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```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 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, Florian&oacute;polis, 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 &times; 9 = 9 &times; 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
101 := 1 + 2 + 34 + 5 + 6 &times; 7 + 8 + 9
= 9&times;8+7+6+5+4+3&times;2+1
102 := 12 + 3 &times; 4 &times; 5 + 6 + 7 + 8 + 9
= 9 + 8 + 7 + 6 + 5 + 43 + 2 + 1
103 := 1 &times; 2 &times; 34 + 5 + 6 + 7 + 8 + 9
= 9 + 8 + 7 &times; 6 + 5 &times; 4 + 3 + 21
104 := 1 + 23 + 4 + 5 + 6 + 7 &times; 8 + 9
= 9 + 8 + 7 + 65 + 4 &times; 3 + 2 + 1
105 := 1 + 2 &times; 3 &times; 4 + 56 + 7 + 8 + 9
= 9+8&times;7+6&times;5+4+3+2+1
106 := 12 + 3 + 4 &times; 5 + 6 + 7 &times; 8 + 9
= 9+8&times;7+6&times;5+4+3&times;2+1
107 := 1 &times; 23 + 4 + 56 + 7 + 8 + 9
= 9 + 8 + 76 + 5 + 4 + 3 + 2 &times; 1
108 := 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9
= 9 + 8 + 76 + 5 + 4 + 3 + 2 + 1.
See more examples,
999 := 12 &times; 3 &times; (4 + 5) + (67 + 8) &times; 9
2535 := 1 + 2345 + (6 + 7 + 8) &times; 9
= 9 + 8 + 7 + 654 + 321.
= 9 + 87 &times; (6 + 5 &times; 4 + 3) + 2 + 1.
2607 := 123 &times; 4 &times; 5 + 6 + (7 + 8) &times; 9
= 987 + 6 &times; 54 &times; (3 + 2) &times; 1.
p
p
10958 := 12 &times; 3 + 4 + 5! &times; (67 + 8 &times; 9) = (9 + 8 &times; 7 &times; 65 + 4) &times; 3 − 2 + 1.
= − 9 + 8 + 7 &times; (6 + 5) &times; (4 &times; 3)2 &times; 1.
11807 := 1 &times; 234 &times; (5 + 6 &times; 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 &times; (222 + 2) + 2/2
:= 3(3+3) − 3 − 3 &times; 3
:= 3 &times; 333 − 3 − 3/3
:= 4 &times; (4 &times; 44 + 4) − 4 + 4/4
:= 4 &times; (44 − 4 − 4) + 4 − 4/4
:= (55 &times; (55 + 5 + 5) + 5 + 5)/5
:= 5 &times; (5 + 5) &times; (5 &times; 5 − 5) − 5
:= (6 &times; 6/(6 + 6))6 − 6 − 6
:= 666 + 6 &times; 66 − 66 − 6/6
:= 777 − 7 &times; 7 − 77/7
:= (7 + 7) &times; (77 − 7) + 7 + 7 + 7/7
:= 8 &times; 88 + (88 + 8 + 8)/8
:= 888 + 88 + 8 + 88/8
:= 9 &times; 9 &times; 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 &times; (222 + 2(2+2) )
:= 33 &times; (33 − 3) − 3 − 3
:= (3 &times; 3 + 3/3)3
:= 4 &times; (4 &times; (44 + 4) + 4) + (4 + 4)/4
:= 4 &times; (44 − 4) − 4 − 4
:= 5 + (55 − 5/5)/(5 − 5/5)
:= 5 &times; (5 + 5) &times; (5 &times; 5 − 5)
:= 66 &times; (6 + 6) − 6
:= ((66 − 6)/6)(6&times;6/(6+6))
:= 777 + 7 + (7 + 7)/7
:= (7 + 7 + 7 − 7/7) &times; (7 &times; 7 + 7/7)
:= 8 &times; (88 + 8) + 8 + (88 − 8)/8
:= 888 + 88 + 8 + 8 + 8
:= 9 &times; 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 &quot;a&quot;, for example,
• Running-Type
5 := (aa − a)/(a + a)
1991 := (aaaaaa/aaa &times; (a + a) − aa)/a
6 := (aa + a)/(a + a)
2020 := (aaaaa − a)/aa &times; (a + a)/a
55 := (aaa − a)/(a + a)
2035 := (aaaa − a)/(a + a + a) &times; aa/(a + a)
56 := (aaa + a)/(a + a)
4477 := (aaa/(a + a + a) &times; aa &times; aa)/(a &times; a)
561 := (aaaa + aa)/(a + a)
4999 := (aaaaa − aaaa − a − a)/(a + a)
666 := aaa &times; (aa + a)/((a + a) &times; 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 &times; (aa + a)
666 :=
(a + a) &times; a
(aa + a) &times; aa
− a) &times; (aa + a)
a
786 :=
(a + a) &times; a
aaaaa − aa
925 :=
aa + a
aaaa − aa − aa
1089 :=
a
aaaaaa
&times; (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
&times; (a + a)
aa
2020 :=
a
aaaa − a
&times; aa
a
+
a
+
a
2035 :=
a+a
aaa
&times; aa &times; aa
a
+
a
+
a
4477 :=
a&times;a
(aaaaa − aaaa − a − a)
(a + a)
(aaaaa − aaaa)
5000 :=
(a + a)
(aaaa − a − a − a) &times; aaa
122988 :=
.
a&times;a
4999 :=
where a ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, and aa = 10 &times; a + a, aaa = 102 &times; a + 10 &times; 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 &times; 9)
98/7 + 6 = 54/3 + 2 &times; 1.
Below are more examples, written in increasing and decreasing ways:
• Increasing Order
12 = 3 + 4 + (5 &times; 6 + 7 + 8)/9
123 = 4 + 5 + 6 &times; 7 + 8 &times; 9
1234 = −5 + 6! + 7 + 8
12 + 3 &times; 4 + 5 &times; (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 &times; (6 + 5) &times; (4 − 3) = 21
p
9 &times; 87 + 6 + 54 = 321
(2)
9 − 8 + 7! − 6 &times; 5! = 4321
9 − 8 + 7 − 6 + 5 + 4 − 3 + 2 = 10
9 &times; (8 + 7) + 6 + 5 + 43 = 210
(9 − 87 + 6!) &times; 5! /4! = 3210
98 = (7 + 6) &times; 5 + 4 &times; 3 + 21
987 = 6! + 5! + (4 + 3) &times; 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 &times; 4 = 5 + 6 + (7 + 8)/ 9
:= (9 + 87)/6 = 5 + 4 + 3 &times; 2 + 1
p
p
18 = 12 + 3!
= 4+5&times;6 = 7+8+ 9
p
p
= 9 + 8 + 7 = 6 &times; 54 = −3 + 21 = 3! + 2 + 10
p
120 := (1 &times; 2 + 3)!
= 4 &times; 5 &times; 6 = ((7 + 8)/ 9)!
p
:= (( 9)! − 8 + 7)! = 6 &times; 5 &times; 4 = (3 &times; 2 − 1)! = 3! &times; 2 &times; 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) &times; F (4) &times; F (5) + 6 − 7 − 8 − 9
(4)
123 = −4 &times; 5 &times; (6 − F (7)) − 8 − 9
1234 = 5 &times; F (6) &times; F (7) + F (8) &times; F (9)
1 + F (23 + F (4)) + (5 − 6)7 = 89
1 &times; 2 &times; 34 &times; 5 − F (F (6)) = 789
1 + 23 + F (4 &times; 5) = 6789.
• Decreasing Order
9 + (−F (8)/7 + 6) &times; 5 − F (4)! + 3 = 21
(5)
−98 − F (7) + F (6) &times; 54 = 321
(F (9) &times; F (8) + 7) &times; 6 − 5 = 4321
98 = (7 − 6) &times; 5 + F (4) &times; (32 − 1)
987 = (6 − 5) &times; F (4 &times; (3 + 2 − 1))
98 = −5 − 4 − 3 + 2 &times; F (10)
987 = (6 − 5)4 &times; F (3 &times; 2 + 10)
p
9876 = ( 5 + 4)! + F (F (3!) &times; 2) &times; 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 &quot;C &quot; represents as &quot;binomial coefficient&quot;.
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) &times; 9
(6)
123 = (−4 + 5) &times; 6 + T (7) + 89
1234 = T (56 &times; 7/8) + 9
1 + 2 + T (3) &times; 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 &times; 8 − T (7) − T (6) + 5 − 4 − 3 = 21
(7)
T (9 + 8) − 7 &times; 6 + T (5 &times; 4) = 321
(−T (9) + T (T (8))) &times; 7 − T (6) − 5 = 4321
98 = (7 − 6) &times; 54 − T (32) + 1
987 = T (6) &times; (5 &times; T (4) − 3) &times; (2 − 1)
9876 = T (5 &times; T (4 + 3)) + T (2 + 1)
98 = (7 − 6) &times; T (5) − 4 + 32 + T (10)
(8)
987 = (6 − 5) &times; 4 &times; (T (T (T (3))) + 2) + T (10)
9876 = (−5 + T (T (T (4)))) &times; T (3) + T (T (−2 + 10))
9 &times; 8 − T (7) − T (6) − T (5) − 4 + 3 &times; 2 = 10
T (9) + 87 &times; (6 − 5) + T (4 &times; 3) = 210
T (9) + 8 + 7 + T (6) &times; T (5) &times; 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.
&Iuml; 1 := 1
&Iuml; 9 := 11 − 1 − 1
&Iuml; 5 := 1 + 1 + 1 + 1 + 1
:= 2/2
:= 2 + 2/2 + 2
:= (2/2 + 2)2
:= 3/3
:= 3 + 3 − 3/3
:= 3 &times; 3
:= 4/4
:= 4 + 4/4
:= 4 + 4 + 4/4
:= 5/5
:= 5
:= 5 + 5 − 5/5
:= 6/6
:= 6 − 6/6
:= 6 + 6 &times; 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 &times; 9 + 9)/(9 + 9)
:= 9
&Iuml; 2 := 1 + 1
&Iuml; 6 := (1 + 1) &times; (1 + 1 + 1)
&Iuml; 10 := 11 − 1
:= 2
:= 2 + 2 + 2
:= 2 + 2 &times; (2 + 2)
:= 3 − 3/3
:= 3 + 3
:= 3 &times; 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
&Iuml; 3 := 1 + 1 + 1
&Iuml; 7 := 1 + (1 + 1) &times; (1 + 1 + 1)
&Iuml; 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 &times; 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
&Iuml; 4 := 1 + 1 + 1 + 1
&Iuml; 8 := (1 + 1)1+1+1
&Iuml; 12 := 1 + 11
:= 2 + 2
:= 2 &times; (2 + 2)
:= 2 &times; (2 + 2 + 2)
:= 3 + 3/3
:= 3 &times; 3 − 3/3
:= 3 + 3 &times; 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 &times; 8/(8 + 8)
:= 8
:= (88 + 8)/8
:= (9 &times; 9 − 9)/(9 + 9)
:= 9 − 9/9
:= (99 + 9)/9
10
Inder J. Taneja
&Iuml; 13 := 1 + 1 + 11
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 18 := (1 + 1) &times; (11 − 1 − 1)
2+2
&Iuml; 23 := 1 + 11 + 11
:= 22 + 2/2
:= 2 + 22/2
:= 2 + 2
:= 3 + 3 &times; 3 + 3/3
:= 3 &times; (3 + 3)
:= 33 − 3 − 3/3
:= 4 + 4/4 + 4 + 4
:= 4 &times; 4 + (4 + 4)/4
:= 4 + 4 + 4 + 44/4
:= (55 + 5 + 5)/5
:= 5 + (55 + 5 + 5)/5
:= 5 &times; 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
&Iuml; 14 := 1 + 1 + 1 + 11
2+2
&Iuml; 19 := (1 + 1) &times; (11 − 1) − 1
&Iuml; 24 := (1 + 1) &times; (1 + 11)
:= 22 − 2/2 − 2
:= 2 + 22
:= 3 + 33/3
:= 3/3 + 3 &times; (3 + 3)
:= 33 − 3
:= 4 + (44 − 4)/4
:= 2
−2
:= 4 + 4 + 44/4
:= 4 + (4 &times; 4 + 4)
:= 5 + 5 − 5/5 + 5
:= 5 &times; 5 − 5 − 5/5
:= 5 &times; 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 &times; 9 + 9)/(9 + 9)
:= 9 + 9/9 + 9
:= 9 + ((9 − ((9 + 9 + 9)/9)) + 9)
&Iuml; 15 := 1 + 1 + 1 + 1 + 11
&Iuml; 25 := 1 + (1 + 1) &times; (1 + 11)
&Iuml; 20 := (1 + 1) &times; (11 − 1)
:= 2 + 2 + 22/2
:= 22 − 2
:= 2 + 22 + 2/2
:= 3 + 3 + 3 &times; 3
:= 3 &times; 3 + 33/3
:= 33 − 3 + 3/3
:= 4 + 44/4
:= 4 + 4 &times; 4
:= 4 + 4 + 4 &times; 4 + 4/4
:= 5 + 5 + 5
:= 5 &times; 5 − 5
:= 5 &times; 5
:= 6 + 6 + 6 &times; 6/(6 + 6)
:= 6 + ((6 + 6)/6 + 6 + 6)
:= 6 &times; 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
&Iuml; 16 := (1 + 1)1+1+1+1
:= 22+2
&Iuml; 26 := (1 + 1) &times; (1 + 1 + 11)
&Iuml; 21 := 11 + 11 − 1
:= 22 − 2/2
:= 2 + 22 + 2
:= 3 − 33/3
:= 3 + 3 &times; (3 + 3)
:= 33 − 3/3
:= 4 &times; 4
:= 4 + 4 &times; 4 + 4/4
:= 4 + 44 &times; 4/(4 + 4)
:= 5 + 55/5
:= 5 + 55/5 + 5
:= 5 &times; 5 + 5/5
:= 6 + (66 − 6)/6
:= 6 &times; (6 &times; 6 + 6)/(6 + 6)
:= 6 &times; 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
&Iuml; 17 := 1 + (1 + 1)1+1+1+1
&Iuml; 27 := (1 + 1 + 1)1+1+1
&Iuml; 22 := 11 + 11
:= 2/2 + 22+2
:= 22
:= 2 + 22 + 2/2 + 2
:= 3 + 33/3 + 3
:= 33 − 33/3
:= 33
:= 4 &times; 4 + 4/4
:= 44 &times; 4/(4 + 4)
:= 4 &times; 4 + 44/4
:= 5 + (55 + 5)/5
:= (55 + 55)/5
:= 5 &times; 5 + (5 + 5)/5
:= 6 + 66/6
:= (66 + 66)/6
:= 66 &times; 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
&Iuml; 28 := 1 + (1 + 1 + 1)1+1+1
&Iuml; 33 := 11 &times; (1 + 1 + 1)
:= 2 + 22 + 2 + 2
:= 22 + 22/2
:= 2 + (2 + 2 + 2)2
:= 3 + 3/3
:= 33
:= 33 + 33/3
:= 44 − 4 &times; 4
:= 4/4 + 4 &times; (4 + 4)
3
:= 5 + 5 &times; 5 − (5 + 5)/5
&Iuml; 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 &times; 6/(6 + 6)
:= 6 &times; 6 + (6 + 6)/6
:= 7 + 7 + 7 + 7
:= 7 + 7 + 7 + (77 + 7)/7
:= 7 &times; 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 &times; 9/(9 + 9 + 9)
:= 9 + 9 + 9 + 99/9
&Iuml; 29 := (11 − 1) &times; (1 + 1 + 1) − 1
&Iuml; 34 := 1 + 11 &times; (1 + 1 + 1)
&Iuml; 39 := (1 + 1 + 1) &times; (1 + 1 + 11)
:= 2 + 2 + 2 + 22 + 2/2
:= 2 + 2 &times; 22+2
:= 2 &times; (22 − 2) − 2/2
:= 3 + 33 − 3/3
:= 3/3 + 33
:= 3 + 3 + 33
:= 44 + 4/4 − 4 &times; 4
:= 44 + (4 − 44)/4
:= 44 − 4 − 4/4
:= 5 + 5 &times; 5 − 5/5
:= 5 + 5 + 5 &times; 5 − 5/5
:= 55 − 5 − 55/5
:= 6 &times; 6 − 6 − 6/6
:= 6 &times; 6 − (6 + 6)/6
:= 6 + 66 &times; 6/(6 + 6)
:= 7 + 7 + 7 + 7 + 7/7
:= 7 &times; 7 − 7 − 7 − 7/7
:= 7 &times; 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
&Iuml; 30 := (11 − 1) &times; (1 + 1 + 1)
:= 22 + 2 &times; (2 + 2)
3
&Iuml; 35 := 1 + 1 + 11 &times; (1 + 1 + 1)
&Iuml; 40 := (1 + 1) &times; (1 + 1) &times; (11 − 1)
:= 2 + 22/2 + 22
:= 2 &times; (22 − 2)
:= 3 + 33 − 3/3
:= 3 + 3 + 33 + 3/3
:= 4 &times; (4 + 4) − (4 + 4)/4
:= 4 + 4 &times; (4 + 4) − 4/4
:= 44 − 4
:= 5 + 5 &times; 5
:= 5 + 5 + 5 &times; 5
:= 5 + 5 + 5 + 5 &times; 5
:= 6 &times; 6 − 6
:= 6 &times; 6 − 6/6
:= 6 + 6 &times; 6 − (6 + 6)/6
:= 7 &times; 7 − 7 − (7 + 7)/7
:= 3 + 3
:= 7 + 7 + 7 + 7 + (7 + 7)/7
:= 7 &times; 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 &times; 9 &times; 9 − 9)/(9 + 9)
&Iuml; 31 := 1 + (11 − 1) &times; (1 + 1 + 1)
:= 22 + (2/2 + 2)
2
3
&Iuml; 36 := (1 + 1 + 1) &times; (1 + 11)
:= (2 + 2 + 2)
2
&Iuml; 41 := 1 + (1 + 1) &times; (1 + 1) &times; (11 − 1)
:= 2/2 + 2 &times; (22 − 2)
:= 3 + 3 + 3/3
:= 3 + 33
:= 3 + 33/3 + 33
:= 4 &times; (4 + 4) − 4/4
:= 4 + 4 &times; (4 + 4)
:= 44 − 4 + 4/4
:= 5 + 5 &times; 5 + 5/5
:= 5 &times; 5 + 55/5
:= 5 + 5 &times; 5 + 55/5
:= 6 &times; 6 + 6/6 − 6
:= 6 &times; 6
:= 6 + 6 &times; 6 − 6/6
:= 7 &times; 7 − 7 − 77/7
:= 7/7 + (7 &times; 7 − (7 + 7))
:= 7 &times; 7 − 7 − 7/7
:= 8 + 8 + 8 + 8 − 8/8
:= (8 &times; 8 + 8) &times; 8/(8 + 8)
:= 8 + 8 + 8 + 8 + 8 + 8/8
:= 9 + (99 + 99)/9
:= 9 + 9 + 9 + 9
:= (9 &times; 9 &times; 9 + 9)/(9 + 9)
&Iuml; 32 := 11 &times; (1 + 1 + 1) − 1
:= 2 &times; 2
2+2
:= 33 − 3/3
&Iuml; 37 := 111/(1 + 1 + 1)
&Iuml; 42 := (1 + 1) &times; (11 + 11 − 1)
:= 2/2 + (2 + 2 + 2)2
:= 2 &times; 22 − 2
:= 3 + 33 + 3/3
:= 3 &times; 3 + 33
:= 4 + 4 &times; (4 + 4) + 4/4
:= 4 &times; (4 + 4)
5
:= 44 − (4 + 4)/4
:= 5 + ((5 + 5)/5)5 + 5
5
:= ((5 + 5)/5)
:= 5 + ((5 + 5)/5)
:= 6 &times; 6 + (6 + 6)/6 − 6
:= 6 &times; 6 + 6/6
:= 6 + 6 &times; 6
:= 7 + 77/7 + 7 + 7
:= 7 &times; 7 − (77 + 7)/7
:= 7 &times; 7 − 7
:= 8 + 8 + 8 + 8
:= 888/(8 + 8 + 8)
:= 8 &times; 8 − (88 + 88)/8
:= 9 + (99 + 99 + 9)/9
:= 9 + 9 + 9 + 9 + 9/9
:= 9 + 99 &times; 9/(9 + 9 + 9)
12
Inder J. Taneja
&Iuml; 43 := (1 + 1) &times; (11 + 11) − 1
:= 2 &times; 22 − 2/2
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 48 := (1 + 1) &times; (1 + 1) &times; (1 + 11)
&Iuml; 53 := (111 − 1)/(1 + 1) − 1 − 1
:= 2/2 + 2 &times; (22 + 2 + 2)
:= 2 &times; (22 + 2)
:= 3 &times; 3 + 3/3 + 33
:= 3 &times; 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 &times; 6 + 6/6
:= 6 + 6 + 6 &times; 6
:= 6 + 6 &times; 6 + 66/6
:= 7 &times; 7 + 7/7 − 7
:= 7 &times; 7 − 7/7
:= 7 &times; 7 − 7 + 77/7
:= 8 + 8 + 8 + 8 + 88/8
:= 8 &times; 8 − 8 − 8
:= 8 &times; 8 − 88/8
:= 9 + 9 + 9 + 9 + 9 − (9 + 9)/9
:= 9 + 9 + 9 + 9 + (99 + 9)/9
:= 9 &times; 9 − 9 − 9 − 9 − 9/9
&Iuml; 44 := (1 + 1) &times; (11 + 11)
:= 2 &times; 22
&Iuml; 49 := 1 + (1 + 1) &times; (1 + 1) &times; (1 + 11)
:= 2/2 + 2 &times; (22 + 2)
&Iuml; 54 := (111 − 1)/(1 + 1) − 1
:= 2 + 2 &times; (22 + 2 + 2)
3
:= 33 + 33/3
:= 3 + 33 − 33/3
:= 3 &times; 3 &times; (3 + 3)
:= 44
:= 4 + 44 + 4/4
:= 44 + (44 − 4)/4
:= 55 − 55/5
:= 55 − 5 − 5/5
:= 55 − 5/5
:= 6 + 6 &times; 6 + (6 + 6)/6
:= 6 + 6 + 6 &times; 6 + 6/6
:= 66 − 6 − 6
:= 7 &times; 7 − 7 + (7 + 7)/7
:= 7 &times; 7
:= 7 + 7 &times; 7 − (7 + 7)/7
:= 88 &times; 8/(8 + 8)
:= 8 &times; 8 − 8 − 8 + 8/8
:= 8 &times; 8 + (8 − 88)/8
:= 9 + 9 + 9 + 9 + 9 − 9/9
:= (9 &times; 99 − 9)/(9 + 9)
:= 9 &times; 9 − 9 − 9 − 9
&Iuml; 45 := 1 + (1 + 1) &times; (11 + 11)
&Iuml; 50 := (11 − 1)1+1 /(1 + 1)
&Iuml; 55 := (111 − 1)/(1 + 1)
:= 2 &times; 22 + 2/2
:= 2 + 2 &times; (22 + 2)
:= 22/2 + 2 &times; 22
:= 3 + 3 &times; 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 &times; (5 + 5)
:= 55
:= 666/6 − 66
:= 6 + 6 + 6 &times; 6 + (6 + 6)/6
:= 66 − 66/6
:= 7 + 7 &times; 7 − 77/7
:= 7 &times; 7 + 7/7
:= 7 + 7 &times; 7 − 7/7
:= 8 &times; 8 − 8 − 88/8
:= 8 &times; 8 − 8 − 8 + (8 + 8)/8
:= 8 &times; 8 − 8 − 8/8
:= 9 + 9 + 9 + 9 + 9
:= (9 &times; 99 + 9)/(9 + 9)
:= (999 − 9)/(9 + 9)
&Iuml; 46 := (1 + 1) &times; (1 + 11 + 11)
:= 2 + 2 &times; 22
&Iuml; 51 := 1 + (11 − 1)1+1 /(1 + 1)
:= 2 + 2 &times; (22 + 2) + 2/2
3
3
&Iuml; 56 := (1 + 111)/(1 + 1)
:= 2 &times; (22 + 2 + 2 + 2)
:= 3 + 3 &times; 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 &times; 6 + (66 − 6)/6
:= 6 − 66 + 666/6
:= 66 + (6 − 66)/6
:= 7 &times; 7 − (7 + 7 + 7)/7
:= 7 &times; 7 + (7 + 7)/7
:= 7 + 7 &times; 7
:= 8 &times; 8 − 8 − (88 − 8)/8
:= 8 &times; 8 − (88 + 8 + 8)/8
:= 8 &times; 8 − 8
:= 9 + 9 + 9 + 9 + 9 + 9/9
:= 9 &times; (9 + 9) − 999/9
:= (999 + 9)/(9 + 9)
&Iuml; 47 := 1 + (1 + 1) &times; (1 + 11 + 11)
:= 2 + 2 &times; 22 + 2/2
&Iuml; 52 := (1 + 1) &times; (1 + 1) &times; (1 + 1 + 11)
:= 2 + 22/2 + 2 &times; 22
:= 2 &times; (22 + 2 + 2)
3
&Iuml; 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 &times; 6 + 66/6
:= ((6 + 6)/6) − 6 − 6
:= 66 + (6 + 6 − 66)/6
:= 7 &times; 7 − (7 + 7)/7
:= 7 &times; 7 + (7 + 7 + 7)/7
:= 7 + 7/7 + 7 &times; 7
:= 8 &times; 8 − 8 − 8 − 8/8
:= 8 + 88 &times; 8/(8 + 8)
:= 8/8 + 8 &times; 8 − 8
:= 9 + 9 + 9 + 9 + 99/9
:= 9 &times; 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
&Iuml; 58 := 1 + 1 + (1 + 111)/(1 + 1)
:= 22 + (2 + 2 + 2)
2
&Iuml; 63 := (1 + 1 + 1) &times; (11 + 11 − 1)
:= 2
2+2+2
&Iuml; 68 := (1 + 1) &times; (1 + 11 &times; (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 &times; 4 &times; 4
:= 5 + 55 − (5 + 5)/5
:= (55 /5 + 5)/(5 + 5)
:= 5 + (55 /5 + 5)/(5 + 5)
:= ((6 + 6)/6)6 − 6
:= 66 − 6 &times; 6/(6 + 6)
:= 66 + (6 + 6)/6
:= 7 + 7 &times; 7 + (7 + 7)/7
:= 7 + 7 + 7 &times; 7
:= 77 − 7 − (7 + 7)/7
:= 8 &times; 8 − 8 + (8 + 8)/8
:= 8 &times; 8 − 8/8
:= 8 &times; 8 + 8 &times; 8/(8 + 8)
:= 9 + (9 &times; 99 − 9)/(9 + 9)
:= 9 &times; 9 − 9 − 9
:= 9 &times; 9 − (99 + 9 + 9)/9
&Iuml; 59 := (111+1 − 1)/(1 + 1) − 1
&Iuml; 64 := (1 + 1)(1+1)&times;(1+1+1)
&Iuml; 69 := (1 + 1 + 1) &times; (1 + 11 + 11)
:= 22/2 + 2 &times; (22 + 2)
:= 22+2+2
:= (2/2 + 2) &times; (22 + 2/2)
:= 33 + 33 − 3/3
:= (3/3 + 3)3
:= 3 + 33 + 33
:= (44 − 4)/4 − 4
:= 4 &times; 4 &times; 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 &times; 6/(6 + 6)
:= 7 &times; 7 + (77 − 7)/7
:= 7 + 7 + 7 &times; 7 + 7/7
:= 77 − 7 − 7/7
:= 8 &times; 8 + 88/8 − 8 − 8
:= 8 &times; 8
:= 88 − 8 − 88/8
:= 9 + (9 &times; 99 + 9)/(9 + 9)
:= 9/9 + 9 &times; 9 − 9 − 9
:= 9 &times; 9 − (99 + 9)/9
6
&Iuml; 60 := (111+1 − 1)/(1 + 1)
&Iuml; 65 := 1 + (1 + 1)(1+1)&times;(1+1+1)
&Iuml; 70 := (11 − 1 − 1)1+1 − 11
:= 2 &times; (2 &times; (2 + 2) + 22)
:= 2/2 + 22+2+2
:= 22 + 2 &times; (22 + 2)
:= 33 + 33
:= (3/3 + 3)3 + 3/3
:= 3 + (((3/3 + 3)3 ) + 3)
:= 4 &times; 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 &times; 7 + 77/7
:= 77 − (77 + 7)/7
:= 77 − 7
:= 8 &times; 8 − 8 &times; 8/(8 + 8)
:= 8/8 + 8 &times; 8
:= 8 + 8 &times; 8 − (8 + 8)/8
:= 9 &times; 9 − 9 − (99 + 9)/9
:= 9 + (999 + 9)/(9 + 9)
:= 9 &times; 9 − 99/9
&Iuml; 61 := (1 + 111+1 )/(1 + 1)
2+2+2
&Iuml; 66 := (1 + 1) &times; 11 &times; (1 + 1 + 1)
&Iuml; 71 := (1 + 11)1+1 /(1 + 1) − 1
:= 2 + 22+2+2
:= 2 &times; (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 &times; 7 + (77 + 7)/7
:= 77 − 77/7
:= 7/7 + 77 − 7
:= 2
− 2/2 − 2
:= 8 + 8 &times; 8 − 88/8
:= 8 &times; 8 + (8 + 8)/8
:= 8 + 8 &times; 8 − 8/8
:= 9 &times; 9 − 9 − 99/9
:= (9 + 9) &times; 99/(9 + 9 + 9)
:= 9 &times; 9 − 9/9 − 9
&Iuml; 62 := 1 + (1 + 111+1 )/(1 + 1)
&Iuml; 67 := 1 + (1 + 1) &times; 11 &times; (1 + 1 + 1)
&Iuml; 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 &times; (2 + 2 + 2)2
3
:= 3 &times; (33 − 3)
:= (4 − 4 − 4)/4
:= 4 + (44 − 4)/4
:= 4 + 4 + 4 &times; 4 &times; 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 &times; 7 − 7/7
:= 7 + 7 &times; 7 + 77/7
:= 77 + (7 + 7)/7 − 7
:= 8 &times; 8 − (8 + 8)/8
:= 8 &times; 8 − 8 + 88/8
:= 8 + 8 &times; 8
:= 9 &times; 9 − 9 − 9 − 9/9
:= 9 + 9 + (9 &times; 99 − 9)/(9 + 9)
:= 9 &times; 9 − 9
14
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 73 := 1 + (1 + 11)1+1 /(1 + 1)
:= 2/2 + 2 &times; (2 + 2 + 2)2
&Iuml; 78 := 111 − 11 &times; (1 + 1 + 1)
:= 2 + (2/2 + 2)2+2
:= 2 &times; 2 &times; (22 − 2) − 2
3
:= ((3 + 3)3 + 3)/3
&Iuml; 83 := 1 + 1 + (11 − 1 − 1)1+1
:= 3 + 3 &times; 33 − 3/3
:= 3 &times; 3 − 3
4
:= 4 + (44 + 44 )/4 + 4
:= 4 + (44 + 4)/4 + 4
:= 4 + (4 − 4 + 44)/4
:= 5 &times; (5 + 5 + 5) − (5 + 5)/5
:= 5 &times; 5 + 55 − (5 + 5)/5
:= 5 + 55 + 5 &times; 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 &times; 8
:= 88 + (8 − 88)/8
:= 8 + 8 &times; 8 + 88/8
:= 9/9 + 9 &times; 9 − 9
:= 9 &times; 9 − (9 + 9 + 9)/9
:= 9 &times; 9 + (9 + 9)/9
&Iuml; 74 := (1 + 1) &times; 111/(1 + 1 + 1)
:= 2 + 2 &times; (2 + 2 + 2)
2
&Iuml; 79 := (11 − 1 − 1)1+1 − 1 − 1
:= (2/2 + 2)
2+2
−2
&Iuml; 84 := 1 + 1 + 1 + (11 − 1 − 1)1+1
:= 2 &times; (2 &times; 22 − 2)
3
:= 3 + ((3 + 3) − 3)/3
:= 3 &times; 3 + 3/3 − 3
:= 3 + 3 &times; 33
:= (44 + 44 − 4)/4
:= 4 + (44 + 44 )/4
:= 4 + 4 &times; (4 &times; 4 + 4)
:= 5 &times; (5 + 5 + 5) − 5/5
:= 5 &times; 5 + 55 − 5/5
:= 5 + (55 − 5/5) + 5 &times; 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 &times; 8 + 8 + (8 + 8)/8
:= 88 − 8 − 8/8
:= 88 − 8 &times; 8/(8 + 8)
:= 9 &times; 9 − 9 + (9 + 9)/9
:= 9 &times; 9 − (9 + 9)/9
:= 9 &times; 9 + (9 + 9 + 9)/9
3
&Iuml; 75 := 1 + (1 + 1) &times; 111/(1 + 1 + 1)
&Iuml; 80 := (11 − 1 − 1)1+1 − 1
&Iuml; 85 := 111 − (1 + 1) &times; (1 + 1 + 11)
:= 2 &times; 2 &times; (22 − 2)
:= 2 + 2 + (2/2 + 2)2+2
:= 3 + 3 &times; (33 − 3)
:= 3 &times; 33 − 3/3
:= 3 + 3 &times; 33 + 3/3
:= (44 + 44 )/4
:= 4 &times; (4 &times; 4 + 4)
:= 4 + (4 − 4/4)4
:= 5 &times; (5 + 5 + 5)
:= 5 &times; 5 + 55
:= 5 + 5 &times; 5 + 55
:= 666/6 − 6 &times; 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 &times; 8 + 88/8
:= 88 − 8
:= 8 + 88 − 88/8
:= 9 &times; 9 + (9 + 9 + 9)/9 − 9
:= 9 &times; 9 − 9/9
:= 9 &times; 9 + (9 &times; 9 − 9)/(9 + 9)
:= 22/2 + 2
2+2+2
&Iuml; 76 := (1 + 1) &times; (1 + 111/(1 + 1 + 1))
2
:= 2 &times; ((2 + 2 + 2) + 2)
3
:= 3 + ((3 + 3) + 3)/3
&Iuml; 81 := (11 − 1 − 1)1+1
&Iuml; 86 := (1 + 1) &times; ((1 + 1) &times; (11 + 11) − 1)
:= (2/2 + 2)2+2
:= 2 &times; 2 &times; 22 − 2
:= 3 &times; 33
:= 3 + 3 + 3 &times; 33 − 3/3
4
:= 44 + 4 &times; (4 + 4)
:= (4 − 4/4)
:= 4 + (4 − 4/4)4 + 4/4
:= 5/5 + 5 &times; (5 + 5 + 5)
:= 5 &times; 5 + 55 + 5/5
:= 555/5 − 5 &times; 5
:= 6 + 6 + ((6 + 6)/6)6
:= 6 + 666/6 − 6 &times; 6
:= 6 + 6 + 6 + 66 + (6 + 6)/6
:= 77 − 7/7
:= 77 − 7 + 77/7
:= 7 + (7 + 7)/7 + 77
:= 8 &times; 8 + (88 + 8)/8
:= 8/8 + 88 − 8
:= 88 − (8 + 8)/8
:= 9 &times; 9 + (9 − 99)/(9 + 9)
:= 9 &times; 9
:= 9 &times; 9 + (9 &times; 9 + 9)/(9 + 9)
&Iuml; 77 := 11 &times; (1 + (1 + 1) &times; (1 + 1 + 1))
:= 2 &times; 2 &times; 22 − 22/2
3
&Iuml; 82 := 1 + (11 − 1 − 1)1+1
&Iuml; 87 := 111 − (1 + 1) &times; (1 + 11)
:= 2 + 2 &times; 2 &times; (22 − 2)
:= 2 &times; 2 &times; 22 − 2/2
3
:= 3 + 3 &times; 33 + 3
:= 3 &times; 3 − 3/3 − 3
:= 3/3 + 3 &times; 3
:= (4 − 4/4)4 − 4
:= 4/4 + (4 − 4/4)4
:= 44 + 44 − 4/4
:= 55 + (55 + 55)/5
:= 5 &times; 5 + 55 + (5 + 5)/5
:= 55 + ((5 + 5)/5)5
:= 66 + 66/6
:= 6 + 6 + 6 + ((6 + 6)/6)6
:= 6 + 666/6 − 6 &times; 6 + 6
:= 77
:= 7 + 77 − (7 + 7)/7
:= 77 + (77 − 7)/7
:= 88 − 88/8
:= 88 − 8 + (8 + 8)/8
:= 88 − 8/8
:= 9 &times; 9 + (9 − 9 &times; 9)/(9 + 9)
:= 9/9 + 9 &times; 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
&Iuml; 93 := ((1 + 1)11−1 − 1)/11
&Iuml; 88 := 11 &times; (1 + 1)1+1+1
&Iuml; 98 := 111 − 11 − 1 − 1
:= 2 + 2 &times; 2 &times; (22 + 2)
:= 2/2 + 2 &times; (2 &times; 22 + 2)
:= 2 &times; 2 &times; 22
3
:= 3 &times; 33 − 33/3
:= 3 &times; 33 − 3/3
:= 3 + 3 &times; (3 + 3)
:= 44 + 44
:= 4 + (444 − 4)/4 − 4 &times; 4
4
:= ((4 + 4) − 4)/44
5
:= 5 &times; 5 &times; 5 − ((5 + 5)/5)5
:= 5 &times; (5 &times; 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 &times; (7 + 7)
:= 88
:= 8 + 88 + 8 − 88/8
:= 8 + 88 + (8 + 8)/8
:= 99 − 99/9
:= 9 &times; 9 + (99 + 9)/9
:= 99 − 9/9
:= 5 &times; 5 + (5 /5 + 5)/(5 + 5)
&Iuml; 89 := 111 − 11 − 11
&Iuml; 94 := 1 + ((1 + 1)11−1 − 1)/11
&Iuml; 99 := 11 &times; (11 − 1 − 1)
:= 2 + 2 &times; (2 &times; 22 + 2)
:= (22/2)2 − 22
:= 3 + 3 &times; (33 + 3) + 3/3
:= 3 &times; 33
:= 4 + 4 + (4 − 4/4)
:= (444 − 4)/4 − 4 &times; 4
:= 4 − 4 &times; 4 + 444/4
:= 5 &times; (5 &times; 5 − 5) − 55/5
:= 5 &times; (5 &times; 5 − 5) − 5 − 5/5
:= 5 &times; (5 &times; 5 − 5) − 5/5
:= 6 + 66 + 6 + 66/6
:= 6 &times; 6 + ((6 + 6)/6)6 − 6
:= 666/6 − 6 − 6
:= 77 + (77 + 7)/7
:= 7 + 77 + (77 − 7)/7
:= 7/7 + 7 &times; (7 + 7)
:= 8/8 + 88
:= 8 + 88 − (8 + 8)/8
:= 88 + 88/8
:= 9 + 9 &times; 9 − 9/9
:= 99 + (9 − 99)/(9 + 9)
:= 99
:= 2/2 + 2 &times; 2 &times; 22
:= 3 &times; (33 + 3) − 3/3
4
&Iuml; 90 := (11 − 1) &times; (11 − 1 − 1)
&Iuml; 100 := (11 − 1)1+1
&Iuml; 95 := 111 − (1 + 1)1+1+1+1
:= (2 &times; (2 + 2) + 2)2
:= 2 + 2 &times; 2 &times; 22
:= 222/2 − 22+2
:= 3 &times; (33 + 3)
:= 3 &times; 33 − 3 − 3/3
:= 3/3 + 3 &times; 33
:= 44 + ((4 + 4)/4 + 44)
:= 444/4 − 4 &times; 4
:= 4 + 4 &times; (4 &times; 4 + 4 + 4)
:= 5 + (5 &times; 5 + 55 + 5)
:= 5 &times; (5 &times; 5 − 5) − 5
:= 5 &times; (5 &times; 5 − 5)
:= 6 + ((66 + 6 + 6) + 6)
:= 6 &times; 6 + 66 − 6 − 6/6
:= 6 &times; 6 + ((6 + 6)/6)6
:= 7 + (77 − 7/7 + 7)
:= 7 + 77 + 77/7
:= (7 + 7)/7 + 7 &times; (7 + 7)
:= 88 + (8 + 8)/8
:= 8 + 88 − 8/8
:= 88 + (88 + 8)/8
:= 9 + 9 &times; 9
:= 99 + (9 − 9 &times; 9)/(9 + 9)
:= 9/9 + 99
&Iuml; 91 := 1 + (11 − 1) &times; (11 − 1 − 1)
:= 2 + 2 &times; 2 &times; 22 + 2/2
3
3
&Iuml; 101 := 1 + (11 − 1)1+1
&Iuml; 96 := (1 + 11) &times; (1 + 1)1+1+1
:= 2222/22
:= 2 &times; 2 &times; (22 + 2)
:= 3 &times; 33 − 3
:= 3 + 3 &times; 33 − 3/3
:= 4 + 44 + 44 − 4/4
:= 4 &times; (4 &times; 4 + 4 + 4)
:= 4444/44
:= 5 − 5 &times; 5 + 555/5
:= 5/5 + 5 &times; (5 &times; 5 − 5) − 5
:= 5/5 + 5 &times; (5 &times; 5 − 5)
:= 6 &times; 6 + 66 − 66/6
:= 6 &times; 6 + 66 − 6
:= 6 &times; 6 + 66 − 6/6
:= 7 + 77 + 7
:= 7 &times; (7 + 7) − (7 + 7)/7
:= 7777/77
:= 88 + (88/8 − 8)
:= 8 + 88
:= 8888/88
:= 9 + 9 &times; 9 + 9/9
:= 99 − (9 + 9 + 9)/9
:= 99 + (9 + 9)/9
:= 3 + (3/3 + 3)
&Iuml; 92 := 11 + (11 − 1 − 1)1+1
:= 2 &times; (2 &times; 22 + 2)
3
:= 3 &times; 3 + 33/3
&Iuml; 102 := 1 + 1 + (11 − 1)1+1
&Iuml; 97 := 111 − 11 − 1 − 1 − 1
:= 2/2 + 2 &times; 2 &times; (22 + 2)
:= 2 + (2 &times; (2 + 2) + 2)2
:= 3/3 + 3 &times; 33 − 3
:= 3 + 3 &times; 33
4
:= 4 + 44 + 44
:= 4 &times; 4 + (4 − 4/4)
5
:= (444 − 4)/4 − 4 − 4
5
:= 5 + 5 + 55 + ((5 + 5)/5)
:= (5 + 5)/5 + 5 &times; (5 &times; 5 − 5)
:= 6 &times; 6 + 66 + (6 − 66)/6
:= 6 &times; 6 + 66 − 6 + 6/6
:= 6 &times; 6 + 66
:= 7 + 7 + 77 + 7/7
:= 7 &times; (7 + 7) − 7/7
:= 7 + 77 + 7 + 77/7
:= 88 + 8 &times; 8/(8 + 8)
:= 8 + 88 + 8/8
:= (888 − 8)/8 − 8
:= 9 &times; 9 + 99/9
:= 99 − (9 + 9)/9
:= 999/9 − 9
:= 5 + 55 + ((5 + 5)/5)
16
Inder J. Taneja
&Iuml; 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
&Iuml; 113 := 1 + 1 + 111
&Iuml; 108 := 111 − 1 − 1 − 1
:= 2 + 2222/22
:= (222 − 2)/2 − 2
:= 2 + 222/2
:= 3 + 3 &times; 33 + 3/3
:= 3 &times; (33 + 3)
:= (333 + 3 + 3)/3
:= 444/4 − 4 − 4
:= 44 + 4 &times; 4 &times; 4
:= (444 + 4 + 4)/4
:= 5 + 5 &times; (5 &times; 5 − 5) − (5 + 5)/5
:= 55 + 55 − (5 + 5)/5
:= (555 + 5 + 5)/5
:= 6 &times; 6 + 66 + 6/6
:= 6 &times; (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
&Iuml; 104 := 1 + 1 + 1 + 1 + (11 − 1)1+1
&Iuml; 114 := 1 + 1 + 1 + 111
&Iuml; 109 := 111 − 1 − 1
:= 2 &times; 2 &times; (22 + 2 + 2)
:= 222/2 − 2
:= 3 + 3 + 3 &times; 33 − 3/3
:= 3/3 + 3 &times; (33 + 3)
:= 2 + (222 + 2)/2
:= 3 + 333/3
4
:= 44 + (4 + 4)/4
:= 4 + (444 − 4)/4
:= (5 − 5)/(5 &times; 5 + 5)
:= 55 + 55 − 5/5
:= 5 &times; 5 &times; 5 − 55/5
:= (666 − 6)/6 − 6
:= 6/6 + 6 &times; (6 + 6 + 6)
:= 6 + 6 &times; (6 + 6 + 6)
:= 777/7 − 7
:= 77/7 + 7 &times; (7 + 7)
:= ((7 + 7)/7)7 − 7 − 7
:= 8 + (88 + 8)
:= (888 − 8 − 8)/8
:= (888 + 88)/8 − 8
:= 99 + (9 &times; 9 + 9)/(9 + 9)
:= 9 + 99 + 9/9
:= (999 + 9 + 9 + 9)/9
:= 4 &times; 4 + 44 + 44
5
&Iuml; 105 := 111 − (1 + 1) &times; (1 + 1 + 1)
&Iuml; 115 := 1 + 1 + 1 + 1 + 111
&Iuml; 110 := 111 − 1
:= 2 + 2 + 222/2
:= 222/2 − 2 − 2 − 2
:= (222 − 2)/2
:= 3 + 3 + 3 &times; 33
:= (333 − 3)/3
:= 3 + (333 + 3)/3
:= 4 + 4444/44
:= (444 − 4)/4
:= 4 + 444/4
:= 5 + 5 &times; (5 &times; 5 − 5)
:= 55 + 55
:= 5 + 55 + 55
:= 666/6 − 6
:= (666 − 6)/6
:= 6 + 6 &times; (6 + 6 + 6) + 6/6
:= 7 + 7 &times; (7 + 7)
:= (777 − 7)/7
:= 77 + 7 &times; 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
&Iuml; 106 := 111 − 1 − 1 − 1 − 1 − 1
&Iuml; 116 := 1 + 1 + 1 + 1 + 1 + 111
&Iuml; 111 := 111
:= 222/2
:= 2 + 2 + (222 + 2)/2
:= 3 + 3 + 3 &times; 33 + 3/3
:= 333/3
:= 3 + 3 + (333 − 3)/3
:= (444 − 4)/4 − 4
:= 444/4
:= 4 + 4 &times; (44 − 4 &times; 4)
:= 555/5 − 5
:= 555/5
:= 5 + 555/5
:= 2 + 2 &times; 2 &times; (22 + 2 + 2)
:= (666 + 6)/6 − 6
:= 666/6
:= 6 + (666 − 6)/6
:= 7 + 7 &times; (7 + 7) + 7/7
:= 777/7
:= 7 + 7 &times; (7 + 7) + 77/7
:= 8 + 8 + 88 + (8 + 8)/8
:= 888/8
:= 8 &times; (8 + 8) − (88 + 8)/8
:= 9 + 99 − (9 + 9)/9
:= 999/9
:= 9 + 9 + 99 − 9/9
&Iuml; 107 := 111 − 1 − 1 − 1 − 1
&Iuml; 117 := 111 + (1 + 1) &times; (1 + 1 + 1)
&Iuml; 112 := 1 + 111
:= 222/2 − 2 − 2
:= (222 + 2)/2
:= (22/2)2 − 2 − 2
:= 3 &times; (33 + 3) − 3/3
:= (333 + 3)/3
:= 3 &times; (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 &times; (6 + 6 + 6) − 6/6
:= (666 + 6)/6
:= 6 + 666/6
:= 7 + (7 + 7)/7 + 7 &times; (7 + 7)
:= (777 + 7)/7
:= 7 + (777 − 7)/7
:= 8 + (88/8 + 88)
:= (888 + 8)/8
:= 8 &times; (8 + 8) − 88/8
:= 9 + 99 − 9/9
:= (999 + 9)/9
:= 9 + (99 + 9)
17
Inder J. Taneja
&Iuml; 118 := 111+1 − 1 − 1 − 1
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 128 := (1 + 1)1+(1+1)&times;(1+1+1)
&Iuml; 123 := 1 + 1 + 11 &times; 11
:= 22 + 2 &times; 2 &times; (22 + 2)
:= 2 + (22/2)2
:= 2 &times; 22+2+2
:= 3 + 3 + (333 + 3)/3
:= 33 − 3 + 3 &times; 33
:= 3 + (3 − 3/3 + 3)3
:= 4 + 4 + (444 − 4)/4
:= 4 + 4 + 4 + 444/4
:= 4 &times; 4 &times; (4 + 4)
:= 5 + (555 + 5 + 5)/5
:= 5 &times; 5 &times; 5 − (5 + 5)/5
:= 5 + 5 &times; 5 &times; 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 &times; (8 + 8)
:= (999 + 99 + 9)/9
:= 99 + 9 + 9 + 99/9
&Iuml; 119 := 111+1 − 1 − 1
2
:= (22/2) − 2
&Iuml; 124 := 1 + 1 + 1 + 11 &times; 11
&Iuml; 129 := 11 &times; (1 + 11) − 1 − 1 − 1
:= 2 &times; (22+2+2 − 2)
:= (22&times;(2+2) + 2)/2
:= 3 &times; 3 + (333 − 3)/3
:= 3 + (33/3)
:= 4 + 4 + 444/4
:= 4 &times; 4 &times; (4 + 4) − 4
:= 4/4 + 4 &times; 4 &times; (4 + 4)
:= 5 &times; 5 &times; 5 − 5 − 5/5
:= 5 &times; 5 &times; 5 − 5/5
:= 5 + (5 &times; 5 &times; 5 − 5/5)
:= 6 + (666 + 6 + 6)/6
:= 6 + 6 + (666 + 6)/6
:= 6 + 6 + 6 + 666/6
:= 77 + 7 &times; 7 − 7
:= 7 + 7 + (777 − 7)/7
:= 8 + 888/8
:= 7/7 + ((7 + 7)/7)7
:= 8 &times; (8 + 8) − 8 &times; 8/(8 + 8)
:= 9 + (99/9 + 99)
:= 8/8 + 8 &times; (8 + 8)
:= 9 + 9 + 9 + 99 − (9 + 9)/9
:= 9 + 9 + 999/9
&Iuml; 120 := 111+1 − 1
:= (2 + 2 + 2) &times; (22 − 2)
:= 3 + 3 &times; (33 + 3 + 3)
:= (4 + 4) &times; (44/4 + 4)
:= 5 &times; 5 &times; 5 − 5
:= 6 + 6 + 6 &times; (6 + 6 + 6)
:= 7 + (777 + 7 + 7)/7
:= 8 &times; (8 + 8) − 8
:= 9 + 999/9
&Iuml; 121 := 11 &times; 11
3−3/3
:= 3 + (3 &times; 33 + 33 )
&Iuml; 125 := 1 + 1 + 1 + 1 + 11 &times; 11
2
:= 2 + 2 &times; 22+2+2
:= 2 + (22/2) + 2
:= (3 − 3/3 + 3)
&Iuml; 130 := (11 − 1) &times; (1 + 1 + 11)
3
:= 3 + 3 &times; 33 + 33 + 3/3
4
:= 44 + (4 − 4/4)
:= (44 + 4)/((4 + 4)/4)
:= 5 &times; 5 &times; 5
:= 5 + 5 &times; 5 &times; 5
:= 66 + 66 − 6 − 6/6
:= 66 + ((6 + 6)/6)6
:= 7 + 7 + 777/7
:= (7 + 7)/7 + ((7 + 7)/7)7
:= 8 + 8 &times; (8 + 8) − 88/8
:= 9 + 99 + 9 + 9 − 9/9
&Iuml; 126 := 1 + 1 + 1 + 1 + 1 + 11 &times; 11
:= (22/2)2
:= 2 &times; 22+2+2 − 2
:= (33/3)3−3/3
:= 3 &times; (3 &times; 3 + 33)
:= 8 &times; (8 + 8) + (8 + 8)/8
:= 9 + 9 + (999 + 9)/9
&Iuml; 131 := 11 &times; (1 + 11) − 1
:= 2 + (22&times;(2+2) + 2)/2
:= 3 + 3 + (3 + 3 − 3/3)3
:= (44/4)(4+4)/4
:= (4 − 4) &times; 4/(4 + 4)
:= 5 + 5 + 555/5
:= 4 + 444/4 + 4 &times; 4
:= 5/5 + 5 &times; 5 &times; 5
:= 66 &times; 66/(6 &times; 6)
:= 5 + 5 &times; 5 &times; 5 + 5/5
:= 66 + 66 − 6
:= ((7 + 7)/7)7 − 7
:= 66 + 66 − 6/6
:= 77 + 7 &times; 7
:= 7 + 7 + 7 + (777 − 7)/7
:= 8 &times; (8 + 8) − 8 + 8/8
:= 8 &times; (8 + 8) − (8 + 8)/8
:= 8 &times; (8 + 8) + 88/8 − 8
:= 9 + (999 + 9)/9
:= 9 + (99 + 9 + 9)
:= 9 + (999 + 99)/9
&Iuml; 122 := 111 + 11
4
&Iuml; 127 := 111 + (1 + 1)1+1+1+1
2&times;(2+2)
− 2)/2
&Iuml; 132 := 11 &times; (1 + 11)
:= (222 + 22)/2
:= (2
:= (333 + 33)/3
3
:= 3 + (3 &times; 33 + 3/3)
:= 33 + 3 &times; 33
:= (444 + 44)/4
:= 4 &times; 4 + 444/4
:= 4 + 4 &times; 4 &times; (4 + 4)
:= (555 + 55)/5
:= 5 &times; 5 &times; 5 + (5 + 5)/5
:= 5 + 5 &times; 5 &times; 5 + (5 + 5)/5
:= (666 + 66)/6
:= 22 &times; (2 + 2 + 2)
:= 6 + 66 &times; 66/(6 &times; 6)
:= 66 + 66
:= (777 + 77)/7
:= 7/7 + (77 + 7 &times; 7)
:= 7 + 7 + 7 + 777/7
:= (888 + 88)/8
:= 8 &times; (8 + 8) − 8/8
:= 88 + 88 &times; 8/(8 + 8)
:= (999 + 99)/9
:= 99 + 9 + 9 + 9 + 9/9
:= 99 &times; (99 + 9)/(9 &times; 9)
18
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 133 := 1 + 11 &times; (1 + 11)
&Iuml; 138 := 111 + (1 + 1 + 1)1+1+1
:= (2/2 + 2) &times; (2 &times; 22 + 2)
:= 22 + 222/2
:= 33 + 3 &times; 33 + 3/3
3
&Iuml; 143 := 11 &times; (1 + 1 + 11)
:= 22 + (22/2)2
:= 33 + (333 − 3)/3
:= 3 + 333/3
:= 4 + 4 &times; 4 &times; (4 + 4) + 4/4
:= 4 + (4 + 4) &times; 4/(4 + 4) + 4
:= 4 &times; (4 + 4) + 444/4
4
:= 5 + 5 + 5 &times; 5 &times; 5 − (5 + 5)/5
:= (5 &times; 5 &times; 55 + 5)/(5 + 5)
:= 5 + (5 &times; 5 &times; 55 + 5)/(5 + 5)
:= 66 + 66 + 6/6
:= 6 + 66 + 66
:= (6 + 6) &times; (6 + 6) − 6/6
:= 7 + 77 + 7 &times; 7
:= 77 + 7 &times; 7 + (77 + 7)/7
:= 77 + 77 − 77/7
:= (8 − 8/8) &times; (88/8 + 8)
:= 8 + 8 &times; (8 + 8) + (8 + 8)/8
:= 8 + 8 + 8 &times; (8 + 8) − 8/8
:= 9 &times; (9 + 9) − 9 − 9 − 99/9
:= 9 + 9 + 9 + 999/9
:= 9 &times; (9 + 9) − 9 − 9 − 9/9
&Iuml; 134 := 1 + 1 + 11 &times; (1 + 11)
&Iuml; 139 := (11 − 1) &times; (1 + 1 + 1 + 11) − 1
&Iuml; 144 := (1 + 11)1+1
:= 2 + 22 &times; (2 + 2 + 2)
:= (22&times;(2+2) + 22)/2
:= (2 &times; (2 + 2 + 2))2
:= 3 &times; 3 + (3 − 3/3 + 3)3
:= 33 + (333 + 3)/3
:= (3 + 3) &times; (33 − 3)
:= 4 + (44 + 4) &times; 4/(4 + 4)
:= 44/4 + 4 &times; 4 &times; (4 + 4)
:= 4 &times; (4 &times; (4 + 4) + 4)
:= 5 + 5 + 5 &times; 5 &times; 5 − 5/5
:= 5 &times; (5 &times; 5 + 5) − 55/5
:= (5/5 + 5) &times; (5 &times; 5 − 5/5)
:= 6 + ((6 + 6)/6)
:= 6 + 66 + 66 + 6/6
:= (6 + 6) &times; (6 + 6)
:= 7 + 77 + 7 &times; 7 + 7/7
:= 77/7 + ((7 + 7)/7)7
:= (7/7 + 7) &times; (77/7 + 7)
:= 8 + 8 &times; (8 + 8) − (8 + 8)/8
:= 8 &times; (8 + 8) + 88/8
:= 8 + 8 &times; (8 + 8) + 8
:= 9 + 9 + 9 + 9 + 99 − 9/9
:= 9 + 9 + 9 + (999 + 9)/9
:= (9 + 9) &times; (9 − 9/9)
6/6+6
&Iuml; 135 := 1 + 1 + 1 + 11 &times; (1 + 11)
&Iuml; 140 := (11 − 1) &times; (1 + 1 + 1 + 11)
&Iuml; 145 := 1 + (1 + 11)1+1
:= 2 + 22 + 222/2
:= 2 &times; (2 &times; (22 + 2) + 22)
:= 2 + (22/2)2 + 22
:= 3 + 33 + 33 &times; 3
:= 3 + (333 − 3)/3 + 33
:= 3/3 + (3 + 3) &times; (33 − 3)
:= (4 − 4/4) &times; (44 + 4/4)
:= 4 &times; (4 &times; (4 + 4) + 4) − 4
:= 44 − 444/4
:= 5 + 5 + 5 &times; 5 &times; 5
:= 5 + 5 + 5 + 5 &times; 5 &times; 5
:= 5 &times; (5 &times; 5 + 5) − 5
:= 6 + 6 + 6 + 6 + 666/6
:= 6 + ((6 + 6)/6)6/6+6 + 6
:= 6/6 + (6 + 6) &times; (6 + 6)
:= 7 + ((7 + 7)/7)
:= 7 &times; (7 + 7 + 7) − 7
:= 7 &times; (7 + 7 + 7) − (7 + 7)/7
:= 8 + 8 &times; (8 + 8) − 8/8
:= 8 &times; (8 + 8) + (88 + 8)/8
:= 8 + 8 + 8 &times; (8 + 8) + 8/8
:= 9 + 9 + 9 + 9 + 99
:= 9 + 9 + (999 + 99)/9
:= 9/9 + (9 + 9) &times; (9 − 9/9)
7
&Iuml; 136 := 1 + 1 + 1 + 1 + 11 &times; (1 + 11)
&Iuml; 141 := (1 + 11)1+1 − 1 − 1 − 1
2
&Iuml; 146 := 1 + 1 + (1 + 11)1+1
:= 2 + 2 + 22 &times; (2 + 2 + 2) + 2
:= 22 + (22/2) − 2
:= 2 + (2 &times; (2 + 2 + 2))2
:= (3/3 + 3) &times; (3/3 + 33)
:= 33 + 3 &times; (33 + 3)
:= 3 + (333 − 3)/3 + 33
:= 4 + 4 + 4 &times; 4 &times; (4 + 4)
:= 44 − 4 − 444/4
:= 44 + (4 − 444)/4
:= 5 &times; 5 + 555/5
:= 5 + 5 &times; 5 + 555/5
:= 5/5 + 5 &times; (5 &times; 5 + 5) − 5
:= 6 &times; 6 + 666/6 − 6
:= 6 &times; 6 + (666 − 6)/6
:= 7 + ((7 + 7)/7) + 7/7
:= 7/7 + 7 &times; (7 + 7 + 7) − 7
:= 7 &times; (7 + 7 + 7) − 7/7
:= 8 + 8 &times; (8 + 8)
:= 8 &times; 8 + 88 − 88/8
:= 8 + 8 + 8 &times; (8 + 8) + (8 + 8)/8
:= (9 − 9/9) &times; (9 + 9 − 9/9)
:= 9 + 99 &times; (99 + 9)/(9 &times; 9)
:= (9 + 9)/9 + (9 + 9) &times; (9 − 9/9)
6
:= 6 + ((6 + 6)/6) + 66
7
&Iuml; 137 := 111 + (1 + 1) &times; (1 + 1 + 11)
:= 22+2 + (22/2)2
:= 33 + (333 − 3)/3
&Iuml; 142 := (1 + 11)1+1 − 1 − 1
&Iuml; 147 := 1 + 1 + 1 + (1 + 11)1+1
:= (2 &times; (2 + 2 + 2))2 − 2
:= 3 + (333 + 3)/3 + 3
3
4
:= (22/2 + 2)2 − 22
:= 3 + (3 + 3) &times; (33 − 3)
:= 4 + 4 &times; 4 &times; (4 + 4) + 4/4 + 4
:= 4 &times; 4 + (4 − 4) &times; 4/(4 + 4)
:= 4 + 444/4 + 4 &times; (4 + 4)
:= 5 &times; 5 + (555 + 5)/5
:= 5 + 5 &times; 5 + (555 + 5)/5
:= 5 &times; 5 + (555 + 55)/5
:= 6 + 66 + 66 − 6/6
:= (6 + 6) &times; (6 + 6) − (6 + 6)/6
:= 6 &times; 6 + 666/6
7
:= 77 + 77/7 + 7 &times; 7
:= 7 + 7 + ((7 + 7)/7)
:= 7 &times; (7 + 7 + 7)
:= 8 + 8 &times; (8 + 8) + 8/8
:= 8 + 8 + 8 &times; (8 + 8) − (8 + 8)/8
:= 8 + 8 &times; (8 + 8) + 88/8
:= 99 + 9 + 9 + 9 + 99/9
:= 9 &times; (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
&Iuml; 148 := 1 + 1 + 1 + 1 + (1 + 11)1+1
:= 2 + 2 + (2 &times; (2 + 2 + 2))2
:= (33 &times; 33 − 3)/(3 + 3)
&Iuml; 153 := 11 &times; (1 + 1 + 1 + 11) − 1
:= 2 &times; ((2/2 + 2)2+2 − 2)
:= 2 &times; 22 + 222/2 − 2
3
&Iuml; 158 := (1 + 1 + 11)1+1 − 11
3
:= 33 + (3 − 3/3 + 3)3
:= 3 &times; (3 − 3 + 3 )
4
:= 4 + 4 &times; (4 &times; (4 + 4) + 4)
:= 4 + 4 + 4 − 444/4
:= 4 &times; (44 − 4) − (4 + 4)/4
:= 5 &times; (5 &times; 5 + 5) − (5 + 5)/5
:= 5 + 5 &times; (5 &times; 5 + 5) − (5 + 5)/5
:= 5 &times; ((5 + 5)/5)5 − (5 + 5)/5
:= 6 &times; 6 + (666 + 6)/6
:= 6 + 666/6 + 6 &times; 6
:= 6 + 6 &times; 6 &times; 6 − ((6 + 6)/6)6
:= 7/7 + 7 &times; (7 + 7 + 7)
:= 77 + 77 − 7/7
:= 77/7 + 7 &times; (7 + 7 + 7)
:= 888/(8 − (8 + 8)/8)
:= 8/8 + 8 &times; 8 + 88
:= 8 + 88 − (8 + 8)/8 + 8 &times; 8
:= 99 + (9 &times; 99 − 9)/(9 + 9)
:= 9 &times; (9 + 9) − 9
:= 9 &times; (9 + 9) + (9 − 9 &times; 9)/(9 + 9)
&Iuml; 149 := 1 + 1 + 1 + 1 + 1 + (1 + 11)1+1
2
:= 2 + (22/2 + 2) − 22
&Iuml; 159 := 1 + (1 + 1 + 11)1+1 − 11
&Iuml; 154 := 11 &times; (1 + 1 + 1 + 11)
:= 22 + 22 &times; (2 + 2 + 2)
:= 222/2 + 2 &times; (22 + 2)
:= (33 &times; 3 + 3)/(3 + 3)
:= 33/3 &times; (33/3 + 3)
:= (3 + 3) &times; 33 − 3
:= 4 + 44 − 444/4
:= 44 + (444 − 4)/4
:= 4 &times; (44 − 4) − 4/4
:= 5 &times; (5 &times; 5 + 5) − 5/5
:= 5 + 5 &times; (5 &times; 5 + 5) − 5/5
:= (55 + 55 )/(5 &times; 5 − 5)
:= 6 + (6 + 6) &times; (6 + 6) − 6/6
:= 6 + (666 + 6)/6 + 6 &times; 6
:= 6 &times; 6 + 6 + 6 + 666/6
:= 7 + ((7 + 7)/7) + 7 + 7
:= 77 + 77
:= 7 &times; 7 + (777 − 7)/7
:= 8 + 8 &times; 8 + 88 − 88/8
:= 8 &times; 8 + (8 + 8)/8 + 88
:= 8 + 8 &times; 8 + 88 − 8/8
:= 99 + (9 &times; 99 + 9)/(9 + 9)
:= 9/9 + 9 &times; (9 + 9) − 9
:= 9 &times; (9 + 9) − (9 + 9 + 9)/9
3
7
&Iuml; 150 := (11 − 1) &times; (1 + 1 + 1 + 1 + 11)
:= 22 + 2 &times; 22+2+2
3
&Iuml; 155 := 11 + (1 + 11)1+1
&Iuml; 160 := (1 + 1) &times; ((11 − 1 − 1)1+1 − 1)
:= 2 &times; 22 + 222/2
:= 2 &times; 2 &times; 2 &times; (22 − 2)
3
3
:= 3/3 + (3 + 3) &times; 33 − 3
:= 3 + (3 + 3) &times; (3 − 3) + 3
:= 3 + (3 − 3/3 + 3) + 3
:= (44 + 44 )/((4 + 4)/4)
:= 44 + 444/4
:= 4 &times; (44 − 4)
:= 5 &times; (5 &times; 5 + 5)
:= 5 + 5 &times; (5 &times; 5 + 5)
:= 5 &times; ((5 + 5)/5)5
:= 6 + (6 + 6) &times; (6 + 6)
:= 66/6 + (6 + 6) &times; (6 + 6)
:= 6 + (666 + 6)/6 + 6 &times; 6 + 6
:= 7 + 77 + 77 − 77/7
:= 7/7 + 77 + 77
:= 7 &times; 7 + 777/7
:= 8 &times; 8 + 88 − (8 + 8)/8
:= 8 + 8 + 8 &times; (8 + 8) + 88/8
:= 8 + 8 &times; 8 + 88
:= 9 &times; (9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + 9 &times; (9 + 9) − 9
:= 9 &times; (9 + 9) − (9 + 9)/9
&Iuml; 151 := 1 + (11 − 1) &times; (1 + 1 + 1 + 1 + 11)
:= 22 + (22&times;(2+2) + 2)/2
&Iuml; 156 := (1 + 11) &times; (1 + 1 + 11)
:= 2 &times; (2 &times; 2 &times; (22 − 2) − 2)
3
&Iuml; 161 := (1 + 1) &times; (11 − 1 − 1)1+1 − 1
:= ((22+2 + 2)2 − 2)/2
:= (3 + 3) &times; 33 − 33/3
:= (3 + 3) &times; (3 − 3/3)
:= (3 + 3) &times; 33 − 3/3
:= 44 + 444/4 − 4
:= 4 &times; (44 − 4) − 4
:= 4/4 + 4 &times; (44 − 4)
5
:= 5/5 + 5 &times; (5 &times; 5 + 5)
:= (5 − 5)/(5 &times; 5 − 5)
:= 5 + (55 − 5)/(5 &times; 5 − 5)
:= 6 + (6 + 6) &times; (6 + 6) + 6/6
:= 6 + (6 + 6) &times; (6 + 6) + 6
:= 6 + (6 + 6) &times; (6 + 6) + 66/6
:= 7 + (7/7 + 7) &times; (77/7 + 7)
:= 77 + 77 + (7 + 7)/7
:= 7 + 77 + 77
:= 8 &times; 8 + (88 − 8/8)
:= 8 + 888/(8 − (8 + 8)/8)
:= 8 + 8 &times; 8 + 88 + 8/8
:= 9 &times; (9 + 9) − 99/9
:= 9 &times; (9 + 9) + (9 + 9 + 9)/9 − 9
:= 9 &times; (9 + 9) − 9/9
&Iuml; 157 := 1 + (1 + 11) &times; (1 + 1 + 11)
&Iuml; 162 := (1 + 1) &times; (11 − 1 − 1)1+1
:= 2 &times; 2 &times; ((2 + 2 + 2)2 + 2)
:= 2 + 222/2 + 2 &times; 22
:= 2 &times; (2/2 + 2)2+2
:= 33 + (3 − 3/3 + 3)3
:= 3 + 33/3 &times; (33/3 + 3)
:= (3 + 3) &times; 33
:= 4 &times; (44 − 4) − 4 − 4
:= 4/4 + 4 &times; (44 − 4) − 4
&Iuml; 152 := 11 &times; (1 + 1 + 1 + 11) − 1 − 1
:= (4 + 4)/4 + 4 &times; (44 − 4)
5
:= 5 &times; 5 &times; 5 + ((5 + 5)/5)
:= 5 + 5 &times; 5 &times; 5 + ((5 + 5)/5)5
:= 6 &times; 6 &times; 6 − ((6 + 6)/6)6
:= 6 + 6 + (6 + 6) &times; (6 + 6) + 6/6
:= 6 + (6 + 6) &times; (6 + 6) + 6 + 6
:= 77 + 77 − (7 + 7)/7
:= 77 + (7 + 7 + 7)/7 + 77
:= 7 + 77 + 77 + 7/7
:= 8 &times; 8 + 88
:= 88 + 88 − 8 − 88/8
:= 8 + 88 + 8 &times; 8 + (8 + 8)/8
:= 9 &times; (9 + 9) − 9/9 − 9
:= 9 &times; (9 + 9) + (9 − 99)/(9 + 9)
:= 9 &times; (9 + 9)
:= (5 + 5)/5 + 5 &times; (5 &times; 5 + 5)
20
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 163 := 1 + (1 + 1) &times; (11 − 1 − 1)1+1
:= ((22+2 + 2)2 + 2)/2
&Iuml; 168 := (1 + 1 + 11)1+1 − 1
&Iuml; 173 := 1 + 1 + 1 + 1 + (1 + 1 + 11)1+1
:= 2 &times; 2 &times; (2 &times; 22 − 2)
3
:= 2 + (22/2 + 2)2 + 2
:= 3/3 + (3 + 3) &times; 33
:= 3 + (3 + 3) &times; 3 + 3
:= (3 + 3) &times; 33 + 33/3
:= 4 + 4 &times; (44 − 4) − 4/4
:= 4 &times; 44 − 4 − 4
:= 4 &times; 44 − 4 + 4/4
:= 5 &times; 55 − (555 + 5)/5
:= 55 + (555 + 5 + 5)/5
:= 5 &times; (5 &times; 5 + 5 + 5) − (5 + 5)/5
:= 6 &times; (6 &times; 6 − 6) − 66/6 − 6
:= 6 &times; 6 + (66 + 66)
:= 6 &times; (6 &times; 6 − 6) − 6/6 − 6
:= 7 + 77 + 77 + (7 + 7)/7
:= 7 + ((77 + 77) + 7)
:= 77 + 7 &times; (7 + 7) − (7 + 7)/7
:= 8 &times; 8 + 88 + 88/8
:= 88 + (88 − 8)
:= 8 + 88 + 88 − 88/8
:= 9/9 + 9 &times; (9 + 9)
:= 9 + (9 &times; (9 + 9) − ((9 + 9 + 9)/9))
:= 99/9 + 9 &times; (9 + 9)
&Iuml; 164 := (1 + 1) &times; (1 + (11 − 1 − 1)1+1 )
:= 2 + 2 &times; (2/2 + 2)2+2
3
&Iuml; 169 := (1 + 1 + 11)1+1
&Iuml; 174 := (1 + 1) &times; (111 − (1 + 1) &times; (1 + 11))
:= (22/2 + 2)2
:= 2 &times; 2 &times; 2 &times; 22 − 2
3
:= 3 + (3 + 3) &times; 3 − 3/3
:= 3 + 3 + (3 + 3) &times; 3 + 3/3
:= 3 + (3 + 3) &times; 33 + 3 &times; 3
:= 4 + 4 &times; (44 − 4)
:= 4 + 4 &times; 44 − 44/4
:= 4 &times; 44 − (4 + 4)/4
:= 5 + 5 &times; 55 − 555/5
:= 5 &times; (5 &times; 5 + 5 + 5) − 5/5
:= 5 &times; 55 − 555/5
:= (6/6 + 6 + 6)
(6+6)/6
:= 6 &times; (6 &times; 6 − 6) − 6
:= 77 + 77 + (77 − 7)/7
:= (7 − 7/7 + 7)
(7+7)/7
:= 77 + 7 &times; (7 + 7) − 7/7
:= 8 &times; 8 + 88 + (88 + 8)/8
:= 8/8 + 88 + 88 − 8
:= 8 &times; 8 + (888 − 8)/8
:= (9 + 9)/9 + 9 &times; (9 + 9)
:= 9 + 9 &times; (9 + 9) − (9 + 9)/9
:= 9 &times; (9 + 9) + (99 + 9)/9
6/6+6
:= 6 &times; 6 + ((6 + 6)/6)
&Iuml; 165 := 11 &times; (1 + 1 + 1 + 1 + 11)
:= 2 &times; 22 + (22/2)2
&Iuml; 175 := 11 &times; (1 + 1)1+1+1+1 − 1
&Iuml; 170 := 1 + (1 + 1 + 11)1+1
:= 2 + 2 &times; (2 &times; (2 &times; 22 − 2))
:= 2 &times; 2 &times; 2 &times; 22 − 2/2
:= 3 + (3 &times; 333 + 3)/(3 + 3)
:= 333/3 + (3/3 + 3)3
:= 4 &times; 44 − 4 − (4 + 4)/4
:= 4 &times; 44 − 4/4
:= 5 + 5 &times; ((5 + 5)/5)
:= 5 &times; (5 &times; 5 + 5 + 5) − 5
:= 5 &times; (5 &times; 5 + 5 + 5)
:= 66 &times; (6 &times; 6 − 6)/(6 + 6)
:= 66 + (666 − 6)/6 − 6
:= 6/6 + 6 &times; (6 &times; 6 − 6) − 6
:= 3 + (3 + 3) &times; 3
3
:= 4 &times; 44 − 44/4
5
7
:= 77 + 7 &times; (7 + 7)
:= 77 + 77 + 77/7
:= 7 &times; 7 + ((7 + 7)/7) − 7
:= 88 + 88 − 88/8
:= 88 + (8 + 8)/8 − 8 + 88
:= 8 &times; 8 + 888/8
:= 9 + 9 &times; (9 + 9) − 9/9
:= 9 &times; (9 + 9) + (99 + 9 + 9)/9
:= 9 &times; (9 + 9) + (9 + 9 + 9)/9
&Iuml; 166 := 1 + 11 &times; (1 + 1 + 1 + 1 + 11)
&Iuml; 171 := 1 + 1 + (1 + 1 + 11)1+1
&Iuml; 176 := 11 &times; (1 + 1)1+1+1+1
:= 2 &times; ((2/2 + 2)2+2 + 2)
:= 2 + (22/2 + 2)2
:= 2 &times; 2 &times; 2 &times; 22
:= 3 + (3 + 3) &times; 33 + 3/3
:= 3 &times; (33 + 33 + 3)
:= 3 + (3 + 3) &times; 33 + 33/3
:= 4 &times; 44 + (4 − 44)/4
:= 4 &times; 44 − 4 − 4/4
:= 4 &times; 44
:= 55 + 555/5
:= 5 + 55 + 555/5
:= 5/5 + 5 &times; (5 &times; 5 + 5 + 5)
:= 6 &times; 6 + 66 + ((6 + 6)/6)6
:= 66 + 666/6 − 6
:= 66 + (666 − 6)/6
:= 7 + (777 − 7)/7 + 7 &times; 7
:= (7 &times; 7 &times; 7 &times; 7 − 7)/(7 + 7)
:= 7/7 + 7 &times; (7 + 7) + 77
:= 8 &times; 8 − 8 + (888 − 8)/8
:= (8/8 + 8) &times; (88/8 + 8)
:= 88 + 88
:= 9 &times; (9 + 9) + (9 &times; 9 − 9)/(9 + 9)
:= 9 + 9 &times; (9 + 9)
:= (9 − 9/9) &times; (99 + 99)/9
&Iuml; 177 := 1 + 11 &times; (1 + 1)1+1+1+1
&Iuml; 167 := (1 + 1 + 11)1+1 − 1 − 1
&Iuml; 172 := 1 + 1 + 1 + (1 + 1 + 11)1+1
:= (22/2 + 2)2 − 2
:= 2 &times; (2 &times; 2 &times; 22 − 2)
:= 2 &times; 2 &times; 2 &times; 22 + 2/2
:= (3 &times; 333 + 3)/(3 + 3)
:= 3 &times; 3 + (3 + 3) &times; 33 + 3/3
:= 3 &times; (33 + 33) − 3
:= 4 &times; 44 − 4 − 4 − 4/4
:= 4 &times; 44 − 4
:= 4 &times; 44 + 4/4
:= 55 + (555 + 5)/5
:= 5 + 55 + (555 + 5)/5
:= 55 + (555 + 55)/5
:= 6 &times; 6 + 66 + 66 − 6/6
:= 66 − 6 + (666 + 6)/6
:= 66 + 666/6
:= 7 + 7 &times; 7 + 777/7
:= (7 &times; 7 &times; 7 &times; 7 + 7)/(7 + 7)
:= 7 &times; 7 + ((7 + 7)/7)7
:= 8 &times; 8 − 8 + 888/8
:= 88 + 88 − 8 &times; 8/(8 + 8)
:= 88 + 88 + 8/8
:= 9 &times; (9 + 9) + (9 &times; 9 + 9)/(9 + 9)
:= 9 + 9 &times; (9 + 9) + 9/9
:= 99 + 9 &times; 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
&Iuml; 178 := (1 + 1) &times; (111 − 11 − 11)
&Iuml; 183 := 1 + (1 + 1 + 11) &times; (1 + 1 + 1 + 11)
:= 2 + 2 &times; 2 &times; 2 &times; 22
:= 2 &times; 2 &times; (2 &times; 22 + 2) − 2/2
&Iuml; 188 := 1 + 1 + ((1 + 1)11 − 1 − 1)/11
:= 2 &times; (2 &times; (2 &times; 22 + 2) + 2)
:= 3/3 + 3 &times; (3 + 33) − 3
:= (3 + 3) − 33
:= (3 + 3)3 − 33 − 3/3
:= 4 &times; 44 + (4 + 4)/4
:= 4 + 4 + 4 &times; 44 − 4/4
:= 444 − 44
:= 55 + 5 &times; 5 &times; 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 &times; (6 &times; 6 − 6) + (6 + 6)/6
:= 7 + (7 &times; 7 &times; 7 &times; 7 − 7)/(7 + 7)
:= 7 + 77 + 7 &times; (7 + 7) + 7/7
:= 77 + 777/7
:= 88 + 88 + 8 + 8/8
:= 8 + 8 &times; 8 + 888/8
:= 88 + 88 + (88 + 8)/8
:= 99 + 9 &times; 9 − (9 + 9)/9
:= 9 &times; 9 − 9 + 999/9
:= 9 + 99 + 9 &times; 9 − 9/9
&Iuml; 179 := 11 + (1 + 1 + 11)1+1 − 1
&Iuml; 184 := (1 + 1) &times; (11 + (11 − 1 − 1)1+1 )
&Iuml; 189 := (11 − 1 − 1) &times; (11 + 11 − 1)
:= 2 + (2 &times; 2 &times; 2 &times; 22) + 2/2
:= 2 &times; 2 &times; (2 &times; 22 + 2)
:= ((22 − 2)2 − 22)/2
:= 3 &times; (33 + 33) − 3/3
:= 3/3 + (3 + 3)3 − 33
:= (3 + 3)3 − 33
:= 4 + 4 &times; 44 − 4/4
:= 4 + 4 + 4 &times; 44
:= 4/4 + 444 − 44
:= 55 + 5 &times; 5 &times; 5 − 5/5
:= 5 + 55 + 5 &times; 5 &times; 5 − 5/5
:= 5 &times; (55 + 5) − 555/5
:= 6 &times; (6 &times; 6 − 6) − 6/6
:= 6 + 66 + (666 + 6)/6
:= 66 + 6 + 6 + 666/6
:= 7 + (7 &times; 7 &times; 7 &times; 7 + 7)/(7 + 7)
:= 7 + 7 &times; 7 + ((7 + 7)/7)
:= (7 + 7) &times; (7 + 7) − 7
:= 8 + (8/8 + 8) &times; (88/8 + 8)
:= 8 + 88 + 88
:= 8 + 8 &times; (8 + 8 + 8) − 88/8
:= 99 + 9 &times; 9 − 9/9
:= 9 &times; 9 − 9 + (999 + 9)/9
:= 9 + 99 + 9 &times; 9
&Iuml; 180 := 11 + (1 + 1 + 11)1+1
7
&Iuml; 185 := (1 + 1 + 1 + 11)1+1 − 11
&Iuml; 190 := (11 − 1) &times; ((1 + 1) &times; (11 − 1) − 1)
:= 2 &times; (2 &times; 2 &times; 22 + 2)
:= 2 &times; 2 &times; (2 &times; 22 + 2) + 2/2
:= 222 − 2 &times; 22+2
:= 3 &times; (33 + 33)
:= 3 + (3 + 3)3 − 33 − 3/3
:= 3/3 + (3 + 3)3 − 33
:= 4 + 4 &times; 44
:= 4 + 4 + 4 &times; 44 + 4/4
:= 44 − (44 + 4 + 4)/4
:= 55 + 5 &times; 5 &times; 5
:= 5 + 55 + 5 &times; 5 &times; 5
:= 5 + 5 &times; 5 &times; 5 + 55 + 5
:= 6 &times; (6 &times; 6 − 6)
:= 6 + 6 &times; (6 &times; 6 − 6) − 6/6
:= (6 − 6/6) &times; ((6 + 6)/6 + 6 &times; 6)
:= 77 + (777 − 7)/7 − 7
:= (7 + 7) &times; (7 + 7) − 77/7
:= 7/7 + (7 + 7) &times; (7 + 7) − 7
:= 88 + 88 + 8 &times; 8/(8 + 8)
:= 8 + 88 + 88 + 8/8
:= 8 &times; (8 + 8 + 8) − (8 + 8)/8
:= 99 + 9 &times; 9
:= 9 + (9 − 9/9) &times; (99 + 99)/9
:= 9 + 99 + 9 &times; 9 + 9/9
&Iuml; 181 := 1 + 11 + (1 + 1 + 11)1+1
&Iuml; 186 := ((1 + 1)11 − 1 − 1)/11
&Iuml; 191 := 1 + (11 − 1) &times; ((1 + 1) &times; (11 − 1) − 1)
:= 2/2 + 2 &times; (2 &times; 2 &times; 22 + 2)
:= 2 + 2 &times; 2 &times; (2 &times; 22 + 2)
:= 22 + (22/2 + 2)2
:= 3/3 + 3 &times; (33 + 33)
:= 3 + (3 + 3)3 − 33
:= 3 &times; (3/3 + 3)3 − 3/3
:= 4 + 4 &times; 44 + 4/4
:= 4 &times; 44 + (44 − 4)/4
:= 44 − (44 + 4)/4
:= 55 + 5 &times; 5 &times; 5 + 5/5
:= 5 + 5 &times; 5 &times; 5 + 55 + 5/5
:= 5 &times; 5 + 55 + 555/5
:= 6/6 + 6 &times; (6 &times; 6 − 6)
:= 6 + 6 &times; (6 &times; 6 − 6)
:= 6 &times; (6 &times; 6 − 6) + 66/6
:= 77 − 7 + 777/7
:= 77 + 7 &times; (7 + 7) + 77/7
:= 7 + ((7 + 7)/7)7 + 7 &times; 7 + 7
:= 8 &times; (8 + 8 + 8) + 88/8
:= 8 + 88 + 88 + (8 + 8)/8
:= 8 &times; (8 + 8 + 8) − 8/8
:= 9/9 + 99 + 9 &times; 9
:= 99 + 99 − (99 + 9)/9
:= 99 + 9 &times; 9 + 99/9
&Iuml; 182 := (1 + 1 + 11) &times; (1 + 1 + 1 + 11)
:= 2 + 2 &times; (2 &times; 2 &times; 22 + 2)
&Iuml; 187 := 1 + ((1 + 1)11 − 1 − 1)/11
&Iuml; 192 := 111 + (11 − 1 − 1)1+1
:= 2 &times; 2 &times; 2 &times; 22 + 22/2
:= 2 &times; 2 &times; 2 &times; (22 + 2)
:= (3 + 3) − 33 − 3/3
:= 33 &times; (3 + 3) − 33/3
:= 3 &times; (3/3 + 3)3
:= 4 + 4 &times; 44 + (4 + 4)/4
:= 4 &times; 44 + 44/4
:= 4 &times; (44 + 4)
:= 55 + 5 &times; 5 &times; 5 + (5 + 5)/5
:= 55/5 &times; ((55 + 5)/5 + 5)
:= (5/5 + 5) &times; ((5 + 5)/5)5
3
:= 6 &times; (6 &times; 6 − 6) + (6 + 6)/6
:= 6 + 6 &times; (6 &times; 6 − 6) + 6/6
:= 6 + 6 &times; (6 &times; 6 − 6) + 6
:= 7 + 77 + 7 &times; (7 + 7)
:= 77 + (777 − 7)/7
:= 7 + (7 + 7) &times; (7 + 7) − 77/7
:= 8 + 8 &times; 8 + (888 − 8)/8
:= 88 + 88 + 88/8
:= 8 &times; (8 + 8 + 8)
:= 9 + 9 &times; (9 + 9) + 99/9
:= 99 + 99 − 99/9
:= 9 &times; 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
&Iuml; 193 := 1 + 111 + (11 − 1 − 1)1+1
&Iuml; 198 := (1 + 1) &times; 11 &times; (11 − 1 − 1)
&Iuml; 203 := 1 + (1 + 1) &times; (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 &times; (3/3 + 3)3
:= 33 &times; (3 + 3)
:= 3 + 3 + 33 &times; (3 + 3) − 3/3
:= 4 &times; (44 + 4) + 4/4
:= 4 + 4 + (4 − 4 + 4)/4
:= 44 + (44 − 44 )/4
:= 5 + (5 − (5 + 5)/5)5 − 55
:= (5 + 5) &times; (5 &times; 5 − 5) − (5 + 5)/5
:= (55 − 5)/(5 + 5 + 5) − 5
:= 6 + 6 + 6 &times; (6 &times; 6 − 6) + 6/6
:= 6 &times; 66 &times; 6/(6 + 6)
:= 6 &times; 6 &times; 6 − 6 − 6 − 6/6
4
4
7
:= (7 + 7) &times; (7 + 7) − (7 + 7 + 7)/7
:= 77 + ((7 + 7)/7) − 7
:= 7 + (7 + 7) &times; (7 + 7)
:= 8/8 + 8 &times; (8 + 8 + 8)
:= 88 + (888 − 8)/8
:= 8 &times; (8 + 8 + 8) + 88/8
:= 9 &times; 9 + (999 + 9)/9
:= 99 + 99
:= 9 &times; 9 + (999 + 99)/9
&Iuml; 194 := (1 + 1 + 1 + 11)1+1 − 1 − 1
&Iuml; 199 := (1 + 1) &times; (11 − 1)1+1 − 1
&Iuml; 204 := (1 + 1) &times; (1 + 1 + (11 − 1)1+1 )
:= (22+2 − 2)2 − 2
:= ((22 − 2)2 − 2)/2
:= 2 + 2 + 222 − 22
:= 33 &times; (3 + 3) − 3 − 3/3
:= 3/3 + 33 &times; (3 + 3)
:= 3 + 33 &times; (3 + 3) + 3
:= 4 + (4 − 4 + 4)/4
:= 4 + 4 + 44 − (44 + 4)/4
:= 44 + 4 &times; (44 − 4)
:= 5 &times; 5 &times; (5 + 5) − 55 − 5/5
:= (5 + 5) &times; (5 &times; 5 − 5) − 5/5
:= 66 + ((6 + 6)/6)6/6+6
:= 6 &times; 6 &times; 6 − 6 − 66/6
:= (7 + 7) &times; (7 + 7) − (7 + 7)/7
:= 77 + (777 + 77)/7
:= (8 + 8)/8 + 8 &times; (8 + 8 + 8)
:= 88 + 888/8
:= 9 &times; 9 + (999 + 9 + 9)/9
:= 99 + 99 + 9/9
4
4
&Iuml; 195 := (1 + 1 + 1 + 11)1+1 − 1
:= 5 + (5 + 5) &times; (5 &times; 5 − 5) − 5/5
:= 6 &times; 6 &times; 6 − 6 − 6
:= 7 + (7 + 7) &times; (7 + 7) + 7/7
:= (88 + 8)/8 + 8 &times; (8 + 8 + 8)
:= (999/9 − 9) &times; (9 + 9)/9
&Iuml; 200 := (1 + 1) &times; (11 − 1)1+1
&Iuml; 205 := (1 + 1 + (1 + 1)11 )/(11 − 1)
:= 2 + 2 + ((22 − 2)2 + 2)/2
:= (22+2 − 2)2 − 2/2
:= 222 − 22
:= 33 &times; (3 + 3) − 3
:= 3 + 33 &times; (3 + 3) − 3/3
:= 4 + 44 − (44 + 4)/4
:= 4 + 4 + 4 &times; (44 + 4)
:= ((4 + 4)4 + 4)/(4 &times; 4 + 4)
:= 5 &times; 5 &times; (5 + 5) − 55
:= (5 + 5) &times; (5 &times; 5 − 5)
:= 5 + (5 + 5) &times; (5 &times; 5 − 5)
:= (6 &times; 66 − 6) &times; 6/(6 + 6)
:= 6 &times; 6 &times; 6 − 6 + (6 − 66)/6
:= 6 &times; 6 &times; 6 − 66/6
:= (7 + 7) &times; (7 + 7) − 7/7
:= (7 + 7) &times; (7 + 7) − 7 + 77/7
:= 77 + ((7 + 7)/7)7
:= 8 + 88 + 88 + 88/8
:= 8 + 8 &times; (8 + 8 + 8)
:= 88 + 8 &times; (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
&Iuml; 206 := 1 + (1 + 1 + (1 + 1)11 )/(11 − 1)
&Iuml; 196 := (1 + 1 + 1 + 11)1+1
&Iuml; 201 := 1 + (1 + 1) &times; (11 − 1)1+1
:= (22+2 − 2)2
:= ((22 − 2)2 + 2)/2
:= 222 − 22+2
:= 3/3 + 33 &times; (3 + 3) − 3
:= 3 + 33 &times; (3 + 3)
:= (3 + 3)3 + (3 − 33)/3
:= 4 + 4 &times; (44 + 4)
:= 44 − 44 − 44/4
:= 44 + 44 + 4 − (4 + 4)/4
:= 5/5 + (5 + 5) &times; (5 &times; 5 − 5)
:= 5 + (5 + 5) &times; (5 &times; 5 − 5) + 5/5
:= 66 + 66 + ((6 + 6)/6)
:= (6 &times; 66 + 6) &times; 6/(6 + 6)
:= 6 &times; 6 &times; 6 + (6 − 66)/6
:= (7 + 7) &times; (7 + 7)
:= 7 + (7 + 7) &times; (7 + 7) − (7 + 7)/7
:= 77 + 7/7 + ((7 + 7)/7)7
:= 8 &times; (8 + 8 + 8) + 8 &times; 8/(8 + 8)
:= 8 + 8 &times; (8 + 8 + 8) + 8/8
:= 8 + 88 + (888 − 8)/8
:= 99 + 99 − (9 + 9)/9
:= 9 + 9 &times; 9 + 999/9
:= 9 + 99 + 99 − 9/9
:= 5/5 + 5 &times; 5 &times; (5 + 5) − 55
6
&Iuml; 197 := 1 + (1 + 1 + 1 + 11)1+1
&Iuml; 202 := (1 + 1) &times; (1 + (11 − 1)1+1 )
&Iuml; 207 := 11 + (1 + 1 + 1 + 11)1+1
:= 2/2 + (22+2 − 2)2
:= 2 + 222 − 22
:= 2/2 + 222 − 22+2
:= 33 &times; (3 + 3) − 3/3
:= 3 + 33 &times; (3 + 3) + 3/3
:= (3 + 3)3 − 3 &times; 3
:= 4 + 44 + (4 − 44 )/4
:= 44 + (4 − 44)/4 − 44
:= 44 − 4 − 44 − 4/4
:= 5 + (5/5 + 5) &times; ((5 + 5)/5)5
:= (5 + 5)/5 + (5 + 5) &times; (5 &times; 5 − 5)
:= (55 − 5 &times; 5 + 5)/(5 + 5 + 5)
:= (66 &times; (6 + 6 + 6) − 6)/6
:= 6 &times; 6 &times; 6 − 6 − 6 − (6 + 6)/6
:= 6 + (6 &times; 66 + 6) &times; 6/(6 + 6)
:= 7/7 + (7 + 7) &times; (7 + 7)
:= 7 + (7 + 7) &times; (7 + 7) − 7/7
:= (7 + 7) &times; (7 + 7) + 77/7
:= 88 + (888 − 8 − 8)/8
:= 8 + 8 &times; (8 + 8 + 8) + (8 + 8)/8
:= 8 + 88 + 888/8
:= 99 + 99 − 9/9
:= 9 + 9 &times; 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
&Iuml; 208 := (1 + 1) &times; (111 − 1) − 1 − 11
:= 2 + 222 − 2
2+2
&Iuml; 213 := (1 + 1) &times; (1 + 111) − 11
&Iuml; 218 := (1 + 1) &times; (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 &times; (44 + 4 + 4)
:= 44 − 44 + 4/4
:= 444 &times; 4/(4 + 4) − 4
:= (55 − 5)/(5 + 5 + 5)
:= 5 + (55 − 5)/(5 + 5 + 5)
:= (5 − (5 + 5)/5)5 − 5 &times; 5
:= 6 &times; 6 &times; 6 − 6 − (6 + 6)/6
:= 6 &times; 6 &times; 6 − 6 &times; 6/(6 + 6)
:= 6 &times; 6 &times; 6 + (6 + 6)/6
3
:= 7 &times; (7 + 7) + (777 − 7)/7
:= 7 + 77 + ((7 + 7)/7) + 7/7
:= 7/7 + 7 &times; (7 &times; 7 − 7) − 77
:= 8 + 8 + 8 &times; (8 + 8 + 8)
:= 8 + 88 + 8 &times; (8 + 8) − 88/8
:= 88 + 8 &times; (8 + 8) + (8 + 8)/8
:= 9 + 99 + 99 + 9/9
:= (9 + 9) &times; (9 + 9) − 999/9
:= 9 + 99 + 99 + 99/9
&Iuml; 209 := 11 &times; ((1 + 1) &times; (11 − 1) − 1)
7
&Iuml; 214 := 1 + (1 + 1) &times; (1 + 111) − 11
:= 222 − 2 &times; (2 + 2)
:= 222 − 22/2 − 2
&Iuml; 219 := (1 + 1) &times; (111 − 1) − 1
:= 222 − 2/2 − 2
3
:= 33/3 + 33 &times; (3 + 3)
:= 3/3 + (3 + 3) − 3
:= 3 + (3 + 3)3
:= 4/4 + 4 &times; (44 + 4 + 4)
:= 44 − 44 + (4 + 4)/4
:= 44 + 4 &times; 44 − 4/4
5
:= (5 + 5 + 5)/(5 + 5 + 5)
:= 5 + (5 + 5 + 5)/(5 + 5 + 5)
:= 5 &times; 55 − 55 − 5/5
:= 6 &times; 6 &times; 6 − 6/6 − 6
:= 6 &times; 6 &times; 6 − (6 + 6)/6
:= 6 &times; 6 &times; 6 + 6 &times; 6/(6 + 6)
:= 7 &times; (7 + 7) + 777/7
:= 7 + (7 + 7) &times; (7 + 7) + 77/7
:= 77 + 7 + 7 + ((7 + 7)/7)7
:= 88/8 &times; (88/8 + 8)
:= 88 + 8 &times; (8 + 8) − (8 + 8)/8
:= 8 + 8 + 8 &times; (8 + 8 + 8) + 88/8
:= 99 + 99 + 99/9
:= (9 + 9)/9 &times; (99 + 9 − 9/9)
:= 9 + 99 + 999/9
&Iuml; 210 := (11 − 1) &times; (11 + 11 − 1)
:= 222 − 2 &times; (2 + 2 + 2)
:= (3 + 3)3 − 3 − 3
5
&Iuml; 215 := (1 + 1) &times; (1 + 1 + 111) − 11
:= 2 + 2 + 222 − 22/2
&Iuml; 220 := (1 + 1) &times; (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 &times; 44
:= 5 + (5 + 5) &times; (5 &times; 5 − 5) + 5
:= 5 &times; 55 − 55 − 5
:= 55 &times; (5 − 5/5)
:= 6 &times; 6 &times; 6 − 6
:= 6 &times; 6 &times; 6 − 6/6
:= 6 + 6 &times; 6 &times; 6 − (6 + 6)/6
:= 7 + 7 + (7 + 7) &times; (7 + 7)
:= 7 &times; 7 &times; 7 − ((7 + 7)/7)7
:= 77/7 &times; (7 − 7/7 + 7 + 7)
:= 88 + (888 + 88)/8
:= 88 + 8 &times; (8 + 8) − 8/8
:= (8 + 8) &times; (888 − 8)/(8 &times; 8)
:= 99 + 999/9
:= ((9 + 9) &times; (99 + 9) − 9)/9
:= 99/9 &times; (99/9 + 9)
4
&Iuml; 211 := (1 + 1) &times; 111 − 11
&Iuml; 216 := (1 + 1) &times; (111 − (1 + 1 + 1))
&Iuml; 221 := (1 + 1) &times; 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 &times; 44 + 4/4
:= 55 + (5 − 5)/(5 &times; 5 − 5)
:= (5 − 5/5) &times; (55 − 5/5)
:= 55 &times; (5 − 5/5) + 5/5
:= 6 &times; 6 &times; 6 − 6 + 6/6
:= 6 &times; 6 &times; 6
:= 6 + (6 &times; 6 &times; 6 − 6/6)
5
:= 7 + 7 + (7 + 7) &times; (7 + 7) + 7/7
:= (7 − 7/7)
:= 8 + 8 &times; (8 + 8 + 8) + 88/8
:= 88 + 8 &times; (8 + 8)
:= (888 − 8 + 888)/8
:= 9 + 9 + 99 + 99
:= 99 + (999 + 99)/9
:= 99 + (999 + 9)/9
&Iuml; 212 := 1 + (1 + 1) &times; 111 − 11
:= 222 + ((2 − 22)/2)
(7+7+7)/7
:= (777 + 777 − 7)/7
&Iuml; 217 := (1 + 1) &times; (111 − 1 − 1) − 1
:= 222 − 2 − 2 − 2/2
3
&Iuml; 222 := (1 + 1) &times; 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 &times; 4/(4 + 4)
5
5
:= (55 + 5 )/(5 + 5 + 5)
:= 5 + (55 + 5 )/(5 + 5 + 5)
:= (5 + 5) &times; 555/(5 &times; 5)
:= 6 &times; 6 &times; 6 − 6 + (6 + 6)/6
:= 6/6 + 6 &times; 6 &times; 6
:= 6 + 6 &times; 6 &times; 6
:= 77 + 7 + ((7 + 7)/7)
:= 7 &times; (7 &times; 7 − 7) − 77
:= ((7 + 7)/7) &times; 777/7
:= (8 &times; 8 &times; 8 − 88) &times; 8/(8 + 8)
:= 8/8 + 8 &times; (8 + 8) + 88
7
:= 99 + (999 + 9 + 9)/9
:= (8 + 8) &times; 888/(8 &times; 8)
9
:= 9 &times; 9 &times; 9 − ((9 + 9)/9)
24
:= (9 + 9) &times; 999/(9 &times; 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 228 := (1 + 1) &times; (1 + 1 + 1 + 111)
&Iuml; 223 := 1 + (1 + 1) &times; 111
&Iuml; 233 := 11 + (1 + 1) &times; 111
:= 222 + 22/2
:= 2 + 222 + 2 + 2
:= 2/2 + 222
:= 3 + (3 + 3) + 3/3 + 3
:= 3 + (3 + 3) + 3 &times; 3
:= 3 &times; (3 &times; 33 − 3) − 3/3
:= 44 + 44/4 − 44
:= 4 + 44 − 4 &times; (4 + 4)
:= 4 + 44 − 4 &times; 4 − 44/4
:= 5 + (5 − (5 + 5)/5)5 − 5 &times; 5
:= (5 − 5/5) &times; ((5 + 5)/5 + 55)
:= (5 − (5 + 5)/5)5 − 5 − 5
:= 6 + 6 &times; 6 &times; 6 + 6/6
:= 6 + 6 + 6 &times; 6 &times; 6
:= 6 + 6 &times; 6 &times; 6 + 66/6
:= 7 + (7 − 7/7)
:= (7 − 7/7) &times; (7 &times; 7 − 77/7)
:= 7 + 7 &times; (7 + 7) + ((7 + 7)/7)7
:= 8 + 88 + 8 &times; (8 + 8) − 8/8
:= (88/8 + 8) &times; (88 + 8)/8
:= 8 + (8 + 8 − 8/8)(8+8)/8
:= 9 &times; (9 + 9 + 9) − 99/9 − 9
:= 9 + 99 + 9 + 999/9
:= 9 &times; (9 + 9 + 9) − 9/9 − 9
3
3
(7+7+7)/7
&Iuml; 229 := 1 + (1 + 1) &times; (1 + 1 + 1 + 111)
&Iuml; 224 := (1 + 1) &times; (1 + 111)
&Iuml; 234 := 1 + 11 + (1 + 1) &times; 111
:= (222 − 22)/2 − 2
:= 22&times;(2+2) − 22
:= 3 &times; 3 + ((3 + 3) − 3/3)
:= (33+3 − 33)/3 − 3
:= 3 &times; (3 &times; 33 − 3)
:= 44 − 4 &times; (4 + 4)
:= 44 − 44/4 − 4 &times; 4
:= 44 − 44 &times; 4/(4 + 4)
:= (5 − 5/5) &times; (55 + 5/5)
:= 5 + (5 − 5/5) &times; (55 + 5/5)
:= (5 − 5/5 + 5) &times; (5 &times; 5 + 5/5)
:= 6 + 6 &times; 6 &times; 6 + (6 + 6)/6
:= 6 + 6 &times; 6 &times; 6 + 6/6 + 6
:= 6 + 6 &times; 6 &times; 6 + 6 + 6
:= 77 + 7 &times; (7 + 7 + 7)
:= 7 + (7 + 7) &times; 777/(7 &times; 7)
:= (77 + 7/7) &times; (7 + 7 + 7)/7
:= 8 + 8 &times; (8 + 8) + 88
:= 8 + (888 + 888 − 8)/8
:= (8 &times; (8 + 8) − 88/8) &times; (8 + 8)/8
:= 9 + ((9 + 9) &times; (99 + 9) − 9)/9
:= 9 + 99/9 &times; (9 + 99/9)
:= 9 &times; (9 + 9 + 9) − 9
:= 2 + 222
3
&Iuml; 225 := 1 + (1 + 1) &times; (1 + 111)
&Iuml; 230 := (11 − 1) &times; (1 + 11 + 11)
:= 2 + 222 + 2/2
:= 222 + 2 &times; (2 + 2)
3
&Iuml; 235 := 11 + (1 + 1) &times; (1 + 111)
:= 2 + 222 + 22/2
3
:= 3 + (3 + 3) + 33/3
:= 3 + (33+3 − 33)/3
:= 4/4 + 44 − 4 &times; (4 + 4)
:= 4 + 4 + 444 &times; 4/(4 + 4)
:= 44 − 4 − 4 &times; 4 − 4/4
:= 5 &times; (55 − 5 − 5)
:= 3 &times; 3 + (3 + 3)
:= 5 + 5 &times; (55 − 5 − 5)
:= 5 + 5 + 5 &times; (55 − 5 − 5)
:= 6 + 6 &times; 6/(6 + 6) + 6 &times; 6 &times; 6
:= 6 + 6 + 6 &times; 6 &times; 6 + (6 + 6)/6
:= 6 + 6 + 6 + 6 &times; 6 &times; 6 + 6/6
:= (7/7 + 7 + 7)(7+7)/7
:= (77 &times; (7 + 7 + 7) − 7)/7
:= 7 + (7 − 7/7) &times; (7 &times; 7 − 77/7)
:= 8 + (8 + 8) &times; 888/(8 &times; 8)
:= (8 &times; 8 &times; 88 + 8)/(8 + 8 + 8)
:= 9 + 99 + (999 + 99)/9
:= 9/9 + (9 &times; (9 + 9 + 9) − 9)
:= (8 − 8/8 + 8)
(8+8)/8
:= (9 + 9) &times; (9 + 9) − 99
&Iuml; 236 := 1 + 11 + (1 + 1) &times; (1 + 111)
&Iuml; 231 := 11 &times; (11 + 11 − 1)
&Iuml; 226 := (1 + 1) &times; (1 + 1 + 111)
:= 2 + 222 + 2
:= (22 − 22)/2
:= 2 + 22&times;(2+2) − 22
:= 3 &times; 3 + (3 + 3)3 + 3/3
:= 33 + 33 &times; (3 + 3)
:= (33+3 − 3)/3 − 3 − 3
:= 4 + 444 &times; 4/(4 + 4)
:= 44 + 4 &times; 44 + 44/4
:= 44 − 4 &times; 4 − 4
:= 5/5 + 5 &times; (55 − 5 − 5)
:= 5 + 5 &times; (55 − 5 − 5) + 5/5
:= 5 &times; 5 &times; 5 + 555/5
:= 6 &times; 6 &times; 6 + (66 − 6)/6
:= 66 &times; (6 &times; 6 + 6)/(6 + 6)
:= 66 &times; 66/(6 + 6 + 6) − 6
:= 7 &times; (7 + 7) + ((7 + 7)/7)7
:= 77 + 77 + 77
:= (7 + 777/7) &times; (7 + 7)/7
:= 8 + 8 &times; (8 + 8) + (8 + 8)/8 + 88
:= 8 &times; (8 + 8) + 888/8 − 8
:= 8 + (88/8 + 8) &times; (88 + 8)/8
:= 9 + (9 + 9) &times; 999/(9 &times; 9)
:= 9 &times; (9 + 9 + 9) − 9 + (9 + 9)/9
2
9
:= 9 + 9 &times; 9 &times; 9 − ((9 + 9)/9)
&Iuml; 227 := 1 + (1 + 1) &times; (1 + 1 + 111)
&Iuml; 232 := 111 + 111+1
&Iuml; 237 := 11 + (1 + 1) &times; (1 + 1 + 111)
:= 2 + 2 + 222 + 2/2
:= 2 + 222 + 2 &times; (2 + 2)
:= (222 − 2)/2 − 2 − 2
:= (3 + 3)3 + 33/3
:= (33+3 − 33)/3
:= 3 + 3 &times; (3 &times; 33 − 3)
:= 4 − 44 + 44 + 44/4
:= 44 − 4 &times; 4 − 4 − 4
:= 4/4 + 44 − (4 &times; 4 + 4)
:= 5 + (5 + 5) &times; 555/(5 &times; 5)
:= (5 + 5)/5 &times; (555/5 + 5)
:= 5 &times; 5 &times; 5 + (555 + 5)/5
:= 6 &times; 6 &times; 6 + 66/6
:= 6 + 6 &times; 6 &times; 6 + (66 − 6)/6
:= 6 + 66 &times; (6 &times; 6 + 6)/(6 + 6)
:= 7 + 77/7 &times; (7 − 7/7 + 7 + 7)
:= 7 &times; 7 &times; 7 − 777/7
:= 7 + (77 &times; (7 + 7 + 7) − 7)/7
:= 88 + 8 &times; (8 + 8) + 88/8
:= 88 + 8 + 8 + 8 &times; (8 + 8)
:= (8 + 8) &times; (8 + 8) − 88/8 − 8
:= ((9 + 9) &times; (99 + 9) + 99)/9
:= 9 &times; (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
&Iuml; 238 := (1 + 1) &times; (111+1 − 1 − 1)
&Iuml; 243 := 1 + (1 + 1) &times; 111+1
:= 2 &times; ((22/2)2 − 2)
:= (222 + 2)/2
:= 2 + 222 /2 + 2 + 2
:= (33+3 + 3)/3 − 3 − 3
:= 3 &times; 3 &times; 33
:= 3 + (33+3 − 3)/3 + 3
:= 44 − 4 &times; 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 &times; 6 &times; 6 + (66 + 66)/6
:= (6 &times; 6/(6 + 6))6−6/6
&Iuml; 248 := (1 + 1) &times; (1 + 1 + 1 + 111+1 )
:= 6 + 66 &times; 66/(6 + 6 + 6)
:= 777 − 7 &times; 77
:= ((7 + 7 + 7)/7)
:= 77 + (7 &times; 7 &times; 7 &times; 7 − 7)/(7 + 7)
:= 8 &times; (8 + 8) + (888 − 8)/8
:= (8 + 8/8) &times; (8 + 8 + 88/8)
:= (8 + 8) &times; (8 + 8) − 8
:= 9 &times; (9 + 9 + 9)
:= 9/9 + ((9 + 9)/9)9−9/9 − 9
&Iuml; 244 := (1 + 1) &times; (1 + 111+1 )
&Iuml; 249 := 1 + (1 + 1) &times; (1 + 1 + 1 + 111+1 )
7−(7+7)/7
9−9/9
:= ((9 + 9)/9)
−9−9
&Iuml; 239 := (1 + 1) &times; (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 &times; 4 − 4/4
:= 4 + 44 − 4 &times; 4
:= 4 + 44 − 44/4
:= 5 &times; 5 &times; (5 + 5) − 55/5
:= 5/5 + (5 − (5 + 5)/5)5
:= 5 &times; 5 &times; (5 + 5) − 5/5
:= 6 + 6 + 6 &times; 6 &times; 6 + 66/6
:= ((6 + 6)/6) + 6 &times; (6 &times; 6 − 6)
:= 6 + (6 &times; 6/(6 + 6))6−6/6
:= 7 + 7 &times; 7 &times; 7 − 777/7
:= 7 &times; (7 &times; 7 − 7 − 7) − 7/7
:= ((7 + 7)/7)7/7+7 − 7
:= 8 &times; (8 + 8) + 888/8
:= (8 + 8) &times; (8 + 8) − (88 + 8)/8
:= 8/8 + (8 + 8) &times; (8 + 8) − 8
:= 9 &times; (9 + 9 + 9) + ((9 − 9 &times; 9)/(9 + 9))
:= 9 &times; (9 + 9 + 9) + 9/9
:= 9 + (999/9 + 9) &times; (9 + 9)/9
&Iuml; 240 := (1 + 1) &times; (111+1 − 1)
2
6
&Iuml; 245 := 1 + (1 + 1) &times; (1 + 111+1 )
2
&Iuml; 250 := (11 − 1) &times; (1 + (1 + 1) &times; (1 + 11))
:= 22 /2 − 2
:= 2 + (22 + 2)/2
:= 2 &times; ((22/2)2 + 2 + 2)
:= 3 &times; 3 &times; 33 − 3
:= 3 + (33+3 − 3)/3
:= 3 + 3 + (33+3 + 3)/3
:= 44 − 4 &times; 4
:= 44 − 44/4
:= 44 − 4 − (4 + 4)/4
:= (5 + 5) &times; (5 &times; 5 − 5/5)
:= 5 &times; 5 &times; (5 + 5) − 5
:= 5 &times; 5 &times; (5 + 5)
:= 6 &times; (6 &times; 6 + 6) − 6 − 6
:= (6 + 6/6) &times; (6 &times; 6 − 6/6)
:= 6 &times; (6 &times; 6 + 6) − (6 + 6)/6
:= (7 − (7 + 7)/7) &times; (7 &times; 7 − 7/7)
:= 7 &times; (7 &times; 7 − 7 − 7)
:= (7 − (7 + 7)/7) &times; (7 &times; 7 + 7/7)
:= 8 &times; 8 + 88 + 88
:= (8 + 8) &times; (8 + 8) − 88/8
:= (8 + 8) &times; (8 + 8) + (8 + 8)/8 − 8
:= (999/9 + 9) &times; (9 + 9)/9
:= 9 &times; (9 + 9 + 9) + (9 + 9)/9
:= 9 + 9 &times; (9 + 9 + 9) − (9 + 9)/9
&Iuml; 241 := (1 + 1) &times; 111+1 − 1
&Iuml; 246 := (1 + 1) &times; (1 + 1 + 111+1 )
&Iuml; 251 := 11 + (1 + 1) &times; (111+1 − 1)
:= (222 − 2)/2
:= 2 + 222 /2 + 2
:= (222 + 22)/2 − 2
:= (33+3 + 3)/3 − 3
:= 3 + 3 &times; 3 &times; 33
:= 3 &times; 3 + (33+3 − 3)/3
:= 4/4 + 44 − 4 &times; 4
:= 44 + (4 − 44)/4
:= 44 − 4/4 − 4
:= 5 + 555/5 + 5 &times; 5 &times; 5
:= 5/5 + 5 &times; 5 &times; (5 + 5) − 5
:= 5/5 + 5 &times; 5 &times; (5 + 5)
:= 6 &times; (6 &times; 6 + 6) − 6/6
:= 6 &times; (6 &times; 6 + 6) − 66/6
:= 6 &times; (6 &times; 6 + 6) − 6
:= 7 + (7/7 + 77) &times; (7 + 7 + 7)/7
:= 7/7 + 7 &times; (7 &times; 7 − 7 − 7)
:= 7 + 7 &times; (7 &times; 7 − 7 − 7) − 7/7
:= 8/8 + 8 &times; 8 + 88 + 88
:= (8 + 8) &times; (8 + 8) + (8 − 88)/8
:= 8 + (8/8 + 8) &times; (88/8 + 8 + 8)
:= 9 &times; (9 + 9 + 9) − (9 + 9)/9
:= 9 &times; (9 + 9 + 9) + (9 + 9 + 9)/9
:= 9 + 9 &times; (9 + 9 + 9) − 9/9
&Iuml; 242 := (1 + 1) &times; 111+1
&Iuml; 247 := 1 + (1 + 1) &times; (1 + 1 + 111+1 )
2
2
:= 22 /2
:= (33+3 − 3)/3
:= 2 + (22 + 2)/2 + 2
:= 22&times;(2+2) − 2 − 2
3+3
:= 3 &times; (3 &times; 33 + 3)
:= 3 + (3
:= 44 &times; 44/(4 + 4)
5
&Iuml; 252 := (1 + 11) &times; (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 &times; 5 &times; (5 + 5)
:= 66 &times; 66/(6 + 6 + 6)
:= 6/6 + 6 &times; (6 &times; 6 + 6) − 6
:= 6 &times; (6 &times; 6 + 6)
:= 77/7 &times; (7 + 7 + 7 + 7/7)
:= (7 + 7)/7 + 7 &times; (7 &times; 7 − 7 − 7)
:= 7 + 7 &times; (7 &times; 7 − 7 − 7)
:= 88 &times; (88 + 88)/(8 &times; 8)
:= (8 + 8) &times; (8 + 8) − 8 − 8/8
:= (8 &times; 8 &times; 8 − 8) &times; 8/(8 + 8)
:= 9 &times; (9 + 9 + 9) − 9/9
9−9/9
:= ((9 + 9)/9)
−9
26
:= 9 + 9 &times; (9 + 9 + 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 253 := 11 &times; (1 + 11 + 11)
2
&Iuml; 258 := 1 + 1 + (1 + 1)(1+1)
1+1+1
&Iuml; 263 := (1 + 1) &times; 11 &times; (1 + 11) − 1
:= 22 + (222 − 2)/2
2&times;(2+2)
:= (22 + 22)/2
:= 2 + 2
:= 3 &times; 3 + (33+3 + 3)/3
:= 3 + 3 &times; (3 &times; 33 + 3) + 3
:= (33 &times; (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 &times; 55 − (55 + 5)/5
:= 6/6 + 6 &times; (6 &times; 6 + 6)
:= 6 + 6 &times; (6 &times; 6 + 6)
:= 66/6 + 6 &times; (6 &times; 6 + 6)
:= 7 + 7 &times; (7 &times; 7 − 7 − 7) + 7/7
:= (7 − 7/7) &times; ((7/7 − 7) + 7 &times; 7)
:= 7 + ((7 + 7)/7)7/7+7
:= 8 + (8 + 8) &times; (8 + 8) − 88/8
:= (8 + 8)/8 + (8 + 8) &times; (8 + 8)
:= 8 + (8 + 8) &times; (8 + 8) − 8/8
:= 9 + 9 &times; (9 + 9 + 9) + 9/9
:= 99 + 9 &times; (9 + 9) − (9 + 9 + 9)/9
:= 9 + 9 &times; (9 + 9 + 9) + 99/9
&Iuml; 254 := 1 + 11 &times; (1 + 11 + 11)
:= 22&times;(2+2) − 2
:= (3
3+3
1+1+1
&Iuml; 259 := 1 + 1 + 1 + (1 + 1)(1+1)
:= 2 + 22&times;(2+2) + 2/2
&Iuml; 264 := (1 + 1) &times; 11 &times; (1 + 11)
:= 2 &times; 22 &times; (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 &times; 5 &times; (5 + 5) − 5/5
:= 5 &times; 55 − 55/5 − 5
:= 5 &times; 55 − 55/5
:= (6 + 6)/6 + 6 &times; (6 &times; 6 + 6)
:= 6 + 6 &times; (6 &times; 6 + 6) + 6/6
:= 6 + 6 &times; (6 &times; 6 + 6) + 6
:= 77 + ((7 + 7)/7) + 7 &times; 7
:= 7 &times; 7 &times; 7 − 77 − 7
:= 7 &times; 7 &times; 7 − (7 + 7)/7 − 77
:= (8 + 8) &times; (8 + 8) − (8 + 8)/8
:= 88/8 + (8 + 8) &times; (8 + 8) − 8
:= 8 + (8 + 8) &times; (8 + 8)
:= 99/9 + 9 &times; (9 + 9 + 9)
:= 99 + 9 &times; (9 + 9) − (9 + 9)/9
:= 9 &times; (9 + 9) + 999/9 − 9
4
7
&Iuml; 255 := 111 + (1 + 11)1+1
2&times;(2+2)
:= 33 &times; (3 &times; 3 − 3/3)
&Iuml; 260 := (1 + 1) &times; (11 − 1) &times; (1 + 1 + 11)
&Iuml; 265 := 1 + (1 + 1) &times; 11 &times; (1 + 11)
:= 2 + 22&times;(2+2) + 2
:= 22 + (222 + 2)/2
:= 3 + 3 &times; (3 &times; 33 + 3)
:= 3 + (33+3 + 33)/3 + 3
:= (33 &times; (33 − 3) + 3)/3
:= 44 − 4/4
:= 4 + 44
:= 4 + 4/4 + 44 + 4
:= 5 + 5 &times; 5 &times; (5 + 5)
:= 5 + 5 &times; 5 &times; (5 + 5) + 5
:= 5 &times; 55 − 5 − 5
:= 6 &times; (6 &times; 6 + 6) + 6 &times; 6/(6 + 6)
:= 6 + 6 &times; (6 &times; 6 + 6) + (6 + 6)/6
:= 6 + 6 &times; (6 &times; 6 + 6) + 6/6 + 6
:= 7 &times; 7 &times; 7 − 77/7 − 77
:= 2
− 2/2
:= 7/7 + 7 &times; 7 &times; 7 − 77 − 7
:= 7 &times; 7 &times; 7 − 7/7 − 77
:= (8 + 8) &times; (8 + 8) − 8/8
:= (8 &times; 8 &times; 8 + 8) &times; 8/(8 + 8)
:= 8 + (8 + 8) &times; (8 + 8) + 8/8
:= 9 &times; (9 + 9 + 9) + (99 + 9)/9
:= 99 + 9 &times; (9 + 9) − 9/9
:= 9 + ((9 + 9)/9)9−9/9
&Iuml; 256 := (1 + 1)(1+1)
1+1+1
&Iuml; 261 := (1 + 1) &times; (11 &times; (1 + 11) − 1) − 1
:= 22&times;(2+2)
:= 2 + 2
:= (3/3 + 3)3/3+3
:= 4
2&times;(2+2)
+ 2/2 + 2
3
4
5−5/5
:= (5 − 5/5)
(6+6)/6+6
:= ((6 + 6)/6)
7/7+7
&Iuml; 266 := (1 + 1) &times; (1 + 11 &times; (1 + 11))
:= 222 + 2 &times; 22
:= 3 &times; (3 &times; 3 + 3 + 3)
:= 3 + (33 &times; (33 − 3) − 3)/3
:= 4 + 4/4 + 44
:= 44 + (44 − 4)/4
:= 5 + (5 − 5/5)5−5/5
:= 5/5 + 5 &times; 55 − (5 + 5)
:= 6 + 6 &times; (6 &times; 6 + 6) + 6 &times; 6/(6 + 6)
:= (6/6 + 6) &times; ((6 + 6)/6 + 6 &times; 6)
7
:= ((7 + 7)/7)
:= 7 + ((7 + 7)/7) + 77 + 7 &times; 7
:= 7 &times; 7 &times; 7 − 77
:= (8 + 8) &times; (8 + 8)
:= (88/8 − 8) &times; (88 − 8/8)
:= 8 + (8 + 8) &times; (8 + 8) + (8 + 8)/8
:= ((9 + 9)/9)9−9/9
:= 99 + 9 &times; (9 + 9)
:= 9 + ((9 + 9)/9)9−9/9 + 9/9
1+1+1
&Iuml; 257 := 1 + (1 + 1)(1+1)
2&times;(2+2)
&Iuml; 262 := (1 + 1) &times; (11 &times; (1 + 11) − 1)
2
&Iuml; 267 := 1 + (1 + 1) &times; (1 + 11 &times; (1 + 11))
:= 22 − 222
:= 22/2 + 22&times;(2+2)
:= 3 + (33+3 + 33)/3
:= 3 + (3/3 + 3)3/3+3 + 3
:= 3 &times; 3 &times; (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 &times; 55 + ((5 + 5)/5 − 5 − 5)
:= 2/2 + 2
:= 6 + 6 &times; (6 &times; 6 + 6) − 6/6
(6+6)/6+6
:= 6 + ((6 + 6)/6)
:= 666 &times; 6/(6 + 6) − 66
:= 7/7 + ((7 + 7)/7)
:= (7 &times; 7 &times; 77 − 7)/(7 + 7) − 7
:= 7/7 + 7 &times; 7 &times; 7 − 77
:= 8/8 + (8 + 8) &times; (8 + 8)
:= 8 + (8 + 8) &times; (8 + 8) − (8 + 8)/8
:= 88/8 + (8 + 8) &times; (8 + 8)
:= 9/9 + ((9 + 9)/9)9−9/9
:= 9/9 + 9 &times; (9 + 9) + 99
:= 99/9 + ((9 + 9)/9)9−9/9
7/7+7
27
Inder J. Taneja
&Iuml; 268 := (1 + 1) &times; (1 + 1 + 11 &times; (1 + 11))
:= 2 + 222 + 2 &times; 22
3
:= 3 + (33 &times; (3 − 3) + 3)/3
4
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 273 := (1 + 1 + 11) &times; (11 + 11 − 1)
:= (22/2 + 2) &times; (22 − 2/2)
3
&Iuml; 278 := (1 + 1111)/(1 + 1 + 1 + 1)
:= 22 + 22&times;(2+2)
:= 3 + (33+3 − 3)/3 + 33
:= 3 + 3 &times; 3 &times; (3 + 3)
4
:= 44 + 44 &times; 4/(4 + 4)
:= 4 + 4 + 4 + 4
:= 4 &times; 4 + 4/4 + 4
:= 5 &times; 5 + (5 − (5 + 5)/5)5
:= 5 &times; 55 − (5 + 5)/5
:= 5 + 5 &times; 55 − (5 + 5)/5
:= (6 − (6 + 6)/6) &times; (66 + 6/6)
:= 6 + 666 &times; 6/(6 + 6) − 66
:= 6 &times; 6 + 66 &times; 66/(6 + 6 + 6)
:= 7 &times; 7 &times; 7 + (7 + 7)/7 − 77
:= 7 + 7 &times; 7 &times; 7 − 77
:= 7 &times; 7 &times; 7 + (77 + 7)/7 − 77
:= 8 + (8 &times; 8 &times; 8 + 8) &times; 8/(8 + 8)
:= 8 + (8 + 8) &times; (8 + 8) + 8/8 + 8
:= 88 + 8 &times; (8 + 8 + 8) − (8 + 8)/8
:= 9 + 9 &times; (9 + 9) − (9 + 9)/9 + 99
:= 9 &times; (9 + 9) + 999/9
:= 99 + 9 &times; 9 − 9/9 + 99
&Iuml; 269 := 1 + (1 + 1) &times; (1 + 1 + 11 &times; (1 + 11))
:= 2 + 2
2&times;(2+2)
+ 22/2
&Iuml; 274 := 11 &times; (1 + (1 + 1) &times; (1 + 11)) − 1
:= 2 + 2
2&times;(2+2)
+2
2+2
&Iuml; 279 := 1 + (1 + 1111)/(1 + 1 + 1 + 1)
:= 2 + ((22 + 2)2 − 22)/2
:= 33 + (33+3 − 3)/3
:= 3 + 3 &times; 3 &times; (33 + 3) + 3/3
:= 3 &times; (3 &times; (33 + 3) + 3)
:= 4 + 4/4 + 44 + 4 + 4
:= 4 &times; 4 + ((4 + 4)/4 + 44 )
:= 4 + 44/4 + 44 + 4 + 4
:= 5 &times; 55 − (5/5 + 5)
:= 5 &times; 55 − 5/5
:= 5 + 5 &times; 55 − 5/5
:= 6 + 6 &times; (6 &times; 6 + 6) + 66/6
6
:= 6 &times; 6 &times; 6 + ((6 + 6)/6) − 6
:= 6 &times; 66 − 666/6 − 6
:= (7 &times; 7 &times; 77 − 7)/(7 + 7)
:= 7 + 7 &times; 7 &times; 7 − 77 + 7/7
:= 7 &times; (7 &times; 7 − 7) − 7/7 − 7 − 7
:= 8 + (88/8 − 8) &times; (88 − 8/8)
:= 8 + (8 + 8) &times; (8 + 8) + (8 + 8)/8 + 8
:= 88 + 8 &times; (8 + 8 + 8) − 8/8
:= 9 + 9 &times; (9 + 9) − 9/9 + 99
:= 9 + ((9 + 9)/9)9−9/9 + 9
:= 99 + 99 + 9 &times; 9
&Iuml; 270 := (11 − 1) &times; (1 + 1 + 1)1+1+1
&Iuml; 275 := 11 &times; (1 + (1 + 1) &times; (1 + 11))
&Iuml; 280 := 111 + (1 + 1 + 11)1+1
:= 222 + 2 &times; (22 + 2)
:= 22 + (222 + 22)/2
:= 2 + 22&times;(2+2) + 22
:= 3 &times; 3 &times; (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 &times; 4 + 44 + 4
:= 5 &times; 55 − 5
:= 5 &times; 55
:= 5 + 5 &times; 55
:= 666 − 6 &times; 66
:= 66 + 6 &times; 6 &times; 6 − 6/6 − 6
:= 6 &times; 6 &times; 6 + ((6 + 6)/6)6
:= (7 &times; 7 &times; 77 + 7)/(7 + 7)
:= 77/7 &times; (77/7 + 7 + 7)
:= 7 &times; (7 &times; 7 − 7) − 7 − 7
:= 8 + 8 + (8 + 8) &times; (8 + 8) − (8 + 8)/8
:= 8 + (8 + 8) &times; (8 + 8) + 88/8
:= 88 + 8 &times; (8 + 8 + 8)
:= 9 + 9 &times; (9 + 9) + 99
:= 9 &times; (9 + 9) + (999 + 9 + 9)/9
:= 9/9 + 99 + 99 + 9 &times; 9
&Iuml; 271 := 1 + (11 − 1) &times; (1 + 1 + 1)1+1+1
&Iuml; 276 := (1 + 11) &times; (1 + 11 + 11)
:= 2 + 22&times;(2+2) + 22/2 + 2
:= 22 + 22&times;(2+2) − 2
:= 2 + ((22 + 2)2 − 22)/2 + 2
:= 33 + (33+3 + 3)/3
:= 33 + 3 &times; 3 &times; 33
:= 33 + (33+3 + 33)/3
:= 4 + 44/4 + 44
:= 4 + 4 &times; 4 + 44
:= 4 + 4 &times; 4 + 44 + 4/4 + 4
:= 5/5 + 5 &times; 55 − 5
:= 5 &times; 55 + 5/5
:= 5 + 5 &times; 55 + 5/5
:= 66 + 6 &times; 6 &times; 6 − 66/6
:= 66 + 6 &times; 6 &times; 6 − 6
:= 66 + 6 &times; 6 &times; 6 − 6/6
:= 7 + 7 &times; 7 &times; 7 − (7 + 7)/7 − 77
:= 7 + (7 &times; 7 &times; 77 − 7)/(7 + 7)
:= 7/7 + 7 &times; (7 &times; 7 − 7) − 7 − 7
:= 8 + (8 + 8) &times; (8 + 8) − 8/8 + 8
:= 8 + (8 &times; 8 &times; 8 + 8) &times; 8/(8 + 8) + 8
:= 8/8 + 8 &times; (8 + 8 + 8) + 88
:= (99/9 + 9 &times; 9) &times; (9 + 9 + 9)/9
:= 9 + 99/9 + 9 &times; (9 + 9) + 99
:= 9 + 9 &times; (9 + 9) + 99 + 9/9
&Iuml; 272 := (1 + 1 + 11) &times; (11 + 11 − 1) − 1
&Iuml; 277 := 1 + (1 + 11) &times; (1 + 11 + 11)
&Iuml; 281 := 1 + 111 + (1 + 1 + 11)1+1
&Iuml; 282 := (1 + 1) &times; ((1 + 11)1+1 − 1 − 1 − 1)
:= 22+2 + 22&times;(2+2)
:= ((22 + 2)2 − 22)/2
:= 2 + 22&times;(2+2) + 22 + 2
:= 3 + (33+3 − 3)/3 + 33
:= 33 + (33+3 + 3)/3
:= 3 + 3 &times; (3 &times; (33 + 3) + 3)
:= 4 &times; 4 + 44
:= 4 + 4 &times; 4 + 44 + 4/4
:= 4 + 44 &times; 4/(4 + 4) + 44
:= 5 &times; 55 + (5 + 5)/5 − 5
:= 5 &times; 55 + (5 + 5)/5
:= 5 + 5 &times; 55 + (5 + 5)/5
:= 6 + (6/6 + 6) &times; ((6 + 6)/6 + 6 &times; 6)
:= 66 + 6 &times; 6 &times; 6 − 6 + 6/6
:= 66 + 6 &times; 6 &times; 6
:= 7 + 7 &times; 7 &times; 7 − 7/7 − 77
:= 7 + (7 &times; 7 &times; 77 + 7)/(7 + 7)
:= 77 + ((7 + 7)/7)7 + 77
:= 8 + (8 + 8) &times; (8 + 8) + 8
:= (8 − 8/8 + 8) &times; (88/8 + 8) − 8
:= 88 + 8 &times; (8 + 8 + 8) + (8 + 8)/8
:= 99 + 99/9 + 9 &times; (9 + 9)
:= 99 + 99 − (9 + 9)/9 + 9 &times; 9
:= 9 + 999/9 + 9 &times; (9 + 9)
28
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 283 := (1 + 1) &times; ((1 + 11)1+1 − 1 − 1) − 1
&Iuml; 288 := (1 + 1) &times; (1 + 11)1+1
&Iuml; 293 := 1 + (1 + 1) &times; (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 &times; (3 &times; 33 − 3)
:= 3 &times; 3 &times; 33 − 3/3 − 3
:= 4 &times; 4 + 44/4 + 44
:= 4 + 4 &times; (4 + 4)
:= 4 + 4 &times; (4 + 4) + 44 + 4/4
:= 5 + 5 &times; 55 − (5 + 5)/5 + 5
:= (55 + 5)/(5 + 5) − 5 &times; 5
:= 5 + (55 + 5)/(5 + 5) − 5 &times; 5
:= 66 + 6 &times; 6 &times; 6 + 6/6
:= 6 &times; (6 &times; 6 + 6 + 6)
:= 6 + 6 &times; (6 &times; 6 + 6 + 6) − 6/6
:= 7 &times; (7 &times; 7 − 7) − 77/7
:= 7/7 + 7 &times; (7 &times; 7 − 7) − 7
:= 7 &times; (7 &times; 7 − 7) − 7/7
:= 8 + 8 + (8 + 8) &times; (8 + 8) + 88/8
:= 8 + 8 &times; (8 + 8 + 8) + 88
:= 8 + (8 − 8/8 + 8) &times; (88/8 + 8)
:= 9 + (99 + 99) + 9 &times; 9
:= ((9 + 9 + 9) &times; (99 − 9/9) − 9)/9
9−9/9
:= 9 + ((9 + 9)/9)
4
+9+9
&Iuml; 284 := (1 + 1) &times; ((1 + 11)1+1 − 1 − 1)
&Iuml; 289 := 1 + (1 + 1) &times; (1 + 11)1+1
&Iuml; 294 := (1 + 1) &times; (1 + 1 + 1 + (1 + 11)1+1 )
:= 2 &times; ((2 &times; (2 + 2 + 2))2 − 2)
:= (22+2 + 2/2)2
:= 2 + (22 + 2)2 /2 + 2 + 2
:= (3/3 + 3) &times; ((3 + 3)3 − 3)/3
:= 3/3 + 3 &times; (3 &times; 33 − 3)
:= 3 &times; 3 &times; 33 − 3
4
:= 44 + 44 − 4 &times; 4
:= 44 + 4 − 44/4
:= 44 + 44 − (4 + 4)/4 − 4
:= 5 + 5 &times; 55 − 5/5 + 5
:= 5 &times; (55 + 5) − 55/5
:= 5 &times; (55 + 5) − 5/5 − 5
:= 66 + 6 &times; 6 &times; 6 + (6 + 6)/6
:= 6/6 + 6 &times; (6 &times; 6 + 6 + 6)
:= 6 + 6 &times; (6 &times; 6 + 6 + 6)
:= (7 − 77)/7 + 7 &times; (7 &times; 7 − 7)
:= (7 + 7)/7 + 7 &times; (7 &times; 7 − 7) − 7
:= 7 &times; (7 &times; 7 − 7)
:= (8 &times; (8 &times; 8 + 8) − 8) &times; 8/(8 + 8)
:= (8/8 + 8 + 8)(8+8)/8
:= 8 + 8 &times; 8 + (8 + 8) &times; 888/(8 &times; 8)
:= 9 &times; (9 + 9) + (999 + 99)/9
:= (9 + 9 − 9/9)(9+9)/9
:= ((9 + 9 + 9)/9) &times; (99 − 9/9)
&Iuml; 285 := (1 + 1) &times; 11 &times; (1 + 1 + 11) − 1
&Iuml; 290 := (1 + 1) &times; (1 + (1 + 11)1+1 )
&Iuml; 295 := 11 &times; (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 &times; (3 &times; 33 − 3) − 3
:= 3 + 3 &times; (3 &times; 33 − 3) − 3/3
4
:= 3/3 + 3 &times; 3 &times; 33 − 3
:= 44 + 44 − 4/4 − 4
:= 44 + 4/4 − 4 &times; 4 + 44
:= 44 + (4 − 44)/4 + 4
:= 5 + 5 &times; 55 + 5
:= 5 + (5 &times; 55 + 5) + 5
:= 5 &times; (55 + 5) − 5
:= 6 &times; 66 − 666/6
:= (6 + 6)/6 + 6 &times; (6 &times; 6 + 6 + 6)
:= 6 + 6 &times; (6 &times; 6 + 6 + 6) + 6/6
:= 7 &times; (7 &times; 7 − 7) − (7 + 7)/7 − 7
:= 7 + 7 &times; (7 &times; 7 − 7) − 77/7
:= 7/7 + 7 &times; (7 &times; 7 − 7)
:= (8 − 8/8 + 8) &times; (88/8 + 8)
:= 8/8 + (8/8 + 8 + 8)(8+8)/8
:= 8 + 88 + 88 + 888/8
:= (99 − 9/9) &times; (9 + 9 + 9)/9 − 9
:= (9/9 + 9) &times; (99/9 + 9 + 9)
:= (99 &times; (9 + 9 + 9) − (9 + 9))/9
&Iuml; 286 := (1 + 1) &times; 11 &times; (1 + 1 + 11)
&Iuml; 291 := 1 + (1 + 1) &times; (1 + (1 + 11)1+1 )
&Iuml; 296 := 11 &times; (1 + 1 + 1)1+1+1 − 1
:= 22 &times; (22/2 + 2)
:= 2 + (22+2 + 2/2)2
:= 2 &times; ((2 &times; (2 + 2 + 2))2 + 2 + 2)
:= 33/3 &times; (33 − 3/3)
:= 3 + 3 &times; (3 &times; 33 − 3)
:= 3 &times; 3 &times; 33 − 3/3
:= 44 + 44 &times; 44/(4 + 4)
:= 4 + 4 &times; (4 + 4) − 4/4 + 44
:= 44 + 44 − 4
:= 5 &times; 55 + 55/5
:= 5 + 5 &times; 55 + 55/5
:= 5/5 + 5 &times; (55 + 5) − 5
:= 6 + ((6 + 6)/6) + 6 &times; 6 &times; 6
:= 6 + 6 &times; 66 − 666/6
:= ((6 + 6)/6 + 6) &times; (6 &times; 6 + 6/6)
:= 7 &times; (7 &times; 7 − 7) − 7/7 − 7
:= 7 &times; (7 &times; 7 − 7) − (7 + 7 + 7)/7
:= (7 + 7)/7 + 7 &times; (7 &times; 7 − 7)
:= 8 &times; 8 + (8 + 8) &times; 888/(8 &times; 8)
:= 88 + 8 &times; (8 + 8 + 8) + 88/8
:= 8 &times; 888/(8 + 8 + 8)
:= 99 &times; (9 − 9/9 + 9 + 9)/9
:= 99 + 9 &times; 9 + 999/9
:= (99 &times; (9 + 9 + 9) − 9)/9
6
&Iuml; 287 := (1 + 1) &times; (1 + 11)1+1 − 1
&Iuml; 292 := (1 + 1) &times; (1 + 1 + (1 + 11)1+1 )
&Iuml; 297 := 11 &times; (1 + 1 + 1)1+1+1
:= ((22 + 2)2 − 2)/2
:= 2 + (22 + 2)2 /2 + 2
:= ((22 + 2)2 + 22)/2 − 2
:= 3 &times; (3 &times; 33 − 3) − 3/3
:= 3 + 3 &times; (3 &times; 33 − 3) + 3/3
:= 3 &times; 3 &times; 33
4
:= 44 + 44 + 4/4 − 4
4
:= 4 + 4 &times; (4 + 4) − 4/4
:= 4 + 4 &times; (4 + 4) + 4
:= 5 &times; 55 + (55 + 5)/5
:= 5 + (55 + 5)/5 + 5 &times; 55
6
:= 5 &times; 55 + (55 + 55)/5
:= 6 &times; (6 &times; 6 + 6 + 6) − 6/6
:= 6 + ((6 + 6)/6) + 6 &times; 6 &times; 6 + 6
:= 6 &times; 6 &times; 66/(6 + (6 + 6)/6)
:= 7 &times; (7 &times; 7 − 7) − 7
:= 7 &times; (7 &times; 7 − 7) − (7 + 7)/7
:= 7 &times; (7 &times; 7 − 7) + (7 + 7 + 7)/7
:= 88 + 88 + 888/8
:= (8 &times; (8 &times; 8 + 8) + 8) &times; 8/(8 + 8)
:= 8 + (8/8 + 8 + 8)(8+8)/8
:= (99 &times; (9 + 9 + 9) − 9)/9 − 9
:= 99 + 9 &times; 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
&Iuml; 298 := 1 + 11 &times; (1 + 1 + 1)1+1+1
:= 2 &times; 22 + 2
2&times;(2+2)
−2
&Iuml; 303 := (1 + 1 + 1) &times; (1 + (11 − 1)1+1 )
:= 222 + (2/2 + 2)
:= 3 + (3 &times; 3 &times; 33 + 3)
:= 3/3 + 3 &times; 3 &times; 33
&Iuml; 308 := (1 + 1) &times; 11 &times; (1 + 1 + 1 + 11)
:= 22 &times; (22+2 − 2)
2+2
:= 33/3 + 3 &times; 3 &times; 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 &times; (6 &times; 6 + 6 + 6)
:= 6 + 6 &times; 6 &times; 66/(6 + (6 + 6)/6)
:= 66 + 66 &times; 66/(6 + 6 + 6)
:= 77/7 + 7 &times; (7 &times; 7 − 7) − 7
:= 7 + 7 &times; (7 &times; 7 − 7) + (7 + 7)/7
:= 7 + 7 &times; (7 &times; 7 − 7) + 7
:= (88 &times; (88/8 + 8 + 8) + 8)/8
:= 8 &times; (8 + 8 + 8) + 888/8
:= 88 &times; (8 &times; 8 − 8)/(8 + 8)
:= (99 &times; (9 + 9 + 9) + 9)/9
:= 9 + (99 − 9/9) &times; (9 + 9 + 9)/9
:= 99/9 &times; (9/9 + 9 + 9 + 9)
&Iuml; 299 := 11 + (1 + 1) &times; (1 + 11)1+1
&Iuml; 304 := 1 + (1 + 1 + 1) &times; (1 + (11 − 1)1+1 )
2+2
2
:= ((22 + 2)2 + 22)/2
:= 2 + (2
:= 3 + (3 &times; 3 &times; 33 − 3/3)
:= 3 + 3 &times; 3 &times; 33 + 3/3 + 3
4
+ 2) − 22
&Iuml; 309 := 1 + (1 + 1) &times; 11 &times; (1 + 1 + 1 + 11)
:= 22 + ((22 + 2)2 − 2)/2
:= 3 + 3 &times; (3 &times; 33 + 3)
:= 44 + (44 − 44)/4
4
:= 44 + 4 − 4/4
:= 4 + 44 + 4
:= 5 &times; (55 + 5) − 5/5
:= 5 + 5 &times; (55 + 5) − 5/5
:= 5 + (5 &times; (55 + 5) − 5/5) + 5
:= 66/6 + 6 &times; (6 &times; 6 + 6 + 6)
:= ((6 + 6)/6 + 6) &times; ((6 + 6)/6 + 6 &times; 6)
:= 66 + (6 &times; 6/(6 + 6))6−6/6
:= 7 + (7 &times; (7 &times; 7 − 7) − (7 + 7)/7)
:= (7/7 + 7) &times; (7 &times; 7 − 77/7)
:= 7 + 7 &times; (7 &times; 7 − 7) + 7/7 + 7
:= (8 &times; (888 + 8) + 8)/(8 + 8 + 8)
:= (8 + 8) &times; (88/8 + 8)
:= 8/8 + 88 &times; (8 &times; 8 − 8)/(8 + 8)
:= (99 &times; (9 + 9 + 9) + 9 + 9)/9
:= (9 + 9) &times; (9 + 9) − 99/9 − 9
:= 99 + 999/9 + 99
&Iuml; 310 := (1 + 1) &times; (11 + (1 + 11)1+1 )
&Iuml; 300 := (1 + 1 + 1) &times; (11 − 1)1+1
&Iuml; 305 := (11 &times; 111 − 1)/(1 + 1 + 1 + 1)
:= 2 &times; 22 + 22&times;(2+2)
:= 2 + (2/2 + 2)2+2 + 222
:= 2 + 22 &times; (22+2 − 2)
:= 3 + 3 &times; 3 &times; 33
:= 333 − 33 − 3/3
:= ((3/3 + 3) + 3)3 − 33
:= 44 + 44
:= 4 + 44 + 44 + 4/4
:= 44 + (44 − 4)/4 + 44
:= 5 &times; (55 + 5)
:= 5 + 5 &times; (55 + 5)
:= 5 + 5 &times; (55 + 5) + 5
:= (6 − 66) &times; (6/6 − 6)
:= (6 − 6/6) &times; (66 − 6 + 6/6)
:= (6 − 6/6) &times; ((6 + 6)/6 − 6 + 66)
:= 7 + 7 &times; (7 &times; 7 − 7) − 7/7
:= 77/7 + 7 &times; (7 &times; 7 − 7)
:= 7 + 7 &times; (7 &times; 7 − 7) + (7 + 7)/7 + 7
:= (8 &times; 8 &times; 8 + 88) &times; 8/(8 + 8)
:= 8/8 + (8 + 8) &times; (8 + 88/8)
:= 88 + (8 + 8) &times; 888/(8 &times; 8)
:= (9/9 + 99) &times; (9 + 9 + 9)/9
:= 9 + (99 &times; (9 + 9 + 9) − 9)/9
:= 99 + 99 + (999 + 9)/9
&Iuml; 301 := 1 + (1 + 1 + 1) &times; (11 − 1)1+1
&Iuml; 306 := (1 + 1) &times; (11 &times; (1 + 1 + 1 + 11) − 1)
&Iuml; 311 := 1 + (1 + 1) &times; (11 + (1 + 11)1+1 )
:= 2 + ((22 + 2)2 + 22)/2
:= 22 &times; (22+2 − 2) − 2
:= 22 + (22+2 + 2/2)2
:= 3 + 3 &times; 3 &times; 33 + 3/3
:= 3 &times; (3 &times; 33 + 3)
:= 3 + 3 &times; 3 &times; 33 + 33/3
4
:= 4 + 44 + 4 + (4 + 4)/4
:= 44 + 44/4 + 44
:= 5/5 + 5 &times; (55 + 5)
:= 5 + 5 &times; (55 + 5) + 5/5
:= 55/5 + 5 &times; (55 + 5)
:= 6/6 + (6 − 66) &times; (6/6 − 6)
:= 6 + (6 − 66) &times; (6/6 − 6)
:= 6 + (6 − 6/6) &times; (66 − 6 + 6/6)
:= 7 + 7 &times; (7 &times; 7 − 7)
:= (7 − 7/7) &times; (7 &times; 7 + (7 + 7)/7)
:= 7 + 7 &times; (7 &times; 7 − 7) + (77 − 7)/7
:= 44 + 4/4 + 4
4
:= 8 + (8 − 8/8 + 8) &times; (88/8 + 8) + 8
:= (8 + 8)/8 + (8 + 8) &times; (8 + 88/8)
:= 8 + 888/8 + 8 &times; (8 + 8 + 8)
:= 9 &times; 9 + (99/9 + 9) &times; 99/9
:= (9 + 9) &times; (9 + 9 − 9/9)
:= (9 + 9) &times; (9 + 9) − (99 + 9 + 9)/9
&Iuml; 302 := 1 + 1 + (1 + 1 + 1) &times; (11 − 1)1+1
&Iuml; 307 := 111 + (1 + 1 + 1 + 11)1+1
2
:= (22+2 + 2)2 − 22
:= 3 + 3 &times; 3 &times; 33 − 3/3 + 3
&Iuml; 312 := (1 + 1) &times; (1 + 11) &times; (1 + 1 + 11)
:= (22 + 2/2) − 222
:= (22 + 2) &times; (22/2 + 2)
:= 3/3 + 3 &times; (3 &times; 33 + 3)
:= 3 + 3 &times; (3 &times; 33 + 3) + 3
:= 4 + 44 − 4/4 + 4 + 4
:= 4 + 4 + 4 + 44 + 44
:= (5 + 5)/5 + 5 &times; (55 + 5)
:= (55 − 55)/(5 + 5)
:= (55 − 5)/(5 + 5)
:= 6 + ((6 + 6)/6 + 6) &times; (6 &times; 6 + 6/6)
:= 6 + (6 − 66) &times; (6/6 − 6) + 6/6
:= 6 &times; ((6 + 6)/6)6 − 6 − 6
:= 44 + (4 + 4)/4 + 4
4
4
:= 7 + 7 &times; (7 &times; 7 − 7) + 7/7
:= 7 + 7 &times; (7 &times; 7 − 7) − 7/7 + 7
:= 7 + 7 &times; (7 &times; 7 − 7) + 77/7
:= (88 − 8/8 + 8 &times; 8) &times; (8 + 8)/8
:= 88 &times; (8 &times; 8 − 8)/(8 + 8) − 8/8
:= 8 + (8 + 8) &times; (88/8 + 8)
:= (9 &times; (9 + 9) − 99/9) &times; (9 + 9)/9
:= 9 + (99 &times; (9 + 9 + 9) + 9)/9
:= (9 + 9) &times; (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
&Iuml; 313 := 1 + (1 + 1) &times; (1 + 11) &times; (1 + 1 + 11)
:= (2
2+2
2
+ 2) − 22/2
&Iuml; 318 := (1 + 1 + 1) &times; (111 − 1) − 1 − 11
:= 2
2+2
&Iuml; 323 := 1 + (1 + 1 + 1) &times; 111 − 11
:= ((22+2 + 2)2 ) − 2/2
&times; (22 − 2) − 2
:= 3 + (3/3 + 3 + 3)3 − 33
:= 3 + 3 &times; 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 &times; (66 − 6 − 6) − 66/6
:= 66 + 6 &times; (6 &times; 6 + 6)
:= (6 &times; (66 − (6 + 6))) − 6/6
:= 7 + 7 &times; (7 &times; 7 − 7) + (77 + 7)/7
:= 7 &times; 7 &times; 7 − 77/7 − 7 − 7
:= 7/7 + (7 &times; 7 &times; 7 − (7 + 7 + 7))
:= 8 + (8 + 8) &times; (88/8 + 8) + 8/8
:= 8 &times; 8 + (8 + 8) &times; (8 + 8) − (8 + 8)/8
:= (8/8 + 8 + 8) &times; (88/8 + 8)
:= (9 + 9) &times; (9 + 9) − 99/9
:= 9 + 999/9 + 99 + 99
:= (9 + 9) &times; (9 + 9) − 9/9
&Iuml; 314 := (1 + 1) &times; (1 + (1 + 11) &times; (1 + 1 + 11))
&Iuml; 319 := 11 &times; ((11 − 1) &times; (1 + 1 + 1) − 1)
&Iuml; 324 := ((1 + 1) &times; (11 − 1 − 1))1+1
:= 2 + (22 + 2) &times; (22/2 + 2)
:= 22+2 &times; (22 − 2) − 2/2
:= (22+2 + 2)2
:= 3 &times; 33 + (3 + 3)3 − 3/3
:= 333 − 33/3 − 3
:= 3 &times; (3 &times; (33 + 3))
4
4
:= 4 + (4 − 4 − 4)/4 − 4
:= 4 + (4 − 4)/4
:= 4 &times; (4 − 4/4)4
:= 5/5 + (55 + 5)/(5 + 5)
:= 5 + (55 + 5)/(5 + 5) + 5/5
:= (5/5 + 5) &times; (55 − 5/5)
:= (6 − 6/6) &times; ((6 + 6)/6) − 6
:= 66 + 6 &times; (6 &times; 6 + 6) + 6/6
:= 6 &times; (66 − (6 + 6))
:= 7 &times; (7 &times; 7 + 7) − (7/7 + 77)
:= 7 + 7 + 7 &times; (7 &times; 7 − 7) + 77/7
:= (77/7 + 7)(7+7)/7
:= 8 + (8 + 8) &times; (88/8 + 8) + (8 + 8)/8
:= 8 &times; 8 + (8 + 8) &times; (8 + 8) − 8/8
:= ((((8 + 8)/8) + 8) + 8)(8+8)/8
:= (9 + 9) &times; (9 + 9) − 9/9 − 9
:= 99/9 &times; (99/9 + 9 + 9)
:= (9 + 9) &times; (9 + 9)
4
6
&Iuml; 315 := ((1 + 1)1+11 − 1)/(1 + 1 + 11)
4
&Iuml; 320 := (11 − 1) &times; (11 &times; (1 + 1 + 1) − 1)
&Iuml; 325 := 1 + ((1 + 1) &times; (11 − 1 − 1))1+1
:= 222 − (22/2 + 2)2
:= 22+2 &times; (22 − 2)
:= 2/2 + ((22+2 + 2)2 )
:= 3 &times; (3 &times; 33 + 3 + 3)
:= 333 + (3 − 33)/3 − 3
:= 3 + (333 − 33/3)
:= 4 + (4 − 4)/4 − 4
:= 4 &times; 4 &times; (4 &times; 4 + 4)
:= 4 + ((44 + 4)/4 + 44 )
:= (5 &times; 5 + 55 )/(5 + 5)
:= (5 + 5) &times; ((5 + 5)/5)5
4
4
:= 5 &times; (55 + 5 + 5)
6
:= (666 − 6 &times; 6) &times; 6/(6 + 6)
:= (6 − 6/6) &times; ((6 + 6)/6)
:= 6/6 + (6 &times; (66 − (6 + 6)))
:= 7 &times; (7 &times; 7 + 7) − 77
:= (7/7 + 7) &times; (7 &times; 7 − (7 + 7)/7 − 7)
:= 7 &times; 7 &times; 7 − (77/7 + 7)
:= 88/8 + (8 + 8) &times; (88/8 + 8)
:= 8 &times; (8 + 8 + 8 + 8 + 8)
:= (8/8 + 8 &times; 8) &times; ((8 − 88/8) + 8)
:= (9 + 9) &times; (9 + 9) − 9
:= (9 − 9/9) &times; (9 &times; 9 &times; 9 − 9)/(9 + 9)
:= 9/9 + (9 + 9) &times; (9 + 9)
&Iuml; 316 := (1 + 1) &times; ((1 + 1 + 11)1+1 − 11)
:= 2 &times; (2 &times; ((2/2) + 2)2+2 − 2)
3
:= (3/3 + 3 + 3) − 3
&Iuml; 321 := (1 + 1 + 1) &times; 111 − 1 − 11
:= 2/2 + (2
3
2+2
&times; (22 − 2))
&Iuml; 326 := (1 + 1 + 1) &times; (111 − 1 − 1) − 1
:= 2 + ((22+2 + 2)2 )
:= 333 − (3 &times; 3 + 3)
:= 333 − ((3/3 + 3) + 3)
:= 44 + (44 + 4)/4
:= 4 + ((((44 + 4) + 4)/4) + 44 )
:= 5/5 + ((5 + 5) &times; ((5 + 5)/5)5 )
:= 5/5 + (5 &times; (55 + 5 + 5))
:= 6 &times; (6 &times; 6 + 6) + ((6 + 6)/6)
:= 6 &times; 6 &times; 6 + (666/6 − 6)
:= 6 + ((6 − 6/6) &times; ((6 + 6)/6)6 )
:= 7/7 + 7 &times; (7 &times; 7 + 7) − 77
:= 7 &times; 7 &times; 7 − (7/7 + 7 + 7 + 7)
:= 7 &times; 7 &times; 7 + (((7 − 77)/7) − 7)
:= 8 + 88 &times; (8 &times; 8 − 8)/(8 + 8)
:= 8/8 + (((8 + 8) &times; (8 + 8)) + 8 &times; 8)
:= 8 + ((((8 + 8) &times; (8 + 8)) − ((8 + 8)/8)) + 8 &times; 8)
:= 9/9 + (9 + 9) &times; (9 + 9) − 9
:= (9 + 9) &times; (9 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 + 9) &times; (9 + 9)
:= 4 &times; 4 + 44 + 4
4
:= 5 + 5 &times; (55 + 5) + 55/5
6
&Iuml; 317 := 1 + (1 + 1) &times; ((1 + 1 + 11)1+1 − 11)
2
:= 2 + 22 − (22/2 + 2)
2
:= 333 + 33/3 − 33
4
&Iuml; 322 := (1 + 1 + 1) &times; 111 − 11
:= ((2
2+2
+ 2) ) − 2
:= 333 − (3 + 3)
:= 333 − 33/3
4
&Iuml; 327 := (1 + 1 + 1) &times; (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 &times; (6 &times; 6 + 6) − 6/6
:= (6 &times; (66 − (6 + 6))) − (6 + 6)/6
:= 6 &times; 6 &times; 6 + 666/6
:= 7 &times; 7 &times; 7 − (7 + 7 + 7)
:= 7 &times; 7 &times; 7 − (((7 + 7)/7 + 7) + 7)
:= (77/7 + 7)
(7+7)/7
−7
4
4
:= 8 + 88 &times; (8 &times; 8 − 8)/(8 + 8) + 8/8
:= 8 &times; 8 + (((8 + 8) &times; (8 + 8)) + ((8 + 8)/8))
:= 8 + ((((8 + 8) &times; (8 + 8)) − 8/8) + 8 &times; 8)
:= (9 + 9)/9 + (9 + 9) &times; (9 + 9) − 9
:= (9 + 9) &times; (9 + 9) − (9 + 9)/9
:= (9 + 9) &times; (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
&Iuml; 328 := 1 + (1 + 1 + 1) &times; (111 − 1 − 1)
&Iuml; 338 := (1 + 1) &times; (1 + 1 + 11)1+1
&Iuml; 333 := (1 + 1 + 1) &times; 111
:= 2 + (((22+2 + 2)2 ) + 2)
:= 222 + 222/2
:= 3/3 + (333 − (3 + 3))
:= 333
:= 2 &times; ((22/2 + 2)2 )
:= 3 + ((333 − 3/3) + 3)
4
4
:= 4 + (4 &times; (4 − 4/4) )
:= 4 + ((4 − 4/4) − 4)
:= 4/4 + ((4 − 4/4)4 + 44 )
:= 5 + (((55 + 5)/(5 + 5) + 5) + 5)
:= 5 &times; 5 + ((55 + 5)/(5 + 5) − 5)
:= 5 &times; 5 + (55 + 5)/(5 + 5)
:= 6 &times; 6 &times; 6 + (666 + 6)/6
:= 666 &times; 6/(6 + 6)
:= 6 + (6 &times; 66 − ((6 + 6)/6)6 )
:= 7 &times; 7 &times; 7 − (7/7 + 7 + 7)
:= 7 &times; 7 &times; 7 + (7 − 77)/7
:= 7 &times; 7 &times; 7 + ((7 + 7)/7 − 7)
:= 8 + (((8 + 8) &times; (8 + 8)) + 8 &times; 8)
:= (88/8 − 8) &times; 888/8
:= 8 &times; (8 &times; 8 − 8) + ((8 − 888)/8)
:= 9 + ((99/9) &times; (99/9 + 9 + 9))
:= 9 + (9 + 9) &times; (9 + 9)
:= 9 + (((9 &times; 9 + 9)/(9 + 9)) + (9 + 9) &times; (9 + 9))
4
&Iuml; 329 := (1 + 1 + 1) &times; (111 − 1) − 1
&Iuml; 334 := 1 + (1 + 1 + 1) &times; 111
&Iuml; 339 := (1 + 1 + 1) &times; (1 + 1 + 111)
:= 2 &times; (((22/2 + 2)2 ) − 2)
:= (((22 + 2 + 2)2 ) + 2)/2
:= 3/3 + 333
:= 3 + (333 + 3)
:= 444 + ((4 − 444)/4)
:= (4 &times; ((4 − 4/4)4 + 4)) − 4/4
:= 55 + (5 &times; 55 − 5/5)
:= 5 + ((5 &times; 55 − 5/5) + 55)
:= ((55 − 55)/5) − 5 &times; 55
:= 6 &times; 66 − (66 + 6/6)
:= 6/6 + 666 &times; 6/(6 + 6)
:= 6 + 666 &times; 6/(6 + 6)
:= 7 &times; 7 &times; 7 − (7 + 7)
:= 7 &times; 7 &times; 7 − ((7 + 7)/7 + 7)
:= 7 + (7 &times; 7 &times; 7 − (77/7))
:= 8 + ((((8 + 8) &times; (8 + 8)) + 8/8) + 8 &times; 8)
:= (((8 + 8 + 8) &times; 888/8) + 8)/8
:= 8 + (((8/8 + 8 + 8) &times; (88/8 + 8)) + 8)
:= (9 + 9) &times; (9 + 9) + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + ((9 + 9) &times; (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)
&Iuml; 335 := 1 + 1 + (1 + 1 + 1) &times; 111
&Iuml; 340 := 1 + (1 + 1 + 1) &times; (1 + 1 + 111)
:= 22 &times; ((22/2 + 2) + 2)
:= 2 + (222/2 + 222)
:= 2 + (2 &times; ((22/2 + 2)2 ))
:= 333 − 3
:= 3 + (333 − 3/3)
:= (((3/3 + 3) + 3)3 ) − 3
&Iuml; 330 := (1 + 1 + 1) &times; (111 − 1)
:= 4 + (((4 − 4) + 44)/4)
:= 4 + (((44 + 44 )/4) + 44 )
:= 55 + 5 &times; 55
:= 5 + (5 &times; 55 + 55)
:= 66 &times; (6 − 6/6)
:= (6 − 6/6) &times; (66 + 6/6)
:= 7/7 + (7 &times; 7 &times; 7 − (7 + 7))
:= 7 &times; 7 &times; 7 − (7/7 + 7)
:= 88/8 &times; ((88 + 88)/8 + 8)
:= ((8 − 8/8)88/8−8 ) − 8
:= ((8 &times; 88 − 8)/((8 + 8)/8)) − 8
:= (999 − 9)/((9 + 9 + 9)/9)
:= 99/9 + (9 + 9) &times; (9 + 9)
:= ((9 − 9/9) + 9) &times; (99/9 + 9)
4
4
&Iuml; 331 := 1 + (1 + 1 + 1) &times; (111 − 1)
&Iuml; 336 := (1 + 1 + 1) &times; (1 + 111)
:= 4 &times; ((4 − 4/4)4 + 4)
:= 5 + ((5 &times; 55 + 55) + 5)
:= (6 − 6/6) &times; (((6 + 6)/6) + 66)
:= 7 &times; 7 &times; 7 − (7 + 7 + 7)/7
&Iuml; 341 := 11 + (1 + 1 + 1) &times; (111 − 1)
:= 222 + (222/2 − 2)
:= 2 &times; (2 &times; (2 &times; ((2 &times; 22) − 2)))
:= 2 + ((((22 + 2 + 2)2 ) + 2)/2)
:= 3/3 + (333 − 3)
:= 3 + 333
:= 3 &times; 3 + (333 − 3/3)
:= 4 &times; ((4 &times; (4 &times; 4 + 4)) + 4)
:= 4 + ((4 − 4/4)4 + 44 )
4
4
:= 4 + ((44 + 4 )/4)
:= 55 + (5 &times; 55 + 5/5)
:= (5/5 + 5) &times; (55 + 5/5)
:= 6/6 + 66 &times; (6 − 6/6)
:= 6 + 66 &times; (6 − 6/6)
:= 7 &times; 7 &times; 7 − (77 + 7)/7
:= 7 &times; 7 &times; 7 − 7
:= 6 + ((6 − 6/6) &times; (66 + 6/6))
:= 8 + ((8/8 + 8 + 8) &times; (88/8 + 8))
:= 88 + (((8 + 8) &times; (8 + 8)) − 8)
:= 9 + ((9 + 9) &times; (9 + 9) − ((9 + 9)/9))
:= (9 + 9) &times; (9 + 9) + (99 + 9)/9
&Iuml; 332 := (1 + 1 + 1) &times; 111 − 1
:= 5 + ((5/5 + 5) &times; (55 + 5/5))
&Iuml; 337 := 1 + (1 + 1 + 1) &times; (1 + 111)
:= 7 &times; 7 &times; 7 − (7 + 7)/7
:= 8 + ((88/8 − 8) &times; 888/8)
:= 9 + (((9 + 9) &times; (9 + 9) − 9/9) + 9)
&Iuml; 342 := (1 + 1 + 1) &times; (1 + (1 + 1 + 111))
:= 2 + (22 &times; ((22/2 + 2) + 2))
:= (((22 + 2 + 2)2 ) − 2)/2
:= 2 &times; (((22/2 + 2)2 ) + 2)
:= 333 − 3/3
:= 3 + (333 + 3/3)
:= 3 &times; 3 + 333
:= 4 + ((4 &times; (4 − 4/4)4 ) + 4)
:= 5 &times; 5 + (55 − 55)/(5 + 5)
4
4
:= 4 + (4 − 4/4)
:= (4 − 44)/4 + ((4 + 4) &times; 44)
5
:= 5 &times; 5 + (5 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5 &times; 5)
:= 6 &times; 66 − ((6 + 6)/6)
:= 6 + (66 &times; (6 − 6/6) + 6/6)
:= 6 + (66 &times; (6 − 6/6) + 6)
:= 7 &times; 7 &times; 7 − 77/7
:= 7/7 + (7 &times; 7 &times; 7 − 7)
:= 7 &times; 7 &times; 7 − 7/7
:= 8 + (((((8 + 8)/8) + 8) + 8)(8+8)/8 )
:= 8 &times; (8 &times; 8 − 8) − 888/8
:= 88 + (((8 + 8) &times; (8 + 8)) − ((8 + 8)/8))
:= 9 + ((9 + 9) &times; (9 + 9) − 9/9)
:= 9 &times; 9 + (((9 + 9)/9)9−9/9 )
:= 9 + ((9 + 9) &times; (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
&Iuml; 343 := 11 + (1 + 1 + 1) &times; 111 − 1
&Iuml; 348 := 1 + 11 + (1 + 1 + 1) &times; (1 + 111)
&Iuml; 353 := 1 + 11 &times; (11 &times; (1 + 1 + 1) − 1)
:= 2 &times; ((2 &times; 2 &times; 2 &times; 22) − 2)
:= 2/2 + (22 &times; 22+2 )
:= ((3/3 + 3) + 3)3
:= 3 + ((333 + 3 &times; 3) + 3)
:= 3 &times; 3 + (333 + 33/3)
:= ((4 − 4/4) + 4)4−4/4
:= ((4 + 4) &times; 44) − 4
:= 4/4 + ((4 + 4) &times; 44)
:= 5 + ((55 + 5)/(5 + 5) + 5 &times; 5)
:= (5 − 5 − 5)/5 − 5 &times; 55
:= ((55 &times; ((5 + 5)/5)5 ) + 5)/5
:= 6 &times; (((6 + 6)/6)6 − 6)
:= 6 &times; (66 − 6) − 6/6 − 6
:= 222 + (22/2)
2
:= (6/6 + 6)6&times;6/(6+6)
:= 7 &times; 7 &times; 7
:= (8 − 8/8)88/8−8
:= 9 + (((9 + 9) &times; (9 + 9) + 9/9) + 9)
&Iuml; 344 := 11 + (1 + 1 + 1) &times; 111
:= 2 &times; (2 &times; (2 &times; 2 &times; 22 − 2))
5
:= 7 + (7 &times; 7 &times; 7 − ((7 + 7)/7))
:= 7 &times; 7 &times; 7 + (77 − 7)/7
:= (8 &times; 88 − 8)/((8 + 8)/8)
:= 8/8 + (8 &times; (88/((8 + 8)/8)))
:= ((99 + 9)/9) &times; (99/9 + 9 + 9)
:= 9 + (((9 + 9) &times; (9 + 9) + (99/9)) + 9)
&Iuml; 349 := 11 + (1 + 1) &times; (1 + 1 + 11)1+1
:= (22 &times; 2
2+2
&Iuml; 354 := 1 + 1 + 11 &times; (11 &times; (1 + 1 + 1) − 1)
:= 2 + (22 &times; 22+2 )
) − 2/2 − 2
:= 333 + 33/3
:= 3 + ((((3/3 + 3) + 3)3 ) + 3)
:= 3 + (333 + (3 &times; (3 + 3)))
:= 44 + (44 + 44 )
:= 4/4 + (((4 + 4) &times; 44) − 4)
:= (4 + 4)/4 + ((4 + 4) &times; 44)
:= (55 − 5)/5 − (5 &times; 55 + 5)
:= (5 − 5)/5 − 5 &times; 55
:= 5 + ((55 − 5)/5 − 5 &times; 55)
:= 6 + ((6 &times; 66 − ((6 + 6)/6)6 ) + 6)
:= 6 &times; (66 − 6) − 66/6
:= 6 &times; (66 − 6) − 6
:= 7 + (7 &times; 7 &times; 7 − 7/7)
:= 7 &times; 7 &times; 7 + 77/7
:= 8/8 + ((8 &times; 88 − 8)/((8 + 8)/8))
:= (8 + 8)/8 + (8 &times; (88/((8 + 8)/8)))
:= 9 + (((9 − 9/9) + 9) &times; (99/9 + 9))
:= 9 &times; (9 + 9 + 9) + 999/9
:= 7/7 + 7 &times; 7 &times; 7
:= 88 + ((8 + 8) &times; (8 + 8))
:= 9 + ((9 + 9) &times; (9 + 9) + (99/9))
&Iuml; 345 := 1 + 11 + (1 + 1 + 1) &times; 111
5
&Iuml; 350 := 11 + (1 + 1 + 1) &times; (1 + 1 + 111)
2+2
&Iuml; 355 := 11 + 11 + (1 + 1 + 1) &times; 111
:= 2 + ((22 &times; 22+2 ) + 2/2)
:= 2 + ((22/2)2 + 222)
:= (22 &times; 2
:= 3 + (333 + 3 &times; 3)
:= 3 + ((333 + 33/3) + 3)
:= 3 + ((((3/3 + 3) + 3)3 ) + 3 &times; 3)
:= 4 + (((4 − 4/4)4 + 44 ) + 4)
:= ((4 + 4) &times; 44) − (4 + 4)/4
:= 4 + (((4 + 4) &times; 44) − 4/4)
:= 5 &times; ((55 + 5 + 5) + 5)
:= 55 + (5 &times; (55 + 5))
:= 55 − (5 &times; 555 + 5)
)−2
6
:= 6 + (666 &times; 6/(6 + 6) + 6)
:= 7 &times; 7 &times; 7 + (7 + 7)/7
:= 8/8 + (((8 + 8) &times; (8 + 8)) + 88)
:= 9 + ((9 + 9) &times; (9 + 9) + ((99 + 9)/9))
&Iuml; 346 := 1 + 1 + 11 + (1 + 1 + 1) &times; 111
:= (6 − 6/6) &times; (((6 + 6)/6) + 6)
:= 6/6 + (6 &times; (66 − 6) − 6)
:= 7 + 7 &times; 7 &times; 7
:= 7 &times; 7 &times; 7 + (77 + 7)/7
:= ((8 + 8)/8) &times; (888/8 + 8 &times; 8)
:= 88 + (((8 + 8) &times; (8 + 8)) + (88/8))
:= (((9 + 9)/9)9 ) − 9 &times; (9 + 9)
:= 99 + (((9 + 9)/9)9−9/9 )
&Iuml; 351 := 11 &times; (11 &times; (1 + 1 + 1) − 1) − 1
&Iuml; 356 := (1 + 1 + 1) &times; (111+1 − (1 + 1)) − 1
:= 22 + ((22+2 + 2)2 )
:= (22 &times; 22+2 ) − 2/2
:= 2 + ((22 &times; 22+2 ) + 2)
:= 3 + (((3/3 + 3) + 3)3 )
:= 3 &times; (3 &times; (33 + 3 + 3))
:= 33 + (333 − (3/3 + 3))
:= ((4 + 4) &times; 44) − 4/4
:= 4 + ((4 + 4) &times; 44)
:= ((4 + 4) &times; 44) − ((4 + 4)/4 + 4)
