9))
Ï 2805 := 11 × (111 + (1 + 11)1+1 )
:= 22/2 × (2
2×(2+2)
− 2/2)
(8+8)/8
Ï 2810 := 1 + (((111 − 1)/(1 + 1) − (1 + 1))1+1 )
:= (2 × (22 × 22+2+2 − 2)) − 2
:= 5 + ((55 − (5 + 5)/5)(5+5)/5 )
:= 6 + (6 × (6 × (66 + 6 + 6)))
:= 77 + 7 × (7 × (7 × 7 + 7)) − 7
:= ((8 + 8)/8) × (88 × (8 + 8) − 8/8)
:= 9 + (9 × (9 + 9) × (9 + 9) − 999/9)
Ï 2815 := 11 × (1 + 1)(1+1)
:= ((22 × 2
2×(2+2)
1+1+1
−1
) − 2)/2
:= 33 + (33 × (3 × 3 + 3))
:= 33 + (((33/3 + 3)3 ) + 33)
:= ((33/3 + 3) ) + (((3 + 3)3 − 3)/3)
:= 44/4 × (44 − 4/4)
:= (4 × 4 × 4 × 44) − ((4 + 4)/4 + 4)
:= (4 × 4 × 4 × 44) − 4/4
:= 5 + (5 × (555 + 5))
:= 5 + ((5 × (555 + 5)) + 5)
:= 5 + (((5 × (555 + 5)) + 5) + 5)
3
3
:= (66 × (6 × 6 + 6)) + (66 × 6/(6 + 6))
:= (6 + 6)/6 + (6 × (6 × (66 + 6 + 6)))
:= 6 + ((6 × (6 × (66 + 6 + 6))) + 6/6)
:= (77 + 7)/7 + 7 × (7 × (7 × 7 + 7) + 7)
:= 77 + (7 × 7 × (7 × 7 + 7) − (77/7))
:= 7 + ((7/7 + 7) × ((7 × 7 × 7 + 7/7) + 7))
:= 88/8 × ((8 + 8) × (8 + 8) − 8/8)
:= 8/8 + ((8 × 8 − 88/8)(8+8)/8 )
:= (88 × ((8 + 8 + 8) + 8)) − 8/8
:= 9 × (9 + 9) × (9 + 9) − 999/9
:= (9 + 9)/9 + ((99 + 9) × (((9 − 9/9) + 9) + 9))
Ï 2806 := (1 + (11 + 11)) × (1 + 111+1 )
Ï 2811 := (1 + 1 + 1) × ((1 + 1)11 − 1111)
:= 9 × (9 + 9) × (9 + 9) − ((9 + 9)/9 + 99)
Ï 2816 := 11 × (1 + 1)(1+1)
1+1+1
:= 2 × 22 × 22+2+2
:= (22 + 2/2) × ((22/2)2 + 2/2)
:= 2 + ((2 × (22 + 2 + 2) + 2/2)2 )
:= 3/3 + ((33 × (3 × 33 + 3)) + 33)
:= 3 + ((33 + 3) × (3 × 33 − 3))
:= 33/3 × ((3/3 + 3)3/3+3 )
:= 4/4 + (44/4 × (44 − 4/4))
:= (4 × 4 × 4 × 44) − (4/4 + 4)
:= 4 × 4 × 4 × 44
:= 5 + ((5 × (555 + 5)) + 5/5)
:= 55/5 + (5 × (555 + 5))
:= 5 + ((5 × (555 + 5)) + (55/5))
:= (6 × (6 × (66 + 6 + 6))) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × (6 × (66 + 6 + 6)))
:= 6 + ((6 × (6 × (66 + 6 + 6))) + ((6 + 6)/6))
:= 7 + ((7 × (7 × (7 × 7 + 7) + 7) − 7/7) + 7)
:= 7 + 7 × (7 × (7 × 7 + 7) + 7) + 77/7
:= 77/7 × (((7 + 7)/7)7/7+7 )
:= (8 − 88)/8 + (88 × ((8 + 8 + 8) + 8))
:= (((8 × 8 × 88) − 8)/((8 + 8)/8)) − 8/8
:= 88 × ((8 + 8 + 8) + 8)
:= ((9 + 9)/9) × ((((9 + 9)/9)9 ) + 9 × 99)
:= 999/9 + ((9 + 9 + 9) × (9/9 + 99))
:= 99/9 × (((9 + 9)/9)9−9/9 )
Ï 2807 := 1 + ((1 + (11 + 11)) × (1 + 111+1 ))
2
:= ((2 × (22 + 2 + 2) + 2/2) ) − 2
Ï 2812 := 1 + ((1 + 1 + 1) × ((1 + 1)11 − 1111))
:= 2 × (22 × 2
2+2+2
− 2)
Ï 2817 := 11 × (1 + 1)(1+1)
:= ((22 × 2
2×(2+2)
1+1+1
+1
) + 2)/2
3
:= ((33 + 3) × (3 × 33 − 3)) − 3/3
:= 3 + (((33 + 3) × (3 × 33 − 3)) + 3/3)
:= 3 × ((3 × (33 + 3)) − 33)
:= (4 × 4 × 4 × 44) − (4/4 + 4 + 4)
:= (4 × 4 × 4 × 44) − 4
:= 4/4 + (4 × 4 × 4 × 44)
:= 55 − ((55 + 5)/(5 + 5) + 5)
:= 55 − (55 + 5)/(5 + 5)
:= 5 + (55 − (55 + 5)/(5 + 5))
:= (6 × (6 × (66 + 6 + 6))) − 6/6
:= 6 + ((6 × (6 × (66 + 6 + 6))) − ((6 + 6)/6))
:= 6 + ((6 × (6 × (66 + 6 + 6))) + (6 × 6/(6 + 6)))
:= 7 + (7 × (7 × (7 × 7 + 7) + 7) + 7)
:= (7/7 − 77) × ((77 + 7)/7 − 7 × 7)
:= 7 + ((7 × 7 × (7 × 7 + 7) − (77/7)) + 77)
:= (88 × ((8 + 8 + 8) + 8)) − (8/8 + 8)
:= ((8 × 8 × 88) − 8)/((8 + 8)/8)
:= 8/8 + (88 × ((8 + 8 + 8) + 8))
:= 9 × (9 + 9) × (9 + 9) − (9/9 + 99 + 9)
:= (9 + 9) × 99 + (9999/9 − 9 × 9)
:= 9 × (9 + 9) × (9 + 9) − 99
197
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2818 := 11 × (1 + 1)(1+1)
:= 2 + (2 × 22 × 2
1+1+1
2+2+2
+1+1
Ï 2823 := 1 + ((1 + (11 × (1 + (1 + 1)11−1−1 )))/(1 + 1))
2
3
3
3+3
1+1+1
)+1
2
:= (2 × (2 × (222 + 22 ))) − 2/2
)
Ï 2828 := 11 × (1 + (1 + 1)(1+1)
:= 2 × ((2 × (222 + 22 )) + 2)
:= 3 + (((33/3 + 3)3 ) + 3 × 33 )
:= 3/3 + (33 × 33 + (3 × 3 + 3) )
:= (3 × (((3 + 3) − 3) + 3
:= (4 + 4)/4 + (4 × 4 × 4 × 44)
:= (44/4 × (4/4 + 44 )) − 4
:= 4 + 4 × 4 × 4 × 44 + 4 + 4
:= 5 + (((5 − 55 )/(5 + 5)) + 55 )
:= 5 + ((((5 − 55 )/(5 + 5)) + 55 ) + 5)
:= 55 + (5 × 555 − ((5 + 5)/5))
:= ((66 − 6)/6) + (6 × (6 × (66 + 6 + 6)))
:= 6 + (((6 × (6 × (66 + 6 + 6))) + (6 × 6/(6 + 6))) + 6)
:= (6/6 + 6) × ((((6 + 6)/6) + 6 × 66) + 6)
)) − 3
:= 7 + (77/7 × (((7 + 7)/7)7+7/7 ))
:= 7 + 7 × 7 × (7 × 7 + 7) + 77
:= (8 + 8)/8 + (88 × ((8 + 8 + 8) + 8))
:= 8 + ((88 × ((8 + 8 + 8) + 8)) − 8/8)
:= 8 + (((8 × 8 × 88) + 8)/((8 + 8)/8))
:= 9/9 + (9 × (9 + 9) × (9 + 9) − 99)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − (99 + 9)/9
:= (9 + 9)/9 + ((9 × ((9 + 9) × (9 + 9) − 9)) − 9)
77/7
:= 777 + ((((7 + 7)/7)
Ï 2819 := 11 × (1 + 1)(1+1)
1+1+1
) − 7)
+1+1+1
Ï 2824 := ((1 + 1 + 111) × (1 + ((1 + 1) × (1 + 11)))) − 1
Ï 2829 := (1 + (11 + 11)) × (1 + (1 + 111+1 ))
:= 2 + (((22 × 22×(2+2) ) + 2)/2)
:= 2 × (2 × (222 + 222 ))
:= (22 + 2/2) × ((22/2)2 + 2)
:= 3 + (((33/3 + 3)3 ) + (3 × (33 − 3)))
:= 3 + (3333 − ((3 − 3/3)3×3 ))
:= 3 + (3 × (((3 + 3)3 − 3) + 33+3 ))
:= 4 + ((4 × 4 × 4 × 44) − 4/4)
:= 4 + ((4 × 4 × 4 × 44) + 4)
:= 4/4 + 4 × (4 × 4 × 44 + 4) − 4
:= 55 + (5 × 555 − (55/5))
:= 55 − ((5 × (55 + 5)) + 5/5)
:= 55 + (5 × 555 − 5/5)
:= 66/6 + (6 × (6 × (66 + 6 + 6)))
:= 6 + ((6 × (6 × (66 + 6 + 6))) + ((66 − 6)/6))
:= ((66/6 + 6) + 6) × ((666/6 + 6) + 6)
:= 77 + (7 × 7 × (7 × 7 + 7) − ((7 + 7)/7))
:= (7/7 + 7) × (((77 − 7)/7) + 7 × 7 × 7)
:= 7 + (7 × 7 × (7 × 7 + 7) + 77 + 7/7)
:= 88/8 + ((88 × ((8 + 8 + 8) + 8)) − 8)
:= 8 + (88 × ((8 + 8 + 8) + 8))
:= 8 + ((((8 × 8 × 88) + 8)/((8 + 8)/8)) + 8/8)
:= (9 + 9)/9 + (9 × (9 + 9) × (9 + 9) − 99)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − 99/9
:= ((99 + 9)/9) + (9 × (9 + 9) × (9 + 9) − 99)
Ï 2820 := (1 + 11) × (11 + ((1 + 1) × (1 + 111)))
Ï 2825 := (1 + 1 + 111) × (1 + ((1 + 1) × (1 + 11)))
Ï 2830 := 1 + ((1 + (11 + 11)) × (1 + (1 + 111+1 )))
:= 2 × (22 × 22+2+2 + 2)
:= 2/2 + (2 × (2 × (222 + 222 )))
:= 2 + (2 × ((2 × (222 + 222 )) + 2))
:= 3 + (33 × 33 + (3 × 3 + 3)3 )
:= 3 × 33 + ((33/3 + 3)3 )
:= 3 + ((3 × (((3 + 3)3 − 3) + 33+3 )) + 3/3)
:= 4 + (4 × 4 × 4 × 44)
:= 4 + (((4 × 4 × 4 × 44) + 4/4) + 4)
:= 4 × (4 × 4 × 44 + 4) − (4 + 4)/4
:= 55 − ((5 × (55 + 5)) + 5)
:= 5 × (555 + 5 + 5)
:= 55 + 5 × 555
:= 6 + ((6 × (6 × (66 + 6 + 6))) + 6)
:= 6 + ((6 × (6 × (66 + 6 + 6))) + (66/6))
:= ((6 + 6)/6)6 + ((66 × (6 × 6 + 6)) − 6)
:= 77 + (7 × 7 × (7 × 7 + 7) − 7/7)
:= 777 + (((7 + 7)/7)77/7 )
:= 7 + ((77/7 × (((7 + 7)/7)7+7/7 )) + 7)
:= ((8 × 8 × 88) + 8)/((8 + 8)/8)
:= 8 + ((88 × ((8 + 8 + 8) + 8)) + 8/8)
:= ((8 + 8)/8) × ((88 × (8 + 8) − 8/8) + 8)
:= (9/9 + 9) × ((999/9 + 9 × (9 + 9)) + 9)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − 9/9 − 9
:= ((9 − 99)/(9 + 9)) + (9 × ((9 + 9) × (9 + 9) − 9))
Ï 2821 := 1 + ((1 + 11) × (11 + ((1 + 1) × (1 + 111))))
Ï 2826 := 11 × (1 + (1 + 1)(1+1)
1+1+1
Ï 2831 := 1 + (1 + ((1 + (11 + 11)) × (1 + (1 + 111+1 ))))
)−1
2
:= 2 + ((((22 × 22×(2+2) ) + 2)/2) + 2)
:= 2 + (2 × (2 × (222 + 22 )))
:= 2 + ((22 + 2/2) × ((22/2)2 + 2))
:= 3333 − ((3 − 3/3)3×3 )
:= 3 × (((3 + 3)3 − 3) + 33+3 )
:= 3 + ((((33/3 + 3)3 ) + 3 × 33 ) + 3)
:= 4 + ((4 × 4 × 4 × 44) + 4/4)
:= (44 − 4)/4 + (4 × 4 × 4 × 44)
:= 4 + 44/4 × (4/4 + 44 )
5
:= 55 + (5/5 − ((5 × (55 + 5)) + 5))
:= 5 + (5/5 − (5 × (55 + 5)))
:= 55 + (5 × 555 + 5/5)
:= (6/6 + 6) × ((6 × 66 + 6/6) + 6)
:= 666 + 6 × 6 × (66 − 6)
:= 66 + ((6/6 + 6) × (6 × 66 − 6/6))
7
:= 77 + 7 × 7 × (7 × 7 + 7)
:= (7 − 7/7) × (((7 + 7)/7) + 7 × 7 × 7)
:= (7 × (7 × (7 × 7 + 7) + 7 + 7)) − 77/7
:= 8/8 + (((8 × 8 × 88) + 8)/((8 + 8)/8))
:= 8 + ((88 × ((8 + 8 + 8) + 8)) + ((8 + 8)/8))
:= 8 + (((88 × ((8 + 8 + 8) + 8)) − 8/8) + 8)
:= ((9/9 + 9 × 9) + 9) × (((99 + 99)/9) + 9)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − 9
:= ((9 − 9 × 9)/(9 + 9)) + (9 × ((9 + 9) × (9 + 9) − 9))
Ï 2822 := (1 + (11 × (1 + (1 + 1)11−1−1 )))/(1 + 1)
:= 2 + (2 × (22 × 2
3
2+2+2
+ 2))
3
:= 3 × 3 + (((33/3 + 3) ) − 3)
:= 4 + ((4 × 4 × 4 × 44) + (4 + 4)/4)
5
5
Ï 2827 := 11 × (1 + (1 + 1)(1+1)
:= 22/2 × (2
2×(2+2)
1+1+1
Ï 2832 := (1 + 1) × ((1 + 11) × (111+1 − (1 + 1 + 1)))
)
:= 2 × (2 × ((222 + 222 ) + 2))
+ 2/2)
3
:= 3/3 + (3 × (((3 + 3) − 3) + 3
3+3
))
4
:= 44/4 × (4/4 + 4 )
5
:= (3 × (33+3 + (3 + 3)3 )) − 3
:= 4 × (4 × 4 × 44 + 4)
:= 5 + ((5 − (5 + 5)/(5 + 5)) + 5)
:= 5 + ((5 + 5)/5 − (5 × (55 + 5)))
:= 55 + (5 × 555 + ((5 + 5)/5))
:= 6 + (((6 × (6 × (66 + 6 + 6))) + ((6 + 6)/6)) + 6)
:= 6 + ((6/6 + 6) × ((6 × 66 + 6/6) + 6))
:= 66 + ((66 × (6 × 6 + 6)) − 6)
:= 7/7 + 7 × 7 × (7 × 7 + 7) + 77
:= 7 + ((7 × 7 × (7 × 7 + 7) − 7/7) + 77)
:= (7/7 + 7) × (7 × 7 × 7 + (77/7))
:= 8 + (((8 + 8)/8) × (88 × (8 + 8) − 8/8))
:= 88/8 + (88 × ((8 + 8 + 8) + 8))
:= 8 + ((88 × ((8 + 8 + 8) + 8)) + 8)
:= 9 + ((99/9 + 9 + 9) × (99 − ((9 + 9)/9)))
:= 9/9 + ((9 × ((9 + 9) × (9 + 9) − 9)) − 9)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − (9 + 9 + 9)/9
198
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2833 := ((1 + 1) × ((1 + 1 + 11) × (111 − 1 − 1))) − 1
2
2
Ï 2838 := 11 × (1 + 1 + (1 + 1)(1+1)
2×(2+2)
1+1+1
Ï 2843 := ((1 + 11) × (11 + ((1 + 1) × (1 + 1 + 111)))) − 1
)
:= 2 + ((((22 × (22×(2+2) + 2)) + 2)/2) + 2)
:= ((22 + 2/2) ) + ((2 × (22 + 2)) )
:= 22 × ((2
:= 3/3 + ((3 × (33+3 + (3 + 3)3 )) − 3)
:= 3 + (3 × (33+3 + (3 + 3)3 ))
:= 3 × 33 + ((33/3 + 3)3 )
:= 4/4 + 4 × (4 × 4 × 44 + 4)
:= 44/4 × ((4 + 4)/4 + 44 )
:= 4 × 4 + (44/4 × (4/4 + 44 ))
:= 5 + ((5 × 555 − ((5 + 5)/5)) + 55)
:= 5 × 5 + (((5 − 55 )/(5 + 5)) + 55 )
:= 55 − ((5 × 55 + ((5 + 5)/5)) + 5)
+ 2)/2)
:= 66 + (((66 × (6 × 6 + 6)) − 6) + 6/6)
:= 66 + (66 × (6 × 6 + 6))
:= (6 × ((6 × (66 + 6 + 6)) + 6)) − 6/6
:= 7 + ((7 − 7/7) × (((7 + 7)/7)7 + 7 × 7 × 7))
:= ((7/7 − 7) + 7 × 7) × (77 − 77/7)
:= 7/7 + (7 × (7 × (7 × 7 + 7) + 7 + 7))
:= 8 + (((88 × ((8 + 8 + 8) + 8)) + 8/8) + 8)
:= ((8 + 8)/8) × (88 × (8 + 8) + 88/8)
:= 8 + (((88 × ((8 + 8 + 8) + 8)) + (88/8)) + 8)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 × ((9 + 9) × (9 + 9) − 9))
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) − 9/9)
Ï 2834 := (1 + 1) × ((1 + 1 + 11) × (111 − 1 − 1))
Ï 2839 := 1111 + ((1 + 11)1+1+1 )
2×(2+2)
Ï 2844 := (1 + 11) × (11 + ((1 + 1) × (1 + 1 + 111)))
:= 2 × (((2 + 2 + 2)2 + 2)2 − 22)
:= (22/2 + 2) × (222 − (2 + 2))
:= ((22 × (2
:= (3 × (33 + 3)) + ((33/3 + 3)3 )
:= (3 × 3 + 3)3 + 3333/3
:= 3 × ((33+3 + (3 + 3)3 ) + 3)
:= (4 + 4)/4 + 4 × (4 × 4 × 44 + 4)
:= 4 + (44 + 4/4) × (44 − 4)/4
:= 44 + (4 × (444 + 44 ))
+ 2)) + 2)/2
:= 5 + ((5 × 555 − 5/5) + 55)
:= 55 − (5 × 55 + (55/5))
:= 55 − ((5 × 55 + 5/5) + 5)
:= (6/6 + 6 + 6) × (6 × 6 × 6 + (6 + 6)/6)
:= 66 + ((66 × (6 × 6 + 6)) + 6/6)
:= 6 × ((6 × (66 + 6 + 6)) + 6)
:= 777 + 7 × 7 × (7 × 7 − 7) − 7/7
:= 7 + ((7/7 + 7) × (7 × 7 × 7 + (77/7)))
:= (7 + 7)/7 + (7 × (7 × (7 × 7 + 7) + 7 + 7))
:= ((8 + 8)/8) × ((88 × (8 + 8) + 8/8) + 8)
:= 8 + ((((88 × ((8 + 8 + 8) + 8)) − 8/8) + 8) + 8)
:= ((8 × (8 × 88 + 8)) − 8)/((8 + 8)/8)
:= (9 × ((9 + 9) × (9 + 9) − 9)) − 9/9
:= 9 × 9 × 9 + (9999/9 + 999)
:= 9 + (9 × ((9 + 9) × (9 + 9) − 9))
Ï 2835 := 1 + ((1 + 1) × ((1 + 1 + 11) × (111 − 1 − 1)))
:= (((22 × (2
2×(2+2)
+ 2)) − 2)/2) − 2
Ï 2840 := 1 + (1111 + ((1 + 11)1+1+1 ))
:= 2 + (22 × ((2
2×(2+2)
Ï 2845 := 1 + ((1 + 11) × (11 + ((1 + 1) × (1 + 1 + 111))))
:= 2/2 + (2 × (((2 + 2 + 2)2 + 2)2 − 22))
+ 2)/2))
:= 3 × (33+3 + (3 + 3)3 )
:= 3 × 33 + (((33/3 + 3)3 ) − 3)
:= 3/3 + (3 × ((33+3 + (3 + 3)3 ) + 3))
:= (44 + 4/4) × (44 − 4)/4
:= 4 + 4 × (4 × 4 × 44 + 4) + 4
:= 444 + (((4 − 4/4) + 4)4 )
:= 5 + (5 × 555 + 55)
:= 55 − ((5 × 55 + 5) + 5)
:= 55 − (5 × 55 + 5)
:= 66 + ((66 × (6 × 6 + 6)) − (6 × 6/(6 + 6)))
:= 66 + ((66 × (6 × 6 + 6)) + ((6 + 6)/6))
:= 6/6 + (6 × ((6 × (66 + 6 + 6)) + 6))
:= 777 + 7 × 7 × (7 × 7 − 7)
:= (7/7 + 7) × ((77 + 7)/7 + 7 × 7 × 7)
:= 7 + (((7/7 − 7) + 7 × 7) × (77 − 77/7))
:= 8 + ((88 × ((8 + 8 + 8) + 8)) + (88/8))
:= 8 + (((88 × ((8 + 8 + 8) + 8)) + 8) + 8)
:= 8/8 + (((8 × (8 × 88 + 8)) − 8)/((8 + 8)/8))
:= 9 × ((9 + 9) × (9 + 9) − 9)
:= 9 × 99 + ((((9 + 9)/9)99/9 ) − 99)
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) + 9/9)
Ï 2836 := (1 + 1) × (1 + ((1 + 1 + 11) × (111 − 1 − 1)))
Ï 2841 := 1 + (1 + (1111 + ((1 + 11)1+1+1 )))
Ï 2846 := ((1 + 1 + 11) × (((1 + 1) × (111 − 1)) − 1)) − 1
:= (22 × 222) − 222/2
:= 2 + (((22 × (2
:= 3/3 + (3 × (33+3 + (3 + 3)3 ))
:= 3 + ((3 × (33+3 + (3 + 3)3 )) + 3)
:= 3 + (((33/3 + 3)3 ) + 3 × 33)
:= 4 + 4 × (4 × 4 × 44 + 4)
:= 444 + ((((4 − 4/4) + 4)4 ) − 4)
:= 4 + ((44/4 × ((4 + 4)/4 + 44 )) + 4)
:= 5 + ((5 × 555 + 55) + 5/5)
:= 55 + (5 × 555 + (55/5))
:= 55 + (5/5 − (5 × 55 + 5))
:= ((6 + 6)/6) + (66 × (6 × 6 + 6))
:= 66 + ((66 × (6 × 6 + 6)) + (6 × 6/(6 + 6)))
:= (6 + 6)/6 + (6 × ((6 × (66 + 6 + 6)) + 6))
:= 7/7 + (7 × 7 × (7 × 7 − 7) + 777)
:= (7 × (7 × (7 × 7 + 7) + 7 + 7)) − 7/7
:= 7 + (((7/7 + 7) × (7 × 7 × 7 + (77/7))) + 7)
:= 8 + ((((8 × 8 × 88) + 8)/((8 + 8)/8)) + 8)
:= 8 + ((((88 × ((8 + 8 + 8) + 8)) + 8/8) + 8) + 8)
:= 8 + (((8 + 8)/8) × (88 × (8 + 8) + 88/8))
:= 9/9 + (9 × ((9 + 9) × (9 + 9) − 9))
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) − ((9 + 9 + 9)/9))
:= 99/9 + (9 × ((9 + 9) × (9 + 9) − 9))
6
Ï 2837 := 1111 + (((1 + 11)1+1+1 ) − (1 + 1))
:= ((22 × (2
2×(2+2)
+ 2)) − 2)/2
3
3
:= 3 + (((33/3 + 3) ) + (3 × (3 + 3)))
:= 4 + 4 × (4 × 4 × 44 + 4) + 4/4
5
5
2×(2+2)
+ 2)) + 2)/2)
Ï 2842 := 1 + (1 + (1 + (1111 + ((1 + 11)1+1+1 ))))
:= 2 + ((22 × ((2
2×(2+2)
+ 2)/2)) + 2)
3
:= 3 + (3333/3 + (3 × 3 + 3) )
Ï 2847 := (1 + 1 + 11) × (((1 + 1) × (111 − 1)) − 1)
:= 2222 + ((2/2 + 2 + 2)2+2 )
:= 3 + (3 × ((33+3 + (3 + 3)3 ) + 3))
4
:= 4 + (44/4 × ((4 + 4)/4 + 4 ))
5
:= 2 + (2 × (((2 + 2 + 2)2 + 2)2 − 22))
:= 4 × (4 × 4 × 44 + 4 + 4) − 4/4
5
:= 5 × 5 + (5 − (5 + 5)/(5 + 5))
:= 5 + ((5 × 5 − (5 + 5)/(5 + 5)) + 5 )
:= 55 + ((5 + 5)/5 − (5 × 55 + 5))
:= 66 + ((66 × (6 × 6 + 6)) − 6/6)
:= 6 + ((66 × (6 × 6 + 6)) + ((6 + 6)/6)6 )
:= 666 + ((66 × 66 + 6)/((6 + 6)/6))
:= 77 + ((7/7 + 7) × (7 × 7 × 7 + ((7 + 7)/7)))
:= 7 × (7 × (7 × 7 + 7) + 7 + 7)
:= 7 + ((7/7 + 7) × ((77 + 7)/7 + 7 × 7 × 7))
:= (8 × ((8 × 88 + 8)/((8 + 8)/8))) − 88/8
:= 8 + (((8 + 8)/8) × ((88 × (8 + 8) + 8/8) + 8))
:= (8 × ((8 × 88 + 8)/((8 + 8)/8))) − 8/8
:= (9 + 9)/9 + (9 × ((9 + 9) × (9 + 9) − 9))
:= (99 − 9/9) × (99/9 + 9 + 9)
:= ((99 + 9)/9) + (9 × ((9 + 9) × (9 + 9) − 9))
199
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2848 := 1 + ((1 + 1 + 11) × (((1 + 1) × (111 − 1)) − 1))
:= 2
2+2
× ((2 × 2 × 2 × 22) + 2)
Ï 2853 := ((1 + 1)1+11 ) − (11 × (1 + 1 + 111))
:= 2/2 + (((2 × 22) + 2) × (2
2+2+2
− 2))
Ï 2858 := (1 + 1) × (((1 + 1 + 11) × (111 − 1)) − 1)
:= 2 + ((22 + 2) × ((22/2)2 − 2))
:= (33 − 3/3) × ((3 × (33 + 3)) − 3/3)
:= 3 × (((33+3 + (3 + 3)3 ) + 3) + 3)
:= 3 + (((33/3 + 3)3 ) + 333/3)
:= 4 × (4 × 4 × 44 + 4 + 4)
:= 4 + (44/4 × ((44 − 4/4) + 4))
:= (44 × (44 + 4)/4) − (4 + 4)/4
:= 55 − (5 × 55 + ((5 + 5)/5))
:= 5 + (55 − (5 × 55 + ((5 + 5)/5)))
:= 5 + ((55 − (5 × 55 + ((5 + 5)/5))) + 5)
:= 6 + (((66 × (6 × 6 + 6)) + ((6 + 6)/6)6 ) + 6)
:= 666 + ((6 × 6/(6 + 6))6/6+6)
:= (6 + 6)/6 + ((6/6 + 6) × (6 × 66 + 6 + 6))
:= 7 + ((7 × (7 × (7 × 7 + 7) + 7 + 7)) − 7/7)
:= 77/7 + (7 × (7 × (7 × 7 + 7) + 7 + 7))
:= 7 × 7 + (((77/7 − 7) + 7 × 7)(7+7)/7 )
:= 8 × ((8 × 88 + 8)/((8 + 8)/8))
:= 8/8 + (((8 × (8 × 88 + 8)) + 8)/((8 + 8)/8))
:= (8 + 8)/8 + ((8 + 8 + 8) × (888/8 + 8))
:= (9 − 9/9) × ((((9 × 9 × 9 × 9) + 9)/(9 + 9)) − 9)
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) + 9)
:= 9 × (9 × 9 + 9) + (((9 + 9)/9)99/9 )
Ï 2849 := 11 × (1 + 1 + 1 + (1 + 1)(1+1)
2×(2+2)
1+1+1
)
Ï 2854 := (1 + 1) × (((1 + 11) × (111+1 − (1 + 1))) − 1)
2
Ï 2859 := ((1 + 1) × ((1 + 1 + 11) × (111 − 1))) − 1
:= ((22 + 2) × ((22/2) − 2)) − 2
:= (22 × ((2 × 22+2+2 ) + 2)) − 2/2
:= 3 + ((((33/3 + 3) ) + 3 × 33) + 3)
:= ((33/3 + 3)3 ) + ((333 − 3)/3)
:= 3 + ((33 + 3/3) × (3 × 33 + 3))
:= 44/4 × ((44 − 4/4) + 4)
:= 4 × 4 + (44/4 × ((4 + 4)/4 + 44 ))
:= (44 × (44 + 4)/4) − 4/4
:= 55 − (5 × 55 + 5/5)
:= 5 + (55 − (5 × 55 + 5/5))
:= 5 + ((55 − (5 × 55 + 5/5)) + 5)
:= (6/6 + 6) × (6 × 66 + (66/6))
:= ((66 − 6)/6) + (6 × ((6 × (66 + 6 + 6)) + 6))
:= 6 + (((6 × 6/(6 + 6))6/6+6) + 666)
:= 7 + (7 × (7 × (7 × 7 + 7) + 7 + 7))
:= 7 × 7 × (7 × 7 + 7) + (777 − 7)/7
:= 77 + (7 × (7 × (7 × 7 + 7) + 7) − (77/7))
:= 8/8 + (8 × ((8 × 88 + 8)/((8 + 8)/8)))
:= ((8 + 8)/8) × ((88 × (8 + 8) + 88/8) + 8)
:= 88/8 + (8 × ((8 × 88 + 8)/((8 + 8)/8)))
:= 99/9 × ((9 × (9 + 9) − ((9 + 9)/9)) + 99)
:= 9 + (((9 × ((9 + 9) × (9 + 9) − 9)) + 9/9) + 9)
:= (99 × (99/9 + 9 + 9)) − (99 + 9)/9
:= 22/2 × ((2
+ 2/2) + 2)
3
Ï 2850 := 11 + (1111 + ((1 + 11)1+1+1 ))
2
:= 2 + 2 × (22 − 2) + 2
22/2
Ï 2855 := 111 + (1 + 1 + 1 + 11)1+1+1
Ï 2860 := (1 + 1) × ((1 + 1 + 11) × (111 − 1))
2
:= (33 + 3) × (3 × 33 − (3/3 + 3))
4
:= 4/4 + (44/4 × ((4 − 4/4) + 4))
5
:= ((22 + 2) × ((22/2) − 2)) − 2/2
:= 22 × ((2 × 22+2+2 ) + 2)
:= 333/3 + ((33/3 + 3)3 )
:= (33 − 3/3) × ((333 − 3)/3)
4
:= (44 × (4 + 4)/4) − (4/4 + 4)
5
:= 44 × (44 + 4)/4
:= 5 − 5 × 55
:= 5 + (5 − 5 × 55)
:= 5 + ((55 − 5 × 55) + 5)
:= 6 + (6 × ((6 × (66 + 6 + 6)) + 6))
:= 6 + ((6/6 + 6) × (6 × 66 + (66/6)))
:= (66 − 6/6) × (((6 + 6)/6 + 6 × 6) + 6)
:= (7/7 + 7 × 7) × ((7/7 + 7 × 7) + 7)
:= 777/7 + 7 × 7 × (7 × 7 + 7)
:= 7 + ((7 × (7 × (7 × 7 + 7) + 7 + 7)) + (77/7))
:= (8 + 8)/8 + (8 × ((8 × 88 + 8)/((8 + 8)/8)))
:= ((8 + 8 + 8) × (888/8 + 8)) − 8/8
:= (8/8 + 8 × 8) × (88/((8 + 8)/8))
:= (9/9 + 9 + 9) × (9 × (9 + 9) − ((99 + 9)/9))
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) + (99/9))
:= 99/9 × ((9 × (9 + 9) − 9/9) + 99)
Ï 2851 := 1 + (11 + (1111 + ((1 + 11)1+1+1 )))
Ï 2856 := (1 + 1) × ((1 + 11) × (111+1 − (1 + 1)))
2
Ï 2861 := 1 + ((1 + 1) × ((1 + 1 + 11) × (111 − 1)))
:= (22 + 2) × ((22/2) − 2)
:= 2/2 + (22 × ((2 × 22+2+2 ) + 2))
:= 3 + ((33 − 3/3) × ((3 × (33 + 3)) − 3/3))
:= (33 + 3/3) × (3 × 33 + 3)
:= ((33/3 + 3)3 ) + (3 × (33 + 3 + 3))
:= 4 + 4 × (4 × 4 × 44 + 4 + 4) − 4/4
:= (44 × (44 + 4)/4) − 4
:= 4/4 + (44 × (44 + 4)/4)
:= 55 + (5/5 − 5 × 55)
:= 5 + ((5/5 − 5 × 55) + 55 )
:= 55 + (55/5 − 5 × 55)
:= 6 + ((6 × ((6 × (66 + 6 + 6)) + 6)) + 6/6)
:= (6/6 + 6) × (6 × 66 + 6 + 6)
:= 6 + (((6/6 + 6) × (6 × 66 + (66/6))) + 6)
:= 7 + ((7 × (7 × (7 × 7 + 7) + 7 + 7)) + ((7 + 7)/7))
:= (7/7 + 7) × (7 × 7 × 7 + 7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7) + (777 − 7)/7)
:= (((8 × (8 × 88 + 8)) + 8)/((8 + 8)/8)) − 8/8
:= (8 + 8 + 8) × (888/8 + 8)
:= 8/8 + ((8/8 + 8 × 8) × (88/((8 + 8)/8)))
:= 9 + ((99 − 9/9) × (99/9 + 9 + 9))
:= ((9/9 + 9 + 9) + 9) × (999/9 − 9)
:= (((9 + 9)/9)9 ) + 9 × (9 × (9 + 9) + 99)
:= 2 + (22/2 × ((2
2×(2+2)
+ 2/2) + 2))
Ï 2852 := (1 + (11 + 11)) × (1 + (1 + (1 + 111+1 )))
2+2+2
Ï 2857 := 1 + ((1 + 1) × ((1 + 11) × (111+1 − (1 + 1))))
2
Ï 2862 := (1 + 1) × (1 + ((1 + 1 + 11) × (111 − 1)))
:= 2/2 + ((22 + 2) × ((22/2) − 2))
:= 2 + (22 × ((2 × 22+2+2 ) + 2))
:= (3 × (33 + 3)) + ((33/3 + 3)3 )
:= 3/3 + ((33 + 3/3) × (3 × 33 + 3))
:= 3 × ((3 + 3) × ((3 + 3) × 33 − 3))
:= 4 + 4 × (4 × 4 × 44 + 4 + 4)
:= 4/4 + ((44 × (44 + 4)/4) − 4)
:= (4 + 4)/4 + (44 × (44 + 4)/4)
:= ((2 × 22) + 2) × (2
− 2)
5
5
:= 5 + ((5 + 5)/5 − 5 × 55)
:= 5 + (((5 + 5)/5 − 5 × 55) + 5 )
:= 55 + (((55 + 5)/5) − 5 × 55)
:= 6 + ((6 × ((6 × (66 + 6 + 6)) + 6)) + ((6 + 6)/6))
:= 6/6 + ((6/6 + 6) × (6 × 66 + 6 + 6))
:= 6 + ((6/6 + 6) × (6 × 66 + 6 + 6))
:= 7 + ((7/7 + 7 × 7) × ((7/7 + 7 × 7) + 7))
:= 7 + (7 × 7 × (7 × 7 + 7) + 777/7)
:= ((8 × (8 × 88 + 8)) + 8)/((8 + 8)/8)
:= 8/8 + ((8 + 8 + 8) × (888/8 + 8))
:= (8 × 8 − 88/8) × ((8 − 88)/8 + 8 × 8)
:= 9 + (((9 × ((9 + 9) × (9 + 9) − 9)) − 9/9) + 9)
:= ((99 + 99)/9) + (9 × ((9 + 9) × (9 + 9) − 9))
:= (99 × (99/9 + 9 + 9)) − 9
7
:= 77 + (((7 + 7 + 7)/7) + 7 × (77 + 7))
200
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2863 := 1 + ((1 + 1) × (1 + ((1 + 1 + 11) × (111 − 1))))
2+2+2
Ï 2868 := (1 + 11) × (((1 + 1) × (111+1 − 1)) − 1)
2
Ï 2873 := (1 + 1 + 11) × ((1 + 1) × 111 − 1)
:= (2 + 2 + 2) × (22 − (2 + 2 + 2))
:= (22/2 + 2) × (222 − 2/2)
:= 3 + ((33 − 3/3) × ((333 − 3)/3))
:= (33 × (3 × 33 + 3 + 3)) − 3
:= 3 + ((33 × (3 × 33 + 3 + 3)) − 3/3)
:= 4 + ((44 × (44 + 4)/4) − 4/4)
:= 4 + ((44 × (44 + 4)/4) + 4)
:= 4 + (((4/4 + 4)4+4/4 ) − 44 )
:= 55 + ((((5 − 55 )/(5 + 5)) − 5) + 55 )
:= 55 + (((5 − 55 )/(5 + 5)) + 55 )
:= 55 + ((5 × (5 − 55)) − ((5 + 5)/5))
:= (6/6 + 6) × (((6 × 66 + 6/6) + 6) + 6)
:= 66 + ((6 × (6 × (66 + 6 + 6))) − 6)
:= 66 + ((6 × (6 × (66 + 6 + 6))) − 6/6)
:= 7 × (7 × 7 × 7 + 77) − 77
:= 77 + (7 × (7 × (7 × 7 + 7) + 7) − ((7 + 7)/7))
:= (7 × 7 × (7 + 7)) + ((7 + 7 + 7)/7)7
:= 8 + (((8 + 8 + 8) × (888/8 + 8)) − 8/8)
:= 8 + ((8/8 + 8 × 8) × (88/((8 + 8)/8)))
:= 8/8 + ((8 × (8 × (8 × 8 − 8) − 88)) − 8)
:= 9/9 + ((99 × (99/9 + 9 + 9)) − 9)
:= (99 × (99/9 + 9 + 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (99 × (99/9 + 9 + 9))
:= 2 + ((22 × ((2 × 2
) + 2)) + 2/2)
Ï 2864 := (1 + 1) × ((111 × (1 + 1 + 11)) − 11)
Ï 2869 := 1 + ((1 + 11) × (((1 + 1) × (111+1 − 1)) − 1))
2
Ï 2874 := 1 + ((1 + 1 + 11) × ((1 + 1) × 111 − 1))
:= (222 × (22/2 + 2)) − 22
:= 2/2 + ((2 + 2 + 2) × (22 − (2 + 2 + 2)))
:= (2/2 + 2) × ((2 × (222 − (2 + 2))) − 2)
:= 3 + (((33/3 + 3)3 ) + (3 × (33 + 3 + 3)))
:= 3/3 + ((33 × (3 × 33 + 3 + 3)) − 3)
:= 3 + (33 × (3 × 33 + 3 + 3))
:= 4 + (44 × (44 + 4)/4)
:= ((4/4 + 4)4+4/4 ) − 44
:= (4 × (4 × (4 × 44 + 4))) − ((4 + 4)/4 + 4)
5
5
5−5/5
:= 55 + ((5 × (5 − 55)) − 5/5)
:= 5 + ((5 × (5 − 55)) − (55/5))
:= 5 − ((5 − 5/5)
:= ((6 + 6)/6 + 6) × (6 × (66 − 6) − ((6 + 6)/6))
:= ((66 − 6) × (6 × 6 + 6 + 6)) − 66/6
:= 7/7 + (7 × (7 × 7 × 7 + 77) − 77)
)
:= 66 + (6 × (6 × (66 + 6 + 6)))
:= 77 + (7 × (7 × (7 × 7 + 7) + 7) − 7/7)
:= (7 − 7/7) × (7 × (77 − 7) − (77/7))
:= 8 + ((8 + 8 + 8) × (888/8 + 8))
:= (88/8 + 8) × ((88 − 8/8) + 8 × 8)
:= (8 + 8)/8 + ((8 × (8 × (8 × 8 − 8) − 88)) − 8)
:= 9 + (((9 × ((9 + 9) × (9 + 9) − 9)) + (99/9)) + 9)
:= (9/9 + 9 + 9) × (9 × (9 + 9) − (99/9))
:= 9 + ((9 + 9) × (9 × (9 + 9) − 9) + 999/9)
Ï 2865 := 1 + ((1 + 1) × ((111 × (1 + 1 + 11)) − 11))
:= 2/2 + ((222 × (22/2 + 2)) − 22)
Ï 2870 := (11 − 1) × (((1 + 1) × (1 + 11)1+1 ) − 1)
Ï 2875 := ((1 + 1)1+11 ) − 11 × 111
:= (2/2 + 2 + 2) × (((22 + 2)2 ) − 2)
:= 2 + ((22/2 + 2) × (222 − 2/2))
:= 3 + (3 × ((3 + 3) × ((3 + 3) × 3 − 3)))
:= (33 × (3 × 33 + 3 + 3)) − 3/3
:= 3 + ((33 × (3 × 33 + 3 + 3)) + 3/3)
:= 4 + ((44 × (44 + 4)/4) + 4/4)
:= 4/4 + (((4/4 + 4)4+4/4 ) − 44 )
:= 4 + (44/4 × ((4/4 + 44 ) + 4))
:= 5 + (((55 − 5 × 55) + 5) + 5)
:= 55 + ((5 × (5 − 55)) − 5)
:= 55 + (5 × (5 − 55))
:= 6 + ((((6 × 6/(6 + 6))6/6+6) + 666) + 6)
:= (6 × 6 − 6/6 + 6) × (((6 + 6)/6)6 + 6)
:= 66 + ((6 × (6 × (66 + 6 + 6))) + 6/6)
:= ((7 + 7)/7) + 7 × (7 × (7 × 7 + 7)) − 7
:= 77 + 7 × (7 × (7 × 7 + 7) + 7)
:= 7777/7 + ((7 + 7) × (77 + 7 × 7))
:= (8 − 8/8 + 8) × (8 × (8 + 8 + 8) − 8/8)
:= (8 − 88)/8 + (8 × (8 × (8 × 8 − 8) − 88))
:= (8 × 8 × 8 × 8) − 88/8 × 888/8
:= 999/9 + (9 + 9) × (9 × (9 + 9) − 9)
:= (99 × (99/9 + 9 + 9)) − 9/9
:= (9 + 9) × 99 + (9999/9 − (9 + 9))
3
7
Ï 2866 := (1 + 1) × (1 + ((111 × (1 + 1 + 11)) − 11))
Ï 2871 := 11 × (((1 + 1) × ((11 × (1 + 11)) − 1)) − 1)
Ï 2876 := 1 + (((1 + 1)1+11 ) − 11 × 111)
:= (2 × ((2 + 2 + 2)2 + 2)2 ) − 22
:= (2/2 + 2) × (2 × 222 − 22/2)
:= 2 × (((2 + 2 + 2)2 + 2)2 − (2 + 2 + 2))
:= 3 + (((33 − 3/3) × ((333 − 3)/3)) + 3)
:= 33 × (3 × 33 + 3 + 3)
:= 33 + (((33/3 + 3)3 ) + 3 × 33)
:= 4 + ((44 × (44 + 4)/4) + (4 + 4)/4)
:= 44/4 × ((4/4 + 44 ) + 4)
:= (4 × (4 × (4 × 44 + 4))) − 4
5
:= 55 + ((5 × (5 − 55)) + 5/5)
:= 5 + ((55/5 − 5 × 55) + 5 )
:= 55 + (((5 × (5 − 55)) − 5) + 5/5)
:= 6 + ((66 − 6/6) × (((6 + 6)/6 + 6 × 6) + 6))
:= 666/6 + ((66 × (6 × 6 + 6)) − (6 + 6))
:= 6 + ((6 × 6 − 6/6 + 6) × (((6 + 6)/6)6 + 6))
:= (7 × 7 × (7 + 7)) + (((7 + 7 + 7)/7)7 − 7)
:= 7/7 + (7 × (7 × (7 × 7 + 7) + 7) + 77)
:= (7 × (7 × 77 − ((7 + 7)/7)7 )) − 7/7
:= 8 + (((8 + 8 + 8) × (888/8 + 8)) + ((8 + 8)/8))
:= (88 − 8/8) × ((8/8 + 8 + 8 + 8) + 8)
:= 8 × 8 + (((8 × 8 × 88) − 8)/((8 + 8)/8))
:= ((999 + 9)/9) + (9 + 9) × (9 × (9 + 9) − 9)
:= 99 × (99/9 + 9 + 9)
:= 99 + ((((9 + 9)/9)99/9 ) + 9 × 9 × 9)
Ï 2867 := ((1 + 11) × (((1 + 1) × (111+1 − 1)) − 1)) − 1
Ï 2872 := ((1 + 1 + 11) × ((1 + 1) × 111 − 1)) − 1
Ï 2877 := 1 + (1 + (((1 + 1)1+11 ) − 11 × 111))
:= 2/2 + ((2 × ((2 + 2 + 2)2 + 2)2 ) − 22)
:= 2 + ((2/2 + 2 + 2) × (((22 + 2)2 ) − 2))
:= (2 × ((2 + 2 + 2)2 + 2)2 ) − 22/2
:= (33/3)3 + (3 × ((3 − 3/3)3×3 ))
:= 3/3 + (33 × (3 × 33 + 3 + 3))
:= ((33 + 3) × (3 × 33 − 3)) − 3
:= (44/4 × ((4/4 + 44 ) + 4)) − 4
:= (4 × (4 × (4 × 44 + 4))) − 4 − 4
:= 4/4 + ((4 × (4 × (4 × 44 + 4))) − 4)
5
5
:= 55 + (5 − (5 + 5)/(5 + 5))
:= 5 − ((((5 − (5 + 5)/5) ) + 5) + 5)
:= 55 − (((5 − (5 + 5)/5)5 ) + 5)
:= (66/6 + 6 × 6) × (66 − 6 + 6/6)
:= ((6 + 6)/6)6 + (6 × (6 × (66 + 6 + 6)))
:= 666/6 + ((66 × (6 × 6 + 6)) − 6)
5
5
7
:= 7 × (7 × 77 − ((7 + 7)/7)7 )
:= 77/7 + ((7/7 + 7) × (7 × 7 × 7 + 7 + 7))
:= ((7 + 7)/7) + 7 × 7 × (7 × 7 + 7)
:= 88/8 + ((8 + 8 + 8) × (888/8 + 8))
:= (8 × (8 × (8 × 8 − 8) − 88)) − 8
:= 8 + ((88/8 + 8) × ((88 − 8/8) + 8 × 8))
:= 9/9 + (99 × (99/9 + 9 + 9))
:= (999/9 × (((9 − 9/9) + 9) + 9)) − 9
99/9
:= 9 + ((((9 + 9)/9)
) + 9 × (9 × 9 + 9))
201
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2878 := (1 + 1) × (((11 − 1) × (1 + 11)1+1 ) − 1)
2
Ï 2883 := 1 + ((1 + 1) × (11 × ((11 × (1 + 11)) − 1)))
Ï 2888 := (1 + 1) × (1 + (111 × (1 + 1 + 11)))
:= ((2 + 2 + 2) × (22 − (2 + 2))) − 2
:= 2/2 + ((22 × (2 + 2 + 2)) − 22)
:= 2 × ((2 + 2 + 2)2 + 2)2
:= 3/3 + (((33 + 3) × (3 × 33 − 3)) − 3)
:= 3 + ((33 + 3) × (3 × 33 − 3))
:= 3 + (((333 × (33 − 3/3)) − 3)/3)
:= (4 × (4 × (4 × 44 + 4))) − (4 + 4)/4
:= 4 + ((4 × (4 × (4 × 44 + 4))) − 4/4)
:= 4 + ((4 × (4 × (4 × 44 + 4))) + 4)
:= 5 + (((5 × (5 − 55)) − ((5 + 5)/5)) + 55 )
:= 55 + (5/5 − ((5 − (5 + 5)/5)5 ))
:= 5 + ((5/5 − ((5 − (5 + 5)/5)5 )) + 55 )
:= ((66 − 6) × (6 × 6 + 6 + 6)) − (6 + 6)/6
:= 666/6 + (66 × (6 × 6 + 6))
:= ((6 + 6)/6 + 6) × (6 × (66 − 6) + 6/6)
:= 7/7 + (7 × (7 × 77 − ((7 + 7)/7) ))
:= (7 × ((7 × 7 × 7 − 7) + 77)) − (7/7 + 7)
:= (7/7 − 77) × (77/7 − 7 × 7)
:= (8 × (8 × (8 × 8 − 8) − 88)) − (8 + 8)/8
:= 88/8 + ((8 × (8 × (8 × 8 − 8) − 88)) − 8)
:= 8 + (8 × (8 × (8 × 8 − 8) − 88))
:= 9 + ((9/9 + 9 + 9) × (9 × (9 + 9) − (99/9)))
:= 99 + (99 × (9 + 9 + 9) + 999/9)
:= (9 − 9/9) × ((9/9 + 9 + 9)(9+9)/9 )
2
7
Ï 2879 := ((1 + 1) × ((11 − 1) × (1 + 11)1+1 )) − 1
2
Ï 2884 := (1 + 1) × (111 + 111+1+1 )
2
2
Ï 2889 := 1 + ((1 + 1) × (1 + (111 × (1 + 1 + 11))))
:= 2/2 + (2 × ((2 + 2 + 2)2 + 2)2 )
:= ((2 + 2 + 2) × (22 − (2 + 2))) − 2/2
:= 2 × (((2 + 2 + 2) + 2) − 2)
:= ((33 + 3) × (3 × 33 − 3)) − 3/3
:= 3 + (((33 + 3) × (3 × 33 − 3)) + 3/3)
:= 3 × (3 × (333 − (3 × 3 + 3)))
:= (4 × (4 × (4 × 44 + 4))) − 4/4
:= 4 + (4 × (4 × (4 × 44 + 4)))
:= 4 + (((4 × (4 × (4 × 44 + 4))) + 4/4) + 4)
:= 5 + (((5 × (5 − 55)) − 5/5) + 55 )
:= 5 + ((((5 × (5 − 55)) − 5/5) + 55 ) + 5)
:= 55 − (555/5 + 5 × 5 × 5)
:= (6 × 666) − (6666 + 6)/6
:= 6 + ((66 × (6 × 6 + 6)) + 666/6)
:= 7 + (7 × 7 × (7 × 7 + 7) + ((7 + 7)/7) )
:= (7 × ((7 × 7 × 7 − 7) + 77)) − 7
:= (7 × ((7 × 7 × 7 − 7) + 77)) − (7 + 7)/7
:= (8 × (8 × (8 × 8 − 8) − 88)) − 8/8
:= 8 × 8 + (((8 × 8 × 88) + 8)/((8 + 8)/8))
:= 8 + ((8 × (8 × (8 × 8 − 8) − 88)) + 8/8)
:= (9 + 9) × 99 + (9999/9 − 9)
:= (9 + 9 + 9) × ((99 − 9/9) + 9)
:= ((66 − 6) × (6 × 6 + 6 + 6)) − 6/6
7
:= 9 + ((99 × (99/9 + 9 + 9)) − 9/9)
Ï 2880 := (1 + 1) × ((11 − 1) × (1 + 11)1+1 )
Ï 2885 := ((1 + 1) × (111 × (1 + 1 + 11))) − 1
Ï 2890 := (1 + 1) × (1 + (1 + (111 × (1 + 1 + 11))))
:= (2 + 2 + 2) × (22 − (2 + 2))
:= (222 × (22/2 + 2)) − 2/2
:= 2 + (2 × ((2 + 2 + 2)2 + 2)2 )
:= (33 + 3) × (3 × 33 − 3)
:= ((333 × (33 − 3/3)) − 3)/3
:= 3 + (((333 × (33 − 3/3)) + 3)/3)
:= 4 × (4 × (4 × 44 + 4))
:= 4 + ((4 × (4 × (4 × 44 + 4))) + 4/4)
:= (44 − 4)/4 + (4 × (4 × (4 × 44 + 4)))
2
5
5
:= (5 × (555 + 5 × 5)) − 5 − 5
:= 5 + ((5 × (5 − 55)) + 5 )
:= 5 + (((5 × (5 − 55)) + 5 ) + 5)
:= (66 − 6) × (6 × 6 + 6 + 6)
:= (6 × 666) − (6666/6)
:= 6 + ((66 × (6 × 6 + 6)) + (666 + 6)/6)
:= (7 × 7 − 7/7) × (77/7 + 7 × 7)
:= 7/7 + ((7 × ((7 × 7 × 7 − 7) + 77)) − 7)
:= (7 × ((7 × 7 × 7 − 7) + 77)) − 7/7
:= 8 × (8 × (8 × 8 − 8) − 88)
:= ((8 − 88/8) + 8) × (8 × (8 × 8 + 8) + 8/8)
:= 8 + ((8 × (8 × (8 × 8 − 8) − 88)) + ((8 + 8)/8))
:= 9 + (99 × (99/9 + 9 + 9))
:= (9 + 9) × 99 + ((9999 + 9)/9 − 9)
:= 9/9 + ((9 + 9 + 9) × ((99 − 9/9) + 9))
Ï 2881 := 1 + ((1 + 1) × ((11 − 1) × (1 + 11)1+1 ))
Ï 2886 := (1 + 1) × (111 × (1 + 1 + 11))
Ï 2891 := ((1 + 11) × (((1 + 1) × 111+1 ) − 1)) − 1
:= 2/2 + ((2 + 2 + 2) × (222 − (2 + 2)))
:= 222 × (22/2 + 2)
:= ((2 + 2 + 2) × (222 − 2)) − 2/2
:= 3/3 + ((33 + 3) × (3 × 33 − 3))
:= (33 − 3/3) × 333/3
:= 33/3 + ((33 + 3) × (3 × 33 − 3))
:= 4/4 + (4 × (4 × (4 × 44 + 4)))
:= 4 + ((4 × (4 × (4 × 44 + 4))) + (4 + 4)/4)
:= 44/4 + (4 × (4 × (4 × 44 + 4)))
:= 5 + (((5 × (5 − 55)) + 5/5) + 55 )
:= 5 × 555 + 555/5
:= 5 + (5 × 555 + 555/5)
:= 6/6 + ((66 − 6) × (6 × 6 + 6 + 6))
:= 6 + ((66 − 6) × (6 × 6 + 6 + 6))
:= 6 + ((6 × 666) − (6666/6))
:= ((7/7 − 77) × (77/7 − 7 × 7)) − 7
:= (7/7 + 77) × (7 × 7 − (77 + 7)/7)
:= 7 × ((7 × 7 × 7 − 7) + 77)
:= 8/8 + (8 × (8 × (8 × 8 − 8) − 88))
:= 888/8 × (((((8 + 8)/8) + 8) + 8) + 8)
:= 88/8 + (8 × (8 × (8 × 8 − 8) − 88))
:= 9 + ((99 × (99/9 + 9 + 9)) + 9/9)
:= 999/9 × (((9 − 9/9) + 9) + 9)
:= (9 − ((9 + 9)/9)) × ((((9 + 9)/9)9 ) − 99)
Ï 2882 := (1 + 1) × (11 × ((11 × (1 + 11)) − 1))
Ï 2887 := 1 + ((1 + 1) × (111 × (1 + 1 + 11)))
Ï 2892 := (1 + 11) × (((1 + 1) × 111+1 ) − 1)
:= (222 × (2 + 2 + 2)) − 22
:= 2/2 + (222 × (22/2 + 2))
:= (2 + 2 + 2) × (222 − 2)
:= 3 + (((33 + 3) × (3 × 33 − 3)) − 3/3)
:= ((333 × (33 − 3/3)) + 3)/3
:= (3/3 + 3) × (33+3 − (3 + 3))
:= (4 + 4)/4 + (4 × (4 × (4 × 44 + 4)))
:= 4 × 444 + 4444/4
:= (4 × ((4 × (4 × 44 + 4)) + 4)) − 4
5
5
:= 5 − ((5 − (5 + 5)/5) )
:= 5 + (5 − ((5 − (5 + 5)/5) ))
:= 5 + ((55 − ((5 − (5 + 5)/5)5 )) + 5)
:= (6 + 6)/6 + ((66 − 6) × (6 × 6 + 6 + 6))
:= 6 + (((66 − 6) × (6 × 6 + 6 + 6)) + 6/6)
:= 6 + (((66 − 6) × (6 × 6 + 6 + 6)) + 6)
5
5
:= 77/7 × (((7 × 7 × 77 − 7)/(7 + 7)) − 7)
:= 7 + ((7 × 7 − 7/7) × (77/7 + 7 × 7))
:= 7/7 + (7 × ((7 × 7 × 7 − 7) + 77))
:= (8 + 8)/8 + (8 × (8 × (8 × 8 − 8) − 88))
:= 8 + ((8 × (8 × (8 × 8 − 8) − 88)) − 8/8)
:= ((8 × (8 × 88 + 8)) + 88)/((8 + 8)/8)
:= 99/9 + (99 × (99/9 + 9 + 9))
:= 9 × (9 + 9) × (9 + 9) − (99/9 + 9 + 9)
:= (9 + 9) × 99 + ((9999 − 9)/9)
202
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2893 := 11 × (((1 + 1) × (11 × (1 + 11))) − 1)
Ï 2898 := ((1 + 1 + 11) × (1 + (1 + 1) × 111)) − 1
Ï 2903 := ((1 + 1) × (11 × (11 × (1 + 11)))) − 1
:= 2/2 + ((2 + 2 + 2) × (22 − 2))
:= (2/2 + 2) × (2 × 22 − 2)
:= (222 × (2 + 2 + 2)) − 2/2
:= 3/3 + ((3/3 + 3) × (33+3 − (3 + 3)))
:= 3 × (3 × 333 − 33)
:= ((3/3 + 3) × (33+3 − 3)) − 3/3
2
2
4
:= 44/4 × (((44 − 4/4) + 4) + 4)
:= ((4 − 4)/4) × ((4 + 4)/4 + 44)
:= (44 × ((44 + 4 + 4)/4)) − 4/4
:= 55 + (55/5 − ((5 − (5 + 5)/5)5 ))
:= (5 × (555 + 5 × 5)) − (5 + 5)/5
:= 55 + (5 − 5555)/(5 × 5)
:= 66/6 × (6 × (6 × 6 + 6) + (66/6))
:= (6/6 + 6) × ((6 × 66 + 6 + 6) + 6)
:= ((6 − 6/6)6−6/6 ) − (6 × 6 × 6 + 6)
:= (7 + 7)/7 + (7 × ((7 × 7 × 7 − 7) + 77))
:= 7 + (7 × ((7 × 7 × 7 − 7) + 77))
:= (77 + 7)/7 + (7 × ((7 × 7 × 7 − 7) + 77))
:= 88/8 × (((8 + 8) × (8 + 8) − 8/8) + 8)
:= (88/8 − 8) × ((88 × 88 − (8 + 8))/8)
:= 88 + ((88 × ((8 + 8 + 8) + 8)) − 8/8)
:= (9 + 9) × 99 + 9999/9
:= (9 + 9) × (9 × (9 + 9) − 9/9)
:= 9 × (9 + 9) × (9 + 9) − ((99 + 9 + 9)/9)
Ï 2894 := 1 + (11 × (((1 + 1) × (11 × (1 + 11))) − 1))
Ï 2899 := (1 + 1 + 11) × (1 + (1 + 1) × 111)
Ï 2904 := (1 + 1) × (11 × (11 × (1 + 11)))
:= 2 + ((2 + 2 + 2) × (22 − 2))
:= (22/2 + 2) × (222 + 2/2)
:= 222 × (2 + 2 + 2)
:= (3 × (3 × 333 − 33)) − (3/3 + 3)
:= 3/3 + (3 × (3 × 333 − 33))
:= (3/3 + 3) × (33+3 − 3)
:= (4 × ((4 × (4 × 44 + 4)) + 4)) − (4 + 4)/4
:= 4 + ((4 × ((4 × (4 × 44 + 4)) + 4)) − 4/4)
:= 44 × ((44 + 4 + 4)/4)
:= (5 × (555 + 5 × 5)) − (5/5 + 5)
:= (5 × (555 + 5 × 5)) − 5/5
:= 6 + (((6 + 6)/6 + 6) × (6 × (66 − 6) + 6/6))
:= (6/6 + 6 + 6) × (6 × 6 × 6 + 6/6 + 6)
:= 7 + (((7 × 7 − 7/7) × (77/7 + 7 × 7)) + 7)
:= 7 + ((7 × ((7 × 7 × 7 − 7) + 77)) + 7/7)
:= 8 + (888/8 × (((((8 + 8)/8) + 8) + 8) + 8))
:= 8 + ((8 × (8 × (8 × 8 − 8) − 88)) + (88/8))
:= ((9 + 9)/9) × (9 × 9 × (9 + 9) − (99/9))
:= 9/9 + ((9 + 9) × (9 × (9 + 9) − 9/9))
2
Ï 2895 := 1 + (1 + (11 × (((1 + 1) × (11 × (1 + 11))) − 1)))
Ï 2900 := 1 + ((1 + 1 + 11) × (1 + (1 + 1) × 111))
2
:= 5 + ((5 × (555 + 5 × 5)) − 5/5)
:= 66 × (((6 + 6)/6 + 6 × 6) + 6)
:= (7 − 7/7) × ((7 × (77 − 7) − 7) + 7/7)
:= 88 + (88 × ((8 + 8 + 8) + 8))
:= 9 × (9 + 9) × (9 + 9) − (99 + 9)/9
Ï 2905 := 1 + ((1 + 1) × (11 × (11 × (1 + 11))))
:= 2 + (((2 + 2 + 2) × (222 − 2)) + 2/2)
:= 2 × (((2/2 + 2) × 22 ) − 2)
:= 2/2 + (222 × (2 + 2 + 2))
:= (3 × (3 × 333 − 33)) − 3
:= (3/3 + 3) × (33+3 − (3/3 + 3))
:= 3/3 + ((3/3 + 3) × (33+3 − 3))
:= (4 × ((4 × (4 × 44 + 4)) + 4)) − 4/4
:= 4 + (4 × ((4 × (4 × 44 + 4)) + 4))
:= 4/4 + (44 × ((44 + 4 + 4)/4))
:= (5 × (555 + 5 × 5)) − 5
:= 5 × (555 + 5 × 5)
:= 5 + (5 × (555 + 5 × 5))
:= 6 + (((66 × (6 × 6 + 6)) + 666/6) + 6)
:= (6 − ((6 + 6)/6)) × ((66 × 66 − 6)/6)
:= 6/6 + (66 × (((6 + 6)/6 + 6 × 6) + 6))
:= 7 + ((7/7 − 77) × (77/7 − 7 × 7))
:= (7/7 + 7 × 7) × (((7 + 7)/7 + 7 × 7) + 7)
:= 7 + ((7 × ((7 × 7 × 7 − 7) + 77)) + 7)
:= (8 − 8/8 + 8) × (8 × (8 + 8 + 8) + 8/8)
:= 88 + (((8 × 8 × 88) − 8)/((8 + 8)/8))
:= 8/8 + ((88 × ((8 + 8 + 8) + 8)) + 88)
:= 9 + (999/9 × (((9 − 9/9) + 9) + 9))
:= (9/9 + 99) × (99/9 + 9 + 9)
:= 9 × (9 + 9) × (9 + 9) − 99/9
Ï 2896 := 11 + (((1 + 1) × (111 × (1 + 1 + 11))) − 1)
Ï 2901 := ((1 + 1) × ((11 × (11 × (1 + 11))) − 1)) − 1
2
Ï 2906 := (1 + 1) × (1 + (11 × (11 × (1 + 11))))
:= 2 + (((2 + 2 + 2) × (22 − 2)) + 2)
:= (2/2 + 2) × (2 × 22 − 2/2)
:= 2 + (222 × (2 + 2 + 2))
:= 3/3 + ((3 × (3 × 333 − 33)) − 3)
:= 3 + (3 × (3 × 333 − 33))
:= (3 + 3) × 33 + ((33/3 + 3)3 )
:= 4 × ((4 × (4 × 44 + 4)) + 4)
:= 4 + ((4 × ((4 × (4 × 44 + 4)) + 4)) + 4/4)
:= (4 + 4)/4 + (44 × ((44 + 4 + 4)/4))
:= 5/5 + ((5 × (555 + 5 × 5)) − 5)
:= 5/5 + (5 × (555 + 5 × 5))
:= 5 + ((5 × (555 + 5 × 5)) + 5/5)
:= (6 × 666) + ((66 − 6666)/6)
:= (6666 × 6/(6 + 6)) − (6 × (66 + 6))
:= 6 + ((6 − ((6 + 6)/6)) × ((66 × 66 − 6)/6))
:= 7 + ((7 × ((7 × 7 × 7 − 7) + 77)) − ((7 + 7)/7))
:= ((77 − 7)/7) + (7 × ((7 × 7 × 7 − 7) + 77))
:= 7 + (((7 × ((7 × 7 × 7 − 7) + 77)) + 7/7) + 7)
:= 8 + ((8 × (8 × (8 × 8 − 8) − 88)) + 8)
:= (88/8 − 8) × ((88 × 88 − 8)/8)
:= 88 + ((88 × ((8 + 8 + 8) + 8)) + ((8 + 8)/8))
:= 9 × (9 + 9) × (9 + 9) − (99/9 + 9)
:= 9 + (((9999 − 9)/9) + (9 + 9) × 99)
:= 9 × (9 + 9) × (9 + 9) − 9/9 − 9
2
Ï 2897 := 11 + ((1 + 1) × (111 × (1 + 1 + 11)))
Ï 2902 := (1 + 1) × ((11 × (11 × (1 + 11))) − 1)
2
Ï 2907 := 1 + ((1 + 1) × (1 + (11 × (11 × (1 + 11)))))
:= 22/2 + (222 × (22/2 + 2))
:= (22 × (2 + 2 + 2)) − 2
:= 2 + ((222 × (2 + 2 + 2)) + 2/2)
:= (3 × (3 × 333 − 33)) − 3/3
:= 3 + ((3 × (3 × 333 − 33)) + 3/3)
:= 3 × ((33 × (33 + 3)) − 3)
:= 4/4 + (4 × ((4 × (4 × 44 + 4)) + 4))
:= 4 + (((44 − 4)/4) × ((4 + 4)/4 + 44))
:= 4 + ((44 × ((44 + 4 + 4)/4)) − 4/4)
:= 5 + (((55 − ((5 − (5 + 5)/5)5 )) + 5) + 5)
:= (5 + 5)/5 + (5 × (555 + 5 × 5))
:= 5 × 5 + (55 − ((5 − (5 + 5)/5)5 ))
:= 6 + (((6 × 666) − (6666/6)) + 6)
:= 66 + ((66 × (6 × 6 + 6)) + ((6 + 6)/6)6 )
:= (66/6 + 6) × ((666/6 − 6) + 66)
:= 7 + ((7 × ((7 × 7 × 7 − 7) + 77)) − 7/7)
:= 77/7 + (7 × ((7 × 7 × 7 − 7) + 77))
:= ((7 + 7)/7 + 7 × 7) × ((7/7 + 7 × 7) + 7)
:= 88 + ((8 × 8 − 88/8)(8+8)/8 )
:= 88 + (((8 + 8)/8) × (88 × (8 + 8) − 8/8))
:= (88/8 − 8) × ((88 × 88 + 8)/8)
:= 9 × (9 + 9) × (9 + 9) − (9/9 + 9 + 9)
:= 9 + (9999/9 + (9 + 9) × 99)
:= 9 × (9 + 9) × (9 + 9) − 9
203
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2908 := (1 + 1) × (1 + (1 + (11 × (11 × (1 + 11)))))
:= 2 + ((222 × (2 + 2 + 2)) + 2)
:= 3/3 + (3 × ((33 × (33 + 3)) − 3))
Ï 2913 := 1 + ((1 + 1) × ((1 + 1 + 11) × (1 + 111)))
:= 2 + ((2 + 2 + 2) × (222 + 2))
:= 2/2 + ((22/2 + 2) × (222 + 2))
3
:= 3 + ((3 × (33 × (33 + 3))) − 3/3)
:= (3 × (3 × (33 + 3))) − 3
4
4
Ï 2918 := 1 + (1 + (((111 − 1)/(1 + 1) − 1)1+1 ))
4
:= 4 + (44 × ((44 + 4 + 4)/4))
:= 4 + ((((4 − 4/4) + 4) ) + 4 )
:= 4 + ((44 − 4/4 + 4) × (44 − 4 − 4)/4)
:= 5 + 55 + (5 − 5555)/(5 × 5)
:= 55 − ((55 + 55 )/(5 + 5 + 5))
:= 5 + (55 − ((55 + 55 )/(5 + 5 + 5)))
:= (6 − ((6 + 6)/6)) × ((66 × 66 + 6)/6)
:= (6 × ((6 + 6) × (6 × 6 + 6))) − 666/6
:= (6 + 6)/6 + ((66 − 6 − 6)(6+6)/6 )
:= 777 + (((7 + 7 + 7)/7)7 − (7 × 7 + 7))
:= 7/7 + (7 × (((77 × 77 − 7)/(7 + 7)) − 7))
:= (77 × (7 × 7 − 77/7)) − (7/7 + 7)
:= 88 + (((8 × 8 × 88) + 8)/((8 + 8)/8))
:= 8 × 8 × 8 + ((8 − 8/8)8×8/(8+8) )
:= 8 + ((88/8 − 8) × (88 × 88 + 8 + 8)/8)
:= 9/9 + (9 × (9 + 9) × (9 + 9) − 9)
:= 9 × (9 + 9) × (9 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 × (9 + 9) × (9 + 9)
Ï 2909 := 1 + ((1 + 1) × (1 + (1 + (11 × (11 × (1 + 11))))))
Ï 2914 := (1 + 1) × (1 + ((1 + 1 + 11) × (1 + 111)))
Ï 2919 := 11111 − ((1 + 1)1+1+11 )
:= 2 + (((222 × (2 + 2 + 2)) + 2/2) + 2)
:= ((2 + 2 + 2) × (222 + 2)) − 2
:= 2 + (((2 + 2 + 2) × (222 + 2)) + 2/2)
:= ((3/3 + 3) × (33+3 − 3/3)) − 3
:= 3/3 + ((3 × (33 × (33 + 3))) − 3)
:= 3 + (3 × (33 × (33 + 3)))
:= 4 + ((44 × ((44 + 4 + 4)/4)) + 4/4)
:= (44 − 4/4 + 4) × (44 − 4 − 4)/4
:= 4 + (44/4 × (((4/4 + 44 ) + 4) + 4))
:= 55 + ((5 − 5/5) × (5/5 − 55))
:= 55 × 55 − 555/5
:= 5 + (55 × 55 − 555/5)
:= ((6 − 6/6)6−6/6 ) − 6 × 6 × 6
:= ((66 − 6 − 6)
:= ((7 × 7 − ((7 + 7)/7) + 7)(7+7)/7 ) − 7