:= (55 + 5)/5 − (5 &times; 55 + 5)
:= 6 + ((6 − 6/6) &times; (((6 + 6)/6) + 66))
:= 7 &times; 7 &times; 7 + (7 + 7 + 7)/7
:= 88 + (((8 + 8) &times; (8 + 8)) + ((8 + 8)/8))
9−9/9
:= 9 + ((((9 + 9)/9)
) + 9 &times; 9)
&Iuml; 347 := 11 + (1 + 1 + 1) &times; (1 + 111)
5
:= (5 + 5)/5 − 5 &times; 55
:= 5 + ((55 + 5)/5 − 5 &times; 55)
:= (666 + 6 &times; 6)/((6 + 6)/6)
:= (6 + 6)/6 + (6 &times; (66 − 6) − 6)
:= 7 + (7 &times; 7 &times; 7 + 7/7)
:= 7 + ((7 &times; 7 &times; 7 − 7/7) + 7)
:= 8 + ((8 − 8/8)88/8−8 )
:= (8 &times; 88 + 8)/((8 + 8)/8)
:= 9 + (((9 + 9) &times; (9 + 9) + 9) + 9)
:= (((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9)) − 9
&Iuml; 352 := 11 &times; (11 &times; (1 + 1 + 1) − 1)
&Iuml; 357 := (1 + 1 + 1) &times; (111+1 − 1 − 1)
:= 2 + (((22/2)2 + 222) + 2)
:= 22 &times; 22+2
:= (2/2 + 2) &times; ((22/2)2 − 2)
:= 3 + (333 + 33/3)
:= 3 &times; 3 + (((3/3 + 3) + 3)3 )
:= 33 + (333 − 3)
:= (4 + 4) &times; 44
:= 4 + (((4 + 4) &times; 44) + 4/4)
:= ((4 + 4) &times; 44) − (4/4 + 4)
5
:= 55/5 &times; ((5 + 5)/5)
:= 5 + (55/5 &times; ((5 + 5)/5)5 )
:= 6 &times; (66 − 6) − (6/6 + 6 + 6)
:= 6 &times; (66 − 6) − ((6 + 6)/6 + 6)
:= 66 &times; 66/(6 + 6) − 6
:= 7 &times; 7 &times; 7 + (77/7 − 7)
:= 7 + (7 &times; 7 &times; 7 + ((7 + 7)/7))
:= 7 + (7 &times; 7 &times; 7 + 7)
:= ((8 &times; 88 − 8)/((8 + 8)/8)) − 8/8
:= 8 &times; (88/((8 + 8)/8))
:= 8/8 + ((8 &times; 88 + 8)/((8 + 8)/8))
:= 9 &times; 9 &times; (9 + 9) − 9999/9
:= ((9/9 + 9 + 9)(9+9)/9 ) − 9
:= ((9 − 9/9) + 9) &times; (((99 + 9)/9) + 9)
5
:= (55/5 &times; ((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
&Iuml; 358 := 1 + (1 + 1 + 1) &times; (111+1 − (1 + 1))
&Iuml; 368 := 1 + 1 + (1 + 1 + 1) &times; (1 + 111+1 )
&Iuml; 363 := 11 &times; 11 &times; (1 + 1 + 1)
:= 2 + (((22 &times; 22+2 ) + 2) + 2)
:= 2 + ((22 − (2/2 + 2))2 )
:= 2 &times; (2 &times; (2 &times; ((2 &times; 22) + 2)))
:= 33 + ((333 − 3) + 3/3)
:= 33 &times; 33/3
:= 3 + (((33 &times; 33 − 3)/3) + 3)
:= 4 + (((4 + 4) &times; 44) + (4 + 4)/4)
:= 4 + (444/4 − 4)
:= 4 &times; (44 + 44 + 4)
:= 5 + (((55 &times; ((5 + 5)/5)5 ) + 5)/5)
:= 55 + ((55 + 5)/(5 + 5) − 5)
:= 55 + (55 + 5)/(5 + 5)
:= 6 &times; (66 − 6) − (6 + 6)/6
:= 66 &times; 66/(6 + 6)
:= 6 + (6 &times; (66 − 6) + ((6 + 6)/6))
:= 7 + ((7 &times; 7 &times; 7 + 7/7) + 7)
:= 7 + (((7 &times; 7 &times; 7 − 7/7) + 7) + 7)
:= 7 + ((7 &times; 7 &times; 7 + (77/7)) + 7)
:= 8 &times; (8 &times; 8 − 8) − ((8 + 8)/8 + 88)
:= 88/8 + (8 &times; (88/((8 + 8)/8)))
:= 8 + (8 &times; (8 &times; 8 − 8) − 88)
:= ((9 + 9)/9) &times; ((9 &times; 9 − 9/9) + 99)
:= (99 &times; 99)/(9 + 9 + 9)
:= 9 + (((((9 + 9)/9)9 ) − 9 &times; (9 + 9)) + 9)
&Iuml; 359 := (1 + 1 + 1) &times; (111+1 − 1) − 1
4
&Iuml; 364 := 1 + 11 &times; 11 &times; (1 + 1 + 1)
&Iuml; 369 := (1 + 1 + 1) &times; (1 + (1 + 111+1 ))
:= ((22 − (2/2 + 2))2 ) − 2
:= 2 &times; ((2 &times; (2 &times; 2 &times; 22 + 2)) + 2)
:= (2/2 + 2) &times; ((22/2)2 + 2)
:= 33 + (333 − 3/3)
:= (33 &times; 33 + 3)/3
:= 3 + (333 + 33)
:= 4 + ((((4 + 4) &times; 44) − 4/4) + 4)
:= 4 + ((((4 + 4) &times; 44) + 4) + 4)
:= (4/4 + 4)4 − 44
:= (5/5 + 5) &times; (55 + 5) − 5/5
:= (5 &times; (5 &times; (5 + 5 + 5))) − 55/5
:= (5 &times; (5 &times; (5 + 5 + 5))) − (5/5 + 5)
:= 6 &times; (66 − 6) − 6/6
:= 6 + (6 &times; (66 − 6) − ((6 + 6)/6))
:= 6 + 66 &times; 66/(6 + 6)
:= 7 + ((7 &times; 7 &times; 7 + ((7 + 7)/7)) + 7)
:= 7 + (7 &times; 7 &times; 7 + 7 + 7)
:= 7 + (((77 + 7)/7 + 7 &times; 7 &times; 7) + 7)
:= 8 &times; (8 &times; 8 − 8) − (8/8 + 88)
:= 8 + ((8 &times; 88 + 8)/((8 + 8)/8))
:= 8 + ((88/8 + 8)(8+8)/8 )
:= 9 + ((((9 + 9)/9)9 ) − 9 &times; (9 + 9))
:= ((9 &times; 9 &times; 9 &times; 9) − 9)/(9 + 9)
:= 9 + ((9 + 9) &times; (99/9 + 9))
&Iuml; 360 := (1 + 1 + 1) &times; (111+1 − 1)
&Iuml; 365 := 1 + 1 + 11 &times; 11 &times; (1 + 1 + 1)
2
:= 2 &times; (2 &times; (2 &times; 2 &times; 22 + 2))
&Iuml; 370 := (1111 − 1)/(1 + 1 + 1)
:= 2 + (((22 − (2/2 + 2)) ) + 2)
:= 2 + (2 &times; (2 &times; (2 &times; ((2 &times; 22) + 2))))
:= 3 + 333
:= 3 + ((33 &times; 33 − 3)/3)
:= 33 + (((3/3 + 3) + 3)3 )
:= 4 + (((4 + 4) &times; 44) + 4)
:= (4/4 + 4)4 − (44 + 4)
:= 4 + ((444 − 4)/4 + 44 )
:= (5/5 + 5) &times; (55 + 5)
:= 5 + (5/5 + 5) &times; (55 + 5)
:= (5 &times; (5 &times; (5 + 5 + 5))) − 5
:= 6 &times; (66 − 6)
:= 6 + (6 &times; (66 − 6) − 6/6)
:= 6 &times; (66 − 6) + (66 − 6)/6
:= 7 + (((77 − 7)/7) + 7 &times; 7 &times; 7)
:= 7 + (((7 &times; 7 &times; 7 + 7/7) + 7) + 7)
:= 77 + (7 &times; (7 &times; 7 − 7) − 7/7)
:= 8 &times; (8 &times; 8 − 8) − 88
:= 888 − (8 &times; 8 &times; 8 + 88/8)
:= (8888 − 8)/(8 + 8 + 8)
:= (9 + 9) &times; (99/9 + 9)
:= ((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9)
:= 9 + ((9/9 + 9 + 9)(9+9)/9 )
3
&Iuml; 361 := (((1 + 1) &times; (11 − 1)) − 1)1+1
:= (22 − (2/2 + 2))2
3
:= 3 + (333 + 3/3)
&Iuml; 366 := (1 + 1 + 1) &times; (1 + 111+1 )
&Iuml; 371 := 1 + (1111 − 1)/(1 + 1 + 1)
:= 222 + ((2 &times; (2 + 2 + 2))2 )
:= 222 − (222/2 + 2)
:= 33 + 333
:= 33 + (333 + 33/3)
4
:= 4 + ((((4 + 4) &times; 44) + 4/4) + 4)
:= 4 + (444 − 4)/4
:= 5/5 + (5/5 + 5) &times; (55 + 5)
:= 4 + (444/4 + 44 )
:= (5/5 + 5) &times; ((55 + 5/5) + 5)
:= 6/6 + 6 &times; (66 − 6)
:= 5/5 + ((5 &times; (5 &times; (5 + 5 + 5))) − 5)
:= 6 + 6 &times; (66 − 6)
:= 7 + (7 &times; 7 &times; 7 + (77/7))
:= 66/6 + 6 &times; (66 − 6)
:= 7 + (((7 &times; 7 &times; 7 + ((7 + 7)/7)) + 7) + 7)
:= (88/8 + 8)(8+8)/8
:= 77 + 7 &times; (7 &times; 7 − 7)
:= ((8 + 8) &times; (8 + 8)) + ((888 − 8)/8)
:= (9/9 + 9 + 9)(9+9)/9
:= (8 − 8/8) &times; (8 &times; 8 − 88/8)
:= 9/9 + (((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9))
:= 99/9 + ((9 + 9) &times; (99/9 + 9))
&Iuml; 367 := 1 + (1 + 1 + 1) &times; (1 + 111+1 )
&Iuml; 362 := 11 &times; 11 &times; (1 + 1 + 1) − 1
&Iuml; 372 := 1 + 1 + (1111 − 1)/(1 + 1 + 1)
:= 2 + (2 &times; (2 &times; (2 &times; 2 &times; 22 + 2)))
:= 222/2 + 22&times;(2+2)
:= 2 &times; ((2 &times; (2 &times; ((2 &times; 22) + 2))) + 2)
:= (33 &times; 33 − 3)/3
:= 3 + ((33 &times; 33 + 3)/3)
:= 44 + ((444 − 4)/4 − 4)
:= 3 + ((333 + 33) + 3)
:= 44 + 444/4
:= 4 + (((4 + 4) &times; 44) + 4 &times; 4)
5
5
:= 55 + (5 − 55)/(5 + 5)
:= 55 + (5 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 55)
:= (6 + 6)/6 + 6 &times; (66 − 6)
:= 6 + (6 &times; (66 − 6) + 6/6)
:= 6 + (6 &times; (66 − 6) + 6)
:= 7 + ((77 + 7)/7 + 7 &times; 7 &times; 7)
:= 7 + ((((77 − 7)/7) + 7 &times; 7 &times; 7) + 7)
:= 7/7 + (7 &times; (7 &times; 7 − 7) + 77)
:= 8/8 + ((88/8 + 8)(8+8)/8 )
:= 888/8 + ((8 + 8) &times; (8 + 8))
:= 8 + (((8 &times; 88 + 8)/((8 + 8)/8)) + 8)
:= 9 + (((9 + 9)/9) &times; ((9 &times; 9 − 9/9) + 99))
:= 9 + ((99 &times; 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
&Iuml; 378 := 1 + 11 + (1 + 1 + 1) &times; (1 + 111+1 )
&Iuml; 373 := (11 + 11)1+1 − 111
2
&Iuml; 383 := (1 + 11) &times; (11 &times; (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 &times; (3 &times; 33 + 33 )
:= ((3 + 3) &times; ((3/3 + 3)3 )) − 3/3
:= 4 + ((4/4 + 4)4 − 44 )
:= 44 + ((444 + 44)/4)
:= ((4 + 4) &times; (44 + 4)) − 4/4
:= 5 + ((55 + 5)/(5 + 5) + 55)
:= ((5 + 5)/5 + 5) &times; (55 − 5/5)
:= 5 + (((5 + 5)/5 + 5) &times; (55 − 5/5))
:= 6 + ((6 &times; (66 − 6) + 6/6) + 6)
:= 6 &times; 66 − 6 − 6 − 6
:= 6 &times; 66 − (6/6 + 6 + 6)
:= 77 + (7 &times; (7 &times; 7 − 7) + ((7 + 7)/7))
:= 7 &times; (7 &times; 7 + 7) − (7 + 7)
:= 7 &times; (7 &times; 7 + 7) − ((7 + 7)/7 + 7)
:= (8 &times; (8 &times; 8 − (8 + 8))) − 88/8
:= (8 − (8 + 8)/8) &times; (8 &times; 8 − 8/8)
:= (8 &times; (8 &times; 8 − (8 + 8))) − 8/8
:= 9 + (((9 &times; 9 &times; 9 &times; 9) − 9)/(9 + 9))
:= (9 + 9) &times; (((99 + 9)/9) + 9)
:= 9 + ((((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9)) + 9)
2
&Iuml; 374 := 11 &times; (1 + 11 &times; (1 + 1 + 1))
&Iuml; 379 := (11 − 1) &times; (1 + 111/(1 + 1 + 1)) − 1
&Iuml; 384 := (1 + 11) &times; (11 &times; (1 + 1 + 1) − 1)
:= 22 + (22 &times; 22+2 )
:= 2/2 + ((22 − 2) − 22)
:= 22+2 &times; (22 + 2)
:= 33/3 &times; (3/3 + 33)
:= 3 + ((((3/3 + 3) + 3)3 ) + 33)
:= (3 + 3) &times; ((3/3 + 3)3 )
:= (44 &times; (4 &times; 4 &times; 4 + 4))/(4 + 4)
:= 444 − (44 + 4)/4
:= (4 + 4) &times; (44 + 4)
:= (5 &times; (5 &times; (5 + 5 + 5))) − 5/5
:= 5 + ((5 &times; (5 &times; (5 + 5 + 5))) − 5/5)
:= ((55 + 5)/5) &times; ((5 + 5)/5)5
:= (66/6) &times; (6 &times; 6 − ((6 + 6)/6))
:= 6 &times; 66 − ((66/6) + 6)
:= 6 &times; ((6 + 6)/6)6
:= 7 &times; (7 &times; 7 + 7) − (77/7 + 7)
:= 7/7 + (7 &times; (7 &times; 7 + 7) − (7 + 7))
:= (7/7 + 7) &times; (7 &times; 7 − 7/7)
:= 888 − (8 &times; 8 &times; 8 + (8 + 8)/8)
:= 8 + ((8 − 8/8) &times; (8 &times; 8 − 88/8))
:= 8 &times; (8 &times; 8 − (8 + 8))
2
:= 9 + (((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9))
&Iuml; 375 := 1 + 11 &times; (1 + 11 &times; (1 + 1 + 1))
2
9
:= 9 &times; 99 − (((9 + 9)/9) )
&Iuml; 380 := (11 − 1) &times; (1 + 111/(1 + 1 + 1))
:= ((9 + 9)/9) &times; (999/9 + 9 &times; 9)
&Iuml; 385 := 11 &times; (1 + 1 + 11 &times; (1 + 1 + 1))
:= 2 + (22 − 222/2)
:= 2 + ((22 − 2) − 22)
:= 2/2 + (22+2 &times; (22 + 2))
:= 3 &times; ((3 − 3/3 + 3)3 )
:= 3 + ((33/3 &times; (3/3 + 33)) + 3)
:= 3/3 + ((3 + 3) &times; ((3/3 + 3)3 ))
:= 4 + ((444/4 + 44 ) + 4)
:= 444 − 4 &times; 4 &times; 4
:= 4/4 + ((4 + 4) &times; (44 + 4))
2
:= 5 &times; (5 &times; (5 + 5 + 5))
:= 5 + (5 &times; (5 &times; (5 + 5 + 5)))
:= 55 &times; ((5 + 5)/5 + 5)
:= 6 + (66 &times; 66/(6 + 6) + 6)
:= 6 &times; 66 + (((6 − 66)/6) − 6)
:= 6 &times; 66 − 66/6
:= 7 + (((7 &times; 7 &times; 7 + (77/7)) + 7) + 7)
:= 7 &times; (7 &times; 7 + 7) − (77 + 7)/7
:= 7 &times; (7 &times; 7 + 7) − 7
:= 888 − (8 &times; 8 &times; 8 + 8/8)
:= (8 &times; (88 + 8) − 8)/((8 + 8)/8)
:= 8/8 + (8 &times; (8 &times; 8 − (8 + 8)))
:= ((9 + 9) &times; (9 + 9 + 9)) − 999/9
:= (9/9 + 9 + 9) &times; (99/9 + 9)
:= 99/9 &times; ((((9 − 9/9) + 9) + 9) + 9)
&Iuml; 376 := 1 + 1 + 11 &times; (1 + 11 &times; (1 + 1 + 1))
:= (22 − 2)2 − (22 + 2)
:= 33 + (((3/3 + 3) + 3)3 )
&Iuml; 381 := 11 + (1111 − 1)/(1 + 1 + 1)
2
:= 2 + (((22 − 2) − 22) + 2/2)
3
:= 3 + (3 &times; (3 &times; 33 + 3 ))
4
&Iuml; 386 := 1 + 11 &times; (1 + 1 + 11 &times; (1 + 1 + 1))
:= 2 + (22+2 &times; (22 + 2))
:= 33+3 − (((3/3 + 3) + 3)3 )
:= (4 + 4) &times; (44 − 4/4 + 4)
:= 444 + ((4 − 4 )/4)
:= 44 + ((44 + 4)/((4 + 4)/4))
:= 5/5 + (5 &times; (5 &times; (5 + 5 + 5)))
:= 5 + ((5 &times; (5 &times; (5 + 5 + 5))) + 5/5)
:= 5 &times; 55 + 555/5
:= 6 + (6 &times; (66 − 6) + ((66 − 6)/6))
:= 6 + ((66 &times; 66/(6 + 6) + 6) + 6)
:= 6 &times; 66 + (6 − 66)/6
:= (7/7 + 7) &times; 7 &times; 7 − (7 + 7)/7
:= 7 &times; (7 &times; 7 + 7) − 77/7
:= 7/7 + (7 &times; (7 &times; 7 + 7) − 7)
:= 888 − 8 &times; 8 &times; 8
:= 8 + ((8 &times; (8 &times; 8 − (8 + 8))) − (88/8))
:= (8 + 8)/8 + (8 &times; (8 &times; 8 − (8 + 8)))
:= 9 &times; 9 + ((99 &times; (9 + 9 + 9) − (9 + 9))/9)
:= 9 + (((99 &times; 99)/(9 + 9 + 9)) + 9)
:= (((9 + 9)/9)9 ) − ((99 + 9 + 9) + 9)
&Iuml; 377 := 11 + (1 + 1 + 1) &times; (1 + 111+1 )
2
:= (22/2) + 2
:= (22+2 &times; (22 + 2)) − 2
2&times;(2+2)
:= 3 + (33/3 &times; (3/3 + 33))
4
&Iuml; 382 := 1 + 11 + (1111 − 1)/(1 + 1 + 1)
4
:= 3333/3 − 3
4
3+3
&Iuml; 387 := 1 + 1 + 11 &times; (1 + 1 + 11 &times; (1 + 1 + 1))
:= (22 − 2)2 − (22/2 + 2)
:= 3 + ((3 + 3) &times; ((3/3 + 3)3 ))
4
:= 4 + (((4/4 + 4) − 4 ) + 4)
:= 4 + ((4 − 4)/((4 + 4)/4))
:= 4 + (((4 + 4) &times; (44 + 4)) − 4/4)
:= (5 + 5)/5 + (5 &times; (5 &times; (5 + 5 + 5)))
:= 55 /5 − ((5 − (5 + 5)/5)5 )
:= 5 &times; 55 + (555 + 5)/5
:= 6 + (6 &times; (66 − 6) + (66/6))
:= 6 &times; 66 − ((6 + 6)/6 + 6 + 6)
:= 6 &times; 66 + (((6 − 66) + 6)/6)
:= 7 &times; (7 &times; 7 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 &times; (7 &times; 7 + 7)
:= ((7 + 7)/7) + (7 &times; (7 &times; 7 + 7) − 7)
:= 8/8 + (888 − 8 &times; 8 &times; 8)
:= (8 &times; (8 &times; 8 − (8 + 8))) − (8 + 8)/8
:= 888 + (88/8 − 8 &times; 8 &times; 8)
:= 9 &times; 9 + ((99 &times; (9 + 9 + 9) − 9)/9)
:= 9999/9 − 9 &times; 9 &times; 9
:= ((9 + 9) &times; (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
&Iuml; 388 := ((1 + 1) &times; (11 − 1))1+1 − 1 − 11
&Iuml; 393 := (1 + 1 + 1) &times; (11 &times; (1 + 11) − 1)
&Iuml; 398 := ((1 + 1) &times; (11 − 1))1+1 − 1 − 1
:= 2 &times; (((22+2 − 2)2 ) − 2)
:= 2/2 + (2 &times; ((22+2 − 2)2 ))
:= (22 − 2)2 − 2
:= 3 + (((3 + 3) &times; ((3/3 + 3)3 )) + 3/3)
:= (33 &times; (3 &times; 3 + 3)) − 3
:= 3 + ((33 &times; (3 &times; 3 + 3)) − 3/3)
:= 4 + ((4 + 4) &times; (44 + 4))
:= 4 + ((((4 + 4) &times; (44 + 4)) + 4/4) + 4)
:= 444 − ((4 + 4)/4 + 44)
:= 5 &times; 55 + (555 + 5 + 5)/5
:= 5 &times; 5 + ((5 + 5)/(5 + 5) + 55)
:= (5 − 5/5)5 − (55 + 5)/5
:= 6 &times; 66 − ((6 + 6)/6 + 6)
:= 6 &times; 66 − 6 &times; 6/(6 + 6)
:= (6 + 6)/6 + 6 &times; 66
5
:= ((7 &times; 777) − 7)/(7 + 7)
:= 7/7 + 7 &times; (7 &times; 7 + 7)
:= 7 + (7 &times; (7 &times; 7 + 7) − 7/7)
:= (8 &times; (88 + 8) + 8)/((8 + 8)/8)
:= 8 + ((8 &times; (8 &times; 8 − (8 + 8))) + 8/8)
:= 8 + ((8 − (8 + 8)/8) &times; (8/8 + 8 &times; 8))
:= 9 + (9 &times; 99 − (((9 + 9)/9)9 ))
:= 9 + (((9 + 9)/9) &times; (999/9 + 9 &times; 9))
:= ((9 + 9)/9) &times; ((9/9 + 99) + 99)
&Iuml; 389 := ((1 + 1) &times; (11 − 1))1+1 − 11
&Iuml; 394 := 1 + (1 + 1 + 1) &times; (11 &times; (1 + 11) − 1)
&Iuml; 399 := ((1 + 1) &times; (11 − 1))1+1 − 1
:= (22 − 2)2 − 2/2
:= (22 − 2)2 − 22/2
:= 2 + (2 &times; ((22+2 − 2)2 ))
:= 33 + ((33 &times; 33 − 3)/3)
:= 3/3 + ((33 &times; (3 &times; 3 + 3)) − 3)
:= 3 + (33 &times; (3 &times; 3 + 3))
:= 4 + (((4 + 4) &times; (44 + 4)) + 4/4)
:= 4 + (((4 + 4)/4 + 4) &times; (4 + 4)/4)
:= 4 &times; 44 − (4/4 + 4)4
:= 5 &times; (5 &times; 5 + 55) − 55/5
:= (5 − 5/5)5 − (55 /5 + 5)
:= (5 − 5/5)5 − 55 /5
:= 6 &times; 66 − 6/6 − 6
:= 6 &times; 66 − (6 + 6)/6
:= 6 &times; 66 + (6 &times; 6/(6 + 6))
:= ((7 &times; 777) + 7)/(7 + 7)
4
:= ((7 + 7)/7) + 7 &times; (7 &times; 7 + 7)
:= 7 + 7 &times; (7 &times; 7 + 7)
:= (((8 − 8/8) &times; 888) + 8)/(8 + 8)
:= 8 + ((8 &times; (8 &times; 8 − (8 + 8))) + ((8 + 8)/8))
:= (8 − 8/8) &times; ((8/8 − 8) + 8 &times; 8)
:= 9 + ((9/9 + 9 + 9) &times; (99/9 + 9))
:= 9 &times; 9 + ((9 + 9) &times; (9 + 9) − (99/9))
:= (9/9 + 9 + 9) &times; (((99 + 9)/9) + 9)
&Iuml; 390 := 1 + ((1 + 1) &times; (11 − 1))1+1 − 11
&Iuml; 395 := 11 &times; (1 + 1 + 1) &times; (1 + 11) − 1
&Iuml; 400 := ((1 + 1) &times; (11 − 1))1+1
:= (2 &times; ((22+2 − 2)2 )) − 2
:= (22 − 2)2 − (2/2 + 2 + 2)
:= (22 − 2)2
:= 33 + (33 &times; 33/3)
:= (33 &times; (3 &times; 3 + 3)) − 3/3
:= 3 + ((33 &times; (3 &times; 3 + 3)) + 3/3)
:= ((4 + 4)/4 + 4) &times; (4 + 4)/4
:= 44 + (((4 + 4) &times; 44) − 4/4)
:= 444 − 44
:= 5 + (55 &times; ((5 + 5)/5 + 5))
:= 5 &times; (5 &times; 5 + 55) − 5
:= 5 &times; (5 &times; 5 + 55)
:= 6 &times; 66 − 6
:= 6 &times; 66 − 6/6
:= 6 + (6 &times; 66 − ((6 + 6)/6))
:= 7 &times; (7 &times; 7 + 7) − (7 + 7)/7
:= 7 + (((7 &times; 777) − 7)/(7 + 7))
:= 7 + (7 &times; (7 &times; 7 + 7) + 7/7)
:= (8 − (8 + 8)/8) &times; (8/8 + 8 &times; 8)
:= 88/8 + (8 &times; (8 &times; 8 − (8 + 8)))
:= 8 + ((8 &times; (8 &times; 8 − (8 + 8))) + 8)
4
:= ((9 &times; 9 &times; 99) − 999)/(9 + 9)
&Iuml; 391 := (1 + 1)11−1−1 − 111+1
9
:= (((9 + 9)/9) ) − (99 + 9 + 9)
&Iuml; 396 := 11 &times; (1 + 1 + 1) &times; (1 + 11)
:= (99/9 + 9)(9+9)/9
&Iuml; 401 := 1 + ((1 + 1) &times; (11 − 1))1+1
:= 2 + ((22 − 2)2 − 22/2)
:= 22 &times; (22+2 + 2)
:= 2/2 + (22 − 2)2
:= 33 + ((33 &times; 33 + 3)/3)
:= 33 &times; (3 &times; 3 + 3)
:= 3 + (((33 &times; (3 &times; 3 + 3)) − 3/3) + 3)
:= 444 + ((44 − 4 )/4)
:= 44 + ((4 + 4) &times; 44)
:= 4/4 + (444 − 44)
:= 5 + (555/5 + 5 &times; 55)
:= 5/5 + (5 &times; (5 &times; 5 + 55) − 5)
:= 5/5 + 5 &times; (5 &times; 5 + 55)
:= 6/6 + (6 &times; 66 − 6)
:= 6 &times; 66
:= 6 + (6 &times; 66 − 6/6)
:= 7 &times; (7 &times; 7 + 7) − 7/7
:= 7 + (((7 &times; 777) + 7)/(7 + 7))
:= 7 + (7 &times; (7 &times; 7 + 7) + ((7 + 7)/7))
:= (88 &times; (8 &times; 8 + 8))/(8 + 8)
:= 8 &times; 8 &times; 8 − 888/8
:= 9 &times; 9 + ((9 + 9) &times; (9 + 9) − 9)
:= (((9 + 9)/9)9 ) − 999/9
4
:= 8 + ((8 &times; (8 &times; 8 − (8 + 8))) − 8/8)
:= ((99/9 + 9)
(9+9)/9
)−9
&Iuml; 392 := (1 + 1) &times; (1 + 1 + 1 + 11)1+1
&Iuml; 397 := 1 + 11 &times; (1 + 1 + 1) &times; (1 + 11)
&Iuml; 402 := 1 + 1 + ((1 + 1) &times; (11 − 1))1+1
:= 2 &times; ((22+2 − 2)2 )
:= (22 − 2)2 − 2/2 − 2
:= 2 + (22 − 2)2
:= (3/3 + 3) &times; (3 &times; 33 − 3/3)
:= 3/3 + (33 &times; (3 &times; 3 + 3))
:= 3 + ((33 &times; (3 &times; 3 + 3)) + 3)
:= 4 + (((4 + 4) &times; (44 + 4)) + 4)
:= 44 + (((4 + 4) &times; 44) + 4/4)
:= 444 + ((4 + 4)/4 − 44)
:= ((5 + 5)/5 + 5) &times; (55 + 5/5)
:= 5 + (((5 + 5)/5 + 5) &times; (55 + 5/5))
:= (5 + 5)/5 + 5 &times; (5 &times; 5 + 55)
:= (6 + 6)/6 + (6 &times; 66 − 6)
:= 6/6 + 6 &times; 66
:= 6 + 6 &times; 66
:= 7 &times; (7 &times; 7 + 7)
:= 7 + (7 &times; (7 &times; 7 + 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 &times; (7 &times; 7 + 7)
:= 8 + (8 &times; (8 &times; 8 − (8 + 8)))
:= 8/8 + ((88 &times; (8 &times; 8 + 8))/(8 + 8))
:= 8 &times; 8 &times; 8 + ((8 − 888)/8)
:= (9 − 9/9) &times; ((9 &times; 99 − 9)/(9 + 9))
:= (((9 + 9) &times; (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
&Iuml; 403 := 1 + 1 + 1 + ((1 + 1) &times; (11 − 1))1+1
:= 2 + (22 − 2)2 + 2/2
&Iuml; 413 := 1 + 1 + 11 + ((1 + 1) &times; (11 − 1))1+1
&Iuml; 408 := 1 + 11 &times; 111/(1 + 1 + 1)
:= 2 &times; (2 + 2) + (22 − 2)2
:= 2 + (22 − 2)2 + 22/2
:= 3 + (33 &times; ((3 &times; 3 + 3) + 3))
:= 3 &times; 33 + (333 − 3/3)
:= 4 + (4 &times; 4 − (4/4 + 4) )
:= 4 + ((444 − 44) + 4)
:= 44 + ((4/4 + 4)4 − 44 )
:= 5 + ((5 − 5/5)5 − (55 + 5)/5)
:= (5 &times; 5 − 5/5) &times; (((55 + 5)/5) + 5)
:= 5 &times; (5 &times; 5 − 5) + (55 + 5)/(5 + 5)
:= 6 + (6 &times; 66 + 6/6)
:= 6 + (6 &times; 66 + 6)
:= 6 + (6 &times; 66 + (66/6))
:= 77/7 + 7 &times; (7 &times; 7 + 7)
:= (7/7 + 7) &times; ((7 + 7)/7 + 7 &times; 7)
:= 77 + (7 &times; 7 &times; 7 − 7)
:= 8 + ((8 &times; (8 &times; 8 − (8 + 8))) + (88/8))
:= 8 &times; 8 &times; 8 − (88 + 8 + 8)
:= 8 &times; 8 &times; 8 − (88/8 + 88)
:= 9 &times; 9 + ((9 + 9) &times; (9 + 9) − ((9 + 9)/9))
:= (99 &times; (9 + 9 + 9) + 999)/9
:= (((9 + 9)/9)9 ) − 99
:= 3 + (((33 &times; (3 &times; 3 + 3)) + 3/3) + 3)
4
4
&Iuml; 404 := (1 + 1) &times; ((1 + 1) &times; (1 + (11 − 1)1+1 ))
2
2
:= 22/2 + (22 − 2) − 2
:= 2 + (22 − 2) + 2
3
:= 333 + (((3 + 3) − 3)/3)
:= 3 + ((3 &times; ((3 &times; 3 + 3) + 3)) + 3/3)
:= 4 + ((4/4 + 4) &times; (4 − 4/4) )
5
:= 5 + ((5 − 5/5) − 5 /5)
:= 6 + (((6 + 6)/6) + 6 &times; 66)
7
5
5
:= 5 + (((5 − 5/5) − 5 /5) + 5)
:= 6 + ((6 &times; 66 + 6/6) + 6)
:= 7 &times; 77 − (((7 + 7)/7) + 7)
:= 77 + (7 &times; 7 &times; 7 − (77/7))
:= 8 + ((88 &times; (8 &times; 8 + 8))/(8 + 8))
:= 8 + (8 &times; 8 &times; 8 − 888/8)
9
:= (((9 + 9)/9) ) − (99 + 9)
&Iuml; 405 := 11 &times; 111/(1 + 1 + 1) − 1 − 1
&Iuml; 414 := (1 + 1) &times; (11 + (1 + 1 + 1 + 11)1+1 )
:= 2 + 2 &times; (222 − 22+2 )
3
4
:= 4 + (444 − 44)
5
&Iuml; 409 := 1 + 1 + 11 &times; 111/(1 + 1 + 1)
:= 9 + ((99/9 + 9)(9+9)/9 )
&Iuml; 410 := 11 + ((1 + 1) &times; (11 − 1))1+1 − 1
2
:= 3 &times; 33 + 333
:= 4 + ((4 + 4)4 + 4)/((44 − 4)/4)
:= (5 − 5/5)5 − (555 + 55)
:= 6 + ((6 &times; 66 + 6) + 6)
:= 7/7 + ((7 &times; 7 &times; 7 − 7) + 77)
:= 8 &times; 8 &times; 8 + ((8 − 88)/8 − 88)
:= 9 + ((9 + 9) &times; (9 + 9) + 9 &times; 9)
&Iuml; 415 := (1 + 1 + 11) &times; (11 &times; (1 + 1 + 1) − 1) − 1
:= 2 + (22 − 2) + 2/2 + 2
:= 2 + (22 − 2) + 2 &times; (2 + 2)
:= 2 + (22 − 2)2 + 22/2 + 2
:= 33 &times; ((3 &times; 3 + 3) + 3)
:= 3 + 33 &times; 333/33
:= 3/3 + (3 &times; 33 + 333)
2
4
:= (((4 + 4) ) + 4)/((44 − 4)/4)
:= 44 + (4 &times; (44 − 4) − 4/4)
:= 5 + 5 &times; (5 &times; 5 + 55)
:= 5 + (5 &times; (5 &times; 5 + 55) + 5)
:= 5 + ((5 &times; (5 &times; 5 + 55) + 5) + 5)
:= 6 + ((6 &times; 6/(6 + 6)) + 6 &times; 66)
:= 6 + ((((6 + 6)/6) + 6 &times; 66) + 6)
:= 6 + (((6 &times; 66 + 6/6) + 6) + 6)
:= 7 + ((7 &times; (7 &times; 7 + 7) − 7/7) + 7)
:= 7 + (7 &times; (7 &times; 7 + 7) + (77/7))
:= 7 + ((7/7 + 7) &times; ((7 + 7)/7 + 7 &times; 7))
:= 8 &times; 8 &times; 8 − ((88/8 + 88) + 8)
:= 8 + (((8 − 888)/8) + 8 &times; 8 &times; 8)
:= 8 &times; 8 &times; 8 − ((8/8 + 88) + 8)
:= 9 &times; (((9 + 9 + 9) + 9) + 9)
:= 9 + ((((9 + 9)/9)9 ) − 999/9)
:= 9 + (((9 + 9) &times; (9 + 9) + 9 &times; 9) + 9/9)
:= (4/4 + 4) &times; (4 − 4/4)4
&Iuml; 406 := 11 &times; 111/(1 + 1 + 1) − 1
2
:= 2 + (22 − 2) + 2 + 2
:= 3/3 + (33 &times; ((3 &times; 3 + 3) + 3))
&Iuml; 411 := 11 + ((1 + 1) &times; (11 − 1))1+1
:= 22/2 + (22 − 2)2
&Iuml; 416 := (1 + 1 + 11) &times; (11 &times; (1 + 1 + 1) − 1)
:= 22+2 + (22 − 2)2
3
:= 3 + ((333 − 3/3) + 3 &times; 33 )
:= 3 &times; 3 + (333 − 3)
4
:= 4 + (((4 + 4)/4 − 44) + 444)
:= 44 + (444/4 + 4 )
:= 44 + (4 &times; (44 − 4))
:= 5 + (5 &times; (5 &times; 5 + 55) + 5/5)
:= 55/5 + 5 &times; (5 &times; 5 + 55)
:= (55/5 + 5) &times; (5 &times; 5 + 5/5)
:= 6 &times; 66 + (66 − 6)/6
:= 6 + (((6 &times; 6/(6 + 6)) + 6 &times; 66) + 6)
:= 6 + (((((6 + 6)/6) + 6 &times; 66) + 6) + 6)
7
:= 7 + (7 &times; (7 &times; 7 + 7) + 7)
:= 7 &times; 77 − ((7 + 7)/7)
:= ((77 &times; 77 − 7)/(7 + 7)) − 7
:= (8 − 8/8) &times; ((((8 + 8)/8) − 8) + 8 &times; 8)
:= 8 &times; 8 &times; 8 − (8888/88)
:= 8 &times; 8 &times; 8 − (88 + 8)
:= 9/9 + ((9 + 9) &times; (9 + 9) + 9 &times; 9)
&Iuml; 407 := 11 &times; 111/(1 + 1 + 1)
2
9
:= (((9 + 9)/9) ) − ((9 + 9)/9 + 99)
&Iuml; 412 := 1 + 11 + ((1 + 1) &times; (11 − 1))1+1
2+2
:= 9 &times; 9 + ((9 + 9) &times; (9 + 9) + (99/9))
&Iuml; 417 := 11 &times; (1 + 111/(1 + 1 + 1)) − 1
:= (22 − 2/2)2 − 22 − 2
:= 2 + (22 − 2) + 2/2 + 2 + 2
:= 2 &times; (222 − 2
:= 33 &times; 333/33
:= 3/3 + ((3 &times; 33 − 3) + 333)
:= 3 + (3 &times; 33 + 333)
:= 4 + ((4 &times; 44 − (4/4 + 4)4 ) + 4)
:= 444 − 4 &times; (4 + 4)
:= 4/4 + (4 &times; (44 − 4) + 44 )
:= 55/5 &times; (((5 + 5)/5)5 + 5)
:= 5 + (55/5 &times; (((5 + 5)/5)5 + 5))
:= ((55 + 55 ) + 5)/(5 + 5 + 5)
:= 6 &times; 66 + 66/6
:= 6 + (((66 − 6)/6) + 6 &times; 66)
:= 6 &times; 66 + ((6 &times; 6 + 6)/((6 + 6)/6))
:= 7 + ((7 &times; (7 &times; 7 + 7) + 7/7) + 7)
:= 77 + (7 &times; 7 &times; 7 − (7/7 + 7))
:= ((77 &times; 77 + 7)/(7 + 7)) − 7
:= 8888/8 − 8 &times; 88
:= (888 − 8 &times; 8)/((8 + 8)/8)
:= 8/8 + (8 &times; 8 &times; 8 − (88 + 8))
:= 9 &times; 9 + ((9 + 9) &times; (9 + 9) + ((9 + 9)/9))
:= (((9 + 9)/9)9 ) − (9/9 + 99)
:= 9 + ((99 &times; (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
&Iuml; 418 := 11 &times; (1 + 111/(1 + 1 + 1))
&Iuml; 423 := 1 + (1 + 1) &times; ((1 + 1) &times; 111 − 11)
2
&Iuml; 428 := (11 &times; ((1 + 1 + 1) &times; (1 + 1 + 11))) − 1
:= 22 + ((22 − 2) + 2/2)
:= 2 &times; (222 − 2 &times; (2 + 2))
:= 33/3 &times; (33/3 + 3 )
:= 3 &times; ((3 &times; (33 + 3)) + 33)
:= ((((3 + 3)3/3+3 ) − 3)/3) − 3
:= 44 + (4 &times; (44 − 4) + (4 + 4)/4)
:= 444 − ((4 &times; 4 + 4/4) + 4)
:= 444 − 4 &times; 4
:= 22 &times; (22 − 2/2 − 2)
3
5
:= 5 + ((55 + 5)/(5 + 5) + 5 &times; (5 &times; 5 − 5))
:= 55 + ((5 + 5)/(5 + 5) + 55)
:= (5 − 5/5) &times; ((555 + 5)/5 − 5)
:= 6 &times; 66 + ((66 + 66)/6)
:= 66 + (66 &times; 66/(6 + 6) − 6)
:= (6 + 6)/6 + ((6 &times; (66 + 6)) − 6)
:= 7 + (7 &times; 77 − ((7 + 7)/7) )
:= (77 &times; 77 − 7)/(7 + 7)
:= 7 &times; 77 − 777/7
:= 8 &times; 8 &times; 8 + (((8 + 8)/8) − (88 + 8))
:= 8 &times; 8 &times; 8 − (8/8 + 88)
:= 888/((8 + 8)/8) − 8 − 8
:= 9 + (((99/9 + 9)(9+9)/9 ) + 9)
:= 99 + (9 + 9) &times; (9 + 9)
:= (((9 &times; 9 &times; 99) + 9)/(9 + 9)) − (9 + 9)
7
&Iuml; 419 := 1 + 11 &times; (1 + 111/(1 + 1 + 1))
:= (22 − 2/2)2 − 22
&Iuml; 424 := (1 + 1) &times; (1 + ((1 + 1) &times; 111 − 11))
&Iuml; 429 := 11 &times; ((1 + 1 + 1) &times; (1 + 1 + 11))
:= (2 &times; (222 − 2)) − 22/2
2
:= 2 + ((22 − 2) + 22)
:= ((33 + 3) &times; (33/3 + 3)) − 3/3
:= 3 &times; 3 + (((3/3 + 3) + 3) )
:= 33 + (33 &times; (3 &times; 3 + 3))
:= (44 &times; 44 − (44 + 4))/4
:= 444 − (4 &times; 4 + 4)
:= 4/4 + (444 − 4 &times; 4)
:= 555 − (555/5 + 5 &times; 5)
:= (5 − 5/5) &times; (555/5 − 5)
:= 555 − (5 &times; 5 &times; 5 + 5/5)
:= 6 + ((6 &times; 66 + (66/6)) + 6)
:= 6 &times; (66 − 6) + ((6 + 6)/6)6
:= 66 + 66 &times; 66/(6 + 6)
:= 77 + (7 &times; 7 &times; 7 − 7/7)
:= (77 &times; 77 + 7)/(7 + 7)
:= 8 + (8 &times; 8 &times; 8 − (8888/88))
:= 8 &times; 8 &times; 8 − 88
:= (((9 &times; 999) + 9)/(9 + 9)) − 9 &times; 9
:= 9/9 + ((9 + 9) &times; (9 + 9) + 99)
&Iuml; 420 := (1 + 1) &times; (11 − 1) &times; (11 + 11 − 1)
3
3
&Iuml; 425 := 1 + (1 + 1) &times; (1 + ((1 + 1) &times; 111 − 11))
:= 22 + (22 − 2)2 − 2
:= ((22 − 2/2)2 ) − 22+2
:= (33 + 3) &times; (33/3 + 3)
:= (((3 + 3) &times; ((3 + 3)3 − 3)) − 3)/3
:= 4 + (4 &times; (44 − 4) + 44 )
:= 4/4 + (444 − (4 &times; 4 + 4))
:= (55 + 5) &times; ((5 + 5)/5 + 5)
:= 5 &times; (5 &times; 5 + 55 + 5)
:= 6 &times; (((6 + 6)/6)6 + 6)
:= (6 &times; (66 + 6)) − 6/6 − 6
:= 77 + 7 &times; 7 &times; 7
:= 7 + ((7 &times; 77 − ((7 + 7)/7)7 ) + 7)
:= 88 &times; 88/(8 + 8) − 8 &times; 8
:= 8/8 + (8 &times; 8 &times; 8 − 88)
:= ((9 + 9)/9) &times; (999/9 + 99)
:= 99 + ((9 + 9) &times; (9 + 9) + ((9 + 9)/9))
&Iuml; 421 := (1 + 1 + 1) &times; (1 + 11)1+1 − 11
&Iuml; 426 := (1 + 1) &times; (1 + 1) &times; (1 + 111) − 11
:= 7 &times; 77 + ((7 − 777)/7)
:= 8 &times; (8 &times; 8 − 8) − (88/8 + 8)
:= (((9 + 9)/9)9 ) − (((9 + 9)/9) + 9 &times; 9)
&Iuml; 430 := ((11 + 11 − 1)1+1 ) − 11
:= 2 + (2 &times; (222 − 2 &times; (2 + 2)))
:= (3 − 3/3) &times; ((3 + 3)3 − 3/3)
:= 44 + ((4 &times; 44) − (4 + 4)/4)
:= 555 − 5 &times; 5 &times; 5
:= (6 &times; (66 + 6)) − (6 + 6)/6
:= 7 + ((77 &times; 77 − 7)/(7 + 7))
:= 8 + (8 &times; 8 &times; 8 − ((8 + 8)/8 + 88))
:= (((9 + 9)/9)9 ) − (9/9 + 9 &times; 9)
&Iuml; 431 := ((1 + 1 + 1) &times; (1 + 11)1+1 ) − 1
:= 2 &times; 222 − (22/2 + 2)
:= 2 + (((22 − 2/2)2 ) − 22)
:= (2 &times; (222 + 2)) − 22
:= (((3 + 3)3/3+3 ) − 33)/3
:= (3 + 3) &times; (((3 + 3)3 − 3)/3)
:= 44 + ((4 &times; 44) − 44/4)
:= 444 − ((4 + 4)/4 + 4 &times; 4)
:= 5 + ((55/5 + 5) &times; (5 &times; 5 + 5/5))
:= 5/5 + (5 &times; (5 &times; 5 + 55 + 5))
:= 5/5 + (555 − 5 &times; 5 &times; 5)
:= (6 &times; (66 + 6)) − 66/6
:= (6 &times; (66 + 6)) − 6
:= (6 &times; (66 + 6)) − 6/6
:= 7/7 + (7 &times; 7 &times; 7 + 77)
:= 7 + ((7 &times; 7 &times; 7 − 7/7) + 77)
:= 7 + ((77 &times; 77 + 7)/(7 + 7))
:= 8 + (8 &times; 8 &times; 8 − (88/8 + 88))
:= 8 &times; 8 &times; 8 + (((8 + 8)/8) − 88)
:= 8 + (8 &times; 8 &times; 8 − (8/8 + 88))
:= 9 + ((((9 + 9)/9)9 ) − (9/9 + 99))
:= (9 + 9) &times; (9 + 9) + (999/9 − 9)
:= (((9 + 9)/9)9 ) − 9 &times; 9
&Iuml; 422 := (1 + 1) &times; (1 + 1) &times; 111 − 11
&Iuml; 427 := 1 + (1 + 1) &times; ((1 + 1) &times; (1 + 111) − 11)
:= (((3 + 3)3/3+3 ) − 3)/3
:= 44 + ((4 &times; 44) − 4/4)
&Iuml; 432 := (1 + 1 + 1) &times; (1 + 11)1+1
:= 22 + (22 − 2)2
:= 2 + (((22 − 2/2)2 ) − 22+2 )
:= 2 &times; ((2 + 2 + 2)2/2+2 )
:= 33 + ((33 &times; (3 &times; 3 + 3)) − 3/3)
:= (((3 + 3) &times; ((3 + 3)3 − 3)) + 3)/3
:= 3 &times; ((3 + 3) &times; (33 − 3))
:= 444 − (44/((4 + 4)/4))
:= 444 − (4 &times; 4 + 4/4)
:= 44 + (4 &times; 44)
:= 5 + (((55 + 55 ) + 5)/(5 + 5 + 5))
:= (5 + 5)/5 + (5 &times; (5 &times; 5 + 55 + 5))
:= 5 &times; 5 &times; 5 + (55 − 55)/(5 + 5)
:= ((6 − 66)/6) + (6 &times; (66 + 6))
:= 6/6 + ((6 &times; (66 + 6)) − 6)
:= 6 &times; (66 + 6)
:= 77 + (7 &times; 7 &times; 7 + ((7 + 7)/7))
:= 7 + (7 &times; 7 &times; 7 + 77)
:= ((7 + 7)/7 + 7) &times; (7 &times; 7 − 7/7)
:= 8 &times; 8 &times; 8 − ((8 + 8)/8 + 88)
:= 8 &times; 8 &times; 8 + (88/8 − (88 + 8))
:= 8 + (8 &times; 8 &times; 8 − 88)
:= (((9 &times; 9 &times; 99) − 9)/(9 + 9)) − (9 + 9)
:= 9 + ((9 + 9) &times; (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
&Iuml; 433 := 1 + ((1 + 1 + 1) &times; (1 + 11)1+1 )
:= (((3 + 3)
3/3+3
&Iuml; 438 := (1 + 1) &times; ((1 + 1) &times; (111 − 1) − 1)
) + 3)/3
:= 444 − 44/4
5
&Iuml; 443 := ((1 + 1) &times; (1 + 1) &times; 111) − 1
:= 2 + ((22 − 2/2)2 )
:= (2 &times; (222 − 2)) − 2
:= 2 &times; 222 − 22/2
3
:= (3 + 3) &times; (((3 + 3) + 3)/3)
:= 333 + ((333 − 3)/3)
:= 444 − ((4 + 4)/4 + 4)
:= 444 − 4/4
5
:= 5 &times; 5 &times; 5 + ((5 + 5)/(5 + 5) − 5)
:= 5 &times; 5 &times; 5 + (5 + 5)/(5 + 5)
:= 555 − (555 + 5)/5
:= 6/6 + (6 &times; (66 + 6))
:= 6 + (6 &times; (66 + 6))
:= 66/6 + (6 &times; (66 + 6))
:= 777 − (7 &times; 7 &times; 7 + 7/7)
:= 7 + (((77 &times; 77 + 7)/(7 + 7)) + 7)
:= ((7 + 7)/7) + 7 &times; (7 &times; 7 + 7 + 7)
:= 8 + ((8 &times; 8 &times; 8 − 88) + 8/8)
:= (8 − 88)/8 + 8 &times; (8 &times; 8 − 8)
:= 888/((8 + 8)/8) − 8/8
:= 9 + (((9 + 9) &times; (9 + 9) + 99) + 9/9)
&Iuml; 434 := 1 + (1 + ((1 + 1 + 1) &times; (1 + 11)1+1 ))
9
:= 9 + ((((9 + 9)/9) ) − (((9 + 9)/9) + 9 &times; 9))
&Iuml; 439 := ((11 + 11 − 1)1+1 ) − 1 − 1
:= 9 + (((9 + 9) &times; (9 + 9) + (99/9)) + 99)
&Iuml; 444 := (1 + 1) &times; (1 + 1) &times; 111
:= ((22 − 2/2)2 ) − 2
:= 2 &times; 222
:= (((3 + 3) &times; ((3 + 3)3 + 3)) + 3)/3
:= 333 + 333/3
:= 444 + (4 − 44)/4
:= 444 − (4/4 + 4)
:= 444
:= 5 + (555 − (5 &times; 5 &times; 5 + 5/5))
:= 555 − (555/5 + 5)
:= (5 − 5/5) &times; 555/5
:= (6 + 6)/6 + (6 &times; (66 + 6))
:= 6 + ((6 &times; (66 + 6)) + 6/6)
:= 6 + ((6 &times; (66 + 6)) + 6)
:= 777 − 7 &times; 7 &times; 7
:= 7 &times; (7 &times; 7 + 7 + 7) − (7 + 7)/7
:= (77/7 − 7) &times; 777/7
:= (8 − 8/8) &times; (8 &times; 8 − ((8 + 8)/8))
:= 8 &times; (8 &times; 8 − 8) − (8/8 + 8)
:= 888/((8 + 8)/8)
:= 99 + ((9 + 9) &times; (9 + 9) + (99/9))
:= 9 + ((((9 + 9)/9)9 ) − (9/9 + 9 &times; 9))
:= (999 &times; (9 − 9/9))/(9 + 9)
:= (2 &times; (222 − (2 + 2))) − 2
:= 3 + ((((3 + 3)
3/3+3
) − 3)/3)
&Iuml; 435 := (1 + 1 + 1) &times; (1 + (1 + 11)1+1 )
&Iuml; 440 := (1 + 1) &times; (1 + 1) &times; (111 − 1)
&Iuml; 445 := 1 + ((1 + 1) &times; (1 + 1) &times; 111)
:= 2 + (2 &times; 222 − 22/2)
:= 2 &times; (222 − 2)
:= 2/2 + 2 &times; 222
:= 3 + (3 &times; 33 + 333)
:= (3/3 + 3) &times; ((333 − 3)/3)
:= 333 + ((333 + 3)/3)
:= 444 − ((4/4 + 4) + 4)
:= 444 − 4
:= 4/4 + 444
:= 5 + (555 − 5 &times; 5 &times; 5)
:= (5 + 5) &times; (55 − (55/5))
:= 555 − (55 + 55)
:= (6 &times; (66 + 6)) + (6 &times; 6/(6 + 6))
:= 6 + ((6 &times; (66 + 6)) + ((6 + 6)/6))
:= 6 + (((6 &times; (66 + 6)) + 6/6) + 6)
:= 7 + (7 &times; 77 − 777/7)
:= 7 &times; (7 &times; 7 + 7 + 7) − 7/7
:= 77/7 + (777 − 7 &times; 7 &times; 7)
:= 8 &times; 8 &times; 8 + (88/8 − 88)
:= 8 &times; (8 &times; 8 − 8) − 8
:= 8 + (8 &times; (8 &times; 8 − 8) − (88/8))
:= (9 + 9) &times; (9 + 9) + 999/9
:= 9 + ((((9 + 9)/9)9 ) − 9 &times; 9)
:= ((9 &times; 9 &times; 99) − 9)/(9 + 9)
&Iuml; 436 := (1 + 1) &times; (1 + 1) &times; (111 − 1 − 1)
&Iuml; 441 := (11 + 11 − 1)1+1
&Iuml; 446 := (1 + 1) &times; (1 + (1 + 1) &times; 111)
:= (22 − 2/2)2
:= 2 + 2 &times; 222
:= 3 &times; (((3 + 3) &times; (33 − 3)) + 3)
:= ((3 &times; (33 &times; 33 )) + 3)/(3 + 3)
:= 444 − 4 − 4
:= 4/4 + (444 − 4)
:= 444 + (4 + 4)/4
:= 5 + ((555 − 5 &times; 5 &times; 5) + 5/5)
:= ((55/5 + 5) + 5)(5+5)/5
:= 5 + (((55/5 + 5) + 5)(5+5)/5 )
:= 6 + ((6 &times; (66 + 6)) − ((6 + 6)/6))
:= 2 &times; (222 − (2 + 2))
:= 3 + ((((3 + 3)
3/3+3
) + 3)/3)
:= 6 + ((6 &times; (66 + 6)) + (6 &times; 6/(6 + 6)))
:= 6 + (((6 &times; (66 + 6)) + ((6 + 6)/6)) + 6)
:= 7 + (((7 − 777)/7) + 7 &times; 77)
:= 7 &times; (7 &times; 7 + 7 + 7)
:= 7 + (7 &times; (7 &times; 7 + 7 + 7) − ((7 + 7)/7))
:= 888/((8 + 8)/8) − 8
:= 8/8 + (8 &times; (8 &times; 8 − 8) − 8)
:= 8 &times; (8 &times; 8 − 8) − (8 + 8)/8
:= (((9 &times; 9 &times; 99) − 9)/(9 + 9)) − 9
:= 9 &times; ((9 &times; 99 − 9)/(9 + 9))
:= ((9 &times; 9 &times; 99) + 9)/(9 + 9)
&Iuml; 437 := 1 + ((1 + 1) &times; (1 + 1) &times; (111 − 1 − 1))
2
:= ((22 − 2/2) ) − 2 − 2
3
&Iuml; 442 := 1 + ((11 + 11 − 1)1+1 )
&Iuml; 447 := 1 + ((1 + 1) &times; (1 + (1 + 1) &times; 111))
:= 2 + (2 &times; 222 + 2/2)
:= 2 &times; 222 − 2
:= 3 &times; 33 + (((3/3 + 3) + 3) )
:= 3 + (333/3 + 333)
:= 444 − (4 + 4)/4
:= 4 + (444 − 4/4)
:= 5 &times; 5 &times; 5 + (5 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5 &times; 5 &times; 5)
:= ((55 − 5/5) + 5)/((5 + 5)/5 + 5)
:= 6 + ((6 &times; (66 + 6)) − 6/6)
:= ((66 − 6)/6) + (6 &times; (66 + 6))
:= ((6/6 + 6) &times; ((6 + 6)/6)6 ) − 6/6
:= 7 + (((77 &times; 77 − 7)/(7 + 7)) + 7)
:= 7/7 + 7 &times; (7 &times; 7 + 7 + 7)
:= 7 + (7 &times; (7 &times; 7 + 7 + 7) − 7/7)
:= 8 &times; (8 &times; 8 − 8) − 88/8
:= (8 + 8)/8 + (8 &times; (8 &times; 8 − 8) − 8)
:= 8 &times; (8 &times; 8 − 8) − 8/8
:= (((9 &times; 9 &times; 99) + 9)/(9 + 9)) − 9
:= 9/9 + (9 &times; ((9 &times; 99 − 9)/(9 + 9)))
:= 9/9 + (((9 &times; 9 &times; 99) + 9)/(9 + 9))
:= (((3 + 3) &times; ((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
&Iuml; 448 := (1 + 1) &times; (1 + 1) &times; (1 + 111)
&Iuml; 453 := 1 + (11 + ((11 + 11 − 1)1+1 ))
&Iuml; 458 := 1 + (11 + ((1 + 1) &times; (1 + (1 + 1) &times; 111)))
:= 222 − (22 + 2 + 2)
:= 2/2 + (2 &times; ((222 + 2) + 2))
:= 2 &times; (222 + 2)
3
:= (3/3 + 3) &times; ((333 + 3)/3)
:= (3 &times; (3 + 3) &times; 3 ) − 33
:= (3 + 3)3 + ((33+3 − 3)/3)
:= 4 + 444
:= 4 + ((444 + 4/4) + 4)
:= 4 + ((44 − 4)/4 + 444)
:= (5 − 5/5) &times; (555 + 5)/5
:= 5 + ((5 − 5/5) &times; (555 + 5)/5)
6
6
:= (5 &times; 5 &times; 5 &times; 55 − 5)/(5 + 5 + 5)
:= (6/6 + 6) &times; ((6 + 6)/6)
:= 6 + (((6/6 + 6) &times; ((6 + 6)/6) ) − 6/6)
:= 66 + (((6 + 6)/6 − 6) + 6 &times; 66)
:= 7 + 7 &times; (7 &times; 7 + 7 + 7)
:= 7 &times; 7 &times; 7 + (777 − 7)/7
:= 7 + (77/7 &times; (7 &times; 7 − (7/7 + 7)))
:= 8 &times; (8 &times; 8 − 8)
:= 8 + ((8 &times; (8 &times; 8 − 8) − (88/8)) + 8)
:= 8 + (8 &times; (8 &times; 8 − 8) + ((8 + 8)/8))
:= (9 − 9/9) &times; ((999 + 9)/(9 + 9))
:= 9 + ((999 &times; (9 − 9/9))/(9 + 9))
:= 9 + ((((((9 + 9)/9)9 ) − 9 &times; 9) + 9) + 9)
&Iuml; 449 := 1 + ((1 + 1) &times; (1 + 1) &times; (1 + 111))
:= 2/2 + (2 &times; (222 + 2))
&Iuml; 454 := 11 + (((1 + 1) &times; (1 + 1) &times; 111) − 1)
:= 2 + (2 &times; ((222 + 2) + 2))
&Iuml; 459 := 11 + ((1 + 1) &times; (1 + 1) &times; (1 + 111))
:= 22/2 + (2 &times; (222 + 2))
:= 3 + (((3 &times; (33 &times; 3 )) + 3)/(3 + 3))
:= 3/3 + ((3 &times; (3 + 3) &times; 3 ) − 33)
:= 3 &times; (3 &times; ((33 − 3) + 33 ))
:= 4 + (444 + 4/4)
:= 444 + (44 − 4)/4
:= 4 + (444 + 44/4)
:= 5 + ((5 − 5/5) &times; 555/5)
:= 5 + (((5 − 5/5) &times; 555/5) + 5)
:= (55 − 555)/5 − 55
:= 6 + ((6 &times; (66 + 6)) + (66/6))
:= 6 + ((6/6 + 6) &times; ((6 + 6)/6)6 )
:= 66 + (6 &times; 66 − (6 &times; 6/(6 + 6)))
:= 7 + (7 &times; (7 &times; 7 + 7 + 7) + 7/7)
:= 7 &times; 7 &times; 7 + 777/7
:= 7 + (7 &times; (7 &times; 7 + 7 + 7) + (77/7))
:= 8/8 + 8 &times; (8 &times; 8 − 8)
:= 8 + (8 &times; (8 &times; 8 − 8) − ((8 + 8)/8))
:= 88/8 + 8 &times; (8 &times; 8 − 8)
:= 9 + (((9 &times; 9 &times; 99) − 9)/(9 + 9))
:= (9 + 9 + 9) &times; ((9 − 9/9) + 9)
3
9
:= 9 + (((((9 + 9)/9) ) − 9 &times; 9) + 9)
&Iuml; 450 := (1 + 1) &times; (1 + (1 + 1) &times; (1 + 111))
3
&Iuml; 455 := 11 + ((1 + 1) &times; (1 + 1) &times; 111)
&Iuml; 460 := (1 + 1) &times; ((11 − 1) &times; (1 + 11 + 11))
:= 2 + (2 &times; (222 + 2))
:= 22/2 + 2 &times; 222
:= 222 − (22 + 2)
:= (3 + 3) &times; ((3 &times; (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 &times; 4 + 444
:= (5 + 5) &times; (55 − 5 − 5)
:= 5 + ((5 + 5) &times; (55 − 5 − 5))
:= 5 + (((5 + 5) &times; (55 − 5 − 5)) + 5)
:= 666 − 6 &times; 6 &times; 6
:= (6/6 + 6) &times; (66 − 6/6)
:= 6 &times; 66 + ((6 + 6)/6)6
:= (7 − 7/7) &times; (77 − (7 + 7)/7)
:= 7 &times; 77 − (77 + 7)
:= 7 &times; 77 − ((7 + 7)/7 + 77)
:= (8 + 8)/8 + 8 &times; (8 &times; 8 − 8)
:= 8 + (8 &times; (8 &times; 8 − 8) − 8/8)
:= 8 + (888/((8 + 8)/8) + 8)
:= 9 &times; ((9 &times; 99 + 9)/(9 + 9))
:= 9 + (((9 &times; 9 &times; 99) + 9)/(9 + 9))
:= (9 &times; (99 + 9)) − (((9 + 9)/9)9 )
&Iuml; 451 := 1 + ((1 + 1) &times; (1 + (1 + 1) &times; (1 + 111)))
&Iuml; 456 := 1 + (11 + ((1 + 1) &times; (1 + 1) &times; 111))
&Iuml; 461 := ((1 + 1) &times; (11 &times; (11 + 11 − 1))) − 1
:= 2 + ((2 &times; (222 + 2)) + 2/2)
:= 2 &times; (((222 + 2) + 2) + 2)
:= 222 − (22 + 2/2)
:= 3 + ((3/3 + 3) &times; ((333 + 3)/3))
:= 3 + ((3 &times; (3 + 3) &times; 33 ) − 33)
:= (33 &times; (33/3 + 3)) − 3/3
:= 4 + ((444 − 4/4) + 4)
:= 4 + (444 + 4 + 4)
:= 4 &times; 4 + (444 + 4/4)
:= 5/5 + ((5 + 5) &times; (55 − 5 − 5))
:= 5 + (((5 + 5) &times; (55 − 5 − 5)) + 5/5)
:= 55/5 + ((5 + 5) &times; (55 − 5 − 5))
:= 66 + (6 &times; 66 − (66/6))
:= 66 + (6 &times; 66 − 6)
:= 66 + (6 &times; 66 − 6/6)
:= 77/7 &times; (7 &times; 7 − (7/7 + 7))
:= (7 − 7/7) &times; (77 − 7/7)
:= 7 &times; 77 − (7/7 + 77)
:= 88/8 + (8 &times; (8 &times; 8 − 8) − 8)
:= 8 + 8 &times; (8 &times; 8 − 8)
:= 8 &times; (8 &times; 8 − 8) + (88 + 8 + 8)/8
:= ((9 &times; 9 &times; 99) + 99)/(9 + 9)
:= (9 &times; (9 &times; 9 − (9 + 9))) − 999/9
:= 9/9 + ((9 &times; (99 + 9)) − (((9 + 9)/9)9 ))
&Iuml; 452 := 11 + ((11 + 11 − 1)1+1 )
:= 2 &times; ((222 + 2) + 2)
&Iuml; 457 := 11 + ((1 + 1) &times; (1 + (1 + 1) &times; 111))
:= 2 + (2 &times; 222 + 22/2)
3
3+3
:= (3/3 + 3) &times; (((333 − 3)/3) + 3)
:= (3 + 3) + (((3
:= 4 + (444 + 4)
:= 4 + (((444 + 4/4) + 4) + 4)
:= (5 + 5)/5 + ((5 + 5) &times; (55 − 5 − 5))
+ 3)/3) − 3)
5−5/5+5
:= (((5 + 5)/5)
) − 55
&Iuml; 462 := (1 + 1) &times; (11 &times; (11 + 11 − 1))
:= 222 − 22
:= 33 &times; (33/3 + 3)
:= 4 &times; 4 + (444 + (4 + 4)/4)
:= 55/5 &times; ((((5 + 5)/5)5 + 5) + 5)
:= 66 + (((6 − 66)/6) + 6 &times; 66)
:= 66 + ((6 &times; 66 − 6) + 6/6)
:= 66 + 6 &times; 66
:= 77/7 + 7 &times; (7 &times; 7 + 7 + 7)
:= 7 + ((7 − 7/7) &times; (77 − (7 + 7)/7))
:= 77 &times; (7 − 7/7)
:= 8 + 888/((8 + 8)/8)
:= 8 + (8 &times; (8 &times; 8 − 8) + 8/8)
:= (8 − 8/8) &times; (((8 + 8)/8) + 8 &times; 8)
:= 9/9 + (((9 &times; 9 &times; 99) + 99)/(9 + 9))
:= ((9999 + 9)/(9 + 9)) − 99
:= 99 + ((99 &times; 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
&Iuml; 463 := 1 + ((1 + 1) &times; (11 &times; (11 + 11 − 1)))
&Iuml; 468 := (1 + 1) &times; (1 + (11 + (1 + 1) &times; 111))
&Iuml; 473 := (11 + 11)1+1 − 11
:= 22 + ((22 − 2/2)2 )
:= 222 − 22+2
:= 222 − 22/2
:= 3/3 + (33 &times; (33/3 + 3))
:= (3 + 3) &times; (3 &times; 33 − 3)
:= 33/3 + (33 &times; (33/3 + 3))
:= 4 + ((444 + 44/4) + 4)
:= 44 + (44 − 44)
:= 44/4 &times; (44 − 4/4)
:= 5 &times; (5 &times; 5 + 5) + (5 + 5)/(5 + 5)
:= (5 − 5/5) &times; ((555 + 5)/5 + 5)
:= 55/5 &times; (55 − ((55 + 5)/5))
:= 66 + (6 &times; 66 + 6/6)
:= 6 &times; (66 + 6 + 6)
:= 6 + ((6 &times; (66 + 6 + 6)) − 6/6)
5
:= 7/7 + (77 &times; (7 − 7/7))
:= (7 − 7/7) &times; (7/7 + 77)
:= 7 &times; 77 + (77/7 − 77)
:= 8 + ((8 &times; (8 &times; 8 − 8) − 8/8) + 8)
:= 8 &times; 8 &times; 8 − (88/((8 + 8)/8))
:= 8 + (((8 &times; (8 &times; 8 − 8) + 8/8) + 8) + 8)
:= 9 + ((((9 &times; 9 &times; 99) − 9)/(9 + 9)) + 9)
:= (9 + 9) &times; (((9 − 9/9) + 9) + 9)
:= 9 &times; 9 &times; 9 − (((9 + 9)/9)9−9/9 )
&Iuml; 464 := (1 + 1) &times; (111 + 111+1 )
&Iuml; 469 := 1 + ((1 + 1) &times; (1 + (11 + (1 + 1) &times; 111)))
&Iuml; 474 := 1 + ((11 + 11)1+1 − 11)
:= 2 + (222 − 22)
:= 2/2 + (222 − 22+2 )
:= 222 + ((2 − 22)/2)
:= 3 + ((33 &times; (33/3 + 3)) − 3/3)
:= 3/3 + ((3 + 3) &times; (3 &times; 33 − 3))
:= (3 &times; ((3 + 3) &times; 33 − 3)) − 3
:= 4 + (444 + 4 &times; 4)
:= 4/4 + ((44 − 44) + 44 )
:= ((44 &times; (44 − 4/4)) + 4)/4
:= (5 − 5/5) &times; (555/5 + 5)
:= (5 − 5/5) − 555
:= 5 + ((5 − 5/5)5 − 555)
:= 66 + (((6 + 6)/6) + 6 &times; 66)
:= 6/6 + (6 &times; (66 + 6 + 6))
:= 6 + (6 &times; (66 + 6 + 6))
:= 7 &times; 77 + ((7 + 7)/7 − 77)
5
:= 7 + (77 &times; (7 − 7/7))
:= (7 − 7/7) &times; ((7 + 7)/7 + 77)
:= 8 + (8 &times; (8 &times; 8 − 8) + 8)
:= (8 − 8/8) &times; ((88/8 − 8) + 8 &times; 8)
:= (8 − (8 + 8)/8) &times; (88 − (8/8 + 8))
:= 9 + ((((9 &times; 9 &times; 99) + 9)/(9 + 9)) + 9)
:= 9 + ((9 &times; (99 + 9)) − (((9 + 9)/9)9 ))
:= ((9 + 9) &times; (9 + 9 + 9)) − (99 + 9)/9
&Iuml; 465 := 1 + ((1 + 1) &times; (111 + 111+1 ))
&Iuml; 470 := (1 + 1) &times; (11 + (1 + 1) &times; (1 + 111))
&Iuml; 475 := 1 + (1 + ((11 + 11)1+1 − 11))
:= 2 + ((222 − 22) + 2/2)
:= 2 + (222 − 22+2 )
:= 2 + (222 − 22/2)
:= 3 + (33 &times; (33/3 + 3))
:= 3 + (((3 + 3) &times; (3 &times; 33 − 3)) − 3/3)
:= (3 &times; (3 + 3) &times; 33 ) − 33/3
:= 4 + ((444 + 4 &times; 4) + 4/4)
:= 4 + ((44/((4 + 4)/4)) + 444)
:= 444 + (4 &times; (4 + 4) − 4/4)
:= (5 &times; (5 &times; (5 &times; 5 − 5) − 5)) − 5
:= 5 &times; (5 &times; (5 &times; 5 − 5) − 5)
:= 5 &times; (5 &times; 5 &times; 5 − ((5 + 5)/5)5 )
:= 66 + ((6 &times; 6/(6 + 6)) + 6 &times; 66)
:= (6 + 6)/6 + (6 &times; (66 + 6 + 6))
:= 6 + ((6 &times; (66 + 6 + 6)) + 6/6)
:= 77 + (((7 &times; 777) − 7)/(7 + 7))
:= 7 + ((77 &times; (7 − 7/7)) + 7/7)
:= 7 + ((7 − 7/7) &times; (7/7 + 77))
:= 8 + ((8 &times; (8 &times; 8 − 8) + 8/8) + 8)
:= 8 + ((8 − 8/8) &times; (((8 + 8)/8) + 8 &times; 8))
:= 8 + ((8 &times; (8 &times; 8 − 8) + (88/8)) + 8)
:= 9 + ((9 &times; (9 &times; 9 − (9 + 9))) − 999/9)
:= (9 + 9)/9 + ((9 + 9) &times; (((9 − 9/9) + 9) + 9))
:= ((9 + 9) &times; (9 + 9 + 9)) − 99/9
&Iuml; 466 := (1 + 1) &times; (11 + (1 + 1) &times; 111)
&Iuml; 471 := (11 + 11)1+1 − (1 + 1 + 11)
2
:= 22 + 2 &times; 222
&Iuml; 476 := (1 + 1) &times; ((1 + 1) &times; (111+1 − (1 + 1)))
:= 222 − 2 &times; (2 + 2)
:= 22 − (22/2 + 2)
:= 3 + ((33 &times; (33/3 + 3)) + 3/3)
:= 3 + ((3 + 3) &times; (3 &times; 3 − 3))
:= (33/3 + 3) &times; (3/3 + 33)
:= 444 + (44/((4 + 4)/4))
:= 4 &times; 4 + (444 + 44/4)
:= (55 + 5)/5 − (5 &times; ((5 + 5)/5)5 )
:= 444 + 4 &times; (4 + 4)
:= 5/5 + ((5 &times; (5 &times; (5 &times; 5 − 5) − 5)) − 5)
:= 5/5 + (5 &times; (5 &times; (5 &times; 5 − 5) − 5))
:= 6 &times; (66 − 6) + 666/6
:= (6/6 + 6) &times; (((6 + 6)/6) + 66)
6
:= 6 + (((6 + 6)/6) + 6 &times; 66)
3
7
:= 77 + (((7 &times; 777) + 7)/(7 + 7))
:= 7 &times; 7 &times; 7 + ((7 + 7)/7)
:= 7 &times; (77 − 7) − (7 + 7)
:= 8 + ((8 &times; (8 &times; 8 − 8) + ((8 + 8)/8)) + 8)
:= 8 + (((8 &times; (8 &times; 8 − 8) − 8/8) + 8) + 8)
:= 88 &times; 88/(8 + 8) − 8
:= ((9 + 9) &times; (9 + 9 + 9)) − (99/9 + 9)
:= 9 + (((99 &times; 99)/(9 + 9 + 9)) + 99)
:= (((9 + 9)/9)9 ) − ((9 + 9 + 9) + 9)
&Iuml; 467 := 1 + ((1 + 1) &times; (11 + (1 + 1) &times; 111))
:= 22 + (2 &times; 222 + 2/2)
3
:= ((3 + 3) &times; (3 &times; 3 − 3)) − 3/3
4
4
:= 4 + (4 − (44 + 4/4))
5
:= 5 + ((5 − 5)/(5 + 5) + 5 &times; (5 &times; 5 + 5))
&Iuml; 472 := (11 + 11)1+1 − 1 − 11
:= 222 − (2 &times; (2 + 2 + 2))
3
:= 3 + (((3 + 3) &times; (3 &times; 3 − 3)) + 3/3)
4
4
:= 4 + ((4 − 44) + 4 )
5
5
:= 55 + (((5 + 5 ) + 5)/(5 + 5 + 5))
&Iuml; 477 := 1 + ((1 + 1) &times; ((1 + 1) &times; (111+1 − (1 + 1))))
:= 2 + ((222 − 22/2) + 2)
:= 3 &times; ((3 + 3) &times; 33 − 3)
:= 4 + (44/4 &times; (44 − 4/4))
:= (5 + 5)/5 + (5 &times; (5 &times; (5 &times; 5 − 5) − 5))
:= (6 &times; (66 + 6 + 6)) − 6/6
:= 6 + ((((6 + 6)/6) + 6 &times; 66) + 6)
:= 6 + (6 &times; (66 − 6) + 666/6)
:= 7 + (7 &times; 77 − ((7 + 7)/7 + 77))
:= 7 &times; (77 − 7) − (77/7 + 7)
:= 7/7 + (7 &times; (77 − 7) − (7 + 7))
:= 8 + (8 &times; (8 &times; 8 − 8) + (88/8))
:= 8 + ((8 &times; (8 &times; 8 − 8) + 8) + 8)
:= (8/8 + 8) &times; (8 &times; 8 − 88/8)
:= 9 + (((((9 &times; 9 &times; 99) − 9)/(9 + 9)) + 9) + 9)
:= ((9 + 9) &times; (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
&Iuml; 478 := (1 + 1) &times; ((1 + 1) &times; (111+1 − 1) − 1)
&Iuml; 483 := (11 + 11)1+1 − 1
&Iuml; 488 := (1 + 1) &times; ((1 + 1) &times; (1 + 111+1 ))
:= 222 − (2 + 2 + 2)
:= 222 − 2/2
:= 2 + 222 + 2
:= 3/3 + (3 &times; ((3 + 3) &times; 33 − 3))
:= (3 &times; (3 + 3) &times; 33 ) − 3
:= 3 + (((3 − 3/3)3&times;3 ) − 33 )
:= 44 + (444/((4 + 4)/4))
:= (44 &times; 44 − 4)/4
:= 44 + 444
:= 5 + (55/5 &times; (55 − ((55 + 5)/5)))
:= 5 &times; 55 + ((55 − 5)/(5 + 5 + 5))
:= (5 − 5/5) &times; ((555 + 55)/5)
:= ((66 − 6)/6) + (6 &times; (66 + 6 + 6))
:= (6/6 + 6) &times; (6 &times; 6/(6 + 6) + 66)
:= ((6 + 6)/6 + 6) &times; ((66 − 6) + 6/6)
:= 7 + (((7 + 7)/7) + 7 &times; 7 &times; 7)
:= 7 &times; (77 − 7) − 7
:= 7 &times; (77 − 7) − (7 + 7)/7
:= (88 &times; 88 − (88 + 8))/(8 + 8)
:= 88 &times; 88/(8 + 8) − 8/8
:= 8 &times; 8 &times; 8 − 8 − 8 − 8
:= 9/9 + (((9 + 9) &times; (9 + 9 + 9)) − 9)
:= ((9 + 9 + 9)/9) &times; (9 &times; (9 + 9) − 9/9)
:= (9 + 9)/9 + ((9 + 9) &times; (9 + 9 + 9))
7
&Iuml; 479 := ((1 + 1) &times; (1 + 1) &times; (111+1 − 1)) − 1
&Iuml; 489 := 1 + ((1 + 1) &times; ((1 + 1) &times; (1 + 111+1 )))
&Iuml; 484 := (11 + 11)1+1
:= 222 − (2/2 + 2 + 2)
:= 222
:= 2 + 222 + 2/2 + 2
:= ((3 − 3/3)3&times;3 ) − 33
:= 3/3 + ((3 &times; (3 + 3) &times; 33 ) − 3)
:= 3 + (3 &times; (3 + 3) &times; 33 )
:= ((44 &times; 44 − 4)/4) − 4
:= 44 &times; 44/4
:= 4 + ((44 &times; 44 + 4)/4)
5
:= 5 + (((5 − 5/5) − 555) + 5)
:= 55/5 &times; (55 − (55/5))
(6+6)/6
:= 555 − (55/5 + 55)
:= 666 − (666/6 + 66)
:= 66/6 + (6 &times; (66 + 6 + 6))
:= ((66 + 66)/6)
:= 7 &times; (77 − 7) − 77/7
:= 7/7 + (7 &times; (77 − 7) − 7)
:= 7 &times; (77 − 7) − 7/7
:= 8 &times; 8 &times; 8 − ((8/8 + 8 + 8 + 8) + 8)
:= 88 &times; 88/(8 + 8)
:= 8/8 + (8 &times; 8 &times; 8 − (8 + 8 + 8))
:= (9 + 9)/9 + (((9 + 9) &times; (9 + 9 + 9)) − 9)
:= ((99 + 99)/9)(9+9)/9
:= ((9 + 9 + 9)/9) + ((9 + 9) &times; (9 + 9 + 9))
&Iuml; 480 := (1 + 1) &times; (1 + 1) &times; (111+1 − 1)
&Iuml; 490 := (1 + 1) &times; (1 + ((1 + 1) &times; (1 + 111+1 )))
&Iuml; 485 := 1 + (11 + 11)1+1
:= 222 − 2 − 2
:= 2/2 + 222
:= 2 + 222 + 2 + 2
:= 3 + (3 &times; ((3 + 3) &times; 33 − 3))
:= ((3 − 3/3)3&times;3 ) − 33
:= 3 + ((3 &times; (3 + 3) &times; 33 ) + 3/3)
:= (4 + 4) &times; (4 &times; 4 + 44)
:= (44 &times; 44 + 4)/4
:= 4 + (((44 &times; 44 + 4) + 4)/4)
:= 5 + (5 &times; (5 &times; (5 &times; 5 − 5) − 5))
:= 5 + ((5 &times; (5 &times; (5 &times; 5 − 5) − 5)) + 5)
:= (5 + 5) &times; (55 − (5/5 + 5))
:= 6 + ((6 &times; (66 + 6 + 6)) + 6)
:= 6 + ((6 &times; (66 + 6 + 6)) + (66/6))
:= (6/6 + 6) &times; (((6 + 6)/6)6 + 6)
:= (7/7 + 7) &times; (77/7 + 7 &times; 7)
:= ((7 + 7)/7) + (7 &times; (77 − 7) − 7)
:= 7 &times; (77 − 7)
:= 8 &times; (8 &times; 8 − (8/((8 + 8)/8)))
:= 8/8 + 88 &times; 88/(8 + 8)
:= (9/9 − 9 &times; 9) &times; (((9 + 9 + 9)/9) − 9)
&Iuml; 481 := (11 + 11)1+1 − 1 − 1 − 1
9
:= (((9 + 9)/9) ) − (9 + 9 + 9)
&Iuml; 486 := 1 + (1 + (11 + 11)1+1 )
:= 8 &times; 8 &times; 8 − (88 + 88)/8
:= (((9 &times; 999) − 9)/(9 + 9)) − 9
&Iuml; 491 := 1 + ((1 + 1) &times; (1 + ((1 + 1) &times; (1 + 111+1 ))))
:= 222 − 2/2 − 2
:= 2 + 222
:= 2 + 222 + 2/2 + 2 + 2
:= 3 + ((3 &times; ((3 + 3) &times; 33 − 3)) + 3/3)
:= 3 &times; (3 + 3) &times; 33
:= 3 + ((((3 − 3/3)3&times;3 ) − 33 ) + 3)
:= ((44 &times; 44 + 4)/4) − 4
:= ((44 &times; 44 + 4) + 4)/4
:= 4 + (((44 &times; 44 − 4)/4) + 4)
:= (5555 − 5 )/5 − 5
:= (5555 − 55 )/5
:= 5 + (5555 − 55 )/5
:= (6/6 + 6 + 6) &times; (6 &times; 6 + 6/6)
:= 666 + (6 &times; (6 − 6 &times; 6))
:= 6 &times; 6 + ((6/6 + 6) &times; (66 − 6/6))
:= 7 &times; (77 − 7) − ((7 + 7)/7 + 7)
:= 7 + (7 &times; (77 − 7) − (77/7))
:= 7/7 + 7 &times; (77 − 7)
:= 8 &times; 8 &times; 8 + ((8 − 88)/8 − (8 + 8))
:= 8 + (88 &times; 88/(8 + 8) − 8/8)
:= (9 + 9) &times; (9 + 9 + 9)
:= (((9 &times; 999) + 9)/(9 + 9)) − 9
5
:= 8/8 + (8 &times; (8 &times; 8 − (8/((8 + 8)/8))))
:= 9 &times; 9 + ((99/9 + 9)
(9+9)/9
)
&Iuml; 482 := (11 + 11)1+1 − 1 − 1
&Iuml; 487 := 1 + (1 + (1 + (11 + 11)1+1 ))
&Iuml; 492 := (1 + 1) &times; ((1 + 1) &times; (1 + (1 + 111+1 )))
:= 222 − 2
:= 2 + 222 + 2/2
:= 2 &times; (2 + 2) + 222
:= 3 + (((3 − 3/3)3&times;3 ) − 33)
:= 3/3 + (3 &times; (3 + 3) &times; 33 )
:= 3 + ((3 &times; (3 + 3) &times; 33 ) + 3)
:= (44 &times; 44 − (4 + 4))/4
:= 4 + ((44 &times; 44 − 4)/4)
:= 4 + (444 + 44)
:= ((5555 − 5 + 5)/5) − 5
:= (5555 − 5 + 5)/5
:= 5 + ((5555 − 55 + 5)/5)
:= 6 + ((6/6 + 6) &times; (((6 + 6)/6) + 66))
:= 6 + ((6/6 + 6 + 6) &times; (6 &times; 6 + 6/6))
:= 66 + ((6 &times; (66 + 6)) − 6)
:= 7 &times; (77 − 7) − (7/7 + 7)
:= 7 &times; (77 − 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + 7 &times; (77 − 7)
:= 8 &times; 8 &times; 8 − ((88 + 88)/8 + 8)
:= 8 &times; 8 &times; 8 − (8/8 + 8 + 8 + 8)
:= 8 + 88 &times; 88/(8 + 8)
:= ((9 &times; ((9 &times; (99 + 9)) − 9)) + 9)/(9 + 9)
:= 9/9 + ((9 + 9) &times; (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
&Iuml; 493 := 11 + ((11 + 11)1+1 − (1 + 1))
&Iuml; 498 := (11 − 1)1+1+1 /(1 + 1) − 1 − 1
&Iuml; 503 := 1 + 1 + (1 + 1)11−1−1 − 11
:= 22/2 + (222 − 2)
:= 22+2 + (222 − 2)
:= 22 + 222 − (2/2 + 2)
:= 3 + (((3 &times; (3 + 3) &times; 33 ) + 3/3) + 3)
:= 3 + (3 &times; ((3 + 3) &times; 33 + 3))
:= ((3 − 3/3)3&times;3 ) − 3 &times; 3
:= 4 + (((44 &times; 44 + 4)/4) + 4)
:= 44 + (44 &times; 44/(4 + 4))
:= (((4 + 4) &times; (44 − 4)) − 4)/4
:= 555 + ((5 − 55 /5)/(5 + 5))
:= 555 − ((5 + 5)/5 + 55)
:= (55 − (555 + 55))/5
:= ((6 + 6) &times; (6 &times; 6 + 6)) − 66/6
:= 66 + (6 &times; (66 + 6))
:= ((6 + 6) &times; (6 &times; 6 + 6)) − 6/6
:= 7 &times; (77 − 7) + (7 + 7 + 7)/7
:= 7 + (7 &times; (77 − 7) + 7/7)
:= 7 + ((7 &times; (77 − 7) − 7/7) + 7)
:= 8 &times; 8 &times; 8 − (88/8 + 8)
:= 8 &times; 8 &times; 8 + (((8 + 8)/8) − (8 + 8))
:= 8 &times; 8 &times; 8 − (8/8 + 8)
:= ((99 + 9)/9) + ((9 + 9) &times; (9 + 9 + 9))
:= (((9 + 9)/9)9 ) − 9
9
:= (((9 + 9)/9) ) − (9/9 + 9 + 9)
&Iuml; 494 := 11 + ((11 + 11)1+1 − 1)
&Iuml; 499 := (11 − 1)1+1+1 /(1 + 1) − 1
&Iuml; 504 := (1 + 1) &times; ((1 + 11) &times; (11 + 11 − 1))
:= 2 + 22/2 + 222 + 2
:= 22 + 222 − 2
:= ((3 − 3/3)3&times;3 ) − (3 &times; (3 + 3))
:= ((3 &times; (3 &times; 333)) − 3)/(3 + 3)
:= (3 + 3) &times; (3 &times; 33 + 3)
:= ((44 &times; 44 − 4) + 44)/4
:= 4 + 44/4 &times; (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) &times; (6 &times; 6 + 6))
:= 66 + ((6 &times; (66 + 6)) + 6/6)
:= (6 + 6) &times; (6 &times; 6 + 6)
:= 77/7 + (7 &times; (77 − 7) − 7)
:= 7 + (7 &times; (77 − 7) + ((7 + 7)/7))
:= 7 + (7 &times; (77 − 7) + 7)
:= 8 &times; 8 &times; 8 + ((8 − 88)/8 − 8)
:= 8 &times; 8 &times; 8 − (88 + 8 + 8)/8
:= 8 &times; 8 &times; 8 − 8
:= ((9 &times; 999) − 9)/(9 + 9)
:= 9 + (((9 + 9) &times; (9 + 9 + 9)) + 9)
:= 2 + 2 &times; (2 + 2) + 22
2
9
:= (((9 + 9)/9) ) − (9 + 9)
&Iuml; 495 := 11 + (11 + 11)1+1
&Iuml; 500 := (11 − 1)1+1+1 /(1 + 1)
&Iuml; 505 := (11111 − 1)/(11 + 11)
:= 22+2 + 222
:= 22 + 222 − 2/2
:= 3 &times; ((3 + 3) &times; 33 + 3)
:= (3/3 + 3) &times; ((3 − 3/3 + 3)3 )
:= 3/3 + ((3 + 3) &times; (3 &times; 33 + 3))
:= 44/4 &times; (44 + 4/4)
:= 4 + ((44 − 4 &times; 4) + 44 )
:= (((4 + 4) &times; (44 − 4)) + 4)/4
:= 55 &times; (5 − 5/5 + 5)
:= 5 &times; 5 &times; (5 &times; 5 − 5)
:= 5 + (5 &times; 5 &times; (5 &times; 5 − 5))
:= 6 &times; 66 + (666/6 − (6 + 6))
:= 66 + ((6 &times; (66 + 6)) + ((6 + 6)/6))
:= 6/6 + ((6 + 6) &times; (6 &times; 6 + 6))
:= 7 + (7 &times; (77 − 7) − ((7 + 7)/7))
:= 7 &times; (77 − 7) + (77 − 7)/7
:= 7 + ((7 &times; (77 − 7) + 7/7) + 7)
:= 8 &times; 8 &times; 8 − (8/8 + 8 + 8)
:= 8 &times; 8 &times; 8 − (88 + 8)/8
:= 8/8 + (8 &times; 8 &times; 8 − 8)
:= 9 + ((9 + 9) &times; (9 + 9 + 9))
:= ((9 &times; 999) + 9)/(9 + 9)
:= (9 + 9)/9 + ((((9 + 9)/9)9 ) − 9)
:= 22/2 + 22
2
&Iuml; 496 := 1 + (11 + (11 + 11)1+1 )
:= 222 + (2 &times; (2 + 2 + 2))
:= 3/3 + (3 &times; ((3 + 3) &times; 33 + 3))
&Iuml; 501 := (1 + 1)11−1−1 − 11
:= 2/2 + 2
2+2
&Iuml; 506 := (1 + 1) &times; 11 &times; (1 + 11 + 11)
2
:= 22 + 222
3
:= ((3 − 3/3)3&times;3 ) − (3 + 3)
+ 22
:= ((3 + 3) &times; (3 &times; 3 + 3)) − 3
4
4
:= 4 &times; (4 &times; 4 &times; (4 + 4) − 4)
:= 4 + (4 − 44/4)
:= 44 + (44 − ((4 + 4)/4 + 4))
:= 5/5 + (55 &times; (5 − 5/5 + 5))
:= 5/5 + (5 &times; 5 &times; (5 &times; 5 − 5))
:= 5 + ((5 &times; 5 &times; (5 &times; 5 − 5)) + 5/5)
:= 6 &times; 66 + (666/6 − 6)
:= (6 + 6)/6 + ((6 + 6) &times; (6 &times; 6 + 6))
:= 7 + (7 &times; (77 − 7) − 7/7)
:= 77/7 + 7 &times; (77 − 7)
:= 7 + ((7 &times; (77 − 7) + ((7 + 7)/7)) + 7)
:= 8 &times; 8 &times; 8 − 8 − 8
:= 8 &times; 8 &times; 8 − 88/8
:= 8 &times; 8 &times; 8 + (((8 + 8)/8) − 8)
:= 9 + (((9 + 9) &times; (9 + 9 + 9)) + 9/9)
:= (((9 + 9)/9)9 ) − 99/9
:= 9 + (((9 + 9) &times; (9 + 9 + 9)) + (99/9))
6
:= ((6 + 6)/6) + (6 &times; (66 + 6))
&Iuml; 497 := 1 + (1 + (11 + (11 + 11)1+1 ))
&Iuml; 502 := 1 + (1 + 1)11−1−1 − 11
&Iuml; 507 := (1 + 1 + 1) &times; (1 + 1 + 11)1+1
:= 2 + (22/2 + 222 )
:= 2 + 22+2 + 222
:= 22 + 222 + 2/2
:= 33/3 + (3 &times; (3 + 3) &times; 33 )
:= 3 + ((3 &times; 3 &times; 333 − 3)/(3 + 3))
:= 3 + ((3 + 3) &times; (3 &times; 33 + 3))
:= 4/4 + ((44 − 4 &times; 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 &times; (66 + 6)) − 6/6)
:= ((6 + 6) &times; (6 &times; 6 + 6)) − (6 + 6)/6
:= 6 &times; 66 + 666/6
:= 7 + 7 &times; (77 − 7)
:= 7 &times; (77 − 7) + (77 + 7)/7
:= 7 + (7 &times; (77 − 7) + ((77 − 7)/7))
:= 8/8 + (8 &times; 8 &times; 8 − (8 + 8))
:= 8 &times; 8 &times; 8 + (8 − 88)/8
:= 8 &times; 8 &times; 8 + (88/8 − (8 + 8))
:= 99/9 + ((9 + 9) &times; (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
&Iuml; 508 := 1 + (1 + 1 + 1) &times; (1 + 1 + 11)1+1
&Iuml; 518 := (1 + 11 + 11)1+1 − 11
&Iuml; 513 := 1 + (1 + 1)11−1−1
:= 2 + 222 + 22
:= 2/2 + 2
(2/2+2)2
:= 2 + (2 &times; (22&times;(2+2) + 2))
:= ((3 − 3/3)3&times;3 ) − (3/3 + 3)
:= 33+3 − (3 + 3)3
:= 3 + (((3 − 3/3)3&times;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 &times; 66 + (666 + 6)/6
:= 6 + (666/6 + 6 &times; 66)
:= 6 + (((6 + 6)/6)6&times;6/(6+6)+6 )
:= 7 + (7 &times; (77 − 7) + (77/7))
:= 7/7 + ((7/7 + 7)(7+7+7)/7 )
:= 7 &times; 77 − (7 + 7 + 7)
:= 8 &times; 8 &times; 8 − (8/((8 + 8)/8))
:= 8/8 + 8 &times; 8 &times; 8
:= 9 + (((9 &times; 999) − 9)/(9 + 9))
&Iuml; 509 := (1 + 1)11−1−1 − 1 − 1 − 1
:= 2 + 222 + 2/2 + 22
:= ((3 − 3/3)3&times;3 ) − 3
:= 4/4 + ((44 − 4) + 44 )
:= (55 − 555)/5 − 5
:= 6 + (((6 + 6) &times; (6 &times; 6 + 6)) − 6/6)
:= 7 + (7 &times; (77 − 7) + (77 + 7)/7)
:= 8 + (8 &times; 8 &times; 8 − 88/8)
:= 9 + (((9 &times; 999) + 9)/(9 + 9))
&Iuml; 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 &times; 5 &times; (5 &times; 5 − 5)) + 5)
:= 6 + ((6 + 6) &times; (6 &times; 6 + 6))
:= (7 − 7/7) &times; (7/7 + 77 + 7)
:= 8 &times; 8 &times; 8 − (8 + 8)/8
:= (((9 + 9)/9)9 ) − (9 + 9)/9
&Iuml; 511 := (1 + 1)11−1−1 − 1
:= 2
(2/2+2)2
− 2/2
3&times;3
:= ((3 − 3/3)
4
) − 3/3
4
:= 4 + (4 − 4/4)
:= 555 + (55/5 − 55)
:= 6 + (((6 + 6) &times; (6 &times; 6 + 6)) + 6/6)
:= 7 &times; (77 + 7) − 77
:= 8 &times; 8 &times; 8 − 8/8
:= (((9 + 9)/9)9 ) − 9/9
&Iuml; 512 := (1 + 1)11−1−1
:= 2
(2/2+2)2
3&times;3
:= (3 − 3/3)
4
:= 4 + 4
4
:= ((5 + 5)/5)5−5/5+5
:= ((6 + 6)/6)6&times;6/(6+6)+6
:= (7/7 + 7)
(7+7+7)/7
:= 8 &times; 8 &times; 8
9
:= ((9 + 9)/9)
:= 9/9 + (((9 + 9)/9) )
&Iuml; 514 := 1 + 1 + (1 + 1)11−1−1
:= 2 + 2(2/2+2)
:= 9 + ((((9 &times; 999) + 9)/(9 + 9)) + 9)
&Iuml; 519 := 1 + (1 + 11 + 11)1+1 − 11
:= 2 + ((2 &times; (22&times;(2+2) + 2)) + 2/2)
2
:= 3 + (((3 − 3/3)3&times;3 ) − 3/3)
:= 33 + (3 &times; (3 + 3) &times; 33 )
:= 44 + ((4 + 4)/4 + 44 )
:= (((4 + 4) &times; (44 + 4)) − 4)/4
:= (55 − 555)/5
:= 5 + (55 − 555)/5
:= 6 + ((666 + 6)/6 + 6 &times; 66)
:= 6 + ((666/6 + 6 &times; 66) + 6)
:= 7 &times; 77 − (77/7 + 7 + 7)
:= 7 + ((7/7 + 7)(7+7+7)/7 )
:= 8 &times; 8 &times; 8 + (8 + 8)/8
:= 8 + (8 &times; 8 &times; 8 − 8/8)
:= (9 + 9)/9 + (((9 + 9)/9)9 )
:= 9 + ((((9 + 9)/9)9 ) − ((9 + 9)/9))
&Iuml; 515 := 1 + 1 + 1 + (1 + 1)11−1−1
&Iuml; 520 := 1 + 1 + (1 + 11 + 11)1+1 − 11
:= 2 + ((2(2/2+2) ) + 2/2)
:= 2 &times; ((22&times;(2+2) + 2) + 2)
:= 3 + ((3 − 3/3)3&times;3 )
:= 3 &times; 3 + (((3 − 3/3)3&times;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) &times; (6 &times; 6 + 6))
:= ((6 + 6)/6 + 6) &times; (66 − 6/6)
:= 7 + ((7 &times; (77 − 7) + (77/7)) + 7)
:= 7 &times; 77 − ((77 + 7)/7 + 7)
:= 8 &times; 8 &times; 8 + (88/8 − 8)
:= 8 + 8 &times; 8 &times; 8
2
3
− ((3 + 3) + 3)
4
:= 8 + (8 &times; 8 &times; 8 − ((8 + 8)/8))
9
9
:= (((9 + 9)/9) ) + ((9 + 9 + 9)/9)
&Iuml; 516 := 1 + 1 + 1 + 1 + (1 + 1)11−1−1
:= 9 + ((((9 + 9)/9)9 ) − 9/9)
&Iuml; 521 := 11 + (1 + 1)11−1−1 − (1 + 1)
:= 2 &times; (22&times;(2+2) + 2)
:= (22/2)2 + (22 − 2)2
:= 3 + (33+3 − (3 + 3)3 )
:= 3 &times; 3 + ((3 − 3/3)3&times;3 )
:= 4 + (44 + 44 )
:= (((4 + 4) &times; (44 + 4)) + 4)/4
:= (55 + 5)/5 − (55 + 55)
:= (55 + 5/5)/(5/5 + 5)
:= 6 + (((6 + 6) &times; (6 &times; 6 + 6)) + 6)
:= 6 + (((6 + 6) &times; (6 &times; 6 + 6)) + (66/6))
:= (7 − 7/7) &times; (((7 + 7)/7 + 77) + 7)
:= 7 &times; 77 − (77/7 + 7)
:= 8 &times; 8 &times; 8 + (8/((8 + 8)/8))
:= 8 + (8 &times; 8 &times; 8 + 8/8)
:= (((9 + 9)/9)9 ) + ((9 &times; 9 − 9)/(9 + 9))
:= 9 + (((9 + 9)/9)9 )
&Iuml; 517 := 11 &times; (1 + (1 + 1) &times; (1 + 11 + 11))
&Iuml; 522 := 11 + (1 + 1)11−1−1 − 1
:= 22 + (22/2 + 222 )
:= 2 + ((2 + 2 + 2)2 + 222 )
:= 3 + ((((3 − 3/3)3&times;3 ) − 3/3) + 3)
:= (3 + 3) &times; (3 &times; 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) &times; (66/6 + 6 &times; 6)
:= 666 − (6 + 6) &times; (6 + 6)
:= 77/7 &times; 7 &times; 7 − (7 + 7)/7
:= 7 &times; 77 + (((7 − 77)/7) − 7)
:= 8 + ((8 &times; 8 &times; 8 − 88/8) + 8)
:= 8 + (8 &times; 8 &times; 8 + (8 + 8)/8)
:= 9 + ((((9 &times; 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
&Iuml; 523 := 11 + (1 + 1)11−1−1
:= 22/2 + (2
(2/2+2)2
)
&Iuml; 533 := 1 + 1 + 1 + 1 + (1 + 11 + 11)1+1
&Iuml; 528 := (1 + 11 + 11)1+1 − 1
:= 2 + (((22 + 2/2)2 ) + 2)
:= 22 &times; (22 + 2)
3
:= 3 + (((3 − 3/3)3&times;3 ) + (3 &times; (3 + 3)))
:= 33/3 + ((3 − 3/3)3&times;3 )
:= 33 &times; (3 − 33/3)
:= 44 + (44/4 + 44 )
:= 44 &times; ((4 + 4) + 4)
:= 555 − ((5 + 5)/5)5
:= 5 + (555 − ((5 + 5)/5) )
:= 555 − (55 + 55)/5
:= 6 + ((66/6) &times; (66/6 + 6 &times; 6))
:= 66 &times; ((6 + 6)/6 + 6)
:= 6 + ((66 &times; ((6 + 6)/6 + 6)) − 6/6)
:= 7 &times; 77 − (((7 + 7)/7 + 7) + 7)
:= 7 &times; 77 − 77/7
:= 7/7 + (7 &times; 77 − 7)
:= 8 &times; 8 &times; 8 + 88/8
:= 8 + (8 &times; 8 &times; 8 + 8)
:= 99/9 + (((9 + 9)/9)9 )
:= 9 + (((((9 + 9)/9) ) − ((9 + 9)/9)) + 9)
&Iuml; 524 := 1 + 11 + (1 + 1)11−1−1
:= 4 + ((44 &times; ((4 + 4) + 4)) + 4/4)
5
:= 8 + ((88 + 8 + 8)/8 + 8 &times; 8 &times; 8)
9
:= 9 + ((((9 + 9)/9)9 ) + ((99 + 9)/9))
&Iuml; 534 := 11 + 11 + (1 + 1)11−1−1
&Iuml; 529 := (1 + 11 + 11)1+1
2
:= 2 &times; (222 − 222)
:= (22 + 2/2)2
:= 22 + (2(2/2+2) )
:= 3 + (((3 − 3/3)3&times;3 ) + 3 &times; 3)
:= (33 − 3/3 − 3)3−3/3
:= 3 + (33 &times; (3 + 3) + 333)
:= (44 &times; ((4 + 4) + 4)) − 4
:= 4/4 + (44 &times; ((4 + 4) + 4))
:= (4 + 4)/4 &times; (44/4 + 44 )
:= 5 + ((55 − 555)/5 + 5)
:= 555 − (5 &times; 5 + 5/5)
:= 5 + (555 − (5 &times; 5 + 5/5))
:= 6 &times; 66 + (((6 + 6)/6)6/6+6)
:= 6/6 + (66 &times; ((6 + 6)/6 + 6))
:= 6 + (66 &times; ((6 + 6)/6 + 6))
:= 7 &times; 77 − (7/7 + 7 + 7)
:= 8 &times; 8 &times; 8 + (88 + 8)/8
:= (((9 + 9)/9)9 ) + (99 + 9)/9
&Iuml; 525 := 1 + 1 + 11 + (1 + 1)11−1−1
:= ((22 + 2/2)2 ) − 2 − 2
:= 3 + ((3 + 3) &times; (3 &times; 33 + 3 + 3))
:= 444 + (4 − 4/4)4
:= 5 &times; (5 &times; (5 &times; 5 − 5) + 5)
:= (6 − 6/6) &times; (666/6 − 6)
:= 7 &times; 77 − (7 + 7)
:= 8 &times; 8 &times; 8 + (88 + 8 + 8)/8
:= (((9 + 9)/9)9 ) + ((99 + 9 + 9)/9)
&Iuml; 526 := (1 + 1) &times; ((1 + 1) &times; 11 &times; (1 + 11) − 1)
:= 7 &times; 77 + (7 − 77)/7
:= 7 &times; 77 + ((7 + 7)/7 − 7)
:= 8 + ((8 &times; 8 &times; 8 + 8/8) + 8)
:= 8 &times; 8 &times; 8 + (88 + 88)/8
:= 9 + (((((9 + 9)/9)9 ) − 9/9) + 9)
:= (((9 + 9)/9)9 ) + ((99 + 99)/9)
&Iuml; 530 := 1 + (1 + 11 + 11)1+1
&Iuml; 535 := 1 + 11 + 11 + (1 + 1)11−1−1
:= 2 + (22 &times; (22 + 2))
:= 2 + ((((22 + 2/2)2 ) + 2) + 2)
:= (3 &times; (3 + 3)) + ((3 − 3/3)3&times;3 )
:= (3333 − (3 &times; 3 + 3)3 )/3
:= (4 + 4)/4 + (44 &times; ((4 + 4) + 4))
:= (4/4 + 4) &times; (444/4 − 4)
:= 555 − 5 &times; 5
:= 5 + (555 − 5 &times; 5)
:= (6 + 6)/6 + (66 &times; ((6 + 6)/6 + 6))
:= 6 + ((66 &times; ((6 + 6)/6 + 6)) + 6/6)
:= 7 &times; 77 − ((7 + 7)/7 + 7)
:= 7 + (7 &times; 77 − (77/7))
:= 8 + ((8 &times; 8 &times; 8 + (8 + 8)/8) + 8)
:= 8 + (((8 &times; 8 &times; 8 − 8/8) + 8) + 8)
9
:= 9 + ((((9 + 9)/9) ) + 9)
:= (((99 &times; 99) − 9)/(9 + 9)) − 9
&Iuml; 531 := 1 + 1 + (1 + 11 + 11)1+1
&Iuml; 536 := (1 + 1) &times; (1 + 1) &times; (1 + 1 + 11 &times; (1 + 11))
2
2
:= (22 &times; (22 + 2)) − 2
:= 2 + ((22 + 2/2) )
:= 2 + ((2(2/2+2) ) + 22)
:= 3 + (((3 − 3/3)3&times;3 ) + 33/3)
:= 3 &times; ((3 &times; (33 + 33)) − 3)
:= 33 + (((3 − 3/3)3&times;3 ) − 3)
:= 4 + ((44 &times; ((4 + 4) + 4)) − 4/4)
:= 4 + ((44 &times; ((4 + 4) + 4)) + 4)
:= (44 &times; ((4 + 4) + 4)) − (4 + 4)/4
:= 5 + ((5 + 5/5)/(5/5 + 5))
:= 5/5 + (555 − 5 &times; 5)
:= 5 + ((555 − 5 &times; 5) + 5/5)
:= 6 + (((6 + 6)/6 + 6) &times; (66 − 6/6))
:= 6 + ((6 − 6/6) &times; (666/6 − 6))
:= ((6 + 6)/6 + 6) &times; (66 + 6/6)
:= 7/7 + (7 &times; 77 − (7 + 7))
:= 7 &times; 77 − (7/7 + 7)
:= 7 &times; 77 − (7 + 7 + 7)/7
:= 8 + ((8 &times; 8 &times; 8 − ((8 + 8)/8)) + 8)
:= 8 + (8 &times; 8 &times; 8 + 88/8)
:= 8 + (8 &times; 8 &times; 8 + 8 + 8)
:= 9 &times; 9 + (((9 &times; 9 &times; 99) − 9)/(9 + 9))
:= 9 &times; 9 &times; 9 − (99 + 99)
:= (((99 &times; 99) + 9)/(9 + 9)) − 9
5
&Iuml; 527 := (1 + 11 + 11)1+1 − 1 − 1
&Iuml; 532 := 1 + 1 + 1 + (1 + 11 + 11)1+1
&Iuml; 537 := 1 + (1 + 1) &times; (1 + 1) &times; (1 + 1 + 11 &times; (1 + 11))
:= ((22 + 2/2)2 ) − 2
:= 2 + ((22 &times; (22 + 2)) + 2)
:= 2 &times; (2 + 2) + ((22 + 2/2)2 )
:= 3 + ((((3 − 3/3)3&times;3 ) + 3 &times; 3) + 3)
:= 3 + ((33 − 3/3 − 3)3−3/3 )
:= (3 &times; (3 &times; (33 + 33))) − 3
:= (44 &times; ((4 + 4) + 4)) − 4/4
:= 4 + (44 &times; ((4 + 4) + 4))
:= (4/4 + 4)4 − (44 + 44)
:= 555 + ((5 + 5)/5 − (5 &times; 5 + 5))
:= 555 + ((5 + 5)/5 − 5 &times; 5)
6
:= 5 + (((5 + 5)/5 − 5 &times; 5) + 555)
:= (66 &times; ((6 + 6)/6 + 6)) − 6/6
:= (6/6 + 6) &times; ((((6 + 6)/6) + 6) + 6)
:= 666/6 + ((6 &times; (66 + 6)) − 6)
:= 7 &times; 77 − (77 + 7)/7
:= 7 &times; 77 − 7
:= 7 &times; 77 − (7 + 7)/7
:= 8 + ((8 &times; 8 &times; 8 − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 8 &times; 8 &times; 8)
:= 8 + (((8 &times; 8 &times; 8 + 8/8) + 8) + 8)
:= 9 &times; 9 + (((9 &times; 9 &times; 99) + 9)/(9 + 9))
:= 9 + ((((9 + 9)/9)9 ) + (99/9))
:= 9 &times; (9 &times; 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
&Iuml; 538 := 11 + ((1 + 11 + 11)1+1 − (1 + 1))
:= 22 + (2 &times; (2
2&times;(2+2)
+ 2))
&Iuml; 543 := (1111 − 1)/(1 + 1) − 1 − 11
:= 2
2+2
2
+ (((22 + 2/2) ) − 2)
&Iuml; 548 := (1111 − 11)/(1 + 1) − 1 − 1
:= 222 + 22+2+2
:= 33 + (((3 − 3/3)3&times;3 ) − 3/3)
:= 3 + (3 &times; (3 &times; (33 + 33)))
:= 3 + (((3 − 3/3)3&times;3 ) + 33)
:= 4 + ((4 + 4)/4 &times; (44/4 + 44 ))
:= ((4 + 4) &times; (4 &times; 4 &times; 4 + 4)) − 4/4
:= 4 + ((4 + 4) &times; (4 &times; 4 &times; 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 &times; (66 + 6))
:= 666 − ((666 + 6)/6 + 6)
:= 7 &times; 77 − 7/7
:= 7 &times; 77 + (77/7 − 7)
:= 7 + (7 &times; 77 + ((7 + 7)/7))
:= 8 + (((8 &times; 8 &times; 8 + (8 + 8)/8) + 8) + 8)
:= 8 + ((((8 &times; 8 &times; 8 − 8/8) + 8) + 8) + 8)
:= 8 &times; 8 + 88 &times; 88/(8 + 8)
:= 9 + ((((((9 + 9)/9)9 ) − 9/9) + 9) + 9)
:= (((99 &times; 99) − 9)/(9 + 9)) − 9/9
:= 9 + ((((((9 + 9)/9)9 ) + 9) + 9) + 9)
&Iuml; 539 := 11 + ((1 + 11 + 11)1+1 − 1)
&Iuml; 544 := (1111 − 1)/(1 + 1) − 11
&Iuml; 549 := (1111 − 11)/(1 + 1) − 1
:= 2 &times; (22&times;(2+2) + 22+2 )
:= 22/2 + (22 &times; (22 + 2))
:= 22 + (((22 + 2/2)2 ) − 2)
3
:= 33 + ((3 − 3/3)3&times;3 )
:= (((3 &times; 3 + 3) + 3)/3) − 33
:= 3 &times; ((3 + 3)3 − 33)
:= 44/4 + (44 &times; ((4 + 4) + 4))
:= (4 + 4) &times; (4 &times; 4 &times; 4 + 4)
:= ((44 &times; 44 + 44 ) + 4)/4
:= 555 − (55/5 + 5)
:= 555 − 55/5
:= 555 − (5/5 + 5)
:= (6/6 + 6) &times; (66/6 + 66)
:= 666 − ((666 + 66)/6)
:= 666 − (666/6 + 6)
:= 7 &times; 77
:= 7 + (7 &times; 77 − ((7 + 7)/7))
:= 7 &times; 77 + (77 − 7)/7
:= 8 + ((8 &times; 8 &times; 8 + 88/8) + 8)
:= 8 + ((8 &times; 8 &times; 8 + 8 + 8) + 8)
:= 8 &times; 8 &times; 8 + 888/(8 + 8 + 8)
:= ((99 &times; 99) − 9)/(9 + 9)
:= 9 &times; (9 &times; 9 − 9) − 99
9
:= 9 + (((((9 + 9)/9) ) + 9) + 9)
&Iuml; 540 := 11 + (1 + 11 + 11)1+1
&Iuml; 545 := (1 + 1111)/(1 + 1) − 11
:= 2 &times; ((2 &times; (22 + 2)) + 222)
3&times;3
3
&Iuml; 550 := (1111 − 11)/(1 + 1)
:= 22+2 + ((22 + 2/2)2 )
:= 22 + (22 &times; (22 + 2))
:= 3/3 + ((3 + 3)3 + 333)
:= 3 &times; (3 &times; (3 + 33))
:= 33 + ((3 − 3/3)
:= (4 − 4/4) &times; ((4 &times; 44) + 4)
:= 4/4 + ((4 + 4) &times; (4 &times; 4 &times; 4 + 4))
:= (4/4 + 4) &times; (444 − 4)/4
:= (5 + 5) &times; (55 − 5/5)
:= 555 − 5 − 5
:= (5 + 5) &times; 55
:= (6 + 6 + 6) &times; (6 &times; 6 − 6)
:= 6 + ((6/6 + 6) &times; (66/6 + 66))
:= (6 − 6/6) &times; ((666 − 6)/6)
:= 7/7 + 7 &times; 77
:= 7 + (7 &times; 77 − 7/7)
:= 7 &times; 77 + 77/7
:= 8 + ((((88 + 8)/8) + 8 &times; 8 &times; 8) + 8)
:= 8 + ((((8 &times; 8 &times; 8 + 8/8) + 8) + 8) + 8)
:= (8888 − 88)/(8 + 8)
:= (9 + 9 + 9) &times; (99/9 + 9)
:= ((99 &times; 99) + 9)/(9 + 9)
:= 9/9 + (9 &times; (9 &times; 9 − 9) − 99)
&Iuml; 541 := 1 + (11 + (1 + 11 + 11)1+1 )
:= 2 + ((22 &times; (22 + 2)) + 22/2)
3
)
&Iuml; 546 := 1 + ((1 + 1111)/(1 + 1) − 11)
:= 222 + ((2
2+2
2
&Iuml; 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) &times; (4 &times; 4 &times; 4 + 4))
:= 444 + (444/4 − 4)
:= 5/5 + ((5 + 5) &times; (55 − 5/5))
:= 5/5 + (555 − 5 − 5)
:= 5/5 + (5 + 5) &times; 55
:= 6/6 + ((6 + 6 + 6) &times; (6 &times; 6 − 6))
:= 6 + ((6 + 6 + 6) &times; (6 &times; 6 − 6))
:= 66/6 + ((6 + 6 + 6) &times; (6 &times; 6 − 6))
:= 7 &times; 77 + (7 + 7)/7
:= 7 + 7 &times; 77
:= 7 &times; 77 + (77 + 7)/7
:= 8 &times; (88 − 8) − (88/8 + 88)
:= (8 − 8/8) &times; ((8 − 88)/8 + 88)
:= 8 &times; (88 − 8) − (8/8 + 88)
:= 9 + (((((9 + 9)/9)9 ) + (99/9)) + 9)
:= ((9999 − 9)/(9 + 9)) − 9
:= (9999 − 9 &times; 9)/(9 + 9)
:= 3/3 + (3 &times; (3 &times; (3 + 33)))
4
&Iuml; 542 := 1 + (1 + (11 + (1 + 11 + 11)1+1 ))
2
:= 2 + (((22 + 2/2) ) + 22/2)
&Iuml; 547 := 1 + (1 + ((1 + 1111)/(1 + 1) − 11))
2
:= 2 + (((22 + 2/2) ) + 2
2+2
)
&Iuml; 552 := 1 + (1 + (1111 − 11)/(1 + 1))
:= (22 + 2) &times; (22 + 2/2)
:= 3 + ((((3 &times; 3 + 3) + 3)/3) − 33)
:= 3 + ((3 + 3)3 + 333)
:= (4 + 4)/4 &times; ((44/4 + 44 ) + 4)
:= ((44 &times; 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) &times; (66 + 6/6))
:= 6 + (((6 + 6 + 6) &times; (6 &times; 6 − 6)) + 6/6)
:= 6 + (((6 + 6 + 6) &times; (6 &times; 6 − 6)) + 6)
:= 7 &times; 77 + (7 + 7 + 7)/7
:= 7 + (7 &times; 77 + 7/7)
:= 7 + ((7 &times; 77 − 7/7) + 7)
:= 8 + ((88 + 88)/8 + 8 &times; 8 &times; 8)
:= (8888 − 8)/(8 + 8) − 8
:= 8 &times; (88 − 8) − 88
:= ((9999 − 9 &times; 9)/(9 + 9)) − 9
:= ((9999 + 9)/(9 + 9)) − 9
:= (9 − 9/9) &times; (9 &times; 9 − ((99 + 9)/9))
3&times;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
&Iuml; 553 := (1111 − 1)/(1 + 1) − 1 − 1
&Iuml; 558 := 1 + (1 + (1 + 1111)/(1 + 1))
&Iuml; 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 &times; (((3 + 3)3 − 33) + 3)
:= ((3 &times; 3 + 3 + 3)3 + 3)/(3 + 3)
:= 4 + (((44 &times; 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) &times; (66 + 6/6 + 6 + 6)
:= 666 − 6 &times; (6 + 6 + 6)
:= 666 − ((6 &times; 6 + 66) + 6/6)
2
4
2
:= 7 + (7 &times; 77 + 7)
:= 7 + ((77 + 7)/7 + 7 &times; 77)
:= 7 + ((7777 + 7)/(7 + 7))
:= 8/8 + (8 &times; (88 − 8) − 88)
:= (8/8 + 8) &times; (8 &times; 8 − ((8 + 8)/8))
:= 8 + (8888 − 8)/(8 + 8)
:= 9 + (((99 &times; 99) − 9)/(9 + 9))
:= (9 &times; (9 &times; 9 − (9 + 9))) − 9
:= 9 + ((((99 &times; 99) + 9)/(9 + 9)) + 9)
&Iuml; 554 := (1111 − 1)/(1 + 1) − 1
&Iuml; 559 := 1 + (1 + (1 + (1 + 1111)/(1 + 1)))
&Iuml; 564 := ((1 + 1) &times; (1 + 11))1+1 − 1 − 11
:= ((22 + 2)2 ) − 22
:= 2 + (((2222 − 2)/(2 + 2)) + 2)
:= 222 + (2 &times; 2 &times; (22 − 2))
:= (3333 − 3 &times; 3)/(3 + 3)
:= 3 + ((3333 + 3)/(3 + 3))
:= (3 &times; ((3 + 3)3 − 33 )) − 3
:= 444 + (444 − 4)/4
:= 4 + (444/4 + 444)
:= (4 &times; (44 − 4)) − 444
:= 555 − 5/5
:= 5 + (555 − 5/5)
:= 5 + (555 − 5/5 + 5)
:= 666 − (666 + 6)/6
:= 6/6 + (666 − 6 &times; (6 + 6 + 6))
:= (6 + 6) &times; (66/6 + 6 &times; 6)
:= 7 + ((7 &times; 77 + 7/7) + 7)
:= 7 + (((7 &times; 77 − 7/7) + 7) + 7)
:= 7 + ((7 &times; 77 + (77/7)) + 7)
:= (8 + 8)/8 + (8 &times; (88 − 8) − 88)
:= 888/8 + 8 &times; (8 &times; 8 − 8)
:= 8 + ((8888 + 8)/(8 + 8))
:= 9 + (((99 &times; 99) + 9)/(9 + 9))
:= 9/9 + ((9 &times; (9 &times; 9 − (9 + 9))) − 9)
:= 9 + ((9999 − 9)/(9 + 9))
&Iuml; 555 := (1111 − 1)/(1 + 1)
&Iuml; 560 := (11 − 1) &times; (1 + 111)/(1 + 1)
&Iuml; 565 := ((1 + 1) &times; (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 &times; 3 + 3)3 − 33)/3
:= 444 + 444/4
:= 4 + ((44 + 44 ) + 44 )
:= 4 + ((4/4 + 4)4 − 4 &times; 4 &times; 4)
:= 555
:= 5 + 555
:= 5 + 555 + 5
:= (6 − 6/6) &times; 666/6
:= (6 − 6/6) &times; (666 + 6)/6
:= 6/6 + ((6 + 6) &times; (66/6 + 6 &times; 6))
:= (7777 − 7)/(7 + 7)
:= 7 + ((7 &times; 77 + 7) + 7)
:= 7 + (((77 + 7)/7 + 7 &times; 77) + 7)
:= (8888 − 8)/(8 + 8)
:= (8 − 8/8) &times; (88 − 8)
:= 8 &times; (8 &times; 8 + 8) − 88/8
:= (9999 − 9)/(9 + 9)
:= (9 − 9/9) &times; (9 &times; 9 − 99/9)
:= 9 + ((9999 + 9)/(9 + 9))
&Iuml; 556 := (1 + 1111)/(1 + 1)
&Iuml; 566 := 11 + (1111 − 1)/(1 + 1)
&Iuml; 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 &times; ((33/3 + 3) + 3)
:= (3 &times; ((3 + 3)3 − 33 )) − 3/3
4
4
4
:= 44 + (4 + 4 )
:= (4/4 + 4) − 4 &times; 4 &times; 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) &times; 666/6)
:= 6 + ((6 − 6/6) &times; (666 + 6)/6)
:= (7777 + 7)/(7 + 7)
:= 7 + (((7 &times; 77 + 7/7) + 7) + 7)
:= 77 + (7 &times; (77 − 7) − 7/7)
:= (8888 + 8)/(8 + 8)
:= 8/8 + ((8 − 8/8) &times; (88 − 8))
:= (8 − 88)/8 + 8 &times; (8 &times; 8 + 8)
:= (9999 + 9)/(9 + 9)
:= (9999 + 99)/(9 + 9)
:= (9 &times; (9 &times; 9 − (9 + 9))) − 9/9
&Iuml; 557 := 1 + (1 + 1111)/(1 + 1)
&Iuml; 562 := 1 + ((11 + 1111)/(1 + 1))
2
2+2
)
&Iuml; 567 := 11 + (1 + 1111)/(1 + 1)
:= 2 + (((22 + 2)2 ) − 22/2)
:= 2 + ((2222 − 2)/(2 + 2))
:= 2 + (((22 + 2) ) − 2
:= (3333 + 3 &times; 3)/(3 + 3)
:= ((3 &times; 3 + 3 + 3)3 − 3)/(3 + 3)
:= 3 &times; ((3 + 3)3 − 33 )
:= 44 + ((4/4 + 44 ) + 44 )
:= ((4 − 44 )/4) + (4/4 + 4)4
:= ((4/4 + 4) + 4) &times; ((44 − 4)/4)
:= 555 + (5 + 5)/5
:= 5 + (555 + (5 + 5)/5)
:= 555 + (55 + 5)/5
:= 6 + (((6 − 6/6) &times; 666/6) + 6)
:= 6/6 + (6666 + 6)/(6 + 6)
:= 6 + (6666 + 6)/(6 + 6)
:= 7 + (7 &times; 77 + (77/7))
:= 7 + ((7777 − 7)/(7 + 7))
:= 77 + 7 &times; (77 − 7)
:= 8 &times; (8 &times; 8 + 8) − (88/8 + 8)
:= (8 + 8)/8 + ((8 − 8/8) &times; (88 − 8))
:= (8/8 + 8) &times; (8 &times; 8 − 8/8)
:= (9 &times; (9 &times; 9 − (9 + 9))) − 9/9 − 9
:= 9 + ((((99 &times; 99) − 9)/(9 + 9)) + 9)
:= 9 &times; (9 &times; 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
&Iuml; 568 := 1 + (11 + (1 + 1111)/(1 + 1))
&Iuml; 573 := ((1 + 1) &times; (1 + 11))1+1 − 1 − 1 − 1
&Iuml; 578 := 1 + (1 + ((1 + 1) &times; (1 + 11))1+1 )
:= ((22 + 2)2 ) − 2 &times; (2 + 2)
:= ((22 + 2)2 ) − 2/2 − 2
:= 2 + ((22 + 2)2 )
:= 3 + (((3 &times; 3 + 3)3 − 33)/3)
:= ((3 &times; 3 + 3)3 /3) − 3
:= 3 + (((3 &times; 3 + 3)3 − 3)/3)
:= 4 + ((4 &times; (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 &times; 5 + (555 − (5 + 5)/5)
:= 6 + ((6666 + 6)/(6 + 6) + 6)
:= 66 + (666/6 + 6 &times; 66)
:= (6 + 6)/6 + (6 &times; ((6 &times; 6 − 6) + 66))
:= 7/7 + (7 &times; (77 − 7) + 77)
:= 7 &times; (77 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 7 &times; (77 + 7)
:= 8 &times; (8 &times; 8 + 8) − 8
:= 8 + (8 &times; (8 &times; 8 + 8) − 88/8)
:= (8 + 8)/8 + 8 &times; (8 &times; 8 + 8)
:= 9/9 + (9 &times; (9 &times; 9 − (9 + 9)))
:= 9 + (((9999 − 9)/(9 + 9)) + 9)
:= 99/9 + (9 &times; (9 &times; 9 − (9 + 9)))
&Iuml; 569 := 1 + (1 + (11 + (1 + 1111)/(1 + 1)))
&Iuml; 574 := ((1 + 1) &times; (1 + 11))1+1 − 1 − 1
&Iuml; 579 := 1 + (1 + (1 + ((1 + 1) &times; (1 + 11))1+1 ))
:= 2 + ((((22 + 2)2 ) − 22/2) + 2)
:= ((22 + 2)2 ) − 2
:= 2 + (((22 + 2)2 ) + 2/2)
:= (((3 &times; 3 + 3)3 − 3)/3) − (3 + 3)
:= (((3 &times; 3 + 3)3 + 3)/3) − 3
:= 3 + ((3 &times; 3 + 3)3 /3)
:= 4 + (((4/4 + 4)4 − 4 &times; 4 &times; 4) + 4)
:= 44 + (((44 − 4 − 4)/4) + 44 )
:= 4 &times; 44 − (444 + 4/4)
:= (55 − 5)/5 − 55
:= 5 + ((55 − 5)/5 − 55)
:= 5 &times; 5 + (555 − 5/5)
:= 66 + (((6 + 6) &times; (6 &times; 6 + 6)) − 6/6)
:= (6 &times; ((6 &times; 6 − 6) + 66)) − (6 + 6)/6
:= 6 &times; 6 &times; 6 + 66 &times; 66/(6 + 6)
:= 7 + (((7777 − 7)/(7 + 7)) + 7)
:= 7 &times; (77 + 7) − (7 + 7)
:= 7 &times; (77 + 7) − ((7 + 7)/7 + 7)
:= 8/8 + (8 &times; (8 &times; 8 + 8) − 8)
:= 8 &times; (8 &times; 8 + 8) − (8 + 8)/8
:= 88/8 + (8 &times; (8 &times; 8 + 8) − 8)
:= (9 + 9)/9 + (9 &times; (9 &times; 9 − (9 + 9)))
:= (9 − ((9 + 9)/9)) &times; (9/9 + 9 &times; 9)
:= ((99 + 9)/9) + (9 &times; (9 &times; 9 − (9 + 9)))
&Iuml; 570 := (11 − 1) &times; (1 + (1 + 111)/(1 + 1))
&Iuml; 575 := ((1 + 1) &times; (1 + 11))1+1 − 1
&Iuml; 580 := (1 + 1) &times; ((1 + 1) &times; (1 + (1 + 11)1+1 ))
:= ((22 + 2)2 ) − (2 + 2 + 2)
:= ((22 + 2)2 ) − 2/2
:= 2 + (((22 + 2)2 ) + 2)
:= 3 + (3 &times; ((3 + 3)3 − 33 ))
:= ((3 &times; 3 + 3)3 − 3)/3
:= 3 + (((3 &times; 3 + 3)3 + 3)/3)
:= 444 + ((44 − 4)/((4 + 4)/4))
:= 44 + (((44 − 4)/4) + 44 )
:= 4 &times; 44 − 444
:= 55 /5 − 55
:= 5 + (55 /5 − 55)
:= 5 &times; 5 + 555
:= 66 + ((6 + 6) &times; (6 &times; 6 + 6))
:= (6 &times; ((6 &times; 6 − 6) + 66)) − 6/6
:= (6 − 6/6) &times; (((666 − 6)/6) + 6)
:= 7 &times; (77 + 7) − (77/7 + 7)
:= 7/7 + (7 &times; (77 + 7) − (7 + 7))
:= 7 &times; (77 + 7) − (7/7 + 7)
:= (8 + 8)/8 + (8 &times; (8 &times; 8 + 8) − 8)
:= 8 &times; (8 &times; 8 + 8) − 8/8
:= 8 &times; (8 &times; 8 + 8) + (8/((8 + 8)/8))
:= 9 + ((9999 + 99)/(9 + 9))
:= 9 + ((9 &times; (9 &times; 9 − (9 + 9))) − 9/9)
:= 9 &times; 9 + (((9 &times; 999) − 9)/(9 + 9))
&Iuml; 571 := 1 + (11 − 1) &times; (1 + (1 + 111)/(1 + 1))
&Iuml; 576 := ((1 + 1) &times; (1 + 11))1+1
&Iuml; 581 := 1 + ((1 + 1) &times; ((1 + 1) &times; (1 + (1 + 11)1+1 )))
:= ((22 + 2)2 ) − (2/2 + 2 + 2)
:= (22 + 2)2
:= 2 + ((((22 + 2)2 ) + 2/2) + 2)
:= (((3 &times; 3 + 3)3 + 3)/3) − (3 + 3)
:= (3 &times; 3 + 3)3 /3
:= 3 + ((((3 &times; 3 + 3)3 − 3)/3) + 3)
:= ((44 &times; (44 + 4 + 4)) − 4)/4
:= 4 &times; (4 &times; (4 &times; (4 + 4) + 4))
:= (4/4 + 4)4 − 44
:= (55 + 5)/5 − 55
:= 5 + ((55 + 5)/5 − 55)
:= 5 &times; 5 + (555 + 5/5)
:= 66 + (((6 + 6) &times; (6 &times; 6 + 6)) + 6/6)
:= 6 &times; ((6 &times; 6 − 6) + 66)
:= 6 + ((6 &times; ((6 &times; 6 − 6) + 66)) − 6/6)
:= 7 + (((7 &times; 77 + (77/7)) + 7) + 7)
:= (77 + 7)/7 &times; (7 &times; 7 − 7/7)
:= 7 &times; (77 + 7) − 7
:= 8 + ((8888 − 8)/(8 + 8) + 8)
:= 8 &times; (8 &times; 8 + 8)
:= 8 + ((8 &times; (8 &times; 8 + 8) − 88/8) + 8)
:= 9 + (((((99 &times; 99) − 9)/(9 + 9)) + 9) + 9)
:= 9 + (9 &times; (9 &times; 9 − (9 + 9)))
:= 9 &times; 9 + (((9 &times; 999) + 9)/(9 + 9))
&Iuml; 572 := 11 + ((11 + 1111)/(1 + 1))
&Iuml; 577 := 1 + ((1 + 1) &times; (1 + 11))1+1
&Iuml; 582 := (1 + 1) &times; (1 + ((1 + 1) &times; (1 + (1 + 11)1+1 )))
:= 22 &times; (22 + 2 + 2)
:= 2/2 + ((22 + 2)2 )
:= 2 + ((((22 + 2)2 ) + 2) + 2)
:= (((3 &times; 3 + 3)3 − 3)/3) − 3
:= ((3 &times; 3 + 3)3 + 3)/3
:= 3 + (((3 &times; 3 + 3)3 /3) + 3)
:= 44 + (44 &times; ((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 &times; 5 + (555 + (5 + 5)/5)
:= (66/6) &times; (((6 + 6)/6) − (6 + 6))
:= 6/6 + (6 &times; ((6 &times; 6 − 6) + 66))
:= 6 + (6 &times; ((6 &times; 6 − 6) + 66))
:= 7777/7 − 7 &times; 77
:= 7 &times; (77 + 7) − 77/7
:= 7/7 + (7 &times; (77 + 7) − 7)
:= 88 + 88 &times; 88/(8 + 8)
:= 8/8 + 8 &times; (8 &times; 8 + 8)
:= 8 + (8 &times; (8 &times; 8 + 8) − (8 + 8)/8)
:= 9 + (((((99 &times; 99) + 9)/(9 + 9)) + 9) + 9)
:= 9 + ((9 &times; (9 &times; 9 − (9 + 9))) + 9/9)
:= 9 &times; 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
&Iuml; 583 := 11 &times; ((111 − 1)/(1 + 1) − 1 − 1)
2
&Iuml; 588 := 1 + (11 + ((1 + 1) &times; (1 + 11))1+1 )
2+2
2
:= 2 + (((((22 + 2) ) + 2/2) + 2) + 2)
:= (2/2 + 2) &times; ((2
− 2) )
:= 3 + ((((3 &times; 3 + 3)3 + 3)/3) + 3)
:= 3 + (3 &times; (33 &times; (3 + 3) − 3))
:= 222 + (222/2 − 2)
:= 3 &times; 33 &times; (3 + 3) − 3/3
:= 44/4 &times; ((44 − 44)/4)
:= 4 + ((4 &times; 4 − 444) + 4)
:= (4/4 + 4)4 − 4 &times; (4 + 4)
:= 55/5 &times; (55 − (5 + 5)/5)
:= 5 &times; 55 + (55 + 5)/(5 + 5)
:= 55 /5 − ((5 + 5)/5)5
:= 6 + ((6 &times; ((6 &times; 6 − 6) + 66)) + 6/6)
:= 666 − (66 + 6 + 6)
:= 666 − (66 + 6/6 + 6)
:= ((7 + 7)/7) + (7 &times; (77 + 7) − 7)
:= 7 &times; (77 + 7)
:= 7 + (7 &times; (77 + 7) − ((7 + 7)/7))
:= 8 + (8 &times; (8 &times; 8 + 8) − 8/8)
:= ((88 + 8)/8) + 8 &times; (8 &times; 8 + 8)
:= 8 &times; 88 − 888/8
:= 9 + ((9 − ((9 + 9)/9)) &times; (9/9 + 9 &times; 9))
:= (99 − 9/9) &times; (9 − ((9 + 9 + 9)/9))
:= 9 &times; 9 + (((9 + 9)/9)9 )
&Iuml; 584 := 1 + 11 &times; ((111 − 1)/(1 + 1) − 1 − 1)
:= 2 &times; (2 + 2) + ((22 + 2)2 )
3
:= 3 &times; 3 + (((3 &times; 3 + 3) − 3)/3)
4
4
&Iuml; 593 := 11 &times; ((111 − 1)/(1 + 1) − 1) − 1
&Iuml; 589 := 1 + (1 + (11 + ((1 + 1) &times; (1 + 11))1+1 ))
&Iuml; 594 := 11 &times; ((111 − 1)/(1 + 1) − 1)
:= 2 + (((22 + 2)2 ) + 22/2)
:= 2 + (((22 + 2)2 ) + 22+2 )
:= 3 + ((3 &times; (33 &times; (3 + 3) − 3)) + 3/3)
:= 3 &times; 33 &times; (3 + 3)
4
:= 4 + (4 &times; 4 − 444)
:= 4 + (((4/4 + 4) − 44) + 4)
:= 4/4 + ((4/4 + 4)4 − 4 &times; (4 + 4))
:= 5 + ((555 − 5/5) + 5 &times; 5)
:= ((55 − 55)/5) − 5 &times; 5
:= 55/5 &times; (55 − 5/5)
:= ((6 + 6)/6 + 6) &times; (66 + 6/6 + 6)
:= 666 − (66/6 + 66)
:= 666 − (66 + 6)
:= 7 + (7 &times; (77 + 7) − (77/7))
:= 7/7 + 7 &times; (77 + 7)
:= 7 + (7 &times; (77 + 7) − 7/7)
:= 8 + 8 &times; (8 &times; 8 + 8)
:= 88 + (8 &times; 8 &times; 8 − 88/8)
:= 8 &times; 88 + ((8 − 888)/8)
:= 9 &times; 9 + ((((9 + 9)/9)9 ) − 9)
:= 9 + ((((9 &times; 999) − 9)/(9 + 9)) + 9 &times; 9)
:= 99 &times; (9 − ((9 + 9 + 9)/9))
&Iuml; 585 := 11 + (((1 + 1) &times; (1 + 11))1+1 − (1 + 1))
&Iuml; 590 := (11 − 1) &times; ((111+1 − 1)/(1 + 1) − 1)
&Iuml; 595 := 111 + (11 + 11)1+1
:= 22/2 + (((22 + 2)2 ) − 2)
:= 22+2 + (((22 + 2)2 ) − 2)
:= 222 + 222/2
:= 3 &times; (33 &times; (3 + 3) − 3)
:= 3 + (((3 &times; 3 + 3)3 + 33)/3)
:= 3/3 + 3 &times; 33 &times; (3 + 3)
4
:= 4 + ((4/4 + 4) − 44)
:= (44 − 4)/4 &times; (((4 − 4)/4) − 4)
:= (44 &times; 44 + 444)/4
:= 5 + (555 + 5 &times; 5)
:= 5 + ((555 + 5 &times; 5) + 5)
:= 55 /5 − (5 &times; 5 + 5)
:= (6 − 6/6) &times; (666/6 + 6)
:= 666 + (((6 − 66)/6) − 66)
:= 6/6 + (666 − (66 + 6))
:= 7 &times; (77 + 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + 7 &times; (77 + 7)
:= 7 + 7 &times; (77 + 7)
:= 8 + (8 &times; (8 &times; 8 + 8) + 8/8)
:= 8 + ((8 &times; (8 &times; 8 + 8) − (8 + 8)/8) + 8)
:= 8 + (8 &times; (8 &times; 8 + 8) + (88/8))
:= 9 + ((9 &times; (9 &times; 9 − (9 + 9))) + 9)
:= 9 + ((((9 &times; 999) + 9)/(9 + 9)) + 9 &times; 9)
:= 9/9 + (99 &times; (9 − ((9 + 9 + 9)/9)))
&Iuml; 586 := 11 + (((1 + 1) &times; (1 + 11))1+1 − 1)
4
&Iuml; 591 := (1 + 1 + 1) &times; (1 + (1 + 1 + 1 + 11)1+1 )
&Iuml; 596 := 1 + (111 + (11 + 11)1+1 )
:= 2 + (((22 + 2)2 ) + 2 &times; (2 + 2))
:= 2 + ((((22 + 2)2 ) + 22/2) + 2)
:= 22 + (((22 + 2)2 ) − 2)
:= 3 &times; 3 + (((3 &times; 3 + 3)3 + 3)/3)
:= 3 &times; 33 &times; (3 + 3) − 3
:= 3 + (3 &times; 33 &times; (3 + 3) − 3/3)
:= 4 + (((4/4 + 4) − 44) + 4/4)
:= 44/4 + (4 &times; 4 − 444)
:= (4 &times; (44 + 4)) − 444
:= 5 + (555 + 5 &times; 5 + 5/5)
:= 5 &times; 5 + (555 + (55/5))
:= (55 + 5)/5 − (5 &times; 5 + 5)
:= ((((6 + 6)/6)6+6 ) + 6)/(6/6 + 6)
:= 6 + ((6 − 6/6) &times; (666/6 + 6))
:= 666 − (((6 + 6)/6)6 + 6)
:= 7 &times; (77 + 7) − (7 + 7)/7
:= 7 &times; (77 + 7) + (7 + 7 + 7)/7
:= 7 + (7 &times; (77 + 7) + 7/7)
:= 8 + (8 &times; (8 &times; 8 + 8) + ((8 + 8)/8))
:= 8 + ((8 &times; (8 &times; 8 + 8) − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 8 &times; (8 &times; 8 + 8))
4
:= 9 + (((9 &times; (9 &times; 9 − (9 + 9))) + 9/9) + 9)
&Iuml; 587 := 11 + ((1 + 1) &times; (1 + 11))1+1
4
9
:= 9 &times; 9 + ((((9 + 9)/9) ) − ((9 + 9)/9))
&Iuml; 592 := (1 + 1) &times; (11 &times; (1 + 1 + 1)1+1+1 − 1)
:= (9999 + 9 &times; 9 &times; 9)/(9 + 9)
&Iuml; 597 := 1 + (1 + (111 + (11 + 11)1+1 ))
:= 22/2 + ((22 + 2)2 )
:= 22+2 + ((22 + 2)2 )
:= 2 + (222/2 + 222 )
:= ((3 &times; 3 + 3)3 + 33)/3
:= 3/3 + (3 &times; 33 &times; (3 + 3) − 3)
:= 3 + 3 &times; 33 &times; (3 + 3)
:= 4 + (44/4 &times; ((4 − 44)/4))
:= 4 &times; ((4 &times; (4 &times; (4 + 4) + 4)) + 4)
:= 4 + ((4/4 + 4)4 − 4 &times; (4 + 4))
:= 555 + ((5 + 5)/5)5
:= 5 + (((5 + 5)/5)5 + 555)
:= (55 + 5 + 5)/5 − (5 &times; 5 + 5)
:= 66/6 + (6 &times; ((6 &times; 6 − 6) + 66))
:= 6 &times; 6 + (6666 + 6)/(6 + 6)
:= 666 − (6 &times; 6/(6 + 6) + 66)
:= 7 &times; (77 + 7) − 7/7
:= 77/7 + (7 &times; (77 + 7) − 7)
:= 7 + (7 &times; (77 + 7) + ((7 + 7)/7))
:= 88/8 + 8 &times; (8 &times; 8 + 8)
:= 8 + (8 &times; (8 &times; 8 + 8) + 8)
:= 8 + ((8 &times; 8 &times; 8 − 88/8) + 88)
:= 9 + ((9 &times; (9 &times; 9 − (9 + 9))) + (99/9))
:= 9 &times; 9 + ((((9 + 9)/9)9 ) − 9/9)
:= 999/9 + ((9 + 9) &times; (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
&Iuml; 598 := 11 + (11 + ((1 + 1) &times; (1 + 11))1+1 )
&Iuml; 603 := 11 &times; (111 − 1)/(1 + 1) − 1 − 1
2
2
2
:= 22+2 &times; ((2 + 2 + 2)2 + 2)
:= 22 + ((22/2) − 2)
:= 22 + ((22 + 2) )
:= 3 + (3 &times; 33 &times; (3 + 3) + 3/3)
4
&Iuml; 608 := (11 &times; 111 − 1)/(1 + 1) − 1 − 1
:= 33 + (((3 &times; 3 + 3)3 − 3)/3)
:= 3 &times; (33 &times; (3 + 3) + 3)
4
:= (4 + 4)/4 &times; ((44 − 4/4) + 4 )
:= 4 + ((((4 + 4) &times; (44 + 4 )) − 4)/4)
:= 44 + ((4 + 4) &times; 44)
:= (55 − 5 − 5)/5 − 5 &times; 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 &times; (77 + 7) + (77 − 7)/7
:= 7 + ((7 &times; (77 + 7) + 7/7) + 7)
:= (7/7 + 7) &times; (77 − 7/7)
:= 88 + (8 &times; 8 &times; 8 − ((8 + 8)/8))
:= 8 + ((8 &times; (8 &times; 8 + 8) + (88/8)) + 8)
:= 8 + (8 &times; 8 &times; 8 + 88)
:= 99 + (((9 &times; 999) − 9)/(9 + 9))
:= 9 + (99 &times; (9 − ((9 + 9 + 9)/9)))
:= 9 &times; 9 &times; 9 − (((999 + 9)/9) + 9)
&Iuml; 599 := 1111 − (1 + 1)11−1−1
:= 22 + (((22 + 2)2 ) + 2/2)
3
:= ((33/3) ) − (3
3+3
&Iuml; 604 := 11 &times; (111 − 1)/(1 + 1) − 1
:= 2 &times; (((2
2+2
2
+ 2) ) − 22)
&Iuml; 609 := (11 &times; 111 − 1)/(1 + 1) − 1
:= 2 + (((22/2)2 + 222 ) + 2)
+ 3)
:= 33 + (((3 &times; 3 + 3)3 + 3)/3)
:= 33 + ((3 &times; 3 + 3)3 /3)
4
:= 44 + (((4 + 4) &times; 44) − 4)
:= (4/4 + 4)4 − 4 &times; 4
:= (5 − 5)/5 − 5 &times; 5
:= (55 &times; 55 − 5)/5
:= ((55 − 55)/5) − 5
:= 666 − (66 + 6/6)
:= 6 + (666 − (((6 + 6)/6) + 66))
:= 6 &times; (6 &times; 6 + 66) − 6 &times; 6/(6 + 6)
:= 77/7 + 7 &times; (77 + 7)
:= 7 + ((7 &times; (77 + 7) + ((7 + 7)/7)) + 7)
:= 77 + (7 &times; 77 − 7)
:= 88 + (8 &times; 8 &times; 8 − 8/8)
:= 88 + ((8/((8 + 8)/8)) + 8 &times; 8 &times; 8)
:= (8 − 8/8) &times; (88 − 8/8)
:= (((4 + 4) &times; (44 + 4 )) − 4)/4
5
:= 99 + (((9 &times; 999) + 9)/(9 + 9))
&Iuml; 600 := (1 + 1) &times; (1 + 1 + 1) &times; (11 − 1)1+1
2
9
:= 9 &times; 9 + ((((9 + 9)/9) ) + (99/9))
&Iuml; 605 := 11 &times; (111 − 1)/(1 + 1)
:= 9 &times; 9 &times; 9 − (999/9 + 9)
&Iuml; 610 := (11 &times; 111 − 1)/(1 + 1)
:= 2 + (((22 + 2) ) + 22)
:= 222 + (22/2)2
:= 2 + ((2 &times; 22+2 ) + ((22 + 2)2 ))
:= 3 + (3 &times; 33 &times; (3 + 3) + 3)
:= 3 + (((33/3)3 ) − 33+3 )
:= 33 + (((3 &times; 3 + 3)3 + 3)/3)
:= (4/4 + 4)4 − (4 &times; 4 + 4)
:= 4/4 + ((4/4 + 4)4 − 4 &times; 4)
:= 55 &times; (55/5)
:= 55 + 555
:= 6 + (666 − 66 − 6/6)
:= 6 &times; (6 &times; 6 + 66) − (6 + 6)/6
:= 77 + (7 &times; 77 − (77/7))
:= 7/7 + (7 &times; 77 − 7 + 77)
:= 8 &times; 88 − (88/8 + 88)
:= 8 + ((8 &times; 8 &times; 8 + (8 + 8)/8) + 88)
:= 99/9 &times; ((999 − 9)/(9 + 9))
:= 99 + ((((9 + 9)/9)9 ) − 9/9)
4
:= (4 + 4) &times; ((44 + 4 )/4)
:= 5 &times; (5 &times; 5 &times; 5 − 5)
:= 666 − 66
:= (7/7 + 7) &times; (77 − (7 + 7)/7)
:= 88 + 8 &times; 8 &times; 8
:= 99 + ((((9 + 9)/9)9 ) − (99/9))
&Iuml; 601 := 1 + (1 + 1) &times; (1 + 1 + 1) &times; (11 − 1)1+1
:= 2 + ((((22 + 2)2 ) + 22) + 2/2)
:= 3 + ((3 &times; 33 &times; (3 + 3) + 3/3) + 3)
4
:= (4/4 + 4) − ((4 &times; 4 + 4) + 4)
5
:= (5 + 5)/5 − 5 &times; 5
:= 6/6 + (666 − 66)
:= 7 + ((7 &times; (77 + 7) − 7/7) + 7)
:= 8/8 + (8 &times; 8 &times; 8 + 88)
:= 9 + (((((9 + 9)/9)9 ) − 9/9) + 9 &times; 9)
&Iuml; 602 := (1 + 1) &times; (1 + (1 + 1 + 1) &times; (11 − 1)1+1 )
2
:= 2 + ((((22 + 2) ) + 22) + 2)
&Iuml; 606 := 1 + 11 &times; (111 − 1)/(1 + 1)
:= 2 + (2 &times; (((2
2+2
2
+ 2) ) − 22))
&Iuml; 611 := (1 + 11 &times; 111)/(1 + 1)
:= 222 + ((22&times;(2+2) − 2)/2)
:= 3 + (3 &times; (33 &times; (3 + 3) + 3))
:= 3 &times; 33 + ((3 − 3/3)3&times;3 )
:= 4/4 + ((4/4 + 4)4 − (4 &times; 4 + 4))
:= 4 + ((((4 + 4) &times; 44) − 4/4) + 44 )
:= (55 &times; 55 + 5)/5
:= 5 + ((55 &times; 55 + 5)/5)
:= 6 + (666 − 66)
:= 6 &times; (6 &times; 6 + 66) − 6/6
:= 7 + (7 &times; (77 + 7) + (77/7))
:= 77 + (((7 + 7)/7 − 7) + 7 &times; 77)
:= 8 + ((8 &times; 8 &times; 8 − ((8 + 8)/8)) + 88)
:= 88 + (8 &times; 8 &times; 8 + 88/8)
:= 9 + (((9 + 9) &times; (9 + 9 + 9)) + 999/9)
:= 99 + (((9 + 9)/9)9 )
&Iuml; 607 := 1 + 1 + 11 &times; (111 − 1)/(1 + 1)
&Iuml; 612 := 1 + (1 + 11 &times; 111)/(1 + 1)
:= 2 + ((22/2)2 + 222 )
:= 2 &times; ((22 &times; (22+2 − 2)) − 2)
:= 3 + ((3 &times; (33 &times; (3 + 3) + 3)) + 3/3)
:= (3 + 3) &times; (3 &times; 33 + 3)
:= (4 + 4)/4 &times; ((44 + 4 ) + 4/4)
:= 44 + (((4 + 4) &times; 44) − 4/4)
:= 4 + (((4 + 4) &times; 44) + 44 )
:= (55 + 5 + 5)/5 − 5 &times; 5
:= ((55 &times; 55 + 5) + 5)/5
:= (55 − (55 + 5 + 5))/5
:= 666 − ((6 + 6)/6)6
:= 6 + ((666 − 66) + 6/6)
:= 6 &times; (6 &times; 6 + 66)
3
:= ((33/3) ) − 3
3+3
4
:= 7 + (7 &times; (77 + 7) + 7)
:= 7 + (7 &times; (77 + 7) + (77 + 7)/7)
:= 7 + ((7 &times; 77 − (77/7)) + 77)
:= 88 + (8 &times; 8 &times; 8 + (8 + 8)/8)
:= 8 + ((8 &times; 8 &times; 8 − 8/8) + 88)
:= 88 + (((88 + 8)/8) + 8 &times; 8 &times; 8)
:= 9 + ((((9 + 9)/9)9 ) + 9 &times; 9)
:= 9 &times; 9 &times; 9 − (999 + 99)/9
:= 9 &times; 9 &times; 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
&Iuml; 613 := 1 + 1 + (1 + 11 &times; 111)/(1 + 1)
&Iuml; 618 := 1 + 1 + 11 &times; (1 + 111)/(1 + 1)
&Iuml; 623 := 111 + (1 + 1)11−1−1
:= 222 + ((22&times;(2+2) + 2)/2)
:= 2 + (2 &times; (22 &times; (22+2 − 2)))
:= ((2/2 + 2 + 2)2+2 ) − 2
:= 3/3 + ((3 + 3) &times; (3 &times; 33 + 3))
:= 3 + (3 &times; (3 + 3)3 − 33)
:= 3 + (3 &times; (3 + 3)3 − (33 + 3/3))
:= 4 + ((4/4 + 4)4 − 4 &times; 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 &times; (6 &times; 6 + 66)
:= 6 + 6 &times; (6 &times; 6 + 66)
:= 66/6 + 6 &times; (6 &times; 6 + 66)
:= 7 + ((7 &times; (77 + 7) + (77/7)) + 7)
:= 77 + (7 &times; 77 + ((7 + 7)/7))
:= 7 + (7 &times; 77 + 77)
:= 8 + (8 &times; 88 − (88/8 + 88))
:= (8 + 8)/8 + 88 &times; (8 − 8/8)
:= 8 &times; 8 &times; 8 + 888/8
:= 9/9 + (9 &times; 9 &times; 9 − (99 + 9 + 9))
:= 9 &times; 9 &times; 9 − 999/9
:= 999/9 + (((9 + 9)/9)9 )
&Iuml; 614 := (1 + (1 + 1) &times; (1 + 11))1+1 − 11
:= (2 &times; (22 &times; (22+2 − 2))) − 2
3
&Iuml; 619 := 1 + 1 + 1 + 11 &times; (1 + 111)/(1 + 1)
:= (2 &times; 22) + (((22 + 2)2 ) − 2/2)
3+3
:= 3 &times; (3 + 3) − (3/3 + 33)
:= 3
:= (4/4 + 4)4 − 44/4
:= 444 + ((4 &times; 44) − 4/4)
+ ((3 − 333)/3)
&Iuml; 624 := (1 + (1 + 1) &times; (1 + 11))1+1 − 1
:= (22 + 2) &times; (22 + 2 + 2)
:= 3 + (3 &times; ((3 + 3)3 − 3 &times; 3))
:= 4 &times; (4 &times; (44 − 4) − 4)
:= (5 − 55)/5
:= (5 − 5)/5 − 5
:= (55 − 5)/5
:= (6 + 6)/6 + 6 &times; (6 &times; 6 + 66)
:= 6 + (6 &times; (6 &times; 6 + 66) + 6/6)
:= 666 − (6 &times; 6 + 6)
5
5
:= 77 + (7 &times; 77 − ((7 + 7)/7))
:= 77 + (((7 + 7 + 7)/7) + 7 &times; 77)
:= (7/7 + 7) &times; (7/7 + 77)
:= 8 &times; 88 − ((8 + 8)/8 + 88)
:= 8 + ((8 &times; 8 &times; 8 + 88/8) + 88)
:= 8 + 88 &times; (8 − 8/8)
:= 999/9 + ((((9 + 9)/9)9 ) − 9)
:= 9 &times; 9 &times; 9 + ((9 − 999)/9)
:= (9 − 9/9) &times; (9 &times; 9 − ((9 + 9 + 9)/9))
&Iuml; 615 := 11 &times; (1 + 111)/(1 + 1) − 1
&Iuml; 620 := 11 + (11 &times; 111 − 1)/(1 + 1) − 1
&Iuml; 625 := (1 + (1 + 1) &times; (1 + 11))1+1
:= (2/2 + 2 + 2) &times; ((22/2)2 + 2)
:= (2 &times; 22) + ((22 + 2)2 )
:= (2/2 + 2 + 2)2+2
:= 3 &times; (3 + 3)3 − 33
:= 3 &times; (3 + 3)3 − (33 + 3/3)
:= (3 − 3/3 + 3)3/3+3
:= (4 − 44)/4 + (4/4 + 4)4
:= 444 + (4 &times; 44)
:= (4/4 + 4)4
:= 55 /5 − 5 − 5
:= 55 /5 − 5
:= 55 /5
:= (66 &times; 66 − 666)/6
:= 6 + (6 &times; (6 &times; 6 + 66) + ((6 + 6)/6))
:= (6 − 6/6)6−(6+6)/6
:= 77 + (7 &times; 77 − 7/7)
:= 77 + ((7 &times; 77 − 7) + (77/7))
:= (7 − ((7 + 7)/7))77/7−7
:= 8 &times; 88 − (8/8 + 88)
:= 8 &times; 8 + ((8888 + 8)/(8 + 8))
:= 8 + (88 &times; (8 − 8/8) + 8/8)
:= 9 &times; 9 &times; 9 − (((999 + 9 + 9) + 9)/9)
:= 9 + ((((9 + 9)/9)9 ) + 99)
:= 9 &times; 9 + (((99 &times; 99) − 9)/(9 + 9))
&Iuml; 616 := 11 &times; ((1 + 111)/(1 + 1))
:= 2 &times; (22 &times; (22+2 − 2))
3
&Iuml; 621 := 11 + (11 &times; 111 − 1)/(1 + 1)
:= ((2/2 + 2 + 2)
2+2
)−2−2
3
&Iuml; 626 := 1 + (1 + (1 + 1) &times; (1 + 11))1+1
:= 2 + ((22 + 2) &times; (22 + 2 + 2))
:= 3/3 + (3 &times; (3 + 3) − 33)
:= 3 &times; ((3 + 3) − 3 &times; 3)
:= 3/3 + ((3 − 3/3 + 3)3/3+3 )
:= (4 + 4) &times; ((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 &times; (6 &times; 6 + 66) − ((6 + 6)/6))
:= 6 + ((66 &times; 66 − 666)/6)
:= 6/6 + ((6 − 6/6)6−(6+6)/6 )
:= 77 + 7 &times; 77
:= 7 + ((7 &times; 77 − ((7 + 7)/7)) + 77)
:= (7 &times; (77 + 7 + 7)) − 77/7
:= 88 &times; (8 − 8/8)
:= 8 &times; (88 − 8) − (88/8 + 8)
:= 8 + ((((8 + 8)/8) − 88) + 8 &times; 88)
:= 99/9 &times; ((999 + 9)/(9 + 9))
:= 9 &times; 9 &times; 9 − (99 + 9)
:= 9 + (9 &times; 9 &times; 9 − ((999 + 9)/9))
&Iuml; 617 := 1 + 11 &times; (1 + 111)/(1 + 1)
2
:= 22 + (222/2 + 22 )
:= 3
3+3
− ((333 + 3)/3)
4
&Iuml; 622 := 11 + (1 + 11 &times; 111)/(1 + 1)
:= 222 + (22 − 2)2
&Iuml; 627 := 11 &times; (1 + ((1 + 111)/(1 + 1)))
:= 2 + ((2/2 + 2 + 2)2+2 )
3
:= 3/3 + (3 &times; ((3 + 3) − 3 &times; 3))
4
:= 33 + 3 &times; 33 &times; (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 &times; (6 &times; 6 + 66) − 6/6)
:= 66 + (6666 + 6)/(6 + 6)
:= 666 − ((66 &times; 6/(6 + 6)) + 6)
:= 7/7 + (7 &times; 77 + 77)
:= 7 + ((7 &times; 77 − 7/7) + 77)
:= 77 + (7 &times; 77 + (77/7))
:= 8/8 + 88 &times; (8 − 8/8)
:= 8 &times; 8 &times; 8 + ((888 − 8)/8)
:= 88/8 + 88 &times; (8 − 8/8)
:= 9 &times; 9 &times; 9 − ((999 + 9)/9)
:= 9/9 + (9 &times; 9 &times; 9 − (99 + 9))
:= 9 + (9 &times; 9 &times; 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
&Iuml; 628 := 1 + (11 &times; (1 + ((1 + 111)/(1 + 1))))
2
2
&Iuml; 633 := (1 + 1 + 1) &times; ((1 + 1) &times; 111 − 11)
2+2
:= (2 &times; (22+2 &times; (22 − 2))) − 2
:= 22 + ((2 &times; (2 + 2 + 2)) )
:= 2 &times; (2 + 2) + ((2/2 + 2 + 2)
:= 3 + ((3 − 3/3 + 3)3/3+3 )
:= 3 + (3 &times; ((3 + 3)3 − (3 + 3)))
:= (3 &times; ((3 + 3)3 − 3)) − 3/3
:= 4 + (4 &times; (4 &times; (44 − 4) − 4))
:= 4 + ((4/4 + 4)4 + 4)
:= (4 &times; (4 &times; (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 &times; 6)
:= 666 − (66 &times; 6/(6 + 6))
:= (66/6) &times; (((6 + 6)/6)6 − 6)
:= 77 + ((77 + 7)/7 + 7 &times; 77)
:= 7 + ((7 &times; (77 + 7 + 7)) − (77/7))
:= 7/7 + (7 &times; (77 + 7 + 7))
:= 8 &times; (88 − 8) − (88 + 8)/8
:= 8/8 + (8 &times; (88 − 8) − 8)
:= 8 &times; (88 − 8) − (8 + 8)/8
:= 9 &times; 9 &times; 9 − ((9 + 9)/9 + 99)
:= 9 &times; 9 &times; 9 + (((9 + 9 + 9)/9) − 99)
:= 9 &times; (9 &times; 9 − 9) − 9/9 − 9
&Iuml; 629 := 1 + (1 + (11 &times; (1 + ((1 + 111)/(1 + 1)))))
2+2
)
&Iuml; 638 := 11 &times; (1 + 1 + (1 + 111)/(1 + 1))
&Iuml; 634 := 1 + ((1 + 1 + 1) &times; ((1 + 1) &times; 111 − 11))
&Iuml; 639 := 1 + 11 &times; (1 + 1 + (1 + 111)/(1 + 1))
:= 2 + (((22 + 2 + 2) ) − (2 &times; 22))
:= 2 + ((((2 + 2 + 2)2+2 ) − 22)/2)
:= 33+3 − (3 &times; 33 + 3/3)
:= 3 &times; (3 + 3)3 − (33/3 + 3)
:= 3 &times; ((3 + 3)3 − 3)
:= 4 + (4/4 + 4)4
:= 4 + (((4/4 + 4)4 + 4/4) + 4)
:= (4 &times; (4 &times; (44 − 4))) − 4/4
:= 5 + (5 − 5)/5
:= 5 + ((55 − 5)/5 + 5)
:= 5 + (((55 − 5)/5 + 5) + 5)
:= 666 − (6 &times; 6 + 6/6)
:= 6 + (666 − (((6 + 6)/6) + 6 &times; 6))
:= 6 + (666 − (66 &times; 6/(6 + 6)))
:= (7 &times; (77 + 7 + 7)) − (7/7 + 7)
:= 7 + ((7 &times; 77 + (77/7)) + 77)
:= ((7 + 7)/7) + (7 &times; (77 + 7 + 7))
:= 8 &times; (88 − 8) − 88/8
:= (8 + 8)/8 + (8 &times; (88 − 8) − 8)
:= 8 &times; (88 − 8) − 8/8
:= 9 &times; 9 &times; 9 − (9/9 + 99)
:= 9 + ((((99 &times; 99) − 9)/(9 + 9)) + 9 &times; 9)
:= 9 &times; (9 &times; 9 − 9) − 9
:= 2 + (((2/2 + 2 + 2)
) + 2)
5
&Iuml; 630 := (11 − 1) &times; ((1 + 1 + 1) &times; (11 + (11 − 1)))
2
&Iuml; 635 := 11 + ((1 + (1 + 1) &times; (1 + 11))1+1 − 1)
&Iuml; 640 := 111 + (1 + 11 + 11)1+1
:= 2 + (((2 &times; (2 + 2 + 2))2 ) + 222 )
:= ((((2 + 2 + 2)2+2 ) − 22)/2) − 2
:= 2 &times; (22+2 &times; (22 − 2))
:= 3 &times; ((3 + 3)3 − (3 + 3))
:= (3 &times; ((3 + 3)3 − 3)) − (3/3 + 3)
:= 3/3 + (3 &times; ((3 + 3)3 − 3))
:= 4 + ((4/4 + 4)4 + 4/4)
:= (4/4 + 4)4 + (44 − 4)/4
:= 4 &times; (4 &times; (44 − 4))
:= 5 + 55 /5
:= 5 + (55 /5 + 5)
:= 5 + (55 /5 + 5 + 5)
:= 666 − 6 &times; 6
:= 6 + (666 − (6 &times; 6 + 6/6))
:= ((66 − 6)/6) &times; ((6 + 6)/6)6
:= (7 &times; (77 + 7 + 7)) − 7
:= (7 &times; (77 + 7 + 7)) − (7 + 7)/7
:= (7 − ((7 + 7)/7)) &times; ((7 + 7)/7)7
:= (8 − 8/8) &times; ((8 + 8)/8 + 88)
:= 8 + (88 &times; (8 − 8/8) + (88/8))
:= 8 &times; (88 − 8)
:= 9 &times; 9 &times; 9 − 99
:= 9 &times; 99 − (((9 + 9)/9)9−9/9 )
:= 9/9 + (9 &times; (9 &times; 9 − 9) − 9)
&Iuml; 631 := 1 + ((11 − 1) &times; ((1 + 1 + 1) &times; (11 + (11 − 1))))
&Iuml; 636 := 11 + (1 + (1 + 1) &times; (1 + 11))1+1
&Iuml; 641 := 1 + 111 + (1 + 11 + 11)1+1
:= 2 + ((((2/2 + 2 + 2)2+2 ) + 2) + 2)
:= 2 &times; ((22+2 &times; (22 − 2)) − 2)
:= 2/2 + (2 &times; (22+2 &times; (22 − 2)))
:= 3/3 + (3 &times; ((3 + 3)3 − (3 + 3)))
:= (3 &times; ((3 + 3)3 − 3)) − 3
:= 3 + ((3 &times; ((3 + 3)3 − 3)) − 3/3)
:= 4 + ((4/4 + 4)4 + (4 + 4)/4)
:= (4 &times; (4 &times; (44 − 4))) − 4
:= 4 &times; 4 + (4/4 + 4)4
:= 5 + (55 + 5)/5
:= (55 + 55 )/5
:= 5 + ((55 + 55 )/5)
:= 6/6 + (666 − 6 &times; 6)
:= 6 + (666 − 6 &times; 6)
:= 666 + ((66/6) − 6 &times; 6)
:= 7 + ((7/7 + 7) &times; (7/7 + 77))
:= (7 &times; (77 + 7 + 7)) − 7/7
:= 77/7 + ((7 &times; (77 + 7 + 7)) − 7)
:= 8 &times; (88 − 8) − (8/8 + 8)
:= 8 &times; (88 − 8) − (8/((8 + 8)/8))
:= 8/8 + 8 &times; (88 − 8)
:= 9/9 + (9 &times; 9 &times; 9 − 99)
:= 9 &times; (9 &times; 9 − 9) − (99 + 9)/9
:= 9 &times; 9 &times; 9 + ((99/9) − 99)
&Iuml; 632 := 111+1 + ((1 + 1)11−1−1 − 1)
2
&Iuml; 637 := 1 + (11 + (1 + (1 + 1) &times; (1 + 11))1+1 )
2+2
) − 22)/2
&Iuml; 642 := (1 + 1) &times; ((1 + 1 + 1) &times; 111 − 1 − 11)
:= 2 + (2 &times; (22+2 &times; (22 − 2)))
:= (22 + 2 + 2) − 2 &times; 22
:= (((2 + 2 + 2)
:= 3 + (33+3 − (3 &times; 33 + 3/3))
:= 3 &times; (3 + 3)3 − 33/3
:= 3 + (3 &times; ((3 + 3)3 − 3))
:= (4 &times; (4 &times; (44 − 4))) − 4 − 4
:= 4 + (((4/4 + 4)4 + 4) + 4)
:= 4 &times; 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 &times; 6)
:= 6 + ((666 − 6 &times; 6) + 6/6)
:= (6 &times; 6 &times; (6 + 6 + 6)) − 6
:= (7/7 + 7) &times; ((7 + 7)/7 + 77)
:= 7 &times; (77 + 7 + 7)
:= 777 − (((7 + 7)/7)7 + 7)
:= 8 &times; (88 − 8) − 8
:= 8 + (8 &times; (88 − 8) − (88/8))
:= (8 + 8)/8 + 8 &times; (88 − 8)
:= 9 &times; 9 &times; 9 + (((9 + 9)/9) − 99)
:= 9 &times; (9 &times; 9 − 9) − 99/9
:= 9 &times; 9 &times; 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
&Iuml; 643 := (1 + 1) &times; ((1 + 1 + 1) &times; 111 − 11) − 1
&Iuml; 648 := (1 + 1) &times; ((1 + 1) &times; (11 − 1 − 1))1+1
&Iuml; 653 := (111+1+1 − 1)/(1 + 1) − 1 − 11
:= (22 − 2)2 + (222 + 2)/2
:= 2 &times; ((22+2 + 2)2 )
:= 222 + ((22/2 + 2)2 )
:= 3 + ((3 &times; ((3 + 3)3 − 3)) + 3/3)
:= 3 &times; (3 + 3)3
:= 3 + ((3 &times; (3 + 3)3 − 3/3) + 3)
:= 4 + ((4 &times; (4 &times; (44 − 4))) − 4/4)
:= (4 + 4) &times; (4 − 4/4)4
:= 44 + ((4/4 + 4)4 − 4 &times; 4)
:= 5 + (((55 + 55 + 5) + 5)/5)
:= 5 &times; 5 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5 &times; 5)
:= 6/6 + ((6 &times; 6 &times; (6 + 6 + 6)) − 6)
:= 6 &times; 6 &times; (6 + 6 + 6)
:= 666 − (6/6 + 6 + 6)
:= 7 + ((7 &times; (77 + 7 + 7)) − 7/7)
:= 77/7 + (7 &times; (77 + 7 + 7))
:= 7 &times; 77 + (((7 + 7)/7)7 − (7 + 7))
:= 88/8 + (8 &times; (88 − 8) − 8)
:= 8 + 8 &times; (88 − 8)
:= 88 + (8 &times; (8 &times; 8 + 8) − 88/8)
:= 99 + (((99 &times; 99) − 9)/(9 + 9))
:= 9 &times; (9 &times; 9 − 9)
:= 9 &times; (9 &times; 9 − 9) + ((9 &times; 9 + 9)/(9 + 9))
&Iuml; 649 := 11 &times; ((111+1 − 1)/(1 + 1) − 1)
&Iuml; 644 := (1 + 1) &times; ((1 + 1 + 1) &times; 111 − 11)
:= 2 &times; (((22+2 + 2)2 ) − 2)
:= (((2 + 2 + 2)2+2 ) + 2)/2
3
3
&Iuml; 654 := (1 + 1) &times; ((1 + 1 + 1) &times; (111 − 1 − 1))
:= ((22 + 2 + 2)2 ) − 22
:= 3 &times; (3 + 3) − (3/3 + 3)
:= 3/3 + 3 &times; (3 + 3)
:= 4 + (4 &times; (4 &times; (44 − 4)))
:= 4 + (((4/4 + 4)4 + 4 &times; 4) + 4)
:= 4 + ((44 − 4)/4 &times; (44 + 4)/4)
:= 5 &times; 5 + ((5 − 5)/5 − 5)
:= 5 &times; 5 + (55 − 5)/5
:= 5 + ((55 − 5)/5 + 5 &times; 5)
:= 666 − ((66 + 66)/6)
:= 6/6 + (6 &times; 6 &times; (6 + 6 + 6))
:= 666 − 6 − 6
:= 7 + (7 &times; (77 + 7 + 7))
:= 777 − ((7 + 7)/7)7
:= 77 + (7 &times; (77 + 7) − (77/7))
:= 8 &times; (88 − 8) + (8/((8 + 8)/8))
:= 8 + (8 &times; (88 − 8) + 8/8)
:= 8 + ((8 &times; (88 − 8) − ((8 + 8)/8)) + 8)
:= 99 + (((99 &times; 99) + 9)/(9 + 9))
:= 9/9 + 9 &times; (9 &times; 9 − 9)
:= 9 + (9 &times; (9 &times; 9 − 9) − ((9 + 9 + 9)/9))
5
&Iuml; 645 := 1 + (1 + 1) &times; ((1 + 1 + 1) &times; 111 − 11)
:= ((((2 + 2 + 2)
2+2
) − 2)/2) − 2
&Iuml; 650 := 1 + 11 &times; ((111+1 − 1)/(1 + 1) − 1)
:= 2 + (2 &times; ((2
3
:= 3 &times; (3 + 3) − 3
:= 3 + (3 &times; (3 + 3)3 + 3)
2+2
2
+ 2) ))
&Iuml; 655 := (1 + 1) &times; (1 + 1 + 1) &times; 111 − 11
:= 2 + (((22/2 + 2)2 ) + 222 )
:= 3 + (3 &times; (3 + 3)3 − 3/3)
:= 3 + ((3 &times; (3 + 3)3 + 3/3) + 3)
:= (44 − 4)/4 &times; (44 + 4)/4
:= (4 &times; (4 &times; (44 − 4) + 4)) − 4/4
:= 5 &times; 5 + (5 /5 − 5)
:= 5 &times; (5 &times; 5 &times; 5 + 5)
:= 5 + (55 /5 + 5 &times; 5)
:= (6 &times; 6 &times; 6 &times; 6 − 6)/((6 + 6)/6)
:= 666 + (((6 − 66)/6) − 6)
:= 666 − 66/6
:= 7 + ((7 &times; (77 + 7 + 7)) + 7/7)
:= 7 &times; 77 + 777/7
:= 7 + ((7 &times; (77 + 7 + 7)) + (77/7))
:= 8 + ((8 &times; (88 − 8) − (88/8)) + 8)
:= 8 + (8 &times; (88 − 8) + ((8 + 8)/8))
:= 8 + ((8 &times; (88 − 8) − 8/8) + 8)
:= 9 &times; (9 &times; 9 − 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 &times; (9 &times; 9 − 9)
:= 9 + (9 &times; (9 &times; 9 − 9) − ((9 + 9)/9))
4
:= 4 + ((4/4 + 4) + 4 &times; 4)
5
&Iuml; 646 := (1 + 1) &times; (1 + (1 + 1 + 1) &times; 111 − 11)
:= (2 &times; ((22+2 + 2)2 )) − 2
&Iuml; 651 := (1 + 1 + 1) &times; ((1 + 1) &times; (111 − 1 − 1) − 1)
:= 2 + ((((2 + 2 + 2)
3
:= 3/3 + (3 &times; (3 + 3) − 3)
4
:= 4 + (((4/4 + 4) + 4 &times; 4) + 4/4)
2+2
) + 2)/2)
&Iuml; 656 := 1 + ((1 + 1) &times; (1 + 1 + 1) &times; 111 − 11)
:= 2 + (((22 + 2 + 2)2 ) − 22)
:= 3 + 3 &times; (3 + 3)3
:= (3 &times; ((3 + 3)3 + 3)) − 3/3
:= 44/4 + (4 &times; (4 &times; (44 − 4)))
:= 4 &times; (4 &times; (44 − 4) + 4)
:= 5 + (((55 + 55 )/5) + 5)
:= 5 &times; 5 + (5 + 5)/5
:= 5 + ((55 + 5)/5 + 5 &times; 5)
:= (6 &times; 6 &times; (6 + 6 + 6)) − (6 + 6)/6
:= (6 &times; 6 &times; 6 &times; 6 + 6)/((6 + 6)/6)
:= 666 + (6 − 66)/6
:= 7 + ((7 &times; (77 + 7 + 7)) + ((7 + 7)/7))
:= 7 + ((7 &times; (77 + 7 + 7)) + 7)
:= 7 + (777 − ((7 + 7)/7)7 )
:= 8 + (8 &times; (88 − 8) − ((8 + 8)/8))
:= 88/8 + 8 &times; (88 − 8)
:= 8 + (8 &times; (88 − 8) + 8)
:= 9 &times; (9 &times; 9 − 9) − (9 + 9)/9
:= 9 &times; (9 &times; 9 − 9) + ((9 + 9 + 9)/9)
:= 9 + (9 &times; (9 &times; 9 − 9) − 9/9)
5
&Iuml; 647 := (1 + 1) &times; ((1 + 1) &times; (11 − 1 − 1))1+1 − 1
:= (((2 + 2 + 2)
2+2
) − 2)/2
&Iuml; 652 := (1 + 1) &times; ((1 + 1 + 1) &times; (111 − 1 − 1) − 1)
:= 2 &times; (((2
3
2+2
2
+ 2) ) + 2)
3
&Iuml; 657 := (1 + 1 + 1) &times; (((1 + 1) &times; (111 − 1)) − 1)
:= (2/2 + 2) &times; (222 − (2/2 + 2))
:= 3 &times; (3 + 3) − 3/3
:= 3 + (3 &times; (3 + 3) + 3/3)
:= 3 &times; ((3 + 3)3 + 3)
:= ((4 + 4) &times; (4 − 4/4)4 ) − 4/4
:= 4 + ((4 + 4) &times; (4 − 4/4)4 )
:= 4 &times; (4 + 4) + (4/4 + 4)4
:= ((55 + 55) + 55 )/5
:= 5 &times; 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)) &times; ((7 + 7)/7) )
:= 7 + (((7 &times; (77 + 7 + 7)) + 7/7) + 7)
:= 7 + (777/7 + 7 &times; 77)
:= 8 + (8 &times; (88 − 8) − 8/8)
:= ((88 + 8)/8) + 8 &times; (88 − 8)
:= 8 + ((8 &times; (88 − 8) + 8/8) + 8)
:= 9 &times; (9 &times; 9 − 9) − 9/9
:= 9 &times; (9 &times; 9 − 9) + ((9 &times; 9 − 9)/(9 + 9))
:= 9 + 9 &times; (9 &times; 9 − 9)
:= (6 &times; 6 &times; (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
&Iuml; 658 := (1 + 1) &times; (((1 + 1 + 1) &times; (111 − 1)) − 1)
&Iuml; 663 := (1 + 1 + 1) &times; ((1 + 1) &times; 111 − 1)
2
&Iuml; 668 := (1 + 1) &times; (1 + (1 + 1 + 1) &times; 111)
:= 222 + ((22 − 2/2) )
:= 2 + ((2/2 + 2) &times; 222)
:= 3/3 + (3 &times; ((3 + 3) + 3))
:= 3 + ((3 &times; ((3 + 3)3 + 3)) + 3)
:= 3 + (33+3 − ((3/3 + 3)3 ))
:= 44 + ((4/4 + 4)4 − 44/4)
:= 4 + 4 &times; 4 &times; 44 − (44 + 4/4)
:= (4 &times; ((4 &times; 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 &times; 6/(6 + 6)
:= 666 + (6 + 6)/6
:= (22 &times; (2 &times; (2 + 2) + 22)) − 2
3
:= 77 + (7 &times; (77 + 7) − 7)
:= 7 + ((777 − ((7 + 7)/7) ) + 7)
:= (7 &times; 7 &times; (7 + 7)) − (77/7 + 7)
:= 8 + ((8 &times; (88 − 8) + ((8 + 8)/8)) + 8)
:= 88 + (8 &times; (8 &times; 8 + 8) − 8/8)
:= 8 + (8 &times; 88 − (88/((8 + 8)/8)))
:= 9 + (9 &times; (9 &times; 9 − 9) + 9/9)
:= 9 + ((9 &times; (9 &times; 9 − 9) − ((9 + 9 + 9)/9)) + 9)
:= 9 + (9 &times; (9 &times; 9 − 9) + (99/9))
&Iuml; 659 := ((1 + 1) &times; ((1 + 1 + 1) &times; (111 − 1))) − 1
2+2
7
&Iuml; 664 := (1 + 1) &times; ((1 + 1 + 1) &times; 111 − 1)
&Iuml; 669 := (1 + 1 + 1) &times; (1 + (1 + 1) &times; 111)
:= ((2/2 + 2) &times; 222) − 2
:= (2/2 + 2) &times; (222 + 2/2)
:= 33/3 + 3 &times; (3 + 3)3
:= (3 − 3/3) &times; (333 − 3/3)
:= 3 + (333 + 333)
:= 4 &times; 4 &times; 44 − (44 + 4/4)
:= 4 + (44 &times; (44/4 + 4))
:= 44 + (4/4 + 4)4
:= ((55 &times; (55 + 5)) − 5)/5
:= 5 + (((55 &times; (55 + 5)) − 5)/5)
:= 55 + ((55 − 55)/5)
:= 666 − 6/6 − 6
:= 666 − (6 + 6)/6
:= 666 + (6 &times; 6/(6 + 6))
:= 777 − (777/7 + 7)
:= 77 + (7 &times; (77 + 7) − 7/7)
:= ((7 − 77)/7) + ((7 &times; 7 &times; (7 + 7)) − 7)
:= 8 + (8 &times; (88 − 8) + (88/8))
:= 88 + 8 &times; (8 &times; 8 + 8)
:= 8 &times; 88 − ((88/8 + 8 + 8) + 8)
:= 99/9 + 9 &times; (9 &times; 9 − 9)
:= (9 − 9/9) &times; (((9 + 9)/9) + 9 &times; 9)
:= 9 + (9 &times; (9 &times; 9 − 9) + ((99 + 9)/9))
:= (((2 + 2 + 2)
) + 22)/2
&Iuml; 660 := (1 + 1) &times; ((1 + 1 + 1) &times; (111 − 1))
:= 22 &times; (2 &times; (2 + 2) + 22)
3
:= 3 + (3 &times; ((3 + 3) + 3))
&Iuml; 665 := (111+1+1 − 1)/(1 + 1)
2
:= ((22 + 2 + 2) ) − 22/2
:= 3
3+3
&Iuml; 670 := 1 + ((1 + 1 + 1) &times; (1 + (1 + 1) &times; 111))
:= 2 + (((2/2 + 2) &times; 222) + 2)
:= 3 + ((((3 + 3) &times; 333) + 3)/3)
3
− ((3/3 + 3) )
:= 44 &times; (44/4 + 4)
:= 44 + ((4/4 + 4) − 4)
:= 44 + ((4/4 + 4)4 + 4/4)
:= 55 &times; ((55 + 5)/5)
:= 5 + (55 &times; ((55 + 5)/5))
:= 55 + (55 /5 − (5 + 5))
:= 666 − 6
:= 666 − 6/6
:= 6 + (666 − ((6 + 6)/6))
4
:= 7 &times; 77 + (((7 + 7)/7)7 − 7)
:= 77 + 7 &times; (77 + 7)
:= ((7 + 7)/7) &times; (7 &times; 7 &times; 7 − (7/7 + 7))
:= 8 &times; 88 − (88/((8 + 8)/8))
:= 8/8 + (8 &times; (8 &times; 8 + 8) + 88)
:= ((88 + 8) &times; (8 − 8/8)) − (8 + 8)/8
:= 9 &times; (9 &times; 9 − 9) + (99 + 9)/9
:= 9 + ((9 &times; (9 &times; 9 − 9) − 9/9) + 9)
:= 9 &times; (9 &times; 9 − 9) + ((99 + 99)/9)
&Iuml; 661 := 1 + ((1 + 1) &times; ((1 + 1 + 1) &times; (111 − 1)))
:= 2 + ((((2 + 2 + 2)
2+2
) + 22)/2)
3
&Iuml; 666 := (1 + 1) &times; (1 + 1 + 1) &times; 111
&Iuml; 671 := 11 &times; ((1 + 111+1 )/(1 + 1))
:= 2 + ((2/2 + 2) &times; (222 + 2/2))
:= (2/2 + 2) &times; 222
:= 3 + ((3 &times; ((3 + 3) + 3)) + 3/3)
:= 3 &times; (((3 + 3) + 3) + 3)
:= 33/3 &times; (((3/3 + 3)3 ) − 3)
:= 4 + ((4/4 + 4)4 + 4 &times; (4 + 4))
:= ((4 + 4)/4 + 4) &times; 444/4
:= 44/4 &times; ((44 + 4)/4 − 4)
:= 5 &times; 5 + ((55 + 55 )/5)
:= 555 + 555/5
:= 5 + (555/5 + 555)
:= 6/6 + (666 − 6)
:= 666
:= 6 + (666 − 6/6)
:= 7 &times; 77 + (777 + 77)/7
:= (7 − 7/7) &times; 777/7
:= (7 &times; 7 &times; (7 + 7)) − (7/7 + 7 + 7)
:= ((88 + 8) &times; (8 − 8/8)) − 88/8
:= (8 − (8 + 8)/8) &times; 888/8
:= ((88 + 8) &times; (8 − 8/8)) − 8/8
:= 9 &times; (9 &times; 9 − 9) + ((99 + 9 + 9)/9)
:= 9 + (9 &times; (9 &times; 9 − 9) + 9)
:= 99/9 &times; (9 &times; 9 − (99/9 + 9))
&Iuml; 662 := (1 + 1) &times; (1 + ((1 + 1 + 1) &times; (111 − 1)))
&Iuml; 667 := 1 + (1 + 1) &times; (1 + 1 + 1) &times; 111
&Iuml; 672 := (1 + 1) &times; ((1 + 1 + 1) &times; (1 + 111))
:= 2 + (22 &times; (2 &times; (2 + 2) + 22))
3
:= 2/2 + ((2/2 + 2) &times; 222)
:= (2/2 + 2) &times; (222 + 2)
:= 3 + (3 &times; (3 + 3) + 33/3)
:= (((3 + 3) &times; 333) + 3)/3
:= 33 + (3 &times; (3 + 3)3 − 3)
:= (4 + 4)/4 + (44 &times; (44/4 + 4))
:= 4444/4 − 444
:= 4 &times; ((4 &times; 44) − (4 + 4))
:= 5 + (((5 + 5)/5) + 5 /5)
:= 555 + (555 + 5)/5
:= (5/5 + 5) &times; (555 + 5)/5
:= 666 + ((6 + 6)/6 − 6)
:= 6/6 + 666
3
5
5
:= 6 + 666
7
:= 7 + (((7 &times; (77 + 7 + 7)) + (77/7)) + 7)
:= 7 &times; 77 + ((7 + 7)/7)
:= (7 + 7) &times; (7 &times; 7 − 7/7)
:= 88 + (8 &times; (8 &times; 8 + 8) − (8 + 8)/8)
:= 8 + ((8 &times; (88 − 8) + (88/8)) + 8)
:= (88 + 8) &times; (8 − 8/8)
:= 9 + (((9 &times; 9 + 9)/(9 + 9)) + 9 &times; (9 &times; 9 − 9))
:= 9 + ((9 &times; (9 &times; 9 − 9) + 9/9) + 9)
:= (9 − 9/9) &times; (((9 + 9 + 9)/9) + 9 &times; 9)
54
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 673 := 1 + ((1 + 1) &times; ((1 + 1 + 1) &times; (1 + 111)))
&Iuml; 678 := 1 + (1 + (((1 + 1) &times; (1 + (1 + 11)))1+1 ))
&Iuml; 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 &times; (3 + 3)3 + 33 )
:= ((33/3)3 ) − 3 &times; (3 + 3)3
:= 4 + ((4/4 + 4)4 + 44)
:= (4 − 44)/4 + (4 &times; ((4 &times; 44) − 4))
:= (4 &times; ((4 &times; 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) &times; 777/7)
:= (7 &times; 7 &times; (7 + 7)) − (7/7 + 7)
:= (7 &times; 7 &times; (7 + 7)) − (7 + 7 + 7)/7
:= 8/8 + ((88 + 8) &times; (8 − 8/8))
:= 8 &times; 88 + ((8 − 88)/8 − (8 + 8))
:= 8 + ((8/8 + 8) &times; (88/8 + 8 &times; 8))
:= 9 + ((9 − 9/9) &times; (((9 + 9)/9) + 9 &times; 9))
:= 999/9 + (9 &times; (9 &times; 9 − (9 + 9)))
:= 9 + ((((9 + 9)/9)9 ) + 9 &times; (9 + 9))
&Iuml; 674 := (1 + 1) &times; (1 + ((1 + 1 + 1) &times; (1 + 111)))
&Iuml; 679 := (((1 + 1)11 ) − 11)/(1 + 1 + 1)
&Iuml; 684 := 1 + ((1 + ((1 + 1)11 ))/(1 + 1 + 1))
:= ((22 + 2 + 2)2 ) − 2
:= 2 + (((22 + 2 + 2)2 ) + 2/2)
:= 2 &times; (2 &times; (((22/2 + 2)2 ) + 2))
:= 33 + (3 &times; (3 + 3)3 − 3/3)
:= 3 + ((33 − 3/3)3−3/3 )
:= 3 + (3 &times; (3 + 3)3 + 33)
:= 4 + (((4/4 + 4)4 + 44) + 4/4)
:= 4 + (44/4 + 4) &times; (44 + 4/4)
:= (4 &times; ((4 &times; 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 &times; 77)
:= (7 &times; 7 &times; (7 + 7)) − 7
:= (7 &times; 7 &times; (7 + 7)) − (7 + 7)/7
:= 8 + ((8 − (8 + 8)/8) &times; 888/8)
:= 8 &times; 88 − (8/8 + 8 + 8 + 8)
:= 8 &times; 88 − (((88 + 8)/8) + 8)
:= 9 &times; (9 + 9) + (((9 + 9)/9)9 )
:= 9 &times; 9 &times; 9 − ((9 &times; 99 + 9)/(9 + 9))
:= (99 &times; (9 − ((9 + 9)/9))) − 9
7
&Iuml; 675 := (((1 + 1) &times; (1 + (1 + 11)))1+1 ) − 1
&Iuml; 680 := 1 + ((((1 + 1)11 ) − 11)/(1 + 1 + 1))
&Iuml; 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 &times; (3 + 3)3 − 3/3)
:= 33+3 − (33/3 + 33)
:= (44/4 + 4) &times; (44 + 4/4)
:= (4 + 4) &times; ((4 − 4/4)4 + 4)
:= 4 &times; 4 + ((4/4 + 4)4 + 44)
:= 5 &times; ((5 &times; 5 &times; 5 + 5) + 5)
:= 55 + 55 /5
:= 5 + (55 /5 + 55)
:= 6 + ((6 &times; 6/(6 + 6)) + 666)
:= 6 + ((666 + ((6 + 6)/6)) + 6)
:= 6 + ((666 + 6/6 + 6) + 6)
:= (7 &times; 7 &times; (7 + 7)) − 77/7
:= 7/7 + ((7 &times; 7 &times; (7 + 7)) − 7)
:= (7 &times; 7 &times; (7 + 7)) − 7/7
:= (8/8 + 8) &times; (88/8 + 8 &times; 8)
:= 8 &times; 88 − 8 − 8 − 8
:= 8 &times; 88 − (88/8 + 8)
:= 9 + ((9 &times; (9 &times; 9 − 9) + 9) + 9)
:= 9 &times; 9 &times; 9 + ((9 − 9 &times; 99)/(9 + 9))
:= 9 + ((((9 − 9/9) + 9) + 9)(9+9)/9 )
:= ((22 + 2 + 2)2 ) − 2/2
:= 3 &times; ((3 + 3)3 + 3 &times; 3)
&Iuml; 676 := ((1 + 1) &times; (1 + (1 + 11)))1+1
&Iuml; 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 &times; (3 + 3)
:= 4 + (4 &times; ((4 &times; 44) − (4 + 4)))
:= 4 + ((((4/4 + 4)4 + 44) + 4) + 4)
5
&Iuml; 686 := (11 + (((1 + 1)11 ) − 1))/(1 + 1 + 1)
:= 2 &times; ((22/2)2 + 222)
:= (3 − 3/3) &times; (((3/3 + 3) + 3)3 )
5
:= (4 &times; ((4 &times; 44) − 4)) − (4 + 4)/4
:= 55 + ((5 + 5)/5 − 5)
:= 55 + (5 + 5)/5
:= 5 + ((55 + 5)/5 + 55)
:= 666 + (66 − 6)/6
:= 66 &times; (6 + 6) − 666/6
:= 6 + (((666 + ((6 + 6)/6)) + 6) + 6)
:= ((7 − 77)/7) + (7 &times; 7 &times; (7 + 7))
:= ((7 + 7)/7) + ((7 &times; 7 &times; (7 + 7)) − 7)
:= 7 &times; 7 &times; (7 + 7)
:= 8 &times; 88 + ((8 − 8 &times; 8)/((8 + 8)/8))
:= 8/8 + (8 &times; 88 − (8 + 8 + 8))
:= 8 &times; 88 + ((8 − 88)/8 − 8)
:= 9 &times; 99 − (999/9 + 99)
:= (99 − 9/9) &times; (9 − ((9 + 9)/9))
:= (((9 − 9/9) + 9) + 9)
(9+9)/9
&Iuml; 677 := 1 + (((1 + 1) &times; (1 + (1 + 11)))1+1 )
&Iuml; 682 := (((1 + 1)11 ) − (1 + 1))/(1 + 1 + 1)
&Iuml; 687 := 11 + (((1 + 1) &times; (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) &times; 333) + 33)/3
:= 3/3 + (3 &times; (3 + 3)3 + 33)
:= 33+3 − (3 &times; 3 + 33)
:= 4 + (((4/4 + 4)4 + 44) + 4)
:= 44/4 &times; ((44 − 4 − 4)/4)
:= (4 &times; ((4 &times; 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 &times; 6/(6 + 6))6 ) − (6 &times; 6 + 6)
:= (7 &times; 7 &times; (7 + 7)) − ((7 + 7)/7 + 7)
:= 7 + ((7 &times; 7 &times; (7 + 7)) − (77/7))
:= 7/7 + (7 &times; 7 &times; (7 + 7))
:= 8 &times; 88 − (88/8 + 8 + 8)
:= 8 &times; 88 − (88 + 88)/8
:= 8 &times; 88 − (8/8 + 8 + 8)
:= 9 + ((9 &times; (9 &times; 9 − 9) + (99/9)) + 9)
:= 99/9 &times; (9 &times; 9 − (9/9 + 9 + 9))
:= 9 + ((9 &times; (9 &times; 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
&Iuml; 688 := (1 + 1) &times; (11 + (1 + 1 + 1) &times; 111)
:= 3 + (3
3+3
&Iuml; 693 := 11 &times; ((1 + 1 + 1) &times; (11 + (11 − 1)))
− (33/3 + 33))
&Iuml; 698 := (1 + 1 + 1) &times; (11 + (1 + 1) &times; 111) − 1
:= (2/2 + 2) &times; ((22 − 22)/2)
:= 22 + ((22 + 2 + 2)2 )
:= 33 &times; ((3 &times; (3 + 3)) + 3)
:= 3 + (33+3 − (3/3 + 33))
2
:= 2 &times; (2 &times; (2 &times; (2 &times; 2 &times; 22 − 2)))
4
:= 4 &times; ((4 &times; 44) − 4)
:= 44/4 &times; ((4 − 4)/4)
:= 44 + (444 − (4 + 4)/4)
:= (5 &times; 5 &times; 5 &times; 55 + 5)/(5 + 5)
:= 5 + ((5 &times; 5 &times; 5 &times; 55 + 5)/(5 + 5))
:= (5 &times; (5 + 5 + 5)) + (55 − 5 − 5)/5
:= 666 + ((66 + 66)/6)
:= ((6 &times; 6/(6 + 6)) ) − 6 &times; 6
:= ((7 + 7)/7) + (7 &times; 7 &times; (7 + 7))
:= ((66/6) &times; ((6 + 6)/6)6 ) − 6
:= 7 + (7 &times; 7 &times; (7 + 7))
:= 8 &times; 88 − 8 − 8
:= 777 − ((7 + 7)/7 + 77)
:= 8 &times; 88 − 88/8
:= 9 &times; 9 &times; 9 − ((9 &times; 9 &times; 9 + 9)/(9 + 9))
:= (8 + 8)/8 + (8 &times; 88 − 8)
:= 99 &times; (9 − ((9 + 9)/9))
:= 9 &times; 9 &times; 9 − (((99 + 99)/9) + 9)
6
&Iuml; 694 := 11 + ((1 + ((1 + 1)11 ))/(1 + 1 + 1))
&Iuml; 699 := (1 + 1 + 1) &times; (11 + (1 + 1) &times; 111)
:= 2 + (((22 + 2 + 2)2 ) + 22+2 )
:= 2 + ((22/2)2 + ((22 + 2)2 ))
:= 3 + ((3 − 3/3) &times; (((3/3 + 3) + 3) ))
:= 3/3 + (33 &times; ((3 &times; (3 + 3)) + 3))
:= 3 + (33+3 − 33)
:= 4/4 + (4 &times; ((4 &times; 44) − 4))
:= (4 − 44)/4 + 4 &times; 4 &times; 44
:= 44 + (444 − 4/4)
&Iuml; 689 := 1 + ((1 + 1) &times; (11 + (1 + 1 + 1) &times; 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 &times; (5 + 5 + 5))
:= 6 + ((666 + (66/6)) + 6)
:= 6 + (((66 + 66)/6) + 666)
:= 777 − (77/7 + 77)
:= 7 + ((7 &times; 7 &times; (7 + 7)) + 7/7)
:= 6 + (((6 &times; 6/(6 + 6))6 ) − 6 &times; 6)
:= 8/8 + (8 &times; 88 − (8 + 8))
:= 8 &times; 88 + (8 − 88)/8
:= 9 &times; 9 &times; 9 + ((9 − 9 &times; 9 &times; 9)/(9 + 9))
:= 9/9 + (99 &times; (9 − ((9 + 9)/9)))
&Iuml; 690 := 11 + ((((1 + 1)11 ) − 11)/(1 + 1 + 1))
:= 2 + (((2/2 + 2) &times; 222) + 22)
:= 33 + (3 &times; ((3 + 3)3 + 3))
:= (4 + 4)/4 + (4 &times; ((4 &times; 44) − 4))
&Iuml; 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 &times; 88 − (8 + 8))
:= 9 &times; (9 &times; 9 + 9) − 999/9
&Iuml; 700 := 1 + (1 + 1 + 1) &times; (11 + (1 + 1) &times; 111)
:= 2 &times; ((22 &times; 22+2 ) − 2)
:= 3 + ((33+3 − 33) + 3/3)
4
:= 4 + (444 − (4/4 + 4))
:= 44 + 444
:= 5 + ((55 /5 + 55) + 5)
:= 5 &times; 5 &times; (5 &times; 5 + 5) − 55
:= 6 + (((666 + 6) + 6) + 6)
:= 5 &times; (((5 &times; 5 &times; 5 + 5) + 5) + 5)
:= 6 &times; 6 + (666 − 6/6 − 6)
:= 77/7 + ((7 &times; 7 &times; (7 + 7)) − 7)
:= 6 &times; 6 + (666 − ((6 + 6)/6))
:= 7 + ((7 &times; 7 &times; (7 + 7)) + ((7 + 7)/7))
:= (8 + 8)/8 + (8 &times; 88 − (8 + 8))
:= 777 − 77
:= 8 &times; 88 − (8/8 + 8)
:= (9/9 + 9) &times; (9 &times; 9 − ((99 + 9)/9))
:= 8 &times; 88 − (8/((8 + 8)/8))
:= 9 + ((99 − 9/9) &times; (9 − ((9 + 9)/9)))
:= (9/9 + 9) &times; (9 &times; 9 − 99/9)
&Iuml; 691 := ((1 + 1)11−1 ) − (1 + 1 + 1) &times; 111
:= 2 + ((((22 + 2 + 2)2 ) + 22/2) + 2)
:= 3
3+3
3
− (33/3 + 3 )
:= 4 + ((4 &times; ((4 &times; 44) − 4)) − 4/4)
5
&Iuml; 696 := (1 + 1 + 1) &times; (111 + 111+1 )
:= 2 &times; (2 &times; ((2 &times; 2 &times; 2 &times; 22) − 2))
:= 3
3+3
− 33
&Iuml; 701 := 1 + 1 + (1 + 1 + 1) &times; (11 + (1 + 1) &times; 111)
:= 2/2 + (2 &times; ((22 &times; 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 &times; (5 + 5 + 5))
:= 6 &times; 6 + (666 − 6/6)
:= 6 &times; 6 + (666 − 66/6)
:= 6 &times; 6 + (666 − 6)
:= 7 + ((7 &times; 7 &times; (7 + 7)) − ((7 + 7)/7))
:= (7 &times; 7 &times; (7 + 7)) + (77 − 7)/7
:= 7/7 + (777 − 77)
:= 8 &times; 88 − (88 + 8 + 8)/8
:= 8 &times; 88 − 8
:= 8 + (8 &times; 88 − (88/8))
:= 9 &times; 9 &times; 9 − ((99/9 + 9 + 9) + 9)
:= (9 − 9/9) &times; (99 − ((99 + 9)/9))
:= 9 &times; 9 &times; 9 − ((9/9 + 9 + 9) + 9)
&Iuml; 692 := (11 &times; ((1 + 1 + 1) &times; (11 + (11 − 1)))) − 1
:= 2 &times; (((2
:= 3
3+3
2+2
2
+ 2) ) + 22)
− (3/3 + 33 + 3)
:= 4 + (4 &times; ((4 &times; 44) − 4))
5
&Iuml; 697 := 1 + (1 + 1 + 1) &times; (111 + 111+1 )
2
2
:= (22/2) + ((22 + 2) )
:= 3/3 + (3
3+3
&Iuml; 702 := (1 + 1) &times; ((11 &times; ((11 &times; (1 + 1 + 1)) − 1)) − 1)
:= (2 &times; (22 &times; 22+2 )) − 2
:= 33+3 − 33
− 33)
4
:= 4 + (44/4 &times; ((4 − 4)/4))
5
:= 4 &times; 4 &times; 44 − (4 + 4)/4
:= 55 + ((55 + 5 + 5)/5)
:= 5 + (((55 + 5 + 5)/5) + 55)
:= 55 + (((55 + 55) + 55 )/5)
:= 6 &times; 6 + (((6 − 66)/6) + 666)
:= 6 &times; 6 + ((666 − 6) + 6/6)
:= 6 &times; 6 + 666
:= 7 + ((7 &times; 7 &times; (7 + 7)) − 7/7)
:= 77/7 + (7 &times; 7 &times; (7 + 7))
:= ((7 + 7)/7) + (777 − 77)
:= 8 &times; 88 − (88 + 8)/8
:= 8/8 + (8 &times; 88 − 8)
:= 8 &times; 88 − (8 + 8)/8
:= 99 + ((((9 + 9)/9)9 ) + 9 &times; 9)
:= 9 + (9 &times; 9 &times; 9 − ((9 &times; 9 &times; 9 + 9)/(9 + 9)))
:= 9 &times; 9 &times; 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
&Iuml; 703 := (11 &times; ((1 + 1)(1+1)&times;(1+1+1) )) − 1
&Iuml; 708 := (1 + 11) &times; ((111+1 − 1)/(1 + 1) − 1)
&Iuml; 713 := (1 + (11 + 11)) &times; (1 + ((11 − 1) &times; (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 &times; 4 &times; 44 − 4/4
:= 4 + 4 &times; 4 &times; 44
:= 4 + (4 &times; 4 &times; 44 + 4/4) + 4
:= 5 &times; 5 + ((55 − 5 − 5)/5 + 55)
:= ((55 + 5)/5) &times; (55 − 5/5 + 5)
:= (55 &times; (55 + 5 + 5) − (5 + 5))/5
:= 6 &times; 6 + (666 + 6/6)
:= 6 + (666 + 6 &times; 6)
:= 6 &times; 6 + (666 + (66/6))
:= (2 &times; (22 &times; 2
2+2
)) − 2/2
2
:= 7 + ((7 &times; 7 &times; (7 + 7)) + ((77 − 7)/7))
:= 7 + ((777 − 77) + 7/7)
:= 7 + ((777 − (7/7 + 77)) + 7)
:= 8 &times; 88 − 8/8
:= 8 &times; 88 + (8/((8 + 8)/8))
:= 8 + (8 &times; 88 + 8/8)
:= 9/9 + (9 &times; 9 &times; 9 − (9 + 9 + 9))
:= 9 &times; 9 &times; 9 − (((99 + 9)/9) + 9)
:= 9 &times; 9 &times; 9 + (((9 + 9)/9) − (9 + 9))
&Iuml; 704 := 11 &times; ((1 + 1)(1+1)&times;(1+1+1) )
&Iuml; 709 := 11 &times; 111 − (1 + 1)11−1−1
&Iuml; 714 := (1 + 1) &times; ((1 + 1 + 1) &times; (111+1 − (1 + 1)))
:= 2 &times; (22 &times; 22+2 )
:= 2 + ((222 + 222 ) + 2/2)
:= (2 + 2 + 2) &times; ((22/2)2 − 2)
:= 33/3 &times; ((3/3 + 3)3 )
:= 33+3 − (33/3 + 3 &times; 3)
:= 3 + (33+3 − (3 &times; (3 + 3)))
:= 4 &times; (4 &times; 44)
:= 4 + 4 &times; 4 &times; 44 + 4/4
:= 4 &times; 4 &times; 44 + (44 − 4)/4
5
5
:= 5 + (((5 − 5)/5 + 55) + 5 &times; 5)
:= (55 &times; (55 + 5 + 5) − 5)/5
:= (66/6) &times; ((6 + 6)/6)
:= 6 + ((666 + 6/6) + 6 &times; 6)
:= (6 + 6) &times; (66 − 6) − 6
:= 7 + ((7 &times; 7 &times; (7 + 7)) + (77/7))
:= 7 + (((7 + 7)/7 − 77) + 777)
:= 7 + ((777 − 77) + 7)
:= 8 &times; 88
:= 8 + ((8 &times; 88 − (88/8)) + 8)
:= 8 + ((8 + 8)/8 + 8 &times; 88)
:= (9 − 9/9) &times; (99 − (99/9))
:= 9 &times; 9 &times; 9 − (99/9 + 9)
:= 9 &times; 9 &times; 9 + (((9 + 9 + 9)/9) − (9 + 9))
:= 5 &times; 5 + ((5 − 5)/5 + 55)
6
&Iuml; 705 := 1 + 11 &times; ((1 + 1)(1+1)&times;(1+1+1) )
&Iuml; 710 := (11 − 1) &times; ((1 + 11)1+1 /(1 + 1) − 1)
2
&Iuml; 715 := 11 &times; (1 + ((1 + 1)(1+1)&times;(1+1+1) ))
:= 2/2 + (2 &times; (22 &times; 22+2 ))
:= 2 + ((222 + 22 ) + 2)
:= 22/2 + (2 &times; (22 &times; 22+2 ))
:= 3 + (33+3 − 33 )
:= 33+3 − ((3 &times; (3 + 3)) + 3/3)
:= 33+3 − (33/3 + 3)
:= 4/4 + 4 &times; 4 &times; 44
:= 4 + 4 &times; 4 &times; 44 + (4 + 4)/4
:= 44/4 + 4 &times; 4 &times; 44
:= 5 &times; 5 + (5 /5 + 55)
:= 5 + ((55 /5 + 55) + 5 &times; 5)
:= 55 &times; (55 + 5 + 5)/5
:= ((66 &times; ((6 + 6)/6)6 ) + 6)/6
:= 6 + ((66/6) &times; ((6 + 6)/6)6 )
:= (66/6) &times; (66 − 6/6)
:= 7 + ((7 &times; 7 &times; (7 + 7)) + (77 + 7)/7)
:= 777 + (((77 − 7)/7) − 77)
:= 7 + (((777 − 77) + 7/7) + 7)
:= 8/8 + 8 &times; 88
:= 8 + (8 &times; 88 − ((8 + 8)/8))
:= 88/8 + 8 &times; 88
:= 9 + ((9 − 9/9) &times; (99 − ((99 + 9)/9)))
:= 9 &times; 9 &times; 9 − (9/9 + 9 + 9)
:= 9 &times; 9 &times; 9 + (((9 − 99)/(9 + 9)) − 9)
5
&Iuml; 706 := 1 + 1 + 11 &times; ((1 + 1)(1+1)&times;(1+1+1) )
&Iuml; 711 := 1111 − ((1 + 1) &times; (11 − 1))1+1
&Iuml; 716 := ((1 + 1)11 ) − (1 + 111+1+1 )
:= 222 + 222
:= (2222/2) − (22 − 2)2
:= 2 + ((2 + 2 + 2) &times; ((22/2)2 − 2))
:= 3 + ((33+3 − 33 ) + 3/3)
:= 33+3 − (3 &times; (3 + 3))
:= 33+3 + (((3 − 33)/3) − 3)
:= (4 + 4)/4 + 4 &times; 4 &times; 44
:= 4 + (4 &times; 4 &times; 44 − 4/4) + 4
:= (4 &times; ((4 &times; 44) + 4)) − 4
:= 5 &times; 5 + ((55 + 5)/5 + 55)
5
:= ((555 + 5 )/5) − 5 &times; 5
:= (55 &times; (55 + 5 + 5) + 5)/5
:= 6 + ((666 − ((6 + 6)/6)) + 6 &times; 6)
:= ((6 &times; 6/(6 + 6)) ) − 6 − 6 − 6
:= ((66 &times; (66 − 6/6)) + 6)/6
:= 7 + (777 − (7/7 + 77))
:= 77/7 + (777 − 77)
:= 7 &times; (77 + 7) + ((7 + 7)/7)7
:= (8 + 8)/8 + 8 &times; 88
:= 8 + (8 &times; 88 − 8/8)
:= 8 &times; 88 + (88 + 8)/8
:= 9 &times; 9 &times; 9 − ((99 + 99 + 9)/9)
:= 9 &times; 9 &times; 9 − (9 + 9)
:= 9 &times; 9 &times; 9 − ((99 + 9 + 9)/9)
&Iuml; 707 := (11 − 1 − 1)1+1+1 − 11 − 11
6
&Iuml; 712 := 1 + (1111 − (((1 + 1) &times; (11 − 1))1+1 ))
&Iuml; 717 := ((1 + 1)11 ) − 111+1+1
:= 2/2 + (222 + 222 )
:= 2 &times; (((22 &times; 22+2 ) + 2) + 2)
:= 222 + (22/2 + 222 )
:= 3 + (33/3 &times; ((3/3 + 3)3 ))
:= 3/3 + (33+3 − (3 &times; (3 + 3)))
:= 33+3 − (3 &times; 3 + 3)
:= 4 + 4 &times; 4 &times; 44 − 4/4
:= 4 + 4 &times; 4 &times; 44 + 4
:= 4/4 + ((4 &times; ((4 &times; 44) + 4)) − 4)
5
5
5
:= 5 &times; 5 + ((5 + 5 + 5)/5 + 55)
:= 55 + (((5 + 5)/5) + 5 /5)
:= ((55 &times; (55 + 5 + 5) + 5) + 5)/5
:= 6 + ((666 − 6/6) + 6 &times; 6)
:= 6 &times; 6 + ((66 − 6)/6 + 666)
:= ((6 &times; 6/(6 + 6))6 ) − 6 − 6
:= 7 + (777 − 77)
:= 777 + ((77 + 7)/7 − 77)
:= 777 − (77/7 + 7 &times; 7)
:= 88/8 + (8 &times; 88 − 8)
:= 8 + 8 &times; 88
:= 8 &times; 88 + (88 + 8 + 8)/8
:= 9 &times; 9 &times; 9 − ((99 + 99)/9)
:= 9/9 + (9 &times; 9 &times; 9 − (9 + 9))
:= 9 &times; 9 &times; 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
&Iuml; 718 := (11 − 1 − 1)1+1+1 − 11
&Iuml; 723 := (1 + 1 + 1) &times; (((1 + 1) &times; 111+1 ) − 1)
&Iuml; 728 := (11 − 1 − 1)1+1+1 − 1
:= 2 &times; (((22 − (2/2 + 2))2 ) − 2)
:= (2/2 + 2) &times; (222 − 2)/2
:= 2 + (222 /2 + 222 )
:= 33+3 − 33/3
:= 33+3 − (3 + 3)
:= 33+3 − 3/3
:= (4 &times; ((4 &times; 44) + 4)) − (4 + 4)/4
:= 4 + ((4 &times; ((4 &times; 44) + 4)) − 4/4)
:= 4 + ((4 &times; ((4 &times; 44) + 4)) + 4)
:= 5 &times; 5 &times; (5 &times; 5 + 5) − ((5 + 5)/5)5
:= 5 &times; (5 &times; 5 − 5) + (55 − 5 − 5)/5
:= (55 + 5/5) &times; (55 + 5 + 5)/5
:= (6 + 6) &times; (66 − 6) − (6 + 6)/6
:= ((6 &times; 6/(6 + 6))6 ) − 6
:= ((6 &times; 6/(6 + 6))6 ) − 6/6
:= 7 + ((77/7 − 77) + 777)
:= 7 + (7 &times; (77 + 7) + ((7 + 7)/7)7 )
:= 777 − 7 &times; 7
:= 8 + ((8 &times; 88 − ((8 + 8)/8)) + 8)
:= 8 + (88/8 + 8 &times; 88)
:= 8 + (8 &times; 88 + 8 + 8)
:= 9 &times; 9 &times; 9 − 99/9
:= 9 &times; 9 &times; 9 + (((9 + 9 + 9)/9) − 9)
:= 9 &times; 9 &times; 9 − 9/9
&Iuml; 719 := 1 + ((11 − 1 − 1)1+1+1 − 11)
&Iuml; 724 := (1 + 1) &times; ((11 &times; (11 &times; (1 + 1 + 1))) − 1)
&Iuml; 729 := (11 − 1 − 1)1+1+1
:= (2/2 + 2)2+2+2
:= ((22 − 2) &times; (2 + 2 + 2)2 ) − 2/2
:= 2 + (2 &times; ((22 − (2/2 + 2))2 ))
:= 33+3 + ((3 − 33)/3)
:= 3/3 + (33+3 − (3 + 3))
:= (4 &times; ((4 &times; 44) + 4)) − 4/4
:= 4 + (4 &times; ((4 &times; 44) + 4))
:= (4 − 4/4)4+(4+4)/4
:= (((55 + 5)(5+5)/5 ) − 5)/5
:= (55 − 5)/5 + 5 &times; (5 &times; 5 − 5)
:= (5 − (5 + 5)/5)5/5+5
:= (6 + 6) &times; (66 − 6) − 6/6
:= (66 &times; 66 − 6 − 6)/6
:= (6 &times; 6/(6 + 6))6
:= 777 − (((7 + 7)/7 + 7 &times; 7) + 7)
:= (7 &times; (7 &times; (7 + 7) + 7)) − 77/7
:= 7/7 + (777 − 7 &times; 7)
:= 8 + ((8 &times; 88 − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 8 &times; 88)
:= (8/8 + 8)88/8−8
:= 9 &times; 9 &times; 9 − 9/9 − 9
:= 9 &times; 9 &times; 9 + ((9 − 99)/(9 + 9))
:= 9 &times; 9 &times; 9
&Iuml; 720 := (1 + 1) &times; ((1 + 1 + 1) &times; (111+1 − 1))
:= (22 − 2) &times; (2 + 2 + 2)2
&Iuml; 725 := ((1 + 1) &times; (11 &times; (11 &times; (1 + 1 + 1)))) − 1
:= 33+3
&Iuml; 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 &times; ((4 &times; 44) + 4)
:= 4 + ((4 &times; ((4 &times; 44) + 4)) + 4/4)
:= 4 + (44/4 &times; (((44 + 4) + 4)/4))
:= (5/5 + 5) &times; (5 &times; 5 &times; 5 − 5)
:= 5 &times; (5 &times; (5 &times; 5 + 5) − 5)
:= 5 + (5 &times; (5 &times; (5 &times; 5 + 5) − 5))
:= (6 + 6) &times; (66 − 6)
:= 3
3+3
−3&times;3
:= (66 &times; 66 − 6)/6
:= 6/6 + ((6 &times; 6/(6 + 6))6 )
:= 777 − ((7/7 + 7 &times; 7) + 7)
:= 777 − (((7 + 7 + 7)/7) + 7 &times; 7)
:= ((7 + 7)/7) + (777 − 7 &times; 7)
:= 8 + (8 &times; 88 + 8)
:= 8 + ((88 + 8 + 8)/8 + 8 &times; 88)
:= 8 + (((((8 + 8)/8) + 8 &times; 88) + 8) + 8)
:= 9 &times; 9 &times; 9 − 9
:= 9 &times; 9 &times; 9 + ((9 − 9 &times; 9)/(9 + 9))
:= 9/9 + 9 &times; 9 &times; 9
&Iuml; 721 := (111 + 111+1+1 )/(1 + 1)
2
2
&Iuml; 726 := (1 + 1) &times; (11 &times; (11 &times; (1 + 1 + 1)))
&Iuml; 731 := 1 + (1 + (11 − 1 − 1)1+1+1 )
:= ((((2 + 2 + 2) + 2) ) − 2)/2
:= 22 &times; (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 &times; ((4 &times; 44) + 4))
:= 44/4 &times; (((44 + 4) + 4)/4)
:= 44/4 + (4 &times; ((4 &times; 44) + 4))
:= 5/5 + (5 &times; (5 &times; (5 &times; 5 + 5) − 5))
:= ((555 + 55 )/5) − 5
:= 6/6 + (6 + 6) &times; (66 − 6)
:= 66 &times; (66/6)
:= 6 + ((66 &times; 66 − 6)/6)
:= 777 − (7 &times; 7 + 7)
:= (7 − 7/7) &times; (((7 + 7)/7)7 − 7)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) − (77/7))
:= 8 + ((8 &times; 88 + 8/8) + 8)
:= 8 &times; 88 + (88 + 88)/8
:= 8 + ((88/8 + 8 &times; 88) + 8)
:= 9/9 + (9 &times; 9 &times; 9 − 9)
:= 9 &times; 9 &times; 9 − (9 + 9 + 9)/9
:= 9 &times; 9 &times; 9 + ((9 + 9)/9)
:= (((55 + 5)
(5+5)/5
) + 5)/5
&Iuml; 722 := (1 + 1) &times; ((((1 + 1) &times; (11 − 1)) − 1)1+1 )
:= 2 &times; ((22 − (2/2 + 2))2 )
&Iuml; 727 := (11 − 1 − 1)1+1+1 − 1 − 1
&Iuml; 732 := 1 + (1 + (1 + (11 − 1 − 1)1+1+1 ))
:= ((2/2 + 2)2+2+2 ) − 2
:= (2/2 + 2) &times; (222 /2 + 2)
:= 3/3 + (33+3 − 3)
:= 3 + 33+3
:= (4 + 4)/4 + (4 &times; ((4 &times; 44) + 4))
:= 4 + (((4 &times; ((4 &times; 44) + 4)) − 4/4) + 4)
:= 44 + (4 &times; ((4 &times; 44) − 4))
(5+5)/5
:= (5 + 5)/5 + (5 &times; (5 &times; (5 &times; 5 + 5) − 5))
:= (((555 + 55 ) + 5)/5) − 5
:= (66 &times; 66 + 6)/6
:= 66 + 666
:= 7/7 + (777 − (7 &times; 7 + 7))
:= 777 − (7/7 + 7 &times; 7)
:= 77/7 + (777 − (7 &times; 7 + 7))
:= 8 + ((((8 + 8)/8) + 8 &times; 88) + 8)
:= 8 + (((8 &times; 88 − 8/8) + 8) + 8)
:= 8 + ((((88 + 8)/8) + 8 &times; 88) + 8)
:= 9 &times; 9 &times; 9 + (((9 + 9)/9) − 9)
:= 9 &times; 9 &times; 9 − (9 + 9)/9
:= 9 &times; 9 &times; 9 + ((9 + 9 + 9)/9)
:= 3
3+3
− ((3/3 + 3) + 3)
:= ((((55 + 5)
) + 5) + 5)/5
:= (6 + 6)/6 + (6 + 6) &times; (66 − 6)
58
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 733 := 1 + (1 + (1 + (1 + (11 − 1 − 1)1+1+1 )))
&Iuml; 738 := 1 + (11 &times; (1 + ((1 + 1) &times; (11 &times; (1 + 1 + 1)))))
&Iuml; 743 := 1 + (1 + (1 + (11 + (11 − 1 − 1)1+1+1 )))
:= 2 + (((2/2 + 2)2+2+2 ) + 2)
:= 2 + (22+2 &times; ((2 &times; 22) + 2))
:= 2 + ((22&times;(2+2) + 222 ) + 2/2)
:= 3 + (33+3 + 3/3)
:= 3 &times; 3 + 33+3
:= 3 + (33+3 + 33/3)
:= 4 + ((4 − 4/4)4+(4+4)/4 )
:= 44 + ((44 &times; 44 − (4 + 4))/4)
:= 4 + (((44 &times; 44 − 4)/4) + 44 )
:= 55 + ((55 − 5 − 5)/5 + 55)
:= (((555 + 55 ) + 5) + 5)/5
:= 5 + ((((555 + 55 ) + 5) + 5)/5)
:= 6 + ((66 &times; 66 + 6)/6)
:= 6 + (666 + 66)
:= 6 + ((66/6) &times; (66 + 6/6))
:= (7 &times; (7 &times; (7 + 7) + 7)) − (7 + 7)/7
:= 777 + (((77 − 7)/7) − 7 &times; 7)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) + 7/7)
:= 8 + (((88 + 8 + 8)/8 + 8 &times; 88) + 8)
:= (8/8 + 8) &times; ((((8 + 8)/8) − 8) + 88)
:= 888/8 + (8 &times; (88 − 8) − 8)
:= 9 &times; 9 &times; 9 + ((9 &times; 9 − 9)/(9 + 9))
:= 9 + 9 &times; 9 &times; 9
:= 9 + (((9 &times; 9 + 9)/(9 + 9)) + 9 &times; 9 &times; 9)
&Iuml; 734 := (1 + 1) &times; (1 + ((1 + 1 + 1) &times; (1 + 111+1 )))
2
&Iuml; 739 := 11 + ((11 − 1 − 1)1+1+1 − 1)
&Iuml; 744 := (1 + 11) &times; (1 + ((1 + 111+1 )/(1 + 1)))
:= 222 + (2(2/2+2) )
:= 222 + (22&times;(2+2) − 2/2)
:= (2 &times; 222 ) − (222 + 2)
:= 3 + ((33+3 − 3/3) + 3)
:= 3 &times; 3 + (33+3 + 3/3)
:= 3 + ((33+3 + 3 &times; 3) + 3)
:= (4 + 4)/4 &times; (444/4 + 44 )
:= 44 + ((44 &times; 44 − 4)/4)
:= 44 + (444 + 44 )
:= 5 + ((5 − (5 + 5)/5)5/5+5 )
:= 5 &times; 5 &times; 5 + ((55 − 55)/5)
:= 5 &times; 5 &times; 5 + ((55 − 5)/5 − 5)
:= 6 + (((6 &times; 6/(6 + 6))6 ) − 6/6)
:= 6 + (((66 &times; 66 + 6)/6) + 6)
:= 6 + ((666 + 66) + 6)
:= (7 &times; (7 &times; (7 + 7) + 7)) − 7/7
:= 77/7 + (777 − 7 &times; 7)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) + ((7 + 7)/7))
:= 8 + ((88 + 88)/8 + 8 &times; 88)
:= 8 + (((88/8 + 8 &times; 88) + 8) + 8)
:= 8 &times; (88 + 8) − 8 − 8 − 8
:= 9 &times; 9 &times; 9 + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + (9 &times; 9 &times; 9 + 9/9)
:= 9 + ((9 &times; 9 &times; 9 − ((9 + 9 + 9)/9)) + 9)
&Iuml; 735 := (1 + 1 + 1) &times; (1 + ((1 + 1) &times; (1 + 111+1 )))
&Iuml; 740 := 11 + (11 − 1 − 1)1+1+1
&Iuml; 745 := (1 + (1 + (1 + ((1 + 1)1+1+11 ))))/11
:= 2 + ((((2/2 + 2)2+2+2 ) + 2) + 2)
:= 222 + 22&times;(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) &times; (44 − 44/4)
:= 4 + (4 &times; (((4 &times; 44) + 4) + 4))
:= 4 + (((44 &times; 44 + 4)/4) + 44 )
:= 55 + (55 /5 + 55)
:= 5 &times; 5 &times; (5 &times; 5 + 5) − 5 − 5
:= 5 &times; 5 &times; (5 &times; 5 + 5) − 5
:= 66/6 + ((6 &times; 6/(6 + 6))6 )
:= 6 + ((((66 &times; 66 + 6)/6) + 6) + 6)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) − ((7 + 7)/7))
:= 777 − ((77/7 + 7 + 7) + 7)
:= 8 &times; 88 + ((8 &times; 8 + 8)/((8 + 8)/8))
:= 8 + (((8/8 + 8)88/8−8 ) + 8)
:= 9 &times; 9 &times; 9 + (99/9)
:= 9 + ((9 &times; 9 &times; 9 − ((9 + 9)/9)) + 9)
:= 6 + ((6 &times; 6/(6 + 6))6 )
:= 7 &times; (7 &times; (7 + 7) + 7)
:= 8 + ((((8 &times; 88 − 8/8) + 8) + 8) + 8)
:= 9 + (9 &times; 9 &times; 9 − ((9 + 9 + 9)/9))
&Iuml; 736 := 111 + (1 + (1 + 1) &times; (1 + 11))1+1
:= 22+2 &times; ((2 &times; 22) + 2)
:= 3 + ((33+3 + 3/3) + 3)
:= 4 &times; (((4 &times; 44) + 4) + 4)
&Iuml; 741 := 1 + (11 + (11 − 1 − 1)1+1+1 )
:= 2/2 + (2
:= 3 + (3
2&times;(2+2)
3+3
2
+ 22 )
:= (2 &times; 222 ) − 222
:= 3 + ((33+3 + 33/3) + 3)
+ 3 &times; 3)
4
:= 4 + ((44 &times; 44 + 4)/4)
5
:= (555 + 55 )/5
&Iuml; 746 := (1 + 1) &times; (((11 + 11)1+1 ) − 111)
:= 44 + 4 &times; 4 &times; 44 − (4 + 4)/4
:= 5 + (((555 + 55 )/5) + 5)
:= 5 + ((555 + 5 )/5)
6
:= 6 + (((6 &times; 6/(6 + 6))6 ) + 6/6)
:= 6 + (((6 &times; 6/(6 + 6)) ) + 6)
:= 6 + (((6 &times; 6/(6 + 6))6 ) + (66/6))
:= 7/7 + (7 &times; (7 &times; (7 + 7) + 7))
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) − 7/7)
:= 77/7 + (7 &times; (7 &times; (7 + 7) + 7))
:= 8 + ((8 &times; 88 + 8 + 8) + 8)
:= 8 &times; 88 + 888/(8 + 8 + 8)
:= 8 &times; (88 + 8) − (88 + 88)/8
:= 9 + (9 &times; 9 &times; 9 − ((9 + 9)/9))
:= 9 &times; 9 &times; 9 + (99 + 9)/9
:= 9 + ((9 &times; 9 &times; 9 − 9/9) + 9)
&Iuml; 737 := 11 &times; (1 + ((1 + 1) &times; (11 &times; (1 + 1 + 1))))
:= 2/2 + (2
2+2
&times; ((2 &times; 22) + 2))
&Iuml; 742 := 1 + (1 + (11 + (11 − 1 − 1)1+1+1 ))
:= 2 + (2
2&times;(2+2)
2
+ 22 )
&Iuml; 747 := ((1 + 1) &times; (11 &times; (1 + (11 &times; (1 + 1 + 1))))) − 1
:= 2/2 + ((2 &times; 222 ) − 222)
:= 3 &times; 3 + (33+3 − 3/3)
:= 3 + ((33+3 + 3/3) + 3 &times; 3)
:= 3 &times; ((3 + 3)3 + 33)
:= 4/4 + (4 &times; (((4 &times; 44) + 4) + 4))
:= 44 + (((44 &times; 44 + 4) + 4)/4)
:= 44 + 4 &times; 4 &times; 44 − 4/4
:= ((555 + 5 ) + 5)/5
:= 5 + (((555 + 5 ) + 5)/5)
:= ((555 + 55) + 55 )/5
:= (66/6) &times; (66 + 6/6)
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + 6/6) + 6)
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + 6) + 6)
:= ((7 + 7)/7) + (7 &times; (7 &times; (7 + 7) + 7))
:= 7 + (7 &times; (7 &times; (7 + 7) + 7))
:= (77 + 7)/7 + (7 &times; (7 &times; (7 + 7) + 7))
:= 8 + (((88 + 88)/8 + 8 &times; 88) + 8)
:= (8/8 + 8) &times; ((88/8 + 8 &times; 8) + 8)
:= 9 &times; 9 &times; 9 + ((99 + 9 + 9)/9)
:= 9 + (9 &times; 9 &times; 9 + 9)
5
5
:= 8 + ((8/8 + 8)
88/8−8
)
:= 9 + (9 &times; 9 &times; 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
&Iuml; 748 := (1 + 1) &times; (11 &times; (1 + (11 &times; (1 + 1 + 1))))
:= 22 &times; ((2 &times; 2
2+2
&Iuml; 753 := (((1 + 11)1+1+1 )/(1 + 1)) − 111
:= 2 + (((2/2 + 2)
) + 2)
2+2+2
) + 22)
&Iuml; 758 := (11 &times; ((1 + 1 + 1) &times; (1 + (11 + 11)))) − 1
:= 2 + (2 &times; ((22 − 2)2 − 22))
:= 3/3 + (33+3 + (3 &times; (3 + 3)))
:= 33 + (33+3 − 3)
:= 3 + ((33+3 − 3/3) + 33 )
:= 44 + 4 &times; 4 &times; 44
:= 4/4 + (4 &times; (444 − 44 ))
:= (4 + 4)/4 + ((4 − 4/4) &times; (44 − 4))
:= 5 &times; 5 &times; 5 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5 &times; 5 &times; 5)
:= 5 + (((55 − 5 − 5)/5 + 5 &times; 5 &times; 5) + 5)
:= (66/6) &times; (((6 + 6)/6) + 66)
:= 6 + (((((6 &times; 6/(6 + 6))6 ) + 6) + 6) + 6)
:= (6 + 6)/6 + (6 &times; ((66 − 6) + 66))
:= 7 + (((7 &times; (7 &times; (7 + 7) + 7)) − 7/7) + 7)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) + (77/7))
:= 777 − ((77 + 7)/7 + 7)
:= 8 &times; 88 + (88/((8 + 8)/8))
:= 8/8 + (8 &times; (88 + 8) − (8 + 8))
:= (8 − 88)/8 + 8 &times; (88 + 8)
:= 9 + ((9 &times; 9 &times; 9 + 9/9) + 9)
:= 9 + (((9 &times; 9 &times; 9 − ((9 + 9 + 9)/9)) + 9) + 9)
:= 9 + ((9 &times; 9 &times; 9 + (99/9)) + 9)
&Iuml; 749 := 1 + ((1 + 1) &times; (11 &times; (1 + (11 &times; (1 + 1 + 1)))))
2+2+2
&Iuml; 754 := 1 + ((((1 + 11)1+1+1 )/(1 + 1)) − 111)
&Iuml; 759 := 11 &times; ((1 + 1 + 1) &times; (1 + (11 + 11)))
:= (2 &times; ((22 − 2)2 − 22)) − 2
:= (2/2 + 2) &times; ((222 + 22)/2)
:= 3 &times; 3 + (33+3 + 33/3)
:= 33 + ((33+3 − 3) + 3/3)
:= 3 + (33+3 + 33 )
:= 44 + 4 &times; 4 &times; 44 + 4/4
:= (4 + 4)/4 + (4 &times; (444 − 44 ))
:= (4 − 4/4) &times; ((4/4 − 4) + 44 )
:= 5 &times; 5 &times; 5 + (55 − 5)/5
:= 5 + ((55 − 5)/5 + 5 &times; 5 &times; 5)
:= 5 + (((55 − 5)/5 + 5 &times; 5 &times; 5) + 5)
:= 6 + (((66/6) &times; (66 + 6/6)) + 6)
:= 6 + ((66/6) &times; (((6 + 6)/6) + 66))
:= 6 &times; 6 + (((6 &times; 6/(6 + 6))6 ) − 6)
:= 22 + (((2/2 + 2)
) − 2)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 7)) + 7)
:= 777 − ((((7 + 7)/7 + 7) + 7) + 7)
:= 777 − (77/7 + 7)
:= 8 &times; (88 + 8) − (88/8 + 8)
:= (8 + 8)/8 + (8 &times; (88 + 8) − (8 + 8))
:= 8 &times; (88 + 8) − (8/8 + 8)
:= 9 + (9 &times; 9 &times; 9 + (99/9))
:= 9 + (((9 &times; 9 &times; 9 − ((9 + 9)/9)) + 9) + 9)
:= 9 &times; (9 &times; 9 − 9) + 999/9
&Iuml; 750 := (1 + 1) &times; (1 + (11 &times; (1 + (11 &times; (1 + 1 + 1)))))
2+2
&Iuml; 755 := 111+1+1 − (((1 + 1) &times; (1 + 11))1+1 )
&Iuml; 760 := 1 + (11 &times; ((1 + 1 + 1) &times; (1 + (11 + 11))))
:= (2 &times; ((22 − 2)2 − 22)) − 2/2
:= 2 &times; (((22 − 2)2 − 22) + 2)
:= 3 + (33+3 + (3 &times; (3 + 3)))
:= 33 + (33+3 − 3/3)
:= 3 + ((33+3 + 3/3) + 33 )
:= 44 + 4 &times; 4 &times; 44 + (4 + 4)/4
:= ((4 − 4/4) &times; (44 − 4)) − 4/4
:= 4 + ((4 − 4/4) &times; (44 − 4))
:= 5 &times; 5 &times; (5 &times; 5 + 5)
:= 5 + 5 &times; 5 &times; (5 &times; 5 + 5)
:= 5 + (5 &times; 5 &times; (5 &times; 5 + 5) + 5)
:= 66 &times; (6 + 6) − (6 &times; 6 + 6)
:= 66 &times; (6 + 6) − (6 &times; 6 + 6/6)
:= 6 &times; 6 + ((66 &times; 66 − 6 − 6)/6)
:= (7/7 + 7 + 7) &times; (7/7 + 7 &times; 7)
:= 2 + (22 &times; ((2 &times; 2
) + 2))
:= 777 − (7/7 + 7 + 7 + 7)
:= 777 + (((7 − 77)/7) − 7)
:= (8 − 88)/8 + (8 &times; (88 + 8) − 8)
:= 8 &times; (88 + 8) − (88 + 8 + 8)/8
:= 8 &times; (88 + 8) − 8
:= 9 + (((99 + 9)/9) + 9 &times; 9 &times; 9)
:= 9 + (((9 &times; 9 &times; 9 − 9/9) + 9) + 9)
:= 9 + (((99 + 99)/9) + 9 &times; 9 &times; 9)
&Iuml; 751 := 11 + (11 + (11 − 1 − 1)1+1+1 )
2+2+2
&Iuml; 756 := (1 + 1 + 1) &times; ((1 + 11) &times; (11 + (11 − 1)))
&Iuml; 761 := 1 + (1 + (11 &times; ((1 + 1 + 1) &times; (1 + (11 + 11)))))
:= 2 &times; ((22 − 2)2 − 22)
:= 2 + ((2/2 + 2) &times; ((222 + 22)/2))
:= 33 + 33+3
:= 33 + (33+3 − 3/3)
:= (4 &times; (4 − 4)) − (4/4 + 4 )
:= (4 − 4/4) &times; (44 − 4)
:= ((4 − 4/4) &times; (44 − 4/4)) − 4
:= 5/5 + 5 &times; 5 &times; (5 &times; 5 + 5)
:= 5 + (5 &times; 5 &times; (5 &times; 5 + 5) + 5/5)
:= 5 &times; 5 + ((555 + 55 )/5)
:= 22 + ((2/2 + 2)
:= 33 + (3
3+3
)
− 33/3)
4
4
:= 6 &times; 6 + ((66/6) &times; (66 − 6/6))
:= 6 &times; ((66 − 6) + 66)
:= 6 &times; 6 + ((66 &times; 66 − 6)/6)
:= 777 − ((77 + 7)/7 + 7 + 7)
:= 777 − (7 + 7 + 7)
:= 777 − (((7 + 7)/7 + 7) + 7)
:= 888/8 + 8 &times; (88 − 8)
:= 8 &times; (88 + 8) − (88 + 8)/8
:= 8/8 + (8 &times; (88 + 8) − 8)
:= 9 &times; 9 &times; 9 + ((99 + 99)/9)
:= 9 + ((9 &times; 9 &times; 9 + 9) + 9)
:= ((99/9) &times; (9 &times; 9 − (99/9))) − 9
&Iuml; 752 := 1 + (11 + (11 + (11 − 1 − 1)1+1+1 ))
2
&Iuml; 757 := 1 + ((1 + 1 + 1) &times; ((1 + 11) &times; (11 + (11 − 1))))
&Iuml; 762 := (1 + 1 + 1) &times; (1 + (11 &times; (1 + (11 + 11))))
:= 2 &times; ((22 − 2) − (22 + 2))
:= 2/2 + (2 &times; ((22 − 2) − 22))
:= (2/2 + 2) &times; (22&times;(2+2) − 2)
:= 33 + (33+3 − (3/3 + 3))
:= 33 + (33+3 + 3/3)
:= 33 + 33+3
:= 4 &times; (444 − 44 )
:= 4/4 + ((4 − 4/4) &times; (44 − 4))
:= (4 − 4/4) &times; (44 − (4 + 4)/4)
:= 5 &times; 5 &times; 5 + (55 + 5 + 5)/5
:= 5 + (5 &times; 5 &times; (5 &times; 5 + 5) + ((5 + 5)/5))
:= 5 &times; 5 + (((555 + 55 ) + 5)/5)
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + (66/6)) + 6)
:= 6/6 + (6 &times; ((66 − 6) + 66))
:= 6 + (6 &times; ((66 − 6) + 66))
:= 777 − (77/7 + 7 + 7)
:= 7/7 + (777 − (7 + 7 + 7))
:= 777 − (7/7 + 7 + 7)
:= 8 &times; (88 + 8) − 8 − 8
:= 8 &times; (88 + 8) − 88/8
:= (8 + 8)/8 + (8 &times; (88 + 8) − 8)
:= 9 &times; 9 &times; 9 + ((99 + 99 + 9)/9)
:= 9 + (((9 &times; 9 &times; 9 + 9/9) + 9) + 9)
:= 9 &times; 9 &times; 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
&Iuml; 763 := (111 − 1 − 1) &times; (1 + ((1 + 1) &times; (1 + 1 + 1)))
&Iuml; 768 := (1 + 1 + 1) &times; (1 + 1)(1+1)
1+1+1
&Iuml; 773 := ((1 + ((1 + 1 + 1)1+1+1 ))1+1 ) − 11
:= 2/2 + ((2/2 + 2) &times; (22&times;(2+2) − 2))
:= 2 &times; (22+2 &times; (22 + 2))
:= (2 &times; 22) + ((2/2 + 2)2+2+2 )
:= 3/3 + (33+3 + 33)
:= 3 + ((33+3 + 33) + 3)
:= 33 + (33+3 + 33/3)
:= 4 &times; 44 − ((4/4 + 44 ) + 4)
:= 4 &times; (4 &times; (44 + 4))
:= 4 + ((4 &times; (4 &times; (44 + 4))) + 4/4)
:= 555 + ((55 − 5)/(5 + 5 + 5))
:= (5 &times; 5 − 5/5) &times; ((5 + 5)/5)5
:= 5 + ((5 &times; 5 − 5/5) &times; ((5 + 5)/5)5 )
:= 6 &times; 6 + ((66 &times; 66 + 6)/6)
:= (6 + 6) &times; ((6 + 6)/6)6
:= 6 + (((6 + 6) &times; ((6 + 6)/6)6 ) − 6/6)
:= 777 − (7 + 7)
:= (7 − 7/7) &times; ((7 + 7)/7)7
:= 7 + (777 − (77/7))
:= 88/8 + (8 &times; (88 + 8) − (8 + 8))
:= 8 &times; (88 + 8)
:= 8 + ((8 &times; (88 + 8) − (88/8)) + 8)
:= (9 − ((9 + 9)/9)) &times; (9/9 + 99 + 9)
:= 9 + (9 &times; (9 &times; 9 − 9) + 999/9)
:= 99 + ((((9 + 9)/9)9 ) + 9 &times; (9 + 9))
&Iuml; 764 := 1 + ((111 − 1 − 1) &times; (1 + ((1 + 1) &times; (1 + 1 + 1))))
&Iuml; 769 := 1 + (1 + 1 + 1) &times; (1 + 1)(1+1)
1+1+1
&Iuml; 774 := 1 + (((1 + ((1 + 1 + 1)1+1+1 ))1+1 ) − 11)
:= 2 &times; ((2
2+2
&times; (22 + 2)) − 2)
:= 2/2 + (2 &times; (22+2 &times; (22 + 2)))
:= (2/2 + 2) &times; (22&times;(2+2) + 2)
:= 3 + ((3
3+3
− 3/3) + 33)
:= 3 + (((33+3 + 3/3) + 33) + 3)
:= 3 + ((33+3 + 33) + 3 &times; 3)
:= 4/4 + (4 &times; (4 &times; (44 + 4)))
:= (4 − 4/4) &times; ((4 + 4)/4 + 44 )
:= (5 &times; (5 − 55)) + ((5 − 5/5)5 − 5)
:= (5 &times; (5 − 55)) + (5 − 5/5)5
:= 6/6 + ((6 + 6) &times; ((6 + 6)/6)6 )
:= 6 + ((6 + 6) &times; ((6 + 6)/6)6 )
:= 7/7 + (777 − (7 + 7))
:= 777 − (7/7 + 7)
:= 777 − (7 + 7 + 7)/7
:= 8 &times; (88 + 8) − (8/((8 + 8)/8))
:= 8/8 + 8 &times; (88 + 8)
:= 8 + (8 &times; (88 + 8) − ((8 + 8)/8))
:= 9 + ((((9 &times; 9 &times; 9 − 9/9) + 9) + 9) + 9)
:= 9 &times; 9 &times; 9 + ((9 &times; 9 &times; 9 − 9)/(9 + 9))
:= 9 &times; 99 − (99 + 9 + 9)
4
4
:= 4 &times; 4 − (4 + 4)
5
:= 5 &times; (5 &times; 5 + 5) + ((5 − 55)/5)
6
:= 6 &times; 6 + (((6 &times; 6/(6 + 6)) ) − 6/6)
&Iuml; 765 := (1 + 1 + 1) &times; (111 + ((1 + 11)1+1 ))
:= (2/2 + 2) &times; (2
:= 3 + (3
3+3
2&times;(2+2)
− 2/2)
&Iuml; 770 := 11 &times; ((11 − 1 − 1)1+1 − 11)
:= 2 + (2 &times; (2
+ 33)
2+2
&times; (22 + 2)))
&Iuml; 775 := 1111 − ((1 + 1 + 1) &times; (1 + 111))
:= 2 + (((22+2 + 2/2)2 ) + 222 )
:= 3 + ((33+3 + 33/3) + 33 )
:= 3333/3 − (333 + 3)
:= (4 + 4)/4 + (4 &times; (4 &times; (44 + 4)))
:= 4 + ((4 − 4/4) &times; (4/4 + 44 ))
:= 55 &times; ((5 − 5/5 + 5) + 5)
:= 5 &times; (5 &times; (5 &times; 5 + 5) + 5)
:= 6 &times; 6 + ((6 &times; 6/(6 + 6)) )
:= (6/6 + 6) &times; ((666 − 6)/6)
:= 66 &times; (6 + 6) − ((66/6) + 6)
:= 777 − (77 + 7)/7
:= 777 − 7
:= 777 − (7 + 7)/7
:= 8 + (8 &times; (88 + 8) − (88/8))
:= (8 + 8)/8 + 8 &times; (88 + 8)
:= 8 + (8 &times; (88 + 8) − 8/8)
:= 9 + (((9 &times; 9 &times; 9 + 9) + 9) + 9)
:= 99/9 &times; (9 &times; 9 − 99/9)
:= ((9 − 9/9) &times; (99 − 9/9)) − 9
4
:= (4 − 4/4) &times; (4 − 4/4)
:= 5 + ((5 &times; 5 &times; (5 &times; 5 + 5) + 5) + 5)
6
&Iuml; 766 := (111 &times; (1 + ((1 + 1) &times; (1 + 1 + 1)))) − 11
:= (2 &times; (2
2+2
&times; (22 + 2))) − 2
&Iuml; 771 := 1 + (11 &times; ((11 − 1 − 1)1+1 − 11))
2
2
:= 22 + ((((22 + 2) ) − 2)/2)
:= 3 + ((33+3 + 3/3) + 33)
:= 3 &times; 3 + (3
:= 4 &times; 44 − ((4 + 4)/4 + 44 )
3+3
:= 5 + (((555 + 5 )/5) + 5 &times; 5)
6
:= 6 &times; 6 + (((6 &times; 6/(6 + 6)) ) + 6/6)
:= 2 &times; (2 &times; (((22+2 − 2)2 ) − 2))
:= 3 + ((33+3 + 33/3) + 33)
+ 33)
4
5
&Iuml; 776 := (111 &times; (1 + ((1 + 1) &times; (1 + 1 + 1)))) − 1
:= (4 − 4/4) &times; (4/4 + 4 )
:= 4 + ((4 &times; (4 &times; (44 + 4))) + 4)
:= 5/5 + (55 &times; ((5 − 5/5 + 5) + 5))
:= 5/5 + (5 &times; (5 &times; (5 &times; 5 + 5) + 5))
6
:= 6 + (((6 &times; 6/(6 + 6)) ) + 6 &times; 6)
:= 666 + (666 − 6)/6
:= 777 − 77/7
:= 7/7 + (777 − 7)
:= 777 − 7/7
:= 8 &times; (88 + 8) − (8 + 8)/8
:= 88/8 + (8 &times; (88 + 8) − 8)
:= 8 + 8 &times; (88 + 8)
:= 9 + ((((9 &times; 9 &times; 9 + 9/9) + 9) + 9) + 9)
:= 9 &times; 99 − (999/9 + 9)
:= (9 − 9/9) &times; (99 − ((9 + 9)/9))
&Iuml; 767 := (1 + 1 + 1) &times; (1 + 1)(1+1)
:= (2 &times; 2 &times; 222) − (22/2)
3
3+3
1+1+1
−1
2
&Iuml; 772 := 1 + (1 + (11 &times; ((11 − 1 − 1)1+1 − 11)))
:= 2 &times; ((2
2+2
&times; (22 + 2)) + 2)
&Iuml; 777 := 111 &times; (1 + ((1 + 1) &times; (1 + 1 + 1)))
:= 222/2 &times; ((2/2 + 2 + 2) + 2)
:= 3 &times; 3 + ((33+3 + 3/3) + 33)
:= (33 &times; (33 + 3)) − 33
:= 4 + (4 &times; (4 &times; (44 + 4)))
:= (4 − 4/4) &times; ((44 − 4/4) + 4)
:= (((5 + 5)/5 + 5) &times; 555/5) − 5
:= ((5 + 5)/5 + 5) &times; 555/5
:= ((6 + 6) &times; ((6 + 6)/6) ) − 6/6
:= 666 + ((666 + 6)/6 − 6)
:= 666 + 666/6
:= 777 + (7 − 77)/7
:= ((7 + 7)/7) + (777 − 7)
:= 777
:= 8 &times; (88 + 8) − 8/8
:= 8 &times; (88 + 8) + (8/((8 + 8)/8))
:= 8 + (8 &times; (88 + 8) + 8/8)
:= 9 + (((9 &times; 9 &times; 9 + (99/9)) + 9) + 9)
:= 9 &times; 99 + (((9 − 999)/9) − 9)
:= (9 − ((9 + 9)/9)) &times; 999/9
:= 3 + (3
+ 33/3)
:= 4 &times; 44 − (4/4 + 44 )
5
:= 5 + ((((555 + 5 ) + 5)/5) + 5 &times; 5)
6
61
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 778 := 1 + (111 &times; (1 + ((1 + 1) &times; (1 + 1 + 1))))
2
&Iuml; 783 := ((1 + ((1 + 1 + 1)1+1+1 ))1+1 ) − 1
2
&Iuml; 788 := 11 + (111 &times; (1 + ((1 + 1) &times; (1 + 1 + 1))))
:= ((22 + 2 + 2 + 2) ) − 2/2
:= 2 + (((22 + 2 + 2 + 2)2 ) + 2)
:= 3 &times; (3 &times; (3 &times; 33 + 3 + 3))
:= (33 &times; (33 − 3)) − (3/3 + 3)
:= 4 + ((4 − 4/4) &times; ((4 + 4)/4 + 4 ))
:= (4 − 4/4) &times; ((4/4 + 44 ) + 4)
:= 4 + (4 &times; ((4 &times; (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 &times; (6 + 6) − 6)
:= 7/7 + 777
:= 7 + (777 − 7/7)
:= 77/7 + 777
:= 8 + (8 &times; (88 + 8) + ((8 + 8)/8))
:= (8/8 + 8) &times; (88 − 8/8)
:= 8 + (((88 + 8)/8) + 8 &times; (88 + 8))
:= 9 &times; 9 &times; 9 + ((9 &times; 99 − 9)/(9 + 9))
:= 9 &times; 99 − (99 + 9)
:= 9 + (9 &times; 99 − ((999 + 9)/9))
:= (2 &times; (22 − 2) ) − 22
:= 3333/3 − 333
4
&Iuml; 779 := 1 + (1 + (111 &times; (1 + ((1 + 1) &times; (1 + 1 + 1)))))
2
&Iuml; 784 := (1 + ((1 + 1 + 1)1+1+1 ))1+1
&Iuml; 789 := (1 + 1 + 1) &times; (((1 + 1) &times; (11 &times; (1 + 11))) − 1)
:= 2/2 + ((2 &times; (22 − 2) ) − 22)
:= (22 + 2 + 2 + 2)2
:= (2 &times; (22 − 2)2 ) − 22/2
:= (33 &times; 33 ) − ((333 + 3)/3)
:= (33 + 3/3)3−3/3
:= (33 &times; (33 − 3)) − 3
:= 44/4 + (4 &times; (4 &times; (44 + 4)))
:= 4 &times; ((4 &times; (44 + 4)) + 4)
:= 4 + ((4 &times; ((4 &times; (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 &times; (6 + 6) − (6/6 + 6 + 6)
:= (6/6 + 6) &times; (666 + 6)/6
:= 66 + (((6 &times; 6/(6 + 6))6 ) − 6)
:= ((7 + 7)/7) + 777
5
:= 7 + 777
:= 777 + (77 + 7)/7
:= 88/8 + 8 &times; (88 + 8)
:= 8 + (8 &times; (88 + 8) + 8)
:= 888 − (88/8 + 88)
:= 9 &times; 99 − ((999 + 9)/9)
:= (9 − 9/9) &times; (99 − 9/9)
:= 9 + (9 &times; 99 − 999/9)
&Iuml; 780 := (11 − 1) &times; (111 − (11 &times; (1 + 1 + 1)))
2
&Iuml; 785 := 1 + ((1 + ((1 + 1 + 1)1+1+1 ))1+1 )
&Iuml; 790 := (1 + 1) &times; ((11 &times; ((1 + 1 + 1) &times; (1 + 11))) − 1)
:= 2 + ((2 &times; (22 − 2) ) − 22)
:= 2/2 + ((22 + 2 + 2 + 2) )
:= (22 &times; (2 + 2 + 2)2 ) − 2
:= (33 + 3) &times; (33 − 3/3)
:= 3/3 + ((33 + 3/3)3−3/3 )
:= 3/3 + ((33 &times; (33 − 3)) − 3)
:= (4 − 4/4) &times; (44 + 4)
:= 4/4 + (4 &times; ((4 &times; (44 + 4)) + 4))
:= 4 + ((4 &times; ((4 &times; (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) &times; (66 − 6/6)
:= 66 &times; (6 + 6) − 6/6 − 6
:= 66 &times; (6 + 6) − (6 + 6)/6
:= 777 + (7 + 7 + 7)/7
:= 7 + (777 + 7/7)
:= 7 + ((777 − 7/7) + 7)
:= ((88 + 8)/8) + 8 &times; (88 + 8)
:= 8 + ((8 &times; (88 + 8) + 8/8) + 8)
:= 88 + (8 &times; 88 − ((8 + 8)/8))
:= 9 &times; 99 − 999/9
:= 9 + ((9 − 9/9) &times; (99 − ((9 + 9)/9)))
:= 9 &times; 99 − ((9 + 9)/9 + 99)
&Iuml; 781 := 11 &times; ((((1 + 11)1+1 )/(1 + 1)) − 1)
&Iuml; 786 := 1 + (1 + ((1 + ((1 + 1 + 1)1+1+1 ))1+1 ))
&Iuml; 791 := (11 &times; (((1 + 11)1+1 )/(1 + 1))) − 1
:= ((22 + 2 + 2 + 2)2 ) − 2/2 − 2
:= 2 + ((22 + 2 + 2 + 2)2 )
:= (22 &times; (2 + 2 + 2)2 ) − 2/2
:= 33/3 &times; (((3 + 3)3 − 3)/3)
:= (33 &times; (33 − 3)) − (3 + 3)
:= (33 &times; (33 − 3)) − 3/3
:= 4/4 + ((4 − 4/4) &times; (44 + 4))
:= (4 + 4)/4 + (4 &times; ((4 &times; (44 + 4)) + 4))
:= 44/4 + ((4 − 4/4) &times; (44 + 4))
:= (55 − 5/5)/(5 − 5/5)
:= 5 + ((55 − 5/5)/(5 − 5/5))
:= 55 + ((555 + 55 )/5)
:= 66 &times; (6 + 6) − 66/6
:= 66 &times; (6 + 6) − 6
:= 66 &times; (6 + 6) − 6/6
:= 77/7 + (777 − 7)
:= 7 + (((7 + 7)/7) + 777)
:= 7 + (777 + 7)
:= 88 + (8 &times; 88 − (88/8))
:= 8 + ((8 &times; (88 + 8) + ((8 + 8)/8)) + 8)
:= 88 + (8 &times; 88 − 8/8)
:= 9 &times; 99 + ((9 − 999)/9)
:= 9 + ((9 − ((9 + 9)/9)) &times; 999/9)
:= 9 &times; 99 − (9/9 + 99)
&Iuml; 782 := 1 + (11 &times; ((((1 + 11)1+1 )/(1 + 1)) − 1))
&Iuml; 787 := 111 + (((1 + 1) &times; (1 + (1 + 11)))1+1 )
&Iuml; 792 := 11 &times; (((1 + 11)1+1 )/(1 + 1))
:= ((22 + 2 + 2 + 2)2 ) − 2
:= 2 + (((22 + 2 + 2 + 2)2 ) + 2/2)
:= 22 &times; (2 + 2 + 2)2
:= 33 + ((33+3 − 3/3) + 33 )
:= 3 + ((33 + 3/3)3−3/3 )
:= 33 &times; (33 − 3)
:= (4 + 4)/4 + ((4 − 4/4) &times; (44 + 4))
:= 4 + ((4 − 4/4) &times; ((4/4 + 44 ) + 4))
:= 44 &times; ((4 + 4)/4 + 4 &times; 4)
:= 5 + (((5 + 5)/5 + 5) &times; 555/5)
:= 5 + ((((5 + 5)/5 + 5) &times; 555/5) + 5)
:= 55 + (((555 + 55 ) + 5)/5)
:= ((6 − 66)/6) + 66 &times; (6 + 6)
:= 6/6 + (66 &times; (6 + 6) − 6)
:= 66 &times; (6 + 6)
:= 7 + (777 − ((7 + 7)/7))
:= 777 + (77 − 7)/7
:= 7 + ((777 + 7/7) + 7)
:= 8 + ((8 &times; (88 + 8) − ((8 + 8)/8)) + 8)
:= 8 + (8 &times; (88 + 8) + (88/8))
:= 88 + 8 &times; 88
:= 9 &times; 99 − (9/9 + 99 + 9)
:= 9999/9 − (9 + 9) &times; (9 + 9)
:= 99 &times; (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
&Iuml; 793 := 1 + (11 &times; (((1 + 11)1+1 )/(1 + 1)))
&Iuml; 798 := (1 + 1) &times; ((((1 + 1) &times; (11 − 1))1+1 ) − 1)
&Iuml; 803 := 11 &times; (1 + (((1 + 11)1+1 )/(1 + 1)))
:= 2/2 + (22 &times; (2 + 2 + 2)2 )
:= (2 &times; (22 − 2)2 ) − 2
:= 2 + ((2 &times; (22 − 2)2 ) + 2/2)
:= 3/3 + (33 &times; (33 − 3))
:= 3 + ((33 &times; (33 − 3)) + 3)
:= 33/3 + (33 &times; (33 − 3))
:= 4/4 + (44 &times; ((4 + 4)/4 + 4 &times; 4))
:= (4 − 4/4) &times; ((44 − 4)/4 + 44 )
:= 4 + (((4 &times; 4 + 4) &times; (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 &times; (6 + 6)
:= 6 + 66 &times; (6 + 6)
:= 66/6 + 66 &times; (6 + 6)
:= 7 + ((((7 + 7)/7) + 777) + 7)
:= 7 + (777 + 7 + 7)
:= 7 + (((77 + 7)/7 + 777) + 7)
:= 8/8 + (8 &times; 88 + 88)
:= 888 − ((8 + 8)/8 + 88)
:= 88 + (88/8 + 8 &times; 88)
:= 9/9 + (99 &times; (9 − 9/9))
:= 9 &times; (9 &times; 9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 &times; (9 &times; 9 + 9) − 9)
&Iuml; 794 := 1 + (1 + (11 &times; (((1 + 11)1+1 )/(1 + 1))))
&Iuml; 799 := ((1 + 1) &times; (((1 + 1) &times; (11 − 1))1+1 )) − 1
&Iuml; 804 := 1 + (11 &times; (1 + (((1 + 11)1+1 )/(1 + 1))))
:= 2 + (22 &times; (2 + 2 + 2)2 )
:= (2 &times; (22 − 2)2 ) − 2/2
:= 2 &times; ((22 − 2)2 + 2)
:= 3 + ((33 &times; (33 − 3)) − 3/3)
:= (33 &times; (33 + 3)) − 33/3
:= (33 &times; (33 + 3)) − (3 + 3)
:= 444 + (((4 + 4) &times; 44) − (4 + 4)/4)
:= ((4 &times; 4 + 4) &times; (44 − 4)) − 4/4
:= 4 + ((4 &times; 4 + 4) &times; (44 − 4))
:= (5 &times; (5 &times; ((5 + 5)/5) )) − (5/5 + 5)
:= (5 &times; (5 &times; ((5 + 5)/5) )) − 5/5
:= 5 + ((5 &times; (5 &times; ((5 + 5)/5)5 )) − 5/5)
:= (6 + 6)/6 + 66 &times; (6 + 6)
:= 6 + (66 &times; (6 + 6) + 6/6)
:= 6 + (66 &times; (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 &times; 88)
:= 888 − (8/8 + 88)
:= 88 + (((88 + 8)/8) + 8 &times; 88)
:= (9 + 9)/9 + (99 &times; (9 − 9/9))
:= 9 &times; (9 &times; 9 + 9) − 99/9
:= 9 &times; 99 + (((99 + 9)/9) − 99)
5
&Iuml; 795 := 11 + ((1 + ((1 + 1 + 1)1+1+1 ))1+1 )
5
&Iuml; 800 := (1 + 1) &times; (((1 + 1) &times; (11 − 1))1+1 )
&Iuml; 805 := ((1 + 1)11 ) − (11 &times; (1 + (1 + 111)))
:= (2 &times; ((22 − 2)2 − 2)) − 2/2
:= 2 &times; (22 − 2)2
:= 2/2 + (2 &times; ((22 − 2)2 + 2))
:= 3 + (33 &times; (33 − 3))
:= 33+3 + (((3 + 3)3 − 3)/3)
:= 3 + ((((3 + 3)3 + 3)/3) + 33+3 )
:= 444 + (((4 + 4) &times; 44) − 4/4)
:= (4 &times; 4 + 4) &times; (44 − 4)
:= 4 + ((4/4 + 4)4 + (4 &times; 44))
:= (55 + 55 )/(5 − 5/5)
:= 5 &times; (5 &times; ((5 + 5)/5)5 )
:= 5 + (5 &times; (5 &times; ((5 + 5)/5)5 ))
:= 66 + ((6 &times; 6/(6 + 6))6 )
:= 6 + (66 &times; (6 + 6) + ((6 + 6)/6))
:= 6 + ((66 &times; (6 + 6) + 6/6) + 6)
:= 7 + (77/7 + 777)
:= 7 + (((((7 + 7)/7) + 777) + 7) + 7)
:= 7 + (777 + 7 + 7 + 7)
:= 8 + ((8 &times; (88 + 8) + (88/8)) + 8)
:= 888 − 88
:= 8 &times; 88 + (8888/88)
:= 9 &times; 99 + (((9 + 9 + 9)/9) − 99)
:= (9 − 9/9) &times; (9/9 + 99)
:= 9 &times; (9 &times; 9 + 9) + ((9 − 99)/(9 + 9))
&Iuml; 796 := (1 + 1) &times; ((((1 + 1) &times; (11 − 1))1+1 ) − (1 + 1))
&Iuml; 801 := 1 + ((1 + 1) &times; (((1 + 1) &times; (11 − 1))1+1 ))
&Iuml; 806 := 1 + (((1 + 1)11 ) − (11 &times; (1 + (1 + 111))))
:= 2 &times; ((22 − 2)2 − 2)
:= 2/2 + (2 &times; (22 − 2)2 )
:= 2 + (2 &times; ((22 − 2)2 + 2))
:= 3 + ((33 &times; (33 − 3)) + 3/3)
:= 3 &times; ((3 &times; (3 &times; (33 + 3))) − 3)
:= (33 &times; (33 + 3)) − (3/3 + 3)
:= 444 + ((4 + 4) &times; 44)
:= (4 &times; 44) + (4/4 + 4)4
:= ((44 − 4)/4 &times; (4 − 4/4)4 ) − 4
:= 5 + (((555 + 55 )/5) + 55)
:= 5/5 + (5 &times; (5 &times; ((5 + 5)/5)5 ))
:= 5 + ((5 &times; (5 &times; ((5 + 5)/5)5 )) + 5/5)
:= 6 + (66 &times; (6 + 6) − ((6 + 6)/6))
:= 6 + (((6 &times; 6/(6 + 6))6 ) + 66)
:= 6 + ((66 &times; (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 &times; 88)
:= 8/8 + (888 − 88)
:= 8 + (888 − ((8 + 8)/8 + 88))
:= 9 &times; (9 &times; 9 + 9) − 9
:= 9 &times; (9 &times; 9 + 9) + ((9 − 9 &times; 9)/(9 + 9))
:= 9 + (9999/9 − (9 + 9) &times; (9 + 9))
&Iuml; 797 := ((11 &times; (1 + ((1 + 11)1+1 ))) − 1)/(1 + 1)
&Iuml; 802 := (1 + 1) &times; (1 + (((1 + 1) &times; (11 − 1))1+1 ))
&Iuml; 807 := (1 + ((1 + 1) &times; (11 &times; (111 − 1))))/(1 + 1 + 1)
:= 2/2 + (2 &times; ((22 − 2) − 2))
:= 2 + (2 &times; (22 − 2) )
:= 2 + ((2 &times; ((22 − 2)2 + 2)) + 2/2)
:= 3 + (((33 &times; (33 − 3)) − 3/3) + 3)
:= 33+3 + (((3 + 3)3 + 3)/3)
:= (33 &times; (33 + 3)) − 3
:= (4 &times; 44) + ((4/4 + 4)4 − 4)
:= 4/4 + ((4/4 + 4)4 + (4 &times; 44))
2
2
:= 4 + ((((4 &times; 4 + 4) &times; (44 − 4)) − 4/4) + 4)
:= 5 + ((((555 + 5 ) + 5)/5) + 55)
:= (5 + 5)/5 + (5 &times; (5 &times; ((5 + 5)/5) ))
:= 5 + ((5 &times; (5 &times; ((5 + 5)/5)5 )) + ((5 + 5)/5))
:= 6 + (66 &times; (6 + 6) − 6/6)
:= 66 &times; (6 + 6) + (66 − 6)/6
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + 66) + 6)
:= 7 + (((777 − 7/7) + 7) + 7)
:= 7 + ((77/7 + 777) + 7)
:= (7 &times; 7 &times; (7 + 7)) + (((7 + 7)/7)7 − 7)
:= 8 + (888 − (88/8 + 88))
:= 888 + (((8 + 8)/8) − 88)
:= 8 + (888 − (8/8 + 88))
:= 9 &times; (9 &times; 9 + 9) − ((99 + 9 + 9)/9)
:= 9/9 + (9 &times; (9 &times; 9 + 9) − 9)
:= 9 &times; (9 &times; 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
&Iuml; 808 := (1 + 1)1+1+1 &times; (1 + (11 − 1)1+1 )
&Iuml; 813 := ((1 + 1) &times; (11 &times; (111/(1 + 1 + 1)))) − 1
&Iuml; 818 := 1 + (1 + (((1 + 1)11 ) − (11 &times; (1 + 111))))
:= 2 &times; (((22 − 2)2 + 2) + 2)
:= 2 + ((2 &times; (22 − 2)2 ) + 22/2)
:= 22 + (2 &times; ((22 − 2)2 − 2))
:= 3/3 + ((33 &times; (33 + 3)) − 3)
:= 3 + (33 &times; (33 + 3))
:= 33 + ((33 &times; (33 − 3)) − 3/3)
:= 4 + (((4 &times; 4 + 4) &times; (44 − 4)) + 4)
:= 44 + ((4 &times; (4 &times; (44 + 4))) + 4/4)
:= 4 + (((44 − 4)/4 &times; (4 − 4/4)4 ) + 4)
:= 5 + ((((5 − (5 + 5)/5)5 ) + 555) + 5)
:= (55 &times; (5 + 5 + 5)) − (55 + 5)/5
:= (55 &times; (5 + 5 + 5)) − ((5 + 5)/5 + 5)
:= 6 + (66 &times; (6 + 6) + ((66 − 6)/6))
:= 6 &times; 6 + (666/6 + 666)
:= (6 − (((6 + 6)/6)6+6 ))/(6/6 − 6)
:= 7 &times; 7 + (777 − (77/7 + 7))
:= 7/7 + ((777 − (7 + 7)) + 7 &times; 7)
:= 7 &times; 7 + (777 − (7/7 + 7))
:= 8 + (888 − 88)
:= 888 − (88/8 + 8 &times; 8)
:= 8 + ((8/8 + 8) &times; ((8 + 8)/8 + 88))
:= 9 &times; (9 &times; 9 + 9) − (9 + 9)/9
:= 9 &times; (9 &times; 9 + 9) + ((9 + 9 + 9)/9)
:= 9 + (9 &times; (9 &times; 9 + 9) − 9/9)
&Iuml; 809 := ((11 − 1) &times; (11 − 1 − 1)1+1 ) − 1
&Iuml; 814 := (1 + 1) &times; (11 &times; (111/(1 + 1 + 1)))
:= 2/2 + (2 &times; (((22 − 2)2 + 2) + 2))
:= 22 + (22 &times; (2 + 2 + 2)2 )
:= (33 &times; (33 + 3)) − 3/3
:= 3 + ((33 &times; (33 + 3)) + 3/3)
4
&Iuml; 819 := 111+1+1 − (1 + 1)11−1−1
:= ((22 + 2)2 ) + (222 + 2)/2
:= 3 &times; ((3 &times; (3 &times; (33 + 3))) + 3)
4
:= 4 + (((4/4 + 4) + (4 &times; 44)) + 4)
:= 4 + ((44 − 4)/4 &times; (4 − 4/4) )
:= (((4 + 4)4 ) − 4/4)/(4/4 + 4)
:= (55 &times; (5 + 5 + 5)) − (55/5 + 5)
:= (55 &times; (5 + 5 + 5)) − 55/5
:= (55 &times; (5 + 5 + 5)) − (5/5 + 5)
:= 6 + (66 &times; (6 + 6) + (66/6))
:= (66/6) &times; ((((6 + 6)/6) + 66) + 6)
:= 7 + (((77/7 + 777) + 7) + 7)
7
:= (6/6 + 6) &times; (666/6 + 6)
:= (7 &times; 7 &times; (7 + 7)) + ((7 + 7)/7)
:= 7 &times; 7 + (777 − 7)
:= 8 + ((888 − 88) + 8/8)
:= 8 &times; 88 + ((888 − 8)/8)
:= 8 + ((888 − 88) + (88/8))
:= 9 &times; (9 &times; 9 + 9) − 9/9
:= 99/9 &times; ((((9 + 9)/9) − 9) + 9 &times; 9)
:= 9 + 9 &times; (9 &times; 9 + 9)
&Iuml; 810 := (11 − 1) &times; (11 − 1 − 1)1+1
&Iuml; 815 := 1 + ((1 + 1) &times; (11 &times; (111/(1 + 1 + 1))))
&Iuml; 820 := (11 − 1) &times; (1 + (11 − 1 − 1)1+1 )
:= 2 + (2 &times; (((22 − 2)2 + 2) + 2))
:= 22/2 + (2 &times; ((22 − 2)2 + 2))
:= 22 + ((2 &times; (22 − 2)2 ) − 2)
:= 33 &times; (33 + 3)
:= 3 + (((33 &times; (33 + 3)) − 3/3) + 3)
:= 33 + ((33 &times; (33 − 3)) + 3/3)
:= (44 − 4)/4 &times; (4 − 4/4)4
:= 4 &times; 4 &times; 44 + 444/4
:= (((4 + 4)4 ) + 4)/(4/4 + 4)
:= (5 + 5 + 5) &times; (55 − 5/5)
:= (55 &times; (5 + 5 + 5)) − 5 − 5
:= (55 &times; (5 + 5 + 5)) − 5
:= 6 + ((66 &times; (6 + 6) + 6) + 6)
:= 6 + ((66 &times; (6 + 6) + (66/6)) + 6)
:= 6 + (((66 + 66)/6) + 66 &times; (6 + 6))
:= (7 − 7/7) &times; (((7 + 7)/7) + 7)
:= 7 &times; 7 + (777 − (77/7))
:= 7/7 + (777 − 7 + 7 &times; 7)
:= (8/8 + 8) &times; ((8 + 8)/8 + 88)
:= 8 &times; 88 + 888/8
:= (((88/8 − 8)8 ) − 8/8)/8
:= 9 &times; (9 &times; 9 + 9)
:= 9 &times; (9 &times; 9 + 9) + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + (9 &times; (9 &times; 9 + 9) + 9/9)
7
&Iuml; 811 := 1 + (11 − 1) &times; (11 − 1 − 1)1+1
&Iuml; 816 := ((1 + 1)11 ) − (11 &times; (1 + 111))
&Iuml; 821 := 1 + ((11 − 1) &times; (1 + (11 − 1 − 1)1+1 ))
:= 22/2 + (2 &times; (22 − 2)2 )
:= 2 &times; ((22 − 2)2 + 2 &times; (2 + 2))
:= 22 + ((2 &times; (22 − 2)2 ) − 2/2)
:= 3/3 + (33 &times; (33 + 3))
:= 3 + ((33 &times; (33 + 3)) + 3)
:= 33/3 + (33 &times; (33 + 3))
:= 44 + (4 &times; 44 − (4/4 + 44 ))
:= 4 &times; (4 &times; (44 − 4) + 44)
:= 4 + ((4 &times; (44 + 4)) + (4/4 + 4)4 )
:= 5/5 + ((5 + 5 + 5) &times; (55 − 5/5))
:= 5/5 + ((55 &times; (5 + 5 + 5)) − (5 + 5))
:= 5/5 + ((55 &times; (5 + 5 + 5)) − 5)
:= 6 + (((66 &times; (6 + 6) + 6/6) + 6) + 6)
:= (6 + 6) &times; (((6 + 6)/6) + 66)
:= 6 &times; 6 + (66 &times; (6 + 6) − (6/6 + 6))
:= 77 + ((7 &times; (7 &times; (7 + 7) + 7)) − 7/7)
:= 7 &times; 7 + (((7 − 77)/7) + 777)
:= 7 + ((7 &times; 7 &times; (7 + 7)) + ((7 + 7)/7)7 )
:= 888 + (88/8 − 88)
:= 888 − (8 &times; 8 + 8)
:= 8 &times; (88 + 8 + 8) − 88/8
:= 9/9 + 9 &times; (9 &times; 9 + 9)
:= (9 − 9/9) &times; (999/9 − 9)
:= 99/9 + 9 &times; (9 &times; 9 + 9)
&Iuml; 812 := 1 + 1 + (11 − 1) &times; (11 − 1 − 1)1+1
&Iuml; 817 := 1 + (((1 + 1)11 ) − (11 &times; (1 + 111)))
:= 2 &times; ((((22 − 2)2 + 2) + 2) + 2)
:= ((22 + 2)2 ) + (222 − 2)/2
:= 3 + ((33 &times; (33 + 3)) − 3/3)
:= 3 + (((33 &times; (33 + 3)) + 3/3) + 3)
&Iuml; 822 := (1 + 1) &times; (11 + (((1 + 1) &times; (11 − 1))1+1 ))
:= 22 + (2 &times; (22 − 2)2 )
4
:= 3 + ((33 &times; (33 − 3)) + 33 )
:= (4 + 4)/4 + ((((4 + 4)4 ) + 4)/(4/4 + 4))
:= 44 + (4 &times; (4 &times; (44 + 4)))
:= (4 &times; (44 + 4)) + (4/4 + 4)
:= ((5 + 5)/5 + 5) &times; (555/5 + 5)
:= 5 + (((5 + 5)/5 + 5) &times; (555/5 + 5))
:= (5 + 5)/5 + ((55 &times; (5 + 5 + 5)) − 5)
:= (6/6 + 6) &times; (((666 − 6)/6) + 6)
:= 6 &times; 6 + (66 &times; (6 + 6) − (66/6))
:= 6 &times; 6 + (66 &times; (6 + 6) − 6)
7
:= 77 + (7 &times; (7 &times; (7 + 7) + 7))
:= 7 + ((7 − 7/7) &times; (((7 + 7)/7) + 7))
:= 888 + ((88 + 8)/8 − 88)
:= 8/8 + (888 − (8 &times; 8 + 8))
:= 888 − (((8 + 8)/8) + 8 &times; 8)
:= (9 + 9)/9 + 9 &times; (9 &times; 9 + 9)
:= 9 + (9 &times; (9 &times; 9 + 9) − ((9 + 9)/9))
:= 9 &times; (9 &times; 9 + 9) + (99 + 9)/9
64
:= 7 + ((777 − (77/7)) + 7 &times; 7)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 823 := 1111 − ((1 + 1) &times; ((1 + 11)1+1 ))
&Iuml; 828 := 1 + (((1 + 1)11 ) − 11 &times; 111)
&Iuml; 833 := 1 + ((1 + (1 + 11)) &times; ((1 + 1)(1+1)&times;(1+1+1) ))
:= 22 + ((2 &times; (22 − 2)2 ) + 2/2)
:= (22+2 + 2) &times; ((2 &times; 22) + 2)
:= 22 + ((2 &times; (22 − 2)2 ) + 22/2)
:= 3 + (((33 &times; (33 − 3)) + 33 ) + 3/3)
:= 3 &times; 33 + 33+3
:= (3333 − 3/3)/(3/3 + 3)
:= 4 + ((((4 + 4)4 ) − 4/4)/(4/4 + 4))
:= (4 &times; (4 &times; (44 + 4 + 4))) − 4
:= 4/4 + (4 &times; (4 &times; (44 + 4 + 4)))
:= (55 &times; (5 + 5 + 5)) − (5 + 5)/5
:= 5 + ((55 &times; (5 + 5 + 5)) − ((5 + 5)/5))
:= 55 /5 + ((55 − 5)/(5 + 5 + 5))
:= 6 &times; 6 + ((66 &times; (6 + 6) − 6) + 6/6)
:= 6 &times; ((66 + 66) + 6)
:= 6 + ((66 &times; (6 + 6) − 6/6) + 6 &times; 6)
:= 7 &times; 7 + (777 − ((7 + 7 + 7)/7))
:= 7 &times; 7 + (((7 + 7)/7) + 777)
8
:= 7 + (777 + 7 &times; 7)
:= 888 − (8/8 + 8 &times; 8)
:= 8 + ((((88/8 − 8) ) − 8/8)/8)
:= 8/8 + 8 &times; (88 + 8 + 8)
:= 9 &times; (9 &times; 9 + 9) + ((99 + 9 + 9)/9)
:= 99 + 9 &times; 9 &times; 9
:= 9 + ((9 − 9/9) &times; (((999 + 9)/9) − 9))
&Iuml; 824 := 1 + (1111 − ((1 + 1) &times; ((1 + 11)1+1 )))
2
&Iuml; 829 := 1 + (1 + (((1 + 1)11 ) − 11 &times; 111))
2+2
:= (22 &times; ((2 + 2 + 2)2 + 2)) − 2
:= 2 + ((2 &times; (22 − 2) ) + 22)
:= 2/2 + ((2
:= 3 + ((33 &times; (33 + 3)) + 33/3)
:= 3/3 + (3 &times; 33 + 33+3 )
:= 3 + ((3 &times; 33 + 33+3 ) + 3)
:= 4 + ((((4 + 4)4 ) + 4)/(4/4 + 4))
:= 4 + (44/4 &times; ((44 + 44 )/4))
:= (4 + 4)/4 + (4 &times; (4 &times; (44 + 4 + 4)))
:= (55 &times; (5 + 5 + 5)) − 5/5
:= 5 + ((55 &times; (5 + 5 + 5)) − 5/5)
:= 5 + (((55 &times; (5 + 5 + 5)) − 5/5) + 5)
:= 6 &times; 6 + (66 &times; (6 + 6) + 6/6)
:= 6 + (66 &times; (6 + 6) + 6 &times; 6)
:= 7 &times; 7 + (777 − ((7 + 7)/7))
:= (77/7 &times; (77 − 7/7)) − 7
:= 7 + ((777 + 7 &times; 7) + 7/7)
:= 888 − 8 &times; 8
:= 8 + (8 &times; (88 + 8 + 8) − 88/8)
:= (8 + 8)/8 + 8 &times; (88 + 8 + 8)
:= (9 − 9/9) &times; (((999 + 9)/9) − 9)
:= 9/9 + (9 &times; 9 &times; 9 + 99)
:= (9999 + 9)/((99 + 9)/9)
6+6
:= 6 + ((6 − (((6 + 6)/6)
))/(6/6 − 6))
&Iuml; 825 := 11 &times; (1 + ((1 + 1) &times; (111/(1 + 1 + 1))))
2
+ 2) &times; ((2 &times; 22) + 2))
&Iuml; 834 := (1 + 1) &times; ((11 &times; (1 + (111/(1 + 1 + 1)))) − 1)
&Iuml; 830 := (11 − 1) &times; (1 + (1 + (11 − 1 − 1)1+1 ))
2+2
:= (22 &times; ((2 + 2 + 2)2 + 2)) − 2/2
:= ((2 &times; 22) ) − (2222/2)
:= 2 + ((2
:= 33 + (33 &times; (33 − 3))
:= 3 + ((33+3 − 3/3) + 3 &times; 33)
:= 3 + ((33 − 3/3) &times; (33 − 3/3))
:= 44/4 &times; ((44 + 44 )/4)
:= (4 &times; (4 &times; (44 + 4 + 4))) − (4 + 4)/4
:= 4 + ((4 &times; (4 &times; (44 + 4 + 4))) − 4/4)
:= 55 &times; (5 + 5 + 5)
:= 5 + (55 &times; (5 + 5 + 5))
:= 5 + ((55 &times; (5 + 5 + 5)) + 5)
:= 6 + ((6/6 + 6) &times; (666/6 + 6))
:= 6 &times; 6 + (66 &times; (6 + 6) + ((6 + 6)/6))
:= 6 + ((66 &times; (6 + 6) + 6 &times; 6) + 6/6)
:= 7 &times; 7 + (777 − 7/7)
:= 7 &times; 7 + ((777 − 7) + (77/7))
:= (77 &times; 77 − (77 + 7))/7
:= 8/8 + (888 − 8 &times; 8)
:= 8 &times; (88 + 8 + 8) − (8 + 8)/8
:= 888 + (88/8 − 8 &times; 8)
:= 9 + ((9 − 9/9) &times; (999/9 − 9))
:= 9 + (9 &times; (9 &times; 9 + 9) + (99/9))
:= 9 + ((9 &times; 9 &times; 9 − ((9 + 9)/9)) + 99)
&Iuml; 826 := ((1 + 1)11 ) − (1 + 11 &times; 111)
+ 2) &times; ((2 &times; 22) + 2))
&Iuml; 835 := 1111 − ((1 + 11) &times; (1 + (11 + 11)))
&Iuml; 831 := 1 + ((11 − 1) &times; (1 + (1 + (11 − 1 − 1)1+1 )))
&Iuml; 836 := (1 + 1) &times; (11 &times; (1 + (111/(1 + 1 + 1))))
:= 22 + (2 &times; ((22 − 2)2 + 2))
:= (2/2 + 2) &times; ((((22 + 2) ) − 22)/2)
:= 22 &times; ((2 + 2 + 2)2 + 2)
:= 3/3 + ((33 &times; (33 − 3)) + 33)
:= 3 + (3 &times; 33 + 33+3 )
:= 33 + ((33 &times; (33 + 3)) − 3/3)
:= ((44 &times; ((44 + 44 )/4)) + 4)/4
:= (4 &times; (4 &times; (44 + 4 + 4))) − 4/4
:= 4 + (4 &times; (4 &times; (44 + 4 + 4)))
:= 5/5 + (55 &times; (5 + 5 + 5))
:= 5 + ((55 &times; (5 + 5 + 5)) + 5/5)
:= 55/5 + (55 &times; (5 + 5 + 5))
:= 6 &times; 6 + (66 &times; (6 + 6) − ((6 + 6)/6))
:= 6 &times; 6 + (((6 &times; 6/(6 + 6)) ) + 66)
:= (66/6) &times; ((((6 + 6)/6)6 + 6) + 6)
:= 7 &times; 7 + 777
:= 7 + ((777 − ((7 + 7)/7)) + 7 &times; 7)
:= 77/7 &times; (77 − 7/7)
:= 888 + (((8 + 8)/8) − 8 &times; 8)
:= 8 &times; (88 + 8 + 8) − 8/8
:= 888 + ((88 + 8)/8 − 8 &times; 8)
:= 99 + (9 &times; 9 &times; 9 − ((9 + 9)/9))
:= 9 &times; 9 &times; 9 + (999/9 − 9)
:= 9 + ((9 &times; 9 &times; 9 − 9/9) + 99)
&Iuml; 827 := ((1 + 1)11 ) − 11 &times; 111
2
6
&Iuml; 832 := (1 + (1 + 11)) &times; ((1 + 1)(1+1)&times;(1+1+1) )
&Iuml; 837 := ((1 + 1)11 ) − (1 + (11 &times; (111 − 1)))
:= 2 + (((2 &times; 22)2 ) − (2222/2))
:= 2 &times; ((22 − 2)2 + 22+2 )
:= 2/2 + (22 &times; ((2 + 2 + 2)2 + 2))
:= 3 &times; 33 + (33+3 − 3/3)
:= (33 − 3/3) &times; (33 − 3/3)
:= 3 &times; (3 &times; ((3 &times; (33 + 3)) + 3))
:= (4 &times; (4 &times; (44 + 4 + 4))) − (4/4 + 4)
:= 4 &times; (4 &times; (44 + 4 + 4))
:= 44 + ((4/4 + 4)4 − 44)
:= (5 + 5)/5 + (55 &times; (5 + 5 + 5))
:= (5 &times; 5 + 5/5) &times; ((5 + 5)/5)5
:= 5 + ((5 &times; 5 + 5/5) &times; ((5 + 5)/5)5 )
:= 6 &times; 6 + (66 &times; (6 + 6) − 6/6)
:= (6/6 + 6 + 6) &times; ((6 + 6)/6)6
:= (66 &times; 66 + 666)/6
:= 7/7 + (777 + 7 &times; 7)
:= 7 + ((777 − 7/7) + 7 &times; 7)
:= 7 &times; 7 + (77/7 + 777)
:= 888 + (88/8 − (8 &times; 8 + 8))
:= 8 &times; (88 + 8 + 8)
:= 8 + ((8 &times; (88 + 8 + 8) − 88/8) + 8)
:= 99 + (9 &times; 9 &times; 9 − 9/9)
:= 9 &times; 9 &times; 9 + (((999 + 9)/9) − 9)
:= 9 + (9 &times; 9 &times; 9 + 99)
65
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 843 := 1 + (1 + ((((11 − 1) &times; (1 + 1 + 1)) − 1)1+1 ))
&Iuml; 838 := ((1 + 1)11 ) − (11 &times; (111 − 1))
2
2
&Iuml; 848 := 1 + (11 &times; (11 &times; (1 + ((1 + 1) &times; (1 + 1 + 1)))))
:= 2 + (22 &times; ((2 + 2 + 2) + 2))
:= 2 + (((((22 + 2/2) + 2) + 2) + 2) )
:= 2 &times; (((22 − 2)2 + 22) + 2)
:= 33 + ((33 &times; (33 + 3)) + 3/3)
:= 33 + (33 &times; (33 + 3))
:= 3 + (((3 − 3/3)3&times;3 ) + 333)
:= (4 − 44)/4 + 4 &times; (44 − 44)
:= 4 &times; (44 − 44) − (4/4 + 4)
:= 4 &times; (44 − 44)
:= 5 + (((55 − 5)/(5 + 5 + 5)) + 55 /5)
:= 55 /5 + (((5 − (5 + 5)/5)5 ) − 5 &times; 5)
:= (55/5 + 5) &times; (55 − (5 + 5)/5)
:= 6 + ((6/6 + 6 + 6) &times; ((6 + 6)/6) )
:= 6 + ((66 &times; 66 + 666)/6)
:= ((66 &times; (66/6 + 66)) + 6)/6
:= ((77 &times; 77 − (7 + 7))/7) − 7
:= 7 + (77/7 &times; (77 − 7/7))
:= (77 &times; 77 + 7)/7
:= 8 + (8 &times; (88 + 8 + 8) − ((8 + 8)/8))
:= 88/8 + 8 &times; (88 + 8 + 8)
:= 8 + (8 &times; (88 + 8 + 8) + 8)
:= 9 + ((9 &times; 9 &times; 9 + 99) + 9/9)
:= 9 + ((9999 + 9)/((99 + 9)/9))
:= 9 + ((9 &times; 9 &times; 9 + (99/9)) + 99)
6
&Iuml; 839 := 1 + (((1 + 1)11 ) − (11 &times; (111 − 1)))
&Iuml; 844 := (1 + 1) &times; ((1 + 1) &times; ((1 + 1) &times; 111 − 11))
&Iuml; 849 := ((1 + 1)11 ) − (11 &times; (111 − 1 − 1))
:= ((2 − 22) &times; (2 − (2 &times; 22))) − 2/2
:= 2 &times; ((22 − 2)2 + 22)
:= 2 + ((2 &times; 222 ) − (22/2)2 )
:= 33+3 + ((333 − 3)/3)
:= 3 + (((333 + 3)/3) + 33+3 )
:= (33 &times; 33 ) − (3 &times; 3 + 33)
:= 4444/4 − (4 &times; 4 + 44 )
:= 4 &times; (44 − 44) − 4
:= 4/4 + 4 &times; (44 − 44)
:= ((5 + 5 + 5) &times; (55 + 5/5)) − 5/5
:= 5 &times; 55 + ((55 − 5)/5 − 55)
:= 5 &times; 5 + ((55 &times; (5 + 5 + 5)) − 5/5)
:= 6 &times; 6 + (66 &times; (6 + 6) + (66/6))
:= 6 + (((6/6 + 6 + 6) &times; ((6 + 6)/6)6 ) + 6)
:= 6 + (((66 &times; 66 + 666)/6) + 6)
:= ((77 &times; 77 − 7)/7) − 7
:= (77 &times; 77 − (7 + 7 + 7))/7
:= ((77 &times; 77 + 7) + 7)/7
:= 8 + (8 &times; (88 + 8 + 8) − 8/8)
:= 888 − (88/((8 + 8)/8))
:= 8 + ((8 &times; (88 + 8 + 8) + 8/8) + 8)
:= 99 + (9 &times; 9 &times; 9 + (99/9))
:= 9 + (((9 &times; 9 &times; 9 − ((9 + 9)/9)) + 99) + 9)
:= 9 + (999/9 + 9 &times; 9 &times; 9)
&Iuml; 840 := 111 + (11 − 1 − 1)1+1+1
&Iuml; 845 := (1 + (1 + (1 + 1 + 1))) &times; ((1 + (1 + 11))1+1 )
&Iuml; 850 := 111+1 + (11 − 1 − 1)1+1+1
:= 2/2 + (2 &times; ((22 − 2)2 + 22))
:= 2 + (2 &times; (((22 − 2)2 + 22) + 2))
:= 333 + ((3 − 3/3)3&times;3 )
:= 33+3 + ((33/3)3−3/3 )
:= 4 &times; (4 − 44) − 4 − 4
:= 4/4 + (4 &times; (44 − 44) − 4)
:= (4 + 4)/4 + 4 &times; (44 − 44)
:= (5 + 5 + 5) &times; (55 + 5/5)
:= 5 + ((5 + 5 + 5) &times; (55 + 5/5))
:= 5 &times; ((5 &times; ((5 &times; 5 + 5) + 5)) − 5)
:= (6 + 6) &times; (((6 + 6)/6)6 + 6)
:= (6/6 + 6 + 6) &times; (66 − 6/6)
:= 66 &times; (6 + 6) + (((6 + 6)/6)6 − 6)
:= (77 &times; (77/7)) − 7
:= (77 &times; 77 − (7 + 7))/7
:= (((77 &times; 77 + 7) + 7) + 7)/7
:= 8 + 8 &times; (88 + 8 + 8)
:= 88 + (8 &times; (88 + 8) − (88/8))
:= 8 + ((8 &times; (88 + 8 + 8) + ((8 + 8)/8)) + 8)
:= 9 &times; 9 &times; 9 + 999/9
:= 9 + (((9 &times; 9 &times; 9 − 9/9) + 99) + 9)
:= 9 + (((999 + 9)/9) + 9 &times; 9 &times; 9)
:= (2 − 22) &times; (2 − (2 &times; 22))
:= 3
3+3
+ 333/3
4
&Iuml; 841 := (((11 − 1) &times; (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 &times; 5 − 5/5 + 5)
(5+5)/5
(6+6)/6
&Iuml; 846 := (11 &times; (11 &times; (1 + ((1 + 1) &times; (1 + 1 + 1))))) − 1
2
&Iuml; 851 := (1 + (11 + 11)) &times; (111/(1 + 1 + 1))
:= 2 + (2 &times; ((22 − 2) + 22))
:= (2 &times; (222 + 2)) − (22/2)2
:= 3 + ((33 &times; (33 + 3)) + 33)
:= 33+3 + ((33+3 + 3)/(3 + 3))
:= 4 &times; (44 − 44) − (4 + 4)/4
:= 4 + (4 &times; (44 − 44) − 4/4)
:= 5 + ((5 &times; 5 − 5/5 + 5)(5+5)/5 )
:= 5 &times; 5 + ((55 &times; (5 + 5 + 5)) + 5/5)
:= 666 + 6 &times; (6 &times; 6 − 6)
:= 6 + ((6/6 + 6 + 6) &times; (66 − 6/6))
:= ((77 &times; 77 + 7)/7) − 7
:= (77 &times; 77 − 7)/7
:= (77/7 &times; (7/7 + 77)) − 7
:= 8 + (8 &times; (88 + 8 + 8) + 8/8)
:= (8/8 + 8) &times; ((88 − ((8 + 8)/8)) + 8)
:= 8 + (8 &times; (88 + 8 + 8) + 88/8)
:= 9 &times; 9 &times; 9 + ((999 + 9)/9)
:= 9 + ((9 &times; 9 &times; 9 + 99) + 9)
:= 9 &times; 9 &times; 9 + (999 + 99)/9
:= (6 &times; 6 − (6/6 + 6))
&Iuml; 842 := 1 + ((((11 − 1) &times; (1 + 1 + 1)) − 1)1+1 )
:= 2 + ((2 − 22) &times; (2 − (2 &times; 22)))
3+3
&Iuml; 847 := 11 &times; (11 &times; (1 + ((1 + 1) &times; (1 + 1 + 1))))
:= (2 &times; 222 ) − (22/2)2
&Iuml; 852 := 11 + ((((11 − 1) &times; (1 + 1 + 1)) − 1)1+1 )
:= 2 &times; ((2 &times; (222 + 2)) − 22)
:= 33/3 &times; (3 &times; 3 − (3/3 + 3))
:= (3/3 + 3) &times; ((3 + 3)3 − 3)
:= 4 &times; (44 − 44) − ((4 + 4)/4 + 4)
:= 4 &times; (44 − 44) − 4/4
:= 4 + 4 &times; (44 − 44)
:= 5 + (((5 &times; 5 + 5/5) &times; ((5 + 5)/5)5 ) + 5)
:= ((555 + 555) + 55 )/5
:= 5 + (((555 + 555) + 55 )/5)
:= 6 + ((66/6) &times; ((((6 + 6)/6)6 + 6) + 6))
:= (66/6) &times; (66/6 + 66)
:= 66 + (66 &times; (6 + 6) − 6)
:= (((77 &times; 77 + 7) + 7)/7) − 7
:= 77 &times; (77/7)
:= 7 + ((77 &times; 77 − (7 + 7))/7)
:= 3 + (((333 − 3)/3) + 3
)
3
:= 8 + (8 &times; (88 + 8 + 8) + ((8 + 8)/8))
:= 88/8 &times; (88 − 88/8)
:= 8 + (888 − (88/((8 + 8)/8)))
:= 9 &times; 9 &times; 9 + ((999 + 9 + 9)/9)
:= 9 + (((9 &times; 9 &times; 9 + 99) + 9/9) + 9)
:= ((99 + 9)/9) &times; (9 &times; 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
&Iuml; 853 := (((1 + 11)1+1+1 )/(1 + 1)) − 11
&Iuml; 858 := 11 &times; (111 − (11 &times; (1 + 1 + 1)))
:= (2 &times; (222 − 2)) − 222/2
:= 22 &times; (2 &times; (22 − 2) − 2/2)
:= 3/3 + ((3/3 + 3) &times; ((3 + 3)3 − 3))
3
&Iuml; 863 := (((1 + 11)1+1+1 )/(1 + 1)) − 1
:= ((22 + 2) &times; (2 + 2 + 2)2 ) − 2/2
:= (33 &times; 33 ) − (33 + 3/3)
:= 33 &times; (3 − 3/3)
4
:= 4 + (4 &times; (44 − 44) + 4/4)
:= 4 + ((4444 − 4)/4 − 4 )
:= (4 &times; (44 − 44 + 4)) − 4/4
:= 5 + ((55/5 + 5) &times; (55 − (5 + 5)/5))
:= 55 /5 + (((5 − (5 + 5)/5)5 ) − (5 + 5))
:= 55 /5 + (((5 − (5 + 5)/5)5 ) − 5)
:= (6 &times; (6 + 6) &times; (6 + 6)) − 66/6
:= 66 + 66 &times; (6 + 6)
:= (6 &times; (6 + 6) &times; (6 + 6)) − 6/6
:= 7 + ((77 &times; 77 − 7)/7)
:= 77/7 &times; (7/7 + 77)
:= 7 + ((((77 &times; 77 + 7) + 7)/7) + 7)
:= 888 − ((88/8 + 8 + 8) + 8)
:= 88 + (8 &times; (88 + 8) + ((8 + 8)/8))
:= 888 − (8/8 + 8 + 8 + 8)
:= 9 &times; 99 − ((99/9 + 9 + 9) + 9)
:= 9 + ((999/9 + 9 &times; 9 &times; 9) + 9)
:= 9 &times; 99 − ((9/9 + 9 + 9) + 9)
&Iuml; 854 := 1 + ((((1 + 11)1+1+1 )/(1 + 1)) − 11)
&Iuml; 859 := 1 + (11 &times; (111 − (11 &times; (1 + 1 + 1))))
:= 2 + (2 &times; ((2 &times; (222 + 2)) − 22))
:= 3
3+3
3
2
:= 2 + ((2 &times; 22 ) − 222/2)
3
+ ((3 − 3/3 + 3) )
:= 3/3 + (33 &times; (3 − 3/3))
&Iuml; 864 := ((1 + 11)1+1+1 )/(1 + 1)
:= (22 + 2) &times; (2 + 2 + 2)2
:= 3 &times; 3 &times; (3 &times; 33 − 3)
:= (4444 − 4)/4 − 44
:= 4 + (4444/4 − 4 )
:= 4 &times; (44 − 44 + 4)
:= 555 + ((5 &times; (55 + 5)) − 5/5)
:= ((55/5 + 5) &times; (55 − 5/5)) − 5
:= (55/5 + 5) &times; (55 − 5/5)
:= (6/6 + 6) &times; ((666 + 66)/6)
:= 66 + (66 &times; (6 + 6) + 6/6)
:= 6 &times; (6 + 6) &times; (6 + 6)
:= 77 + 777
:= ((77 &times; 77 + 77) + 7)/7
:= (77/7 + 7) &times; (7 &times; 7 − 7/7)
4
:= 88 + (8 &times; (88 + 8) − ((8 + 8)/8))
:= 8 + ((8 &times; (88 + 8 + 8) + 88/8) + 8)
:= (88 + 8) &times; (8/8 + 8)
:= 9 &times; 99 − (((9/9 + 9 + 9) + 9) + 9)
:= 9 + ((((999 + 9)/9) + 9 &times; 9 &times; 9) + 9)
:= 9 &times; 99 − (9 + 9 + 9)
&Iuml; 855 := 1111 − (1 + 1)(1+1)
1+1+1
&Iuml; 860 := (1 + 1) &times; (((11 + (11 − 1))1+1 ) − 11)
&Iuml; 865 := 1 + (((1 + 11)1+1+1 )/(1 + 1))
:= (2 &times; 222 ) − (222/2 + 2)
:= (2 − 22) &times; (2/2 − (2 &times; 22))
:= 2/2 + ((22 + 2) &times; (2 + 2 + 2)2 )
:= 3 &times; (3 &times; (3 &times; 33 − 3) − 3)
:= (3/3 + 3) &times; ((3 + 3)3 − 3/3)
:= 3/3 + (3 &times; 3 &times; (3 &times; 33 − 3))
:= (4/4 + 4) &times; ((4 &times; 44) − 4)
:= 4/4 + (4 &times; (44 − 44 + 4))
:= 555 + (5 &times; (55 + 5))
:= 5 + ((5 &times; (55 + 5)) + 555)
:= (5 &times; (5 &times; ((5 &times; 5 + 5) + 5))) − 5 − 5
:= 6 &times; 6 + ((6/6 + 6) &times; (666/6 + 6))
:= 66 + (66 &times; (6 + 6) + ((6 + 6)/6))
:= 6/6 + (6 &times; (6 + 6) &times; (6 + 6))
:= 7 + ((77 &times; 77 + 7)/7)
:= 7 + (((77 &times; 77 − 7)/7) + 7)
:= 7 + (77/7 &times; (7/7 + 77))
:= 88 + (8 &times; (88 + 8) − 8/8)
:= 888 + ((8 − 8 &times; 8)/((8 + 8)/8))
:= 8/8 + (88 + 8) &times; (8/8 + 8)
:= 9 &times; 99 − ((9 + 9 + 9) + 9)
:= 9 &times; 99 − (((99 + 99)/9) + 9)
:= 9/9 + (9 &times; 99 − (9 + 9 + 9))
:= 4444/4 − 4
4
&Iuml; 856 := (11 − 1)1+1+1 − ((1 + 11)1+1 )
:= 2 &times; (2 &times; (222 − 2 &times; (2 + 2)))
:= (3 &times; 3
3+3
3
&Iuml; 861 := (1 + 1 + 1) &times; (((1 + 1) &times; ((1 + 11)1+1 )) − 1)
2
:= (2/2 + 2) &times; ((22 + 2) − 2)/2
3
) − ((33/3) )
4
:= 4 + (4 &times; (4 − 44) + 4)
:= 5/5 + ((5 &times; (55 + 5)) + 555)
6
&Iuml; 866 := 1 + (1 + (((1 + 11)1+1+1 )/(1 + 1)))
:= (2 &times; 2 &times; 222) − 22
:= 3 + (33 &times; (3 − 3/3))
:= 3 + ((33 &times; 33 ) − (33 + 3/3))
:= 4/4 + ((4/4 + 4) &times; ((4 &times; 44) − 4))
:= (4 + 4)/4 + (4 &times; (44 − 44 + 4))
5
:= 5 &times; 5 &times; 5 + ((555 + 5 )/5)
6
:= 5 + (((555 + 55 )/5) + 5 &times; 5 &times; 5)
:= 66 &times; (6 + 6) + ((6 + 6)/6)
:= 66 + (((6 &times; 6/(6 + 6)) ) + 66)
:= (6 + 6)/6 + (6 &times; (6 + 6) &times; (6 + 6))
:= 7 + (((77 &times; 77 + 7) + 7)/7)
:= 7 + (777 + 77)
:= 7 + (((77 &times; 77 + 77) + 7)/7)
:= 88 + 8 &times; (88 + 8)
:= 888 − (88/8 + 8 + 8)
:= 888 − (88 + 88)/8
:= (9 − 9/9) &times; ((99 − 9/9) + 9)
:= (9 &times; (99 + 9)) − 999/9
:= (9 + 9)/9 + (9 &times; 99 − (9 + 9 + 9))
&Iuml; 857 := ((1 + 1) &times; ((11 + 11)1+1 )) − 111
2
:= (2 &times; 22 ) − 222/2
3
:= (33 &times; 3 ) − (3/3 + 33)
&Iuml; 862 := (((1 + 11)1+1+1 )/(1 + 1)) − 1 − 1
:= 2 &times; (2 &times; 222 − 2) − 22
3
:= 3 + ((33 &times; (3 − 3/3)) + 3/3)
&Iuml; 867 := 1 + (1 + (1 + (((1 + 11)1+1+1 )/(1 + 1))))
:= 2/2 + ((2 &times; 2 &times; 222) − 22)
:= 3 + (3 &times; 3 &times; (3 &times; 33 − 3))
:= 4 + ((4 &times; (4 − 44) + 4/4) + 4)
:= (4 &times; (4 − 44 + 4)) − (4 + 4)/4
:= 4 + ((4 &times; (44 − 44 + 4)) − 4/4)
:= ((5 + 5)/5)5 + (55 &times; (5 + 5 + 5))
:= 555 + (55 − 55)/(5 + 5)
:= 555 + (55 − 5)/(5 + 5)
:= 66 + (66 &times; (6 + 6) − 6/6)
:= (6 &times; (6 + 6) &times; (6 + 6)) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + (6 &times; (6 + 6) &times; (6 + 6))
4
4
:= ((77 &times; 77 − 7) + 77)/7
:= 7 + (((77 &times; 77 + 7)/7) + 7)
:= 7 + ((((77 &times; 77 − 7)/7) + 7) + 7)
:= 8/8 + (8 &times; (88 + 8) + 88)
:= 888 + ((8 − 88)/8 − (8 + 8))
:= 88 + (8 &times; (88 + 8) + (88/8))
:= 9 + (((9 &times; 9 &times; 9 + (99/9)) + 99) + 9)
:= 9 &times; 99 − (99/9 + 9 + 9)
:= 9 + (((999/9 + 9 &times; 9 &times; 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
&Iuml; 868 := 1111 − (1 + ((1 + 1) &times; 111+1 ))
:= 2 + ((2 &times; 2 &times; 222) − 22)
3
:= (3/3 + 3) &times; ((3 + 3) + 3/3)
4
&Iuml; 873 := 1 + ((1 + 1)1+1+1 &times; (111 − 1 − 1))
&Iuml; 878 := 1 + ((111 &times; (1 + 1)1+1+1 ) − 11)
:= 2 + (((((2 &times; 22) − 2)2 ) − 22)/2)
:= 2 &times; (((22 − 2/2)2 ) − 2)
:= 3 &times; (3 &times; (3 &times; 33 − 3) + 3)
:= (33 &times; 33 ) + (((3 − 33)/3) − 3)
4
4
:= 4 + (4 &times; (4 − 44 + 4))
:= 4 + ((4/4 + 4) − (4 + 4))
:= 44 &times; (4 &times; 4 + 4) − (4 + 4)/4
:= 55 /5 + ((5 − (5 + 5)/5)5 )
:= 5 + (((5 − (5 + 5)/5)5 ) + 55 /5)
:= (55 &times; (55/5 + 5)) − (5 + 5)/5
:= 6 + ((6 &times; (6 + 6) &times; (6 + 6)) − ((6 + 6)/6))
:= (6 + 6) &times; (6 + 6) + ((6 &times; 6/(6 + 6))6 )
:= 6 + (((6 &times; (6 + 6) &times; (6 + 6)) + ((6 + 6)/6)) + 6)
:= 7 + (777 + 77 + 7)
:= 777 + (7 &times; (7 + 7) − ((7 + 7)/7))
:= 7 + (7 &times; (77 + 7 &times; 7) − (77/7))
:= 888 − (((88 + 8)/8) + 8)
:= 8/8 + (888 − (8 + 8))
:= 888 + (8 − 88)/8
:= 9 &times; 99 − ((99 + 99 + 9)/9)
:= 9 &times; 99 − (9 + 9)
:= 9 &times; 99 − ((99 + 9 + 9)/9)
&Iuml; 869 := 11 &times; ((11 − 1 − 1)1+1 − (1 + 1))
&Iuml; 874 := 11 + ((((1 + 11)1+1+1 )/(1 + 1)) − 1)
&Iuml; 879 := (11 − 1)1+1+1 − 111+1
:= (2222 − 222 )/2
:= 2 + (2 &times; (2 &times; (222 − (2 + 2))))
:= (2 &times; (2 &times; (222 − 2))) − 2/2
:= 33/3 + (33 &times; (33 − 3/3))
:= 3/3 + (3 &times; (3 &times; (3 &times; 33 − 3) + 3))
:= (33 &times; 33 ) − (3 &times; 3 + 3)
:= 44 &times; (4 &times; 4 + 4) − 44/4
:= 44 &times; (4 &times; 4 + 4) − ((4 + 4)/4 + 4)
:= 44 &times; (4 &times; 4 + 4) − 4/4
:= 5 + ((55/5 + 5) &times; (55 − 5/5))
:= (5 &times; 5 &times; (5 + 5)) + (5 − 5)/5
:= (55 &times; (55/5 + 5)) − 5/5
:= 6 + ((6 &times; (6 + 6) &times; (6 + 6)) − 6/6)
:= ((66 − 6)/6) + (6 &times; (6 + 6) &times; (6 + 6))
:= 6 + (((6 &times; 6/(6 + 6))6 ) + (6 + 6) &times; (6 + 6))
:= 77/7 &times; ((7 + 7)/7 + 77)
:= 777 + (7 &times; (7 + 7) − 7/7)
:= 7 &times; (77 + 7 &times; 7) − (7 + 7 + 7)/7
:= 888 − (88/8 + 8)
:= 888 + (((8 + 8)/8) − (8 + 8))
:= 888 − (8/8 + 8)
:= 9 &times; 99 − ((99 + 99)/9)
:= 9/9 + (9 &times; 99 − (9 + 9))
:= 9 &times; 99 − (99 + 9)/9
&Iuml; 875 := 11 + (((1 + 11)1+1+1 )/(1 + 1))
&Iuml; 880 := 11 &times; ((11 − 1 − 1)1+1 − 1)
&Iuml; 870 := 1 + (11 &times; ((11 − 1 − 1)1+1 − (1 + 1)))
5
:= (2 &times; (2 &times; 222 + 2)) − 22
:= (2 &times; 2 &times; 222) − (22/2 + 2)
:= 2 &times; (2 &times; (222 − 2))
:= 3 + ((3 &times; 3 &times; (3 &times; 33 − 3)) + 3)
:= 33/3 + (3 &times; 3 &times; (3 &times; 33 − 3))
:= (33 &times; 33 ) − 33/3
:= 44 + ((4/4 + 4)4 − 44/4)
:= (4/4 + 4) &times; ((4 &times; 44) − 4/4)
:= 44 &times; (4 &times; 4 + 4)
:= (5 &times; (5 &times; ((5 &times; 5 + 5) + 5))) − 5
:= 5 &times; (5 &times; ((5 &times; 5 + 5) + 5))
:= 55 &times; (55/5 + 5)
:= 6 + (6 &times; (6 + 6) &times; (6 + 6))
:= 66/6 + (6 &times; (6 + 6) &times; (6 + 6))
:= 666 + (6 &times; 6 &times; 6 − (6 + 6)/6)
:= 7 &times; (77 + 7 &times; 7) − (77 + 7)/7
:= 777 + 7 &times; (7 + 7)
:= (7/7 + 7) &times; (777 − 7)/7
:= 888 + ((8 − 88)/8 − 8)
:= 888 − (88 + 8 + 8)/8
:= 888 − 8
:= 9 &times; 99 − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + (9 &times; 99 − (9 + 9))
:= 9 &times; 99 − 99/9
&Iuml; 871 := 1111 − ((1 + 1) &times; (111+1 − 1))
&Iuml; 876 := 1 + (11 + (((1 + 11)1+1+1 )/(1 + 1)))
&Iuml; 881 := 1 + (11 &times; ((11 − 1 − 1)1+1 − 1))
:= ((((2 &times; 22) − 2)2 ) − 22)/2
:= 2 &times; ((2 &times; (222 − 2)) − 2)
:= ((((2 &times; 22) − 2)2 ) − 2)/2
:= 3 + ((3/3 + 3) &times; ((3 + 3)3 + 3/3))
:= (3/3 + 3) &times; ((3 + 3)3 + 3)
:= (33 &times; 33 ) + ((3 − 33)/3)
:= 4 &times; 4 + (4444/4 − 44 )
:= 44 &times; (4 &times; 4 + 4) − 4
:= 44 + (4/4 + 4)4
:= 5/5 + ((5 &times; (5 &times; ((5 &times; 5 + 5) + 5))) − 5)
:= 5/5 + (5 &times; (5 &times; ((5 &times; 5 + 5) + 5)))
:= 5/5 + (55 &times; (55/5 + 5))
:= 6 + ((6 &times; (6 + 6) &times; (6 + 6)) + 6/6)
:= 6 + ((6 &times; (6 + 6) &times; (6 + 6)) + 6)
:= 666 + (6 &times; 6 &times; 6 − 6/6)
:= 7 &times; (77 + 7 &times; 7) − 77/7
:= 7/7 + (777 + 7 &times; (7 + 7))
:= 7 &times; (77 + 7 &times; 7) − 7/7
:= 888 − (8/8 + 8 + 8)
:= 888 − (88 + 8)/8
:= 8/8 + 888 − 8
:= 9 &times; 99 − (99/9 + 9)
:= 9 &times; 99 + (((9 + 9 + 9)/9) − (9 + 9))
:= 9 &times; 99 − 9/9 − 9
&Iuml; 872 := (1 + 1)1+1+1 &times; (111 − 1 − 1)
:= 2 &times; (2 &times; (222 − (2 + 2)))
3
:= (3/3 + 3) &times; (((3 + 3) − 3/3) + 3)
&Iuml; 877 := (111 &times; (1 + 1)1+1+1 ) − 11
:= (2 &times; 2 &times; 222) − 22/2
3
:= (33 &times; 3 ) − (33/3 + 3)
4
4
&Iuml; 882 := (1 + 1) &times; ((11 + (11 − 1))1+1 )
:= 2 &times; ((22 − 2/2)2 )
:= 3 &times; (3 &times; 3 &times; 33 − 3)
:= 44 &times; (4 &times; 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 &times; (5 &times; ((5 &times; 5 + 5) + 5)))
:= (5 + 5)/5 + (55 &times; (55/5 + 5))
:= 6 + ((6 &times; (6 + 6) &times; (6 + 6)) + ((6 + 6)/6))
:= 6 + (((6 &times; (6 + 6) &times; (6 + 6)) + 6/6) + 6)
:= 666 + 6 &times; 6 &times; 6
:= 7 + ((77/7 &times; (7/7 + 77)) + 7)
:= ((7 + 7)/7) + (777 + 7 &times; (7 + 7))
:= 7 &times; (77 + 7 &times; 7)
:= 888 − 8 − 8
:= 888 − 88/8
:= 888 + (((8 + 8)/8) − 8)
:= 9 &times; 99 − (9/9 + 9 + 9)
:= 9 &times; 99 + (((9 − 99)/(9 + 9)) − 9)
:= 9 &times; 99 − 9
68
Inder J. Taneja
&Iuml; 883 := 1 + ((1 + 1) &times; ((11 + (11 − 1))1+1 ))
:= ((((2 &times; 22) − 2)2 ) + 2)/2
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 888 := 111 &times; (1 + 1)1+1+1
&Iuml; 893 := 1 + (1 + (11 &times; (11 − 1 − 1)1+1 ))
:= 2 &times; 2 &times; 222
:= 2/2 + (2 &times; (2 &times; 222 + 2))
:= 3 + ((33 &times; 33 ) − 33/3)
:= (33 &times; 3 ) − 3
:= 3 + ((33 &times; 33 ) − 3/3)
:= 4 + (44 &times; (4 &times; 4 + 4) − 4/4)
:= (4 + 4) &times; 444/4
:= 4 + (((4 + 4) &times; 444 + 4)/4)
3
:= 5 + ((55 &times; (55/5 + 5)) − ((5 + 5)/5))
:= 5 &times; 55 + ((5 − (55 + 5))/5)
:= 5 &times; 55 + ((55 − 5 − 5)/5 − 5)
:= 6/6 + (666 + 6 &times; 6 &times; 6)
:= 6 + (666 + 6 &times; 6 &times; 6)
:= 666 + (6 &times; 6 &times; 6 + 66/6)
:= 7/7 + 7 &times; (77 + 7 &times; 7)
:= 777 + 777/7
:= 77/7 + 7 &times; (77 + 7 &times; 7)
:= 888 + (88/8 − (8 + 8))
:= 888
:= 8 + (888 − 88/8 + 8)
:= 9/9 + (9 &times; 99 − 9)
:= 9 &times; 99 − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 &times; 99
&Iuml; 884 := (1 + 1) &times; (1 + ((11 + (11 − 1))1+1 ))
5
&Iuml; 889 := 1 + (111 &times; (1 + 1)1+1+1 )
&Iuml; 894 := 1 + (1 + (1 + (11 &times; (11 − 1 − 1)1+1 )))
:= 2 &times; (2 &times; 222 − 2)
:= 2/2 + (2 &times; 2 &times; 222)
:= 2 + (2 &times; (2 &times; 222 + 2))
:= (33 − 3/3) &times; (3/3 + 33)
:= 3/3 + ((33 &times; 33 ) − 3)
:= 3 + (33 &times; 33 )
:= 4 + 44 &times; (4 &times; 4 + 4)
:= ((4 + 4) &times; 444 + 4)/4
:= 4 + ((4 + 4)/4 &times; (444 + 4/4))
:= 5 + ((55 &times; (55/5 + 5)) − 5/5)
:= 5 &times; 55 + ((55 − 55)/5)
:= 5 &times; 55 + ((55 − 5)/5 − 5)
:= 666 + (6 &times; 6 &times; 6 + (6 + 6)/6)
:= 6 + ((666 + 6 &times; 6 &times; 6) + 6/6)
:= (6 &times; ((6 + 6) &times; (6 + 6) + 6)) − 6
:= ((7 + 7)/7) + 7 &times; (77 + 7 &times; 7)
:= 7 + 7 &times; (77 + 7 &times; 7)
:= (7 &times; ((7 + 7)/7)7 ) − (7 + 7)/7
:= 888 − (8/((8 + 8)/8))
:= 8/8 + 888
:= 8 + (888 − ((8 + 8)/8))
:= (9 + 9)/9 + (9 &times; 99 − 9)
:= 9 &times; 99 − (9 + 9)/9
:= 9 &times; 99 + ((9 + 9 + 9)/9)
&Iuml; 890 := (11 &times; (11 − 1 − 1)1+1 ) − 1
&Iuml; 895 := ((1 + 111) &times; (1 + 1)1+1+1 ) − 1
&Iuml; 885 := 1 + ((1 + 1) &times; (1 + ((11 + (11 − 1))1+1 )))
:= 2 + (((((2 &times; 22) − 2)2 ) + 2)/2)
:= (2 &times; (2 &times; (222 + 2))) − 2/2
:= 2 + (2 &times; 2 &times; 222)
:= (33 &times; 33 ) − (3 + 3)
:= (33 &times; 3 ) − 3/3
:= 3 + ((33 &times; 33 ) + 3/3)
:= 4 + ((4/4 + 4)4 + 44 )
:= (4 + 4)/4 &times; (444 + 4/4)
:= 4 + (44/4 &times; (4 − 4/4)4 )
:= 5 + (55 &times; (55/5 + 5))
:= 5 + ((55 &times; (55/5 + 5)) + 5)
:= 55 /5 + (5 &times; 55 − 5)
:= (6 − 6/6) &times; (666/6 + 66)
:= 6 + ((666 + 6 &times; 6 &times; 6) + ((6 + 6)/6))
:= (6666/6) − 6 &times; 6 &times; 6
:= (7 &times; ((7 + 7)/7)7 ) − 77/7
:= 7 + (7 &times; (77 + 7 &times; 7) + 7/7)
:= (7 &times; ((7 + 7)/7)7 ) − 7/7
:= 8 + (888 − 88/8)
:= 9 &times; 99 + (((9 + 9 + 9)/9) − 9)
&Iuml; 886 := (1 + 1) &times; (((1 + 1) &times; (1 + 1) &times; 111) − 1)
:= (2 &times; 2 &times; 222) − 2
:= 3/3 + ((33 &times; 33 ) − (3 + 3))
3
:= 888 + (8 + 8)/8
:= 8 + (888 − 8/8)
:= 9 &times; 99 − 9/9
:= 9 &times; 99 + ((9 &times; 9 − 9)/(9 + 9))
&Iuml; 891 := 11 &times; (11 − 1 − 1)1+1
:= 2 + ((2 &times; 2 &times; 222) + 2/2)
:= 33 &times; 3
&Iuml; 896 := (1 + 111) &times; (1 + 1)1+1+1
:= 2 &times; (2 &times; (222 + 2))
:= 3 + (((33 &times; 33 ) − 3/3) + 3)
3
:= 44/4 &times; (4 − 4/4)4
:= 4 &times; (44 − 4 &times; (4 + 4))
:= 5 + ((55 &times; (55/5 + 5)) + 5/5)
:= 5 &times; 55 + ((55 + 5)/5 − (5 + 5))
:= 5 &times; 55 + ((55 + 5)/5 − 5)
:= 6 + ((666 − ((6 + 6)/6)) + 6 &times; 6 &times; 6)
:= 6 + ((6 − 6/6) &times; (666/6 + 66))
:= (6/6 + 6) &times; (((6 + 6)/6)6/6+6)
:= 77/7 + (777 + 7 &times; (7 + 7))
:= 7 + (7 &times; (77 + 7 &times; 7) + ((7 + 7)/7))
:= 7 &times; ((7 + 7)/7)7
:= 888 − (8 + 8)/8
:= 888 + (88/8 − 8)
:= 8 + 888
:= 9 &times; 99 + ((9 − 99)/(9 + 9))
:= 9 &times; 99
:= 9 &times; 99 + ((9 &times; 9 + 9)/(9 + 9))
:= (4 + 4)/4 &times; (444 − 4/4)
&Iuml; 887 := (111 &times; (1 + 1)1+1+1 ) − 1
:= (2 &times; 2 &times; 222) − 2/2
&Iuml; 892 := 1 + (11 &times; (11 − 1 − 1)1+1 )
:= 2 &times; (2 &times; 222 + 2)
&Iuml; 897 := 1 + ((1 + 111) &times; (1 + 1)1+1+1 )
:= 2/2 + (2 &times; (2 &times; (222 + 2)))
3
:= (33 &times; 33 ) − (3/3 + 3)
:= 3/3 + (33 &times; 3 )
:= 3 + ((33 &times; 33 ) + 3)
:= ((4 + 4) &times; 444 − 4)/4
:= 4 + (444 + 444)
:= 4 &times; 4 + ((4/4 + 4)4 + 44 )
5
:= 5 + ((55 &times; (55/5 + 5)) + ((5 + 5)/5))
:= 5 &times; 5 + ((5 − 5)/(5 + 5) + 555)
:= 5 &times; 55 + ((55 + 5 + 5)/5 − 5)
:= 6 + ((6 &times; 6 &times; 6 − 6/6) + 666)
:= 6 &times; 6 + (66 &times; (6 + 6) + ((6 + 6)/6)6 )
:= 666/6 + (66 &times; (6 + 6) − 6)
7
:= 7/7 + (7 &times; ((7 + 7)/7)7 )
:= 777 + (777 − 7)/7
:= 7 + ((7 &times; ((7 + 7)/7) ) − (77/7))
:= 888 − 8/8
:= 888 + (8/((8 + 8)/8))
:= 8 + (888 + 8/8)
:= 9 &times; 99 + ((9 − 9 &times; 9)/(9 + 9))
:= 9/9 + 9 &times; 99
:= 9 + (9 &times; 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
&Iuml; 898 := (((11 − 1) &times; (1 + 1 + 1))1+1 ) − 1 − 1
&Iuml; 903 := (1 + 1)11−1 − 111+1
&Iuml; 908 := ((11 − 1 − 1) &times; (1 + (11 − 1)1+1 )) − 1
:= 2 + (((2 &times; (2 + 2) + 22)2 ) + 2/2)
:= 2 &times; ((2 &times; ((222 + 2) + 2)) + 2)
:= 3 + (((33 &times; 3 ) + 3/3) + 3)
:= 3 + ((33 &times; 33 ) + 3 &times; 3)
:= 3 + (((33 &times; 33 ) + 33/3) + 3)
:= (4 + 4)/4 + (4 &times; (44 − 4 &times; (4 + 4)))
:= 4 + (((4 + 4) &times; 444 + 44)/4)
:= 44 + (4 &times; (44 − 44 + 4))
:= 5 &times; 55 + (55 − 5 − 5)/5
:= 5 + ((55 − 5 − 5)/5 + 5 &times; 55)
:= (5 − 5/5)5 − (555/5 + 5)
:= 666/6 + 66 &times; (6 + 6)
:= 6 + ((6 &times; ((6 + 6) &times; (6 + 6) + 6)) + ((6 + 6)/6))
:= 2 + (2 &times; (2 &times; (222 + 2)))
3
:= (6 &times; ((6 + 6) &times; (6 + 6) + 6)) − (6 + 6)/6
7
7
:= ((7 + 7)/7) + (7 &times; ((7 + 7)/7) )
:= 7 + (7 &times; ((7 + 7)/7) )
:= (77 + 7)/7 + (7 &times; ((7 + 7)/7)7 )
:= 8 + (888 + ((8 + 8)/8))
:= 8 + (888 − 8/8 + 8)
:= 8 + (((88 + 8)/8) + 888)
:= 9 + (9 &times; 99 − ((9 + 9)/9))
:= 9 &times; 99 + (99 + 9)/9
:= 9 + ((9 &times; 99 − 9/9) + 9)
&Iuml; 899 := (((11 − 1) &times; (1 + 1 + 1))1+1 ) − 1
&Iuml; 904 := 1 + ((1 + 1)11−1 − 111+1 )
&Iuml; 909 := (11 − 1 − 1) &times; (1 + (11 − 1)1+1 )
:= 22/2 + (2 &times; 2 &times; 222)
3
:= 3 &times; 3 + ((33 &times; 3 ) − 3/3)
:= 2 &times; (2 &times; ((222 + 2) + 2))
3
:= 3 + (((33 &times; 3 ) + 3 &times; 3) + 3/3)
:= 22 + ((2 &times; 2 &times; 222) − 2/2)
:= 3 &times; ((3 &times; 3 &times; 33 + 3) + 3)
:= ((4 + 4) &times; 444 + 44)/4
:= 4 + ((4 &times; (4 − 4 &times; (4 + 4))) + 4)
:= 4 &times; 44 − (444/4 + 4)
:= 5 &times; 55 + (55 − 5)/5
:= 5 + ((55 − 5)/5 + 5 &times; 55)
:= 5 + (((55 − 5)/5 + 5 &times; 55) + 5)
:= (6 &times; ((6 + 6) &times; (6 + 6) + 6)) − 6/6
:= 66 &times; (6 + 6) + (666 + 6)/6
:= 6 + (666/6 + 66 &times; (6 + 6))
4
:= 777 + (777 + 77)/7
:= 7 + ((7 &times; ((7 + 7)/7) ) + 7/7)
:= 7 + (((7 &times; ((7 + 7)/7)7 ) − 7/7) + 7)
:= 888 + 88/8
:= 8 + (888 + 8)
:= 8 + ((88 + 8 + 8)/8 + 888)
:= 9 + (9 &times; 99 − 9/9)
:= 9 &times; 99 + ((99 + 9 + 9)/9)
:= 9 + (9 &times; 99 + 9)
&Iuml; 900 := ((11 − 1) &times; (1 + 1 + 1))1+1
&Iuml; 905 := 1 + (1 + ((1 + 1)11−1 − 111+1 ))
7
&Iuml; 910 := 1 + ((11 − 1 − 1) &times; (1 + (11 − 1)1+1 ))
:= (2 &times; (2 + 2) + 22)2
:= 22 + (((((2 &times; 22) − 2)2 ) + 2)/2)
:= 22 + (2 &times; 2 &times; 222)
:= 3 &times; (3 &times; 3 &times; 33 + 3)
:= 3 + ((33 &times; 33 ) + 33/3)
:= 3/3 + ((33 &times; 33 ) + (3 &times; (3 + 3)))
:= 4 + (4 &times; (44 − 4 &times; (4 + 4)))
:= 4 &times; 4 + (((4 + 4) &times; 444 + 4)/4)
:= 4 &times; 44 + (((4 − 444)/4) − 4)
:= 5 &times; (5 &times; 5 &times; 5 + 55)
:= 5 + (55 /5 + 5 &times; 55)
:= 5 + ((55 /5 + 5 &times; 55) + 5)
:= 6 &times; ((6 + 6) &times; (6 + 6) + 6)
:= 6 + ((6 &times; ((6 + 6) &times; (6 + 6) + 6)) − 6/6)
:= (6/6 + 6) &times; (((6 + 6)/6)6 + 66)
:= (77/7 + 7) &times; (7/7 + 7 &times; 7)
:= 777 + ((7 + 7)/7)7
:= 7 + ((7 &times; ((7 + 7)/7)7 ) + 7)
:= 888 + (88 + 8)/8
:= 8 + (888 + 8/8 + 8)
:= 888 + (88 + 88)/8
:= 9 + 9 &times; 99
:= 9 + (((9 &times; 9 + 9)/(9 + 9)) + 9 &times; 99)
:= 9 + ((9 &times; 99 + 9/9) + 9)
&Iuml; 901 := 1 + (((11 − 1) &times; (1 + 1 + 1))1+1 )
&Iuml; 906 := 1 + (1 + (1 + ((1 + 1)11−1 − 111+1 )))
&Iuml; 911 := 11 + (((11 − 1) &times; (1 + 1 + 1))1+1 )
:= 2 + (2 &times; (2 &times; ((222 + 2) + 2)))
:= 22 + ((2 &times; 2 &times; 222) + 2/2)
:= 3 &times; 3 + ((33 &times; 3 ) + 3/3)
3
:= 3 + (((33 &times; 3 ) + 3 &times; 3) + 3)
:= 3 &times; 3 + ((33 &times; 33 ) + 33/3)
:= 4 + ((4 &times; (44 − 4 &times; (4 + 4))) + 4/4)
:= 4 + (4 &times; 44 − ((444 + 44)/4))
:= 4 &times; 44 − ((444 + 4 + 4)/4)
:= 5 &times; 55 + (55 + 5)/5
:= 5 + ((55 + 5)/5 + 5 &times; 55)
:= 5 &times; 55 + ((55 + 55 )/5)
:= 6 + (6 &times; ((6 + 6) &times; (6 + 6) + 6))
:= 66/6 + (6 &times; ((6 + 6) &times; (6 + 6) + 6))
:= 2/2 + ((2 &times; (2 + 2) + 22)2 )
3
:= 6/6 + (6 &times; ((6 + 6) &times; (6 + 6) + 6))
7
:= 7 + ((7 &times; ((7 + 7)/7) ) − ((7 + 7)/7))
:= 7/7 + (((7 + 7)/7) + 777)
:= 7 + (((7 &times; ((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 &times; 99 + 9/9)
:= 9 + ((9 &times; 99 − ((9 + 9 + 9)/9)) + 9)
:= 9 + ((99/9) + 9 &times; 99)
&Iuml; 902 := 11 &times; (1 + (11 − 1 − 1)1+1 )
:= 2 + ((2 &times; (2 + 2) + 22)2 )
3
7
&Iuml; 907 := 11 + ((1 + 111) &times; (1 + 1)1+1+1 )
:= 22/2 + (2 &times; (2 &times; (222 + 2)))
&Iuml; 912 := (1 + 1)11−1 − (1 + 111)
:= 2 + ((2 &times; 2 &times; 222) + 22)
:= 33/3 + (33 &times; 3 )
:= 3 + ((33 &times; 3 ) − 33/3)
:= 3 + ((33 &times; 33 ) + (3 &times; (3 + 3)))
:= 4 &times; 44 − ((444 + 44)/4)
:= 4 &times; 4 + (44/4 &times; (4 − 4/4)4 )
:= 4 &times; ((44 − 4 &times; (4 + 4)) + 4)
:= 5 &times; 55 + (55 + 5 + 5)/5
:= 5 + ((55 + 5 + 5)/5 + 5 &times; 55)
:= 5 &times; 55 + ((55 + 55 + 5)/5)
:= (6 + 6)/6 + (6 &times; ((6 + 6) &times; (6 + 6) + 6))
:= 6 + ((6 &times; ((6 + 6) &times; (6 + 6) + 6)) + 6/6)
:= 6 + ((6 &times; ((6 + 6) &times; (6 + 6) + 6)) + 6)
3
7
3
7
:= 7 + (((7 + 7)/7)7 + 777)
:= 7 + ((7 &times; ((7 + 7)/7) ) − 7/7)
:= 77/7 + (7 &times; ((7 + 7)/7) )
:= 8 + ((888 − ((8 + 8)/8)) + 8)
:= 8 + (888 + 88/8)
:= 8 + (888 + 8 + 8)
:= 99/9 + 9 &times; 99
:= 9 + ((9 &times; 99 − ((9 + 9)/9)) + 9)
:= 9 + (((99 + 9)/9) + 9 &times; 99)
70
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 913 := (1 + 1)11−1 − 111
&Iuml; 918 := (11 − 1 − 1) &times; (1 + (1 + (11 − 1)1+1 ))
:= (222/2 − 222)/2
:= 22 + (2 &times; (2 &times; (222 + 2)))
:= 33 + ((33 &times; 33 ) − 33/3)
:= 3 &times; (3 &times; (3 &times; 33 + 3))
:= (2 &times; (222 − 22)) − 2/2
:= 33 + ((33 &times; 33 ) − 3/3)
4
:= 4 + (((4 − 444)/4) + 4 &times; 4 )
:= 4 &times; 44 − 444/4
&Iuml; 923 := 11 + ((1 + 1)11−1 − (1 + 111))
5
:= 44 + (44 &times; (4 &times; 4 + 4) − 4/4)
:= (5 − 5/5)5 − 555/5
:= 5 + ((5 − 5/5) − 555/5)
:= (5 &times; (55 + 5)) + (55 − 5 − 5)/5
:= (66/6) &times; (66/6 + 66 + 6)
:= 666 + 6 &times; (6 &times; 6 + 6)
:= (6/6 + 6 + 6) &times; ((66 − 6/6) + 6)
:= 77/7 &times; (77 − 7/7 + 7)
:= ((77 &times; (77 + 7) + 7)/7) − 7
:= (77 &times; (77 + 7) − 7)/7
:= 8 + ((888 + 8/8 + 8) + 8)
:= 8 + ((88 + 88)/8 + 888)
:= 8 + (((888 + 88/8) + 8) + 8)
:= 9 &times; 99 + ((99 + 99)/9)
:= 999 − 9 &times; 9
:= 9 + (((99 + 99 + 9)/9) + 9 &times; 99)
&Iuml; 914 := 1 + ((1 + 1)11−1 − 111)
&Iuml; 919 := 1 + ((11 − 1 − 1) &times; (1 + (1 + (11 − 1)1+1 )))
&Iuml; 924 := 11 + ((1 + 1)11−1 − 111)
:= 22 + (2 &times; (2 &times; 222 + 2))
:= (2 &times; (222 − (22 + 2))) − 2/2
:= 2 &times; (222 − 22)
:= 33 + ((33 &times; 33 ) − (3/3 + 3))
:= 33 + ((33 &times; 33 ) + 3/3)
:= 33 + (33 &times; 33 )
:= 4 &times; 44 + ((4 − 444)/4)
:= 4444/4 − (4 &times; (44 + 4))
:= 44 + 44 &times; (4 &times; 4 + 4)
5
:= (5 − 5/5) − (55 + 55)
:= 5 + ((5 − 5/5) − (55 + 55))
:= (5 &times; (5 − 5 &times; 5)) + (5 − 5/5)5
:= 66 &times; (6 + 6) + ((666 + 66)/6)
:= 6/6 + (6 &times; (6 &times; 6 + 6) + 666)
:= 66 &times; ((6 + 6)/6 + 6 + 6)
:= 7 + ((7 &times; ((7 + 7)/7) ) + (77/7))
:= 7 + ((((7 + 7)/7)7 + 777) + 7)
:= 77 &times; (77 + 7)/7
:= 8 + (((888 + ((8 + 8)/8)) + 8) + 8)
:= 8 + (((888 − 8/8 + 8) + 8) + 8)
:= 888 + ((8 &times; 8 + 8)/((8 + 8)/8))
:= 9 &times; 99 + ((99 + 99 + 9)/9)
:= 9/9 + (999 − 9 &times; 9)
:= 9 &times; 99 + (99/((9 + 9 + 9)/9))
5
7
&Iuml; 915 := 1 + (1 + ((1 + 1)11−1 − 111))
&Iuml; 920 := (11 − 1) &times; (11 + (11 − 1 − 1)1+1 )
&Iuml; 925 := (11111 − 11)/(1 + 11)
:= 2 + (222/2 − 222)/2
:= 2 &times; (222 − (22 + 2))
:= 2/2 + (2 &times; (222 − 22))
:= 33 + ((33 &times; 33 ) − 3)
:= 3 + (((33 &times; 33 ) − 3/3) + 33 )
:= 3/3 + ((33 &times; 33 ) + 33)
:= 4 &times; 44 + (((4 − 444) + 4)/4)
:= (4 + 4) &times; (444/4 + 4)
:= 44 + ((4/4 + 4)4 + 44 )
5
:= (5 + 5 + 5) &times; ((55 + 5/5) + 5)
:= 5 /5 + ((5 &times; (55 + 5)) − 5)
:= 5 &times; ((5 &times; 5 &times; 5 + 55) + 5)
:= 6 + ((666/6 + 66 &times; (6 + 6)) + 6)
:= 666 + (6 &times; (6 &times; 6 + 6) + ((6 + 6)/6))
:= 6/6 + (66 &times; ((6 + 6)/6 + 6 + 6))
:= 7777/7 − (7 + 7) &times; (7 + 7)
:= 7 + (77/7 &times; (77 − 7/7 + 7))
:= (77 &times; (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 &times; 9)
:= 9 + (((99/9) + 9 &times; 99) + 9)
:= (((9 + 9)/9)9/9+9 ) − 99
&Iuml; 916 := 1 + (1 + (1 + ((1 + 1)11−1 − 111)))
&Iuml; 921 := 1 + ((11 − 1) &times; (11 + (11 − 1 − 1)1+1 ))
2
&Iuml; 926 := (1 + 11111)/(1 + 11)
:= 2 &times; (22 − (22 + 2 + 2))
:= 2/2 + (2 &times; (22 − (22 + 2)))
:= 2 + (2 &times; (222 − 22))
:= 33 + (((33 &times; 33 ) − 3) + 3/3)
:= 3 + ((33 &times; 33 ) + 33 )
:= 3 + (((33 &times; 33 ) − 3/3) + 33)
:= (4 &times; (44 − 4 &times; 4)) − 44
:= 4 + ((4 &times; 44 − 444/4) + 4)
:= 44 + (((4/4 + 4)4 + 44 ) + 4/4)
:= 5 + (((55 + 55 )/5) + 5 &times; 55)
:= (5 &times; (55 + 5)) + ((55 + 5)/5 − 5)
:= (5 &times; (55 + 5)) + (55 + 5)/5
:= 6 + ((6/6 + 6) &times; (((6 + 6)/6)6 + 66))
:= 6 + (((666/6 + 66 &times; (6 + 6)) + 6) + 6)
:= (6 + 6)/6 + (66 &times; ((6 + 6)/6 + 6 + 6))
:= ((77 &times; (77 + 7) − 7)/7) − 7
:= (77 &times; (77 + 7) − (7 + 7 + 7))/7
:= ((77 &times; (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 &times; 9)
:= 9 &times; (9 &times; 9 + 9) + 999/9
:= 9 + (999 − (9/9 + 9 &times; 9))
2
&Iuml; 917 := 1 + (1 + (1 + (1 + ((1 + 1)11−1 − 111))))
&Iuml; 922 := 11 + (11 + (((11 − 1) &times; (1 + 1 + 1))1+1 ))
2
&Iuml; 927 := 1 + ((1 + 11111)/(1 + 11))
:= (2 &times; (22 − 22)) − 2
:= 2 + ((2 &times; (222 − 22)) + 2/2)
:= 33 + ((33 &times; 33 ) − 3/3)
:= 3 + (((33 &times; 33 ) + 33 ) + 3/3)
:= 3 + ((33 &times; 33 ) + 33)
:= 4 + (4 &times; 44 − 444/4)
:= 4 + ((((4 − 444)/4) + 4 &times; 44 ) + 4)
:= (4 &times; (44 − 4)) − (4 − 4/4)4
:= 5 + (((55 + 55 + 5)/5) + 5 &times; 55)
:= 55 + ((55 − 5)/(5 + 5) + 555)
:= (5 &times; (55 + 5)) + (55 + 5 + 5)/5
:= 666 + (6 &times; (6 &times; 6 + 6) − 6/6)
:= 66 + (66 &times; (6 + 6) + ((6 + 6)/6)6 )
:= 66 + ((((6 &times; 6/(6 + 6))6 ) + 66) + 66)
:= (77 &times; (77 + 7)/7) − 7
:= (77 &times; (77 + 7) − (7 + 7))/7
:= (((77 &times; (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) &times; (888/8 − 8)
:= 999 − (9/9 + 9 &times; 9)
:= 9 + (((99 + 99)/9) + 9 &times; 99)
:= 9 + (999 − 9 &times; 9)
71
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 928 := 1 + (1 + ((1 + 11111)/(1 + 11)))
:= 2 &times; ((222 − 22) + 2)
:= 3 + (((33 &times; 33 ) + 33) + 3/3)
&Iuml; 933 := 1 + ((1 + 1) &times; ((1 + 1) &times; (11 + (1 + 1) &times; 111)))
:= 2/2 + (2 &times; (2 &times; 222 + 22))
3
:= 3 &times; 3 + ((33 &times; 3 ) + 33)
4
&Iuml; 938 := 1 + (((1 + 1)11 ) − 1111)
:= 2 + (2 &times; (222 − 22+2 ))
:= 3 + ((3 &times; 333) − ((3/3 + 3)3 ))
:= 4 &times; (44 − ((4 &times; 4 + 4) + 4))
:= 4 + ((4 &times; (4 + 4)) − 444/4)
:= 4 + ((4444 − 4)/4 − (4 &times; 44))
:= ((5 + 5)/5)5 &times; (5 &times; 5 − 5/5 + 5)