:= 7 + (((7 + 7)/7 + 7 × 7) × ((7/7 + 7 × 7) + 7))
:= (77 × (7 × 7 − 77/7)) − 7
:= (8 × 8 − ((8 + 8)/8)) × (8 × 8 − (8/8 + 8 + 8))
:= (8 − 8/8) × ((8 × 8 × 8 − (88 + 8)) + 8/8)
:= 9 × (9 + 9) × (9 + 9) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + 9 × (9 + 9) × (9 + 9)
:= 8 + ((88/8 − 8) × ((88 × 88 − 8)/8))
:= (9 + 9)/9 + (9 × (9 + 9) × (9 + 9) − 9)
Ï 2910 := (1 + 1) × (((1 + 1 + 11) × (1 + 111)) − 1)
:= (2/2 + 2) × (2 × 222 + 2)
:= 3 + (3 × ((33 × (33 + 3)) − 3))
:= ((4 + 4)/4 + 4) × ((44 × 44 + 4)/4)
:= 5 + ((5 × (555 + 5 × 5)) + 5)
:= ((66 − 6 − 6)(6+6)/6 ) − 6
:= (7 − 7/7) × ((7 × (77 − 7) − 7) + ((7 + 7)/7))
:= (88/8 − 8) × (88 × 88 + 8 + 8)/8
:= ((9 + 9 + 9)/9) + (9 × (9 + 9) × (9 + 9) − 9)
Ï 2911 := ((1 + 1) × ((1 + 1 + 11) × (1 + 111))) − 1
:= 2/2 + ((2/2 + 2) × (2 × 222 + 2))
:= 3 + ((3 × ((33 × (33 + 3)) − 3)) + 3/3)
:= 444/4 + (4 × (444 + 44 ))
(6+6)/6
) − (6 + 6)/6
:= 666/6 + (6 × (6 × (66 + 6 + 6)))
Ï 2915 := 11 × (1 + ((1 + 1) × (11 × (1 + 11))))
Ï 2920 := 1 + (11111 − ((1 + 1)1+1+11 ))
:= 22/2 + (222 × (2 + 2 + 2))
:= 2 + (((2 + 2 + 2) × (222 + 2)) + 2)
:= (3 × (33 × (33 + 3))) − 3/3
:= (3/3 + 3) × (33+3 + 3/3)
:= 44/4 × (((4/4 + 44 ) + 4) + 4)
:= 4 + (4 × ((4 − 4/4)4+(4+4)/4 ))
:= 55 × (55 − (5 + 5)/5)
:= 5 + (55 × (55 − (5 + 5)/5))
:= ((66 − 6 − 6)
(6+6)/6
:= 6 + (((66 − 6 − 6)(6+6)/6 ) − ((6 + 6)/6))
) − 6/6
:= 777 + (((7 + 7 + 7)/7)7 − 7 × 7)
:= 7/7 + ((77 × (7 × 7 − 77/7)) − 7)
:= 88/8 × (((8 + 8) × (8 + 8) + 8/8) + 8)
:= (8 × (888 − 8 × 8 × 8)) − 88
:= 9 × (9 + 9) × (9 + 9) − 9/9
:= (9 − 9/9) × (((9 × 9 × 9 × 9) + 9)/(9 + 9))
Ï 2916 := ((111 − 1)/(1 + 1) − 1)1+1
Ï 2921 := 1 + (1 + (11111 − ((1 + 1)1+1+11 )))
:= (2 + 2 + 2) × (222 + 2)
:= (22 + 2/2) × ((22×(2+2) − 2)/2)
3
:= 3 + (((3 × (33 × (33 + 3))) − 3/3) + 3)
:= 3 × (3 × (33 + 3))
4+(4+4)/4
:= 4 × ((4 − 4/4)
:= ((44 + 4/4) × (44 + 4)/4) − 4
)
(5+5)/5
:= (55 − 5/5)
:= 5 + ((55 − 5/5)(5+5)/5 )
:= (66 − 6 − 6)(6+6)/6
:= 6 + (((66 − 6 − 6)(6+6)/6 ) − 6/6)
:= (7 × 7 − ((7 + 7)/7) + 7)(7+7)/7
:= ((7 + 7)/7)7 + 7 × (7 × (7 × 7 + 7) + 7)
:= 8 + (((88 × ((8 + 8 + 8) + 8)) − 8/8) + 88)
:= ((8 − 88)/8 + 8 × 8)(8+8)/8
:= ((8 − 8/8 + 8) + 8) × (8 × (8 + 8) − 8/8)
:= ((9 − 99)/(9 + 9)) + 9 × (9 + 9) × (9 + 9)
:= 9 × (9 + 9) × (9 + 9)
:= 9 × (9 + 9) × (9 + 9) + ((9 × 9 + 9)/(9 + 9))
:= ((55 − 5/5)(5+5)/5 ) − 5
:= 6/6 + (((66 − 6 − 6)(6+6)/6 ) − 6)
:= (7 × 7 − (7/7 + 7)) × ((7/7 − 7) + 77)
Ï 2912 := (1 + 1) × ((1 + 1 + 11) × (1 + 111))
:= (22/2 + 2) × (222 + 2)
:= (3/3 + 3) × (33+3 − 3/3)
Ï 2917 := 1 + (((111 − 1)/(1 + 1) − 1)1+1 )
Ï 2922 := (1 + 1) × ((11 × (1 + (11 × (1 + 11)))) − (1 + 1))
:= 2/2 + ((2 + 2 + 2) × (222 + 2))
:= (2/2 + 2) × ((2 × (222 + 2)) + 2)
3
:= 3 + ((3 × (33 × (33 + 3))) + 3)
:= 3/3 + (3 × (3 × (33 + 3)))
4
4
4
:= 4 × (((4 × (4 × 44 + 4)) + 4) + 4)
:= 4 + (((((4 − 4/4) + 4) ) + 4 ) + 4 )
:= (44/4 × ((44 − 4)/4 + 44 )) − 4
:= 55 + (((5 − 55 )/(5 + 5 + 5)) − 5)
:= 55 + ((5 − 55 )/(5 + 5 + 5))
:= 5 + (((5 − 55 )/(5 + 5 + 5)) + 55 )
:= (6 + 6)/6 + (((66 − 6 − 6)(6+6)/6 ) − 6)
:= 6/6 + ((66 − 6 − 6)(6+6)/6 )
:= 6 + ((66 − 6 − 6)(6+6)/6 )
:= 7 × (((77 × 77 − 7)/(7 + 7)) − 7)
:= 7/7 + ((7 × 7 − ((7 + 7)/7) + 7)(7+7)/7 )
:= (7 × (7 × (7 + 7) + 7)) + ((7 + 7 + 7)/7)7
:= 8 + ((88 × ((8 + 8 + 8) + 8)) + 88)
:= 8/8 + (((8 − 88)/8 + 8 × 8)(8+8)/8 )
:= (8 × (8 × 8 × 8 − 8)) + ((8 − 8888)/8)
:= (9 − 9/9) × (((9 × 9 × 9 × 9) − 9)/(9 + 9))
:= 9/9 + 9 × (9 + 9) × (9 + 9)
:= 9 + (9 × (9 + 9) × (9 + 9) − ((9 + 9 + 9)/9))
204
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2923 := 11 + ((1 + 1) × ((1 + 1 + 11) × (1 + 111)))
Ï 2933 := 1 + ((1 + 1) × (1 + (1 + ((1 + 11) × (1 + 111+1 )))))
Ï 2928 := (1 + 1) × ((1 + 11) × (1 + 111+1 ))
:= 2 + ((22 + 2/2) × ((22×(2+2) − 2)/2))
:= (2 + 2 + 2) × (222 + 2 + 2)
:= 2/2 + (2 × (((2 + 2 + 2)2 + 2)2 + 22))
:= 3 + ((3/3 + 3) × (33+3 + 3/3))
:= (3/3 + 3) × (33+3 + 3)
:= 3 × 3 × 333 − ((3/3 + 3)3 )
:= 44 + ((4 × (4 × (4 × 44 + 4))) − 4/4)
:= 4 × ((4 × (4 × 44 − 4)) + 44)
:= (44/4 × (44/4 + 44 )) − 4
:= 55 + (((5 + 5) × (5 − 5 × 5)) − ((5 + 5)/5))
:= 55 − (((5/5 + 5) × ((5 + 5)/5)5 ) + 5)
:= 55 − ((5/5 + 5) × ((5 + 5)/5)5 )
:= 6 + (((66 − 6 − 6)(6+6)/6 ) + 6/6)
:= 6 + (((66 − 6 − 6)(6+6)/6 ) + 6)
:= 6 + (((66 − 6 − 6)(6+6)/6 ) + (66/6))
:= 7 + ((7 × 7 − ((7 + 7)/7) + 7)(7+7)/7 )
:= (7 + 7)/7 + (77 × (7 × 7 − 77/7))
:= 7 × (7 × 7 × 7 + 77) − 7
:= 8 + (88/8 × (((8 + 8) × (8 + 8) + 8/8) + 8))
:= (8 + 8) × ((888/8 + 8 × 8) + 8)
:= ((8 + 8) × ((88 + 88) + 8)) − 88/8
:= 9 + (9 × (9 + 9) × (9 + 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + 9 × (9 + 9) × (9 + 9)
:= 9 + ((9 × (9 + 9) × (9 + 9) − 9/9) + 9)
Ï 2924 := (1 + 1) × ((11 × (1 + (11 × (1 + 11)))) − 1)
:= 2 × (2 × (((2/2 + 2)2+2+2 ) + 2))
:= (3/3 + 3) × ((33+3 − 3/3) + 3)
:= 44 + (4 × (4 × (4 × 44 + 4)))
Ï 2929 := 1 + ((1 + 1) × ((1 + 11) × (1 + 111+1 )))
2
:= 2 + (2 × (((2 + 2 + 2)2 + 2)2 + 22))
:= 2/2 + ((2 + 2 + 2) × (22 + 2 + 2))
:= 3/3 + ((3/3 + 3) × (3
3+3
Ï 2934 := (1 + 1) × (11 + ((1 + 1 + 11) × (1 + 111)))
:= 3 × (((33 × (33 + 3)) + 3) + 3)
+ 3))
4
:= 4 + ((44/4 × ((44 − 4)/4 + 44 )) + 4)
:= 4 + ((44 + 4/4) × (4 + 4)/4)
5
:= 55 + (((5 + 5) × (5 − 5 × 5)) − 5/5)
:= 5 + ((((5 + 5) × (5 − 5 × 5)) − 5/5) + 5 )
:= ((6 × 6 + 6/6) + 6) × (((6 + 6)/6) + 66)
:= 6 + ((((66 − 6 − 6)(6+6)/6 ) + 6/6) + 6)
:= ((6 − 6/6) × 666) − 6 × 66
:= (77 × (7 × 7 − 77/7)) − (7 + 7)/7
:= 7 × (7 × 7 × 7 + 77) − 77/7
:= 7/7 + (7 × (7 × 7 × 7 + 77) − 7)
:= 8 + (((8 − 88)/8 + 8 × 8)(8+8)/8 )
:= 8 + (((8 − 8/8 + 8) + 8) × (8 × (8 + 8) − 8/8))
:= 888 + ((8 × (8 + 8) × (8 + 8)) − ((8 + 8)/8))
:= 9 + (9 × (9 + 9) × (9 + 9) − 9/9)
:= 9 × (9 + 9) × (9 + 9) + ((99 + 9 + 9)/9)
:= 9 + (9 × (9 + 9) × (9 + 9) + 9)
Ï 2925 := (1 + 1 + 11) × (1 + ((1 + 1) × (1 + 111)))
Ï 2930 := (1 + 1) × (1 + ((1 + 11) × (1 + 111+1 )))
2
:= 55 − ((555/5 + 55) + 5 × 5)
Ï 2935 := 1 + ((1 + 1) × (11 + ((1 + 1 + 11) × (1 + 111))))
:= 22 + ((22 × (2 + 2 + 2)) − 2/2)
:= 2 + ((2 + 2 + 2) × (22 + 2 + 2))
:= 222/2 + 2 × 2 × 222 − 2/2
:= 3 × ((33 × (33 + 3)) + 3)
:= 3 + ((3 × (33 × (33 + 3))) + 33/3)
:= 3 + ((3/3 + 3) × ((33+3 + 3/3) + 3))
:= (44 + 4/4) × (44 + 4)/4
:= 4 + (44/4 × ((44 − 4)/4 + 44 ))
:= ((4 + 4) × (444/4 + 44 )) − 4/4
:= 5 × ((555 + 5 × 5) + 5)
:= 5 + (((5 + 5) × (5 − 5 × 5)) + 55 )
:= 5 × (((5 + 5)/5)5 + 555)
:= (6/6 − 66) × (66 − 666/6)
:= 6 + (((6 × 6 + 6/6) + 6) × (((6 + 6)/6) + 66))
:= 6/6 + (((6 − 6/6) × 666) − 6 × 66)
:= (77 × (7 × 7 − 77/7)) − 7/7
:= ((7 − 77)/7) + 7 × (7 × 7 × 7 + 77)
:= (7 + 7)/7 + (7 × (7 × 7 × 7 + 77) − 7)
:= (8/8 + 8 × 8) × (8 × 8 − (88/8 + 8))
:= 8 + (((8 − 8888)/8) + (8 × (8 × 8 × 8 − 8)))
:= 888 + ((8 × (8 + 8) × (8 + 8)) − 8/8)
2
99/9
:= 9 + 9 × (9 + 9) × (9 + 9)
:= 9 × 99 + ((((9 + 9)/9)
Ï 2926 := (1 + 1) × (11 × (1 + (11 × (1 + 11))))
2
:= 9 + ((9 × (9 + 9) × (9 + 9) + 9/9) + 9)
) − 9)
Ï 2931 := 1 + ((1 + 1) × (1 + ((1 + 11) × (1 + 111+1 ))))
Ï 2936 := (1 + 1) × (((1 + 1 + 11) × (1 + 1 + 111)) − 1)
:= 22 + (22 × (2 + 2 + 2))
:= 2 + (((2 + 2 + 2) × (22 + 2 + 2)) + 2/2)
:= 222/2 + 2 × 2 × 222
:= 3/3 + (3 × ((33 × (33 + 3)) + 3))
:= 3 + ((3/3 + 3) × (33+3 + 3))
:= 3 + (3 × 3 × 333 − ((3/3 + 3)3 ))
:= 44/4 × ((44 − 4)/4 + 44 )
:= 4 + ((4 × 4 × 4 × 44) + 444/4)
:= (4 + 4) × (444/4 + 44 )
:= 5 + (((55 − 5/5)(5+5)/5 ) + 5)
:= 5 + ((((55 − 5/5)(5+5)/5 ) + 5) + 5)
:= 5 × 5 + (((55 − 5/5)(5+5)/5 ) − 5)
:= ((6 + 6)/6 + 6) × ((6 × (66 − 6) + 6/6) + 6)
2
:= ((6 + 6)/6 + 6 × 6) × (66/6 + 66)
:= 6 + ((6/6 − 66) × (66 − 666/6))
:= 77 × (7 × 7 − 77/7)
:= 7 × (7 × 7 × 7 + 77) − ((7 + 7)/7 + 7)
:= 7 + (7 × (7 × 7 × 7 + 77) − (77/7))
:= 8 × 88 + (((8 + 8)/8) × 8888/8)
:= (88/8 − 8) × (((88 × 88 + 8)/8) + 8)
:= 888 + (8 × (8 + 8) × (8 + 8))
:= 9 + (9 × (9 + 9) × (9 + 9) + 9/9)
:= 9 + ((9 × (9 + 9) × (9 + 9) − ((9 + 9 + 9)/9)) + 9)
:= 9 + (9 × (9 + 9) × (9 + 9) + (99/9))
Ï 2927 := 1 + ((1 + 1) × (11 × (1 + (11 × (1 + 11)))))
2
Ï 2932 := (1 + 1) × (1 + (1 + ((1 + 11) × (1 + 111+1 ))))
2
2
Ï 2937 := 11 × (1 + ((1 + 1) × (1 + (11 × (1 + 11)))))
:= 22 + ((22 × (2 + 2 + 2)) + 2/2)
:= 2 × (((2 + 2 + 2) + 2) + 22)
:= (2/2 + 2) × (2 × 222 + 22/2)
:= 33/3 + (3 × (33 × (33 + 3)))
:= (3/3 + 3) × ((33+3 + 3/3) + 3)
:= 33 × ((3 × (33 + 3)) − 3/3)
:= 444/4 + (4 × 4 × 4 × 44)
:= 4 + (4 × ((4 × (4 × 44 − 4)) + 44))
:= 44/4 × (44/4 + 44 )
:= 5 + ((((5 − 55 )/(5 + 5 + 5)) + 55 ) + 5)
:= 55 + (55 − (((5 − (5 + 5)/5)5 ) + 5))
:= 55 + (55 − ((5 − (5 + 5)/5)5 ))
:= 6 + (((6 + 6)/6 + 6 × 6) × (66/6 + 66))
:= (6666 × 6/(6 + 6)) − 6 × 66
:= 7 × (7 × 7 × 7 + 77) − (7/7 + 7)
:= 77/7 + (77 × (7 × 7 − 77/7))
:= 66/6 + ((66 − 6 − 6)
(6+6)/6
)
:= 7/7 + (77 × (7 × 7 − 77/7))
(8+8)/8
:= 888/8 + (88 × ((8 + 8 + 8) + 8))
:= 8 + ((((8 − 88)/8 + 8 × 8)
:= 99/9 + 9 × (9 + 9) × (9 + 9)
:= 9 + ((9 × (9 + 9) × (9 + 9) − ((9 + 9)/9)) + 9)
205
) + 8)
:= 8/8 + ((8 × (8 + 8) × (8 + 8)) + 888)
:= 9 + (9 × (9 + 9) × (9 + 9) + ((99 + 9)/9))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2938 := (1 + 1) × ((1 + 1 + 11) × (1 + 1 + 111))
2
Ï 2943 := (111 − 1 − 1) × (1 + 1 + 1)1+1+1
2+2+2
Ï 2948 := (1 + 1) × (11 × (1 + (1 + (11 × (1 + 11)))))
:= 22 × (22 × (2 + 2 + 2) + 2)
:= 22 + ((2 + 2 + 2) × (22 + 2))
:= (((2 × 22) + 2) × 2
:= 3/3 + (33 × ((3 × (33 + 3)) − 3/3))
:= 3 × (3 × (333 − (3 + 3)))
:= 33 + ((3 × (33 × (33 + 3))) − 3/3)
:= 4/4 + (44/4 × (44/4 + 44 ))
:= (4 × (4 × ((4 × 44 + 4) + 4))) − 4/4
:= 44 × (((44 − 4)/4) + 4)
:= 5 + (55 − ((5/5 + 5) × ((5 + 5)/5)5 ))
:= 55 − ((5 × 5 × 5 + ((5 + 5)/5)) + 55)
:= 55 − (((555 + 55)/5) + 55)
:= (6/6 + 6 + 6) × (((66 − 6)/6) + 6 × 6 × 6)
:= 6 + ((6666 × 6/(6 + 6)) − 6 × 66)
:= (66 + 6/6) × (((6 + 6)/6 + 6 × 6) + 6)
:= 7 × (7 × 7 × 7 + 77) − (7 + 7)/7
:= (7 + 7 + 7)/7 + 7 × (7 × 7 × 7 + 77)
:= 7 + (7 × (7 × 7 × 7 + 77) + 7/7)
:= 888 + ((8 × (8 + 8) × (8 + 8)) + ((8 + 8)/8))
:= ((8 + 8) × ((88 + 88) + 8)) − 8/8
:= 88 + ((8/8 + 8 × 8) × (88/((8 + 8)/8)))
:= ((9 + 9)/9) × (9 × 9 × (9 + 9) + (99/9))
:= 9 + ((9 × (9 + 9) × (9 + 9) + 9) + 9)
:= 9 + ((((9 + 9)/9)99/9 ) + 9 × 99)
Ï 2939 := 1 + ((1 + 1) × ((1 + 1 + 11) × (1 + 1 + 111)))
:= 2 + ((2/2 + 2) × (2 × 222 + 22/2))
) − 2/2
Ï 2944 := 1 + ((111 − 1 − 1) × (1 + 1 + 1)1+1+1 )
Ï 2949 := 11 × 111 + ((1 + 11)1+1+1 )
:= ((2 × 22) + 2) × 22+2+2
:= 2/2 + (22 × (22 × (2 + 2 + 2) + 2))
:= 3/3 + (3 × (3 × (333 − (3 + 3))))
:= 33 + (3 × (33 × (33 + 3)))
:= (44 × (4 − 4/4)4 ) − (4/4 + 4)4
:= 4 × (4 × ((4 × 44 + 4) + 4))
:= 4/4 + (44 × (((44 − 4)/4) + 4))
:= 5 × 5 + (55 × 55 − 555/5)
:= 55 − ((5 × 5 × 5 + 55) + 5/5)
:= 55 − ((5 × ((5 × 5 + 5) + 5)) + 5/5)
:= ((6 × 6 + 6) × (((6 + 6)/6)6 + 6)) − 6/6
:= ((6 + 6)/6)6 × (((66 − 6)/6) + 6 × 6)
:= 66 + ((66 × (6 × 6 + 6)) + 666/6)
:= 7 × (7 × 7 × 7 + 77) − 7/7
:= 77/7 + (7 × (7 × 7 × 7 + 77) − 7)
:= 7 + (7 × (7 × 7 × 7 + 77) + ((7 + 7)/7))
:= 8 + ((88/8 − 8) × (((88 × 88 + 8)/8) + 8))
:= (8 + 8) × ((88 + 88) + 8)
:= 8 + ((((8 + 8) × ((88 + 88) + 8)) − (88/8)) + 8)
:= 9 + (((9 × (9 + 9) × (9 + 9) + 9/9) + 9) + 9)
:= 9 + ((99 − 9/9) × ((((99 + 9)/9) + 9) + 9))
3
:= 3 + ((3/3 + 3) × (3
3+3
− 3/3))
99/9
:= 9 × 99 + (((9 + 9)/9)
)
Ï 2940 := (1 + 11) × (1 + ((1 + 1) × (1 + 111+1 )))
:= (2 + 2 + 2) × ((222 + 2 + 2) + 2)
Ï 2945 := 1 + (1 + ((111 − 1 − 1) × (1 + 1 + 1)1+1+1 ))
:= 2/2 + (((2 × 22) + 2) × 22+2+2 )
Ï 2950 := 1 + (11 × 111 + ((1 + 11)1+1+1 ))
:= 2 + (22 × (22 × (2 + 2 + 2) + 2))
:= 3 + (((33/3 + 3) ) + 33 × (3 + 3))
:= 3/3 + ((3 × (33 × (33 + 3))) + 33)
:= (4 − 4/4) × (4 × 44 − 44)
:= 4/4 + (4 × (4 × ((4 × 44 + 4) + 4)))
:= (4 + 4)/4 + (44 × (((44 − 4)/4) + 4))
:= 55 − ((5 × 5 × 5 + 55) + 5)
:= 55 − (5 × 5 × 5 + 55)
:= (5 + 5) × ((5 × (55 + 5)) − 5)
:= (3/3 + 3) × ((3
3+3
+ 3) + 3)
6
3
6−6/6
:= (6 × (6 − 6 × 6)) + ((6 − 6/6)
:= 7 × (7 × 7 × 7 + 77)
:= 7 + (7 × (7 × 7 × 7 + 77) − ((7 + 7)/7))
:= 777 + (((7 + 7 + 7)/7)7 − (7 + 7))
:= 88 + (((8 × (8 × 88 + 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8) × ((88 + 88) + 8))
:= 8 + (((8 + 8) × ((88 + 88) + 8)) − ((8 + 8)/8))
:= (99 − 9/9) × ((((99 + 9)/9) + 9) + 9)
:= 9 + ((9 × (9 + 9) × (9 + 9) + (99/9)) + 9)
:= 9 + (((9 − 9/9) + 9) × (99/9 + 9 × (9 + 9)))
Ï 2941 := 1 + ((1 + 11) × (1 + ((1 + 1) × (1 + 111+1 ))))
2
)
:= ((6 + 6) × (6 × (6 × 6 + 6) − 6)) − (6 + 6)/6
:= (6 × 6 + 6) × (((6 + 6)/6) + 6)
Ï 2946 := (1 + 1) × ((11 × (1 + 11)1+1 ) − 111)
Ï 2951 := (1 + 1 + 11) × (1 + ((1 + 1) × (1 + 1 + 111)))
:= 2222 + ((2/2 + 2)2+2+2 )
:= 2/2 + ((2 + 2 + 2) × ((22 + 2 + 2) + 2))
:= 2 + (((2 × 22) + 2) × 2
:= 3/3 + ((3/3 + 3) × ((33+3 + 3) + 3))
:= 3 + (3 × (3 × (333 − (3 + 3))))
:= (3 + 3)3 + (((33/3 + 3)3 ) − 3 × 3)
:= 4 + (44/4 × (44/4 + 44 ))
:= (4 + 4)/4 + (4 × (4 × ((4 × 44 + 4) + 4)))
:= 4 + ((44 × (((44 − 4)/4) + 4)) − 4/4)
:= 5 × 5 + ((55 − 5/5)(5+5)/5 )
:= 55 + (5/5 − (5 × 5 × 5 + 55))
:= 55 + (5/5 − (5 × ((5 × 5 + 5) + 5)))
:= 6/6 + ((6 × 6 + 6) × (((6 + 6)/6)6 + 6))
:= ((6 + 6) × (6 × (6 × 6 + 6) − 6)) − 6
:= ((6 + 6) × (6 × (6 × 6 + 6) − 6)) − 6/6
:= 7/7 + 7 × (7 × 7 × 7 + 77)
:= 7 + (7 × (7 × 7 × 7 + 77) − 7/7)
:= 77/7 + 7 × (7 × 7 × 7 + 77)
:= 8 + (((8 + 8) × ((88 + 88) + 8)) − (88/8))
:= (8 + 8)/8 + ((8 + 8) × ((88 + 88) + 8))
:= 8 + (((8 + 8) × ((88 + 88) + 8)) − 8/8)
:= ((9 − 9/9) + 9) × (99/9 + 9 × (9 + 9))
:= 999/9 + (9 × ((9 + 9) × (9 + 9) − 9))
:= ((9 + 9) × (((9 + 9)/9) + 9 × (9 + 9))) − 9/9
Ï 2942 := ((111 − 1 − 1) × (1 + 1 + 1)1+1+1 ) − 1
2+2+2
)
Ï 2947 := ((1 + 1) × (11 × (1 + (1 + (11 × (1 + 11)))))) − 1
Ï 2952 := (1 + 1) × ((1 + 11) × (1 + (1 + 111+1 )))
:= (((2 × 22) + 2) × 22+2+2 ) − 2
:= (22 × (22 × (2 + 2 + 2) + 2)) − 2/2
:= (22 + 2) × ((22/2)2 + 2)
:= 33 × (3 + 3) + ((33/3 + 3)3 )
:= 3 + ((3 × (3 × (333 − (3 + 3)))) + 3/3)
:= 3 × ((3 × (333 − (3 + 3))) + 3)
4
:= (4 × (4 × ((4 × 44 + 4) + 4))) − (4 + 4)/4
:= (44 × (((4 − 4)/4) + 4)) − 4/4
:= 4 + (44 × (((44 − 4)/4) + 4))
:= 5 + ((55 − ((5 − (5 + 5)/5)5 )) + 55 )
:= 55 + ((5 + 5)/5 − (5 × 5 × 5 + 55))
:= 55 + ((5 + 5)/5 − (5 × ((5 × 5 + 5) + 5)))
:= (6 + 6)/6 + ((6 × 6 + 6) × (((6 + 6)/6)6 + 6))
:= 6/6 + (((6 + 6) × (6 × (6 × 6 + 6) − 6)) − 6)
:= (6 + 6) × (6 × (6 × 6 + 6) − 6)
:= (7 + 7)/7 + 7 × (7 × 7 × 7 + 77)
:= 7 + 7 × (7 × 7 × 7 + 77)
:= (7 − 7/7) × (7 × (77 − 7) + ((7 + 7)/7))
:= ((8 + 8) × ((88 + 88) + 8)) − (8 + 8)/8
:= 888 + ((8 × (8 + 8) × (8 + 8)) + (88/8))
:= 8 + ((8 + 8) × ((88 + 88) + 8))
:= 9 + (((9 × (9 + 9) × (9 + 9) − 9/9) + 9) + 9)
:= 9 + (9 × (9 + 9) × (9 + 9) + ((99 + 99)/9))
:= (9 + 9) × (((9 + 9)/9) + 9 × (9 + 9))
206
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2953 := 1 + ((1 + 1) × ((1 + 11) × (1 + (1 + 111+1 ))))
Ï 2958 := ((1 + 1 + 1) × ((1 + 1)11−1 − 1)) − 111
Ï 2963 := ((1 + 1) × ((1 + 1)11 − 11)) − 1111
:= 2/2 + ((22 + 2) × ((22/2)2 + 2))
:= 2 × ((2 × (22 − 2))2 − (22/2)2 )
:= 2 + (((222/2 + 2)/2) + (2 × 22)2 )
:= 3/3 + (3 × ((3 × (333 − (3 + 3))) + 3))
:= (3 × ((3 × (333 − 3)) − 3)) − 3
:= 3 + (((33/3 + 3)3 ) + (3 + 3)3 )
4
4
:= 4 + ((44 × (((4 − 4)/4) + 4)) + 4/4)
:= 4 × 4 + (44 × 44 − (4 + 4)/4)
:= 4 + ((44 × 44 − 4/4) + 4 × 44 )
:= 55 − (((555 + 5)/5 + 55) + 5)
:= 55 − ((555 + 5)/5 + 55)
:= 5 + (55 − ((555 + 5)/5 + 55))
:= 6/6 + ((6 + 6) × (6 × (6 × 6 + 6) − 6))
:= 6 + ((6 + 6) × (6 × (6 × 6 + 6) − 6))
:= 66/6 + ((6 + 6) × (6 × (6 × 6 + 6) − 6))
:= 7 + ((7 × (7 × 7 × 7 + 77) − 7/7) + 7)
:= 7 + (7 × (7 × 7 × 7 + 77) + (77/7))
:= 777 + (((7 + 7 + 7)/7)7 − 7/7)
:= 8 + (((8 + 8) × ((88 + 88) + 8)) + 8/8)
:= ((8 + 8)/8) × ((8/8 + 8 + 8) × (88 − 8/8))
:= (8888 + 8/8)/(88/8 − 8)
:= 9/9 + ((9 + 9) × (((9 + 9)/9) + 9 × (9 + 9)))
:= (99/9 + 9 + 9) × (999/9 − 9)
:= 99/9 + ((9 + 9) × (((9 + 9)/9) + 9 × (9 + 9)))
Ï 2954 := (1 + 1) × (1 + ((1 + 11) × (1 + (1 + 111+1 ))))
Ï 2964 := (1 + 1) × ((1 + 1 + 11) × (1 + 1 + 1 + 111))
:= (2 × 22)2 + ((222/2 − 2)/2)
:= 2 × ((2 × (22×(2+2) + 222 )) + 2)
3
:= (3 × (3 × (333 − 3))) − 33/3
:= 3 × 3 × 333 − 33
:= 2 + ((22 + 2) × ((22/2) + 2))
3
Ï 2959 := 11 × (1 + ((1 + 1) × (1 + (1 + (11 × (1 + 11))))))
2
:= (3 + 3) + (((33/3 + 3) ) − (3 + 3))
4
:= 4 + ((44 × (((4 − 4)/4) + 4)) + (4 + 4)/4)
4
:= 4 + (44 × 44 + 4 × 44 )
:= 4 × 4 + (44 × 44 − 4/4)
5
5
:= 5 − ((555/5 + 55) + 5)
:= 5 − (555/5 + 55)
:= 5 + (55 − (555/5 + 55))
:= (6 + 6)/6 + ((6 + 6) × (6 × (6 × 6 + 6) − 6))
:= 6 + (((6 + 6) × (6 × (6 × 6 + 6) − 6)) + 6/6)
:= 6 + (((6 + 6) × (6 × (6 × 6 + 6) − 6)) + 6)
:= 7 + (7 × (7 × 7 × 7 + 77) + 7)
:= 77/7 × ((7 × 7 × 77 − 7)/(7 + 7))
:= 777 + ((7 + 7 + 7)/7)7
:= 8 + (((8 + 8) × ((88 + 88) + 8)) + ((8 + 8)/8))
:= 8 + ((((8 + 8) × ((88 + 88) + 8)) − 8/8) + 8)
:= ((88 + 8)/8) × ((8 + 8) × (8 + 8) − (8/8 + 8))
:= 9 + (((9 × (9 + 9) × (9 + 9) + (99/9)) + 9) + 9)
:= 999 + ((99 − 9/9) × (99/9 + 9))
:= ((9 + 9 + 9)/9) × (999 − (99/9))
Ï 2955 := 1 + ((1 + 1) × (1 + ((1 + 11) × (1 + (1 + 111+1 )))))
2
Ï 2960 := (11 − 1) × ((11 × (1 + 1 + 1)1+1+1 ) − 1)
2×(2+2)
2
:= 2 + (((22 + 2) × ((22/2) + 2)) + 2/2)
:= 2 × (2 × (2
:= (3 × (3 × 333 − 3)) − 33
:= (3 + 3)3 + ((33/3 + 3)3 )
Ï 2965 := 1 + ((1 + 1) × ((1 + 1 + 11) × (1 + 1 + 1 + 111)))
:= ((22 + 2/2) × ((22×(2+2) + 2)/2)) − 2
+ 22 ))
:= 3/3 + (3 × 3 × 333 − 33)
4
:= 44/4 + (4 × (4 × ((4 × 44 + 4) + 4)))
:= 4 × 4 + 44 × 44
:= 4 + ((44 × 44 + 4 × 44 ) + 4/4)
:= 5 + ((5 + 5) × ((5 × (55 + 5)) − 5))
:= 55 + (55 × ((5 + 5)/5 − 5))
:= 55 − (5 × ((5 + 5)/5)5 )
:= 666/6 + (6 × ((6 × (66 + 6 + 6)) + 6))
:= (6 × ((6 + 6) × (6 × 6 + 6))) − ((6 + 6)/6)6
:= 6 + ((((6 + 6) × (6 × (6 × 6 + 6) − 6)) + 6/6) + 6)
:= 7 + ((7 × (7 × 7 × 7 + 77) + 7/7) + 7)
:= (7 × ((77 × 77 − 7)/(7 + 7))) − 7/7
:= (77 × 77 + 7/7)/((7 + 7)/7)
:= 88/8 + ((8 + 8) × ((88 + 88) + 8))
:= 8 + (((8 + 8) × ((88 + 88) + 8)) + 8)
:= (8/8 − (8888 + 8))/(8 − 88/8)
:= 9 + ((9 × ((9 + 9) × (9 + 9) − 9)) + 999/9)
:= (9/9 + 9) × ((99 × (9 + 9 + 9) − 9)/9)
:= 9 × (9 + 9) × (9 + 9) + ((9 × 99 − 9)/(9 + 9))
Ï 2956 := (1 + 1) × (1 + (1 + ((1 + 11) × (1 + (1 + 111+1 )))))
:= 2 + (((22 + 2) × ((22/2)2 + 2)) + 2)
:= 3/3 + ((3 × (3 × 333 − 3)) − 33)
4
:= 4 × 4 + (44 × 44 − 4)
:= 5 + ((5/5 − (5 × ((5 × 5 + 5) + 5))) + 55 )
:= 6 + (((6 + 6) × (6 × (6 × 6 + 6) − 6)) − ((6 + 6)/6))
:= 7 + ((7 × (7 × 7 × 7 + 77) + ((7 + 7)/7)) + 7)
:= ((88 + 8)/8) + ((8 + 8) × ((88 + 88) + 8))
:= ((9 + 9)/9) × ((9 × 9 × (9 + 9) + (99/9)) + 9)
Ï 2957 := ((1 + 1 + 1) × ((1 + 1)11−1 − 1)) − (1 + 111)
2
:= (2 × 22) + (((2
3
22/2
− 2)/2) − 2)
3
:= (3 + 3) + (((33/3 + 3) ) − 3)
4
Ï 2961 := ((1 + 1 + 1) × (1 + 1)11−1 ) − 111
Ï 2966 := ((1 + 1 + 1) × ((11 − 1)1+1+1 − 11)) − 1
:= (2 × 22)2 + ((222/2 + 2)/2)
:= 22 + (((2 × 22) + 2) × 22+2+2 )
:= 3 × ((3 × (333 − 3)) − 3)
:= (3 × (3 × (333 − 3))) − (3/3 + 3)