:= 55 /5 + ((55 + 5)/(5 + 5) − 5)
:= 55 /5 + (55 + 5)/(5 + 5)
:= ((6 + 6)/6)6 + (6 &times; (6 + 6) &times; (6 + 6))
:= 6 &times; 6 &times; 6 + (((6 &times; 6/(6 + 6))6 ) − (6 + 6))
:= (6 + 6)/6 + ((6 + 6) &times; (66 + 6 + 6))
:= (77/7 &times; (7/7 + 77 + 7)) − 7
:= 7 + (((77 &times; (77 + 7) + 7) + 7)/7)
:= 7 + (7 &times; (77 + 7 &times; 7 + 7))
:= 8 &times; 8 &times; (8 + 8) − (88 + 8)
:= 8 &times; 8 + (888 − (88/8 + 8))
:= (8 + 8)/8 + (8 &times; 8 &times; (8 + 8) − 88)
:= 9 + ((999 − 9 &times; 9) + 9/9)
:= 9 + ((99/((9 + 9 + 9)/9)) + 9 &times; 99)
:= 9 + (((99/9) − 9 &times; 9) + 999)
&Iuml; 929 := ((11 &times; ((1 + (1 + 11))1+1 )) − 1)/(1 + 1)
&Iuml; 934 := ((1 + 1)11 ) − (1 + (1 + (1 + 1111)))
&Iuml; 939 := 1 + (1 + (((1 + 1)11 ) − 1111))
:= 2/2 + (2 &times; 222 + 222 )
:= 2 + (2 &times; (2 &times; 222 + 22))
:= (((2/2 + 2)2 + 22)2 ) − 22
:= 33 + ((33 &times; 33 ) + 33/3)
:= 33+3 + ((3 + 3)3 − 33/3)
:= (33 &times; (33 + 3)) − 33
:= (4 &times; (44 + 4)) − 444/4
:= (4444 − 4)/4 − (4 &times; 44)
:= 4 &times; 44 − ((4 − 4/4)4 + 4)
:= 5 + ((5 &times; (5 − 5 &times; 5)) + (5 − 5/5)5 )
:= (55 &times; (((55 + 5)/5) + 5)) − 5/5
:= (55 + 5)/5 + (55 + 5)/(5 + 5)
:= ((6 + 6) &times; (66 + 6 + 6)) − (6 + 6)/6
:= 6 &times; 6 &times; 6 + (((6 &times; 6/(6 + 6))6 ) − 6)
:= 7 + ((77 &times; (77 + 7) − (7 + 7))/7)
:= (7 &times; (((7 + 7)/7)7 + 7)) − 77/7
:= 7 + (((77 &times; (77 + 7) + 7)/7) + 7)
:= 8/8 + (8 &times; 8 &times; (8 + 8) − (88 + 8))
:= 8 &times; 8 &times; (8 + 8) − ((8 + 8)/8 + 88)
:= 8 &times; 8 + 888 − (88 + 8 + 8)/8
:= 99/9 + (999 − 9 &times; 9)
:= 9 + ((((9 + 9)/9)9/9+9 ) − 99)
:= 99 + (999/9 + 9 &times; 9 &times; 9)
:= 66 + ((6 &times; (6 + 6) &times; (6 + 6)) − 6/6)
&Iuml; 930 := (11 − 1) &times; (((1 + 1)11−1 − 1)/11)
&Iuml; 935 := ((1 + 1)11 ) − (1 + (1 + 1111))
&Iuml; 940 := 1 + (1 + (1 + (((1 + 1)11 ) − 1111)))
:= 2 + (2 &times; 222 + 222 )
:= 22/2 + (2 &times; (222 − 22))
:= 2 &times; ((222 − 22+2 ) + 2)
:= 3 + (((33 &times; 33 ) + 33) + 3)
:= (3 &times; 333) − ((3/3 + 3)3 )
:= 3/3 + ((33 &times; (33 + 3)) − 33)
:= (4 &times; (44 + 4)) + ((4 − 444)/4)
:= 44/4 &times; ((4 − 4/4)4 + 4)
:= (4 &times; (44 − (4 &times; 4 + 4))) − 4
:= 5 + ((5 &times; (55 + 5)) + 55 /5)
:= 55 &times; (((55 + 5)/5) + 5)
:= 5 + (55 &times; (((55 + 5)/5) + 5))
:= 66 + (6 &times; (6 + 6) &times; (6 + 6))
:= ((6 + 6) &times; (66 + 6 + 6)) − 6/6
:= 6 + (((6 + 6) &times; (66 + 6 + 6)) − ((6 + 6)/6))
:= 7 + ((77 &times; (77 + 7) − 7)/7)
:= 77/7 &times; (7/7 + 77 + 7)
:= 7 + ((((77 &times; (77 + 7) + 7) + 7)/7) + 7)
:= (8 − 8/8 + 8) &times; (8 &times; 8 − ((8 + 8)/8))
:= 8 &times; 8 &times; (8 + 8) − (8/8 + 88)
:= 8 + ((88/((8 + 8)/8)) + 888)
:= 9 + (9 &times; (9 &times; 9 + 9) + 999/9)
:= 9 + ((999 − (9/9 + 9 &times; 9)) + 9)
:= 9 &times; 99 + ((9 &times; 99 − 9)/(9 + 9))
&Iuml; 941 := 1111 − (1 + ((1 + (1 + 11))1+1 ))
&Iuml; 931 := (1 + ((11 − 1) &times; (1 + 1)11−1 ))/11
&Iuml; 936 := ((1 + 1)11 ) − (1 + 1111)
:= (2 &times; (2 &times; 222 + 22)) − 2/2
:= 2 &times; (222 − 22+2 )
:= (2 &times; (222 − 2)) − (22 + 2/2)
:= 3 + ((((33 &times; 33 ) + 33) + 3/3) + 3)
:= (3 &times; 3 + 3) &times; (3 &times; 33 − 3)
:= 33+3 + ((3 + 3)3 − (3/3 + 3))
:= 4 + ((4 &times; (44 − 4)) − (4 − 4/4)4 )
:= 4 &times; 44 − (44 + 44)
:= 4/4 + ((4 &times; (44 − (4 &times; 4 + 4))) − 4)
:= 5 + ((5 &times; (55 + 5)) + (55 + 5)/5)
:= 5/5 + (55 &times; (((55 + 5)/5) + 5))
:= 5 + ((55 &times; (((55 + 5)/5) + 5)) + 5/5)
:= 66 + ((6 &times; (6 + 6) &times; (6 + 6)) + 6/6)
:= (6 + 6) &times; (66 + 6 + 6)
:= 6 + (((6 + 6) &times; (66 + 6 + 6)) − 6/6)
:= 7 &times; (77 + 7 &times; 7 + 7)
:= (77 + 7)/7 &times; (7/7 + 77)
:= 7 + ((7 &times; (((7 + 7)/7)7 + 7)) − (77/7))
:= 8 + ((((888 + 88/8) + 8) + 8) + 8)
:= 8 &times; 8 &times; (8 + 8) − 88
:= 8 &times; 8 + (888 − 88/8)
:= 9 &times; 99 + ((9 &times; 9 &times; 9 − 9)/(9 + 9))
:= 9 + ((999 − 9 &times; 9) + 9)
:= 9 &times; 99 + ((9 &times; 99 + 9)/(9 + 9))
&Iuml; 932 := (1 + 1) &times; ((1 + 1) &times; (11 + (1 + 1) &times; 111))
&Iuml; 937 := ((1 + 1)11 ) − 1111
&Iuml; 942 := 1111 − ((1 + (1 + 11))1+1 )
:= 2 &times; (2 &times; 222 + 22)
:= 2/2 + (2 &times; (222 − 22+2 ))
:= (2 &times; (222 − 2)) − 22
:= (3 &times; 333) − (((3/3 + 3)3 ) + 3)
:= 3/3 + ((3 &times; 3 + 3) &times; (3 &times; 33 − 3))
:= 33+3 + ((3 + 3)3 − 3)
:= 4 + (4 &times; (44 − ((4 &times; 4 + 4) + 4)))
:= 4/4 + (4 &times; 44 − (44 + 44))
:= 4 &times; 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 &times; (6 + 6) &times; (6 + 6)) + ((6 + 6)/6))
:= 6/6 + ((6 + 6) &times; (66 + 6 + 6))
:= 6 + ((6 + 6) &times; (66 + 6 + 6))
:= 7 + ((77 &times; (77 + 7) + 7)/7)
:= 7 + (((77 &times; (77 + 7) − 7)/7) + 7)
:= 7 + (77/7 &times; (7/7 + 77 + 7))
:= 888 + (88/((8 + 8)/8))
:= 8/8 + (8 &times; 8 &times; (8 + 8) − 88)
:= 8 &times; 8 + ((8 − 88)/8 + 888)
:= 9 &times; 99 + ((9 &times; 9 &times; 9 + 9)/(9 + 9))
:= 9 + (((999 − 9 &times; 9) + 9/9) + 9)
:= (9 &times; (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
&Iuml; 943 := 1 + (1111 − ((1 + (1 + 11))1+1 ))
&Iuml; 948 := 11 + (((1 + 1)11 ) − 1111)
&Iuml; 953 := 1 + ((1 + 1)1+1+1 &times; (111+1 − (1 + 1)))
:= 2/2 + ((2 &times; (222 − 2)) − 22)
:= 2 + ((2 &times; 222 ) − 22)
:= (2 &times; (222 − 2)) − 22/2
:= 3/3 + (((3 + 3)3 − 3) + 33+3 )
:= 3 + (33+3 + (3 + 3)3 )
:= ((3 + 3) &times; ((3 + 3) &times; 33 − 3)) − 3/3
:= 4 &times; 44 − (4 − 4/4)4
:= 4 + (4 &times; (44 − (4 &times; 4 + 4)))
:= 4 + (4 &times; 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 &times; 55)
:= 6 + (((6 + 6) &times; (66 + 6 + 6)) + 6/6)
:= 6 + (((6 + 6) &times; (66 + 6 + 6)) + 6)
:= 6 + (((6 + 6) &times; (66 + 6 + 6)) + (66/6))
:= 7 + ((77 + 7)/7 &times; (7/7 + 77))
:= (77 + 7)/7 &times; ((7 + 7)/7 + 77)
:= 7 + ((7 &times; (((7 + 7)/7)7 + 7)) + 7/7)
:= 8 &times; 8 + (888 − (8/8 + 8))
:= 8 &times; 8 + (888 − (8/((8 + 8)/8)))
:= 8/8 + (888 + 8 &times; 8)
:= ((99 + 9)/9) &times; (9 &times; 9 − ((9 + 9)/9))
:= (9 &times; (99 + 9)) − (9/9 + 9 + 9)
9/9+9
:= (((9 + 9)/9)
)−9&times;9
&Iuml; 944 := (1 + 1) &times; (((11 + 11)1+1 ) − (1 + 11))
&Iuml; 949 := 1 + (11 + (((1 + 1)11 ) − 1111))
&Iuml; 954 := (11 − 1 − 1) &times; (111 − (1 + (1 + (1 + 1 + 1))))
:= (2 &times; 22 ) − (22 + 2)
:= 2 + (((2 &times; 22 ) − 22) + 2/2)
:= 2 + (2 &times; (222 − 2 &times; (2 + 2)))
:= 33+3 + ((3 + 3)3 − 3/3)
:= 3 + ((33+3 + (3 + 3)3 ) + 3/3)
:= (3 + 3) &times; ((3 + 3) &times; 33 − 3)
:= 4 &times; (44 − (4 &times; 4 + 4))
:= 4 &times; 44 − ((44 + 44 )/4)
:= 4 &times; 44 − ((((44 + 4) + 4)/4) + 4)
:= (5 − 5/5)5 − (5 &times; 5 + 55)
:= (5 − 5/5)5 − (5 &times; (5 + 5 + 5))
:= 5 + ((5 − 5/5)5 − (5 &times; (5 + 5 + 5)))
:= 6 + (((6 + 6) &times; (66 + 6 + 6)) + ((6 + 6)/6))
:= (6/6 + 6 + 6) &times; (66 + 6/6 + 6)
:= 666 + 6 &times; (6 &times; 6 + 6 + 6)
:= (7/7 + 7) &times; (777/7 + 7)
:= 7 + ((77/7 &times; (7/7 + 77 + 7)) + 7)
:= 7 &times; 7 + (((7 + 7)/7)7 + 777)
:= 8 &times; 8 + 888 − 8
:= (8 &times; (8 &times; (8 + 8) − 8)) − 88/8
:= 8 &times; 8 + (888 + ((8 + 8)/8))
:= 999 + ((9 − 999)/(9 + 9))
:= 9999/9 − 9 &times; (9 + 9)
:= (9 &times; (99 + 9)) − (9 + 9)
2
2
&Iuml; 945 := ((1 + 1) &times; (((11 + 11)1+1 ) − 11)) − 1
2
&Iuml; 950 := ((1 + ((11 − 1) &times; (1 + 1 + 1)))1+1 ) − 11
&Iuml; 955 := 1111 − ((1 + 11) &times; (1 + (1 + 11)))
:= (2 &times; 22 ) − (22 + 2/2)
:= (2 &times; (22 + 2)) − 22
:= (2 &times; 222 ) − (22/2 + 2)
:= 33+3 + (3 + 3)3
:= 3 + (((33+3 − 3/3) + (3 + 3)3 ) + 3)
:= (33 &times; 33 ) + ((3/3 + 3)3 )
:= 4/4 + (4 &times; (44 − (4 &times; 4 + 4)))
:= 4 + ((44 &times; ((4 &times; 44) − 4))/(4 + 4))
:= 4 &times; 44 − ((44 + 4)/4 + 4)
:= (5 &times; (5 &times; (55 + 5))) − 555
:= (5 + 5) &times; (5 &times; (5 &times; 5 − 5) − 5)
:= 5 + ((5 + 5) &times; (5 &times; (5 &times; 5 − 5) − 5))
:= (6 &times; 6 − (66/6)) &times; (((6 + 6)/6) + 6 &times; 6)
:= (((6 &times; 6 − 6) + 6/6)(6+6)/6 ) − 6
2
6
:= 6 &times; 6 &times; 6 + ((6 &times; 6/(6 + 6)) )
7
:= 7 &times; (((7 + 7)/7)7 + 7)
:= 77 &times; (7 + 7) − ((7 + 7)/7)
:= 7 + ((77 + 7)/7 &times; ((7 + 7)/7 + 77))
:= 8/8 + ((888 − 8) + 8 &times; 8)
:= 8 &times; 8 + (888 − ((8 + 8)/8))
:= 8 &times; 8 + (888 − 8 + 88/8)
:= (9 &times; (99 + 9)) − (9 + 9 + 9)
:= 999 + ((9 − 9 &times; 99)/(9 + 9))
:= 9/9 + ((9 &times; (99 + 9)) − (9 + 9))
&Iuml; 946 := (1 + 1) &times; (((11 + 11)1+1 ) − 11)
&Iuml; 951 := 1 + (((1 + ((11 − 1) &times; (1 + 1 + 1)))1+1 ) − 11)
&Iuml; 956 := 1111 − (11 + ((1 + 11)1+1 ))
:= (2 &times; 222 ) − 22
:= 2/2 + ((2 &times; (222 + 2)) − 22)
:= 2 &times; (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 &times; ((4 &times; 44) − 4))/(4 + 4)
:= 4 + ((4 &times; 44 − (4 − 4/4)4 ) + 4)
:= (4 &times; (44 − 4 &times; 4)) − 4
:= 55/5 &times; (555/5 − 5 &times; 5)
:= 5/5 + ((5 + 5) &times; (5 &times; (5 &times; 5 − 5) − 5))
:= 55 + ((55 + 5)/5 + 5 &times; 55)
:= 6/6 + (((6 &times; 6/(6 + 6))6 ) + 6 &times; 6 &times; 6)
:= 6 + (((6 &times; 6/(6 + 6))6 ) + 6 &times; 6 &times; 6)
:= 6 + ((6 &times; 6 − (66/6)) &times; (((6 + 6)/6) + 6 &times; 6))
:= 7/7 + (7 &times; (((7 + 7)/7) + 7))
:= 7 + ((7/7 + 7) &times; (777/7 + 7))
:= 77/7 + (7 &times; (((7 + 7)/7)7 + 7))
:= 88/8 &times; (88 − ((8 + 8)/8))
:= 8 &times; 8 + (888 − 8/8)
:= (88 &times; 88 − (88 + 8))/8
:= (9 &times; (99 + 9)) − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + ((9 &times; (99 + 9)) − (9 + 9))
7
9
:= 9 &times; 9 &times; (9 + 9) − (((9 + 9)/9) )
&Iuml; 947 := 1 + ((1 + 1) &times; (((11 + 11)1+1 ) − 11))
&Iuml; 952 := (1 + 1)1+1+1 &times; (111+1 − (1 + 1))
&Iuml; 957 := 11 &times; (111 − ((1 + 1) &times; (1 + 11)))
:= 2/2 + ((2 &times; 22 ) − 22)
:= 2 &times; (22 − 2 &times; (2 + 2))
:= (2 &times; 222 ) − 22/2
:= 3 + ((33+3 − 3/3) + (3 + 3)3 )
:= (33 + 3/3) &times; (3/3 + 33)
:= 33 &times; ((33 − 3/3) + 3)
:= 4 + (4 &times; 44 − (4 − 4/4)4 )
:= (4 &times; (44 − 4 &times; 4)) − 4 − 4
:= 4/4 + ((4 &times; (44 − 4 &times; 4)) − 4)
:= 5 + (((55 − 5)/(5 + 5) + 55 /5) + 5)
:= (5 − 5/5) &times; (((5 − (5 + 5)/5)5 ) − 5)
:= 55/5 &times; (((5 + 5)/5)5 + 55)
:= 66/6 + ((6 + 6) &times; (66 + 6 + 6))
:= 6 &times; 66 + (6666 + 6)/(6 + 6)
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + 6 &times; 6 &times; 6) + 6)
:= ((7 + 7)/7) + (7 &times; (((7 + 7)/7)7 + 7))
:= 7 + (7 &times; (((7 + 7)/7)7 + 7))
:= 7 + (77 &times; (7 + 7) − ((7 + 7)/7)7 )
:= 88/8 + (8 &times; 8 &times; (8 + 8) − 88)
:= 8 &times; 8 + 888
:= 88/8 &times; (88 − 8/8)
:= 9 &times; 99 + ((999 + 9)/(9 + 9))
:= (9 &times; (99 + 9)) − (99/9 + 9)
:= 99/9 &times; (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
&Iuml; 958 := 1 + (11 &times; (111 − ((1 + 1) &times; (1 + 11))))
&Iuml; 963 := 1 + (1 + ((1 + ((11 − 1) &times; (1 + 1 + 1)))1+1 ))
&Iuml; 968 := (1 + 1) &times; ((11 + 11)1+1 )
:= (2 &times; (222 − (2 + 2))) − 2
:= (2 &times; (222 − 2)) − 2/2
:= 2 &times; 222
:= 3 + ((33 &times; 33 ) + ((3/3 + 3)3 ))
:= 3 &times; (333 − (3 &times; 3 + 3))
:= (3/3 + 3) &times; ((33+3 − 3)/3)
4
4
:= 4 &times; 44 − (((44 + 4) + 4)/4)
:= 4 + (4 &times; 4 − (4 + 4)/4)
:= 4 + ((4 &times; (44 − 4 &times; 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 &times; 6 &times; 6 &times; 6 − 666 &times; 6/(6 + 6)
:= (66/6) &times; (((66 + 66)/6) + 66)
:= (77 &times; 77 + 777)/7
:= 7 + ((7 &times; (((7 + 7)/7)7 + 7)) + (77/7))
:= (7/7 + 7) &times; (((7 + 7)/7)7 − 7)
:= (8 &times; (8 &times; (8 + 8) − 8)) − (8 + 8)/8
:= 8 &times; 8 + (888 + 88/8)
:= 88 &times; (88/8)
:= 9 + (9999/9 − 9 &times; (9 + 9))
:= (9 &times; (99 + 9)) − 9
:= 99/9 &times; (99 − (99/9))
&Iuml; 959 := ((1 + 1)11 ) − ((11 &times; (1 + 1 + 1))1+1 )
2
&Iuml; 964 := (1 + 1) &times; (((11 + 11)1+1 ) − (1 + 1))
2
:= 2 + ((2 &times; 22 ) − 22/2)
:= 2/2 + (2 &times; 222 )
:= 2 &times; (22 − 2)
3
&Iuml; 969 := 1 + ((1 + 1) &times; ((11 + 11)1+1 ))
3
:= ((3 &times; 3 + 3/3) ) − (33 + 3)
:= (33 &times; (33 + 3)) − 3
4
:= 4 &times; 4 − (4 + 4)/4
:= 4 + (4 &times; (44 − 4 &times; 4))
:= 4 &times; 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) &times; ((66 − 6/6 + 66) + 6)
:= 6 + ((((6 + 6)/6)(66−6)/6) − 66)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 666/6
:= 77 + 7 &times; (77 + 7 &times; 7)
:= 7777/7 − 7 &times; (7 + 7 + 7)
:= (7 + 7) &times; (77 − 7) − 77/7
:= (8 &times; (8 &times; (8 + 8) − 8)) − 8/8
:= 8 + ((88 &times; 88 − (88 + 8))/8)
:= (88 &times; 88 + 8)/8
:= 999 + ((9 − 9 &times; 9 &times; 9)/(9 + 9))
:= 9/9 + ((9 &times; (99 + 9)) − 9)
:= (9 &times; (99 + 9)) − (9 + 9 + 9)/9
:= ((3 + 3) &times; (33 − 3/3)) − 3/3
4
&Iuml; 960 := (1 + 11) &times; ((11 − 1 − 1)1+1 − 1)
:= 2 &times; (222 − (2 + 2))
:= (33 + 3) &times; (33 − 3/3)
:= 4 &times; (44 − 4 &times; 4)
&Iuml; 965 := ((1 + 1) &times; (((11 + 11)1+1 ) − 1)) − 1
:= 2/2 + (2 &times; (222 − 2))
:= 2 + (2 &times; 222 )
:= (3 &times; 333) − (3/3 + 33)
:= 3 + (((3 &times; 3 + 3/3)3 ) − 33)
:= 4 + ((4 &times; (44 − 4 &times; 4)) + 4/4)
5
&Iuml; 970 := (1 + 1) &times; (1 + ((11 + 11)1+1 ))
:= (44 &times; 44 + 4)/((4 + 4)/4)
:= (5 &times; 5 + 5) &times; ((5 + 5)/5)
:= 5 + ((5 &times; 5 + 5) &times; ((5 + 5)/5) )
:= 5/5 + ((5 − 5/5)5 − 55)
5
:= (6 − 66) &times; (((6 − 66)/6) − 6)
:= 66 + ((6 &times; ((6 + 6) &times; (6 + 6) + 6)) − 6/6)
:= ((6 − 666)/6) + (6 &times; 6 &times; (6 &times; 6 − 6))
:= 7/7 + (7 &times; (77 + 7 &times; 7) + 77)
:= 7 + ((77 &times; 77 + 777)/7)
:= ((7 − 77)/7) + (7 + 7) &times; (77 − 7)
:= 8 &times; (8 &times; (8 + 8) − 8)
:= 8 + (88/8 &times; (88 − 8/8))
:= ((88 &times; 88 + 8) + 8)/8
:= (9 − 9/9) &times; (999/9 + 9)
:= (9 + 9)/9 + ((9 &times; (99 + 9)) − 9)
:= (9 &times; (99 + 9)) − (9 + 9)/9
&Iuml; 961 := (1 + ((11 − 1) &times; (1 + 1 + 1)))1+1
:= ((2/2 + 2)2 + 22)2
:= ((33 + 3/3) + 3)3−3/3
4
:= 4/4 + (4 &times; (4 − 4 &times; 4))
&Iuml; 966 := (1 + 1) &times; (((11 + 11)1+1 ) − 1)
&Iuml; 971 := 1 + ((1 + 1) &times; (1 + ((11 + 11)1+1 )))
:= (2 &times; 222 ) − 2
:= 2 + ((2 &times; 222 ) + 2/2)
:= (3 &times; 333) − 33
:= (33 &times; (33 + 3)) − 3/3
:= (44 &times; 44 − 4)/((4 + 4)/4)
:= 44/4 + (4 &times; (44 − 4 &times; 4))
(5+5)/5
:= 5 + (((5 &times; 5 + 5/5) + 5)(5+5)/5 )
:= ((5 − 5/5)5/5+5 ) − 55
(6+6)/6
:= 66 + (6 &times; ((6 + 6) &times; (6 + 6) + 6))
:= ((6 + 6 + 6) &times; (66 − (6 + 6))) − 6/6
:= (7 + 7) &times; (77 − (7/7 + 7))
:= (7 + 7) &times; (77 − 7) − ((7 + 7)/7 + 7)
:= 8/8 + (8 &times; (8 &times; (8 + 8) − 8))
:= (88 &times; 88 − (8 + 8))/8
:= 88/8 + (8 &times; (8 &times; (8 + 8) − 8))
:= (9 &times; (99 + 9)) − 99/9
:= 999 − (99/((9 + 9 + 9)/9))
:= (9 &times; (99 + 9)) − 9/9
:= ((5 &times; 5 + 5/5) + 5)
:= ((6 &times; 6 − 6) + 6/6)
:= (7 &times; 7 − (77/7 + 7))
(7+7)/7
&Iuml; 962 := 1 + ((1 + ((11 − 1) &times; (1 + 1 + 1)))1+1 )
&Iuml; 967 := 1111 − ((1 + 11)1+1 )
&Iuml; 972 := (1 + 11) &times; (11 − 1 − 1)1+1
:= (2 &times; (222 − 2)) − 2
:= (2 &times; 222 ) − 2/2
:= 2 &times; (222 + 2)
:= (33 &times; 33 ) + (((3 + 3)3 − 3)/3)
:= ((3 &times; 3 + 3/3)3 ) − 33
:= 33 &times; (33 + 3)
:= (4 + 4)/4 + (4 &times; (44 − 4 &times; 4))
:= ((44 &times; (44 + 44)) − 4)/4
:= 4 &times; ((4 − 4/4)4+4/4 )
:= (5 &times; 5 + 5/5) &times; (((5 + 5)/5)5 + 5)
:= (5 − 5/5)5 − ((5 + 5)/5 + 55)
:= (5 − 5/5) &times; ((5 − (5 + 5)/5)5 )
:= 6/6 + (((6 &times; 6 − 6) + 6/6)(6+6)/6 )
:= 6 + (((6 &times; 6 − 6) + 6/6)(6+6)/6 )
:= (6 + 6 + 6) &times; (66 − (6 + 6))
:= (7 + 7) &times; (77 − 7) − (77/7 + 7)
:= 77 &times; (7 + 7) − 777/7
:= (7 + 7) &times; (77 − 7) − (7/7 + 7)
:= (8 + 8)/8 + (8 &times; (8 &times; (8 + 8) − 8))
:= (88 &times; 88 − 8)/8
:= 88 + (888 − (8/((8 + 8)/8)))
:= (9 &times; (99 + 9)) − 9/9 − 9
:= (9 &times; (99 + 9)) + ((9 − 99)/(9 + 9))
:= 9 &times; (99 + 9)
74
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 973 := 1 + ((1 + 11) &times; (11 − 1 − 1)1+1 )
&Iuml; 978 := 1111 − (1 + 11 &times; (1 + 11))
&Iuml; 983 := 1 + 1 + (11 − 1 − 1) &times; (111 − 1 − 1)
2
:= 2/2 + (2 &times; (22 + 2))
:= 2 + (2 &times; (22 + 2 + 2))
:= 22/2 + (2 &times; (222 + 2))
:= 3/3 + (33 &times; (33 + 3))
:= 3 + ((33 &times; (33 + 3)) + 3)
:= 33/3 + (33 &times; (33 + 3))
:= 4 + (4 &times; 44 − (44/4 + 44))
:= 4 &times; 44 − ((4 + 4)/4 + 44)
:= 4 + 4 &times; 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 &times; 5) + 5)
2
:= 6/6 + ((6 + 6 + 6) &times; (66 − (6 + 6)))
:= 6 + ((6 + 6 + 6) &times; (66 − (6 + 6)))
:= 66/6 + ((6 + 6 + 6) &times; (66 − (6 + 6)))
:= (7 + 7) &times; (77 − 7) − 7
:= (7 + 7) &times; (77 − 7) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 + 7) &times; (77 − 7)
:= 8 + ((88/8 &times; (88 − 8/8)) + 8)
:= 8 + (((88 &times; 88 + 8) + 8)/8)
:= 8 + (((88 &times; 88 − 8)/8) + 8)
:= 9/9 + (9 &times; (99 + 9))
:= 999 − (((99 + 9)/9) + 9)
:= 99/9 + (9 &times; (99 + 9))
&Iuml; 974 := 1 + (1 + ((1 + 11) &times; (11 − 1 − 1)1+1 ))
:= 2 + (2 &times; (222 + 2))
3
:= 3 + ((3 &times; (33 + 3)) − 3/3)
&Iuml; 984 := (1 + 11) &times; (1 + (11 − 1 − 1)1+1 )
&Iuml; 979 := 11 &times; (111 − 11 − 11)
:= 22/2 + (2 &times; 222 )
:= 2 &times; (2 &times; (2 + 2) + 222 )
:= (3 &times; (333 − 3)) − 33/3
:= 3 + (3 &times; (333 − (3 + 3)))
4
:= 4 + ((44 &times; 44 + 4)/((4 + 4)/4))
:= 4 &times; 4 − (44 + 4/4)
:= 4 + (4 &times; 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) &times; (66 − (6 + 6)))
:= (6666/6) − (66 + 66)
:= 6 + (((6 + 6 + 6) &times; (66 − (6 + 6))) + 6)
:= 7/7 + ((7 + 7) &times; (77 − 7) − 7)
:= (7 + 7) &times; (77 − 7) − 7/7
:= 77/7 + ((7 + 7) &times; (77 − 7) − 7)
:= 8 + ((88 &times; 88 − (8 + 8))/8)
:= 88/8 &times; (8/8 + 88)
:= 8 + (888 + 88)
:= (9 + 9)/9 + (9 &times; (99 + 9))
:= 999 − (99/9 + 9)
:= (9 &times; (99 + 9)) + (99 + 9)/9
&Iuml; 975 := 111 + (((1 + 11)1+1+1 )/(1 + 1))
&Iuml; 980 := 1 + 11 &times; (111 − 11 − 11)
&Iuml; 985 := 1 + (1 + 11) &times; (1 + (11 − 1 − 1)1+1 )
:= 2 + ((2 &times; (222 + 2)) + 2/2)
:= 2 &times; ((222 + 2 + 2) + 2)
:= 2 + ((2 &times; (222 + 2)) + 22/2)
:= 3 + (33 &times; (33 + 3))
:= (3 &times; (333 − (3 + 3))) − 3/3
:= (3 &times; 333) − (33/3 + 3)
4
4
:= 4 &times; 4 − 44
:= 4 + ((4 &times; 44 − 44) + 4/4)
:= 5 &times; ((5 &times; 5 &times; (5 + 5)) − 55)
:= 5 + (5 &times; ((5 &times; 5 &times; (5 + 5)) − 55))
:= 5 + ((5 &times; ((5 &times; 5 &times; (5 + 5)) − 55)) + 5)
:= 666/6 + (6 &times; (6 + 6) &times; (6 + 6))
:= (6 &times; 6 − 6/6) &times; (((66 + 66)/6) + 6)
:= 6 + ((6666/6) − (66 + 66))
:= 7 + ((7/7 + 7) &times; (((7 + 7)/7) − 7))
:= (7 + 7) &times; (77 − 7)
:= 7 + ((7 + 7) &times; (77 − 7) − ((7 + 7)/7))
:= 8 + ((88 &times; 88 − 8)/8)
:= ((88 &times; 88 + 88) + 8)/8
:= 8 + (((88 &times; 88 + 8)/8) + 8)
:= (9 &times; (99 + 9)) + ((9 + 9 + 9)/9)
:= 9 + ((9 &times; (99 + 9)) − 9/9)
:= 9 &times; 9 &times; 9 + (((9 + 9)/9)9−9/9 )
:= 4 &times; 4 − ((44 + 4/4) + 4)
7
&Iuml; 976 := (1 + 1)1+1+1 &times; (1 + 111+1 )
&Iuml; 986 := (11 − 1)1+1+1 − 1 − 1 − 1 − 11
&Iuml; 981 := (11 − 1 − 1) &times; (111 − 1 − 1)
2
2
:= 2 + ((2 &times; 22 ) + 22/2)
:= 22 + (2 &times; (222 − 2))
3
:= 3 &times; (333 − (3 + 3))
:= (3 &times; (333 − 3)) − (3/3 + 3)
4
:= 4 &times; ((4 − 4 &times; 4) + 4)
:= 4/4 + (4 &times; 44 − 44)
:= 4 + (((4 + 4)/4 − 44) + 4 &times; 44 )
:= 5 + (((5 − 5/5)5/5+5 ) − 55 )
:= (5555/5) − (5 &times; 5 &times; 5 + 5)
:= 2 &times; (22 + 2 + 2)
:= 3 + ((3 &times; (33 + 3)) + 3/3)
:= (5555/5) − 5 &times; 5 &times; 5
6
:= ((6 + 6)/6 + 6) &times; ((666 + 66)/6)
:= 6 &times; (6 &times; 6 + 6) + ((6 &times; 6/(6 + 6)) )
:= ((66/6) + 6) &times; (((6 + 6)/6)6 − 6)
:= 7 + ((7 + 7) &times; (77 − 7) − (77/7))
:= 7/7 + (7 + 7) &times; (77 − 7)
:= 7 + ((7 + 7) &times; (77 − 7) − 7/7)
:= 88 + 888
:= (8/8 + 8) &times; ((888 − (8 + 8))/8)
:= 8 + ((((88 &times; 88 + 8) + 8)/8) + 8)
:= 999 − ((99 + 99 + 9)/9)
:= 9 + (9 &times; (99 + 9))
:= 999 − ((99 + 9 + 9)/9)
&Iuml; 977 := 1 + (1 + 1)1+1+1 &times; (1 + 111+1 )
&Iuml; 982 := 1 + (11 − 1 − 1) &times; (111 − 1 − 1)
&Iuml; 987 := (11 − 1)1+1+1 − 1 − 1 − 11
:= 2/2 + (2 &times; (222 + 2 + 2))
:= 2 + (2 &times; ((222 + 2 + 2) + 2))
:= 22 + ((2 &times; (222 − 2)) + 2/2)
:= 3 + (((33 &times; (33 + 3)) − 3/3) + 3)
:= 3/3 + (3 &times; (333 − (3 + 3)))
:= (3 &times; (333 − 3)) − 3
4
:= 4/4 + (4 &times; ((4 − 4 &times; 4) + 4))
:= (4 + 4)/4 + (4 &times; 4 − 44)
:= (4 &times; (44 − 4 − 4)) − (4/4 + 4)
:= 5 + ((5 − 5/5) &times; ((5 − (5 + 5)/5)5 ))
:= 5 + (((5 − 5/5) &times; ((5 − (5 + 5)/5)5 )) + 5)
:= (5 − 5/5)5 − (((5 + 5)/5)5 + 5)
:= 6 + (((6 + 6 + 6) &times; (66 − (6 + 6))) − 6/6)
:= (((6 + 6)/6)(66−6)/6) − (6 &times; 6 + 6)
:= ((66 &times; (6 &times; 6 − 6)) − 6)/((6 + 6)/6)
:= (7 + 7) &times; (77 − 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + (7 + 7) &times; (77 − 7)
:= 7 + (7 + 7) &times; (77 − 7)
4
:= 8 + ((88 &times; 88 + 8)/8)
:= 8 + (((88 &times; 88 − (8 + 8))/8) + 8)
:= 8 + (88/8 &times; (8/8 + 88))
:= 999 − ((99 + 99)/9)
:= 9 + ((9 &times; (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
&Iuml; 988 := (11 − 1)1+1+1 − 1 − 11
&Iuml; 993 := 1 + 1 + 1 + (11 − 1 − 1) &times; (111 − 1)
&Iuml; 998 := (11 − 1)1+1+1 − 1 − 1
:= 22 + ((2 &times; 222 ) − 2)
:= 2 + (((2 &times; 222 ) + 22) + 2/2)
:= 22 + (2 &times; (222 + 2 + 2))
:= (3 &times; 333) − 33/3
:= 3 + (3 &times; (333 − 3))
:= (3 &times; 333) − 3/3
4
:= (4 &times; (4 − 4 − 4)) − 4
:= 4/4 + (4 &times; (4 − 4 − 4))
:= (4 &times; (44 − 4)) + (4 − 44)/4
:= (5 − 5/5)5 − (55/5 + 5 &times; 5)
:= (5 − 5/5)5 − ((5 &times; 5 + 5/5) + 5)
:= (5 − 5/5)5 − (5 &times; 5 + 5/5)
:= (((6 + 6)/6)(66−6)/6) − 6 &times; 6
:= ((66 &times; (6 &times; 6 − 6)) + 6)/((6 + 6)/6)
:= 666 + (6 &times; 66 − ((6 + 6)/6)6 )
:= 7 + ((7 + 7) &times; (77 − 7) + 7/7)
:= 7 + (((7 + 7) &times; (77 − 7) − 7/7) + 7)
:= 7 + ((7 + 7) &times; (77 − 7) + (77/7))
:= 8 + (((88 &times; 88 + 88) + 8)/8)
:= 8 + ((((88 &times; 88 + 8)/8) + 8) + 8)
:= 888 + ((888 − 8)/8)
:= 999 − 99/9
:= 999 + (((9 + 9 + 9)/9) − 9)
:= 999 − 9/9
&Iuml; 989 := (11 − 1)1+1+1 − 11
&Iuml; 994 := (1 + 1)11−1 − (11 − 1) &times; (1 + 1 + 1)
&Iuml; 999 := 111 &times; (11 − 1 − 1)
4
:= 22 + (2 &times; (222 + 2))
2
:= 22 + ((2 &times; 22 ) − 2/2)
:= (2/2 + 2)2 &times; 222/2
3
:= ((3 &times; 3 + 3/3) ) − (3 + 3)
:= (3 &times; (333 − 3)) − 3/3
:= 3 &times; 333
4
:= 4/4 + ((4 &times; (44 − 4 − 4)) − 4)
:= ((4/4 + 4) + 4) &times; 444/4
:= (4 + 4)/4 + (4 &times; (4 − 4 − 4))
:= (5 − 5/5)5 − ((5 &times; 5 + 5) + 5)
:= (5 − 5/5) − (5 &times; 5 + 5)
:= (5 − 5/5)5 − 5 &times; 5
:= 666 + ((6 &times; (66 − (6 + 6))) − 6/6)
:= (((66 − 6)/6)6&times;6/(6+6) ) − 6
:= (6 &times; 666)/(6 − ((6 + 6)/6))
:= 7 + ((7 + 7) &times; (77 − 7) + ((7 + 7)/7))
:= 7 + ((7 + 7) &times; (77 − 7) + 7)
:= ((7 + 7)/7 + 7) &times; 777/7
:= 888 + (8888/88)
:= 8 + (((((88 &times; 88 + 8) + 8)/8) + 8) + 8)
:= 888 + 888/8
:= 999 − 9/9 − 9
:= 999 + ((9 − 99)/(9 + 9))
:= 999
5
&Iuml; 1000 := (11 − 1)1+1+1
&Iuml; 995 := (11 − 1)1+1+1 − 1 − 1 − 1 − 1 − 1
&Iuml; 990 := (11 − 1 − 1) &times; (111 − 1)
:= 22 + (2 &times; 222 )
:= 22 + ((2 &times; (222 + 2)) + 2/2)
:= 2 &times; (22+2 + 222 )
:= 3 &times; (333 − 3)
:= (3 &times; 333) − (3/3 + 3)
:= (3 &times; 3 + 3/3)3
:= (4 &times; (44 − 4 − 4)) − (4 + 4)/4
:= 4 + ((4 &times; (44 − 4 − 4)) − 4/4)
:= (4 &times; (44 − 4)) − 4 − 4
:= (5 + 5) &times; (5 &times; (5 &times; 5 − 5) − 5/5)
:= (5 &times; (5 + 5) &times; (5 &times; 5 − 5)) − 5
:= 5 &times; (5 + 5) &times; (5 &times; 5 − 5)
:= 6 &times; ((66 &times; (6 &times; 6 − 6))/(6 + 6))
:= 666 + (6 &times; 66 − (66 + 6/6))
:= ((66 − 6)/6)6&times;6/(6+6)
:= 77/7 &times; ((77 − 7/7 + 7) + 7)
:= 7 + (((7 + 7) &times; (77 − 7) + 7/7) + 7)
:= ((77 − 7)/7)(7+7+7)/7
:= 88/8 &times; ((8 + 8)/8 + 88)
:= 8 + ((88/8 &times; (8/8 + 88)) + 8)
:= 8 &times; 8 &times; (8 + 8) − 8 − 8 − 8
:= 999 − 9
:= 999 + ((9 − 9 &times; 9)/(9 + 9))
:= 9/9 + 999
&Iuml; 996 := (1 + 1 + 1) &times; ((1 + 1 + 1) &times; 111 − 1)
&Iuml; 991 := 1 + (11 − 1 − 1) &times; (111 − 1)
2
&Iuml; 1001 := 1 + (11 − 1)1+1+1
:= 22 + ((2 &times; 222 ) + 2/2)
:= 2 + ((2 &times; (22 + 2)) + 22)
:= ((222/2 − 2)/2) − 22
:= 3/3 + (3 &times; (333 − 3))
:= (3 &times; 333) − 3
:= 3 + (3 &times; 333 − 3/3)
:= (4 &times; (4 − 4 − 4)) − 4/4
:= 4 + (4 &times; (44 − 4 − 4))
:= 5 + ((5555/5) − 5 &times; 5 &times; 5)
:= 5/5 + ((5 &times; (5 + 5) &times; (5 &times; 5 − 5)) − 5)
:= 5/5 + 5 &times; (5 + 5) &times; (5 &times; 5 − 5)
:= 6/6 + ((6 &times; (66 − (6 + 6))) + 666)
:= (6 + 6) &times; (66/6 + 66 + 6)
:= (6/6 + 6) &times; ((6 + 6) &times; (6 + 6) − 6/6)
:= 77/7 + (7 + 7) &times; (77 − 7)
:= (77 + 7)/7 &times; (77 − 7/7 + 7)
:= 77 &times; (7 − 7/7 + 7)
:= 888 + (888/8 − 8)
:= 8 &times; 8 &times; 8 + 88 &times; 88/(8 + 8)
:= 8/8 + (8 &times; 8 &times; (8 + 8) − (8 + 8 + 8))
:= 9/9 + (999 − 9)
:= 999 − (9 + 9 + 9)/9
:= (9 + 9)/9 + 999
4
&Iuml; 992 := 1 + 1 + (11 − 1 − 1) &times; (111 − 1)
&Iuml; 997 := (11 − 1)1+1+1 − (1 + 1 + 1)
:= 2 + ((2 &times; 22 ) + 22)
:= ((2/2 + 2)2 &times; 222/2) − 2
:= 3 + ((3 &times; (333 − 3)) − 3/3)
:= ((3 &times; 3 + 3/3)3 ) − 3
2
&Iuml; 1002 := 1 + 1 + (11 − 1)1+1+1
:= 22&times;(2+2)+2 − 22
:= 3 + 3 &times; 333
4
4
:= (4 &times; (4 − 4)) − 44/4
:= 4 &times; (4 − 4 − 4)
5
:= 4 + (4 &times; (44 − 4) − 44/4)
5
5
:= 4 &times; 44 − (44/((4 + 4)/4))
5
:= 5 + ((5 − 5/5) − ((5 + 5)/5) )
:= (5 − 5/5) − ((5 + 5)/5)
6
:= 6 + (((66/6) + 6) &times; (((6 + 6)/6) − 6))
(6+6)/6
:= 6 &times; 6 + (((6 &times; 6 − 6) + 6/6)
(7+7)/7
:= (5 + 5)/5 + 5 &times; (5 + 5) &times; (5 &times; 5 − 5)
)
:= 66 + ((6 + 6) &times; (66 + 6 + 6))
:= (77 + 7)/7 + (7 + 7) &times; (77 − 7)
:= (((77 + 7)
:= 8 + ((888 + 88) + 8)
:= 888 + ((888 − (8 + 8))/8)
:= ((8 + 8)/8) &times; (8 &times; 8 &times; 8 − 88/8)
:= (9 + 9)/9 + (999 − 9)
:= 999 − (9 + 9)/9
:= 999 + ((9 + 9 + 9)/9)
) − 77)/7
76
:= 7/7 + 77 &times; (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
&Iuml; 1003 := 1 + 1 + 1 + (11 − 1)1+1+1
&Iuml; 1008 := (11 − 1 − 1) &times; (1 + 111)
&Iuml; 1013 := (1 + 1)11−1 − 11
:= (222/2 + 2)/2 − 22
:= 2 &times; (222 − 2 + 22)
:= (222/2 − 22)/2
:= 3 + ((3 &times; 3 + 3/3)3 )
:= 3 &times; (333 + 3)
:= 3 + (3 &times; 333 + 33/3)
4
:= 4 &times; (44 − 4) − (4/4 + 4)
:= 4 &times; (4 − 4)
:= 4 &times; 44 − 44/4
:= 5 + ((5 − 5/5)5 − (5 &times; 5 + 5/5))
:= (5 − 5/5)5 − (55/5 + 5)
:= (5 − 5/5)5 − 55/5
:= (66/6 + 6) &times; (66 − (6/6 + 6))
:= 6 &times; ((6 &times; 6 + 66) + 66)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − (66 + 6/6)
:= ((7 + 7)/7) + 77 &times; (7 − 7/7 + 7)
:= 7 + 77 &times; (7 − 7/7 + 7)
:= 7777/7 − 7 &times; (7 + 7)
:= 8 + ((88/8 &times; (8/8 + 88) + 8) + 8)
:= (8 + 8) &times; (8 &times; 8 − 8/8)
:= 8 &times; 8 &times; (8 + 8) − 88/8
:= 999 + ((9 &times; 9 − 9)/(9 + 9))
:= 9 + 999
:= (9999 + 9)/9 − 99
&Iuml; 1004 := 1 + 1 + 1 + 1 + (11 − 1)1+1+1
&Iuml; 1009 := 1 + (11 − 1 − 1) &times; (1 + 111)
&Iuml; 1014 := 1 + (1 + 1)11−1 − 11
:= 2 + 22&times;(2+2)+2 − 22
:= 2/2 + 2 &times; (222 − 2 + 22)
:= 2 + 2 &times; (222 + 22)
:= 3 + ((3 &times; 333 − 3/3) + 3)
:= 3 &times; 3 + ((3 &times; 3 + 3/3)3 )
:= 3 + (3 &times; (333 + 3) + 3)
:= 4 &times; (4 − 4) − 4
:= 4/4 + 4 &times; (4 − 4)
:= 4 &times; 44 + (4 − 44)/4
:= 5 + ((5 − 5/5)5 − 5 &times; 5)
:= (5 − 5/5)5 − (5 + 5 + 5)
:= (5 − 5/5)5 − 5 − 5
:= (6 − ((6 + 6)/6)) &times; (6 &times; (6 &times; 6 + 6) − 6/6)
:= 6/6 + (66 &times; (6 + 6) + 6 &times; 6 &times; 6)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 66
4
4
:= 7 + ((((77 + 7)
(7+7)/7
) − 77)/7)
:= (((77 + 7)
(7+7)/7
) + 7)/7
:= (7 − 7/7 + 7) &times; (7/7 + 77)
:= 8 &times; 8 &times; (8 + 8) − ((88 + 8)/8 + 8)
:= 8/8 + ((8 + 8) &times; (8 &times; 8 − 8/8))
:= (8 − 88)/8 + 8 &times; 8 &times; (8 + 8)
:= 999 + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + (999 + 9/9)
:= 9 + ((999 − ((9 + 9 + 9)/9)) + 9)
&Iuml; 1005 := 1 + 1 + 1 + 1 + 1 + (11 − 1)1+1+1
&Iuml; 1015 := 1 + 1 + (1 + 1)11−1 − 11
&Iuml; 1010 := (11111 − 1)/11
:= (2 + 222/2 + 2)/2 − 22
:= 2 &times; (222 + 22) − 2
:= 2 + (222/2 − 22)/2
:= 3 + (3 &times; 333 + 3)
:= 33/3 + 3 &times; 333
:= 3 + ((3 &times; (333 + 3) + 3/3) + 3)
:= 4/4 + (4 &times; (44 − 4) − 4)
:= (4 + 4)/4 + 4 &times; (44 − 4)
:= 5 + 5 &times; (5 + 5) &times; (5 &times; 5 − 5)
:= 5 + (5 &times; (5 + 5) &times; (5 &times; 5 − 5) + 5)
:= 5 &times; (((55 − 5)/(5 + 5 + 5)) − 5)
:= 6 + ((6 &times; 666)/(6 − ((6 + 6)/6)))
:= 6 &times; 6 &times; 6 + (66 &times; (6 + 6) + ((6 + 6)/6))
:= 6/6 + ((6 &times; 6 &times; (6 &times; 6 − 6)) − 66)
:= 7 + (((7 + 7) &times; (77 − 7) + (77/7)) + 7)
:= ((7 + 7)/7)7 + 7 &times; (77 + 7 &times; 7)
:= 7 &times; 7 &times; (7 + 7 + 7) − (7 + 7)
:= 8 &times; 8 &times; (8 + 8) − (88/8 + 8)
:= 888 + ((888 + 88)/8)
:= 8 &times; 8 &times; (8 + 8) − (8/8 + 8)
:= 9 + (999 − ((9 + 9 + 9)/9))
:= 99/9 + 999
:= (((9 + 9)/9)9/9+9 ) − 9
&Iuml; 1006 := (11 − 1 − 1) &times; (1 + 111) − 1 − 1
&Iuml; 1011 := 11 + (11 − 1)1+1+1
:= 4 &times; 44 − ((4/4 + 4) + 4)
&Iuml; 1016 := 1 + 1 + 1 + (1 + 1)11−1 − 11
:= (2 &times; ((222 − 2) + 22)) − 2
:= 2 &times; (222 + 22) − 2/2
:= 2 &times; (222 + 22 + 2)
:= 3 + (((3 &times; 3 + 3/3)3 ) + 3)
:= 3 + (3 &times; (333 + 3))
:= (3 &times; ((333 + 3) + 3)) − 3/3
:= 4 &times; (4 − 4) − (4 + 4)/4
:= 4 + (4 &times; (4 − 4) − 4/4)
:= 4 &times; 44 − 4 − 4
:= 5 + (5 &times; (5 + 5) &times; (5 &times; 5 − 5) + 5/5)
:= 5 &times; 55 + (555 + 55 )/5
:= (5 + 5)/5 + ((5 − 5/5)5 − (5 + 5))
:= 6 + (((66 − 6)/6)6&times;6/(6+6) )
:= 6 + (((6 &times; 666)/(6 − ((6 + 6)/6))) + 6)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − ((6 + 6)/6)6
:= 7 + (((7 + 7)/7 + 7) &times; 777/7)
:= 7 &times; 7 &times; (7 + 7 + 7) − (77/7 + 7)
:= 7 + ((((77 + 7)(7+7)/7 ) + 7)/7)
:= 8 + ((888 − 8)/8 + 888)
:= 8 &times; 8 &times; (8 + 8) − (88 + 8 + 8)/8
:= 8 &times; 8 &times; (8 + 8) − 8
:= 9 + (999 − ((9 + 9)/9))
:= 999 + (99 + 9)/9
:= 9 + ((999 − 9/9) + 9)
4
&Iuml; 1007 := (11 − 1 − 1) &times; (1 + 111) − 1
:= (2
22/2
− 2)/2 − 2
2+2
:= (3 &times; (333 + 3)) − 3/3
4
&Iuml; 1012 := (1 + 1)11−1 − 1 − 11
&Iuml; 1017 := (11 − 1 − 1) &times; (1 + 1 + 111)
:= 2 &times; (22 + 22)
:= 2 + (222/2 − 22)/2 + 2
:= 3 + (3 &times; (333 + 3) + 3/3)
:= 3 &times; ((333 + 3) + 3)
2
:= 4 &times; (4 − 4) − 4/4
:= 4 + 4 &times; (4 − 4)
:= 4 + (4 &times; 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 &times; 6 &times; 6 + (66 &times; (6 + 6) − 6/6)
:= (((6 + 6)/6)(66−6)/6) − 6 − 6
:= 6 &times; (6 &times; 6 + 6 + 6) + ((6 &times; 6/(6 + 6))6 )
:= (((77 + 7)(7+7)/7 ) − 7)/7
:= 77/7 + 77 &times; (7 − 7/7 + 7)
:= ((7/7 + 7) &times; ((7 + 7)/7)7 ) − 7
:= 8 + (888/8 + 888)
:= 8 &times; 8 &times; (8 + 8) − (88 + 8)/8
:= 8/8 + (8 &times; 8 &times; (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
&Iuml; 1018 := ((1 + 1)11 − (1 + 11))/(1 + 1)
&Iuml; 1023 := (1 + 1)11−1 − 1
&Iuml; 1028 := 1 + (1 + 1 + 1 + (1 + 1)11−1 )
:= 2 + (2 &times; ((222 + 22) + 2))
:= (222/2 − 2)/2
:= 2 + ((22&times;(2+2)+2 ) + 2)
:= 3/3 + (3 &times; ((333 + 3) + 3))
:= 33 + (3 &times; 333 − 3)
:= (3 &times; (((3/3 + 3) + 3)3 )) − 3/3
:= 4 &times; 4 − ((4 + 4)/4 + 4)
:= 4 &times; 44 − 4/4
:= 4 + 4 &times; 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 &times; (66 − 6) + ((6 + 6)/6))
:= 7 &times; 7 &times; (7 + 7 + 7) − 77/7
:= 7/7 + (7 &times; 7 &times; (7 + 7 + 7) − 7)
:= 7 &times; 7 &times; (7 + 7 + 7) − 7/7
:= (8 + 8)/8 + (8 &times; 8 &times; (8 + 8) − 8)
:= 8 &times; 8 &times; (8 + 8) − 8/8
:= 8 &times; 8 &times; (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
&Iuml; 1019 := (1 + (1 + 1)11 − 11)/(1 + 1)
&Iuml; 1029 := (11 + (1 + 1)11 − 1)/(1 + 1)
&Iuml; 1024 := (1 + 1)11−1
:= ((222/2 − 2)/2) − 2 − 2
:= 22&times;(2+2)+2
:= 2 + (((222/2 + 2)/2) + 2)
:= 33/3 + (3 &times; (333 + 3))
:= (3 − 3/3)3&times;3+3/3
:= 3 &times; (((3/3 + 3) + 3)3 )
:= 4 &times; 44 − (4/4 + 4)
:= 4 &times; 44
:= 4 + (4 &times; 44 + 4/4)
:= (5 − 5/5)5 − 5
:= (5 − 5/5)5
:= 5 + (5 − 5/5)5
:= ((66 − 6) &times; (66/6 + 6)) − 6/6
:= ((6 + 6)/6)(66−6)/6
:= 666 + 66 &times; 66/(6 + 6)
:= ((7 − 77)/7) + 7 &times; 7 &times; (7 + 7 + 7)
:= (7/7 + 7) &times; ((7 + 7)/7)7
:= 7 &times; 7 &times; (7 + 7 + 7)
:= 88/8 + ((8 + 8) &times; (8 &times; 8 − 8/8))
:= 9 + (99/9 + 999)
&Iuml; 1020 := (1 + 1)11−1 − 1 − 1 − 1 − 1
2
:= 8 &times; 8 &times; (8 + 8)
:= ((9999 − 9)/9) − 9 &times; 9
:= ((9 + 9)/9)9/9+9
&Iuml; 1030 := (1 + 11 + (1 + 1)11 )/(1 + 1)
&Iuml; 1025 := 1 + (1 + 1)11−1
:= 2 + (((22&times;(2+2)+2 ) + 2) + 2)
:= 2 &times; ((2(2/2+2) ) − 2)
:= (222/2 + 2)/2
:= 3 + (3 &times; ((333 + 3) + 3))
:= 33 + (3 &times; 333 − 3/3)
:= 4 &times; 44 − 4
:= 5/5 + ((5 − 5/5)5 − 5)
:= (66 − 6) &times; (66/6 + 6)
:= 7 + (7777/7 − 7 &times; (7 + 7))
:= 8 &times; 8 &times; (8 + 8) − (8/((8 + 8)/8))
:= 9 + (((99 + 9)/9) + 999)
&Iuml; 1021 := (1 + 1)11−1 − 1 − 1 − 1
:= ((222/2 − 2)/2) − 2
:= 8 + ((8 &times; 8 &times; (8 + 8) − (88/8)) + 8)
:= 4/4 + 4 &times; 44
:= 3 + (((3 &times; 3 + 3/3)3 ) + 33 )
:= 4 + ((4 + 4)/4 + 4 &times; 44 )
:= 5 + ((5 − 5/5)5 + 5/5)
:= 5/5 + (5 − 5/5)5
:= 6/6 + (((6 + 6)/6)(66−6)/6)
:= 7 + (7 &times; 7 &times; (7 + 7 + 7) − (77/7))
:= 8/8 + 8 &times; 8 &times; (8 + 8)
:= 9 + (((999 − 9/9) + 9) + 9)
&Iuml; 1026 := 1 + 1 + (1 + 1)11−1
:= 2 + (22&times;(2+2)+2 )
:= 6 + (((6 + 6)/6)(66−6)/6)
:= 7/7 + 7 &times; 7 &times; (7 + 7 + 7)
:= 8 + (8 &times; 8 &times; (8 + 8) − ((8 + 8)/8))
:= 9999/9 − 9 &times; 9
&Iuml; 1031 := 1 + (1 + 11 + (1 + 1)11 )/(1 + 1)
:= 2 + ((((222/2 + 2)/2) + 2) + 2)
:= 33 + (3 &times; 333 − 3/3)
:= ((3 − 3/3)3&times;3+3/3 ) − 3
:= 3 &times; (333 + 3 &times; 3)
:= 4/4 + (4 &times; 44 − 4)
:= (4 + 4)/4 + 4 &times; 44
:= (5 + 5)/5 + ((5 − 5/5)5 − 5)
:= (5 + 5)/5 + (5 − 5/5)5
:= 6/6 + ((66 − 6) &times; (66/6 + 6))
:= 666 + 6 &times; (66 − 6)
:= 7 &times; 7 &times; (7 + 7 + 7) − (7/7 + 7)
:= ((7 + 7 + 7)/7) &times; (7 &times; 7 &times; 7 − 7/7)
:= 7 + ((7/7 + 7) &times; ((7 + 7)/7)7 )
:= 8 + (8 &times; 8 &times; (8 + 8) − (88/8))
:= (8 + 8)/8 + 8 &times; 8 &times; (8 + 8)
:= 8 + (8 &times; 8 &times; (8 + 8) − 8/8)
:= 9 + (9999/9 − 99)
:= 9 + (999 + 9 + 9)
:= (9999 + 9)/9 − 9 &times; 9
&Iuml; 1022 := (1 + 1)11−1 − 1 − 1
:= 4 + ((4 &times; 44 − 4/4) + 4)
&Iuml; 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)
&Iuml; 1032 := 11 + (1 + 1)11−1 − 1 − 1 − 1
:= (22&times;(2+2)+2 ) − 2
:= 2 + ((222/2 + 2)/2)
:= 2 &times; (2 &times; (22&times;(2+2) + 2))
:= 3 + (3 &times; (333 + 3) + 33/3)
:= 33 + ((3 &times; 3 + 3/3)3 )
:= 33 + 3 &times; 333
:= 4 &times; 4 − (4 + 4)/4
:= 4 + (4 &times; 44 − 4/4)
:= 4 + (4 &times; 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 &times; 6 &times; (6 &times; 6 − 6)) − ((6 + 6)/6)6 )
:= 6/6 + (6 &times; (66 − 6) + 666)
:= 6 + (6 &times; (66 − 6) + 666)
:= 7 &times; 7 &times; (7 + 7 + 7) − 7
:= 7 &times; 7 &times; (7 + 7 + 7) − (7 + 7)/7
:= (7/7 + 7) &times; (((7 + 7)/7)7 + 7/7)
:= 8 &times; 8 &times; (8 + 8) − (8 + 8)/8
:= 88/8 + (8 &times; 8 &times; (8 + 8) − 8)
:= 8 + 8 &times; 8 &times; (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
&Iuml; 1033 := 11 + (1 + 1)11−1 − 1 − 1
&Iuml; 1038 := 1 + 1 + 1 + 11 + (1 + 1)11−1
&Iuml; 1043 := (1 + 1) &times; (11 − 1) + (1 + 1)11−1 − 1
:= ((222/2 + 22)/2) − 2
:= 2 + (2 &times; ((2 &times; (22&times;(2+2) + 2)) + 2))
:= 22 + (((222/2 − 2)/2) − 2)
:= 33 + ((3 &times; 3 + 3/3)3 )
:= 3 &times; ((((3/3 + 3) + 3)3 ) + 3)
:= 33 + (3 &times; 333 + 33/3)
:= 4 + ((4 &times; 44 + 4/4) + 4)
:= 4 &times; (44 + 4) − (4 + 4)/4
:= 4 + (4 &times; (44 + 4) − 4/4)
:= 5 + (((5 − 5/5)5 − 5/5) + 5)
:= 5 + ((((5 − 5/5)5 − 5/5) + 5) + 5)
:= 5 &times; 5 + ((5 − 5/5)5 − (5/5 + 5))
:= (6666/6) − (66 + 6 + 6)
:= (6 &times; (6 &times; (6 &times; 6 − 6) − 6)) − 6
:= (6 &times; (6 &times; (6 &times; 6 − 6) − 6)) − 6/6
:= (7777 − 7)/7 − 77
:= 7 + (((7/7 + 7) &times; ((7 + 7)/7)7 ) + 7)
:= 7 + (7 &times; 7 &times; (7 + 7 + 7) + 7)
:= 8 + (8 &times; 8 &times; (8 + 8) + 8/8)
:= 8 + ((8 &times; 8 &times; (8 + 8) − ((8 + 8)/8)) + 8)
:= 8 + (8 &times; 8 &times; (8 + 8) + (88/8))
:= 9 + (((9999 − 9)/9) − 9 &times; 9)
:= (9 &times; (99 + 9 + 9)) − 9/9 − 9
9/9+9
:= 9 + (((9 + 9)/9)
)
&Iuml; 1034 := 11 + (1 + 1)11−1 − 1
&Iuml; 1039 := 1 + 1 + 1 + 1 + 11 + (1 + 1)11−1
&Iuml; 1044 := ((1 + 1) &times; (11 − 1)) + (1 + 1)11−1
:= 2 + (2 &times; (2 &times; (22&times;(2+2) + 2)))
:= 2 + (((222/2 + 22)/2) + 2)
:= 22 + ((22&times;(2+2)+2 ) − 2)
:= ((33/3)3 ) − 3 &times; 3 &times; 33
:= (3333 − (3 + 3)3 )/3
:= 3 + (3 &times; (333 + 3) + 33)
:= 4 &times; 44 + (44 − 4)/4
:= 4 &times; (44 + 4) − 4/4
:= 4 + 4 &times; (44 + 4)
:= 5 + ((5 − 5/5)5 + 5)
:= 5 + (((5 − 5/5)5 + 5) + 5)
:= 5 &times; 5 + ((5 − 5/5)5 − 5)
:= ((6666 − 66)/6) − 66
:= (6666/6) − (66 + 6)
:= 6 &times; (6 &times; (6 &times; 6 − 6) − 6)
:= 7777/7 − 77
:= ((77 − 7)/7) + 7 &times; 7 &times; (7 + 7 + 7)
:= 7 + ((7 &times; 7 &times; (7 + 7 + 7) + 7/7) + 7)
:= 8 + (8 &times; 8 &times; (8 + 8) + ((8 + 8)/8))
:= 8 + ((8 &times; 8 &times; (8 + 8) − 8/8) + 8)
:= ((88 + 8)/8) &times; (88 − 8/8)
:= 9 + ((((999 − 9/9) + 9) + 9) + 9)
:= 9 + (9999/9 − 9 &times; 9)
:= (9 &times; (99 + 9 + 9)) − 9
&Iuml; 1035 := 11 + (1 + 1)11−1
&Iuml; 1040 := 11 + ((11 + ((1 + 1)11 − 1))/(1 + 1))
&Iuml; 1045 := 11 + (11 + ((1 + 1)11−1 − 1))
:= (222/2 + 22)/2
:= 2 &times; ((2 + 2 + 2)2 + 222 )
:= 22 + ((222/2 − 2)/2)
:= 3 + (3 &times; 333 + 33)
:= 33 + (3 &times; (333 + 3) − 3/3)
:= 33/3 &times; (3 &times; 33 − (3/3 + 3))
:= 4 &times; (4 + 4)
:= 4 + (4 &times; (44 + 4) + 4/4)
:= 55/5 + (5 − 5/5)5
:= 5 &times; ((55 − 5)/(5 + 5 + 5))
:= 55 &times; (5 &times; 5 − (5/5 + 5))
:= 6 + (66 &times; 66/(6 + 6) + 666)
:= (6/6 − 66) &times; (((6 − 66)/6) − 6)
:= (6666/6) − 66
:= 7 + (7 &times; 7 &times; (7 + 7 + 7) − 7/7)
:= 77/7 + 7 &times; 7 &times; (7 + 7 + 7)
:= 77/7 &times; ((77/7 + 77) + 7)
:= 88/8 + 8 &times; 8 &times; (8 + 8)
:= 8 + (8 &times; 8 &times; (8 + 8) + 8)
:= 88/8 &times; (88 − 8/8 + 8)
:= 9 + ((999 + 9 + 9) + 9)
:= 9 + ((9999 + 9)/9 − 9 &times; 9)
:= 9/9 + ((9 &times; (99 + 9 + 9)) − 9)
:= 44/4 + 4 &times; 4
4
&Iuml; 1036 := 1 + 11 + (1 + 1)11−1
4
&Iuml; 1041 := 11 + ((1 + (11 + (1 + 1)11 ))/(1 + 1))
&Iuml; 1046 := 11 + 11 + (1 + 1)11−1
:= 2 &times; ((2 &times; (22&times;(2+2) + 2)) + 2)
:= 22+2 + ((222/2 + 2)/2)
:= 22 + (22&times;(2+2)+2 )
:= 3 + (((3 &times; 3 + 3/3)3 ) + 33)
:= 33 + (3 &times; (333 + 3))
:= 3 + ((3 &times; 333 + 33/3) + 33)
4
:= 4 + (4 &times; (44 + 4) + (4 + 4)/4)
:= 4 &times; (44 + 4) − 4
:= 4/4 + 4 &times; (4 + 4)
:= (5 − 5/5)5 + (55 + 5)/5
:= 5 + ((5 − 5/5)5 + ((55 + 5)/5))
:= 5/5 + (55 &times; (5 &times; 5 − (5/5 + 5)))
:= 6 + ((((6 + 6)/6)(66−6)/6) + 6)
:= (6666/6) − (((6 + 6)/6) + 6)
:= (6666 + 6)/6 − 66
:= 7 + 7 &times; 7 &times; (7 + 7 + 7)
:= 7 + (7777/7 − 77)
:= 7 + (7 &times; 7 &times; (7 + 7 + 7) + ((77 − 7)/7))
:= ((88 + 8)/8) + 8 &times; 8 &times; (8 + 8)
:= 8 + ((8 &times; 8 &times; (8 + 8) + 8/8) + 8)
:= (8888 − 8)/8 − 8 &times; 8
:= 9 + ((((999 + 9/9) + 9) + 9) + 9)
:= (9 &times; (99 + 9 + 9)) − (99 + 9)/9
:= (9 + 9)/9 + ((9 &times; (99 + 9 + 9)) − 9)
6
&Iuml; 1042 := (1 + 1)11−1 + (1 + 1) &times; (11 − 1 − 1)
&Iuml; 1047 := 1 + (11 + 11 + (1 + 1)11−1 )
:= 2 + ((222/2 + 22)/2)
:= 2 + ((22&times;(2+2)+2 ) + 22+2 )
:= 22 + ((222/2 + 2)/2)
:= 3 + (((33/3)3 ) − 3 &times; 3 &times; 33)
:= 3 + ((3333 − (3 + 3)3 )/3)
:= 3 &times; (((((3/3 + 3) + 3)3 ) + 3) + 3)
:= 4/4 + (4 &times; (44 + 4) − 4)
:= (4 + 4)/4 + 4 &times; (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) &times; ((66 − 6) + 6/6)
:= (6 &times; (6 &times; (6 &times; 6 − 6) − 6)) − (6 + 6)/6
:= (6666/6) − ((6 + 6)/6)6
:= 7 + (7 &times; 7 &times; (7 + 7 + 7) + 7/7)
:= 7 + ((7 &times; 7 &times; (7 + 7 + 7) − 7/7) + 7)
:= 7 + (7 &times; 7 &times; (7 + 7 + 7) + (77/7))
:= 8 &times; 8 &times; (8 + 8) + (88 + 8 + 8)/8
:= 8 + ((8 &times; 8 &times; (8 + 8) + ((8 + 8)/8)) + 8)
:= 8888/8 − 8 &times; 8
:= 9 + (((99/9 + 999) + 9) + 9)
:= 9 + ((((9 + 9)/9)9/9+9 ) + 9)
:= 9 + ((((9999 − 9)/9) − 9 &times; 9) + 9)
&Iuml; 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
&Iuml; 1048 := (1 + 1)11 − (11 − 1)1+1+1
:= 2 + ((22&times;(2+2)+2 ) + 22)
:= 3 + (33/3 &times; (3 &times; 33 − (3/3 + 3)))
&Iuml; 1053 := (1 + 1 + 11) &times; (11 − 1 − 1)1+1
:= 2/2 + (2 &times; ((22 &times; (22 + 2)) − 2))
3
:= 3 &times; (33 + 3 + 3)
&Iuml; 1058 := (1 + 1) &times; (1 + 11 + 11)1+1
:= 2 &times; ((22 + 2/2)2 )
:= 3 + (33 &times; 33 − (3/3 + 33))
4
:= 4 + (4 &times; (44 + 4) + 4)
:= 44 + (4 &times; (4 − 4) + 4/4)
:= (4 + 4)/4 + 4 &times; (44 + 4 + 4)
:= 5 &times; 5 + ((5 − 5/5)5 − 5/5)
:= 5 + (((5 − 5/5)5 − 5/5) + 5 &times; 5)
:= (5555 + 5 + 5)/5 − 55
:= 6 + ((6 &times; (6 &times; (6 &times; 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 &times; (7 + 7) − (77/7 + 7 + 7)
:= 7/7 + ((7 + 7) &times; (77 − 7) + 77)
:= 8 + ((8 &times; 8 &times; (8 + 8) + 8) + 8)
:= 8 + (88/8 &times; (88 − 8/8 + 8))
:= ((88 &times; (88 + 8) + 8) + 8)/8
:= 9 + ((9999/9 − 9 &times; 9) + 9)
:= 9 &times; (99 + 9 + 9)
:= ((99/9) &times; (99 − ((9 + 9)/9))) − 9
&Iuml; 1049 := 1 + ((1 + 1)11 − (11 − 1)1+1+1 )
&Iuml; 1054 := 1 + ((1 + 1 + 11) &times; (11 − 1 − 1)1+1 )
&Iuml; 1059 := 1 + ((1 + 1) &times; (1 + 11 + 11)1+1 )
:= 2 + (((222/2 + 2)/2) + 22)
:= 2 &times; (((22 + 2/2)2 ) − 2)
:= ((((2 &times; 22) + 2)2 ) + 2)/2
:= 33 &times; (33 + 3 + 3) − (3/3 + 3)
:= 3/3 + 33 &times; (33 + 3 + 3)
:= 3 + (33 &times; (33 − 3/3))
:= 4 + ((4 &times; (44 + 4) + 4/4) + 4)
:= 4 &times; (44 + 4 + 4) − (4 + 4)/4
:= 5 &times; 5 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 5 &times; 5)
:= 5 + (((5 − 5/5)5 + 5 &times; 5) + 5)
:= 6 + ((6 &times; (6 &times; (6 &times; 6 − 6) − 6)) − 6/6)
:= 6 &times; 6 + ((((6 + 6)/6)(66−6)/6) − 6)
:= 666 + (6 &times; 66 − (6 &times; 6/(6 + 6)))
:= ((7 + 7) &times; (77 − (7 + 7)/7)) − 7/7
:= (7777 − 7)/7 − (7 &times; 7 + 7)
:= ((88 &times; (88 + 8) + 8)/8) − 8
:= (88 &times; (88 + 8) − (8 + 8))/8
:= 999 + ((9 &times; 99 + 9)/(9 + 9))
:= 9/9 + (9 &times; (99 + 9 + 9))
&Iuml; 1050 := (11 − 1) &times; (111 − ((1 + 1) &times; (1 + 1 + 1)))
&Iuml; 1055 := 1111 − (1 + 111)/(1 + 1)
:= 4 + (4 &times; (44 + 4 + 4) − 4/4)
:= 77 &times; (7 + 7) − ((77 + 7)/7 + 7)
:= (((88 &times; (88 + 8) + 8) + 8) + 8)/8
:= 9 + ((9 &times; (99 + 9 + 9)) − ((9 + 9 + 9)/9))
&Iuml; 1060 := (1 + 1) &times; (1 + (1 + 11 + 11)1+1 )
:= 2 + (2 &times; ((22 + 2/2)2 ))
:= 2 &times; (((22 + 2/2)2 ) − (2 + 2))
:= (2 &times; (22 &times; (22 + 2))) − 2/2
:= 33 &times; (33 + 3 + 3) − 3
:= 33 &times; 33 − 3/3 − 33
:= 4 &times; (44 + 4) + (44 − 4)/4
:= 4 &times; (44 + 4 + 4) − 4/4
:= (5 + 5) &times; (5 &times; (5 &times; 5 − 5) + 5)
:= 5 + ((5 + 5) &times; (5 &times; (5 &times; 5 − 5) + 5))
:= 6 + (6 &times; (6 &times; (6 &times; 6 − 6) − 6))
:= 666 + (6 &times; 66 − (6/6 + 6))
:= 6 &times; 6 + (((6 + 6)/6)(66−6)/6)
:= (7 + 7) &times; (77 − (7 + 7)/7)
:= 7777/7 − (7 &times; 7 + 7)
:= 77 &times; (7 + 7) − (77/7 + 7)
:= (((88 &times; (88 + 8) + 8) + 8)/8) − 8
:= (88 &times; (88 + 8) − 8)/8
:= ((88 &times; (8 + 8 + 8)) + 8)/((8 + 8)/8)
:= (9 &times; (99 + 9 + 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 &times; (99 + 9 + 9))
:= 9 + ((9 &times; (99 + 9 + 9)) − ((9 + 9)/9))
&Iuml; 1051 := 1111 − (111+1 − 1)/(1 + 1)
&Iuml; 1056 := (1 + 1) &times; ((1 + 11 + 11)1+1 − 1)
:= 3 + ((33 &times; (33 − 3/3)) + 3/3)
:= 4 + 4 &times; (44 + 4 + 4)
:= (5 + 5) &times; (555/5 − 5)
&Iuml; 1061 := 1 + ((1 + 1) &times; (1 + (1 + 11 + 11)1+1 ))
:= (2 &times; ((22 &times; (22 + 2)) − 2)) − 2/2
:= 2 &times; (22 &times; (22 + 2))
:= 2 + (((((2 &times; 22) + 2)2 ) + 2)/2)
:= 33 + ((3 − 3/3)3&times;3+3/3 )
:= 33 &times; (33 − 3/3)
:= 33+3 + (333 − 3/3)
:= 44/4 + 4 &times; (44 + 4)
:= 4 &times; (44 + 4 + 4)
:= 4 + (4 &times; (44 + 4 + 4) + 4/4)
:= 5555/5 − (55 + 5)
:= 5555/5 − 55
:= 5 + (5555/5 − 55)
:= 6 + ((6666/6) − 66)
:= 66 &times; (((66 − 6)/6) + 6)
:= 666 + (6 &times; 66 − 6/6)
:= 7/7 + ((7 + 7) &times; (77 − (7 + 7)/7))
:= 77 + ((7 + 7) &times; (77 − 7) − 7/7)
:= (7777 − 7)/7 − 7 &times; 7
:= 8 + ((8 &times; 8 &times; (8 + 8) + (88/8)) + 8)
:= 88 &times; (88 + 8)/8
:= 8 + ((88/8 &times; (88 − 8/8 + 8)) + 8)
:= 99/9 &times; (99 − ((9 + 9 + 9)/9))
:= 9 + ((9 &times; (99 + 9 + 9)) − 9/9)
:= (9 &times; (99 + 9 + 9)) − (9 + 9)/9
&Iuml; 1052 := 11 &times; 111 − (1 + 1 + 11)1+1
&Iuml; 1057 := (11 &times; (1 + 1 + 1)) + (1 + 1)11−1
&Iuml; 1062 := (1 + 1) &times; (1 + (1 + (1 + 11 + 11)1+1 ))
:= 2 &times; ((22 &times; (22 + 2)) − 2)
:= ((((2 &times; 22) + 2)2 ) − 2)/2
:= 2 &times; (((22 + 2/2)2 ) + 2)
:= 33 &times; (33 + 3 + 3) − 3/3
:= 3/3 + (33 &times; (33 − 3/3))
:= 33+3 + 333
:= 44 + 4 &times; (44 − 4)
:= 4/4 + 4 &times; (44 + 4 + 4)
:= 4 + (4 &times; (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 &times; 66
:= 7 + (77/7 &times; ((77/7 + 77) + 7))
:= 77 + (7 + 7) &times; (77 − 7)
:= 7777/7 − 7 &times; 7
:= 8 + (((88 + 8)/8) &times; (88 − 8/8))
:= (88 &times; (88 + 8) + 8)/8
:= 8 + ((88 &times; (88 + 8) − (8 + 8))/8)
:= (9 &times; (99 + 9 + 9)) − 9/9
:= 9 + (((9999/9 − 9 &times; 9) + 9) + 9)
:= 9 + (9 &times; (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
&Iuml; 1063 := 1111 − ((1 + 1) &times; ((1 + 1) &times; (1 + 11)))
2
&Iuml; 1068 := (1 + 11) &times; (111 − 11 − 11)
(2/2+2)2
&Iuml; 1073 := 1111 − (1 + (111/(1 + 1 + 1)))
:= ((22/2 + 22)2 ) − 22+2
:= 2 + ((((((2 &times; 22) + 2) ) + 2)/2) + 2)
:= 2 &times; ((2
:= 3/3 + (33+3 + 333)
:= 3 + ((33+3 + 333) + 3)
:= 33+3 + (333 + 33/3)
:= 44 + (4 &times; 44 − (4/4 + 4))
:= 44 + 4 &times; 44
:= 4 + ((4 &times; 44 + 4/4) + 44)
:= 5 &times; 5 &times; 55 + ((5 − 55 )/(5 + 5))
:= 55 + ((5 − 5/5)5 − 55/5)
:= 55 + ((5 − 5/5)5 − (5/5 + 5))
:= 6/6 + (666 + 6 &times; 66)
) + 22)
:= 6 + (666 + 6 &times; 66)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 6/6 − 6
:= 77 &times; (7 + 7) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + 77 &times; (7 + 7)
:= ((7 + 7)/7) + (77 &times; (7 + 7) − 7)
:= 8 + ((88 &times; (88 + 8) − 8)/8)
:= (8/8 + 88) &times; ((88 + 8)/8)
:= 8 + (((88 &times; (88 + 8) + 8)/8) + 8)
:= 9 + ((9 &times; (99 + 9 + 9)) + 9/9)
:= 9 &times; 9 + (999 − ((99 + 9)/9))
:= 9 + ((9 &times; (99 + 9 + 9)) + (99/9))
&Iuml; 1064 := (1 + 1)1+1+1 &times; (1 + (11 &times; (1 + 11)))
&Iuml; 1069 := 1 + ((1 + 11) &times; (111 − 11 − 11))
&Iuml; 1074 := 1111 − (111/(1 + 1 + 1))
:= 2 + (2 &times; (((22 + 2/2) ) + 2))
:= ((((2 &times; 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 &times; ((333 − 3) + 33 ))
:= 44 + (4 &times; 44 − 4)
:= 44 + (4 &times; 44 + 4/4)
:= 4 + (((4 + 4)/4 + 4 &times; 44 ) + 44)
:= 5 + ((((5 − 5/5)5 + 5 &times; 5) + 5) + 5)
:= 55 + ((5 − 5/5)5 − (5 + 5))
:= 55 + ((5 − 5/5)5 − 5)
2
2
:= 666 + (((6 + 6)/6) + 6 &times; 66)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 66/6
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 6
:= (7 + 7) &times; (77 − 7/7)
:= 7 + (7777/7 − 7 &times; 7)
:= 7 + (77 &times; (7 + 7) − (77/7))
:= 8 + (88 &times; ((88 + 8)/8))
:= (8 &times; (8 &times; (8 + 8) + 8)) − (88/8 + 8)
:= 8 + ((((88 &times; (88 + 8) + 8) + 8)/8) + 8)
:= 99/9 + (9 &times; (99 + 9 + 9))
:= 9 &times; 9 + (999 − (99/9))
:= (9 &times; (99 + 9)) + (999/9 − 9)
&Iuml; 1065 := 1111 − ((1 + 1) &times; (1 + (11 + 11)))
:= (2222/2) − ((2 &times; 22) + 2)
:= 3 + (3
3+3
&Iuml; 1070 := (11 − 1) &times; (111 − (1 + 1 + 1 + 1))
:= 2 + ((22&times;(2+2)+2 ) + 2 &times; 22)
3
:= 3 &times; 333 + (((3 + 3) − 3)/3)
+ 333)
4
4
&Iuml; 1075 := 1111 − ((1 + 1 + 1) &times; (1 + 11))
:= 222/2 + (2 &times; (222 − 2))
:= 33 &times; 33 − (33/3 + 3)
:= 44 + ((4 &times; 4 − 4) + 4/4)
:= 44 + ((4 + 4)/4 + 4 &times; 4 )
:= ((44/4)4−4/4 ) − 44
:= 5 + ((5 + 5) &times; (555/5 − 5))
:= (5 + 5) &times; ((555 + 5)/5 − 5)
:= 5 &times; (5 &times; 55 − (55 + 5))
:= 66 + ((6 &times; 666)/(6 − ((6 + 6)/6)))
:= (6 − 6/6) &times; (6 &times; 6 &times; 6 − (6 + 6)/6)
:= (6666/6) − 6 &times; 6
:= 7/7 + ((7 + 7) &times; (77 − 7/7))
:= 77 &times; (7 + 7) − (7/7 + 7)
:= 77 &times; (7 + 7) − (7 + 7 + 7)/7
:= 8 + ((88 &times; (88 + 8) + 8)/8)
:= 8 + (((88 &times; (88 + 8) − (8 + 8))/8) + 8)
:= 8 + (88/8 &times; ((8/8 + 88) + 8))
:= ((99 + 9)/9) + (9 &times; (99 + 9 + 9))
:= 99 + ((9 &times; (99 + 9)) − 9/9)
:= 9999/9 − ((9 + 9 + 9) + 9)
&Iuml; 1066 := (1 + 1 + 11) &times; (1 + (11 − 1 − 1)1+1 )
2
&Iuml; 1071 := (11 − 1 − 1) &times; (111+1 − (1 + 1))
2
&Iuml; 1076 := 1 + (1111 − ((1 + 1 + 1) &times; (1 + 11)))
:= 2 &times; ((((22 + 2/2) ) + 2) + 2)
:= 2 + (((((2 &times; 22) + 2) ) + 22)/2)
:= 2 + ((2 + 2 + 2)2+2 − 222)
:= 3 + ((33+3 + 333) + 3/3)
:= 3 &times; ((333 − 3) + 33 )
:= ((3 − 33)/3) + (33 &times; 33 − 3)
4
:= 44 + (4 &times; 4 − (4 + 4)/4)
:= 4 + ((4 &times; 4 − 4/4) + 44)
:= 4 + ((4 &times; 44 + 44) + 4)
:= 5 + (5555/5 − 55 + 5)
:= 5 + (5555/5 − 55 + 5 + 5)
:= 5/5 + (5 &times; (5 &times; 55 − (55 + 5)))
:= (6/6 + 6) &times; (666/6 + 6 &times; 6 + 6)
:= (6666 + 6)/6 − 6 &times; 6
:= 77 &times; (7 + 7) − 7
:= 77 &times; (7 + 7) − (7 + 7)/7
:= 8 + (((88 &times; (88 + 8) + 8) + 8)/8)
:= (8/8 + 8) &times; (888/8 + 8)
:= 8 + ((8/8 + 88) &times; ((88 + 8)/8))
:= (9/9 + 9 &times; 9) &times; ((99 + 9 + 9)/9)
:= 99 + (9 &times; (99 + 9))
:= 9 + ((99/9) &times; (99 − ((9 + 9)/9)))
4
(66−6)/6)
:= 6 + ((((6 + 6)/6)
+ 6 &times; 6)
:= 77 &times; (7 + 7) − (77 + 7)/7
&Iuml; 1067 := 11 &times; (111 − (1 + (1 + 1 + 11)))
&Iuml; 1072 := 1 + ((11 − 1 − 1) &times; (111+1 − (1 + 1)))
(2/2+2)2
:= (2222/2) − (2 &times; 22)
:= 2 &times; (((2
) + 22) + 2)
:= 33/3 + (33 &times; (33 − 3/3))
:= 33+3 + (((3/3 + 3) + 3)3 )
&Iuml; 1077 := 1111 − (1 + 11 &times; (1 + 1 + 1))
:= 222/2 + ((2 &times; 222 ) − 2)
:= 33 &times; 33 − (3 &times; 3 + 3)
:= 44 + (4 &times; 44 − 4/4)
:= 4 + (4 &times; 4 + 44)
:= 4 &times; 44 + ((44 − 44)/4)
:= ((5555 + 55)/5) − 55
:= 55 − ((((5 + 5)/5)55/5 ) + 5)
:= 55 − (((5 + 5)/5)55/5 )
:= 6 + ((6 &times; 66 − 6/6) + 666)
:= (66 + 6/6) &times; (((66 − 6)/6) + 6)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 6 &times; 6/(6 + 6)
:= 77 &times; (7 + 7) − 77/7
4
:= 7/7 + (77 &times; (7 + 7) − 7)
:= 77 &times; (7 + 7) − 7/7
:= 88/8 &times; ((8/8 + 88) + 8)
:= (8 &times; (8 &times; (8 + 8) + 8)) − 8 − 8
:= (8 &times; (8 &times; (8 + 8) + 8)) − 88/8
:= 99/9 &times; (99 − ((9 + 9)/9))
:= 9/9 + ((999 − 9) + 9 &times; 9)
:= ((99 &times; 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
&Iuml; 1083 := (1 + 1 + 1) &times; ((1 + 1) &times; (11 − 1) − 1)1+1
&Iuml; 1078 := 11 &times; (111 − 1 − 1 − 11)
&Iuml; 1088 := (11 &times; (1 + 1 + 1))1+1 − 1
:= 22 + (2 &times; (22 &times; (22 + 2)))
:= (2/2 + 2) &times; ((22 − (2/2 + 2))2 )
:= ((22/2 + 22)2 ) − 2/2
:= 33 &times; 33 − 33/3
:= 33 &times; 33 − (3 + 3)
:= 33 &times; 33 − 3/3
4
:= 44 + ((44 − 4)/4 + 4 &times; 4 )
:= 44 + (4 &times; (4 + 4) − 4/4)
:= 4 &times; (4 &times; 4 + 44 )
:= 55 + ((5 − 5/5)5 − 5/5)
:= 5 + (((5 − 5/5)5 − 5/5) + 55)
:= (5 + 5)/5 &times; (555 − (55/5))
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + (6 &times; 6 &times; (6 &times; 6 − 6))
:= (66/6 + 6) &times; ((6 + 6)/6)6
:= 77 &times; (7 + 7)
:= 7 + (77 &times; (7 + 7) − ((7 + 7)/7))
:= 77 &times; (7 + 7) + (77 − 7)/7
:= 88/8 &times; (((8 + 8)/8 + 88) + 8)
:= (88/8 + 8) &times; ((8/8 − 8) + 8 &times; 8)
:= 8 &times; (8 &times; (8 + 8) + 8)
:= 99/9 &times; (99 − 9/9)
:= (9 &times; (99 + 9)) + 999/9
:= ((99 &times; 99) − 9)/9
&Iuml; 1079 := 1 + (11 &times; (111 − 1 − 1 − 11))
:= 222/2 + (2 &times; 222 )
4
&Iuml; 1084 := 1111 − ((1 + 1 + 1)1+1+1 )
&Iuml; 1089 := (11 &times; (1 + 1 + 1))1+1
:= 22 + (2 &times; (((22 + 2/2)2 ) + 2))
:= (22/2 + 22)2
:= 3333/3 − 33
:= 33 &times; 33
:= 44 + (44/4 + 4 &times; 4 )
:= 44 + 4 &times; (44 + 4)
:= 4/4 + (4 &times; (4 &times; 4 + 44 ))
:= 55 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 55)
:= 5 + (((5 − 5/5)5 + 55) + 5)
:= (6 &times; 6 &times; (6 &times; 6 − 6)) − 6/6
:= 6 + ((6 &times; 6 &times; (6 &times; 6 − 6)) − ((6 + 6)/6))
:= 66 &times; 66/(6 − ((6 + 6)/6))
:= 7/7 + 77 &times; (7 + 7)
:= 7 + (77 &times; (7 + 7) − 7/7)
:= 77/7 + 77 &times; (7 + 7)
:= (8 &times; (8 &times; (8 + 8) + 8)) − (8/8 + 8)
:= (8 &times; (8 &times; (8 + 8) + 8)) − (8/((8 + 8)/8))
:= 8/8 + (8 &times; (8 &times; (8 + 8) + 8))
:= 9 &times; 9 + (999 − 9/9)
:= 9999/9 − (9 + 9 + 9)
:= 99 &times; (99/9)
:= ((3 − 33)/3) + 33 &times; 33
4
&Iuml; 1080 := (11 − 1) &times; (111 − 1 − 1 − 1)
&Iuml; 1085 := 1111 − ((1 + 1) &times; (1 + 1 + 11))
&Iuml; 1090 := 1 + (11 &times; (1 + 1 + 1))1+1
:= 22 + (2 &times; ((22 + 2/2)2 ))
:= ((22/2 + 22)2 ) − 2 − 2
:= 2/2 + ((22/2 + 22)2 )
:= 3 &times; (333 + 33 )
:= 33 &times; 33 − (3/3 + 3)
:= 3/3 + 33 &times; 33
4
:= 44 + (4 &times; (4 + 4) − 4)
:= 44 + (4 &times; (4 + 4) + 4/4)
:= (4 + 4)/4 + (4 &times; (4 &times; 4 + 44 ))
:= (5 − 5/5) &times; (5 &times; 55 − 5)
:= 5 + ((5 − 5/5) &times; (5 &times; 55 − 5))
:= 55 &times; (5 &times; 5 − 5) − 5 − 5
:= 6 &times; 6 &times; (6 &times; 6 − 6)
:= 6 + ((6 &times; 6 &times; (6 &times; 6 − 6)) − 6/6)
:= 66 + (((6 + 6)/6)(66−6)/6)
:= ((7 + 7)/7) + 77 &times; (7 + 7)
:= 7 + 77 &times; (7 + 7)
:= 77 &times; (7 + 7) + (77 + 7)/7
:= (8 &times; (8 &times; (8 + 8) + 8)) − 8
:= 8 + ((8 &times; (8 &times; (8 + 8) + 8)) − (88/8))
:= (8 + 8)/8 + (8 &times; (8 &times; (8 + 8) + 8))
:= 9 &times; 9 + 999
:= (9999 + 9)/9 − (9 + 9 + 9)
:= ((99 &times; 99) + 9)/9
4
&Iuml; 1081 := 1 + ((11 − 1) &times; (111 − 1 − 1 − 1))
&Iuml; 1086 := 1111 − (1 + ((1 + 1) &times; (1 + 11)))
&Iuml; 1091 := 1 + 1 + (11 &times; (1 + 1 + 1))1+1
:= 2 + (222/2 + (2 &times; 222 ))
:= ((22/2 + 22)2 ) − 2/2 − 2
:= 2 + ((22/2 + 22)2 )
:= 33 &times; 33 − 3
:= 3 + (33 &times; 33 − 3/3)
:= 3 + (33 &times; 33 − 33/3)
4
4
:= 4 + (((4 − 44)/4) + 4 &times; 4 )
4
4
:= 4 &times; 4 + (4 − 4 − 4)/4
:= 4 + (((44 − 4)/4) + 4 &times; 44 )
:= 5555/5 − (5 &times; 5 + 5)
:= 5555/5 − 5 &times; 5
:= 5 + (5555/5 − 5 &times; 5)
:= 6/6 + (6 &times; 6 &times; (6 &times; 6 − 6))
:= 6 + (6 &times; 6 &times; (6 &times; 6 − 6))
:= 66/6 + (6 &times; 6 &times; (6 &times; 6 − 6))
:= 77 &times; (7 + 7) + (7 + 7 + 7)/7
:= 7 + (77 &times; (7 + 7) + 7/7)
:= 7 + ((77 &times; (7 + 7) − 7/7) + 7)
:= 8/8 + ((8 &times; (8 &times; (8 + 8) + 8)) − 8)
:= (8 &times; (8 &times; (8 + 8) + 8)) − (8 + 8)/8
:= 88/8 + ((8 &times; (8 &times; (8 + 8) + 8)) − 8)
:= 9/9 + (999 + 9 &times; 9)
:= ((99 &times; 99) − (9 + 9 + 9))/9
:= (((99 &times; 99) + 9) + 9)/9
&Iuml; 1082 := 1 + (1 + ((11 − 1) &times; (111 − 1 − 1 − 1)))
2
&Iuml; 1087 := 1111 − (1 + 1) &times; (1 + 11)
&Iuml; 1092 := 1 + 1 + 1 + (11 &times; (1 + 1 + 1))1+1
:= 2 + ((2 &times; ((22 + 2/2) )) + 22)
:= ((22/2 + 22)2 ) − 2
:= (22 + 2 + 2) &times; ((2 &times; 22) − 2)
:= 33 &times; 33 − ((3/3 + 3) + 3)
:= 3/3 + (33 &times; 33 − 3)
:= 3 + 33 &times; 33
:= 44 + (4 &times; (44 + 4) − (4 + 4)/4)
:= 4 &times; 44 + ((44 − 4)/4)
:= 4 + (4 &times; (4 &times; 4 + 44 ))
:= 5 + (55 − (((5 + 5)/5)55/5 ))
:= (5555 + 5)/5 − 5 &times; 5
:= 5 + ((5555 + 5)/5 − 5 &times; 5)
:= (6 + 6)/6 + (6 &times; 6 &times; (6 &times; 6 − 6))
:= 6 + ((6 &times; 6 &times; (6 &times; 6 − 6)) + 6/6)
:= 6 + ((6 &times; 6 &times; (6 &times; 6 − 6)) + 6)
:= 77/7 + (77 &times; (7 + 7) − 7)
:= 7 + (77 &times; (7 + 7) + ((7 + 7)/7))
:= 7 + (77 &times; (7 + 7) + 7)
:= (8 + 8)/8 + ((8 &times; (8 &times; (8 + 8) + 8)) − 8)
:= (8 &times; (8 &times; (8 + 8) + 8)) − 8/8
:= ((8888 − 88)/8) − 8
:= 9 &times; 9 + (((9 + 9)/9) + 999)
:= ((99 &times; 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
&Iuml; 1093 := 1111 − (1 + 1) &times; (11 − 1 − 1)
2
&Iuml; 1103 := 1 + (1 + (1 + (1111 − 11)))
&Iuml; 1098 := 1111 − 1 − 1 − 11
:= 2 + (((22/2 + 22) ) + 2)
:= (22 &times; ((2 &times; (22 + 2)) + 2)) − 2
:= (2222/2) − 2 &times; (2 + 2)
:= 3 + (33 &times; 33 + 3/3)
:= 3 &times; (333 + 33)
:= 3 + (33 &times; 33 + 33/3)
:= 4 + ((4444 − 4)/4 − 4 &times; 4)
:= 4444/4 − 4 − 4
4
:= 4 + ((4 &times; (4 &times; 4 + 4 )) + 4/4)
:= 55 &times; (5 &times; 5 − 5) − ((5 + 5)/5 + 5)
:= 55 &times; (5 &times; 5 − 5) − (5 + 5)/5
:= 5 + (55 &times; (5 &times; 5 − 5) − ((5 + 5)/5))
:= (6666/6) − 6 − 6 − 6
:= 666 + (6 &times; (66 + 6))
:= ((6666 − (6 + 6))/6) − 6
:= 7 + ((77 &times; (7 + 7) + 7/7) + 7)
:= (7777 + 7)/7 − (7 + 7)
:= (7777 − 7)/7 − 7
:= ((8888 − (8 + 8))/8) − 8 − 8
:= 8 + ((8 &times; (8 &times; (8 + 8) + 8)) + ((8 + 8)/8))
:= 8888/8 − 8
:= 9999/9 − (9 + 9)
:= 99 + 999
:= (9999 + 9)/9 − 9
&Iuml; 1094 := 1 + (1111 − (1 + 1) &times; (11 − 1 − 1))
&Iuml; 1104 := 1 + (1 + (1 + (1 + (1111 − 11))))
&Iuml; 1099 := 1111 − 1 − 11
:= (2222 − (22 + 2))/2
:= (22 + 2) &times; ((2 &times; 22) + 2)
:= 3 + ((33 &times; 33 − 3/3) + 3)
:= 3 &times; 3 + (33 &times; 33 + 3/3)
:= 3 + (33 &times; 33 + 3 &times; 3 + 3)
:= (4444 − 4)/4 − 4 &times; 4
:= 4 + (4444/4 − 4 &times; 4)
:= 4 &times; ((4 &times; 4 + 44 ) + 4)
:= 55 &times; (5 &times; 5 − 5) − (5/5 + 5)
:= 55 &times; (5 &times; 5 − 5) − 5/5
:= 5 + (55 &times; (5 &times; 5 − 5) − 5/5)
:= ((6666 − 66)/6) − 6
:= (6666/6) − 6 − 6
:= ((6666 − 6)/6) − 6
:= 7 + ((77 &times; (7 + 7) + ((7 + 7)/7)) + 7)
:= 7 + ((77 &times; (7 + 7) + 7) + 7)
:= 7777/7 − 7
:= (8888 − 8)/8 − 8 − 8
:= 88/8 + (8 &times; (8 &times; (8 + 8) + 8))
:= 8 + ((8 &times; (8 &times; (8 + 8) + 8)) + 8)
:= (9999 + 9)/9 − (9 + 9)
:= 9 + (((99 &times; 99) + 9)/9)
:= (((9999 + 9) + 9)/9) − 9
:= 2 + ((22 + 2 + 2) &times; ((2 &times; 22) − 2))
&Iuml; 1095 := 1111 − ((1 + 1)1+1+1+1 )
:= (2222/2) − 2
&Iuml; 1100 := 1111 − 11
2+2
:= 3 + (33 &times; 33 + 3)
&Iuml; 1105 := 1111 − ((1 + 1) &times; (1 + 1 + 1))
:= 22 &times; ((2 &times; (22 + 2)) + 2)
:= (2222/2) − (2 + 2 + 2)
:= 33/3 + 33 &times; 33
:= 3333/3 − (3 + 3)
4
:= 4444/4 − 4 &times; 4
:= 44 + 4 &times; (4 + 4 + 4)
:= 4 &times; 44 + (4 − 4/4)4
:= 55 &times; (5 &times; 5 − 5) − 5
:= 55 &times; (5 &times; 5 − 5)
:= 5 + 55 &times; (5 &times; 5 − 5)
:= 6 + (66 &times; 66/(6 − ((6 + 6)/6)))
:= (6666 − 66)/6
:= (6666/6) − 6
:= 7 + (77 &times; (7 + 7) + ((77 − 7)/7))
:= (7777 − 77)/7
:= (7777 + 7)/7 − 7
:= 8888/8 − 8 − 8
:= (8888 − 88)/8
:= (8/8 + 8 + 8) &times; (8/8 + 8 &times; 8)
:= 9 + (((99 &times; 99) − (9 + 9 + 9))/9)
:= 99/9 &times; (9/9 + 99)
:= 9 &times; 9 + (((9 + 9)/9)9/9+9 )
&Iuml; 1096 := 1111 − (1 + 1 + 1 + 1 + 11)
&Iuml; 1101 := 1 + (1111 − 11)
&Iuml; 1106 := 1111 − 1 − 1 − 1 − 1 − 1
:= 2 &times; (22+2+2 + 222 )
:= ((2222 − 22) + 2)/2
:= 2 + ((22 + 2) &times; ((2 &times; 22) + 2))
:= 3 + ((33 &times; 33 + 3/3) + 3)
:= 3 + (33 &times; 33 + 3 &times; 3)
:= ((3333 + 3)/3) − (3 + 3)
:= 4 + ((4 &times; (4 &times; 4 + 44 )) + 4)
:= 4 &times; 44 + ((4 − 4/4)4 − 4)
:= (4444 − 4)/4 − 4
:= 5/5 + (55 &times; (5 &times; 5 − 5) − 5)
:= 5/5 + 55 &times; (5 &times; 5 − 5)
:= 5555/5 − 5
:= ((6666 − 66) + 6)/6
:= (6666 + 6)/6 − 6
(66−6)/6)
:= 6 + ((((6 + 6)/6)
+ 66)
:= 7 + (77 &times; (7 + 7) + (77/7))
:= (7777 − 77 + 7)/7
:= 77 + 7 &times; 7 &times; (7 + 7 + 7)
:= 8 + (8 &times; (8 &times; (8 + 8) + 8))
:= ((8888 − (8 + 8))/8) − 8
:= ((8888 + 88)/8) − 8 − 8
:= 9 + (((99 &times; 99) − (9 + 9))/9)
:= ((9999 − 9)/9) − 9
:= 9 + ((((99 &times; 99) − 9)/9) + 9)
&Iuml; 1097 := 1111 − 1 − 1 − 1 − 11
&Iuml; 1102 := 1 + (1 + (1111 − 11))
&Iuml; 1107 := 1111 − 1 − 1 − 1 − 1
:= 2 + ((2222/2) − 22+2 )
:= 2 + (22 &times; ((2 &times; (22 + 2)) + 2))
:= (2222/2) − 2 − 2
:= 3 &times; 3 + (33 &times; 33 − 3/3)
:= 3333/3 − 3 &times; 3
:= 3 &times; ((333 + 33) + 3)
:= 4 + (((4 &times; (4 &times; 4 + 4 )) + 4/4) + 4)
:= (4444 − 4)/4 − 4 − 4
:= 4444/4 − 4
:= (5 + 5)/5 + (55 &times; (5 &times; 5 − 5) − 5)
:= (5 + 5)/5 + 55 &times; (5 &times; 5 − 5)
:= (5555 + 5)/5 − 5
4
:= 6 + ((6 &times; 6 &times; (6 &times; 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 &times; (8 &times; (8 + 8) + 8)) + 8/8)
:= (8888 − 8)/8 − 8
:= 8 + ((8 &times; (8 &times; (8 + 8) + 8)) + (88/8))
:= 9 + (((99 &times; 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
&Iuml; 1118 := (11 − 1) &times; (1 + 111) − 1 − 1
&Iuml; 1113 := 1 + 1 + 1111
&Iuml; 1108 := 1111 − 1 − 1 − 1
2
:= 2 &times; (((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 &times; ((4 &times; 4 + 4 ) + 4))
:= ((4444 + 4) + 4)/4
:= 4 + ((4444 − 4)/4 + 4)
:= (5 + 5)/5 &times; (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 &times; 7 &times; 7 + (777 − 7)
:= 7 + 7777/7
:= 8 + ((8888 − 88)/8)
:= ((8888 + 8) + 8)/8
:= 8 + (8888 − 8)/8
:= 9 + ((((99 &times; 99) + 9)/9) + 9)
:= ((9999 + 9) + 9)/9
:= 9 + ((9999 − (9 + 9))/9)
4
&Iuml; 1109 := 1111 − 1 − 1
&Iuml; 1119 := (11 − 1) &times; (1 + 111) − 1
&Iuml; 1114 := 1 + 1 + 1 + 1111
:= (2222/2) − 2
:= 2 + ((2222 + 2)/2)
:= 2 &times; (2 + 2) + (2222/2)
:= ((3333 + 3)/3) − 3
:= 3 + 3333/3
:= 3 + (33 &times; 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) &times; (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)
&Iuml; 1110 := 1111 − 1
&Iuml; 1115 := 1 + 1 + 1 + 1 + 1111
&Iuml; 1120 := (11 − 1) &times; (1 + 111)
:= (2222 − 2)/2
:= 2 + ((2222/2) + 2)
:= 2 &times; (((22 + 2)2 ) − 22+2 )
:= (3333 − 3)/3
:= 3 + ((3333 + 3)/3)
:= 3 &times; 3 + 3333/3
:= (4444 − 4)/4
:= 4 + 4444/4
:= 4 &times; (((4 &times; 4 + 44 ) + 4) + 4)
:= (5 + 5) &times; 555/5
:= 5 + (555 + 555)
:= (5 + 5) &times; (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 &times; 7 &times; 7 + 777
:= (8888 − 8)/8
:= (8/((8 + 8)/8)) + 8888/8
:= 8 + (8 &times; 8 &times; (8 + 8) + 88)
:= (9999 − 9)/9
:= 9 + (((((99 &times; 99) − 9)/9) + 9) + 9)
:= 9 + 9999/9
&Iuml; 1111 := 1111
:= 2222/2
&Iuml; 1116 := 1 + 1 + 1 + 1 + 1 + 1111
:= 2 + (((2222 + 2)/2) + 2)
3
&Iuml; 1121 := 11 + 1111 − 1
:= ((2222 − 2) + 22)/2
:= 3333/3
:= 3 + 33 &times; 33
:= 3 &times; 3 + ((3333 + 3)/3)
:= 4444/4
:= 4 + (4444 + 4)/4
:= 4 &times; (44 + 4) + (4 − 4/4)4
:= 5555/5
:= 5 + 5555/5
:= 5 + (5555/5 + 5)
:= 6666/6
:= 6 &times; (6 &times; (6 &times; 6 − 6) + 6)
:= (66 &times; (66/6 + 6)) − 6/6
:= 7777/7
:= 7 + ((7777 − 7 − 7)/7)
:= 7/7 + (7 &times; 7 &times; 7 + 777)
:= 8888/8
:= 8 + (((8888 − 88)/8) + 8)
:= 8 + (((8888 + 8) + 8)/8)
:= 9999/9
:= 9 + (999 + 99 + 9)
:= 9 + (9999 + 9)/9
&Iuml; 1112 := 1 + 1111
&Iuml; 1117 := 1111 + (1 + 1) &times; (1 + 1 + 1)
&Iuml; 1122 := 11 + 1111
:= (2222 + 2)/2
:= 2 + (((2222/2) + 2) + 2)
:= (2222 + 22)/2
:= (3333 + 3)/3
:= 3 + 3333/3 + 3
:= 33 + 33 &times; 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 &times; (66/6 + 6)
:= (7777 + 7)/7
:= 7 + (7777 − 7)/7
:= (7777 + 77)/7
:= 88 + 8 &times; 8 &times; (8 + 8)
:= 8 + ((8888 − (8 + 8))/8)
:= (8888 + 88)/8
:= (9999 + 9)/9
:= 9 + (((((99 &times; 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
&Iuml; 1128 := (11 − 1) &times; (1 + 1 + 111) − 1 − 1
&Iuml; 1123 := 1 + 11 + 1111
2
&Iuml; 1133 := 11 + 11 + 1111
:= ((2222 + 22) + 2)/2
:= (2 &times; ((22 + 2) )) − (22 + 2)
:= 22 + (2222/2)
:= 3 + (3333/3 + 3 &times; 3)
:= 3 + ((3 &times; 3 + 3 + 3)3 /3)
:= ((33/3)3 ) − 33 &times; (3 + 3)
:= 4 + (4444/4 + 4 + 4)
:= 4 &times; 4 + (4444 + 4)/4
:= 44 + ((4 &times; (4 &times; 4 + 44 )) + 4/4)
:= ((5555 + 55) + 5)/5
:= 5 + (((5555 + 55) + 5)/5)
:= ((5555 + 55) + 55)/5
:= 6 + ((6666/6) + 6)
:= 6 + (66 &times; (66/6 + 6))
:= 66/6 + (66 &times; (66/6 + 6))
:= (7777 + 77 + 7)/7
:= 7/7 + (77 &times; (7 + 7) + 7 &times; 7)
:= 7 + (((7777 + 7)/7 + 7) + 7)
:= ((8888 + 88) + 8)/8
:= 8 &times; (8 &times; 8 + 88) − 88
:= 88/8 &times; (888/8 − 8)
:= 9 + (((9999 − 9)/9) + 9)
:= (9 &times; ((99 + 9 + 9) + 9)) − 9/9
9/9+9
:= 99 + (((9 + 9)/9)
)
&Iuml; 1124 := 1 + 1 + 11 + 1111
&Iuml; 1129 := (11 − 1) &times; (1 + 1 + 111) − 1
:= 2 + ((2222 + 22)/2)
:= 2 + ((2222/2) + 2
3
2+2
)
&Iuml; 1134 := 1 + 11 + 11 + 1111
:= 22 + ((2222 + 2)/2)
:= ((3 &times; 3 + 3 + 3) − 3)/3
:= 3 + (((3 &times; 3 + 3 + 3) + 3)/3)
:= 33 &times; (3 &times; 3 + 33)
:= 4 + (4 &times; (((4 &times; 4 + 44 ) + 4) + 4))
:= 4 &times; 4 + (((4444 + 4) + 4)/4)
:= 4 &times; 44 + (444 − 4)/4
:= (5 − 5/5)5 + 5 &times; (5 &times; 5 − 5)
:= 5 + ((5 − 5/5)5 + 5 &times; (5 &times; 5 − 5))
:= 55 + ((5 − 5/5)5 + 55)
:= 6 + ((66 &times; (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 &times; (7 + 7) + 7 &times; 7)
:= (((8888 + 88) + 8) + 8)/8
:= 8 + ((((8888 + 8) + 8)/8) + 8)
:= 8 + (((8888 − 8)/8 + 8) + 8)
:= ((9 &times; 999) + 9/9)/(9 − 9/9)
:= 9 + (9999/9 + 9)
:= 9 &times; ((99 + 9 + 9) + 9)
&Iuml; 1130 := (11 − 1) &times; (1 + 1 + 111)
&Iuml; 1125 := 1 + 1 + 1 + 11 + 1111
:= 2 + (((2222 + 22) + 2)/2)
3
:= (2 &times; ((22 + 2)2 )) − 22
&Iuml; 1135 := 111 + (1 + 1)11−1
:= 2 + ((2222/2) + 22)
:= (3 &times; 3 + 3 + 3) /3
:= 3 + ((((3 &times; 3 + 3 + 3) − 3)/3) + 3)
:= 33 + (3333/3 − 3)
:= 4 + (4 &times; (44 + 4) + (4 − 4/4)4 )
:= 4 + ((4444 + 44)/4 + 4)
:= 4 &times; 44 + 444/4
3
:= 5 + (5 &times; 5 &times; (55 − 5 − 5))
:= 5 + ((5 &times; 5 &times; (55 − 5 − 5)) + 5)
:= 6 &times; 66 + ((6 &times; 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 &times; (7 + 7) + 7 &times; 7) + 7/7)
:= 8 + (((8888 − (8 + 8))/8) + 8)
:= 8 + ((8888 + 88)/8)
:= 8 + ((8888/8 + 8) + 8)
:= (9 &times; ((99 + 9 + 9) + 9)) − 9
:= 9 + ((9999 + 9)/9 + 9)
:= 9/9 + (9 &times; ((99 + 9 + 9) + 9))
:= 5 &times; 5 &times; (55 − 5 − 5)
6
&Iuml; 1126 := 1 + 1 + 1 + 1 + 11 + 1111
&Iuml; 1131 := 1 + (11 − 1) &times; (1 + 1 + 111)
&Iuml; 1136 := 1 + 111 + (1 + 1)11−1
:= (2 &times; (((22 + 2)2 ) − 2)) − 22
:= 22 + ((2222/2) − 2)
:= 2 &times; (((22 + 2)2 ) − 2 &times; (2 + 2))
:= ((3 &times; 3 + 3 + 3)3 + 3)/3
:= (33 &times; (3 &times; 3 + 33)) − 3
:= ((3 &times; 3 + 3 + 3)3 + 33)/3
:= 4 + (4444 + 44)/4
:= 4 + (4444/4 + 4 &times; 4)
:= 4 &times; ((44 − 4 &times; 4) + 44)
:= 5 + ((5555/5 + 5) + 5)
:= 5 &times; 5 + (5555/5 − 5)
:= 5 &times; 5 + 5555/5
6
:= 6 + (((((6666 + 6) + 6) + 6)/6) + 6)
:= 6 + (((6 &times; 6/(6 + 6)) ) + 6 &times; 66)
:= 6 &times; 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) &times; ((8 &times; 8 − 8/8) + 8)
:= 9/9 + ((9 &times; ((99 + 9 + 9) + 9)) − 9)
:= 9 + ((9999 + 99)/9)
:= (9 + 9)/9 + (9 &times; ((99 + 9 + 9) + 9))
&Iuml; 1127 := 1111 + (1 + 1)1+1+1+1
:= 22+2 + (2222/2)
&Iuml; 1137 := 1 + 1 + 111 + (1 + 1)11−1
&Iuml; 1132 := 11 + 11 + 1111 − 1
:= 2 + ((2 &times; ((22 + 2)2 )) − 22)
3
:= 2 + (((2222/2) + 22) + 2)
:= 3 + (((3 &times; 3 + 3 + 3)3 − 3)/3)
:= 3 + ((3 &times; 3 + 3 + 3) + 3)/3 + 3
:= 3 + (33 &times; (3 &times; 3 + 33))
:= 4 &times; 4 + 4444/4
:= 44 + (4 &times; (4 &times; 4 + 44 ))
:= 44 + ((4/4 + 4)4 + 44 )
:= 5 + ((5555 + 55)/5)
:= 5 + (((5555 + 55)/5) + 5)
:= 5 &times; 5 + (5555 + 5)/5
:= 6 + ((66 &times; (66/6 + 6)) − 6/6)
:= ((66 − 6)/6) + (66 &times; (66/6 + 6))
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + 6 &times; 66) + 6)
:= 7 &times; ((77 + 77) + 7)
:= 7 + ((7777/7 + 7) + 7)
:= (77/7 &times; (777/7 − 7)) − 7
:= 8 + (8888/8 + 8)
:= ((88 &times; (888/8 − 8)) − 8)/8
:= 8/8 + ((8 + 8) &times; ((8 &times; 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
&Iuml; 1138 := 1111 + (1 + 1 + 1)1+1+1
&Iuml; 1143 := 1111 + 11 &times; (1 + 1 + 1) − 1
&Iuml; 1148 := 1111 + (111/(1 + 1 + 1))
:= 2 + (2 &times; (((22 + 2) ) − 2 &times; (2 + 2)))
:= 2 + ((2 &times; ((22 + 2) )) − 22/2)
:= 2 &times; (((22 + 2)2 ) − 2)
:= 33 + 3333/3
:= 3 &times; ((3 &times; (3 &times; 33 + 33 )) + 3)
:= 3 + (((3333 + 3)/3) + 33)
:= 4 + ((444 − 4)/4 + 4 &times; 44 )
:= 4 &times; (4 + 4) + 4444/4
:= 444 + 4 &times; 4 &times; 44
:= 5 &times; 5 + (5555 + 5 + 5)/5
:= ((5 + 5)/5)5 + 5555/5
:= 5 &times; 5 &times; 5 + ((5 − 5/5)5 − 5/5)
:= ((6 + 6)/6)6 + ((6 &times; 6 &times; (6 &times; 6 − 6)) − 6)
:= 6 &times; 6 + ((((6666 + 6) + 6)/6) − 6)
:= 6 &times; 6 + (6666 + 6)/6
:= 7 &times; 7 + (77 &times; (7 + 7) + (77/7))
:= ((77 &times; (777/7 − 7)) − 7)/7
:= 77 + (77 &times; (7 + 7) − 7)
:= 8 + (((8888 + 88)/8) + 8)
:= (8/8 + 8) &times; (8 &times; (8 + 8) − 8/8)
:= 8 &times; 88 + 888/((8 + 8)/8)
:= 9 + ((9999/9 + 9) + 9)
:= 9 + (9 &times; ((99 + 9 + 9) + 9))
:= 9 + ((((9999 + 9)/9 + 9) + 9) + 9)
2
&Iuml; 1139 := (11 − 1) &times; (1 + 1 + 1 + 111) − 1
2
&Iuml; 1144 := 1111 + 11 &times; (1 + 1 + 1)
&Iuml; 1149 := 1 + (1111 + (111/(1 + 1 + 1)))
:= (2 &times; ((22 + 2)2 )) − (22/2 + 2)
:= 2 &times; (22 &times; (22 + 2 + 2))
:= 2/2 + (2 &times; (((22 + 2)2 ) − 2))
:= 33 + ((3333 + 3)/3)
:= 33 + 3333/3
:= 33 + (33 &times; 33 + 33)
:= 4 + (444/4 + 4 &times; 44 )
:= 44 + (444 + 444)
:= 44 + ((4 − 4/4)4 + 4 &times; 44 )
:= ((55 − 555) + 55 )/5
:= 5 &times; 5 &times; 5 + ((5 − 5/5)5 − 5)
:= 5 &times; 5 &times; 5 + (5 − 5/5)5
:= (66/6 + 6) &times; (66 + 6/6)
:= 66/6 &times; (((666 − 6)/6) − 6)
:= 6 &times; 6 + (((6666 + 6) + 6)/6)
:= 7 + (((7777/7 + 7) + 7) + 7)
:= 77/7 &times; (777/7 − 7)
:= 7/7 + ((77 &times; (7 + 7) − 7) + 77)
:= 8 + ((((8888 + 88) + 8)/8) + 8)
:= (8 + 8) &times; (8 &times; 8 + 8) − 8
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) − (88/8))
:= 9 + (((9999 + 9)/9 + 9) + 9)
:= 9 + ((9 &times; ((99 + 9 + 9) + 9)) + 9/9)
:= 9 + ((((9999 + 99)/9) + 9) + 9)
&Iuml; 1140 := (11 − 1) &times; (1 + 1 + 1 + 111)
&Iuml; 1145 := 111+1 + (1 + 1)11−1
&Iuml; 1150 := (1 + 1) &times; ((((1 + 1) &times; (1 + 11))1+1 ) − 1)
:= 2 &times; (((22 + 2)2 ) − (2 + 2 + 2))
:= 2/2 + (2 &times; (22 &times; (22 + 2 + 2)))
:= (2 &times; ((22 + 2)2 )) − 2
:= 3 + ((33 &times; (3 &times; 3 + 33)) + 3)
:= 33 + ((3333 + 3)/3)
:= 3 + ((3333/3 + 33) + 3)
:= 4 + (44 &times; (4 &times; 4 + 4) + 44 )
:= 4 + (((4 + 4) &times; 444 + 4)/4)
:= 44 + ((4444 − 4)/4 − 4)
:= (5 + 5) &times; (5 &times; 5 &times; 5 − (55/5))
:= (5 + 5) &times; (5 &times; 5 &times; 5 − 5) − 55
:= 5 &times; (5 &times; (55 − 5 − 5) + 5)
:= 66 + ((6 &times; 6 &times; (6 &times; 6 − 6)) − 6)
:= 6 + ((66/6 + 6) &times; (66 + 6/6))
:= ((6 &times; 6 − ((6 + 6)/6))(6+6)/6 ) − 6
:= (7/7 + 7 + 7) &times; (77 − 7/7)
:= (((7 + 7)/7 + 7) &times; ((7 + 7)/7)7 ) − 7
:= 7 + (((77 &times; (777/7 − 7)) − 7)/7)
:= ((88 + 8)/8) &times; (88 − 8/8 + 8)
:= 8/8 + ((8 + 8) &times; (8 &times; 8 + 8) − 8)
:= (8 + 8) &times; (8 &times; 8 + 8) − (8 + 8)/8
:= 9 + (((9999 + 99)/9) + 9)
:= 99/9 + (9 &times; ((99 + 9 + 9) + 9))
:= 999 + (9 &times; (9 + 9) − (99/9))
&Iuml; 1141 := 1 + (11 − 1) &times; (1 + 1 + 1 + 111)
4
&Iuml; 1146 := 1 + (111+1 + (1 + 1)11−1 )
&Iuml; 1151 := ((1 + 1) &times; (((1 + 1) &times; (1 + 11))1+1 )) − 1
:= (2 &times; ((22 + 2)2 )) − 22/2
:= (2 &times; (((22 + 2)2 ) − 2)) − 2
:= (2 &times; ((22 + 2)2 )) − 2/2
:= 3 + (3333/3 + 33 )
:= 3333 − (3 &times; 33+3 )
:= 33 + (((3 &times; 3 + 3 + 3)3 − 3)/3)
:= 4 + (((4/4 + 4)4 + 44 ) + 44 )
:= 4 &times; 44 + ((444 + 44)/4)
:= 44 + (4444/4 − 4)
:= 5 + (5555/5 + 5 &times; 5)
:= 5 + ((5555/5 + 5 &times; 5) + 5)
:= 5/5 + (5 &times; (5 &times; (55 − 5 − 5) + 5))
:= 6 &times; 6 + ((6666/6) − 6)
:= 66 + (6 &times; 6 &times; (6 &times; 6 − 6))
:= (66 &times; (6 + 6 + 6)) − (6 &times; 6 + 6/6)
:= 7 + ((77 &times; (7 + 7) + 7 &times; 7) + 7)
:= 7 &times; 7 + (7777/7 − (7 + 7))
:= 7 + (77/7 &times; (777/7 − 7))
:= (8 + 8) &times; (8 &times; 8 + 8) − 88/8
:= (8 + 8)/8 + ((8 + 8) &times; (8 &times; 8 + 8) − 8)
:= (8 + 8) &times; (8 &times; 8 + 8) − 8/8
:= (9 − ((9 + 9)/9)) &times; (9 &times; (9 + 9) + 9/9)
:= 9 + (((((9999 − 9)/9) + 9) + 9) + 9)
:= 99 + ((9 &times; (99 + 9 + 9)) − 9/9)
&Iuml; 1142 := 1 + 1 + (11 − 1) &times; (1 + 1 + 1 + 111)
&Iuml; 1147 := 1111 + ((1 + 1 + 1) &times; (1 + 11))
&Iuml; 1152 := (1 + 1) &times; (((1 + 1) &times; (1 + 11))1+1 )
:= 2 &times; ((22 + 2)2 )
:= (2 &times; (22 &times; (22 + 2 + 2))) − 2
:= (2 &times; (((22 + 2)2 ) − 2)) − 2/2
:= 3 + (((3333 + 3)/3) + 33 )
:= 3 + (3333/3 + 33)
:= 3 &times; ((3 + 3) &times; ((3/3 + 3)3 ))
:= 4 &times; (4 + 4) + (4444 − 4)/4
:= 4 + (4444/4 + 4 &times; (4 + 4))
:= 4 &times; (4 &times; (4 + 4) + 44 )
:= 5 + ((5555 + 5)/5 + 5 &times; 5)
:= 5 &times; 5 + ((5555 + 55)/5)
:= ((5 + 5)/5)5 &times; (55/5 + 5 &times; 5)
:= 6 &times; 6 + ((6666 + 6)/6 − 6)
:= 6 &times; 6 + (6666/6)
:= 6 &times; ((6 &times; (6 &times; 6 − 6) + 6) + 6)
:= 7 + (((77 &times; (7 + 7) + 7 &times; 7) + 7/7) + 7)
:= 7 + ((7/7 + 7 + 7) &times; (77 − 7/7))
:= ((7 + 7)/7 + 7) &times; ((7 + 7)/7)7
:= (8 − 88)/8 + (8 + 8) &times; (8 &times; 8 + 8)
:= 88/8 + ((8 + 8) &times; ((8 &times; 8 − 8/8) + 8))
:= (8 + 8) &times; (8 &times; 8 + 8)
:= 9 + ((9 &times; ((99 + 9 + 9) + 9)) − 9/9)
:= 9 + (((9999/9 + 9) + 9) + 9)
:= 99 + (9 &times; (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
&Iuml; 1153 := 1 + ((1 + 1) &times; (((1 + 1) &times; (1 + 11))1+1 ))
&Iuml; 1158 := 1 + (1 + ((1 + 11 &times; (1 + 1 + 1))1+1 ))
&Iuml; 1163 := 11 + ((1 + 1) &times; (((1 + 1) &times; (1 + 11))1+1 ))
:= 2/2 + (2 &times; ((22 + 2)2 ))
:= 2 + (2 &times; (((22 + 2)2 ) + 2))
:= 22/2 + (2 &times; ((22 + 2)2 ))
:= ((3/3 + 3)3 ) + 33 &times; 33
:= 3 + (33 &times; ((33 − 3/3) + 3))
:= 3 + (3 − 3/3)3&times;3 + 3 &times; (3 + 3)3
:= 4/4 + (4 &times; (4 &times; (4 + 4) + 44 ))
:= 4 + ((4444 − 4)/4 + 44)
:= 4 + ((4444/4 + 44) + 4)
5
:= 5 + (((5 − 5/5)5 − 5/5) + 5 &times; 5 &times; 5)
:= 555 + ((5 − (55 + 55))/5)
:= 55 + ((5 + 5)/5 &times; (555 − 5/5))
:= 6 + ((6666/6) + 6 &times; 6)
:= 6 + (6 &times; ((6 &times; (6 &times; 6 − 6) + 6) + 6))
:= 66/6 + (6 &times; ((6 &times; (6 &times; 6 − 6) + 6) + 6))
:= 7 &times; 7 + (7777/7 − 7)
:= 7 &times; 7 + ((7777 − 7 − 7)/7)
:= 7 + ((77 &times; (7 + 7) + 77) + 7/7)
:= 8/8 + (8 + 8) &times; (8 &times; 8 + 8)
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) − ((8 + 8)/8))
:= 88/8 + (8 + 8) &times; (8 &times; 8 + 8)
:= 9/9 + ((9 &times; (99 + 9 + 9)) + 99)
:= 9 &times; 9 + (((99 &times; 99) − (99 + 9))/9)
:= (9 + 9)/9 + (999 + 9 &times; (9 + 9))
&Iuml; 1154 := (1 + 1) &times; (1 + (((1 + 1) &times; (1 + 11))1+1 ))
&Iuml; 1159 := 1 + (1 + (1 + ((1 + 11 &times; (1 + 1 + 1))1+1 )))
2
&Iuml; 1164 := (1 + 11) &times; (111 − (1 + (1 + 1 + 11)))
:= 2 + (2 &times; ((22 + 2) ))
:= 2 + ((2 &times; (((22 + 2) ) + 2)) + 2/2)
:= 2 &times; (((((22 + 2)2 ) + 2) + 2) + 2)
:= (3 − 3/3) &times; (((3 &times; 3 + 3)3 + 3)/3)
:= 3 + ((3/3 + 33)3−3/3 )
:= 3 + ((33 &times; (33 + 3)) − 33 )
:= 44 + (4444 − 4)/4
:= 4 + (4444/4 + 44)
:= 444 + (4 &times; (4 &times; 44 + 4))
2
5
5
:= 5 + ((5 − 5/5) + 5 &times; 5 &times; 5)
:= 5 + (((5 − 5/5) + 5 &times; 5 &times; 5) + 5)
:= 55 + ((5555 − (5 + 5))/5)
:= 6 + ((6666 + 6)/6 + 6 &times; 6)
:= 6 + (((6666/6) + 6 &times; 6) + 6)
:= 6 &times; 6 &times; 6 &times; 6 − (66 + 66)
:= 77 + (77 &times; (7 + 7) − 7/7)
:= 7 &times; 7 + (7777 − 7)/7
:= (77 + 7)/7 &times; (7 &times; (7 + 7) − 7/7)
:= (8 + 8)/8 + (8 + 8) &times; (8 &times; 8 + 8)
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) − 8/8)
:= 8 &times; 8 + ((8888 − 88)/8)
:= 9 + ((9 &times; ((99 + 9 + 9) + 9)) + (99/9))
:= 9 &times; 9 + ((99/9) &times; (99 − 9/9))
:= 999/9 + (9 &times; (99 + 9 + 9))
&Iuml; 1155 := ((1 + 11 &times; (1 + 1 + 1))1+1 ) − 1
&Iuml; 1160 := (11 − 1) &times; (1 + 1 + 1 + 1 + 1 + 111)
&Iuml; 1165 := 1111 + ((111 − 1)/(1 + 1) − 1)
:= 2 + ((2 &times; ((22 + 2) )) + 2/2)
:= 2 &times; ((((22 + 2) ) + 2) + 2)
:= 2 + ((2 &times; ((22 + 2)2 )) + 22/2)
:= 33 &times; ((33 − 3/3) + 3)
:= 33 &times; 33 + (((3 + 3)3 − 3)/3)
:= 33 + (3333/3 + 33 )
:= 44 + 4444/4
:= 4 + ((4 &times; (4 &times; (4 + 4) + 44 )) + 4)
:= 4 + ((44 − 4/4) &times; (44/4 + 4 &times; 4))
:= 55 + 55 &times; (5 &times; 5 − 5)
:= (5 + 5) &times; (555/5 + 5)
:= 5 + ((5 + 5) &times; (555/5 + 5))
2
2
:= 66/6 &times; (666/6 − 6)
:= 6 + (((6666 + 6)/6 + 6 &times; 6) + 6)
:= 66 + ((6666/6) − (6 + 6))
:= 77 + 77 &times; (7 + 7)
:= 7 &times; 7 + 7777/7
:= (7 + 7) &times; (77 + 7) − 77/7
:= 88/8 + ((8 + 8) &times; (8 &times; 8 + 8) − 8)
:= 8 + (8 + 8) &times; (8 &times; 8 + 8)
:= 8 + ((8/8 + 88) &times; (88 + 8 + 8)/8)
:= 99/9 &times; ((99 − ((9 + 9 + 9)/9)) + 9)
:= 999 + (9 &times; (9 + 9) − 9/9)
:= 9 + (((999/9 − 9)(9+9)/9 )/9)
&Iuml; 1156 := (1 + 11 &times; (1 + 1 + 1))1+1
2
:= 2 &times; (((22 + 2) ) + 2)
&Iuml; 1161 := 1111 + ((11 − 1)1+1 /(1 + 1))
&Iuml; 1166 := 11 &times; (111 − (1 + (1 + 1 + 1 + 1)))
:= 2/2 + (2 &times; ((((22 + 2) ) + 2) + 2))
:= 2 + (2 &times; (((((22 + 2)2 ) + 2) + 2) + 2))
:= (33 &times; (33 + 3)) − 33
:= ((33/3)3 ) − ((3 + 3) &times; 33 + 3)
:= 4 + (4 &times; (4 &times; (4 + 4) + 4 ))
:= (44 − 4/4) &times; (44/4 + 4 &times; 4)
:= (44 &times; (44 − 44))/(4 + 4)
:= 55 + (55 &times; (5 &times; 5 − 5) + 5/5)
:= 55 + (5555/5 − 5)
:= 55 + 5555/5
2
:= (3/3 + 33)3−3/3
4
(6+6)/6
:= (6 &times; 6 − ((6 + 6)/6))
:= 6 + ((66/6) &times; (666/6 − 6))
:= 66 + ((6666 − 66)/6)
:= 7/7 + (77 &times; (7 + 7) + 77)
:= 7 &times; 7 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 7 &times; 7)
:= (8/((8 + 8)/8)) + (8 + 8) &times; (8 &times; 8 + 8)
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) + 8/8)
:= 8 &times; 8 + ((8888 − 8)/8 − 8)
:= ((999/9 − 9)(9+9)/9 )/9
:= 999 + 9 &times; (9 + 9)
:= 99/9 &times; ((99 − ((9 + 9)/9)) + 9)
&Iuml; 1157 := 1 + ((1 + 11 &times; (1 + 1 + 1))1+1 )
&Iuml; 1162 := 111+1+1 − (1 + 1 + 11)1+1
&Iuml; 1167 := 11 + ((1 + 11 &times; (1 + 1 + 1))1+1 )
:= 2/2 + (2 &times; (((22 + 2)2 ) + 2))
:= 2 + (2 &times; ((((22 + 2)2 ) + 2) + 2))
:= 22/2 + (2 &times; (((22 + 2)2 ) + 2))
:= 3/3 + ((3/3 + 33)3−3/3 )
:= 3 + (((3/3 + 33)3−3/3 ) + 3)
:= 33 + (33 &times; (3 &times; 3 + 33))
:= 4 + ((4 &times; (4 &times; (4 + 4) + 44 )) + 4/4)
:= (4 + 4)/4 &times; ((4/4 + 4)4 − 44)
:= 444/4 + 4 &times; (44 + 4 + 4)
:= 5 + (((5 + 5)/5)5 &times; (55/5 + 5 &times; 5))
:= 55 + ((5555 + 5)/5 − 5)
:= 55 + (5555 + 5)/5
:= 6 + ((66 &times; (6 + 6 + 6)) − (6 &times; 6 + 6/6))
(6+6)/6
:= 6 + ((6 &times; 6 − ((6 + 6)/6))
)
:= ((6 &times; (6 &times; 66 − 6)) − 6)/((6 + 6)/6)
:= 77 + (77 &times; (7 + 7) + ((7 + 7)/7))
:= 7 + (77 &times; (7 + 7) + 77)
:= 7 + (7777/7 + 7 &times; 7)
:= (8/8 + 88) &times; (88 + 8 + 8)/8
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) + ((8 + 8)/8))
:= 8 &times; 8 + (8888/8 − 8)
:= (((9 + 9)/9)99/9 ) − 9 &times; 99
:= 9/9 + (999 + 9 &times; (9 + 9))
:= ((99 − 9/9) &times; ((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
&Iuml; 1168 := (1 + 11)1+1 + (1 + 1)11−1
2
&Iuml; 1173 := 1 + (1111 + ((1 + 111+1 )/(1 + 1)))
&Iuml; 1178 := 1 + (11 &times; (111 − (1 + 1 + 1 + 1)))
:= 2 &times; (((22 + 2) ) + 2 &times; (2 + 2))
:= 22 + ((2 &times; ((22 + 2) )) − 2/2)
:= 22 + (2 &times; (((22 + 2)2 ) + 2))
:= 3 + (3/3 + 33)3−3/3 + 3 &times; 3
:= 3 + (33 &times; 33 + 3 &times; 33 )
:= ((3 − 33)/3) + (33 &times; (33 + 3))
:= 4 &times; ((4 &times; (4 + 4) + 44 ) + 4)
:= (4 &times; 4 + 4/4) &times; ((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 &times; 6 − ((6 + 6)/6))(6+6)/6 ) + 6)
:= ((6 &times; (6 &times; 66 − 6)) + 6)/((6 + 6)/6)
:= 66 + (6666 + 6)/6
:= (7 + 7) &times; (77 + 7) − (7/7 + 7)
:= (7 + 7) &times; (77 + 7) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + (7 + 7) &times; (77 + 7)
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) + 8)
:= 8 &times; 8 + ((8888 − (8 + 8))/8)
:= (88/8 + 8) &times; (8 &times; 8 − ((8 + 8)/8))
:= 9 &times; 9 + (((99 &times; 99) − (9 + 9))/9)
:= 9 + ((9 &times; (99 + 9 + 9)) + 999/9)
:= (((99 &times; (99 + 9)) − 9)/9) − 9
&Iuml; 1169 := 1 + ((1 + 11)1+1 + (1 + 1)11−1 )
2
&Iuml; 1174 := (1 + 1) &times; (11 + (((1 + 1) &times; (1 + 11))1+1 ))
&Iuml; 1179 := (11 − 1 − 1) &times; ((11 &times; (1 + 11)) − 1)
:= ((((2 &times; 22) + 2)2 ) + 222)/2
:= 22 + (2 &times; ((22 + 2)2 ))
:= 2 + (22/2 &times; (222/2 − (2 + 2)))
:= ((33/3)3 ) − (3 + 3) &times; 33
:= (33 &times; (33 + 3)) − (33/3 + 3)
:= 3 &times; ((33 &times; (3 &times; 3 + 3)) − 3)
:= 4/4 + (4 &times; ((4 &times; (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) &times; (66 − 6/6)) − 6/6
:= 66 + ((6666 − (6 + 6 + 6))/6)
:= 66 + (((6666 + 6) + 6)/6)
:= (7 + 7) &times; (77 + 7) − 7
:= (7 + 7) &times; (77 + 7) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 + 7) &times; (77 + 7)
:= 8 + (((8 + 8) &times; (8 &times; 8 + 8) + 8/8) + 8)
:= 8 &times; 8 + (8888 − 8)/8
:= 8 + (((8 + 8) &times; (8 &times; 8 + 8) + (88/8)) + 8)
:= 9 &times; 9 + (((99 &times; 99) − 9)/9)
:= 9 &times; 9 + (9999/9 − (9 + 9))
:= 99 + (999 + 9 &times; 9)
&Iuml; 1170 := (11 − 1) &times; (111 + ((1 + 1) &times; (1 + 1 + 1)))
&Iuml; 1175 := 1111 + ((1 + 1)(1+1)&times;(1+1+1) )
&Iuml; 1180 := (11 − 1) &times; (111+1 − (1 + 1 + 1))
:= 22 + (2 &times; (((22 + 2)2 ) − 2))
:= 22 + ((2 &times; ((22 + 2)2 )) + 2/2)
:= 2 + ((2 &times; (((22 + 2)2 ) + 2)) + 22)
:= (3 + 3) &times; (33 &times; (3 + 3) − 3)
:= ((3/3 + 3)3 ) + 3333/3
:= 3 + ((33 &times; (33 + 3)) − 33/3)
:= 4 + ((44 &times; (4 − 44))/(4 + 4))
:= (4444 + 44 )/4
:= (4 &times; ((44 − 4) + 44)) − 4
:= (5 + 5) &times; ((555 + 5)/5 + 5)
:= 55 /5 + (5 + 5) &times; 55
:= 55 /5 + 555
:= (6 + 6 + 6) &times; (66 − 6/6)
:= 66 + ((6666 − (6 + 6))/6)
:= (66 &times; (6 + 6 + 6)) − ((6 + 6)/6 + 6)
:= 7/7 + ((7 + 7) &times; (77 + 7) − 7)
:= (7 + 7) &times; (77 + 7) − 7/7
:= 77 + ((7777 − 7)/7 − 7)
:= (8/8 + 8) &times; (8 &times; (8 + 8) + ((8 + 8)/8))
:= 8 &times; 8 + 8888/8
:= 88 + (((8888 − 88)/8) − 8)
:= 9 + (999 + 9 &times; (9 + 9))
:= (((99 + 9) &times; (99 − 9/9)) − 9)/9
:= (((99 &times; (99 + 9)) + 9)/9) − 9
4
&Iuml; 1171 := 1111 + (111+1 − 1)/(1 + 1)
&Iuml; 1176 := (1 + 11) &times; (111 − 1 − 1 − 11)
&Iuml; 1181 := 1 + (11 − 1) &times; (111+1 − 1 − 1 − 1)
:= 2 + (((((2 &times; 22) + 2)2 ) + 222)/2)
:= 2 + ((2 &times; ((22 + 2)2 )) + 22)
:= (((22/2 + 22) + 2)2 ) − (2 &times; 22)
:= 33 + (3333/3 + 33)
:= (3/3 + 3) &times; (3 &times; 3 &times; 33 − 3)
:= (33 &times; (33 + 3)) − ((3/3 + 3) + 3)
4
4
:= ((4444 + 4 )/4) − 4
:= (4 &times; ((4 − 4) + 44)) − 4 − 4
:= 5 + (5555/5 + 55)
:= 555 + ((55 + 5)/5 − 5)
:= 66 + ((6666/6) − 6)
:= (66 &times; (6 + 6 + 6)) − 6 − 6
:= 7 &times; 7 + ((7777 + 77)/7)
:= (7 + 7) &times; (77 + 7)
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) + (88/8))
:= 88 + (8 &times; (8 &times; (8 + 8) + 8))
:= 9 &times; 9 + (((99 &times; 99) + 9)/9)
&Iuml; 1172 := 1111 + ((1 + 111+1 )/(1 + 1))
:= 4 + (44/4 &times; (444/4 − 4))
:= 555 + (55 + 5)/5
:= (66 &times; (6 + 6 + 6)) − 6/6 − 6
:= 77 + (7777/7 − 7)
:= (99 − 9/9) &times; ((99 + 9)/9)
&Iuml; 1177 := 11 &times; (111 − (1 + 1 + 1 + 1))
:= 8 + (((8888 − (8 + 8))/8) + 8 &times; 8)
:= 9 &times; 9 + ((99/9) &times; (9/9 + 99))
&Iuml; 1182 := 1 + 1 + (11 − 1) &times; (111+1 − 1 − 1 − 1)
:= 22 + ((2 &times; ((22 + 2)2 )) − 2)
:= 22/2 &times; (222/2 − (2 + 2))
:= 22 + (2 &times; ((((22 + 2)2 ) + 2) + 2))
:= 3 + (33/3)3 − (3 + 3) &times; 33
:= (33 &times; (33 + 3)) − 33/3
:= (33 &times; (33 + 3)) − (3 + 3)
:= 4 + (4 &times; ((4 &times; (4 + 4) + 44 ) + 4))
:= 44/4 &times; (444/4 − 4)
:= (4 &times; ((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 &times; (6 + 6 + 6)) − 6
:= 7 + ((7 + 7) &times; (77 + 7) − (77/7))
:= 7/7 + (7 + 7) &times; (77 + 7)
:= 7 + ((7 + 7) &times; (77 + 7) − 7/7)
:= 8 + (((8888 − 88)/8) + 8 &times; 8)
:= 8/8 + ((8 &times; (8 &times; (8 + 8) + 8)) + 88)
:= 8 + ((8888 − 8)/8 + 8 &times; 8)
:= 9 &times; 9 + ((((99 &times; 99) + 9) + 9)/9)
:= 99/9 &times; ((99 − 9/9) + 9)
:= 9 &times; 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
&Iuml; 1183 := 1111 + (1 + 11)1+1 /(1 + 1)
&Iuml; 1188 := 11 &times; (111 − 1 − 1 − 1)
&Iuml; 1193 := 1 + 1111 + (11 − 1 − 1)1+1
:= 222 + (((2/2 + 2)2 + 22)2 )
:= 22 &times; (2 &times; (22 + 2 + 2) + 2)
:= 2 + (((2 &times; 222 ) + 222) + 2/2)
:= (3333 + (3 + 3)3 )/3
:= 33 &times; (33 + 3)
:= 3 + (((33 &times; (33 + 3)) − 3/3) + 3)
4
:= (4 &times; ((44 − 4) + 44)) − 4/4
:= 4 + (4 &times; ((4 − 4) + 44))
:= 4 + (4 &times; (44 + 44 ) − 44/4)
:= 5 + ((55 − 5 − 5)/5 + 555)
:= (55 − 5/5) &times; (55 + 55)/5
:= ((5 + 5)/5 &times; (55 − 5)/5) − 55
:= 6 + ((6666/6) + 66)
:= 66 &times; (6 + 6 + 6)
:= 6 + ((66 &times; (6 + 6 + 6)) − 6/6)
:= 7 + (7 + 7) &times; (77 + 7)
:= 77 + 7777/7
:= 7 + (((7777 − 7 − 7)/7) + 77)
:= 8 + (8888/8 + 8 &times; 8)
:= 88 + ((8888 − 88)/8)
:= 8/8 + ((8 + 8) &times; (88 − 8) − 88)
:= 9 &times; 9 + (9999/9 − 9)
:= 99 &times; ((99 + 9)/9)
:= 9 &times; 9 + (9999 + 9)/9
&Iuml; 1189 := 1 + 11 &times; (111 − 1 − 1 − 1)
&Iuml; 1194 := 11 &times; 111 − ((1 + 1 + 1)1+1+1 )
&Iuml; 1184 := 11 &times; 111 − 111/(1 + 1 + 1)
:= 2 &times; (((22 + 2)2 ) + 22+2 )
:= 222 + ((2 &times; 222 ) − 2/2)
:= 222 + (2 &times; (222 + 2))
:= (33 &times; (33 + 3)) − (3/3 + 3)
:= 3/3 + (33 &times; (33 + 3))
:= 3 + ((33 &times; (33 + 3)) + 3)
4
:= 4 &times; ((4 − 4) + 44)
:= 4 &times; (44 + 4 ) − 44/4
:= 4 &times; (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 &times; (6 + 6 + 6))
:= 6 + (66 &times; (6 + 6 + 6))
:= 7 + ((7 + 7) &times; (77 + 7) + 7/7)
:= 77 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 77)
:= 8 + ((8 &times; (8 &times; (8 + 8) + 8)) + 88)
:= ((88/8 + 8) &times; (8 &times; 8 − 8/8)) − 8
:= 8 + (((8888 + 88)/8) + 8 &times; 8)
:= 9 &times; 9 + ((9999 + 9)/9 − 9)
:= ((99 &times; (99 + 9)) + 9)/9
:= 9 &times; 9 + (((9999 + 9) + 9)/9)
4
&Iuml; 1185 := 11 &times; 111 − (1 + 1 + 1) &times; (1 + 11)
&Iuml; 1190 := (11 − 1) &times; (111+1 − 1 − 1)
2
&Iuml; 1195 := 11 &times; 111 − ((1 + 1) &times; (1 + 1 + 11))
:= 222 + (2 &times; 22 )
:= (2 &times; (((22 + 2)2 ) + 22)) − 2/2
:= 3 + ((33 &times; (33 + 3)) − 3/3)
:= 3 + (((33 &times; (33 + 3)) + 3/3) + 3)
:= 4/4 + (4 &times; ((4 − 4) + 44))
:= (4 − 44)/4 + 4 &times; (44 + 44 )
:= 4 &times; (44 + 44 ) − (4/4 + 4)
:= 5 + (55 /5 + 555)
:= 5 &times; (((5 − (5 + 5)/5)5 ) − 5)
:= 5 &times; 5 &times; 5 &times; (5 + 5) − 55
:= 6 &times; 6 &times; 6 &times; 6 − 666/6
:= (6 + 6)/6 + (66 &times; (6 + 6 + 6))
:= 6 + ((66 &times; (6 + 6 + 6)) + 6/6)
:= 7 + ((7 + 7) &times; (77 + 7) + ((7 + 7)/7))
:= 7 + ((7 + 7) &times; (77 + 7) + 7)
:= 7 + (7777/7 + 77)
:= (8 − 8/8 + 8) &times; (88 − (8/8 + 8))
:= 88 + ((8888 − 8)/8 − 8)
:= 8 + ((((8888 + 88) + 8)/8) + 8 &times; 8)
:= 9 + ((99 − 9/9) &times; ((99 + 9)/9))
:= (((99 &times; (99 + 9)) + 9) + 9)/9
:= 9 + (((99 &times; (99 + 9)) − (9 + 9))/9)
:= (2 + 2 + 2)
2+2
− 222/2
:= (33 &times; (33 + 3)) − 3
4
&Iuml; 1186 := 11 &times; (111 − 1 − 1 − 1) − 1 − 1
&Iuml; 1191 := 1 + (11 − 1) &times; (111+1 − 1 − 1)
2
&Iuml; 1196 := (11 &times; (111 − 1 − 1)) − 1 − 1 − 1
:= 222 + (2 &times; (222 − 2))
:= 2/2 + ((2 &times; 22 ) + 222)
:= 2 &times; (((22 + 2)2 ) + 22)
:= 3/3 + ((33 &times; (33 + 3)) − 3)
:= 3 + (33 &times; (33 + 3))
:= 3 &times; 3 + ((33 &times; (33 + 3)) − 3/3)
4
:= (4 + 4)/4 + (4 &times; ((44 − 4) + 44))
:= 4 &times; 4 + ((4444 + 4 )/4)
:= 4 &times; (44 + 44 ) − 4
:= 5 + ((55 + 5)/5 + 555)
:= 555 + ((55 + 55 )/5)
:= ((55 + 55 + 5)/5) − 55
:= (66 &times; (6 + 6 + 6)) − (6 + 6)/6
:= (6 &times; 6 &times; 66 + 6)/((6 + 6)/6)
:= 6 + ((66 &times; (6 + 6 + 6)) + ((6 + 6)/6))
:= 77 + ((7777 − 7 − 7)/7)
:= 7 + (((7 + 7) &times; (77 + 7) + 7/7) + 7)
:= 7 + ((7777 + 7)/7 + 77)
:= 8 &times; 8 + ((8888 + 88)/8)
:= 88 + (8888/8 − 8)
:= 8 + (((8888 − 88)/8) + 88)
:= ((99 &times; (99 + 9)) − (9 + 9))/9
:= 9 &times; 9 + ((9999 − 9)/9)
:= 9 + (((99 &times; (99 + 9)) − 9)/9)
&Iuml; 1187 := 11 &times; (111 − 1 − 1 − 1) − 1
:= 2 + ((2 + 2 + 2)
2+2
− 222/2)
:= (33 &times; (33 + 3)) − 3/3
&Iuml; 1192 := 1111 + (11 − 1 − 1)1+1
2
&Iuml; 1197 := (11 &times; (111 − 1 − 1)) − 1 − 1
:= 2 + ((2 &times; 22 ) + 222)
:= 2/2 + (2 &times; (((22 + 2)2 ) + 22))
:= 3 + ((33 &times; (33 + 3)) + 3/3)
:= 3 &times; ((33 &times; (3 &times; 3 + 3)) + 3)
4
:= 4 + ((4 &times; ((4 − 4) + 44)) − 4/4)
:= 4 &times; (44 + 4 ) − 4 − 4
:= 4/4 + (4 &times; (44 + 44 ) − 4)
:= 5 + ((55 + 5 + 5)/5 + 555)
:= 555 + ((55 + 55 + 5)/5)
:= ((5 + 5)/5 &times; (55 + 5)/5) − 55
:= (66 &times; (6 + 6 + 6)) − 6/6
:= 6 + ((66 &times; (6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + ((6 &times; 6 &times; 66 + 6)/((6 + 6)/6))
4
:= 77 + (7777 − 7)/7
:= 7 + (((7 + 7) &times; (77 + 7) + (7 + 7)/7) + 7)
:= 7 + (((7 + 7) &times; (77 + 7) + 7) + 7)
:= 8 &times; 8 + (((8888 + 88) + 8)/8)
:= (8 + 8) &times; (88 − 8) − 88
:= (88/8 + 8) &times; (8 &times; 8 − 8/8)
:= ((99 &times; (99 + 9)) − 9)/9
:= 9 &times; 9 + 9999/9
:= 9 + (99 &times; ((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
&Iuml; 1198 := (11 &times; (111 − 1 − 1)) − 1
&Iuml; 1203 := 1 + (1 + (1 + (1 + (11 &times; (111 − 1 − 1)))))
&Iuml; 1208 := (11 &times; (111 − 1)) − 1 − 1
:= 2 + (2 &times; (((22 + 2) ) + 22))
:= (2/2 + 2) &times; ((22 − 2) + 2/2)
:= 2 + ((2/2 + 2) &times; ((22 − 2)2 + 2))
:= 3 &times; 3 + ((33 &times; (33 + 3)) + 3/3)
:= ((3 + 3) &times; (33 &times; (3 + 3) + 3)) − 3
:= 3 + (((33/3)3 ) − (3 &times; 33 + 33 ))
2
4
2
4
:= 4 &times; (44 + 4 ) − (4 + 4)/4
:= 4 + (4 &times; (44 + 4 ) − 4/4)
:= 4 + (4 &times; (44 + 44 ) + 4)
:= (5 + 5)/5 &times; ((55 − 5)/5 − 5 &times; 5)
:= 5 + ((5 + 5)/5 &times; ((55 − 5)/5 − 5 &times; 5))
:= (5 + 5)/5 &times; ((55 &times; 55 − 5)/5)
:= ((66 − 6)/6) + (66 &times; (6 + 6 + 6))
:= ((6 &times; (6 &times; 66 + 6)) − 6)/((6 + 6)/6)
:= (6 + 6)/6 + ((6 + 6 + 6) &times; (66 + 6/6))
:= 7 + ((((7 + 7) &times; (77 + 7) + 7/7) + 7) + 7)
:= (77/7 &times; (777 − 7)/7) − 7
:= 7 &times; (7 + 7) + (7777 − 7)/7
:= 88 + (8888 − 8)/8
:= 88/8 + ((8 + 8) &times; (88 − 8) − 88)
:= 8 &times; (8 &times; 8 + 88) − 8
:= 9 + (((99 &times; (99 + 9)) + 9)/9)
:= 9 &times; 9 + ((9999 + 99)/9)
:= 9 + ((99/9) &times; (9/9 + 99 + 9))
&Iuml; 1204 := (11 &times; (111 − 1)) − ((1 + 1) &times; (1 + 1 + 1))
&Iuml; 1199 := 11 &times; (111 − 1 − 1)
:= 2 + (((2/2 + 2) &times; (22 − 2)2 ) + 2)
:= 22/2 &times; (222/2 − 2)
:= 33/3 + (33 &times; (33 + 3))
3
&Iuml; 1209 := (11 &times; (111 − 1)) − 1
:= (2 &times; 222 ) + (222 − 2)/2
:= (33 &times; ((3 + 3) + 3) − 3)/(3 + 3)
:= 3 + ((3 + 3) &times; (33 &times; (3 + 3) + 3))
:= 4 &times; (44 + 4 ) − 4/4
:= 4 + 4 &times; (44 + 44 )
:= 4 + ((4 &times; (44 + 44 ) + 4/4) + 4)
:= (5 + 5) &times; (5 &times; 5 &times; 5 − 5) − 5/5
:= 5 + ((5 + 5) &times; (5 &times; 5 &times; 5 − 5) − 5/5)
:= (55 &times; (55 + 55) − 5)/5
:= 66/6 + (66 &times; (6 + 6 + 6))
:= ((66/6) &times; ((666 − 6)/6)) − 6
:= ((6 &times; (6 &times; 66 + 6)) + 6)/((6 + 6)/6)
:= 77 + ((7777 + 77)/7)
:= 7 &times; ((7 &times; 7 &times; 7 &times; 7 + 7)/(7 + 7))
:= 7 &times; (7 + 7) + 7777/7
:= 88 + 8888/8
:= 8 &times; (8 &times; 8 + 88) − (88 + 8)/8
:= 8/8 + (8 &times; (8 &times; 8 + 88) − 8)
:= 99/9 &times; (9/9 + 99 + 9)
:= 9 + ((((99 &times; (99 + 9)) − (9 + 9))/9) + 9)
:= 99 + ((9999 − 9)/9)
4
&Iuml; 1200 := 1 + (11 &times; (111 − 1 − 1))
:= (2/2 + 2) &times; (22 − 2)
2
:= 3 + ((33 &times; (33 + 3)) + 3 &times; 3)
4
&Iuml; 1205 := 11 &times; 111 − ((1 + 1)1+1+1+1 )
2
&Iuml; 1210 := 11 &times; (111 − 1)
:= (2/2 + 2 + 2) &times; (22 − 2)/2
:= 2 &times; ((22/2)2 + 222 )
:= ((33/3)3 ) − (3 &times; 33 + 33 )
:= 3 &times; 33 + 3333/3
4
:= 4 &times; (44 + 4 )
:= 4 + (4 &times; (44 + 4 ) + 4/4)
:= 44/4 &times; (444 − 4)/4
:= (5 + 5) &times; (5 &times; 5 &times; 5 − 5)
:= 5 + (5 + 5) &times; (5 &times; 5 &times; 5 − 5)
:= 55 &times; (55 + 55)/5
:= 6 + ((66 &times; (6 + 6 + 6)) + 6)
:= 6 + ((66 &times; (6 + 6 + 6)) + (66/6))
:= 66/6 &times; ((666 − 6)/6)
:= (7 &times; 7 − 7/7) &times; (77/7 + 7 + 7)
:= 7/7 + ((77 &times; (7 + 7) + 77) + 7 &times; 7)
:= 77/7 &times; (777 − 7)/7
:= (8 + 8) &times; (88/8 + 8 &times; 8)
:= 8 &times; (8 &times; 8 + 88) − 88/8
:= 88/8 &times; (888 − 8)/8
:= (9/9 + 9) &times; (999/9 + 9)
:= 9 + ((((99 &times; (99 + 9)) − 9)/9) + 9)
:= 99 + 9999/9
&Iuml; 1201 := 1 + (1 + (11 &times; (111 − 1 − 1)))
&Iuml; 1206 := (11 &times; (111 − 1)) − 1 − 1 − 1 − 1
&Iuml; 1211 := 1 + (11 &times; (111 − 1))
:= 2 + (22/2 &times; (222/2 − 2))
:= (2/2 + 2) &times; ((22 − 2)2 + 2)
:= (2 &times; 222 ) + (222 + 2)/2
:= (3 &times; (33 + 3)) + 3333/3
:= (3 + 3) &times; (33 &times; (3 + 3) + 3)
:= 3 &times; 33 + ((3333 + 3)/3)
:= 4/4 + 4 &times; (44 + 44 )
:= 4 + (4 &times; (44 + 44 ) + (4 + 4)/4)
:= 44/4 + 4 &times; (44 + 44 )
:= 5/5 + (5 + 5) &times; (5 &times; 5 &times; 5 − 5)
:= 5 + ((5 + 5) &times; (5 &times; 5 &times; 5 − 5) + 5/5)
:= (55 &times; (55 + 55) + 5)/5
:= 6 + (((66 &times; (6 + 6 + 6)) + 6/6) + 6)
:= (6 + 6 + 6) &times; (66 + 6/6)
:= ((66 &times; ((666 − 6)/6)) + 6)/6
:= (7 &times; 7 &times; 7 &times; 7 + 7/7)/((7 + 7)/7)
:= 77 &times; (7 + 7) + ((7 + 7)/7)
:= (7 &times; (7 &times; (7 + 7) + 77)) − (7 + 7)
:= 8/8 + ((8 + 8) &times; (88/8 + 8 &times; 8))
:= 8 + ((8888 − 8)/8 + 88)
:= ((88 &times; (888 − 8)/8) + 8)/8
:= 9 + (9999/9 + 9 &times; 9)
:= 9 + ((99 &times; ((99 + 9)/9)) + 9)
:= 99 + (9999 + 9)/9
&Iuml; 1202 := 1 + (1 + (1 + (11 &times; (111 − 1 − 1))))
7
&Iuml; 1207 := (11 &times; (111 − 1)) − 1 − 1 − 1
&Iuml; 1212 := 1 + (1 + (11 &times; (111 − 1)))
:= 2 + ((2/2 + 2) &times; (22 − 2)2 )
:= 2 + ((2/2 + 2 + 2) &times; (222 − 2)/2)
:= 2 + ((2 &times; 222 ) + 222 /2)
:= 3 + ((33 &times; (33 + 3)) + 33/3)
:= 3/3 + ((3 + 3) &times; (33 &times; (3 + 3) + 3))
:= 33 + ((33 &times; (33 + 3)) − 3)
:= (4 + 4)/4 + 4 &times; (44 + 44 )
:= 4 + ((4 &times; (44 + 44 ) − 4/4) + 4)
:= (4 &times; ((44 + 44 ) + 4)) − 4
:= (5 + 5)/5 + (5 + 5) &times; (5 &times; 5 &times; 5 − 5)
:= 5 + ((5 + 5) &times; (5 &times; 5 &times; 5 − 5) + ((5 + 5)/5))
:= (5 + 5)/5 &times; ((55 &times; 55 + 5)/5)
:= 6 + (((66 &times; (6 + 6 + 6)) + ((6 + 6)/6)) + 6)
:= 6/6 + ((6 + 6 + 6) &times; (66 + 6/6))
:= 6 + ((6 + 6 + 6) &times; (66 + 6/6))
:= 7 + ((7777/7 + 77) + 7)
:= 77/7 &times; 777/7 − (7 + 7)
:= 7/7 + ((7 &times; (7 &times; (7 + 7) + 77)) − (7 + 7))
:= (88/8 &times; (888 − 8)/8) − 8
:= 8 + (8888/8 + 88)
:= 8 &times; (8 &times; 8 + 88) − (8/((8 + 8)/8))
:= 9 + ((9999 + 9)/9 + 9 &times; 9)
:= 9 + ((((99 &times; (99 + 9)) + 9)/9) + 9)
:= ((99/9) &times; 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
&Iuml; 1213 := 1 + (1 + (1 + (11 &times; (111 − 1))))
2
2+2+2
&Iuml; 1218 := 11 &times; 111 − 1 − 1 − 1
2
&Iuml; 1223 := 1 + (1 + 11 &times; 111)
:= 22 + (2 &times; (((22 + 2) ) + 22))
:= 2 + (22/2 &times; 222/2)
:= 3 + (3333/3 + 3 &times; 33)
:= 3 + ((33 &times; (33 + 3)) + 33 )
:= ((33/3)3 ) − (3 &times; (33 + 3))
:= 4/4 + ((4 &times; ((44 + 44 ) + 4)) − 4)
:= (4 + 4)/4 + (4 &times; ((44 + 44 ) + 4))
:= (((44 &times; 444/4) + 4) + 4)/4
:= (5 &times; ((5 − (5 + 5)/5)5 )) − (5 + 5)/5
:= (5 + 5)/5 &times; (((55 − 55)/5) − 5)
:= (5 &times; (5 &times; 5 &times; (5 + 5) − 5)) − (5 + 5)/5
:= 6 &times; 6 + ((6666/6) + 66)
:= (6 &times; (6 &times; 6 &times; 6 − (6 + 6))) − 6
:= (6 &times; (6 &times; 6 &times; 6 − (6 + 6))) − 6/6
:= 7 + (77 &times; (7 + 7) + ((7 + 7)/7) )
:= (7 &times; (7 &times; (7 + 7) + 77)) − 7
:= (7 &times; (7 &times; (7 + 7) + 77)) − (7 + 7)/7
:= 88/8 &times; 888/8 − 8
:= (8 + 8)/8 + 8 &times; (8 &times; 8 + 88)
:= 8 + (8 &times; (8 &times; 8 + 88) − 8/8)
:= (((99 &times; 999/9) + 9)/9) − 9
:= 9 + (((9999 − 9)/9) + 99)
:= 22 + ((2/2 + 2)
)
7
&Iuml; 1214 := 1 + (1 + (1 + (1 + (11 &times; (111 − 1)))))
:= 9 &times; 9 &times; 9 + ((((9 + 9)/9)9 ) − (9 + 9))
&Iuml; 1224 := 1 + (1 + (1 + 11 &times; 111))
&Iuml; 1219 := 11 &times; 111 − 1 − 1
:= 2 &times; (((22/2) + 22 ) + 2)
:= (22/2 &times; 222/2) − 2
:= (2 &times; 222 ) + 22&times;(2+2)
:= 33 + ((33 &times; (33 + 3)) − 3/3)
:= 3 + (((3 &times; 3 + 3/3)3 ) + (3 + 3)3 )
:= 333 + (33 &times; 33 )
:= 4 + (44/4 &times; (444 − 4)/4)
:= 4 + ((44/4 + 4) &times; (4 − 4/4)4 )
:= 4 + ((4 &times; ((44 + 44 ) + 4)) + 4)
:= (5 &times; ((5 − (5 + 5)/5)5 )) − 5/5
:= 5 + ((5 &times; ((5 − (5 + 5)/5)5 )) − 5/5)
:= (5 &times; (5 &times; 5 &times; (5 + 5) − 5)) − 5/5
:= (((66 &times; 666/6) − 6)/6) − 6
:= ((6 &times; 6 − 6/6)(6+6)/6 ) − 6
:= 6 &times; (6 &times; 6 &times; 6 − (6 + 6))
2
2
:= 77/7 &times; 777/7 − 7
:= 7/7 + ((7 &times; (7 &times; (7 + 7) + 77)) − 7)
:= 8 &times; (8 &times; 8 + 88) − (8 + 8)/8
:= 88/8 + (8 &times; (8 &times; 8 + 88) − 8)
:= 9 &times; 99 + ((9 + 9) &times; (9 + 9) − 9/9)
:= 9 + (9999/9 + 99)
&Iuml; 1215 := 11 &times; 111 − ((1 + 1) &times; (1 + 1 + 1))
2
:= (2/2 + 2 + 2) &times; (22 + 2)/2
:= 3 &times; (33 &times; ((3 &times; 3 + 3) + 3))
:= (44/4 + 4) &times; (4 − 4/4)4
5
:= 5 &times; ((5 − (5 + 5)/5) )
:= ((7 + 7)/7 + 7) &times; (((7 + 7)/7)7 + 7)
:= ((22 &times; 222/2) − 2)/2
:= 4 + (4 &times; ((44 + 44 ) + 4))
:= 4 + (44/4 &times; 444/4)
:= ((66 &times; 666/6) − 6)/6
:= (77 &times; 777/7 − 7)/7
:= 9 &times; (((99 + 9 + 9) + 9) + 9)
:= ((99 &times; 999/9) − 9)/9
:= 2 &times; ((2 &times; 2
2
) + ((22 + 2) ))
:= (3 + 3)3 + ((3 &times; 3 + 3/3)3 )
:= 5 &times; (5 &times; 5 &times; (5 + 5) − 5)
:= (6 &times; 6 − 6/6)(6+6)/6
:= 7 &times; (7 &times; (7 + 7) + 77)
:= 8 + (8 &times; (8 &times; 8 + 88) + 8/8)
:= 9/9 + ((9 − 9/9) &times; (9 &times; (9 + 9) − 9))
&Iuml; 1226 := 1 + (1 + (1 + (1 + (1 + 11 &times; 111))))
&Iuml; 1221 := 11 &times; 111
:= 2 + ((2 &times; 222 ) + 22&times;(2+2) )
:= 22/2 &times; 222/2
:= 33 + (33 &times; (33 + 3))
:= 4 &times; ((44 + 44 ) + 4)
:= ((22/2 + 22) + 2)2
:= ((33 − 3/3) + 3)3−3/3
:= ((88 &times; 888/8) − 8)/8
2+2
:= (9 − 9/9) &times; (9 &times; (9 + 9) − 9)
:= ((33/3)3 ) − 333/3
:= 8 &times; (8 &times; 8 + 88) − 8/8
&Iuml; 1216 := 11 &times; 111 − 1 − 1 − 1 − 1 − 1
:= 8 + 8 &times; (8 &times; 8 + 88)
&Iuml; 1225 := 1 + (1 + (1 + (1 + 11 &times; 111)))
&Iuml; 1220 := 11 &times; 111 − 1
:= 5 + (5 &times; ((5 − (5 + 5)/5)5 ))
:= ((66/6) &times; 666/6) − 6
:= (7 &times; (7 &times; (7 + 7) + 77)) − 7/7
:= 3 + (((33/3)3 ) − (3 &times; (33 + 3)))
:= 44/4 &times; 444/4
:= 4 + (((44 &times; 444/4) + 4)/4)
:= 5/5 + (5 &times; ((5 − (5 + 5)/5) ))
:= 55/5 &times; 555/5
:= 5 + 55/5 &times; 555/5
:= 6 + ((66/6) &times; ((666 − 6)/6))
:= 66/6 &times; 666/6
:= 6 + (((66 &times; 666/6) − 6)/6)
:= 7 + (7777/7 + 7 &times; (7 + 7))
:= 77/7 &times; 777/7
:= 7/7 + (7 &times; (7 &times; (7 + 7) + 77))
:= 8 &times; (8 &times; 8 + 88)
:= 88/8 &times; 888/8
:= 8 + (8 &times; (8 &times; 8 + 88) + ((8 + 8)/8))
:= 99/9 &times; 999/9
:= (9 + 9)/9 + ((9 − 9/9) &times; (9 &times; (9 + 9) − 9))
5
:= 9/9 + ((9 + 9) &times; (9 + 9) + 9 &times; 99)
&Iuml; 1217 := 11 &times; 111 − 1 − 1 − 1 − 1
&Iuml; 1227 := 11 &times; 111 + ((1 + 1) &times; (1 + 1 + 1))
&Iuml; 1222 := 1 + 11 &times; 111
:= (22/2 &times; 222/2) − 2 − 2
:= ((22 &times; 222/2) + 2)/2
:= 2 + (((22/2 + 22) + 2)2 )
:= ((33/3)3 ) − (333/3 + 3)
:= ((33 &times; 333/3) + 3)/3
:= 3 + ((33 &times; 33 ) + 333)
:= 4/4 + (4 &times; ((44 + 44 ) + 4))
:= ((44 &times; 444/4) + 4)/4
:= 44/4 + (4 &times; ((44 + 44 ) + 4))
:= (5 + 5)/5 + (5 &times; ((5 − (5 + 5)/5)5 ))
:= ((55 &times; 555/5) + 5)/5
:= 5 + (((55 &times; 555/5) + 5)/5)
:= (6 &times; (6 &times; 6 &times; 6 − (6 + 6))) − 6/6 − 6
:= ((66 &times; 666/6) + 6)/6
:= 6 + ((66/6) &times; 666/6)
:= 7 + (77/7 &times; (777 − 7)/7)
:= (77 &times; 777/7 + 7)/7
:= 7 + ((77 &times; 777/7 − 7)/7)
:= 8/8 + 8 &times; (8 &times; 8 + 88)
:= ((88 &times; 888/8) + 8)/8
:= 88/8 + 8 &times; (8 &times; 8 + 88)
:= 9 + (((99/9) &times; (9/9 + 99 + 9)) + 9)
:= ((99 &times; 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
&Iuml; 1228 := (11 &times; (1 + 111)) − 1 − 1 − 1 − 1
:= 2 &times; ((2 &times; (22 &times; (2
2+2
− 2))) − 2)
&Iuml; 1238 := ((1 + 1) &times; (1 + 1 + 1)) + (11 &times; (1 + 111))
&Iuml; 1233 := 1 + (11 &times; (1 + 111))
:= ((22 &times; (222 + 2)/2) + 2)/2
3
:= (2 &times; (((22 + 2)2 ) + 2 &times; 22)) − 2
:= 3 + (((33 − 3/3) + 3)3−3/3 )
:= 3 &times; ((3 &times; 3 − 3) + 333)
:= 3 + ((((33/3)3 ) − 3 &times; 33) + 3)
:= 44 &times; (44 − 4 &times; 4) − 4
:= 4/4 + 44 &times; (44 − 4 &times; 4)
:= ((4 + 4)/4 &times; ((4/4 + 4)4 − 4)) − 4
5
:= (5 + 5)/5 &times; ((55 − 55)/5)
:= 5 + ((5 + 5)/5 &times; ((5 − 55)/5))
:= (5 + 5)/5 &times; ((55 − 5)/5 − 5)
:= 6 + (((66 &times; 666/6) + 6)/6)
:= 6 + (((66/6) &times; 666/6) + 6)
:= 6 + ((66/6) &times; (666 + 6)/6)
:= 7 + 77/7 &times; 777/7
:= 7 + ((7 &times; (7 &times; (7 + 7) + 77)) + 7/7)
:= (7 &times; (((7 + 7)/7)7 + 7 &times; 7)) − 7/7
:= 8 + (((88 &times; 888/8) − 8)/8)
:= 8 + ((8 &times; (8 &times; 8 + 88) + 8/8) + 8)
:= 8 &times; (8 + 8) + (8888 − 8)/8
:= 9 + ((9999/9 + 99) + 9)
:= 9 + ((9 − 9/9) &times; (9 &times; (9 + 9) − 9))
:= 9 + ((((99 &times; 999/9) − 9)/9) + 9)
&Iuml; 1229 := (11 &times; (1 + 111)) − 1 − 1 − 1
2
&Iuml; 1234 := 1 + (1 + (11 &times; (1 + 111)))
2+2
− 2))))
&Iuml; 1239 := 11 &times; 111 + (1 + 1) &times; (11 − 1 − 1)
:= (2222 + 22&times;(2+2) )/2
:= 2 + ((((22/2 + 22) + 2) ) + 2)
:= 2 + (2 &times; (2 &times; (22 &times; (2
:= ((33/3)3 ) − (3 &times; 33 + 3)
:= (33 &times; 33 ) + (((3/3 + 3) + 3)3 )
:= (3 &times; (3 &times; 33 + 333)) − 3
:= 4 + ((44/4 &times; 444/4) + 4)
:= (4 + 4)/4 + 44 &times; (44 − 4 &times; 4)
:= 4 &times; 4 &times; (4 + 4) + 4444/4
:= ((55 &times; 55 − 5) + 5 )/5
:= (((5 − 55) + 5 )/5) − 5
:= ((55 − 55) + 55 )/5
:= 6 &times; 6 &times; 6 &times; 6 − (66 + 6/6)
:= 6 + ((((66 &times; 666/6) + 6)/6) + 6)
:= (6/6 + 6) &times; (666/6 + 66)
:= 7 + ((77 &times; 777/7 + 7)/7)
:= 7 + (((77 &times; 777/7 − 7)/7) + 7)
:= 7 &times; (((7 + 7)/7)7 + 7 &times; 7)
:= 8 + 88/8 &times; 888/8
:= 8 + ((8 &times; (8 &times; 8 + 88) + ((8 + 8)/8)) + 8)
:= 8 &times; (8 + 8) + 8888/8
:= 9 + (((99 &times; 999/9) − 9)/9)
:= 9 + (((9 − 9/9) &times; (9 &times; (9 + 9) − 9)) + 9/9)
:= 9 + (((99/9) &times; 999/9) + 9)
5
5
&Iuml; 1230 := (11 &times; (1 + 111)) − 1 − 1
5
&Iuml; 1235 := 1 + (1 + (1 + (11 &times; (1 + 111))))
&Iuml; 1240 := (11 − 1) &times; (1 + (1 + (1 + 111+1 )))
:= (2 &times; (2 + 2) + 2) &times; ((22/2)2 + 2)
:= 2 + (((22 &times; (222 + 2)/2) + 2)/2)
:= 2 &times; (((22 + 2)2 ) + 2 &times; 22)
:= (3 + 3) &times; ((3 + 3)3 − 33/3)
:= 3 + (((33/3)3 ) − 3 &times; 33)
:= 3/3 + ((3 &times; (3 &times; 33 + 333)) − 3)
:= (4/4 + 4) &times; ((4 − 44)/4 + 44 )
:= 4 + (44 &times; (44 − 4 &times; 4) − 4/4)
:= (4/4 + 4) &times; (44 − 4 − 4)
:= 5 + (5 &times; (5 &times; 5 &times; (5 + 5) − 5))
:= 5 + ((5 &times; (5 &times; 5 &times; (5 + 5) − 5)) + 5)
:= 5 &times; (((5 − (5 + 5)/5)5 ) + 5)
:= 6 &times; 6 &times; 6 &times; 6 − 66
:= 6 + (6 &times; 6 &times; 6 &times; 6 − (66 + 6/6))
:= 6 &times; 6 &times; 6 + (((6 + 6)/6)(66−6)/6)
:= (7 − 7/7) &times; (((7 + 7)/7)7 + 77)
:= 7 + (77/7 &times; 777/7 + 7)
:= 7/7 + (7 &times; (((7 + 7)/7)7 + 7 &times; 7))
:= 8 + (((88 &times; 888/8) + 8)/8)
:= (88/8 + 8) &times; (8/8 + 8 &times; 8)
:= 88 + (8 + 8) &times; (8 &times; 8 + 8)
:= 9 + ((99/9) &times; 999/9)
:= 99/9 + ((9 − 9/9) &times; (9 &times; (9 + 9) − 9))
:= 9 &times; 9 &times; 9 + ((((9 + 9)/9)9 ) − 9/9)
&Iuml; 1236 := 1 + (1 + (1 + (1 + (11 &times; (1 + 111)))))
&Iuml; 1231 := (11 &times; (1 + 111)) − 1
:= ((22 &times; (222 + 2)/2) − 2)/2
3
:= 2 &times; ((2 &times; (22 &times; (2
2+2
− 2))) + 2)
&Iuml; 1241 := (11 &times; (1 + 1 + 111)) − 1 − 1
:= (22/2 &times; (222/2 + 2)) − 2
:= ((33/3) ) − (3 &times; 33 + 3/3)
:= 3 + (3 &times; ((3 &times; 33 − 3) + 333))
:= 33+3 + ((3 − 3/3)3&times;3 )
:= 44 &times; (44 − 4 &times; 4) − 4/4
:= 4 + 44 &times; (44 − 4 &times; 4)
:= 4/4 + ((4/4 + 4) &times; (44 − 4 − 4))
:= 5 &times; 5 &times; 5 + 5555/5
:= ((5/5 + 5)5−5/5 ) − 55
:= 6 + (6 &times; 6 &times; 6 &times; 6 − 66)
:= 66/6 + (6 &times; 6 &times; 6 &times; 6 − 66)
:= 7 + ((7 &times; (7 &times; (7 + 7) + 77)) − 7/7)
:= 77/7 + (7 &times; (7 &times; (7 + 7) + 77))
:= ((7 + 7)/7) + (7 &times; (((7 + 7)/7)7 + 7 &times; 7))
:= 8 + ((8 &times; (8 &times; 8 + 88) − 8/8) + 8)
:= ((88 + 8)/8) &times; (888/8 − 8)
:= 8/8 + ((8 + 8) &times; (8 &times; 8 + 8) + 88)
:= 9 + (((99 &times; 999/9) + 9)/9)
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 − 999/9)
:= 9 &times; 9 &times; 9 + (((9 + 9)/9)9 )
5
:= ((55 &times; 55 + 5 ) + 5)/5
(6+6)/6
:= 6 + ((6 &times; 6 − 6/6)
)
&Iuml; 1237 := 1 + (1 + (1 + (1 + (1 + (11 &times; (1 + 111))))))
&Iuml; 1232 := 11 &times; (1 + 111)
:= 2 &times; (2 &times; (22 &times; (2
2+2
− 2)))
:= 2
2+2
&Iuml; 1242 := (11 &times; (1 + 1 + 111)) − 1
:= 2 + (2 &times; (((22 + 2)2 ) + 2 &times; 22))
+ (22/2 &times; 222/2)
:= ((33/3)3 ) − 3 &times; 33
:= 3 + ((((3/3 + 3) + 3)3 ) + (33 &times; 33 ))
:= 3 &times; (3 &times; 33 + 333)
:= 44 &times; (44 − 4 &times; 4)
:= 4 + (44 &times; (44 − 4 &times; 4) + 4/4)
:= (4 + 4)/4 &times; ((4/4 + 4)4 − 4)
:= 55/5 &times; (555 + 5)/5
:= 5 + (55/5 &times; (555 + 5)/5)
:= (5 + 5)/5 &times; ((55 + 5)/5 − 5)
:= 66/6 &times; (666 + 6)/6
(6+6)/6
:= 6 + (((6 &times; 6 − 6/6)
) + 6)
:= 6 + ((6 &times; 6 &times; 6 &times; 6 − 66) + 6)
:= 7 + (7 &times; (7 &times; (7 + 7) + 77))
:= 77 + (7777/7 + 7 &times; 7)
:= (77/7 + 7) &times; (77 − (7/7 + 7))
:= 8 + (8 &times; (8 &times; 8 + 88) + 8)
:= 8 + (88/8 &times; 888/8 + 8)
:= 88 + ((8 + 8) &times; (8 &times; 8 + 8) + ((8 + 8)/8))
:= 9 &times; 9 &times; 9 + ((((9 + 9)/9)9 ) − 9)
:= 9 + (((9999/9 + 99) + 9) + 9)
:= 999 + 9 &times; (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
&Iuml; 1248 := (1 + 1) &times; ((1 + (1 + 1) &times; (1 + 11))1+1 − 1)
&Iuml; 1243 := 11 &times; (1 + 1 + 111)
:= 22/2 &times; (222/2 + 2)
3
:= 3/3 + (3 &times; (3 &times; 3 + 333))
:= 2 + (((((2 &times; (22 + 2)) + 2)2 ) + 2)/2)
:= (33 + 3 + 3) &times; (33 − 3/3)
:= 3 + (((33/3)3 ) − 3 &times; 33 )
4
4
4
:= 4 + (4 &times; (4 − 4 − 4))
:= 44 + (4 &times; (44 + 4 ) − 4/4)
&Iuml; 1253 := (11 &times; (1 + (1 + 1 + 111))) − 1
:= 2 &times; ((22 + 2) &times; (22 + 2 + 2))
:= 4 + ((((4 + 4) &times; (4/4 + 4)4 ) − 4)/4)
5
:= ((5 + 5)/5 &times; (55 − 5)/5) − 5
:= (5 + 5)/5 &times; (5 − 5)/5
:= 66 + ((6666/6) + 66)
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − 6 − 6
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − 6/6 − 6
:= 77/7 &times; (777 + 7 + 7)/7
:= (77 + 7)/7 &times; (777/7 − 7)
:= 77 + (7 + 7) &times; (77 + 7)
:= 8 + ((88/8 + 8) &times; (8/8 + 8 &times; 8))
:= 8 + ((8 + 8) &times; (8 &times; 8 + 8) + 88)
:= (8 + 8) &times; (88 − 8) − (88/8 + 8 + 8)
:= 9/9 + (9 &times; (9 + 9 + 9) + 999)
:= 9 + ((((99/9) &times; 999/9) + 9) + 9)
:= 99/9 + (9 &times; (9 + 9 + 9) + 999)
:= 5 + ((5 + 5)/5 &times; (55 − 5)/5)
&Iuml; 1249 := ((1 + 1) &times; (1 + (1 + 1) &times; (1 + 11))1+1 ) − 1
&Iuml; 1244 := 1 + (11 &times; (1 + 1 + 111))
:= ((((2 &times; (22 + 2)) + 2)2 ) − 2)/2
2
:= 2 &times; ((22 − 2) + 222)
3
3
&Iuml; 1254 := 11 &times; (1 + (1 + 1 + 111))
:= 2 &times; (((2/2 + 2 + 2)2+2 ) + 2)
:= 3 + (((3 − 3/3)3&times;3 ) + 33+3 )
:= ((33/3) ) − (3 &times; 3 + 3/3)
:= 33 &times; (33/3 + 33 )
:= 44 + 4 &times; (44 + 44 )
:= (((4 + 4) &times; (4/4 + 4)4 ) − 4)/4
:= 4 + (4 + 4)/4 &times; (4/4 + 4)4
:= ((55 − 5 + 55 )/5) − 5
:= (55 − 5 + 55 )/5
:= 5 + ((55 − 5 + 55 )/5)
:= 6 + (((66/6) &times; (666 + 6)/6) + 6)
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − 66/6
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − 6
:= (7 &times; (7 + 7) &times; (7 + 7)) − ((7 + 7)/7)
:= 7 + ((77/7 + 7) &times; (77 − (7/7 + 7)))
:= 7/7 + ((7 + 7) &times; (77 + 7) + 77)
:= 8 + (((88 + 8)/8) &times; (888/8 − 8))
:= (((88 + 8) &times; (88 + 8 + 8)) + 8)/8
:= (88/8 + 8) &times; (((8 + 8)/8) + 8 &times; 8)
:= (9 + 9)/9 + (9 &times; (9 + 9 + 9) + 999)
:= 9 + (((((9 + 9)/9)9 ) − 9/9) + 9 &times; 9 &times; 9)
:= 9 &times; 99 + ((99 &times; 99)/(9 + 9 + 9))
7
&Iuml; 1245 := 1 + (1 + (11 &times; (1 + 1 + 111)))
:= 2 + (22/2 &times; (222/2 + 2))
&Iuml; 1250 := (1 + 1) &times; (1 + (1 + 1) &times; (1 + 11))1+1
:= 2 &times; ((2/2 + 2 + 2)2+2 )
3
3
:= ((33/3) ) − 3 &times; 3
:= 3 + (3 &times; (3 &times; 3 + 333))
&Iuml; 1255 := 1 + (11 &times; (1 + (1 + 1 + 111)))
:= 2 + ((((((2 &times; (22 + 2)) + 2)2 ) + 2)/2) + 2)
3
:= 3/3 + (33 &times; (33/3 + 33 ))
4
:= 44 + (4 &times; (44 + 44 ) + 4/4)
:= (4 + 4)/4 &times; (4/4 + 4)
:= 5 &times; 5 &times; 5 &times; (5 + 5) − 5
:= 5 &times; 5 &times; 5 &times; (5 + 5)
:= 6 + ((6/6 + 6) &times; (666/6 + 66))
:= ((6 − 66)/6) + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 6/6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) − 6)
:= (7/7 + 7 + 7) &times; (77 − 7/7 + 7)
:= (7/7 + 7 &times; 7) &times; (77/7 + 7 + 7)
:= 7 + ((77 + 7)/7 &times; (777/7 − 7))
:= 8 + ((88/8 &times; 888/8 + 8) + 8)
:= ((((8 + 8)/8) + 8)8&times;8/(8+8) )/8
:= 8 + ((8888/8 + 8 &times; (8 + 8)) + 8)
:= 999/9 + (9 &times; ((99 + 9 + 9) + 9))
&Iuml; 1246 := 1 + (1 + (1 + (11 &times; (1 + 1 + 111))))
:= 2 &times; (((2/2 + 2 + 2)
2+2
) − 2)
3
:= 3 + ((3 &times; (3 &times; 3 + 333)) + 3/3)
4
:= (4 + 4)/4 &times; (4/4 + 4) − 4
5
5
:= (4/4 + 4) &times; (44 − 4/4 − 4)
:= 5 + 5 &times; 5 &times; 5 &times; (5 + 5)
9
:= 9 + ((((9 + 9)/9) ) + 9 &times; 9 &times; 9)
&Iuml; 1251 := 1 + ((1 + 1) &times; (1 + (1 + 1) &times; (1 + 11))1+1 )
2
:= ((((2 &times; (22 + 2)) + 2) ) + 2)/2
3
&Iuml; 1256 := 1 + (1 + (11 &times; (1 + (1 + 1 + 111))))
:= 2 + (2 &times; (((2/2 + 2 + 2)2+2 ) + 2))
:= 3 + ((((33/3)3 ) − 3 &times; 33 ) + 3)
:= 3 &times; ((3 &times; 3 + 333) + 3)
4
:= (((4 + 4) &times; (4/4 + 4) ) + 4)/4
5
:= 999 + (((9 + 9)/9)9−9/9 )
5
:= (4/4 + 4) &times; (44 − 4) − 4
:= ((5 + 5 + 5)/5) − 5
:= (5 + 5 + 5)/5
:= 6 &times; 6 + ((66/6) &times; ((666 − 6)/6))
:= 6 &times; 6 + (((66/6) &times; 666/6) − 6)
:= (6 + 6)/6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) − 6)
:= 7 + (7 &times; (((7 + 7)/7) + 7 &times; 7))
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) − ((7 + 7)/7)7 )
:= 777 + (7 &times; (77 − 7) − (77/7))
:= (8/8 + 88) &times; ((8 − (8 + 8)/8) + 8)
:= (8/8 + 8) &times; (8 &times; (8 + 8) + (88/8))
:= 88 &times; (8 + 8) − (8 &times; 8 + 88)
:= 9 + (9 &times; (9 + 9 + 9) + 999)
:= 9 &times; 99 + (((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9))
7
9−9/9
:= 999 + ((((9 + 9)/9)
) − 9)
&Iuml; 1247 := 1 + (1 + (1 + (1 + (11 &times; (1 + 1 + 111)))))
2
:= 5 + ((55 + 55 + 5)/5)
&Iuml; 1252 := (11 &times; (1 + (1 + 1 + 111))) − 1 − 1
2+2
&Iuml; 1257 := 1 + (1 + (1 + (11 &times; (1 + (1 + 1 + 111)))))
:= 222 + ((222/2 + 22)/2)
:= 22 + (((22/2 + 22) + 2) )
:= 2 + (2 &times; ((2/2 + 2 + 2)
:= ((33/3)3 ) − (3 &times; 33 + 3)
:= ((3/3 + 3)3 ) + (33 &times; (33 + 3))
:= 3 + (33 &times; (33/3 + 33 ))
:= 44 + ((4 &times; (44 − 4 − 4)) − 4/4)
:= ((4 + 4)/4 + 4)4 − 44
:= 4/4 + ((4/4 + 4) &times; (44 − 4) − 4)
:= ((5 + 5)/5 &times; (55 + 5)/5) − 5
:= (5 + 5)/5 &times; (55 + 5)/5
:= 5 + ((5 + 5)/5 &times; (55 + 5)/5)
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − (6/6 + 6 + 6)
:= ((6 + 6)/6)6 + (66 &times; (6 + 6 + 6))
:= 6 &times; 6 + ((66/6) &times; 666/6)
:= 77 + ((7 + 7) &times; (77 + 7) − 7/7)
:= 7 + ((7/7 + 7 &times; 7) &times; (77/7 + 7 + 7))
:= 8 + (8888/8 + 8 &times; (8 + 8))
:= 8 &times; (88 + 8) + 88 &times; 88/(8 + 8)
:= 8 + ((((88 + 8) &times; (88 + 8 + 8)) + 8)/8)
:= 9 + (((((99 &times; 999/9) − 9)/9) + 9) + 9)
:= 9 &times; (9 + 9) + (((99 &times; 99) + 9)/9)
:= 9 &times; 9 + ((99 − 9/9) &times; ((99 + 9)/9))
77/7−7
:= (7 − 7/7)
−7&times;7
93
))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1258 := 11 &times; 111 + (111/(1 + 1 + 1))
2+2
&Iuml; 1263 := (11 &times; (1 + (1 + (1 + 1 + 111)))) − 1 − 1
2
&Iuml; 1268 := 1 + (1 + (1 + (11 &times; (1 + (1 + (1 + 1 + 111))))))
:= 222/2 + (2 &times; ((22 + 2) ))
:= 2 + ((222/2 + 222 )/2)
:= ((3 + 3)3/3+3 ) − 33
:= 3 + (((33/3)3 ) − (33 + 33))
:= (4 + 4)/4 &times; ((4/4 + 4) + 4)
:= 44 + (4 &times; (44 − 4) − 4/4)
:= 4 + (4 &times; (44 − 4) + 44 )
:= (5 + 5)/5 &times; ((55 − 5)/5 + 5)
:= 5 + ((5 + 5)/5 &times; ((55 − 5)/5 + 5))
:= (5 + 5)/5 &times; (((55 − 5)/5 + 5) + 5)
:= 2 &times; ((((2/2 + 2 + 2)
) + 2) + 2)
:= (3/3 + 33) &times; (3/3 + 33 + 3)
4
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) + ((6 + 6)/6))
:= 7 &times; (7 + 7 + 7) + 7777/7
:= (7 + 7) &times; (77 + 7 + 7) − 77/7
:= 7/7 + (7 &times; (77 − 7) + 777)
:= 8 + (((((8 + 8)/8) + 8)8&times;8/(8+8) )/8)
:= 8 &times; 8 + (8888/8 + 88)
:= (8 + 8) &times; (88 − 8) − (88 + 8)/8
:= 9 &times; 9 + ((99/9) &times; ((99 − 9/9) + 9))
:= 9 &times; (9 + 9) + (((9999 − 9)/9) − 9)
:= 9 &times; 9 + (((99 &times; (99 + 9)) − 9)/9)
&Iuml; 1259 := 1 + (11 &times; 111 + (111/(1 + 1 + 1)))
&Iuml; 1264 := (11 &times; (1 + (1 + (1 + 1 + 111)))) − 1
&Iuml; 1269 := 1111 + ((1 + 1 + 11)1+1 − 11)
:= 222/2 + (2 &times; (((22 + 2)2 ) − 2))
:= 2 &times; (((22 + 2 + 2)2 ) − (2 &times; 22))
:= 2 + (((222/2 + 222 ) + 2)/2)
:= (3 &times; (3 − 33 )) + ((33/3)3 )
:= 3/3 + (((3 + 3)3/3+3 ) − 33)
:= ((3 + 3)3/3+3 ) − 33
:= (4/4 + 4) &times; (44 − 4) − 4/4
:= 44 + 4 &times; (44 − 4)
:= 44 + (4 &times; 44 − 44/4)
:= 5 + (((55 − 5 + 55 )/5) + 5)
:= 5 &times; 5 &times; 55 − 555/5
:= 5 + (5 &times; 5 &times; 55 − 555/5)
:= (6 &times; (6 &times; 6 &times; 6 − 6)) − 6/6
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) − ((6 + 6)/6))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) + (6 &times; 6/(6 + 6)))
:= ((77 − 7) &times; (77/7 + 7)) − 7/7
:= 77 + ((7777 − 7)/7 + 77)
:= 7 &times; 7 + ((77 &times; 777/7 − 7)/7)
:= 8 + ((8/8 + 8) &times; (8 &times; (8 + 8) + (88/8)))
:= (8 + 8) &times; (88 − (8/8 + 8))
:= (8 + 8) &times; (88 − 8) − 88/8
9
:= 9 &times; (9 + 9) + (9999/9 − 9)
:= 9 + ((9 + 9) &times; (9 &times; 9 − (99/9)))
:= 9 + (((((9 + 9)/9) ) + 9 &times; 9 &times; 9) + 9)
&Iuml; 1260 := (1 + 11) &times; (111 − ((1 + 1) &times; (1 + 1 + 1)))
&Iuml; 1265 := 11 &times; (1 + (1 + (1 + 1 + 111)))
&Iuml; 1270 := (11 − 1) &times; (111 + ((1 + 1)1+1+1+1 ))
:= (2 − 22) &times; (2/2 − 22+2+2 )
:= 22/2 &times; ((222/2 + 2) + 2)
:= (2/2 + 2 + 2) &times; (22&times;(2+2) − 2)
:= (3 + 3) &times; ((3 + 3)3 − (3 + 3))
:= ((33/3)3 ) − (33 + 33)
:= 3 + (((33/3)3 ) − ((3/3 + 3)3 ))
:= (4/4 + 4) &times; (44 − 4)
:= 4/4 + (4 &times; (44 − 4) + 44 )
:= (4/4 + 4) &times; (44 − (4 + 4)/4)
:= 5 + (5 &times; 5 &times; 5 &times; (5 + 5) + 5)
:= 55 &times; (5 &times; 5 − ((5 + 5)/5))
:= (5 &times; (5 &times; 5 &times; (5 + 5) + 5)) − 5
:= 6 &times; (6 &times; 6 &times; 6 − 6)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) − 6/6)
:= ((66 − 6)/6) + (6 &times; (6 &times; 6 &times; 6 − 6))
:= (77 − 7) &times; (77/7 + 7)
:= 77 + (7777/7 + 77)
:= 7 &times; 7 + 77/7 &times; 777/7
:= (8 &times; 8 − 8/8) &times; ((88 + 8)/8 + 8)
:= 8/8 + ((8 + 8) &times; (88 − (8/8 + 8)))
:= (8 − 88)/8 + (8 + 8) &times; (88 − 8)
:= (9 + 9) &times; (9 &times; 9 − 99/9)
:= 9 &times; (9 + 9) + ((9999 + 9)/9 − 9)
:= (9 + 9) &times; 99 − (((9 + 9)/9)9 )
&Iuml; 1261 := 1 + ((1 + 11) &times; (111 − ((1 + 1) &times; (1 + 1 + 1))))
&Iuml; 1266 := 1 + (11 &times; (1 + (1 + (1 + 1 + 111))))
&Iuml; 1271 := 11 &times; 111 + ((11 − 1)1+1 /(1 + 1))
:= ((((2 &times; (22 + 2)) + 2)2 ) + 22)/2
:= (222/2 + 222 )/2
:= 22 + (((((2 &times; (22 + 2)) + 2)2 ) − 2)/2)
:= 3/3 + ((3 + 3) &times; ((3 + 3)3 − (3 + 3)))
:= 3 + (((3 + 3)3/3+3 ) − 33)
:= ((33/3)3 ) − (33 + 33)
:= 4/4 + (4/4 + 4) &times; (44 − 4)
:= 4 &times; 4 + (4 + 4)/4 &times; (4/4 + 4)4
:= (4/4 + 4) &times; (44 − 4/4) − 4
:= ((55 + 55 ) + 55 )/5
:= 5 + (((55 + 55 ) + 55 )/5)
:= ((5/5 + 5)5−5/5 ) − 5 &times; 5
:= 6/6 + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 6 + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 66/6 + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 7/7 + ((77 − 7) &times; (77/7 + 7))
:= 777 + (7 &times; (77 − 7) − 7/7)
:= (((7 + 7)/7)77/7 ) − 777
:= (8 + 8) &times; (88 − 8) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) &times; (88 − (8/8 + 8)))
:= (8 + 8) &times; (88 − 8) − (8/8 + 8)
:= 9/9 + ((9 + 9) &times; (9 &times; 9 − (99/9)))
:= 9 &times; (9 &times; (9 + 9) − 9) − 999/9
:= 9/9 + ((9 + 9) &times; 99 − (((9 + 9)/9)9 ))
&Iuml; 1262 := (11 &times; (1 + 111)) + ((11 − 1) &times; (1 + 1 + 1))
:= 2 + ((2 − 22) &times; (2/2 − 2
2+2+2
))
&Iuml; 1267 := 1 + (1 + (11 &times; (1 + (1 + (1 + 1 + 111)))))
:= ((2
22/2
2
+ 22 ) + 2)/2
&Iuml; 1272 := (1 + 11) &times; (111 − (1 + (1 + 1 + 1 + 1)))
:= (2 + 2 + 2)2+2 − (22 + 2)
:= 3 + ((3 &times; (3 − 33 )) + ((33/3)3 ))
:= ((33/3)3 ) − ((3/3 + 3)3 )
:= 3 + (((3 + 3)3/3+3 ) − 33 )
:= 4 + ((4 + 4)/4 &times; ((4/4 + 4)4 + 4))
:= 4 + ((4 &times; (44 − 4) − 4/4) + 44 )
:= 44 + (4 &times; 44 − (4 + 4))
:= (5 + 5)/5 &times; ((55 + 5)/5 + 5)
:= 5 + ((5 + 5)/5 &times; ((55 + 5)/5 + 5))
:= (5 + 5)/5 &times; ((55 + 55 )/5)
:= (6 + 6)/6 + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) + 6/6)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) + 6)
:= ((7 + 7)/7) + ((77 − 7) &times; (77/7 + 7))
:= 777 + 7 &times; (77 − 7)
:= (7 + 7) &times; (77 + 7 + 7) − (7 + 7)/7
88/8−8
:= 8 + ((88/8 + 8) &times; (((8 + 8)/8) + 8 &times; 8))
:= ((88/8)
:= (9 + 9)/9 + ((9 + 9) &times; (9 &times; 9 − (99/9)))
:= 9 &times; 9 + (((99 &times; (99 + 9)) − (9 + 9))/9)
)−8&times;8
94
:= (8 + 8) &times; (88 − 8) − 8
:= 9 &times; (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
&Iuml; 1273 := 1 + ((1 + 11) &times; (111 − (1 + (1 + 1 + 1 + 1))))
2
2
&Iuml; 1278 := (11 − 1 − 1) &times; ((1 + 11)1+1 − (1 + 1))
2+2+2
&Iuml; 1283 := ((1 + 11) &times; (111 − (1 + 1 + 1 + 1))) − 1
:= (2 + 2 + 2)2+2 − (22/2 + 2)
:= (22/2) + (2 &times; ((22 + 2) ))
:= ((22 − 2) &times; 2
:= (3 + 3) &times; 33 + 3333/3
:= (3 + 3) &times; ((3 + 3)3 − 3)
:= 33 + (((33/3)3 ) − 3 &times; 33 )
:= 4 + ((4 &times; 44 − 44/4) + 44 )
:= 44 + (4 &times; 44 − (4 + 4)/4)
:= 4 + ((4 &times; 44 − 4/4) + 44 )
:= 5 &times; 5 + ((5 + 5)/5 &times; (55 − 5)/5)
:= 5 + (((5 + 5)/5 &times; (55 − 5)/5) + 5 &times; 5)
:= 5 &times; 5 + ((5 + 5)/5 &times; ((55 − 5)/5 + 5))
)−2
:= 6 + (((6 &times; (6 &times; 6 &times; 6 − 6)) + 6/6) + 6)
:= 6 &times; 6 &times; 6 &times; 6 − 6 − 6 − 6
:= 6 &times; 6 &times; 6 &times; 6 − (6/6 + 6 + 6)
:= (7 + 7) &times; (77 + 7 + 7) − 7/7
:= (77/7 + 7) &times; ((7/7 − 7) + 77)
:= 7 + ((7 + 7) &times; (77 + 7 + 7) + ((7 + 7)/7))
:= 8/8 + ((8 + 8) &times; (88 − 8) − 8)
:= (8 + 8) &times; (88 − 8) − (8 + 8)/8
:= 88/8 + ((8 + 8) &times; (88 − 8) − 8)
:= 9 &times; (9 + 9) + 9999/9
:= (9 + 9) &times; (9 &times; 9 − (9/9 + 9))
:= 9 + ((9999 + 9)/9 + 9 &times; (9 + 9))
&Iuml; 1274 := (1 + 1 + 11) &times; (111 − 1 − 1 − 11)
:= (2 + 2 + 2)
2+2
:= ((22 − 2) &times; 2
− 22
3
&Iuml; 1279 := 1111 + ((1 + 1 + 11)1+1 − 1)
3
:= 3 + (((33/3) ) − (3 + 33))
2+2+2
&Iuml; 1284 := (1 + 11) &times; (111 − (1 + 1 + 1 + 1))
:= 222 + 2 &times; (22 − 2)2
) − 2/2
3
:= 3/3 + ((3 + 3) &times; ((3 + 3) − 3))
4
:= (33 &times; (33 + 3 + 3)) − 3
4
:= 4 + ((4/4 + 4) &times; (44 − (4 + 4)/4))
:= 4 + (4 &times; 4 − 4/4)
:= 4 + (4 &times; 44 + 44 )
:= 5 &times; 5 + ((55 − 5 + 55 )/5)
:= 5 + (((55 − 5 + 55 )/5) + 5 &times; 5)
:= (((55 &times; (55 + 5)) − 5) + 55 )/5
:= 6 &times; 6 &times; 6 &times; 6 − ((66 + 66)/6)
:= 6 &times; 6 &times; 6 &times; 6 − (66/6 + 6)
:= 6 &times; 6 &times; 6 &times; 6 − 6 − 6
7
:= (7 + 7) &times; (77 + 7 + 7)
:= (((77 − 7) &times; ((7 + 7)/7) ) − 7)/7
:= ((77 − 7)/7) + (7 + 7) &times; (77 + 7 + 7)
:= (8 + 8)/8 + ((8 + 8) &times; (88 − 8) − 8)
:= (8 + 8) &times; (88 − 8) − 8/8
:= (8/((8 + 8)/8)) + (8 + 8) &times; (88 − 8)
:= 9 &times; (9 + 9) + (9999 + 9)/9
&Iuml; 1275 := 111+1+1 − (1 + 111)/(1 + 1)
9
:= 9 + ((9 + 9) &times; 99 − (((9 + 9)/9) ))
:= ((99 + 9)/9) &times; ((99 − 9/9) + 9)
&Iuml; 1285 := ((1 + 1 + 1) &times; (1 + 11))1+1 − 11
&Iuml; 1280 := 1111 + (1 + 1 + 11)1+1
:= 2/2 + ((2 + 2 + 2)2+2 − 22)
:= (22 − 2) &times; 22+2+2
:= (2 + 2 + 2)2+2 − 22/2
:= ((3 + 3) &times; ((3 + 3)3 − 3)) − 3
:= 3 + (((33/3)3 ) − (33 + 33 ))
:= ((3 + 3)3/3+3 ) − 33/3
:= (4/4 + 4) &times; (44 − 4/4)
:= 44 + 4 &times; 44
:= (4/4 + 4) &times; (4/4 + 44 )
:= 5 &times; (5 &times; 5 &times; (5 + 5) + 5)
:= 5 + (5 &times; (5 &times; 5 &times; (5 + 5) + 5))
:= 5 + ((5 &times; (5 &times; 5 &times; (5 + 5) + 5)) + 5)
:= 6 &times; 6 &times; 6 &times; 6 + ((6 − (66 + 66))/6)
:= ((6 − 66)/6) + (6 &times; 6 &times; 6 &times; 6 − 6)
:= 6 &times; 6 &times; 6 &times; 6 − 66/6
7
:= 7/7 + (7 + 7) &times; (77 + 7 + 7)
:= ((77 − 7)/7) &times; ((7 + 7)/7)
:= 77/7 + (7 + 7) &times; (77 + 7 + 7)
:= (8/8 + 8 + 8) &times; (88/8 + 8 &times; 8)
:= (8 + 8) &times; (88 − 8)
:= 8 &times; 8 + 88/8 &times; 888/8
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) − 999/9)
:= (9 − 9/9) &times; (9 &times; (9 + 9) − ((9 + 9)/9))
:= ((9 + 9) &times; (9 &times; 9 − 9)) − 99/9
&Iuml; 1276 := 11 &times; (1 + 1 + 1 + 1 + 1 + 111)
:= 2 + ((2 + 2 + 2)
2+2
− 22)
:= 3/3 + (((3 + 3) &times; ((3 + 3)3 − 3)) − 3)
&Iuml; 1281 := 1 + (1111 + (1 + 1 + 11)1+1 )
&Iuml; 1286 := 1 + (((1 + 1 + 1) &times; (1 + 11))1+1 − 11)
:= 2/2 + ((22 − 2) &times; 22+2+2 )
:= 2 + 2 &times; (22 − 2)2 + 222
:= 3 + ((3 + 3) &times; ((3 + 3)3 − 3))
:= (33 &times; (33 + 3 + 3)) − 3/3
:= 4 + (4 &times; 4 − 4)
:= 4/4 + (4 &times; 4 + 4 )
:= 4 + (((4 + 4)/4 + 4 &times; 44 ) + 44 )
:= 5 &times; 5 + ((55 + 55 + 5)/5)
:= 5 + (((55 + 55 + 5)/5) + 5 &times; 5)
:= ((5/5 + 5)5−5/5 ) − 5 − 5
:= 66/6 &times; (((666 − 6)/6) + 6)
:= ((66/6) &times; (666/6 + 6)) − 6
:= ((6 − 66)/6) + 6 &times; 6 &times; 6 &times; 6
:= ((7 + 7)/7) + (7 + 7) &times; (77 + 7 + 7)
:= 7 + (7 + 7) &times; (77 + 7 + 7)
:= 7 + ((((77 − 7) &times; ((7 + 7)/7)7 ) − 7)/7)
:= (8 + 8) &times; (88 − 8) − (8/((8 + 8)/8))
:= 8/8 + (8 + 8) &times; (88 − 8)
:= 8 + ((8 + 8) &times; (88 − 8) − ((8 + 8)/8))
:= 99/9 &times; (((99 − 9/9) + 9) + 9)
:= 9 + (((9999 − 9)/9) + 9 &times; (9 + 9))
:= ((9 + 9) &times; (9 &times; 9 − 9)) − 9/9 − 9
4
4
4
&Iuml; 1277 := 1 + 11 &times; (1 + 1 + 1 + 1 + 1 + 111)
:= 2 + (((2 + 2 + 2)2+2 − 22) + 2/2)
3
3
3
4
&Iuml; 1282 := 1 + (1 + (1111 + (1 + 1 + 11)1+1 ))
:= 2 + ((22 − 2) &times; 22+2+2 )
3
:= ((33/3) ) − (3 + 3 )
&Iuml; 1287 := 11 &times; (111 + ((1 + 1) &times; (1 + 1 + 1)))
:= 2 + ((2 + 2 + 2)2+2 − 22/2)
:= 3 + (((3 + 3) &times; ((3 + 3) − 3)) + 3/3)
:= 33 &times; (33 + 3 + 3)
:= 44 + ((4 + 4)/4 + 4 &times; 44 )
:= 4 &times; 44 + 4444/4
:= 5 + ((5 + 5)/5 &times; ((55 + 5 )/5))
:= (5 + 5)/5 &times; (((55 + 55 )/5) + 5)
:= 55/5 &times; ((555 + 5)/5 + 5)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) + (66/6))
:= 6 &times; 6 &times; 6 &times; 6 − ((6 + 6)/6 + 6 + 6)
:= 66/6 &times; (666/6 + 6)
:= 7 + (77/7 &times; 777/7 + 7 &times; 7)
:= 7 + ((7 + 7) &times; (77 + 7 + 7) + 7/7)
:= 7 + (((77 − 7)/7) &times; ((7 + 7)/7)7 )
:= 8 + ((8 + 8) &times; (88 − 8) − (88/8))
:= (8 + 8)/8 + (8 + 8) &times; (88 − 8)
:= 8 + ((8 + 8) &times; (88 − 8) − 8/8)
:= 9 + (9999/9 + 9 &times; (9 + 9))
:= ((9 + 9) &times; (9 &times; 9 − 9)) − 9
4
4
:= 4/4 + ((4 &times; 4 − 4) + 4 )
5
:= ((9 + 9) &times; (9 &times; 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
&Iuml; 1288 := 1 + 11 &times; (111 + (1 + 1) &times; (1 + 1 + 1))
&Iuml; 1293 := ((1 + 1 + 1) &times; (1 + 11))1+1 − 1 − 1 − 1
&Iuml; 1298 := 11 &times; (111+1 − (1 + 1 + 1))
:= 2 &times; (2 &times; (((22+2 + 2)2 ) − 2))
:= (2 + 2 + 2)2+2 − 2/2 − 2
:= 2 + (2 + 2 + 2)2+2
:= 3/3 + (33 &times; (33 + 3 + 3))
:= ((3 + 3)3/3+3 ) − 3
:= ((33/3)3 ) − 33
:= 4 + (4 &times; 4 + 4 + 4)
:= 4/4 + (((4 + 4)/4 + 4)4 − 4)
:= (4 + 4)/4 + ((4 + 4)/4 + 4)4
:= (55 + 5/5) &times; (5 &times; 5 − ((5 + 5)/5))
:= 5 + ((55 + 5/5) &times; (5 &times; 5 − ((5 + 5)/5)))
:= 5 &times; 55 + ((5 − 5/5)5 − 5/5)
:= 6 &times; 6 &times; 6 &times; 6 − ((6 + 6)/6 + 6)
:= 6 &times; 6 &times; 6 &times; 6 − 6 &times; 6/(6 + 6)
:= (6 + 6)/6 + 6 &times; 6 &times; 6 &times; 6
:= 7 + ((7 + 7) &times; (77 + 7 + 7) + 7)
:= (7 &times; (7 + 7) &times; (7 + 7)) − ((7 + 7)/7 + 77)
:= 77/7 &times; (777/7 + 7)
:= 8 + (8 + 8) &times; (88 − 8)
:= 8 + (88/8 &times; 888/8 + 8 &times; 8)
:= 88/8 &times; ((888 − 8)/8 + 8)
:= 9/9 + (((9 + 9) &times; (9 &times; 9 − 9)) − 9)
:= ((9 + 9) &times; (9 &times; 9 − 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + ((9 + 9) &times; (9 &times; 9 − 9))
4
4
&Iuml; 1289 := (1 + 1 + 11) &times; (11 − 1)1+1 − 11
&Iuml; 1294 := ((1 + 1 + 1) &times; (1 + 11))1+1 − 1 − 1
&Iuml; 1299 := 1 + (11 &times; (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 &times; 3 + 33)
:= 3/3 + (((3 + 3)3/3+3 ) − 3)
:= 3 + ((3 + 3)3/3+3 )
:= 4 + ((4/4 + 4) &times; (4/4 + 44 ))
:= ((4 + 4)/4 + 4)4 − (4 + 4)/4
:= 4 + (((4 + 4)/4 + 4)4 − 4/4)
5
5
:= 5 &times; 55 + ((5 − 5/5) − (5 + 5))
:= 5 &times; 55 + ((5 − 5/5) − 5)
:= 5 &times; 55 + (5 − 5/5)5
:= 6 &times; 6 &times; 6 &times; 6 − 6/6 − 6
:= 6 &times; 6 &times; 6 &times; 6 − (6 + 6)/6
:= 6 &times; 6 &times; 6 &times; 6 + (6 &times; 6/(6 + 6))
77/7−7
:= (7 &times; (7 + 7) &times; (7 + 7)) − (7/7 + 77)
:= ((77 &times; (777/7 + 7)) + 7)/7
:= 8 + ((8 + 8) &times; (88 − 8) + 8/8)
:= 8 + (((8 + 8) &times; (88 − 8) − ((8 + 8)/8)) + 8)
:= 8 + ((8 + 8) &times; (88 − 8) + (88/8))
:= (9 + 9)/9 + (((9 + 9) &times; (9 &times; 9 − 9)) − 9)
:= ((9 + 9) &times; (9 &times; 9 − 9)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + ((9 + 9) &times; (9 &times; 9 − 9))
:= (7 − 7/7)
−7
&Iuml; 1290 := (11 − 1) &times; (11 &times; (1 + 11) − 1 − 1 − 1)
&Iuml; 1295 := ((1 + 1 + 1) &times; (1 + 11))1+1 − 1
2+2
&Iuml; 1300 := (1 + 1 + 11) &times; (11 − 1)1+1
:= 2 + ((2 + 2 + 2)2+2 + 2)
:= (2 + 2 + 2)2+2 − (2 + 2 + 2)
:= (2 + 2 + 2)
:= 3 + (33 &times; (33 + 3 + 3))
:= ((33/3)3 ) − (33 + 3)
:= 3 + (((3 + 3)3/3+3 ) + 3/3)
:= (4/4 + 4) &times; ((4 + 4)/4 + 44 )
:= ((4 + 4)/4 + 4)4 − 4/4
:= 4 + ((4 + 4)/4 + 4)4
:= 5 &times; ((55 + 5)/(5 + 5) − 55)
:= (5 &times; (5 &times; 55 − 5)) − 55
:= (5 + 5) &times; (5 &times; 5 &times; 5 + 5)
:= 6 &times; 6 &times; 6 &times; 6 − 6
:= 6 &times; 6 &times; 6 &times; 6 − 6/6
:= 6 + (6 &times; 6 &times; 6 &times; 6 − ((6 + 6)/6))
:= 7 &times; 7 &times; 7 &times; 7 − 7777/7
:= (7 &times; (7 + 7) &times; (7 + 7)) − 77
:= ((7 + 7)/7) &times; (777/7 + 7 &times; 77)
:= 8 + ((8 + 8) &times; (88 − 8) + ((8 + 8)/8))
:= 8 + (((8 + 8) &times; (88 − 8) − 8/8) + 8)
:= (8/8 + 8 &times; 8) &times; ((88 + 8)/8 + 8)
:= (9/9 + 9) &times; ((999/9 + 9) + 9)
:= ((9 + 9) &times; (9 &times; 9 − 9)) − 9/9
:= (9/9 + 99) &times; ((99 + 9 + 9)/9)
&Iuml; 1291 := 1 + ((11 − 1) &times; (11 &times; (1 + 11) − 1 − 1 − 1))
− 2/2
&Iuml; 1296 := ((1 + 1 + 1) &times; (1 + 11))1+1
&Iuml; 1301 := 1 + (1 + 1 + 11) &times; (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 &times; (33 + 3 + 3)) + 3/3)
:= (3 + 3)3/3+3
:= 3 + (((33/3)3 ) − 33)
:= 44 + (44/4 + 4 &times; 44 )
:= ((4 + 4)/4 + 4)4
:= 4 + (((4 + 4)/4 + 4)4 + 4/4)
:= ((5/5 + 5)
5−5/5
)−5
:= 6/6 + (6 &times; 6 &times; 6 &times; 6 − 6)
:= (5/5 + 5)
5−5/5
:= 5 + ((5/5 + 5)5−5/5 )
:= 6 &times; 6 &times; 6 &times; 6
:= 6 + (6 &times; 6 &times; 6 &times; 6 − 6/6)
77/7−7
:= (7 − 7/7)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) − (7/7 + 77))
:= 88/8 + (8 + 8) &times; (88 − 8)
:= 8 + ((8 + 8) &times; (88 − 8) + 8)
:= (88/8 &times; (888/8 + 8)) − 8
:= 99 + (9999/9 + 9 &times; 9)
:= (9 + 9) &times; (9 &times; 9 − 9)
:= 9 &times; 9 + (((99 &times; 999/9) − 9)/9)
:= 77/7 &times; (777/7 + 7) − 7
&Iuml; 1292 := 111+1+1 − ((1 + 1 + 1) &times; (1 + 1 + 11))
&Iuml; 1297 := 1 + ((1 + 1 + 1) &times; (1 + 11))1+1
&Iuml; 1302 := 1 + 1 + (1 + 1 + 11) &times; (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 &times; 6 &times; 6 &times; 6 − 6)
:= 6/6 + 6 &times; 6 &times; 6 &times; 6
:= 6 + 6 &times; 6 &times; 6 &times; 6
:= (7/7 − 77) &times; (((7 − 77)/7) − 7)
77/7−7
:= 7/7 + (7 − 7/7)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) − 77)
:= ((88 + 8)/8) + (8 + 8) &times; (88 − 8)
:= 88 &times; (8 + 8) − 888/8
:= 88 + (8 &times; (8 &times; 8 + 88) − ((8 + 8)/8))
:= 99 + ((9999 + 9)/9 + 9 &times; 9)
:= 9/9 + ((9 + 9) &times; (9 &times; 9 − 9))
:= 9 &times; 9 + ((99/9) &times; 999/9)
96
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1303 := 1 + 1 + 1 + (1 + 1 + 11) &times; (11 − 1)1+1
&Iuml; 1308 := (1 + 11) &times; (111 − 1 − 1)
&Iuml; 1313 := (1 + 1 + 11) &times; (1 + (11 − 1)1+1 )
:= 2 + ((((2 + 2 + 2)2+2 + 2/2) + 2) + 2)
:= 2 &times; (((22 + 2 + 2)2 ) − 22)
:= 2 + ((22/2 &times; ((22/2)2 − 2)) + 2)
:= ((33/3)3 ) − (33 + 3/3)
:= 3 + (((3 + 3)3/3+3 ) + 3 &times; 3)
:= ((33/3)3 ) − (3 &times; (3 + 3))
:= 4 + ((((4 + 4)/4 + 4)4 − 4/4) + 4)
:= 4 + ((((4 + 4)/4 + 4)4 + 4) + 4)
:= 4 &times; 4 + (((4 + 4)/4 + 4)4 + 4/4)
:= 55 + ((5 + 5)/5 &times; (55 − 5)/5)
:= 5 + (((5 + 5)/5 &times; (55 − 5)/5) + 55)
:= 55 + ((5 + 5)/5 &times; ((55 − 5)/5 + 5))
:= 6 + (6 &times; 6 &times; 6 &times; 6 + 6/6)
:= 6 + (6 &times; 6 &times; 6 &times; 6 + 6)
:= 6 + (6 &times; 6 &times; 6 &times; 6 + (66/6))
:= 7 + (7 − 7/7)
:= 7 &times; 77 + (777 − (7/7 + 7))
:= 7 &times; 77 + (777 − ((7 + 7 + 7)/7))
:= 88 + (8 &times; (8 &times; 8 + 88) − 8/8)
:= (88 &times; (888/8 + 8) − 8)/8
:= 8 + ((8 − 8/8 + 8) &times; (88 − 8/8))
:= 9 + (((9 + 9) &times; (9 &times; 9 − 9)) − ((9 + 9)/9))
:= ((99 + 9)/9) + ((9 + 9) &times; (9 &times; 9 − 9))
:= 9 + ((((9 + 9) &times; (9 &times; 9 − 9)) − 9/9) + 9)
77/7−7
&Iuml; 1304 := 111+1+1 − (1 + 1 + 1)1+1+1
&Iuml; 1309 := 11 &times; (111+1 − 1 − 1)
:= 2 &times; (2 &times; (((22+2 + 2)2 ) + 2))
:= 22/2 &times; ((22/2)2 − 2)
:= 2 + ((2 + 2 + 2)2+2 + 22+2 )
:= ((33/3)3 ) − 33
:= 33/3 + (((33/3)3 ) − 33)
:= (3 + 3) &times; ((3 + 3)3 + 3)
:= 4 + (((4 + 4)/4 + 4)4 + 4)
:= 4 + ((4/4 + 4) &times; ((4/4 + 44 ) + 4))
:= (((4 + 4) &times; (4/4 + 4)4 ) + 44 )/4
:= 5 + ((5 − 5/5)5 + 5 &times; 55)
:= 5 + (((5 − 5/5)5 + 5 &times; 55) + 5)
:= 5 &times; 5 &times; 55 − ((55 + 5/5) + 5)
:= 6 + (6 &times; 6 &times; 6 &times; 6 + ((6 + 6)/6))
:= 6 + ((6 &times; 6 &times; 6 &times; 6 + 6/6) + 6)
:= 6 + ((6 &times; 6 &times; 6 &times; 6 + 6) + 6)
:= ((7 + 7)/7)7 + (7 + 7) &times; (77 + 7)
:= 77 &times; ((77 − 7)/7 + 7)
:= 7 &times; 77 + (777 − ((7 + 7)/7))
:= 88 + 8 &times; (8 &times; 8 + 88)
:= 88/8 &times; (888/8 + 8)
:= (8 + 8)/8 + (88 &times; (8 + 8) − (88 + 8))
:= 9 + (((9 + 9) &times; (9 &times; 9 − 9)) − 9/9)
:= 99 + (9999/9 + 99)
:= 9 + (((9 + 9) &times; (9 &times; 9 − 9)) + 9)
&Iuml; 1305 := (11 − 1 − 1) &times; (1 + (1 + 11)1+1 )
&Iuml; 1310 := 1 + 11 &times; (111+1 − 1 − 1)
&Iuml; 1314 := 1 + (1 + 1 + 11) &times; (1 + (11 − 1)1+1 )
&Iuml; 1315 := 111+1+1 − (1 + 1)1+1+1+1
:= 22/2 + ((2 + 2 + 2)2+2 − 2)
:= ((2 + 2 + 2) &times; 222) − 22
:= (22/2)2/2+2 − 22+2
:= 3 &times; 3 + ((3 + 3)3/3+3 )
:= 3 + ((((33/3)3 ) − 33 ) + 3)
:= 3/3 + ((3 + 3) &times; ((3 + 3)3 + 3))
:= (4/4 + 4) &times; ((4/4 + 44 ) + 4)
:= (4/4 + 4) &times; (((4 + 4)/4 + 44 ) + 4)
:= ((44/4)4−4/4 ) − 4 &times; 4
:= 5 + ((5 + 5) &times; (5 &times; 5 &times; 5 + 5))
:= 5 + (((5 + 5) &times; (5 &times; 5 &times; 5 + 5)) + 5)
:= 5 &times; 5 &times; 55 − (55 + 5)
:= 6 + (6 &times; 6 &times; 6 &times; 6 + (6 &times; 6/(6 + 6)))
:= 6 + ((6 &times; 6 &times; 6 &times; 6 + ((6 + 6)/6)) + 6)
:= 6 + (((6 &times; 6 &times; 6 &times; 6 + 6/6) + 6) + 6)
:= 7 + 77/7 &times; (777/7 + 7)
:= 7 + ((7 − 7/7)77/7−7 + 7)
:= 7 &times; 77 + (777 − 7/7)
:= (8 − 8/8 + 8) &times; (88 − 8/8)
:= (88 &times; (888/8 + 8) + 8)/8
:= ((88/8)88/8−8 ) − 8 − 8
:= 9 + ((9 + 9) &times; (9 &times; 9 − 9))
:= 99 + ((9999 + 9)/9 + 99)
:= 9 + ((((9 + 9) &times; (9 &times; 9 − 9)) + 9/9) + 9)
&Iuml; 1306 := (1 + 11) &times; (111 − 1 − 1) − 1 − 1
:= 2 + ((2 + 2 + 2)2+2 + 2 &times; (2 + 2))
:= 3 + (((33/3)3 ) − (33 + 3/3))
&Iuml; 1311 := 1 + 1 + 11 &times; (111+1 − 1 − 1)
2
:= 2 + (22/2 &times; ((22/2) − 2))
3
:= ((3 + 3) &times; ((3 + 3) + 3)) − 3
4
&Iuml; 1316 := 111+1+1 − 1 − 1 − 1 − 1 − 11
:= 22 + ((2 + 2 + 2)2+2 − 2)
:= 3 + (((33/3)3 ) − (3 &times; (3 + 3)))
:= (44 − 4)/4 + ((4 + 4)/4 + 4)4
:= 44 &times; 44 − (4/4 + 4)
:= 5 + (((5/5 + 5)5−5/5 ) + 5)
:= 5 + ((((5/5 + 5)5−5/5 ) + 5) + 5)
:= 5 &times; 5 + (((5/5 + 5)5−5/5 ) − 5)
:= ((66 − 6)/6) + 6 &times; 6 &times; 6 &times; 6
:= 6 + ((6 &times; 6 &times; 6 &times; 6 + (6 &times; 6/(6 + 6))) + 6)
:= 6 + (((6 &times; 6 &times; 6 &times; 6 + ((6 + 6)/6)) + 6) + 6)
:= 7 + (((77 &times; (777/7 + 7)) + 7)/7)
:= 7 + ((7 + 7) &times; (77 + 7) + ((7 + 7)/7)7 )
:= 7 &times; 77 + 777
:= 8 + (((8 − 888)/8) + 88 &times; (8 + 8))
:= 88 &times; (8 + 8) − ((8/8 + 88) + 8)
:= 8 + ((88 &times; (888/8 + 8) − 8)/8)
:= 9 + (((9 + 9) &times; (9 &times; 9 − 9)) + 9/9)
:= 99/9 &times; (999/9 + 9) − 9
:= 9 + (((9 + 9) &times; (9 &times; 9 − 9)) + (99/9))
&Iuml; 1307 := (1 + 11) &times; (111 − 1 − 1) − 1
&Iuml; 1312 := 1 + 1 + 1 + 11 &times; (111+1 − 1 − 1)
:= 4 + (4 &times; (44 + 4 + 4) + 44 )
&Iuml; 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) &times; ((3 + 3)3 + 3)) − 3)
:= 3 + ((3 + 3) &times; ((3 + 3)3 + 3))
:= 44/4 + ((4 + 4)/4 + 4)
4
4
:= 4 &times; ((4 &times; (4 − 4/4) ) + 4)
:= (4 − 4/4) &times; (444 − (4/4 + 4))
:= 55 + ((5 + 5)/5 &times; (55 + 5)/5)
:= 5 + (((5 + 5)/5 &times; (5 + 5)/5) + 55)
:= 55 + ((5 + 5)/5 &times; ((55 + 5)/5 + 5))
:= 66/6 + 6 &times; 6 &times; 6 &times; 6
:= 6 + (((66 − 6)/6) + 6 &times; 6 &times; 6 &times; 6)
:= 6 &times; 6 &times; 6 &times; 6 + ((6 &times; 6 + 6)/((6 + 6)/6))
:= (7 + 7) &times; (7 + 7) + 7777/7
:= (7 &times; ((7 + 7) &times; (7 + 7) − 7)) − 77/7
:= 7/7 + (7 &times; 77 + 777)
5
:= 8 + (((8 + 8) &times; (88 − 8) + (88/8)) + 8)
:= 88 &times; (8 + 8) − (88 + 8)
:= 8 + (88/8 &times; (888/8 + 8))
:= 99/9 + ((9 + 9) &times; (9 &times; 9 − 9))
:= (9 − 9/9) &times; (((9 + 9)/9) + 9 &times; (9 + 9))
:= 9 + (((9 + 9) &times; (9 &times; 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
&Iuml; 1318 := 111+1+1 − 1 − 1 − 11
&Iuml; 1323 := 1 + 1 + 1 + 11 &times; (111+1 − 1)
&Iuml; 1328 := 111+1+1 − 1 − 1 − 1
:= 22 + (2 + 2 + 2)2+2
:= (2/2 + 2) &times; ((22 − 2/2)2 )
:= 2 &times; (((2/2 + 2) &times; 222) − 2)
:= 3 + (((3 + 3) &times; ((3 + 3)3 + 3)) + 3/3)
:= 33 + ((3 + 3)3/3+3 )
:= ((33/3)3 ) − 3
:= 4 + ((((4 + 4) &times; (4/4 + 4)4 ) + 44 )/4)
:= ((44/4)4−4/4 ) − 4 − 4
:= ((4 − 4/4) &times; 444) − 4
:= 5 &times; 5 &times; 55 − ((5 + 5)/5 + 55)
:= (5 &times; (5 &times; 55 − (5 + 5))) − (5 + 5)/5
:= 5 + ((5 &times; (5 &times; 55 − (5 + 5))) − ((5 + 5)/5))
:= ((6 + 6)/6) &times; (666 − 6/6 − 6)
:= 6 &times; 6 + ((66/6) &times; (666/6 + 6))
:= 6 &times; 6 &times; 6 + (6666 + 6)/6
:= 7 &times; 77 + (((7 + 7)/7) + 777)
:= 7 &times; ((7 + 7) &times; (7 + 7) − 7)
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) − 7)) − ((7 + 7)/7))
:= 88 &times; (8 + 8) − ((8 + 8)/8 + 88)
:= ((88/8)88/8−8 ) − 8
:= 8 + 88 &times; (8 − 8/8 + 8)
:= 9 + ((9999/9 + 99) + 99)
:= 999 + (9 + 9) &times; (9 + 9)
&Iuml; 1319 := 111+1+1 − 1 − 11
&Iuml; 1324 := 1 + 1 + 1 + 1 + 11 &times; (111+1 − 1)
:= 9 + ((((9 + 9)/9)99/9 ) − 9 &times; 9 &times; 9)
&Iuml; 1329 := 111+1+1 − 1 − 1
:= 22 + ((2 + 2 + 2)2+2 + 2/2)
:= 2 + (((2 + 2 + 2) &times; (222 − 2)) + 2)
:= (22/2)2/2+2 − 2
:= ((33/3)3 ) − (3 &times; 3 + 3)
:= ((33/3)3 ) − ((3/3 + 3) + 3)
:= 3/3 + (((33/3)3 ) − 3)
:= 4 + (((44/4)4−4/4 ) − 4 &times; 4)
:= 44 + (4 &times; 44 + 44 )
:= (4 − 4/4) &times; (444 − 4/4)
:= 5 &times; 5 &times; 55 − (55 + 5/5)
:= (5 &times; (55 + 5)) + (5 − 5/5)5
:= 5 + ((5 &times; (55 + 5)) + (5 − 5/5)5 )
:= (((6 + 6) &times; (666 − 6)) − 6)/6
:= ((6 + 6)/6)6 + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 6 &times; 6 &times; 6 &times; 6 + (66 &times; 6/(6 + 6))
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) − 7)) − (77/7))
:= 7/7 + (7 &times; ((7 + 7) &times; (7 + 7) − 7))
:= 88 &times; (8 + 8) − (8/8 + 88)
:= (((88 + 8)/8) &times; 888/8) − 8
:= (((9 + 9)/9)99/9 ) − 9 &times; 9 &times; 9
:= 9/9 + ((9 + 9) &times; (9 + 9) + 999)
&Iuml; 1325 := 111+1+1 − (1 + 1) &times; (1 + 1 + 1)
&Iuml; 1320 := 11 &times; (111+1 − 1)
:= (2 + 2 + 2) &times; (222 − 2)
:= 2 + ((2/2 + 2) &times; ((22 − 2/2)2 ))
:= (3/3 + 3) &times; (333 − 3)
:= ((33/3)3 ) − (3 + 3)
:= (4 − 4/4) &times; (444 − 4)
:= 44 + ((4 &times; 44 + 4/4) + 44 )
:= 55 &times; (5 &times; 5 − 5/5)
:= 5 &times; (5 &times; 55 − (5 + 5))
:= (6 + 6) &times; ((666 − 6)/6)
:= (6 &times; (6 &times; 6 &times; 6 + 6)) − 6/6 − 6
:= (77 + 7)/7 &times; (777 − 7)/7
:= ((7 + 7)/7) + (7 &times; ((7 + 7) &times; (7 + 7) − 7))
:= 88 &times; (8 − 8/8 + 8)
:= 8 + ((88/8 &times; (888/8 + 8)) + 8)
:= 99/9 &times; (999/9 + 9)
:= 9 + ((((9 + 9) &times; (9 &times; 9 − 9)) + (99/9)) + 9)
&Iuml; 1321 := 1 + 11 &times; (111+1 − 1)
&Iuml; 1326 := 111+1+1 − 1 − 1 − 1 − 1 − 1
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) − 7)) − 7/7)
:= 8 + (88 &times; (8 − 8/8 + 8) + 8/8)
:= 9 + 99/9 &times; (999/9 + 9)
&Iuml; 1330 := 111+1+1 − 1
:= ((2 + 2 + 2) &times; 222) − 2
:= ((33/3)3 ) − 3/3
:= ((44/4)4−4/4 ) − 4/4
:= 5 + (5 &times; (5 &times; 55 − (5 + 5)))
:= ((6 + 6)/6) &times; (666 − 6/6)
:= 7 + (7 &times; ((7 + 7) &times; (7 + 7) − 7))
:= ((88/8)88/8−8 ) − 8/8
:= (9/9 + 9 + 9) &times; (9 &times; 9 − 99/9)
&Iuml; 1331 := 111+1+1
:= (22/2)2/2+2
:= 2/2 + ((2 + 2 + 2) &times; (222 − 2))
:= (2/2 + 2) &times; (2 &times; 222 − 2)
:= ((3 − 33)/3) + ((33/3)3 )
:= 3 + (((3 + 3)3/3+3 ) + 33 )
:= 4/4 + ((4 − 4/4) &times; (444 − 4))
:= (4 − 4/4) &times; (444 − (4 + 4)/4)
:= 5 &times; 5 + ((5/5 + 5)5−5/5 )
:= 5/5 + (5 &times; (5 &times; 55 − (5 + 5)))
:= (55/5)5−(5+5)/5
:= (6 &times; (6 &times; 6 &times; 6 + 6)) − 66/6
:= (6 &times; (6 &times; 6 &times; 6 + 6)) − 6
:= (66/6)6&times;6/(6+6)
:= (7 &times; ((7 + 7) &times; (7 + 7) − 7)) − (7 + 7)/7
:= (7/7 + 77) &times; ((77 − 7)/7 + 7)
:= (77/7)(7+7+7)/7
:= 8/8 + 88 &times; (8 − 8/8 + 8)
:= 8 + (88 &times; (8 + 8) − ((8 + 8)/8 + 88))
:= (88/8)88/8−8
:= 9 + ((9 − 9/9) &times; (((9 + 9)/9) + 9 &times; (9 + 9)))
:= ((9 − 9/9) + 9) &times; (9 &times; 9 − ((9 + 9 + 9)/9))
:= (99/9)(9+9+9)/9
&Iuml; 1322 := 1 + 1 + 11 &times; (111+1 − 1)
&Iuml; 1327 := 111+1+1 − 1 − 1 − 1 − 1
:= 2 + ((2 + 2 + 2) &times; (222 − 2))
:= (22/2)
3
2/2+2
−2−2
3
:= ((33/3) ) − 3 &times; 3
:= (33/3)3
:= (44/4)4−4/4
&Iuml; 1332 := 1 + 111+1+1
:= (2 + 2 + 2) &times; 222
:= ((33/3) ) − (3/3 + 3)
:= 3/3 + ((33/3)3 )
:= 44 + ((4 &times; 4 − (4 + 4)/4) + 4 )
:= ((44/4)4−4/4 ) − 4
:= (4 − 4/4) &times; 444
:= (5 + 5)/5 + (55 &times; (5 &times; 5 − 5/5))
:= (5 + 5)/5 + (5 &times; (5 &times; 55 − (5 + 5)))
:= ((55 + 5)/5) &times; 555/5
:= ((6 + 6)/6) &times; ((666 − 6) + 6/6)
:= 6/6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) − 6)
:= 6 &times; (6 &times; 6 &times; 6 + 6)
:= (7 &times; ((7 + 7) &times; (7 + 7) − 7)) − 7/7
4
4
:= 7 &times; 77 + (77/7 + 777)
:= (77 + 7)/7 &times; 777/7
:= (8 + 8)/8 + 88 &times; (8 − 8/8 + 8)
:= 8 + (88 &times; (8 + 8) − (8/8 + 88))
:= ((88 + 8)/8) &times; 888/8
:= 9 &times; (9 &times; 9 + 9) + (((9 + 9)/9)9 )
:= 9 + (((9999/9 + 99) + 99) + 9)
:= 9 + ((9 + 9) &times; (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
&Iuml; 1338 := 11 + 111+1+1 − 1 − 1 − 1 − 1
&Iuml; 1333 := 1 + 1 + 111+1+1
:= 2 + (22/2)2/2+2
:= (2/2 + 2) &times; (2 &times; 222 + 2)
&Iuml; 1343 := 1 + 11 + 111+1+1
:= 22/2 + ((2 + 2 + 2) &times; 222)
:= 3 + (((33/3)3 ) − 3/3)
:= 3 + ((((33/3) ) + 3/3) + 3)
:= 3 + (((33/3)3 ) + 3 &times; 3)
:= 4/4 + ((4 − 4/4) &times; 444)
:= (4 + 4)/4 &times; ((4/4 + 4)4 + 44)
:= 4 + ((((44/4)4−4/4 ) + 4) + 4)
:= ((555 &times; ((55 + 5)/5)) + 5)/5
:= 5 &times; 5 &times; 55 − (((5 + 5)/5)5 + 5)
:= 5 &times; 5 &times; 55 − ((5 + 5)/5)5
:= 6/6 + (6 &times; (6 &times; 6 &times; 6 + 6))
:= 6 + (6 &times; (6 &times; 6 &times; 6 + 6))
:= 66/6 + (6 &times; (6 &times; 6 &times; 6 + 6))
3
:= ((777 &times; (77 + 7)/7) + 7)/7
:= 7 + ((77/7)
:= 88 &times; (8 + 8) − (88/8 + 8 &times; 8)
:= 8 + (((88/8)88/8−8 ) − 8/8)
:= 88 &times; (8 + 8) − (8/8 + 8 &times; 8)
:= 9 + (((9 + 9) &times; (9 + 9) + 999) + 9/9)
:= 9 &times; 9 &times; (9 + 9) − (999/9 + 9)
:= ((9 − 9/9) + 9) &times; (9 &times; 9 − ((9 + 9)/9))
&Iuml; 1334 := 1 + 1 + 1 + 111+1+1
(7+7+7)/7
)
&Iuml; 1339 := 11 + 111+1+1 − 1 − 1 − 1
:= 2 + ((2 + 2 + 2) &times; 222)
:= ((77 − 7)/7 + 7) &times; ((7 + 7)/7 + 77)
&Iuml; 1344 := (1 + 11) &times; (1 + 111)
:= 2 &times; (2 + 2) + (22/2)2/2+2
:= (2 + 2 + 2) &times; (222 + 2)
:= 3 &times; 3 + (((33/3)3 ) − 3/3)
:= (3/3 + 3) &times; (333 + 3)
:= 4 + (((44/4)4−4/4 ) + 4)
:= 4 &times; ((4 &times; (4 &times; 4 + 4)) + 44 )
:= (5 &times; (5 &times; 55 − 5)) − (55/5 + 5)
:= (5 &times; (5 &times; 55 − 5)) − 55/5
:= (5 &times; 5 − 5/5) &times; (55 + 5/5)
:= (6 + 6)/6 + (6 &times; (6 &times; 6 &times; 6 + 6))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + 6/6)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + 6)
:= 77/7 + (7 &times; ((7 + 7) &times; (7 + 7) − 7))
:= 7 + ((77 + 7)/7 &times; 777/7)
:= (7 + 7) &times; (7 &times; (7 + 7) − ((7 + 7)/7))
:= 888 + (8 &times; (8 &times; 8 − 8) − ((8 + 8)/8))
:= 8 + ((88/8)88/8−8 )
:= 8 &times; (88 − 8 + 88)
3
:= 3 + ((33/3) )
:= 4 + (((44/4)
4−4/4
) − 4/4)
:= 99/9 + ((9 + 9) &times; (9 + 9) + 999)
&Iuml; 1335 := 1 + 1 + 1 + 1 + 111+1+1
:= 2 + ((22/2)
2/2+2
+ 2)
:= 9 + ((9/9 + 9 + 9) &times; (9 &times; 9 − (99/9)))
&Iuml; 1340 := 11 + 111+1+1 − 1 − 1
:= (2 &times; 22) + (2 + 2 + 2)2+2
:= ((99 + 9)/9) &times; ((999 + 9)/9)
&Iuml; 1345 := 1 + (1 + 11) &times; (1 + 111)
:= 2/2 + ((2 + 2 + 2) &times; (222 + 2))
:= 3 + (((33/3)3 ) + 33/3)
:= 3 + (((33/3)3 ) + 3/3)
:= 3 &times; 3 + ((33/3)3 )
:= 4 + ((44/4)4−4/4 )
:= 44 + ((4 + 4)/4 + 4)4
:= 5 + ((5 &times; (5 &times; 55 − (5 + 5))) + 5)
:= (5 &times; (5 &times; 55 − 5)) − 5 − 5
:= (6 &times; 6/(6 + 6)) + (6 &times; (6 &times; 6 &times; 6 + 6))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + ((6 + 6)/6))
:= (77 + 7)/7 + (7 &times; ((7 + 7) &times; (7 + 7) − 7))
:= 7 + (((777 &times; (77 + 7)/7) + 7)/7)
:= 7 + (((77/7)(7+7+7)/7 ) + 7)
:= (8 − 8/8 + 8) &times; (8/8 + 88)
:= 8 + (((88 + 8)/8) &times; 888/8)
:= 8/8 + 8 &times; (88 − 8 + 88)
:= (9 − (9 &times; 9 &times; 99))/(((9 + 9 + 9)/9) − 9)
:= 9 + (99/9)(9+9+9)/9
:= 9 &times; (9 + 9 + 9) + (9999/9 − 9)
&Iuml; 1336 := 1 + 1 + 1 + 1 + 1 + 111+1+1
:= 2 + (((2 + 2 + 2) &times; 222) + 2)
:= 4/4 + ((4 &times; (4 &times; 4 + 44 )) + 44 )
:= (5 &times; (5 &times; 55 − 5)) − 5
:= (((6 + 6) &times; (666 + 6)) + 6)/6
&Iuml; 1346 := 1 + 1 + (1 + 11) &times; (1 + 111)
&Iuml; 1341 := 11 + 111+1+1 − 1
:= (2 &times; ((22 + 2 + 2)2 )) − 22/2
:= 2 + ((2 + 2 + 2) &times; (222 + 2))
:= 3 + ((((33/3) ) − 3/3) + 3)
:= 33 + ((3 + 3) &times; ((3 + 3)3 + 3))
:= 3 + ((((33/3)3 ) + 3 &times; 3) + 3)
:= 4 + ((4 − 4/4) &times; 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 &times; (5 &times; 55 − 5)) − 5)
3
:= 5 + ((55/5)5−(5+5)/5 )
:= 6 + (((6 + 6)/6) &times; (666 − 6/6))
:= 7 + (((7 &times; ((7 + 7) &times; (7 + 7) − 7)) − 7/7) + 7)
:= 888 + 8 &times; (8 &times; 8 − 8)
:= 9 &times; 99 + (((9 &times; 9 &times; 99) − 9)/(9 + 9))
&Iuml; 1337 := 1 + 1 + 1 + 1 + 1 + 1 + 111+1+1
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + (6 &times; 6/(6 + 6)))
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) − 7)) + (77/7))
:= 8 + (88 &times; (8 + 8) − (88/8 + 8 &times; 8))
:= 9 &times; 9 &times; (9 + 9) − (99 + 9 + 9)
:= ((6 + 6)/6) &times; (666 + 6/6 + 6)
:= 7 + (((77 + 7)/7 &times; 777/7) + 7)
:= (8 + 8)/8 + 8 &times; (88 − 8 + 88)
:= 9 &times; 9 &times; (9 + 9) − ((999 + 9)/9)
&Iuml; 1347 := 1 + 1 + 1 + (1 + 11) &times; (1 + 111)
&Iuml; 1342 := 11 + 111+1+1
:= ((2 + 2 + 2) &times; (222 + 2)) − 2
:= 22+2 + (22/2)2/2+2
:= 3 + (((33/3)3 ) + 3)
:= 33/3 + ((33/3)3 )
:= 3 + ((3/3 + 3) &times; (333 + 3))
:= 4 + (((4 − 4/4) &times; 444) + 4/4)
:= 44/4 + ((44/4)4−4/4 )
:= 4 &times; 4 + ((44/4)4−4/4 )
:= 5 + (((55 + 5)/5) &times; 555/5)
:= 55/5 &times; ((555 + 55)/5)
:= (5 + 5)/5 + ((5 &times; (5 &times; 55 − 5)) − 5)
:= 2 + (((22/2)
:= 6 + ((66/6)
2/2+2
+ 2) + 2)
6&times;6/(6+6)
)
:= 66/6 &times; ((666 + 66)/6)
:= ((66/6) &times; ((666/6 + 6) + 6)) − 6
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) − 7)) + 7)
:= 77/7 &times; (777 + 77)/7
:= 7 &times; 7 + 77/7 &times; (777/7 + 7)
:= 8/8 + (8 &times; (8 &times; 8 − 8) + 888)
:= 88/8 &times; ((888 + 88)/8)
:= 8 + (((88/8)88/8−8 ) + 8)
:= 9 &times; 99 + (((9 &times; 9 &times; 99) + 9)/(9 + 9))
:= 99/9 &times; (999 + 99)/9
:= 9 &times; 9 &times; (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
&Iuml; 1348 := 1 + 1 + 1 + 1 + (1 + 11) &times; (1 + 111)
2
&Iuml; 1353 := 11 &times; (1 + 1 + 111+1 )
&Iuml; 1358 := 1 + 1 + (1 + 11) &times; (1 + 1 + 111)
:= 2 + (2 &times; (((22 + 2 + 2)2 ) + 2))
2/2+2
:= 2 &times; (((22 + 2 + 2) ) − 2)
:= 22 + (22/2)
:= 3 + ((((33/3)3 ) + 33/3) + 3)
:= 3 + ((3 + 3) &times; ((3 + 3)3 + 3 &times; 3))
:= 33 + ((33/3)3 )
:= 4 &times; ((4 − 4/4)4 + 44 )
:= 4 + ((4 &times; ((4 − 4/4)4 + 44 )) + 4/4)
:= 4 + ((44 &times; (44 − 4) − 44 )/(4 + 4))
:= (5 &times; (5 &times; 55 − 5)) − (5 + 5)/5
:= 5 + ((5 &times; (5 &times; 55 − 5)) − ((5 + 5)/5))
:= (5 + 5)/5 &times; ((55 − 5)/5 + 55)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + ((66 − 6)/6))
:= 66/6 &times; ((666/6 + 6) + 6)
:= 66 + ((6 &times; 6 &times; 6 &times; 6 − 6) + ((6 + 6)/6))
:= (77/7 − 7) &times; ((7 &times; 7 &times; 7 − 7) + 7/7)
:= 77/7 &times; ((777 + 77 + 7)/7)
:= (7 + 7) &times; (7 &times; (7 + 7) − 7/7)
:= 8 + ((((88 + 8)/8) &times; 888/8) + 8)
:= 8 + (8 &times; (88 − 8 + 88) + 8/8)
:= 8 + ((8 − 8/8 + 8) &times; ((8 + 8)/8 + 88))
:= 9 &times; 9 &times; (9 + 9) + ((9 − 999)/9)
:= 9 + (((99 + 9)/9) &times; ((999 + 9)/9))
:= 9 &times; 9 &times; (9 + 9) − (9/9 + 99)
&Iuml; 1349 := 111+1+1 + (1 + 1) &times; (11 − 1 − 1)
2
&Iuml; 1354 := 1 + 11 &times; (1 + 1 + 111+1 )
&Iuml; 1359 := 1 + 1 + 1 + (1 + 11) &times; (1 + 1 + 111)
:= 2/2 + (2 &times; (((22 + 2 + 2) ) − 2))
:= 2 + (2 &times; ((22 + 2 + 2)2 ))
:= 2 + ((2 &times; (((22 + 2 + 2)2 ) + 2)) + 2/2)
:= (3 &times; (3 + 3)) + ((33/3)3 )
:= (3333 + 33+3 )/3
:= 3 &times; ((3 &times; (3 + 3) &times; 33 ) − 33)
:= 4/4 + (4 &times; ((4 − 4/4) + 4 ))
:= (44 &times; (44 − 4) − 44 )/(4 + 4)
:= 44 + (4444/4 − (4 + 4))
:= (5 &times; (5 &times; 55 − 5)) − 5/5
:= 5 + ((5 &times; (5 &times; 55 − 5)) − 5/5)
:= 5 &times; 5 &times; 55 − (55/5 + 5)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + (66/6))
:= ((6 + 6)/6) &times; (666 + (66/6))
:= 6 + ((66/6) &times; ((666/6 + 6) + 6))
:= 7 + (77/7 &times; (777 + 77)/7)
:= (7 &times; (7 + 7) &times; (7 + 7)) − (77/7 + 7)
:= 7/7 + ((7 + 7) &times; (7 &times; (7 + 7) − 7/7))
4
4
:= (88/8 + 8) &times; ((8 &times; 8 − 8/8) + 8)
:= 8 + (8 &times; (88 − 8 + 88) + ((8 + 8)/8))
:= (8/8 + 8) &times; ((88 − 8/8) + 8 &times; 8)
:= 9 + ((99/9)(9+9+9)/9 + 9)
:= 9 &times; (9 + 9 + 9) + 9999/9
:= 9 &times; 9 &times; (9 + 9) − 99
&Iuml; 1350 := 111+1+1 + (1 + 1) &times; (11 − 1) − 1
&Iuml; 1355 := 11 + (1 + 11) &times; (1 + 111)
&Iuml; 1360 := 1 + 1 + 1 + 1 + (1 + 11) &times; (1 + 1 + 111)
:= (2 &times; ((22 + 2 + 2) )) − 2
:= 2 + ((2 &times; ((22 + 2 + 2) )) + 2/2)
:= 2 &times; ((((22 + 2 + 2)2 ) + 2) + 2)
:= (3 + 3) &times; ((3 + 3)3 + 3 &times; 3)
:= 33 + (((33/3)3 ) − 3)
:= 3 + ((((33/3)3 ) − 3/3) + 33 )
:= 44 + ((4444 − 4)/4 − 4 &times; 4)
:= (4/4 + 4) &times; ((44/4 + 44 ) + 4)
:= 4 &times; (4 &times; ((4 − 4/4)4 + 4))
:= 5 &times; (5 &times; 55 − 5)
:= 5 + (5 &times; (5 &times; 55 − 5))
:= 5 + ((5 &times; (5 &times; 55 − 5)) + 5)
:= 6 + (((6 &times; (6 &times; 6 &times; 6 + 6)) + 6) + 6)
:= 66 + (6 &times; 6 &times; 6 &times; 6 − (6/6 + 6))
:= ((6 + 6)/6)6 + 6 &times; 6 &times; 6 &times; 6
:= ((77 − 7)/7) &times; (((7 + 7)/7)7 + 7)
:= ((7 − 77)/7) + ((7 &times; (7 + 7) &times; (7 + 7)) − 7)
:= (7 &times; (7 + 7) &times; (7 + 7)) − (77 + 7)/7
:= (8 − 8/8 + 8) &times; ((8 + 8)/8 + 88)
:= 88/8 + 8 &times; (88 − 8 + 88)
:= (88 − 8) &times; (8/8 + 8 + 8)
:= 9 &times; 9 &times; (9 + 9) − (99 + 9)
:= 9 + (9 &times; 9 &times; (9 + 9) − ((999 + 9)/9))
:= 9/9 + (9 &times; 9 &times; (9 + 9) − 99)
2
&Iuml; 1351 := 111+1+1 + (1 + 1) &times; (11 − 1)
2
&Iuml; 1356 := (1 + 11) &times; (1 + 1 + 111)
&Iuml; 1361 := 111+1+1 + (11 − 1) &times; (1 + 1 + 1)
:= (2 &times; ((22 + 2 + 2)2 )) − 2/2
:= 2 &times; (((22 + 2 + 2)2 ) + 2)
:= 2/2 + (2 &times; ((((22 + 2 + 2)2 ) + 2) + 2))
:= 3 &times; 3 + (((33/3)3 ) + 33/3)
:= (3/3 + 3) &times; ((333 + 3) + 3)
:= 3 + (((33/3)3 ) + 33 )
:= 4 + (((44/4)4−4/4 ) + 4 &times; 4)
:= (4 − 4/4) &times; (444 + 4 + 4)
:= 4/4 + (4 &times; (4 &times; ((4 − 4/4)4 + 4)))
:= 5/5 + (5 &times; (5 &times; 55 − 5))
:= 5 + ((5 &times; (5 &times; 55 − 5)) + 5/5)
:= 55/5 + (5 &times; (5 &times; 55 − 5))
:= 6 + ((((6 + 6) &times; (666 + 6)) + 6)/6)
:= 66 + (6 &times; 6 &times; 6 &times; 6 − 6)
:= 66 + (6 &times; 6 &times; 6 &times; 6 − 6/6)
7
:= 77 + (7 + 7) &times; (77 + 7 + 7)
:= (7 − 7/7) &times; (((7 + 7)/7) + 7 &times; (7 + 7))
:= (7 &times; (7 + 7) &times; (7 + 7)) − 77/7
:= 8 + (88 &times; (8 + 8) − (8/8 + 8 &times; 8))
:= ((88 + 8)/8) + 8 &times; (88 − 8 + 88)
:= 8/8 + (88 − 8) &times; (8/8 + 8 + 8)
:= 9/9 + (9 &times; 9 &times; (9 + 9) − (99 + 9))
:= 9 + (9 &times; 9 &times; (9 + 9) − 999/9)
:= (9 + 9)/9 + (9 &times; 9 &times; (9 + 9) − 99)
&Iuml; 1352 := 11 &times; (1 + (1 + 111+1 )) − 1
&Iuml; 1357 := 1 + (1 + 11) &times; (1 + 1 + 111)
&Iuml; 1362 := 11 &times; (1 + 1 + 1 + 111+1 ) − 1 − 1
:= 2 &times; ((22 + 2 + 2)2 )
:= 2/2 + (2 &times; (((22 + 2 + 2)2 ) + 2))
:= (22 &times; (22+2+2 − 2)) − 2
:= 3 + (((33/3)3 ) + (3 &times; (3 + 3)))
:= 33 + (((33/3)3 ) − 3/3)
:= (3 + 3) &times; ((3 + 3)3 + 33/3)
:= 4 + (4 &times; ((4 − 4/4)4 + 44 ))
:= 4/4 + ((4 − 4/4) &times; (444 + 4 + 4))
:= 44 + ((4444 − 4)/4 − 4)
:= (5 + 5)/5 + (5 &times; (5 &times; 55 − 5))
:= 5 + ((5 &times; (5 &times; 55 − 5)) + ((5 + 5)/5))
:= (5 + 5)/5 &times; ((55 + 5)/5 + 55)
:= 6 + (((6 + 6)/6) &times; (666 + 6/6 + 6))
:= 66 + ((6 &times; 6 &times; 6 &times; 6 − 6) + 6/6)
:= 66 + 6 &times; 6 &times; 6 &times; 6
:= (7 − 7/7 + 7) &times; (777/7 − 7)
:= (7 &times; (7 + 7) &times; (7 + 7)) − (7/7 + 7 + 7)
:= ((7 − 77)/7) + (7 &times; (7 + 7) &times; (7 + 7))
:= 8 + 8 &times; (88 − 8 + 88)
:= 8 + ((88/8 + 8) &times; ((8 &times; 8 − 8/8) + 8))
:= (8 + 8)/8 + (88 − 8) &times; (8/8 + 8 + 8)
:= 9 + (((9 − 9/9) + 9) &times; (9 &times; 9 − ((9 + 9)/9)))
:= 9 &times; 9 &times; (9 + 9) − ((9 + 9)/9 + 99)
:= 999 + ((99 &times; 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
&Iuml; 1363 := 11 &times; (1 + 1 + 1 + 111+1 ) − 1
&Iuml; 1368 := (1 + 11) &times; (1 + 1 + 1 + 111)
&Iuml; 1373 := 1 + (1 + 1 + 1 + 11)1+1+1 /(1 + 1)
:= 22/2 + (2 &times; ((22 + 2 + 2)2 ))
:= (2 &times; 222 ) + (22 − 2)2
:= 2 + ((((2 + 2 + 2)2 + 2/2)2 ) + 2)
:= 33 + (((33/3)3 ) − 3/3)
:= (3 + 3) &times; (((3 + 3)3 + 3 &times; 3) + 3)
:= 3 &times; 3 + (((33/3)3 ) + 33)
:= 44 + (4444/4 − 4)
:= 4 + (44 &times; (4 &times; (4 + 4) − 4/4))
:= 4/4 + (((4 + 4) &times; (4 &times; 44 − 4)) − 4)
:= 5 &times; 5 &times; 55 − (55 + 5)/5
:= 5 &times; 5 &times; 55 − ((5 + 5)/5 + 5)
:= 5 &times; 5 &times; 55 − (5 + 5)/5
:= 66 + (6 &times; 6 &times; 6 &times; 6 + 6/6)
:= 6 &times; (6 &times; 6 &times; 6 + 6 + 6)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6 + 6)) − 6/6)
:= (7 &times; (7 + 7) &times; (7 + 7)) − ((7 + 7)/7 + 7)
:= (77/7 + 7) &times; (77 − 7/7)
:= 7/7 + (7 &times; (7 + 7) &times; (7 + 7))
:= 8 + (8 &times; (88 − 8 + 88) + (88/8))
:= 88 + (8 + 8) &times; (88 − 8)
:= 8 &times; 8 + (88/8 &times; (888/8 + 8))
:= 9 + (9999/9 + 9 &times; (9 + 9 + 9))
:= 9 &times; (9 &times; (9 + 9) − 9) − 9
:= ((9 − 9 &times; 9)/(9 + 9)) + 9 &times; (9 &times; (9 + 9) − 9)
&Iuml; 1364 := 11 &times; (1 + 1 + 1 + 111+1 )
&Iuml; 1369 := (111/(1 + 1 + 1))1+1
&Iuml; 1374 := 11 &times; (1 + 1 + 1 + 1 + 111+1 ) − 1
:= 22 &times; (22+2+2 − 2)
:= ((2 + 2 + 2)2 + 2/2)2
:= 22 + (2 &times; ((22 + 2 + 2)2 ))
:= 33 + ((33/3)3 )
:= (3/3 + 33 + 3)3−3/3
:= 33+3 + (3 &times; (3 + 3)3 − 3)
:= 44 &times; (4 &times; (4 + 4) − 4/4)
:= 4 + ((((4 + 4)4 ) − 4/4)/(4 − 4/4))
:= ((4 + 4) &times; (4 &times; 44 − 4)) − (4 + 4)/4
:= 5 &times; 5 &times; 55 − 55/5
:= 5 &times; 5 &times; 55 − (5/5 + 5)
:= 5 &times; 5 &times; 55 − 5/5
:= 66 + (6 &times; 6 &times; 6 &times; 6 + ((6 + 6)/6))
:= (6 &times; 6 + 6/6)
:= 6 + (6 &times; (6 &times; 6 &times; 6 + 6 + 6))
:= (7 &times; (7 + 7) &times; (7 + 7)) − (7/7 + 7)
:= (7 &times; (7 + 7) &times; (7 + 7)) − (7 + 7 + 7)/7
:= ((7 + 7)/7) + (7 &times; (7 + 7) &times; (7 + 7))
:= 88 &times; (8 + 8) − (88/((8 + 8)/8))
:= 8/8 + ((8 + 8) &times; (88 − 8) + 88)
:= ((8 + 8)/8) &times; (8 &times; 88 − (8/8 + 8 + 8))
:= 999 + (((9 &times; 9 &times; 9 &times; 9) + 9)/(9 + 9))
:= 9/9 + (9 &times; (9 &times; (9 + 9) − 9) − 9)
:= 9 &times; (9 &times; (9 + 9) − 9) − (9 + 9 + 9)/9
&Iuml; 1365 := 1 + 11 &times; (1 + 1 + 1 + 111+1 )
(6+6)/6
&Iuml; 1375 := 11 &times; (1 + 1 + 1 + 1 + 111+1 )
&Iuml; 1370 := 1 + (111/(1 + 1 + 1))1+1
:= 2/2 + (22 &times; (22+2+2 − 2))
:= 2 + ((2 &times; 222 ) + (22 − 2)2 )
:= (2 &times; 22) + (22/2)2/2+2
:= 3/3 + (((33/3)3 ) + 33)
:= 3 + ((((33/3)3 ) + 33) + 3)
:= 33/3 &times; ((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 &times; 5 &times; 55 − 5 − 5
:= 5 &times; 5 &times; 55 − 5
:= 5 &times; 5 &times; 55
:= 6 + ((6 &times; 6 + 6/6)(6+6)/6 )
:= (6/6 + 6 + 6) &times; (666/6 − 6)
:= 6/6 + ((6 &times; 6 + 6/6)
:= (7 &times; (7 + 7) &times; (7 + 7)) − 7
:= (7 &times; (7 + 7) &times; (7 + 7)) − (7 + 7)/7
:= 7 + ((77/7 + 7) &times; (77 − 7/7))
:= (8 − 8/8 + 8) &times; ((88/8 − 8) + 88)
:= 88 + ((8 + 8) &times; (88 − 8) + ((8 + 8)/8))
:= ((8 + 8) &times; (88 − ((8 + 8)/8))) − 8/8
:= 9 &times; (9 &times; (9 + 9) − 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 &times; (9 &times; (9 + 9) − 9) − 9)
:= 9 &times; (9 &times; (9 + 9) − 9) − (9 + 9)/9
&Iuml; 1366 := 11 + (1 + 11) &times; (1 + 1 + 111) − 1
2+2+2
(6+6)/6
)
&Iuml; 1371 := 1 + 1 + (111/(1 + 1 + 1))1+1
&Iuml; 1376 := 1 + (11 &times; (1 + 1 + 1 + 1 + 111+1 ))
:= 2 + (((2 + 2 + 2)2 + 2/2)2 )
:= 22+2 &times; (2 &times; 2 &times; 22 − 2)
:= 3 + ((3 &times; ((3 + 3)3 − 3)) + 33+3 )
:= 3 + ((((33/3)3 ) + 33) + 3 &times; 3)
:= 4 + (4444 − 4)/4
:= 4 + (4444/4 + 44 )
:= (4 + 4) &times; (4 &times; 44 − 4)
:= 5/5 + (5 &times; 5 &times; 55 − (5 + 5))
:= 5/5 + (5 &times; 5 &times; 55 − 5)
:= 5/5 + 5 &times; 5 &times; 55
:= 6 + (((6 + 6)/6) + 6 &times; 6 &times; 6 &times; 6)
:= 666/6 + (6 &times; (6 &times; 6 &times; 6 − 6))
:= 6 + (((6 &times; 6 + 6/6)(6+6)/6 ) + 6/6)
:= 7/7 + ((7 &times; (7 + 7) &times; (7 + 7)) − 7)
:= (7 &times; (7 + 7) &times; (7 + 7)) − 7/7
:= 77/7 + ((7 &times; (7 + 7) &times; (7 + 7)) − 7)
:= 88 + ((8 + 8) &times; (88 − 8) − ((8 + 8)/8))
:= 88/8 + (88 − 8) &times; (8/8 + 8 + 8)
:= (8 + 8) &times; (88 − ((8 + 8)/8))
:= 9 &times; (9 &times; (9 + 9) − 9) − 99/9
:= 9 &times; 9 &times; (9 + 9) + (((99 + 9)/9) − 99)
:= 9 &times; (9 &times; (9 + 9) − 9) − 9/9
:= 2 + (22 &times; (2
− 2))
3
:= 3 + ((((33/3) ) − 3/3) + 33)
4
6
&Iuml; 1367 := 11 + (1 + 11) &times; (1 + 1 + 111)
2
2
:= (((2 + 2 + 2) + 2/2) ) − 2
3
&Iuml; 1372 := (1 + 1 + 1 + 11)1+1+1 /(1 + 1)
2
&Iuml; 1377 := 1 + 1 + 11 &times; (1 + 1 + 1 + 1 + 111+1 )
:= 2 + ((22/2)2/2+2 + 2 &times; 22)
:= 22 + (2 &times; 2 &times; 222)
3
:= 3 + (((33/3) ) + 33)
:= (3/3 + 3) &times; (((3/3 + 3) + 3) )
:= 33 &times; ((33 − 3) + 33 )
:= 44 + 4444/4
:= ((4 + 4) &times; (4 &times; 44 − 4)) − 4
:= 4/4 + ((4 + 4) &times; (4 &times; 44 − 4))
:= (5 + 5)/5 + (5 &times; 5 &times; 55 − (5 + 5))
:= (5 + 5)/5 + (5 &times; 5 &times; 55 − 5)
:= (5 + 5)/5 + 5 &times; 5 &times; 55
6
:= (6 &times; (6 &times; 6 &times; 6 + 6 + 6)) − 6/6
:= 6 + ((((6 + 6)/6) + 6 &times; 6 &times; 6 &times; 6) + 6)
:= ((7 + 7)/7) + ((7 &times; (7 + 7) &times; (7 + 7)) − 7)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6)) + 666/6)
:= 7 &times; (7 + 7) &times; (7 + 7)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) − ((7 + 7)/7))
:= 88 + ((8 + 8) &times; (88 − 8) − 8/8)
:= 888 + 88 &times; 88/(8 + 8)
:= 8/8 + ((8 + 8) &times; (88 − ((8 + 8)/8)))
:= 9 &times; (9 &times; (9 + 9) − 9) − 9/9 − 9
:= 99 + (9999/9 + 9 &times; (9 + 9))
:= 9 &times; (9 &times; (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
&Iuml; 1378 := 11 + 11 + (1 + 11) &times; (1 + 1 + 111)
&Iuml; 1383 := 11 + ((1 + 1 + 1 + 11)1+1+1 /(1 + 1))
&Iuml; 1388 := 1 + 111+1+1 + (1 + 111)/(1 + 1)
:= (2 &times; (22 − 2))2 − 222
:= (2/2 + 2) &times; (222 − 22 − 2/2)
:= 2 + (22 &times; (22+2+2 − 2/2))
:= 3 + (33/3 &times; ((3 − 3/3 + 3)3 ))
:= (33 &times; (3 &times; 3 + 33)) − 3
:= 3 + ((((33/3)3 ) + 33 ) + 33 )
4
:= 44 + (4444 + 44)/4
:= 4 &times; 4 + (4444/4 + 4 )
:= (4 &times; ((4 + 4) &times; 44 − 4)) − 4
:= 5 + (5 &times; 5 &times; 55 − ((5 + 5)/5))
:= 5 + ((5 &times; 5 &times; 55 − ((5 + 5)/5)) + 5)
:= 5 &times; 5 &times; 55 + (55 + 5 + 5)/5
:= ((66 − 6)/6) + (6 &times; (6 &times; 6 &times; 6 + 6 + 6))
:= ((66 &times; (6 &times; 6 + 6)) − 6)/((6 + 6)/6)
:= 6 + ((((6 + 6)/6)66/6 ) − 666)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) − 7/7)
:= 77/7 + (7 &times; (7 + 7) &times; (7 + 7))
:= ((7 + 7)/7) + (77 &times; (77/7 + 7))
:= (8 + 8)/8 + ((8 + 8) &times; (88 − ((8 + 8)/8)))
:= 88 &times; (8 + 8) − (8/8 + 8 + 8 + 8)
:= 88 &times; (8 + 8) − ((88 + 8)/8 + 8)
:= 9/9 + 9 &times; (9 &times; (9 + 9) − 9)
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) − ((9 + 9 + 9)/9))
:= 99/9 + 9 &times; (9 &times; (9 + 9) − 9)
&Iuml; 1379 := 11 + (1 + 11) &times; (1 + 1 + 1 + 111)
2
:= 2/2 + (2 &times; (22 − 2)) − 222
3
3
&Iuml; 1384 := 1 + 11 + (1 + 1 + 1 + 11)1+1+1 /(1 + 1)
2
:= 2 &times; (((22 + 2 + 2) ) + 2
2+2
)
&Iuml; 1389 := (1 + 11111)/(1 + 1)1+1+1
:= 22 + ((((2 + 2 + 2)2 + 2/2)2 ) − 2)
:= 3 &times; 3 + (((33/3) ) − 33)
:= 3/3 + ((33 &times; (3 &times; 3 + 33)) − 3)
:= 3 + (33 &times; (3 &times; 3 + 33))
:= 4 + (((44/4)4−4/4 ) + 44)
:= 4 + (((4 + 4) &times; (4 &times; 44 − 4)) + 4)
:= 4/4 + ((4 &times; ((4 + 4) &times; 44 − 4)) − 4)
:= 5 + (5 &times; 5 &times; 55 − 5/5)
:= 5 + ((5 &times; 5 &times; 55 − 5/5) + 5)
:= 5 &times; (5 &times; 55 + 5) − 55/5
:= 66/6 + (6 &times; (6 &times; 6 &times; 6 + 6 + 6))
:= ((66 &times; ((66 − 6) + 66)) − (6 + 6))/6
:= 666 + (((6 &times; 6/(6 + 6))6 ) − 6)
:= 7 + (7 &times; (7 + 7) &times; (7 + 7))
:= (77 + 7)/7 + (7 &times; (7 + 7) &times; (7 + 7))
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + ((77 − 7)/7))
:= 88 + ((8 + 8) &times; (88 − 8) + (88/8))
:= 88 &times; (8 + 8) − 8 − 8 − 8
:= 88 &times; (8 + 8) − (88/8 + 8)
:= (9 + 9)/9 + 9 &times; (9 &times; (9 + 9) − 9)
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + 9 &times; (9 &times; (9 + 9) − 9)
&Iuml; 1380 := 11 + ((111/(1 + 1 + 1))1+1 )
&Iuml; 1385 := 111+1+1 + ((111 − 1)/(1 + 1) − 1)
&Iuml; 1390 := 11 &times; 111 + (1 + 1 + 11)1+1
:= 2 + (2 &times; (22 − 2))2 − 222
:= 22+2 + (((2 + 2 + 2)2 + 2/2)2 )
:= (2 &times; (222 + 222 )) − 22
:= 3 + (3 &times; (3 + 3)3 + 33+3 )
:= 33 + (((33/3)3 ) + 33 )
:= 3 + ((33 &times; (3 &times; 3 + 33)) + 3/3)
4
:= (44 &times; (4 − 4)/(4 + 4)) − 4/4
:= 4 + (44 &times; (44 − 4)/(4 + 4))
:= 5 + 5 &times; 5 &times; 55
:= 5 + (5 &times; 5 &times; 55 + 5)
:= 5 + ((5 &times; 5 &times; 55 + 5) + 5)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6 + 6)) + 6)
:= ((66 &times; ((66 − 6) + 66)) − 6)/6
:= 666 + ((66 &times; 66 − 6 − 6)/6)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + 7/7)
:= (77 &times; (77/7 + 7)) − 7/7
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + (77/7))
:= 8 + (88 &times; 88/(8 + 8) + 888)
:= 8/8 + (88 &times; (8 + 8) − (8 + 8 + 8))
:= ((8 + 8)/8) &times; (8 &times; 88 − (8/8 + 8))
:= ((9 + 9 + 9)/9) + 9 &times; (9 &times; (9 + 9) − 9)
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) − 9/9)
:= 9 &times; 99 + (((9 &times; 999) − 9)/(9 + 9))
:= 4 + ((4 + 4) &times; (4 &times; 44 − 4))
&Iuml; 1381 := 1 + 11 + ((111/(1 + 1 + 1))1+1 )
2
:= 2 + (2 &times; (22 − 2)) − 222 + 2/2
3
&Iuml; 1386 := 11 &times; (1 + 1 + 1 + 1 + 1 + 111+1 )
&Iuml; 1391 := (1 + 1 + 11) &times; (111 − 1 − 1 − 1 − 1)
2+2+2
:= 22 + (((2 + 2 + 2)2 + 2/2)2 )
:= 22 &times; (2
− 2/2)
:= 3 + ((33/3 &times; ((3 − 3/3 + 3) )) + 3)
:= 33 &times; (3 &times; 3 + 33)
:= 33 + (((33/3)3 ) + 33)
:= 4 + (((4 + 4) &times; (4 &times; 44 − 4)) + 4/4)
:= 44 &times; (44 − 4)/(4 + 4)
:= (4 &times; ((4 + 4) &times; 44 − 4)) − 4/4
:= 5 + (5 &times; 5 &times; 55 + 5/5)
:= 55/5 + 5 &times; 5 &times; 55
:= 5 + (5 &times; 5 &times; 55 + (55/5))
:= 6 + (((6 &times; 6 + 6/6)(6+6)/6 ) + 6)
:= 6 &times; ((66 &times; (6 &times; 6 + 6))/(6 + 6))
:= 666 + ((66 &times; 66 − 6)/6)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + ((7 + 7)/7))
:= 77 &times; (77/7 + 7)
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + (77 + 7)/7)
:= 88 &times; (8 + 8) − (88/8 + 8 + 8)
:= ((8 + 8)/8) &times; (8 &times; 88 − (88/8))
:= 88 &times; (8 + 8) − (8/8 + 8 + 8)
:= 9 &times; (9 &times; (9 + 9) − 9) + ((9 &times; 9 − 9)/(9 + 9))
:= 9 + 9 &times; (9 &times; (9 + 9) − 9)
:= 9 &times; 99 + (((9 &times; 999) + 9)/(9 + 9))
&Iuml; 1382 := 1 + 1 + 11 + ((111/(1 + 1 + 1))1+1 )
2
:= 2 + (2 &times; (22 − 2)) − 222 + 2
3
3
3
:= 3 + ((((33/3) ) − 3) + 3 )
4
&Iuml; 1387 := 111+1+1 + (1 + 111)/(1 + 1)
&Iuml; 1392 := (1 + 11) &times; (1 + 1 + 1 + 1 + 1 + 111)
:= 2/2 + (22 &times; (22+2+2 − 2/2))
:= 2 &times; ((((22 + 2 + 2)2 ) − 2) + 22)
:= 3/3 + (33 &times; (3 &times; 3 + 33))
:= 3 + ((33 &times; (3 &times; 3 + 33)) + 3)
4
:= (44 &times; (4 − 4)/(4 + 4)) − 4
:= 4/4 + (44 &times; (4 − 4)/(4 + 4))
:= 4 &times; ((4 + 4) &times; 44 − 4)
:= 5 + (5 &times; 5 &times; 55 + ((5 + 5)/5))
:= 5 &times; 5 &times; 55 + (55 + 5)/5
:= 5 + (5 &times; 5 &times; 55 + ((55 + 5)/5))
:= ((66 &times; ((66 − 6) + 66)) + 6)/6
:= 66 + ((6 &times; (6 &times; 6 &times; 6 + 6)) − 6)
:= ((77 − 7)/7) + (7 &times; (7 + 7) &times; (7 + 7))
:= 7/7 + (77 &times; (77/7 + 7))
:= 7 + ((77 &times; (77/7 + 7)) − 7/7)
:= (8 − 88)/8 + ((8 + 8) &times; (88 − 8/8))
:= (88/8 + 8) &times; ((8/8 + 8 &times; 8) + 8)
:= (8 + 8) &times; (88 − 8/8)
:= 9 &times; (9 &times; (9 + 9) − 9) + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) + 9/9)
:= 9 &times; 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
&Iuml; 1393 := 1 + ((1 + 11) &times; (1 + 1 + 1 + 1 + 1 + 111))
&Iuml; 1398 := 1 + (11 &times; (111 + (1 + 1)1+1+1+1 ))
&Iuml; 1403 := (1 + 1 + 11) &times; (111 − 1 − 1 − 1) − 1
:= 2 + ((((2 + 2 + 2) + 2/2) ) + 22)
:= 2 + (2 &times; (((22 + 2 + 2) ) + 22))
:= 22 &times; 22+2+2 − (2/2 + 2 + 2)
:= (((3 + 3) &times; (33+3 − 33)) + 3)/3
:= 3 + ((3 &times; 3 + 3)3 − 333)
:= (3 &times; (33 − 3)) + ((33/3)3 )
:= 4/4 + (4 &times; ((4 + 4) &times; 44 − 4))
:= (4 − 44)/4 + 4 &times; (4 + 4) &times; 44
:= 4 &times; (4 + 4) &times; 44 − (4/4 + 4)
:= 5 &times; (5 &times; 55 + 5) − ((5 + 5)/5 + 5)
:= 5 &times; (5 &times; 55 + 5) − (5 + 5)/5
:= 5 + (5 &times; (5 &times; 55 + 5) − ((5 + 5)/5))
:= 666 + ((66 &times; 66 + 6)/6)
:= 66 + (6 &times; (6 &times; 6 &times; 6 + 6))
:= (6 &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6)) − 6/6
:= 7 + (77 &times; (77/7 + 7))
:= ((7 + 7)/7) &times; (777 − (7/7 + 77))
:= (((7 + 7 + 7)/7)7 ) − (777 + 7)
:= 8/8 + ((8 + 8) &times; (88 − 8/8))
:= (8 − 88)/8 + 88 &times; (8 + 8)
:= 88/8 + ((8 + 8) &times; (88 − 8/8))
:= 9 + ((9 &times; (9 &times; (9 + 9) − 9) − ((9 + 9)/9)) + 9)
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) + ((99 + 9)/9))
:= 9 &times; 99 + (((9 + 9)/9)9 )
2
2
2
&Iuml; 1394 := (1 + 1 + (1 + 111)1+1 )/(11 − 1 − 1)
&Iuml; 1399 := 1111 + ((1 + 1) &times; (1 + 11)1+1 )
:= 2 &times; ((22/2)2 + ((22 + 2)2 ))
:= (2222 + ((22 + 2)2 ))/2
3
3
:= 3 + ((((33/3) ) + 33) + 3 )
&Iuml; 1404 := (1 + 1 + 11) &times; (111 − 1 − 1 − 1)
:= 2 &times; ((2 &times; (22 &times; 22+2 )) − 2)
3
:= 3 + (((3 &times; 3 + 3) − 333) + 3/3)
:= 3 &times; ((3 + 3) &times; (3 &times; 33 − 3))
:= 4 + ((44 &times; (4 − 4)/(4 + 4)) + 4)
:= 4 &times; (4 + 4) &times; 44 − ((4/4 + 4) + 4)
:= 4 &times; (4 + 4) &times; 44 − 4
:= 5 &times; (5 &times; 55 + 5) − (5/5 + 5)
:= 5 &times; (5 &times; 55 + 5) − 5/5
:= 5 + (5 &times; (5 &times; 55 + 5) − 5/5)
:= 666 + (((6 &times; 6/(6 + 6)) ) − 6/6)
:= 66 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + 6/6)
:= 6 &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6)
:= 7 + ((77 &times; (77/7 + 7)) + 7/7)
:= (((7 + 7) &times; (777 − 77)) − 7)/7
:= (77/7 + 7) &times; (7/7 + 77)
:= (8 + 8)/8 + ((8 + 8) &times; (88 − 8/8))
:= 88 &times; (8 + 8) − (8/8 + 8)
:= 88 &times; (8 + 8) − (8/((8 + 8)/8))
4
6
9
:= 9 &times; 99 + ((((9 + 9)/9) ) − 9)
&Iuml; 1395 := (11 − 1 − 1) &times; (11 + (1 + 11)1+1 )
:= 2
2+2+2
+ (22/2)
:= 999 + ((99/9 + 9)
(9+9)/9
)
:= 9 + ((9 &times; (9 &times; (9 + 9) − 9) + 9) + 9)
&Iuml; 1400 := (1 + (1 + 1 + 11)) &times; (11 − 1)1+1
&Iuml; 1405 := 1 + ((1 + 1 + 11) &times; (111 − 1 − 1 − 1))
:= 2 &times; (2 &times; ((22 &times; 22+2 ) − 2))
2/2+2
:= 22 &times; 22+2+2 − 2/2 − 2
3
:= (3 &times; 3 + 3)3 − 333
:= 3 + ((((33/3) ) + 33) + 33)
:= 3/3 + (3 &times; ((3 + 3) &times; (3 &times; 33 − 3)))
:= 4 + ((4 &times; ((4 + 4) &times; 44 − 4)) − 4/4)
:= (4 + 4) &times; (4 &times; 44 − 4/4)
:= 4/4 + (4 &times; (4 + 4) &times; 44 − 4)
:= 5 &times; (5 &times; 55 + 5) − 5
:= 5 &times; (5 &times; 55 + 5)
:= 5 + 5 &times; (5 &times; 55 + 5)
:= 666 + ((6 &times; 6/(6 + 6)) )
:= 66 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + ((6 + 6)/6))
:= 6 &times; 6 + ((6 &times; 6 + 6/6)(6+6)/6 )
:= 7 + ((77 &times; (77/7 + 7)) + ((7 + 7)/7))
:= 7 + ((77 &times; (77/7 + 7)) + 7)
:= 7 &times; (7 &times; 7 − 7) + 7777/7
:= 8 &times; 8 + ((88/8)88/8−8 )
:= 88 &times; (8 + 8) − 8
:= 8 + (88 &times; (8 + 8) − (88/8))
:= (99/9 + 9) &times; (9 &times; 9 − 99/9)
:= 9 + (((9 &times; (9 &times; (9 + 9) − 9) + 9/9) + 9) + 9)
6
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) + 9)
&Iuml; 1396 := 1 + ((11 − 1 − 1) &times; (11 + (1 + 11)1+1 ))
2
:= 2 &times; (((22 + 2 + 2) ) + 22)
:= 3/3 + ((3 &times; 3 + 3)3 − 333)
:= 4 + (4 &times; ((4 + 4) &times; 44 − 4))
:= 5/5 + (5 &times; (5 &times; 55 + 5) − 5)
:= ((6 + 6)/6)6 + (6 &times; (6 &times; 6 &times; 6 + 6))
&Iuml; 1401 := 111+1+1 + (11 − 1 − 1)1+1 − 11
&Iuml; 1406 := ((1 + 1 + 11) &times; (111 − 1 − 1)) − 11
:= 2 + ((2222 + ((22 + 2)2 ))/2)
:= 22 &times; 22+2+2 − 2
3
:= (3 &times; ((3 + 3) &times; (3 &times; 3 − 3))) − 3
:= 3 + ((3 &times; (33 − 3)) + ((33/3)3 ))
:= 4 + (4 &times; (4 + 4) &times; 44 − 44/4)
:= 4 &times; (4 + 4) &times; 44 − (4 + 4)/4
:= 5/5 + 5 &times; (5 &times; 55 + 5)
:= 5 + (5 &times; (5 &times; 55 + 5) + 5/5)
:= 6 + (((6 &times; 6/(6 + 6))6 ) + 666)
:= 6 &times; 6 &times; 6 &times; 6 + (666 − 6)/6
:= ((77 − 7)/7) + (77 &times; (77/7 + 7))
:= (77/7 &times; ((7 + 7)/7)7 ) − 7
:= ((77 &times; ((7 + 7)/7)7 ) − (7 + 7))/7
:= 88 &times; (8 + 8) − (88 + 8)/8
:= 8/8 + (88 &times; (8 + 8) − 8)
:= 88 &times; (8 + 8) − (8 + 8)/8
:= 9 + ((9 &times; (9 &times; (9 + 9) − 9) + 9/9) + 9)
:= 9 &times; 9 + 99/9 &times; (999/9 + 9)
:= 9 + ((9 &times; (9 &times; (9 + 9) − 9) + (99/9)) + 9)
&Iuml; 1397 := 11 &times; (111 + (1 + 1)1+1+1+1 )
&Iuml; 1402 := (1 + 1 + 11) &times; (111 − 1 − 1 − 1) − 1 − 1
2+2
:= 22/2 &times; ((22&times;(2+2) − 2)/2)
:= 2 + (2 &times; (2 &times; ((22 &times; 2
:= 33 + (((33/3)3 ) + 33)
:= ((33/3) ) + (((3 + 3) − 3)/3)
3
) − 2)))
:= 22 &times; 22+2+2 − 2/2
3
:= 4 &times; (4 + 4) &times; 44 − 44/4
:= 4 &times; 4 + (44 &times; (44 − 4)/(4 + 4))
:= (5 + 5)/5 + (5 &times; (5 &times; 55 + 5) − 5)
:= (5 + 5)/5 + 5 &times; (5 &times; 55 + 5)
&Iuml; 1407 := 111 + ((1 + 1 + 1) &times; (1 + 11))1+1
:= 3 + (3 &times; ((3 + 3) &times; (3 &times; 33 − 3)))
:= 4 &times; (4 + 4) &times; 44 − 4/4
6
:= 5 &times; 5 &times; 55 + ((5 + 5)/5)5
:= 66 + ((66/6)6&times;6/(6+6) )
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 6)) + ((6 + 6)/6) )
:= 666/6 + 6 &times; 6 &times; 6 &times; 6
:= 77/7 + (77 &times; (77/7 + 7))
:= (((77 &times; ((7 + 7)/7)7 ) + 7)/7) − 7
:= (7 &times; ((7 + 7) &times; (7 + 7) + 7)) − (7 + 7)
:= 88 &times; (8 + 8) − 88/8
:= (8 + 8)/8 + (88 &times; (8 + 8) − 8)
:= 88 &times; (8 + 8) − 8/8
:= 9 + (9 &times; (9 &times; (9 + 9) − 9) + (99/9))
9
:= 9 &times; 99 + ((((9 + 9)/9) ) − 9/9)
103
:= 999/9 + ((9 + 9) &times; (9 &times; 9 − 9))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1408 := 11 &times; ((1 + 1)1+(1+1)&times;(1+1+1) )
&Iuml; 1413 := 1 + (111+1+1 + (11 − 1 − 1)1+1 )
&Iuml; 1418 := 1 + ((1 + 1 + 11) &times; (111 − 1 − 1))
:= 2/2 + (2 &times; (222 + 22 ))
:= 2 + (2 &times; ((222 + 222 ) + 2))
:= 33/3 &times; (((3 − 3/3 + 3)3 ) + 3)
:= 3 &times; (((3 + 3) &times; (3 &times; 33 − 3)) + 3)
:= 3 + ((((33/3)3 ) + 3 &times; 33 ) + 3)
:= 4 &times; (4 + 4) &times; 44
:= 4 + (4 &times; (4 + 4) &times; 44 + 4/4)
:= (44 − 4)/4 + 4 &times; (4 + 4) &times; 44
:= ((5 + 5)/5)5 &times; (55 − (55/5))
:= 5 + (((5 + 5)/5)5 &times; (55 − (55/5)))
:= 55 + (5 &times; 5 &times; 55 − ((55 + 5)/5))
:= 66/6 &times; (((6 + 6)/6)6/6+6)
:= 6 + (666/6 + 6 &times; 6 &times; 6 &times; 6)
:= 6 &times; 6 &times; 6 &times; 6 + ((666 + 66)/6)
:= 77/7 &times; ((7 + 7)/7)7
:= (7 &times; ((7 + 7) &times; (7 + 7) + 7)) − (7/7 + 7)
:= 7 + (((77/7 + 7) &times; (7/7 + 77)) + 7)
:= 88 &times; (8 + 8)
:= 8 + ((88 &times; (8 + 8) − (88/8)) + 8)
:= 8 + (88 &times; (8 + 8) + ((8 + 8)/8))
:= 99/9 &times; (((99/9 + 99) + 9) + 9)
:= ((9 + 9) &times; (9 &times; 9 − ((9 + 9)/9))) − 9
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 − (99/9 + 9))
:= 22 &times; 2
2
2+2+2
&Iuml; 1409 := 1 + 11 &times; ((1 + 1)1+(1+1)&times;(1+1+1) )
&Iuml; 1414 := (1 + (1 + 1 + 11)) &times; (1 + (11 − 1)1+1 )
2
&Iuml; 1419 := 11 &times; (11 &times; (1 + 11) − 1 − 1 − 1)
:= 2/2 + 22 &times; 22+2+2
:= 2 + (2 &times; (222 + 22 ))
:= 22/2 + 22 &times; 22+2+2
:= 3 &times; 33 + (((33/3)3 ) − 3)
:= 3 + ((((33/3)3 ) − 3/3) + 3 &times; 33 )
:= 33 + (33 &times; (3 &times; 3 + 33))
:= 4/4 + 4 &times; (4 + 4) &times; 44
:= 4 + (4 &times; (4 + 4) &times; 44 + (4 + 4)/4)
:= 44/4 + 4 &times; (4 + 4) &times; 44
:= 5 + ((5 &times; (5 &times; 55 + 5) − 5/5) + 5)
:= (5 &times; ((5 &times; 55 + 5) + 5)) − 55/5
:= 55 + (5 &times; 5 &times; 55 − (55/5))
:= 6 + ((6 &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6)) − 6/6)
:= 6 + ((666 + 6)/6 + 6 &times; 6 &times; 6 &times; 6)
:= 6 + ((666/6 + 6 &times; 6 &times; 6 &times; 6) + 6)
:= ((77 &times; ((7 + 7)/7)7 ) + 7)/7
:= (7 &times; ((7 + 7) &times; (7 + 7) + 7)) − 7
:= 77/7 &times; (((7 + 7)/7)7 + 7/7)
:= 8/8 + 88 &times; (8 + 8)
:= 8 + (88 &times; (8 + 8) − ((8 + 8)/8))
:= 88/8 + 88 &times; (8 + 8)
:= 9 + ((99/9 + 9) &times; (9 &times; 9 − (99/9)))
:= 9/9 + (((9 + 9) &times; (9 &times; 9 − ((9 + 9)/9))) − 9)
:= 99/9 &times; ((999/9 + 9) + 9)
&Iuml; 1410 := (11 − 1) &times; ((1 + 11)1+1 − 1 − 1 − 1)
&Iuml; 1415 := ((1 + 1 + 11) &times; (111 − 1 − 1)) − 1 − 1
&Iuml; 1420 := (11 − 1) &times; ((1 + 11)1+1 − (1 + 1))
:= 2 + 22 &times; 22+2+2
:= 2 + ((2 &times; (222 + 222 )) + 2/2)
:= 2 &times; (((222 + 222 ) + 2) + 2)
:= 3 + ((3 &times; ((3 + 3) &times; (3 &times; 33 − 3))) + 3)
:= 3 + (((33/3)3 ) + 3 &times; 33 )
:= 3/3 + ((33 &times; (3 &times; 3 + 33)) + 33)
:= (4 + 4)/4 + 4 &times; (4 + 4) &times; 44
:= 4 + ((4 &times; (4 + 4) &times; 44 − 4/4) + 4)
:= (4 &times; ((4 + 4) &times; 44 + 4)) − 4
:= 5 + (5 &times; (5 &times; 55 + 5) + 5)
:= 5 + ((5 &times; (5 &times; 55 + 5) + 5) + 5)
:= (5 &times; ((5 &times; 55 + 5) + 5)) − 5
:= 6 + (6 &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6))
:= 66/6 + (6 &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6))
:= 6 &times; 66 + (((6 + 6)/6)(66−6)/6)
:= (((7 + 7 + 7)/7)7 ) − 777
:= 7 + (77/7 &times; ((7 + 7)/7)7 )
:= (7 &times; ((7 + 7) &times; (7 + 7) + 7)) − 7/7
:= (8 + 8)/8 + 88 &times; (8 + 8)
:= 8 + (88 &times; (8 + 8) − 8/8)
:= ((88 + 8)/8) + 88 &times; (8 + 8)
:= ((99/9) &times; ((999/9 + 9) + 9)) − 9
:= 9 + (((9 &times; (9 &times; (9 + 9) − 9) + (99/9)) + 9) + 9)
:= (99/9 + 9) &times; (9 &times; 9 − (9/9 + 9))
&Iuml; 1411 := 1 + (11 − 1) &times; ((1 + 11)1+1 − 1 − 1 − 1)
&Iuml; 1416 := (1 + 11) &times; (111+1 − (1 + 1 + 1))
2
:= 2 + (22 &times; 22+2+2 + 2/2)
&Iuml; 1421 := 1 + (11 − 1) &times; ((1 + 11)1+1 − 1 − 1)
:= 2 + (22 &times; 22+2+2 + 22/2)
:= 2 &times; ((222 + 22 ) + 2)
3
:= (3 &times; (33 + 3)) + ((33/3)3 )
:= 3 &times; 33 + (((33/3)3 ) − 3/3)
:= 3 + ((33 &times; (3 &times; 3 + 33)) + 3 )
:= 4 + (4 &times; (4 + 4) &times; 44 − 4/4)
:= 4 + (4 &times; (4 + 4) &times; 44 + 4)
:= 4/4 + ((4 &times; ((4 + 4) &times; 44 + 4)) − 4)
:= 55/5 + 5 &times; (5 &times; 55 + 5)
:= 5 + (5 &times; (5 &times; 55 + 5) + (55/5))
:= 5/5 + ((5 &times; ((5 &times; 55 + 5) + 5)) − 5)
:= 6 + ((6 &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6)) + 6/6)
:= (6 + 6) &times; ((666 + 6)/6 + 6)
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 − (6/6 + 6 + 6))
7
:= 7 + ((77/7 + 7) &times; (7/7 + 77))
:= 7 + (((77 &times; ((7 + 7)/7) ) + 7)/7)
:= 7 &times; ((7 + 7) &times; (7 + 7) + 7)
:= 88/8 + (88 &times; (8 + 8) − 8)
:= 8 + 88 &times; (8 + 8)
:= 88 &times; (8 + 8) + (88 + 8 + 8)/8
:= ((9 − 9/9) + 9) &times; (((9 + 9)/9) + 9 &times; 9)
:= 9 + (((9 + 9) &times; (9 &times; 9 − 9)) + 999/9)
:= 9 + (((((9 + 9)/9)9 ) + 9 &times; 99) + 9)
&Iuml; 1412 := 111+1+1 + (11 − 1 − 1)1+1
&Iuml; 1417 := (1 + 1 + 11) &times; (111 − 1 − 1)
2
2
:= (22/2) + (2 + 2 + 2)
:= 2 &times; (222 + 22 )
3
&Iuml; 1422 := 1 + 1 + (11 − 1) &times; ((1 + 11)1+1 − (1 + 1))
2+2
:= ((2 + 2 + 2)2 + 2)2 − 22
3
3
:= 3 &times; 3 + ((33/3) )
:= 333 + (3333/3 − 3 )
:= 4 + 4 &times; (4 + 4) &times; 44
:= 4 + ((4 &times; (4 + 4) &times; 44 + 4/4) + 4)
5
:= 333 + 33 &times; 33
5
:= (4 &times; ((4 + 4) &times; 44 + 4)) − (4 + 4)/4
:= 5 + (5 &times; 5 &times; 55 + ((5 + 5)/5) )
:= 5 + ((5 &times; 5 &times; 55 + ((5 + 5)/5) ) + 5)
:= (5 + 5)/5 + ((5 &times; ((5 &times; 55 + 5) + 5)) − 5)
:= 6 + (((666 − 6)/6) + 6 &times; 6 &times; 6 &times; 6)
:= 6/6 + ((6 + 6) &times; ((666 + 6)/6 + 6))
:= 6 + ((6 + 6) &times; ((666 + 6)/6 + 6))
7
:= 7 + (7777/7 + 7 &times; (7 &times; 7 − 7))
:= 7 + ((((7 + 7 + 7)/7) ) − 777)
:= 7/7 + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= 88 &times; (8 + 8) + (8/((8 + 8)/8))
:= 8 + (88 &times; (8 + 8) + 8/8)
:= 8 + ((88 &times; (8 + 8) − ((8 + 8)/8)) + 8)
:= 9 &times; 9 &times; (9 + 9) − ((9 &times; 9 &times; 9 + 9)/(9 + 9))
:= (9 + 9) &times; (9 &times; 9 − ((9 + 9)/9))
9
:= 9 + ((((9 + 9)/9) ) + 9 &times; 99)
104
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1423 := (1 + 1)11 − (1 + (1 + 1) &times; (1 + 11))1+1
&Iuml; 1428 := (1 + 11) &times; (111+1 − 1 − 1)
&Iuml; 1433 := 1 + 111 &times; (1 + 1 + 11) − 11
:= 2/2 + (((2 + 2 + 2) + 2) − 22)
:= 2 &times; ((2 + 2 + 2) &times; ((22/2) − 2))
:= ((2 + 2 + 2)2 + 2)2 − 22/2
:= 3/3 + (33 &times; 33 + 333)
:= (33/3 + 3) &times; (3 &times; 33 + 3)
:= 3 + (((33/3)3 ) + 3 &times; 33)
:= 4 + (4 &times; ((4 + 4) &times; 44 + 4))
:= 4 + (((4 &times; ((4 + 4) &times; 44 + 4)) + 4/4) + 4)
:= (((5 + 5)/5)55/5 ) − 55 /5
:= 55 + (5 &times; 5 &times; 55 − ((5 + 5)/5))
:= 5 + ((5 &times; 5 &times; 55 − ((5 + 5)/5)) + 55)
:= 6 + (((6 + 6) &times; ((666 + 6)/6 + 6)) + 6/6)
:= 66 + (6 &times; 6 &times; 6 &times; 6 + 66)
:= 66 + ((6 &times; (6 &times; 6 &times; 6 + 6 + 6)) − 6/6)
:= ((7 + 7)/7) + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= 7 + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= (77 + 7)/7 + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= 8 + ((88 &times; (8 + 8) − 8/8) + 8)
:= 8 + (((88 + 8)/8) + 88 &times; (8 + 8))
:= 8 + (((88 &times; (8 + 8) + 8/8) + 8) + 8)
:= 9/9 + ((9 + 9) &times; (9 &times; 9 − ((9 + 9)/9)))
:= (9 &times; (9 &times; (9 + 9) + 9)) − 999/9
:= (9 + 9)/9 + (9 &times; 9 &times; (9 + 9) − (9 + 9 + 9))
2
2
4
:= (4 &times; (4 + 4)) − (4/4 + 4)
4
&Iuml; 1424 := (1 + 1)11−1 + ((1 + 1) &times; (11 − 1))1+1
2
&Iuml; 1429 := ((1 + 1 + 11) &times; (111 − 1)) − 1
2+2+2
:= 22 + (2 &times; (222 + 222 ))
:= 22 + (22 &times; 2
:= 3 + ((3 &times; (33 + 3)) + ((33/3)3 ))
:= 3 &times; 33 + (((33/3)3 ) − 3/3)
:= 3 + (3 &times; (3 &times; ((3 + 3) &times; 33 − 3)))
:= 4 &times; ((4 + 4) &times; 44 + 4)
:= 4 + ((4 &times; ((4 + 4) &times; 44 + 4)) + 4/4)
:= 4 + ((44 &times; (44 − 4)/(4 + 4)) + 44)
:= (5 &times; ((5 &times; 55 + 5) + 5)) − 5/5
:= 55 + (5 &times; 5 &times; 55 − 5/5)
:= 5 + ((5 &times; 5 &times; 55 − 5/5) + 55)
:= 66 + ((6 &times; 6 &times; 6 &times; 6 + 66) + 6/6)
:= 66 + (6 &times; (6 &times; 6 &times; 6 + 6 + 6))
2
6/6+6)
:= 6 &times; 6 &times; 6 &times; 6 + (((6 + 6)/6)
:= 7 + (((((7 + 7 + 7)/7)7 ) − 777) + 7)
:= 8 + (88 &times; (8 + 8) + 8)
:= (9 + 9)/9 + ((9 + 9) &times; (9 &times; 9 − ((9 + 9)/9)))
&Iuml; 1425 := ((1 + 11) &times; (111+1 − (1 + 1))) − 1 − 1 − 1
2
:= (2/2 + 2) &times; ((22 − 22/2) + 2)
:= 3 + (33 &times; 33 + 333)
− 2/2)
&Iuml; 1434 := 1 + 1 + 111 &times; (1 + 1 + 11) − 11
:= 2 + (((2 + 2 + 2) + 2) − 22)
2
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) + 7)) + 7/7)
:= (7 &times; (((7 + 7)/7)7 + 77)) − 7/7
:= 8 + ((88 + 8 + 8)/8 + 88 &times; (8 + 8))
:= 8 + (((88 &times; (8 + 8) + ((8 + 8)/8)) + 8) + 8)
:= 9 &times; 9 &times; (9 + 9) − (99/9 + 9 + 9)
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 − ((99 + 9)/9))
&Iuml; 1430 := (1 + 1 + 11) &times; (111 − 1)
&Iuml; 1435 := 1 + 1 + 1 + (111 &times; (1 + 1 + 11) − 11)
:= 22 + 22 &times; 22+2+2
:= 2 + (((2 + 2 + 2)2 + 2)2 − 22/2)
3
:= 3 &times; 33 + ((33/3) )
:= 3 + ((3 &times; (3 &times; ((3 + 3) &times; 33 − 3))) + 3/3)
4
:= 4/4 + (4 &times; ((4 + 4) &times; 44 + 4))
:= 44 + (44 &times; (4 − 4)/(4 + 4))
:= 44/4 + (4 &times; ((4 + 4) &times; 44 + 4))
:= 5 &times; ((5 &times; 55 + 5) + 5)
:= 55 + 5 &times; 5 &times; 55
:= 5 + (5 &times; 5 &times; 55 + 55)
6
:= (6 − 6/6) &times; (6 &times; 66 − 666/6)
:= 66/6 &times; (((6 + 6)/6) + 66)
:= 66 + ((6 &times; 6 + 6/6)(6+6)/6 )
:= 77/7 + ((7 &times; ((7 + 7) &times; (7 + 7) + 7)) − 7)
:= (7 − 7/7 + 7) &times; (777 − 7)/7
:= 7 &times; (((7 + 7)/7)7 + 77)
:= 8 + ((88 &times; (8 + 8) + 8/8) + 8)
:= ((8 + 8)/8) &times; (88/8 + 8 &times; 88)
:= 8 + ((88 &times; (8 + 8) + 88/8) + 8)
:= 9 &times; 9 &times; (9 + 9) − (99/((9 + 9 + 9)/9))
&Iuml; 1426 := ((1 + 11) &times; (111+1 − (1 + 1))) − 1 − 1
:= (22 + 2/2) &times; (22+2+2 − 2)
3+3
:= 99 + (99/9)
(9+9+9)/9
:= (9 + 9) &times; (9 + 9) + 9999/9
&Iuml; 1431 := 1 + (1 + 1 + 11) &times; (111 − 1)
:= 22 + (22 &times; 22+2+2 + 2/2)
3
&Iuml; 1436 := (1 + 1) &times; ((11 − 1 − 1)1+1+1 − 11)
:= 2 &times; (2 &times; (((22 − (2/2 + 2))2 ) − 2))
:= 3 &times; (3 &times; ((3 + 3) &times; 3 − 3))
:= 3 + ((((33/3)3 ) + 3 &times; 33) + 3)
:= (4 + 4)/4 + (4 &times; ((4 + 4) &times; 44 + 4))
:= 44 + ((4444 + 44 )/4)
:= (4 + 4) &times; (4 &times; 44 + 4) − 4
:= 5/5 + (5 &times; ((5 &times; 55 + 5) + 5))
:= 55 + (5 &times; 5 &times; 55 + 5/5)
:= ((((3 + 3) &times; 3
) + 3)/3) − 33
6
:= 5 + ((5 &times; 5 &times; 55 + 5/5) + 55)
:= 66 + (((6 + 6)/6) + 6 &times; 6 &times; 6 &times; 6)
:= 6 &times; 6 + (((6 &times; 6/(6 + 6)) ) + 666)
:= 6 + ((66/6) &times; (((6 + 6)/6)6 + 66))
:= 7 + (77/7 &times; (((7 + 7)/7)7 + 7/7))
:= ((77 − 7)/7) + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= 7/7 + (7 &times; (((7 + 7)/7)7 + 77))
:= 8 + ((88 &times; (8 + 8) + ((8 + 8)/8)) + 8)
:= 8 + (((88 &times; (8 + 8) − 8/8) + 8) + 8)
:= 8 + ((((88 + 8)/8) + 88 &times; (8 + 8)) + 8)
:= (9 + 9) &times; (9 + 9) + (9999/9 − 9)
:= 9 &times; 9 &times; (9 + 9) − (9 + 9 + 9)
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 − (99/9))
&Iuml; 1427 := ((1 + 11) &times; (111+1 − (1 + 1))) − 1
6
&Iuml; 1432 := 111 &times; (1 + 1 + 11) − 11
&Iuml; 1437 := ((11 − 1) &times; (1 + 11)1+1 ) − 1 − 1 − 1
:= ((2/2 + 2) &times; (222 − 2/2)) − 22
:= 2 + (22 &times; 22+2+2 + 22)
:= (2/2 + 2) &times; (222 − (2/2 + 2 + 2))
:= 3 &times; 33 + (((33/3)3 ) − 3)
:= 3/3 + (3 &times; (3 &times; ((3 + 3) &times; 33 − 3)))
:= 3 &times; (((3 − 3/3)3&times;3 ) − 33)
:= 4 + ((44 &times; (4 + 4)) − (4/4 + 4)4 )
:= 4 + ((4 &times; ((4 + 4) &times; 44 + 4)) + 4)
:= 4/4 + ((4 + 4) &times; (4 &times; 44 + 4) − 4)
:= (5 + 5)/5 + (5 &times; ((5 &times; 55 + 5) + 5))
:= 55 + (5 &times; 5 &times; 55 + ((5 + 5)/5))
:= 5 + ((5 &times; 5 &times; 55 + ((5 + 5)/5)) + 55)
6
:= 66 + ((6 &times; 6 &times; 6 &times; 6 − 6/6) + 66)
:= ((6 + 6)/6) + (6 &times; (6 &times; 6 &times; 6 + 6 + 6))
:= ((6/6 + 6 + 6) &times; 666/6) − 6
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) + 7)) − 7/7)
:= 77/7 + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= ((7 &times; 7 − 77/7)(7+7)/7 ) − 7
:= 8 + (88 &times; (8 + 8) + 88/8)
:= 8 + ((88 &times; (8 + 8) + 8) + 8)
:= 8 &times; 8 &times; (8 + 8 + 8) − (88/8 + 88)
:= 9 &times; 9 &times; (9 + 9) − (((99 + 99)/9) + 9)
:= 9/9 + (9 &times; 9 &times; (9 + 9) − (9 + 9 + 9))
:= 9 &times; 9 &times; (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
&Iuml; 1438 := ((11 − 1) &times; (1 + 11)1+1 ) − 1 − 1
2
2
:= ((2 + 2 + 2) + 2) − (2 + 2 + 2)
&Iuml; 1448 := 11 &times; (11 &times; (1 + 11)) − 1 − 1 − 1 − 1
&Iuml; 1443 := 111 &times; (1 + 1 + 11)
:= 2 + (((2 + 2 + 2)2 + 2)2 + 2)
:= (22/2 + 2) &times; 222/2
3
:= 3/3 + (3 &times; (((3 − 3/3)3&times;3 ) − 33))
:= 3 + ((3 + 3) &times; (3 &times; 3 &times; 3 − 3))
:= (3/3 + 3) &times; ((33 &times; 33 − 3)/3)
:= (4 + 4) &times; (4 &times; 44 + 4) − (4 + 4)/4
:= 4 + ((4 + 4) &times; (4 &times; 44 + 4) − 4/4)
:= 4 + ((4 + 4) &times; (4 &times; 44 + 4) + 4)
:= 5 &times; 5 &times; 55 + (55 /5 + 5)/(5 + 5)
:= 555/5 &times; (55 + 5 + 5)/5
:= 5 + (555/5 &times; (55 + 5 + 5)/5)
:= ((((6 + 6)/6) + 6 &times; 6)(6+6)/6 ) − 6
:= (6/6 + 6 + 6) &times; 666/6
:= 6 &times; 6 &times; (6 &times; 6 + 6) − ((6 + 6)/6)6
:= 77 + ((7 &times; (7 + 7) &times; (7 + 7)) − (77/7))
:= (7 − 7/7 + 7) &times; 777/7
:= 77 + ((7 &times; (7 + 7) &times; (7 + 7)) − 7/7)
:= 8 + ((88 + 88)/8 + 88 &times; (8 + 8))
:= 888/8 &times; (88 + 8 + 8)/8
:= 8 &times; 8 &times; (8 + 8 + 8) − 88
:= 9 &times; 9 &times; (9 + 9) − (99/9 + 9)
:= 999/9 &times; ((99 + 9 + 9)/9)
:= 9 &times; 9 &times; (9 + 9) − 9/9 − 9
&Iuml; 1439 := ((11 − 1) &times; (1 + 11)1+1 ) − 1
&Iuml; 1444 := 1 + 111 &times; (1 + 1 + 11)
&Iuml; 1449 := (1 + 1 + 1) &times; ((11 + 11)1+1 − 1)
:= ((2 + 2 + 2)2 + 2)2 − (2/2 + 2 + 2)
:= ((2 + 2 + 2)2 + 2)2
:= (2/2 + 2) &times; (222 − 2/2)
:= (3 &times; (33 + 3)) + ((33/3)3 )
:= 333 + 3333/3
:= 3 &times; ((3 &times; (3 + 3) &times; 33 ) − 3)
:= 4 + (4 + 4) &times; (4 &times; 44 + 4)
:= (4 − 4/4) &times; ((44 &times; 44 − 4)/4)
:= (5 &times; (5 &times; (55 + 5))) − (55 + 5/5)
:= (55 − 5)/5 + (55 &times; (5 + 5 + 5))
:= (4 + 4) &times; (4 &times; 44 + 4) − 4/4
:= ((55 + 5) &times; (5 &times; 5 − 5/5)) − 5/5
(6+6)/6
:= (6 &times; (6 &times; (6 &times; 6 + 6) − (6 + 6))) − 6/6
:= (((6 + 6)/6) + 6 &times; 6)
:= 7 + ((7 &times; ((7 + 7) &times; (7 + 7) + 7)) + (77/7))
:= (7 &times; 7 − 77/7)(7+7)/7
:= 77 + (7 &times; (7 + 7) &times; (7 + 7))
:= ((8 + 8) &times; ((8 + 8)/8 + 88)) − 8/8
:= 888 + ((8888 + 8)/(8 + 8))
:= 8/8 + (8 &times; 8 &times; (8 + 8 + 8) − 88)
:= 9 &times; 9 &times; (9 + 9) − (9/9 + 9 + 9)
&Iuml; 1440 := (11 − 1) &times; (1 + 11)1+1
:= ((99/9 + 9 + 9) + 9)
(9+9)/9
&Iuml; 1445 := 1 + 1 + 111 &times; (1 + 1 + 11)
:= 6 + ((6/6 + 6 + 6) &times; 666/6)
:= 9 &times; 9 &times; (9 + 9) − 9
&Iuml; 1450 := (11 − 1) &times; (1 + (1 + 11)1+1 )
:= 2 &times; ((22 − 2) &times; (2 + 2 + 2)2 )
:= 2/2 + ((2 + 2 + 2)2 + 2)2
:= ((2/2 + 2) &times; 222 ) − 2
:= (3 + 3) &times; (3 &times; 3 &times; 33 − 3)
:= 3 + (333/3 + ((33/3)3 ))
:= 3 + ((((3 + 3) &times; 33+3 ) − 33)/3)
:= (4 + 4) &times; (4 &times; 44 + 4)
:= 4 + ((4 + 4) &times; (4 &times; 44 + 4) + 4/4)
:= 44 + (4 &times; (4 + 4) &times; 44 − (4 + 4)/4)
:= (55 + 5) &times; (5 &times; 5 − 5/5)
:= (5 &times; (5 &times; (55 + 5))) − 55
:= 5 &times; (((5 &times; 55 + 5) + 5) + 5)
:= 6 &times; (6 &times; (6 &times; 6 + 6) − (6 + 6))
:= 6 &times; 6 &times; (6 &times; 6 + 6) − (66 + 6/6)
:= 6 + ((((6 + 6)/6) + 6 &times; 6)(6+6)/6 )
:= 7 &times; 7 &times; 7 + (7777/7 − (7 + 7))
:= 7/7 + ((7 &times; 7 − 77/7)(7+7)/7 )
:= 7 + ((7 − 7/7 + 7) &times; 777/7)
:= (8 + 8) &times; ((8 + 8)/8 + 88)
:= 88 &times; (8 + 8) + 888/(8 + 8 + 8)
:= (8 + 8)/8 + (8 &times; 8 &times; (8 + 8 + 8) − 88)
:= (9 + 9) &times; (9 &times; 9 − 9/9)
:= 9 &times; 9 &times; (9 + 9) − ((99 + 9 + 9)/9)
:= 9/9 + (9 &times; 9 &times; (9 + 9) − 9)
&Iuml; 1441 := 11 &times; ((11 &times; (1 + 11)) − 1)
2
2
&Iuml; 1446 := 1 + 1 + 1 + 111 &times; (1 + 1 + 11)
2
2
&Iuml; 1451 := 11 &times; 11 &times; (1 + 11) − 1
:= ((2/2 + 2) &times; 222 ) − 2/2
:= ((2 + 2 + 2) + 2) − 2/2 − 2
:= 2 + ((2 + 2 + 2) + 2)
:= 3/3 + ((3 + 3) &times; (3 &times; 3 &times; 33 − 3))
:= (3 + 3) &times; (((33+3 + 3)/3) − 3)
:= (((3 + 3) &times; (33+3 − 3)) − 3)/3
:= 4/4 + (4 + 4) &times; (4 &times; 44 + 4)
:= 4 + ((4 + 4) &times; (4 &times; 44 + 4) + (4 + 4)/4)
:= 44 + (4 &times; (4 + 4) &times; 44 − 4/4)
:= 55 + (5 &times; 5 &times; 55 + (55/5))
:= 5/5 + ((5 &times; (5 &times; (55 + 5))) − 55)
:= 5/5 + (5 &times; (((5 &times; 55 + 5) + 5) + 5))
:= 6/6 + (6 &times; (6 &times; (6 &times; 6 + 6) − (6 + 6)))
:= 6 &times; 6 &times; (6 &times; 6 + 6) − 66
:= ((66 &times; (66 + 66)) − 6)/6
:= 7 + ((7 &times; (((7 + 7)/7) + 77)) − 7/7)
:= 7 &times; 7 &times; 7 + ((7777 − 7)/7 − 7)
:= 7 + ((7 &times; 7 − 77/7)(7+7)/7 )
:= 8/8 + ((8 + 8) &times; ((8 + 8)/8 + 88))
:= ((8 + 8)/8) &times; ((88/8 + 8 &times; 88) + 8)
:= 8 + (888/8 &times; (88 + 8 + 8)/8)
:= 9/9 + ((9 + 9) &times; (9 &times; 9 − 9/9))
:= 9 &times; 9 &times; (9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 &times; 9 &times; (9 + 9) − 9)
7
&Iuml; 1442 := 111 + 111+1+1
&Iuml; 1447 := 1 + 1 + 1 + 1 + 111 &times; (1 + 1 + 11)
:= ((2 + 2 + 2)2 + 2)2 − 2
2
2
&Iuml; 1452 := 11 &times; 11 &times; (1 + 11)
:= 2 + (((2 + 2 + 2) + 2) + 2/2)
:= (2/2 + 2) &times; 222
:= 333/3 + ((33/3) )
:= (((3 + 3) &times; 33+3 ) − 33)/3
:= 33 &times; (33/3 + 33)
:= (4 + 4)/4 + (4 + 4) &times; (4 &times; 44 + 4)
:= 4 + (((4 + 4) &times; (4 &times; 44 + 4) − 4/4) + 4)
:= 44 + 4 &times; (4 + 4) &times; 44
:= 55 + (5 &times; 5 &times; 55 + ((55 + 5)/5))
:= (5 + 5)/5 + ((5 &times; (5 &times; (55 + 5))) − 55)
:= (5 + 5)/5 + (5 &times; (((5 &times; 55 + 5) + 5) + 5))
:= ((666 &times; (6/6 + 6 + 6)) − 6)/6
:= 6/6 + (6 &times; 6 &times; (6 &times; 6 + 6) − 66)
:= 66 &times; ((66 + 66)/6)
:= 7 + (7 &times; (((7 + 7)/7) + 77))
:= 7 &times; 7 &times; 7 + (7777/7 − 7)
:= (77 + 7)/7 &times; (((7 + 7)/7)7 − 7)
:= (8 + 8)/8 + ((8 + 8) &times; ((8 + 8)/8 + 88))
:= 8 &times; 8 &times; (8 + 8 + 8) − (8/8 + 88)
:= 88 &times; (8 + 8) + (88/((8 + 8)/8))
:= (9 + 9)/9 + ((9 + 9) &times; (9 &times; 9 − 9/9))
:= 9 &times; 9 &times; (9 + 9) − 99/9
:= 9 &times; 9 &times; (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
&Iuml; 1453 := 1 + 11 &times; 11 &times; (1 + 11)
2
&Iuml; 1458 := (1 + 1) &times; (11 − 1 − 1)1+1+1
2+2+2
&Iuml; 1463 := 11 &times; (1 + 11 &times; (1 + 11))
:= 22/2 + ((2/2 + 2) &times; 222 )
:= 2/2 + ((2/2 + 2) &times; 22 )
:= 2 &times; ((2/2 + 2)
)
:= (((3 + 3) &times; (33+3 − 3)) + 3)/3
:= 3 &times; (3 &times; (3 + 3) &times; 33 )
:= (((3 + 3) &times; (33+3 + 3)) − 3)/3
:= 44 + (4 &times; (4 + 4) &times; 44 + 4/4)
:= (4 − 4/4)4 &times; ((4 + 4)/4 + 4 &times; 4)
:= 444 + (4 &times; 44 − (4/4 + 4))
:= ((5/5 + 5) &times; ((5 − (5 + 5)/5)5 )) − 5
:= (5/5 + 5) &times; ((5 − (5 + 5)/5)5 )
:= 5 + ((5/5 + 5) &times; ((5 − (5 + 5)/5)5 ))
:= ((66 &times; (66 + 66)) + 6)/6
:= 666 + 66 &times; (6 + 6)
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 − (6/6 + 6))
:= 7 &times; 7 &times; 7 + (7777 − 7)/7
:= 7 + (((7 &times; 7 − 77/7)
:= 8 &times; 8 + (88 &times; (8 + 8) − (88/8 + 8))
:= ((8 + 8)/8) &times; ((8/8 + 8)88/8−8 )
:= 8 &times; 8 + (88 &times; (8 + 8) − (8/8 + 8))
:= 9 &times; 9 &times; (9 + 9)
:= 9 &times; 9 &times; (9 + 9) + ((9 &times; 9 + 9)/(9 + 9))
:= 9 &times; 9 &times; (9 + 9) + ((9 − 99)/(9 + 9))
&Iuml; 1454 := 1 + 1 + 11 &times; 11 &times; (1 + 11)
(7+7)/7
) + 7)
&Iuml; 1459 := 1 + (1 + 1) &times; (11 − 1 − 1)1+1+1
:= 77 &times; ((77 + 7)/7 + 7)
&Iuml; 1464 := (1 + 11) &times; (1 + 111+1 )
:= 2 + ((2/2 + 2) &times; 222 )
:= 2/2 + (2 &times; ((2/2 + 2)2+2+2 ))
:= (2/2 + 2) &times; (222 + 2 + 2)
:= ((((3 + 3) &times; 33+3 ) − 3)/3) − 3
:= (((3 + 3) &times; 33+3 ) + 3)/3
:= (3 + 3) &times; ((33+3 + 3)/3)
:= 44 + (4 &times; (4 + 4) &times; 44 + (4 + 4)/4)
:= 4 + ((4 &times; (44 + 44 ) − 4/4) + 44 )
:= 444 + (4 &times; 44 − 4)
:= 55 + (5 &times; (5 &times; 55 + 5) − 5/5)
:= 5 + ((5 &times; (5 &times; 55 + 5) − 5/5) + 55)
:= (5 &times; 5 − 5/5) &times; ((55 + 5/5) + 5)
:= ((6 + 6)/6) &times; ((66 &times; 66 + 6)/6)
:= 6 + (((66 &times; (66 + 66)) + 6)/6)
:= 6 + (66 &times; (6 + 6) + 666)
:= 7 &times; 7 &times; 7 + 7777/7
:= ((7 + 7) &times; (7 &times; (7 + 7) + 7)) − 77/7
:= 7/7 + ((7 &times; 7 &times; (7 + 7)) + 777)
:= ((8/8 + 8 + 8) &times; (88 − ((8 + 8)/8))) − 8
:= 8 &times; (8 + 8) + ((88/8)88/8−8 )
:= 8 &times; 8 + (88 &times; (8 + 8) − 8)
:= 9/9 + 9 &times; 9 &times; (9 + 9)
:= 9 + (9 &times; 9 &times; (9 + 9) − ((9 + 9 + 9)/9))
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 − ((9 + 9)/9))
&Iuml; 1455 := 1 + 1 + 1 + 11 &times; 11 &times; (1 + 11)
&Iuml; 1460 := (1 + 1) &times; (1 + (11 − 1 − 1)1+1+1 )
&Iuml; 1465 := 1 + (1 + 11) &times; (1 + 111+1 )
:= (2/2 + 2) &times; (222 + 2/2)
:= 2 + (2 &times; ((2/2 + 2)2+2+2 ))
:= 2 + (((2/2 + 2) &times; 222 ) + 22/2)
:= 33+3 + (33+3 − 3)
:= 3 + ((((3 + 3) &times; 33+3 ) − 3)/3)
:= (((3 + 3) &times; (33+3 + 3)) + 3)/3
:= 44 + (4 &times; (44 + 44 ) − 4/4)
:= 4 + (4 &times; (44 + 44 ) + 44 )
:= 4/4 + ((4 &times; 44 − 4) + 444)
:= 55 + 5 &times; (5 &times; 55 + 5)
:= 5 + (5 &times; (5 &times; 55 + 5) + 55)
:= (5 &times; ((5 &times; (55 + 5)) − 5)) − 5 − 5
:= 6 + (((6/6 + 6 + 6) &times; 666/6) + 6)
:= 6 + (((6 + 6)/6) &times; ((66 &times; 66 + 6)/6))
:= (6 &times; (6 &times; (6 &times; 6 + 6) − 6)) − 66/6
:= 7 &times; 7 &times; 7 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 7 &times; 7 &times; 7)
:= 7 &times; 7 &times; 7 + ((7777 + 77)/7)
:= (8 − 8/8 + 8) &times; ((8/8 + 88) + 8)
:= 8 + ((88/((8 + 8)/8)) + 88 &times; (8 + 8))
:= 8/8 + ((88 &times; (8 + 8) − 8) + 8 &times; 8)
:= 9 &times; 9 &times; (9 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + 9 &times; 9 &times; (9 + 9)
:= 9 + (9 &times; 9 &times; (9 + 9) − ((9 + 9)/9))
&Iuml; 1456 := (1 + 1 + 11) &times; (1 + 111)
&Iuml; 1461 := 11 &times; (1 + 11 &times; (1 + 11)) − 1 − 1
&Iuml; 1466 := 11 + (1 + 1 + 11) &times; (1 + 111) − 1
:= 2 + (((2/2 + 2) &times; 222 ) + 2)
:= (2/2 + 2) &times; ((222 + 2/2) + 2)
:= 22 + ((2 + 2 + 2)2 + 2)2
:= (3 − 3/3) &times; (33+3 − 3/3)
:= 3 + (33+3 + 33+3 )
:= 3 + ((((3 + 3) &times; (33+3 + 3)) − 3)/3)
:= 44 + 4 &times; (44 + 44 )
:= 4 + ((4 &times; (44 + 44 ) + 44 ) + 4/4)
:= 444 + (4 &times; 44 − (4 + 4)/4)
:= (5 &times; 5 + 5/5) &times; (55 + 5/5)
:= 5 + ((5 &times; 5 + 5/5) &times; (55 + 5/5))
:= 5 + (((5 &times; 5 + 5/5) &times; (55 + 5/5)) + 5)
:= (6/6 + 6 + 6) &times; (666 + 6)/6
:= 66 + (((6 &times; 6/(6 + 6))6 ) + 666)
:= ((6 + 6)/6) &times; (((66 &times; 66 + 6)/6) + 6)
:= (7 + 7) &times; (777/7 − 7)
:= 7 + (7777/7 + 7 &times; 7 &times; 7)
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) − (77/7))
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) − 88)
:= 8 &times; 8 + (88 &times; (8 + 8) − (88/8))
:= 8 &times; 8 + ((88 &times; (8 + 8) − 8) + ((8 + 8)/8))
:= 9 &times; 9 &times; (9 + 9) − (9 + 9)/9
:= 9 &times; 9 &times; (9 + 9) + ((9 + 9 + 9)/9)
:= 9 + (9 &times; 9 &times; (9 + 9) − 9/9)
&Iuml; 1457 := 1 + (1 + 1 + 11) &times; (1 + 111)
2
&Iuml; 1462 := 11 &times; (1 + 11 &times; (1 + 11)) − 1
2+2+2
) + 2)
&Iuml; 1467 := 11 + (1 + 1 + 11) &times; (1 + 111)
:= 22 + (((2 + 2 + 2)2 + 2)2 + 2/2)
:= 2 + ((2/2 + 2) &times; (22 + 2/2))
:= 2 &times; (((2/2 + 2)
:= (((3 + 3) &times; 33+3 ) − 3)/3
:= 3 + ((((3 + 3) &times; 33+3 ) + 3)/3)
:= 3 &times; ((3 &times; (3 + 3) &times; 33 ) + 3)
:= 4/4 + (4 &times; (44 + 44 ) + 44 )
:= 4 + ((4 − 4/4)4 &times; ((4 + 4)/4 + 4 &times; 4))
:= 444 + (4 &times; 44 − 4/4)
:= 55 + (5 &times; (5 &times; 55 + 5) + ((5 + 5)/5))
:= 55 + (5 &times; 5 &times; 55 + ((5 + 5)/5)5 )
:= ((5 + 5)/5 &times; (555 + 55 )/5) − 5
:= 6 + (((66 &times; (66 + 66)) − 6)/6)
:= 6 + ((6/6 + 6 + 6) &times; (666 + 6)/6)
:= 6 + ((((6 &times; 6/(6 + 6))6 ) + 666) + 66)
:= 7/7 + ((7 + 7) &times; (777/7 − 7))
:= 777 + ((7 &times; 7 &times; (7 + 7)) − 7/7)
:= 77/7 + ((7 + 7) &times; (777/7 − 7))
:= 8 + ((8 &times; 8 &times; (8 + 8 + 8) − 88) + 8/8)
:= (8/8 + 8 + 8) &times; (88 − ((8 + 8)/8))
:= (8/8 + 8) &times; ((88/8 + 88) + 8 &times; 8)
:= 9 &times; 9 &times; (9 + 9) − 9/9
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 + ((9 + 9)/9))
:= 9 + 9 &times; 9 &times; (9 + 9)
107
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1468 := (1 + 1 + 11) &times; (1 + 1 + 111) − 1
2
2
&Iuml; 1473 := 11 &times; (1 + 1 + 11 &times; (1 + 11)) − 1
2
&Iuml; 1478 := 11 + (1 + 1 + 11) &times; (1 + 1 + 111) − 1 − 1
:= 2 + (((2 + 2 + 2) + 2) + 22)
:= 22 + (((2/2 + 2) &times; 22 ) − 2/2)
:= (2 &times; (22&times;(2+2) + 222 )) − 2
:= 3 + ((((3 + 3) &times; (33+3 + 3)) + 3)/3)
:= ((3 + 3) &times; (3 &times; 3 &times; 33 + 3)) − 3
:= 3 + (((3 + 3) &times; (33 − 3)) + ((33/3)3 ))
:= 444 + 4 &times; 44
:= 4 + ((444 + 4 &times; 44 ) + 4/4)
:= 44 + (((44 &times; 444/4) + 4)/4)
:= (5 &times; (5 &times; (55 + 5))) − ((5 + 5)/5)5
:= (5 &times; ((5 &times; (55 + 5)) − 5)) − (5 + 5)/5
:= 5 + ((5 &times; ((5 &times; (55 + 5)) − 5)) − ((5 + 5)/5))
:= (6 &times; (6 &times; (6 &times; 6 + 6) − 6)) − ((6 + 6)/6 + 6)
:= 66 + (666/6 + 6 &times; 6 &times; 6 &times; 6)
:= (6 + 6)/6 + (6 &times; (6 &times; (6 &times; 6 + 6) − 6))
:= ((7 + 7) &times; (7 &times; (7 + 7) + 7)) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + ((7 + 7) &times; (7 &times; (7 + 7) + 7))
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) + 7/7)
:= 8 &times; 8 + (88 &times; (8 + 8) − (8/((8 + 8)/8)))
:= 8/8 + (88 &times; (8 + 8) + 8 &times; 8)
:= 8 + ((88 &times; (8 + 8) − ((8 + 8)/8)) + 8 &times; 8)
:= 9 + (9 &times; 9 &times; (9 + 9) + 9/9)
:= 9 + ((9 &times; 9 &times; (9 + 9) − ((9 + 9 + 9)/9)) + 9)
:= 9 + (9 &times; 9 &times; (9 + 9) + (99/9))
&Iuml; 1469 := (1 + 1 + 11) &times; (1 + 1 + 111)
&Iuml; 1474 := 11 &times; (1 + 1 + 11 &times; (1 + 11))
2
&Iuml; 1479 := 11 + (1 + 1 + 11) &times; (1 + 1 + 111) − 1
:= (22/2 + 2) &times; (222/2 + 2)
:= 22 + ((2/2 + 2) &times; 22 )
:= (2 &times; (22 − 2))2 − (22/2)2
:= (((3 + 3) &times; 33+3 ) + 33)/3
:= (33 &times; 33 + 3333)/3
:= 3 + ((3 + 3) &times; (3 &times; 3 &times; 33 + 3))
:= 4/4 + (444 + 4 &times; 44 )
:= 4 + ((444 + 4 &times; 44 ) + (4 + 4)/4)
:= 444 + (44/4 + 4 &times; 44 )
:= (5 &times; ((5 &times; (55 + 5)) − 5)) − (5/5 + 5)
:= (5 &times; ((5 &times; (55 + 5)) − 5)) − 5/5
:= 5 + ((5 &times; ((5 &times; (55 + 5)) − 5)) − 5/5)
:= (6 &times; (6 &times; (6 &times; 6 + 6) − 6)) − 6/6 − 6
:= (66 + 6/6) &times; ((66 + 66)/6)
:= 6 &times; 6 + ((6/6 + 6 + 6) &times; 666/6)
:= ((7 + 7) &times; (7 &times; (7 + 7) + 7)) − 7/7
:= 77/7 + ((7 &times; 7 &times; (7 + 7)) + 777)
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) + ((7 + 7)/7))
:= 8 + ((88 &times; (8 + 8) − (88/8)) + 8 &times; 8)
:= 8 &times; 8 + (88 &times; (8 + 8) + ((8 + 8)/8))
:= (8/8 + 8 + 8) &times; (88 − 8/8)
:= 99/9 + 9 &times; 9 &times; (9 + 9)
:= 9 + ((9 &times; 9 &times; (9 + 9) − ((9 + 9)/9)) + 9)
:= 9 + (9 &times; 9 &times; (9 + 9) + ((99 + 9)/9))
&Iuml; 1470 := 1 + (1 + 1 + 11) &times; (1 + 1 + 111)
2
&Iuml; 1475 := 1 + 11 &times; (1 + 1 + 11 &times; (1 + 11))
2
&Iuml; 1480 := 11 + (1 + 1 + 11) &times; (1 + 1 + 111)
:= (2/2 + 2) &times; ((22 + 2 + 2) + 2)
:= 22 + (((2/2 + 2) &times; 22 ) + 2/2)
:= 2 &times; (22&times;(2+2) + 222 )
:= 3 + (3 &times; ((3 &times; (3 + 3) &times; 33 ) + 3))
:= ((33/3)3 ) + ((3 + 3) &times; (33 − 3))
:= (3 − 3/3) &times; (33+3 + 33/3)
:= 444 + ((4 + 4)/4 + 4 &times; 44 )
:= (4 &times; ((4/4 + 4)4 − 44 )) − 4/4
:= 4 + (4 &times; ((4/4 + 4)4 − 44 ))