:= 4/4 + (44 × 44 + 4 × 44 )
:= ((44 + 4/4) × ((44 + 4 + 4)/4)) − 4
:= 55 + (555/5 − 5 × 55)
:= 55 + (5/5 − (5 × ((5 + 5)/5)5 ))
:= (66/6 + 6 × 6) × (66 − (6 × 6/(6 + 6)))
:= 6 + ((6 × ((6 + 6) × (6 × 6 + 6))) − ((6 + 6)/6)6 )
:= 7 × ((77 × 77 − 7)/(7 + 7))
:= 7 + (77/7 × ((7 × 7 × 77 − 7)/(7 + 7)))
:= (8 − 8/8) × (8 × 8 × 8 − (8/8 + 88))
:= ((8 × 8 − 8) × (8 × 8 − 88/8)) − (8 + 8)/8
:= 9 + ((9 + 9) × (((9 + 9)/9) + 9 × (9 + 9)))
:= 999 + ((((9 + 9)/9)99/9 ) − 9 × 9)
Ï 2962 := 1 + (((1 + 1 + 1) × (1 + 1)11−1 ) − 111)
Ï 2967 := (1 + 1 + 1) × ((11 − 1)1+1+1 − 11)
:= 2 + ((22×(2+2)+2 ) + (2 × 22)2 )
:= (22 + 2/2) × ((22×(2+2) + 2)/2)
:= 3/3 + (3 × ((3 × (333 − 3)) − 3))
:= (3 × (3 × (333 − 3))) − 3
:= 4/4 + ((44 × 44 − 4) + 4 × 4 )
:= (4 + 4)/4 + (44 × 44 + 4 × 4 )
:= (44 − 4/4) × ((44 + 4)/4 + 4)
:= 55 + (((5 + 5)/5 + 5) × (5/5 − 5 × 5))
:= 55 + ((555 + 5)/5 − 5 × 55)
:= 55 + ((5 + 5)/5 − (5 × ((5 + 5)/5)5 ))
:= 6 + (((6 + 6) × (6 × (6 × 6 + 6) − 6)) − 6/6)
:= ((66 − 6)/6) + ((6 + 6) × (6 × (6 × 6 + 6) − 6))
:= ((6 × 6 + 6/6) + 6) × (6 × 6/(6 + 6) + 66)
:= 777 + (((7 + 7 + 7)/7)7 − 7)
4
:= 7/7 + (7 × ((77 × 77 − 7)/(7 + 7)))
:= ((7/7 − 7) + 7 × 7) × (77 − (7/7 + 7))
:= ((8 × 8 − 8) × (8 × 8 − 88/8)) − 88/8
:= ((8 − 888)/8) + (8 × (8 × (8 × 8 − (8 + 8))))
:= ((8 − 8/8 + 8) + 8) × (8 × (8 + 8) + 8/8)
:= 9 + (((((9 + 9)/9)99/9 ) + 9 × 99) + 9)
:= 9 + (((9 + 9) × (((9 + 9)/9) + 9 × (9 + 9))) + 9/9)
:= ((9 + 9 + 9)/9) × (999 − (9/9 + 9))
207
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2968 := 1 + ((1 + 1 + 1) × ((11 − 1)1+1+1 − 11))
Ï 2973 := (1 + 1 + 1) × (1 + ((11 − 1 − 1) × (111 − 1)))
Ï 2978 := ((1 + ((1 + 1)1+1+1+1+11 ))/11) − 1
:= 2 × (2 × ((22×(2+2) + 222 ) + 2))
:= (2/2 + 2) × ((2 × 222 + 22) + 2/2)
:= (2 × (2 × (22 − 2))2 ) − 222
:= 3/3 + ((3 × (3 × (333 − 3))) − 3)
:= 3 + (3 × (3 × (333 − 3)))
:= (3 × ((3 × (333 − 3)) + 3)) − 3/3
4
:= 4 + ((44 × 44 + 4 × 44 ) + 4)
:= 4/4 + ((((4 + 4) + 4) × (4 − 4 − 4)) − 4)
:= (4 + 4)/4 + (((4 + 4) + 4) × (44 − 4 − 4))
:= 55 − (((5 + 5)/5)5 + 5 × 5 × 5)
:= 55 − (5 × (5 × 5 + 5) + ((5 + 5)/5))
:= 5 + (55 − (5 × (5 × 5 + 5) + ((5 + 5)/5)))
:= (6/6 + 6) × (6 × (66 − 6) + ((6 + 6)/6)6 )
:= (6 × (6 − 66)) + (6666 × 6/(6 + 6))
:= 666 + (6 × 6 × 66 − ((6 + 6)/6)6 )
:= 7 × ((77 × 77 + 7)/(7 + 7))
:= 7777/7 + (7 × (7 × 7 × 7 − 77))
:= 7 + ((((7 + 7 + 7)/7)7 + 777) + 7)
:= (8 × (8 × (8 × 8 − (8 + 8)))) − (88/8 + 88)
:= 8 + ((88/8 + 8 + 8) × (888 − 8)/8)
:= ((9 + 9 + 9)/9) × ((999 − 9) + 9/9)
:= 9 × ((9 + 9) × (9 + 9) + 9) − (9/9 + 9 + 9)
:= (8 × 8 − 8) × (8 × 8 − 88/8)
9
:= ((9 + 9)/9) × ((9 × (99 + 9)) + (((9 + 9)/9) ))
Ï 2969 := ((111 − 1) × (1 + 1 + 1)1+1+1 ) − 1
Ï 2974 := ((1 + 1)1+11 ) − (11 + 1111)
Ï 2979 := (1 + ((1 + 1)1+1+1+1+11 ))/11
:= 2 + ((22 + 2/2) × ((22×(2+2) + 2)/2))
:= 22 + ((22 + 2) × ((22/2)2 + 2))
:= 2/2 + ((2 × (2 × (22 − 2))2 ) − 222)
:= (3 × (3 × (333 − 3))) − 3/3
:= 3 + ((3 × (3 × (333 − 3))) + 3/3)
:= 3 × ((3 × (333 − 3)) + 3)
:= 44 + ((44 + 4/4) × (44 + 4)/4)
:= 4 + ((44 + 4/4) × ((4 + 4 + 4)/4))
:= 55 + ((55 − 5)/(5 − 5 × 5))
:= 5 − (5 × (5 × 5 + 5) + 5/5)
:= 5 + (55 − (5 × (5 × 5 + 5) + 5/5))
:= (66 × (666/6 − 66)) − 6/6
:= (((6 + 6)/6)6+6 ) − (66 × (66/6 + 6))
:= 66 × (6 + 6) + ((6 × 6/(6 + 6))6/6+6)
:= 7/7 + (7 × ((77 × 77 + 7)/(7 + 7)))
:= 7 + (((7/7 − 7) + 7 × 7) × (77 − (7/7 + 7)))
:= 7 + (((77 × 77 + 7/7)/((7 + 7)/7)) + 7)
:= 8/8 + ((8 × 8 − 8) × (8 × 8 − 88/8))
:= (8 × 8 × 8 × 8) − ((8888 + 88)/8)
:= 88/8 + ((8 × 8 − 8) × (8 × 8 − 88/8))
:= (99 × ((((99 + 9)/9) + 9) + 9)) − 9/9
:= 9 × 9 + (9999/9 + (9 + 9) × 99)
:= 9 × ((9 + 9) × (9 + 9) + 9) − (9 + 9)
Ï 2970 := (111 − 1) × (1 + 1 + 1)1+1+1
4
5
Ï 2975 := 1 + (((1 + 1)1+11 ) − (11 + 1111))
:= 4 + ((4 × 4 + 4/4) × (4 × 44 − 4/4))
Ï 2980 := 1 + ((1 + ((1 + 1)1+1+1+1+11 ))/11)
:= 22 × (222/2 + 22 + 2)
:= ((22 + 2/2) + 2) × ((22/2)2 − 2)
:= 2 + ((2 × (2 × (22 − 2))2 ) − 222)
:= 3 × (3 × (333 − 3))
:= 3 + (((3 × (3 × (333 − 3))) − 3/3) + 3)
:= 3/3 + (3 × ((3 × (333 − 3)) + 3))
:= (44 + 4/4) × ((44 + 4 + 4)/4)
:= (4 × 4 + 4/4) × (4 × 44 − 4/4)
:= 4 + (((4 + 4) + 4) × (44 − 4 − 4))
:= 55 × (55 − 5/5)
:= 55 − 5 × (5 × 5 + 5)
:= 5 + (55 − 5 × (5 × 5 + 5))
:= 66 × (666/6 − 66)
:= (6/6 + 6) × ((6 × (66 + 6)) − (6/6 + 6))
:= (6 − 6/6) × (666 − (((6 + 6)/6)6 + 6))
:= 77/7 × ((7 × 7 × 77 + 7)/(7 + 7))
:= 7 + (7 × ((77 × 77 + 7)/(7 + 7)))
:= (7 − ((7 + 7)/7)) × ((7 × (77 + 7) + 7/7) + 7)
:= (88/8 + 8 + 8) × (888 − 8)/8
:= (8 − 8/8) × ((8 × 8 × 8 − 88) + 8/8)
:= 8 × 8 + (((8 − 88)/8 + 8 × 8)(8+8)/8 )
:= 99 × ((((99 + 9)/9) + 9) + 9)
:= 9 × ((9 + 9) × (9 + 9) + 9) − ((99 + 99)/9)
:= 9/9 + (9 × ((9 + 9) × (9 + 9) + 9) − (9 + 9))
Ï 2971 := 1 + ((111 − 1) × (1 + 1 + 1)1+1+1 )
Ï 2976 := (1 + 1) × ((1 + 11) × (1 + (1 + (1 + 111+1 ))))
Ï 2981 := 11 + ((111 − 1) × (1 + 1 + 1)1+1+1 )
:= 2/2 + (22 × (222/2 + 22 + 2))
:= 2 × ((22 + 2) × (22+2+2 − 2))
:= ((22/2 + 2 × 22)2 ) − (2 × 22)
:= 3/3 + (3 × (3 × (333 − 3)))
:= 3 + ((3 × (3 × (333 − 3))) + 3)
:= 33/3 + (3 × (3 × (333 − 3)))
4
:= 4 + ((44 − 4/4) × ((4 + 4)/4 + 4))
:= ((4 + 4) + 4) × (4 − 4 − 4)
:= ((4 + 4)4 ) − (4444/4 + 4)
:= 5/5 + (55 × (55 − 5/5))
:= 55 + (5/5 − 5 × (5 × 5 + 5))
:= 5 + ((5/5 − 5 × (5 × 5 + 5)) + 55 )
:= 6/6 + (66 × (666/6 − 66))
:= 6 + (66 × (666/6 − 66))
:= ((6 − 6/6)6−6/6 ) − (6 + 6) × (6 + 6)
4
7
:= 7 + (((7 + 7 + 7)/7) + 777)
:= (7 − 7/7) × ((7 × (77 − 7) − 7/7) + 7)
:= (7 × (7 × 7 × 7 + 77 + 7)) − (7/7 + 7)
:= 8 + ((8888 + 8/8)/(88/8 − 8))
:= 8 + ((8 × 8 − 8) × (8 × 8 − 88/8))
:= 88/8 × ((((8 + 8) × (8 + 8) − 8/8) + 8) + 8)
:= 9/9 + (99 × ((((99 + 9)/9) + 9) + 9))
:= 9 + (((9 + 9 + 9)/9) × (999 − (9/9 + 9)))
:= 9 × 9 + ((9/9 + 99) × (99/9 + 9 + 9))
Ï 2972 := 1 + (1 + ((111 − 1) × (1 + 1 + 1)1+1+1 ))
Ï 2977 := 1 + ((1 + 1) × ((1 + 11) × (1 + (1 + (1 + 111+1 )))))
Ï 2982 := ((1 + 1)1+11 ) − (1 + (1 + (1 + 1111)))
:= 2 + (22 × (222/2 + 22 + 2))
:= 2 + (((22 + 2/2) + 2) × ((22/2)2 − 2))
:= 222 + ((2 × (22 + 2) + 2)2 − 2)
:= 3 + ((3 × (3 × (333 − 3))) − 3/3)
:= (3 × (3 × 333 − 3)) − 33/3
:= 3 + (3 × ((3 × (333 − 3)) + 3))
4
:= (((4 + 4) + 4) × (4 − 4 − 4)) − 4
:= 4/4 + (((4 + 4) + 4) × (4 − 4 − 4))
:= ((4 + 4)4 ) + (((4 − 4444)/4) − 4)
:= (5 + 5)/5 + (55 × (55 − 5/5))
:= 55 + ((5 + 5)/5 − 5 × (5 × 5 + 5))
:= 5 + (((5 + 5)/5 − 5 × (5 × 5 + 5)) + 55 )
:= (6 + 6)/6 + (66 × (666/6 − 66))
:= (6 × ((6 × (66 + 6)) + 66)) − 66/6
:= (6/6 + 6) × ((6 × (66 + 6)) − 6)
:= (7 − 7/7) × (7 × (77 − 7) + 7)
4
:= 7 + ((77 × 77 + 7/7)/((7 + 7)/7))
:= 7 + (77/7 × ((7 × 7 × 77 + 7)/(7 + 7)))
:= 8 + (((88 + 8)/8) × ((8 + 8) × (8 + 8) − (8/8 + 8)))
:= (8 × 8 × 8 × 8) − (8888/8 + 8)
:= (8 − 8/8) × ((8 × 8 × 8 − 88) + ((8 + 8)/8))
:= (9 + 9)/9 + (99 × ((((99 + 9)/9) + 9) + 9))
:= 9 × ((9 + 9) × (9 + 9) + 9) − (99/9 + 9)
:= 9 + (((9 + 9 + 9)/9) × ((999 − 9) + 9/9))
208
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2983 := ((1 + 1)1+11 ) − (1 + (1 + 1111))
Ï 2988 := (11 − 1 − 1) × ((1 + 1 + 1) × 111 − 1)
Ï 2993 := ((1 + 1 + 1) × ((11 − 1)1+1+1 − (1 + 1))) − 1
:= 2 + (((22/2 + 2 × 22)2 ) − (2 × 22))
:= 2 × ((2 × (2 × 222 − 222)) + 2)
:= 2 × 222 + (2 × 22 + 2/2)2
:= 3 × 3 × 333 − (33/3 + 3)
:= 3 × (3 × 333 − 3)
:= 3 × 3 × 333 − (3/3 + 3)
:= (44 × (4 × 4 × 4 + 4)) − 4
:= 4/4 + (44 × (4 × 4 × 4 + 4))
4
4
:= 4 + (((44 × (4 − 4 − 4)) − 4)/4)
:= 55 + (((5 − 5 × 5 × 55)/(5 + 5)) − 5)
:= 5 + ((5 − 5 × 5 × 55)/(5 + 5))
:= 55 × 55 − ((5 + 5)/5)5
:= 6/6 + ((6/6 + 6) × ((6 × (66 + 6)) − 6))
:= 6 × ((6 × (66 + 6)) + 66)
:= 6 + ((6 × ((6 × (66 + 6)) + 66)) − 6/6)
:= 7/7 + ((7 − 7/7) × (7 × (77 − 7) + 7))
:= (7 × (7 × 7 × 7 + 77 + 7)) − 7/7
:= ((7/7 + 7 × 7) × (77/7 + 7 × 7)) − 7
:= (8 × (8 × (8 × 8 − (8 + 8)))) − (8/8 + 88)
:= ((88 + 8)/8) × (((8 + 8) × (8 + 8) − 8) + 8/8)
:= 8 + ((8 × 8 × 8 × 8) − 8888/8)
:= 9 + ((9999/9 + (9 + 9) × 99) + 9 × 9)
:= 9 × ((9 + 9) × (9 + 9) + 9) − 9
:= ((9 − 9 × 9)/(9 + 9)) + 9 × ((9 + 9) × (9 + 9) + 9)
5
Ï 2989 := ((1 + 1 + 1) × (11 − 1)1+1+1 ) − 11
Ï 2984 := ((1 + 1)1+11 ) − (1 + 1111)
Ï 2994 := (1 + 1 + 1) × ((11 − 1)1+1+1 − (1 + 1))
:= 2 × (2 × (2 × 222 − 222))
:= (2 × (222 − 2)) + (2 × 22 + 2/2)2
:= 2 + (2 × (2 × (22 × ((2 × 22+2 ) + 2))))
:= (3 × (3 × 333 − 3)) − (3/3 + 3)
:= 3/3 + (3 × (3 × 333 − 3))
:= 3 × 3 × 333 − 3
4
:= (44 × (4 × 4 × 4 + 4)) − 4 − 4
:= 4 + (((4 + 4) ) − 4444/4)
5
5
:= 5 − ((555/5 + 5 × 5) + 5)
:= 5 − (555/5 + 5 × 5)
:= (6 + 6)/6 + ((6/6 + 6) × ((6 × (66 + 6)) − 6))
:= (4 + 4)/4 + (44 × (4 × 4 × 4 + 4))
:= 55 − ((5 × 5 × 5 + 5/5) + 5)
:= 6/6 + (6 × ((6 × (66 + 6)) + 66))
:= 6 + (6 × ((6 × (66 + 6)) + 66))
:= (7 + 7)/7 + ((7 − 7/7) × (7 × (77 − 7) + 7))
:= 7 × (7 × 7 × 7 + 77 + 7)
:= 7 × 7 × 77 − (((7 + 7)/7) + 777)
:= (8 × (8 × (8 × 8 − (8 + 8)))) − 88
:= (888/8 × (88/8 + 8 + 8)) − 8
:= 8 + (((8 − 8888)/8) + (8 × 8 × 8 × 8))
:= (9 − 9/9) × ((((9 × 9 × 9 × 9) − 9)/(9 + 9)) + 9)
:= 9/9 + (9 × ((9 + 9) × (9 + 9) + 9) − 9)
:= ((9 + 9 + 9)/9) × (999 − 9/9)
Ï 2985 := ((1 + 1)1+11 ) − 1111
Ï 2990 := 1 + (((1 + 1 + 1) × (11 − 1)1+1+1 ) − 11)
Ï 2995 := (111 × (1 + 1 + 1)1+1+1 ) − 1 − 1
:= (22×(2+2+2) ) − 2222/2
:= 2 × ((2 × 22)2 − ((22 − 2/2)2 ))
:= 2 + ((2 × 22 + 2/2)2 + 2 × 222 )
:= (3 × (3 × 333 − 3)) − 3
:= 3 + ((3 × (3 × 333 − 3)) − 3/3)
:= 3/3 + (3 × 3 × 333 − 3)
:= ((4 + 4)4 ) − 4444/4
4
:= (4 + 4)/4 × ((4 + 4)/4 + 44)
5
:= 4 + ((44 × (4 × 4 × 4 + 4)) − 4/4)
:= 5 − ((5 × 5 × 5 + 5) + 5)
:= 55 − (5 × 5 × 5 + 5)
:= (((6 + 6)/6)6+6 ) − (6666/6)
:= (6 + 6)/6 + (6 × ((6 × (66 + 6)) + 66))
:= 6 + ((6 × ((6 × (66 + 6)) + 66)) + 6/6)
:= 7 × 7 × 77 − (77/7 + 777)
:= 7/7 + (7 × (7 × 7 × 7 + 77 + 7))
:= 7 × 7 × 77 − (777 + 7/7)
:= (8 × 8 × 8 × 8) − 8888/8
:= ((8 + 8)/8) × ((88 × (8 + 8) − 8/8) + 88)
:= 88/8 + ((8 × (8 × (8 × 8 − (8 + 8)))) − 88)
:= 9 × ((9 + 9) × (9 + 9) + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 × ((9 + 9) × (9 + 9) + 9) − 9)
:= 9 × ((9 + 9) × (9 + 9) + 9) − (9 + 9)/9
5
:= 5 + ((5 − 5 × (5 × 5 + 5)) + 5)
Ï 2986 := 1 + (((1 + 1)1+11 ) − 1111)
Ï 2991 := (1 + 1 + 1) × ((11 − 1)1+1+1 − (1 + 1 + 1))
:= 2 + ((2 × (22 + 2) + 2)2 + 222 )
Ï 2996 := (111 × (1 + 1 + 1)1+1+1 ) − 1
:= 2 × 222 + ((2 × 22 + 2/2)2 − 2)
:= 2 × ((2 × (22 × ((2 × 22+2 ) + 2))) + 2)
:= 3 × 3 × 333 − 33/3
:= 3 + (3 × (3 × 333 − 3))
:= 3 × 3 × 333 − 3/3
:= ((4 + 4)4 ) + ((4 − 4444)/4)
:= (44 × (4 × 4 × 4 + 4)) − 4/4
:= 4 + (44 × (4 × 4 × 4 + 4))
:= 55 + (5/5 − ((5 × 5 × 5 + 5) + 5))
:= 55 + (5/5 − (5 × 5 × 5 + 5))
5
:= 5 + (55/5 − 5 × (5 × 5 + 5))
:= (6 × ((6 × (66 + 6)) + 66)) − (6 + 6)/6
(7+7)/7
6+6
:= 6 + ((((6 + 6)/6)
) − (6666/6))
:= 6 + ((6 × ((6 × (66 + 6)) + 66)) + ((6 + 6)/6))
:= (7 + 7)/7 + (7 × (7 × 7 × 7 + 77 + 7))
:= 7 × 7 × 77 − 777
:= (8 × 8 × 8 × 8) + ((8 − 8888)/8)
:= ((8/8 + 8 + 8) × (88 + 88)) − 8/8
:= (8 − 8/8) × (888/((8 + 8)/8) − (8 + 8))
:= 9 × ((9 + 9) × (9 + 9) + 9) − 99/9
:= ((9 + 9 + 9)/9) × (999 − ((9 + 9)/9))
:= 9 × ((9 + 9) × (9 + 9) + 9) − 9/9
:= 777 + ((7 × 7 − (7 + 7)/7)
)
Ï 2987 := 1 + (1 + (((1 + 1)1+11 ) − 1111))
Ï 2992 := 11 × (((1 + 1 + 11) × (11 + (11 − 1))) − 1)
Ï 2997 := 111 × (1 + 1 + 1)1+1+1
:= 2 + ((22×(2+2+2) ) − (2222/2))
:= 2 × (2 × (22 × ((2 × 22+2 ) + 2)))
:= 222/2 × (((22 + 2/2) + 2) + 2)
:= (3 × (3 × 333 − 3)) − 3/3
:= 3 + ((3 × (3 × 333 − 3)) + 3/3)
:= 3 × 3 × 333
:= (44 × (4 × 4 × 4 + 4)) − (4/4 + 4)
:= 44 × (4 × 4 × 4 + 4)
:= 4 + ((44 × (4 × 4 × 4 + 4)) + 4/4)
:= 5 − ((5 × 5 × 55 + 5)/(5 + 5))
:= 5 + (5 − ((5 × 5 × 55 + 5)/(5 + 5)))
:= 55 + ((5 + 5)/5 − (5 × 5 × 5 + 5))
:= (6 × ((6 × (66 + 6)) + 66)) − 6/6
:= 6 + ((6 × ((6 × (66 + 6)) + 66)) − ((6 + 6)/6))
:= 666/6 × ((66 × 6/(6 + 6)) − 6)
:= (7 × (7 × 7 × 7 + 77 + 7)) − (7 + 7)/7
:= 7 + (7 × 7 × 77 − (77/7 + 777))
:= 7/7 + (7 × 7 × 77 − 777)
:= (8 × 8 × 8 × 8) + (((8 − 8888) + 8)/8)
:= (8/8 + 8 + 8) × (88 + 88)
:= 888/8 × (88/8 + 8 + 8)
:= 9 × ((9 + 9) × (9 + 9) + 9) − 9/9 − 9
:= 99 + (9999/9 + (9 + 9) × 99)
:= 9 × ((9 + 9) × (9 + 9) + 9)
5
5
209
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 2998 := 1 + (111 × (1 + 1 + 1)1+1+1 )
Ï 3003 := (1 + 1 + 1) × (1 + (11 − 1)1+1+1 )
Ï 3008 := 11 + (111 × (1 + 1 + 1)1+1+1 )
:= 2 × (((2 × (22 − 2) − 2/2)2 ) − 22)
:= ((22/2 + 2 × 22)2 ) − 22
:= 2 × (2 × (2 × ((22 − 2)2 − (22 + 2))))
:= 3/3 + 3 × 3 × 333
:= 3 + (3 × 3 × 333 + 3)
:= 33/3 + 3 × 3 × 333
:= 4 + ((44 × (4 × 4 × 4 + 4)) + (4 + 4)/4)
:= 44/4 + (44 × (4 × 4 × 4 + 4))
:= 4 × (4 × (444 − 44 ))
:= 55 − (5 × 5 × 5 + ((5 + 5)/5))
:= 55 − ((555 + 55)/5)
:= 55 − ((555 + 5)/5 + 5)
:= ((66 − 6)/6) + (6 × ((6 × (66 + 6)) + 66))
:= 666666/(6 × 6 × 6 + 6)
:= ((6 + 6)/6)6 × (66/6 + 6 × 6)
:= (7 + 7)/7 + (7 × 7 × 77 − 777)
:= 7 + (7 × 7 × 77 − 777)
:= ((7 × 7 + 7)(7+7)/7 ) − ((7 + 7)/7)7
:= 8/8 + (888/8 × (88/8 + 8 + 8))
:= 88/8 + ((8/8 + 8 + 8) × (88 + 88))
:= 8 × (888 − 8 × 8 × 8)
:= 9/9 + 9 × ((9 + 9) × (9 + 9) + 9)
:= 9 + (((9 + 9 + 9)/9) × (999 − 9/9))
:= 99/9 + 9 × ((9 + 9) × (9 + 9) + 9)
Ï 2999 := ((1 + 1 + 1) × (11 − 1)1+1+1 ) − 1
Ï 3004 := 1 + ((1 + 1 + 1) × (1 + (11 − 1)1+1+1 ))
2+2+2
:= 2 + (222/2 × (((22 + 2/2) + 2) + 2))
:= 2 × (2 × (((2/2 + 2)
:= 3 + (3 × 3 × 333 − 3/3)
:= 3 + ((3 × 3 × 333 + 3/3) + 3)
:= 4 + (((44 × (4 × 4 × 4 + 4)) − 4/4) + 4)
5
:= 5 − (5 × 5 × 5 + 5/5)
) + 22))
4
:= (4 × (4 × (444 − 4 ))) − 4
5
Ï 3009 := 1 + (11 + (111 × (1 + 1 + 1)1+1+1 ))
:= ((22/2 + 2 × 22)2 ) − 22+2
:= 3 + 3 × (3 × 333 + 3)
:= 4/4 + (4 × (4 × (444 − 44 )))
:= 5 + (5 − (5 × 5 × 5 + 5/5))
:= 55 − (555/5 + 5)
:= 66/6 + (6 × ((6 × (66 + 6)) + 66))
:= 666 + (6 × (6 × 66 − 6) − ((6 + 6)/6))
:= (66/6 + 6) × (666/6 + 66)
:= 777 + (((7 + 7)/7) × 7777/7)
:= 7 + ((7 × 7 × 77 − 777) + 7/7)
:= 7 × 7 × (7 × 7 + 7 + 7) − 7/7 − 77
:= 888 + ((88 × (8 + 8 + 8)) − 8/8)
:= 88 + (((8 − 88)/8 + 8 × 8)(8+8)/8 )
:= 8/8 + (8 × (888 − 8 × 8 × 8))
:= (9 + 9)/9 + 9 × ((9 + 9) × (9 + 9) + 9)
:= 9 + (9 × ((9 + 9) × (9 + 9) + 9) − ((9 + 9)/9))
:= 9 + (((9 + 9 + 9)/9) × (999 + 9/9))
Ï 3000 := (1 + 1 + 1) × (11 − 1)1+1+1
Ï 3005 := 1 + (1 + ((1 + 1 + 1) × (1 + (11 − 1)1+1+1 )))
Ï 3010 := 11 + (((1 + 1 + 1) × (11 − 1)1+1+1 ) − 1)
:= (2 + 2 + 2) × (22+2 + 222 )
:= 2 + (((22/2 + 2 × 22)2 ) − 22)
:= 222/2 + ((2 × (222 − 2)) − 2)
:= 3 + 3 × 3 × 333
:= 3 × (3 × 333 + 3) − 3/3
:= 3 + (3 × (3 × 333 + 3) + 3/3)
:= 4 + ((44 × (4 × 4 × 4 + 4)) + 4)
4
:= 4/4 + ((4 × (4 × (444 − 4 ))) − 4)
:= (4 + 4)/4 + (4 × (4 × (444 − 44 )))
:= 5 − 5 × 5 × 5
:= 5 + (55 − 5 × 5 × 5)
:= 5 + ((55 − 5 × 5 × 5) + 5)
:= (6 − 6/6) × (666 − 66)
:= 666 + (6 × (6 × 66 − 6) − 6/6)
:= (6 − 6/6) × (666 − ((6 + 6)/6)6 )
:= (7/7 + 7 × 7) × (77/7 + 7 × 7)
:= (7 × 7 × (77 + 7)) − 7777/7
:= (77 − 7) × ((7/7 − 7) + 7 × 7)
:= 888 + (88 × (8 + 8 + 8))
:= 8 + (888/8 × (88/8 + 8 + 8))
:= (8 + 8)/8 + (8 × (888 − 8 × 8 × 8))
:= ((9 + 9 + 9)/9) × (999 + 9/9)
:= 9 + (9 × ((9 + 9) × (9 + 9) + 9) − 9/9)
:= ((99 + 9 + 9)/9) + 9 × ((9 + 9) × (9 + 9) + 9)
5
Ï 3001 := 1 + ((1 + 1 + 1) × (11 − 1)1+1+1 )
2
Ï 3006 := (1 + 1 + 1) × (1 + 1 + (11 − 1)1+1+1 )
Ï 3011 := 11 + ((1 + 1 + 1) × (11 − 1)1+1+1 )
:= ((22/2 + 2 × 22) ) − (22 + 2)
:= (2/2 + 2) × ((22×(2+2)+2 ) − 22)
:= 2 + (((22/2 + 2 × 22)2 ) − 22+2 )
:= 3 + (3 × 3 × 333 + 3/3)
:= 3 × (3 × 333 + 3)
:= 3 + (3 × 3 × 333 + 33/3)
:= 4 + (((44 × (4 × 4 × 4 + 4)) + 4/4) + 4)
4
4
:= (4 × ((4 + 4) + 4)) − ((4 + 4 + 4)/4)
:= 4 + ((44 × ((4 + 4) + 4)) − (44 + 4)/4)
:= 5 + (5/5 − 5 × 5 × 5)
:= 5 + ((5/5 − 5 × 5 × 5) + 55 )
:= 55 + (55/5 − 5 × 5 × 5)
5
:= 6/6 + ((6 − 6/6) × (666 − 66))
:= 666 + 6 × (6 × 66 − 6)
:= (6 × (6 + 6) × (6 × 6 + 6)) − (6/6 + 6 + 6)
:= 7 + (7 × 7 × 77 − (((7 + 7)/7) + 777))
:= (7 − 7/7) × (7 × (77 − 7) + (77/7))
:= 7/7 + ((77 − 7) × ((7/7 − 7) + 7 × 7))
:= 8/8 + ((88 × (8 + 8 + 8)) + 888)
:= (8 − (8 + 8)/8) × (8 × 8 × 8 − 88/8)
:= 888 + ((88 × (8 + 8 + 8)) + (88/8))
:= 9/9 + (((9 + 9 + 9)/9) × (999 + 9/9))
:= 9 + 9 × ((9 + 9) × (9 + 9) + 9)
:= 9 × (99 + 9) + ((((9 + 9)/9)99/9 ) − 9)
Ï 3002 := 1 + (1 + ((1 + 1 + 1) × (11 − 1)1+1+1 ))
Ï 3007 := 1 + ((1 + 1 + 1) × (1 + 1 + (11 − 1)1+1+1 ))
Ï 3012 := 1 + (11 + ((1 + 1 + 1) × (11 − 1)1+1+1 ))
:= 2 + ((2 + 2 + 2) × (22+2 + 222 ))
:= 2 + ((((22/2 + 2 × 22)2 ) − 22) + 2)
:= 222/2 + (2 × (222 − 2))
:= 3 + ((3 × 3 × 333 − 3/3) + 3)
:= 3/3 + 3 × (3 × 333 + 3)
:= 3 + (3 × (3 × 333 + 3) + 3)
4
4
:= (44 − 4)/4 + (44 × (4 × 4 × 4 + 4))
:= (4 × ((4 + 4) + 4)) − (4 + 4)/4
:= 4 + (4 × (4 × (444 − 44 )))
:= 55 + ((5 + 5)/5 − 5 × 5 × 5)
:= 5 + (((5 + 5)/5 − 5 × 5 × 5) + 55 )
:= 55 − (555 + 5 + 5)/5
:= (((6 + 6)/6)6 × (66/6 + 6 × 6)) − 6
:= 6/6 + (6 × (6 × 66 − 6) + 666)
:= (6 + 6) × (6 × (6 × 6 + 6) − 6/6)
:= 7 + (7 × 7 × 77 − (777 + 7/7))
:= 7 + ((7/7 + 7 × 7) × (77/7 + 7 × 7))
:= 7 + ((7 × 7 × (77 + 7)) − 7777/7)
:= 888 + ((88 × (8 + 8 + 8)) + ((8 + 8)/8))
:= (8 × (888 − 8 × 8 × 8)) − 8/8
:= (8/((8 + 8)/8)) + (8 × (888 − 8 × 8 × 8))
:= ((9 × 9 + 9)/(9 + 9)) + 9 × ((9 + 9) × (9 + 9) + 9)
:= 9 + (9 × ((9 + 9) × (9 + 9) + 9) + 9/9)
:= 9 + ((((9 + 9 + 9)/9) × (999 − 9/9)) + 9)
210
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3018 := 1 + (1 + ((1 + 1 + 11) × (111 + 111+1 )))
Ï 3013 := (1 + (11 + 11)) × ((11 × (1 + 11)) − 1)
Ï 3023 := ((111 − 1)/(1 + 1))1+1 − 1 − 1
:= 2/2 + ((2 × (222 − 2)) + 222/2 )
:= 2 + (2 × 222 + 222/2 )
:= ((22/2 + 2 × 22)2 ) − 2
:= (3 × (3 × (333 + 3))) − 33/3
:= 3 + (3 × ((3 × 333 + 3) + 3))
:= (3 × (3 × (333 + 3))) − 3/3
:= (((4 + 4) + 4) × (44 − 4)) − 44/4
:= (4 − 4/4) × (4 × (44 − 4) − (4 + 4)/4)
:= (((4 + 4) + 4) × (44 − 4)) − 4/4
:= 55 − (555 + 5)/5
:= 5 + (55 − (555 + 5)/5)
:= 55 × 55 − (5 + 5)/5
:= (6 × (6 + 6) × (6 × 6 + 6)) − 6
:= (6 × (6 + 6) × (6 × 6 + 6)) − 6/6
:= (6 × (6 + 6) × (6 × 6 + 6)) − 66/6
7
(7+7)/7
:= (((7 + 7)/7 + 7) × (7 × 7 × 7 − 7)) − 7/7
:= 7 × 7 + (((7 + 7 + 7)/7) + 777)
:= (((7 × 7 − 7/7) + 7)
:= 8 + ((888/8 × (88/8 + 8 + 8)) + 8)
:= (8 − (8 + 8)/8) × (8 × 8 × 8 − (8/8 + 8))
:= ((8 − (8 + 8)/8) × (8 × 8 × 8 − 8)) − 8/8
:= 99 + (9 × (9 + 9) × (9 + 9) − ((9 + 9)/9))
:= 999/9 + (9 × (9 + 9) × (9 + 9) − 9)
:= 9 + ((9 × (9 + 9) × (9 + 9) − 9/9) + 99)
Ï 3014 := ((111 − 1)/(1 + 1))1+1 − 11
)−7
Ï 3019 := 11 + (11 + (111 × (1 + 1 + 1)1+1+1 ))
:= 22 × ((22/2)2 + 22+2 )
:= (3 × ((3 × 333 + 3) + 3)) − 3/3
Ï 3024 := ((111 − 1)/(1 + 1))1+1 − 1
:= ((22/2 + 2 × 22)2 ) − (2 + 2 + 2)
:= 2 × (2 × (2 × ((22 − 2)2 − 22)))
:= 3 + ((3 × ((3 × 333 + 3) + 3)) + 3/3)
:= 3 × (3 × (333 + 3))
:= 44/4 × (((4 + 4)/4 + 4 ) + 4 × 4)
:= (((4 + 4) + 4) × (4 − 4)) − (4/4 + 4)
:= ((4 + 4) + 4) × (44 − 4)
:= 55 − 555/5
:= 5 + (55 − 555/5)
:= 55 × 55 − 5/5
:= 6 + (((6 + 6)/6)6 × (66/6 + 6 × 6))
:= 6/6 + ((6 × (6 + 6) × (6 × 6 + 6)) − 6)
:= 6 × (6 + 6) × (6 × 6 + 6)
:= 7 + (((7/7 + 7 × 7) × (77/7 + 7 × 7)) + 7)