:= (5 &times; ((5 &times; (55 + 5)) − 5)) − 5
:= 5 &times; ((5 &times; (55 + 5)) − 5)
:= 5 + (5 &times; ((5 &times; (55 + 5)) − 5))
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 − 6)
:= (6 &times; (6 &times; (6 &times; 6 + 6) − 6)) − 6/6
:= 6 + ((66 + 6/6) &times; ((66 + 66)/6))
:= (7 + 7) &times; (7 &times; (7 + 7) + 7)
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) − ((7 + 7)/7))
:= (7/7 + 7) &times; ((7 + 7) &times; (7 + 7) − (77/7))
:= 8 &times; 8 + (88 &times; (8 + 8) − ((8 + 8)/8))
:= 8 &times; 8 + ((88 &times; (8 + 8) − 8) + (88/8))
:= 8 + (88 &times; (8 + 8) + 8 &times; 8)
:= 9 &times; 9 &times; (9 + 9) + (99 + 9)/9
:= 9 + ((9 &times; 9 &times; (9 + 9) − 9/9) + 9)
:= ((9 + 9)/9) &times; (9 &times; 9 &times; 9 + (99/9))
&Iuml; 1471 := 1 + 1 + (1 + 1 + 11) &times; (1 + 1 + 111)
:= 2 + ((22/2 + 2) &times; (222/2 + 2))
&Iuml; 1476 := (1 + 11) &times; (1 + (1 + 111+1 ))
2
&Iuml; 1481 := ((1 + 1 + 11) &times; (1 + 1 + 1 + 111)) − 1
:= 2 + (((2/2 + 2) &times; 22 ) + 22)
:= 2 + ((2 &times; (22 − 2))2 − (22/2)2 )
:= (3 + 3) &times; (3 &times; 3 &times; 33 + 3)
:= 33 + (((((3 + 3) &times; 33+3 ) − 3)/3) − 3)
:= 4 + ((4 &times; 4 − 4/4) + 444)
:= 4 &times; ((4/4 + 4)4 − 44 )
:= (4444 − 4/4)/(4 − 4/4)
:= 5/5 + ((5 &times; ((5 &times; (55 + 5)) − 5)) − 5)
:= 5/5 + (5 &times; ((5 &times; (55 + 5)) − 5))
:= 5 + ((5 &times; ((5 &times; (55 + 5)) − 5)) + 5/5)
:= 6/6 + ((6/6 + 6) &times; (6 &times; 6 &times; 6 − 6))
:= 6 &times; (6 &times; (6 &times; 6 + 6) − 6)
:= 6 + ((6 &times; (6 &times; (6 &times; 6 + 6) − 6)) − 6/6)
:= 7/7 + ((7 + 7) &times; (7 &times; (7 + 7) + 7))
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) − 7/7)
:= 77/7 + ((7 + 7) &times; (7 &times; (7 + 7) + 7))
:= (((3 + 3) &times; ((3
3+3
+ 3) + 3)) + 3)/3
4
:= 8 &times; 8 + (88 &times; (8 + 8) − 8/8)
:= 8 &times; 8 + (88 &times; (8 + 8) + (8/((8 + 8)/8)))
:= 8 + ((88 &times; (8 + 8) + 8/8) + 8 &times; 8)
:= 9 &times; 9 &times; (9 + 9) + ((99 + 9 + 9)/9)
:= 9 + (9 &times; 9 &times; (9 + 9) + 9)
:= 9 &times; 9 &times; (9 + 9) + ((99 + 99 + 9)/9)
&Iuml; 1472 := 11 &times; (1 + 1 + 11 &times; (1 + 11)) − 1 − 1
:= 2 &times; (2
2+2
&times; ((2 &times; 22) + 2))
&Iuml; 1477 := 1 + (1 + 11) &times; (1 + 1 + 111+1 )
2
&Iuml; 1482 := (1 + 1 + 11) &times; (1 + 1 + 1 + 111)
:= 22 + ((2/2 + 2) &times; (22 + 2/2))
:= 2 + (2 &times; (22&times;(2+2) + 222 ))
:= 3 + ((((3 + 3) &times; 33+3 ) + 33)/3)
:= 3/3 + ((3 + 3) &times; (3 &times; 3 &times; 33 + 3))
:= (3 + 3) &times; (((33+3 + 3)/3) + 3)
:= 4 + (444 + 4 &times; 44 )
:= 4/4 + (4 &times; ((4/4 + 4)4 − 44 ))
:= 444 + (4 &times; (44 + 4) − (4 + 4)/4)
:= (5 + 5)/5 + (5 &times; ((5 &times; (55 + 5)) − 5))
:= (5 &times; 5 + 5/5) &times; ((5 + 5)/5 + 55)
:= 6/6 + (6 &times; (6 &times; (6 &times; 6 + 6) − 6))
:= 6 + (6 &times; (6 &times; (6 &times; 6 + 6) − 6))
:= ((7 + 7)/7) + ((7 + 7) &times; (7 &times; (7 + 7) + 7))
:= 7 + ((7 + 7) &times; (7 &times; (7 + 7) + 7))
:= (7/7 + 77) &times; ((77 + 7)/7 + 7)
:= 8 &times; ((88 + 88) + 8)
:= 88 + (88 &times; (8 + 8) − (88/8 + 8))
:= 8 + ((88 &times; (8 + 8) + ((8 + 8)/8)) + 8 &times; 8)
:= 9 + (((9 &times; 9 + 9)/(9 + 9)) + 9 &times; 9 &times; (9 + 9))
:= 9 + ((9 &times; 9 &times; (9 + 9) + 9/9) + 9)
:= ((9 + 9)/9) &times; (((99 + 9)/9) + 9 &times; 9 &times; 9)
5
:= (5 + 5)/5 &times; (555 + 5 )/5
6
:= ((6 + 6)/6) &times; ((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
&Iuml; 1483 := 1 + ((1 + 1 + 11) &times; (1 + 1 + 1 + 111))
&Iuml; 1488 := (1 + 11) &times; (1 + 1 + 1 + 111+1 )
&Iuml; 1493 := (1 + 1)11 − (1111 − 1)/(1 + 1)
:= ((2/2 + 2) &times; (22/2 + 222 )) − 2
:= (22 + 2) &times; (22+2+2 − 2)
:= 2/2 + (2 &times; ((2 &times; 222 ) − 222))
:= 3 + ((3 − 3/3) &times; (33+3 + 33/3))
:= 3 + (3 &times; (3 &times; ((3 + 3) &times; 33 + 3)))
:= (3 + 3) &times; 33 + ((33/3)3 )
:= 444 + (4 &times; (44 + 4) − 4/4)
:= 4 + (4 &times; (44 + 4) + 444)
:= 4/4 + (44 &times; 44 − 444)
:= 5 &times; 5 + ((5/5 + 5) &times; ((5 − (5 + 5)/5)5 ))
:= (5/5 + 5) &times; (((5 − (5 + 5)/5)5 ) + 5)
:= (5 &times; (5 &times; (55 + 5))) − ((5 + 5)/5 + 5)
:= 6 + ((6 &times; (6 &times; (6 &times; 6 + 6) − 6)) + 6/6)
:= 6 + ((6 &times; (6 &times; (6 &times; 6 + 6) − 6)) + 6)
:= 6 + ((6 &times; (6 &times; (6 &times; 6 + 6) − 6)) + (66/6))
:= 777/7 + (7 &times; (7 + 7) &times; (7 + 7))
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) + (77/7))
:= 7 + (((77 &times; (((7 + 7)/7)7 + 7)) + 7)/7)
:= 8 &times; 8 + (88 &times; (8 + 8) + 88/8)
:= 88 + (88 &times; (8 + 8) − 8)
:= 8 + ((88 &times; (8 + 8) − (88/8)) + 88)
:= 9 + (((9 &times; 9 &times; (9 + 9) − ((9 + 9)/9)) + 9) + 9)
:= 999/9 + 9 &times; (9 &times; (9 + 9) − 9)
:= 9 + ((9 &times; (99 + 9)) + (((9 + 9)/9)9 ))
&Iuml; 1484 := 11 &times; (1 + 1 + 1 + 11 &times; (1 + 11)) − 1
2&times;(2+2)
2
&Iuml; 1489 := 1 + (1 + 11) &times; (1 + 1 + 1 + 111+1 )
&Iuml; 1494 := 1 + (1 + 1)11 − (1111 − 1)/(1 + 1)
:= (2 &times; (22 − 2))2 − 222/2
:= 2 + (2 &times; ((2 &times; 222 ) − 222))
:= 3 + ((3 &times; (3 &times; ((3 + 3) &times; 33 + 3))) + 3/3)
:= (3 + 3) &times; ((3 + 3)3 + 33)
:= 444 + 4 &times; (4 + 4)
:= (4 − 4/4)4 + 4 &times; (4 + 4) &times; 44
:= (4 + 4)/4 + (44 &times; 44 − 444)
:= (5 &times; (5 &times; (55 + 5))) − (55/5 + 5)
:= (5 &times; (5 &times; (55 + 5))) − 55/5
:= (5 &times; (5 &times; (55 + 5))) − (5/5 + 5)
:= 6 + ((6 &times; (6 &times; (6 &times; 6 + 6) − 6)) + ((6 + 6)/6))
:= 6 + (((6 &times; (6 &times; (6 &times; 6 + 6) − 6)) + 6/6) + 6)
:= 2 &times; ((2
+ 22 ) + 2)
3
:= 3 + ((((3 + 3) &times; 3
3+3
) − 3)/3)
4
7
:= (6 &times; 6 &times; (66 − 6)) − 666
:= 7 + (((7 + 7) &times; (7 &times; (7 + 7) + 7)) + 7)
:= (77 &times; (7 + 7 + 7)) − ((7 + 7)/7)
:= (77/7 + 7) &times; (77 − 7/7 + 7)
:= 8 &times; 8 + (((88 + 8)/8) + 88 &times; (8 + 8))
:= 8/8 + ((88 &times; (8 + 8) − 8) + 88)
:= 88 + (88 &times; (8 + 8) − ((8 + 8)/8))
:= 9 + (((99 + 99)/9) + 9 &times; 9 &times; (9 + 9))
:= (9 + 9) &times; (((9 + 9)/9) + 9 &times; 9)
9
:= (9 &times; (99 + 9)) + (((9 + 9)/9) )
&Iuml; 1490 := 1 + 1 + (1 + 11) &times; (1 + 1 + 1 + 111+1 )
&Iuml; 1495 := (1 + 1 + 11) &times; (1 + 1 + 1 + 1 + 111)
:= (2/2 + 2) &times; (22/2 + 22 )
:= 2 + ((22 + 2) &times; (22+2+2 − 2))
:= ((2 &times; 22)2 ) − ((22 − 2/2)2 )
:= 3 &times; (3 &times; ((3 + 3) &times; 33 + 3))
:= 33 + ((((3 + 3) &times; 33+3 ) − 3)/3)
:= 3/3 + ((3 + 3) &times; ((3 + 3)3 + 33))
:= 4/4 + (4 &times; (44 + 4) + 444)
:= (4/4 + 4) &times; ((44 − (4 + 4)/4) + 44)
:= (4/4 + 4) &times; ((44 − 4/4) + 44 )
:= 55 &times; ((5 + 5)/5 + 5 &times; 5)
:= (5 &times; (5 &times; (55 + 5))) − 5 − 5
:= (5 &times; (5 &times; (55 + 5))) − 5
:= 6 + (((6/6 + 6 + 6) &times; 666/6) + 6 &times; 6)
:= 6 &times; 6 &times; (6 &times; 6 + 6) − ((66 + 66)/6)
:= (66 − 6/6) &times; ((66/6 + 6) + 6)
&Iuml; 1485 := 11 &times; (1 + 1 + 1 + 11 &times; (1 + 11))
2
7
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + 777/7)
:= 7 + ((((7 + 7) &times; (7 &times; (7 + 7) + 7)) + (77/7)) + 7)
:= 88 + (88 &times; (8 + 8) − (88/8))
:= 88 + ((88 &times; (8 + 8) − 8) + ((8 + 8)/8))
:= 88 + (88 &times; (8 + 8) − 8/8)
:= 9 + ((9 &times; 9 &times; (9 + 9) + 9) + 9)
:= 9 + (((99 + 99 + 9)/9) + 9 &times; 9 &times; (9 + 9))
:= 9/9 + ((9 + 9) &times; (((9 + 9)/9) + 9 &times; 9))
:= 77/7 &times; (((7 + 7)/7) + 7)
&Iuml; 1486 := 1 + 11 &times; (1 + 1 + 1 + 11 &times; (1 + 11))
2+2+2
&Iuml; 1491 := (1 + 1)11 − 1 − (1 + 1111)/(1 + 1)
&Iuml; 1496 := 11 &times; (1 + 1 + 1 + 1 + 11 &times; (1 + 11))
:= 2 + ((2 &times; (22 − 2)) − 222/2)
:= 2 &times; (22 &times; ((2 &times; 22+2 ) + 2))
:= 33 + ((((3 + 3) &times; 33+3 ) + 3)/3)
:= 33 + (33+3 + 33+3 )
:= 3 + ((3 + 3) &times; 33 + ((33/3)3 ))
:= 444 + (4 &times; (44 + 4) + (4 + 4)/4)
:= 44 &times; 44 − (444 + 4/4)
:= 4 + (44 &times; 44 − 444)
:= 5 &times; 5 &times; 55 + 555/5
:= 5 + (5 &times; 5 &times; 55 + 555/5)
:= 5/5 + ((5 &times; (5 &times; (55 + 5))) − 5)
:= ((66 − 6)/6) + (6 &times; (6 &times; (6 &times; 6 + 6) − 6))
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 − (6 &times; 6/(6 + 6)))
:= 6 &times; 66 + ((6666 − 66)/6)
:= ((77 &times; (((7 + 7)/7) + 7)) + 7)/7
:= (7 + 7 + 7) &times; ((7/7 − 7) + 77)
:= 7 + ((77 &times; (7 + 7 + 7)) − ((7 + 7)/7)7 )
:= 88 + ((8 − 88)/8 + 88 &times; (8 + 8))
:= 8 + ((88 &times; (8 + 8) + 88/8) + 8 &times; 8)
:= 88 + 88 &times; (8 + 8)
:= 9 + (((9 &times; 9 &times; (9 + 9) + 9/9) + 9) + 9)
:= 9 &times; 9 &times; (9 + 9) + (99/((9 + 9 + 9)/9))
:= ((9 − 9/9) + 9) &times; (99 − (99/9))
:= ((22 + 2) &times; (2
2
− 2)) − 2
7
&Iuml; 1487 := 11 + (1 + 11) &times; (1 + 1 + 111+1 )
&Iuml; 1492 := (1 + 1)11 − (1 + 1111)/(1 + 1)
:= 2 + ((2/2 + 2) &times; (22/2 + 222 ))
:= 3 + (((((3 + 3) &times; 3
3+3
:= 2 &times; ((2 &times; 222 ) − 222)
3
) − 3)/3) + 3 )
4
:= 33 + ((((3 + 3) &times; 3
3+3
&Iuml; 1497 := 1 + 11 &times; (1 + 1 + 1 + 1 + 11 &times; (1 + 11))
:= (2 &times; 222 ) + ((22 + 2/2)2 )
) + 3)/3)
:= 3 + ((3 + 3) &times; ((3 + 3)3 + 33))
:= 44 &times; 44 − 444
:= 4 + ((44 &times; 44 − 444) + 4/4)
:= 5 &times; 5 &times; 55 + (555 + 5)/5
:= 5 + ((555 + 5)/5 + 5 &times; 5 &times; 55)
:= (5 + 5)/5 + ((5 &times; (5 &times; (55 + 5))) − 5)
:= 66/6 + (6 &times; (6 &times; (6 &times; 6 + 6) − 6))
:= ((6/6 + 6) &times; (6 &times; 6 &times; 6 − (6 + 6)/6)) − 6
:= 6 + ((6/6 + 6) &times; (6 &times; 6 &times; 6 − (6 &times; 6/(6 + 6))))
:= (44 &times; (44 + 4)) − (4/4 + 4)
7
7
:= 77 + ((((7 + 7 + 7)/7) ) − 777)
:= 7 + (77/7 &times; (((7 + 7)/7) + 7))
:= 77 + ((7 &times; ((7 + 7) &times; (7 + 7) + 7)) − 7/7)
:= 8 + ((8/8 + 8 + 8) &times; (88 − 8/8))
:= 88 + (88 &times; (8 + 8) − (8/((8 + 8)/8)))
:= 8/8 + (88 &times; (8 + 8) + 88)
:= 9 + ((9 &times; 9 &times; (9 + 9) + (99/9)) + 9)
:= ((9 + 9)/9) &times; (((9 &times; 9 &times; 9 − 9/9) + 9) + 9)
:= 9 + (9 &times; (9 &times; (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
&Iuml; 1498 := (1 + 1)11 − (1111 − 11)/(1 + 1)
&Iuml; 1508 := (1 + 1)11−1 + (11 + 11)1+1
&Iuml; 1503 := (1 + 1 + 1) &times; ((1 + 1)11−1−1 − 11)
:= 2 + (2 &times; (22 &times; ((2 &times; 22+2 ) + 2)))
:= (2/2 + 2)2 &times; (((22/2 + 2)2 ) − 2)
:= 222 + (22&times;(2+2)+2 )
:= ((33 &times; 333) − 3)/(3 + 3)
:= 3 &times; (((3 + 3) &times; (3 &times; 33 + 3)) − 3)
:= 3 + (((3/3 + 3) + 3) &times; ((3 + 3)3 − 3/3))
:= 44 + ((4 + 4)/4 &times; ((4/4 + 4)4 − 4))
:= (4 &times; (444 − 4)) − (4/4 + 44 )
:= 4 + ((4 + 4) &times; (444 − 44 ))
:= (5 &times; (5 &times; (55 + 5))) − (5 + 5)/5
:= 5 + ((5 &times; (5 &times; (55 + 5))) − ((5 + 5)/5))
:= ((55 + 5/5)/((5 + 5)/5)) − 55
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 − (6 + 6)/6)
:= (((6 − 66) + 6)/6) + 6 &times; 6 &times; (6 &times; 6 + 6)
:= (6 + 6)/6 + (6 &times; 6 &times; (6 &times; 6 + 6) − 6)
:= 77 + (7 &times; ((7 + 7) &times; (7 + 7) + 7))
:= 7 &times; (7 &times; 7 + 7) + 7777/7
:= 7 + ((((7 + 7 + 7)/7)7 ) − (7 &times; 7 &times; (7 + 7)))
:= 88 + (88 &times; (8 + 8) + ((8 + 8)/8))
:= 8 + ((88 &times; (8 + 8) − 8/8) + 88)
:= 88 + (((88 + 8)/8) + 88 &times; (8 + 8))
:= 999 + (((9 &times; 999) − 9)/(9 + 9))
:= 9 + ((9 + 9) &times; (((9 + 9)/9) + 9 &times; 9))
:= 9 &times; 9 &times; (9 + 9) + ((9 &times; 99 + 9)/(9 + 9))
&Iuml; 1499 := 11 + (1 + 11) &times; (1 + 1 + 1 + 111+1 )
:= ((2 &times; (22 − 2) − 2/2)2 ) − 22
3
&Iuml; 1504 := 1 + (1 + 1 + 1) &times; ((1 + 1)11−1−1 − 11)
&Iuml; 1509 := 1 + (1 + 1)11−1 + (11 + 11)1+1
:= 2 &times; (2 &times; ((22 − 2)2 − (22 + 2)))
:= 222 + ((222/2 + 2)/2)
3
:= ((3 &times; 333) + 3)/(3 + 3)
:= 3 &times; (((3 − 3/3)3&times;3 ) − 3 &times; 3)
:= ((3 &times; 333) + 33)/(3 + 3)
4
4
:= ((4/4 + 4) &times; (44 + 4 )) − 4/4
:= (4 + 4) &times; (444 − 4 )
:= (5 &times; (5 &times; (55 + 5))) − 5/5
:= 5 + ((5 &times; (5 &times; (55 + 5))) − 5/5)
:= 5 + (((5 &times; (5 &times; (55 + 5))) − 5/5) + 5)
:= 6 &times; 6 &times; (6 &times; 6 + 6) − (6/6 + 6 + 6)
:= 6 &times; 6 &times; (6 &times; 6 + 6) − ((6 + 6)/6 + 6)
:= 6 &times; 6 &times; (6 &times; 6 + 6) − 6 &times; 6/(6 + 6)
:= 7 + ((77/7 &times; (((7 + 7)/7)7 + 7)) + 7)
:= (7/7 + 7) &times; (777/7 + 77)
:= (((7 + 7)/7)77/7 ) − 7 &times; 77
:= 88 + ((88 &times; (8 + 8) − 8) + (88/8))
:= 8 + (88 &times; (8 + 8) + 88)
:= 8 + ((88/8 + 8) &times; (88 − (8/8 + 8)))
:= 999 + (((9 &times; 999) + 9)/(9 + 9))
:= 9 + (((9 + 9) &times; (((9 + 9)/9) + 9 &times; 9)) + 9/9)
:= ((9 + 9 + 9)/9) &times; ((((9 + 9)/9)9 ) − 9)
&Iuml; 1500 := (1 + 1 + 1) &times; (11 − 1)1+1+1 /(1 + 1)
:= 2 &times; ((22 &times; ((2 &times; 22+2 ) + 2)) + 2)
3
:= (3 &times; 3 + 3) &times; ((3 − 3/3 + 3) )
4
:= 4 &times; 44 + ((44 &times; 44 + 4)/4)
&Iuml; 1505 := (1 + 11)1+1+1 − 1 − (1 + 1) &times; 111
&Iuml; 1510 := ((1 + 1 + 1) &times; (1 + 1 + 11))1+1 − 11
:= 222 + (((222/2 − 2)/2) − 2)
:= 2 + ((22&times;(2+2)+2 ) + 222 )
3
:= ((3/3 + 3) + 3) &times; ((3 + 3) − 3/3)
:= 3/3 + (3 &times; (((3 − 3/3)3&times;3 ) − 3 &times; 3))
4
:= (4/4 + 4) &times; (44 + 4 )
:= 4/4 + ((4 + 4) &times; (444 − 4 ))
:= 4 + ((4 + 4)/4 &times; (4/4 + 4)4 + 44 )
:= 5 &times; (5 &times; (55 + 5))
:= 5 + (5 &times; (5 &times; (55 + 5)))
:= 5 + ((5 &times; (5 &times; (55 + 5))) + 5)
:= 6 &times; 6 &times; (6 &times; 6 + 6) − 6 − 6
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 − 6/6)
:= ((7 + 7)/7)7 + (7 &times; (7 + 7) &times; (7 + 7))
:= 7 &times; (7 &times; 7 &times; 7 − ((7 + 7)/7)7 )
:= 6 &times; 6 &times; (6 &times; 6 + 6) − (6 + 6)/6
:= 7 + (7777/7 + 7 &times; (7 &times; 7 + 7))
:= 88 + (88 &times; (8 + 8) + (8/((8 + 8)/8)))
:= 8 + ((88 &times; (8 + 8) + 88) + 8/8)
:= 88 &times; (8 + 8) + ((888 − 8)/8 − 8)
:= 999 + ((((9 + 9)/9)9 ) − (99/9))
:= 9 + (((9 − 9/9) + 9) &times; (99 − (99/9)))
:= 999 + ((((9 + 9)/9)9 ) − 9/9)
&Iuml; 1501 := 1 + (1 + 1 + 1) &times; ((11 − 1)1+1+1 /(1 + 1))
&Iuml; 1506 := (1 + 11)1+1+1 − (1 + 1) &times; 111
&Iuml; 1511 := 1111 + ((1 + 1) &times; (11 − 1))1+1
:= 2 + (((2 &times; (22 − 2) − 2/2)2 ) − 22)
:= 222 + ((22&times;(2+2)+2 ) − 2)
:= (22 − 2)2 + 2222/2
:= 3 + (((33 &times; 333) − 3)/(3 + 3))
:= 3 + (3 &times; (((3 + 3) &times; (3 &times; 33 + 3)) − 3))
:= 3 &times; 333 + ((3 − 3/3)3&times;3 )
:= 4/4 + ((4/4 + 4) &times; (44 + 44 ))
:= 44 + (4 + 4)/4 &times; (4/4 + 4)4
:= ((44 − 4) &times; ((4 + 4)/4 + 4)) − 4/4
:= 5/5 + (5 &times; (5 &times; (55 + 5)))
:= 5 + ((5 &times; (5 &times; (55 + 5))) + 5/5)
:= 55/5 + (5 &times; (5 &times; (55 + 5)))
:= 6 &times; 6 &times; (6 &times; 6 + 6) − 66/6
:= 6 &times; 6 &times; (6 &times; 6 + 6) − 6
:= 6 &times; 6 &times; (6 &times; 6 + 6) − 6/6
:= (((7 + 7 + 7)/7)7 ) − (7 &times; 7 &times; (7 + 7))
:= (77 &times; (7 + 7 + 7)) − 777/7
:= ((77 + 7) &times; (77/7 + 7)) − 7/7
:= (88/8 + 8) &times; (88 − (8/8 + 8))
:= 8 + ((88 &times; (8 + 8) + ((8 + 8)/8)) + 88)
:= 888/8 + (88 &times; (8 + 8) − 8)
:= (9/9 + 9 + 9) &times; (9 &times; 9 − ((9 + 9)/9))
:= 9 + ((9 &times; (9 &times; (9 + 9) − 9) + 999/9) + 9)
:= 999 + (((9 + 9)/9)9 )
&Iuml; 1502 := (1 + 1 + 1) &times; ((1 + 1)11−1−1 − 11) − 1
:= 2 &times; (((2/2 + 2)
2+2+2
) + 22)
3
:= 3 + (((3 &times; 333) + 3)/(3 + 3))
4
4
&Iuml; 1507 := 11 &times; (111 + ((1 + 1) &times; (1 + 1 + 11)))
2
:= 22 + ((2
22/2
:= 2 &times; (2 &times; ((22 − 2)2 − 22))
− 2)/2)
3
:= 3 &times; ((3 + 3) &times; (3 &times; 33 + 3))
:= 3 + (((3 &times; 333) + 33)/(3 + 3))
4−4/4
:= 4 + ((4 + 4)/4 &times; (4/4 + 4) − 4)
:= 4 &times; 44 + ((44/4)
:= (5 + 5)/5 + (5 &times; (5 &times; (55 + 5)))
:= 5 + ((5 &times; (5 &times; (55 + 5))) + ((5 + 5)/5))
:= ((6 − 66)/6) + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 6/6 + (6 &times; 6 &times; (6 &times; 6 + 6) − 6)
:= 7 &times; (7 &times; 7 + 7) + (7777 − 7)/7
&Iuml; 1512 := (11 − 1 − 1) &times; ((1 + 1 + 11)1+1 − 1)
:= (44 − 4) &times; ((4 + 4)/4 + 4)
)
:= ((55 + 5)/5) + (5 &times; (5 &times; (55 + 5)))
:= 6 &times; 6 &times; (6 &times; 6 + 6)
7
:= 7 + ((7 &times; (7 + 7) &times; (7 + 7)) + ((7 + 7)/7) )
:= (77 + 7) &times; (77/7 + 7)
:= 8 + ((88 &times; (8 + 8) − ((8 + 8)/8)) + 88)
:= 88 + (88 &times; (8 + 8) + 88/8)
:= (8 + 8 + 8) &times; (8 &times; 8 − 8/8)
:= 999 + ((((9 + 9)/9)9 ) − 9)
:= 9 &times; 9 &times; (9 + 9) + ((9 &times; 99 − 9)/(9 + 9))
:= (9 + 9) &times; (((9 + 9 + 9)/9) + 9 &times; 9)
110
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1513 := (1 + ((111 − 1)/(1 + 1))1+1 )/(1 + 1)
&Iuml; 1518 := 11 &times; (111 + (1 + 1 + 1)1+1+1 )
&Iuml; 1523 := 1 + 1 + ((1 + 1 + 1) &times; (1 + 1 + 11))1+1
:= 2 + 2222/2 + (22 − 2)2
:= 222 + (2 + 2 + 2)2+2
:= 2 + ((2 &times; (22 − 2) − 2/2)2 )
:= 3/3 + (3 &times; ((3 + 3) &times; (3 &times; 33 + 3)))
:= 3 &times; (((3 − 3/3)3&times;3 ) − (3 + 3))
:= ((33/3)3 ) + (3 &times; ((3/3 + 3)3 ))
:= 4/4 + ((44 − 4) &times; ((4 + 4)/4 + 4))
:= ((4 + 4)/4 + 4) &times; ((4/4 − 4) + 44 )
:= 4 + (4 &times; 444 − (4/4 + 44 ))
:= (55 &times; 55 + 5/5)/((5 + 5)/5)
:= 5 + ((55 &times; 55 + 5/5)/((5 + 5)/5))
:= (5 &times; ((5 &times; (55 + 5)) + 5)) − (5 + 5)/5
:= 6/6 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 6 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 66/6 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 7/7 + ((77 + 7) &times; (77/7 + 7))
:= (7 &times; (7 &times; (7 &times; 7 − 7) − 77)) − 7/7
:= 77/7 + ((77 + 7) &times; (77/7 + 7))
:= (8/8 + 8 + 8) &times; (8/8 + 88)
:= 88 &times; (8 + 8) + (888 − 8)/8
:= ((9 − 9/9) + 9) &times; ((9 &times; 9 − 9/9) + 9)
&Iuml; 1514 := 1 + (1 + ((111 − 1)/(1 + 1))1+1 )/(1 + 1)
:= 88/8 + ((8 + 8 + 8) &times; (8 &times; 8 − 8/8))
9
:= 9 + (((9 + 9 + 9)/9) &times; ((((9 + 9)/9) ) − 9))
:= (9 + 9)/9 + ((9 &times; (99 + 9 &times; 9)) − 99)
&Iuml; 1524 := 1 + 1 + 1 + ((1 + 1 + 1) &times; (1 + 1 + 11))1+1
&Iuml; 1519 := (1 + 1)11 − (1 + 11 + 11)1+1
:= 2 + 2 &times; 2 &times; ((22 − 2)2 − 22)
:= ((2 &times; (22 − 2) − 2/2)2 ) − 2
:= (2 + 2 + 2) &times; (22&times;(2+2) − 2)
:= (3 + 3)3 + (((33/3)3 ) − 33)
:= ((3/3 + 3) + 3) &times; ((3 + 3)3 + 3/3)
:= 3 + ((33 + 3 + 3)3−3/3 )
:= 44 + ((4 + 4)/4 &times; ((4/4 + 4)4 + 4))
:= 4 &times; 444 − (4/4 + 44 )
:= 4 + (4 &times; 444 − 44 )
:= (5 &times; ((5 &times; (55 + 5)) + 5)) − 55/5
:= (5 &times; ((5 &times; (55 + 5)) + 5)) − (5/5 + 5)
:= (5 &times; ((5 &times; (55 + 5)) + 5)) − 5/5
:= (6 + 6)/6 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + 6/6)
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + 6)
:= ((7 + 7)/7) + ((77 + 7) &times; (77/7 + 7))
:= 7 &times; (7 &times; (7 &times; 7 − 7) − 77)
:= ((7 + 7)/7) &times; (777 − (7/7 + 7 + 7))
:= 8/8 + ((8/8 + 8 + 8) &times; (8/8 + 88))
:= 888/8 + 88 &times; (8 + 8)
:= ((88 + 8)/8) &times; (8 &times; (8 + 8) − 8/8)
:= 9 &times; 9 &times; (9 + 9) + ((999 + 9)/(9 + 9))
:= (9 &times; (9 &times; (9 + 9) + 9)) − (99/9 + 9)
:= 9 &times; 9 + (999/9 &times; ((99 + 9 + 9)/9))
&Iuml; 1515 := (1 + 1 + 1 + 1 + 11) &times; (1 + (11 − 1)1+1 )
&Iuml; 1520 := ((1 + 1 + 1) &times; (1 + 1 + 11))1+1 − 1
&Iuml; 1525 := (1 + 1 + 1) &times; (1 + 1)11−1−1 − 11
:= (2/2 + 2) &times; (222 − 2/2 + 22)
:= 2 + ((2 + 2 + 2)2+2 + 222)
:= 2 + (((2 &times; (22 − 2) − 2/2)2 ) + 2)
:= 3 + (3 &times; ((3 + 3) &times; (3 &times; 33 + 3)))
:= (3 + 3)3 + (((33/3)3 ) − 33 )
:= (3 &times; ((3 − 3/3)3&times;3 )) − 33/3
:= 4 &times; 444 − ((4/4 + 44 ) + 4)
:= 4 &times; 444 − 44
:= 4 + 4 &times; 444 − 44 + 4/4
:= 5 + (((5 &times; (5 &times; (55 + 5))) + 5) + 5)
:= (5 &times; ((5 &times; (55 + 5)) + 5)) − 5
:= 5 &times; ((5 &times; (55 + 5)) + 5)
:= (6 &times; 6/(6 + 6)) + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + ((6 + 6)/6))
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6) + 6/6) + 6)
:= (7/7 + 7 + 7) &times; (7777/77)
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 77))
:= (((7 + 7) &times; (777 − (7 + 7))) − 7)/7
:= (8 + 8) &times; (88 − 8/8 + 8)
:= 8 &times; 8 &times; (8 + 8 + 8) − 88/8
:= 8 + ((88 &times; (8 + 8) + 88/8) + 88)
9
:= 9 &times; 9 &times; (9 + 9) + (((((9 + 9)/9) ) + 9/9)/9)
&Iuml; 1516 := 1 + ((1 + 1 + 1 + 1 + 11) &times; (1 + (11 − 1)1+1 ))
9
:= 9 &times; 9 + (((99/9 + 9 + 9) + 9)(9+9)/9 )
:= 9 + ((((9 + 9)/9) ) + 999)
&Iuml; 1521 := ((1 + 1 + 1) &times; (1 + 1 + 11))1+1
2
&Iuml; 1526 := (1 + (1 + 1 + 11)) &times; (111 − 1 − 1)
:= 2 &times; ((2 &times; ((22 − 2)2 − 22)) + 2)
:= (2 &times; (22 − 2) − 2/2)
:= 2 + ((2 + 2 + 2) &times; (22&times;(2+2) − 2))
:= 3 + ((3 &times; ((3 + 3) &times; (3 &times; 33 + 3))) + 3/3)
:= (33 + 3 + 3)3−3/3
:= 33 &times; (3 + 3) + (((33/3)3 ) − 3)
:= 4 &times; 444 − (44 + 4)
:= 4/4 + (4 &times; 444 − 44 )
:= (((4 + 4)/4 + 4) &times; (44 − 4/4)) − 4
:= 5 + ((5 &times; (5 &times; (55 + 5))) + (55/5))
:= 5/5 + ((5 &times; ((5 &times; (55 + 5)) + 5)) − 5)
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) − ((6 + 6)/6))
77/7
:= 7 + ((((7 + 7)/7)
) − 7 &times; 77)
6
:= 66 &times; (6 + 6) + ((6 &times; 6/(6 + 6)) )
:= (((7 − 77)/7) + 7 &times; 7)
(7+7)/7
:= 5/5 + (5 &times; ((5 &times; (55 + 5)) + 5))
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 + (6 + 6)/6)
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 77))
:= 8 &times; 8 &times; (8 + 8 + 8) − ((88 + 8)/8 + 8)
:= 8 + ((8/8 + 8 + 8) &times; (8/8 + 88))
:= (8 − 88)/8 + 8 &times; 8 &times; (8 + 8 + 8)
:= 9 + (((9 &times; 99 − 9)/(9 + 9)) + 9 &times; 9 &times; (9 + 9))
:= (9 &times; (99 + 9 &times; 9)) − 99
:= (9 + 9) &times; 99 − (((9 + 9)/9)9−9/9 )
&Iuml; 1517 := (11 &times; (111 + (1 + 1 + 1)1+1+1 )) − 1
&Iuml; 1522 := 1 + ((1 + 1 + 1) &times; (1 + 1 + 11))1+1
2
&Iuml; 1527 := 1 + (1 + 1 + 1 + 11) &times; (111 − 1 − 1)
:= ((2 &times; (22 − 2) − 2/2) ) − 2 − 2
:= 2/2 + ((2 &times; (22 − 2) − 2/2) )
:= 2 + ((((2 &times; (22 − 2) − 2/2)2 ) + 2) + 2)
:= 3 + ((((33/3)3 ) − 33) + (3 + 3)3 )
:= 3/3 + ((33 + 3 + 3)3−3/3 )
:= 3 &times; (((3 − 3/3)3&times;3 ) − 3)
:= 4/4 + (4 &times; 444 − (44 + 4))
:= (4 + 4)/4 + (4 &times; 444 − 44 )
:= 4 + ((4 &times; 444 − (4/4 + 44 )) + 4)
:= 5 + ((5 &times; (5 &times; (55 + 5))) + ((55 + 5)/5))
:= (5 + 5)/5 + ((5 &times; ((5 &times; (55 + 5)) + 5)) − 5)
:= (5 + 5)/5 + (5 &times; ((5 &times; (55 + 5)) + 5))
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) − 6/6)
:= ((66 − 6)/6) + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 6 + (((6 &times; 6/(6 + 6))6 ) + 66 &times; (6 + 6))
:= (7 &times; (7 &times; (7 &times; 7 − 7) − 77)) − (7 + 7)/7
:= ((77 + 7)/7 &times; ((7 + 7)/7)7 ) − (7 + 7)
:= (((7 + 7) &times; (777 − (7 + 7))) + 7)/7
2
:= 8 &times; 8 &times; (8 + 8 + 8) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) &times; (88 − 8/8 + 8))
:= 8 &times; 8 &times; (8 + 8 + 8) − (8/8 + 8)
:= (9 &times; (9 &times; (9 + 9) + 9)) − ((99 + 99)/9)
:= 9/9 + ((9 &times; (99 + 9 &times; 9)) − 99)
:= (9 &times; (9 &times; (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
&Iuml; 1528 := 1 + (1 + (1 + 1 + 1 + 11) &times; (111 − 1 − 1))
&Iuml; 1533 := (1 + 1 + 1) &times; ((1 + 1)11−1−1 − 1)
&Iuml; 1538 := 1 + (1 + ((1 + 1 + 1) &times; (1 + 1)11−1−1 ))
:= 2 &times; (2 &times; ((22+2 &times; (22 + 2)) − 2))
:= (2/2 + 2) &times; ((2
:= 3/3 + (3 &times; (((3 − 3/3)3&times;3 ) − 3))
:= (3 &times; ((3 − 3/3)3&times;3 )) − 3
:= 3 + ((3 &times; ((3 − 3/3)3&times;3 )) − 3/3)
:= 4 + ((4 &times; 444 − 44 ) + 4)
:= 4/4 + ((4 &times; (4 + 4) &times; (44 + 4)) − 4)
:= (4 + 4)/4 + (4 &times; (4 + 4) &times; (44 + 4))
(2/2+2)2
) − 2/2)
:= 2 + ((2 + 2 + 2) &times; 22&times;(2+2) )
:= 5 + ((5 &times; ((5 &times; (55 + 5)) + 5)) − ((5 + 5)/5))
:= ((5/5 + 5) − 555/5)/5
:= ((55 + 5/5)/((5 + 5)/5)) − 5 &times; 5
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + ((66 − 6)/6))
:= (6/6 + 6) &times; ((6 &times; 6/(6 + 6)) + 6 &times; 6 &times; 6)
:= ((6 − 66)/6) + 6 &times; 6 &times; (6 &times; 6 + 6 + 6)
:= (77 &times; (7 + 7 + 7)) − (77 + 7)
:= ((7 + 7)/7) &times; (777 − (7/7 + 7))
:= 8 &times; 8 &times; (8 + 8 + 8) − 8
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) − (88/8))
:= (8 + 8)/8 + 8 &times; 8 &times; (8 + 8 + 8)
:= (9 &times; (9 &times; (9 + 9) + 9)) − 99/9
:= ((9 + 9 + 9)/9) + ((9 &times; (9 &times; (9 + 9) + 9)) − 9)
:= (9 &times; (9 &times; (9 + 9) + 9)) − 9/9
:= 7 + ((((7 − 77)/7) + 7 &times; 7)
(7+7)/7
)
&Iuml; 1529 := 11 &times; (((11 − 1) &times; (1 + (1 + 1 + 11))) − 1)
5
&Iuml; 1534 := 1 + ((1 + 1 + 1) &times; ((1 + 1)11−1−1 − 1))
&Iuml; 1539 := (1 + 1 + 1) &times; (1 + (1 + 1)11−1−1 )
:= 2 &times; (2 + 2) + ((2 &times; (22 − 2) − 2/2)2 )
:= ((2 + 2 + 2) &times; 22&times;(2+2) ) − 2
:= 2 + (((2 + 2 + 2) &times; 22&times;(2+2) ) + 2/2)
:= 33 &times; (3 + 3) + ((33/3)3 )
:= 3/3 + ((3 &times; ((3 − 3/3)3&times;3 )) − 3)
:= 3 + (3 &times; ((3 − 3/3)3&times;3 ))
4
:= 4 + (((4 &times; 444 − 44 ) + 4/4) + 4)
:= 4 + (((4 + 4)/4 + 4) &times; (4 − 4/4))
:= 4 + ((4 &times; (4 + 4) &times; (44 + 4)) − 4/4)
:= 5 + ((5 &times; ((5 &times; (55 + 5)) + 5)) − 5/5)
:= (5 &times; 5 + 5/5) &times; (55 − 5/5 + 5)
:= (55 &times; (5 &times; 5 + 5)) − 555/5
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + (66/6))
:= (6/6 + 6 + 6) &times; ((666 + 6)/6 + 6)
:= 6 + ((6/6 + 6) &times; ((6 &times; 6/(6 + 6)) + 6 &times; 6 &times; 6))
:= ((77 + 7)/7 &times; ((7 + 7)/7) ) − 7
:= (7 − 7/7 + 7) &times; (777/7 + 7)
:= (((7 + 7) &times; (777 − 7)) − 7)/7
:= 8/8 + (8 &times; 8 &times; (8 + 8 + 8) − 8)
:= 8 &times; 8 &times; (8 + 8 + 8) − (8 + 8)/8
:= 88/8 + (8 &times; 8 &times; (8 + 8 + 8) − 8)
:= (9 &times; (9 &times; (9 + 9) + 9)) − 9/9 − 9
:= ((9 − 99)/(9 + 9)) + (9 &times; (9 &times; (9 + 9) + 9))
:= 9 &times; (9 &times; (9 + 9) + 9)
7
&Iuml; 1530 := (11 − 1) &times; ((11 &times; (1 + (1 + 1 + 11))) − 1)
:= (2/2 + 2) &times; ((2
(2/2+2)2
) − 2)
&Iuml; 1535 := ((1 + 1 + 1) &times; (1 + 1)11−1−1 ) − 1
:= ((2 + 2 + 2) &times; 22&times;(2+2) ) − 2/2
3&times;3
:= (3 &times; ((3 − 3/3)
:= ((4 + 4)/4 + 4) &times; (44 − 4/4)
:= (4 &times; (4 + 4) &times; (44 + 4)) − 4/4
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6) + 6) + 6)
:= 22 &times; ((2 &times; (22 + 2)) + 22)
:= 3 + ((3 &times; ((3 − 3/3)3&times;3 )) + 3/3)
:= 3 + (3 &times; (((3 − 3/3)3&times;3 ) − 3))
:= 5 + (5 &times; ((5 &times; (55 + 5)) + 5))
&Iuml; 1540 := (111 − 1) &times; (1 + (1 + 1 + 11))
)) − 3/3
5
:= 4 + (4 &times; (4 + 4) &times; (44 + 4))
:= 5 &times; ((55 + 5)/(5 + 5) − 5)
:= 5 &times; (5 − 55)/(5 + 5)
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6) + (66/6)) + 6)
7
:= (6/6 + 6) &times; ((6 &times; 6 &times; 6 − (6 + 6)/6) + 6)
:= 77 &times; ((7 − 7/7 + 7) + 7)
:= (77/7 + 7) &times; (7/7 + 77 + 7)
:= (((77 + 7) &times; ((7 + 7)/7) ) − 7)/7
:= (8 + 8)/8 + (8 &times; 8 &times; (8 + 8 + 8) − 8)
:= 8 &times; 8 &times; (8 + 8 + 8) − 8/8
:= (8/((8 + 8)/8)) + 8 &times; 8 &times; (8 + 8 + 8)
:= ((9 − 9 &times; 9)/(9 + 9)) + (9 &times; (9 &times; (9 + 9) + 9))
:= 9/9 + (9 &times; (9 &times; (9 + 9) + 9))
:= (9 &times; (9 &times; (9 + 9) + 9)) − 9
&Iuml; 1531 := 1 + ((11 − 1) &times; ((11 &times; (1 + (1 + 1 + 11))) − 1))
:= 2 + (((2 &times; (22 − 2) − 2/2)2 ) + 2 &times; (2 + 2))
&Iuml; 1536 := (1 + 1 + 1) &times; (1 + 1)11−1−1
:= (2 + 2 + 2) &times; 2
&Iuml; 1541 := 1 + ((111 − 1) &times; (1 + (1 + 1 + 11)))
2&times;(2+2)
:= ((2 &times; 22 + 2/2)2 ) − 222
:= 3 + ((3 &times; (((3 − 3/3)3&times;3 ) − 3)) + 3/3)
:= 3 &times; ((3 − 3/3)3&times;3 )
:= (3 + 3)3 + (((33/3)3 ) − (3 + 3))
:= 44 + (4/4 + 4) &times; (44 − 4/4)
:= 4 &times; (4 + 4) &times; (44 + 4)
:= 4 + ((4 &times; (4 + 4) &times; (44 + 4)) + 4/4)
:= 5 + ((5 &times; ((5 &times; (55 + 5)) + 5)) + 5/5)
:= 5/5 + (5 &times; (5 − 55)/(5 + 5))
:= 5/5 + (5 &times; ((55 + 5)/(5 + 5) − 5))
:= 6 &times; (((6 + 6)/6)6+(6+6)/6 )
:= (66 + 6/6) &times; ((66/6 + 6) + 6)
:= 6 + (((6 &times; 6 &times; (6 &times; 6 + 6) + 6/6) + 6) + 6)
5
7
:= (77 + 7)/7 &times; ((7 + 7)/7)
:= (((7 + 7) &times; (777 − 7)) + 7)/7
:= 88/8 + ((8 + 8) &times; (88 − 8/8 + 8))
:= 8 &times; 8 &times; (8 + 8 + 8)
:= 8 + ((8 &times; 8 &times; (8 + 8 + 8) − (88/8)) + 8)
:= 9/9 + ((9 &times; (9 &times; (9 + 9) + 9)) − 9)
:= ((9 + 9 + 9)/9) &times; (((9 + 9)/9)9 )
:= (9 + 9)/9 + (9 &times; (9 &times; (9 + 9) + 9))
:= 77 + (7777/7 + 7 &times; 7 &times; 7)
&Iuml; 1532 := 11 + ((1 + 1 + 1) &times; (1 + 1 + 11))1+1
:= 2 &times; ((2 &times; (2
2+2
&times; (22 + 2))) − 2)
&Iuml; 1537 := 1 + ((1 + 1 + 1) &times; (1 + 1)11−1−1 )
:= 2/2 + ((2 + 2 + 2) &times; 2
2&times;(2+2)
)
&Iuml; 1542 := 1 + (1 + ((111 − 1) &times; (1 + (1 + 1 + 11))))
:= (2 &times; 22 − 2)2 − 222
:= 3 + (33 &times; (3 + 3) + ((33/3)3 ))
:= 3/3 + (3 &times; ((3 − 3/3)3&times;3 ))
:= 3 + ((3 &times; ((3 − 3/3)3&times;3 )) + 3)
:= (4 &times; (4 + 4) &times; (44 + 4)) − 4
:= 4/4 + (4 &times; (4 + 4) &times; (44 + 4))
:= ((4 + 4)/4 + 4) &times; (4/4 + 44 )
:= ((5 + 5)/5)5 + (5 &times; (5 &times; (55 + 5)))
:= ((55 − 5/5)/((5 + 5)/5)) − 5 &times; 5
:= 5 + (((55 − 5/5)/((5 + 5)/5)) − 5 &times; 5)
:= 6 + ((6/6 + 6) &times; (6 &times; 6 &times; 6 + (6 + 6)/6))
:= 6 &times; 6 &times; (6 &times; 6 + 6 + 6) − 66/6
:= 6 &times; 6 &times; (6 &times; 6 + 6 + 6) − 6
7
:= ((7 + 7)/7) &times; (777 − (77/7))
:= (((77 + 7) &times; ((7 + 7)/7) ) + 7)/7
:= ((7 + 7)/7) &times; ((777 − 7) + 7/7)
:= 8 &times; 8 &times; (8 + 8 + 8) − (8/((8 + 8)/8))
:= 8/8 + 8 &times; 8 &times; (8 + 8 + 8)
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) − ((8 + 8)/8))
:= (9 + 9)/9 + ((9 &times; (9 &times; (9 + 9) + 9)) − 9)
:= (9 &times; (9 &times; (9 + 9) + 9)) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 &times; (9 &times; (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
&Iuml; 1548 := 1 + ((1 + 1 + 11) &times; (111+1 − (1 + 1)))
&Iuml; 1543 := (111 &times; (1 + (1 + 1 + 11))) − 11
2
2&times;(2+2)
&Iuml; 1553 := (111 &times; (1 + (1 + 1 + 11))) − 1
:= 222 + (22/2)2/2+2
:= 22 + ((2 &times; (22 − 2) − 2/2) )
:= (2 + 2 + 2) &times; (2
:= ((((3 &times; (3 + 3)) + 3)3 ) − 3)/(3 + 3)
:= 3 + (3 &times; (((3 − 3/3)3&times;3 ) + 3))
:= 3 + (33/3)3 + (3 + 3)3 + 3
:= 4 + (((4 &times; (4 + 4) &times; (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 &times; 5)
:= (((5/5 + 5)5 − (55/5))/5) − 5
:= ((5/5 + 5)5 − (55/5))/5
:= 6/6 + (6 &times; 6 &times; (6 &times; 6 + 6 + 6) − 6)
:= 6 &times; (6 &times; (6 &times; 6 + 6) + 6)
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) − 6/6
:= (((7 + 7) &times; 777) − 77)/7
:= ((((7 + 7) &times; 777) + 7)/7) − 7
:= (((7 + 7) &times; 777) − 7)/7
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) − 8/8)
:= ((88 + 8)/8) + 8 &times; 8 &times; (8 + 8 + 8)
:= 8 + ((8 &times; 8 &times; (8 + 8 + 8) + 8/8) + 8)
:= 999 + (((99 &times; 99) − 9)/(9 + 9))
:= 9 + (9 &times; (9 &times; (9 + 9) + 9))
:= 9 + ((((99 &times; 99) + 9)/(9 + 9)) + 999)
&Iuml; 1544 := 1 + ((111 &times; (1 + (1 + 1 + 11))) − 11)
2
2
3
3
+ 2)
&Iuml; 1549 := ((1 + 1 + 11) &times; (111+1 − 1)) − 11
2
:= 2 &times; 22 + ((22 + 2) )
&Iuml; 1554 := 111 &times; (1 + (1 + 1 + 11))
:= 2 + ((22/2 + 2) &times; ((22/2) − 2))
:= 222 &times; ((2/2 + 2 + 2) + 2)
:= (3 + 3) + (((33/3) ) − 3)
:= ((((3 &times; (3 + 3)) + 3)3 ) + 33)/(3 + 3)
:= 3 &times; ((((3 − 3/3)3&times;3 ) + 3) + 3)
:= 4 + ((4 &times; (4 + 4) &times; (44 + 4)) + 4)
:= 4/4 + ((((4 + 4)/4 + 4)4 − 4) + 44 )
:= 444 + (4444 − 4)/4
:= ((5/5 + 5) − (55 + 5/5))/5
:= (5 &times; (55 − 5)/(5 + 5)) − 55/5
:= ((5/5 + 5)5 − (5/5 + 5))/5
:= (6 + 6)/6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) − 6
:= 6/6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6)
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6)
5
:= ((((7 + 7) &times; 777) − 77) + 7)/7
:= (((7 + 7)/7) &times; (777 + 7/7)) − 7
:= (7 + 7) &times; 777/7
:= 8 + 8 &times; 8 &times; (8 + 8 + 8)
:= 8 &times; 8 &times; (8 + 8 + 8) + (88 + 8 + 8)/8
:= 888/8 &times; ((8 − (8 + 8)/8) + 8)
:= 999 + (((99 &times; 99) + 9)/(9 + 9))
:= 9 + ((9 &times; (9 &times; (9 + 9) + 9)) + 9/9)
:= 9 + ((((9 + 9 + 9)/9) &times; (((9 + 9)/9)9 )) + 9)
&Iuml; 1545 := 1 + (1 + ((111 &times; (1 + (1 + 1 + 11))) − 11))
2
:= 2 + (((2 &times; (22 − 2) − 2/2) ) + 22)
&Iuml; 1550 := (11 − 1) &times; (11 + (1 + 11)1+1 )
&Iuml; 1555 := 1 + (111 &times; (1 + (1 + 1 + 11)))
:= 2 + ((2 + 2 + 2) &times; (22&times;(2+2) + 2))
:= 2 &times; 222 + 2222/2
:= 3 + (((33/3)3 ) + (3 + 3)3 )
:= ((333 &times; (33/3 + 3)) + 3)/3
:= 4 + ((((4 + 4)/4 + 4) &times; (4/4 + 44 )) + 4)
:= 444 + 4444/4
:= 5 + (5 &times; ((5 + 5)/(5 + 5) − 5))
:= 5 &times; (((5 &times; (55 + 5)) + 5) + 5)
:= 55 + (5 &times; (5 &times; (55 + 5)))
:= 6 &times; 6 &times; (6 &times; 6 + 6) + (66 &times; 6/(6 + 6))
:= (6 + 6)/6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6)
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 6/6
:= (((7 + 7)/7) &times; (777 − 7/7)) − 7
:= 7 + ((((7 + 7) &times; 777) − 77)/7)
:= (((7 + 7) &times; 777) + 7)/7
3&times;3
:= 3 &times; (((3 − 3/3)
) + 3)
:= 4 + (((4 &times; (4 + 4) &times; (44 + 4)) + 4/4) + 4)
5
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) + 8/8)
9
:= 9 + (((9 + 9 + 9)/9) &times; (((9 + 9)/9) ))
&Iuml; 1546 := ((1 + 1 + 11) &times; (111+1 − (1 + 1))) − 1
:= 8 + ((8 &times; 8 &times; (8 + 8 + 8) − ((8 + 8)/8)) + 8)
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) + (88/8))
:= 99/9 + (9 &times; (9 &times; (9 + 9) + 9))
:= 99 + (9 &times; 9 &times; (9 + 9) − ((9 + 9)/9))
&Iuml; 1551 := 11 &times; ((1 + 11)1+1 − (1 + 1 + 1))
&Iuml; 1556 := 1 + (1 + (111 &times; (1 + (1 + 1 + 11))))
:= 2 + (2 &times; 222 + ((22 + 2)2 ))
:= 222 + ((22/2)2/2+2 − 2)
:= 2 &times; ((2 &times; (22 − 2))2 − 22)
:= 3 + (((((3 &times; (3 + 3)) + 3)3 ) − 3)/(3 + 3))
:= 33 &times; (33/3 + 33 + 3)
:= 3 &times; 3 + (((33/3)3 ) + (3 + 3)3 )
:= 4 + (((4 + 4)/4 + 4) &times; (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 &times; (((5 &times; (55 + 5)) + 5) + 5))
:= (((5/5 + 5)5 − 5/5) + 5)/5
:= 6 &times; (6 &times; (6 &times; 6 + 6) + 6) − (6 + 6)/6
:= 66/6 &times; ((666/6 − 6) + 6 &times; 6)
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + (6 + 6)/6
:= ((((7 + 7) &times; 777) − 7)/7) − 7
:= (((7 + 7) &times; (777 − 7)) + 77)/7
:= ((7 + 7)/7) &times; (777 + 7/7)
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) + ((8 + 8)/8))
:= 8 + ((8 &times; 8 &times; (8 + 8 + 8) − 8/8) + 8)
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) + ((88 + 8)/8))
:= ((99 + 9)/9) + (9 &times; (9 &times; (9 + 9) + 9))
:= 99 + (9 &times; 9 &times; (9 + 9) − 9/9)
:= 9 + ((9 &times; (9 &times; (9 + 9) + 9)) − ((9 + 9)/9))
&Iuml; 1547 := (1 + 1 + 11) &times; (111+1 − 1 − 1)
2
:= (22/2 + 2) &times; ((22/2) − 2)
&Iuml; 1552 := (111 &times; (1 + (1 + 1 + 11))) − 1 − 1
:= 2 &times; (2 &times; (2 &times; (((2
2+2
2
&Iuml; 1557 := 1 + (1 + (1 + (111 &times; (1 + (1 + 1 + 11)))))
− 2) ) − 2)))
:= 2 + 2222/2 + 2 &times; 222
3
:= (3 + 3)3 + ((33/3)3 )
:= 3 + ((3 &times; (3 + 3) + 3) + 33)/(3 + 3)
:= 3 &times; ((3 &times; (3 + 3) &times; 33 ) + 33)
:= 44/4 + (4 &times; (4 + 4) &times; (44 + 4))
:= 4 &times; ((4 + 4) &times; (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 &times; (6 &times; (6 &times; 6 + 6) + 6) − 6/6
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) − (6 + 6)/6
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 6 &times; 6/(6 + 6)
:= 777 + (777 − 7)
:= ((7 + 7)/7) &times; (777 − 7/7)
:= (((((7 + 7) &times; 777) + 7) + 7) + 7)/7
:= 88/8 + 8 &times; 8 &times; (8 + 8 + 8)
:= 8 + (8 &times; 8 &times; (8 + 8 + 8) + 8)
:= (88/8 − 8) &times; ((8 &times; 8 &times; 8 − 8/8) + 8)
:= 9 + ((9 &times; (9 &times; (9 + 9) + 9)) − 9/9)
:= (9 &times; (9 &times; (9 + 9) + 9)) + ((99 + 9 + 9)/9)
:= 99 + 9 &times; 9 &times; (9 + 9)
113
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1558 := ((1 + 1 + 11) &times; (111+1 − 1)) − 1 − 1
2
&Iuml; 1563 := 1 + (11 &times; ((1 + 11)1+1 − (1 + 1)))
22/2
2
&Iuml; 1568 := (1 + 111) &times; (1 + (1 + 1 + 11))
:= 2 &times; ((22 + 2 + 2 + 2)2 )
:= 2 + 2 &times; (2 &times; (22 − 2) − 22)
:= 2
:= 3 + (((333 &times; (33/3 + 3)) + 3)/3)
:= 3 &times; (((3 − 3/3)3&times;3 ) + 3 &times; 3)
:= (33/3 + 3) &times; ((333 + 3)/3)
:= 4 + ((4444 − 4)/4 + 444)
:= 4 + ((4444/4 + 444) + 4)
:= (4 + 4) &times; ((4 &times; (44 + 4)) + 4)
:= ((5 + 5/5)/((5 + 5)/5)) − 5
:= (55 + 5/5)/((5 + 5)/5)
:= 5 + ((55 + 5/5)/((5 + 5)/5))
:= 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + (66 − 6)/6
:= ((66 &times; (66 + 66)) + 666)/6
:= (6/6 + 6) &times; ((6 &times; 6 &times; 6 + (6 + 6)/6) + 6)
5
− (22 + 2/2)
:= ((7 + 7)/7) &times; (((7 + 7)/7) + 777)
:= 7 + (((7 + 7)/7) &times; (777 + 7/7))
:= 7 &times; (7 &times; (7 + 7 + 7) + 77)
:= ((8 + 8)/8) &times; (8 &times; (88 + 8) + (88/8))
:= 8 + ((8 &times; 8 &times; (8 + 8 + 8) + (88/8)) + 8)
:= 8 + ((8 + 8 + 8) &times; (8/8 + 8 &times; 8))
:= 9/9 + (9 &times; 9 &times; (9 + 9) + 99)
:= ((9 + 9 + 9)/9) &times; ((((9 + 9)/9)9 ) + 9)
:= 99 + (9 &times; 9 &times; (9 + 9) + (99/9))
&Iuml; 1559 := ((1 + 1 + 11) &times; (111+1 − 1)) − 1
&Iuml; 1564 := (1 + 1)11 − ((11 + 11)1+1 )
&Iuml; 1569 := 1 + ((1 + 111) &times; (1 + (1 + 1 + 11)))
:= 2/2 + (2 &times; ((22 + 2 + 2 + 2)2 ))
:= (2 &times; (222 + 2)) + 2222/2
:= 2
:= 3 + ((((33/3)3 ) + (3 + 3)3 ) + 3 &times; 3)
:= 33 + ((3 &times; ((3 − 3/3)3&times;3 )) + 3/3)
:= 33 + (3 &times; ((3 − 3/3)3&times;3 ))
:= 4 + (4444/4 + 444)
:= 4 + ((44 + 4) &times; ((4 + 4)/4 + 4))
:= 4/4 + ((4 + 4) &times; ((4 &times; (44 + 4)) + 4))
:= (5 &times; (55 − 5)/(5 + 5)) − 5/5
:= 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 66/6
:= 7 + (((7 + 7)/7) &times; (777 − 7/7))
:= 8 &times; (8 &times; 8 − 8) + 8888/8
:= 9 + ((9 &times; (9 &times; (9 + 9) + 9)) + (99/9))
&Iuml; 1560 := (1 + 1 + 11) &times; (111+1 − 1)
2
22/2
− 22
2
:= (5 &times; (55 + 5)/(5 + 5)) − 5/5
:= 5 + ((5 &times; (55 + 5)/(5 + 5)) − 5/5)
:= (6 − ((6 + 6)/6)) &times; ((6 &times; 66 − 6) + 6/6)
:= 6 + (((66 &times; (66 + 66)) + 666)/6)
:= ((((7 + 7) &times; 777) − 7) + 77)/7
:= (((7 + 7) &times; (777 + 7)) + 7)/7
:= (8/8 + 8 + 8) &times; ((8/((8 + 8)/8)) + 88)
:= 8 + (((8 + 8 + 8) &times; (8/8 + 8 &times; 8)) + 8/8)
:= ((9 − 9/9) + 9) &times; ((99/9) + 9 &times; 9)
:= 9 &times; 9 &times; (9 + 9) + 999/9
&Iuml; 1565 := 11 + (111 &times; (1 + (1 + 1 + 11)))
&Iuml; 1570 := 1 + (1 + ((1 + 111) &times; (1 + (1 + 1 + 11))))
:= 2/2 + (222/2 − 222 )
:= 2 + (2 &times; ((22 + 2 + 2 + 2)2 ))
:= ((33/3)3 ) + (3 &times; (3 &times; 33 − 3))
:= 33 + ((3 &times; (3 + 3) + 3)3 − 3)/(3 + 3)
:= 4 + (((4 + 4)/4 + 4) &times; ((4/4 + 44 ) + 4))
:= 5 &times; (55 − 5)/(5 + 5)
:= 4 + (((44 + 4) &times; ((4 + 4)/4 + 4)) + 4/4)
:= 5 &times; (55 + 5)/(5 + 5)
:= 5 + (5 &times; (55 + 5)/(5 + 5))
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 6
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 66/6
:= ((6 + 6)/6) &times; (66 &times; (6 + 6) − (6/6 + 6))
:= 7 + ((((7 + 7) &times; 777) − 7)/7)
:= (((7 + 7) &times; 777) + 77)/7
:= ((7 + 7)/7) &times; ((777 + 7/7) + 7)
:= (8 + 8 + 8) &times; (8/8 + 8 &times; 8)
:= ((88/8 + 8) &times; (88 − 8/8)) − 88
:= 8 + ((888/8 &times; ((8 − (8 + 8)/8) + 8)) + 8)
:= 9 &times; 9 &times; (9 + 9) + (999/9 − 9)
:= 9 + ((9 &times; 9 &times; (9 + 9) − 9/9) + 99)
:= 9 &times; 9 &times; (9 + 9) + ((999 + 9)/9)
:= 2 &times; (2 &times; (22 − 2) − 22 + 2)
:= (33 − 3/3) &times; (33 + 33)
:= (44 + 4) &times; ((4 + 4)/4 + 4)
&Iuml; 1561 := 1 + ((1 + 1 + 11) &times; (111+1 − 1))
22/2
2
&Iuml; 1566 := 1 + (11 + (111 &times; (1 + (1 + 1 + 11))))
&Iuml; 1571 := (11 &times; (11 &times; (1 + 1 + 11))) − 1 − 1
:= 2 + (222/2 − 222 )
:= ((22/2 + 2) &times; (22/2)2 ) − 2
:= 3/3 + ((33 − 3/3) &times; (33 + 33))
:= 3 &times; ((3 + 3) &times; (3 &times; 33 + 3 + 3))
:= 3 &times; 3 &times; 33 + (33/3)3 − 3
:= 4/4 + ((44 + 4) &times; ((4 + 4)/4 + 4))
:= ((4 + 4)/4 + 4) &times; ((4/4 + 44 ) + 4)
:= 4 + (((4 + 4) &times; ((4 &times; (44 + 4)) + 4)) − 4/4)
:= 2
− ((22 + 2/2) + 2)
:= 5/5 + (5 &times; (5 − 5)/(5 + 5))
:= 5/5 + (5 &times; (55 + 5)/(5 + 5))
:= 5 + ((5 &times; (55 + 5)/(5 + 5)) + 5/5)
:= (6/6 + 6) &times; (6 &times; 6 &times; 6 + 6/6 + 6)
:= 6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 6 + 6
:= (((6 + 6) &times; (66 &times; (6 + 6) − 6)) − 6)/6
:= 7 + (777 + 777)
:= ((7 + 7)/7) &times; ((777 − 7/7) + 7)
:= 7 + (((((7 + 7) &times; 777) − 7) + 77)/7)
:= 8/8 + ((8 + 8 + 8) &times; (8/8 + 8 &times; 8))
:= (8/8 − 88) &times; ((8 − 88)/8 − 8)
:= 88/8 + ((8 + 8 + 8) &times; (8/8 + 8 &times; 8))
:= 9 &times; 9 &times; (9 + 9) + (((999 + 9)/9) − 9)
:= 9 + (9 &times; 9 &times; (9 + 9) + 99)
:= 9 &times; 9 &times; (9 + 9) + ((999 + 9 + 9)/9)
5
&Iuml; 1562 := 11 &times; ((1 + 11)1+1 − (1 + 1))
&Iuml; 1567 := ((1 + 111) &times; (1 + (1 + 1 + 11))) − 1
&Iuml; 1572 := (1 + 11) &times; ((11 &times; (1 + 11)) − 1)
:= (2 &times; ((22 + 2 + 2 + 2)2 )) − 2/2
:= 2 &times; (((22 + 2 + 2 + 2)2 ) + 2)
:= 33 + ((3 &times; ((3 − 3/3)3&times;3 )) − 3/3)
:= 3/3 + (3 &times; ((3 + 3) &times; (3 &times; 33 + 3 + 3)))
:= 3 + ((3 &times; ((3 − 3/3)3&times;3 )) + 33)
:= 4 &times; 44 + (44 &times; (44 − 4)/(4 + 4))
:= ((4 + 4) &times; ((4 &times; (44 + 4)) + 4)) − 4/4
:= 4 + ((4 + 4) &times; ((4 &times; (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 &times; ((6 + 6) &times; (6 + 6) − ((6 + 6)/6))
:= 6 + ((6/6 + 6) &times; (6 &times; 6 &times; 6 + 6/6 + 6))
:= 66 + (6 &times; 6 &times; (6 &times; 6 + 6) − 6)
:= 7 + ((((7 + 7) &times; 777) + 7)/7)
:= (((7 + 7) &times; (777 + 7)) − 7)/7
:= 7 + ((((7 + 7) &times; 777) + 77)/7)
:= 8 + (888/8 &times; ((8 − (8 + 8)/8) + 8))
:= 8 + (8888/8 + 8 &times; (8 &times; 8 − 8))
:= (((8 &times; 8 − 8)(8+8)/8 ) + 8)/((8 + 8)/8)
:= 99/9 &times; (9 &times; (9 + 9) − (99/9 + 9))
:= 9 + ((9 &times; 9 &times; (9 + 9) + 99) + 9/9)
:= 9 + (((9 + 9 + 9)/9) &times; ((((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
&Iuml; 1573 := 11 &times; (11 &times; (1 + 1 + 11))
&Iuml; 1578 := 11 + (((1 + 111) &times; (1 + (1 + 1 + 11))) − 1)
&Iuml; 1583 := (11 &times; (1 + 11)1+1 ) − 1
:= (22/2 + 2) &times; (22/2)2
:= (2 &times; (22 − 2))2 − 22
:= (2 &times; (22 &times; (2 + 2 + 2)2 )) − 2/2
:= ((33/3)3 ) + ((33+3 − 3)/3)
:= 33 + (3 &times; (((3 − 3/3)3&times;3 ) + 3))
:= ((33/3)3 ) + (3 &times; (3 &times; 33 + 3))
:= (44 &times; (4 &times; (4 + 4) + 4)) − 44/4
:= ((4 + 4)/4 + 4) &times; (((44 − 4/4) + 4) + 4)
:= (44 &times; (4 &times; (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 &times; ((6 + 6) &times; (6 + 6) − 6/6)
:= 66 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= (66 &times; ((6 + 6 + 6) + 6)) − 6/6
:= (7 − 7/7 + 7) &times; (((7 + 7)/7) − 7)
:= (7 − 7/7) &times; ((((7 + 7)/7)7+7/7 ) + 7)
:= (((7 + 7) &times; (777 + 7 + 7)) + 7)/7
:= 888 + (8 &times; 88 − (88/8 + 8))
:= 888 + ((((8 + 8)/8) − (8 + 8)) + 8 &times; 88)
:= 888 + (8 &times; 88 − (8/8 + 8))
:= 9 + (((9 − 9/9) + 9) &times; ((99/9) + 9 &times; 9))
:= 9 + (9 &times; 9 &times; (9 + 9) + 999/9)
:= ((9 + 9) &times; (99 − (99/9))) − 9/9
7
&Iuml; 1574 := 1 + (11 &times; (11 &times; (1 + 1 + 11)))
&Iuml; 1579 := 11 + ((1 + 111) &times; (1 + (1 + 1 + 11)))
&Iuml; 1584 := 11 &times; (1 + 11)1+1
:= 2 + (2 &times; (((22 + 2 + 2 + 2)2 ) + 2))
:= 2/2 + ((2 &times; (22 − 2))2 − 22)
:= 2 &times; (22 &times; (2 + 2 + 2)2 )
:= 3 &times; 3 &times; 33 + ((33/3)3 )
:= (((3 + 3) &times; ((33 &times; (33 − 3)) − 3)) + 3)/3
:= 33 &times; (3 &times; 33 − 33)
:= 44 + (((4 + 4)/4 + 4) &times; (44 − 4/4))
:= (44 &times; (4 &times; (4 + 4) + 4)) − (4/4 + 4)
:= 44 &times; (4 &times; (4 + 4) + 4)
:= 555 + ((5 − 5/5)5 − 5)
:= 555 + (5 − 5/5)5
:= 5 + ((5 − 5/5)5 + 555)
:= ((6 + 6)/6) &times; ((66 &times; (6 + 6) − 6) + 6/6)
:= 66 + (6 &times; 6 &times; (6 &times; 6 + 6) + 6/6)
:= 66 &times; ((6 + 6 + 6) + 6)
:= 7 + ((((7 + 7) &times; (777 + 7)) − 7)/7)
:= (((7 + 7) &times; (777 + 7)) + 77)/7
:= (77/7 + 7) &times; (77/7 + 77)
:= 8 + ((8/8 − 88) &times; ((8 − 88)/8 − 8))
:= 8 + (((8 + 8 + 8) &times; (8/8 + 8 &times; 8)) + (88/8))
:= 88 &times; ((((8 + 8)/8) + 8) + 8)
:= 9 + (((9 &times; 9 &times; (9 + 9) − 9/9) + 99) + 9)
:= 9 + (((999 + 9)/9) + 9 &times; 9 &times; (9 + 9))
:= (9 + 9) &times; (99 − (99/9))
&Iuml; 1585 := 1 + (11 &times; (1 + 11)1+1 )
&Iuml; 1575 := 1 + (1 + (11 &times; (11 &times; (1 + 1 + 11))))
&Iuml; 1580 := (11 − 1) &times; ((1 + 1 + 11)1+1 − 11)
:= 2 + ((22/2 + 2) &times; (22/2)2 )
:= 2 + ((2 &times; (22 − 2))2 − 22)
:= 2/2 + (2 &times; (22 &times; (2 + 2 + 2)2 ))
:= 3 &times; (((3 + 3) &times; (3 &times; 33 + 3 + 3)) + 3)
:= 33 + (((33/3)3 ) + (3 + 3)3 )
:= 3/3 + (33 &times; (3 &times; 33 − 33))
:= ((4/4 + 4) + 4) &times; (4 &times; 44 − 4/4)
:= (44 &times; (4 &times; (4 + 4) + 4)) − 4
:= 4/4 + (44 &times; (4 &times; (4 + 4) + 4))
:= 5 &times; ((5 &times; 5 + 55 )/(5 + 5))
:= 5 + (5 &times; ((5 &times; 5 + 55 )/(5 + 5)))
:= 5 &times; ((55 − 5)/(5 + 5) + 5)
:= (6 &times; 6 − 6/6) &times; (666/6 − 66)
:= (6 − ((6 + 6)/6)) &times; (6 &times; 66 − 6/6)
:= 6/6 + (66 &times; ((6 + 6 + 6) + 6))
:= 7 + (7 &times; (7 &times; (7 + 7 + 7) + 77))
:= 7 + ((7 − 7/7 + 7) &times; (((7 + 7)/7)7 − 7))
:= 7 &times; 7 + ((77 + 7)/7 &times; ((7 + 7)/7)7 )
:= (8/8 + 8) &times; (888/8 + 8 &times; 8)
:= 888 + (8 &times; 88 − ((88 + 8)/8))
:= 8/8 + ((888 − 8) + 8 &times; 88)
:= 9 + ((9 &times; 9 &times; (9 + 9) + 99) + 9)
:= (99/9 + 9) &times; (9 &times; 9 − ((9 + 9)/9))
:= 9/9 + ((9 + 9) &times; (99 − (99/9)))
&Iuml; 1576 := 1 + (1 + (1 + (11 &times; (11 &times; (1 + 1 + 11)))))
&Iuml; 1581 := (11 &times; (1 + 11)1+1 ) − 1 − 1 − 1
&Iuml; 1586 := 1 + (1 + (11 &times; (1 + 11)1+1 ))
:= (2 &times; (22 − 2))2 − (22 + 2)
:= (2/2 + 2) &times; (((22 + 2/2)2 ) − 2)
:= 2 + (2 &times; (22 &times; (2 + 2 + 2)2 ))
:= 3 + (((33+3 − 3)/3) + ((33/3)3 ))
:= (33 &times; (3 &times; 33 − 33)) − 3
:= (33 − 3/3) &times; (((3/3 + 3)3 ) − 3)
:= (44 &times; (4 &times; (4 + 4) + 4)) − 4 − 4
:= 4/4 + ((44 &times; (4 &times; (4 + 4) + 4)) − 4)
:= (4 + 4)/4 + (44 &times; (4 &times; (4 + 4) + 4))
:= 5/5 + (5 &times; ((5 &times; 5 + 55 )/(5 + 5)))
:= 5 + ((5 &times; ((5 &times; 5 + 55 )/(5 + 5))) + 5/5)
:= 5/5 + (5 &times; ((55 − 5)/(5 + 5) + 5))
:= ((6 + 6)/6)6 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= (66 &times; ((6 + 6 + 6) + 6)) − 6 &times; 6/(6 + 6)
:= (6 + 6)/6 + (66 &times; ((6 + 6 + 6) + 6))
:= 7 + ((((7 + 7) &times; (777 + 7)) + 7)/7)
:= (((7 + 7) &times; (777 + 7 + 7)) − 7)/7
:= (7 &times; 7 &times; 77) − (((7 + 7 + 7)/7)7 )
:= 888 + (8 &times; 88 − (8 + 8))
:= 888 + (8 &times; 88 − (88/8))
:= 888 + ((((8 + 8)/8) − 8) + 8 &times; 88)
:= ((9 − 9/9) + 9) &times; (((99 + 9)/9) + 9 &times; 9)
:= 9 + ((9/9 + 9 + 9) &times; (((9 + 9)/9) + 9 &times; 9))
:= (9 − 9/9) &times; ((99 − 9/9) + 99)
&Iuml; 1577 := 1 + (1 + (1 + (1 + (11 &times; (11 &times; (1 + 1 + 11))))))
&Iuml; 1582 := (11 &times; (1 + 11)1+1 ) − 1 − 1
&Iuml; 1587 := 1 + (1 + (1 + (11 &times; (1 + 11)1+1 )))
:= (2 &times; (22 − 2))2 − (22 + 2/2)
:= (2 &times; (22 &times; (2 + 2 + 2)2 )) − 2
:= (2/2 + 2) &times; ((22 + 2/2)2 )
:= 3 + (3 &times; 3 &times; 33 + ((33/3)3 ))
:= 3/3 + ((33 &times; (3 &times; 33 − 33)) − 3)
:= 3 + (33 &times; (3 &times; 33 − 33))
:= 4 + ((44 &times; (4 &times; (4 + 4) + 4)) − 44/4)
:= (44 &times; (4 &times; (4 + 4) + 4)) − (4 + 4)/4
:= 44 + ((44/4)4−4/4 )
:= 5 + ((((55 − 5/5)/((5 + 5)/5)) + 5) + 5)
:= 5 &times; 5 + ((55 − (55/5))/((5 + 5)/5))
:= 5 &times; 5 + ((55 − 5/5)/((5 + 5)/5))
:= 66 + (6 &times; 6 &times; (6 &times; 6 + 6) − 6/6)
:= ((6 + 6)/6) &times; (66 &times; (6 + 6) − 6/6)
:= 666/6 + (6 &times; (6 &times; (6 &times; 6 + 6) − 6))
7
:= (7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − ((7 + 7)/7)7
:= 7 + (((7 + 7)/7) &times; ((777 + 7/7) + 7))
:= 7 &times; (((7 + 7)/7) + 7 &times; (7 + 7))
:= 8/8 + ((888 − (8 + 8)) + 8 &times; 88)
:= 888 + ((8 − 88)/8 + 8 &times; 88)
:= 888 + ((88/8 − (8 + 8)) + 8 &times; 88)
:= (9/9 + 9 + 9) &times; (((9 + 9)/9) + 9 &times; 9)
:= ((9 + 9)/9) &times; (9 &times; 99 − (9/9 + 99))
:= 9 + ((9 &times; 9 &times; (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
&Iuml; 1588 := 1 + (1 + (1 + (1 + (11 &times; (1 + 11)1+1 ))))
&Iuml; 1593 := (11 &times; (1 + (1 + 11)1+1 )) − 1 − 1
&Iuml; 1598 := 1 + (1 + (1 + (11 &times; (1 + (1 + 11)1+1 ))))
:= 2 &times; ((22 &times; (2 + 2 + 2)2 ) + 2)
:= 2/2 + (2 &times; (2 &times; ((22 − 2)2 − 2)))
:= (2 &times; (22 − 2))2 − 2
:= 3 + ((33 &times; (3 &times; 33 − 33)) + 3/3)
:= 3 &times; (33 &times; (3 + 3) + 333)
:= 33 &times; 33 + (((3 − 3/3)3&times;3 ) − 3)
:= 4 + (44 &times; (4 &times; (4 + 4) + 4))
:= ((4/4 + 4) + 4) &times; (4 &times; 44 + 4/4)
:= (4 &times; (444 − 44)) − (4 + 4)/4
:= 5 &times; 5 + ((55 + 5/5)/((5 + 5)/5))
:= (6 − ((6 + 6)/6)) &times; (6 &times; 66 + 6/6)
5
:= (5 &times; ((5 + 5) &times; ((5 + 5)/5)5 )) − (5 + 5)/5
:= 5 + (((5 + 5/5)/((5 + 5)/5)) + 5 &times; 5)
6
:= (6 &times; (6 + 6) &times; (6 + 6)) + ((6 &times; 6/(6 + 6)) )
7
:= ((6 + 6)/6) &times; ((66 &times; (6 + 6) + 6/6) + 6)
:= 7 + ((((7 + 7) &times; (777 + 7 + 7)) − 7)/7)
:= 7 + ((7 &times; 7 &times; 77) − (((7 + 7 + 7)/7) ))
:= ((7 + 7)/7) + ((7 + 7 + 7) &times; (77 − 7/7))
:= 888 + (8 &times; 88 − (8/((8 + 8)/8)))
:= 8/8 + (888 + 8 &times; 88)
:= 8 &times; (8 &times; (8 + 8 + 8) + 8) − (8 + 8)/8
:= 9 + ((((999 + 9)/9) + 9 &times; 9 &times; (9 + 9)) + 9)
:= 9 + ((9 + 9) &times; (99 − (99/9)))
:= ((9 + 9)/9) &times; (9 &times; (9 &times; 9 + 9) − (99/9))
&Iuml; 1589 := (((1 + 1) &times; ((1 + 1) &times; (11 − 1)))1+1 ) − 11
&Iuml; 1594 := (11 &times; (1 + (1 + 11)1+1 )) − 1
&Iuml; 1599 := (1 + 1 + 11) &times; (1 + (1 + 111+1 ))
:= (2 &times; (22 − 2))2 − 22/2
:= 2 + (2 &times; (2 &times; ((22 − 2)2 − 2)))
:= (2 &times; (22 − 2))2 − 2/2
:= 3 + ((33 − 3/3) &times; (((3/3 + 3)3 ) − 3))
:= 3/3 + (3 &times; (33 &times; (3 + 3) + 333))
:= ((3 + 3) &times; ((3 &times; (3 &times; (33 + 3))) − 3)) − 3
:= 4 + ((44 &times; (4 &times; (4 + 4) + 4)) + 4/4)
:= (44 − 4)/4 + (44 &times; (4 &times; (4 + 4) + 4))
:= (4 &times; (444 − 44)) − 4/4
:= 5 + (((5 − 5/5) + 555) + 5)
:= 55 /5 + ((5 − 5/5)5 − 55)
:= (5 &times; ((5 + 5) &times; ((5 + 5)/5)5 )) − 5/5
:= 6 + ((66 &times; ((6 + 6 + 6) + 6)) − 6/6)
:= 6 + ((6 − ((6 + 6)/6)) &times; (6 &times; 66 + 6/6))
:= (6/6 + 6 + 6) &times; ((666/6 + 6) + 6)
:= ((7 + 7 + 7) &times; (77 − 7/7)) − 7
:= 7 &times; (77 − 7) + (7777/7 − 7)
:= (77 &times; (7 + 7 + 7)) − (77/7 + 7)
:= 8 &times; (8 &times; (8 + 8 + 8) + 8) − 88/8
:= 888 + ((8 + 8)/8 + 8 &times; 88)
:= 8 &times; (8 &times; (8 + 8 + 8) + 8) − 8/8
:= 9 + ((99/9 + 9) &times; (9 &times; 9 − ((9 + 9)/9)))
:= 9 + (((9 + 9) &times; (99 − (99/9))) + 9/9)
:= (9 &times; (99 + 9 &times; 9)) − (((99 + 9)/9) + 9)
5
&Iuml; 1590 := (1 + 1 + 1) &times; (1 + (1 + 11 + 11)1+1 )
&Iuml; 1595 := 11 &times; (1 + (1 + 11)1+1 )
&Iuml; 1600 := ((1 + 1) &times; ((1 + 1) &times; (11 − 1)))1+1
:= (2 &times; (22 − 2))2
:= (2 &times; (2 &times; ((22 − 2)2 − 2))) − 2
:= 222 + 2222/2
:= 3 + ((33 &times; (3 &times; 33 − 33)) + 3)
:= 33/3 + (33 &times; (3 &times; 33 − 33))
:= ((3/3 + 3)3 ) &times; ((3/3 − 3) + 33 )
:= 4 + ((44 &times; (4 &times; (4 + 4) + 4)) + (4 + 4)/4)
:= 44/4 + (44 &times; (4 &times; (4 + 4) + 4))
:= 4 &times; (444 − 44)
:= 5 &times; ((55 + 5)/(5 + 5) + 5)
:= 55 &times; (5 &times; 5 − 5/5 + 5)
:= 5 &times; ((5 + 5) &times; ((5 + 5)/5)5 )
:= 6 + (66 &times; ((6 + 6 + 6) + 6))
:= 66/6 + (66 &times; ((6 + 6 + 6) + 6))
:= ((6 + 6)/6)6 &times; (6 &times; 6 − 66/6)
:= 7 + ((((7 + 7) &times; (777 + 7 + 7)) + 7)/7)
:= ((7 + 7 + 7) &times; (77 − 7/7)) − 7/7
:= (7 &times; 7 − ((7 + 7)/7 + 7))(7+7)/7
:= 888 + (8 &times; 88 − ((8 + 8)/8))
:= 888 + ((8 &times; 88 − 8) + (88/8))
:= 8 &times; (8 &times; (8 + 8 + 8) + 8)
:= (9/9 + 9) &times; (9 &times; (9 + 9) − ((9 + 9 + 9)/9))
:= 99/9 + ((9 + 9) &times; (99 − (99/9)))
:= (99/9 + 9) &times; (9 &times; 9 − 9/9)
&Iuml; 1591 := 1 + ((1 + 1 + 1) &times; (1 + (1 + 11 + 11)1+1 ))
2
&Iuml; 1596 := 1 + (11 &times; (1 + (1 + 11)1+1 ))
&Iuml; 1601 := 1 + (((1 + 1) &times; ((1 + 1) &times; (11 − 1)))1+1 )
:= 2 &times; ((2 &times; (22 − 2))2 − 2)
:= 2/2 + (2 &times; (22 − 2))2
:= 3 + (3 &times; (33 &times; (3 + 3) + 333))
:= 33 &times; 33 + ((3 − 3/3)3&times;3 )
:= (4 &times; (444 − 44)) − 4
:= 4/4 + (4 &times; (444 − 44))
:= 5/5 + (5 &times; ((5 + 5)/(5 + 5) + 5))
:= 5/5 + (55 &times; (5 &times; 5 − 5/5 + 5))
:= 5/5 + (5 &times; ((5 + 5) &times; ((5 + 5)/5)5 ))
:= 6 + ((66 &times; ((6 + 6 + 6) + 6)) + 6/6)
:= 6 + ((66 &times; ((6 + 6 + 6) + 6)) + 6)
:= 6 + ((66 &times; ((6 + 6 + 6) + 6)) + (66/6))
:= 7 + ((77/7 + 7) &times; (77/7 + 77))
:= (7 + 7 + 7) &times; (77 − 7/7)
:= 7 &times; (77 − 7) + 7777/7
:= 888 + (8 &times; 88 − 8/8)
:= 888 + ((8/((8 + 8)/8)) + 8 &times; 88)
:= 8/8 + 8 &times; (8 &times; (8 + 8 + 8) + 8)
:= ((99/9 + 9) &times; (9 &times; 9 − 9/9)) − 9
:= (9/9 + 9 + 9) &times; (((9 + 9 + 9)/9) + 9 &times; 9)
:= (((9 + 9)/9)9 ) + (99 &times; (99/9))
:= 2 + ((2 &times; (22 − 2)) − 22/2)
3
:= (((3 + 3) &times; ((33 &times; (3 − 3)) + 3)) + 3)/3
:= 4 + (((44/4)
4−4/4
4
)+4 )
5
&Iuml; 1592 := (11 &times; (1 + (1 + 11)1+1 )) − 1 − 1 − 1
&Iuml; 1597 := 1 + (1 + (11 &times; (1 + (1 + 11)1+1 )))
&Iuml; 1602 := 1 + (1 + (((1 + 1) &times; ((1 + 1) &times; (11 − 1)))1+1 ))
:= 2 &times; (2 &times; ((22 − 2)2 − 2))
:= (2 &times; (22 − 2))2 − 2/2 − 2
:= 2 + (2 &times; (22 − 2))2
:= ((33/3)3 ) + (3 &times; (3 &times; 33 + 3 + 3))
:= (3 &times; (3 + 3) &times; 33 ) + 3333/3
:= (3 + 3) &times; ((3 &times; (3 &times; (33 + 3))) − 3)
:= 4 + ((44 &times; (4 &times; (4 + 4) + 4)) + 4)
:= 4/4 + ((4 &times; (444 − 44)) − 4)
:= (4 + 4)/4 + (4 &times; (444 − 44))
:= 5 + (((5 − 5/5)/((5 + 5)/5)) + 5 &times; 5)
:= (5 + 5)/5 + (55 &times; (5 &times; 5 − 5/5 + 5))
:= (5 + 5)/5 + (5 &times; ((5 + 5) &times; ((5 + 5)/5)5 ))
:= 6 + ((66 &times; ((6 + 6 + 6) + 6)) + ((6 + 6)/6))
5
:= (((6 + 6) &times; (66 &times; (6 + 6) + 6)) + 6)/6
:= 6 + (((66 &times; ((6 + 6 + 6) + 6)) + 6) + 6)
:= 7 &times; 7 + ((((7 + 7) &times; 777) − 77)/7)
:= 7/7 + ((7 + 7 + 7) &times; (77 − 7/7))
:= 7 &times; 7 + ((((7 + 7) &times; 777) − 7)/7)
:= 888 + 8 &times; 88
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) − (88/8))
:= (8 + 8)/8 + 8 &times; (8 &times; (8 + 8 + 8) + 8)
:= 9 &times; 9 + ((((9 + 9)/9)9 ) + 999)
:= ((9 + 9) &times; (9 + 9 + 9)) + 9999/9
:= (9 + 9) &times; ((9 &times; 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
&Iuml; 1603 := 1 + (1 + (1 + (((1 + 1) &times; ((1 + 1) &times; (11 − 1)))1+1 )))
&Iuml; 1608 := (1 + 11) &times; (1 + (1 + (11 &times; (1 + 11))))
&Iuml; 1613 := 1 + ((1 + 1 + 11) &times; (1 + (1 + (1 + 111+1 ))))
:= 2 + ((2 &times; (22 − 2))2 + 2/2)
:= 2 &times; (2 &times; ((22 − 2)2 + 2))
:= 2 + ((2 &times; (22 − 2))2 + 22/2)
:= (3 &times; 3 + 3)3 − ((3 − 3/3 + 3)3 )
:= (33 − 3) &times; (((3/3 + 3)3 ) + 3)
:= 3 + ((((3 &times; (3 + 3))3 − 3)/3) − 333)
:= 4 + ((4 &times; (444 − 44)) − 4/4)
:= 4 + ((4 &times; (444 − 44)) + 4)
:= 4/4 + ((4 &times; ((444 − 44) + 4)) − 4)
:= 5 + ((5 &times; ((5 + 5) &times; ((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) &times; (66 &times; (6 + 6) + 6)) + 6)/6)
:= (6 − ((6 + 6)/6)) &times; (6 &times; 66 + 6)
:= 66 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) − 6/6
:= (77 &times; (7 + 7 + 7)) − (7 + 7)
:= (77 &times; (7 + 7 + 7)) − ((7 + 7)/7 + 7)
:= 7 + ((77 &times; (7 + 7 + 7)) − (77/7))
:= 888 + (88/8 + 8 &times; 88)
:= 8 + 8 &times; (8 &times; (8 + 8 + 8) + 8)
:= 88 + (8 &times; 8 &times; (8 + 8 + 8) − (88/8))
:= 9/9 + ((9 + 9) &times; ((9 &times; 9 − 9/9) + 9))
:= (9 &times; (99 + 9 &times; 9)) − (99 + 9)/9
:= (9 + 9)/9 + ((9 &times; (99 + 9 &times; 9)) − 9)
&Iuml; 1604 := (1 + 1)11 − ((1 + 1) &times; (1 + 1) &times; 111)
&Iuml; 1609 := 1 + ((1 + 11) &times; (1 + (1 + (11 &times; (1 + 11)))))
&Iuml; 1614 := ((1 + 11)1+1+1 ) − (1 + 1 + 1 + 111)
:= 2 + ((2 &times; (22 − 2))2 + 2)
:= 2/2 + (2 &times; (2 &times; ((22 − 2)2 + 2)))
:= 2 + (2 &times; ((2 &times; ((22 − 2)2 + 2)) + 2))
:= 3 + ((3 &times; (3 &times; (33 + 3))) + ((33/3)3 ))
:= (33 &times; (33 + 33)) − 33/3
:= (33 &times; (33 + 33)) − (3 + 3)
:= 4 + (4 &times; (444 − 44))
:= 4 + (((4 &times; (444 − 44)) + 4/4) + 4)
:= (4 &times; ((444 − 44) + 4)) − (4 + 4)/4
5
5
:= 5 &times; 5 + ((5 − 5/5) + 555)
:= 5 + (((5 − 5/5) + 555) + 5 &times; 5)
:= (5/5 + 5) &times; (5 &times; 55 − (5/5 + 5))
:= (6 − ((6 + 6)/6)) &times; ((6 &times; 66 − 6/6) + 6)
:= 6/6 + ((6 − ((6 + 6)/6)) &times; (6 &times; 66 + 6))
:= 66 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6)
:= 7/7 + ((77 &times; (7 + 7 + 7)) − (7 + 7))
:= (77 &times; (7 + 7 + 7)) − (7/7 + 7)
:= ((7 + 7 + 7)/7) &times; (7 &times; 77 − 7/7)
:= 888 + (((88 + 8)/8) + 8 &times; 88)
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) + 8/8)
:= 8 &times; 8 &times; 8 + ((8888 − 8)/8 − 8)
:= (9 + 9)/9 + ((9 + 9) &times; ((9 &times; 9 − 9/9) + 9))
:= (9 &times; (99 + 9 &times; 9)) − 99/9
:= (((9 + 9)/9)9 ) + (9999/9 − 9)
&Iuml; 1605 := (11 &times; (1 + (1 + (1 + 11)1+1 ))) − 1
&Iuml; 1610 := 1 + (1 + ((1 + 11) &times; (1 + (1 + (11 &times; (1 + 11))))))
&Iuml; 1615 := ((1 + 11)1+1+1 ) − (1 + 1 + 111)
:= 2 + (((2 &times; (22 − 2))2 + 2/2) + 2)
:= 2 + (2 &times; (2 &times; ((22 − 2)2 + 2)))
:= 2 + (((2 &times; (22 − 2))2 + 22/2) + 2)
:= 3333 − (3 &times; 3 + 3)3
:= (((3 &times; (3 + 3))3 − 3)/3) − 333
:= 3 + ((((3 &times; (3 + 3))3 + 3)/3) − 333)
:= 4 + ((4 &times; (444 − 44)) + 4/4)
:= (4/4 + 4) &times; (((44 + 4 + 4)/4) + 44 )
:= (4 &times; ((444 − 44) + 4)) − 4/4
5
:= 5 &times; (((5 − 5)/(5 + 5) + 5) + 5)
:= 5 &times; (((55 + 5)/(5 + 5) + 5) + 5)
:= 6 + ((6/6 + 6 + 6) &times; ((666/6 + 6) + 6))
:= (6 &times; 6 − 6/6) &times; (((66 − 6)/6) + 6 &times; 6)
:= 66 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6) + 6/6
:= (77 &times; (7 + 7 + 7)) − (77 + 7)/7
:= (77 &times; (7 + 7 + 7)) − 7
:= (77 &times; (7 + 7 + 7)) − (7 + 7)/7
:= (8 − 8/8 + 8) &times; ((88/8 + 88) + 8)
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) + ((8 + 8)/8))
:= 8 &times; 8 &times; 8 + (8888/8 − 8)
:= 9 + ((9/9 + 9 + 9) &times; (((9 + 9 + 9)/9) + 9 &times; 9))
:= (9/9 + 9) &times; (9 &times; (9 + 9) − 9/9)
:= ((9 − 99)/(9 + 9)) + (9 &times; (99 + 9 &times; 9))
5
:= 5 + (5 &times; ((5 + 5) &times; ((5 + 5)/5) ))
&Iuml; 1606 := 11 &times; (1 + (1 + (1 + 11)1+1 ))
&Iuml; 1611 := 11 + (((1 + 1) &times; ((1 + 1) &times; (11 − 1)))1+1 )
2
&Iuml; 1616 := ((1 + 11)1+1+1 ) − (1 + 111)
:= 22/2 + (2 &times; (22 − 2))2
:= 2 &times; (2 &times; (((22 − 2)2 + 2) + 2))
:= 3 + ((3 &times; 3 + 3) − ((3 − 3/3 + 3) ))
:= 3 &times; ((3 &times; (3 &times; (33 + 33))) − 3)
:= (3 &times; 3 + 3)3 − ((333 + 3)/3)
:= 4 + ((4 &times; (444 − 44)) + (4 + 4)/4)
:= 44/4 + (4 &times; (444 − 44))
:= 4 &times; ((444 − 44) + 4)
:= 2 + (((2 &times; (22 − 2)) + 2) + 2)
3
3
5
:= 5 + ((5 &times; ((5 + 5) &times; ((5 + 5)/5) )) + 5/5)
:= 555/5 + (5 &times; (5 &times; (55 + 5)))
:= 5 + ((5 &times; (5 &times; (55 + 5))) + 555/5)
6
:= 6 + (((6 + 6)/6) &times; (6 &times; 6 − 66/6))
:= (6 &times; (6 &times; 66 − 6)) − ((6 &times; 6/(6 + 6)) )
:= (((6 + 6)/6)66/6 ) − (6 &times; (66 + 6))
:= (77 &times; (7 + 7 + 7)) − 77/7
:= 7/7 + ((77 &times; (7 + 7 + 7)) − 7)
:= (77 &times; (7 + 7 + 7)) − 7/7
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) − ((8 + 8)/8))
:= 88/8 + 8 &times; (8 &times; (8 + 8 + 8) + 8)
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) + 8)
:= ((9/9 − 9) + 9 &times; 9) &times; ((99 + 99)/9)
:= (9 &times; (99 + 9 &times; 9)) − 9
:= ((9 + 9)/9) &times; (9 &times; (9 &times; 9 + 9) − ((9 + 9)/9))
6
&Iuml; 1607 := 1 + (11 &times; (1 + (1 + (1 + 11)1+1 )))
&Iuml; 1612 := (1 + 1 + 11) &times; (1 + (1 + (1 + 111+1 )))
&Iuml; 1617 := ((1 + 11)1+1+1 ) − 111
:= 222/2 − ((22 − 2/2)2 )
:= 2 &times; ((2 &times; ((22 − 2)2 + 2)) + 2)
:= 22 + ((2222/2) + 222 )
:= 33 + (3 &times; 3 &times; 33 + ((33/3)3 ))
:= (((3 &times; (3 + 3))3 + 3)/3) − 333
:= (33 &times; (33 + 33)) − 3
:= (4 &times; ((444 − 44) + 4)) − 4
:= 4/4 + (4 &times; ((444 − 44) + 4))
:= 4 + (((4 &times; (444 − 44)) − 4/4) + 4)
5
:= 5 + ((5 &times; ((5 + 5) &times; ((5 + 5)/5) )) + ((5 + 5)/5))
:= 55 + ((5 − (55/5))/((5 + 5)/5))
:= 55 + ((55 − 5/5)/((5 + 5)/5))
:= ((6 − ((6 + 6)/6)) &times; (6 &times; 66 + 6)) − 6/6
:= ((6 + 6)/6)6 + 6 &times; (6 &times; (6 &times; 6 + 6) + 6)
:= 66/6 &times; (666/6 + 6 &times; 6)
:= ((7 − 77)/7) + (77 &times; (7 + 7 + 7))
:= ((7 + 7)/7) + ((77 &times; (7 + 7 + 7)) − 7)
:= 77 &times; (7 + 7 + 7)
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) − 8/8)
:= 8 &times; 8 &times; 8 + ((8888 − 88)/8)
:= 8 + ((8 &times; (8 &times; (8 + 8 + 8) + 8) + 8/8) + 8)
:= (9 &times; (99 + 9 &times; 9)) − ((99 + 9 + 9)/9)
:= 9/9 + ((9 &times; (99 + 9 &times; 9)) − 9)
:= (9 &times; (99 + 9 &times; 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
&Iuml; 1618 := 1 + (((1 + 11)1+1+1 ) − 111)
&Iuml; 1623 := 1111 + (1 + 1)11−1−1
&Iuml; 1628 := 11 + (((1 + 11)1+1+1 ) − 111)
:= 2 + ((2 &times; (22 − 2))2 + 22+2 )
:= 22 + ((2 &times; (22 − 2))2 + 2/2)
:= 22 &times; (2 &times; (2 + 2 + 2)2 + 2)
:= 3/3 + ((33 &times; (33 + 33)) − 3)
:= 3 + (33 &times; (33 + 33))
:= ((33/3)3 ) + 3 &times; 3 &times; 33
:= (4 + 4)/4 + (4 &times; ((444 − 44) + 4))
:= 44 + (4444/4 + 44 )
:= 44 + (44 &times; (4 &times; (4 + 4) + 4))
:= 55 + ((55 + 5/5)/((5 + 5)/5))
:= (5 &times; (5 &times; (55 + 5 + 5))) − (5 + 5)/5
:= 5 + ((5 &times; (5 &times; (55 + 5 + 5))) − ((5 + 5)/5))
:= 6 &times; (666 − 6 &times; 66) − (6 + 6)/6
:= 666/6 + 6 &times; 6 &times; (6 &times; 6 + 6)
:= 66/6 &times; ((666 + 6)/6 + 6 &times; 6)
:= 7/7 + (77 &times; (7 + 7 + 7))
:= 7 + ((77 &times; (7 + 7 + 7)) − 7/7)
:= 77/7 + (77 &times; (7 + 7 + 7))
:= 8 + ((8 &times; (8 &times; (8 + 8 + 8) + 8) + ((8 + 8)/8)) + 8)
:= 8 &times; 8 &times; 8 + 8888/8
:= 88/8 &times; (888/(8 − (8 + 8)/8))
:= (9 &times; (99 + 9 &times; 9)) − (9 + 9)/9
:= (((9 + 9)/9)9 ) + 9999/9
:= 9 + ((9 &times; (99 + 9 &times; 9)) − 9/9)
&Iuml; 1619 := 1 + (1 + (((1 + 11)1+1+1 ) − 111))
&Iuml; 1624 := 1 + (1111 + (1 + 1)11−1−1 )
&Iuml; 1629 := 1 + (11 + (((1 + 11)1+1+1 ) − 111))
:= 22 + ((2 &times; (22 − 2))2 − (2/2 + 2))
:= 2 + ((2 &times; (22 − 2))2 + 22)
:= 2 + ((((2 &times; 22) − 2/2)2 ) − 222)
:= (33 &times; (33 + 33)) − 3/3
:= 3 + ((33 &times; (33 + 33)) + 3/3)
:= (3 &times; 3 + 3)3 − 3 &times; 33
:= ((4 &times; 4 + 4) &times; (4 − 4/4)4 ) − 4/4
:= 4 + ((4 &times; 4 + 4) &times; (4 − 4/4)4 )
:= 4 &times; (44 − 4) + ((4/4 + 4)4 − 4)
:= (5 &times; (5 &times; (55 + 5 + 5))) − (5/5 + 5)
:= (5 &times; (5 &times; (55 + 5 + 5))) − 5/5
:= 5 + ((5 &times; (5 &times; (55 + 5 + 5))) − 5/5)
:= 6 &times; (666 − 6 &times; 66) − 6/6
:= 6 &times; 6 &times; (6 &times; 6 + 6) + (666 + 6)/6
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + 666/6)
:= ((7 + 7)/7) + (77 &times; (7 + 7 + 7))
:= 7 + (77 &times; (7 + 7 + 7))
:= (77 + 7)/7 + (77 &times; (7 + 7 + 7))
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 8) + (88/8))
:= 88 + 8 &times; 8 &times; (8 + 8 + 8)
:= (8/8 + 8) &times; (8 &times; (8 + 8 + 8) − (88/8))
:= (9 &times; (99 + 9 &times; 9)) − 9/9
:= ((9 + 9)/9) &times; (9 &times; (9 &times; 9 + 9) + ((9 + 9)/9))
:= 9 + (9 &times; (99 + 9 &times; 9))
&Iuml; 1620 := (1 + 1) &times; ((11 − 1) &times; (11 − 1 − 1)1+1 )
&Iuml; 1625 := 1 + (1 + (1111 + (1 + 1)11−1−1 ))
&Iuml; 1630 := (11 − 1) &times; (1 + ((1 + 1) &times; (11 − 1 − 1)1+1 ))
:= 22 + ((2 &times; (22 − 2))2 − 2)
:= 2 + (((2 &times; (22 − 2))2 + 22) + 2/2)
:= 2 + (22 &times; (2 &times; (2 + 2 + 2)2 + 2))
:= 33 &times; (33 + 33)
:= ((33/3)3 ) + (3 &times; 3 &times; 33 − 3)
:= 3/3 + ((3 &times; 3 + 3)3 − 3 &times; 33)
:= (4 &times; 4 + 4) &times; (4 − 4/4)4
:= 4 + (((4 &times; 4 + 4) &times; (4 − 4/4)4 ) + 4/4)
:= 4 + (((4 + 4)/4 + 4) &times; ((44/4 + 44 ) + 4))
:= (5/5 + 5) &times; (5 &times; 55 − 5)
:= 5 &times; (5 &times; (55 + 5 + 5))
:= 5 + (5 &times; (5 &times; (55 + 5 + 5)))
:= 6 &times; (666 − 6 &times; 66)
:= (6/6 − 66) &times; ((66/6) − 6 &times; 6)
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6) + (666 + 6)/6)
:= ((7 + 7 + 7)/7) + (77 &times; (7 + 7 + 7))
:= 7 + ((77 &times; (7 + 7 + 7)) + 7/7)
:= 7 + (((77 &times; (7 + 7 + 7)) − 7/7) + 7)
:= ((88 + 8)/8) &times; ((8 &times; (8 + 8) − 8/8) + 8)
:= 8/8 + (8 &times; 8 &times; (8 + 8 + 8) + 88)
:= 8 + ((8888 − 8)/8 + 8 &times; 8 &times; 8)
:= 9 &times; (99 + 9 &times; 9)
:= (9 &times; (99 + 9 &times; 9)) + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + ((9 &times; (99 + 9 &times; 9)) + 9/9)
&Iuml; 1621 := 1 + ((1 + 1) &times; ((11 − 1) &times; (11 − 1 − 1)1+1 ))
&Iuml; 1626 := 1 + (1 + (1 + (1111 + (1 + 1)11−1−1 )))
&Iuml; 1631 := 1 + (11 − 1) &times; (1 + (1 + 1) &times; (11 − 1 − 1)1+1 )
:= 22 + ((2 &times; (22 − 2))2 − 2/2)
:= 2 + (((2 &times; (22 − 2))2 + 22) + 2)
:= (((2 &times; 22) + 2)2 ) − (222 + 2/2)
:= 3/3 + (33 &times; (33 + 33))
:= 3 + ((33 &times; (33 + 33)) + 3)
:= 3 + (((33/3)3 ) + 3 &times; 3 &times; 33)
:= 4/4 + ((4 &times; 4 + 4) &times; (4 − 4/4)4 )
:= ((4 + 4)/4 + 4) &times; ((44/4 + 44 ) + 4)
:= 44 + (((44/4)4−4/4 ) + 44 )
:= 5/5 + ((5/5 + 5) &times; (5 &times; 55 − 5))
:= 5/5 + (5 &times; (5 &times; (55 + 5 + 5)))
:= 5 + ((5 &times; (5 &times; (55 + 5 + 5))) + 5/5)
:= 6/6 + 6 &times; (666 − 6 &times; 66)
:= 6 + 6 &times; (666 − 6 &times; 66)
:= 6 + ((6/6 − 66) &times; ((66/6) − 6 &times; 6))
:= 77/7 + ((77 &times; (7 + 7 + 7)) − 7)
:= 7 + ((77 &times; (7 + 7 + 7)) + ((7 + 7)/7))
:= 7 + ((77 &times; (7 + 7 + 7)) + 7)
:= 8 &times; 8 &times; 8 + ((8888 − (8 + 8))/8)
:= 88 + (8 &times; 8 &times; (8 + 8 + 8) + ((8 + 8)/8))
:= 8 + (8888/8 + 8 &times; 8 &times; 8)
:= 9 + ((9 &times; (99 + 9 &times; 9)) − ((9 + 9 + 9)/9))
:= 99/9 + (9 &times; (99 + 9 &times; 9))
:= 9/9 + (9 &times; (99 + 9 &times; 9))
&Iuml; 1622 := 1111 + ((1 + 1)11−1−1 − 1)
&Iuml; 1627 := 11 + (((1 + 11)1+1+1 ) − (1 + 111))
:= 22 + (2 &times; (22 − 2))2
:= (((2 &times; 22) − 2/2)2 ) − 222
:= (((2 &times; 22) + 2)2 ) − 222
:= 3 + ((33 &times; (33 + 33)) − 3/3)
:= 3 + (((33 &times; (33 + 33)) + 3/3) + 3)
:= 3 + ((3 &times; 3 + 3)3 − 3 &times; 33)
:= 44 + ((4444 − 4)/4 + 44 )
:= 4 + ((4444/4 + 44 ) + 44 )
:= 4 &times; (((444 − 44) + 4) + 4)
:= 5 + (((5 − 5/5)/((5 + 5)/5)) + 55)
:= (5 + 5)/5 + (5 &times; (5 &times; (55 + 5 + 5)))
:= ((5 + 5)/5)5 &times; ((5/5 − 5) + 55)
:= (6 + 6)/6 + 6 &times; (666 − 6 &times; 66)
:= 6 + (6 &times; (666 − 6 &times; 66) + 6/6)
:= 6 + (6 &times; (666 − 6 &times; 66) + 6)
:= 7 + ((77 &times; (7 + 7 + 7)) − ((7 + 7)/7))
:= ((77 − 7)/7) + (77 &times; (7 + 7 + 7))
:= 7 + (((77 &times; (7 + 7 + 7)) + 7/7) + 7)
:= 8 &times; 8 &times; 8 + (8888 − 8)/8
:= ((88/8 + 8) &times; (8/8 + 88)) − 8 &times; 8
:= (88 + 8) &times; (8/8 + 8 + 8)
:= (9 + 9)/9 + (9 &times; (99 + 9 &times; 9))
:= 9 + ((9 &times; (99 + 9 &times; 9)) − ((9 + 9)/9))
:= ((99 + 9)/9) + (9 &times; (99 + 9 &times; 9))
5
118
&Iuml; 1632 := (1 + 1) &times; ((1 + 1)11 − (11 &times; (1 + 111)))
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1633 := (1 + 1 + 1 + 11)1+1+1 − 1111
&Iuml; 1638 := ((1 + 1) &times; 111+1+1 ) − (1 + 1)11−1
&Iuml; 1643 := ((1 + 1)1+11 ) − (11 &times; (1 + (1 + 1) &times; 111))
:= 22 + ((2 &times; (22 − 2))2 + 22/2)
:= 2 + ((2 &times; ((22 + 2)2 )) + 222 )
:= (2 &times; 22 − 2)2 − (22/2)2
:= ((3 &times; (3 &times; 33 &times; 33)) − 3)/(3 + 3)
:= 3 &times; (((3 + 3)3 − 3) + 333)
:= (3 &times; 3 + 3)3 − ((3 &times; 33 + 3/3) + 3)
:= 44 + ((44 &times; (44 − 4)/(4 + 4)) − 4)
:= (((4 − 4/4)4+4 ) + 44/4)/4
:= (5 − 5/5)5 + (((55 − 55)/5) − 5)
:= (5/5 + 5) &times; (5 &times; 55 − ((5 + 5)/5))
:= (5 − 5/5)5 + ((55 − 5)/5 − 5)
:= ((66/6 + 6) + 6) &times; ((66 − 6/6) + 6)
:= (6/6 + 6) &times; ((6 &times; 6 &times; 6 + 6 + 6) + 6)
:= (66 &times; (6 &times; 6 − 66/6)) − 6/6 − 6
:= 7 + (((77 &times; (7 + 7 + 7)) + ((7 + 7)/7)) + 7)
:= (7 + 7 + 7) &times; (7/7 + 77)
:= 7 &times; 77 + (7777/7 − 7)
:= 8/8 + ((88 + 8) &times; (8/8 + 8 + 8))
:= (8/8 − 8 &times; 8) &times; ((8 − 88)/8 − (8 + 8))
:= 88/8 + ((88 + 8) &times; (8/8 + 8 + 8))
:= (9 &times; (99 + 9 &times; 9)) + ((99 + 9 + 9)/9)
:= 9 + ((9 &times; (99 + 9 &times; 9)) + 9)
:= (9 &times; (99 + 9 &times; 9)) + ((99 + 99 + 9)/9)
4
:= 4 &times; (4 − 4) + (4/4 + 4)
4
&Iuml; 1634 := 11 + (1111 + (1 + 1)11−1−1 )
2
&Iuml; 1639 := 11 &times; (1 + (1 + (1 + (1 + (1 + (1 + 11)1+1 )))))
2
&Iuml; 1644 := (1 + 11) &times; (111 + ((1 + 1) &times; (1 + 1 + 11)))
:= 2 + ((((2 &times; 22) + 2) ) − 22 )
:= (22 &times; (22 + 2)) + 2222/2
:= 2 &times; (2 &times; (22 − 2)2 + 22)
:= ((3 &times; (3 &times; 33 &times; 33)) + 3)/(3 + 3)
:= 3/3 + ((3 &times; 3 + 3)3 + (3 &times; (3 − 33)))
:= (3 &times; 3 + 3)3 − (3 &times; 33 + 3)
:= 4/4 + (4 &times; (44 − 4) + (4/4 + 4)4 )
:= (((4 − 4/4)4+4 ) − (4/4 + 4))/4
:= 44 + (4 &times; (444 − 44))
:= 55 + ((5 − 5/5) + 555)
:= (55 &times; (5 &times; 5 + 5)) − 55/5
:= 55 /5 + ((5 − 5/5)5 − 5)
:= (((6 + 6)/6) + 6 &times; 6) &times; ((6 &times; 6 + 6/6) + 6)
:= 66/6 &times; (((6 + 6) &times; (6 + 6) − 6/6) + 6)
:= (66 &times; (6 &times; 6 − 66/6)) − 6
:= 7 + ((77 &times; (7 + 7 + 7)) + ((77 − 7)/7))
:= 7/7 + ((7 + 7 + 7) &times; (7/7 + 77))
:= 7 + (((7 + 7 + 7) &times; (7/7 + 77)) − 7/7)
:= (88/8 + 8) &times; (88 − ((8 + 8)/8))
:= 8 + ((8888/8 + 8 &times; 8 &times; 8) + 8)
:= ((88 + 8)/8) &times; ((8 &times; (8 + 8) + 8/8) + 8)
:= 9 + ((9 &times; (99 + 9 &times; 9)) + ((9 &times; 9 + 9)/(9 + 9)))
:= 9 + (((9 &times; (99 + 9 &times; 9)) + 9/9) + 9)
:= 9 + ((((9 + 9 + 9)/9) &times; (((9 + 9)/9)9 )) + 99)
5
&Iuml; 1635 := (111 − 1 − 1) &times; (1 + 1 + 1 + 1 + 11)
2
&Iuml; 1640 := 1111 + (1 + 11 + 11)1+1
&Iuml; 1645 := 1 + ((1 + 11) &times; (111 + ((1 + 1) &times; (1 + 1 + 11))))
:= 2 &times; (2 &times; (22 − 2)2 − 2 + 22)
2
:= 22 + ((2 &times; ((22 + 2) )) − 2/2)
3
:= 2 + ((2 &times; 22 − 2)2 − (22/2)2 )
3
:= 3 &times; (((3 − 3/3)3&times;3 ) + 33)
:= 3 + ((((33/3) ) − 3 ) + 333)
:= 3/3 + ((3 &times; 3 + 3)3 − (3 &times; 33 + 3))
:= (44/4 + 4) &times; ((44 + 4)/4 + 44)
:= (44 − 4) &times; ((4/4 − 4) + 44)
:= 4 &times; 44 + ((4/4 + 4)4 − 4)
:= 5 + ((5 &times; (5 &times; (55 + 5 + 5))) + 5)
:= (55 &times; (5 &times; 5 + 5)) − 5 − 5
:= (55 &times; (5 &times; 5 + 5)) − 5
:= (6 − 6/6) &times; (666/6 + 6 &times; 6 &times; 6)
:= ((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 − (66/6))
:= (7/7 + 7) &times; (((7 + 7)/7)7 + 77)
:= (6 &times; 6 − 6/6) &times; (66/6 + 6 &times; 6)
:= 7 + ((77 &times; (7 + 7 + 7)) + (77/7))
:= 88 + (8 &times; 8 &times; (8 + 8 + 8) + (88/8))
:= 8 + ((88 + 8) &times; (8/8 + 8 + 8))
:= ((88/8 + 8) &times; (88 − 8/8)) − 8
:= (99/9 + 9) &times; (9/9 + 9 &times; 9)
:= 999 + (9 &times; (9 &times; 9 − 9) − ((9 + 9)/9))
9
:= 99 + (((9 + 9 + 9)/9) &times; (((9 + 9)/9) ))
&Iuml; 1636 := 1 + ((111 − 1 − 1) &times; (1 + 1 + 1 + 1 + 11))
:= 7 + ((7 + 7 + 7) &times; (7/7 + 77))
&Iuml; 1641 := 1 + (1111 + (1 + 11 + 11)1+1 )
2
&Iuml; 1646 := 11 + ((111 − 1 − 1) &times; (1 + 1 + 1 + 1 + 11))
2
:= 222 + (2 &times; ((22 + 2)2 ))
:= (2 &times; 22 − 2) − ((22/2) + 2)
:= 2222 − ((22 + 2)2 )
:= 3 + (((3 &times; (3 &times; 33 &times; 33)) − 3)/(3 + 3))
:= 3 + ((3 &times; 3 + 3)3 + (3 &times; (3 − 33)))
:= (3 &times; 3 + 3)3 − (3 &times; 33 + 3/3)
:= 44 &times; 44 − (44 + 44 )
:= 4 &times; 44 + ((4/4 + 4)4 − (4 + 4))
:= 4 + ((44 &times; (44 − 4)/(4 + 4)) + 44 )
:= 55/5 + (5 &times; (5 &times; (55 + 5 + 5)))
:= 5/5 + ((55 &times; (5 &times; 5 + 5)) − (5 + 5))
:= 5/5 + ((55 &times; (5 &times; 5 + 5)) − 5)
6
6
:= 6 &times; 6 + (((6 + 6)/6) &times; (6 &times; 6 − 66/6))
:= 6 &times; 6 &times; 66 − (((6 &times; 6/(6 + 6)) ) + 6)
:= (((6 + 6)/6)66/6 ) − (6 &times; 66 + 6)
:= 7 + ((77 &times; (7 + 7 + 7)) + (77 + 7)/7)
:= (((7 + 7 + 7)/7)7 ) − (7 &times; 77 + 7)
:= 7 + (((7 + 7 + 7) &times; (7/7 + 77)) + 7/7)
:= 8 + (88/8 &times; (888/(8 − (8 + 8)/8)))
:= 8 + (((88 + 8) &times; (8/8 + 8 + 8)) + 8/8)
:= ((8 + 8)/8) &times; (888 − (8/8 + 8 &times; 8))
:= 9 + (((9 &times; (99 + 9 &times; 9)) − ((9 + 9)/9)) + 9)
:= 9 + ((9 &times; (99 + 9 &times; 9)) + ((99 + 9)/9))
:= 999 + (9 &times; (9 &times; 9 − 9) − 9/9)
&Iuml; 1637 := 1 + (1 + ((111 − 1 − 1) &times; (1 + 1 + 1 + 1 + 11)))
&Iuml; 1642 := 1 + (1 + (1111 + (1 + 11 + 11)1+1 ))
&Iuml; 1647 := 111 + ((1 + 1 + 1) &times; (1 + 1)11−1−1 )
:= 2/2 + ((2 &times; ((22 + 2)2 )) + 222 )
:= (2 &times; 22) + ((2 &times; (22 − 2))2 − 2)
:= 2/2 + 2222 − (22 + 2)2
:= 333 + (((33/3)3 ) − 33 )
:= 3 + (((3 &times; 3 + 3)3 + (3 &times; (3 − 33))) + 3/3)
:= 3 &times; ((3 + 3)3 + 333)
:= 4 + (4 &times; (44 − 4) + (4/4 + 4)4 )
:= 44 + (44 &times; (44 − 4)/(4 + 4))
:= (4 &times; (444 − 4 &times; (4 + 4))) − 4/4
5
:= 5 + (((5 + 5)/5) &times; ((5/5 − 5) + 55))
:= (5 + 5)/5 + ((55 &times; (5 &times; 5 + 5)) − (5 + 5))
:= (5 + 5)/5 + ((55 &times; (5 &times; 5 + 5)) − 5)
:= 6 + (((6/6 − 66) &times; ((66/6) − 6 &times; 6)) + 6)
:= 66 + (6 &times; 6 &times; (6 &times; 6 + 6) + ((6 + 6)/6) )
:= 6 &times; 6 &times; 66 − ((6 &times; 6/(6 + 6))6 )
:= ((7 + 7 + 7) &times; (7/7 + 77)) − 7/7
:= 7 + (((77 &times; (7 + 7 + 7)) + (77/7)) + 7)
:= 7 + ((7/7 + 7) &times; (((7 + 7)/7)7 + 77))
6
:= ((8 + 8) &times; (888/8 − 8)) − 88/8
:= 8 + ((88/8 + 8) &times; (88 − ((8 + 8)/8)))
:= 888/8 + 8 &times; 8 &times; (8 + 8 + 8)
:= 9 + (((9 &times; (99 + 9 &times; 9)) − 9/9) + 9)
:= ((9 + 9)/9) &times; (9 &times; (9 &times; 9 + 9) + (99/9))
:= 999 + 9 &times; (9 &times; 9 − 9)
119
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1648 := (1 + 1)11 − ((1 + 1) &times; (11 − 1))1+1
&Iuml; 1653 := (1 + 1 + 1) &times; (1 + ((1111 − 11)/(1 + 1)))
&Iuml; 1658 := ((1 + 1 + 111)1+1 ) − 11111
:= 222/2 − (22 − 2)2
:= (2 &times; 22 − 2)2 − 222/2
:= 2 + (((2 &times; 22) + 2) &times; (2 + 2 + 2)2 )
:= 3/3 + (3 &times; ((3 + 3)3 + 333))
:= 33 + (33 &times; (33 + 33))
:= 333 + (((33/3)3 ) − (3 + 3))
:= 4 &times; (444 − 4 &times; (4 + 4))
:= 4 + ((4/4 + 4)4 + 4 &times; 44 )
:= 4 + ((((4/4 + 4)4 + 4 &times; 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 &times; ((6 + 6 + 6) + 6))
:= 6 + (6 &times; 6 &times; 66 − ((6 &times; 6/(6 + 6))6 ))
:= 6 + ((((6 + 6)/6)66/6 ) − 6 &times; 66)
:= (((7 + 7 + 7)/7)7 ) − 7 &times; 77
:= 7/7 + ((7 + 7) &times; (777/7 + 7))
:= ((7/7 + 7 + 7) &times; 777/7) − 7
:= (8 + 8) &times; (888/8 − 8)
:= (88/8 + 8) &times; (88 − 8/8)
:= 8 + ((8 − 8/8 + 8) &times; (888 − 8)/8)
:= 9/9 + (9 &times; (9 &times; 9 − 9) + 999)
:= (9/9 + 9 + 9) &times; (99 − ((99 + 9)/9))
:= 9 + (((99/9 + 9) &times; (9/9 + 9 &times; 9)) + 9)
&Iuml; 1649 := ((111 − 1) &times; (1 + 1 + 1 + 1 + 11)) − 1
&Iuml; 1654 := (1 + 1) &times; ((1 + 1)11 − 11 &times; 111)
&Iuml; 1659 := (1 + 1 + 1) &times; (((1111 − 1)/(1 + 1)) − (1 + 1))
:= (((2 &times; 22) + 2) &times; (2 + 2 + 2) ) − 2
:= ((2 &times; (22 − 2) + 2/2)2 ) − 22
:= 3 + ((3 &times; 3 + 3)3 − (3 &times; 33 + 3/3))
:= 3/3 + ((33 &times; (33 + 33)) + 33)
:= 3 + ((3 &times; (3 − 33 )) + (3 &times; 3 + 3)3 )
:= 4 &times; 44 + (4/4 + 4)4
:= 4 + (((4/4 + 4)4 + 4 &times; 44 ) + 4/4)
:= 44 + (4 &times; (4 + 4) &times; 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 &times; (6 &times; 6 − 66/6)) − 6/6
:= (6 &times; (6 &times; 6 &times; 6 − 6 + 66)) − (6 + 6)/6
:= 666/6 + 6 &times; 6 &times; (6 &times; 6 + 6 + 6)
:= 2/2 + (2
22/2
2
− (22 − 2) )
2
:= 7 &times; 77 + (7777 − 7)/7
:= ((7 + 7)/7) + ((7 + 7) &times; (777/7 + 7))
:= 7 + ((7 + 7) &times; (777/7 + 7))
:= 8/8 + ((8 + 8) &times; (888/8 − 8))
:= 8/8 + ((88/8 + 8) &times; (88 − 8/8))
:= 88/8 + ((8 + 8) &times; (888/8 − 8))
:= 9 + ((99/9 + 9) &times; (9/9 + 9 &times; 9))
:= ((9 + 9)/9) &times; ((9 &times; 9 &times; 9 − 9/9) + 99)
:= 9 + ((9 &times; (9 &times; (9 + 9) + 9)) + 999/9)
&Iuml; 1650 := (111 − 1) &times; (1 + 1 + 1 + 1 + 11)
&Iuml; 1655 := 1 + ((1 + 1) &times; ((1 + 1)11 − 11 &times; 111))
:= 2 + (222/2 − (22 − 2)2 )
:= 2 + ((2 &times; 22 − 2)2 − 222/2)
:= (22 − 2) &times; (((2/2 + 2)2+2 ) + 2)
:= 3 + (3 &times; ((3 + 3)3 + 333))
:= 333 + (((33/3)3 ) − 3 &times; 3)
:= (3 − 3/3 + 3) &times; (333 − 3/3)
:= 4/4 + ((4/4 + 4)4 + 4 &times; 44 )
:= (4/4 + 4) &times; (((44 + 44 )/4) + 44 )
:= 44 + (4 &times; (4 + 4) &times; 44 − 4)
:= 55 &times; (5 &times; 5 + 5)
:= 5 + (55 &times; (5 &times; 5 + 5))
:= 5 + ((55 &times; (5 &times; 5 + 5)) + 5)
:= 66 &times; (6 &times; 6 − 66/6)
:= (6 &times; (6 &times; 6 &times; 6 − 6 + 66)) − 6/6
:= (6 − 6/6) &times; (6 &times; 66 − ((6 + 6)/6)6 )
:= 7 &times; 77 + 7777/7
:= 7 + ((((7 + 7 + 7)/7)7 ) − 7 &times; 77)
:= 7 + (((7 + 7) &times; (777/7 + 7)) + 7/7)
:= (8 − 8/8 + 8) &times; (888 − 8)/8
:= 888 + (8 &times; (88 + 8) − 8/8)
:= ((8 + 8) &times; (88 + 8 + 8)) − (8/((8 + 8)/8))
:= 999/9 + (9 &times; (9 &times; (9 + 9) + 9))
:= ((9 + 9) &times; ((99/9) + 9 &times; 9)) − 9/9
:= (99/9 + 9) &times; (((9 + 9)/9) + 9 &times; 9)
&Iuml; 1651 := 1 + ((111 − 1) &times; (1 + 1 + 1 + 1 + 11))
:= (22/2 + 2) &times; ((2
2&times;(2+2)
− 2)/2)
&Iuml; 1656 := (1 + 1) &times; (1 + ((1 + 1)11 − 11 &times; 111))
:= ((2 &times; 22) + 2) &times; (2 + 2 + 2)
&Iuml; 1660 := (11 − 1) &times; (1 + (11 &times; (1 + 1 + 1 + 1 + 11)))
&Iuml; 1661 := (((1 + 1 + 1) &times; 1111) − 11)/(1 + 1)
:= 2 + (((2 &times; (22 − 2) + 2/2)2 ) − 22)
2
:= 3 + ((3 &times; ((3 + 3)3 + 333)) + 3/3)
:= 3 &times; (((3 + 3)3 + 333) + 3)
:= 333 + (((33/3)3 ) − 3)
:= 4 + ((4 &times; (444 − 4 &times; (4 + 4))) − 4/4)
:= 4 + ((4 &times; (444 − 4 &times; (4 + 4))) + 4)
:= 4 &times; (44 + 4) + ((4/4 + 4)4 − 4)
:= 5/5 + (55 &times; (5 &times; 5 + 5))
:= 5 + ((55 &times; (5 &times; 5 + 5)) + 5/5)
:= 55/5 + (55 &times; (5 &times; 5 + 5))
:= 6/6 + (66 &times; (6 &times; 6 − 66/6))
:= 6 &times; (6 &times; 6 &times; 6 − 6 + 66)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6 + 66)) − 6/6)
:= 7 &times; 77 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 7 &times; 77)
:= 7 &times; 77 + ((7777 + 77)/7)
:= (88 + 8 + 8)/8 &times; (8 &times; (8 + 8) − 8/8)
:= 888 + 8 &times; (88 + 8)
:= 8 + ((88/8 + 8) &times; (88 − 8/8))
:= (9 &times; (9 &times; (9 + 9) + 9)) + ((999 + 9)/9)
:= (9 + 9) &times; ((99/9) + 9 &times; 9)
:= 99/9 &times; (9 &times; (9 + 9) − (99/9))
&Iuml; 1652 := 1 + (1 + ((111 − 1) &times; (1 + 1 + 1 + 1 + 11)))
&Iuml; 1657 := ((1 + 1 + 1) &times; ((1 + 1111)/(1 + 1))) − 11
&Iuml; 1662 := (1 + 1 + 1) &times; (((1111 − 1)/(1 + 1)) − 1)
:= 2 &times; ((2 &times; ((22 − 2) + 2)) + 22)
:= 2/2 + (((2 &times; 22) + 2) &times; (2 + 2 + 2) )
:= (2/2 + 2) &times; (((22 + 2)2 ) − 22)
:= 33 + ((33 &times; (33 + 33)) − 3/3)
:= (3 &times; 3 + 3)3 + ((3 − (3 + 3)3 )/3)
:= (3 &times; 3 + 3)3 − (33 + 33)
:= 4 + (4 &times; (444 − 4 &times; (4 + 4)))
:= 4 + (((4/4 + 4)4 + 4 &times; 44 ) + 4)
:= 44 + (4 &times; (4 + 4) &times; 44 − (4 + 4)/4)
:= (5 + 5)/5 + (55 &times; (5 &times; 5 + 5))
:= 5 + ((55 &times; (5 &times; 5 + 5)) + ((5 + 5)/5))
:= (5/5 + 5) &times; (5 &times; 55 + ((5 + 5)/5))
:= 6/6 + (6 &times; (6 &times; 6 &times; 6 − 6 + 66))
:= 6 + (6 &times; (6 &times; 6 &times; 6 − 6 + 66))
:= 7 + (7777/7 + 7 &times; 77)
:= 7 + (((((7 + 7 + 7)/7)7 ) − 7 &times; 77) + 7)
:= ((8 + 8) &times; (88 + 8 + 8)) − (88 + 8)/8
:= 8/8 + (8 &times; (88 + 8) + 888)
:= ((8 + 8) &times; (88 + 8 + 8)) − (8 + 8)/8
:= ((99/9) &times; (9 &times; (9 + 9) − (99/9))) − 9
:= 9/9 + ((9 + 9) &times; ((99/9) + 9 &times; 9))
:= (9 + 9) &times; 99 − (999/9 + 9)
2
66/6
:= (((6 + 6)/6)
) − 6 &times; 66
:= (7 + 7) &times; (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
&Iuml; 1663 := (111 &times; (1 + 1 + 1 + 1 + 11)) − 1 − 1
:= ((22 + 2 + 2) &times; 2
2+2+2
&Iuml; 1673 := 1 + (11 &times; ((11 &times; (1 + (1 + 1 + 11))) − (1 + 1)))
&Iuml; 1668 := (1 + 1 + 1) &times; ((1 + 1111)/(1 + 1))
:= 2 &times; ((22 &times; ((2 + 2 + 2) + 2)) − 2)
:= 2/2 + (2 &times; (22 &times; ((2 + 2 + 2)2 + 2)))
:= 3 + (333 &times; (3 − 3/3 + 3))
:= 3 &times; 3 + (((33/3)3 ) + 333)
2
) − 2/2
:= 333 + (((33/3)3 ) − 3/3)
4
:= 4 + ((4 &times; (44 + 4) + (4/4 + 4)4 ) + 4)
:= 44 + (4 &times; (4 + 4) &times; 44 − 4/4)
:= 4 + (4 &times; (4 + 4) &times; 44 + 4 )
:= (((5 + 5 + 5) &times; 555) − (5 + 5))/5
:= (5 − (5 + 5)/5) &times; (555 + 5/5)
:= 5 + ((5 − (5 + 5)/5) &times; (555 + 5/5))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6 + 66)) + 6/6)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 − 6 + 66)) + 6)
:= (6 &times; (6 &times; 66 − 6)) − (666 + 6/6)
:= 77/7 + ((7 + 7) &times; (777/7 + 7))
:= 7 &times; 7 + ((77 &times; (7 + 7 + 7)) + ((7 + 7)/7))
:= 7 + ((77 &times; (7 + 7 + 7)) + 7 &times; 7)
:= ((8 + 8) &times; (88 + 8 + 8)) − 8/8
:= (8/((8 + 8)/8)) + ((8 + 8) &times; (88 + 8 + 8))
:= 8/8 + 88 &times; (88/8 + 8)
:= (9 + 9) &times; 99 + (((9 − 999)/9) − 9)
:= 99 + (9 &times; 9 &times; (9 + 9) + 999/9)
:= (9 + 9) &times; 99 − (9/9 + 99 + 9)
&Iuml; 1669 := 1 + ((1 + 1 + 1) &times; ((1 + 1111)/(1 + 1)))
&Iuml; 1664 := (111 &times; (1 + 1 + 1 + 1 + 11)) − 1
&Iuml; 1674 := (1 + 1 + 1) &times; (1 + (1 + ((1 + 1111)/(1 + 1))))
:= 2 + ((222/2 &times; ((22/2 + 2) + 2)) + 2)
:= 2 + (2 &times; (22 &times; ((2 + 2 + 2)2 + 2)))
:= 3 + (((3 &times; 3333) − 3)/(3 + 3))
:= 3 &times; (3 &times; (((3 + 3)3 − 33) + 3))
:= 4 &times; (4 &times; (44 − 4) + 44 )
:= 4 + (4 &times; (44 + 4) + (4/4 + 4)4 )
:= 44 &times; 44 − (((4 + 4)/4 + 44 ) + 4)
:= (((5 + 5 + 5) &times; 555) − 5)/5
:= 5 + ((((5 + 5 + 5) &times; 555) − 5)/5)
:= 5 &times; 5 + ((5 − 5/5)5 + 55 /5)
:= (6/6 + 6 + 6) &times; (((6 + 6)/6)6/6+6)
:= (((6 &times; 6 − 6/6) + 6)
:= (7 − 7/7 + 7) &times; ((7 + 7)/7)7
:= 7 + ((((((7 + 7 + 7)/7) ) − 7 &times; 77) + 7) + 7)
:= ((7 &times; 7 − (7/7 + 7))(7+7)/7 ) − 7
:= (8 + 8) &times; (88 + 8 + 8)
:= 8 + (((88/8 + 8) &times; (88 − 8/8)) + 8)
:= (8 + 8)/8 + 88 &times; (88/8 + 8)
:= 9 + (((9 + 9) &times; ((99/9) + 9 &times; 9)) − 9/9)
:= 9 + ((99/9 + 9) &times; (((9 + 9)/9) + 9 &times; 9))
:= (9 + 9) &times; 99 − (99 + 9)
:= (22 + 2 + 2) &times; 2
2+2+2
:= 333 + ((33/3)3 )
(6+6)/6
)−6−6
:= (6 &times; (6 &times; 66 − 6)) − 666
7
&Iuml; 1670 := (11 − 1) &times; ((1 + 1 + 11)1+1 − (1 + 1))
&Iuml; 1665 := 111 &times; (1 + 1 + 1 + 1 + 11)
&Iuml; 1675 := 11 + ((111 &times; (1 + 1 + 1 + 1 + 11)) − 1)
2
:= 222/2 &times; ((22/2 + 2) + 2)
:= (2 &times; (22 &times; ((2 + 2 + 2) + 2))) − 2
:= (((2 &times; 22) + 2)2 ) − ((22 − 2/2)2 )
:= 333 &times; (3 − 3/3 + 3)
:= 3 + ((((33/3)3 ) + 333) + 3)
:= (3 &times; 33+3 ) − ((3 − 3/3)3&times;3 )
:= 4 &times; (44 + 4) + (4/4 + 4)4
:= 4 + (((4 − 444)/4) + 4 &times; 444)
:= 44 &times; 44 − ((4/4 + 44 ) + 4)
:= (5 + 5 + 5) &times; 555/5
:= 5 + ((5 + 5 + 5) &times; 555/5)
:= 5 &times; ((5 &times; 55 + 55) + 5)
6/6+6)
:= (6 − 6/6) &times; 666 &times; 6/(6 + 6)
:= 6 + ((6/6 + 6 + 6) &times; (((6 + 6)/6)
:= (7/7 + 7 + 7) &times; 777/7
:= 7 + (((7 + 7) &times; (777/7 + 7)) + (77/7))
)
:= (7 − ((7 + 7)/7)) &times; (7 &times; 7 &times; 7 − (7/7 + 7))
:= 8/8 + ((8 + 8) &times; (88 + 8 + 8))
:= 88 &times; (88/8 + 8) − (8 + 8)/8
:= 88/8 + ((8 + 8) &times; (88 + 8 + 8))
:= 9 + ((9 + 9) &times; ((99/9) + 9 &times; 9))
:= (9 + 9) &times; 99 − ((999 + 9)/9)
:= 9 + ((99 − 9/9) &times; ((9 − 9/9) + 9))
&Iuml; 1671 := (1 + 1 + 1) &times; (1 + ((1 + 1111)/(1 + 1)))
&Iuml; 1666 := 1 + (111 &times; (1 + 1 + 1 + 1 + 11))
2
:= (66 + 6/6) &times; (6 &times; 6 − 66/6)
&Iuml; 1676 := 11 + (111 &times; (1 + 1 + 1 + 1 + 11))
:= (2 &times; (22 &times; ((2 + 2 + 2) + 2))) − 2/2
:= 2 &times; ((22 &times; ((2 + 2 + 2)2 + 2)) + 2)
:= ((3 &times; 3333) − 3)/(3 + 3)
:= 3 + ((333 &times; (3 − 3/3 + 3)) + 3)
:= 3 + ((((33/3)3 ) + 333) + 3 &times; 3)
:= 4 &times; 444 + ((4 − 444)/4)
:= 44 &times; 44 − (((4/4 + 44 ) + 4) + 4)
:= 44 &times; 44 − (44 + 4)
:= 5 + (5555/5 + 555)
:= 5 &times; 5 + ((55 &times; (5 &times; 5 + 5)) + 5/5)
:= 6 + ((6 − 6/6) &times; (6 &times; 66 − ((6 + 6)/6) ))
:= 6 + ((6 − 6/6) &times; 666 &times; 6/(6 + 6))
:= 6/6 + ((66 + 6/6) &times; (6 &times; 6 − 66/6))
:= 7 &times; (777 − 7 &times; 77)
:= 7 + ((7 − 7/7 + 7) &times; ((7 + 7)/7)7 )
:= (77/7 &times; ((77 − 7/7) + 77)) − 7
:= (8 + 8)/8 + ((8 + 8) &times; (88 + 8 + 8))
:= 88 &times; (88/8 + 8) − 8/8
:= ((88 + 8)/8) + ((8 + 8) &times; (88 + 8 + 8))
:= (99 − 9/9) &times; ((9 − 9/9) + 9)
:= (9 + 9) &times; 99 − 999/9
:= (9 + 9)/9 + ((9 + 9) &times; 99 − (99 + 9))
:= 2 + ((22 + 2 + 2) &times; 2
2+2+2
)
:= 555 + 5555/5
6
&Iuml; 1667 := (1 + ((1 + 1 + 1) &times; 1111))/(1 + 1)
&Iuml; 1672 := 11 &times; ((11 &times; (1 + (1 + 1 + 11))) − (1 + 1))
&Iuml; 1677 := 1 + (11 + (111 &times; (1 + 1 + 1 + 1 + 11)))
:= 2 &times; (22 &times; ((2 + 2 + 2)2 + 2))
:= ((2 &times; (22 − 2) + 2/2)2 ) − 2 − 2
:= ((3 &times; 3333) + 33)/(3 + 3)
:= 3 + (3 &times; (3 &times; (((3 + 3)3 − 33) + 3)))
:= 4 + ((4 &times; (4 + 4) &times; 44 − 4/4) + 4 )
:= 44 &times; (44 − ((4 + 4)/4 + 4))
:= 4/4 + (44 &times; 44 − (44 + 4))
:= 555 + (5555 + 5)/5
:= 5 + ((5555 + 5)/5 + 555)
:= 2 + (222/2 &times; ((22/2 + 2) + 2))
:= 3 + (((33/3)3 ) + 333)
4
:= 66/6 + (6 &times; (6 &times; 6 &times; 6 − 6 + 66))
:= 5 &times; 5 + ((55 &times; (5 &times; 5 + 5)) + ((5 + 5)/5))
6
:= 66/6 &times; (6 &times; 6 &times; 6 − ((6 + 6)/6) )
:= 6 + (((6 − 6/6) &times; 666 &times; 6/(6 + 6)) + 6)
:= 7/7 + ((77 &times; (7 + 7 + 7)) + 7 &times; 7)
:= 7 + ((7/7 + 7 + 7) &times; 777/7)
:= 7 &times; 7 + ((77 &times; (7 + 7 + 7)) + (77/7))
:= 888 + (8 &times; (88 + 8) + (88/8))
:= 88 &times; (88/8 + 8)
:= (88 + 8 + 8)/8 &times; (8 &times; (8 + 8) + 8/8)
:= 9/9 + ((99 − 9/9) &times; ((9 − 9/9) + 9))
:= 99/9 &times; (9 &times; (9 + 9) − (9/9 + 9))
:= 9 + ((9 &times; 9 &times; (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
&Iuml; 1678 := 11 + ((1 + ((1 + 1 + 1) &times; 1111))/(1 + 1))
&Iuml; 1683 := 11 &times; ((11 &times; (1 + (1 + 1 + 11))) − 1)
&Iuml; 1688 := ((11 − 1) &times; (1 + 1 + 11)1+1 ) − 1 − 1
:= (2 &times; 22)2 − (22&times;(2+2) + 2)
:= 2 + ((2 &times; (22 − 2) + 2/2)2 )
:= 2 &times; (2 &times; ((22 − 2)2 + 22))
:= 3 + 3 &times; 33+3 − (3 − 3/3)3&times;3
:= 33 &times; ((33 − 3) + 33 )
:= ((3 &times; (3 &times; 3 + 3 + 3)3 ) + 3)/(3 + 3)
:= 44 &times; 44 − ((4 + 4)/4 + 44 )
:= 4 + (44 &times; 44 − (4/4 + 44 ))
:= 4 + ((44 &times; 44 − 44 ) + 4)
:= ((5/5 + 5) &times; (5 &times; 55 + 5)) − (5 + 5)/5
:= 5 &times; 5 &times; 55 + ((55 + 5)/(5 + 5) − 5)
:= 5 &times; 5 &times; 55 + (55 + 5)/(5 + 5)
:= 6 + ((66/6) &times; (6 &times; 6 &times; 6 − ((6 + 6)/6)6 ))
:= 66/6 &times; (666/6 + 6 &times; 6 + 6)
:= (6 + 6)/6 + ((6 &times; (6 &times; 6 &times; 6 + 66)) − 6)
:= 7 + (((7 − 7/7 + 7) &times; ((7 + 7)/7) ) + 7)
:= 77/7 &times; ((77 − 7/7) + 77)
:= 7 + ((7 &times; 7 − (7/7 + 7))(7+7)/7 )
:= 8 + (88 &times; (88/8 + 8) − ((8 + 8)/8))
:= 88/8 + 88 &times; (88/8 + 8)
:= 8 + (88 &times; (88/8 + 8) + 8)
:= (9 &times; (9 &times; 9 − (9 + 9))) + 9999/9
:= 99 &times; ((9 − 9/9) + 9)
:= (9 − 9/9) &times; (((999 + 9)/9) + 99)
7
&Iuml; 1679 := ((11 − 1) &times; (1 + 1 + 11)1+1 ) − 11
&Iuml; 1684 := 1 + (11 &times; ((11 &times; (1 + (1 + 1 + 11))) − 1))
2
2
2
:= 2 + ((((2 + 2 + 2) + 22) )/2)
:= ((2 &times; (22 − 2) + 2/2) ) − 2
3
3
&Iuml; 1689 := ((11 − 1) &times; (1 + 1 + 11)1+1 ) − 1
:= 2 + ((2222/2) + ((22 + 2)2 ))
:= (33 − 3/3 − 3) &times; (((3 + 3)3 + 3)/3)
:= 3/3 + (33 &times; ((3 − 3) + 3 ))
:= (3 &times; 3 + 3)3 − (33 + 3 + 3)
:= 44 &times; 44 − (4/4 + 44 )
:= 4 + (44 &times; 44 − 44 )
:= 4 + (((44 &times; 44 − 44 ) + 4/4) + 4)
:= ((5/5 + 5) &times; (5 &times; 55 + 5)) − 5/5
:= 5 + (((5/5 + 5) &times; (5 &times; 55 + 5)) − 5/5)
:= 5 &times; 5 + ((((5 + 5 + 5) &times; 555) − 5)/5)
:= ((66/6 + 6) + 6) &times; (66 + 6/6 + 6)
:= (((6 + 6)/6)6+6 ) − (6 &times; (6 &times; 66 + 6))
:= (6 − 6 &times; 6) &times; (6 − 66) − 666/6
:= 7 + (((7/7 + 7 + 7) &times; 777/7) + 7)
:= 7 + (((77 &times; (7 + 7 + 7)) + (77/7)) + 7 &times; 7)
:= (((7 + 7)/7) &times; ((77 &times; 77 + 7)/7)) − 7
:= 8 + (88 &times; (88/8 + 8) − 8/8)
:= ((88 + 8)/8) + 88 &times; (88/8 + 8)
:= 8 + ((88 &times; (88/8 + 8) + 8/8) + 8)
:= 9 + ((9 + 9) &times; 99 − ((999 + 9)/9))
:= 9/9 + (99 &times; ((9 − 9/9) + 9))
:= 9 + (((9 + 9) &times; 99 − 999/9) + 9)
&Iuml; 1680 := (11 − 1) &times; ((1 + 1 + 11)1+1 − 1)
&Iuml; 1685 := (1 + 1)11 − (11 &times; (11 &times; (1 + 1 + 1)))
:= 2 &times; ((2 − 22) &times; (2 − (2 &times; 22)))
&Iuml; 1690 := (11 − 1) &times; (1 + 1 + 11)1+1
:= 2 + (((2 &times; (22 − 2) + 2/2)2 ) + 2)
:= 2 + (2 &times; (2 &times; ((22 − 2)2 + 22)))
:= 3 + ((33 &times; ((33 − 3) + 33 )) − 3/3)
:= (3 &times; 3 + 3)3 − (33/3 + 33 )
:= 4 + ((44 &times; 44 − 44 ) + 4/4)
:= 44 &times; 44 + ((44 − 4)/4 − 44 )
:= 5 + ((5/5 + 5) &times; (5 &times; 55 + 5))
:= 5 &times; ((55 + 5)/(5 + 5) + 5 &times; 5)
:= 6 &times; (((6 + 6)/6) + 6 &times; 6 &times; 6)
:= (6 &times; (6 &times; 6 &times; 6 + 66)) − 6/6 − 6
:= (6 &times; (6 &times; 6 &times; 6 + 66)) − (6 + 6)/6
:= (77 + 7) &times; ((7 − 7/7 + 7) + 7)
:= (7 − ((7 + 7)/7)) &times; ((7 &times; 7 &times; 7 − 7) + 7/7)
:= 7 + (77/7 &times; ((77 − 7/7) + 77))
:= 8 + 88 &times; (88/8 + 8)
:= 8 + ((88 + 8 + 8)/8 &times; (8 &times; (8 + 8) + 8/8))
:= (8/8 + 8 &times; 8) &times; (((((8 + 8)/8) + 8) + 8) + 8)
:= 9 + ((9 + 9) &times; 99 − 999/9)
:= (9 + 9)/9 + (99 &times; ((9 − 9/9) + 9))
:= (9 + 9) &times; 99 − ((99/9) + 9 &times; 9)
3
3
:= (3 + 3/3) &times; (3 + 33)
:= 44 &times; 44 − 4
4
:= (5/5 + 5) &times; (5 &times; 55 + 5)
6
&Iuml; 1681 := (1 + ((1 + 1) &times; ((1 + 1) &times; (11 − 1))))1+1
2
:= (2 &times; (22 − 2) + 2/2)
3
:= ((33/3 + 3 ) + 3)
&Iuml; 1686 := 1 + ((1 + 1)11 − (11 &times; (11 &times; (1 + 1 + 1))))
2
:= (2 &times; (2 &times; ((22 − 2) + 22))) − 2
3−3/3
3
3
&Iuml; 1691 := 1 + ((11 − 1) &times; (1 + 1 + 11)1+1 )
:= 2222 − (((22 + 2/2)2 ) + 2)
:= 3 + (33 &times; ((3 − 3) + 3 ))
:= 33 + (((33/3)3 ) + 333)
:= 4/4 + (44 &times; 44 − 4 )
:= 4 + ((44 &times; 44 − 44 ) + (4 + 4)/4)
:= 44/4 + (44 &times; 44 − 44 )
:= 5/5 + ((5/5 + 5) &times; (5 &times; 55 + 5))
:= (5/5 + 5) &times; ((5 &times; 55 + 5/5) + 5)
:= 5 + ((5/5 + 5) &times; ((5 &times; 55 + 5/5) + 5))
:= (6 &times; (6 &times; 6 &times; 6 + 66)) − 6
:= (6 &times; (6 &times; 6 &times; 6 + 66)) − 6/6
:= 77 + ((77 &times; (7 + 7 + 7)) − (7/7 + 7))
:= 77 + (((7 + 7 + 7)/7) &times; (7 &times; 77 − 7/7))
:= 8 + (88 &times; (88/8 + 8) + 8/8)
:= 8 &times; (8 &times; 8 + 8) + (8888 − 8)/8
:= (88/8 + 8) &times; (8/8 + 88)
:= (9 + 9) &times; 99 − ((9 + 9)/9 + 99)
:= (9 + 9) &times; 99 + (((9 + 9 + 9)/9) − 99)
:= 9 + ((9 + 9) &times; 99 − (9/9 + 99))
4
:= ((6 &times; 6 − 6/6) + 6)
(6+6)/6
:= (7 &times; 7 − (7/7 + 7))
(7+7)/7
&Iuml; 1682 := 1 + ((1 + ((1 + 1) &times; ((1 + 1) &times; (11 − 1))))1+1 )
2
2
&Iuml; 1687 := 1111 + (((1 + 1) &times; (1 + 11))1+1 )
&Iuml; 1692 := 1 + (1 + ((11 − 1) &times; (1 + 1 + 11)1+1 ))
:= (((2 + 2 + 2) + 22) )/2
:= ((22 + 2)2 ) + 2222/2
:= 2 &times; ((2 &times; ((22 − 2)2 + 22)) + 2)
:= (33 &times; ((33 − 3) + 33 )) − 3/3
:= ((3 &times; (3 &times; 3 + 3 + 3)3 ) − 3)/(3 + 3)
:= (3 &times; 3 + 3)3 − (33 + 3)
:= (4 + 4)/4 + (44 &times; 44 − 44 )
:= 4 + ((44 &times; 44 − (4/4 + 44 )) + 4)
:= ((4 + 4) &times; (44 − 44)) − 4
:= ((5 + 5)/5)5 + (55 &times; (5 &times; 5 + 5))
:= 5 &times; 5 &times; 55 + (55 − 5)/(5 + 5)
:= 5 + ((55 − 5)/(5 + 5) + 5 &times; 5 &times; 55)
:= 6/6 + (((6 &times; 6 − 6/6) + 6)(6+6)/6 )
:= 6 + (((6 &times; 6 − 6/6) + 6)(6+6)/6 )
:= 6 &times; (6 &times; 6 &times; 6 + 66)
:= 77 + ((77 &times; (7 + 7 + 7)) − 7)
:= ((7 + 7)/7) &times; ((77 &times; 77 − 7)/7)
:= 8 + (88 &times; (88/8 + 8) + ((8 + 8)/8))
:= 8 &times; (8 &times; 8 + 8) + 8888/8
:= 8/8 + ((88/8 + 8) &times; (8/8 + 88))
:= (9 + 9) &times; 99 − (9/9 + 99)
:= 9 + ((9 &times; (9 &times; 9 − (9 + 9))) + 9999/9)
:= 9 + (99 &times; ((9 − 9/9) + 9))
:= 7/7 + ((7 &times; 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
&Iuml; 1693 := (11 &times; (11 &times; (1 + (1 + 1 + 11)))) − 1
2
&Iuml; 1698 := (1 + 1 + 1) &times; (11 + ((1111 − 1)/(1 + 1)))
2+2
2
:= 22 + ((2 &times; (22 − 2) + 2/2)2 )
:= 2222 − ((22 + 2/2) )
:= 2 + (((2
:= 3/3 + ((3 &times; 3 + 3)3 − (33 + 3))
:= 3 + ((3 &times; 3 + 3)3 − 33)
:= 3 + ((3 &times; 3 + 3)3 − (33 + 3/3))
:= 44 + ((4/4 + 4)4 + 4 &times; 44 )
:= (4 + 4)/4 + ((4 + 4) &times; (44 − 44))
:= 4 + ((((4 + 4)4 ) − (((4 − 4/4) + 4)4 )) + 4)
:= 5 + ((55 + 5)/(5 + 5) + 5 &times; 5 &times; 55)
:= (5 − (5 + 5)/5) &times; (555 + (55/5))
:= 55 + ((5 − 5/5)5 + (55 − 5)/5)
:= 6/6 + (6 &times; (6 &times; 6 &times; 6 + 66))
:= 6 + (6 &times; (6 &times; 6 &times; 6 + 66))
:= 66/6 + (6 &times; (6 &times; 6 &times; 6 + 66))
:= 77 + ((77 &times; (7 + 7 + 7)) − 7/7)
:= (7 − 7/7) &times; (7 &times; (7 &times; 7 − 7) − (77/7))
:= 7 + (((7 + 7)/7) &times; ((77 &times; 77 + 7)/7))
:= ((8 &times; 8 − 8/8) &times; (88/8 + 8 + 8)) − 8
:= 8 + ((8/8 + 8 &times; 8) &times; (((((8 + 8)/8) + 8) + 8) + 8))
:= 888 + (888/8 + 8 &times; 88)
:= 9 + ((99 &times; ((9 − 9/9) + 9)) + 9/9)
:= (9 + 9) &times; 99 − (((9 + 9 + 9)/9) + 9 &times; 9)
:= (9 + 9)/9 + (9 &times; ((99 + 9 &times; 9) + 9))
&Iuml; 1694 := 11 &times; (11 &times; (1 + (1 + 1 + 11)))
− 2) &times; (22/2) ) + 2)
&Iuml; 1703 := (1 + 1 + 11) &times; ((11 &times; (1 + 11)) − 1)
&Iuml; 1699 := ((11 − 1) &times; (1 + (1 + 1 + 11)1+1 )) − 1
:= ((22 − 2/2) &times; ((2/2 + 2)2+2 )) − 2
:= (22+2 − 2) &times; (22/2)2
3
&Iuml; 1704 := (1 + 11) &times; ((1 + 11)1+1 − (1 + 1))
:= 2 &times; (2 &times; ((2 &times; (222 + 2)) − 22))
:= (3 &times; 3 + 3) − 3/3 − 33
:= 3 + (((3 &times; 3 + 3) − 33) + 3/3)
:= 3 + ((3 &times; 3 + 3)3 − 33 )
:= ((4 + 4) &times; (44 − 44)) − (4 + 4)/4
:= 4 + (((4 + 4)4 ) − (((4 − 4/4) + 4)4 ))
:= 4 + (((4 + 4) &times; (44 − 44)) + 4)
:= 55 + ((55 &times; (5 &times; 5 + 5)) − (55/5))
:= 55 + (((5 − 5/5)5 − 5) + 55 /5)
:= 55 + ((5 − 5/5)5 + 55 /5)
:= (6 + 6)/6 + (6 &times; (6 &times; 6 &times; 6 + 66))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 66)) + 6/6)
:= 6 + 6 &times; (6 &times; 6 &times; 6 + 66) + 6
:= 77 + (77 &times; (7 + 7 + 7))
:= 7 &times; (77 + 7) + 7777/7
:= (7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − 77/7
3
:= 88/8 &times; (((8 + 8)/8 + 88) + 8 &times; 8)
:= 8 + ((88/8 + 8) &times; (8/8 + 88))
:= (8 + 8 + 8) &times; ((8 &times; 8 − 8/8) + 8)
:= 99/9 + (99 &times; ((9 − 9/9) + 9))
:= (9 + 9) &times; 99 − (((9 + 9)/9) + 9 &times; 9)
:= (9 + 9) &times; 99 + (((9 + 9 + 9)/9) − 9 &times; 9)
&Iuml; 1695 := 1 + (11 &times; (11 &times; (1 + (1 + 1 + 11))))
&Iuml; 1700 := (11 − 1) &times; (1 + (1 + 1 + 11)1+1 )
:= (2 &times; 22)2 + ((2 − 222 )/2)
:= (2 &times; 22 − 2)2 − 22+2+2
:= 2 + (((2 &times; (22 − 2) + 2/2)2 ) + 22)
:= (3 &times; 3 + 3)3 − 33
:= (3 &times; 3 + 3)3 − (33 + 3/3)
:= 3 + (((3 &times; 3 + 3)3 − 33 ) + 3/3)
:= ((4 + 4)4 ) − (((4 − 4/4) + 4)4 )
:= 4 + ((4 + 4) &times; (44 − 44))
:= 44/4 &times; (444/4 + 44)
:= 5 − (5 &times; 5 &times; 55 + 55)
:= 5 &times; (((5 &times; 55 + 55) + 5) + 5)
:= 55 + (55 &times; (5 &times; 5 + 5))
:= (6 − 6/6) &times; (666 &times; 6/(6 + 6) + 6)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 66)) + ((6 + 6)/6))
:= 6 + 6 &times; (6 &times; 6 &times; 6 + 66) + 6/6 + 6
:= 7/7 + ((77 &times; (7 + 7 + 7)) + 77)
:= 7 + (((77 &times; (7 + 7 + 7)) − 7/7) + 77)
:= 77/7 &times; (7/7 + 77 + 77)
:= 8 + (8888/8 + 8 &times; (8 &times; 8 + 8))
:= (8/8 + 8 + 8) &times; (((88 + 8)/8) + 88)
:= 8/8 + ((8 + 8 + 8) &times; ((8 &times; 8 − 8/8) + 8))
:= (9 + 9) &times; 99 + (((99 + 9)/9) − 99)
:= ((9 − 9/9) + 9) &times; (9/9 + 99)
:= 99/9 &times; ((((9 + 9)/9) − 9) + 9 &times; (9 + 9))
5
&Iuml; 1696 := 1 + (1 + (11 &times; (11 &times; (1 + (1 + 1 + 11)))))
&Iuml; 1701 := 1 + ((11 − 1) &times; (1 + (1 + 1 + 11)1+1 ))
&Iuml; 1705 := 11 &times; (11 + (1 + 11)1+1 )
&Iuml; 1706 := 1 + (11 &times; (11 + (1 + 11)1+1 ))
:= 2 + ((22+2 − 2) &times; (22/2)2 )
:= (22 − 2/2) &times; ((2/2 + 2)2+2 )
:= (2 &times; (2 &times; (222 − 2))) − 222
:= 3/3 + ((3 &times; 3 + 3)3 − 33)
:= (3 &times; 3 + 3)3 − 33
:= 33/3 + ((3 &times; 3 + 3)3 − 33)
:= (4 + 4) &times; (44 − 44)
:= 4 + (((4 + 4) &times; (44 − 44)) + 4/4)
:= (44 − 4)/4 + ((4 + 4) &times; (44 − 44))
:= (55/5 + 5) &times; (555/5 − 5)
:= ((5 + 5)/5 + 5) &times; ((5 − (5 + 5)/5)5 )
:= 55 + ((55 &times; (5 &times; 5 + 5)) + 5/5)
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 66)) − ((6 + 6)/6))
:= (6/6 + 6) &times; ((6 &times; 6/(6 + 6))6−6/6 )
:= 6 + 6 &times; (6 &times; 6 &times; 6 + 66) + (6 + 6)/6 + 6
:= ((7 + 7)/7) &times; ((77 &times; 77 + 7)/7)
:= 7 + ((77 &times; (7 + 7 + 7)) + 77)
:= 7 + (7777/7 + 7 &times; (77 + 7))
:= 8 + ((88 &times; (88/8 + 8) + 8) + 8)
:= (8 &times; 8 − 8/8) &times; (88/8 + 8 + 8)
:= (8 + 8)/8 + ((8 + 8 + 8) &times; ((8 &times; 8 − 8/8) + 8))
:= (9 − 9/9) &times; (((999 + 9 + 9)/9) + 99)
:= 9 &times; ((99 + 9 &times; 9) + 9)
:= 9 &times; 9 &times; 9 + (999 − ((99 + 99)/9))
&Iuml; 1697 := 1 + (1 + (1 + (11 &times; (11 &times; (1 + (1 + 1 + 11))))))
2
2
&Iuml; 1702 := ((1 + 1 + 11) &times; ((11 &times; (1 + 11)) − 1)) − 1
&Iuml; 1707 := 1 + (1 + (11 &times; (11 + (1 + 11)1+1 )))
:= 2 + ((2 &times; 22 − 2)2 − 22+2+2 )
:= 2 + ((((2 &times; (22 − 2) + 2/2)2 ) + 22) + 2)
:= 3/3 + ((3 &times; 3 + 3)3 − 33 )
:= 3 + (((3 &times; 3 + 3)3 − 33 ) + 3)
:= 44/4 + ((4 + 4) &times; (44 − 44))
:= 5 + (((55 − 5)/(5 + 5) + 5 &times; 5 &times; 55) + 5)
:= 4 + (((4 + 4) &times; (44 − 44)) + (4 + 4)/4)
:= 55 + (((55 &times; (5 &times; 5 + 5)) − 5) + ((5 + 5)/5))
:= 55 + ((55 &times; (5 &times; 5 + 5)) + ((5 + 5)/5))
:= 6 + ((6 &times; (6 &times; 6 &times; 6 + 66)) − 6/6)
:= ((66 − 6)/6) + (6 &times; (6 &times; 6 &times; 6 + 66))
:= 6 + (6/6 + 6) &times; (6 &times; 6/(6 + 6))6−6/6
:= (7 &times; (7 − 77)) + (((7 + 7 + 7)/7)7 )
:= 7 + (((77 &times; (7 + 7 + 7)) + 77) + 7/7)
:= (7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − (7/7 + 7)
:= 2 + (((2 − 22 )/2) + (2 &times; 22) )
:= 3 + ((3 &times; 3 + 3)3 − (3/3 + 33))
:= 4/4 + ((4 + 4) &times; (44 − 44))
8&times;8/(8+8)
:= ((8 − 8/8)
) − 8 &times; 88
:= 9 + ((9 − 9/9) &times; (((999 + 9)/9) + 99))
:= ((88/8 + 8) &times; ((8 + 8)/8 + 88)) − 8
:= 8 + (((88/8 + 8) &times; (8/8 + 88)) + 8)
:= 9/9 + (9 &times; ((99 + 9 &times; 9) + 9))
:= 9 + ((9 + 9) &times; 99 − (((9 + 9 + 9)/9) + 9 &times; 9))
123
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1708 := (1 + (1 + 1 + 11)) &times; (1 + 111+1 )
&Iuml; 1713 := 1 + ((1 + 1)11 − ((1 + 1 + 1) &times; (1 + 111)))
&Iuml; 1718 := 1 + (((1 + 11)1+1+1 ) − 11)
:= 2 + ((2 &times; (2 &times; (222 − 2))) − 222)
:= (2 &times; 22)2 − (222 + 2/2)
:= 2 + (((2 &times; 22)2 − 222) + 2)
:= ((3/3 + 3) + 3) &times; ((33+3 + 3)/3)
:= 3 + ((3 &times; 3 + 3)3 − (3 &times; (3 + 3)))
:= ((3 − 33)/3) + (3 &times; 3 + 3)3
:= (4 &times; (444 − 4 &times; 4)) − 4
:= 4/4 + (4 &times; (444 − 4 &times; 4))
:= 4 + ((4 &times; (444 − 4 &times; 4)) + (4 + 4)/4)
:= 5 + (((5 − 5/5)5 + (55 − 5)/5) + 55)
:= (5 − (5 + 5)/5) &times; ((55 + 5)/5 − 55)
:= 55 − (5 &times; 5 &times; 55 + ((5 + 5)/5)5 )
:= 6 &times; 6 &times; 66 − (666 + (6 + 6)/6)
:= 66 + (6 &times; 6 &times; 66 − ((6 &times; 6/(6 + 6))6 ))
:= ((6 − 66)/6) + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= (7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − 7
:= (7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= 8 + ((8/8 + 8 + 8) &times; (((88 + 8)/8) + 88))
:= 8/8 + ((888 − 8 &times; 8) + 888)
:= 8 + ((88/8 + 8) &times; ((8 + 8)/8 + 88))
:= 9 + ((9 + 9) &times; 99 − (((9 + 9)/9) + 9 &times; 9))
:= (9 + 9) &times; 99 + (((99 + 9)/9) − 9 &times; 9)
:= 99 + ((9 &times; (99 + 9 &times; 9)) − 9/9)
&Iuml; 1709 := 1 + (1 + 1 + 1 + 11) &times; (1 + 111+1 )
2
2
&Iuml; 1714 := (1 + 1)11 − (1 + (1 + 1 + 1) &times; 111)
&Iuml; 1719 := 1 + (1 + (((1 + 11)1+1+1 ) − 11))
:= (2 &times; 22)2 − 222
:= (2/2 + 2) &times; (((22 + 2)2 ) − (2/2 + 2))
:= (3 &times; 3 + 3) − ((3 &times; (3 + 3)) + 3/3)
:= (3 &times; 3 + 3)3 − (33/3 + 3)
:= (3 &times; 3 + 3)3 − 3 &times; 3
:= 4/4 + ((4 &times; (444 − 4 &times; 4)) − 4)
:= (4 + 4)/4 + (4 &times; (444 − 4 &times; 4))
:= ((44 − 4) &times; (44 − 4/4)) − 4/4
:= 22 + (((22/2 + 22) + 2) )
3
5
5
5
:= 5 + (((5 − 5/5) + 5 /5) + 55)
:= 5 − (5 &times; (5 &times; 55 + 5) + (55/5))
:= 55 + ((((5 + 5 + 5) &times; 555) − 5)/5)
:= 6 &times; 6 &times; 66 − (666 + 6/6)
:= (((6 + 6)/6)6+6 ) − (6 &times; 6 &times; 66 + 6)
:= (((6 − 66) + 6)/6) + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= 7/7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − 7)
:= (7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − 7/7
:= 77/7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − 7)
:= 8 + ((8 &times; 8 − 8/8) &times; (88/8 + 8 + 8))
:= (((8 + 8)/8) &times; (888 + 8/8)) − 8 &times; 8
:= (8/8 + 8) &times; (8 &times; (8 + 8 + 8) − 8/8)
:= 9 + (((9 − 9/9) + 9) &times; (9/9 + 99))
:= 9 &times; 9 &times; (9 + 9) + (((9 + 9)/9)9−9/9 )
:= 99 + (9 &times; (99 + 9 &times; 9))
&Iuml; 1710 := (11 − 1) &times; (1 + (1 + (1 + 1 + 11)1+1 ))
2
:= 2222 − (2(2/2+2) )
&Iuml; 1715 := (1 + 1)11 − (1 + 1 + 1) &times; 111
:= 2/2 + ((2 &times; 22)2 − 222)
&Iuml; 1720 := 1 + (1 + (1 + (((1 + 11)1+1+1 ) − 11)))
:= (2 − 22) &times; (2 − 2 &times; 2 &times; 22)
:= (3 &times; 3 + 3) − (3 &times; (3 + 3))
:= ((3 − 33)/3) + ((3 &times; 3 + 3)3 − 3)
:= 3 + ((3 &times; 3 + 3)3 − 33/3)
:= 4 &times; 444 − ((44 + 4 + 4)/4)
:= 4 + (4 &times; 444 − (44 + 4)/4)
:= (44 − 4) &times; (44 − 4/4)
:= 5 + ((55 &times; (5 &times; 5 + 5)) + 55)
:= 5 + (((55 &times; (5 &times; 5 + 5)) + 55) + 5)
:= 6 &times; 6 &times; 66 − 666
:= 6 + (6 &times; 6 &times; 66 − (666 + 6/6))
:= ((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 − 6/6)
3
:= 55 − (5 &times; (5 &times; 55 + 5) + 5)
:= (7 − ((7 + 7)/7)) &times; (7 &times; 7 &times; 7 − 7/7)
:= 7 &times; (7 &times; (7 &times; 7 − (7 + 7)))
:= (7 − ((7 + 7)/7)) &times; (7 &times; 7 &times; 7 + 7/7)
:= (88/8 + 8) &times; ((8 + 8)/8 + 88)
:= (8 − 8/8) &times; ((8 + 8) &times; (8 + 8) − (88/8))
:= 8 &times; (8 &times; (8 + 8) + 88) − 8
:= 9 + (9 &times; ((99 + 9 &times; 9) + 9))
:= 9 &times; 9 &times; 9 + (999 − ((99 + 9 + 9)/9))
:= 9/9 + ((9 &times; (99 + 9 &times; 9)) + 99)
&Iuml; 1711 := 1 + ((11 − 1) &times; (1 + (1 + (1 + 1 + 11)1+1 )))
&Iuml; 1716 := 11 &times; ((1 + 11) &times; (1 + 1 + 11))
&Iuml; 1721 := 1 + (1 + (1 + (1 + (((1 + 11)1+1+1 ) − 11))))
:= 222/2 + (2 &times; (22 − 2))2
:= 2 + ((2 &times; 22)2 − 222)
:= (22/2)2 + (2 &times; (22 − 2))2
:= 3/3 + ((3 &times; 3 + 3)3 − (3 &times; (3 + 3)))
:= (3 &times; 3 + 3)3 − (3 &times; 3 + 3)
:= (3 &times; 3 + 3)3 − ((3/3 + 3) + 3)
:= 4 &times; 444 − (44 + 4)/4
:= 4 + (4 &times; (444 − 4 &times; 4))
:= 4/4 + ((44 − 4) &times; (44 − 4/4))
:= (5555 + 5 )/5 − 5 &times; 5
:= (5/5 + 5) &times; (5 &times; 55 + (55/5))
:= 5 + ((5/5 + 5) &times; (5 &times; 55 + (55/5)))
:= 6/6 + (6 &times; 6 &times; 66 − 666)
:= 6 + (6 &times; 6 &times; 66 − 666)
:= (6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6/6 − 6
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − (77/7))
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) − 7/7)
:= 88 + (8888/8 + 8 &times; 8 &times; 8)
:= 8 &times; (8 &times; (8 + 8) + 88) − (88 + 8)/8
:= 8/8 + (8 &times; (8 &times; (8 + 8) + 88) − 8)
:= 9 + ((9 &times; ((99 + 9 &times; 9) + 9)) + 9/9)
:= 9 &times; 9 &times; 9 + (999 − ((99 + 9)/9))
:= 9 + ((9 &times; ((99 + 9 &times; 9) + 9)) + (99/9))
5
&Iuml; 1712 := (1 + 1)11 − ((1 + 1 + 1) &times; (1 + 111))
&Iuml; 1717 := ((1 + 11)1+1+1 ) − 11
&Iuml; 1722 := (1 + (1 + 1 + 11)) &times; (1 + (1 + 111+1 ))
:= (2 &times; 22)2 − (222 + 2)
:= 2 + (((2 &times; 22)2 − 222) + 2/2)
:= (2/2 + 2) &times; (((22 + 2)2 ) − 2)
:= 33/3 + ((3 &times; 3 + 3)3 − 33 )
:= (3 &times; 3 + 3)3 − 33/3
:= (3 &times; 3 + 3)3 − (3 + 3)
:= 4 &times; (444 − 4 &times; 4)
:= 4 + ((4 &times; (444 − 4 &times; 4)) + 4/4)
:= (4 + 4)/4 + ((44 − 4) &times; (44 − 4/4))
:= (55/5 + 5) &times; ((555 + 5)/5 − 5)
:= ((55 &times; 55 + 5555) + 5)/5
:= (5/5 + 5) &times; (((55 + 5)/5) + 5 &times; 55)
:= (6 + 6)/6 + (6 &times; 6 &times; 66 − 666)
:= (6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 66/6
:= (6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6
:= 7 + (77/7 &times; (7/7 + 77 + 77))
:= ((7 + 7)/7) + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= 7 + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= (8 + 8) &times; ((88/8 + 88) + 8)
:= 8 &times; (8 &times; (8 + 8) + 88) − 88/8
:= (8 + 8)/8 + (8 &times; (8 &times; (8 + 8) + 88) − 8)
:= 99/9 + (9 &times; ((99 + 9 &times; 9) + 9))
:= 9 &times; 9 &times; 9 + (999 − (99/9))
:= (9/9 + 9 &times; 9) &times; (((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
&Iuml; 1723 := ((1 + 11)1+1+1 ) − 1 − 1 − 1 − 1 − 1
&Iuml; 1728 := (1 + 11)1+1+1
&Iuml; 1733 := 1 + (1 + (1 + (1 + (1 + ((1 + 11)1+1+1 )))))
:= 2 + ((2 &times; (22 − 2))2 + (22/2)2 )
:= (2/2 + 2) &times; ((22 + 2)2 )
:= 2 + ((2/2 + 2) &times; (((22 + 2)2 ) + 2/2))
:= 3/3 + ((3 &times; 3 + 3)3 − (3 + 3))
:= (3 &times; 3 + 3)3
:= 3 + (((3 &times; 3 + 3)3 − 3/3) + 3)
:= 44/4 + (4 &times; (444 − 4 &times; 4))
:= 4 &times; (4 &times; 44 + 44 )
:= 4/4 + (4 &times; 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 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6)
:= 6 &times; 6 &times; (6 &times; 6 + 6 + 6)
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6/6)
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + 7/7)
:= ((77 + 7)/7)(7+7+7)/7
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + (77/7))
:= 8 + ((8 − 8/8) &times; ((8 + 8) &times; (8 + 8) − (88/8)))
:= 8 &times; (8 &times; (8 + 8) + 88)
:= 8 &times; 8 &times; 8 + 88/8 &times; 888/8
:= 9 &times; 9 &times; 9 + 999
:= (9 + 9) &times; 99 + ((9 − 9 &times; 99)/(9 + 9))
9−9/9
:= 9 + ((((9 + 9)/9)
) + 9 &times; 9 &times; (9 + 9))
&Iuml; 1724 := ((1 + 11)1+1+1 ) − 1 − 1 − 1 − 1
&Iuml; 1734 := ((1 + 1) &times; (1 + 1 + 1)) + ((1 + 11)1+1+1 )
&Iuml; 1729 := 1 + ((1 + 11)1+1+1 )
:= 2 + ((2/2 + 2) &times; (((22 + 2)2 ) − 2))
:= 2/2 + ((2/2 + 2) &times; ((22 + 2)2 ))
:= (2/2 + 2) &times; (((22 + 2)2 ) + 2)
:= (3 &times; 3 + 3)3 − (3/3 + 3)
:= 3/3 + (3 &times; 3 + 3)3
:= 3 + ((3 &times; 3 + 3)3 + 3)
:= (4 &times; (4 &times; 44 + 44 )) − 4
:= 4/4 + (4 &times; (4 &times; 44 + 44 ))
:= (4 + 4)/4 + (4 &times; 444 − 44)
:= 5 − (5 &times; (5 &times; 55 + 5) + 5/5)
:= 5 + (55 &times; (5 &times; 5 − 5) + (55 − 5)/5)
:= ((5555 − (5 + 5)) + 55 )/5
:= (6 + 6)/6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6)
:= 6/6 + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= 6 + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + ((7 + 7)/7))
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + 7)
:= (7 &times; (7 &times; 7 &times; 7 − 77)) − ((7 + 7)/7)7
:= 8 &times; (8 &times; (8 + 8) + 88) − (8/((8 + 8)/8))
:= 8/8 + 8 &times; (8 &times; (8 + 8) + 88)
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) − ((8 + 8)/8))
:= 9/9 + (9 &times; 9 &times; 9 + 999)
:= ((9 − 9/9) + 9) &times; (999/9 − 9)
5
99/9
:= (((9 + 9)/9)
) − (9 + 9) &times; (9 + 9)
&Iuml; 1725 := ((1 + 11)1+1+1 ) − 1 − 1 − 1
&Iuml; 1730 := 1 + (1 + ((1 + 11)1+1+1 ))
&Iuml; 1735 := 1 + (((1 + 1) &times; (1 + 1 + 1)) + ((1 + 11)1+1+1 ))
:= (2/2 + 2) &times; (((22 + 2)2 ) − 2/2)
:= 2 + ((2/2 + 2) &times; ((22 + 2)2 ))
:= 2/2 + ((2/2 + 2) &times; (((22 + 2)2 ) + 2))
:= (3 &times; 3 + 3)3 − 3
:= 3 + ((3 &times; 3 + 3)3 − 3/3)
:= 3 + (((3 &times; 3 + 3)3 + 3/3) + 3)
:= 4/4 + ((4 &times; (4 &times; 44 + 44 )) − 4)
:= (4 + 4)/4 + (4 &times; (4 &times; 44 + 44 ))
:= 4 + 4 &times; 444 − (44 + 4/4)
:= 5 + (55 − 5 &times; (5 &times; 55 + 5))
:= (55 &times; ((5 + 5)/5)5 ) − 5 &times; 5
:= (6 + 6)/6 + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) + 6/6)
:= 77 + ((((7 + 7 + 7)/7) ) − 7 &times; 77)
:= 7 + (((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + 7/7) + 7)
:= 7 + (((77 + 7)/7)(7+7+7)/7 )
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) − (88/8))
:= (8 + 8)/8 + 8 &times; (8 &times; (8 + 8) + 88)
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) − 8/8)
:= 9 &times; 9 &times; 9 + (999 − ((9 + 9 + 9)/9))
:= 9 &times; 9 &times; 9 + (((9 + 9)/9) + 999)
:= 9 + ((9 &times; 9 &times; 9 − ((9 + 9)/9)) + 999)
&Iuml; 1731 := 1 + (1 + (1 + ((1 + 11)1+1+1 )))
&Iuml; 1736 := 11 + (((1 + 11)1+1+1 ) − (1 + 1 + 1))
5
:= 5 − 5 &times; (5 &times; 55 + 5)
:= (6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6 &times; 6/(6 + 6)
7
&Iuml; 1726 := ((1 + 11)1+1+1 ) − 1 − 1
:= ((2/2 + 2) &times; ((22 + 2)2 )) − 2
:= (2/2 + 2) &times; (((22 + 2)2 ) + 2/2)
:= 2222 − (222 + 2)
:= 3/3 + ((3 &times; 3 + 3)3 − 3)
:= 3 + (3 &times; 3 + 3)3
:= 3 &times; 3 + ((3 &times; 3 + 3)3 − 3/3)
:= (4 &times; (4 &times; 44 + 44 )) − (4 + 4)/4
:= 4 &times; 444 − (44 + 4/4)
:= 4 + (4 &times; 444 − 44)
5
:= (5555 + 55 )/5
5
:= 5 + (5/5 − 5 &times; (5 &times; 55 + 5))
:= (5555 + 5 )/5 − 5
:= (6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= ((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 + 6/6)
:= 77/7 + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= 7 + (((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + ((7 + 7)/7)) + 7)
:= 7 + (((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + 7) + 7)
:= 8 &times; (8 &times; (8 + 8) + 88) − (8 + 8)/8
:= 88/8 + (8 &times; (8 &times; (8 + 8) + 88) − 8)
:= 8 + 8 &times; (8 &times; (8 + 8) + 88)
:= 9 &times; 9 &times; 9 + (999 − ((9 + 9)/9))
:= 999/9 + (9 &times; (99 + 9 &times; 9))
:= 9 + ((999 − 9/9) + 9 &times; 9 &times; 9)
&Iuml; 1727 := ((1 + 11)1+1+1 ) − 1
&Iuml; 1732 := 1 + (1 + (1 + (1 + ((1 + 11)1+1+1 ))))
&Iuml; 1737 := 11 + (((1 + 11)1+1+1 ) − (1 + 1))
:= ((2/2 + 2) &times; ((22 + 2)2 )) − 2/2
:= 2 &times; ((2 &times; 2 &times; 222) − 22)
:= 2222 − (222 + 2/2)
:= (3 &times; 3 + 3)3 − 3/3
:= 3 + ((3 &times; 3 + 3)3 + 3/3)
:= 3 &times; 3 + (3 &times; 3 + 3)3
:= 4 &times; 444 − 44
:= 4 + ((4 &times; 444 − 44) + 4/4)
4
:= (4 &times; (4 &times; 44 + 4 )) − 4/4
5
:= 55 + ((5 + 5)/5 − 5 &times; (5 &times; 55 + 5))
:= (((5555 + 5 ) + 5)/5) − 5
:= ((5555 + 55 ) + 5)/5
:= (6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − 6/6
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) + (6 &times; 6/(6 + 6)))
7
:= 7 + (((((7 + 7 + 7)/7) ) − 7 &times; 77) + 77)
:= 7 + ((((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + 7/7) + 7) + 7)
:= 8 &times; (8 &times; (8 + 8) + 88) − 8/8
:= (8/((8 + 8)/8)) + 8 &times; (8 &times; (8 + 8) + 88)
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) + 8/8)
:= 9 &times; 9 &times; 9 + (999 − 9/9)
:= (9 + 9) &times; 99 − ((9 &times; 99 + 9)/(9 + 9))
:= 9 + (9 &times; 9 &times; 9 + 999)
:= 7 + ((7 − ((7 + 7)/7)) &times; (7 &times; 7 &times; 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
&Iuml; 1743 := 1 + (1 + (1 + (1 + (11 + ((1 + 11)1+1+1 )))))
&Iuml; 1738 := 11 + (((1 + 11)1+1+1 ) − 1)
&Iuml; 1748 := ((1 + 1) &times; (11 − 1)) + ((1 + 11)1+1+1 )
:= 2222 − 222
:= 2/2 + ((2 &times; 22 − 2)2 − 22)
:= (2 &times; 22 − 2)2 − 22+2
:= 3 &times; 3 + ((3 &times; 3 + 3)3 + 3/3)
:= 3 + (((3 &times; 3 + 3)3 + 3 &times; 3) + 3)
:= 3 &times; 3 + ((3 &times; 3 + 3)3 + 33/3)
:= 4 + ((4 &times; 444 − 44) + (4 + 4)/4)
:= (4 &times; (444 − (4 + 4))) − 4/4
:= 4 + (4 &times; (444 − (4 + 4)))
:= (((5555 + 55 ) + 5) + 5)/5
:= 5 + ((((5555 + 55 ) + 5) + 5)/5)
:= 55 − (5 &times; 5 &times; 55 + ((5 + 5)/5))
:= ((66 − 6)/6) + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= (6/6 + 6) &times; (((6 &times; 6/(6 + 6))6−6/6 ) + 6)
:= 6 + ((((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 + 6/6)) + 6)
:= 77/7 &times; (7 &times; (7 + 7 + 7) + (77/7))
:= 7 &times; ((((7 + 7)/7)7+7/7 ) − 7)
:= (77 − 7/7) &times; ((((7 + 7)/7 + 7) + 7) + 7)
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) + ((8 + 8)/8))
:= 8 + ((8 &times; (8 &times; (8 + 8) + 88) − 8/8) + 8)
:= 8 + ((88 − 8/8) &times; ((88 + 8)/8 + 8))
:= 9 + ((9 &times; 9 &times; 9 + 999) + 9/9)
:= 9 + (((9 − 9/9) + 9) &times; (999/9 − 9))
:= (9/9 + 9 + 9) &times; ((99/9) + 9 &times; 9)
&Iuml; 1739 := 11 + ((1 + 11)1+1+1 )
&Iuml; 1744 := (111 − 1 − 1) &times; (1 + 1)1+1+1+1
&Iuml; 1749 := 11 + (11 + (((1 + 11)1+1+1 ) − 1))
:= 2/2 + (2222 − 222 )
:= 2 + ((2 &times; 22 − 2)2 − 22)
:= 2/2 + ((2 &times; 22 − 2)2 − 22+2 )
:= 33/3 + (3 &times; 3 + 3)3
:= 33 + ((3 &times; 3 + 3)3 − 33/3)
:= 3 + ((3 &times; 3 + 3)3 + (3 &times; (3 + 3)))
:= 44/4 + (4 &times; (4 &times; 44 + 44 ))
:= 4 &times; (444 − (4 + 4))
:= (4 &times; (444 − 4)) − 44/4
:= 5 − (5 &times; 5 &times; 55 + (55/5))
:= 55 − ((5 &times; 5 &times; 55 + 5/5) + 5)
:= 55 − (5 &times; 5 &times; 55 + 5/5)
:= 66/6 + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= ((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 + (6 + 6)/6)
:= 66/6 &times; ((666/6 + 6 &times; 6 + 6) + 6)
5
7+7/7
:= 77/7 &times; ((777 − 7)/7 + 7 &times; 7)
:= ((77/7 + 7) &times; (7 &times; (7 + 7) − 7/7)) − 7
:= 7/7 + (7 &times; ((((7 + 7)/7)
:= 88/8 + 8 &times; (8 &times; (8 + 8) + 88)
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) + 8)
:= 88/8 &times; (((88 − 8/8) + 8 &times; 8) + 8)
:= 9 &times; 9 &times; 9 + (99/9 + 999)
:= 9 &times; 9 &times; 9 + ((((9 + 9)/9)9/9+9 ) − 9)
:= 99/9 &times; (9 &times; (9 + 9) − ((9 + 9 + 9)/9))
&Iuml; 1740 := 1 + (11 + ((1 + 11)1+1+1 ))
) − 7))
&Iuml; 1745 := 1 + ((111 − 1 − 1) &times; (1 + 1)1+1+1+1 )
&Iuml; 1750 := 11 + (11 + ((1 + 11)1+1+1 ))
:= 2 + (2222 − 222 )
:= 2 + (((2 &times; 22 − 2)2 − 22) + 2/2)
:= 2 + ((2 &times; 22 − 2)2 − 22+2 )
:= 3 + ((3 &times; 3 + 3)3 + 3 &times; 3)
:= 3 + (((3 &times; 3 + 3)3 + 33/3) + 3)
:= 33 + ((3 &times; 3 + 3)3 − 33/3)
:= (4 &times; (444 − (4 + 4))) − 4
:= 4/4 + (4 &times; (444 − (4 + 4)))
:= (4 − 44)/4 + (4 &times; (444 − 4))
5
:= 5 − ((5 &times; 5 &times; 55 + 5) + 5)
:= 5 − (5 &times; 5 &times; 55 + 5)
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) + 6)
:= 6 + ((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) + (66/6))
5
:= 55 − 5 &times; 5 &times; 55
:= (6/6 + 6) &times; (6 &times; (6 &times; 6 + 6) − ((6 + 6)/6))
:= 7 + (((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + (77/7)) + 7)
:= ((7 + 7)/7) + (77 &times; (7 + 7 + 7))
:= (7 + 7) &times; (777/7 + 7 + 7)
:= (88 − 8/8) &times; ((88 + 8)/8 + 8)
:= 8 + ((8 &times; (8 &times; (8 + 8) + 88) + 8/8) + 8)
:= 8 &times; (88 − 8) + (8888 − 8)/8
:= 9 + ((9 &times; (99 + 9 &times; 9)) + 999/9)
:= ((9 + 9) &times; (99 − ((9 + 9)/9))) − 9/9
:= 9 &times; (9 &times; 9 − 9) + (9999/9 − 9)
&Iuml; 1741 := 1 + (1 + (11 + ((1 + 11)1+1+1 )))
7
&Iuml; 1746 := ((1 + 11)1+1+1 ) + (1 + 1) &times; (11 − 1 − 1)
2
&Iuml; 1751 := 1 + (11 + (11 + ((1 + 11)1+1+1 )))
:= 2 + (((2 &times; 22 − 2)2 − 22) + 2)
:= (2 &times; 22 − 2)2 − (22/2 + 2)
:= 3 + (((3 &times; 3 + 3) + 3/3) + 3 &times; 3)
:= (3 &times; (3 + 3)) + (3 &times; 3 + 3)3
:= 33 + ((3 &times; 3 + 3)3 − (3/3 + 3))
:= 4/4 + ((4 &times; (444 − (4 + 4))) − 4)
:= (4 + 4)/4 + (4 &times; (444 − (4 + 4)))
:= (4 &times; (444 − 4)) − ((4/4 + 4) + 4)
:= (2 &times; 22 − 2) − (22 + 2/2)
3
:= 5 + (5555 + 55 )/5
:= 5 + ((5555 + 5 )/5 + 5)
:= 55 + (5/5 − 5 &times; 5 &times; 55)
:= 666 + (6 &times; 6 &times; (6 &times; 6 − 6))
:= (66/6 + 6) &times; ((6 &times; 6 + 66) + 6/6)
:= 77 + ((7 − 7/7 + 7) &times; ((7 + 7)/7) )
:= (77/7 + 7) &times; (7 &times; (7 + 7) − 7/7)
:= 7/7 + ((7 + 7) &times; (777/7 + 7 + 7))
:= 88 + ((88/8 + 8) &times; (88 − 8/8))
:= ((8 + 8)/8) &times; ((888 − (8 + 8)) + 8/8)
:= 8 &times; (88 − 8) + 8888/8
:= 9 &times; 9 &times; 9 + (9999/9 − 99)
:= (9 + 9) &times; (99 − ((9 + 9)/9))
:= ((9 − 9/9) + 9) &times; (((999 + 9)/9) − 9)
5
:= 6 + (((6 &times; 6 &times; (6 &times; 6 + 6 + 6)) + 6/6) + 6)
7
&Iuml; 1742 := 1 + (1 + (1 + (11 + ((1 + 11)1+1+1 ))))
:= (2 &times; 22 − 2)2 − 22
&Iuml; 1747 := ((1 + 1) &times; (11 − 1)) + (((1 + 11)1+1+1 ) − 1)
:= (2 &times; 22 − 2)2 − (22+2 + 2/2)
&Iuml; 1752 := (1 + 11) &times; (1 + (1 + (1 + 11)1+1 ))
:= 2 &times; (2 &times; ((2 &times; (222 − 2)) − 2))
:= 3 + ((3 &times; 3 + 3) + 33/3)
:= 3/3 + ((3 &times; 3 + 3) + (3 &times; (3 + 3)))
:= 33 + ((3 &times; 3 + 3)3 − 3)
:= (4 &times; (444 − (4 + 4))) − (4 + 4)/4
:= 4 + ((4 &times; (444 − (4 + 4))) − 4/4)
:= (4 &times; (444 − 4)) − 4 − 4
3
3
:= 5 + (((5555 + 5 ) + 5)/5)
:= ((5555 + 55) + 5 )/5
:= 55 + ((5 + 5)/5 − 5 &times; 5 &times; 55)
:= 6 + (((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 + 6/6))
:= 66 + (((6 &times; 6 − 6/6) + 6)(6+6)/6 )
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6 − 6
:= 7/7 + ((77/7 + 7) &times; (7 &times; (7 + 7) − 7/7))
:= 7 + ((77 &times; (7 + 7 + 7)) + ((7 + 7)/7)7 )
:= ((8 + 8)/8) &times; (888 − (8/8 + 8 + 8))
:= 8 + (8 &times; (8 &times; (8 + 8) + 88) + (88/8))
:= 88 + ((8 + 8) &times; (88 + 8 + 8))
:= ((9 + 9)/9) &times; (9 &times; 99 − (99/9 + 9))
:= 9/9 + ((9 + 9) &times; (99 − ((9 + 9)/9)))
:= 9 &times; 9 + ((9 + 9) &times; 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
&Iuml; 1753 := 1 + ((1 + 11) &times; (1 + (1 + (1 + 11)1+1 )))
&Iuml; 1758 := (1 + 1) &times; ((11 − 1)1+1+1 − 111+1 )
&Iuml; 1763 := (((1 + 1) &times; (11 + (11 − 1)))1+1 ) − 1
:= (2 &times; 22 − 2)2 − 22/2
:= (2 &times; 22 − 2)2 − (2 + 2 + 2)
:= (2 &times; 22 − 2)2 − 2/2
:= 33 + (((3 &times; 3 + 3)3 − 3) + 3/3)
:= 3 + ((3 &times; 3 + 3)3 + 33 )
:= 3 + (((3 &times; 3 + 3)3 − 3/3) + 33)
:= 4 + ((4 &times; (444 − 4)) − 44/4)
:= (4 &times; (444 − 4)) − (4 + 4)/4
:= 4 + ((4 &times; (444 − 4)) − 4/4)
:= 5 + (55 − (5 &times; 5 &times; 55 + ((5 + 5)/5)))
:= (55 &times; ((5 + 5)/5)5 ) − (5 + 5)/5
:= 5 + ((55 &times; ((5 + 5)/5)5 ) − ((5 + 5)/5))
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 66/6
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6/6
:= ((7 + 7) &times; (77 + 7 &times; 7)) − 77/7
:= 7/7 + (((7 + 7) &times; (77 + 7 &times; 7)) − 7)
:= ((7 + 7) &times; (77 + 7 &times; 7)) − 7/7
:= 8/8 + (((8 + 8) &times; (88 + 8 + 8)) + 88)
:= ((8 + 8)/8) &times; (888 − 8/8 − 8)
:= 8 + ((8/8 + 8 &times; 8) &times; (88/8 + 8 + 8))
:= ((9 + 9)/9) &times; (9 &times; 99 − ((99 + 9)/9))
:= (9 + 9) &times; 99 − (9/9 + 9 + 9)
9/9+9
:= 9 &times; 9 &times; 9 + (((9 + 9)/9)
)
&Iuml; 1754 := 1 + (1 + ((1 + 11) &times; (1 + (1 + (1 + 11)1+1 ))))
&Iuml; 1759 := ((111 − 1) &times; (1 + 1)1+1+1+1 ) − 1
&Iuml; 1764 := ((1 + 1) &times; (11 + (11 − 1)))1+1
:= (2 &times; (2 &times; 2 &times; 222)) − 22
:= (2 &times; 22 − 2)2 − (2/2 + 2 + 2)
:= ((2 &times; 22) − 2)2
:= 33 + ((3 &times; 3 + 3)3 − 3/3)
:= 3 + (((3 &times; 3 + 3)3 + 33 ) + 3/3)
:= 3 + ((3 &times; 3 + 3)3 + 33)
:= 4 &times; 444 − (44/((4 + 4)/4))
:= (4 &times; (444 − 4)) − 4/4
:= 4 + (4 &times; (444 − 4))
5
:= 5 + (5 − (5 &times; 5 &times; 55 + 5/5))
:= (55 &times; ((5 + 5)/5) ) − 5/5
:= 5 + ((55 &times; ((5 + 5)/5)5 ) − 5/5)
:= ((6 − 66)/6) + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= 6/6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6)
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= ((7 − 77)/7) + ((7 + 7) &times; (77 + 7 &times; 7))
:= ((7 + 7)/7) + (((7 + 7) &times; (77 + 7 &times; 7)) − 7)
:= (7 + 7) &times; (77 + 7 &times; 7)
:= ((8 + 8)/8) &times; (888 − 88/8)
:= (88 &times; ((88 + 8)/8 + 8)) − 8/8
:= 888 + (888 − ((88 + 8)/8))
:= (9 + 9) &times; 99 − ((9/9 + 9 + 9) + 9)
:= 9 &times; (9 &times; 9 − 9) + 9999/9
:= (9 + 9) &times; (99 − 9/9)
5
&Iuml; 1755 := (11 − 1 − 1) &times; ((1 + 1 + 1 + 11)1+1 − 1)
&Iuml; 1765 := 1 + (((1 + 1) &times; (11 + (11 − 1)))1+1 )
&Iuml; 1760 := (111 − 1) &times; (1 + 1)1+1+1+1
:= 2 + ((2 &times; 22 − 2)2 − 22/2)
:= 2 &times; (2 &times; (2 &times; (222 − 2)))
:= 2/2 + (2 &times; 22 − 2)2
:= 33 + (3 &times; 3 + 3)3
:= 33 + ((3 &times; 3 + 3)3 − 3/3)
:= 3 + (((3 &times; 3 + 3)3 + 33) + 3/3)
:= (4 &times; (444 − 4)) − (4/4 + 4)
:= 4 &times; (444 − 4)
:= 4 &times; 444 − 44/4
5
5
:= 55 &times; ((5 + 5)/5)
:= 5 + (55 &times; ((5 + 5)/5)5 )
:= (6 + 6)/6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6)
:= 6/6 + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= (7 − 7/7 + 7) &times; (((7 + 7)/7) + 7)
:= 77/7 &times; (777/7 + 7 &times; 7)
:= 7/7 + ((7 + 7) &times; (77 + 7 &times; 7))
:= (8/8 + 8 &times; 8) &times; (88/8 + 8 + 8)
:= 88 &times; ((88 + 8)/8 + 8)
:= 888 + (888 − 88/8)
:= (9 + 9) &times; 99 − (9 + 9 + 9)
:= 99/9 &times; (9 &times; (9 + 9) − ((9 + 9)/9))
:= 9/9 + ((9 + 9) &times; (99 − 9/9))
:= 5 + (5 − 5 &times; 5 &times; 55)
:= (6/6 − 66) &times; (6 − (66 &times; 6/(6 + 6)))
7
&Iuml; 1756 := 1 + (11 − 1 − 1) &times; ((1 + 1 + 1 + 11)1+1 − 1)
&Iuml; 1761 := 1 + ((111 − 1) &times; (1 + 1)1+1+1+1 )
&Iuml; 1766 := 1 + (1 + (((1 + 1) &times; (11 + (11 − 1)))1+1 ))
:= 2 &times; (2 &times; (((22 − 2/2)2 ) − 2))
:= (2 &times; 22 − 2)2 − 2/2 − 2
:= 2 + (2 &times; 22 − 2)2
:= 33 + ((3 &times; 3 + 3)3 + 3/3)
:= 33 + (3 &times; 3 + 3)3
:= 33 + ((3 &times; 3 + 3)3 + 33/3)
:= (4 &times; (444 − 4)) − 4
:= 4/4 + (4 &times; (444 − 4))
:= (4 − 44)/4 + 4 &times; 444
:= 5 + ((5 − 5 &times; 5 &times; 55) + 5/5)
:= 5/5 + (55 &times; ((5 + 5)/5)5 )
:= 5 + ((55 &times; ((5 + 5)/5)5 ) + 5/5)
:= ((6 + 6)/6)6 + (6 &times; (6 &times; 6 &times; 6 + 66))
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6 &times; 6/(6 + 6)
:= (6 + 6)/6 + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= ((7 + 7) &times; (77 + 7 &times; 7)) − (7/7 + 7)
:= ((7 + 7 + 7)/7) &times; (7 &times; (77 + 7) − 7/7)
:= ((7 + 7)/7) + ((7 + 7) &times; (77 + 7 &times; 7))
:= ((8 + 8)/8) &times; ((8 − 88)/8 + 888)
:= 8/8 + (88 &times; ((88 + 8)/8 + 8))
:= (((8 + 8)/8) &times; (888 − 8/8)) − 8
:= 9/9 + ((9 + 9) &times; 99 − (9 + 9 + 9))
:= (9 + 9) &times; 99 − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + ((9 + 9) &times; (99 − 9/9))
5
&Iuml; 1757 := (1 + 1)11 − (1 + ((1 + 1) &times; (1 + (1 + 11)1+1 )))
&Iuml; 1762 := (((1 + 1) &times; (11 + (11 − 1)))1+1 ) − 1 − 1
&Iuml; 1767 := 1 + (1 + (1 + (((1 + 1) &times; (11 + (11 − 1)))1+1 )))
:= 2 + (((2 &times; 22 − 2)2 − 22/2) + 2)
:= (2 &times; 22 − 2)2 − 2
:= 2 + ((2 &times; 22 − 2)2 + 2/2)
:= 3 + (((3 &times; 3 + 3)3 − 3/3) + 33 )
:= 3/3 + ((3 &times; 3 + 3)3 + 33)
:= 3 + (((3 &times; 3 + 3)3 + 33) + 3)
:= (4 + 4)/4 + (4 &times; (444 − 4))
:= 4/4 + ((4 &times; (444 − 4)) − 4)
:= 4 &times; 444 − ((4/4 + 4) + 4)
5
:= 5 + (((5 + 5)/5 − 5 &times; 5 &times; 55) + 5 )
:= (5 + 5)/5 + (55 &times; ((5 + 5)/5) )
:= 5 + ((55 &times; ((5 + 5)/5)5 ) + ((5 + 5)/5))
:= (6/6 + 6) &times; (6 &times; (6 &times; 6 + 6) − 6/6)
:= (6 &times; 6 + 6) &times; (6 &times; 6 + 6) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
5
:= ((7 + 7) &times; (77 + 7 &times; 7)) − 7
:= ((7 + 7) &times; (77 + 7 &times; 7)) − (7 + 7)/7
:= 7 + (77/7 &times; (777/7 + 7 &times; 7))
:= 888 + (888 − (88/8 + 8))
:= ((8 + 8)/8) &times; ((888 − 8) + 8/8)
:= 888 + (888 − 8/8 − 8)
:= 9 + ((9/9 + 9 + 9) &times; ((99/9) + 9 &times; 9))
:= (9 + 9) &times; 99 − (99/9 + 9)
:= (9 + 9) &times; 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
&Iuml; 1768 := (1 + 1)1+1+1 &times; ((1 + 1) &times; 111 − 1)
&Iuml; 1773 := (1 + 1)11 − (11 &times; (1 + ((1 + 1) &times; (1 + 11))))
&Iuml; 1778 := 1 + (1 + (111 &times; (1 + 1)1+1+1+1 ))
:= 2 + ((2 &times; 22 − 2)2 + 2)
:= 2/2 + (2 &times; ((2 &times; 2 &times; 222) − 2))
:= 2 + (2 &times; (2 &times; 2 &times; 222))
:= 3 + ((((3 &times; 3 + 3)3 + 33) + 3/3) + 3)
:= 3 &times; (3 &times; 33 &times; (3 + 3) − 3)
:= (3 &times; 3 &times; 33 &times; (3 + 3)) − (3/3 + 3)
:= 4 &times; 444 − 4 − 4
:= 4/4 + (4 &times; 444 − 4)
:= (4 + 4)/4 + 4 &times; 444
55/5
:= (((5 + 5)/5)
:= (((5 + 5)/5)55/5 ) − (5 &times; 55 + 5)
) − 5 &times; 55
:= 5 + ((((5 + 5)/5)55/5 ) − 5 &times; 55)
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) − ((6 + 6)/6))
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + (6 &times; 6/(6 + 6)))
:= 666 + (6666 + 6)/6
:= 77/7 + (((7 + 7) &times; (77 + 7 &times; 7)) − 7)
:= 7 + (((7 + 7) &times; (77 + 7 &times; 7)) + ((7 + 7)/7))
:= 7 + (((7 + 7) &times; (77 + 7 &times; 7)) + 7)
:= 888 + 888 − 8
:= 8 + ((888 − 88/8) + 888)
:= ((8 + 8)/8) &times; (888 + 8/8)
:= 9 + (9999/9 + 9 &times; (9 &times; 9 − 9))
:= (9 + 9) &times; 99 − 9
:= ((9 + 9)/9) &times; (9 &times; 99 − ((9 + 9)/9))
&Iuml; 1769 := 1 + ((1 + 1)1+1+1 &times; ((1 + 1) &times; 111 − 1))
&Iuml; 1774 := (1 + 1) &times; ((111 &times; (1 + 1)1+1+1 ) − 1)
&Iuml; 1779 := 1 + (1 + (1 + (111 &times; (1 + 1)1+1+1+1 )))
:= 2 + (((2 &times; 22 − 2)2 + 2/2) + 2)
:= (2 &times; (2 &times; 2 &times; 222)) − 2
:= 2 + ((2 &times; (2 &times; 2 &times; 222)) + 2/2)
:= 3 + (((3 &times; 3 + 3)3 + 33/3) + 33 )
:= 3/3 + (3 &times; (3 &times; 33 &times; (3 + 3) − 3))
:= (3 &times; 3 &times; 33 &times; (3 + 3)) − 3
:= 4 + (4 &times; 444 − 44/4)
:= 4 &times; 444 − (4 + 4)/4
:= 4 + (4 &times; 444 − 4/4)
:= 5 + (((55 &times; ((5 + 5)/5) ) − 5/5) + 5)
:= 5 + ((5 &times; (5 − 5 &times; 55)) − 5/5)
:= 5 + (((5 &times; (5 − 5 &times; 55)) − 5/5) + 55 )
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6/6)
:= ((66 − 6)/6) + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= 666 + (((6666 + 6) + 6)/6)
:= 7 + (((7 + 7) &times; (77 + 7 &times; 7)) − ((7 + 7)/7))
:= ((77 − 7)/7) + ((7 + 7) &times; (77 + 7 &times; 7))
:= 7 + ((((7 + 7) &times; (77 + 7 &times; 7)) + 7/7) + 7)
:= 8/8 + ((888 − 8) + 888)
:= ((8 + 8)/8) &times; (888 − 8/8)
:= 8/8 + (((8 + 8)/8) &times; (888 + 8/8))
:= (9 + 9) &times; 99 − ((99 + 9 + 9)/9)
:= 9/9 + ((9 + 9) &times; 99 − 9)
:= (9 + 9) &times; 99 − (9 + 9 + 9)/9
5
&Iuml; 1770 := (11 − 1) &times; (1 + (11 &times; (1 + 1)1+1+1+1 ))
5
&Iuml; 1775 := (111 &times; (1 + 1)1+1+1+1 ) − 1
:= 22/2 + (2 &times; 22 − 2)2
:= 2 + (((2 &times; 22 − 2)2 + 2) + 2)
&Iuml; 1780 := (1 + 1) &times; ((11 &times; (11 − 1 − 1)1+1 ) − 1)
:= 2 &times; ((2 &times; 2 &times; 222) + 2)
:= 3 &times; 3 + ((3 &times; 3 + 3) + 33)
:= (((3 + 3) &times; ((33 &times; 3 ) − 3)) − 3)/3
:= 3/3 + ((3 &times; 3 &times; 33 &times; (3 + 3)) − 3)
:= 4 &times; 444 − ((4 + 4)/4 + 4)
:= 4 &times; 444 − 4/4
:= 4 + 4 &times; 444
:= 5 + ((55 &times; ((5 + 5)/5)5 ) + 5)
:= 5 &times; ((5 &times; (55 + 5)) + 55)
:= 5 + ((5 &times; (5 − 5 &times; 55)) + 55 )
:= 6 + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= 66/6 + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + ((66 − 6)/6))
:= 7 + (((7 + 7) &times; (77 + 7 &times; 7)) − 7/7)
:= 77/7 + ((7 + 7) &times; (77 + 7 &times; 7))
:= ((7 + 7) &times; ((7 + 7)/7)7 ) − (77 + 7)/7
:= (((8 + 8)/8) &times; (888 + 8/8)) − 8
:= 888 + (888 − 8/8)
:= ((8 + 8)/8) &times; (888 + ((8 + 8)/8))
:= (9 + 9) &times; 99 − (99 + 9)/9
:= (9 + 9)/9 + ((9 + 9) &times; 99 − 9)
:= (9 + 9) &times; 99 − (9 + 9)/9
3
3
&Iuml; 1771 := 11 &times; (((1 + 1) &times; (11 − 1 − 1)1+1 ) − 1)
&Iuml; 1776 := 111 &times; (1 + 1)1+1+1+1
&Iuml; 1781 := ((1 + 1) &times; (11 &times; (11 − 1 − 1)1+1 )) − 1
:= 2 &times; (2 &times; 2 &times; 222)
:= 2/2 + (2 &times; ((2 &times; 2 &times; 222) + 2))
:= 33/3 &times; ((3 + 3) &times; 3 − 3/3)
:= 3 + (3 &times; (3 &times; 33 &times; (3 + 3) − 3))
:= (3 &times; 3 &times; 33 &times; (3 + 3)) − 3/3
:= 4 &times; 444 − (4/4 + 4)
:= 4 &times; 444
:= 4 + (4 &times; 444 + 4/4)
:= 55/5 + (55 &times; ((5 + 5)/5)5 )
:= (55/5 + 5) &times; 555/5
:= 5 + ((55/5 + 5) &times; 555/5)
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + 6/6)
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + 6)
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + (66/6))
:= 7 + ((7 + 7) &times; (77 + 7 &times; 7))
:= 777/7 &times; (((7 + 7)/7 + 7) + 7)
:= ((7 + 7) &times; ((7 + 7)/7)7 ) − 77/7
:= 88/8 + (88 &times; ((88 + 8)/8 + 8))
:= (8 + 8) &times; 888/8
:= (8 &times; ((8 &times; (8 + 8) + 88) + 8)) − 88/8
:= (9 + 9) &times; 99 − 99/9
:= (9 + 9) &times; 99 + (((9 + 9 + 9)/9) − 9)
:= (9 + 9) &times; 99 − 9/9
:= (2 &times; ((2 &times; 2 &times; 222) − 2)) − 2/2
3
&Iuml; 1772 := 1 + (11 &times; (((1 + 1) &times; (11 − 1 − 1)1+1 ) − 1))
:= 2 &times; ((2 &times; 2 &times; 222) − 2)
&Iuml; 1777 := 1 + (111 &times; (1 + 1)1+1+1+1 )
:= 2/2 + (2 &times; (2 &times; 2 &times; 222))
3
3
:= 33 + ((3 &times; 3 + 3) + 33/3)
&Iuml; 1782 := (1 + 1) &times; (11 &times; (11 − 1 − 1)1+1 )
:= 22 &times; ((2/2 + 2)2+2 )
:= (((3 + 3) &times; ((33 &times; 3 ) − 3)) + 3)/3
:= 3 &times; 3 &times; 33 &times; (3 + 3)
:= 4/4 + 4 &times; 444
:= 4 + (4 &times; 444 + (4 + 4)/4)
:= ((55 + 5)/5) + (55 &times; ((5 + 5)/5) )
:= (555 &times; (55/5 + 5) + 5)/5
:= 5 + ((555 &times; (55/5 + 5) + 5)/5)
:= 6 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + ((6 + 6)/6))
:= 666 + (6666/6)
:= 66 &times; ((66 &times; 6/(6 + 6)) − 6)
:= 7 + (((7 + 7) &times; (77 + 7 &times; 7)) + 7/7)
:= 7 + ((((7 + 7) &times; (77 + 7 &times; 7)) − 7/7) + 7)
:= (77/7 + 7) &times; (7 &times; (7 + 7) + 7/7)
:= ((8 + 8)/8) &times; (888 − ((8 + 8)/8))
:= 8/8 + (888 + 888)
:= 8 + (((8 + 8)/8) &times; (888 − 8/8))
:= (9 + 9) &times; 99 − 9/9 − 9
:= (9 + 9) &times; 99 + ((9 − 99)/(9 + 9))
:= (9 + 9) &times; 99
:= 4 &times; 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
&Iuml; 1783 := 1 + ((1 + 1) &times; (11 &times; (11 − 1 − 1)1+1 ))
&Iuml; 1788 := 1 + (11 + (111 &times; (1 + 1)1+1+1+1 ))
&Iuml; 1793 := 1 + ((1 + 111) &times; (1 + 1)1+1+1+1 )
:= 2/2 + (22 &times; ((2/2 + 2)2+2 ))
:= 2 + ((2 &times; 22 − 2)2 + 22)
:= 2/2 + (2 &times; (2 &times; (2 &times; (222 + 2))))
:= 3/3 + (3 &times; 3 &times; 33 &times; (3 + 3))
:= 3 + ((3 &times; 3 &times; 33 &times; (3 + 3)) + 3)
:= 33/3 + (3 &times; 3 &times; 33 &times; (3 + 3))
:= 4 + ((4 &times; 444 − 4/4) + 4)
:= (4 &times; (444 + 4)) − 4
:= 4/4 + (4 &times; (444 + 4))
5−(5+5)/5
:= 55 + (((55 + 5)/5)
:= (5/5 + 5) &times; (((5 − (5 + 5)/5) ) + 55)
:= 5/5 + (((5 + 5)/5)5 &times; (55 + 5/5))
:= (6 − 6 &times; 6) &times; (6 − 66) − 6 − 6
:= (6 − 6 &times; 6) &times; (6 − 66) − 6/6 − 6
5
)
:= 6 + ((6666/6) + 666)
:= 7 + (777/7 &times; (((7 + 7)/7 + 7) + 7))
:= 7 + (((7 + 7) &times; ((7 + 7)/7) ) − (77/7))
:= 7/7 + ((7 + 7) &times; ((7 + 7)/7)7 )
:= 8 + ((888 − 8/8) + 888)
:= ((8 &times; 8 &times; (8 &times; 8 − 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 &times; ((8 &times; (8 + 8) + 88) + 8))
:= 9/9 + (9 + 9) &times; 99
:= 9 + ((9 + 9) &times; 99 − ((9 + 9 + 9)/9))
:= 99/9 + (9 + 9) &times; 99
7
&Iuml; 1784 := (1 + 1) &times; (1 + (11 &times; (11 − 1 − 1)1+1 ))
&Iuml; 1789 := ((1 + 1) &times; (((11 − 1) &times; (1 + 1 + 1))1+1 )) − 11
&Iuml; 1794 := 1 + (1 + ((1 + 111) &times; (1 + 1)1+1+1+1 ))
:= 2 &times; (2 &times; (2 &times; 222 + 2))
:= 2 + (((2 &times; 22 − 2)2 + 22) + 2/2)
:= 2 + (2 &times; (2 &times; (2 &times; (222 + 2))))
:= 3 + ((3 &times; 3 &times; 33 &times; (3 + 3)) − 3/3)
:= ((3/3 + 3)3 ) + ((3 &times; 3 + 3)3 − 3)
:= 3 + (3 &times; (3 &times; 33 &times; (3 + 3) + 3))
:= 4 + (4 &times; 444 + 4)
:= 4/4 + ((4 &times; (444 + 4)) − 4)
:= (4 + 4)/4 + (4 &times; (444 + 4))
:= (5 &times; 5 + 5) &times; (55 + 5) − 55/5
:= 55 − ((55/5)5−(5+5)/5 )
:= (6 − 6 &times; 6) &times; (6 − 66) − 66/6
:= (6 − 6 &times; 6) &times; (6 − 66) − 6
:= ((7 + 7) &times; ((7 + 7)/7) ) − (7/7 + 7)
:= 7 + ((77/7 + 7) &times; (7 &times; (7 + 7) + 7/7))
:= ((7 + 7)/7) + ((7 + 7) &times; ((7 + 7)/7)7 )
:= 8 + (888 + 888)
:= 8 + ((8 &times; ((8 &times; (8 + 8) + 88) + 8)) − (88/8))
:= ((8 + 8)/8) &times; (888 + 8/8 + 8)
:= (9 + 9)/9 + (9 + 9) &times; 99
:= 9 + ((9 + 9) &times; 99 − ((9 + 9)/9))
:= (9 + 9) &times; 99 + (99 + 9)/9
:= 5 &times; 5 + ((55 &times; ((5 + 5)/5)5 ) − 5/5)
:= 6 + ((6666 + 6)/6 + 666)
7
&Iuml; 1785 := 1 + ((1 + 1) &times; (1 + (11 &times; (11 − 1 − 1)1+1 )))
&Iuml; 1790 := (1 + 1) &times; (((1 + 111) &times; (1 + 1)1+1+1 ) − 1)
&Iuml; 1795 := (1 + 1)11 − (11 &times; (1 + (11 + 11)))
:= 22 + ((2 &times; 22 − 2)2 − 2/2)
:= (2 &times; (2 &times; (2 &times; (222 + 2)))) − 2
:= 2 + ((2 &times; (2 &times; (2 &times; (222 + 2)))) + 2/2)
:= 3 + (3 &times; 3 &times; 33 &times; (3 + 3))
:= (3 &times; (3 &times; 33 &times; (3 + 3) + 3)) − 3/3
:= 3 + (((3/3 + 3)3 ) + (3 &times; 3 + 3)3 )
:= 4 + ((4 &times; 444 + 4/4) + 4)
:= (4 &times; (444 + 4)) − (4 + 4)/4
:= 4 + ((4 &times; (444 + 4)) − 4/4)
:= 5 &times; 5 + (55 &times; ((5 + 5)/5) )
:= (5 &times; 5 + 5) &times; (55 + 5) − 5 − 5
:= (5 &times; 5 + 5) &times; (55 + 5) − 5
:= (66/6 + 6) &times; (666/6 − 6)
:= (6 − 6/6) &times; (6 &times; (66 − 6) − ((6 + 6)/6))
:= 6/6 + ((6 − 6 &times; 6) &times; (6 − 66) − 6)
5
7
7
:= ((7 + 7) &times; ((7 + 7)/7) ) − 7
:= ((7 + 7) &times; ((7 + 7)/7) ) − (7 + 7)/7
:= (((7 + 7 + 7)/7)7 ) − 7 &times; (7 &times; 7 + 7)
:= 8 + ((888 + 888) + 8/8)
:= ((8 + 8)/8) &times; (888 − 8/8 + 8)
:= 8 + ((888 + 888) + (88/8))
:= (9 + 9) &times; 99 + ((9 + 9 + 9)/9)
:= 9 + ((9 + 9) &times; 99 − 9/9)
:= (9 + 9) &times; 99 + ((99 + 9 + 9)/9)
&Iuml; 1791 := ((1 + 111) &times; (1 + 1)1+1+1+1 ) − 1
&Iuml; 1796 := 1 + ((1 + 1)11 − (11 &times; (1 + (11 + 11))))
&Iuml; 1786 := 11 + ((111 &times; (1 + 1)1+1+1+1 ) − 1)
2
:= (2 &times; (2 &times; (2 &times; (222 + 2)))) − 2/2
:= 2 &times; ((2 &times; (2 &times; (222 + 2))) + 2)
:= 3 + ((3 &times; 3 &times; 33 &times; (3 + 3)) + 3/3)
:= 3 &times; (3 &times; 33 &times; (3 + 3) + 3)
:= 3 + ((3 &times; 3 &times; 33 &times; (3 + 3)) + 33/3)
:= 4 &times; 444 + (44 − 4)/4
:= (4 &times; (444 + 4)) − 4/4
:= 4 + (4 &times; (444 + 4))
:= 22 + (2 &times; 22 − 2)
5
:= 5 + (((55/5 + 5) &times; 555/5) + 5)
:= 55 + (5555 + 5 )/5
:= 5/5 + ((5 &times; 5 + 5) &times; (55 + 5) − 5)
:= (((6 + 6)/6) + 6 &times; 6) &times; (66/6 + 6 &times; 6)
:= 6 + ((66/6 + 6) &times; (666/6 − 6))
:= (6 + 6)/6 + ((6 − 6 &times; 6) &times; (6 − 66) − 6)
:= 7/7 + (((7 + 7) &times; ((7 + 7)/7) ) − 7)
:= ((7 + 7) &times; ((7 + 7)/7)7 ) − 7/7
:= 77/7 + (((7 + 7) &times; ((7 + 7)/7)7 ) − 7)
:= 8 + (((8 + 8)/8) &times; (888 + 8/8))
:= (8/8 + 8) &times; (888/8 + 88)
:= ((8 &times; 8 &times; (8 &times; 8 − 8)) + 8)/((8 + 8)/8)
:= ((9 + 9)/9) &times; (((9 + 9)/9) + 9 &times; 99)
:= 9 + (9 + 9) &times; 99
:= 9 + (((9 &times; 9 + 9)/(9 + 9)) + (9 + 9) &times; 99)
7
&Iuml; 1792 := (1 + 111) &times; (1 + 1)1+1+1+1
&Iuml; 1787 := 11 + (111 &times; (1 + 1)1+1+1+1 )
&Iuml; 1797 := 11 &times; 111 + (((1 + 1) &times; (1 + 11))1+1 )
:= 22 + ((2 &times; 22 − 2)2 + 2/2)
:= 2 &times; (2 &times; (2 &times; (222 + 2)))
:= (((3 + 3) &times; ((33 &times; 33 ) + 3)) − 3)/3
:= ((3/3 + 3)3 ) + (3 &times; 3 + 3)3
:= 44/4 + 4 &times; 444
:= 4 &times; (444 + 4)
:= 4 + ((4 &times; (444 + 4)) + 4/4)
:= (((5 + 5)/5) &times; (55 + 5/5)) − 5
:= ((5 + 5)/5)5 &times; (55 + 5/5)
:= 5 + (((5 + 5)/5)5 &times; (55 + 5/5))
:= (6 − 6 &times; 6) &times; (6 − 66) − (6/6 + 6 + 6)
:= (6/6 + 6) &times; (((6 + 6)/6)6+(6+6)/6 )
:= (6 − 6 &times; 6) &times; (6 − 66) − 6 &times; 6/(6 + 6)
5
7
7
:= 22 + ((2 &times; 22 − 2)2 + 22/2)
:= ((3 + 3) &times; (3 &times; 3 &times; 33 + 3)) − 3
:= ((7 + 7)/7) + (((7 + 7) &times; ((7 + 7)/7) ) − 7)
:= (7 + 7) &times; ((7 + 7)/7)
:= (7 &times; 7 &times; (7 + 7)) + 7777/7
:= 888 + (888 + 88/8)
:= 8 &times; ((8 &times; (8 + 8) + 88) + 8)
:= ((88/8 + 8) &times; (88 − 8/8 + 8)) − 8
:= (9 + 9) &times; 99 + ((9 &times; 9 + 9)/(9 + 9))
:= 9 + ((9 + 9) &times; 99 + 9/9)
:= 9 + (((9 + 9) &times; 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
&Iuml; 1798 := (1 + 1) &times; ((((11 − 1) &times; (1 + 1 + 1))1+1 ) − 1)
&Iuml; 1803 := 1 + ((1 + 1) &times; (1 + (((11 − 1) &times; (1 + 1 + 1))1+1 )))
&Iuml; 1808 := (1 + 1)11 − ((1 + 1) &times; (111+1 − 1))
:= ((2 &times; 22 + 2/2)2 ) − 222
:= 2 &times; (2 &times; (2 &times; ((222 + 2) + 2)))
:= 3 + ((((3/3 + 3) ) + (3 &times; 3 + 3) ) + 3)
:= 3 + ((3 + 3) &times; (3 &times; 3 &times; 33 + 3))
:= 3 &times; 33 + ((3 &times; 3 + 3)3 − 3/3)
:= 4 + ((4 &times; (444 + 4)) + (4 + 4)/4)
:= 44/4 + (4 &times; (444 + 4))
:= 4 &times; (444 + 4 + 4)
:= (5 &times; 5 + 5) &times; (55 + 5) − (5 + 5)/5
:= 5 + ((5 &times; 5 + 5) &times; (55 + 5) − ((5 + 5)/5))
:= (55/5 + 5) &times; (555 + 5 + 5)/5
:= (6 − 6 &times; 6) &times; (6 − 66) − (6 + 6)/6
:= 666/6 + (6 &times; (6 &times; 6 &times; 6 + 66))
:= 6 + ((6 − 6 &times; 6) &times; (6 − 66) + ((6 + 6)/6))
:= 22 + (2 &times; (2 &times; 2 &times; 222))
3
3
:= 7 + (((7 + 7) &times; ((7 + 7)/7) ) − 7/7)
:= 77/7 + ((7 + 7) &times; ((7 + 7)/7) )
:= (7/7 + 7) &times; (((7 + 7)/7)7 + 7 &times; (7 + 7))
:= ((8 + 8)/8) &times; (888 + 88/8)
:= 88/8 + (8 &times; ((8 &times; (8 + 8) + 88) + 8))
:= (8 + 8) &times; ((888 + 8 + 8)/8)
:= 9 + (((9 + 9) &times; 99 − ((9 + 9)/9)) + 9)
:= 9 + ((9 + 9) &times; 99 + ((99 + 9)/9))
:= 9 + ((((9 + 9) &times; 99 − 9/9) + 9) + 9)
7
&Iuml; 1799 := ((1 + 1) &times; (((11 − 1) &times; (1 + 1 + 1))1+1 )) − 1
7
&Iuml; 1804 := (1 + 1) &times; (11 &times; (1 + (11 − 1 − 1)1+1 ))
&Iuml; 1809 := 1 + ((1 + 1)11 − ((1 + 1) &times; (111+1 − 1)))
:= (2 &times; ((2 &times; (2 + 2) + 22)2 )) − 2/2
:= 2 &times; (((2 &times; (2 + 2) + 22)2 ) + 2)
:= 2 + (((2 − 222 )/2) + 222/2 )
:= (3 &times; 3 + 3)3 + (((3 + 3)3 − 3)/3)
:= 3 + (((3 + 3) &times; (3 &times; 3 &times; 33 + 3)) + 3/3)
:= 3 &times; (3 &times; (33 &times; (3 + 3) + 3))
:= ((4 − 4/4) + 4) &times; (4/4 + 4 )
:= 44 + (4 &times; (444 − 4))
:= 4/4 + (4 &times; (444 + 4 + 4))
:= (5 &times; 5 + 5) &times; (55 + 5) − 5/5
:= 5 + ((5 &times; 5 + 5) &times; (55 + 5) − 5/5)
:= 5 + (((5 &times; 5 + 5) &times; (55 + 5) − 5/5) + 5)
:= (6 − 6 &times; 6) &times; (6 − 66) − 6/6
:= 6 + ((6 − 6 &times; 6) &times; (6 − 66) − ((6 + 6)/6))
:= (66 + 6/6) &times; ((66 &times; 6/(6 + 6)) − 6)
:= 7 + ((7 + 7) &times; ((7 + 7)/7)7 )
:= 7 + (7777/7 + (7 &times; 7 &times; (7 + 7)))
:= 7 + (((((7 + 7 + 7)/7)7 ) − 7 &times; (7 &times; 7 + 7)) + 7)
:= (8 − 8/8) &times; ((8 + 8) &times; (8 + 8) + 8/8)
:= 8 + (((8 &times; 8 &times; (8 &times; 8 − 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8) &times; ((888 + 8 + 8)/8))
:= 9 + (((9 + 9) &times; 99 − 9/9) + 9)
:= 99/9 &times; (((9 + 9)/9) + 9 &times; (9 + 9))
:= 9 + ((9 + 9) &times; 99 + 9 + 9)
4
&Iuml; 1800 := (1 + 1) &times; (((11 − 1) &times; (1 + 1 + 1))1+1 )
&Iuml; 1805 := (1 + 1)11 − (1 + ((1 + 1) &times; 111+1 ))
22/2
2
− (22 + 2)/2
&Iuml; 1810 := (11 − 1) &times; (1 + (11 + (1 + 1 + 11)1+1 ))
:= 2 &times; ((2 &times; (2 + 2) + 22)2 )
:= 2
:= (3 + 3) &times; (3 &times; 3 &times; 33 + 3)
:= 3 &times; 3 + ((3 &times; 3 + 3) − (3/3 + 3))
:= 3/3 + ((3 &times; 3 + 3)3 + 3 &times; 33 )
:= 4 + ((4 &times; (444 + 4)) + 4)
:= 44 + ((4 &times; (444 − 4)) + 4/4)
:= (4 + 4)/4 + (4 &times; (444 + 4 + 4))
:= (5 &times; 5 + 5) &times; (55 + 5)
:= 5 + (5 &times; 5 + 5) &times; (55 + 5)
:= 5 + ((5 &times; 5 + 5) &times; (55 + 5) + 5)
:= (6 − 6 &times; 6) &times; (6 − 66)
:= 6 + ((6 − 6 &times; 6) &times; (6 − 66) − 6/6)
3
:= 2 + ((2 &times; 22 − 2)2 + 2 &times; 22)
3
7
:= (6 − 6/6) &times; (6 &times; (66 − 6) + ((6 + 6)/6))
:= 7 + (((7 + 7) &times; ((7 + 7)/7) ) + 7/7)
:= 7 + ((((7 + 7) &times; ((7 + 7)/7) ) − 7/7) + 7)
:= 7 + (((7 + 7) &times; ((7 + 7)/7)7 ) + (77/7))
:= 8 + (8 &times; ((8 &times; (8 + 8) + 88) + 8))
:= (88/8 + 8) &times; (88 − 8/8 + 8)
:= ((8 + 8)/8) &times; ((888 + 8/8 + 8) + 8)
:= 9 + ((9 + 9) &times; 99 + 9)
:= (9 + 9) &times; 99 + ((99 + 99 + 9)/9)
:= 9 + ((((9 + 9) &times; 99 + 9/9) + 9) + 9)
7
&Iuml; 1801 := 1 + ((1 + 1) &times; (((11 − 1) &times; (1 + 1 + 1))1+1 ))
:= 2/2 + (2 &times; ((2 &times; (2 + 2) + 22)2 ))
:= 3/3 + ((3 + 3) &times; (3 &times; 3 &times; 33 + 3))
&Iuml; 1806 := (1 + 1)11 − ((1 + 1) &times; 111+1 )
:= 2
22/2
&Iuml; 1811 := 11 + ((1 + 1) &times; (((11 − 1) &times; (1 + 1 + 1))1+1 ))
2
− 22 /2
3
:= 3 &times; (((33/3) ) − 3
3+3
:= 22/2 + (2 &times; ((2 &times; (2 + 2) + 22)2 ))
)
4
:= 3 + (((3 &times; 3 + 3)3 − 3/3) + 3 &times; 33 )
:= 4 + (((4 &times; (444 + 4)) + 4/4) + 4)
:= ((4 − 4/4) + 4) &times; ((4 + 4)/4 + 4 )
:= 4 + ((4 &times; (444 + 4 + 4)) − 4/4)
:= 5/5 + (5 &times; 5 + 5) &times; (55 + 5)
:= 5 + ((5 &times; 5 + 5) &times; (55 + 5) + 5/5)
:= 55/5 + (5 &times; 5 + 5) &times; (55 + 5)
:= 6/6 + (6 − 6 &times; 6) &times; (6 − 66)
:= 6 + (6 − 6 &times; 6) &times; (6 − 66)
:= 66/6 + (6 − 6 &times; 6) &times; (6 − 66)
7
7
:= 7 + (((7 + 7) &times; ((7 + 7)/7) ) + ((7 + 7)/7))
:= 7 + (((7 + 7) &times; ((7 + 7)/7) ) + 7)
:= 777 + (7777/7 − 77)
:= 8 + ((8 &times; ((8 &times; (8 + 8) + 88) + 8)) + 8/8)
:= 8 + (((8 + 8)/8) &times; (888 + 88/8))
:= 8 + ((8 &times; ((8 &times; (8 + 8) + 88) + 8)) + (88/8))
:= 9 + (((9 + 9) &times; 99 + 9/9) + 9)
:= ((9 + 9)/9) &times; (((99 + 9)/9) + 9 &times; 99)
:= 9 + (((9 + 9) &times; 99 + (99/9)) + 9)
&Iuml; 1802 := (1 + 1) &times; (1 + (((11 − 1) &times; (1 + 1 + 1))1+1 ))
&Iuml; 1807 := 1 + ((1 + 1)11 − ((1 + 1) &times; 111+1 ))
&Iuml; 1812 := (1 + 1 + 1) &times; (11 &times; (111 − 1)/(1 + 1) − 1)
:= 2 + (2 &times; ((2 &times; (2 + 2) + 22)2 ))
:= 222/2 + ((2 − 222 )/2)
:= 2 &times; ((2 &times; (2 &times; ((222 + 2) + 2))) + 2)
:= 3 + ((((3 + 3)3 − 3)/3) + (3 &times; 3 + 3)3 )
:= 3/3 + (((3 &times; 3 + 3)3 − 3) + 3 &times; 33 )
:= 3 + ((3 &times; 3 + 3)3 + 3 &times; 33 )
:= (44 − 4)/4 + (4 &times; (444 + 4))
:= (4 &times; (444 + 4 + 4)) − 4/4
:= 4 + (4 &times; (444 + 4 + 4))
:= (5 + 5)/5 + (5 &times; 5 + 5) &times; (55 + 5)
:= 5 + ((5 &times; 5 + 5) &times; (55 + 5) + ((5 + 5)/5))
:= (5/5 + 5) &times; ((5 &times; (55 + 5)) + ((5 + 5)/5))
:= (6 + 6)/6 + (6 − 6 &times; 6) &times; (6 − 66)
:= 6 + ((6 − 6 &times; 6) &times; (6 − 66) + 6/6)
:= 6 + ((6 − 6 &times; 6) &times; (6 − 66) + 6)
7
7
:= 7 + ((((7 + 7 + 7)/7) ) − 7 &times; (7 &times; 7 + 7))
:= 7 + ((((7 + 7) &times; ((7 + 7)/7) ) + 7/7) + 7)
:= (7 &times; (7 &times; 7 &times; 7 − 77 − 7)) − 7/7
:= 8 + (((8 + 8)/8) &times; (888 + 8/8 + 8))
:= 8 &times; 88 + (8888/8 − 8)
:= ((88 + 8)/8) &times; ((88 − 8/8) + 8 &times; 8)
:= 9 + ((9 + 9) &times; 99 + (99/9))
:= 9 + ((((9 + 9) &times; 99 − ((9 + 9)/9)) + 9) + 9)
:= 9 + (((9 + 9) &times; 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
&Iuml; 1813 := (1 + 1)11 − (11 + ((1 + 1) &times; (1 + 111)))
&Iuml; 1818 := 1 + ((1 + 1)11 − (11 &times; (11 + (11 − 1))))
&Iuml; 1823 := (1 + 1)11 − (1 + ((1 + 1) &times; (1 + 111)))
:= (2 &times; 22)2 − ((22/2)2 + 2)
:= ((2 &times; (22 + 2))2 ) − (222 + 2)
:= (2 &times; 22)2 − (222/2 + 2)
:= 3 + (((3 &times; 3 + 3)3 + 3 &times; 33 ) + 3/3)
:= 33+3 + 33 &times; 33
:= 3 &times; 33 + ((3 &times; 3 + 3)3 − (3/3 + 3))
:= 4 + ((4 &times; (444 + 4 + 4)) + 4/4)
:= 44 + (4 &times; 444 − (4 + 4)/4)
:= 4 + ((4 &times; 444 − 4/4) + 44)
:= (5 &times; (5 &times; (55 + 5))) + (55 + 5)/(5 + 5)
:= (5/5 + 5) &times; ((55 + 5)/(5 + 5) − (5 + 5))
:= ((5/5 + 5) &times; (55 + 5)/(5 + 5)) − 55
:= 6 + (((6 − 6 &times; 6) &times; (6 − 66) + 6/6) + 6)
:= 6 + (((6 − 6 &times; 6) &times; (6 − 66) + 6) + 6)
:= 6 + (((6 − 6 &times; 6) &times; (6 − 66) + (66/6)) + 6)
:= 7 &times; (7 &times; 7 &times; 7 − 77 − 7)
:= (77/7 + 7) &times; (7777/77)
:= 77 + ((77/7 + 7) &times; (7 &times; (7 + 7) − 7/7))
:= 8 + ((88/8 + 8) &times; (88 − 8/8 + 8))
:= 8 + (((8 + 8)/8) &times; ((888 + 8/8 + 8) + 8))
:= 8 + (8888/8 + 8 &times; 88)
:= 9 + (((99 + 99)/9) + (9 + 9) &times; 99)
:= (9 + 9) &times; ((9 + 9)/9 + 99)
:= (9 + 9) &times; 99 + ((9 &times; 9 &times; 9 + 9)/(9 + 9))
&Iuml; 1814 := (11 &times; (11 &times; (1 + 1 + 1 + 1 + 11))) − 1
&Iuml; 1819 := 1 + (1 + ((1 + 1)11 − (11 &times; (11 + (11 − 1)))))
&Iuml; 1824 := (1 + 1)11 − ((1 + 1) &times; (1 + 111))
:= 22 + (2 &times; (2 &times; (2 &times; (222 + 2))))
:= 2 + (((2 &times; 22)2 − (22/2)2 ) + 2)
:= 222/2 − (222 + 2)
:= ((33 &times; ((3 + 3) &times; 33 + 3)) − 3)/3
:= 3/3 + (33 &times; 33 + 33+3 )
:= 3 &times; 33 + ((3 &times; 3 + 3)3 − 3)
:= 4 + ((4 &times; (444 + 4 + 4)) + (4 + 4)/4)
:= 44 + (4 &times; 444 − 4/4)
:= 4 + (4 &times; 444 + 44)
5
5
5−5/5
:= 55 + ((55 &times; ((5 + 5)/5) ) − 5/5)
:= 5 − ((((5/5 + 5)
:= 6 + (((6 − 6 &times; 6) &times; (6 − 66) + ((6 + 6)/6)) + 6)
:= (66/6 + 6) &times; (6 &times; (6 + 6 + 6) − 6/6)
:= 7/7 + (7 &times; (7 &times; 7 &times; 7 − 77 − 7))
) + 5) + 5)
:= 55 − (((5/5 + 5)5−5/5 ) + 5)
:= 66 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6)
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77 − 7)) − 7/7)
:= (7 &times; 7 − 7/7) &times; (7 &times; 7 − 77/7)
:= 8 &times; 88 + (8888 − 8)/8
:= (8/8 + 8 + 8) &times; ((88/8 + 88) + 8)
:= (88 + 8) &times; (88/8 + 8)
:= 9 + (((99 + 99 + 9)/9) + (9 + 9) &times; 99)
:= 9/9 + ((9 + 9) &times; ((9 + 9)/9 + 99))
:= ((99 + 9)/9) &times; (9 &times; (9 + 9) − (9/9 + 9))
&Iuml; 1815 := 11 &times; (11 &times; (1 + 1 + 1 + 1 + 11))
&Iuml; 1820 := (11 − 1) &times; ((1 + 1 + 11) &times; (1 + (1 + 1 + 11)))
&Iuml; 1825 := (1 + 1)11 − (1 + (1 + 1) &times; 111)
:= (2 &times; 22)2 − (22/2)2
:= 2 &times; ((2 &times; 2 &times; 222) + 22)
:= (2 &times; 22)2 − 222/2
:= 33 + (3 &times; 3 &times; 33 &times; (3 + 3))
:= 3 + ((33 &times; 33 − 3/3) + 33+3 )
:= 3/3 + (((3 &times; 3 + 3)3 − 3) + 3 &times; 33)
:= 44 + 4 &times; 444
:= 44 &times; 44 − 444/4
:= 44 + (4 &times; 444 − (4/4 + 4))
5
:= (5 &times; (5 &times; (5 &times; (5 + 5 + 5)))) − 55
:= 55 + (55 &times; ((5 + 5)/5) )
6
:= 5 &times; ((5/5 + 5) &times; (55 + 5) + 5)
:= (6 &times; 6 − 6/6) &times; (((6 + 6)/6) − (6 + 6))
:= (6 − 6/6) &times; ((6 &times; (66 − 6) − 6/6) + 6)
:= (7/7 + 7 + 7) &times; (((7 + 7)/7) − 7)
:= 7 + (7 &times; (7 &times; 7 &times; 7 − 77 − 7))
:= 7 + ((77/7 + 7) &times; (7777/77))
:= 8 &times; 88 + 8888/8
:= (8 − 8/8) &times; ((8 &times; 8 &times; 8 + 8)/((8 + 8)/8))
:= 8/8 + ((88 + 8) &times; (88/8 + 8))
:= (99/9 + 9) &times; ((9/9 + 9 &times; 9) + 9)
:= ((9 + 9) &times; (999/9 − 9)) − 99/9
:= (6 − 6/6) &times; 66 &times; 66/(6 + 6)
7
:= 99/9 &times; (((9 + 9 + 9)/9) + 9 &times; (9 + 9))
&Iuml; 1826 := (1 + 1)11 − (1 + 1) &times; 111
&Iuml; 1816 := 1 + (11 &times; (11 &times; (1 + 1 + 1 + 1 + 11)))
&Iuml; 1821 := (1 + 1)11 − (1 + ((1 + 1) &times; (1 + 1 + 111)))
:= 2 &times; (((2 &times; 2 &times; 222) − 2) + 22)
:= 2/2 + (2 &times; ((2 &times; 2 &times; 222) + 22))
:= 222/2 − 222
:= ((33 &times; ((3 + 3) &times; 33 + 3)) + 3)/3
:= 3 + (33 &times; 33 + 33+3 )
:= 3 &times; 33 + ((3 &times; 3 + 3)3 − 3/3)
:= 44 + (4 &times; 444 − 4)
:= 44 + (4 &times; 444 + 4/4)
:= 44 &times; 44 + ((4 − 444)/4)
:= 5 + ((5 &times; 5 + 5) &times; (55 + 5) + (55/5))
:= 5 + (((5 − (5 + 5 ))/5) − 55)
:= 55 − ((5 − 5/5)5 + 5 &times; 55)
:= ((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 + 66/6)
:= 6 + ((6 − 6/6) &times; 66 &times; 66/(6 + 6))
:= (((6 + 6)/6)66/6 ) − (6 &times; 6 &times; 6 + 6)
:= 7/7 + ((7/7 + 7 + 7) &times; (((7 + 7)/7)7 − 7))
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77 − 7)) + 7/7)
:= 777/7 + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= 88 + 8 &times; (8 &times; (8 + 8) + 88)
:= 8 + (((88/8 + 8) &times; (88 − 8/8 + 8)) + 8)
:= (8 + 8)/8 + ((88 + 8) &times; (88/8 + 8))
:= 9 &times; 9 &times; 9 + (((99 &times; 99) − (9 + 9))/9)
:= 9 + ((((9 + 9) &times; 99 + ((99 + 9)/9)) + 9) + 9)
:= 9 + ((((99 &times; 99) − 9)/9) + 9 &times; 9 &times; 9)
&Iuml; 1817 := (1 + 1)11 − (11 &times; (11 + (11 − 1)))
5
5
5
&Iuml; 1822 := (1 + 1)11 − ((1 + 1) &times; (1 + 1 + 111))
&Iuml; 1827 := 1 + ((1 + 1)11 − (1 + 1) &times; 111)
:= 2 + ((2 &times; 22)2 − (22/2)2 )
:= 222 + (2 &times; (22 − 2))2
:= (((2 &times; 22) − 2/2)2 ) − 22
:= 33+3 + (33 &times; 33 − 3/3)
:= 3 + ((33 &times; 33 + 33+3 ) + 3/3)
:= 3 &times; 33 + (3 &times; 3 + 3)3
:= 44 + ((4 &times; 444 − 4) + 4/4)
:= 44 + (4 &times; 444 + (4 + 4)/4)
:= (4 − 4/4) &times; ((4/4 + 4)4 − 4 &times; 4)
:= 5 &times; 5 + (((5 + 5)/5)5 &times; (55 + 5/5))
:= 5 + ((((5 + 5)/5)5 &times; (55 + 5/5)) + 5 &times; 5)
:= (5 − (5 + 5)/5) &times; (((55 − 55)/5) − 5)
:= 6 + ((6 − 6 &times; 6) &times; (6 − 66) + (66/6))
:= 6 + (((6 + 6)/6 + 6) &times; (6 &times; 6 &times; 6 + 66/6))
:= 6 + (((6 − 6/6) &times; 66 &times; 66/(6 + 6)) + 6)
7
:= ((7 &times; 7 − 7/7) &times; (7 &times; 7 − 77/7)) − 7
:= 7 + ((7/7 + 7 + 7) &times; (((7 + 7)/7) − 7))
:= 8/8 + (8 &times; (8 &times; (8 + 8) + 88) + 88)
:= 8 + ((8888 − 8)/8 + 8 &times; 88)
:= (8/8 + 8) &times; (8 &times; (8 + 8 + 8) + (88/8))
:= 9 &times; 9 &times; 9 + (((99 &times; 99) − 9)/9)
:= 9 &times; 9 &times; 9 + (9999/9 − (9 + 9))
:= 9 + ((9 + 9) &times; ((9 + 9)/9 + 99))
131
:= 7 + ((7 &times; (7 &times; 7 &times; 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
&Iuml; 1828 := (1 + 1)11 − ((1 + 1) &times; (111 − 1))
&Iuml; 1833 := (1 + 1 + 1) &times; (1 + 11 &times; 111)/(1 + 1)
&Iuml; 1838 := 111 + (((1 + 11)1+1+1 ) − 1)
:= 2 + (222/2 − 222)
:= (((2 &times; 22) − 2/2)2 ) − 22+2
:= (((2 &times; 22) − 2/2)2 ) − 22/2
:= 3/3 + ((3 &times; 3 + 3)3 + 3 &times; 33)
:= (3 &times; ((3 + 3) &times; (3 &times; 33 + 3))) − 3
:= (3 &times; 3 + 3)3 + ((333 − 3)/3)
:= 4 + ((4 &times; 444 + 44) + 4)
:= ((44 − 4/4)(4+4)/4 ) − 4 &times; 4
:= 4 &times; 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 &times; 55 + ((55 + 5/5)/((5 + 5)/5))
:= ((6 + 6)/6)6 + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= 6 &times; 6 &times; (66 − 6 − 6) − 666/6
:= 6 + ((((6 + 6)/6)66/6 ) − 6 &times; 6 &times; 6)
:= 7 + (((7 &times; (7 &times; 7 &times; 7 − 77 − 7)) + 7/7) + 7)
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − (7 + 7)))) + 777/7)
:= 7 + (((7 &times; 7 − 7/7) &times; (7 &times; 7 − 77/7)) + 7)
:= ((8 &times; (8 &times; (8 &times; 8 − 8) + 8)) + 8)/((8 + 8)/8)
:= 8 + (((88 + 8) &times; (88/8 + 8)) + 8/8)
:= 8 &times; 8 + (((8 + 8)/8) &times; (888 − 8/8))
:= 9 + (((9 + 9) &times; ((9 + 9)/9 + 99)) + 9/9)
:= (9 + 9) &times; (99 + 9) − 999/9
:= 9 &times; 9 &times; 9 + ((9999 − (9 + 9))/9)
&Iuml; 1829 := 1 + ((1 + 1)11 − ((1 + 1) &times; (111 − 1)))
&Iuml; 1834 := (1 + (1 + 1 + 11)) &times; ((11 &times; (1 + 11)) − 1)
&Iuml; 1839 := 111 + ((1 + 11)1+1+1 )
:= 2 + ((((2 &times; 22) − 2/2)2 ) − 22)
:= 2 + (2 &times; (2 &times; (222 − (22 + 2 + 2))))
:= 2 + (22/2 &times; (((22/2 + 2)2 ) − 2))
:= ((33 &times; 333 + 3)/(3 + 3)) − 3
:= 3 + ((33 &times; 333 − 3)/(3 + 3))
:= 3 + (3 &times; ((3 + 3) &times; (3 &times; 33 + 3)))
:= 4 + (44 &times; 44 − 444/4)
:= 44 + ((4 &times; (444 + 4)) − (4 + 4)/4)
:= 4 &times; 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) &times; (6 &times; 6 + 66) − (6 + 6)/6
:= 666/6 + (6 &times; 6 &times; (6 &times; 6 + 6 + 6))
:= 7 + (((7/7 + 7 + 7) &times; (((7 + 7)/7) − 7)) + 7)
:= 7 &times; ((((7 &times; 7 &times; 77) − 7)/(7 + 7)) − 7)
:= 777 + (7777/7 − 7 &times; 7)
:= 8 &times; 8 + ((888 − 88/8) + 888)
:= 8 + (((88 + 8) &times; (88/8 + 8)) + ((8 + 8)/8))
:= 8 &times; 8 + ((888 − 8/8) + 888)
:= 9 + ((99/9 + 9) &times; ((9/9 + 9 &times; 9) + 9))
:= ((9 + 9)/9) &times; (999 − (9/9 + 9 &times; 9))
:= 9 &times; 9 &times; 9 + ((9999 − 9)/9)
:= 66 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) − 6/6)
7
&Iuml; 1830 := (1 + 1 + 1) &times; (11 &times; 111 − 1)/(1 + 1)
&Iuml; 1835 := 1 + (1 + 1 + 1 + 11) &times; (11 &times; (1 + 11) − 1)
&Iuml; 1840 := 1 + (111 + ((1 + 11)1+1+1 ))
:= 2 + ((222/2 − 222) + 2)
:= 2 + ((((2 &times; 22) − 2/2)2 ) − 22+2 )
:= 2 &times; (2 &times; (222 − (22 + 2)))
:= 3 + ((3 &times; 3 + 3)3 + 3 &times; 33)
:= 3 + ((33 &times; 333 + 3)/(3 + 3))
:= 33+3 + 3333/3
:= 4 + (((4 − 444)/4) + 44 &times; 44)
:= 44 + ((4 &times; (444 + 4)) − 4/4)
:= 4 &times; (444 + 4 &times; 4)
5
:= (5/5 + 5) &times; ((5 &times; (55 + 5)) + 5)
:= 5 &times; ((5 − 5)/(5 + 5) + 55)
:= 5 &times; ((55 + 5)/(5 + 5) + 55)
:= 66 + (6 &times; 6 + 6) &times; (6 &times; 6 + 6)
:= (6 + 6 + 6) &times; (6 &times; 6 + 66) − 6/6
:= ((6 &times; 6/(6 + 6))6 ) + (6666/6)
:= (7 − 7/7) &times; (7 &times; (7 &times; 7 − 7) + (77/7))
:= 7 &times; 77 + (7 − 7/7)77/7−7
:= 77 + (((7 + 7) &times; (77 + 7 &times; 7)) − 7/7)
:= (8 − 8/8 + 8) &times; ((888 + 88)/8)
:= 88/8 + ((88 + 8) &times; (88/8 + 8))
:= 8 &times; 8 + (888 + 888)
:= 9 &times; 9 &times; 9 + (((9999 − 9)/9) − 9)
:= ((9 + 9) &times; (999/9 − 9)) − 9/9
:= 9 &times; 9 &times; 9 + 9999/9
&Iuml; 1831 := ((11 &times; (1 + 1 + 1) &times; 111) − 1)/(1 + 1)
&Iuml; 1836 := (1 + 11) &times; ((11 &times; (1 + (1 + 1 + 11))) − 1)
&Iuml; 1841 := 1 + (1 + (111 + ((1 + 11)1+1+1 )))
:= 2 + (((((2 &times; 22) − 2/2)2 ) − 22) + 2)
:= 2 &times; ((2 &times; (2 &times; (222 + 2))) + 22)
:= (((2 &times; 22) − 2/2)2 ) − 2 &times; (2 + 2)
:= (33 &times; 333 − 3)/(3 + 3)
:= 3 &times; ((3 + 3) &times; (3 &times; 33 + 3))
:= 33+3 + ((3333 + 3)/3)
:= 44 + (4 &times; 444 + 44/4)
:= 44 + (4 &times; (444 + 4))
:= 4/4 + (4 &times; (444 + 4 &times; 4))
5
5
5
:= 5 + (5 − ((5 − 5/5) + 5 &times; 55))
:= 5/5 + (5 &times; ((5 − 5)/(5 + 5) + 55))
:= 5/5 + (5 &times; ((55 + 5)/(5 + 5) + 55))
:= 66 + ((6 &times; 6 + 6) &times; (6 &times; 6 + 6) + 6/6)
:= (6 + 6 + 6) &times; (6 &times; 6 + 66)
:= 6 + ((6 + 6 + 6) &times; (6 &times; 6 + 66) − 6/6)
:= 7 + ((7 &times; 7 − 7/7) &times; (7 &times; 7 − 77/7))
:= (77 + 7)/7 &times; ((77 − 7/7) + 77)
:= 7 &times; ((((7 + 7)/7)7+7/7 ) + 7)
:= 8 + ((8888/8 + 8 &times; 88) + 8)
:= ((88 + 8)/8) &times; ((8 &times; 8 + 88) + 8/8)
:= 8/8 + ((888 + 888) + 8 &times; 8)
:= 9 &times; 9 &times; 9 + (9999/9 − 9)
:= (9 + 9) &times; (999/9 − 9)
:= 9 &times; 9 &times; 9 + (9999 + 9)/9
&Iuml; 1832 := (1 + (11 &times; (1 + 1 + 1) &times; 111))/(1 + 1)
&Iuml; 1837 := 11 &times; ((1 + 1 + 11)1+1 − (1 + 1))
&Iuml; 1842 := 1 + (1 + (1 + (111 + ((1 + 11)1+1+1 ))))
:= 2 &times; (2 &times; (222 − (22 + 2 + 2)))
:= 22/2 &times; (((22/2 + 2)2 ) − 2)
:= 2 + (2 &times; (2 &times; (222 − (22 + 2))))
:= (33 &times; 333 + 3)/(3 + 3)
:= 3/3 + (3 &times; ((3 + 3) &times; (3 &times; 33 + 3)))
:= (3 &times; (3 &times; (3 + 3)3 − 33)) − 3
:= 44 + ((4 &times; (444 + 4)) − 4)
:= 44 + ((4 &times; (444 + 4)) + 4/4)
:= (4 + 4)/4 + (4 &times; (444 + 4 &times; 4))
5
5
:= ((5 + 5)/5) + (5 &times; 5 + 5) &times; (55 + 5)
:= 5 &times; 55 + ((5 − 5/5)/((5 + 5)/5))
:= (5/5 + 5) &times; (55 − 55)/(5 + 5)
:= (((6 + 6)/6)66/6 ) − 6 &times; 6 &times; 6
:= 6/6 + (6 + 6 + 6) &times; (6 &times; 6 + 66)
:= 6 + (6 + 6 + 6) &times; (6 &times; 6 + 66)
7
7
:= (((7/7 − 7) + 7 &times; 7)(7+7)/7 ) − 7
:= 77 + ((7 − 7/7 + 7) &times; (((7 + 7)/7) + 7))
:= (((7 + 7 + 7)/7) ) − (7 &times; 7 &times; 7 + 7)
:= 8 + ((88 + 8) &times; (88/8 + 8))
:= 88/8 &times; ((888/8 − 8) + 8 &times; 8)
:= 8 &times; 8 + (((8 + 8)/8) &times; (888 + 8/8))
:= 9 &times; 9 &times; 9 + ((9999 + 9)/9 − 9)
:= 9/9 + ((9 + 9) &times; (999/9 − 9))
:= 9 + ((9 + 9) &times; (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
&Iuml; 1843 := 111+1+1 + (1 + 1)11−1−1
2
&Iuml; 1848 := 11 &times; ((1 + 1 + 11)1+1 − 1)
&Iuml; 1853 := 1 + ((1 + 1) &times; ((1 + 11111)/(1 + 11)))
:= (((2 &times; 22) − 2/2) ) − (2 + 2 + 2)
:= 2 &times; (2 &times; (22 − 22))
:= 2 + ((((2 &times; 22) − 2/2)2 ) + 2)
:= 3 + (3333/3 + 33+3 )
:= 3 + (3 &times; (3 &times; (3 + 3)3 − 33))
:= (3 &times; 3 + 3)3 + ((3 − 3/3 + 3)3 )
:= 4 + (4 &times; 444 + ((44 − 4)/4))
:= 44 &times; (44 − (4 + 4)/4)
:= 4 + ((44 − 4/4)(4+4)/4 )
:= (5 &times; (5 &times; (5 &times; (5 + 5 + 5)))) − ((5 + 5)/5)5
:= (5/5 + 5) &times; ((55 + 5)/(5 + 5) − 5)
:= 5 + ((5/5 + 5) &times; ((55 + 5)/(5 + 5) − 5))
:= (((6 &times; 6 + 6/6) + 6)(6+6)/6 ) − 6
:= 66 &times; (((66 + 66)/6) + 6)
:= 666 + ((66 &times; (6 + 6 + 6)) − 6/6)
:= ((7 + 7)/7) + (7 &times; (7 &times; (7 &times; 7 − (7 + 7))))
:= 77 &times; (((77 − 7)/7 + 7) + 7)
:= 777 + (77 &times; (7 + 7) − ((7 + 7)/7))
:= (88/8 + 8) &times; ((8/8 + 88) + 8)
:= 88 &times; ((88 + 8 + 8)/8 + 8)
:= (8/8 + 8 + 8) &times; ((888 − (8 + 8))/8)
:= (9/9 + 9 + 9) &times; (99 − ((9 + 9)/9))
:= 9 + (((9999 − 9)/9) + 9 &times; 9 &times; 9)
:= 9 &times; 9 + ((9 + 9) &times; 99 − (9/9 + 9))
7
&Iuml; 1844 := 1 + (111+1+1 + (1 + 1)11−1−1 )
2
&Iuml; 1849 := (((1 + 1) &times; (11 + 11)) − 1)1+1
2
&Iuml; 1854 := (1 + 1) &times; (1 + ((1 + 11111)/(1 + 11)))
:= 2 &times; ((2 &times; (222 − 22)) − 2)
:= ((2 &times; 22) − 2/2)
:= 2 + (2 &times; ((2 &times; (222 − 22)) + 2))
:= (3 &times; 33+3 ) − (((3/3 + 3) + 3)3 )
:= ((3 &times; 3 + 33) + 3/3)3−3/3
:= (3 &times; 33+3 ) − 333
:= 4 + (4 &times; (444 + 4 &times; 4))
:= (44 − 4/4)(4+4)/4
:= 4 + (((44 − 4/4)(4+4)/4 ) + 4/4)
:= (5 − 5/5)5 + ((55 &times; (5 + 5 + 5)) − 5)
5
:= (5 − 5/5) + (55 &times; (5 + 5 + 5))
(6+6)/6
:= 5 + ((55 &times; (5 + 5 + 5)) + (5 − 5/5)5 )
:= 6 + (((((6 + 6)/6)66/6 ) − 6 &times; 6 &times; 6) + 6)
:= ((6 &times; 6 + 6/6) + 6)
:= (((7 + 7 + 7)/7)7 ) − 7 &times; 7 &times; 7
:= ((7/7 − 7) + 7 &times; 7)(7+7)/7
:= 777 + (77 &times; (7 + 7) − 7/7)
:= 8 + (((88 + 8)/8) &times; ((8 &times; 8 + 88) + 8/8))
:= 8/8 + ((8 &times; (8 &times; (8 + 8) − 8)) + 888)
:= 888 + ((88 &times; 88 − (8 + 8))/8)
:= (9 + 9) &times; (99 + 9) − (9/9 + 99)
:= 9 + (9999/9 + 9 &times; 9 &times; 9)
:= 9 &times; 9 + ((9 + 9) &times; 99 − 9)
&Iuml; 1845 := (1 + 1 + 1) &times; (11 &times; (1 + 111)/(1 + 1) − 1)
&Iuml; 1850 := 1 + ((((1 + 1) &times; (11 + 11)) − 1)1+1 )
:= 666 + (66 &times; (6 + 6 + 6))
&Iuml; 1855 := (11 &times; (1 + 1 + 11)1+1 ) − 1 − 1 − 1 − 1
:= (((2 &times; 22) − 2/2)2 ) − 2 − 2
:= 2 + (2 &times; (2 &times; (222 − 22)))
:= 2 + (((((2 &times; 22) − 2/2)2 ) + 2) + 2)
:= 3 &times; (3 &times; (3 + 3)3 − 33)
:= (3 &times; 3 + 3)3 + ((33+3 + 3)/(3 + 3))
:= 3/3 + ((3 &times; 33+3 ) − 333)
:= ((44 − 4/4)(4+4)/4 ) − 4
:= 4/4 + ((44 − 4/4)(4+4)/4 )
:= 44 &times; 44 − (4 − 4/4)4
:= (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) − 5)) − 5
:= 5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) − 5)
:= 5 + (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) − 5))
:= (6 − 6/6) &times; (66 &times; 66/(6 + 6) + 6)
:= 6/6 + (((6 &times; 6 + 6/6) + 6)(6+6)/6 )
:= 6 + (((6 &times; 6 + 6/6) + 6)(6+6)/6 )
:= 7/7 + ((((7 + 7 + 7)/7)7 ) − 7 &times; 7 &times; 7)
:= 7/7 + (((7/7 − 7) + 7 &times; 7)(7+7)/7 )
:= 777 + 77 &times; (7 + 7)
:= 888 + (88/8 &times; (88 − 8/8))
:= 8 + ((((8 + 8)/8) &times; (888 + 8/8)) + 8 &times; 8)
:= 888 + ((88 &times; 88 − 8)/8)
:= (9 + 9) &times; (99 + 9) − 99
:= 9 + ((9999 + 9)/9 + 9 &times; 9 &times; 9)
:= 9/9 + (((9 + 9) &times; 99 − 9) + 9 &times; 9)
&Iuml; 1846 := (1 + 1 + 11) &times; ((1 + 11)1+1 − (1 + 1))
&Iuml; 1851 := 1 + (1 + ((((1 + 1) &times; (11 + 11)) − 1)1+1 ))
&Iuml; 1856 := (11 &times; (1 + 1 + 11)1+1 ) − 1 − 1 − 1
:= 2 &times; (2 &times; 222 + 222 )
:= (2 &times; (2 &times; (222 − 22))) − 2
:= 2 + (((2 &times; 22) − 2/2)2 )
:= 3/3 + (3 &times; (3 &times; (3 + 3)3 − 33))
:= (3 &times; 33+3 ) − (333 + 3)
:= ((3/3 + 3)3 ) &times; ((33 − 3/3) + 3)
:= 4 + ((4 &times; (444 + 4 &times; 4)) + (4 + 4)/4)
:= 44/4 + (4 &times; (444 + 4 &times; 4))
:= 4 &times; ((444 + 4 &times; 4) + 4)
:= ((55 &times; 555/5) + 55 )/5
:= 5/5 + (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) − 5))
:= (55/5 + 5) &times; (555/5 + 5)
:= 6 + (((6 &times; 6/(6 + 6))6 ) + (6666/6))
:= ((66/6 + 6) &times; 666/6) − 6 &times; 6
:= ((6 + 6)/6)6 &times; (6 &times; 6 − (6/6 + 6))
:= ((7 + 7)/7) &times; ((77 &times; (77 + 7) − 7)/7)
:= 7 + ((((7 + 7 + 7)/7)7 ) − 7 &times; 7 &times; 7)
:= 7 + (((7/7 − 7) + 7 &times; 7)(7+7)/7 )
:= 8 + ((((8 + 8)/8) &times; (888 − 8/8)) + 8 &times; 8)
:= 8 + ((88/8 + 8) &times; ((8/8 + 88) + 8))
:= 8 &times; (((8 &times; (8 + 8) + 88) + 8) + 8)
:= 9/9 + ((9 + 9) &times; (99 + 9) − 99)
:= 9 &times; 9 + ((9 + 9) &times; 99 − ((99 + 9)/9))
:= 9 &times; 9 + (((9 + 9) &times; 99 − 9) + ((9 + 9)/9))
&Iuml; 1857 := (11 &times; (1 + 1 + 11)1+1 ) − 1 − 1
&Iuml; 1847 := (11 &times; ((1 + 1 + 11)1+1 − 1)) − 1
&Iuml; 1852 := (1 + 1) &times; ((1 + 11111)/(1 + 11))
:= (((2 &times; 22) − 2/2)2 ) − 2
:= 2 &times; ((2 &times; (222 − 22)) + 2)
:= 2 &times; (2 + 2) + (((2 &times; 22) − 2/2)2 )
:= 3 + ((3 &times; 33+3 ) − (((3/3 + 3) + 3)3 ))
:= 3 + (((3 &times; 3 + 33) + 3/3)3−3/3 )
:= 3 + ((3 &times; 33+3 ) − 333)
:= (44 &times; (44 − (4 + 4)/4)) − 4/4
:= 4 + (44 &times; (44 − (4 + 4)/4))
:= 4 &times; 444 + (4 − 4/4)4
:= 5 + ((5/5 + 5) &times; (55 − 55)/(5 + 5))
:= (5 + 5)/5 + (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) − 5))
:= (5 − (5 + 5)/5) &times; ((55 − 5)/5 − 5)
:= 66/6 + (6 + 6 + 6) &times; (6 &times; 6 + 66)
:= 666 + ((66 &times; (6 + 6 + 6)) − ((6 + 6)/6))
:= 6 + (((66/6 + 6) &times; 666/6) − 6 &times; 6)
:= 777 + (77 &times; (7 + 7) − (7/7 + 7))
:= ((7 − 77)/7) + (7 &times; (7 &times; 7 &times; 7 − 77))
:= ((7 + 7)/7) + (77 &times; (7 + 7) + 777)
:= 888 + ((8 &times; (8 &times; (8 + 8) − 8)) − 8/8)
:= 88 &times; (8 + 8) + 888/((8 + 8)/8)
:= 888 + ((88 &times; 88 + 8)/8)
:= (9 + 9)/9 + ((9 + 9) &times; (99 + 9) − 99)
:= 9 &times; 9 + ((9 + 9) &times; 99 − (99/9))
:= 9 + ((((9999 − 9)/9) + 9 &times; 9 &times; 9) + 9)
133
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1858 := (11 &times; (1 + 1 + 11)1+1 ) − 1
&Iuml; 1863 := (1 + (11 + 11)) &times; (11 − 1 − 1)1+1
&Iuml; 1868 := (11 &times; (1 + (1 + 1 + 11)1+1 )) − 1 − 1
:= 2 + (2 &times; (2 &times; 222 + 222 ))
:= (22 + 2/2) &times; ((2/2 + 2)2+2 )
:= 2 &times; ((2 &times; (2 &times; 222 + 22)) + 2)
:= 3 + (((3 &times; 33+3 ) − 333) + 3/3)
:= 3 &times; (3 &times; ((3 + 3)3 − 3 &times; 3))
:= ((33 − 3) &times; (3 &times; 33 − 3)) − (3/3 + 3)
:= 4/4 + (4 &times; 444 + (4 − 4/4)4 )
:= (4 − 4/4) &times; ((4/4 + 4)4 − 4)
:= (44 &times; (4 + 4)) − (4 &times; 44 + 4)
:= ((5 − (5 + 5)/5) &times; ((55 + 5)/5 − 5)) − 5
:= (5 − (5 + 5)/5) &times; ((55 + 5)/5 − 5)
:= 5 + ((5 − (5 + 5)/5) &times; ((55 + 5)/5 − 5))
:= ((6 + 6)/6)6 + ((6 − 6 &times; 6) &times; (6 − 66) − 6)
:= (6 &times; (6 &times; (66 + 6))) − ((6 &times; 6/(6 + 6))6 )
:= (6 &times; (6 − 6 &times; 6)) + (((6 + 6)/6)66/6 )
:= 7 + (((((7 + 7 + 7)/7)7 ) − 7 &times; 7 &times; 7) + 7)
:= 7/7 + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) − 7/7)
:= 888 + (88 &times; 88 + 8 + 8)/8
:= 8 + (((88 &times; 88 − 8)/8) + 888)
:= 88 + (((8 + 8)/8) &times; (888 + ((8 + 8)/8)))
:= 9 + ((9999/9 + 9 &times; 9 &times; 9) + 9)
:= 9 &times; (99 + 99 + 9)
:= (((9 + 9)/9)99/9 ) − (99 + 9 &times; 9)
&Iuml; 1859 := 11 &times; (1 + 1 + 11)1+1
&Iuml; 1864 := 1 + ((1 + (11 + 11)) &times; (11 − 1 − 1)1+1 )
:= 22/2 &times; ((22/2 + 2)2 )
&Iuml; 1869 := (11 &times; (1 + (1 + 1 + 11)1+1 )) − 1
:= 22 + ((((2 &times; 22) − 2/2)2 ) − 2)
:= 2 &times; (2 &times; (2 &times; 222 + 22))
3
:= 3 + (((3/3 + 3)3 ) &times; ((33 − 3/3) + 3))
:= 3/3 + (3 &times; (3 &times; ((3 + 3) − 3 &times; 3)))
:= ((33 − 3) &times; (3 &times; 33 − 3)) − 3
:= 4 + (44 &times; 44 − (4 − 4/4)4 )
:= 44 + (4 &times; 444 + 44)
:= ((4 − 44 )/4) + (44 &times; 44 − 4)
:= (((5 + 5 + 5) &times; (55 /5 − 5)) − 5)/5
:= 66/6 &times; ((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 &times; 6) &times; (6 − 66)
:= 66 &times; (6 &times; 6 − 6) − 666/6
:= ((7 + 7)/7) + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 7 + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 88/8 &times; ((88 − 8 + 88) + 8/8)
:= 88 + (888 + 888)
:= (8 − 8/8) &times; ((8 + 8) &times; (8 + 8) + (88/8))
:= 99/9 &times; ((9 &times; (9 + 9) − ((9 + 9)/9)) + 9)
:= 9/9 + ((9 + 9) &times; 99 + 9 &times; 9)
:= 99 + ((9 + 9) &times; 99 − ((99 + 9)/9))
:= 77/7 &times; ((7 − 7/7 + 7)
(7+7)/7
)
&Iuml; 1860 := 1 + (11 &times; (1 + 1 + 11)1+1 )
&Iuml; 1865 := 1 + (1 + ((1 + (11 + 11)) &times; (11 − 1 − 1)1+1 ))
&Iuml; 1870 := 11 &times; (1 + (1 + 1 + 11)1+1 )
:= 2 &times; ((2 &times; 222 + 222 ) + 2)
:= 22+2 + (((2 &times; 22) − 2/2)2 )
:= 22 + (2 &times; (2 &times; (222 − 22)))
:= (3 &times; (3 &times; ((3 + 3)3 − 3 &times; 3))) − 3
:= 3 + ((((3 &times; (3 + 3))3 − 3)/3) − 3 &times; 33 )
:= 3/3 + (((33 − 3) &times; (3 &times; 33 − 3)) − 3)
(4+4)/4
)
:= (4 &times; 4 + 4/4) &times; (444 − 4)/4
:= 4 + (4 &times; ((444 + 4 &times; 4) + 4))
:= 4 &times; 4 + ((44 − 4/4)
:= (5 + 5 + 5) &times; (5 &times; 5 &times; 5 − 5/5)
:= (5 &times; (5 &times; (5 &times; (5 + 5 + 5)))) − 5 − 5
:= (5 &times; (5 &times; (5 &times; (5 + 5 + 5)))) − 5
:= (6 − 66) &times; (6 − (6 &times; 6 + 6/6))
:= 66 + ((6 − 6 &times; 6) &times; (6 − 66) − 6/6)
:= (66/6 + 6) &times; ((666 − 6)/6)
:= (7 &times; (7 &times; 7 &times; 7 − 77)) − (7 + 7)/7
:= ((7 + 7 + 7)/7) + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) + 7/7)
:= 8 + (888/((8 + 8)/8) + 88 &times; (8 + 8))
:= 8 + (((88 &times; 88 + 8)/8) + 888)
:= (8/8 + 8 + 8) &times; (888 − 8)/8
:= 9 &times; 9 + ((9 + 9) &times; 99 − ((9 + 9 + 9)/9))
:= 9 &times; 9 + ((9 + 9) &times; 99 + ((9 + 9)/9))
:= 99 + ((9 + 9) &times; 99 − (99/9))
&Iuml; 1861 := 1 + (1 + (11 &times; (1 + 1 + 11)1+1 ))
&Iuml; 1866 := (1 + 1 + 1) &times; (11 + (1 + 11 &times; 111)/(1 + 1))
&Iuml; 1871 := 1 + (11 &times; (1 + (1 + 1 + 11)1+1 ))
:= 2 + (22/2 &times; ((22/2 + 2)2 ))
:= 2 + (2 &times; (2 &times; (2 &times; 222 + 22)))
:= 22 + (((2 &times; 22) − 2/2)2 )
:= 3/3 + ((3 &times; (3 &times; ((3 + 3)3 − 3 &times; 3))) − 3)
:= 3 + (3 &times; (3 &times; ((3 + 3)3 − 3 &times; 3)))
:= ((33 − 3) &times; (3 &times; 33 − 3)) − 3/3
:= 4 + (4 &times; 444 + (4 − 4/4)4 )
:= (4 − 4/4) &times; (((4/4 + 4)4 − 4) + 4/4)
:= 44 &times; 44 − (44 + 4)/4
:= (((5 + 5 + 5) &times; (55 /5 − 5)) + 5)/5
:= 5 + ((((5 + 5 + 5) &times; (55 /5 − 5)) + 5)/5)
:= 55 + (((5 − (55 + 55 ))/5) − 5)
:= 6 + ((((6 &times; 6 + 6/6) + 6)(6+6)/6 ) + 6)
:= 66 + (6 − 6 &times; 6) &times; (6 − 66)
:= (6 &times; ((6 &times; (6 &times; 6 + 6) − 6) + 66)) − 6/6
:= (7 &times; (7 &times; 7 &times; 7 − 77)) − 7/7
:= 77/7 + (77 &times; (7 + 7) + 777)
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) + ((7 + 7)/7))
:= 8 + ((8/8 + 8 + 8) &times; ((888 − (8 + 8))/8))
:= 88 + (((8 + 8)/8) &times; (888 + 8/8))
:= 8 &times; (88 + 8) + (8888/8 − 8)
:= 9 &times; 9 + ((9 + 9) &times; 99 − ((9 + 9)/9))
:= 9 &times; 9 + ((9 + 9) &times; 99 + ((9 + 9 + 9)/9))
:= 9 + (((9 + 9) &times; 99 − 9/9) + 9 &times; 9)
&Iuml; 1862 := 1 + (1 + (1 + (11 &times; (1 + 1 + 11)1+1 )))
&Iuml; 1867 := (11 &times; (1 + (1 + 1 + 11)1+1 )) − 1 − 1 − 1
:= (2 &times; (2 &times; (2 &times; 222 + 22))) − 2
:= 2 + ((((2 &times; 22) − 2/2)2 ) + 22+2 )
:= 2 &times; (2 &times; (222 − 22+2 ))
:= (((3 &times; (3 + 3))3 − 3)/3) − 3 &times; 33
:= 3 + ((3 &times; (3 &times; ((3 + 3)3 − 3 &times; 3))) + 3/3)
:= (33 − 3) &times; (3 &times; 33 − 3)
:= 44 &times; 44 + ((4 − (44 + 44 ))/4)
:= 4 + ((4 − 4/4) &times; ((4/4 + 4)4 − 4))
:= 4 &times; ((44 − 44) + 44 )
:= 5 + ((5 − (5 + 5)/5) &times; ((55 − 5)/5 − 5))
:= ((5/5 + 5) &times; (55 − 5)/(5 + 5)) − 5
:= (5/5 + 5) &times; (55 − 5)/(5 + 5)
:= 6 + (((6 + 6)/6)6 &times; (6 &times; 6 − (6/6 + 6)))
:= 66 + ((6 − 6 &times; 6) &times; (6 − 66) + 6/6)
:= 6 &times; ((6 &times; (6 &times; 6 + 6) − 6) + 66)
:= 7 &times; (7 &times; 7 &times; 7 − 77)
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) − ((7 + 7)/7))
:= (77/7 + 7) &times; (777/7 − 7)
:= 88 + (((8 + 8)/8) &times; (888 − 8/8))
:= 888 + 88/8 &times; (8/8 + 88)
:= (8 + 8) &times; (8 &times; (8 + 8) − (88/8))
:= 9 &times; 9 + ((9 + 9) &times; 99 − 9/9)
:= 9 + (((9999/9 + 9 &times; 9 &times; 9) + 9) + 9)
:= 9 + ((9 + 9) &times; 99 + 9 &times; 9)
134
&Iuml; 1872 := (1 + 1 + 11) &times; (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
&Iuml; 1873 := 1 + ((1 + 1 + 11) &times; (1 + 11)1+1 )
&Iuml; 1878 := (1 + 1)11 − (1 + (1 + 1 + 11)1+1 )
&Iuml; 1883 := 11 + ((1 + 1 + 11) &times; (1 + 11)1+1 )
:= 2 + ((((2 &times; 22) − 2/2)2 ) + 22)
:= 2 + (2 &times; ((2 &times; (222 − 22+2 )) + 2))
:= 2 + (22/2 &times; (((22/2 + 2)2 ) + 2))
:= 3/3 + ((33 − 3) &times; (3 &times; 33 − 3))
:= (33 &times; ((33 + 33 ) + 3)) − 3
:= 3 + ((3 &times; 3 &times; (3 + 3)3 ) − ((3/3 + 3)3 ))
:= ((4 − 44 )/4) + 44 &times; 44
:= (4 − 4/4) &times; ((4/4 + 4)4 + 4/4)
:= 4 + (((4 − 4/4) &times; (4/4 + 4)4 ) + 4)
:= 55 − ((5 + 5)/5 &times; (55 + 5)/5)
:= (5/5 + 5) &times; (55 + 5)/(5 + 5)
:= 5 + ((5/5 + 5) &times; (55 + 5)/(5 + 5))
:= 6/6 + (6 &times; ((6 &times; (6 &times; 6 + 6) − 6) + 66))
:= 6 &times; 6 &times; (66 − 6 − 6) − 66
:= 6 + (6 &times; 6 &times; (66 − 6 − 6) − (66 + 6/6))
:= 77/7 + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 7 + (((7 &times; (7 &times; 7 &times; 7 − 77)) + ((7 + 7)/7)) + 7)
:= 7 &times; (((7 &times; 7 &times; 77) − 7)/(7 + 7))
:= 8/8 + ((8 + 8) &times; (8 &times; (8 + 8) − (88/8)))
:= 8 + ((8/8 + 8 + 8) &times; (888 − 8)/8)
:= 8 + ((8/8 + 8 + 8 + 8) &times; (88/8 + 8 &times; 8))
:= 9 + (((9 + 9) &times; 99 + 9 &times; 9) + 9/9)
:= (((9 − 9/9) + 9) &times; 999/9) − 9
:= 99 + ((9 + 9) &times; 99 + ((9 + 9)/9))
&Iuml; 1874 := (1 + 1) &times; ((1 + 1)11 − 1111)
2
2+2
&Iuml; 1879 := (1 + 1)11 − (1 + 1 + 11)1+1
&Iuml; 1884 := (1 + 11) &times; (1 + ((1 + 11) &times; (1 + 1 + 11)))
:= 222/2 − ((22/2 + 2)2 )
:= 2 &times; ((2 &times; (222 − 2)) − 22)
:= 3 + ((33 + 3/3) &times; (((3/3 + 3)3 ) + 3))
:= 3 + (33 &times; ((33 + 33 ) + 3))
:= 4 + ((4 − 4/4) &times; (4/4 + 4)4 )
:= 44 + (4 &times; (444 + 4 &times; 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 &times; (6 − 6 &times; 6)))
:= 6/6 + (6 &times; 6 &times; (66 − 6 − 6) − 66)
:= 6 + (6 &times; 6 &times; (66 − 6 − 6) − 66)
:= 777 + (7777/7 − (7 + 7))
:= 7 + ((77/7 + 7) &times; (777/7 − 7))
:= 7/7 + (7 &times; (((7 &times; 7 &times; 77) − 7)/(7 + 7)))
:= 8 + ((((8 + 8)/8) &times; (888 + 8/8)) + 88)
:= 8 &times; (88 + 8) + 8888/8
:= 88 + (((8 &times; 8 &times; (8 &times; 8 − 8)) + 8)/((8 + 8)/8))
:= 9 &times; 9 + ((9 + 9) &times; 99 + (99/9))
:= 99 + ((9 + 9) &times; 99 − ((9 + 9)/9))
:= (9 + 9) &times; 99 + (999/9 − 9)
:= 2 + (2 &times; (2 &times; (22 − 2
)))
:= 3 + (((33 − 3) &times; (3 &times; 33 − 3)) − 3/3)
4
:= 44 &times; 44 + (((4 − 4 ) + 4)/4)
5
5
5
&Iuml; 1875 := 1 + ((1 + 1) &times; ((1 + 1)11 − 1111))
&Iuml; 1885 := (1 + 1 + 11) &times; (1 + (1 + 11)1+1 )
&Iuml; 1880 := 1 + ((1 + 1)11 − (1 + 1 + 11)1+1 )
:= (2/2 + 2) &times; ((2/2 + 2 + 2)2+2 )
:= 2 &times; (2 &times; ((222 − 22+2 ) + 2))
:= (22/2)2 + (2 &times; 22 − 2)2
:= 3 + ((33 − 3) &times; (3 &times; 33 − 3))
:= (3 &times; 3 &times; (3 + 3)3 ) − ((3/3 + 3)3 )
:= 3 + ((33 &times; ((33 + 33 ) + 3)) + 3/3)
:= (4 − 4/4) &times; (4/4 + 4)4
:= (44 − 4) &times; (44 − 4/4 + 4)
:= 4 + (44/4 &times; (4 &times; 44 − (4/4 + 4)))
:= 5 &times; (5 &times; (5 &times; (5 + 5 + 5)))
:= 5 + (5 &times; (5 &times; (5 &times; (5 + 5 + 5))))
:= 5 + ((5 &times; (5 &times; (5 &times; (5 + 5 + 5)))) + 5)
:= 6 + (66 &times; (6 &times; 6 − 6) − 666/6)
:= 6 &times; 6 &times; (66 − 6 − 6) − ((6 + 6)/6)
:= (6/6 − 66) &times; ((6/6 − 6 &times; 6) + 6)
:= 7 + (((7 &times; (7 &times; 7 &times; 7 − 77)) − 7/7) + 7)
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) + (77/7))
:= ((7 + 7)/7) + (7 &times; (((7 &times; 7 &times; 77) − 7)/(7 + 7)))
:= (8/8 + 8 + 8 + 8) &times; (88/8 + 8 &times; 8)
:= 8 + ((8 + 8) &times; (8 &times; (8 + 8) − (88/8)))
:= 8 + ((8/8 − (8 &times; 8 &times; 88))/(8 − 88/8))
:= 9 &times; 9 + ((9 + 9) &times; 99 + ((99 + 9)/9))
:= 99 + ((9 + 9) &times; 99 − 9/9)
:= (9 + 9) &times; 99 + (((999 + 9)/9) − 9)
&Iuml; 1876 := (1 + 1) &times; (1 + ((1 + 1)11 − 1111))
6
&Iuml; 1881 := 11 &times; (1 + (1 + (1 + 1 + 11)1+1 ))
&Iuml; 1886 := 1 + ((1 + 1 + 11) &times; (1 + (1 + 11)1+1 ))
:= 2 &times; ((2 &times; (222 − 22+2 )) + 2)
:= 22/2 &times; (((22/2 + 2)2 ) + 2)
:= 2 + (2 &times; ((2 &times; (222 − 2)) − 22))
:= (33 + 3/3) &times; (((3/3 + 3)3 ) + 3)
:= 33 &times; ((33 + 33 ) + 3)
:= (33 − 3/3 − 3) &times; (3 &times; 33 + 3/3)
:= 4 + (4 &times; ((44 − 44) + 44 ))
:= 44/4 &times; (4 &times; 44 − (4/4 + 4))
:= 4 &times; 444 + (444 − 4)/4
:= 55 + ((5 − (55 + 55 ))/5)
:= 5 + ((5 &times; (5 &times; (5 &times; (5 + 5 + 5)))) + 5/5)
:= 55 + ((55 − (55 + 55 ))/5)
:= 6 + ((66/6 + 6) &times; ((666 − 6)/6))
:= ((66/6 + 6) &times; 666/6) − 6
:= ((66/6 + 6) &times; 666/6) − 6/6
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) + 7)
:= 777 + (7777/7 − 7)
:= ((7 &times; 7 &times; 77) − 7/7)/((7 + 7)/7)
:= 88 + (((8 &times; 8 &times; (8 &times; 8 − 8)) − 8)/((8 + 8)/8))
:= (88/8 + 8) &times; (88/8 + 88)
:= ((8/8 + 8 + 8) &times; 888/8) − 8/8
:= 99 + (((9 − 99)/(9 + 9)) + (9 + 9) &times; 99)
:= 99 + (9 + 9) &times; 99
:= (((9 + 9)/9)99/9 ) − 9 &times; (9 + 9)
&Iuml; 1877 := 1 + ((1 + 1) &times; (1 + ((1 + 1)11 − 1111)))
&Iuml; 1882 := 1 + (11 &times; (1 + (1 + (1 + 1 + 11)1+1 )))
&Iuml; 1887 := 111 &times; (1 + (1 + 1)1+1+1+1 )
:= 2 + ((2 &times; 22 − 2)2 + 222/2)
:= (2 &times; ((2 &times; (222 − 2)) − 22)) − 2
:= 222/2 &times; (22+2 + 2/2)
:= (3 &times; 3 &times; (3 + 3)3 ) − (((3/3 + 3)3 ) + 3)
:= 3/3 + (33 &times; ((33 + 33 ) + 3))
:= (3 &times; (3 &times; ((3 + 3)3 − (3 + 3)))) − 3
:= 4 + (((4 − 44 )/4) + 44 &times; 44)
:= 4 + ((4 − 4/4) &times; ((4/4 + 4)4 + 4/4))
:= (4 − 4/4) &times; ((4/4 + 4)4 + 4)
:= 5 + ((5/5 + 5) &times; (55 − 5)/(5 + 5))
:= 5 + (((5/5 + 5) &times; (55 − 5)/(5 + 5)) + 5)
:= (5 − (5 + 5)/5) &times; ((55 − 5)/5 + 5)
:= 6 &times; 6 &times; (66 − 6 − 6) − (66 + 6/6)
:= 6 + (((66/6 + 6) &times; ((666 − 6)/6)) + 6)
:= (66/6 + 6) &times; 666/6
:= 7 + (((7 &times; (7 &times; 7 &times; 7 − 77)) + 7/7) + 7)
:= 777 + ((7777 + 7)/7 − 7)
:= 777 + (7777 − 7)/7
:= (8/8 − (8 &times; 8 &times; 88))/(8 − 88/8)
:= 8/8 + ((88/8 + 8) &times; (88/8 + 88))
:= (8/8 + 8 + 8) &times; 888/8
:= (((9 + 9)/9)99/9 ) − (9 &times; (9 + 9) + 9)
:= 9/9 + ((9 + 9) &times; 99 + 99)
:= ((9 − 9/9) + 9) &times; 999/9
135
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 1888 := 1 + (111 &times; (1 + (1 + 1)1+1+1+1 ))
&Iuml; 1898 := (1 + 1 + 11) &times; (1 + (1 + (1 + 11)1+1 ))
&Iuml; 1893 := (1 + 1)11 − (11 + (1 + 11)1+1 )
:= 2 &times; (2 &times; 222 − (22 + 2))
:= 2/2 + (2 &times; (2 &times; 222 − 22))
:= 2 + ((2 &times; 22)2 + (2 &times; (2 − 22)))
:= 3/3 + ((3 &times; (3 &times; ((3 + 3)3 − (3 + 3)))) − 3)
:= 3 + (3 &times; (3 &times; ((3 + 3)3 − (3 + 3))))
:= (33 − 3/3) &times; (((3 + 3)3 + 3)/3)
:= (4 + 4) &times; (44 − (4 &times; 4 + 4))
:= 4/4 + (44 &times; (44 − 4/4))
:= 4 + ((44 &times; (44 − 4/4)) + (4 + 4)/4)
:= ((5 + 5)/5)5 &times; (55 − 5/5 + 5)
:= (5 − (5 + 5)/5) &times; ((55 + 5)/5 + 5)
:= 5 + ((5 − (5 + 5)/5) &times; ((55 + 5)/5 + 5))
:= 6/6 + ((66/6 + 6) &times; 666/6)
:= 6 + ((66/6 + 6) &times; 666/6)
:= ((66/6 + 6) &times; (666 + 6)/6) − 6
7
:= 777 + 7777/7
:= (7 &times; (7 − 7 &times; 7)) + (((7 + 7 + 7)/7) )
:= 7 + (((7 + 7) &times; (((7 + 7)/7)7 + 7)) + 7/7)
:= (8 + 8) &times; ((888 − 8)/8 + 8)
:= ((8 + 8) &times; (888/8 + 8)) − 88/8
:= 8 + ((8 − 8/8 + 8) &times; (8 &times; (8 + 8) − ((8 + 8)/8)))
:= 999 + (9 &times; 99 − ((9 + 9)/9))
:= (9 + 9) &times; 99 + 999/9
:= 9 + ((9 &times; 99 − 9/9) + 999)
&Iuml; 1889 := ((1 + 1) &times; (11 − 1)1+1+1 ) − 111
&Iuml; 1894 := (1 + 1)11 − (11 &times; (1 + (1 + 1 + 11)))
:= 2 + (222/2 &times; (22+2 + 2/2))
2
&Iuml; 1899 := (11 − 1 − 1) &times; ((1 + 1) &times; 111 − 11)
:= 2 + (2 &times; (2 &times; 22 − 22))
:= (2/2 + 2)2 &times; (222 − 22/2)
:= (3 &times; (3 &times; ((3 + 3) − (3 + 3)))) − 3/3
:= 3 + ((3 &times; (3 &times; ((3 + 3)3 − (3 + 3)))) + 3/3)
:= 3 &times; ((3 &times; ((3 + 3)3 − (3 + 3))) + 3)
:= 4/4 + ((4 + 4) &times; (44 − (4 &times; 4 + 4)))
:= (4 + 4)/4 + (44 &times; (44 − 4/4))
:= (4 − 4/4) &times; (((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 &times; ((5 &times; (5 &times; (5 + 5 + 5))) + 5)) − 5/5
:= ((6 + 6 + 6) &times; (666/6 − 6)) − 6/6
:= 6 + (((66/6 + 6) &times; 666/6) + 6/6)
:= 6 + (((66/6 + 6) &times; 666/6) + 6)
:= 777 + (7777 + 7)/7
:= 7 + ((7777 − 7)/7 + 777)
:= 777 + ((7777 + 77)/7)
:= 8 + ((88/8 + 8) &times; (88/8 + 88))
:= 8 + (((8/8 + 8 + 8) &times; 888/8) − 8/8)
:= 88/8 + ((8 + 8) &times; ((888 − 8)/8 + 8))
:= 999 + (9 &times; 99 − 9/9)
:= (9 + 9) &times; 99 + ((999 + 9)/9)
:= 9 + (999 + 9 &times; 99)
&Iuml; 1890 := 1 + (((1 + 1) &times; (11 − 1)1+1+1 ) − 111)
&Iuml; 1895 := 1 + ((1 + 1)11 − (11 &times; (1 + (1 + 1 + 11))))
&Iuml; 1900 := (11 − 1)1+1 &times; (((1 + 1) &times; (11 − 1)) − 1)
:= (2 &times; 22)2 − ((2 &times; 22) + 2)
:= 2 + ((2 &times; (2 &times; 222 − 22)) + 2/2)
:= 2 &times; ((2 &times; (222 + 2)) − 22)
:= 3 &times; (3 &times; ((3 + 3)3 − (3 + 3)))
:= ((33 − 3/3) &times; (((3 + 3)3 + 3)/3)) − 3
:= (((3 &times; (3 + 3))3 − 33)/3) − 33
:= (44 + 4/4) &times; (44 − (4 + 4)/4)
:= 4 + 44 &times; 44 − (44 + 4/4)
:= 4 + ((44 &times; (44 − 4/4)) + 4)
:= (5 + 5 + 5) &times; (5 &times; 5 &times; 5 + 5/5)
:= (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) + 5)) − 5
:= 5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) + 5)
:= (6 + 6 + 6) &times; (666/6 − 6)
:= (6 − 6/6) &times; (6 &times; 66 − (66/6 + 6))
:= (6 − 6/6) &times; ((((6 − 66)/6) − 6) + 6 &times; 66)
:= (7 + 7) &times; (((7 + 7)/7)7 + 7)
:= 7 + (7777/7 + 777)
:= (7/7 + 7 &times; 7) &times; (7 &times; 7 − 77/7)
:= (8 − 8/8 + 8) &times; (8 &times; (8 + 8) − ((8 + 8)/8))
:= 8 + ((8/8 + 8 + 8) &times; 888/8)
:= (88/8 + 8) &times; (((88 + 8)/8) + 88)
99/9
:= 9 + ((((9 + 9)/9)
:= 999 + 9 &times; 99
&Iuml; 1891 := (1 + 1)11 − (1 + ((1 + 11) &times; (1 + 1 + 11)))
2
) − 9 &times; (9 + 9))
&Iuml; 1896 := (1 + 11) &times; ((1 + 1 + 11)1+1 − 11)
:= (9/9 + 9 + 9) &times; (9/9 + 99)
&Iuml; 1901 := 1 + ((11 − 1)1+1 &times; (((1 + 1) &times; (11 − 1)) − 1))
:= (2 &times; 22) − (2 &times; 22 + 2/2)
:= 2 &times; ((2 &times; 222 − 22) + 2)
:= 2/2 + (2 &times; ((2 &times; (222 + 2)) − 22))
:= 3/3 + (3 &times; (3 &times; ((3 + 3)3 − (3 + 3))))
:= 3 + ((3 &times; (3 &times; ((3 + 3)3 − (3 + 3)))) + 3)
:= 3 + ((33 − 3/3) &times; (((3 + 3)3 + 3)/3))
:= 44 &times; 44 − (44 + 4/4)
:= 4 + 44 &times; (44 − 4/4)
5
:= 4 + (((44 &times; (44 − 4/4)) + 4/4) + 4)
5
:= (((5 + 5 + 5) &times; (5 /5 + 5)) + 5)/5
:= 5 + ((((5 + 5 + 5) &times; (5 /5 + 5)) + 5)/5)
:= 5/5 + (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) + 5))
:= 6 + ((6/6 − 66) &times; ((6/6 − 6 &times; 6) + 6))
:= 6 + ((6 + 6 + 6) &times; (666/6 − 6))
:= (6 &times; (6 &times; (6 &times; 6 + 6) + 66)) − 6/6 − 6
:= 7 + ((7777 + 7)/7 + 777)
:= 77/7 + ((7 + 7) &times; (((7 + 7)/7)7 + 7))
:= 8 + (((8/8 + 8 + 8 + 8) &times; (88/8 + 8 &times; 8)) + 8)
:= (8 + 8 + 8) &times; (88 − (8/8 + 8))
:= 888 + (8 &times; 8 &times; (8 + 8) − (88/8))
:= 9/9 + (999 + 9 &times; 99)
:= 9 + (((9 − 9/9) + 9) &times; 999/9)
:= 99/9 + (999 + 9 &times; 99)
7
:= 7/7 + ((7 + 7) &times; (((7 + 7)/7) + 7))
&Iuml; 1892 := 11 &times; (1 + (1 + (1 + (1 + 1 + 11)1+1 )))
&Iuml; 1897 := 1 + ((1 + 11) &times; ((1 + 1 + 11)1+1 − 11))
&Iuml; 1902 := (1 + 1)11 − (1 + (1 + (1 + 11)1+1 ))
:= 2 &times; (2 &times; 222 − 22)
:= 2/2 + ((2 &times; 22)2 + (2 &times; (2 − 22)))
:= 2 + (2 &times; ((2 &times; (222 + 2)) − 22))
:= 3 + ((3 &times; (3 &times; ((3 + 3)3 − (3 + 3)))) − 3/3)
:= (((3 &times; (3 + 3))3 − 33)/3) − (33 + 3)
:= (3 &times; (3 &times; (3 + 3)3 − 3)) − 33
:= 44 &times; (44 − 4/4)
:= 4 + ((44 &times; (44 − 4/4)) + 4/4)
:= 44 &times; 44 + ((44 − 4)/4 − 44)
5
5
:= 5 + ((5 − (5 + 5)/5) &times; ((5 − 5)/5 + 5))
:= 5 &times; 5 + ((5/5 + 5) &times; (5 − 5)/(5 + 5))
:= (5/5 + 5) &times; ((55 − 5)/(5 + 5) + 5)
:= 6 + (((66/6 + 6) &times; 666/6) − 6/6)
:= 66 &times; (6 + 6) + ((6666/6) − 6)
:= (6 &times; (6 &times; (6 &times; 6 + 6) + 66)) − 6
7
7
:= 7 + ((7 + 7) &times; (((7 + 7)/7) + 7))
:= 7 + ((7777/7 + 777) + 7)
:= 88 &times; (8 + 8) + 88 &times; 88/(8 + 8)
:= 8/8 + ((8 + 8 + 8) &times; (88 − (8/8 + 8)))
:= ((8 + 8)/8) &times; ((888 − 8/8) + 8 &times; 8)
:= 99 + ((9 + 9) &times; 99 + (99/9))
:= 9 + ((999 − ((9 + 9)/9)) + 9 &times; 99)
:= 9 + ((9 + 9) &times; 99 + 999/9)
:= ((7 + 7)/7) + ((7 + 7) &times; (((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
&Iuml; 1908 := (1 + 11) &times; (1 + ((1 + 1 + 11)1+1 − 11))
&Iuml; 1903 := (1 + 1)11 − (1 + (1 + 11)1+1 )
&Iuml; 1913 := (1 + 1)11 − (1 + (1 + (1 + (11 &times; (1 + 11)))))
:= (2 &times; 22)2 − (22/2 + 22)
:= 2 + ((2 &times; (2 &times; (222 − 2))) − 22)
:= (2 &times; 22)2 − (22 + 2/2)
:= 3 + ((((3 &times; (3 + 3))3 − 33)/3) − 33)
:= 3 &times; ((3 &times; ((3 + 3)3 − 3)) − 3)
:= 3 + ((((3 &times; (3 + 3))3 − 3)/3) − 33)
:= 44/4 + (44 &times; (44 − 4/4))
:= 4 + (4 &times; (444 + 4 &times; (4 + 4)))
:= 4 + (44 &times; 44 − (44/4 + 4 &times; 4))
:= 5 &times; 5 + ((5/5 + 5) &times; (55 + 5)/(5 + 5))
:= (5/5 + 5) &times; ((55 + 5)/(5 + 5) + 5)
:= 5 + ((5/5 + 5) &times; ((55 + 5)/(5 + 5) + 5))
:= 66 &times; (6 + 6) + (6666/6)
:= 6 &times; (6 &times; (6 &times; 6 + 6) + 66)
:= 66 &times; (6 &times; 6 − 6) − (66 + 6/6)
:= 777/7 + ((7 + 7) &times; ((7 + 7)/7) )
:= (77/7 + 7) &times; ((7 &times; (7 + 7) + 7/7) + 7)
:= ((7/7 + 7 + 7) &times; ((7 + 7)/7)7 ) − 7
:= ((8 + 8) &times; (888/8 + 8)) − 8/8
:= ((8 + 8) &times; (8 &times; (8 + 8) − 8)) − (88 + 8)/8
:= 8/8 + (8 &times; 8 &times; (8 + 8) + 888)
:= 99/9 &times; (99/9 + 9 &times; (9 + 9))
:= 9 + ((999 + 9 &times; 99) + 9)
:= 9 + (((9 − 9/9) + 9) &times; ((999 + 9)/9))
7
&Iuml; 1909 := 1 + ((1 + 11) &times; (1 + ((1 + 1 + 11)1+1 − 11)))
&Iuml; 1904 := (1 + 1)11 − (1 + 11)1+1
&Iuml; 1914 := (1 + 1)11 − (1 + (1 + (11 &times; (1 + 11))))
:= 2 &times; (2 &times; (222 − 2 &times; (2 + 2)))
:= (22 + 2/2) &times; (((2/2 + 2)2+2 ) + 2)
:= (2 &times; 22)2 − 22
:= ((33/3)3 ) + (((3 &times; 3 + 3)3 /3) − 3)
:= 3/3 + (3 &times; ((3 &times; ((3 + 3)3 − 3)) − 3))
:= (3 &times; (3 &times; ((3 + 3)3 − 3))) − 3
:= 4 &times; (444 + 4 &times; (4 + 4))
:= 44 &times; 44 − (44/4 + 4 &times; 4)
:= 44/4 &times; (4 &times; 44 − (4 + 4)/4)
5
5
:= 5 + (5 − 55/5 &times; 555/5)
:= 55 &times; 55 − 5555/5
:= (66/6 + 6) &times; (666 + 6)/6
:= 6/6 + (6 &times; (6 &times; (6 &times; 6 + 6) + 66))
:= 66 &times; (6 &times; 6 − (6/6 + 6))
:= (7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) − 7
:= (7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) − (7 + 7)/7
:= 77/7 &times; ((7 &times; (7 + 7) − 7/7) + 77)
:= (8 + 8) &times; (888/8 + 8)
:= ((8 + 8) &times; (8 &times; (8 + 8) − 8)) − 88/8
:= (88 − 8/8) &times; (88 + 88)/8
:= ((9 − 9/9) + 9) &times; ((999 + 9)/9)
:= 9 + ((9/9 + 9 + 9) &times; (9/9 + 99))
:= 99/9 &times; (((99 + 9)/9) + 9 &times; (9 + 9))
:= 5 − 55/5 &times; 555/5
&Iuml; 1905 := 1 + ((1 + 1)11 − (1 + 11)1+1 )
&Iuml; 1910 := 11 + ((11 − 1 − 1) &times; ((1 + 1) &times; 111 − 11))
&Iuml; 1915 := (1 + 1)11 − (1 + (11 &times; (1 + 11)))
:= 2 + ((2 &times; 22)2 − (22/2 + 22))
:= (2 &times; 22)2 − (22 + 2 + 2)
:= 2/2 + ((2 &times; 22)2 − 22)
:= (3 &times; ((3 &times; ((3 + 3)3 − 3)) − 3)) − 3
:= (((3 &times; (3 + 3))3 − 3)/3) − 33
:= 3 + ((((3 &times; (3 + 3))3 + 3)/3) − 33)
:= 4/4 + (4 &times; (444 + 4 &times; (4 + 4)))
:= (4 − 44)/4 + ((4 + 4) &times; (44 − 4 &times; 4))
:= 44 &times; 44 − ((4 &times; 4 + 4/4) + 4)
:= 5 + (5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) + 5))
:= 55 − (5 &times; ((5 − (5 + 5)/5)5 ))
:= (55 &times; ((5 &times; 5 + 5) + 5)) − 5 − 5
:= 6 + ((((66/6 + 6) &times; 666/6) + 6) + 6)
:= 6 + ((66/6 + 6) &times; (666 + 6)/6)
:= 6/6 + (66 &times; (6 &times; 6 − (6/6 + 6)))
:= 7/7 + ((7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) − 7)
:= (7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) − 7/7
:= 7 + ((77/7 + 7) &times; ((7 &times; (7 + 7) + 7/7) + 7))
:= 8/8 + ((8 + 8) &times; (888/8 + 8))
:= 888 + (8 &times; 8 &times; (8 + 8) − ((8 + 8)/8))
:= 88/8 + ((8 + 8) &times; (888/8 + 8))
:= 9 + ((((9 − 9/9) + 9) &times; 999/9) + 9)
:= 9 + ((999 + 9 &times; 99) + (99/9))
:= 9 &times; 99 + (((9 + 9)/9)9/9+9 )
&Iuml; 1906 := 1 + (1 + ((1 + 1)11 − (1 + 11)1+1 ))
&Iuml; 1911 := (1 + 1 + 11) &times; (1 + (1 + (1 + (1 + 11)1+1 )))
&Iuml; 1916 := (1 + 1)11 − (11 &times; (1 + 11))
:= (2 &times; (2 &times; (222 − 2))) − 22
:= (2 &times; 22)2 − ((22 + 2/2) + 2)
:= 2 + ((2 &times; 22)2 − 22)
:= (3 &times; (3 &times; ((3 + 3)3 − 3))) − 33/3
:= (3 &times; 3 &times; (3 + 3)3 ) − 33
:= (((3 &times; (3 + 3))3 − 3)/3) − 33
:= (4 + 4)/4 + (4 &times; (444 + 4 &times; (4 + 4)))
:= 4 &times; 44 + (((4 + 4) &times; 444 − 4)/4)
:= 44 &times; 44 − (4 &times; 4 + 4)
:= 5 + ((5 &times; ((5 &times; (5 &times; (5 + 5 + 5))) + 5)) + 5/5)
:= 55 + (5/5 − (5 &times; ((5 − (5 + 5)/5)5 )))
:= 55 + ((5 − (55 &times; (55 + 55)))/5)
:= (6 &times; (6 &times; (6 &times; 6 + 6) + 66)) − (6 + 6)/6
:= 666/6 + (6 − 6 &times; 6) &times; (6 − 66)
:= 66 &times; (6 &times; 6 − 6) − ((6 + 6)/6)6
:= ((7/7 + 7 + 7) &times; ((7 + 7)/7)7 ) − (7 + 7)
:= 7 &times; ((7 &times; 7 &times; 7 − 77) + 7)
:= 7 + ((7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) − ((7 + 7)/7))
:= (8 + 8)/8 + ((8 + 8) &times; (888/8 + 8))
:= 888 + (8 &times; 8 &times; (8 + 8) − 8/8)
:= ((8 + 8) &times; (8 &times; (8 + 8) − 8)) − (8/((8 + 8)/8))
:= 9 + (((9 + 9) &times; 99 + 999/9) + 9)
:= 9 &times; 9 &times; 9 + (((99 &times; (99 + 9)) − 9)/9)
9/9+9
:= 9 &times; 99 + ((((9 + 9)/9)
) − 9)
&Iuml; 1907 := 1 + (1 + (1 + ((1 + 1)11 − (1 + 11)1+1 )))
&Iuml; 1912 := 1 + ((1 + 1 + 11) &times; (1 + (1 + (1 + (1 + 11)1+1 ))))
&Iuml; 1917 := 1 + ((1 + 1)11 − (11 &times; (1 + 11)))
:= 2/2 + ((2 &times; (2 &times; (222 − 2))) − 22)
:= (2 &times; 22)2 − (22 + 2)
:= 2 + (((2 &times; 22)2 − 22) + 2/2)
:= ((33/3)3 ) + ((3 &times; 3 + 3)3 /3)
:= (((3 &times; (3 + 3))3 + 3)/3) − 33
:= 3 &times; (3 &times; ((3 + 3)3 − 3))
:= 4 + ((44 &times; (44 − 4/4)) + 44/4)
:= (4 + 4) &times; (44 − (4 &times; 4 + 4/4))
:= 4/4 + (44 &times; 44 − (4 &times; 4 + 4))
:= 5 + ((5/5 + 5) &times; ((5 − 5)/(5 + 5) + 5))
:= 5 + (((5/5 + 5) &times; ((5 − 5)/(5 + 5) + 5)) + 5)
:= 55 + ((5 + 5)/5 &times; ((5 − 55 &times; 55)/5))
:= (6 &times; (6 &times; (6 &times; 6 + 6) + 66)) − 6/6
:= 66 &times; (6 &times; 6 − 6) − (((6 + 6)/6) + 66)
:= ((6 &times; 6/(6 + 6))6 ) + (66 &times; (6 + 6 + 6))
:= 7 + ((7/7 + 7 &times; 7) &times; (7 &times; 7 − 77/7))
:= 7/7 + (7 &times; ((7 &times; 7 &times; 7 − 77) + 7))
:= 7 + ((7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) − 7/7)
:= 8 &times; 8 + ((88/8 + 8) &times; ((8/8 + 88) + 8))
:= 888 + 8 &times; 8 &times; (8 + 8)
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) − (88/8))
:= 9 + (((9 &times; 99 − 9/9) + 999) + 9)
:= 9 + ((99/9) &times; (99/9 + 9 &times; (9 + 9)))
:= 999 + (999 − 9 &times; 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
&Iuml; 1918 := 1 + (1 + ((1 + 1)11 − (11 &times; (1 + 11))))
&Iuml; 1923 := 1 + ((1 + 1) &times; ((1 + ((11 − 1) &times; (1 + 1 + 1)))1+1 ))
&Iuml; 1928 := 1 + ((1 + 1)11 − 111+1 )
:= 2 + (((2 &times; 22)2 − 22) + 2)
:= (2 &times; 22)2 − (22/2 + 2)
:= 2 &times; (2 &times; (222 − 2))
:= 3/3 + (3 &times; (3 &times; ((3 + 3)3 − 3)))
:= 3 + ((3 &times; (3 &times; ((3 + 3)3 − 3))) + 3)
:= 33/3 + (3 &times; (3 &times; ((3 + 3)3 − 3)))
:= 44 &times; 44 − ((4 + 4)/4 + 4 &times; 4)
:= 4 + (44 &times; 44 − (4 &times; 4 + 4/4))
:= 44 &times; 44 − 4 − 4
:= ((5 + 5)/5 + 5) &times; (5 &times; 55 − 5/5)
:= (((5 + 5)/5)55/5 ) − 5 &times; 5 &times; 5
:= 5 + ((((5 + 5)/5)55/5 ) − 5 &times; 5 &times; 5)
6
:= ((6 &times; 6 − 6) &times; ((6 + 6)/6) ) − (6 + 6)/6
:= 6 + ((66 &times; (6 &times; 6 − 6) − ((6 + 6)/6)6 ) + 6)
:= 6 &times; 6 + ((66/6 + 6) &times; 666/6)
7
:= 7 + (7 &times; ((7 &times; 7 &times; 7 − 77) + 7))
:= (7 &times; 7 &times; (7 &times; 7 − 7)) − (((7 + 7)/7) + 7)
:= 7 + (((7/7 + 7 + 7) &times; ((7 + 7)/7)7 ) + 7/7)
:= ((8 + 8) &times; (8 &times; (8 + 8) − 8)) − (8 + 8)/8
:= 888 + (8 &times; 8 &times; (8 + 8) + (88/8))
:= 8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8))
:= 9 + (((9/9 + 9 + 9) &times; (9/9 + 99)) + 9)
:= (9 + 9) &times; (99 + 9) − (((99 + 9)/9) + 9)
:= 9 + ((9/9 + 9 + 9) &times; ((9 + 9)/9 + 99))
&Iuml; 1919 := 1 + (1 + (1 + ((1 + 1)11 − (11 &times; (1 + 11)))))
2
:= (2 &times; 22) − (2
2+2
+ 2/2)
&Iuml; 1924 := (1 + 1)11 − (1 + (1 + (1 + 111+1 )))
&Iuml; 1929 := 1 + (1 + ((1 + 1)11 − 111+1 ))
:= 2 &times; ((2 &times; (222 − 2)) − 2)
:= 2/2 + (2 &times; (2 &times; (222 − 2)))
:= 3 + ((((3 &times; (3 + 3)) − 3)/3) − 3 )
:= (3 &times; (3 &times; (3 + 3)3 − 3)) − 33/3
:= 44 &times; 44 − (4 &times; 4 + 4/4)
:= 3 + (3 &times; ((3 &times; ((3 + 3)3 − 3)) + 3))
:= 4 + ((4 + 4) &times; (44 − 4 &times; 4))
:= 4 + (44 &times; 44 − 44/4)
3
3
:= (55 &times; ((5 &times; 5 + 5) + 5)) − (5/5 + 5)
:= ((6 &times; 6 − 6) &times; ((6 + 6)/6)6 ) − 6/6
:= (55 &times; ((5 &times; 5 + 5) + 5)) − 5/5
:= 5 + ((55 &times; ((5 &times; 5 + 5) + 5)) − 5/5)
6
:= 7 + ((7 &times; ((7 &times; 7 &times; 7 − 77) + 7)) + 7/7)
:= ((8 + 8) &times; (8 &times; (8 + 8) − 8)) − 8/8
:= (9/9 + 9 + 9) &times; ((9 + 9)/9 + 99)
&Iuml; 1920 := (111+1 − 1) &times; (1 + 1)1+1+1+1
:= (6 &times; 6 + 6/6) &times; (((6 + 6)/6) − (6 + 6))
:= (((6 &times; 666) − 6)/((6 + 6)/6)) − 66
:= (77 &times; (77/7 + 7 + 7)) − 7/7
:= (7 &times; 7 &times; (77 + 7)) − (((7 + 7 + 7)/7)7 )
:= (8/((8 + 8)/8)) + ((8 + 8) &times; (8 &times; (8 + 8) − 8))
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) + 8/8)
:= (9 + 9) &times; (99 + 9) − (99/9 + 9)
:= 9 + ((9/9 + 9) &times; (999/9 + 9 &times; 9))
&Iuml; 1925 := (1 + 1)11 − (1 + (1 + 111+1 ))
2
:= 2 &times; (2 &times; (22 − (2 + 2)))
&Iuml; 1930 := 1 + (1 + (1 + ((1 + 1)11 − 111+1 )))
2
:= (2 &times; 22) − 22/2
:= 2 + (2 &times; (2 &times; (222 − 2)))
:= ((33/3)3 ) + 3 &times; 33 &times; (3 + 3)
:= (((3 &times; (3 + 3))3 − 33)/3) − 3
:= 44 &times; 44 − 44/4
:= 44 &times; 44 − ((4 + 4)/4 + 4)
:= (6 &times; 6 − 6) &times; ((6 + 6)/6)6
:= 55 &times; ((5 &times; 5 + 5) + 5)
:= 5 + (55 &times; ((5 &times; 5 + 5) + 5))
:= (6 − 6/6) &times; (6 &times; 66 − (66/6))
:= (6 − 6/6) &times; (((6 − 66)/6) + 6 &times; 66)
:= (7/7 + 7 + 7) &times; ((7 + 7)/7)7
:= 77 &times; (77/7 + 7 + 7)
:= (7 &times; 7 &times; (7 &times; 7 − 7)) − ((7 + 7)/7)7
:= (8 + 8) &times; (8 &times; (8 + 8) − 8)
:= 88/8 &times; (888/8 + 8 &times; 8)
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) + ((8 + 8)/8))
:= (9/9 + 9) &times; (999/9 + 9 &times; 9)
:= (9 + 9) &times; (99 + 9) − (9/9 + 9 + 9)
:= 9 + (9999/9 + 9 &times; (9 &times; 9 + 9))
:= 3 + (3 &times; (3 &times; ((3 + 3)3 − 3)))
:= (4 + 4) &times; (44 − 4 &times; 4)
:= (55 + 5) &times; ((5 + 5)/5)5
&Iuml; 1921 := 1 + ((111+1 − 1) &times; (1 + 1)1+1+1+1 )
:= (((2
2+2+2
2
− 2) ) − 2)/2
&Iuml; 1926 := (1 + 1)11 − (1 + 111+1 )
&Iuml; 1931 := 1 + (1 + (1 + (1 + ((1 + 1)11 − 111+1 ))))
:= (2 &times; (2 &times; (222 − 2))) − 2
:= (2 &times; 22)2 − (2/2 + 2 + 2)
:= 3 &times; ((3 &times; ((3 + 3)3 − 3)) + 3)
:= (((3 &times; (3 + 3))3 − 3)/3) − (3 &times; 3 + 3)
:= (4 − 44)/4 + 44 &times; 44
:= 44 &times; 44 − (4/4 + 4)
:= 5/5 + ((55 + 5) &times; ((5 + 5)/5) )
:= 5/5 + (55 &times; ((5 &times; 5 + 5) + 5))
:= 5 + ((55 &times; ((5 &times; 5 + 5) + 5)) + 5/5)
:= 6/6 + ((6 &times; 6 − 6) &times; ((6 + 6)/6)6 )
:= 6 + ((6 &times; 6 − 6) &times; ((6 + 6)/6)6 )
:= 6 + ((6 − 6/6) &times; (6 &times; 66 − (66/6)))
:= 7/7 + ((7/7 + 7 + 7) &times; ((7 + 7)/7)7 )
:= 7/7 + (77 &times; (77/7 + 7 + 7))
:= 7 + ((77 &times; (77/7 + 7 + 7)) − 7/7)
:= 8/8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8))
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) − ((8 + 8)/8))
:= 88/8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8))
:= 9 &times; (9 &times; 9 + 9) + 9999/9
:= (9 + 9) &times; ((99 − 9/9) + 9)
:= (9 + 9) &times; (99 + 9) − ((99 + 9 + 9)/9)
3
:= 3 + ((3 &times; (3 &times; ((3 + 3) − 3))) + 3/3)
4
:= 4/4 + ((4 + 4) &times; (4 − 4 &times; 4))
5
&Iuml; 1922 := (1 + 1) &times; ((1 + ((11 − 1) &times; (1 + 1 + 1)))1+1 )
:= ((2
2+2+2
2
&Iuml; 1927 := (1 + 1)11 − 111+1
&Iuml; 1932 := (1 + 1) &times; ((1 + 1) &times; (((11 + 11)1+1 ) − 1))
2
− 2) )/2
:= 2 + ((2 &times; 22) − 22/2)
3
:= 2 &times; (2 &times; 222 − 2)
3
:= (((3 &times; (3 + 3)) + 33)/3) − 33
4
:= (4 + 4)/4 + ((4 + 4) &times; (4 − 4 &times; 4))
5
:= (5 + 5)/5 + ((55 + 5) &times; ((5 + 5)/5) )
6
:= 6 + (66 &times; (6 &times; 6 − 6) − ((6 + 6)/6) )
:= 3/3 + (3 &times; ((3 &times; ((3 + 3) − 3)) + 3))
:= (3 &times; (3 &times; (3 + 3)3 − 3)) − 3
:= 44 &times; 44 − ((4/4 + 4) + 4)
:= 44 &times; 44 − 4
:= (5 + 5)/5 + (55 &times; ((5 &times; 5 + 5) + 5))
6
:= 6 + (((6 &times; 6 − 6) &times; ((6 + 6)/6) ) + 6/6)
7
:= ((5 + 5)/5 + 5) &times; (5 &times; 55 + 5/5)
:= 6 &times; 6 &times; (66 − 6 − 6) − 6 − 6
:= 7 + ((7/7 + 7 + 7) &times; ((7 + 7)/7) )
:= 7 + (77 &times; (77/7 + 7 + 7))
:= (8 + 8)/8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8))
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) − 8/8)
:= ((8 + 8)/8) &times; ((88 &times; 88 − (8 + 8))/8)
:= ((9 + 9)/9) &times; ((9 &times; (99 + 9)) − (99/9))
:= 9/9 + ((9 + 9) &times; ((99 − 9/9) + 9))
:= ((99 + 9)/9) &times; (9 &times; (9 + 9) − 9/9)
:= 77/7 + (7 &times; ((7 &times; 7 &times; 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
&Iuml; 1938 := 1 + ((1 + 1)11 − 111)
&Iuml; 1933 := (1 + 1)11 − (1 + (1 + 1 + 1 + 111))
&Iuml; 1943 := (1 + 1)11 − (111 − ((1 + 1) &times; (1 + 1 + 1)))
:= (2 &times; 22)2 − 2/2 − 2
:= 2 + (2 &times; 22)2
:= (2 &times; (2 &times; (222 + 2))) − 2/2
:= ((3 &times; (3 + 3))3 − 33)/3
:= 3 + (3 &times; (3 &times; (3 + 3)3 − 3))
:= ((3 &times; (3 + 3))3 − 3)/3
:= 4/4 + (44 &times; 44 − 4)
:= (4 + 4)/4 + 44 &times; 44
:= 4 + ((44 &times; 44 − 4/4) + 4)
:= 55 + ((5/5 + 5) &times; (55 + 5)/(5 + 5))
:= (5 + 5)/5 &times; ((5 − 5/5)5 − 55)
:= ((5/5 + 5)5 /(5 − 5/5)) − 5/5
:= 6 &times; 6 &times; (66 − 6 − 6) − 66/6
:= 6 &times; 6 &times; (66 − 6 − 6) − 6
:= 6 &times; 6 &times; (66 − 6 − 6) − 6/6
:= 7 + ((77 &times; (77/7 + 7 + 7)) + 7/7)
:= 77 + ((7 &times; (7 &times; 7 &times; 7 − 77)) − 7/7)
:= 7 + ((((7 + 7)/7 − 7) + 7 &times; 7)(7+7)/7 )
:= 8 + (88/8 &times; (888/8 + 8 &times; 8))
:= ((8 + 8)/8) &times; ((88 &times; 88 + 8)/8)
:= 8 + ((8 − 8/8 + 8) &times; (8 &times; (8 + 8) + 8/8))
:= (9 + 9) &times; (99 + 9) − 99/9
:= (9/9 + 9 + 9) &times; (999/9 − 9)
:= (9 + 9) &times; (99 + 9) − 9/9
&Iuml; 1934 := (1 + 1)11 − (1 + 1 + 1 + 111)
&Iuml; 1939 := 1 + (1 + ((1 + 1)11 − 111))
&Iuml; 1944 := (1 + 1) &times; ((1 + 11) &times; (11 − 1 − 1)1+1 )
:= (2 &times; 22)2 − 2
:= 2 + ((2 &times; 22)2 + 2/2)
:= 2 &times; (2 &times; (222 + 2))
:= (((3 &times; (3 + 3))3 − 3)/3) − 3 &times; 3
:= (((3 &times; (3 + 3))3 + 3)/3) − (3 + 3)
:= 3 &times; 3 &times; (3 + 3)3
:= 44 &times; 44 − (4 + 4)/4
:= 4 + (44 &times; 44 − 4/4)
:= 4 + (44 &times; 44 + 4)
5
5
:= ((5/5 + 5) /(5 − 5/5)) − 5
:= (5/5 + 5)5 /(5 − 5/5)
:= 6/6 + (6 &times; 6 &times; (66 − 6 − 6) − 6)
:= 6 &times; (6 &times; (66 − 6 − 6))
:= 7 + (((7/7 + 7 + 7) &times; ((7 + 7)/7) ) + 7)
:= 77 + (7 &times; (7 &times; 7 &times; 7 − 77))
:= (7 − 7/7) &times; ((77/7 + 7)(7+7)/7 )
:= ((8 + 8)/8) &times; ((88 &times; 88 − 8)/8)
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) + (88/8))
:= 8 + (88 &times; (88 + 88)/8)
:= (9 + 9) &times; (99 + 9) − 9/9 − 9
:= 99 + (9999/9 + 9 &times; 9 &times; 9)
:= (9 + 9) &times; (99 + 9)
:= ((5/5 + 5) /(5 − 5/5)) − 5 − 5
:= ((6 − 66)/6) + 6 &times; 6 &times; (66 − 6 − 6)
7
&Iuml; 1940 := 1 + (1 + (1 + ((1 + 1)11 − 111)))
&Iuml; 1935 := (1 + 1)11 − (1 + 1 + 111)
&Iuml; 1945 := 1 + ((1 + 1) &times; ((1 + 11) &times; (11 − 1 − 1)1+1 ))
:= (2 &times; 22)2 − 2/2
:= 2 + ((2 &times; 22)2 + 2)
:= 2/2 + (2 &times; (2 &times; (222 + 2)))
:= 3 &times; (3 &times; (3 + 3)3 − 3)
:= (((3 &times; (3 + 3))3 − 3)/3) − 3
:= ((3 &times; (3 + 3))3 + 3)/3
:= 44 &times; 44 − 4/4
:= 4 + 44 &times; 44
:= 4 + ((44 &times; 44 + 4/4) + 4)
:= 5 + ((55 &times; ((5 &times; 5 + 5) + 5)) + 5)
:= 5 + (((55 &times; ((5 &times; 5 + 5) + 5)) + 5) + 5)
:= 55 − (55 /5 + 555)
:= (6 &times; 6/(6 + 6) + 6) &times; (6 &times; 6 &times; 6 − 6/6)
:= (6 + 6)/6 + (6 &times; 6 &times; (66 − 6 − 6) − 6)