:= 7/7 + ((((7 × 7 − 7/7) + 7)(7+7)/7 ) − 7)
:= ((7 + 7)/7 + 7) × (7 × 7 × 7 − 7)
:= 8 + ((8 − (8 + 8)/8) × (8 × 8 × 8 − 88/8))
:= 88/8 + (8 × (888 − 8 × 8 × 8))
:= 99 + (9 × (9 + 9) × (9 + 9) − 9/9)
:= 9 × (9 + 9) × (9 + 9) + (((999 + 9)/9) − 9)
4
4
Ï 3015 := ((111 − 1)/(1 + 1))1+1 − 11 + 1
Ï 3020 := (1 + 1) × (((1 + 1 + 1) × (1 + 1 + 11))1+1 − 11)
:= (8 − (8 + 8)/8) × (8 × 8 × 8 − 8)
:= 9 + (9 × (9 + 9) × (9 + 9) + 99)
Ï 3025 := ((111 − 1)/(1 + 1))1+1
:= (22/2 + 2 × 22)2
:= 222/2 + (222 × (2 + 2 + 2))
:= 222/2 + (2 × (222 + 2))
:= 3 × ((3 × 333 + 3) + 3)
:= (3 × (3 × (333 + 3))) − (3/3 + 3)
:= (44 + 4/4) × (((44 − 4)/4) + 4)
:= (((4 + 4) + 4) × (44 − 4)) − 4
:= 55 − (55 + 55)
:= 55 × 55 − 5
:= (66 + 6/6) × (666/6 − 66)
:= (6 + 6)/6 + ((6 × (6 + 6) × (6 × 6 + 6)) − 6)
:= ((7 + 7)/7 + 7) × (7 × 7 × 7 − (7/7 + 7))
:= 7 + ((((7 + 7 + 7)/7)7 + 777) + 7 × 7)
:= 8 + ((8 × (888 − 8 × 8 × 8)) − 8/8)
:= ((88 + 8)/8) + (8 × (888 − 8 × 8 × 8))
:= (8 × 8 − (8/8 + 8))(8+8)/8
:= 99 + 9 × (9 + 9) × (9 + 9)
:= 9 × (99 + 9) + (((9 + 9)/9)99/9 )
:= ((999 − 9)/(9 + 9))(9+9)/9
Ï 3016 := (1 + 1 + 11) × (111 + 111+1 )
2
Ï 3021 := (1 + 1 + 1) × ((11 − 1 − 1) × (1 + 111) − 1)
22/2
:= 3/3 + (3 × (3 × (333 + 3)))
:= 4/4 + (((4 + 4) + 4) × (44 − 4))
:= 55 × 55
:= 6/6 + (6 × (6 + 6) × (6 × 6 + 6))
:= ((7 × 7 − 7/7) + 7)(7+7)/7
Ï 3026 := ((111 − 1)/(1 + 1))1+1 + 1
:= ((22/2 + 2 × 22)2 ) − 2 − 2
:= 2/2 + ((22/2 + 2 × 22)2 )
:= (3 × (3 × (333 + 3))) − 3
:= 3 + ((3 × (3 × (333 + 3))) − 3/3)
:= 4/4 + ((((4 + 4) + 4) × (44 − 4)) − 4)
:= (4 + 4)/4 + (((4 + 4) + 4) × (44 − 4))
:= 5/5 + (55 × 55 − 5)
:= 5/5 + 55 × 55
:= (6 × (6 + 6) × (6 × 6 + 6)) − 6 × 6/(6 + 6)
:= (6 + 6)/6 + (6 × (6 + 6) × (6 × 6 + 6))
:= 77 + (7 × (7 × 7 × 7 + 77) − 7/7)
:= 77/7 + ((77 − 7) × ((7/7 − 7) + 7 × 7))
:= 7/7 + (((7 × 7 − 7/7) + 7)(7+7)/7 )
:= 8 + (8 × (888 − 8 × 8 × 8))
:= (8 × 8 − 88/8) × ((8/8 − 8) + 8 × 8)
:= 8/8 + ((8 × 8 − (8/8 + 8))(8+8)/8 )
:= 9/9 + (9 × (9 + 9) × (9 + 9) + 99)
:= ((9 + 9 + 9)/9) × ((999 − 9/9) + 9)
:= 99 + (9 × (9 + 9) × (9 + 9) + (99/9))
:= 2 × 22 + 2
:= 3/3 + (3 × ((3 × 333 + 3) + 3))
4
:= (((4 + 4) + 4) × (4 − 4)) − 4 − 4
:= 55 + (5/5 − (55 + 55))
6+6
:= (6 × (6 × (6 − 6 × 6))) + (((6 + 6)/6)
)
Ï 3017 := 1 + ((1 + 1 + 11) × (111 + 111+1 ))
2
:= 2/2 + (2 × 22 + 2
22/2
)
Ï 3022 := ((111 − 1)/(1 + 1))1+1 − 1 − 1 − 1
:= 2222 + (2 × (22 − 2))
:= 33/3 + 3 × (3 × 333 + 3)
2
:= 3/3 + ((3 × (3 × (333 + 3))) − 3)
4
4
Ï 3027 := ((111 − 1)/(1 + 1))1+1 + 1 + 1
:= 2 + ((22/2 + 2 × 22)2 )
:= 3 + (3 × (3 × (333 + 3)))
:= (((4 + 4) + 4) × (4 − 4)) − (4 + 4)/4
:= 4 + ((((4 + 4) + 4) × (44 − 4)) − 4/4)
:= 5 + (5 − (555 + 5 + 5)/5)
:= (5 + 5)/5 + (55 × 55 − 5)
:= (5 + 5)/5 + 55 × 55
:= (6/6 + 6) × ((6 × (66 + 6)) − 6/6)
:= (6 × (6 + 6) × (6 × 6 + 6)) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (6 × (6 + 6) × (6 × 6 + 6))
:= 7 + (((7 + 7)/7 + 7) × (7 × 7 × 7 − (7/7 + 7)))
:= 7 × (777 − 7 × 7 × 7) − 77/7
:= 4 + ((((4 + 4) + 4) × (4 − 4)) − 44/4)
5
:= 77 + 7 × (7 × 7 × 7 + 77)
:= ((8 × 8 − (8/8 + 8))
(8+8)/8
)−8
:= 9 + (9 × ((9 + 9) × (9 + 9) + 9) + (99/9))
:= ((8 − (8 + 8)/8) × (8 × 8 × 8 − 8)) − (8 + 8)/8
:= 8 + ((8 × (888 − 8 × 8 × 8)) + (88/8))
:= ((9 + 9)/9) × ((((9 + 9)/9)9 ) + 999)
:= 999/9 + 9 × (9 + 9) × (9 + 9)
211
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3028 := ((111 − 1)/(1 + 1))1+1 + 1 + 1 + 1
Ï 3033 := (1 + 1 + 1) × (11 + (11 − 1)1+1+1 )
Ï 3038 := ((1 + 1 + 1) × ((1 + 1)11−1 − 11)) − 1
:= 2 + (((22/2 + 2 × 22)2 ) + 2/2)
:= 2 × (2 + 2) + ((22/2 + 2 × 22)2 )
:= 2 + ((2 + 2 + 2) × (222 + 22))
:= 3 + ((3 × (3 × (333 + 3))) + 3/3)
:= 3 × (3 × (333 + 3) + 3)
:= (33/3 + 3) × ((3 + 3)3 + 3/3)
4
4
:= 4 + (((4 + 4) + 4) × (4 − 4))
:= 4 + (((((4 + 4) + 4) × (4 − 4)) + 4/4) + 4)
:= (4 + 4)/4 + (44 × ((44 + 4)/4 + 4))
:= 5 + (55 × 55 − ((5 + 5)/5))
:= 5 + ((55 × 55 − ((5 + 5)/5)) + 5)
:= 55 − (((5 + 5)/5)5 + 55)
:= 6 + ((6 × (6 + 6) × (6 × 6 + 6)) − ((6 + 6)/6))
:= 6 + ((6 × (6 + 6) × (6 × 6 + 6)) + (6 × 6/(6 + 6)))
:= (6/6 + 6) × ((6 × (66 + 6)) + ((6 + 6)/6))
:= 77 + (7 × (7 × 7 × 7 + 77) + (77/7))
:= ((7 + 7)/7 + 7) × ((7 × 7 × 7 − 7) + 7/7)
:= 7 × (777 − 7 × 7 × 7)
:= ((8 × 8 × (88 + 8)) − 88)/((8 + 8)/8)
:= 8 + ((8 × 8 − (8/8 + 8))(8+8)/8 )
:= 8 + ((8 − (8 + 8)/8) × ((8 × 8 × 8 − 8) + 8/8))
:= 9 × (9 + 9) × (9 + 9) + ((999 + 9)/9)
:= 9 + ((9 × (9 + 9) × (9 + 9) + 99) + 9)
:= 999 + ((((9 + 9)/9)99/9 ) − 9)
Ï 3029 := (1 + 1 + 11) × (11 + (1 + 1) × 111)
2
Ï 3034 := 1 + ((1 + 1 + 1) × (11 + (11 − 1)1+1+1 ))
2
Ï 3039 := (1 + 1 + 1) × ((1 + 1)11−1 − 11)
:= 2 + (((22/2 + 2 × 22) ) + 2)
:= ((2 + 2 + 2) × (22 + 22)) − 2
:= (2/2 + 2) × ((222/2 − 22)/2)
:= 33 + (3 × 3 × 333 − 3/3)
:= 3/3 + (3 × (3 × (333 + 3) + 3))
:= 33 + 3 × (3 × 333 + 3)
:= 4 + ((((4 + 4) + 4) × (4 − 4)) + 4/4)
:= (44 − 4)/4 + (((4 + 4) + 4) × (44 − 4))
:= (4 − 4/4) × (4 × 44 − 44/4)
:= 5 + (55 × 55 − 5/5)
:= 5 + ((55 × 55 − 5/5) + 5)
:= 5 × 5 + (55 − 555/5)
:= 6 + ((6 × (6 + 6) × (6 × 6 + 6)) − 6/6)
:= ((66 − 6)/6) + (6 × (6 + 6) × (6 × 6 + 6))
4
:= 77 × (7 × 7 − 7) − (((7 + 7)/7)7 + 77)
:= 8 + ((8 × 8 − 88/8) × ((8/8 − 8) + 8 × 8))
:= 9 + ((((9 + 9)/9)99/9 ) + 9 × (99 + 9))
Ï 3030 := (1 + 1 + 1) × ((11111 − 1)/11)
7
:= (77 × (77/7)) + ((7 + 7 + 7)/7)
:= 8 + (((8 × 8 − (8/8 + 8))
(8+8)/8
:= 666 + (6 × 6 × 66 − (6 × 6/(6 + 6)))
:= 7/7 + 7 × (777 − 7 × 7 × 7)
) + 8/8)
:= (88/8 − 8) × (8 × 8 × (8 + 8) − (88/8))
)
:= 9 + (((9 + 9 + 9)/9) × (99/9 + 999))
Ï 3035 := 11 + ((111 − 1)/(1 + 1))1+1 − 1
Ï 3040 := 1 + ((1 + 1 + 1) × ((1 + 1)11−1 − 11))
:= 9 + (((999 − 9)/(9 + 9))
(9+9)/9
:= (2/2 + 2) × ((2 × (222 + 22)) − 2)
:= ((2 + 2 + 2) × (222 + 22)) − 2/2
:= (2 × ((2 × (22 − 2) − 2/2)2 )) − 2
:= 33 + 3 × 3 × 333
:= 33/3 + (3 × (3 × (333 + 3)))
:= 3/3 + (3 × (3 × 333 + 3) + 33)
:= 4 + ((((4 + 4) + 4) × (4 − 4)) + (4 + 4)/4)
:= 44/4 + (((4 + 4) + 4) × (44 − 4))
:= 4 + (44 × ((44 + 4)/4 + 4))
:= 5 + 55 × 55
4
:= 5 + (55 × 55 + 5)
:= 5 + ((55 × 55 + 5) + 5)
:= 6 + (6 × (6 + 6) × (6 × 6 + 6))
:= 66/6 + (6 × (6 + 6) × (6 × 6 + 6))
:= 666 + (6 × 6 × 66 − ((6 + 6)/6))
:= 7 × (777 − 7 × 7 × 7) − (7/7 + 7)
:= 77/7 + (((7 + 7)/7 + 7) × (7 × 7 × 7 − 7))
:= (7 + 7)/7 + 7 × (777 − 7 × 7 × 7)
:= (8 − (8 + 8)/8) × ((8 × 8 × 8 − 8) + 8/8)
:= 88/8 + ((8 − (8 + 8)/8) × (8 × 8 × 8 − 8))
:= 8 × ((8 × (88 + 8) − 8)/((8 + 8)/8))
:= ((9 + 9 + 9)/9) × (99/9 + 999)
:= 9 + ((9 × (9 + 9) × (9 + 9) + (99/9)) + 99)
:= (9/9 + 9 + 9) × (9 × (9 + 9) − ((9 + 9)/9))
Ï 3031 := 1 + ((1 + 1 + 1) × ((11111 − 1)/11))
Ï 3036 := 11 + ((111 − 1)/(1 + 1))1+1
Ï 3041 := 1 + (1 + ((1 + 1 + 1) × ((1 + 1)11−1 − 11)))
:= ((((2 × 2 × (22 − 2)) − 2)2 ) − 2)/2
:= 2 + ((((22/2 + 2 × 22)2 ) + 2) + 2)
:= (2 + 2 + 2) × (222 + 22)
:= 3/3 + (3 × 3 × 333 + 33)
:= 3 + (3 × (3 × (333 + 3) + 3))
:= 3 × 3 × 33 + ((33/3 + 3)3 )
:= 4 + (((((4 + 4) + 4) × (44 − 4)) − 4/4) + 4)
:= 44 × ((44 + 4)/4 + 4)
:= 4 + ((44 × ((44 + 4)/4 + 4)) + 4/4)
:= 5 + (55 × 55 + 5/5)
:= 55/5 + 55 × 55
:= 5 + (55 × 55 + (55/5))
:= 6 + ((6 × (6 + 6) × (6 × 6 + 6)) + 6/6)
:= 6 + ((6 × (6 + 6) × (6 × 6 + 6)) + 6)
:= 666 + (6 × 6 × 66 − 6/6)
:= 7 × (777 − 7 × 7 × 7) − 7
:= 7 × (777 − 7 × 7 × 7) − (7 + 7)/7
:= 77 + (((7 + 7 + 7)/7)7 + 777)
:= ((8 × (8 × (88 + 8) − 8)) − 8)/((8 + 8)/8)
:= 8 + (((8 × 8 − (8/8 + 8))(8+8)/8 ) + 8)
:= 9 + (9 × (9 + 9) × (9 + 9) + 999/9)
:= 9 + ((9 − 9/9) × (9 × 99 − ((9 + 9)/9)9 ))
:= 8 + (((8 − (8 + 8)/8) × (8 × 8 × 8 − 8)) − 8/8)
9
:= 9 + (((9 + 9)/9) × ((((9 + 9)/9) ) + 999))
Ï 3032 := ((1 + 1 + 1) × (11 + (11 − 1)1+1+1 )) − 1
Ï 3037 := 11 + ((111 − 1)/(1 + 1))1+1 + 1
Ï 3042 := (1 + 1) × ((1 + 1 + 1) × (1 + 1 + 11))1+1
:= 2 × (((2 + 2 + 2)2+2 − 2) + 222)
:= 2/2 + ((2 + 2 + 2) × (222 + 22))
:= 2 × ((2 × (22 − 2) − 2/2)2 )
:= (3 × (3 × (333 + 3) + 3)) − 3/3
:= 3 + ((3 × (3 × (333 + 3) + 3)) + 3/3)
:= 3 × ((3 × (333 + 3) + 3) + 3)
4
4
:= 4 + ((((4 + 4) + 4) × (4 − 4)) + 4)
:= 4/4 + (44 × ((4 + 4)/4 + 4))
:= (4 − 4/4) × ((4 − 44)/4 + 4 × 44 )
:= 5 + (55 × 55 + ((5 + 5)/5))
:= 55 × 55 + (55 + 5)/5
:= 5 + (55 × 55 + ((55 + 5)/5))
:= 6 + ((6 × (6 + 6) × (6 × 6 + 6)) + ((6 + 6)/6))
:= 6 + (((6 × (6 + 6) × (6 × 6 + 6)) + 6/6) + 6)
:= 666 + 6 × 6 × 66
:= 7 + (((7 × 7 − 7/7) + 7)(7+7)/7 )
:= 7 × (777 − 7 × 7 × 7) − 7/7
:= (7/7 + 77) × (((7 − 77)/7) + 7 × 7)
:= 8 + ((8 − (8 + 8)/8) × (8 × 8 × 8 − 8))
9
:= (9 − 9/9) × (9 × 99 − ((9 + 9)/9) )
:= ((8 × 8 − 8)
(8+8)/8
) − (88/8 + 88)
:= 9 + (9 × (9 + 9) × (9 + 9) + ((999 + 9)/9))
212
:= 8 + ((((8 × 8 − (8/8 + 8))(8+8)/8 ) + 8/8) + 8)
:= (9 + 9) × ((9 × (9 + 9) − ((9 + 9)/9)) + 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3043 := 1 + ((1 + 1) × ((1 + 1 + 1) × (1 + 1 + 11))1+1 )
2
Ï 3048 := (1 + 1)11 + (11 − 1)1+1+1
2×(2+2)
Ï 3053 := (1 + (11 × ((1111 − 1)/(1 + 1))))/(1 + 1)
:= ((((2 × 2 × (22 − 2)) − 2)2 ) + 22)/2
:= ((((2 × 2 × (22 − 2)) − 2) ) + 2)/2
:= 2 × ((2 + 2 + 2) × (2
:= ((3 + 3) × 33+3 ) − (33/3)3
:= (3/3 + 3) × (33+3 + 33)
:= (((((3 × (3 + 3)) + 3)3 ) − 3)/3) − 33
:= 4 + ((4 − 4/4) × (4 × 44 − 44/4))
:= (4 − 4/4) × (4 × 44 − (4 + 4))
:= 4/4 + (((4 + 4)4 ) − (4 × (44 + 4) + 4))
:= 5 + (55 − (((5 + 5)/5)5 + 55))
:= 5 × 5 + (55 × 55 − ((5 + 5)/5))
:= 55 − ((((55 + 5)/5) + 55) + 5)
:= 6/6 + (6 × 6 × 66 + 666)
:= 6 + (6 × 6 × 66 + 666)
:= ((6 − 6/6)6−6/6 ) − (66 + 6)
:= 7 + (7 × (777 − 7 × 7 × 7) − ((7 + 7)/7))
:= (7/7 + 7) × (7 × (7 × 7 + 7) − (77/7))
:= ((7/7 − 7) + 7 × 7) × ((7/7 − 7) + 77)
:= 8 + (((8 − (8 + 8)/8) × (8 × 8 × 8 − 8)) + (88/8))
:= (8 + 8 + 8) × (8 × (8 + 8) − 8/8)
:= (8 × (8 × (8 × 8 − 8 − 8))) − (88/8 + 8)
:= ((9 − 9/9) + 9) × ((9 × 9 − 9/9) + 99)
:= 9/9 + ((((9 + 9)/9)99/9 ) + 999)
:= 9/9 + (((9/9 + 9 + 9) + 9) × (9/9 + 99 + 9))
Ï 3049 := 1 + ((1 + 1)11 + (11 − 1)1+1+1 )
Ï 3054 := (1 + 1 + 1) × (((1 + 1)11 − (1 + 11))/(1 + 1))
Ï 3044 := (1 + 1) × (1 + ((1 + 1 + 1) × (1 + 1 + 11))1+1 )
2
− 2))
:= 2 + (((22/2 + 2 × 22)2 ) + 22)
:= 2 + ((22+2 − 2) × (222 − (2 + 2)))
:= 3/3 + ((3/3 + 3) × (33+3 + 33))
:= 3 + (3 × (3 × (333 + 3 + 3)))
:= 4 + ((44 × ((4 + 4)/4 + 4)) + 4)
:= 4 + ((4/4 + 4) × ((4/4 + 4)4 − 4 × 4))
:= (4 − 4/4) × (4 × 44 − ((4 + 4)/4 + 4))
:= 55 − ((5 × 5 + 55) + 5/5)
:= 5 × 5 + (55 × 55 − 5/5)
:= 55 − ((55/5 + 55) + 5)
:= 666 + (6 × 6 × 66 + ((6 + 6)/6))
:= 6 + ((6 × 6 × 66 + 666) + 6/6)
:= (6 × ((6 + 6) × (6 × 6 + 6) + 6)) − 6
:= 7 + (7 × (777 − 7 × 7 × 7) − 7/7)
:= 77/7 + 7 × (777 − 7 × 7 × 7)
:= ((7 + 7)/7)7 + (77 × (7 × 7 − 77/7))
:= ((8 × (8 × (88 + 8) − 8)) + 8)/((8 + 8)/8)
:= 8/8 + ((8 + 8 + 8) × (8 × (8 + 8) − 8/8))
:= ((8 + 8)/8) × (8 × 8 × (8 + 8 + 8) − (8/8 + 8))
:= 9 + (((9 × (9 + 9) × (9 + 9) + (99/9)) + 99) + 9)
:= 9 + ((9/9 + 9 + 9) × (9 × (9 + 9) − ((9 + 9)/9)))
:= ((9 + 9 + 9)/9) × (((999 + 9/9) + 9) + 9)
:= 2 + (2 × ((2 × (22 − 2) − 2/2) ))
3
:= 3 + (((33/3 + 3) ) + 3 × 3 × 33)
4
Ï 3045 := (11 + (11 − 1)) × (1 + (1 + 11)1+1 )
2
:= 2 + (((((2 × 2 × (22 − 2)) − 2) ) + 2)/2)
:= 3 + (3 × ((3 × (333 + 3) + 3) + 3))
4
:= (4/4 + 4) × ((4/4 + 4) − 4 × 4)
:= 55 − (5 × 5 + 55)
Ï 3050 := (1 + 111+1 ) × (1 + ((1 + 1) × (1 + 11)))
Ï 3055 := ((11 × 1111) − 1)/(1 + 1 + 1 + 1)
:= 2 + (2 × ((2 + 2 + 2) × (22×(2+2) − 2)))
:= 2 + (((((2 × 2 × (22 − 2)) − 2)2 ) + 22)/2)
:= (3 × (3 × (333 + 3 + 3))) − 3/3
:= (((((3 × (3 + 3)) + 3)3 ) + 3)/3) − 33
:= 44 + (44/4 × (44 − (4 + 4)/4))
:= ((4 + 4)4 ) − (4 × (44 + 4) + 4/4)
:= 5 × (555 + 55)
:= 5 + (55 × 55 + 5 × 5)
:= (6 × 6 − (6/6 + 6)) × (666/6 − 6)
:= 6 + ((6 × 6 × 66 + 666) + ((6 + 6)/6))
:= (66 − 6/6) × (66/6 + 6 × 6)
:= 7 + 7 × (777 − 7 × 7 × 7)
:= (7/7 + 7 × 7) × ((77 + 7)/7 + 7 × 7)
:= 7 + ((7/7 + 7) × (7 × (7 × 7 + 7) − (77/7)))
:= (88 − 8/8) × ((88/8 + 8 + 8) + 8)
:= (8 + 8)/8 + ((8 + 8 + 8) × (8 × (8 + 8) − 8/8))
:= (8/8 + 8 × 8) × (8 × 8 − (8/8 + 8 + 8))
:= 9 + ((9 × (9 + 9) × (9 + 9) + 999/9) + 9)
:= ((9/9 + 9 + 9) × (9 × (9 + 9) − 9/9)) − 9
:= (9 + 9) × (99 + 9) + 9999/9
Ï 3046 := 1 + ((11 + (11 − 1)) × (1 + (1 + 11)1+1 ))
2
:= 2 × (((2 × (22 − 2) − 2/2) ) + 2)
:= 3 + (((3 + 3) × 33+3 ) − (33/3)3 )
:= 44 × 44 + (4444 − 4)/4
5
Ï 3051 := (1 + 1 + 111) × (1 + 1 + 1)1+1+1
Ï 3056 := 1 + (((11 × 1111) − 1)/(1 + 1 + 1 + 1))
:= 2 + ((((22/2 + 2 × 22)2 ) + 22) + 2)
:= 2 × (2 × (2 × ((22+2 × (22 + 2)) − 2)))
:= 3 × (3 × (333 + 3 + 3))
:= (33/3)3 + ((3 × 3 + 3)3 − 3)
:= 4 + (4444/4 + 44 × 44)
:= 4 × (4 × 44 − (44 + 4))
:= 5 + ((5/5 − (5 × (5 + 5 + 5))) + 55 )
:= 5 + (5/5 − (5 × 5 + 55))
:= 5 × 5 + (55 × 55 + 5/5)
:= 6 + ((6 × 6 × 66 − ((6 + 6)/6)) + 666)
:= 6 + ((6 × 6 − (6/6 + 6)) × (666/6 − 6))
:= 6/6 + ((66 − 6/6) × (66/6 + 6 × 6))
:= 7 + (7 × (777 − 7 × 7 × 7) + 7/7)
:= 777/7 + 7 × (7 × 7 × 7 + 77)
:= ((8 × 8 − 8)(8+8)/8 ) − ((8 + 8)/8 + 88)
:= 7 + (7 × (777 − 7 × 7 × 7) + (77/7))
:= (88/8 − 8) × ((8 × 8 × (8 + 8) − 8) + 8/8)
:= (8 + 8) × (8 × (8 + 8 + 8) − 8/8)
:= (9 + 9 + 9) × ((999 + 9 + 9)/9)
:= 9 + ((((9 + 9)/9)99/9 ) + 999)
99/9
:= 999 + ((((9 + 9)/9)
) − 9/9)
Ï 3047 := 11 × (1 + ((1 + 11) × (1 + (11 + 11))))
:= 22 + ((22/2 + 2 × 22)2 )
:= 33/3 × (((3
3+3
+ 3)/3) + 33)
Ï 3052 := 1 + ((1 + 1 + 111) × (1 + 1 + 1)1+1+1 )
Ï 3057 := (1 + 1 + 1) × ((1 + (1 + 1)11 − 11)/(1 + 1))
:= (22+2 − 2) × (222 − (2 + 2))
:= ((22 × (22×(2+2) + 22)) − 2)/2
:= 3/3 + (3 × (3 × (333 + 3 + 3)))
:= 33 + (3 × (3 × (333 + 3)))
:= 44 × 44 + 4444/4
:= ((4 + 4) ) − (4 × (4 + 4) + 4)
:= 4/4 + (4 × (4 × 44 − (44 + 4)))
:= 55 + ((5 + 5)/5 − (5 × 5 + 55))
:= 5 × 5 + (55 × 55 + ((5 + 5)/5))
:= 55 × 55 + ((5 + 5)/5)5
:= 6 + ((6 × 6 × 66 − 6/6) + 666)
4
4
:= 666 + (6 × 6 × 66 + ((66 − 6)/6))
:= (66 × (6 × 6 + 6 + 6)) − 666/6
:= 7 + (7 × (777 − 7 × 7 × 7) + ((7 + 7)/7))
:= 7 + (7 × (777 − 7 × 7 × 7) + 7)
:= 7 + ((7/7 + 7 × 7) × ((77 + 7)/7 + 7 × 7))
:= ((8 × 8 − 8)(8+8)/8 ) − (8/8 + 88)
:= (8 − 8/8) × (888/((8 + 8)/8) − 8)
:= 8/8 + ((8 + 8) × (8 × (8 + 8 + 8) − 8/8))
:= 999 + (((9 + 9)/9)99/9 )
:= ((9/9 + 9 + 9) + 9) × (9/9 + 99 + 9)
:= ((9 + 9 + 9)/9) × ((99/9 + 999) + 9)
213
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3058 := 11 × ((1 + 1111)/(1 + 1 + 1 + 1))
Ï 3063 := (1 + 1 + 1) × ((1 + 1)11−1 − 1 − 1 − 1)
Ï 3068 := ((1 + 1 + 1) × ((1 + 1)11−1 − 1)) − 1
:= 22 × ((22×(2+2) + 22)/2)
:= (2/2 + 2) × (((222/2 − 2)/2) − 2)
:= 2 × (((2 + 2 + 2) × 22×(2+2) ) − 2)
:= 3 + ((((((3 × (3 + 3)) + 3)3 ) + 3)/3) − 33)
:= 3 + ((33 + 3) × (3 × 33 + 3))
:= (33 × ((3 × (33 + 3)) + 3)) − 3/3
:= ((4 + 4)4 ) + ((4 + 4)/4 − 4 × (44 + 4))
:= (4 − 4/4) × ((4 × 44 − 4) + 4/4)
:= (44 × ((4 + 4) + 4)) − 4
:= 55 − (((55 + 5)/5) + 55)
:= 55 + ((5 − 55 /5)/(5 + 5))
:= 55 − ((5 + 5)/5 + 55)
:= 6 + ((66 × (6 × 6 + 6 + 6)) − 666/6)
:= 6 + ((6 × ((6 + 6) × (6 × 6 + 6) + 6)) + ((6 + 6)/6))
:= ((77/7 − 7) × (777 + 7/7)) − 7 × 7
:= ((7 + 7)/7)7 + 7 × (7 × 7 × 7 + 77)
:= ((6 − 6/6)6−6/6 ) − (66 + 6/6)
:= ((7 × 7 + 7)
(7+7)/7
) − 7/7 − 77
:= ((8 − (8 + 8)/8) × (8 × 8 × 8 − 8/8)) − 8
:= (8 × (8 × (8 × 8 − 8 − 8))) − (8/8 + 8)
:= ((8 × 8 × (88 + 8)) − 8)/((8 + 8)/8)
:= ((9 + 9) × (9 × (9 + 9) + 9)) − (99/9 + 9)
:= ((((9 + 9)/9)9 ) × (9 − ((9 + 9 + 9)/9))) − 9
:= ((9 + 9) × (9 × (9 + 9) + 9)) − 9/9 − 9
Ï 3069 := (1 + 1 + 1) × ((1 + 1)11−1 − 1)
Ï 3059 := 111+1+1 + ((1 + 11)1+1+1 )
Ï 3064 := ((1 + 1 + 1) × (1 + (1 + 1)11−1 )) − 11
:= (22 + 2/2) × (222/2 + 22)
:= 2 × (2 × ((2 × (22+2 × (22 + 2))) − 2))
:= (2/2 + 2) × ((222/2 − 2)/2)
:= (33/3)3 + (3 × 3 + 3)3
:= 3 + (((33 + 3) × (3 × 33 + 3)) + 3/3)
:= 33 × ((3 × (33 + 3)) + 3)
:= ((4 − 4/4) × (4 × 44 − 4)) − 4/4
:= (44 × ((4 + 4) + 4)) − 4 − 4
:= 4/4 + ((44 × ((4 + 4) + 4)) − 4)
:= 55 − (55/5 + 55)
:= 55 − ((55 + 5/5) + 5)
:= 55 − (55 + 5/5)
:= ((6 − 6/6)6−6/6 ) − 66
:= 6 + (((6 − 6/6)6−6/6 ) − (66 + 6/6))
:= 6 × (6 × 66 − 6) + ((6 × 6/(6 + 6))6 )
:= ((7 × 7 + 7)(7+7)/7 ) − 77
:= (77/7 − 7) × (777 − (77/7))
:= ((7 + 7)/7 + 7) × (7 × 7 × 7 − ((7 + 7)/7))
:= (8 − 8/8) × (8 × (8 × 8 − 8) − (88/8))
:= (8 × (8 × (8 × 8 − 8 − 8))) − 8
:= 8 + ((8 × (8 × (8 × 8 − 8 − 8))) − (88/8))
:= (9/9 + 9 + 9) × (9 × (9 + 9) − 9/9)
:= 9 + ((9 + 9) × (99 + 9) + 9999/9)
:= ((9 + 9) × (9 × (9 + 9) + 9)) − 9
Ï 3060 := (1 + 11) × (111 + (1 + 11)1+1 )
Ï 3065 := ((1 + 1 + 1) × ((1 + 1)11−1 − (1 + 1))) − 1
Ï 3070 := 1 + ((1 + 1 + 1) × ((1 + 1)11−1 − 1))
:= (2 + 2 + 2) × (2(2/2+2) ) − 2)
:= 2 + ((2/2 + 2) × (((222/2 − 2)/2) − 2))
:= (2 × ((2 + 2 + 2) × 22×(2+2) )) − 2
:= (33 + 3) × (3 × 33 + 3)
:= 3 + (((33/3)3 + (3 × 3 + 3)3 ) + 3)
:= 3/3 + (33 × ((3 × (33 + 3)) + 3))
:= (4 − 4/4) × (4 × 44 − 4)
:= 4 + ((44 × ((4 + 4) + 4)) − 44/4)
:= (44 × ((4 + 4) + 4)) − (4 + 4)/4
:= 55 − (55 + 5 + 5)
:= 55 − (55 + 5)
:= 55 − 55
:= 6 × ((6 + 6) × (6 × 6 + 6) + 6)
:= 6 + (((6 − 6/6)6−6/6 ) − 66)
:= 66/6 + (((6 − 6/6)6−6/6 ) − 66)
:= 7/7 + (((7 × 7 + 7)(7+7)/7 ) − 77)
:= 7 + (((7 × 7 + 7)(7+7)/7 ) − (7/7 + 77))
:= 77/7 + (((7 × 7 + 7)(7+7)/7 ) − 77)
:= (8 − (8 + 8)/8) × (8 × 8 × 8 − ((8 + 8)/8))
:= 8/8 + ((8 × (8 × (8 × 8 − 8 − 8))) − 8)
2
:= (9 + 9) × ((9 × (9 + 9) − 9/9) + 9)
Ï 3061 := ((1 + 1 + 1) × (1 + 1)11−1 ) − 11
99/9
:= 9 + (((((9 + 9)/9)
) + 999) + 9)
Ï 3066 := (1 + 1 + 1) × ((1 + 1)11−1 − (1 + 1))
:= (8 × (8 × (8 × 8 − 8 − 8))) − (8 + 8)/8
:= 9/9 + (((9 + 9) × (9 × (9 + 9) + 9)) − 9)
Ï 3071 := ((1 + 1 + 1) × (1 + 1)11−1 ) − 1
:= (((2/2 + 2) × 222/2 ) − 22)/2
:= (2/2 + 2) × ((22×(2+2)+2 ) − 2)
:= (((2/2 + 2) × 222/2 ) − 2)/2
:= 3/3 + ((33 + 3) × (3 × 33 + 3))
:= (33/3 + 3) × ((3 + 3)3 + 3)
:= ((3 + 3) × ((3 − 3/3)3×3 )) − 3/3
:= (44 × ((4 + 4) + 4)) − 44/4
:= (4 − 4/4) × (4 × 44 − (4 + 4)/4)
:= (44 × ((4 + 4) + 4)) − 4/4
:= 55 + (5/5 − (55 + 5 + 5))
:= 55 + (5/5 − (55 + 5))
:= 55 + (5/5 − 55)
:= 6/6 + (6 × ((6 + 6) × (6 × 6 + 6) + 6))
:= 6 + (6 × ((6 + 6) × (6 × 6 + 6) + 6))
:= 6 + ((((6 − 6/6)6−6/6 ) − 66) + 6)
:= (7 + 7)/7 + (((7 × 7 + 7)(7+7)/7 ) − 77)
:= 7 + (((7 × 7 + 7)(7+7)/7 ) − 77)
:= (((7 + 7)/7 + 7) × (7 × 7 × 7 − 7/7)) − 7
:= (8 × (8 × (8 × 8 − 8 − 8))) − 88/8
:= (8 − (8 + 8)/8) × (8 × 8 × 8 − 8/8)
:= (8 × (8 × (8 × 8 − 8 − 8))) − 8/8
:= 9/9 + ((9 + 9) × ((9 × (9 + 9) − 9/9) + 9))
:= ((9 + 9) × (9 × (9 + 9) + 9)) − (99 + 9)/9
:= (9 + 9)/9 + (((9 + 9) × (9 × (9 + 9) + 9)) − 9)
Ï 3062 := 1 + (((1 + 1 + 1) × (1 + 1)11−1 ) − 11)
Ï 3067 := 1 + ((1 + 1 + 1) × ((1 + 1)11−1 − (1 + 1)))
Ï 3072 := (1 + 1 + 1) × (1 + 1)11−1
:= 2 + ((2 + 2 + 2) × (2(2/2+2) ) − 2))
:= ((2/2 + 2) × ((222/2 − 2)/2)) − 2
:= 2 × ((2 + 2 + 2) × 22×(2+2) )
:= 3 + ((33/3)3 + (3 × 3 + 3)3 )
:= 3/3 + ((33/3 + 3) × ((3 + 3)3 + 3))
:= (3 + 3) × ((3 − 3/3)3×3 )
:= (4 − 44)/4 + (44 × ((4 + 4) + 4))
:= (44 × ((4 + 4) + 4)) − (4/4 + 4)
:= 44 × ((4 + 4) + 4)
:= 55 − (55 /5 + 5)/(5 + 5)
:= 55 + ((5 + 5)/5 − (55 + 5))
:= 55 + ((5 + 5)/5 − 55)
:= (6 + 6)/6 + (6 × ((6 + 6) × (6 × 6 + 6) + 6))
:= 6 + ((6 × ((6 + 6) × (6 × 6 + 6) + 6)) + 6/6)
:= (6 × 6 + 6 + 6) × ((6 + 6)/6)6
:= 777 + (((7 + 7 + 7)/7)7 + 7 × (7 + 7))
:= 7 + ((((7 × 7 + 7)(7+7)/7 ) − 77) + 7/7)
:= (7 − 7/7) × ((7/7 + 7)(7+7+7)/7 )
:= (8 − 88)/8 + (8 × (8 × (8 × 8 − 8 − 8)))
:= 8 + ((8 − 8/8) × (8 × (8 × 8 − 8) − (88/8)))
:= 8 × (8 × (8 × 8 − 8 − 8))
:= (9 + 9)/9 + ((9 + 9) × ((9 × (9 + 9) − 9/9) + 9))
:= ((9 + 9) × (9 × (9 + 9) + 9)) − 99/9
:= (((9 + 9)/9)9 ) × (9 − ((9 + 9 + 9)/9))
2
214
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3073 := 1 + ((1 + 1 + 1) × (1 + 1)11−1 )
Ï 3078 := (1 + 1 + 1) × (1 + 1 + (1 + 1)11−1 )
Ï 3083 := 11 + ((1 + 1 + 1) × (1 + 1)11−1 )
:= (((2/2 + 2) × 222/2 ) + 2)/2
:= (2/2 + 2) × ((22×(2+2)+2 ) + 2)
:= (((2/2 + 2) × 222/2 ) + 22)/2
:= 3/3 + ((3 + 3) × ((3 − 3/3)3×3 ))
:= 3 × (3 × 333 + 33 )
:= (((((3 × (3 + 3)) + 3)3 ) − 3)/3) − 3
:= 4/4 + (44 × ((4 + 4) + 4))
:= 4 + ((44 × ((4 + 4) + 4)) + (4 + 4)/4)
:= 44/4 + (44 × ((4 + 4) + 4))
:= 5 + (55 − ((5 + 5)/5 + 55))
:= 5 + ((55 − ((5 + 5)/5 + 55)) + 5)
:= 55 − ((((5 + 5)/5)5 + 5) + 5)
:= 6/6 + ((6 × 6 + 6 + 6) × ((6 + 6)/6)6 )
:= 666 + 6 × (6 × 66 + 6)
:= ((6 − 6/6)6−6/6 ) − (6 × 6 + 6)
:= 7 × 7 × (7 × 7 + 7 + 7) − (7 + 7)
:= ((7 + 7)/7 + 7) × (7 × 7 × 7 − 7/7)
:= 7 + (7 × 7 × (7 × 7 + 7 + 7) − (77/7))
:= 8/8 + (8 × (8 × (8 × 8 − 8 − 8)))
:= (8 − (8 + 8)/8) × (8 × 8 × 8 + 8/8)
:= 88/8 + (8 × (8 × (8 × 8 − 8 − 8)))
:= (9 + 9) × (9 × (9 + 9) + 9)
:= ((9 × 9 + 9)/(9 + 9)) + ((9 + 9) × (9 × (9 + 9) + 9))
9
:= 9/9 + ((((9 + 9)/9) ) × (9 − ((9 + 9 + 9)/9)))
Ï 3074 := 1 + (1 + ((1 + 1 + 1) × (1 + 1)11−1 ))
Ï 3079 := 1 + ((1 + 1 + 1) × (1 + 1 + (1 + 1)11−1 ))
Ï 3084 := 1 + (11 + ((1 + 1 + 1) × (1 + 1)11−1 ))
:= 2 + (2 × ((2 + 2 + 2) × 22×(2+2) ))
:= (2 × (2 × (22 − 2))2 ) − (22/2)2
:= 2 × ((2 × 22 − 2)2 − 222)
:= 333 + (((33/3 + 3)3 ) − 3)
:= 3 + (((((3 × (3 + 3)) + 3)3 ) − 33)/3)
:= ((((3 × (3 + 3)) + 3)3 )/3) − 3
:= (4 + 4)/4 + (44 × ((4 + 4) + 4))
4
:= 4 + (((4 × ((4 + 4) + 4)) − 4/4) + 4)
5
:= (4 − 4/4) × (4 × 44 + 4)
:= 5 + (55 − (55 + 5/5))
:= 5 + ((5 − (55 + 5/5)) + 5)
:= 55 − (55/5 + 5 × 5 + 5)
:= 66 + (((6 + 6)/6)6 × (66/6 + 6 × 6))
:= 6/6 + (6 × (6 × 66 + 6) + 666)
:= 6 + (6 × (6 × 66 + 6) + 666)
:= 7/7 + (7 × 7 × (7 × 7 + 7 + 7) − (7 + 7))
:= 7 × 7 × (7 × 7 + 7 + 7) − (7/7 + 7)
:= (77/7 − 7) × ((777 − 7) + 7/7)
:= (8 + 8)/8 + (8 × (8 × (8 × 8 − 8 − 8)))
:= 8 + ((8 × (8 × (8 × 8 − 8 − 8))) − 8/8)
:= 8 + (((8 × 8 × (88 + 8)) + 8)/((8 + 8)/8))
:= ((9 + 9)/9) × ((9 × (9 × (9 + 9) + 9)) − ((9 + 9)/9))
:= 9/9 + ((9 + 9) × (9 × (9 + 9) + 9))
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) − ((9 + 9 + 9)/9))
Ï 3080 := ((1111+1 ) − 1)/(1 + 1 + 1 + 1)
Ï 3085 := 1 + (1 + (11 + ((1 + 1 + 1) × (1 + 1)11−1 )))
:= (2/2 + 2) × ((222/2 + 2)/2)
:= (2 − 222) × (2 − 22+2 )
:= 2 + ((((2/2 + 2) × 222/2 ) + 22)/2)
:= 3 + ((3 + 3) × ((3 − 3/3)3×3 ))
:= 3 + (((33/3 + 3)3 ) + 333)
:= ((((3 × (3 + 3)) + 3)3 ) − (3 + 3))/3
:= 4 + ((44 × ((4 + 4) + 4)) − 4/4)
:= 4 + ((44 × ((4 + 4) + 4)) + 4)
:= 4 + (((4/4 + 4)4+4/4 ) − 44)
:= 5 + (55 − 55)
:= 5 + ((55 − 55) + 5)
:= 5 + (((55 − 55) + 5) + 5)
:= 6 + (6 × (6 × 66 − 6) + ((6 × 6/(6 + 6))6 ))
:= 66/6 × (((6 + 6)/6)6 + 6 × 6 × 6)
:= ((6 + 6) × (6 × (6 × 6 + 6) + 6)) − 66/6
:= (7/7 + 7 + 7) × (((7 + 7)/7) + 77)
:= 7 × 7 × (7 × 7 + 7 + 7) − 7
:= 7 × 7 × (7 × 7 + 7 + 7) − (7 + 7)/7
:= 888 + ((88/8 − 8)8−8/8 )
:= 8 + (8 × (8 × (8 × 8 − 8 − 8)))
:= 88 + (888/8 × (88/8 + 8 + 8))
:= (9 + 9)/9 + ((9 + 9) × (9 × (9 + 9) + 9))
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) − ((9 + 9)/9))
Ï 3075 := (1 + 1 + 1) × (1 + (1 + 1)11−1 )
7
:= ((9 + 9) × (9 × (9 + 9) + 9)) − (9 + 9 + 9)/9
Ï 3076 := 1 + ((1 + 1 + 1) × (1 + (1 + 1)11−1 ))
Ï 3081 := 1 + (((1111+1 ) − 1)/(1 + 1 + 1 + 1))
Ï 3086 := 11 + ((1 + 1 + 1) × (1 + (1 + 1)11−1 ))
:= 2 × (((2 + 2 + 2) × 22×(2+2) ) + 2)
:= 2/2 + ((2 − 222) × (2 − 22+2 ))
:= (222 × (22+2 − 2)) − 22
:= ((((3 × (3 + 3)) + 3)3 ) − 33)/3
:= 3 + (3 × (3 × 333 + 33 ))
:= ((((3 × (3 + 3)) + 3)3 ) − 3)/3
:= 4 + (44 × ((4 + 4) + 4))
:= ((4/4 + 4)4+4/4 ) − 44
:= 4 + ((44 × ((4 + 4) + 4)) + (44 − 4)/4)
:= 5 + ((55 − 55) + 5/5)
:= 55 + (55/5 − 55)
:= 5 + ((55/5 − 55) + 55 )
:= 666 + (6 × (6 × 66 + 6) − ((6 + 6)/6))
:= 6666 × 6/(6 + 6) − 6 × (6 × 6 + 6)
:= 6 + ((66/6) × (((6 + 6)/6)6 + 6 × 6 × 6))
:= 7 × 7 × (7 × 7 + 7 + 7) − 77/7
:= 7/7 + (7 × 7 × (7 × 7 + 7 + 7) − 7)
:= 7 × 7 × (7 × 7 + 7 + 7) − 7/7
:= ((8 × 8 × (88 + 8)) + 8)/((8 + 8)/8)
:= 8 + ((8 × (8 × (8 × 8 − 8 − 8))) + 8/8)
:= 8 + ((8 − (8 + 8)/8) × (8 × 8 × 8 + 8/8))
9
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) − 9/9)
:= ((9 + 9) × (9 × (9 + 9) + 9)) − (9 + 9)/9
:= 9 + ((((9 + 9)/9) ) × (9 − ((9 + 9 + 9)/9)))
Ï 3087 := ((11 + (11 − 1))1+1+1 )/(1 + 1 + 1)
Ï 3077 := 1 + (1 + ((1 + 1 + 1) × (1 + (1 + 1)11−1 )))
Ï 3082 := 11 + (((1 + 1 + 1) × (1 + 1)11−1 ) − 1)
:= 2 + ((2/2 + 2) × ((222/2 + 2)/2))
:= 2 + ((2 − 222) × (2 − 22+2 ))
:= ((2/2 + 2 + 2) + 2) × ((22 − 2/2)2 )
:= 333 + ((33/3 + 3)3 )
:= (((((3 × (3 + 3)) + 3)3 ) + 3)/3) − 3 − 3
:= (((3 × (3 + 3)) + 3)3 )/3
:= 4 + ((44 × ((4 + 4) + 4)) + 4/4)
:= (44 − 4)/4 + (44 × ((4 + 4) + 4))
:= 4 + ((44 × ((4 + 4) + 4)) + 44/4)
:= 5 + (((5 + 5)/5 − 55) + 55 )
:= 55 + (((55 + 5)/5) − 55)
:= 5 + ((((55 + 5)/5) − 55) + 55 )
:= 666 + (6 × (6 × 66 + 6) − 6/6)
:= (66 + 6/6) × (((66 − 6)/6) + 6 × 6)
:= ((6 − 6/6)6−6/6 ) − ((6 + 6)/6 + 6 × 6)
:= ((7 − 77)/7) + 7 × 7 × (7 × 7 + 7 + 7)
:= (7 + 7)/7 + (7 × 7 × (7 × 7 + 7 + 7) − 7)
:= 7 × 7 × (7 × 7 + 7 + 7)
:= 8 + (((8 × (8 × (8 × 8 − 8 − 8))) − (88/8)) + 8)
:= 8 + ((8 × (8 × (8 × 8 − 8 − 8))) + ((8 + 8)/8))
:= 8 + (((8 × (8 × (8 × 8 − 8 − 8))) − 8/8) + 8)
:= ((9 + 9) × (9 × (9 + 9) + 9)) − 9/9
:= ((9 + 9)/9) × ((9 × (9 × (9 + 9) + 9)) + ((9 + 9)/9))
:= 9 + ((9 + 9) × (9 × (9 + 9) + 9))
215
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3088 := 1 + (((11 + (11 − 1))1+1+1 )/(1 + 1 + 1))
Ï 3093 := (1 + 1 + 1 + 11) × ((1 + 1) × 111 − 1) − 1
Ï 3098 := 1 + (1 + (((1 + 1)1+11 ) − (11 − 1)1+1+1 ))
:= 2 × (2 × 222 + ((22 + 2)2 ))
:= 22 + ((((2/2 + 2) × 222/2 ) − 2)/2)
:= 2 + (2 × ((2 + 2 + 2) × (22×(2+2) + 2)))
:= ((((3 × (3 + 3)) + 3)3 ) + 3)/3
:= 3 + (((((3 × (3 + 3)) + 3)3 )/3) + 3)
:= ((((3 × (3 + 3)) + 3)3 ) + 33)/3
:= 4 × ((4 × (4 × (44 + 4))) + 4)
:= ((4/4 + 4)4+4/4 ) − 4 × (4 + 4)
:= 4 + ((44/4 × ((4 + 4)/4 + 44 )) + 44 )
:= 55 − (((5 + 5)/5)5 + 5)
:= 55 − ((5 + 5)/5)5
:= 5 + (55 − ((5 + 5)/5)5 )
:= ((6 + 6)/6)6 + (6 × (6 + 6) × (6 × 6 + 6))
:= 6 × 6 × 66 + (((6 × 6/(6 + 6))6 ) − (6 + 6))
:= (6 + 6)/6 + ((6 + 6) × (6 × (6 × 6 + 6) + 6))
:= 7/7 + 7 × 7 × (7 × 7 + 7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7 + 7) − 7/7)
:= 77/7 + 7 × 7 × (7 × 7 + 7 + 7)
:= 8 + ((8 × (8 × (8 × 8 − 8 − 8))) + 8)
:= (88/8 − 8) × ((8 × 8 × (8 + 8) − 8/8) + 8)
:= (8 + 8)/8 + ((8 + 8 + 8) × (8 × (8 + 8) + 8/8))
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) + 9/9)
:= 99 + (((9 + 9 + 9)/9) × (999 − 9/9))
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) + (99/9))
Ï 3089 := 11 + ((1 + 1 + 1) × (1 + 1 + (1 + 1)11−1 ))
:= (2 × (2 × (22 − 2))2 ) − 222/2
Ï 3094 := (1 + (1 + 1 + 11)) × ((1 + 1) × 111 − 1)
Ï 3099 := (1 + 1 + 1) × (11 + ((1 + 1)11−1 − (1 + 1)))
:= (22 + 2 + 2) × ((22/2)2 − 2)
:= 2 + ((222 × (22+2 − 2)) − 22/2)
:= 3 + ((((((3 × (3 + 3)) + 3)3 ) + 3)/3) + 3)
:= 3 + (3 × (3 × 333 + 33))
:= 4 × 4 + ((4 × ((4 + 4) + 4)) + 4/4)
:= 44 + (44/4 × ((4 + 4)/4 + 44 ))
:= 4 × 4 + ((44 × ((4 + 4) + 4)) + 44/4)
:= 55 − (55/5 + 5 × 5)
:= 55 − ((5 × 5 + 5/5) + 5)
:= 55 − (5 × 5 + 5/5)
:= ((6 − 6/6)6−6/6 ) − 6 × 6
:= ((6 + 6) × (6 × (6 × 6 + 6) + 6)) − (6 + 6)/6
:= 6 × 6 × 66 + (((6 × 6/(6 + 6))6 ) − 6)
:= (7 + 7)/7 + 7 × 7 × (7 × 7 + 7 + 7)
:= 7 + 7 × 7 × (7 × 7 + 7 + 7)
:= (77 + 7)/7 + 7 × 7 × (7 × 7 + 7 + 7)
:= 8 + (((8 × (8 × (8 × 8 − 8 − 8))) + 8/8) + 8)
:= 8 + (((8 − (8 + 8)/8) × (8 × 8 × 8 + 8/8)) + 8)
:= 8 + (((8 × (8 × (8 × 8 − 8 − 8))) + (88/8)) + 8)
:= 99/9 + ((9 + 9) × (9 × (9 + 9) + 9))
:= 9 + ((((9 + 9) × (9 × (9 + 9) + 9)) − ((9 + 9)/9)) + 9)
:= (999/9 × ((9/9 + 9 + 9) + 9)) − 9
3
:= 3 + (((((3 × (3 + 3)) + 3) ) − 3)/3)
4
Ï 3090 := (1 + 1 + 1) × ((1 + (11 + (1 + 1)11 ))/(1 + 1))
Ï 3095 := 1 + (1 + 1 + 1 + 11) × ((1 + 1) × 111 − 1)
Ï 3100 := (1 + 1) × ((11 − 1) × (11 + (1 + 11)1+1 ))
:= 2 + ((2 × ((22 + 2)2 )) + (2 × 22)2 )
:= (((22 + 2) × (22×(2+2) + 2)) − 2)/2
:= 2 × (((2 + 2 + 2) × (22×(2+2) + 2)) + 2)
:= 3 + ((((3 × (3 + 3)) + 3)3 )/3)
:= 3 × 3 + (((((3 × (3 + 3)) + 3)3 ) − 3)/3)
:= 3 + ((3 × (3 × 333 + 33)) + 3/3)
:= 4 × 4 + ((4 × ((4 + 4) + 4)) + (4 + 4)/4)
:= 44/4 + ((4 − 4/4) × (4 × 4 + 4))
:= 4 + ((4 − 4/4) × ((4 × 44 + 4) + 4))
:= 55 − ((5 × 5 + 5) + 5)
:= 55 − (5 × 5 + 5)
:= 55 − 5 × 5
:= 66 + (6 × (6 + 6) × (6 × 6 + 6))
:= 6 + (((6 − 6/6)6−6/6 ) − 6 × 6)
:= (66 × (66/6 + 6 × 6)) − (6 + 6)/6
:= (7 + 7 + 7)/7 + 7 × 7 × (7 × 7 + 7 + 7)
:= 7 + (7 × 7 × (7 × 7 + 7 + 7) + 7/7)
:= 7 + ((7 × 7 × (7 × 7 + 7 + 7) − 7/7) + 7)
:= 8 + (((8 × (8 × (8 × 8 − 8 − 8))) + ((8 + 8)/8)) + 8)
:= ((8 + 8 + 8) × (8 × (8 + 8) + 8/8)) − 8/8
:= ((8 × (8 × (88 + 8) + 8)) − 8)/((8 + 8)/8)
:= (99 + 9)/9 + ((9 + 9) × (9 × (9 + 9) + 9))
:= 9 + ((((9 + 9) × (9 × (9 + 9) + 9)) − 9/9) + 9)
:= (9/9 + 99) × (((99 + 99)/9) + 9)
4
Ï 3091 := 11 + (((1111+1 ) − 1)/(1 + 1 + 1 + 1))
4
Ï 3096 := ((1 + 1)1+11 ) − (11 − 1)1+1+1
Ï 3101 := ((1 + 1 + 1) × (11 + ((1 + 1)11−1 − 1))) − 1
:= 2 + ((2 × (2 × (22 − 2))2 ) − 222/2)
:= 2 × ((2 + 2 + 2) × (22×(2+2) + 2))
:= ((2/2 + 2 + 2) + 2) × (((22 − 2/2)2 ) + 2)
:= 3 + (((((3 × (3 + 3)) + 3)3 ) + 3)/3)
:= 3 × (3 × 333 + 33)
:= 3 + (((((3 × (3 + 3)) + 3)3 ) + 33)/3)
:= 4 + (((44 × ((4 + 4) + 4)) + 44/4) + 4)
:= (4 − 4/4) × ((4 × 44 + 4) + 4)
:= ((4 − 4/4) + 4) × (444 − 4/4)
5
:= 5 + (5/5 − ((5 × 5 + 5) + 5))
5
:= 5 + (5/5 − (5 × 5 + 5))
:= 55 + (5/5 − 5 × 5)
:= 66 + ((6 × (6 + 6) × (6 × 6 + 6)) + 6/6)
:= (6 + 6) × (6 × (6 × 6 + 6) + 6)
:= (66 × (66/6 + 6 × 6)) − 6/6
:= 77/7 + (7 × 7 × (7 × 7 + 7 + 7) − 7)
:= ((7 + 7)/7 + 7) × (7 × 7 × 7 + 7/7)
:= 7 + (7 × 7 × (7 × 7 + 7 + 7) + 7)
:= 8 + ((8 × (8 × (8 × 8 − 8 − 8))) + (88/8))
:= (8 + 8 + 8) × (8 × (8 + 8) + 8/8)
:= (8 − 8/8) × (888/((8 + 8)/8) − 8/8)
:= 9999/9 + (99 × (99/9 + 9))
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) + 9)
:= 9 + ((99999/9) − 9 × 9 × 99)
Ï 3092 := 1 + (11 + (((1111+1 ) − 1)/(1 + 1 + 1 + 1)))
Ï 3097 := 1 + (((1 + 1)1+11 ) − (11 − 1)1+1+1 )
Ï 3102 := (1 + 1 + 1) × (11 + ((1 + 1)11−1 − 1))
:= 2 × ((2 × 222 + ((22 + 2)2 )) + 2)
:= (222 × (22+2 − 2)) − 22/2
:= 22 × (((22/2)2 − 2) + 22)
:= 3 + ((((((3 × (3 + 3)) + 3)3 ) − 3)/3) + 3)
:= 3 × 3 + (((((3 × (3 + 3)) + 3)3 ) + 3)/3)
:= 3 + ((3 × (3 × 333 + 33)) + 3)
4
4+4/4
:= (4 − 4/4) × ((44 − 4)/4 + 4 × 44 )
:= 4 + ((4 × ((4 + 4) + 4)) + 4 × 4)
:= 4 + (((4/4 + 4)
:= 55 − (((5 + 5)/5)5 + 5/5)
:= 55 + ((5 + 5)/5 − (5 × 5 + 5))
:= 55 + ((5 + 5)/5 − 5 × 5)
:= 66 + ((6 × (6 + 6) × (6 × 6 + 6)) + ((6 + 6)/6))
:= 6/6 + ((6 + 6) × (6 × (6 × 6 + 6) + 6))
:= 66 × (66/6 + 6 × 6)
:= 7 + (7 × 7 × (7 × 7 + 7 + 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 × 7 × (7 × 7 + 7 + 7)
:= 7 + ((7 × 7 × (7 × 7 + 7 + 7) + 7/7) + 7)
:= 8 + ((((8 × 8 × (88 + 8)) + 8)/((8 + 8)/8)) + 8)
:= 8/8 + ((8 + 8 + 8) × (8 × (8 + 8) + 8/8))
:= (((8 + 8)/8) + 8 × 8) × (8 × 8 − (8/8 + 8 + 8))
:= (99999/9) − 9 × 9 × 99
:= (9/9 + 9 + 9) × (9 × (9 + 9) + 9/9)
:= 99/9 × ((999/9 + 9 × (9 + 9)) + 9)
) − 4 × (4 + 4))
216
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3103 := 1 + ((1 + 1 + 1) × (11 + ((1 + 1)11−1 − 1)))
Ï 3108 := (1 + 1) × (111 × (1 + (1 + 1 + 11)))
Ï 3113 := 11 × (((1 + 1) × ((1 + 11)1+1 − (1 + 1))) − 1)
:= ((2/2 + 2 + 2)2/2+2+2 ) − 22
:= 222 × (22+2 − 2)
:= 2 + (((222 × (22+2 − 2)) + 2/2) + 2)
:= 33 + (((((3 × (3 + 3)) + 3)3 ) − 33)/3)
:= (33 + 3/3) × 333/3
:= 33 + (((((3 × (3 + 3)) + 3)3 ) − 3)/3)
:= 4 × (4 + 4) + ((44 × ((4 + 4) + 4)) − 4/4)
:= 444 × ((4 − 4/4) + 4)
:= 4 + (((4/4 + 4)4+4/4 ) − 4 × 4)
:= 55 − (55 + 55)/5
:= 55 − (((55 + 5)/5) + 5)
:= 55 − (55 + 5)/5
:= 6/6 + (66 × (66/6 + 6 × 6))
:= 6 + (66 × (66/6 + 6 × 6))
:= ((6 − 6/6)6−6/6 ) − 6 − 6
:= 7 + (((7 + 7)/7 + 7) × (7 × 7 × 7 + 7/7))
:= 777 × (77/7 − 7)
:= 7 + (77 × (7 × 7 − 7) − ((7 + 7)/7)7 )
:= 8 + (((8 + 8 + 8) × (8 × (8 + 8) + 8/8)) − 8/8)
:= (8 − 8/8) × 888/((8 + 8)/8)
:= (99/9 + 9 + 9) × ((99 − 9/9) + 9)
:= 999/9 × ((9/9 + 9 + 9) + 9)
:= 88 + ((8 × 8 − (8/8 + 8))(8+8)/8 )
Ï 3104 := ((1 + 1 + 1) × (11 + (1 + 1)11−1 )) − 1
:= 2
2+2
× (((2
2+2
2
Ï 3109 := 1 + ((1 + 1) × (111 × (1 + (1 + 1 + 11))))
:= 2/2 + (222 × (2
− 2) ) − 2)
3
:= 3 + ((((((3 × (3 + 3)) + 3) ) + 33)/3) + 3)
:= 4 × (((4 × (4 × (44 + 4))) + 4) + 4)
5
2+2
− 2))
3
:= ((333 × (3 + 3/3)) + 3)/3
:= ((4/4 + 4)
4+4/4
)−4×4
5
:= ((9 + 9) × (99/9 + 9 × (9 + 9))) − 9/9
Ï 3114 := ((1 + 111)/(1 + 1))1+1 − 11 − 11
:= ((2 × (22 + 2 + 2 + 2))2 ) − 22
:= 3 × (3 × (((3/3 + 3 + 3)3 ) + 3))
:= ((4/4 + 4)4/4+4 ) − 44/4
:= 5 + (5 − (5 × 5 + 5/5))
:= 5 − (55/5 + 5)
:= 55 − 55/5
:= (6 + 6)/6 + (66 × (66/6 + 6 × 6))
:= 6 + ((66 × (66/6 + 6 × 6)) + 6/6)
:= 6 + ((66 × (66/6 + 6 × 6)) + 6)
:= (77/7 − 7) × (777 − 7/7)
:= 7/7 + (777 × (77/7 − 7))
:= 7 + ((777 × (77/7 − 7)) − 7/7)
:= 8 + ((8 + 8 + 8) × (8 × (8 + 8) + 8/8))
:= 8/8 + ((8 − 8/8) × 888/((8 + 8)/8))
:= (8 − (8 + 8)/8) × ((8 × 8 × 8 − 8/8) + 8)
:= 9 + (((((9 + 9) × (9 × (9 + 9) + 9)) − 9/9) + 9) + 9)
:= 9 + ((9/9 + 99) × (((99 + 99)/9) + 9))
:= (9 + 9) × (99/9 + 9 × (9 + 9))
Ï 3105 := (1 + 1 + 1) × (11 + (1 + 1)11−1 )
Ï 3110 := (1 + 1) × (1 + (111 × (1 + (1 + 1 + 11))))
Ï 3115 := ((1 + 111)/(1 + 1))1+1 − 11 − 11 + 1
:= (2/2 + 2) × ((222/2 + 22)/2)
:= 2 + (222 × (22+2 − 2))
:= 2/2 + (((2 × (22 + 2 + 2 + 2))2 ) − 22)
:= 3 × ((3 × 333 + 33) + 3)
:= 3 + (((333 × (33 + 3/3)) − 3)/3)
:= 33 + (((((3 × (3 + 3)) + 3)3 ) + 3)/3)
:= 4/4 + (((4/4 + 4)4+4/4 ) − 4 × 4)
:= 44 + ((44 × ((4 + 4) + 4)) − 4/4)
:= 5 + (5 − 5 × 5)
:= 55 − (5 + 5 + 5)
:= 55 − 5 − 5
:= 6 × 6 × 66 + ((6 × 6/(6 + 6))6 )
:= 6 + ((66 × (66/6 + 6 × 6)) + ((6 + 6)/6))
:= (6 − 66)/6 + ((6 − 6/6)6−6/6 )
:= 7 + (7 × 7 × (7 × 7 + 7 + 7) + (77/7))
:= (7 + 7)/7 + (777 × (77/7 − 7))
:= 7 + (777 × (77/7 − 7))
:= 8 × 88 + ((8 − 8/8)8×8/(8+8) )
:= 888 + (((8 + 8)/8) × 8888/8)
:= (8/8 + 88) × ((88/8 + 8 + 8) + 8)
:= 9 + ((((9 + 9) × (9 × (9 + 9) + 9)) + 9) + 9)
:= 9 + (((99999/9) − 9 × 9 × 99) + 9)
:= 9/9 + ((9 + 9) × (99/9 + 9 × (9 + 9)))
4
:= (4/4 + 4) × ((4/4 + 4) − 4)
5
Ï 3106 := 1 + ((1 + 1 + 1) × (11 + (1 + 1)11−1 ))
Ï 3111 := 1 + ((1 + 1) × (1 + (111 × (1 + (1 + 1 + 11)))))
Ï 3116 := 11 + ((1 + 1 + 1) × (11 + (1 + 1)11−1 ))
:= (222 × (22+2 − 2)) − 2
:= 2 + ((222 × (22+2 − 2)) + 2/2)
:= 2 + (((2 × (22 + 2 + 2 + 2))2 ) − 22)
:= 3/3 + (3 × ((3 × 333 + 33) + 3))
:= 3 + ((33 + 3/3) × 333/3)
:= 3333 − ((3 + 3)3 + 3/3)
:= 4/4 + ((4/4 + 4) × ((4/4 + 4)4 − 4))
:= 4 + ((444 × ((4 − 4/4) + 4)) − 4/4)
:= 44 + (44 × ((4 + 4) + 4))
:= 5 + ((5/5 − 5 × 5) + 55 )
:= 55 + (5/5 − (5 + 5 + 5))
:= 55 + (5/5 − 5 − 5)
6
:= 6 + (((6 × 6/(6 + 6)) ) + 6 × 6 × 66)
:= (((6 − 66) + 6)/6) + ((6 − 6/6)6−6/6 )
:= 77 × (7 × 7 − 7) − ((7 + 7)/7)
:= 7 + ((77/7 − 7) × (777 − 7/7))
:= (7/7 − 77) × ((7/7 − 7 × 7) + 7)
:= ((8 − (8 + 8)/8) × ((8 × 8 × 8 − 8/8) + 8)) − 8
:= (8/8 + 8 + 8) × ((888/8 + 8 × 8) + 8)
:= 8 + ((8 − 8/8) × 888/((8 + 8)/8))
:= 9 + ((9/9 + 9 + 9) × (9 × (9 + 9) + 9/9))
:= 9 × 99 + ((99/9 + 9) × 999/9)
:= (9/9 + 9 + 9) × (((9 + 9)/9) + 9 × (9 + 9))
:= 6 + ((66 × (66/6 + 6 × 6)) − ((6 + 6)/6))
7
Ï 3107 := ((1 + 1) × (111 × (1 + (1 + 1 + 11)))) − 1
Ï 3112 := (1 + 1) × (1 + (1 + (111 × (1 + (1 + 1 + 11)))))
Ï 3117 := 1 + (11 + ((1 + 1 + 1) × (11 + (1 + 1)11−1 )))
:= (222 × (22+2 − 2)) − 2/2
:= 2 + ((222 × (22+2 − 2)) + 2)
:= 22/2 + ((222 × (22+2 − 2)) − 2)
:= ((333 × (33 + 3/3)) − 3)/3
:= 3 + (((333 × (33 + 3/3)) + 3)/3)
:= 3333 − (3 + 3)3
:= (444 × ((4 − 4/4) + 4)) − 4/4
:= 4 + (444 × ((4 − 4/4) + 4))
:= ((4/4 + 4)4+4/4 ) − 4 − 4
:= 5 + (((5 + 5)/5 − 5 × 5) + 55 )
:= 55 − (55 + 5 + 5)/5
:= 55 + ((5 + 5)/5 − 5 − 5)
:= ((6 − 6/6)6−6/6 ) − 6 − 6 − 6
:= ((6 + 6)/6 + 6) × (6 × 66 − (6/6 + 6))
:= ((6 − 6/6)6−6/6 ) − ((6 + 6)/6 + 6)
:= (777 × (77/7 − 7)) − 7/7
:= (77/7 − 7) × (777 + 7/7)
:= 7 + ((777 × (77/7 − 7)) + ((7 + 7)/7))
(8+8)/8
:= 88/8 + ((8 + 8 + 8) × (8 × (8 + 8) + 8/8))
:= ((8 × 8 − 8)
)−8−8−8
:= 9 + ((((9 + 9) × (9 × (9 + 9) + 9)) + (99/9)) + 9)
:= 9 + ((99/9 + 9 + 9) × ((99 − 9/9) + 9))
217
:= ((8 × 8 − 8)(8+8)/8 ) − (88/8 + 8)
:= 9 + (999/9 × ((9/9 + 9 + 9) + 9))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3118 := (1 + 1) × (((1 + 1 + 11) × (111+1 − 1)) − 1)
Ï 3123 := 1 + (1 + 1 + 1 + 11) × (1 + (1 + 1) × 111)
2/2+2+2
:= 2222 + (2 × (2 × (222 + 2)))
:= ((2/2 + 2 + 2)
:= 3/3 + (3333 − (3 + 3)3 )
:= 3 × (3 × (333 + 3) + 33)
:= (((4 + 4) + 4) × (44 + 4)) − (4 + 4)/4
5
:= ((6 − 6/6)
7−(7+7)/7
:= ((7 − ((7 + 7)/7))
:= (8 − 88)/8 + (((8 × 8 − 8)
:= 4 + ((((4 + 4) + 4) × (4 + 4)) − 4/4)
6−6/6
:= ((6 − 6/6)
) − 8)
Ï 3119 := 11 + ((1 + 1) × (111 × (1 + (1 + 1 + 11))))
− 2))
:= 33 + (((((3 × (3 + 3)) + 3)3 ) − 3)/3)
4
:= (((4 + 4) + 4) × (4 + 4)) − 4/4
:= 4 + ((((4 + 4) + 4) × (44 + 4)) + 4)
:= 5 + (55 − ((5 + 5)/5))
) − (6 + 6)/6
:= 77 × (7 × 7 − 7) − 777/7
)−7
(8+8)/8
:= (99 × (9 × 9 − 9 − 9) − 9/9)/((9 + 9)/9)
:= 22/2 + (222 × (2
:= 3 + ((3 − 3/3 + 3)3−3/3+3 )
4
:= 5 − (5 + 5)/5
) − 6/6 − 6
2+2
:= 2 × (222/2 − 222 )
5
:= 5 − ((5 + 5)/5 + 5)
6−6/6
)−2
Ï 3128 := (1 + 1) × ((1 + 1)11 − ((11 + 11)1+1 ))
:= 8 + ((8/8 + 88) × ((88/8 + 8 + 8) + 8))
:= 9 + ((9 + 9) × (99/9 + 9 × (9 + 9)))
Ï 3124 := (1 + 1) × (11 × ((1 + 11)1+1 − (1 + 1)))
2
:= 22 × (((2 × (2 + 2 + 2)) ) − 2)
3
:= (6 × 6/(6 + 6)) + ((6 − 6/6)6−6/6 )
:= (7/7 + 7) × (7 × (7 × 7 + 7) − 7/7)
:= ((8 × 8 − 8)(8+8)/8 ) − 8
:= 9 × 9 + ((((9 + 9)/9)99/9 ) + 999)
Ï 3129 := 1 + ((1 + 1) × ((1 + 1)11 − ((11 + 11)1+1 )))
:= 2 + (((2/2 + 2 + 2)2/2+2+2 ) + 2)
:= 3 + ((((((3 × (3 + 3)) + 3) ) + 3)/3) + 33)
:= 33 + (3 × (3 × 333 + 33))
:= 4 + (((4 + 4) + 4) × (44 + 4))
:= 4 + ((4/4 + 4)4+4/4 )
5
:= 55 − (5/5 + 5)
:= 5 − 5/5
:= 5 + (55 − 5/5)
:= ((6 − 6/6)6−6/6 ) − 6
:= ((6 − 6/6)6−6/6 ) − 6/6
:= 6 + (((6 − 6/6)6−6/6 ) − ((6 + 6)/6))
:= 7 + ((77/7 − 7) × (777 + 7/7))
:= ((7 − 777)/7) + 77 × (7 × 7 − 7)
:= ((7 × 7 + 7)(7+7)/7 ) − 7
:= ((8 × 8 − 8)
(8+8)/8
) − (8/8 + 8 + 8)
:= (99 × (9 × 9 − 9 − 9) + 9/9)/((9 + 9)/9)
:= (88/((8 + 8)/8)) × ((8 × 8 − 8/8) + 8)
:= 9 + (((9 + 9) × (99/9 + 9 × (9 + 9))) + 9/9)
Ï 3120 := (1 + 1) × ((1 + 1 + 11) × (111+1 − 1))
Ï 3125 := (1 + (1 + 1 + 1 + 1))1+1+1+1+1
:= (22 + 2) × ((2 × 22+2+2 ) + 2)
:= (2/2 + 2 + 2)2/2+2+2
3
:= (3 − 3/3 + 3)
:= 3 + (3333 − (3 + 3) )
4
:= ((4 + 4) + 4) × (4 + 4)
:= (4/4 + 4)
5
:= 8/8 + (((8 × 8 − 8)(8+8)/8 ) − 8)
:= ((9 + 9) × (99 + 9 × 9)) − 999/9
Ï 3130 := (1 + 1) × (11 + (111 × (1 + (1 + 1 + 11))))
:= 2 + (2 × (222/2 − 222 ))
3−3/3+3
:= 33+3 + ((3/3 + 3 + 3)3/3+3 )
4+4/4
:= 4 + (((4/4 + 4)4+4/4 ) + 4/4)
5
:= 5 + 55
:= 5 − 5
:= 5
:= ((6 + 6)/6 + 6) × (6 × 66 − 6)
:= (6 − 6/6)6−6/6
:= 6 + (((6 − 6/6)6−6/6 ) − 6/6)
:= (7/7 + 7) × (7 × (7 × 7 + 7) − ((7 + 7)/7))
:= (7 − ((7 + 7)/7))7−(7+7)/7
:= (8 − (8 + 8)/8) × (8 × 8 × 8 + 8)
:= 7/7 + (((7 × 7 + 7)(7+7)/7 ) − 7)
:= ((8 × 8 − 8)(8+8)/8 ) − 88/8
:= (8 + 8)/8 + (((8 × 8 − 8)(8+8)/8 ) − 8)
:= (((9 − 9/9) + 9) + 9) × (999/9 + 9)
Ï 3121 := 1 + ((1 + 1) × ((1 + 1 + 11) × (111+1 − 1)))
:= ((2/2 + 2 + 2)2/2+2+2 ) − 2 − 2
3
:= 33 + (((((3 × (3 + 3)) + 3) ) + 3)/3)
:= ((4/4 + 4)
4+4/4
)−4
5
:= 5 + 5/5 − 5
:= 6/6 + (((6 + 6)/6 + 6) × (6 × 66 − 6))
:= ((7 × 7 + 7)(7+7)/7 ) − (7/7 + 7 + 7)
:= 8/8 + ((8 − (8 + 8)/8) × (8 × 8 × 8 + 8))
:= ((99 + 9) × (99/9 + 9 + 9)) − 99/9
:= ((9 × 9 + 9)/(9 + 9))
(9×9+9)/(9+9)
Ï 3126 := 1 + ((1 + (1 + 1 + 1 + 1))1+1+1+1+1 )
:= (2
− 2) × (222 + 2/2)
:= ((3 − 3/3 + 3)3−3/3+3 ) − 3
:= 4/4 + (((4/4 + 4)
4+4/4
) − 4)
5
:= 5 + ((5 + 5)/5 − 5)
6−6/6
:= ((6 − 6/6)
:= ((7 × 7 + 7)
) − 6 × 6/(6 + 6)
(7+7)/7
) − (7 + 7)
:= (8 − 8/8) × (8 × (8 × 8 − 8) − ((8 + 8)/8))
:= 9 + (((9 + 9) × (99/9 + 9 × (9 + 9))) − 9/9)
Ï 3131 := 1 + ((1 + 1) × (11 + (111 × (1 + (1 + 1 + 11)))))
:= 2 + (22 × (((2 × (2 + 2 + 2))2 ) − 2))
:= ((22/2 + 2) × (222 − 2)/2) − 2
:= 3 + (3 × (3 × (333 + 3) + 33))
:= 3 + (((3 − 3/3 + 3)3−3/3+3 ) + 3)
:= 4/4 + ((4/4 + 4)4+4/4 )
:= 44/4 + (((4 + 4) + 4) × (44 + 4))
:= 55 + 5/5
:= 5 + (55 + 5/5)
:= 6/6 + ((6 − 6/6)6−6/6 )
:= 6 + ((6 − 6/6)6−6/6 )
:= 7/7 + ((7 − ((7 + 7)/7))7−(7+7)/7 )
:= (7 + 7)/7 + (((7 × 7 + 7)(7+7)/7 ) − 7)
:= (8 − 88)/8 + ((8 × 8 − 8)(8+8)/8 )
:= 88/8 + ((8 − (8 + 8)/8) × (8 × 8 × 8 + 8))
:= (9 − ((9 + 9 + 9)/9)) × (((9 + 9)/9)9 + 9)
:= ((99 + 9) × (99/9 + 9 + 9)) − 9/9
Ï 3122 := (1 + (1 + 1 + 11)) × (1 + (1 + 1) × 111)
2+2
:= (9/9 + 9) × ((9 + 9) × (9 + 9) − (99/9))
Ï 3127 := 1 + (1 + ((1 + (1 + 1 + 1 + 1))1+1+1+1+1 ))
:= 2 + ((2/2 + 2 + 2)
2/2+2+2
)
:= 3 + (((((((3 × (3 + 3)) + 3)3 ) + 3)/3) + 33) + 3)
:= (4 + 4)/4 + ((4/4 + 4)4+4/4 )
:= 55 + (5 + 5)/5
:= (6 + 6)/6 + ((6 − 6/6)6−6/6 )
Ï 3132 := (1 + 11) × (((1 + 1) × ((11 × (1 + 11)) − 1)) − 1)
:= 2 × ((222/2 − 222 ) + 2)
:= (33 + 3) × (3 × 33 + 3 + 3)
:= (4 − 4/4) × (4 × (44 + 4) + 4)
:= 5 + ((5 + 5)/5 + 55 )
:= 6 × (666 − (6 + 6) × (6 + 6))
:= 7 + ((7 − ((7 + 7)/7))7−(7+7)/7 )
:= ((7 × 7 + 7)
(7+7)/7
) − ((7 + 7)/7 + 7)
:= (8/8 + 8) × ((8 × 88 − 8)/((8 + 8)/8))
:= ((8 × 8 − 8)
(8+8)/8
) − (8/8 + 8)
:= (99 + 9) × (99/9 + 9 + 9)
:= (((999 + 9)/(9 + 9))
(9+9)/9
218
)−9
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3133 := (1 + 1 + 11) × (((1 + 1) × 111+1 ) − 1)
2
:= (22/2 + 2) × (22 − 2)/2
:= 3/3 + ((33 + 3) × (3 × 3 + 3 + 3))
:= 4 + (((4/4 + 4)
) + 4)
5
:= 5 + ((5 − ((5 + 5)/5)) + 5)
6−6/6
:= 6 + (((6 − 6/6)
:= ((7 × 7 + 7)
(7+7)/7
:= 8 + (((8 × 8 − 8)
) − (7 + 7 + 7)/7
(8+8)/8
:= (((222 × (22/2 + 2)) − 2)/2) − 2
:= 3 + (33 × (3 × 33 − (3/3 + 3)))
:= 3 + (((33/3 + 3)3 ) + (33 × (3 × 3 + 3)))
:= (4 + 4)/4 + (4 × (4 × ((4 × (44 + 4)) + 4)))
:= ((4 − 4/4) + 4) × ((444 + 4/4) + 4)
:= 5 + (55 + 5 + 5)/5
:= 5 + ((55 + 5 + 5)/5 + 55 )
:= 6 + (6 × (666 − (6 + 6) × (6 + 6)))
:= 6 + ((((6 − 6/6)6−6/6 ) + 6) + 6)
:= (7 + 7)/7 + ((7 × 7 + 7)(7+7)/7 )
:= 7 + ((7 × 7 + 7)(7+7)/7 )
:= (8 + 8)/8 + ((8 × 8 − 8)(8+8)/8 )
:= 8 + (((8 × 8 − 8)(8+8)/8 ) − 8/8)
:= 9 + (((9 + 9) × (99 + 9 × 9)) − 999/9)
:= (9 + 9)/9 + (((9 + 9) × (99 + 9 × 9)) − 99)
5
) + ((6 + 6)/6))
) − (88/8))
:= 9/9 + ((99 + 9) × (99/9 + 9 + 9))
Ï 3134 := ((1 + 111)/(1 + 1))1+1 − 1 − 1
Ï 3139 := ((1 + 111)/(1 + 1))1+1 + 1 + 1 + 1
2
Ï 3143 := ((1 + 1) × ((1 + 11) × ((11 × (1 + 11)) − 1))) − 1
:= 2 + ((2 × (22 + 2 + 2 + 2)) )
2
3
4+4/4
Ï 3138 := ((1 + 111)/(1 + 1))1+1 + 1 + 1
Ï 3144 := (1 + 1) × ((1 + 11) × ((11 × (1 + 11)) − 1))
:= ((2 × (22 + 2 + 2 + 2)) ) − 2
:= 2 + (((2 × (22 + 2 + 2 + 2))2 ) + 2/2)
:= 2 × (2 × (((22 + 2 + 2 + 2)2 ) + 2))
:= 3 × 3 + ((3 − 3/3 + 3)3+3−3/3 )
:= 3 + ((33 + 3/3) × ((333 + 3)/3))
:= 3 + ((33 × (3 × 33 − 3)) − 33 )
:= 4 + ((((4/4 + 4)4/4+4 ) + 4/4) + 4)
:= 4 + ((44 × ((4 + 4) + 4)) + ((44 − 4)/4))
:= 4 + ((4 × (4 × ((4 × (44 + 4)) + 4))) + 4)
:= 5 + ((5 − 5/5) + 5)
:= 5 + (((55 − 5/5) + 5) + 5)
:= 5 × 5 + (55 − (5/5 + 5))
:= 6 + (((6 − 6/6)6−6/6 ) + (6 × 6/(6 + 6)))
:= ((6 × 6 + 6/6) + 6) × (66 + 6/6 + 6)
:= (6 − ((6 + 6)/6)) × (66 × (6 + 6) − 6)
:= ((7 × 7 + 7)(7+7)/7 ) − (7 + 7)/7
:= 77 + (7 − (7 + 7)/7)7−(7+7)/7 + 7
:= ((8 × 8 − 8)(8+8)/8 ) − (8 + 8)/8
:= 88/8 + (((8 × 8 − 8)(8+8)/8 ) − 8)
:= 8 + ((8 × 8 − 8)(8+8)/8 )
:= 9 + ((9 × 9 + 9)/(9 + 9))(9×9+9)/(9+9)
:= 9 + ((9/9 + 9) × ((9 + 9) × (9 + 9) − (99/9)))
:= (99 + 9)/9 × ((9 × (9 + 9) + 99) + 9/9)
5
Ï 3135 := ((1 + 111)/(1 + 1))1+1 − 1
Ï 3140 := ((1 + 111)/(1 + 1))1+1 + 1 + 1 + 1 + 1
:= ((2 × (22 + 2 + 2 + 2))2 ) − 2/2
:= 2 + (((2 × (22 + 2 + 2 + 2))2 ) + 2)
:= 33 × (3 × 33 − (3/3 + 3))
:= ((33/3 + 3)3 ) + (33 × (3 × 3 + 3))
:= ((44 − 4)/4) + (44 × ((4 + 4) + 4))
:= 4 + (4 × (4 × ((4 × (44 + 4)) + 4)))
:= 5 + (55 + 5)
:= 5 + (55 + 5 + 5)
:= 66/6 × (6 × 66 − 666/6)
:= (6 − 6/6) × (666 − ((6 + 6)/6 + 6 × 6))
:= ((7 × 7 + 7)(7+7)/7 ) − 7/7
:= 77/7 + (((7 × 7 + 7)(7+7)/7 ) − 7)
:= ((8 × 8 − 8)(8+8)/8 ) − 8/8
:= (8/((8 + 8)/8)) + ((8 × 8 − 8)(8+8)/8 )
:= 9 + ((9 − ((9 + 9 + 9)/9)) × (((9 + 9)/9)9 + 9))
:= (9/9 + 9) × ((9 + 9) × (9 + 9) − (9/9 + 9))
Ï 3136 := ((1 + 111)/(1 + 1))1+1
:= (2 × (22 + 2 + 2 + 2))
Ï 3141 := ((1 + 111)/(1 + 1))1+1 + 1 + 1 + 1 + 1 + 1
2
2
:= 2 + ((((2 × (22 + 2 + 2 + 2)) ) + 2/2) + 2)
3
:= (3 + 3/3) × ((333 + 3)/3)
:= (33 × (3 × 33 − 3)) − 33
:= 4 × (4 × ((4 × (44 + 4)) + 4))
:= 4 × 4 + ((4/4 + 4)4/4+4 )
:= 55 + 55/5
:= 5 + (55/5 + 55 )
:= 66/6 + ((6 − 6/6)6−6/6 )
:= 6 × (6 × 66 + 6) + ((6 × 6/(6 + 6))6 )
:= (7 × 7 + 7)(7+7)/7
:= 7 + (((7 × 7 + 7)(7+7)/7 ) − ((7 + 7)/7))
:= 8 + ((((8 × 8 − 8)
:= ((9 + 9) × (99 + 9 × 9)) − 99
Ï 3137 := ((1 + 111)/(1 + 1))
+1
) − (88/8)) + 8)
Ï 3142 := (1 + 1) × ((11 × (11 × (1 + 1 + 11))) − (1 + 1))
:= 2/2 + ((2 × (22 + 2 + 2 + 2)) )
33/3
:= 33 × 33 + (3 − 3/3)
:= 4 + ((((4/4 + 4)
:= 3 + (((33 × (3 × 33 − 3)) − 33 ) + 3/3)
:= (4/4 + 4) × ((4/4 + 4)4 + 4)
:= 5 × 5 + (55 − 5)
:= (6 − 6/6) × (666 − (6 × 6 + 6/6))
:= 7 + (((7 × 7 + 7)(7+7)/7 ) + ((7 + 7)/7))
:= 8 + (((8 × 8 − 8)(8+8)/8 ) + 8/8)
:= 9 + (((999 + 9)/(9 + 9))(9+9)/9 )
Ï 3146 := (1 + 1) × (11 × (11 × (1 + 1 + 11)))
:= 22 × ((22/2)2 + 22)
:= 33/3 × 33/3 × (33 − 3/3)
:= 66/6 × ((((6 + 6)/6)6 + 6 × 6 × 6) + 6)
:= 77/7 × (7 × (7 × 7 − 7) − (7/7 + 7))
:= 8 + (((8 × 8 − 8)(8+8)/8 ) + ((8 + 8)/8))
:= 99 + ((((9 + 9)/9)99/9 ) + 999)
Ï 3147 := ((1 + 111)/(1 + 1))1+1 + 11
2
4/4+4
:= ((222 × (22/2 + 2)) − 2)/2
:= 5 + ((55/5 + 55 ) + 5)
:= ((999 + 9)/(9 + 9))(9+9)/9
1+1
Ï 3145 := ((1 + 1) × (11 × (11 × (1 + 1 + 11)))) − 1
:= 4/4 + ((4/4 + 4) × ((4/4 + 4)4 + 4))
:= (8 × 8 − 8)(8+8)/8
(8+8)/8
:= 7 + (((7 × 7 + 7)(7+7)/7 ) + 7/7)
) + 4) + 4)
:= 2 + ((((2 × (22 + 2 + 2 + 2))2 ) + 2) + 2)
:= (3 × (33 × (33 + 3 + 3) − 3)) − 3
:= 3/3 + ((33 × (3 × 33 − 3)) − 33 )
:= 44/4 + (4 × (4 × ((4 × (44 + 4)) + 4)))
:= 4 × 4 + (((4/4 + 4)
5
:= ((222 × (22/2 + 2)) + 2)/2
4+4/4
) + 4/4)
:= 55 + (55 + 55)/5
:= 5 + (55 + 5)/5
:= 5 + (((55 + 5)/5) + 55 )
:= 6 + (6 × (6 × 66 + 6) + ((6 × 6/(6 + 6))6 ))
:= 6 + (((6 − 6/6)6−6/6 ) + 6)
:= 6 + (((6 − 6/6)6−6/6 ) + (66/6))
:= 77/7 + ((7 × 7 + 7)(7+7)/7 )
:= 7/7 + ((7 × 7 + 7)(7+7)/7 )
:= 7 + (((7 × 7 + 7)(7+7)/7 ) − 7/7)
:= 88/8 + ((8 × 8 − 8)(8+8)/8 )
:= 8/8 + ((8 × 8 − 8)(8+8)/8 )
:= 8 + (((8 × 8 − 8)(8+8)/8 ) − ((8 + 8)/8))
:= ((9 + 9 + 9) × (99 + 9 + 9)) − (99 + 9)/9
:= 9/9 + (((999 + 9)/(9 + 9))
(9+9)/9
)
:= 9/9 + (((9 + 9) × (99 + 9 × 9)) − 99)
219
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3148 := 1111 + ((1 + 1)11 − 11)
Ï 3153 := 1111 + ((1 + 1)11 − ((1 + 1) × (1 + 1 + 1)))
Ï 3158 := 1111 + ((1 + 1)11 − 1)
:= 2 + (22 × ((22/2)2 + 22))
:= 2/2 + (2 × ((2 × (22 − 2))2 − (22 + 2)))
:= 2 + (2 × ((2 × (22 − 2))2 − 22))
:= (3 × 33 × (33 + 3 + 3)) − 33/3
:= 3 + (3 × (33 × (33 + 3 + 3) − 3))
:= (3 × 33 × (33 + 3 + 3)) − 3/3
:= 4444 − ((4 + 4)/4 + 4)4
:= (4 − 4/4)4 + (44 × ((4 + 4) + 4))
:= 44 × (4 + 4) + (4444 − 4)/4
:= 5 × 5 + (55 − ((5 + 5)/5))
:= 5 + ((55 − ((5 + 5)/5)) + 5 × 5)
:= 55 + (((5 + 5)/5)5 + 5/5)
:= 6 + ((((6 − 6/6)6−6/6 ) + (66/6)) + 6)
:= 666 + (666/6 + 6 × 6 × 66)
:= (77 + 7)/7 + ((7 × 7 + 7)
(7+7)/7
:= ((88 + 8)/8) + ((8 × 8 − 8)
:= 7 + (((7 × 7 + 7)
)
(8+8)/8
)
:= ((9 + 9 + 9) × (99 + 9 + 9)) − 99/9
Ï 3149 := 1 + (1111 + ((1 + 1)11 − 11))
(7+7)/7
:= 8 + ((((8 × 8 − 8)
) + ((77 − 7)/7))
(8+8)/8
) + 8/8) + 8)
:= 9 × 9 + ((((9 + 9)/9)9 ) × (9 − ((9 + 9 + 9)/9)))
Ï 3154 := 1111 + ((1 + 1)11 − (1 + (1 + 1 + 1 + 1)))
:= (6 − 66)/6 + (66 × (6 × 6 + 6 + 6))
:= 7/7 + (77 × (7 × 7 − (7/7 + 7)))
:= 8 + ((8 − 8/8) × (8 × (8 × 8 − 8) + ((8 + 8)/8)))
:= ((9 + 9 + 9) × (99 + 9 + 9)) − 9/9
Ï 3159 := 1111 + (1 + 1)11
:= 2 + (((222 × (22/2 + 2)) + 2)/2)
:= (2 × ((2 × (22 − 2))2 − 22)) − 2
:= 222/2 + 2222/2
:= (3 × (33 × (33 + 3 + 3) − 3)) − 3/3
:= (33 × (3 × 33 − 3)) − (33/3 + 3)
:= 3 × 33 × (33 + 3 + 3)
:= 4 + ((4/4 + 4) × ((4/4 + 4)4 + 4))
:= 4 + ((4/4 + 4) × (((4/4 + 4)4 + 4/4) + 4))
:= 44 × (4 + 4) + 4444/4
:= 5 × 5 + (55 − 5/5)
:= 5 + ((55 − 5/5) + 5 × 5)
:= 5 + (((55 − 5/5) + 5 × 5) + 5)
:= (66 + 6/6) × (66/6 + 6 × 6)
:= (((6 + 6)/6 + 6) × (6 × 66 − 6/6)) − 6
:= 6 × 6 + (((6 − 6/6)6−6/6 ) − ((6 + 6)/6))
:= 7 + ((((7 × 7 + 7)(7+7)/7 ) − 7/7) + 7)
:= 7 + (((7 × 7 + 7)(7+7)/7 ) + (77/7))
:= (7 + 7)/7 + (77 × (7 × 7 − (7/7 + 7)))
:= 88 + ((8 × (8 × (8 × 8 − 8 − 8))) − (88/8))
:= 8 + ((((8 × 8 − 8)(8+8)/8 ) + ((8 + 8)/8)) + 8)
:= 88 + ((8 × (8 × (8 × 8 − 8 − 8))) − 8/8)
:= ((9 + 9 + 9) × (99 + 9 + 9)) − 9/9 − 9
:= 9 + ((((999 + 9)/(9 + 9))(9+9)/9 ) + 9)
:= (9 + 9 + 9) × (99 + 9 + 9)
Ï 3150 := 1 + (1 + (1111 + ((1 + 1)11 − 11)))
:= 2 + ((22 × ((22/2)2 + 22)) + 2)
:= 3 × (33 × (33 + 3 + 3) − 3)
Ï 3155 := 1111 + ((1 + 1)11 − (1 + 1 + 1 + 1))
:= (2 × ((2 × (22 − 2))2 − 22)) − 2/2
Ï 3160 := 1 + (1111 + (1 + 1)11 )
:= 2 × (((2 × (22 − 2))2 − 22) + 2)
:= (3 × 33 × 33) − ((333 + 3)/3)
:= 3/3 + (3 × 33 × (33 + 3 + 3))
:= (4/4 + 4) × (((4/4 + 4)4 + 4/4) + 4)
:= 44 × (4 + 4) + (4444/4 − 4)
:= 44 + ((44 × ((4 + 4) + 4)) + 44)
:= 5 × 5 + 55
:= 5 + (5 × 5 + 55 )
:= 5 + ((5 × 5 + 55 ) + 5)
:= 6 × 6 + (((6 − 6/6)6−6/6 ) − 6)
:= ((6 + 6)/6 + 6) × (6 × 66 − 6/6)
:= 77 × (7 × 7 − 7) − ((7 + 7)/7 + 77)
:= 777 + (7 × 7 × 7 × 7 − (77/7 + 7))
:= 8 + (((8 × 8 − 8)(8+8)/8 ) + (88/8))
:= 88 + (8 × (8 × (8 × 8 − 8 − 8)))
:= (6 − 6 × 6) × (6 − 666/6)
:= 7 + (((7 × 7 + 7)
(7+7)/7
) + 7)
:= (8 − 8/8) × (8 × (8 × 8 − 8) + ((8 + 8)/8))
:= 9 × ((((9 + 9)/9)9 ) − 9 × (9 + 9))
Ï 3151 := 1 + (1 + (1 + (1111 + ((1 + 1)11 − 11))))
:= 2 + ((((222 × (22/2 + 2)) + 2)/2) + 2)
:= 3/3 + (3 × (33 × (33 + 3 + 3) − 3))
:= (4 × ((4 × ((4 × (44 + 4)) + 4)) + 4)) − 4/4
5
:= 5 × 5 + (5 + 5/5)
:= 6/6 + ((6 − 6 × 6) × (6 − 666/6))
:= 7 + ((((7 × 7 + 7)
(7+7)/7
:= 8 + ((((8 × 8 − 8)
(8+8)/8
:= 9 + (((((9 + 9)/9)99/9 ) + 999) + 99)
Ï 3156 := (1 + 11) × (((1 + 1) × (11 × (1 + 11))) − 1)
:= 2 × ((2 × (22 − 2))2 − 22)
:= (3 × 33 × (33 + 3 + 3)) − 3
:= 4 + (4 × ((4 × ((4 × (44 + 4)) + 4)) + 4))
:= 5 + ((5 × 5 + 55 ) + 5/5)
:= 9/9 + ((9 + 9 + 9) × (99 + 9 + 9))
Ï 3161 := 1 + (1 + (1111 + (1 + 1)11 ))
:= 2 + ((2222/2) + 222/2 )
:= 3 + ((3 × 33 × (33 + 3 + 3)) − 3/3)
:= 4 + (((4/4 + 4)4+4/4 ) + 4 × (4 + 4))
:= 5 × 5 + (55/5 + 55 )
:= (66 × (6 × 6 + 6 + 6)) − 6 − 6
:= 6 × 6 + ((6 − 6/6)6−6/6 )
) + 7/7) + 7)
:= 77 × (7 × 7 − 7) − 7/7 − 77
:= ((7 − 7/7) × (7 × 77 − (77/7))) − 7
) − 8/8) + 8)
:= (((88 × (8 × 8 + 8)) − 8)/((8 + 8)/8)) − 8
:= 8/8 + ((8 × (8 × (8 × 8 − 8 − 8))) + 88)
:= ((9 + 9 + 9)/9) × ((9 × (99 + 9 + 9)) − 9/9)
:= (9 + 9)/9 + ((9 + 9 + 9) × (99 + 9 + 9))
:= 9/9 + (9 × ((((9 + 9)/9)9 ) − 9 × (9 + 9)))
Ï 3152 := (1 + 1) × (1 + (1 + (1 + (11 × (11 × (1 + 1 + 11)))))) Ï 3157 := 11 × (((1 + 1) × (1 + 11)1+1 ) − 1)
Ï 3162 := 1 + (1 + (1 + (1111 + (1 + 1)11 )))
:= 2 × ((2 × (22 − 2))2 − (22 + 2))
:= 22/2 × ((((22 + 2)2 ) − 2)/2)
:= (2 + 2 + 2) × ((22 + 2/2)2 − 2)
:= 33 + ((3 − 3/3 + 3)3−3/3+3 )
:= (33 × (3 × 33 − 3)) − 33/3
:= 3 + (3 × 33 × (33 + 3 + 3))
:= 4 × ((4 × ((4 × (44 + 4)) + 4)) + 4)
:= 4 × (4 + 4) + ((4/4 + 4)4+4/4 )
:= 4 + ((4444 − 4)/4 + 44 × (4 + 4))
:= 5 × 5 + ((5 + 5)/5 + 55 )
:= 55 + ((5 + 5)/5)5
:= 5 + (((5 + 5)/5)5 + 55 )
:= ((6 + 6)/6 + 6) × (6 × 66 − ((6 + 6)/6))
:= (66 × (6 × 6 + 6 + 6)) − 66/6
:= (66 × (6 × 6 + 6 + 6)) − 6
:= (7/7 + 7) × (7 × (7 × 7 + 7) + ((7 + 7)/7))
:= 77 × (7 × 7 − (7/7 + 7))
:= (7 − 7/7) × (7 × 77 − (77 + 7)/7)
:= 88/8 × ((888/8 + 88) + 88)
:= 88 + ((8 × (8 × (8 × 8 − 8 − 8))) + ((8 + 8)/8))
:= ((9 + 9 + 9) × (99 + 9 + 9)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + ((9 + 9 + 9) × (99 + 9 + 9))
:= 8 + (((8 × 8 − 8)
(8+8)/8
) + 8)
9
:= (9 + 9)/9 + (9 × ((((9 + 9)/9) ) − 9 × (9 + 9)))
220
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3163 := 1 + (1 + (1 + (1 + (1111 + (1 + 1)11 ))))
Ï 3168 := (1 + 1) × (11 × (1 + 11)1+1 )
Ï 3173 := 1 + ((1 + 1) × (1 + (1 + (11 × (1 + 11)1+1 ))))
:= 2 + (((2222/2) + 222/2 ) + 2)
:= 22 × ((2 × (2 + 2 + 2))2 )
:= ((2 + 2 + 2) × ((22 + 2/2)2 )) − 2/2
:= 3 + ((3 × 33 × (33 + 3 + 3)) + 3/3)
:= 33 × (3 × 33 − 3)
:= 3 + (((33 × (3 × 33 − 3)) − 3/3) + 3)
:= 4 + (4444/4 + 44 × (4 + 4))
:= 44 × ((4 × 4 × 4 + 4) + 4)
:= 4 + (((4/4 + 4)4+4/4 ) + 44)
5
:= 5 + ((((5 + 5)/5)5 + 55 ) + 5/5)
:= 55 + (5 − ((55 + 5)/5))
:= 55 + (55 − ((5 + 5)/5 + 5))
:= 6/6 + ((66 × (6 × 6 + 6 + 6)) − 6)
:= 66 × (6 × 6 + 6 + 6)
:= 6 + ((66 × (6 × 6 + 6 + 6)) − 6/6)
:= 7 + (77 × (7 × 7 − 7) − (7/7 + 77))
:= (7 − 7/7) × (7 × 77 − (77/7))
:= 77 + (((7 + 7)/7 + 7) × (7 × 7 × 7 + 7/7))
:= 8 + ((((8 × 8 − 8)(8+8)/8 ) + (88/8)) + 8)
:= 8 × ((88 × (8 × 8 + 8))/(8 + 8))
:= 8/8 + (((88 × (8 × 8 + 8)) + 8)/((8 + 8)/8))
:= 9 + (((((999 + 9)/(9 + 9))(9+9)/9 ) + 9) + 9)
:= 9 + ((9 + 9 + 9) × (99 + 9 + 9))
:= (9 × (9 − 9 × 99)) + (99999/9)
Ï 3164 := (1 + 1) × ((11 × (1 + 11)1+1 ) − (1 + 1))
:= 2 × ((2 × (22 × (2 + 2 + 2)2 )) − 2)
:= (33 × (3 × 33 − 3)) − (3/3 + 3)
:= 44 + (((4 + 4) + 4) × (44 + 4))
:= 55 + (55 − (55/5 + 5))
:= (6 + 6)/6 + ((66 × (6 × 6 + 6 + 6)) − 6)
:= 7 + (77 × (7 × 7 − (7/7 + 7)))
:= ((88 × (8 × 8 + 8)) − 8)/((8 + 8)/8)
:= ((9/9 + 9 + 9) + 9) × ((999 + 9 + 9)/9)
Ï 3165 := ((1 + 1) × ((11 × (1 + 11)1+1 ) − 1)) − 1
:= (22 × ((2 × (2 + 2 + 2))2 )) − 2/2 − 2
:= (33 × (3 × 33 − 3)) − 3
Ï 3169 := 1 + ((1 + 1) × (11 × (1 + 11)1+1 ))
Ï 3174 := (1 + 1) × (1 + (1 + (1 + (11 × (1 + 11)1+1 ))))
:= 2/2 + (22 × ((2 × (2 + 2 + 2))2 ))
:= (2 + 2 + 2) × ((22 + 2/2)2 )
:= 3/3 + (33 × (3 × 33 − 3))
:= 3 + ((33 × (3 × 33 − 3)) + 3)
:= 44 + ((4/4 + 4)
4+4/4
:= ((44 + 4)/4 + 4) × ((4 + 4)/4 + 44)
)
:= 55 + (55 − (55/5))
:= 55 + (55 − (5/5 + 5))
:= 6/6 + (66 × (6 × 6 + 6 + 6))
:= 6 + (66 × (6 × 6 + 6 + 6))
:= 7777/7 + 7 × 7 × (7 × 7 − 7)
:= 7 × 7 + ((7 − ((7 + 7)/7))7−(7+7)/7 )
:= 8/8 + (8 × ((88 × (8 × 8 + 8))/(8 + 8)))
:= (8 − (8 + 8)/8) × (((8 × 8 × 8 + 8/8) + 8) + 8)
:= 9 + (((9 + 9 + 9) × (99 + 9 + 9)) + 9/9)
:= 9 + (((99 × 99 − 9)/((9 + 9 + 9)/9)) − 99)
Ï 3170 := 11 + (1111 + (1 + 1)11 )
Ï 3175 := 1 + ((1 + 1) × (1 + (1 + (1 + (11 × (1 + 11)1+1 )))))
:= 2 + (22 × ((2 × (2 + 2 + 2))2 ))
:= 2/2 + ((2 + 2 + 2) × ((22 + 2/2)2 ))
:= 3 + ((33 × (3 × 33 − 3)) − 3/3)
:= 3 + (((33 × (3 × 33 − 3)) + 3/3) + 3)
4+4/4
:= 44 + (((4/4 + 4)4+4/4 ) − 4)
:= 44 + (((4/4 + 4)
:= 5 + (((5 × 5 + 55 ) + 5) + 5)
:= 55 + (5 − 5 − 5)
:= 55 + (55 − 5)
:= (66 × (6 × 6 + 6 + 6)) − 6 × 6/(6 + 6)
:= 7 + ((77 × (7 × 7 − (7/7 + 7))) + 7/7)
:= 8/8 + (((88 × (8 × 8 + 8)) − 8)/((8 + 8)/8))
:= ((99 × 99 − 9)/((9 + 9 + 9)/9)) − 99
Ï 3166 := (1 + 1) × ((11 × (1 + 11)1+1 ) − 1)
) + 4/4)
5
:= 4 + (((44 × ((4 × 4 × 4 + 4) + 4)) − 4/4) + 4)
:= (6 + 6)/6 + (66 × (6 × 6 + 6 + 6))
:= 6 + ((66 × (6 × 6 + 6 + 6)) + 6/6)
:= 777 + (7 × 7 × 7 × 7 − (7/7 + 7))
:= 7 + ((7 − 7/7) × (7 × 77 − (77/7)))
:= (8 + 8)/8 + (8 × ((88 × (8 × 8 + 8))/(8 + 8)))
:= (8/8 + 8 + 8 + 8) × (8 × (8 + 8) − 8/8)
:= 99/9 + ((9 + 9 + 9) × (99 + 9 + 9))
:= 99 + (((9 + 9) × (9 × (9 + 9) + 9)) − ((9 + 9)/9))
Ï 3171 := 1 + (11 + (1111 + (1 + 1)11 ))
Ï 3176 := 11 + (((1 + 1) × ((11 × (1 + 11)1+1 ) − 1)) − 1)
:= (22 × ((2 × (2 + 2 + 2))2 )) − 2
:= 2 + ((22 × ((2 × (2 + 2 + 2))2 )) + 2/2)
:= 2 + ((2 + 2 + 2) × ((22 + 2/2)2 ))
:= 3/3 + ((33 × (3 × 33 − 3)) − 3)
:= 3 + (33 × (3 × 33 − 3))
:= 3 × 3 + ((33 × (3 × 33 − 3)) − 3/3)
:= (44 × ((4 × 4 × 4 + 4) + 4)) − (4 + 4)/4
:= 4 + ((44 × ((4 × 4 × 4 + 4) + 4)) − 4/4)
:= 5 + ((55/5 + 55 ) + 5 × 5)
:= 55 + ((5/5 − 5 − 5) + 5 )
:= 55 + ((5/5 − 5) + 55 )
:= (66 × (6 × 6 + 6 + 6)) − (6 + 6)/6
:= (6 × 6/(6 + 6)) + (66 × (6 × 6 + 6 + 6))
:= ((6 + 6)/6 + 6) × (6 × 66 + 6/6)
:= 7 + ((77 × (7 × 7 − (7/7 + 7))) + ((7 + 7)/7))
:= 777 + (7 × 7 × 7 × 7 − 7)
:= 777 + (7 × 7 × 7 × 7 − ((7 + 7)/7))
:= 88 + ((8 − (8 + 8)/8) × (8 × 8 × 8 + 8/8))
:= 88 + ((8 × (8 × (8 × 8 − 8 − 8))) + (88/8))
:= 8 + (8 × ((88 × (8 × 8 + 8))/(8 + 8)))
:= 99 + ((((9 + 9)/9)9 ) × (9 − ((9 + 9 + 9)/9)))
:= 99 + (((9 + 9) × (9 × (9 + 9) + 9)) − 9/9)
:= 9 + (((9 + 9 + 9) × (99 + 9 + 9)) − ((9 + 9)/9))
Ï 3167 := ((1 + 1) × (11 × (1 + 11)1+1 )) − 1
5
Ï 3172 := (1 + 1) × (1 + (1 + (11 × (1 + 11)1+1 )))
:= 4 + ((44 × ((4 × 4 × 4 + 4) + 4)) + 4)
Ï 3177 := 11 + ((1 + 1) × ((11 × (1 + 11)1+1 ) − 1))
:= (22 × ((2 × (2 + 2 + 2))2 )) − 2/2
:= 2 + ((22 × ((2 × (2 + 2 + 2))2 )) + 2)
:= (2/2 + 2) × (((2 × 22 + 2)2 + 2)/2)
:= (33 × (3 × 33 − 3)) − 3/3
:= 3 + ((33 × (3 × 33 − 3)) + 3/3)
:= 3 × ((33 × (33 − 3/3)) + 3)
:= (44 × ((4 × 4 × 4 + 4) + 4)) − 4/4
:= 4 + (44 × ((4 × 4 × 4 + 4) + 4))
:= 4 + ((((4/4 + 4)4+4/4 ) + 44) + 4)
:= 5 + ((((5 + 5)/5)5 + 55 ) + 5)
:= 5 + ((5 + 5)/5)5 + 55 + 5 + 5
:= 55 + (((5 + 5)/5 − 5) + 55 )
:= (66 × (6 × 6 + 6 + 6)) − 6/6
:= 6 + ((66 × (6 × 6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + ((66 × (6 × 6 + 6 + 6)) + (6 × 6/(6 + 6)))
:= 777 + (7 × 7 × 7 × 7 − (77/7))
:= 7/7 + ((7 × 7 × 7 × 7 − 7) + 777)
:= 777 + (7 × 7 × 7 × 7 − 7/7)
:= (8 × ((88 × (8 × 8 + 8))/(8 + 8))) − 8/8
:= ((88 × (8 × 8 + 8)) + 8)/((8 + 8)/8)
:= 8 + ((8 × ((88 × (8 × 8 + 8))/(8 + 8))) + 8/8)
:= 9 + (((9 + 9 + 9) × (99 + 9 + 9)) − 9/9)
:= 9 × (9 + 9) × (9 + 9) + (((9 + 9)/9)9−9/9 )
:= 99 + ((9 + 9) × (9 × (9 + 9) + 9))
221
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3178 := (11 × (1 + ((1 + 1) × (1 + 11)1+1 ))) − 1
Ï 3183 := 111 + ((1 + 1 + 1) × (1 + 1)11−1 )
Ï 3188 := (1 + 1) × ((11 × (1 + (1 + 11)1+1 )) − 1)
:= (2 × (2 × (22 − 2))2 ) − 22
:= (2 × (2 × (2 × ((22 − 2)2 − 2)))) − 2/2
:= 2 × ((2 × (2 × ((22 − 2)2 − 2))) + 2)
:= 3 × 3 + ((33 × (3 × 33 − 3)) + 3/3)
:= (3 × (33+3 + 333)) − 3
:= 3 + ((3 × (33+3 + 333)) − 3/3)
:= 4 + (((4 + 4)/4 + 4) × ((4 + 4)/4 + 44))
:= 444/4 + (44 × ((4 + 4) + 4))
:= 4 + (4 × ((4 + 4) × 44 + 444))
:= 55 + (55 − ((5 + 5)/5))
:= 5 + ((55 − ((5 + 5)/5)) + 55)
:= 55 + (55 /5 + 5)/(5 + 5)
:= ((66 − 6)/6) + (66 × (6 × 6 + 6 + 6))
:= ((6 + 6)/6)6 + (((6 − 6/6)6−6/6 ) − 6)
:= 6 + ((((6 + 6)/6 + 6) × (6 × 66 + 6/6)) + 6)
:= 777 + 7 × 7 × 7 × 7
:= 77 × (7 × 7 − 7) − ((7 + 7)/7 + 7 × 7)
:= 7 + (((7 − 7/7) × (7 × 77 − 7)) − (77/7))
:= (8 − 8/8) × ((8 × (8 × 8 − 8) − ((8 + 8)/8)) + 8)
:= 888/8 + (8 × (8 × (8 × 8 − 8 − 8)))
:= 8 + ((((88 × (8 × 8 + 8)) + 8)/((8 + 8)/8)) + 8)
:= 9/9 + (((9 + 9) × (9 × (9 + 9) + 9)) + 99)
:= ((99 × 99 − 9)/((9 + 9 + 9)/9)) − 9 × 9
:= 9 + ((99/9) × (((9 − 9/9) + 9)(9+9)/9 ))
Ï 3184 := 1 + (111 + ((1 + 1 + 1) × (1 + 1)11−1 ))
Ï 3189 := ((1 + 1) × (11 × (1 + (1 + 11)1+1 ))) − 1
4
Ï 3179 := 11 × (1 + ((1 + 1) × (1 + 11)1+1 ))
:= 22/2 × ((22+2 + 2/2)2 )
:= 2 × (2 × (2 × ((22 − 2)2 − 2)))
:= (2 × (2 × (22 − 2))2 ) − 22/2
:= 33/3 + (33 × (3 × 33 − 3))
:= 3/3 + ((3 × (33+3 + 333)) − 3)
:= 3 + (3 × (33+3 + 333))
:= 44/4 + (44 × ((4 × 4 × 4 + 4) + 4))
:= 4 × ((4 + 4) × 44 + 444)
:= 4 × 4 × 4 + ((4/4 + 4)4/4+4 )
:= 55 + (55 − 5/5)
:= 5 + ((55 − 5/5) + 55 )
:= 5 + (((55 − 5/5) + 55 ) + 5)
:= 66/6 + (66 × (6 × 6 + 6 + 6))
:= ((6 + 6)/6 + 6) × (((6 + 6)/6) + 6 × 66)
:= ((6 + 6)/6)6 + ((6 − 6/6)6−6/6 )
:= 7/7 + (7 × 7 × 7 × 7 + 777)
:= 77 × (7 × 7 − 7) − (7/7 + 7 × 7)
:= 77/7 + (7 × 7 × 7 × 7 + 777)
:= (8 + 8) × (888/8 + 88)
:= ((8 + 8) × (8 × (8 + 8 + 8) + 8)) − 88/8
:= 999 + (9 × 9 × (9 + 9 + 9) − ((9 + 9)/9))
:= 999/9 + ((9 + 9) × (9 × (9 + 9) + 9))
:= 88/8 × ((8/8 + 8 + 8)(8+8)/8 )
:= 99/9 × (((9 − 9/9) + 9)
(9+9)/9
)
Ï 3180 := 1 + (11 × (1 + ((1 + 1) × (1 + 11)1+1 )))
Ï 3185 := (1 + 1 + 11) × (1 + ((1 + 1) × (1 + 111+1 )))
Ï 3190 := (1 + 1) × (11 × (1 + (1 + 11)1+1 ))
:= 2 + ((2 × (2 × (22 − 2))2 ) − 22)
:= 2/2 + (2 × (2 × (2 × ((22 − 2)2 − 2))))
:= 2 × 222 + 2222
:= 3 + ((33 × (3 × 33 − 3)) + 3 × 3)
:= (3 × (33+3 + 333)) − 3/3
:= 3 + ((3 × (33+3 + 333)) + 3/3)
:= (44/4 + 4) × (44 − 44)
:= (44 + 4)/4 × ((44 + 4/4) + 4)
:= (44 + 4)/4 + ((4/4 + 4)4+4/4 )
:= 55 + 55
:= 5 + (55 + 55 )
:= 5 + (55 + 55 + 5)
:= 6 + ((66 × (6 × 6 + 6 + 6)) + 6)
:= 66 + (((6 − 6/6)6−6/6 ) − 6)
:= 66 + (((6 − 6/6)6−6/6 ) − 6/6)
:= (7 + 7)/7 + (7 × 7 × 7 × 7 + 777)
:= 7 × (7 × 77 − (77 + 7))
:= ((7 − 7/7) × (7 × 77 − 7)) − (7 + 7)/7
:= 8 + (((88 × (8 × 8 + 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8) × (888/8 + 88))
:= (8 − 88)/8 + ((8 + 8) × (8 × (8 + 8 + 8) + 8))
:= 999/9 + (((9 + 9) × (9 × (9 + 9) + 9)) − 9)
Ï 3181 := 1 + (1 + (11 × (1 + ((1 + 1) × (1 + 11)1+1 ))))
9
:= (99/9 + 9 + 9) × (99/9 + 99)
:= (((9 + 9)/9) ) + 99 × (9 + 9 + 9)
Ï 3186 := (1 + 1) × ((11 × (1 + (1 + 11)1+1 )) − (1 + 1))
Ï 3191 := 1 + ((1 + 1) × (11 × (1 + (1 + 11)1+1 )))
:= 2 + (22/2 × ((22+2 + 2/2)2 ))
:= 2 + (2 × (2 × (2 × ((22 − 2)2 − 2))))
:= 2 + ((2 × (2 × (22 − 2))2 ) − 22/2)
:= 3 + (((33 × (3 × 33 − 3)) + 3 × 3) + 3/3)
:= 3 × (33+3 + 333)
:= 3 + (((3 × (33+3 + 333)) − 3/3) + 3)
:= 4/4 + ((44/4 + 4) × (44 − 44))
:= ((4 + 4)/4 + 4 × 4) × (4 × 44 + 4/4)
:= (4 + 4) × (44 + 4) + 4444/4
:= 55 + (55 + 5/5)
:= 5 + ((55 + 55 ) + 5/5)
:= 55 + (55/5 + 55 )
:= 6 + (((66 × (6 × 6 + 6 + 6)) + 6/6) + 6)
:= 6 + (((66 × (6 × 6 + 6 + 6)) + 6) + 6)
:= 66 + ((6 − 6/6)6−6/6 )
:= ((7 − 7/7) × (7 × 77 − 7)) − 77/7
:= 7/7 + (7 × (7 × 77 − (77 + 7)))
:= ((7 − 7/7) × (7 × 77 − 7)) − 7/7
:= ((8 + 8) × (8 × (8 + 8 + 8) + 8)) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) × (888/8 + 88))
:= ((8 + 8) × (8 × (8 + 8 + 8) + 8)) − (8/8 + 8)
:= ((9 × 9 × 9 × 9 − 9/9)/((9 + 9)/9)) − 99
:= 999 + 9 × 9 × (9 + 9 + 9)
:= ((9 − 9/9) × ((99/9 + 9)(9+9)/9 )) − 9
Ï 3182 := 111 + (((1 + 1 + 1) × (1 + 1)11−1 ) − 1)
Ï 3187 := ((1 + 1) × ((11 × (1 + (1 + 11)1+1 )) − 1)) − 1
Ï 3192 := (1 + 1) × (1 + (11 × (1 + (1 + 11)1+1 )))
:= (2 × ((2 × (22 − 2))2 + 2)) − 22
:= (2 × (2 × (22 − 2))2 ) − (22/2 + 2)
:= 2 × (2 × (2 × (22 − 2)2 − 2))
:= 3 + ((33 × (3 × 33 − 3)) + 33/3)
:= 3/3 + (3 × (33+3 + 333))
:= 3 + ((3 × (33+3 + 333)) + 3)
:= (44 − 4/4) × (((44 − 4) + 44)/4)
:= 4 + ((44 × ((4 + 4) + 4)) + 444/4)
:= (4 + 4) × (4 × 44 − (4/4 + 4)4 )
5
5
:= 55 + ((5 + 5)/5 + 5 )
:= 5 + (((5 + 5)/5 + 55) + 5 )
:= 55 + (((55 + 5)/5) + 55 )
:= 6 + (((6 + 6)/6 + 6) × (6 × 66 + 6/6))
:= 6 + ((((66 × (6 × 6 + 6 + 6)) + 6/6) + 6) + 6)
:= (6/6 + 6) × ((6 × 66 − 6) + 66)
:= 7 + (((7 − 7/7) × (7 × 77 − (77/7))) + 7)
:= (7 + 7)/7 + (7 × (7 × 77 − (77 + 7)))
:= (7 − 7/7) × (7 × 77 − 7)
(8+8)/8
:= ((8 + 8)/8) × ((888 − 8/8) + 8 × 88)
:= 8 + (88/8 × ((8/8 + 8 + 8)
:= ((9 × 9 × 9 × 9 + 9/9)/((9 + 9)/9)) − 99
:= 9/9 + (9 × 9 × (9 + 9 + 9) + 999)
222
))
:= (8 × 8 − 8) × ((8/8 − 8) + 8 × 8)
:= ((99 + 9)/9 + 9) × (9 × (9 + 9) − (9/9 + 9))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3193 := 1 + ((1 + 1) × (1 + (11 × (1 + (1 + 11)1+1 ))))
Ï 3198 := (1 + 1) × ((1 + 1 + 11) × (1 + (1 + 111+1 )))
Ï 3203 := 1 + (1 + (11 × (1 + ((1 + 1) × (1 + (1 + 11)1+1 )))))
:= 2/2 + (2 × (2 × (2 × (22 − 2)2 − 2)))
:= (2 × (2 × (22 − 2))2 ) − 2
:= 2 + ((2 × (2 × (22 − 2))2 ) + 2/2)
:= 3 + (((3 × (33+3 + 333)) + 3/3) + 3)
:= 3 + ((33 × (3 × 33 − 3)) + 33 )
:= (3 × 33 × 33) − ((3/3 + 3)3 )
:= 4 + (((4/4 + 4)4+4/4 ) + 4 × 4 × 4)
:= (4 × ((4 × 4 + 4) × (44 − 4))) − (4 + 4)/4
:= 4 + ((4 × ((4 × 4 + 4) × (44 − 4))) − 4/4)
:= 5 + ((55 /5 + 5)/(5 + 5) + 55 )
:= 55 + ((5 × (5 + 5 + 5)) − ((5 + 5)/5))
:= 5 × 5 + ((55 − ((5 + 5)/5)) + 55)
:= 6 × 6 + ((66 × (6 × 6 + 6 + 6)) − (66/6))
:= 6 × 6 + ((66 × (6 × 6 + 6 + 6)) − 6)
:= 6 + ((((6 − 6/6)6−6/6 ) + 66) + 6)
:= 7/7 + ((7 − 7/7) × (7 × 77 − 7))
:= (7 − 7/7) × ((7 × 77 − 7) + 7/7)
:= 77/7 + ((7 − 7/7) × (7 × 77 − 7))
:= 8/8 + ((8 × 8 − 8) × ((8/8 − 8) + 8 × 8))
:= ((8 + 8) × (8 × (8 + 8 + 8) + 8)) − (8 + 8)/8
:= 88/8 + ((8 × 8 − 8) × ((8/8 − 8) + 8 × 8))
:= 9 + ((9 × 9 × (9 + 9 + 9) − ((9 + 9)/9)) + 999)
:= 9 + (((9 + 9) × (9 × (9 + 9) + 9)) + 999/9)
:= 9 + ((99 × (9 + 9 + 9) + ((9 + 9)/9)9 ) + 9)
Ï 3194 := (1 + 1) × (1 + (1 + (11 × (1 + (1 + 11)1+1 ))))
:= (2 × ((2 × (22 − 2))2 − 2)) − 2
3
:= 3 + ((33 × (3 × 33 − 3)) − 3/3)
:= 4 + (((4/4 + 4)
4+4/4
4
) + (4 + 4)/4)
Ï 3199 := ((1 + 1) × (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 )) − 1
:= (2 × (2 × (22 − 2))2 ) − 2/2
Ï 3204 := (1 + 11) × (1 + ((1 + 1) × (1 + (11 × (1 + 11)))))
:= 2 × ((2 × (22 − 2))2 + 2)
3
:= (((((3 × (3 + 3)) + 3) ) + 333) + 3)/3
:= 3 + ((33 × (3 × 33 − 3)) + 33)
:= (4 × ((4 × 4 + 4) × (44 − 4))) − 4/4
:= 4 + (4 × ((4 × 4 + 4) × (44 − 4)))
5
:= 5 + ((((55 − 5/5) + 55 ) + 5) + 5)
:= 5 + ((5 × (5 + 5 + 5)) − 5/5)
:= 5 × 5 + ((55 − 5/5) + 55 )
:= 6 + (((((6 + 6)/6 + 6) × (6 × 66 + 6/6)) + 6) + 6)
:= 6 × 6 + (((66 × (6 × 6 + 6 + 6)) − 6) + 6/6)
:= 6 × ((66 × ((6 + 6)/6 + 6)) + 6)
:= (7 + 7)/7 + ((7 − 7/7) × (7 × 77 − 7))
:= 7 + ((7 − 7/7) × (7 × 77 − 7))
:= (7 − 7/7) × (((7 + 7)/7 − 7) + 7 × 77)
:= (8 + 8)/8 + ((8 × 8 − 8) × ((8/8 − 8) + 8 × 8))
:= ((8 + 8) × (8 × (8 + 8 + 8) + 8)) − 8/8
:= (8/8 + 8) × ((8 × 88 + 8)/((8 + 8)/8))
:= 9 + (99 × (9 + 9 + 9) + ((9 + 9)/9)9 )
:= 9 + ((99/9 + 9 + 9) × (99/9 + 99))
:= 9 × (((9 × 9 × 9 × 9 + 9)/(9 + 9)) − 9)
Ï 3195 := 1 + ((1 + 1) × (1 + (1 + (11 × (1 + (1 + 11)1+1 )))))
Ï 3200 := (1 + 1) × (((1 + 1) × ((1 + 1) × (11 − 1)))1+1 )
Ï 3205 := 1 + ((1 + 11) × (1 + ((1 + 1) × (1 + (11 × (1 + 11))))))
:= (2 × ((2 × (22 − 2))2 − 2)) − 2/2
:= 2 × (2 × (22 − 2))2
:= 2/2 + (2 × ((2 × (22 − 2))2 + 2))
:= 3 × ((33+3 + 333) + 3)
:= (33 − 3/3) × (3 × 33 + 3/3)
:= ((3/3 + 3)3+3 ) − (33 × 33 )
:= (4/4 + 4) × ((4 × (4 × (44 − 4))) − 4/4)
:= 4 × ((4 × 4 + 4) × (44 − 4))
:= (4/4 + 4) × ((4/4 + 4)4 + 4 × 4)
:= 5 + ((55 + 55 + 5) + 5)
5
:= 5 × 5 + (55 + 55 )
:= 5 + (5 × (5 + 5 + 5))
6
:= 6 + (((6 − 6/6)6−6/6 ) + ((6 + 6)/6)6 )
:= (6 − 6/6) × (((66 − 6)/6) × ((6 + 6)/6) )
:= 6 × 6 + ((66 × (6 × 6 + 6 + 6)) + 6/6)
:= 77 + (((7 − ((7 + 7)/7))7−(7+7)/7 ) − 7)
:= ((7 + 7)/7)7 × (77/7 + 7 + 7)
:= 7 + ((7 − 7/7) × ((7 × 77 − 7) + 7/7))
:= 88/8 + ((8 + 8) × (888/8 + 88))
:= (8 + 8) × (8 × (8 + 8 + 8) + 8)
:= 8 + (8 × 8 × 8 × 8 − (888 + 88/8))
:= 9 + (9 × 9 × (9 + 9 + 9) + 999)
Ï 3196 := (1 + 1) × (1 + (1 + (1 + (11 × (1 + (1 + 11)1+1 )))))
:= (9 − 9/9) × ((99/9 + 9)
(9+9)/9
:= ((9 − 9/9) × ((9 + 9)/9)9 ) − 9 × 99
)
Ï 3201 := 11 × (1 + ((1 + 1) × (1 + (1 + 11)1+1 )))
Ï 3206 := ((1 + 1 + 1) × (1111 − (1 + 1))) − 111+1
:= 2 × ((2 × (22 − 2))2 − 2)
:= 2/2 + (2 × (2 × (22 − 2))2 )
:= 2 + (2 × ((2 × (22 − 2))2 + 2))
:= 33 + ((33 × (3 × 33 − 3)) + 3/3)
:= 33 + (33 × (3 × 33 − 3))
:= 3 + ((3 × 33 × 33) − ((3/3 + 3)3 ))
:= (4 × ((4 × 4 + 4) × (44 − 4))) − 4
:= 4/4 + (4 × ((4 × 4 + 4) × (44 − 4)))
:= (4 − 4/4)4 + ((4/4 + 4)4+4/4 )
:= 5 + ((55/5 + 55) + 55 )
:= 55 + ((5 × (5 + 5 + 5)) + 5/5)
:= 5 × 5 + ((55 + 55 ) + 5/5)
:= (66/6 + 6 × 6) × (((6 + 6)/6) + 66)
:= 66/6 × ((6 × 66 − 666/6) + 6)
:= 6 × 6 + ((66 × (6 × 6 + 6 + 6)) + ((6 + 6)/6))
:= 77/7 + (7 × (7 × 77 − (77 + 7)))
:= (77 × (7 × 7 + 7)) − 7777/7
:= 7 + (((7 − 7/7) × (7 × 77 − 7)) + 7)
:= ((8 × (888 − 88)) − 8)/((8 + 8)/8)
:= 8/8 + ((8 + 8) × (8 × (8 + 8 + 8) + 8))
:= 8 × 8 × 8 × 8 − (888 + ((8 + 8)/8))
:= 9 + ((9 × 9 × (9 + 9 + 9) + 999) + 9/9)
:= 99/9 × ((999/9 + 99) + 9 × 9)
:= 9 + ((9 × 9 × (9 + 9 + 9) + (99/9)) + 999)
Ï 3197 := (11 − 1)1+1+1 + ((1 + 1 + 11)1+1+1 )
Ï 3202 := 1 + (11 × (1 + ((1 + 1) × (1 + (1 + 11)1+1 ))))
Ï 3207 := ((1 + 1) × (111 + (1 + 1)11 )) − 1111
:= 2/2 + (2 × ((2 × (22 − 2))2 − 2))
:= 2 + (2 × (2 × (22 − 2))2 )
:= 2 + ((2 × ((2 × (22 − 2))2 + 2)) + 2/2)
:= 33/3 + (3 × (33+3 + 333))
:= 3/3 + ((33 × (3 × 33 − 3)) + 33)
:= 3333 − (3 × 33 + 33 )
:= 4/4 + ((4 × ((4 × 4 + 4) × (44 − 4))) − 4)
:= (4 + 4)/4 + (4 × ((4 × 4 + 4) × (44 − 4)))
:= (4 − 4/4) × ((4 × 44 + 4/4) + 44)
:= 5 + ((((55 + 5)/5) + 55 ) + 55)
:= 55 + ((55 + 55)/5 + 55 )
:= 5 × 5 + (((5 + 5)/5 + 55) + 55 )
:= 6 + (((6 − 6/6)6−6/6 ) + 66)
:= 6 + ((66/6 + 6 × 6) × (((6 + 6)/6) + 66))
:= 666/6 + ((6 + 6) × (6 × (6 × 6 + 6) + 6))
:= 7 + (((7 − 7/7) × (7 × 77 − 7)) − ((7 + 7)/7))
7−(7+7)/7
:= 77 + ((7 − ((7 + 7)/7))
)
:= 7 + (((7 + 7)/7)7 × (77/7 + 7 + 7))
:= 8 × 8 × 8 × 8 − (888 + 88/8)
:= (8 + 8)/8 + ((8 + 8) × (8 × (8 + 8 + 8) + 8))
:= 8 × 8 × 8 × 8 − (888 + 8/8)
:= 99/9 + (9 × 9 × (9 + 9 + 9) + 999)
:= ((99/9 + 9) × (9 × (9 + 9) − 9/9)) − (9 + 9)
:= 99 + (999/9 × ((9/9 + 9 + 9) + 9))
223
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3208 := (111 × (((11 − 1) × (1 + 1 + 1)) − 1)) − 11
2
Ï 3213 := 1 + ((1 + 1) × (11 × (1 + (1 + (1 + 11)1+1 ))))
2
Ï 3218 := (111 × (((11 − 1) × (1 + 1 + 1)) − 1)) − 1
:= 2 × (((2 × (22 − 2)) + 2) + 2)
:= 2 + ((2 × (2 × (22 − 2)) ) + 22/2)
:= 2 + (2 × (2 × (2 × ((22 − 2)2 + 2))))
:= 3333 − ((3 − 3/3 + 3)3 )
:= 3 × (3 × ((333 − 3) + 33 ))
:= 3333 − (((333 + 3)/3) + 3)
:= 4 + ((4 × ((4 × 4 + 4) × (44 − 4))) + 4)
:= 44 + (((4/4 + 4)4+4/4 ) + 44)
:= 444 + (((44 × (44 − 4) + 4) + 4)/4)
:= 5 + (((55 − ((5 + 5)/5)) + 55) + 5 × 5)
:= 5 × 5 + ((55 /5 + 5)/(5 + 5) + 55 )
:= 55 + (5 × 5 × 5 − ((5 + 5)/5)5 )
:= ((6 + 6)/6 + 6) × ((6 × 66 − 6/6) + 6)
:= (666/6 × (6 × 6 − (6/6 + 6))) − 6
:= (6 + 6)/6 + (((6 + 6)/6 + 6) × (6 × 66 + 6))
:= 7 + ((77 × (7 × 7 + 7)) − 7777/7)
:= 77 + ((7 × 7 + 7)(7+7)/7 )
:= 77 × (7 × 7 − 7) − (((7 + 7)/7 + 7) + 7)
:= 8 × 8 × 8 × 8 − 888
:= (8 − 8/8) × (8 × (8 × 8 − 8) + (88/8))
:= 8 + ((8 × 8 × 8 × 8 − 888) + ((8 + 8)/8))
:= (9 + 9 + 9) × ((99/9 + 99) + 9)
:= ((9 + 9)/9) × ((9 × (99 + 9 × 9)) − (99/9))
9
:= (9 − 9/9) × ((((9 + 9)/9) ) − 999/9)
Ï 3209 := ((1 + 1 + 1) × (1111 − 1)) − 111+1
Ï 3214 := (1 + 1) × (1 + (11 × (1 + (1 + (1 + 11)1+1 ))))
2
Ï 3219 := 111 × (((11 − 1) × (1 + 1 + 1)) − 1)
:= 2/2 + (2 × (((2 × (22 − 2)) + 2) + 2))
:= 2 + (22 × (((2 × (2 + 2 + 2)) ) + 2))
:= 2 + ((2 × (2 × (2 × ((22 − 2)2 + 2)))) + 2/2)
:= 3 + (((3 × 33 × 33) − ((3/3 + 3)3 )) + 3)
:= 3/3 + (3 × 3 × 333 + (3 + 3)3 )
:= 33 + (3 × (33+3 + 333))
:= 4 + ((4/4 + 4) × ((4/4 + 4)4 + 4 × 4))
:= 4 + ((44 − 4)/4 × ((44 + 4)/4 + 44 ))
:= 4 + (((4 + 4)4 ) − ((4/4 + 4)4 + 44 ))
:= 5 + (((55 − 5/5) + 55 ) + 5 × 5)
:= 55 + (5 × (5 × 5 − 5) − (55/5))
:= 55 + (5 × (5 × 5 − 5) − (5/5 + 5))
:= 6 + (((((6 − 6/6)6−6/6 ) + 66) + 6) + 6)
:= 6 + (((6 + 6)/6 + 6) × ((6 × 66 − 6/6) + 6))
:= 666/6 × (6 × 6 − (6/6 + 6))
:= 77 × (7 × 7 − 7) − (77/7 + 7 + 7)
:= 7/7 + (((7 × 7 + 7)(7+7)/7 ) + 77)
:= 77 × (7 × 7 − 7) − (7/7 + 7 + 7)
:= 8/8 + (8 × 8 × 8 × 8 − 888)
:= 8 + (8 × 8 × 8 × 8 − (888 + ((8 + 8)/8)))
:= 88/8 + (8 × 8 × 8 × 8 − 888)
:= 9 + ((9 − 9/9) × ((99/9 + 9)
(9+9)/9
))
Ï 3210 := (1 + 1) × ((11 × (1 + (1 + (1 + 11)1+1 ))) − 1)
2
9
:= 9 + (((9 − 9/9) × ((9 + 9)/9) ) − 9 × 99)
Ï 3215 := ((1 + 1 + 1) × (1 + 1111)) − 111+1
:= 999/9 × (99/9 + 9 + 9)
Ï 3220 := 1 + (111 × (((11 − 1) × (1 + 1 + 1)) − 1))
:= 2 + (2 × (((2 × (22 − 2)) + 2) + 2))
:= 22/2 + (2 × ((2 × (22 − 2))2 + 2))
:= 2 × ((2 × (2 × ((22 − 2)2 + 2))) + 2)
:= (3 + 3)3 + (3 × 3 × 333 − 3)
:= 3 + ((3 × 3 × 333 − 3/3) + (3 + 3)3 )
:= 3333 + (((3 − 333)/3) − 3)
:= ((4 + 4)4 ) − ((4/4 + 4)4 + 44 )
:= 4 + (4 × (((4 × 4 + 4) × (44 − 4)) + 4))
2
4
4
:= (44 − 4)/4 × ((4 + 4)/4 + 4 )
5
:= 55 + (5 × (5 × 5 − 5) − 5)
:= 5 + ((55 + 5 ) + 5 × 5)
:= 5 + (((55 + 55 ) + 5 × 5) + 5)
:= 6 + ((66 × (6 × 6 + 6 + 6)) + 6 × 6)
:= (((6 + 6)/6 + 6) × (6 × 66 + 6)) − 6/6
:= (6/6 + 6) × (((6 + 6)/6)6 + 6 × 66)
:= (7 − 7/7) × ((7 × 77 − (77/7)) + 7)
:= ((7 + 7)/7)7 + 7 × 7 × (7 × 7 + 7 + 7)
:= 77 × (7 × 7 − 7) − (7 + 7)
:= (8 + 8)/8 + (8 × 8 × 8 × 8 − 888)
:= 8 + (8 × 8 × 8 × 8 − (888 + 8/8))
:= 8 + ((88/((8 + 8)/8)) × ((8/8 + 8 × 8) + 8))
:= (999/9 × (99/9 + 9 + 9)) − 9
:= 99 + ((9/9 + 9 + 9) × (((9 + 9)/9) + 9 × (9 + 9)))
:= (99/9 + 9) × (9 × (9 + 9) − 9/9)
Ï 3216 := (1 + 1) × ((1 + 11) × (1 + (1 + (11 × (1 + 11)))))
Ï 3221 := ((1 + 1 + 1) × 1111) − (1 + 111)
Ï 3211 := (((1 + 1) × (11 − 1)) − 1) × (1 + 1 + 11)1+1
2
:= 22/2 + (2 × (2 × (22 − 2)) )
:= 2 × (2 × (2 × ((22 − 2) + 2)))
:= 22 + ((2 × (2 × (22 − 2))2 ) − 2/2)
:= 3 + (3333 − ((3 − 3/3 + 3)3 ))
:= 3 + (3 × 3 × 333 + (3 + 3)3 )
:= 3333 − ((333 + 3)/3)
:= 44/4 + (4 × ((4 × 4 + 4) × (44 − 4)))
:= 4 × (((4 × 4 + 4) × (44 − 4)) + 4)
:= ((44 − 4/4) × ((44 + 44 )/4)) − 4
:= 5 + (555/5 − 5 × 5)
:= 5 + ((555/5 − 5 × 5) + 55 )
:= 55 + ((5 × (5 × 5 − 5) − 5) + 5/5)
:= 6 + (((66 × (6 × 6 + 6 + 6)) + 6 × 6) + 6/6)
:= ((6 + 6)/6 + 6) × (6 × 66 + 6)
:= 6 + ((((6 + 6)/6 + 6) × (6 × 66 + 6)) − 6/6)
:= 7 + ((7 − 7/7) × (((7 + 7)/7 − 7) + 7 × 77))
:= 77 × (7 × 7 − 7) − (77/7 + 7)
:= 7/7 + (77 × (7 × 7 − 7) − (7 + 7))
:= 88/8 + ((8 + 8) × (8 × (8 + 8 + 8) + 8))
:= 8 + (8 × 8 × 8 × 8 − 888)
:= 8 + ((8 − 8/8) × (8 × (8 × 8 − 8) + (88/8)))
:= ((99/9 + 9) × (9 × (9 + 9) − 9/9)) − 9
:= 9 + ((999/9 × ((9/9 + 9 + 9) + 9)) + 99)
:= ((9 + 9) × (99 + 9 × 9)) − (9/9 + 9 + 9)
5
Ï 3212 := (1 + 1) × (11 × (1 + (1 + (1 + 11)1+1 )))
2
2
Ï 3217 := (111 × (((11 − 1) × (1 + 1 + 1)) − 1)) − 1 − 1
Ï 3222 := ((1 + 1 + 1) × 1111) − 111
:= 22 × (((2 × (2 + 2 + 2)) ) + 2)
:= 2/2 + (2 × (2 × (2 × ((22 − 2) + 2))))
:= 22 + (2 × (2 × (22 − 2))2 )
:= (3 + 3)3 + (3 × 3 × 333 − 3/3)
:= 3 + ((3 × 3 × 333 + (3 + 3)3 ) + 3/3)
:= 3333 − 333/3
:= 4/4 + (4 × (((4 × 4 + 4) × (44 − 4)) + 4))
:= 44 × (44 + 4) + (4444 − 4)/4
4
:= 44 × (((4 + 4)/4 + 4) + 4)
5
5
2
:= 55 + (((5 + 5)/5) + 5 )
:= 5 + ((((5 + 5)/5) + 5 ) + 55)
:= 5 + (((((5 + 5)/5)5 + 55 ) + 55) + 5)
:= 6 × 6 + (((6 + 6)/6 + 6) × (6 × 66 + 6/6))
:= 6/6 + (((6 + 6)/6 + 6) × (6 × 66 + 6))
:= 6 + (((6 + 6)/6 + 6) × (6 × 66 + 6))
:= ((7 − 77)/7) + (77 × (7 × 7 − 7) − 7)
:= (7 − 7/7) × (7 × 77 − ((7 + 7)/7))
:= (88/((8 + 8)/8)) × ((8/8 + 8 × 8) + 8)
:= 8 + ((8 × 8 × 8 × 8 − 888) + 8/8)
:= 88 + (((8 × 8 − 8)(8+8)/8 ) − ((8 + 8)/8))
:= (((9 + 9)/9)9 ) + ((9 + 9 + 9) × (9/9 + 99))
:= 9 × 9 + (((999 + 9)/(9 + 9))(9+9)/9 )
:= (9 + 9) × ((9 × 9 − 9/9) + 99)
:= 77 + (((7 × 7 + 7)
(7+7)/7
5
) − 7/7)
5
224
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
Ï 3223 := 1 + (((1 + 1 + 1) × 1111) − 111)
Ï 3228 := ((1 + 1 + 1) × (1 + (1 + 1111))) − 111
Ï 3233 := 11 + (((1 + 1 + 1) × 1111) − 111)
:= 22 + ((2 × (2 × (22 − 2)) ) + 2/2)
:= 2 + ((2 × ((2 × (22 − 2)) + 2)) + 22)
:= 22 + ((2 × (2 × (22 − 2))2 ) + 22/2)
:= 3333 + ((3 − 333)/3)
:= 3 + (3333 − (3 × (33 + 3)))
:= 3333 − (3 × 33 + 3/3)
:= 44 × (44 + 4) + 4444/4
:= ((4 + 4) × ((444 − 44) + 4)) − 4
2
5
2
:= 4/4 + ((4 + 4) × ((444 − 44) + 4))
5
:= 55 + ((55 − ((5 + 5)/5)) + 55)
:= 5 + (5 × (5 × 5 − 5) − ((5 + 5)/5))
:= 5 + 5 × (5 × 5 − 5) − (5 + 5)/5 + 5
:= 6 + ((((6 + 6)/6 + 6) × (6 × 66 + 6)) + 6/6)
:= 66 + ((66 × (6 × 6 + 6 + 6)) − 6)
:= 66 + ((66 × (6 × 6 + 6 + 6)) − 6/6)
:= 77 × (7 × 7 − 7) − 77/7
:= 88 + (((8 × 8 − 8)
(8+8)/8
) − 8/8)
:= 9/9 + ((9 + 9) × ((9 × 9 − 9/9) + 99))
Ï 3224 := 1 + (1 + (((1 + 1 + 1) × 1111) − 111))
:= 7/7 + (77 × (7 × 7 − 7) − 7)
:= 77 × (7 × 7 − 7) − 7/7
:= 8 × 8 + (((88 × (8 × 8 + 8)) − 8)/((8 + 8)/8))
:= (((8/8 − 8) + 8 × 8)(8+8)/8 ) − 8 − 8
:= 9 + (999/9 × (99/9 + 9 + 9))
:= (9 + 9)/9 + (((9 + 9) × (99 + 9 × 9)) − 9)
Ï 3229 := 1 + (((1 + 1 + 1) × (1 + (1 + 1111))) − 111)
Ï 3234 := (1 + 1) × (((1 + 11)1+1+1 ) − 111)
:= 2 + ((2 × (2 × (22 − 2))2 ) + 22)
:= 2 + ((2222/2) + (2 × 22 + 2)2 )
:= 22 × (((22/2 + 2)2 ) − 22)
:= 3 + (3333 − ((333 + 3)/3))
:= (3 × 33 × 33) − (33/3 + 33 )
:= 33 × (3 × 33 − 3/3)
4
4
:= ((4 − 4 ) + 4) × (4 − (4 × 4 + 4/4))
:= 4 + ((44 − 4/4) × ((44 + 4 )/4))
:= (44 − (4 + 4)/4) × ((4 − 4/4)4 − 4)
:= 55 + (5 × (5 × 5 − 5) − 5/5)
:= 55 + (55 − 5)/(5 × 5 + 5)
:= 55 + ((55 − 5/5) + 55 )
:= ((6 + 6)/6 + 6) × ((6 × 66 + 6/6) + 6)
:= (6 × ((6 + 6 + 6) × (6 × 6 − 6))) − 66/6
:= 66 + (66 × (6 × 6 + 6 + 6))
:= ((7 − 77)/7) + 77 × (7 × 7 − 7)
:= (7 + 7)/7 + (77 × (7 × 7 − 7) − 7)
:= 77 × (7 × 7 − 7)
:= 8 + (((8 − 8/8) × (8 × (8 × 8 − 8) + (88/8))) + 8)
:= (8 − 8/8) × ((8 − 8/8) × (((8 + 8)/8) + 8 × 8))
:= ((9 + 9) × (99 + 9 × 9)) − 99/9
:= (99 − 9/9) × (99/((9 + 9 + 9)/9))
:= 88 + ((8 × 8 − 8)
(8+8)/8
)
:= (9 + 9)/9 + ((9 + 9) × ((9 × 9 − 9/9) + 99))
Ï 3225 := ((1 + 1 + 1) × (1 + 1111)) − 111
Ï 3230 := 11 + (111 × (((11 − 1) × (1 + 1 + 1)) − 1))
Ï 3235 := 1 + ((1 + 1) × (((1 + 11)1+1+1 ) − 111))
:= 2 + (((2