:= 6/6 + 6 &times; 6 &times; (66 − 6 − 6)
7
:= (7/7 + 7 + 7) &times; (((7 + 7)/7) + 7/7)
:= 7/7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) + 77)
:= 7 + (((7 &times; (7 &times; 7 &times; 7 − 77)) − 7/7) + 77)
:= (8 − 8/8 + 8) &times; (8 &times; (8 + 8) + 8/8)
:= ((8 + 8)/8) &times; (88 &times; 88 + 8 + 8)/8
:= 8 + ((88 &times; (88 + 88)/8) + 8/8)
:= (9 + 9) &times; (99 + 9) − 9
:= (99/9 + 9) &times; (99 − ((9 + 9)/9))
:= 9/9 + (9 + 9) &times; (99 + 9)
&Iuml; 1941 := 1 + (1 + (1 + (1 + ((1 + 1)11 − 111))))
&Iuml; 1936 := (1 + 1)11 − (1 + 111)
&Iuml; 1946 := 11 + ((1 + 1)11 − (1 + 1 + 111))
2
2
:= 2 + (((2 &times; 22) + 2/2) + 2)
:= (2 &times; 22)
:= 2 + (2 &times; (2 &times; (222 + 2)))
3
3
:= 3 + (((3 &times; (3 + 3)) − 33)/3)
:= (3 &times; 3 &times; (3 + 3) ) − 3
:= 3 + (((3 &times; (3 + 3))3 − 3)/3)
:= 44 &times; 44
:= 4 + (44 &times; 44 + 4/4)
:= 44 &times; 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 &times; 6) + 6)
(6+6)/6
:= ((6 − (6
6−6/6
)) + 6)/((6 + 6)/6 − 6)
:= (6 + 6)/6 + 6 &times; 6 &times; (66 − 6 − 6)
7
(7+7)/7
:= 7 + ((((7/7 + 7 + 7) &times; ((7 + 7)/7) ) + 7) + 7)
:= 7 + ((7 &times; (7 &times; 7 &times; 7 − 77)) + 77)
:= 88 &times; (88 + 88)/8
:= 8 + ((88/8 &times; (888/8 + 8 &times; 8)) + 8)
:= 8 + (((8 + 8)/8) &times; ((88 &times; 88 + 8)/8))
:= 9/9 + ((9 + 9) &times; (99 + 9) − 9)
:= (9 + 9) &times; (99 + 9) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 + 9) &times; (99 + 9)
:= (((7 + 7)/7 − 7) + 7 &times; 7)
&Iuml; 1942 := 1 + (1 + (1 + (1 + (1 + ((1 + 1)11 − 111)))))
&Iuml; 1937 := (1 + 1)11 − 111
&Iuml; 1947 := 11 + ((1 + 1)11 − (1 + 111))
:= 2/2 + (2 &times; 22)2
:= 2 + (((2 &times; 22)2 + 2) + 2)
:= 22/2 + (2 &times; 22)2
:= (((3 &times; (3 + 3))3 − 3)/3) − (3 + 3)
:= ((3 &times; (3 + 3))3 − (3 + 3))/3
:= 3 + (3 &times; 3 &times; (3 + 3)3 )
:= 4/4 + 44 &times; 44
:= 4 + (44 &times; 44 + (4 + 4)/4)
:= 44/4 + 44 &times; 44
5
:= 55 + ((((5 − 55 ) + 5)/5) − 555)
:= ((5/5 + 5) /(5 − 5/5)) − (5 + 5)/5
:= 5 + (((5 + 5)/5 + 5) &times; (5 &times; 55 + 5/5))
:= 6 &times; 6 &times; (66 − 6 − 6) − (6 + 6)/6
:= 6 &times; 6 &times; (66 − 6 − 6) − 6/6 − 6
:= 66/6 &times; (666/6 + 66)
7
:= (7 &times; ((7 − 7 &times; 7) + 7)) + (((7 + 7 + 7)/7) )
:= 77/7 &times; (((7 + 7)/7)7 + 7 &times; 7)
:= 8/8 + (88 &times; (88 + 88)/8)
:= 8 + (((8 + 8)/8) &times; ((88 &times; 88 − 8)/8))
:= 88/8 &times; ((88 + 88) + 8/8)
:= (9 + 9)/9 + ((9 + 9) &times; (99 + 9) − 9)
:= (9 + 9) &times; (99 + 9) − (9 + 9)/9
:= ((9 + 9 + 9)/9) + (9 + 9) &times; (99 + 9)
7
:= 7 + ((7 &times; 7 &times; (7 &times; 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
&Iuml; 1953 := 1 + ((1 + 111+1 ) &times; (1 + 1)1+1+1+1 )
&Iuml; 1948 := 11 + ((1 + 1)11 − 111)
2
2
2+2
:= 22 + (2 &times; 22)2
:= 2 &times; ((2 &times; (22 + 2)) + 2)
:= 2/2 + ((2 &times; 22) + 2
:= 3 + (((3 &times; (3 + 3))3 + 3)/3)
:= 3 &times; (3 &times; (3 + 3)3 + 3)
:= 3 + (((3 &times; (3 + 3))3 + 33)/3)
:= 4 + ((44 &times; 44 + 4) + 4)
:= 4 &times; 4 + (44 &times; 44 + 4/4)
:= 44/4 &times; (4 &times; 44 + (4 + 4)/4)
:= 5 + (((5/5 + 5)5 /(5 − 5/5)) − 5/5)
:= (5 − (5 + 5)/5) &times; ((55 + 5)/5 + 5 &times; 5)
:= 55 − ((5555 + 5)/5 + 55)
:= 6 + (6 &times; 6 &times; (66 − 6 − 6) − ((6 + 6)/6))
:= 6 + ((66/6) &times; (666/6 + 66))
:= 66/6 &times; ((666 + 6)/6 + 66)
:= ((7 − 777)/7) + (7 &times; 7 &times; (7 &times; 7 − 7))
:= 777 + (7 + 7) &times; (77 + 7)
:= ((7 + 7)/7) &times; ((7 + 7) &times; (77 − 7) − 7/7)
:= 8 + (((8 + 8)/8) &times; (88 &times; 88 + 8 + 8)/8)
:= 8/8 + ((8 + 8) &times; ((888 + 88)/8))
:= (8/8 + 88) &times; (88 + 88)/8
:= ((9 + 9)/9) &times; ((9 &times; (99 + 9)) + ((9 + 9)/9))
:= 9 + (9 + 9) &times; (99 + 9)
:= 9 + ((((9 + 9)/9)99/9 ) − 99)
&Iuml; 1949 := 1 + (11 + ((1 + 1)11 − 111))
)
&Iuml; 1958 := (1 + 1) &times; (11 &times; (111 − 11 − 11))
&Iuml; 1954 := 1 + (1 + ((1 + 111+1 ) &times; (1 + 1)1+1+1+1 ))
&Iuml; 1959 := 11 + (11 + ((1 + 1)11 − 111))
:= 2 + ((2 &times; 22)2 + 22/2)
:= 2 + ((2 &times; 22)2 + 22+2 )
:= 22 + ((2 &times; 22)2 + 2/2)
:= 3 + ((((3 &times; (3 + 3))3 − 3)/3) + 3)
:= 3 &times; 3 + (((3 &times; (3 + 3))3 + 3)/3)
:= 3 + ((3 &times; (3 &times; (3 + 3)3 + 3)) + 3)
:= 4 + (((44 &times; 44 + 4/4) + 4) + 4)
:= 4 &times; 4 + (44 &times; 44 + (4 + 4)/4)
:= ((4 + 4) &times; (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 &times; 6 &times; (66 − 6 − 6) − 6/6)
:= ((66 − 6)/6) + 6 &times; 6 &times; (66 − 6 − 6)
:= ((6 &times; (666 − (6 + 6))) − 6)/((6 + 6)/6)
7
:= (7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7))) − 77/7
:= 7 + (77/7 &times; (((7 + 7)/7) + 7 &times; 7))
:= (7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7))) − 7/7
:= (8 &times; (8 + 8) &times; (8 + 8)) − (88/8 + 88)
:= ((8 + 8)/8) &times; (((88 &times; 88 + 8)/8) + 8)
:= (8 &times; (8 + 8) &times; (8 + 8)) − (8/8 + 88)
:= 9 + ((9 + 9) &times; (99 + 9) + 9/9)
:= ((99 − 9/9) &times; (99/9 + 9)) − 9/9
99/9
:= (((9 + 9)/9)
) − 99
&Iuml; 1955 := (1 + 1)11 − (((1 + 1)11−1 − 1)/11)
&Iuml; 1950 := 1 + (1 + (11 + ((1 + 1)11 − 111)))
&Iuml; 1960 := (11 − 1) &times; (1 + 1 + 1 + 11)1+1
:= 2 + (2 &times; ((2 &times; (222 + 2)) + 2))
:= 22 + ((2 &times; 22)2 − (2/2 + 2))
:= 2 + ((2 &times; 22)2 + 22)
:= 3 + ((3 &times; 3 &times; (3 + 3)3 ) + 3)
:= ((3 &times; (3 + 3))3 + 33)/3
:= 33 + (((3 &times; (3 + 3))3 − 33)/3)
:= 4 + (44 &times; 44 + (44 − 4)/4)
:= 4 + ((44 &times; 44 + 44/4) + 4)
:= (4 + 4) &times; (44 − 44/4)
:= (5 + 5 + 5) &times; (5 &times; 5 &times; 5 + 5)
:= 5 + ((5 + 5 + 5) &times; (5 &times; 5 &times; 5 + 5))
:= ((5 + 5)/5 + 5) &times; (5 &times; 55 + 5)
:= 666 + (6 &times; 6 &times; 6 &times; 6 − ((6 + 6)/6))
:= 6 + 6 &times; 6 &times; (66 − 6 − 6)
:= 66/6 + 6 &times; 6 &times; (66 − 6 − 6)
:= (7/7 + 77) &times; (77/7 + 7 + 7)
:= (7 − ((7 + 7)/7)) &times; (7 &times; (7 &times; 7 + 7) − 7/7)
:= 7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7))
:= ((8 + 8)/8) &times; (((88 &times; 88 − 8)/8) + 8)
:= 8 + (88/8 &times; ((88 + 88) + 8/8))
:= (8 &times; (8 + 8) &times; (8 + 8)) − 88
:= 9 + ((9 + 9) &times; (99 + 9) − ((9 + 9 + 9)/9))
:= 99/9 + (9 + 9) &times; (99 + 9)
:= (99 − 9/9) &times; (99/9 + 9)
&Iuml; 1951 := 1 + (1 + (1 + (11 + ((1 + 1)11 − 111))))
&Iuml; 1956 := (1 + 1) &times; (1111 − (1 + (11 &times; (1 + 11))))
&Iuml; 1961 := 1 + (11 − 1) &times; (1 + 1 + 1 + 11)1+1
:= 2 + (((2 &times; 22)2 + 22/2) + 2)
:= 22 + ((2 &times; 22)2 − 2)
:= 2 + (((2 &times; 22)2 + 22) + 2/2)
:= 3 + ((((3 &times; (3 + 3))3 + 3)/3) + 3)
:= 3 + (3 &times; (3 &times; (3 + 3)3 + 3))
:= 3 + ((((3 &times; (3 + 3))3 + 33)/3) + 3)
:= 4 + (44 &times; 44 + 44/4)
:= 4 + (44 &times; 44 + 4 &times; 4)
:= 4 + ((((4 − 4/4) + 4)4 ) − 444)
:= 5/5 + ((5 + 5 + 5) &times; (5 &times; 5 &times; 5 + 5))
:= 5 + (((5 + 5 + 5) &times; (5 &times; 5 &times; 5 + 5)) + 5/5)
:= 5 &times; 5 + ((55 − (55/5))(5+5)/5 )
:= 6 + (6 &times; 6 &times; (66 − 6 − 6) + 6/6)
:= 6 + (6 &times; 6 &times; (66 − 6 − 6) + 6)
:= 666 + (6 &times; 6 &times; 6 &times; 6 − 6/6)
:= (77/7 − 7) &times; (7 &times; (77 − 7) − 7/7)
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7)))
:= 8 &times; 8 + ((8/8 + 8 + 8) &times; 888/8)
:= (((8 &times; 8 &times; 8 &times; 8) − 8)/((8 + 8)/8)) − 88
:= 8/8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 88)
:= 9 + ((9 + 9) &times; (99 + 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + (9 + 9) &times; (99 + 9)
:= 9 + (((9 + 9) &times; (99 + 9) − 9/9) + 9)
:= 7 + ((7 − 7/7) &times; ((77/7 + 7)
(7+7)/7
))
&Iuml; 1952 := (1 + 111+1 ) &times; (1 + 1)1+1+1+1
2
&Iuml; 1957 := (1 + 1)11 − (1 + (11 − 1) &times; (11 − 1 − 1))
2
&Iuml; 1962 := (1 + 1) &times; ((11 − 1 − 1) &times; (111 − 1 − 1))
:= 2 &times; (2 &times; (22 + 2 + 2))
:= 22 + ((2 &times; 22) − 2/2)
:= 2 + (((2 &times; 22)2 + 22) + 2)
:= 3 &times; 3 + (((3 &times; (3 + 3))3 − 3)/3)
:= 3 + ((((3 &times; (3 + 3))3 + 3)/3) + 3 &times; 3)
:= 3 &times; ((3 &times; (3 + 3)3 + 3) + 3)
:= 4 &times; (444 + 44)
5
4
:= 4 + ((44/((4 + 4)/4)) + 44 &times; 44)
5
:= (((4 − 4/4) + 4) ) − 444
:= ((5 + 5)/5) &times; ((55 + 5/5) + 5)
:= 5 + (((5 + 5)/5) &times; ((55 + 5/5) + 5))
:= (55 − 5)/(5 + 5) + (55 &times; (5 &times; 5 + 5))
:= 6 + (6 &times; 6 &times; (66 − 6 − 6) + ((6 + 6)/6))
:= 6 + ((6 &times; 6 &times; (66 − 6 − 6) + 6/6) + 6)
:= 666 + 6 &times; 6 &times; 6 &times; 6
:= 777 + ((7 + 7) &times; (77 + 7) − 7/7)
:= 7 + ((7/7 + 77) &times; (77/7 + 7 + 7))
:= ((7 + 7)/7) + (7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7)))
:= (8 + 8) &times; ((888 + 88)/8)
:= (88/8 + 8) &times; (888/8 − 8)
:= (8 + 8)/8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 88)
:= 9 + ((9 + 9) &times; (99 + 9) − 9/9)
:= (9/9 + 9 + 9) &times; (((999 + 9)/9) − 9)
:= 9 + ((9 + 9) &times; (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
&Iuml; 1963 := 1 + ((1 + 1) &times; ((11 − 1 − 1) &times; (111 − 1 − 1)))
2
&Iuml; 1968 := 1 + ((1 + 1)11 − (11 − 1 − 1)1+1 )
2
2+2
&Iuml; 1973 := 1 + (1 + ((11 − 1 − 1) &times; (((1 + 1) &times; (111 − 1)) − 1)))
:= 2/2 + ((2 &times; 22)2 + (2 + 2 + 2)2 )
:= 2 + ((((2 &times; 22) + 22) + 2/2) + 2)
:= 2 &times; (2 &times; 22 + 2
:= 3/3 + (3 &times; ((3 &times; (3 + 3)3 + 3) + 3))
:= (3 &times; (3 &times; ((3 + 3)3 + 3))) − 3
:= 3 + ((((3 &times; (3 + 3))3 − 3)/3) + 33 )
:= 4 &times; 4 + (44 &times; 44 + 44/4)
:= 4 &times; ((444 + 44) + 4)
:= 4/4 + (((4 + 4) &times; (44 − 4)) − 44)
:= (55 + 5)/(5 + 5) + (55 &times; (5 &times; 5 + 5))
:= (55/5 + 5) &times; (5 &times; 5 &times; 5 − ((5 + 5)/5))
:= (5 &times; (5 &times; 55 − 5)) + (55 − 5 − 5)/5
:= 6/6 + (6 &times; 6 &times; 6 &times; 6 + 666)
:= 66 &times; (6 &times; 6 − 6) − 6 − 6
:= 66 &times; (6 &times; 6 − 6) − 6/6 − 6
:= 7 + ((77/7 − 7) &times; (7 &times; (77 − 7) − 7/7))
:= (7 &times; 7 − 7/7) &times; (7 &times; 7 − (7/7 + 7))
:= 777/7 + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 8 + ((88/8 &times; ((88 + 88) + 8/8)) + 8)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 88)
:= (8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 88/8
:= 9 + (((9 + 9) &times; (99 + 9) + 9/9) + 9)
&Iuml; 1964 := (1 + 1) &times; (1 + ((11 − 1 − 1) &times; (111 − 1 − 1)))
2
:= 2 + ((((2 &times; 22) + 22) + 2) + 2)
3
)
99/9
:= 9/9 + ((((9 + 9)/9)
) − 9 &times; 9)
:= 9 + (((9 + 9) &times; (99 + 9) + (99/9)) + 9)
&Iuml; 1969 := 11 &times; (11 + ((1 + 1 + 11)1+1 − 1))
&Iuml; 1974 := (1 + 1) &times; ((11 − 1)1+1+1 − (1 + 1 + 11))
2
:= 22 + ((2 &times; 22) + 22/2)
3
:= 2 + ((2 &times; 22)2 + (2 + 2 + 2)2 )
3
:= 3 + (((3 &times; (3 + 3)) − (3 + 3))/3)
:= 3 + (3 &times; (3 &times; ((3 + 3)3 + 3)))
:= 44 + (44 &times; 44 − 44/4)
:= 44 + (44 &times; 44 − ((4 + 4)/4 + 4))
:= 5 + (5 − (5555/5 + 55))
:= 5 &times; 5 + ((5/5 + 5)5 /(5 − 5/5))
:= (55 − 5)/5 + (5 &times; (5 &times; 55 − 5))
:= 666 + (6 &times; 6 &times; 6 &times; 6 + ((6 + 6)/6))
:= 66 &times; (6 &times; 6 − 6) − 66/6
:= 3 &times; 3 + (((3 &times; (3 + 3)) + 33)/3)
4
:= 4 + ((4 + 4) &times; (4 − 44/4))
5
:= 66 &times; (6 &times; 6 − 6) − 6
7
:= 7 &times; 7 + ((7/7 + 7 + 7) &times; ((7 + 7)/7) )
:= (7 − 7/7) &times; (7 &times; 7 &times; 7 − (7 + 7))
:= (((8 &times; 8 &times; 8 &times; 8) + 8)/((8 + 8)/8)) − 88
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − 88) + 8/8)
:= (8 − 88)/8 + (8 &times; ((8 + 8) &times; (8 + 8) − 8))
:= 9 + ((9 + 9) &times; (99 + 9) + (99/9))
:= 9 + ((99 − 9/9) &times; (99/9 + 9))
:= 9 &times; 9 + ((9 + 9) &times; 99 + 999/9)
77/7
:= (((7 + 7)/7)
) − (77 + 7)
&Iuml; 1965 := (1 + 1)11 − (1 + (1 + (11 − 1 − 1)1+1 ))
&Iuml; 1970 := 111 + (11 &times; (1 + 1 + 11)1+1 )
2
2+2
:= 22 + ((2 &times; (2 &times; (222 + 2))) − 2/2)
:= 2 + ((2 &times; 22) + 2 &times; 2
:= ((3 + 3) &times; 333) − 33
:= 3 + (((3 &times; (3 + 3)) − 3)/3)
4
:= 4 + (((((4 − 4/4) + 4) ) − 444) + 4)
3
&Iuml; 1975 := 1111 + (((1 + 11)1+1+1 )/(1 + 1))
)
:= (2 &times; 22)2 + (2 &times; (22 − 2) − 2/2)
3
:= 3 + ((3 &times; (3 &times; ((3 + 3)3 + 3))) + 3/3)
4
:= 4 + ((4444/((4 + 4)/4)) − 4 )
5
:= 44 + (44 &times; 44 − (4/4 + 4))
5
:= 5 + (((5 + 5)/5 + 5) &times; (5 &times; 55 + 5))
:= (5 &times; ((5 − 5/5) − 5)) − 5
:= ((6 &times; 666) − 66)/((6 + 6)/6)
:= (6 − 6/6) &times; (6 &times; 66 − ((6 + 6)/6))
:= 77 + (7777/7 + 777)
:= ((77 − 7)/7) &times; ((7 + 7) &times; (7 + 7) + 7/7)
:= 7 + ((7 &times; 7 − 7/7) &times; (7 &times; 7 − (7/7 + 7)))
:= 8 + ((88/8 + 8) &times; (888/8 − 8))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − 88) + ((8 + 8)/8))
:= (8 &times; ((8 + 8) &times; (8 + 8) − 8)) − (8/8 + 8)
:= 9 + ((9 + 9) &times; (99 + 9) + ((99 + 9)/9))
:= 9 &times; 9 &times; (9 + 9) + (((9 + 9)/9)9 )
:= 9 + ((9 + 9) &times; (99 + 9) + ((99 + 99)/9))
&Iuml; 1966 := (1 + 1)11 − (1 + (11 − 1 − 1)1+1 )
:= 22 + (2 &times; (2 &times; (222 + 2)))
:= 3/3 + (((3 + 3) &times; 333) − 33)
&Iuml; 1971 := (11 − 1 − 1) &times; (((1 + 1) &times; (111 − 1)) − 1)
2
:= 6/6 + (66 &times; (6 &times; 6 − 6) − 6)
&Iuml; 1976 := (1 + 1) &times; ((11 − 1)1+1+1 − (1 + 11))
:= 2 &times; ((2 &times; 222 − 2) + 22)
:= 2 + (((2 &times; 22) + 22/2) + 22)
3
:= 33 + (((3 &times; (3 + 3))3 − 3)/3)
:= 3 &times; (3 &times; ((3 + 3) + 3))
4
:= (4444/((4 + 4)/4)) − 44
:= 5 &times; (5 &times; (5 &times; 5 + 55) − 5)
4
:= 4 + ((4 &times; (4 + 4)) − (4 − 4/4) )
5
:= 44 + (44 &times; 44 − 4)
5
:= 5 + ((((5 + 5)/5 + 5) &times; (5 &times; 55 + 5)) + 5/5)
:= 5/5 + ((5 &times; ((5 − 5/5) − 5)) − 5 )
:= 5/5 + (5 &times; (5 &times; (5 &times; 5 + 55) − 5))
:= 66 &times; (6 &times; 6 − 6) − ((6 + 6)/6 + 6 + 6)
:= 6 + (((6 &times; 666) − 66)/((6 + 6)/6))
:= (6 + 6)/6 + (66 &times; (6 &times; 6 − 6) − 6)
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7))) − 7/7)
:= (((7 + 7)/7)
:= 8 + ((8/8 + 88) &times; (88 + 88)/8)
:= 88/8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 88)
:= (8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 8
:= ((9 + 9)/9) &times; ((9 &times; (99 + 9)) + (99/9))
:= 999 + (9 &times; (99 + 9))
:= 9 + ((((9 + 9)/9)99/9 ) − 9 &times; 9)
&Iuml; 1967 := (1 + 1)11 − (11 − 1 − 1)1+1
:= 2
22/2
− ((2/2 + 2)
3
2+2
:= (77 − 7/7) &times; ((77 + 7)/7 + 7 + 7)
) − 77
&Iuml; 1972 := 1 + ((11 − 1 − 1) &times; (((1 + 1) &times; (111 − 1)) − 1))
2
)
:= 2 &times; ((2 &times; (22 − 2)) + 22)
3
:= 3 + ((((3 &times; (3 + 3)) − 3)/3) − 3)
4
77/7
4
3
3
:= 3 + (((3 &times; (3 + 3)) + 3)/3)
4
&Iuml; 1977 := ((1 + 1) &times; ((11 − 1)1+1+1 − 11)) − 1
:= 2/2 + ((2 &times; 22)2 + 2 &times; (22 − 2))
:= 33 + (3 &times; 3 &times; (3 + 3)3 )
:= (4 &times; (4 + 4)) − (4 − 4/4)
:= ((4 + 4) &times; (4 − 4)) − 44
:= 44 + ((44 &times; 44 − 4) + 4/4)
:= ((5 + 5)/5 + 5) &times; ((5 &times; 55 + 5/5) + 5)
:= (((55 + 5)/5) + 5) &times; (555/5 + 5)
:= (5 + 5)/5 + (5 &times; (5 &times; (5 &times; 5 + 55) − 5))
:= 66 &times; (6 &times; 6 − 6) − (6/6 + 6 + 6)
:= 66 &times; (6 &times; 6 − 6) − ((6 + 6)/6 + 6)
:= 66 &times; (6 &times; 6 − 6) − 6 &times; 6/(6 + 6)
77/7
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − (7 + 7)))
:= 7/7 + ((((7 + 7)/7)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) − (8/8 + 88))
:= (8 &times; ((8 + 8) &times; (8 + 8) − 8)) − (88 + 8)/8
:= 8/8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 8)
:= 9/9 + ((9 &times; (99 + 9)) + 999)
:= (99 &times; (99/9 + 9)) − (9 + 9 + 9)/9
99/9
:= (((9 + 9)/9)
)−9&times;9
141
) − 77)
:= 7 + (((77 − 7)/7) &times; ((7 + 7) &times; (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
&Iuml; 1978 := (1 + 1) &times; ((11 − 1)1+1+1 − 11)
&Iuml; 1983 := 1 + ((1 + 1) &times; (1 + ((11 − 1 − 1) &times; (111 − 1))))
&Iuml; 1988 := (1 + 1)11 − (111+1 − 1)/(1 + 1)
:= (2 &times; 22) + ((2 &times; 22)2 − 2)
:= 2 + (((2 &times; 22)2 + 2 &times; 22) + 2/2)
:= 2 &times; ((2 &times; (222 + 2)) + 22)
:= 33 + (((3 &times; (3 + 3))3 + 3)/3)
:= 3 + (33 &times; (33 + 33))
:= 33 + (((3 &times; (3 + 3))3 + 33)/3)
:= 44 + (44 &times; 44 − (4 + 4)/4)
:= (44 &times; (4 + 4)) − (44 + 4)/4
:= 4 + ((4 + 4) &times; (44 − 4 − 4))
:= (55 &times; (55/5 + 5 &times; 5)) − (5 + 5)/5
:= (((5 + 5)/5)55/5 ) − (55 + 5 + 5)
:= (((5 + 5)/5)55/5 ) − (55 + 5)
:= 66 &times; (6 &times; 6 − 6) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + 66 &times; (6 &times; 6 − 6)
:= 6 + (66 &times; (6 &times; 6 − 6) + ((6 + 6)/6))
:= 7 + ((((7 + 7)/7)77/7 ) − 77)
:= ((7 + 7)/7) + ((7 &times; 7 &times; (7 &times; 7 − 7)) − 77)
:= 7 + ((7 &times; 7 &times; (7 &times; 7 − 7)) − 77)
:= (8 + 8)/8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 8)
:= (8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 8/8
:= 888 + ((8888 − 88)/8)
:= ((9 + 9)/9) &times; (999 − (9/9 + 9))
:= ((9 + 9 + 9)/9) + (99 &times; (99/9 + 9))
:= 9 + ((99 &times; (99/9 + 9)) − 9/9)
&Iuml; 1979 := 1 + ((1 + 1) &times; ((11 − 1)1+1+1 − 11))
:= (2 &times; 22) + ((2 &times; 22)2 − 2/2)
&Iuml; 1984 := (1 + 1)11 − ((1 + 1)(1+1)&times;(1+1+1) )
2
:= 2 &times; ((2 &times; 22 + 22) + 2)
3
&Iuml; 1989 := (11 − 1 − 1) &times; ((1 + 1) &times; 111 − 1)
:= (2/2 + 2)2 &times; (222 − 2/2)
:= ((33/3)3 ) + 3 &times; (3 + 3)3
:= 3 + ((33 &times; (3 + 33)) + 3/3)
:= ((3 + 3) &times; 333) − 3 &times; 3
:= 44 + (44 &times; 44 − 4/4)
:= (4 + 4) &times; (44 − 4 − 4)
:= 4 + (((4 + 4) &times; (44 − 4 − 4)) + 4/4)
:= (55 &times; (55/5 + 5 &times; 5)) − 5/5
:= (55/5 + 5) &times; (5 &times; 5 &times; 5 − 5/5)
:= (((5 + 5)5−5/5 ) − 55)/5
:= 66 &times; (6 &times; 6 − 6) − 6/6
:= 6 + (66 &times; (6 &times; 6 − 6) − ((6 + 6)/6))
:= (66/6 + 6) &times; (666/6 + 6)
7
:= (7 &times; 7 &times; (7 &times; 7 − 7)) − ((7 + 7)/7 + 77)
:= (((7 + 7 + 7)/7) ) − ((7 + 7) &times; (7 + 7) + 7)
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − 88) + (88/8))
:= 8 &times; ((8 + 8) &times; (8 + 8) − 8)
:= (8/8 + 8 + 8) &times; (8 &times; (8 + 8) − (88/8))
:= 9 &times; 99 + (9999/9 − (9 + 9))
:= 9 + (99 &times; (99/9 + 9))
:= (99 &times; (99/9 + 9)) − 9/9
&Iuml; 1980 := (1 + 1) &times; ((11 − 1 − 1) &times; (111 − 1))
&Iuml; 1985 := 1 + ((1 + 1)11 − ((1 + 1)(1+1)&times;(1+1+1) ))
2
:= 7 + (((7 &times; 7 &times; (7 &times; 7 − 7)) − 77) + 7/7)
&Iuml; 1990 := 1 + ((11 − 1 − 1) &times; ((1 + 1) &times; 111 − 1))
:= 2 &times; (2 &times; 22 + 22)
:= 2/2 + ((2 &times; 22) + 2 &times; (22 + 2))
:= 2 + (2 &times; ((2 &times; (222 + 2)) + 22))
:= 33 &times; (33 + 33)
:= 3 + ((((33/3)3 ) + 3 &times; (3 + 3)3 ) + 3)
:= 3 + (((3 + 3) &times; 333) − 33/3)
2
:= 44 + 44 &times; 44
:= 4/4 + ((4 + 4) &times; (4 − 4 − 4))
:= 4 + (((4 + 4) &times; (44 − 4 − 4)) + (4 + 4)/4)
:= 55 &times; (55/5 + 5 &times; 5)
:= 5 + (55 &times; (55/5 + 5 &times; 5))
:= (5 &times; (5 − 5/5)5 ) − (55 + 5)
:= 66 &times; (6 &times; 6 − 6)
:= 6 + (66 &times; (6 &times; 6 − 6) − 6/6)
:= (6 − 6/6) &times; (((6 + 6)/6) + 6 &times; 66)
4
77/7
:= (77/7 + 7) &times; (777 − 7)/7
:= 7 + (((((7 + 7)/7)
:= ((((8 + 8)/8) + 8) + 8) &times; (888 − 8)/8
:= 8/8 + (8 &times; ((8 + 8) &times; (8 + 8) − 8))
:= 99 &times; (99/9 + 9)
&Iuml; 1981 := 1 + ((1 + 1) &times; ((11 − 1 − 1) &times; (111 − 1)))
99/9
:= 9 + (((((9 + 9)/9)
) − 77) + 7)
) − 9 &times; 9) + 9)
&Iuml; 1986 := (1 + 1)11 − (1 + ((1 + 111+1 )/(1 + 1)))
:= ((7 + 7)/7)7 + (7 &times; (7 &times; 7 &times; 7 − 77))
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) − ((8 + 8)/8))
:= 9 + ((99 &times; (99/9 + 9)) + 9/9)
&Iuml; 1991 := 11 &times; (1 + (11 + (1 + 1 + 11)1+1 ))
:= 2/2 + ((2 &times; 22)2 + 2 &times; 22)
:= 222 + (2 &times; 22 − 2)2
:= 2 + ((2/2 + 2)2 &times; (222 − 2/2))
:= 3/3 + (33 &times; (33 + 33))
:= 3 + ((33 &times; (33 + 33)) + 3)
:= 33/3 + (33 &times; (33 + 33))
:= 44 + (44 &times; 44 + 4/4)
:= (4 + 4)/4 + ((4 + 4) &times; (44 − 4 − 4))
:= 44 + (44 &times; 44 + 44/4)
:= 5/5 + (55 &times; (55/5 + 5 &times; 5))
:= 5 + ((55 &times; (55/5 + 5 &times; 5)) + 5/5)
:= ((((5 + 5)5−5/5 ) + 5)/5) − 5 − 5
:= 6/6 + 66 &times; (6 &times; 6 − 6)
:= 6 + 66 &times; (6 &times; 6 − 6)
:= 66/6 + 66 &times; (6 &times; 6 − 6)
:= (7 &times; 7 &times; (7 &times; 7 − 7)) − 77
:= (7 − 7/7) &times; (7 &times; 7 &times; 7 − (77 + 7)/7)
:= (((7 + 7 + 7)/7)7 ) − (7 + 7) &times; (7 + 7)
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) − (88/8))
:= (8 + 8)/8 + (8 &times; ((8 + 8) &times; (8 + 8) − 8))
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 8/8)
:= 9/9 + (99 &times; (99/9 + 9))
:= 99 + (((9 − 9/9) + 9) &times; 999/9)
:= 99/9 + (99 &times; (99/9 + 9))
&Iuml; 1982 := (1 + 1) &times; (1 + ((11 − 1 − 1) &times; (111 − 1)))
&Iuml; 1987 := (1 + 1)11 − ((1 + 111+1 )/(1 + 1))
2
&Iuml; 1992 := (1 + 1)11 − (1 + 111)/(1 + 1)
:= 2 + ((2 &times; 22)2 + 2 &times; 22)
:= 2/2 + ((2 &times; 22 − 2) + 222)
:= 2 &times; (((2 &times; (222 + 2)) + 22) + 2)
:= 3 + (((33/3)3 ) + 3 &times; (3 + 3)3 )
:= ((3 + 3) &times; 333) − 33/3
:= (3 + 3) &times; (333 − 3/3)
:= 44 + (44 &times; 44 + (4 + 4)/4)
:= 4 + ((44 &times; (4 + 4)) − (44 + 4)/4)
:= 4 + (((4 + 4) &times; (44 − 4 − 4)) + 4)
5−5/5
:= (5 + 5)/5 + (55 &times; (55/5 + 5 &times; 5))
:= (((5 + 5)
:= (6 + 6)/6 + 66 &times; (6 &times; 6 − 6)
:= 6 + (66 &times; (6 &times; 6 − 6) + 6/6)
:= 6 + (66 &times; (6 &times; 6 − 6) + 6)
:= 7/7 + ((7 &times; 7 &times; (7 &times; 7 − 7)) − 77)
) − (55 + 5 + 5))/5
:= ((55 + 5)/5) &times; (555/5 + 55)
:= 7 + ((77/7 + 7) &times; (777 − 7)/7)
:= (7 − 7/7) &times; (7 &times; 7 &times; 7 − (77/7))
:= (8 &times; ((8 + 8) &times; (8 + 8) − 8)) − (8 + 8)/8
:= 88/8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) − 8)
:= 8 + (8 &times; ((8 + 8) &times; (8 + 8) − 8))
:= (9 + 9)/9 + (99 &times; (99/9 + 9))
:= 999 + (999 − (99/9))
:= 99 + ((9 + 9) &times; 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
&Iuml; 1993 := (1 + 1)11 − (111 − 1)/(1 + 1)
&Iuml; 1998 := (1 + 1) &times; (111 &times; (11 − 1 − 1))
&Iuml; 2003 := 1 + ((1 + 1) &times; (1 + (11 − 1)1+1+1 ))
:= 2 &times; 222 + ((222/2 + 2)/2)
:= 222 &times; (2/2 + 2)2
:= (2 &times; 22 + 2/2)2 − 22
:= 3/3 + ((3 + 3) &times; (333 − 3/3))
:= (3 + 3) &times; 333
:= 3 + (((3 + 3) &times; 333 − 3/3) + 3)
:= (44 &times; (4 + 4)) − (44/4 + 44)
:= 44 &times; 44 + (4 − 4 − 4)/4
:= 44 &times; (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 &times; (6 &times; 6 − 6) + 6/6) + 6)
:= 6 &times; 666 &times; 6/(6 + 6)
:= 6 + ((66 &times; (6 &times; 6 − 6) + (66/6)) + 6)
:= 7777/7 + 7 &times; (77 + 7 &times; 7)
:= (77/7 + 7) &times; 777/7
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) − 7)
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) + 8/8)
:= 888 + (8888 − 8)/8
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) + (88/8))
:= 9 &times; 99 + (9999/9 − 9)
:= (9 + 9) &times; 999/9
:= 9 &times; 99 + (9999 + 9)/9
4
&Iuml; 1994 := 1 + (1 + 1)11 − (111 − 1)/(1 + 1)
&Iuml; 1999 := ((1 + 1) &times; (11 − 1)1+1+1 ) − 1
&Iuml; 2004 := (1 + 1) &times; (1 + 1 + (11 − 1)1+1+1 )
:= (222 &times; (2/2 + 2)2 ) − 2 − 2
:= 2/2 + (222 &times; (2/2 + 2)2 )
:= 222/2 − (2 &times; 22)
:= ((3 + 3) &times; 333) − (3/3 + 3)
:= 3/3 + ((3 + 3) &times; 333)
:= 3 + ((3 + 3) &times; 333 + 3)
:= (4 + 4)/4 &times; (4 &times; (4 − 4) − 44/4)
:= 44 &times; 44 + ((4 − 4)/4)
:= 44 &times; (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 &times; (6 &times; 6 − 6) + ((6 + 6)/6)) + 6)
:= 6/6 + (6 &times; 666 &times; 6/(6 + 6))
:= 6 + (6 &times; 666 &times; 6/(6 + 6))
4
4
77/7
:= (7 + 7)/7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) − 7)
:= (7 &times; (7 &times; (7 &times; 7 − 7) − 7)) − (7/7 + 7 + 7)
:= (((7 + 7)/7)
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) + ((8 + 8)/8))
:= 888 + 8888/8
:= ((8 &times; 8 &times; 8 &times; 8) − 88)/((8 + 8)/8)
:= ((9 + 9)/9) &times; (999 − ((9 + 9)/9))
:= 9/9 + (999 + 999)
:= 9 &times; 99 + (((9999 + 9) + 9)/9)
&Iuml; 1995 := (1 + 1 + 1) &times; ((111+1+1 − 1)/(1 + 1))
)−7&times;7
&Iuml; 2000 := (1 + 1) &times; (11 − 1)1+1+1
&Iuml; 2005 := 1 + ((1 + 1) &times; (1 + 1 + (11 − 1)1+1+1 ))
:= (((2 &times; 22) + 2)2 ) − (22/2)2
:= 2222 − 222
:= 2 + ((2 &times; 22 + 2/2)2 − 22)
:= ((3 + 3) &times; 333) − 3
:= 3 + (((3 + 3) &times; 333) − 3/3)
:= 3 + (((3 + 3) &times; 333 + 3/3) + 3)
4
:= 44/4 + ((4 + 4) &times; (4 − 4 − 4))
:= 4 &times; 4 &times; 4 + 44 &times; 44
:= 4/4 + (44 &times; (4 + 4) − 44)
:= (5 &times; (5 − 5/5)5 ) − 55
:= 5 &times; 5 &times; (5 &times; 5 + 55)
:= 5 + (5 &times; 5 &times; (5 &times; 5 + 55))
:= ((6 &times; 666) − 6)/((6 + 6)/6)
:= (6 + 6)/6 + (6 &times; 666 &times; 6/(6 + 6))
:= (6 − 6/6) &times; ((6 &times; 66 − 6/6) + 6)
:= (7 &times; (7 &times; (7 &times; 7 − 7) − 7)) − (7 + 7)
:= 7/7 + ((((7 + 7)/7)77/7 ) − 7 &times; 7)
:= 7 + (77/7 + 7) &times; 777/7
:= 88/8 + (8 &times; ((8 + 8) &times; (8 + 8) − 8))
:= 8 + ((8 &times; ((8 + 8) &times; (8 + 8) − 8)) + 8)
:= 8 + (((8888 − (8 + 8))/8) + 888)
:= ((9 + 9)/9) &times; (999 + 9/9)
:= 9 + (((9 + 9)/9) &times; (999 − 9/9))
:= 999 + (999 − ((9 + 9 + 9)/9))
&Iuml; 1996 := (1 + 1) &times; ((11 − 1)1+1+1 − (1 + 1))
&Iuml; 2001 := 1 + ((1 + 1) &times; (11 − 1)1+1+1 )
&Iuml; 2006 := (1 + 1) &times; (1 + 1 + 1 + (11 − 1)1+1+1 )
:= (222 &times; (2/2 + 2)2 ) − 2
:= 2/2 + 2222 − 222
:= 2 + (222/2 − (2 &times; 22))
:= 3/3 + (((3 + 3) &times; 333) − 3)
:= 3 + (3 + 3) &times; 333
:= 3 &times; 3 + ((3 + 3) &times; 333 − 3/3)
:= 4 &times; 4 + (44 &times; 44 + 44)
:= 44 &times; 44 + (4 + 4)/4
:= (4 + 4)/4 + (44 &times; (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 &times; (6 &times; 6 − 6) + ((66 − 6)/6))
:= ((6 &times; 666) + 6)/((6 + 6)/6)
:= (((6 + 6)/6)66/6 ) − (6 &times; 6 + 6)
:= 7/7 + ((7 &times; (7 &times; (7 &times; 7 − 7) − 7)) − (7 + 7))
:= 7 &times; (7 &times; (7 &times; 7 − 7) − 7) − (7/7 + 7)
:= 7 + ((((7 + 7)/7)77/7 ) − 7 &times; 7)
:= ((88 + 8)/8) + (8 &times; ((8 + 8) &times; (8 + 8) − 8))
:= (8/8 − 88) &times; (8/8 − (8 + 8 + 8))
:= 8 + ((8888 − 8)/8 + 888)
:= ((9 + 9)/9) &times; (999 − 9/9)
:= 9 &times; 99 + ((9999 − 9)/9)
:= 9 + ((999 − 9/9) + 999)
4
&Iuml; 1997 := ((1 + 1) &times; (111 &times; (11 − 1 − 1))) − 1
2
&Iuml; 2002 := (1 + 1) &times; (1 + (11 − 1)1+1+1 )
&Iuml; 2007 := (11 − 1 − 1) &times; (1 + (1 + 1) &times; 111)
:= (222 &times; (2/2 + 2) ) − 2/2
:= 2 + 2222 − 222
:= (2/2 + 2)2 &times; (222 + 2/2)
:= ((3 + 3) &times; 333) − 3/3
:= 3 + ((3 + 3) &times; 333 + 3/3)
:= 3 &times; 3 + (3 + 3) &times; 333
4
4
:= 44 &times; 44 + ((4 + 4)/4 − 4)
:= 44 &times; 44 + ((4 + 4 + 4)/4)
:= 4 + 44 &times; (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 &times; (6 &times; 6 − 6) + (66/6))
:= 6/6 + (((6 &times; 666) + 6)/((6 + 6)/6))
:= 6 + (((6 &times; 666) + 6)/((6 + 6)/6))
:= 77 + ((7/7 + 7 + 7) &times; ((7 + 7)/7) )
:= 7 &times; (7 &times; (7 &times; 7 − 7) − 7) − 7
:= 7 &times; (7 &times; (7 &times; 7 − 7) − 7) − (7 + 7)/7
:= 888 + ((8888 − (8 + 8))/8)
:= ((8 − 888)/8) + (88 &times; (8 + 8 + 8))
:= 8 + (8888/8 + 888)
:= 999 + (999 − 9/9)
:= 9 &times; 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
&Iuml; 2008 := 1 + ((11 − 1 − 1) &times; (1 + (1 + 1) &times; 111))
&Iuml; 2013 := 1 + ((1 + 1)11 − ((1 + 1 + 1) &times; (1 + 11)))
&Iuml; 2018 := (1 + 1)11 − (11 − 1) &times; (1 + 1 + 1)
:= 222/2 + (2 &times; (2 − 22))
:= 2/2 + (222/2 − (2 + 2 + 2)2 )
:= 2 + (2 &times; (2 &times; ((222 − 2) + 22)))
:= 3 &times; 3 + ((3 + 3) &times; 333 + 3/3)
:= 33 &times; (((3/3 + 3)3 ) − 3)
:= 3 + (((33/3 + 3)3 ) − 33+3 )
:= 4 + (44 &times; (4 + 4) − 44)
:= 4/4 + ((4 + 4) &times; (44 − 4) − 4)
:= (4 + 4)/4 + (4 + 4) &times; (44 − 4)
:= 55 − ((5555 + 5)/5 + 5)
:= 55 − (5555 + 5)/5
:= 5 + (55 − (5555 + 5)/5)
:= ((6 + 6)/6)6 + 6 &times; 6 &times; (66 − 6 − 6)
:= (6 &times; (666 + 6) − 6)/((6 + 6)/6)
:= 6 + ((((6 + 6)/6)66/6 ) − 6 &times; 6)
:= 7 &times; (7 &times; (7 &times; 7 − 7) − 7) − 7/7
:= 7 + (((((7 + 7)/7)77/7 ) − 7 &times; 7) + 7)
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) + ((7 + 7)/7))
:= 88 + ((8 + 8) &times; (8 &times; (8 + 8) − 8))
:= 88/8 &times; ((888/8 + 8 &times; 8) + 8)
:= 8 + (((8888 + 88)/8) + 888)
:= 9 + ((999 + 999) + 9/9)
:= 9 &times; 99 + ((9999 + 99)/9)
:= ((9 + 9)/9) &times; ((999 + 9/9) + 9)
&Iuml; 2014 := (1 + 1)11 − (1 + 11 &times; (1 + 1 + 1))
&Iuml; 2019 := 1 + ((1 + 1)11 − ((11 − 1) &times; (1 + 1 + 1)))
&Iuml; 2009 := 11 + ((1 + 1) &times; (111 &times; (11 − 1 − 1)))
:= (2 &times; 22 + 2/2)2 − 22+2
:= 2 + (222/2 − (2 + 2 + 2)2 )
:= (2 &times; 22 + 2/2)2 − (2 + 2 + 2)
:= 33/3 + (3 + 3) &times; 333
:= 3/3 + (33 &times; (((3/3 + 3)3 ) − 3))
:= 3 + (3 + 3) &times; (333 + 3)
:= (4 + 4) &times; (44 − 4) − (4 + 4)/4
:= 4 + ((4 + 4) &times; (44 − 4) − 4/4)
:= 5 − (5555/5 + 5)
:= 55 − 5555/5
:= 5 + (55 − 5555/5)
:= ((6 − 6/6) &times; (6 &times; 66 + 6)) − 6/6
:= 6 &times; 6 + (66 &times; (6 &times; 6 − 6) − ((6 + 6)/6))
:= (6 &times; (666 + 6) + 6)/((6 + 6)/6)
:= 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) − ((7 + 7)/7))
:= ((77 − 7)/7) + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= 8 + ((8/8 − 88) &times; (8/8 − (8 + 8 + 8)))
:= 8 + (((8888 − 8)/8 + 888) + 8)
:= 88 + (((8 + 8) &times; (8 &times; (8 + 8) − 8)) + (88/8))
:= 9 + (((9 + 9)/9) &times; (999 + 9/9))
:= ((9 + 9)/9) &times; ((999 − 9/9) + 9)
:= 9 + ((((9999 − 9)/9) + 9 &times; 99) + 9)
4
:= 4 + ((4 &times; (4 + 4) − 44) + 4/4)
5
&Iuml; 2010 := 11 + (((1 + 1) &times; (11 − 1)1+1+1 ) − 1)
&Iuml; 2015 := (1 + 1)11 − (11 &times; (1 + 1 + 1))
&Iuml; 2020 := (1 + 1) &times; ((11111 − 1)/11)
:= 222/2 − (22/2 + 22)
:= 2 &times; (2 &times; (222 + 22) − 2)
:= ((33/3 + 3)3 ) − 33+3
:= 3 + ((3 + 3) &times; (333 + 3) + 3/3)
:= (4 + 4) &times; (4 − 4) − ((4 + 4)/4 + 4)
:= (4 + 4) &times; (44 − 4) − 4/4
:= 4 + (4 + 4) &times; (44 − 4)
:= 5 + ((5 &times; 5 &times; (5 &times; 5 + 55)) + 5)
:= 55 + ((5 − 5555)/5)
:= 55 + ((55 &times; (5 − 5 &times; 5)) − 5)
:= (6 − 6/6) &times; (6 &times; 66 + 6)
:= 6 &times; 6 + (66 &times; (6 &times; 6 − 6) − 6/6)
:= (6 − 6/6) &times; ((((6 + 6)/6) + 6 &times; 66) + 6)
:= 7/7 + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) − 7/7)
:= 77/7 + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= 888 + ((8888 + 88)/8)
:= 8 + ((8888/8 + 888) + 8)
:= ((8 &times; (8 &times; 8 &times; 8 − 8)) + 8)/((8 + 8)/8)
:= 9 + (((9999 − 9)/9) + 9 &times; 99)
:= ((9 + 9) &times; ((999 + 9)/9)) − 9/9
:= ((9 + 9)/9) &times; (99/9 + 999)
:= 2 + (2
22/2
+ (2 &times; (2 − 22)))
:= 3 + ((3 + 3) &times; 333 + 3 &times; 3)
4
&Iuml; 2011 := 11 + ((1 + 1) &times; (11 − 1)1+1+1 )
&Iuml; 2016 := 1 + ((1 + 1)11 − (11 &times; (1 + 1 + 1)))
&Iuml; 2021 := (1 + 1)11 − (1 + 1 + 1)1+1+1
:= 2 + ((2 &times; 22 + 2/2)2 − 22+2 )
:= 2 &times; (2 &times; ((222 − 2) + 22))
:= (2 &times; 22 + 2/2)2 − 2 − 2
:= 3 + (((3 + 3) &times; 333 + 3 &times; 3) + 3/3)
:= (3 + 3) &times; (333 + 3)
:= (3 − 3/3)33/3 − 33
:= (4 + 4) &times; (44 − 4) − (4/4 + 4)
:= (4 + 4) &times; (44 − 4)
:= 4 + ((4 + 4) &times; (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) &times; (6 &times; 66 + 6))
:= 6 &times; (66 &times; (6 − 6/6) + 6)
:= 6 + ((66 &times; (6 &times; 6 − 6) − 6/6) + 6 &times; 6)
:= (7 + 7)/7 + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= 7 + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= (77 + 7)/7 + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= 8 + (((8 &times; ((8 + 8) &times; (8 + 8) − 8)) + (88/8)) + 8)
:= 8 &times; ((8 &times; 8 &times; 8 − 8)/((8 + 8)/8))
:= (8 &times; (8 + 8) &times; (8 + 8)) − (88/8 + 8 + 8)
:= 9 + (9999/9 + 9 &times; 99)
:= (9 + 9) &times; ((999 + 9)/9)
:= (((9 + 9)/9)99/9 ) − (9 + 9 + 9)
:= (((5 + 5)
5−5/5
) + 55)/5
&Iuml; 2012 := (1 + 1)11 − ((1 + 1 + 1) &times; (1 + 11))
&Iuml; 2017 := 1 + (1 + ((1 + 1)11 − (11 &times; (1 + 1 + 1))))
&Iuml; 2022 := (1 + 1) &times; (11 + (11 − 1)1+1+1 )
:= 222/2 − (2 + 2 + 2)2
:= (2 &times; 22)2 + (2/2 + 2)2+2
:= 222/2 − 22 − 2 − 2
:= 3 + ((3 + 3) &times; 333 + 33/3)
:= 3/3 + (3 + 3) &times; (333 + 3)
:= 3 + ((3 + 3) &times; (333 + 3) + 3)
4
4
:= (4 + 4) &times; (4 − 4) − 4
:= 4/4 + (4 + 4) &times; (4 − 4)
:= 4 + ((4 + 4) &times; (44 − 4) + (4 + 4)/4)
:= 55 − (5555 + 5 + 5)/5
:= 5 + (55 − (5555 + 5 + 5)/5)
:= (5/5 + 5) &times; ((55 − 5)/(5 + 5) + 5 &times; 5)
:= (((6 + 6)/6)66/6 ) − 6 &times; 6
:= 6 &times; 6 + (66 &times; (6 &times; 6 − 6) + 6/6)
:= 6 + (66 &times; (6 &times; 6 − 6) + 6 &times; 6)
:= 7 + ((77/7 + 7) &times; 777/7 + 7)
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) + 7/7)
:= (7 − 7/7) &times; ((7 &times; 7 &times; 7 − 7) + 7/7)
:= ((8 &times; (8 &times; 8 &times; 8 − 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 &times; ((8 &times; 8 &times; 8 − 8)/((8 + 8)/8)))
:= (88 &times; (8 + 8 + 8)) − ((8 + 8)/8 + 88)
:= 9 + ((9999 + 9)/9 + 9 &times; 99)
:= 9/9 + ((9 + 9) &times; ((999 + 9)/9))
:= ((9 + 9)/9) &times; (((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
&Iuml; 2023 := (1 + 1)11 − (1 + ((1 + 1) &times; (1 + 11)))
&Iuml; 2028 := (1 + 11) &times; (1 + 1 + 11)1+1
&Iuml; 2033 := (1 + 1)11 − 1 − 1 − 1 − 1 − 11
:= (2 &times; 22 + 2/2)2 − 2
:= 2 + (222/2 − 22)
:= 2/2 + (222/2 − 22+2 )
:= 3 + (((3 + 3) &times; (333 + 3) + 3/3) + 3)
:= 3 + ((3 + 3) &times; 333 + 33 )
:= 33+3 + ((33/3)3 − 33 )
:= 4 + (((4 + 4) &times; (4 − 4) − 4/4) + 4)
:= 44 + (4 &times; 444 − 4)
:= 4/4 + (4 &times; 444 + 44 )
:= (((5 + 5)/5)55/5 ) − 5 &times; 5
:= 5 + ((((5 + 5)/5)55/5 ) − 5 &times; 5)
:= (((5 + 5)/5)55/5 ) − (5 + 5 + 5)
:= 6 + ((66 &times; (6 &times; 6 − 6) + 6 &times; 6) + 6/6)
:= 6 + ((66 &times; (6 &times; 6 − 6) + 6 &times; 6) + 6)
:= (6 &times; (666 &times; 6/(6 + 6) + 6)) − 6/6
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) + 7)
:= (7 − 7/7) &times; ((7 &times; 7 &times; 7 − 7) + ((7 + 7)/7))
:= ((7 + 7 + 7)/7)7 − 77 − 77
:= (8/8 + 8 + 8) &times; (888/8 + 8)
:= 8 + (((8 &times; (8 &times; 8 &times; 8 − 8)) + 8)/((8 + 8)/8))
:= 8/8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8/8))
:= (((9 + 9)/9)99/9 ) − (99/9 + 9)
:= (9/9 + 9 + 9) &times; ((99 − 9/9) + 9)
4
9/9+9
:= 999 + (((9 + 9)/9)
)
&Iuml; 2024 := (1 + 1)11 − ((1 + 1) &times; (1 + 11))
&Iuml; 2029 := 1 + ((1 + 11) &times; (1 + 1 + 11)1+1 )
&Iuml; 2034 := (1 + 1)11 − 1 − 1 − 1 − 11
:= 2 &times; (2 &times; (222 + 22))
:= 2 + ((2 &times; 22 + 2/2)2 + 2)
:= 2 + (222/2 − 22+2 )
:= 3 + ((3 − 3/3)33/3 − 33 )
:= (((3 &times; 33 − 3)3−3/3 ) + 3)/3
:= (3 + 3) &times; (333 + 3 + 3)
:= 4 + ((4 + 4) &times; (4 − 4) + 4)
:= 4 + ((44 + 4/4)(4+4)/4 )
:= 44 + (4 &times; 444 + (4 + 4)/4)
:= 5 + (55 − 5555/5 + 5)
:= 5 + ((55 − 5555/5 + 5) + 5)
:= (55 &times; (((5 + 5)/5)5 + 5)) − 5/5
:= 6 + (((((6 + 6)/6)66/6 ) − 6 &times; 6) + 6)
:= ((6 − 6/6) &times; (6 &times; 66 + (66/6))) − 6
:= 6 &times; (666 &times; 6/(6 + 6) + 6)
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − 7) − 7) + 7/7) + 7)
:= 7 + ((7 − 7/7) &times; ((7 &times; 7 &times; 7 − 7) + 7/7))
:= (((7 + 7)/7)77/7 ) − (7 + 7)
:= 88 &times; ((8 − 8/8 + 8) + 8)
:= (8 &times; (8 + 8) &times; (8 + 8)) − (88/8 + 8)
:= (8 + 8)/8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8/8))
:= 9 &times; 9 + ((9 + 9) &times; (99 + 9) − 9/9)
:= 9 + (((9 + 9)/9) &times; (99/9 + 999))
:= 9 + ((9 + 9) &times; (99 + 9) + 9 &times; 9)
4
&Iuml; 2025 := (1 + 1)11 − (1 + (11 + 11))
&Iuml; 2030 := (1 + 1)11 − (1 + 1) &times; (11 − 1 − 1)
&Iuml; 2035 := (1 + 1)11 − 1 − 1 − 11
:= (2 &times; 22 + 2/2)2
:= 2 + ((222/2 − 22) + 2)
:= 222/2 − (22/2 + 2)
:= 3 &times; (3 &times; (3 + 3)3 + 33 )
:= 33 + ((3 + 3) &times; 333 − 3/3)
:= 3/3 + ((3 + 3) &times; (333 + 3 + 3))
:= (44 + 4/4)
:= 4 + ((4 + 4) − 44)/((4 + 4)/4)
:= 4 + ((4 &times; 444 − 4/4) + 44 )
:= 5 &times; (5 &times; (5 &times; 5 + 55) + 5)
:= 5 + ((55 &times; (5 − 5 &times; 5)) + 55 )
:= 55 &times; (((5 + 5)/5)5 + 5)
:= 6 &times; 6 &times; 6 &times; 6 + ((6 &times; 6/(6 + 6))6 )
:= (6 &times; 6 − 6/6) &times; (((6 + 6)/6)6 − 6)
:= (6 − 6/6) &times; (6 &times; 66 + (66/6))
:= (7/7 + 7 + 7) &times; (((7 + 7)/7)7 + 7)
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7) − 77
:= 7/7 + ((((7 + 7)/7)77/7 ) − (7 + 7))
:= 8/8 + (88 &times; ((8 − 8/8 + 8) + 8))
:= ((8 + 8)/8) &times; (8 &times; 8 &times; (8 + 8) − (8/8 + 8))
:= 88/8 + (88 &times; ((8 − 8/8 + 8) + 8))
:= 9 &times; ((9 + 9) &times; (9 + 9) − 99)
:= (((9 + 9)/9)99/9 ) − (9 + 9)
:= 9 + (((9 + 9) &times; (99 + 9) + 9 &times; 9) + 9/9)
4
(4+4)/4
&Iuml; 2036 := (1 + 1)11 − 1 − 11
&Iuml; 2026 := (1 + 1)11 − 11 − 11
&Iuml; 2031 := 1 + ((1 + 1)11 − (1 + 1) &times; (11 − 1 − 1))
:= 222/2 − 22
:= 2 + (((2 &times; 22 + 2/2)2 + 2) + 2)
:= 222/2 − (2 &times; (2 + 2 + 2))
:= 33 + ((3 + 3) &times; 333 + 3/3)
:= 33 + (3 + 3) &times; 333
:= (3/3 + 3) &times; ((3 − 3/3)3&times;3 − 3)
:= (((4 + 4)4 ) − 44)/((4 + 4)/4)
:= 44 + (4 &times; 444 − 4/4)
:= 4 + (4 &times; 444 + 44 )
:= 5 &times; 5 + ((((5 + 5)5−5/5 ) + 5)/5)
:= 5 + (((55 &times; (5 − 5 &times; 5)) + 5/5) + 55 )
:= 5/5 + (55 &times; (((5 + 5)/5)5 + 5))
:= 6 + ((6 − 6/6) &times; ((((6 + 6)/6) + 6 &times; 66) + 6))
:= ((6 &times; 666) + 66)/((6 + 6)/6)
:= (((6 + 6)/6)66/6 ) − 6 − 6
:= ((7 + 7 + 7)/7) − ((77 + 77) + 7)
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 − 7) + 7) − 77)
:= 7 &times; 7 &times; (7 &times; 7 − 7) − (7/7 + 7 + 7 + 7)
:= (8 + 8)/8 + (88 &times; ((8 − 8/8 + 8) + 8))
:= (8 &times; (8 + 8) &times; (8 + 8)) − (8/8 + 8 + 8)
:= (8 &times; (8 + 8) &times; (8 + 8)) − (88 + 8)/8
7
:= 9/9 + ((9 + 9) &times; (99 + 9) + 9 &times; 9)
&Iuml; 2027 := 1 + ((1 + 1)11 − (11 + 11))
99/9
:= 9/9 + ((((9 + 9)/9)
) − (9 + 9))
&Iuml; 2032 := (1 + 1)11 − (1 + 1)1+1+1+1
:= 9 &times; 9 + ((9 + 9) &times; (99 + 9) + (99/9))
&Iuml; 2037 := (1 + 1)11 − 11
:= 222/2 − 22/2
:= 2 + (2 &times; 22 + 2/2)2
:= 222/2 − 22+2
:= 33+3 + ((33/3)3 − 33)
:= 3/3 + ((3 + 3) &times; 333 + 33)
:= 3 + (3 + 3) &times; (333 + 3 + 3)
:= 44/4 + (4 + 4) &times; (4 − 4)
:= 4 &times; ((4 − 4) + 4 )
:= 44 &times; (4 + 4) − 44/4
:= 5 &times; 5 + (((((5 + 5)5−5/5 ) + 5) + 5)/5)
:= ((5 + 5)/5)5 + (5 &times; 5 &times; (5 &times; 5 + 55))
:= (((5 + 5)/5)55/5 ) − 55/5
:= 6 &times; 6 + (66 &times; (6 &times; 6 − 6) + (66/6))
:= 6/6 + (((6 &times; 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) &times; (((7 + 7)/7)7 + 7))
:= (7 + 7 + 7) &times; (7 &times; (7 + 7) − 7/7)
:= 88/8 + (8 &times; ((8 &times; 8 &times; 8 − 8)/((8 + 8)/8)))
:= (8 + 8) &times; (8 &times; (8 + 8) − 8/8)
:= (8 &times; (8 + 8) &times; (8 + 8)) − 88/8
:= 9 + (((9 + 9)/9) &times; ((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
&Iuml; 2038 := 1 + ((1 + 1)11 − 11)
&Iuml; 2043 := (1 + 1)11 − 1 − 1 − 1 − 1 − 1
&Iuml; 2048 := (1 + 1)11
:= ((2 − 22)/2) + 222/2
:= 222/2 − (2/2 + 2 + 2)
:= 222/2
:= 3 + (3 + 3) &times; (333 + 3 + 3) + 3/3
:= 3 &times; (3 &times; (3 + 3)3 + 33)
:= (3 − 3/3)33/3
:= (4 − 44)/4 + 44 &times; (4 + 4)
:= 44 &times; (4 + 4) − (4/4 + 4)
:= 44 &times; (4 + 4)
:= (5 + 5)/5 &times; ((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) &times; (7 &times; (7 + 7) − 7/7))
:= 7 &times; 7 &times; (7 &times; 7 − 7) − (7/7 + 7 + 7)
:= ((7 + 7)/7)77/7
:= (8 − 88)/8 + (8 &times; (8 + 8) &times; (8 + 8))
:= 88/8 + ((8 + 8) &times; (8 &times; (8 + 8) − 8/8))
:= 8 &times; (8 + 8) &times; (8 + 8)
:= (((9 + 9)/9)99/9 ) − 9/9 − 9
:= 99 + (9 + 9) &times; (99 + 9)
:= ((9 + 9)/9)99/9
&Iuml; 2039 := 1 + (1 + (1 + 1)11 − 11)
:= 2 + (2
22/2
&Iuml; 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&times;3
:= 3/3 + (3 &times; (3 &times; (3 + 3) + 33))
4
4
:= 4 &times; (4 + 4) − (4/4 + 4 + 4)
&Iuml; 2049 := 1 + (1 + 1)11
:= 3/3 + (3 − 3/3)33/3
:= 4 &times; (4 + 4) − 4
:= 4/4 + 44 &times; (4 + 4)
:= 5 &times; 5 + (5 − 5555/5)
:= 5/5 + ((((5 + 5)/5)55/5 ) − 5)
:= 5/5 + (((5 + 5)/5)55/5 )
:= 66 + (66 &times; (6 &times; 6 − 6) − (6/6 + 6))
:= ((6 + 6)/6)6 + 66 &times; (6 &times; 6 − 6)
:= 7 &times; 7 &times; (7 &times; 7 − 7) − ((77 + 7)/7 + 7)
:= 6/6 + (((6 + 6)/6)66/6 )
:= 7 &times; 7 &times; (7 &times; 7 − 7) − (7 + 7)
:= (8 &times; (8 + 8) &times; (8 + 8)) − (8/8 + 8)
:= 7/7 + (((7 + 7)/7)77/7 )
:= ((8 &times; 8 &times; 8 &times; 8) − 8)/((8 + 8)/8)
:= (((9 + 9)/9)99/9 ) − 9
:= 8/8 + (8 &times; (8 + 8) &times; (8 + 8))
:= 9/9 + ((9 + 9) &times; (99 + 9) + 99)
:= 9/9 + (((9 + 9)/9)99/9 )
5
&Iuml; 2040 := 1 + (1 + (1 + (1 + 1)11 − 11))
:= 2
22/2
&Iuml; 2045 := (1 + 1)11 − 1 − 1 − 1
− 2 &times; (2 + 2)
:= 2
3
:= (3 + 3) &times; ((((3/3 + 3) + 3) ) − 3)
4
22/2
&Iuml; 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) &times; (4 − 4/4)
:= 4/4 + (4 &times; (4 + 4) − 4)
5
5
:= (4 + 4)/4 + 44 &times; (4 + 4)
:= 5 + (55 &times; (((5 + 5)/5) + 5))
:= 5 + ((55 &times; (((5 + 5)/5) + 5)) + 5)
:= 5 &times; ((5 &times; (5 &times; 5 + 55) + 5) + 5)
:= 66 + (66 &times; (6 &times; 6 − 6) − 6)
:= 66 + (66 &times; (6 &times; 6 − 6) − 6/6)
:= (6 + 6)/6 + (((6 + 6)/6)66/6 )
:= 7 &times; 7 &times; (7 &times; 7 − 7) − (77/7 + 7)
:= 7/7 + (7 &times; 7 &times; (7 &times; 7 − 7) − (7 + 7))
:= (8 &times; (8 + 8) &times; (8 + 8)) − 8
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) − (88/8))
:= 9/9 + ((((9 + 9)/9)99/9 ) − 9)
:= 99 + ((9 + 9) &times; (99 + 9) + ((9 + 9)/9))
&Iuml; 2041 := 1 + (1 + (1 + (1 + (1 + 1)11 − 11)))
:= 2 + ((2
22/2
− 22/2) + 2)
&Iuml; 2046 := (1 + 1)11 − 1 − 1
:= 2
3
:= 3/3 + (3 + 3) &times; ((3/3 + 3 + 3) − 3)
4
22/2
−2
3
:= 3 + (3 &times; (3 &times; (3 + 3) + 33))
:= 44 &times; (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 &times; (6 &times; 6 − 6)
:= (((7 + 7)/7)77/7 ) − 7
:= (7 − 7/7) &times; (7 &times; 7 &times; 7 − ((7 + 7)/7))
:= 8/8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 8)
:= (8 &times; (8 + 8) &times; (8 + 8)) − (8 + 8)/8
:= 99 + ((9 + 9) &times; (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 &times; (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 &times; (8 + 8) &times; (8 + 8)) − 8)
:= 99 + ((9 + 9) &times; (99 + 9) − 9/9)
:= 2 + (222/2 + 2/2)
:= 3 + (3 − 3/3)33/3
:= 4 + (44 &times; (4 + 4) − 4/4)
:= 5 + (55 − ((5 − 5/5)5 + 55))
:= ((((6 + 6)/6)6+6 ) + 6)/((6 + 6)/6)
:= 7 &times; 7 &times; (7 &times; 7 − 7) − 7
:= 88/8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 8)
:= (((9 + 9)/9)9 ) + (9 &times; (9 &times; (9 + 9) + 9))
:= 2 + 222/2 + 2
− 3/3
4
:= 4 &times; (4 + 4) − 4/4
:= (3 + 3) &times; (333 + 3 &times; 3)
:= 4 + 44 &times; (4 + 4)
55/5
) − 5/5
:= 5 + ((((5 + 5)/5)55/5 ) − 5/5)
66/6
) − 6/6
:= 6 + (66 &times; (6 &times; 6 − 6) + 66)
:= (((5 + 5)/5)
:= (((6 + 6)/6)
)−6
:= (9 + 9)/9 + (((9 + 9)/9)99/9 )
&Iuml; 2052 := 1 + (1 + (1 + (1 + (1 + 1)11 )))
− 2/2
33/3
4
55/5
&Iuml; 2047 := (1 + 1)11 − 1
22/2
:= (8 + 8)/8 + (8 &times; (8 + 8) &times; (8 + 8))
&Iuml; 2051 := 1 + (1 + (1 + (1 + 1)11 ))
:= 4 + (4 &times; (4 + 4) − 44/4)
&Iuml; 2042 := (1 + 1)11 − ((1 + 1) &times; (1 + 1 + 1))
:= 7 &times; 7 &times; (7 &times; 7 − 7) − (7/7 + 7)
:= 7 &times; 7 &times; (7 &times; 7 − 7) − 77/7
:= 7/7 + (7 &times; 7 &times; (7 &times; 7 − 7) − 7)
:= (8 &times; (8 + 8) &times; (8 + 8)) − 8/8
:= ((8 &times; 8 &times; 8 &times; 8) + 8)/((8 + 8)/8)
99/9
:= (((9 + 9)/9)
) − 9/9
146
:= (99 + 9) &times; (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
&Iuml; 2053 := 1 + (1 + (1 + (1 + (1 + (1 + 1)11 ))))
&Iuml; 2058 := 11 + ((1 + 1)11 − 1)
&Iuml; 2063 := 1 + (1 + (1 + (1 + (11 + (1 + 1)11 ))))
:= 2 + ((222/2 + 2/2) + 2)
:= 2 + (222/2 + 2 &times; (2 + 2))
:= 2 + ((222/2 + 22/2) + 2)
:= 3/3 + ((3 + 3) &times; (333 + 3 &times; 3))
:= (3 + 3) &times; (((3/3 + 3) + 3)3 )
:= 3 + ((33/3)3 + 33+3 )
:= 4 + (44 &times; (4 + 4) + 4/4)
:= 44 &times; (4 + 4) + (44 − 4)/4
:= 4 + (44 &times; (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 &times; ((6/6 + 6)6&times;6/(6+6) )
:= 6 + (((6 + 6)/6)6+6 + 6)/((6 + 6)/6) + 6
:= (7 + 7)/7 + (7 &times; 7 &times; (7 &times; 7 − 7) − 7)
:= 7 &times; 7 &times; (7 &times; 7 − 7)
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) − ((7 + 7)/7))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − (88/8)) + 8)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) + ((8 + 8)/8))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − 8/8) + 8)
:= 9/9 + ((99 + 9) &times; (9/9 + 9 + 9))
&Iuml; 2054 := (1 + 1)11 + ((1 + 1) &times; (1 + 1 + 1))
99/9
:= 9 + ((((9 + 9)/9)
) + 9/9)
:= 99/9 + ((99 + 9) &times; (9/9 + 9 + 9))
&Iuml; 2064 := (1 + 1)11 + (1 + 1)1+1+1+1
&Iuml; 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) &times; (3/3 + 3 + 3)3
:= 3 + ((3 &times; 3 + 3)3 + 333)
:= 4 + (44 &times; (4 + 4) + (4 + 4)/4)
:= 44/4 + 44 &times; (4 + 4)
:= 4 &times; ((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 &times; ((6/6 + 6)6&times;6/(6+6) ))
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) − (77/7))
:= 7/7 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 7 + 7 &times; 7 &times; (7 &times; 7 − 7) − 7/7
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) − ((8 + 8)/8))
:= 88/8 + (8 &times; (8 + 8) &times; (8 + 8))
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) + 8)
:= 99 + ((9 + 9) &times; (99 + 9) + (99/9))
:= 99/9 + (((9 + 9)/9)99/9 )
:= 9 + ((9 + 9) &times; (99 + 9) + 999/9)
&Iuml; 2055 := 1 + ((1 + 1)11 + ((1 + 1) &times; (1 + 1 + 1)))
&Iuml; 2065 := 1 + ((1 + 1)11 + (1 + 1)1+1+1+1 )
&Iuml; 2060 := 1 + (11 + (1 + 1)11 )
:= 2 + (((222/2 + 2/2) + 2) + 2)
:= 222/2 + (2 &times; (2 + 2 + 2))
:= 2/2 + (222/2 + 22+2 )
:= 3 + ((3 + 3) &times; (333 + 3 &times; 3))
:= 33+3 + (33/3)3
:= 3 + (((3 &times; 3 + 3)3 + 333) + 3/3)
:= 4 + ((4 &times; (4 + 4) − 4/4) + 4)
:= 4 + ((44 &times; (4 + 4) + 4) + 4)
:= 4 &times; 4 + (44 &times; (4 + 4) + 4/4)
:= 55 + (5 &times; 5 &times; (5 &times; 5 + 55))
:= 5 + ((5 &times; 5 &times; (5 &times; 5 + 55)) + 55)
:= 55 + ((5 + 5) &times; (5 − 555/5))
:= 6 + ((6 + 6)/6)66/6 + 6
:= (6 &times; 6 − 6/6) &times; (66 − (6/6 + 6))
:= (7 + 7)/7 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 7 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) − 8/8)
:= 8 + (((8 &times; 8 &times; 8 &times; 8) + 8)/((8 + 8)/8))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + 8/8) + 8)
:= 999/9 + (9 + 9) &times; (99 + 9)
:= 9 + ((9 &times; (9 &times; (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)
)
&Iuml; 2056 := 11 + ((1 + 1)11 − (1 + 1 + 1))
&Iuml; 2061 := 1 + (1 + (11 + (1 + 1)11 ))
&Iuml; 2066 := (1 + 1)11 + (1 + 1) &times; (11 − 1 − 1)
:= 2 &times; (2 + 2) + 222/2
:= 2 + (222/2 + 22/2)
:= 2 + (222/2 + 22+2 )
:= (((3 &times; (3 + 3))3 + 333) + 3)/3
:= 333 + (3 &times; 3 + 3)3
:= 3 + (((33/3)3 + 33+3 ) + 3)
:= 4 + (44 &times; (4 + 4) + 4)
:= 4 + (((44 &times; (4 + 4) + 4/4) + 4) + 4)
:= 4 &times; 4 + (44 &times; (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 &times; 7 &times; (7 &times; 7 − 7) − (7 + 7)/7
:= (7 + 7 + 7)/7 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) + 7/7)
:= 8 + (8 &times; (8 + 8) &times; (8 + 8))
:= (8 &times; (8 + 8) &times; (8 + 8)) + (88 + 8 + 8)/8
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + ((8 + 8)/8)) + 8)
:= 9 + ((99 + 9) &times; (9/9 + 9 + 9))
:= 9 + ((((9 + 9)/9)99/9 ) + 9)
99/9
:= 9 + ((((9 + 9)/9)
) − 9/9)
&Iuml; 2057 := 11 + ((1 + 1)11 − (1 + 1))
&Iuml; 2062 := 1 + (1 + (1 + (11 + (1 + 1)11 )))
&Iuml; 2067 := (1 + 1)11 + (((1 + 1) &times; (11 − 1)) − 1)
:= 22/2 + 222/2 − 2
:= 22+2 + 222/2 − 2
:= 22 + (222/2 − (2/2 + 2))
:= 3 &times; 3 + (3 − 3/3)33/3
:= 3/3 + ((3 &times; 3 + 3)3 + 333)
:= 3 + (((3 &times; 3 + 3)3 + 333) + 3)
:= 4 + ((44 &times; (4 + 4) + 4/4) + 4)
:= 4 + (44 &times; (4 + 4) + (44 − 4)/4)
:= 4 + ((44 &times; (4 + 4) + 44/4) + 4)
:= (((5 + 5) &times; ((5 − 5/5)5 + 5)) − 5)/5
:= 5 + ((((5 + 5) &times; ((5 − 5/5)5 + 5)) − 5)/5)
:= ((55 + 5/5) &times; (((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 &times; 666) + 6)/((6 + 6)/6))
77/7
:= 7 &times; 7 &times; (7 &times; 7 − 7) − 7/7
:= 7 + ((((7 + 7)/7)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) + 8/8)
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − ((8 + 8)/8)) + 8)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) + (88/8))
:= 9 + (((99 + 9) &times; (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 &times; 7 &times; (7 &times; 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
&Iuml; 2068 := (1 + 1)11 + ((1 + 1) &times; (11 − 1))
&Iuml; 2073 := 1 + ((1 + 1)11 + ((1 + 1) &times; (1 + 11)))
&Iuml; 2078 := (1 + 1)11 + ((11 − 1) &times; (1 + 1 + 1))
:= 22 + 222/2 − 2
:= 2 + 222/2 + 22 + 2/2
:= 22 + 222/2 + 2 &times; (2 + 2)
:= 33/3 &times; ((3 + 3)3 − (33 + 3/3))
:= (3 &times; (33+3 − 33 )) − 33
:= 3 + ((3 − 3/3)33/3 + 33 )
:= 4 + (44 &times; (4 + 4) + 4 &times; 4)
:= 4 + ((4 + 4) &times; (44 + 4) − 44/4)
:= (4 + 4) &times; (44 + 4) − (4 + 4)/4
:= 5 &times; 5 + ((((5 + 5)/5)55/5 ) − 5)
:= 5 &times; 5 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 &times; 5)
:= 66/6 &times; ((6 &times; (6 &times; 6 − 6) + ((6 + 6)/6)) + 6)
:= 6 + ((((6 &times; 666) + 6)/((6 + 6)/6)) + 66)
:= 6 &times; 6 + ((((6 + 6)/6)66/6 ) − 6)
:= 77/7 &times; (777/7 + 77)
:= 7 + ((7 &times; 7 &times; (7 &times; 7 − 7) + 7/7) + 7)
:= 7 + ((7 &times; 7 &times; (7 &times; 7 − 7) − 7/7 + 7) + 7)
:= 8 + ((((8 &times; 8 &times; 8 &times; 8) + 8)/((8 + 8)/8)) + 8)
:= 8 + ((((8 &times; (8 + 8) &times; (8 + 8)) + 8/8) + 8) + 8)
:= 8 + ((88 + 88)/8 + (8 &times; (8 + 8) &times; (8 + 8)))
:= 9 + ((((9 + 9)/9)99/9 ) + (99/9))
:= 9 + (((9 + 9) &times; (99 + 9) + 999/9) + 9)
:= (99 &times; (((99 + 9)/9) + 9)) − 9/9
&Iuml; 2074 := (1 + 1)11 + ((1 + 1) &times; (1 + 1 + 11))
&Iuml; 2069 := 11 + (11 + ((1 + 1)11 − 1))
:= 22 + (222/2 − 2/2)
:= 2 + 2
22/2
&Iuml; 2079 := 1 + ((1 + 1)11 + ((11 − 1) &times; (1 + 1 + 1)))
:= 2 &times; 222 + 2222/2
+ 22 + 2
3
:= 3 &times; 3 + ((33/3)3 + 33+3 )
:= (3/3 + 33) &times; (((3/3 + 3) ) − 3)
:= 3 &times; (33 &times; ((3 &times; (3 + 3)) + 3))
:= (4 + 4) &times; (44 + 4) − 44/4
:= 4 + ((4 + 4)4 + 44)/((4 + 4)/4)
:= (4 + 4) &times; (44 + 4) − 4/4
:= 55 + (55 − 5555/5)
:= 5 + ((55 − 5555/5) + 55 )
:= ((5 + 5) &times; ((55 − 5)/(5 + 5 + 5))) − 5/5
:= 66/6 + (6 &times; ((6/6 + 6)6&times;6/(6+6) ))
:= (66/6 + 6) &times; ((666 + 66)/6)
:= 6 &times; 6 + (((((6 + 6)/6)66/6 ) − 6) + 6/6)
:= 77/7 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 7 + ((7 &times; 7 &times; (7 &times; 7 − 7) + ((7 + 7)/7)) + 7)
:= 7 + ((7 &times; 7 &times; (7 &times; 7 − 7) + 7) + 7)
:= (8/8 + 8 + 8) &times; ((888 + 88)/8)
:= (88/8 − 8) &times; (8 &times; 88 − (88/8))
:= (9 &times; (99 + 9)) + (9999/9 − 9)
:= 99 &times; (((99 + 9)/9) + 9)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) + (88 + 8 + 8)/8)
9
:= 99 + (9 &times; 9 &times; (9 + 9) + (((9 + 9)/9) ))
&Iuml; 2070 := 11 + (11 + (1 + 1)11 )
&Iuml; 2075 := (1 + 1)11 + (1 + 1 + 1)1+1+1
&Iuml; 2080 := (1 + 1)11 + ((11 &times; (1 + 1 + 1)) − 1)
:= 22 + 222/2
:= 2 + 222/2 + 22 + 2/2 + 2
:= 222/2 + 2 &times; 22+2
:= 3 &times; ((3 &times; ((3 + 3)3 + 3)) + 33)
:= 33 + (3 − 3/3)33/3
:= 3/3 + (3 &times; (33 &times; ((3 &times; (3 + 3)) + 3)))
:= (((4 + 4)4 ) + 44)/((4 + 4)/4)
:= (4 + 4) &times; (4 + 4) − (4/4 + 4)
:= (4 + 4) &times; (44 + 4)
:= (5 &times; (5 &times; (5 &times; 5 + 55 + 5))) − 55
:= 5 &times; (((5 &times; (5 &times; 5 + 55) + 5) + 5) + 5)
:= (5 + 5) &times; ((55 − 5)/(5 + 5 + 5))
:= 6 &times; ((666 &times; 6/(6 + 6) + 6) + 6)
:= 66 + (((6 − 6/6) &times; (6 &times; 66 + 6)) − 6/6)
:= (6/6 − 66) &times; (6 − ((6 + 6)/6 + 6 &times; 6))
:= (7 − 7/7) &times; (7 &times; 7 &times; 7 + ((7 + 7)/7))
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) + ((77 − 7)/7))
:= 7 + (((7 &times; 7 &times; (7 &times; 7 − 7) + 7/7) + 7) + 7)
:= ((8 + 8)/8) &times; (8 &times; 8 &times; (8 + 8) + (88/8))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + (88/8)) + 8)
:= 8 &times; ((8 &times; 8 &times; 8 + 8)/((8 + 8)/8))
99/9
:= 9/9 + (99 &times; (((99 + 9)/9) + 9))
:= (9/9 + 9) &times; (99 + 99 + 9)
&Iuml; 2071 := 1 + (11 + (11 + (1 + 1)11 ))
4
:= 9 + (((((9 + 9)/9)
) + 9) + 9)
&Iuml; 2076 := 1 + ((1 + 1)11 + (1 + 1 + 1)1+1+1 )
&Iuml; 2081 := (1 + 1)11 + (11 &times; (1 + 1 + 1))
:= 22 + 222/2 + 2/2
:= 2 + 222/2 + 22 + 2 + 2
:= 22 + (222/2 + 22/2)
:= (3 &times; 3 + 3)3 + (((3/3 + 3) + 3)3 )
:= (3 + 3) &times; ((((3/3 + 3) + 3)3 ) + 3)
:= 33 + (3 − 3/3)33/3
:= (4 + 4) &times; (44 + 4) − (4/4 + 4 + 4)
:= (4 + 4) &times; (44 + 4) − 4
:= 4/4 + (4 + 4) &times; (44 + 4)
:= 55 − (((5 − 5/5)5 + 5 &times; 5) + 5)
:= 55 − ((5 − 5/5)5 + 5 &times; 5)
:= 5 + (55 − ((5 − 5/5)5 + 5 &times; 5))
:= 6 + ((6 &times; 6 − 6/6) &times; (66 − (6/6 + 6)))
:= 66 + ((6 − 6/6) &times; (6 &times; 66 + 6))
:= (6 &times; (6 &times; (((6 + 6)/6)6 − 6))) − 6/6 − 6
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) − 7/7 + 7)
:= 7 + 7 &times; 7 &times; (7 &times; 7 − 7) + 77/7
:= 7 + (((7 &times; 7 &times; (7 &times; 7 − 7) + ((7 + 7)/7)) + 7) + 7)
:= 8 + ((((8 &times; (8 + 8) &times; (8 + 8)) − 8/8) + 8) + 8)
:= ((8 &times; (8 &times; 8 &times; 8 + 8)) − 8)/((8 + 8)/8)
:= 8/8 + (8 &times; ((8 &times; 8 &times; 8 + 8)/((8 + 8)/8)))
:= (9/9 + 9 + 9) &times; (9/9 + 99 + 9)
:= 9 &times; 9 &times; (9 + 9 + 9) − 999/9
:= 9 &times; 9 + (((9 + 9)/9) &times; (999 + 9/9))
&Iuml; 2072 := (1 + 1)11 + ((1 + 1) &times; (1 + 11))
&Iuml; 2077 := (1 + 1)11 + (((11 − 1) &times; (1 + 1 + 1)) − 1)
&Iuml; 2082 := 1 + ((1 + 1)11 + (11 &times; (1 + 1 + 1)))
:= 2 + 222/2 + 22
:= 2 &times; 222 + 2222/2 − 2
:= ((2 &times; (22 + 2))2 ) − 222
:= 33 + ((3 − 3/3)33/3 − 3)
:= 3 &times; 33+3 + ((3 − 333)/3)
:= 3 + (3 &times; (33 &times; ((3 &times; (3 + 3)) + 3)))
:= (4 + 4) &times; ((4 − 4/4) + 4)
:= 4/4 + ((4 + 4) &times; (4 + 4) − 4)
:= (4 + 4)/4 + (4 + 4) &times; (44 + 4)
:= (55 + 5/5) &times; (((5 + 5)/5)5 + 5)
:= 5 + ((55 + 5/5) &times; (((5 + 5)/5)5 + 5))
:= 5 + (((55 + 5/5) &times; (((5 + 5)/5)5 + 5)) + 5)
:= 6 + ((((((6 + 6)/6)66/6 ) + 6) + 6) + 6)
:= (66 + 6/6) &times; ((6 &times; 6 − 6) + 6/6)
:= (6 &times; (6 &times; (((6 + 6)/6)6 − 6))) − 6
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) + 7)
:= 7 + 7 &times; 7 &times; (7 &times; 7 − 7) + (77 + 7)/7
:= ((7 + 7 + 7)/7)7 − (7 &times; (7 + 7) + 7)
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + 8) + 8)
:= ((8 + 8 + 8) &times; (88 − 8/8)) − 88/8
:= ((88/8 + 8) &times; (888 − 8)/8) − 8
:= 9 + (((99 + 9) &times; (9/9 + 9 + 9)) + (99/9))
:= 9 &times; 9 + (((9 + 9)/9) &times; (999 − 9/9))
:= (9 &times; (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
&Iuml; 2083 := 1 + (1 + ((1 + 1)11 + (11 &times; (1 + 1 + 1))))
&Iuml; 2088 := (1 + 1)11 + ((1 + 1) &times; ((1 + 1) &times; (11 − 1)))
&Iuml; 2093 := 1 + ((1 + 1)11 + ((1 + 1) &times; (11 + 11)))
:= 2 + ((222/2 + 22/2) + 22)
:= 222/2 + 2 &times; (22 − 2)
:= 2/2 + (222/2 + 2 &times; 22)
:= 3 + 3 &times; 33 &times; (3 &times; (3 + 3) + 3) + 3/3
:= 3 &times; (33+3 − 33)
:= 33 + ((33/3)3 + 33+3 )
:= 4 + ((4 + 4) &times; (44 + 4) − 4/4)
:= 4 + ((4 + 4) &times; (44 + 4) + 4)
:= 44 + (44 &times; (4 + 4) + 4/4)
:= 5 + (((((5 + 5)/5)55/5 ) + 5 &times; 5) + 5)
:= (5 + 5)/5 &times; (((5 − 5/5)5 − 5) + 5 &times; 5)
:= 55 + ((5 + 5)/5 &times; ((5 − 5/5)5 − 5))
:= 6 + ((66 + 6/6) &times; ((6 &times; 6 − 6) + 6/6))
:= 6 &times; (6 &times; (((6 + 6)/6)6 − 6))
:= 6 &times; 6 &times; (66 − 6) − (66 + 6/6)
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) + 77/7 + 7)
:= (7 − 7/7) &times; ((7 &times; 7 &times; 7 − ((7 + 7)/7)) + 7)
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7) − (7 + 7)
:= 8 + ((((8 &times; (8 + 8) &times; (8 + 8)) + (88/8)) + 8) + 8)
:= (8 + 8 + 8) &times; (88 − 8/8)
:= (88 &times; (8 + 8 + 8)) − (88/8 + 8)
:= (9 &times; (99 + 9)) + 9999/9
:= 9 &times; 9 &times; (9 + 9 + 9) − 99
:= ((99 + 9 + 9)/9) &times; (9 &times; (9 + 9) − 9/9)
&Iuml; 2084 := (1 + 1)11 + ((1 + 1 + 1) &times; (1 + 11))
&Iuml; 2089 := ((1 + 1) &times; (1111 − 11)) − 111
&Iuml; 2094 := (1 + 1)11 + ((1 + 1) &times; (1 + (11 + 11)))
:= 222 + (2 &times; (22 − 2))2
:= 2/2 + (222/2 + 2 &times; (22 − 2))
:= (2 &times; 22 + 2)2 − 22
:= 3 + ((3 − 3/3)33/3 + 33)
:= 3/3 + (3 &times; (33+3 − 33))
:= 3 + ((3 &times; (33+3 − 33)) + 3)
:= 4 + (4 + 4) &times; (44 + 4)
:= 4 + (((4 + 4) &times; (44 + 4) + 4/4) + 4)
:= 44 + (44 &times; (4 + 4) + (4 + 4)/4)
:= (5 − 5/5) &times; ((55 + 5/5)/(5/5 + 5))
:= 55 − ((5 − 5/5)5 + ((55 + 5)/5))
:= 55 − (((5 − 5/5)5 + ((5 + 5)/5)) + 5)
:= 6 &times; 6 + (((6 + 6)/6)66/6 )
:= 6/6 + (6 &times; (6 &times; (((6 + 6)/6)6 − 6)))
:= 6 &times; 6 &times; (66 − 6) − 66
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) + (77 + 7)/7 + 7)
:= ((7 + 7 + 7)/7)7 − 7 &times; (7 + 7)
:= (7 − 7/7) &times; ((7 &times; 7 &times; 7 − 7/7) + 7)
:= ((8 &times; (8 &times; 8 &times; 8 + 8)) + 8)/((8 + 8)/8)
:= 8/8 + ((8 + 8 + 8) &times; (88 − 8/8))
:= ((8 + 8)/8) &times; (8888/8 − 8 &times; 8)
:= 9/9 + (9 &times; 9 &times; (9 + 9 + 9) − 99)
:= (9 &times; (9 &times; (9 + 9 + 9) − 9)) − (99 + 9)/9
99/9
:= 9 + ((((((9 + 9)/9)
) + 9) + 9) + 9)
&Iuml; 2085 := (1 + 1)11 + (111/(1 + 1 + 1))
&Iuml; 2090 := (111 − 1) &times; (((1 + 1) &times; (11 − 1)) − 1)
&Iuml; 2095 := 1 + ((1 + 1)11 + ((1 + 1) &times; (1 + (11 + 11))))
:= 2/2 + ((2 &times; (22 − 2))2 + 222 )
:= 2 &times; 22 + 222/2 − 2
:= 2/2 + (2 &times; 22 + 2)2 − 22
:= (3 &times; (33+3 − 33)) − 3
:= 3 + ((3 &times; (33+3 − 33)) − 3/3)
:= (3 &times; (33+3 − 33 )) − 33/3
:= 4 + ((4 + 4) &times; (44 + 4) + 4/4)
:= 44 + (44 &times; (4 + 4) − (4 + 4)/4)
:= 4 + ((44 &times; (4 + 4) − 4/4) + 44)
:= 5 + ((5 + 5) &times; ((55 − 5)/(5 + 5 + 5)))
:= 55 + (55 &times; (((5 + 5)/5)5 + 5))
:= ((55 + 5) &times; ((5 &times; 5 + 5) + 5)) − 5
:= 6 + ((((6 + 6)/6)66/6 ) + 6 &times; 6)
:= 6/6 + (6 &times; 6 &times; (66 − 6) − 66)
66/6
:= 6 &times; 6 + ((((6 + 6)/6)
) + 6/6)
:= 77 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) − 7/7)
:= 7 &times; 7 + ((((7 + 7)/7)77/7 ) − 7)
:= (88/8 − 8) &times; (8 &times; 88 − (8/8 + 8))
:= (88/8 + 8) &times; (888 − 8)/8
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) − 999/9)
&Iuml; 2086 := ((1 + 1 + 11)1+1+1 ) − 111
:= (9/9 + 9 + 9) &times; (99/9 + 99)
&Iuml; 2091 := 1 + ((111 − 1) &times; (((1 + 1) &times; (11 − 1)) − 1))
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7) − (77 + 7)/7
:= (88 &times; (8 + 8 + 8)) − (8/8 + 8 + 8)
:= (9 &times; (9 &times; (9 + 9 + 9) − 9)) − 99/9
&Iuml; 2096 := (1 + 1)11 + ((1 + 1) &times; ((1 + 1) &times; (1 + 11)))
:= 2 + (2 &times; 22 + 2)2 − 22
:= 2 + ((2 &times; (22 − 2))2 + 222 )
:= 2 &times; 22 + (222/2 − 2/2)
:= 3/3 + 3 &times; (33+3 − 33) − 3
:= 3 + (3 &times; (33+3 − 33))
:= 4 + ((4 + 4) &times; (44 + 4) + (4 + 4)/4)
:= 44 + (44 &times; (4 + 4) − 4/4)
:= 55 − ((((5 − 5/5)5 + 5) + 5) + 5)
:= 55 − (((5 − 5/5)5 + 5) + 5)
:= (6 &times; (6 &times; (((6 + 6)/6)6 − 6))) − (6 + 6)/6
:= 666/6 + 66 &times; (6 &times; 6 − 6)
:= 77 + 7 &times; (7 &times; (7 &times; 7 − 7) − 7)
:= (7 + 7) &times; (77 − 7) + 7777/7
:= 7 + (((7 + 7 + 7)/7)7 − 7 &times; (7 + 7))
:= ((8 + 8 + 8) &times; (88 − 8/8)) − (8 + 8)/8
:= (88/8 − 8) &times; ((8 &times; 88 − 8) + 8/8)
:= (88 &times; (8 + 8 + 8)) − 8 − 8
:= 999 + (((99 &times; 99) − (9 + 9))/9)
:= 999/9 + (99 &times; (99/9 + 9))
:= (9 &times; (9 &times; (9 + 9 + 9) − 9)) − 9/9 − 9
&Iuml; 2087 := 1 + (((1 + 1 + 11)1+1+1 ) − 111)
:= 2
22/2
+ (2 &times; (22 − 2) − 2/2)
&Iuml; 2092 := (1 + 1)11 + ((1 + 1) &times; (11 + 11))
:= 2 &times; 22 + 2
:= ((33/3 + 3)3 ) − 3 &times; (3 + 3)3
:= 4 + (44 &times; (4 + 4) + 44)
:= 55 − ((5 − 5/5)5 + 5)
:= 6 &times; 6 &times; (66 − 6) − ((6 + 6)/6)6
&Iuml; 2097 := (11 − 1 − 1) &times; (11 + (1 + 1) &times; 111)
:= 2 + (2 &times; 22 + 2)2 − 22 + 2/2
22/2
:= (3 &times; (33+3 − 33)) − 3/3
:= 3 + ((3 &times; (33+3 − 33)) + 3/3)
:= 3 &times; ((33+3 − 33) + 3)
:= 4 + (((4 + 4) &times; (44 + 4) − 4/4) + 4)
:= 44 + 44 &times; (4 + 4)
:= 4 + ((44 &times; (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 &times; (6 &times; (((6 + 6)/6)6 − 6))) − 6/6
:= 66 &times; (6 &times; 6 − 6) + (666 + 6)/6
:= 6 + (66 &times; (6 &times; 6 − 6) + 666/6)
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 − 7) − 7) + 77)
:= 7 &times; 77 + ((((7 + 7) &times; 777) − 7)/7)
:= 7 &times; 7 + (((7 + 7)/7)77/7 )
:= ((8 + 8 + 8) &times; (88 − 8/8)) − 8/8
:= ((8 &times; 8 &times; 8 &times; 8) + 88)/((8 + 8)/8)
:= 8/8 + ((88 &times; (8 + 8 + 8)) − (8 + 8))
:= 999 + (((99 &times; 99) − 9)/9)
:= 9 + (9999/9 + (9 &times; (99 + 9)))
:= (9 &times; (9 &times; (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
&Iuml; 2098 := (1 + 1)11 + ((11 − 1)1+1 /(1 + 1))
2
&Iuml; 2103 := (1 + 1)11 + (111 − 1)/(1 + 1)
2
&Iuml; 2108 := (111 &times; (((1 + 1) &times; (11 − 1)) − 1)) − 1
:= 2 &times; (2 &times; (((22 + 2/2)2 ) − 2))
:= 2 + (2 &times; 22 + 2) − 22 + 2
:= 2 + 2222 − (22/2)
:= 3/3 + (3 &times; ((33+3 − 33) + 3))
:= (3 &times; (33+3 − 33 )) − 3
:= 3 + ((3 &times; (33+3 − 33 )) − 3/3)
:= 4 + ((44 &times; (4 + 4) + (4 + 4)/4) + 44)
:= 44 + (44 &times; (4 + 4) + 44/4)
:= 44 &times; (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) &times; (6/6 − 6 &times; 6)) − (6 + 6)/6
:= ((6 &times; (666 + 6 &times; 6)) − 6)/((6 + 6)/6)
:= 66 + ((((6 + 6)/6)66/6 ) − 6)
:= 7/7 + ((((7 + 7)/7)77/7 ) + 7 &times; 7)
:= ((7 + 7 + 7)/7)7 − 77 − 7
:= 7/7 + 7 &times; (7 &times; (7 &times; 7 − 7) + 7)
:= 8 + ((88/8 + 8) &times; (888 − 8)/8)
:= (88 &times; (8 + 8 + 8)) − (8/8 + 8)
:= (88 &times; (8 + 8 + 8)) − (8/((8 + 8)/8))
:= 9/9 + ((9 &times; (9 &times; (9 + 9 + 9) − 9)) − 9)
:= (9 &times; (9 &times; (9 + 9 + 9) − 9)) − (9 + 9 + 9)/9
:= (9 + 9)/9 + (9 &times; (9 &times; (9 + 9 + 9) − 9))
&Iuml; 2099 := ((1 + 1) &times; (1111 − 1)) − 111+1
2
:= 2 &times; (2 &times; (22 &times; (22 + 2) − 2))
:= 2222 − (22/2) − 2
:= 33/3 + (3 &times; (3
3+3
&Iuml; 2104 := (1 + 1)11 + (1 + 111)/(1 + 1)
− 33))
:= 3/3 + ((3 &times; (3
3+3
3
− 3 )) − 3)
&Iuml; 2109 := 111 &times; (((1 + 1) &times; (11 − 1)) − 1)
:= 2222 − (222/2 + 2)
:= 3 + (3 &times; (33+3 − 33 ))
:= 4 + (((44 &times; (4 + 4) − 4/4) + 44) + 4)
:= 44 &times; (44 + 4) − 4 − 4
:= 4/4 + (44 &times; (44 + 4) − 4)
:= 55 − ((5 − 5/5)5 + ((5 + 5)/5))
:= 5 + (55 − ((5 − 5/5)5 + ((5 + 5)/5)))
:= 555/5 &times; (5 &times; 5 − (5/5 + 5))
:= ((6 − 66) &times; (6/6 − 6 &times; 6)) − 6/6
:= 6 + (((6 − 66) &times; (6/6 − 6 &times; 6)) − ((6 + 6)/6))
:= 666/6 &times; ((6/6 + 6 + 6) + 6)
77/7
:= 7 + ((((7 + 7)/7)
:= ((88/8 − 8)8−8/8 ) − 88
:= (88 &times; (8 + 8 + 8)) − 8
:= (88/8 + 8) &times; 888/8
:= 9 + ((9/9 + 9 + 9) &times; (99/9 + 99))
:= (9 &times; (9 &times; (9 + 9 + 9) − 9)) − (9 + 9)/9
:= (9/9 + 9 + 9) &times; 999/9
&Iuml; 2100 := (11 + (11 − 1)) &times; (11 − 1)1+1
) + 7 &times; 7)
:= (7 + 7)/7 + 7 &times; (7 &times; (7 &times; 7 − 7) + 7)
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7) − (7/7 + 7)
&Iuml; 2105 := 1 + ((1 + 1)11 + (1 + 111)/(1 + 1))
&Iuml; 2110 := 1 + (111 &times; (((1 + 1) &times; (11 − 1)) − 1))
:= (2 &times; 22 + 2)2 − 22+2
:= (2 &times; 22 + 2)2 − 22/2
:= (2 &times; 22 + 2)2 − (2 + 2 + 2)
:= 3 + (3 &times; ((33+3 − 33) + 3))
:= (3 &times; (33+3 − 33 )) − 3/3
:= 3 + ((3 &times; (33+3 − 33 )) + 3/3)
:= 4 + ((44 &times; (4 + 4) + 44) + 4)
:= 4 + (44 &times; (44 + 4) − 44/4)
:= 44 &times; (44 + 4) − (4 + 4)/4
:= (55 + 5) &times; ((5 &times; 5 + 5) + 5)
:= 5 + ((55 + 5) &times; ((5 &times; 5 + 5) + 5))
:= 5 + (((55 + 5) &times; ((5 &times; 5 + 5) + 5)) + 5)
:= (6 − 66) &times; (6/6 − 6 &times; 6)
:= 6 + (((6 − 66) &times; (6/6 − 6 &times; 6)) − 6/6)
:= ((66 − 6)/6) &times; (6 &times; 6 &times; 6 − 6 + 6/6)
:= (7 − 7/7) &times; (7 &times; 7 &times; 7 + 7)
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7) − (7 + 7)/7
:= ((7 + 7 + 7)/7)7 − 77
:= (88 &times; (8 + 8 + 8)) − (88 + 8)/8
:= 8/8 + ((88 &times; (8 + 8 + 8)) − 8)
:= (88 &times; (8 + 8 + 8)) − (8 + 8)/8
:= (9/9 + 9) &times; (999/9 + 99)
:= (9 &times; (9 &times; (9 + 9 + 9) − 9)) − 9/9
:= 999 + 9999/9
&Iuml; 2106 := 1 + (1 + ((1 + 1)11 + (1 + 111)/(1 + 1)))
&Iuml; 2111 := ((1 + 1) &times; 1111) − 111
&Iuml; 2101 := ((1 + 1) &times; 1111) − 111+1
:= 2222 − (22/2)
2
:= 2 + (2 &times; (2 &times; ((22 &times; (22 + 2)) − 2)))
3
3+3
3
:= 2222 − 222/2
:= (33 &times; ((3/3 + 3)3 )) − 3/3
:= (33 &times; ((3/3 + 3) )) − 33/3
:= 3 &times; (3
−3 )
:= 44 &times; (44 + 4) − 44/4
:= 44 &times; (44 + 4) − ((4 + 4)/4 + 4)
5
5
:= 44 &times; (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) &times; (6/6 − 6 &times; 6))
:= 6 + ((6 − 66) &times; (6/6 − 6 &times; 6))
:= 66/6 + ((6 − 66) &times; (6/6 − 6 &times; 6))
:= 7/7 + ((7 − 7/7) &times; (7 &times; 7 &times; 7 + 7))
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7) − 7/7
:= 7/7 + (((7 + 7 + 7)/7)7 − 77)
:= (88 &times; (8 + 8 + 8)) − 88/8
:= (8 + 8)/8 + ((88 &times; (8 + 8 + 8)) − 8)
:= (88 &times; (8 + 8 + 8)) − 8/8
:= 999 + (9999/9 − 9)
:= 9 &times; (9 &times; (9 + 9 + 9) − 9)
:= 999 + (9999 + 9)/9
&Iuml; 2102 := 1 + (((1 + 1) &times; 1111) − 111+1 )
2
:= 2 + (2 &times; 22 + 2) − 2
:= (3 &times; (3
3+3
2+2
3
− 3 )) − (3/3 + 3)
:= (4 − 44)/4 + 44 &times; (44 + 4)
5
5
&Iuml; 2107 := ((1 + 1) &times; (1111 − (1 + 1))) − 111
:= 2 + ((2 &times; 22 + 2)2 − 22/2)
:= 3/3 + (3 &times; (3
3+3
:= 2 &times; (2 &times; (22 &times; (22 + 2)))
3
:= 33 &times; ((3/3 + 3)3 )
− 3 ))
:= 44 &times; (44 + 4) − (4/4 + 4)
5
&Iuml; 2112 := 1 + (((1 + 1) &times; 1111) − 111)
:= 44 &times; (44 + 4)
5
:= 5 + ((5/5 − (5 − 5/5) ) + 5 )
:= 55 + (55/5 − (5 − 5/5)5 )
:= 6 + (6 &times; 6 &times; (66 − 6) − ((6 + 6)/6)6 )
:= 6 + (((6 − 66) &times; (6/6 − 6 &times; 6)) + 6/6)
:= 66 &times; (((6 + 6)/6 − 6) + 6 &times; 6)
:= (7 + 7)/7 + ((7 − 7/7) &times; (7 &times; 7 &times; 7 + 7))
:= 7 &times; (7 &times; (7 &times; 7 − 7) + 7)
:= 5 + (5/5 − (5 − 5/5) )
8−8/8
:= (8 − 88)/8 + (88 &times; (8 + 8 + 8))
:= 8 + (((88/8 − 8)
:= 999 + ((9999 + 9)/9 − 9)
:= 9/9 + (9 &times; (9 &times; (9 + 9 + 9) − 9))
) − 88)
150
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) + 7) − ((7 + 7)/7))
:= 88 &times; (8 + 8 + 8)
:= 99/9 &times; (999/9 + 9 &times; 9)
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 2113 := ((1 + 1) &times; (1 + 1111)) − 111
2
&Iuml; 2118 := 1 + (1 + (((1 + 1) &times; (1 + (11 + 11)))1+1 ))
2
&Iuml; 2123 := 1 + (11 + (((1 + 1) &times; 1111) − 111))
:= 2 + (2 &times; 22 + 2)2 + 2/2 + 2 + 2
:= (2 &times; 22 + 2) − 2/2 − 2
:= 2 + (2 &times; 22 + 2)
:= 3/3 + (33 &times; ((3/3 + 3)3 ))
:= 3 + ((33 &times; ((3/3 + 3)3 )) + 3)
:= 3 &times; 33+3 − ((3/3 + 3)3 )
:= 4/4 + 44 &times; (44 + 4)
:= 4 + (44 &times; (44 + 4) + (4 + 4)/4)
:= 44/4 + 44 &times; (44 + 4)
:= 5 + (((((5 + 5)/5)55/5 ) + 55) + 5)
:= 555 + ((55 + 5/5)/((5 + 5)/5))
:= (5 &times; (5 + 5 + 5)) + (((5 + 5)/5)55/5 )
:= (6 &times; (6 &times; (66 − 6) − 6)) − 66/6
:= (6 &times; (6 &times; (66 − 6) − 6)) − 6
:= (6 &times; (6 &times; (66 − 6) − 6)) − 6/6
77/7
:= 7 + ((7 &times; 7 − ((7 + 7 + 7)/7))(7+7)/7 )
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) + 7) − 7/7)
:= 77 + ((((7 + 7)/7)
:= 8/8 + (88 &times; (8 + 8 + 8))
:= 8 + ((88 &times; (8 + 8 + 8)) − ((8 + 8)/8))
:= 88/8 + (88 &times; (8 + 8 + 8))
:= 9 + ((9 &times; (9 &times; (9 + 9 + 9) − 9)) − ((9 + 9)/9))
:= 9 + ((9/9 + 9 + 9) &times; 999/9)
:= 9 + (((9 &times; (9 &times; (9 + 9 + 9) − 9)) − 9/9) + 9)
&Iuml; 2114 := 1 + (((1 + 1) &times; (1 + 1111)) − 111)
) − 7)
&Iuml; 2119 := 1 + (1 + (1 + (((1 + 1) &times; (1 + (11 + 11)))1+1 )))
&Iuml; 2124 := 1111 + (1 + 1)11−1 − 11
:= (2 &times; 22 + 2)2 − 2
:= 2 + (2 &times; 22 + 2)2 + 2/2
:= 2 &times; 2 &times; ((22 + 2/2)2 + 2)
:= 3 + ((33 &times; ((3/3 + 3)3 )) − 3/3)
:= 3 + (((33 &times; ((3/3 + 3)3 )) + 3/3) + 3)
:= 3 &times; (((33+3 − 33 ) + 3) + 3)
:= (4 + 4)/4 + 44 &times; (44 + 4)
:= 4 + ((44 &times; (44 + 4) − 4/4) + 4)
:= 44 + (4 + 4) &times; (44 + 4)
:= 5 + (555/5 &times; (5 &times; 5 − (5/5 + 5)))
:= (5 − 5/5)5 + (55 &times; (5 &times; 5 − 5) − 5)
:= (5 − 5/5)5 + 55 &times; (5 &times; 5 − 5)
:= 6/6 + ((6 &times; (6 &times; (66 − 6) − 6)) − 6)
:= 6 &times; (6 &times; (66 − 6) − 6)
:= 7 + 7 &times; (7 &times; (7 &times; 7 − 7) + 7)
:= (77 + 7)/7 + 7 &times; (7 &times; (7 &times; 7 − 7) + 7)
:= 7 + ((((7 + 7 + 7)/7)7 − 77) + 7)
:= (8 + 8)/8 + (88 &times; (8 + 8 + 8))
:= 8 + ((88 &times; (8 + 8 + 8)) − 8/8)
:= ((88 + 8)/8) + (88 &times; (8 + 8 + 8))
:= 9 + ((9 &times; (9 &times; (9 + 9 + 9) − 9)) − 9/9)
:= 9 + (9999/9 + 999)
:= 9 + ((9 &times; (9 &times; (9 + 9 + 9) − 9)) + 9)
66/6
:= 66 + (((6 + 6)/6)
)
&Iuml; 2115 := (((1 + 1) &times; (1 + (11 + 11)))1+1 ) − 1
&Iuml; 2120 := 11 + (111 &times; (((1 + 1) &times; (11 − 1)) − 1))
&Iuml; 2125 := 1 + (1111 + (1 + 1)11−1 − 11)
:= (2 &times; 22 + 2)2 − 2/2
:= 2 + (2 &times; 22 + 2)2 + 2
:= 22/2 + (2 &times; 22 + 2)2 − 2
:= 3 + (33 &times; ((3/3 + 3)3 ))
:= 3 &times; 33+3 − (((3/3 + 3)3 ) + 3)
:= 3/3 + (3 &times; (((33+3 − 33 ) + 3) + 3))
:= 4 + (44 &times; (44 + 4) − 4/4)
:= 4 + (44 &times; (44 + 4) + 4)
:= 44 + ((4 + 4) &times; (44 + 4) + 4/4)
:= (5 − 5/5) &times; (555 − 5 &times; 5)
:= 5 &times; (5 &times; (5 &times; 5 + 55 + 5))
5
:= 555 + (5 &times; (5 − 5)/(5 + 5))
66/6
:= 6 + (666/6 &times; ((6/6 + 6 + 6) + 6))
:= 6 + ((((6 + 6)/6)
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) + 7) + 7/7)
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − 7) + 7) − 7/7) + 7)
:= 88/8 + ((88 &times; (8 + 8 + 8)) − 8)
:= 8 + (88 &times; (8 + 8 + 8))
:= 8 + (((88/8 + 8) &times; 888/8) + 8)
:= 9 + (9 &times; (9 &times; (9 + 9 + 9) − 9))
:= 9 &times; 9 + ((((9 + 9)/9)99/9 ) − 9)
:= 9 + (((9 &times; (9 &times; (9 + 9 + 9) − 9)) + 9/9) + 9)
&Iuml; 2116 := ((1 + 1) &times; (1 + (11 + 11)))1+1
:= (2 &times; 22 + 2)
2
) + 66)
:= 6/6 + (6 &times; (6 &times; (66 − 6) − 6))
&Iuml; 2121 := (11 + (11 − 1)) &times; (1 + (11 − 1)1+1 )
:= 77 + (((7 + 7)/7)77/7 )
&Iuml; 2126 := 11 + ((((1 + 1) &times; (1 + (11 + 11)))1+1 ) − 1)
:= 2 + (2 &times; 22 + 2)2 + 2/2 + 2
:= 2 + ((2 &times; 22 + 2)2 + 2 &times; (2 + 2))
:= 3 + ((33 &times; ((3/3 + 3) )) + 3/3)
:= 3 &times; 3 + (33 &times; ((3/3 + 3)3 ))
:= 3 + (3 &times; 33+3 − ((3/3 + 3)3 ))
:= 4 + 44 &times; (44 + 4)
:= 4 + ((44 &times; (44 + 4) + 4/4) + 4)
:= 4 + (44 &times; (44 + 4) + (44 − 4)/4)
:= 5 &times; 5 + (55 − ((5 − 5/5)5 + 5))
:= 5 &times; 5 + (55 − (5 − 5/5)5 )
:= ((6 &times; 6/(6 + 6))6/6+6) − 66
:= (6 + 6)/6 + (6 &times; (6 &times; (66 − 6) − 6))
:= (7 &times; 7 − ((7 + 7 + 7)/7))
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) + 7) + 7)
:= 7/7 + ((((7 + 7)/7)77/7 ) + 77)
:= (8/((8 + 8)/8)) + (88 &times; (8 + 8 + 8))
:= 8 + ((88 &times; (8 + 8 + 8)) + 8/8)
:= 8 + (((88 &times; (8 + 8 + 8)) − ((8 + 8)/8)) + 8)
:= 9 + ((9 &times; (9 &times; (9 + 9 + 9) − 9)) + 9/9)
:= 9 + ((99/9) &times; (999/9 + 9 &times; 9))
:= 9 + ((9 &times; (9 &times; (9 + 9 + 9) − 9)) + (99/9))
3
5
5
:= 5 + (((5 − (5 − 5/5) ) + 5) + 5)
:= (((66 − 6)/6) + 6 &times; 6)
(6+6)/6
(7+7)/7
&Iuml; 2117 := 1 + (((1 + 1) &times; (1 + (11 + 11)))1+1 )
&Iuml; 2122 := 11 + (((1 + 1) &times; 1111) − 111)
&Iuml; 2127 := 11 + (((1 + 1) &times; (1 + (11 + 11)))1+1 )
:= 2/2 + (2 &times; 22 + 2)2
:= 2 + (2 &times; 22 + 2)2 + 2 + 2
:= 22/2 + (2 &times; 22 + 2)2
:= 33/3 + (3 &times; (33+3 − 33 ))
:= 3 &times; 3 + ((33 &times; ((3/3 + 3)3 )) + 3/3)
:= 3 &times; 33+3 − (33 + 33)
:= (44 − 4)/4 + 44 &times; (44 + 4)
:= 4 + (44 &times; (44 + 4) + 44/4)
:= 4 + (44 &times; (44 + 4) + 4/4)
5
5
:= 5 + ((55/5 − (5 − 5/5) ) + 5 )
:= 5 + (((55/5 − (5 − 5/5) ) + 5 ) + 5)
:= 5 &times; 5 + ((5/5 − (5 − 5/5)5 ) + 55 )
:= (6 &times; (6 &times; (66 − 6) − 6)) − 6/6 − 6
:= (6 &times; (6 &times; (66 − 6) − 6)) − (6 + 6)/6
:= 6 + (((6 &times; 6/(6 + 6))6/6+6) − 66)
:= 7 + (((7 + 7 + 7)/7)7 − 77)
:= 7 + ((7 &times; (7 &times; (7 &times; 7 − 7) + 7) + 7/7) + 7)
:= 77 + (7 &times; 7 &times; (7 &times; 7 − 7) − (7/7 + 7))
:= 8 + ((88/8 + 8) &times; 888/8)
:= 8 + ((88 &times; (8 + 8 + 8)) + ((8 + 8)/8))
:= 8 + (((88 &times; (8 + 8 + 8)) − 8/8) + 8)
:= 99/9 + (9 &times; (9 &times; (9 + 9 + 9) − 9))
:= 9 + (((9 &times; (9 &times; (9 + 9 + 9) − 9)) − ((9 + 9)/9)) + 9)
:= 9 + (((9/9 + 9 + 9) &times; 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
&Iuml; 2128 := (1 + 111) &times; (((1 + 1) &times; (11 − 1)) − 1)
:= 2 &times; ((2 &times; (((22 + 2/2)2 ) + 2)) + 2)
:= 3/3 + (3 &times; 33+3 − (33 + 33))
&Iuml; 2133 := 1111 + ((1 + 1)11−1 − (1 + 1))
:= 22 + (2 &times; 22 + 2)2
:= 22 + (2222 − 222/2)
:= 3 &times; (3
3+3
:= 33 + ((33 &times; ((3/3 + 3)3 )) − 3/3)
− (3 &times; (3 + 3)))
4
&Iuml; 2138 := (1 + 1)11 + (11 − 1) &times; (11 − 1 − 1)
4
:= 4 &times; ((44 &times; ((4 + 4) + 4)) + 4)
:= 4 + ((4 − 4/4) + 4 &times; (4 + 4))
:= 4 + ((4444 − 4)/4 + 4 &times; 44 )
:= 5 &times; 5 + ((((5 + 5)/5)55/5 ) + 55)
:= 55 + (((5 + 5)/5)5 − (5 − 5/5)5 )
:= 5 &times; 555 − ((55 + 55 + 5)/5)
:= 6 + ((6 &times; (6 &times; (66 − 6) − 6)) − ((6 + 6)/6))
:= 6 + ((((6 &times; 6/(6 + 6))6/6+6) − 66) + 6)
:= 6 &times; 6 &times; (66 − 6) − ((66 + 66)/6)
:= 77 + (7 &times; 7 &times; (7 &times; 7 − 7) − 7)
:= 77 + (7 &times; 7 &times; (7 &times; 7 − 7) − ((7 + 7)/7))
:= ((7 + 7 + 7)/7)7 − 7 &times; 7
:= 8 + ((88 &times; (8 + 8 + 8)) + 8)
:= (88/8 − 8) &times; ((8 &times; 88 − 8/8) + 8)
:= 88 + ((8 &times; (8 + 8) &times; (8 + 8)) + ((8 + 8)/8))
:= (9/9 + 9 + 9) &times; ((999 + 9)/9)
:= (9 + 9 + 9) &times; (9 &times; 9 − ((9 + 9)/9))
:= 9 + ((((9 + 9)/9)99/9 ) + 9 &times; 9)
&Iuml; 2129 := (1 + 1)11 + (11 − 1 − 1)1+1
&Iuml; 2134 := 1111 + ((1 + 1)11−1 − 1)
2
:= 2 + ((2 &times; 22 + 2) + 22/2)
3
:= 2222 − 2 &times; 2 &times; 22
33/3
:= 3 &times; 3 + (3 − 3/3)
4
&Iuml; 2139 := 1 + ((1 + 1)11 + (11 − 1) &times; (11 − 1 − 1))
:= 3/3 + (3 &times; (3
4
3+3
:= 22 + ((2 &times; 22 + 2)2 + 2/2)
− (3 &times; (3 + 3))))
4
:= (4 − 4/4) + 4 &times; (4 + 4)
:= 4 &times; 4 + (4444 − 4)/4
5
:= 33 + (33 &times; ((3/3 + 3)3 ))
:= 4 + (4444/4 + 4 &times; 44 )
5
:= 5 + ((5 &times; (5 &times; (55 + 5))) + (5 − 5)/5)
:= 555 + ((5 − 5/5) + 555)
:= 5 &times; 555 − ((55 + 55 )/5)
:= 6 + ((6 &times; (6 &times; (66 − 6) − 6)) − 6/6)
:= ((66 − 6)/6) + (6 &times; (6 &times; (66 − 6) − 6))
:= ((66 &times; 66 − 6)/((6 + 6)/6)) − 6 &times; 6
:= 7/7 + ((7 &times; 7 &times; (7 &times; 7 − 7) − 7) + 77)
:= 77 + 7 &times; 7 &times; (7 &times; 7 − 7) − 7/7
:= 7/7 + (((7 + 7 + 7)/7)7 − 7 &times; 7)
:= 88 + ((8 &times; (8 + 8) &times; (8 + 8)) − ((8 + 8)/8))
:= 8 + (((88 &times; (8 + 8 + 8)) + (88/8)) + 8)
:= 9/9 + ((9 + 9 + 9) &times; (9 &times; 9 − ((9 + 9)/9)))
:= ((99/9 + 9) &times; 999/9) − 9 &times; 9
:= 8 + (((88 &times; (8 + 8 + 8)) + 8/8) + 8)
99/9
:= 9 &times; 9 + (((9 + 9)/9)
)
&Iuml; 2130 := 1 + ((1 + 1)11 + (11 − 1 − 1)1+1 )
&Iuml; 2135 := 1111 + (1 + 1)11−1
&Iuml; 2140 := 11 + ((1 + 1)11 + (11 − 1 − 1)1+1 )
:= 22+2 + ((2 &times; 22 + 2)2 − 2)
:= (222/2 + 2222)/2
:= 2 + ((2 &times; 22 + 2)2 + 22)
:= (33 + 3) &times; (((3 + 3)3 − 3)/3)
:= 3 + ((33 − 3/3) &times; (3 &times; 33 + 3/3))
:= 33 + ((33 &times; ((3/3 + 3)3 )) + 3/3)
:= 4 &times; 4 + (44 &times; (44 + 4) + (4 + 4)/4)
:= 4 &times; 44 + 4444/4
:= 4 + ((4 + 4) &times; (44/4 + 44 ))
:= 5 + (5 &times; (5 &times; (5 &times; 5 + 55 + 5)))
:= 5 + ((5 &times; (5 &times; (5 &times; 5 + 55 + 5))) + 5)
:= (5 − 5/5) &times; ((555 − 5 &times; 5) + 5)
:= 6 + (6 &times; (6 &times; (66 − 6) − 6))
:= 66/6 + (6 &times; (6 &times; (66 − 6) − 6))
:= ((66 − 6)/6) &times; (6 &times; 6 &times; 6 − (6 + 6)/6)
:= (7 − 7/7) &times; ((77 + 7)/7 + 7 &times; 7 &times; 7)
:= 77 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 8 + (((88 &times; (8 + 8 + 8)) + ((8 + 8)/8)) + 8)
:= (7 + 7)/7 + (((7 + 7 + 7)/7)7 − 7 &times; 7)
:= 88 + ((8 &times; (8 + 8) &times; (8 + 8)) − 8/8)
:= 88 + (((8 &times; 8 &times; 8 &times; 8) + 8)/((8 + 8)/8))
:= 9 + (((9 &times; (9 &times; (9 + 9 + 9) − 9)) + (99/9)) + 9)
:= (99/9 + 9) &times; ((99 − 9/9) + 9)
99/9
:= 9/9 + ((((9 + 9)/9)
) + 9 &times; 9)
&Iuml; 2131 := 1 + (1 + ((1 + 1)11 + (11 − 1 − 1)1+1 ))
2
:= 2 + (((2 &times; 22 + 2) + 22/2) + 2)
3
3
:= (((3 + 3) &times; ((3 + 3) − 3)) + 3)/3
4
4
:= 4 + ((4 − 4/4) &times; (4/4 + 4) )
5
5
&Iuml; 2136 := 1 + (1111 + (1 + 1)11−1 )
&Iuml; 2141 := (1 + 1)11 + (((1 + 1)11−1 − 1)/11)
2
:= 2222 − (2/2 + 2)2+2
:= 22 + ((2 &times; 22 + 2) − 2)
:= 3 + (3 &times; (3
3+3
− (3 &times; (3 + 3))))
4
:= (33/3)3 + (33 &times; (33 + 3))
:= (4 − 4/4 + 4)4 − 44 − 4
:= (4 + 4) &times; (44/4 + 4 )
5
5
:= 5 + ((5 &times; 5 − (5 − 5/5) ) + 5 )
:= 5 + (((5 &times; 5 − (5 − 5/5) ) + 5 ) + 5)
:= 6 + ((6 &times; (6 &times; (66 − 6) − 6)) + 6/6)
:= 6 + ((6 &times; (6 &times; (66 − 6) − 6)) + 6)
:= 6 + ((6 &times; (6 &times; (66 − 6) − 6)) + (66/6))
:= 7/7 + (7 &times; 7 &times; (7 &times; 7 − 7) + 77)
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) − 7/7 + 77)
:= 8 + ((88 &times; (8 + 8 + 8)) + (88/8))
:= 88 + (8 &times; (8 + 8) &times; (8 + 8))
:= 8 + ((88/8 − 8) &times; ((8 &times; 88 − 8/8) + 8))
:= ((99/9 + 9) &times; ((99 − 9/9) + 9)) − 9
:= 9 + ((((9/9 + 9 + 9) &times; 999/9) + 9) + 9)
:= 9 + ((9 &times; (99 + 9 &times; 9)) + (((9 + 9)/9)9 ))
7
:= ((7 + 7 + 7)/7) − (7 &times; 7 + 7)
&Iuml; 2132 := 1111 + ((1 + 1)11−1 − 1 − 1 − 1)
&Iuml; 2137 := 1 + (1 + (1111 + (1 + 1)11−1 ))
:= ((5 − 55 )/5) + (5 &times; 555 − (5 + 5))
&Iuml; 2142 := (1 + 1) &times; ((11 − 1 − 1) &times; (111+1 − (1 + 1)))
:= 22+2 + (2 &times; 22 + 2)2
:= 22 + ((2 &times; 22 + 2)2 − 2/2)
:= 2 + (((2 &times; 22 + 2)2 + 22) + 2)
:= (33 − 3/3) &times; (3 &times; 33 + 3/3)
:= 3 + ((3 &times; (33+3 − (3 &times; (3 + 3)))) + 3/3)
:= 3 &times; ((33+3 − (3 &times; (3 + 3))) + 3)
:= 4 + (44 &times; (44 + 4) + 4 &times; 4)
:= 4/4 + ((4 + 4) &times; (44/4 + 44 ))
:= 4/4 + (4 − 4/4 + 4)4 − 44 − 4
(5+5)/5+5
:= ((5 − (5 + 5)/5)
) − 55
:= 6 + ((6 &times; (6 &times; (66 − 6) − 6)) + ((6 + 6)/6))
77/7
:= 7 + ((((7 + 7)/7)
) + 77)
:= 8 + ((88 &times; (8 + 8 + 8)) + ((88 + 8)/8))
9
:= (((9 + 9)/9) ) + (9 &times; (99 + 9 &times; 9))
(5+5)/5+5
:= 5 + (((5 − (5 + 5)/5)
) − 55)
:= 6 + (((6 &times; (6 &times; (66 − 6) − 6)) + 6/6) + 6)
7
:= (5 − 5/5 + 5) &times; (((5 − (5 + 5)/5)5 ) − 5)
:= 6 &times; (66 &times; 66/(6 + 6) − 6)
:= ((7 + 7 + 7)/7) − (7/7 + 7 &times; 7)
:= 7 + (7 &times; 7 &times; (7 &times; 7 − 7) + 77)
:= 8/8 + ((8 &times; (8 + 8) &times; (8 + 8)) + 88)
:= (8/8 + 8 + 8) &times; (8 &times; (8 + 8) − ((8 + 8)/8))
:= 9 + ((9/9 + 9 + 9) &times; ((999 + 9)/9))
:= (9 + 9) &times; ((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
&Iuml; 2143 := 11 &times; ((1 + 1 + 1 + 11)1+1 − 1) − 1 − 1
&Iuml; 2148 := 111 + ((1 + 1)11 − 11)
&Iuml; 2153 := 111 + ((1 + 1)11 − ((1 + 1) &times; (1 + 1 + 1)))
:= 2 + (2222 − (2/2 + 2)2+2 )
:= 2 &times; ((2 + 2 + 2)2+2 − 222)
:= ((22/2 + 2)2/2+2 ) − (2 &times; 22)
:= 3 &times; 33+3 − (33/3 + 33)
:= 3 + ((3 &times; (33+3 − 3)) − 33)
:= 3 &times; 33+3 − 3/3 − 33
:= (44 − 4)/4 + (4 + 4) &times; (44 + 4)
:= 4 + ((4 + 4) &times; ((44 + 4 + 4) + 4))
:= 4 + (((((4 − 4/4) + 4)4 ) − 44 ) + 4)
:= 5 &times; 555 − ((55 + 5 + 5)/5 + 5)
:= 5 &times; 555 − (55 + 5 + 5)/5
:= 5 + (5 &times; 555 − (55 + 5 + 5)/5)
:= 6 &times; 6 &times; (66 − 6) − (66/6 + 6)
:= 6 &times; 6 &times; (66 − 6) − 6 − 6
:= 6 &times; 6 &times; (66 − 6) − 6/6 − 6
:= 7 + ((7 &times; 7 &times; (7 &times; 7 − 7) + 77) + 7/7)
:= 77 &times; (7 + 7 + 7 + 7) − (7/7 + 7)
:= 7 + ((((7 + 7)/7)77/7 ) + 7 &times; (7 + 7))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) − 8/8) + 88)
:= 8 + ((((8 &times; 8 &times; 8 &times; 8) + 8)/((8 + 8)/8)) + 88)
:= 8 + ((((8 &times; (8 + 8) &times; (8 + 8)) + 88) + 8/8) + 8)
:= 9/9 + ((9 + 9) &times; ((99/9 + 99) + 9))
:= 9/9 + ((((9 + 9)/9)99/9 ) + 99)
:= (9 + 9)/9 + (((9 + 9) &times; (999/9 + 9)) − 9)
&Iuml; 2144 := 11 &times; ((1 + 1 + 1 + 11)1+1 − 1) − 1
:= 2
2+2
&times; (22 &times; (2 + 2 + 2) + 2)
3
:= (33 − 3/3) &times; (((3/3 + 3) ) + 3)
4
:= (4 + 4) &times; ((4 + 4 + 4) + 4)
5
:= 5 &times; 555 − ((5 + 5)/5 + 5)
&Iuml; 2149 := 1 + (111 + ((1 + 1)11 − 11))
&Iuml; 2154 := (11 &times; (1 + 1 + 1 + 11)1+1 ) − 1 − 1
:= 22 + ((2 &times; 22 + 2)2 + 22/2)
:= 222 + (2 &times; (2 &times; 222 − 2))
:= 3 &times; 33+3 − (33/3 + 33 )
:= 3 &times; 33+3 − 33
:= 4 + ((((4 − 4/4) + 4)4 ) − 44 )
:= 44 + (44 &times; (44 + 4) − (4 + 4)/4)
:= 5 &times; 555 − (5 + 5)/5
:= 5 + (5 &times; 555 − (55 + 5)/5)
:= (6 − 66)/6 + (6 &times; 6 &times; (66 − 6) − 6)
:= 6 &times; 6 &times; (66 − 6) − 66/6
:= 6 &times; 6 &times; (66 − 6) − 6
:= 7 + (((7 + 7 + 7)/7)7 − (7/7 + 7 &times; 7))
:= 77 &times; (7 + 7 + 7 + 7) − 7
:= ((7 + 7)/7) &times; (77 &times; (7 + 7) − 7/7)
:= 8 + ((8 &times; (8 + 8) &times; (8 + 8)) + 88)
:= (88 &times; (8 + 8 + 8)) + 888/(8 + 8 + 8)
:= 8 &times; 8 + ((88/8 + 8) &times; (888 − 8)/8)
:= (9 + 9)/9 + ((9 + 9) &times; ((99/9 + 99) + 9))
:= 9 + ((99/9 + 9) &times; ((99 − 9/9) + 9))
:= ((9 + 9 + 9)/9) &times; (9 &times; 9 &times; 9 − (99/9))
5
&Iuml; 2145 := 11 &times; ((1 + 1 + 1 + 11)1+1 − 1)
:= 22/2 &times; (((2
2+2
2
− 2) ) − 2/2)
&Iuml; 2150 := 1 + (1 + (111 + ((1 + 1)11 − 11)))
&Iuml; 2155 := (11 &times; (1 + 1 + 1 + 11)1+1 ) − 1
:= 222 + (2 &times; (2 &times; (222 − 2)))
:= ((22 &times; ((22+2 − 2)2 )) − 2)/2
:= (3 &times; 3 + 3/3) &times; ((3 + 3)3 − 3/3)
:= 3/3 + (3 &times; 33+3 − 33)
:= 4 + (4 − 4/4 + 4)4 − 44 + 4/4
:= 44 + (44 &times; (44 + 4) − 4/4)
:= 55 &times; (55 − (55/5 + 5))
:= 5 &times; (555 − 5 &times; 5 &times; 5)
:= 5 + 5 &times; (555 − 5 &times; 5 &times; 5)
:= (66 − 6/6) &times; (66 &times; 6/(6 + 6))
:= (6 − 66)/6 + 6 &times; 6 &times; (66 − 6)
:= 6/6 + (6 &times; 6 &times; (66 − 6) − 6)
:= 7 + (((7 + 7 + 7)/7)7 − 7 &times; 7)
:= 7/7 + (77 &times; (7 + 7 + 7 + 7) − 7)
:= 77 &times; (7 + 7 + 7 + 7) − 7/7
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + 88) + 8/8)
:= (8/8 + 8 + 8 + 8) &times; (88 − ((8 + 8)/8))
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + (88/8)) + 88)
:= (9/9 + 9) &times; (((9 + 9) &times; (99 + 9) − 9)/9)
:= 9 + (((((9 + 9)/9)99/9 ) − 9/9) + 99)
:= (3 &times; (3
3+3
− 3)) − 33
4
:= (((4 − 4/4) + 4) ) − 4
4
99/9
:= 99 + ((((9 + 9)/9)
) − ((9 + 9)/9))
&Iuml; 2146 := 1 + 11 &times; ((1 + 1 + 1 + 11)1+1 − 1)
:= 2 + (2 &times; 2 &times; (22 + 2) + 2
:= 3/3 + ((3 &times; (3
3+3
22/2
)
− 3)) − 33)
4
4
:= 4/4 + ((((4 − 4/4) + 4) ) − 4 )
&Iuml; 2151 := 1 + (1 + (1 + (111 + ((1 + 1)11 − 11))))
2
2
&Iuml; 2156 := 11 &times; (1 + 1 + 1 + 11)1+1
:= (2/2 + 2) &times; ((22 − 2)/2 − 2)
:= 22 &times; (2 &times; 2 &times; (22 + 2) + 2)
:= 3 &times; (33+3 − (3 &times; 3 + 3))
:= 3 + (3 &times; 33+3 − (3/3 + 33))
4
:= 4 &times; (4 + 4) + 4444/4
:= 44 + 44 &times; (44 + 4)
:= ((5 − 5 )/5) + (5 &times; 555 − 5)
:= ((5 − 55 )/5) + 5 &times; 555
:= 55 + (55 − (5 − 5/5)5 )
:= (6 &times; 6 + 6/6) &times; (((6 + 6)/6)6 − 6)
:= ((6 &times; 6/(6 + 6))6/6+6) − 6 &times; 6
:= (6 + 6)/6 + (6 &times; 6 &times; (66 − 6) − 6)
:= (7 + 7)/7 + (77 &times; (7 + 7 + 7 + 7) − 7)
:= 77 &times; ((7 + 7 + 7) + 7)
:= (8/8 + 8) &times; (888/8 + 8 &times; (8 + 8))
:= (8 − 8/8) &times; ((88 &times; (8 &times; 8 − 8))/(8 + 8))
:= ((9 + 9) &times; (999/9 + 9)) − 9
:= 9 + ((((9 + 9)/9)99/9 ) + 99)
5
77/7
:= 7 &times; (7 + 7) + (((7 + 7)/7)
)
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + ((8 + 8)/8)) + 88)
99/9
:= 99 + ((((9 + 9)/9)
) − 9/9)
&Iuml; 2147 := 111 + ((1 + 1)11 − (1 + 11))
&Iuml; 2152 := 1 + (1 + (1 + (1 + (111 + ((1 + 1)11 − 11)))))
&Iuml; 2157 := 111 + (1 + 1)11 − (1 + 1)
:= 222 + ((2 &times; 22)2 − 22/2)
:= (2 + 2 + 2)2 + (2 &times; 22 + 2)2
:= 222 + ((2 &times; 22)2 − 2/2)
:= 3 &times; 33 + (3 − 3/3)33/3
:= 3/3 + (3 &times; (33+3 − (3 &times; 3 + 3)))
:= 3 + (3 &times; 33+3 − 33)
:= 44 + (44 &times; (44 + 4) − 4)
:= 44 + (44 &times; (44 + 4) + 4/4)
4
4
:= 4 + (4 + 4) &times; (4 + 4) + (4 − 4)/4
5
5
:= 5 &times; 555 + ((((5 − 5 ) + 5)/5) − 5)
:= 5 &times; 555 + (((5 − 5 ) + 5)/5)
:= 5 + ((((5 − 55 ) + 5)/5) + 5 &times; 555)
:= 6 &times; 6 &times; (66 − 6) − (6/6 + 6 + 6)
:= 6 &times; 6 &times; (66 − 6) − ((6 + 6)/6 + 6)
:= 6 &times; 6 &times; (66 − 6) − 6 &times; 6/(6 + 6)
:= 77 &times; (7 + 7 + 7 + 7) − ((7 + 7)/7 + 7)
:= 88 + ((8 &times; (8 + 8) &times; (8 + 8)) + (88/8))
99/9
:= 99 + (((9 + 9)/9)
)
7
:= 7 + ((((7 + 7 + 7)/7) − 7 &times; 7) + 7)
:= 7/7 + 77 &times; (7 + 7 + 7 + 7)
:= 8 + (((8 &times; (8 + 8) &times; (8 + 8)) + 88) + 8)
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − (88/8 + 8)
:= 9/9 + (((9 + 9) &times; (999/9 + 9)) − 9)
:= ((9 + 9 + 9)/9) &times; (9 &times; 9 &times; 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
&Iuml; 2158 := 111 + (1 + 1)11 − 1
&Iuml; 2163 := 1 + (1 + (1 + (1 + (111 + (1 + 1)11 ))))
&Iuml; 2168 := 111+1 + ((1 + 1)11 − 1)
:= 222 + (2 &times; 22)2
:= 2 + ((222/2 + 222/2 ) + 2)
:= 2 + ((2 &times; (2 &times; (222 + 2))) + 222)
:= 3 + ((3 &times; 33+3 − 33) + 3/3)
:= 3 + (3 &times; (33+3 − 3 &times; 3))
:= (3 &times; (33+3 − (3 + 3))) − 3/3
:= 44 &times; (4 + 4) + (444 − 4)/4
:= 4 + (444/4 + 44 &times; (4 + 4))
:= (4 + 4) &times; ((44/4 + 44 ) + 4)
:= (5 + 5)/5 &times; ((5 − 5/5)5 + 55)
:= 5 + ((5 + 5)/5 &times; ((5 − 5/5)5 + 55))
:= 5 &times; 5 &times; 5 + ((((5 + 5)/5)55/5 ) − 5)
:= 6 &times; 6 &times; (66 − 6) − (6 + 6)/6
:= (6 &times; 6/(6 + 6)) + 6 &times; 6 &times; (66 − 6)
:= 6 + (6 &times; 6 &times; (66 − 6) + ((6 + 6)/6))
:= (7 + 7)/7 + 77 &times; (7 + 7 + 7 + 7)
:= 7 + 77 &times; (7 + 7 + 7 + 7)
:= ((7 + 7)/7) &times; ((77 &times; (7 + 7) − 7/7) + 7)
:= (888 − 8)/8 + (8 &times; (8 + 8) &times; (8 + 8))
:= ((88/8 − 8)8−8/8 ) − 8 − 8 − 8
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 8
:= ((9 + 9)/9) &times; ((999 − 9/9) + 9 &times; 9)
:= ((9 + 9 + 9)/9) &times; ((9 &times; 9 &times; 9 − 9) + 9/9)
:= 9 &times; 9 &times; (9 + 9 + 9) − (9/9 + 9 + 9)
&Iuml; 2159 := 111 + (1 + 1)11
:= 222/2 + 2
:= 3 &times; 3
3+3
&Iuml; 2164 := 1 + (1 + (1 + (1 + (1 + (111 + (1 + 1)11 )))))
22/2
3
− (3 + 3/3)
4
:= (22 − 2)2 + (2 &times; 22 − 2)2
:= (22/2)2 + 222/2
:= (((3 &times; 3 + 3/3) + 3)3 ) − 33
:= 3 &times; (33+3 − (3 + 3))
:= 4 + (((4 + 4)4 ) − 44 &times; 44)
:= 444/4 + 4 &times; (4 + 4)
5
&Iuml; 2169 := 111+1 + (1 + 1)11
5
(5+5)/5
:= 4 + ((4/4 + 4) &times; (444 − 44/4))
:= 5 + ((5 &times; 555 − (5 + 5)/5) + 5)
:= 5 − (((5 &times; 5 + 5/5) + 5)
:= 6 &times; 6 &times; (66 − 6) − 6/6
:= 6 + (6 &times; 6 &times; (66 − 6) − ((6 + 6)/6))
)
:= 5 &times; 555 − ((55 &times; 55 + 5)/5)
:= ((66 &times; 66 − 6)/((6 + 6)/6)) − 6
:= 7 + (77 &times; (7 + 7 + 7 + 7) + 7/7)
:= 777/7 + 7 &times; 7 &times; (7 &times; 7 − 7)
:= 888/8 + (8 &times; (8 + 8) &times; (8 + 8))
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − (88 + 8)/8
:= 8/8 + (((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 8)
:= ((9 + 9) &times; (999/9 + 9)) − 9/9
:= 9999/9 + (9 &times; (99 + 9 + 9))
:= 9 &times; 9 &times; (9 + 9 + 9) − (9 + 9)
7
:= 7 &times; 7 + (((7 + 7 + 7)/7) − 77)
&Iuml; 2160 := 1 + (111 + (1 + 1)11 )
&Iuml; 2165 := 111 + ((1 + 1)11 + ((1 + 1) &times; (1 + 1 + 1)))
&Iuml; 2170 := 1 + (111+1 + (1 + 1)11 )
:= 2 + ((2 &times; 22)2 + 222)
:= 222 + ((2 &times; (22 − 2) + 2/2)2 )
:= 2 &times; ((22/2 + 22)2 − (2 + 2))
:= 3 &times; (33+3 − 3 &times; 3)
:= 33/3 + (3 &times; 33+3 − 33)
:= 3/3 + (3 &times; (33+3 − (3 + 3)))
:= ((4 + 4)4 ) − 44 &times; 44
:= (4/4 + 4) &times; (444 − 44/4)
:= (4/4 + 4) &times; ((4 − 44)/4 + 444)
:= (55 + 5) &times; (55/5 + 5 &times; 5)
:= ((5 − 5/5) &times; 555) − 55
:= 5 + (((5 − 5/5) &times; 555) − 55)
:= 6 &times; 6 &times; (66 − 6)
:= 6 + (6 &times; 6 &times; (66 − 6) − 6/6)
:= ((66 − 6)/6) + 6 &times; 6 &times; (66 − 6)
:= (77/7 − 7) &times; (7 &times; 77 + 7/7)
:= 7 + (77 &times; (7 + 7 + 7 + 7) + ((7 + 7)/7))
:= 7 + (77 &times; (7 + 7 + 7 + 7) + 7)
:= (8 + 8) &times; ((8 &times; (8 + 8) − 8/8) + 8)
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 88/8
:= (8 + 8)/8 + (((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 8)
:= (9 + 9) &times; (999/9 + 9)
&Iuml; 2161 := 1 + (1 + (111 + (1 + 1)11 ))
99/9
:= 9 + (((((9 + 9)/9)
) + 99) + 9)
&Iuml; 2166 := 111+1 + ((1 + 1)11 − (1 + 1 + 1))
:= 9/9 + (9 &times; 9 &times; (9 + 9 + 9) − (9 + 9))
&Iuml; 2171 := 1 + (1 + (111+1 + (1 + 1)11 ))
:= 2 + (222/2 + 222/2 )
:= 222 + (2 &times; (2 &times; (222 + 2)))
:= 2 + ((22/2)2 + 222/2 )
:= 3/3 + (3 &times; (33+3 − 3 &times; 3))
:= (3 &times; (33+3 − (3 + 3))) − 3
:= 3 + ((3 &times; (33+3 − (3 + 3))) − 3/3)
:= 4 &times; 4 + ((((4 − 4/4) + 4)4 ) − 44 )
:= (4 − 44)/4 + (4 + 4) &times; (4 &times; 4 + 44 )
:= (4 + 4) &times; (4 &times; 4 + 44 ) − (4/4 + 4)
:= 5 + ((55 − (5 − 5/5)5 ) + 55 )
:= 5 + (((55 − (5 − 5/5)5 ) + 55 ) + 5)
:= ((5 − 55 )/5) + ((5 &times; (555 + 5)) − 5)
:= 6/6 + 6 &times; 6 &times; (66 − 6)
:= 6 + 6 &times; 6 &times; (66 − 6)
:= 66/6 + 6 &times; 6 &times; (66 − 6)
:= 7 + (((7 + 7)/7) &times; (77 &times; (7 + 7) − 7/7))
:= ((7 + 7 + 7)/7)7 − (7 + 7 + 7)
:= 7 + ((77 &times; (7 + 7 + 7 + 7) + 7/7) + 7)
:= 8/8 + ((8 + 8) &times; ((8 &times; (8 + 8) − 8/8) + 8))
:= (8 − 88)/8 + ((8 + 8) &times; (8 &times; (8 + 8) + 8))
:= ((88/8 − 8)8−8/8 ) − 8 − 8
:= 9/9 + ((9 + 9) &times; (999/9 + 9))
:= 9 &times; 9 &times; (9 + 9 + 9) − (((99 + 9)/9) + 9)
:= (9 + 9)/9 + (9 &times; 9 &times; (9 + 9 + 9) − (9 + 9))
&Iuml; 2162 := 1 + (1 + (1 + (111 + (1 + 1)11 )))
:= 2 + (((2 &times; 22)2 + 222) + 2)
&Iuml; 2167 := 11 &times; (1 + (1 + 1 + 1 + 11)1+1 )
&Iuml; 2172 := 1 + (1 + (1 + (111+1 + (1 + 1)11 )))
:= (22/2)2 + 222/2 − 2
:= 2222 − ((2 &times; (22 + 2)) + 2)
:= (3 &times; (33+3 − 3)) − 33/3
:= 3 + (3 &times; (33+3 − (3 + 3)))
:= 4 + ((444 − 4)/4 + 44 &times; (4 + 4))
:= 4 + ((444/4 + 44 &times; (4 + 4)) + 4)
:= (4 + 4) &times; (4 &times; 4 + 44 ) − 4
:= 5 &times; 555 + (((55 − 55 ) + 5)/5)
:= ((5 + 5)/5 &times; 5555/5) − 55
:= (5 + 5)/5 &times; (5555/5 − 5 &times; 5)
:= (6 + 6)/6 + 6 &times; 6 &times; (66 − 6)
:= 6 + (6 &times; 6 &times; (66 − 6) + 6/6)
:= 6 + (6 &times; 6 &times; (66 − 6) + 6)
:= 7 + (77 &times; (7 + 7 + 7 + 7) − 7/7)
:= 77/7 + 77 &times; (7 + 7 + 7 + 7)
:= ((7 + 7 + 7)/7)7 − (7/7 + 7 + 7)
:= (8 + 8)/8 + ((8 + 8) &times; ((8 &times; (8 + 8) − 8/8) + 8))
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − (8/8 + 8)
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − (8/((8 + 8)/8))
:= (9 + 9)/9 + ((9 + 9) &times; (999/9 + 9))
:= 99/9 &times; ((99 − 9/9) + 99)
:= ((99 + 9)/9) &times; ((9/9 + 99) + 9 &times; 9)
:= 3 + (3 &times; 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
&Iuml; 2173 := 1 + (1 + (1 + (1 + (111+1 + (1 + 1)11 ))))
&Iuml; 2178 := (1 + 1) &times; (11 &times; (1 + 1 + 1))1+1
&Iuml; 2183 := 1 + (1 + (1 + ((1 + 1)11 + (11 &times; (1 + 11)))))
:= 2 + (((22/2)2 + 222/2 ) + 2)
:= 2 &times; (22/2 + 22)2
:= ((2 &times; (22 + 2))2 ) − (22/2)2
:= 3 &times; 33+3 − (33/3 + 3)
:= 3 &times; (33+3 − 3)
:= 3 &times; 33+3 − (3/3 + 3)
:= 4/4 + ((4 + 4) &times; (4 &times; 4 + 44 ) − 4)
:= (4 + 4)/4 + (4 + 4) &times; (4 &times; 4 + 44 )
:= ((4 − 4/4)4+4−4/4 ) − 4
:= 5 &times; 5 &times; 5 + (((5 + 5)/5)55/5 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 &times; 5 &times; 5)
:= (((5 + 5)/5)5 + 5) &times; (55 − 5/5 + 5)
:= 6 + ((6 &times; 6 &times; (66 − 6) + 6/6) + 6)
:= 6 &times; 66 &times; 66/(6 + 6)
:= (6 &times; 6 + 6/6) &times; (66 − (6/6 + 6))
7
:= (77/7 + 7) &times; (((7 + 7)/7) − 7)
:= 7 + (((7 + 7 + 7)/7)7 − (77/7))
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) + 8)) − (88/8))
:= (8 + 8)/8 + ((8 + 8) &times; (8 &times; (8 + 8) + 8))
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 8/8)
:= 9 + (9999/9 + (9 &times; (99 + 9 + 9)))
:= 9 &times; 9 &times; (9 + 9 + 9) − 9
:= ((9 − 9 &times; 9)/(9 + 9)) + 9 &times; 9 &times; (9 + 9 + 9)
7
:= ((7 + 7 + 7)/7) − (7 + 7)
&Iuml; 2174 := (1 + 1) &times; (1111 − ((1 + 1) &times; (1 + 11)))
:= 2 &times; ((22/2 + 22)2 − 2)
&Iuml; 2179 := 1 + ((1 + 1) &times; (11 &times; (1 + 1 + 1))1+1 )
:= (((22+2+2 + 2)2 ) + 2)/2
3+3
&Iuml; 2184 := (1 + 1 + 1) &times; ((11 − 1 − 1)1+1+1 − 1)
:= 2 + (2222 + (2 &times; (2 − 22)))
:= 3 &times; 33+3 − 3
:= (3 &times; (33+3 − 3)) − (3/3 + 3)
:= 3/3 + (3 &times; (3
:= ((((4 + 4)4 ) − 4) + 44 )/((4 + 4)/4)
:= 4 + ((4 + 4) &times; (4 &times; 4 + 44 ) − 4/4)
:= 4 + ((4 + 4) &times; (4 &times; 4 + 44 ) + 4)
:= (5 &times; (555 + 5)) − (55 + 5)/5
:= 5 + ((5 &times; (555 + 5)) − (55 + 5)/5)
:= ((5 + 5)/5 + 5) &times; (55 − 5)/(5 + 5)
:= 6 + ((6 &times; 6 &times; (66 − 6) + ((6 + 6)/6)) + 6)
:= 6/6 + (6 &times; 66 &times; 66/(6 + 6))
:= 6 + (6 &times; 66 &times; 66/(6 + 6))
− 3))
7
:= 7/7 + (((7 + 7 + 7)/7)7 − (7 + 7))
:= ((7 + 7 + 7)/7) − (7/7 + 7)
:= 77 + 7 &times; (7 &times; (7 &times; 7 − 7) + 7)
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − (8 + 8)/8
:= ((88/8 − 8)8−8/8 ) − 8
:= 8 + ((8 + 8) &times; (8 &times; (8 + 8) + 8))
:= ((9 + 9)/9) &times; (((99 &times; 99) − (9 + 9))/9)
:= 9/9 + (9 &times; 9 &times; (9 + 9 + 9) − 9)
:= ((9 + 9 + 9)/9) &times; (9 &times; 9 &times; 9 − 9/9)
&Iuml; 2175 := ((1 + 1 + 11)1+1+1 ) − 11 − 11
&Iuml; 2180 := (1 + 1)11 + (11 &times; (1 + 11))
&Iuml; 2185 := ((1 + 1 + 11)1+1+1 ) − 1 − 11
:= ((22/2 + 2)2/2+2 ) − 22
:= 2 + (2 &times; (22/2 + 22)2 )
:= ((2/2 + 2)2/2+2+2+2 ) − 2
:= (3 &times; (33+3 − 3)) − 3
:= 3 + ((3 &times; (33+3 − 3)) − 3/3)
:= 3/3 + (3 &times; 33+3 − 3)
:= (4 + 4) &times; (4 &times; 4 + 44 ) − 4/4
:= 4 + (4 + 4) &times; (4 &times; 4 + 44 )
:= 4 + (((4 + 4) &times; (4 &times; 4 + 44 ) + 4/4) + 4)
:= 5 &times; ((555 − 5 &times; 5 &times; 5) + 5)
:= (5 − 5/5) &times; (555 − 5 − 5)
:= 5 + ((5 − 5/5) &times; (555 − 5 − 5))
:= (66 &times; 66 − 6)/((6 + 6)/6)
:= 66 + ((((6 + 6)/6)66/6 ) + 66)
:= 6 &times; (6 &times; (66 − 6) + 6) − 66/6
:= ((7 + 7 + 7)/7)7 − (77 + 7)/7
:= ((7 + 7 + 7)/7)7 − 7
:= ((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 8/8
:= 8/8 + (((88/8 − 8)8−8/8 ) − 8)
:= 9 &times; 9 &times; (9 + 9 + 9) − (99 + 9)/9
:= (9 + 9)/9 + (9 &times; 9 &times; (9 + 9 + 9) − 9)
&Iuml; 2176 := (1 + 1) &times; ((11 &times; (1 + 1 + 1))1+1 − 1)
&Iuml; 2181 := 1 + ((1 + 1)11 + (11 &times; (1 + 11)))
:= ((7 + 7 + 7)/7)7 − (7 + 7)/7
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) + 8)) + 8/8)
:= 9 &times; 9 &times; (9 + 9 + 9) − (9 + 9)/9
&Iuml; 2186 := ((1 + 1 + 11)1+1+1 ) − 11
:= 2222 − (2 + 2 + 2)2
:= 2222 − ((2 &times; 22) + 2)
:= 2 + ((((22+2+2 + 2)2 ) + 2)/2)
:= 3 &times; 33+3 − 33/3
:= 3 + (3 &times; (33+3 − 3))
:= (4 + 4) &times; (4 &times; 4 + 44 )
:= 4 + ((4 + 4) &times; (4 &times; 4 + 44 ) + 4/4)
:= (5 − 5/5) &times; (555 − (55/5))
:= 5 + ((5 &times; (555 + 5)) + ((5 − 55 )/5))
:= ((6 + 6)/6)6 &times; (6 &times; 6 − ((6 + 6)/6))
:= (66 &times; 66 + 6)/((6 + 6)/6)
:= ((7 + 7 + 7)/7)7 − 77/7
:= 7/7 + (((7 + 7 + 7)/7)7 − 7)
:= (8 + 8) &times; (8 &times; (8 + 8) + 8)
:= (88/8 &times; (888/8 + 88)) − 8
:= ((88/8 − 8)8−8/8 ) − 8/8
:= 9 &times; 9 &times; (9 + 9 + 9) − 99/9
:= ((9 + 9 + 9)/9) + (9 &times; 9 &times; (9 + 9 + 9) − 9)
:= 9 &times; 9 &times; (9 + 9 + 9) − 9/9
&Iuml; 2177 := ((1 + 1) &times; (11 &times; (1 + 1 + 1))1+1 ) − 1
&Iuml; 2182 := 1 + (1 + ((1 + 1)11 + (11 &times; (1 + 11))))
:= 3 &times; 33+3 − 3/3
:= ((4 − 4/4)4+4−4/4 ) − 4/4
:= (55 + 5)/5 + (5 &times; (55 − 5)/(5 + 5))
:= ((6 &times; 6/(6 + 6))6/6+6) − 6/6
:= ((7 + 7 + 7)/7)7 − 7/7
&Iuml; 2187 := (1 + 1 + 1)1+(1+1)&times;(1+1+1)
:= (((22+2+2 + 2)2 ) − 2)/2
:= 2 &times; ((22/2 + 22)2 + 2)
:= (2/2 + 2)2/2+2+2+2
:= (3 &times; (33+3 − 3)) − 3/3
:= 3 + ((3 &times; (33+3 − 3)) + 3/3)
:= 3 &times; 33+3
:= 4/4 + (4 + 4) &times; (4 &times; 4 + 44 )
:= 4 + ((4 + 4) &times; (4 &times; 4 + 44 ) + (4 + 4)/4)
:= (4 − 4/4)4+4−4/4
:= (((5 − 55 ) + 5)/5) + (5 &times; (555 + 5))
:= ((5 − (5 + 5)/5)(5+5)/5+5 ) − 5
:= (5 − (5 + 5)/5)(5+5)/5+5
:= 6 + (6 &times; 6 &times; (66 − 6) + (66/6))
:= 6 + (((6 + 6)/6)6 &times; (6 &times; 6 − ((6 + 6)/6)))
:= (6 &times; 6/(6 + 6))6+6/6
:= 7 + ((77 &times; (7 + 7 + 7 + 7) + 7) + 7)
:= (7 + 7)/7 + (((7 + 7 + 7)/7)7 − 7)
:= ((7 + 7 + 7)/7)7
:= 8/8 + ((8 + 8) &times; (8 &times; (8 + 8) + 8))
:= 8 + (((8 + 8) &times; (8 &times; (8 + 8) + 8)) − ((8 + 8)/8))
:= (88/8 − 8)8−8/8
:= 9 &times; 9 &times; (9 + 9 + 9) − 9/9 − 9
:= ((9 + 9)/9) &times; ((((99 &times; 99) + 9) + 9)/9)
:= 9 &times; 9 &times; (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
&Iuml; 2188 := 1 + ((1 + 1 + 1)1+(1+1)&times;(1+1+1) )
&Iuml; 2193 := 1 + ((1 + 1)11 + (1 + 11)1+1 )
&Iuml; 2198 := 1 + ((1 + 1 + 11)1+1+1 )
:= 2 + (2222 − (2 + 2 + 2)2 )
:= ((2 &times; (22 + 2))2 ) − 222/2
:= 2222 − (22 + 2)
:= 3/3 + 3 &times; 33+3
:= 3 + (3 &times; 33+3 + 3)
:= 33/3 + 3 &times; 33+3
:= 44 + (44 &times; 44 − 4)
:= 4/4 + (44 &times; 44 + 44 )
:= 4 + ((44 &times; 44 + (4 + 4)/4) + 44 )
:= 55 + (((5 − 55 )/(5 + 5)) − 55 /5)
:= 5 + ((((5 − 55 )/(5 + 5)) − 55 /5) + 55 )
:= (5 + 5)/5 &times; (55 &times; (5 &times; 5 − 5) − 5/5)
:= 6/6 + ((6 &times; 6/(6 + 6))6/6+6)
:= 6 + ((6 &times; 6/(6 + 6))6/6+6)
:= 6 &times; (6 &times; (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) &times; (8 &times; (8 + 8) + 8/8)
:= 88 + ((88 &times; (8 + 8 + 8)) − ((8 + 8)/8))
:= 9/9 + 9 &times; 9 &times; (9 + 9 + 9)
:= 9 + (((9 + 9 + 9)/9) &times; (9 &times; 9 &times; 9 − 9/9))
:= 99/9 + 9 &times; 9 &times; (9 + 9 + 9)
&Iuml; 2189 := 11 &times; (((1 + 1) &times; (11 − 1)1+1 ) − 1)
&Iuml; 2194 := ((1 + 1 + 11)1+1+1 ) − 1 − 1 − 1
&Iuml; 2199 := 1 + (1 + ((1 + 1 + 11)1+1+1 ))
:= 2 + ((2/2 + 2)2/2+2+2+2 )
:= 2 + ((2 &times; 22)2 + 22&times;(2+2) )
:= 2 + ((22/2 + 2)2/2+2 )
:= 3 + (3 &times; 33+3 − 3/3)
:= (((3 &times; 3 + 3/3) + 3)3 ) − 3
:= 3 + (3 &times; (33+3 + 3))
:= 44 + ((((4 − 4/4) + 4)4 ) − 44 )
:= 44 + (44 &times; 44 + (4 + 4)/4)
:= ((4/4 + 4) &times; (444 − 4)) − 4/4
:= 5 + (((5 + 5)/5 + 5) &times; (55 − 5)/(5 + 5))
:= (5 &times; (5 + 5) &times; 55) − (555 + 5/5)
:= ((5 + 5) &times; (55 &times; (5 − 5/5))) − 5/5
:= 6 &times; 6 &times; (66 − 6 + 6) − 6/6 − 6
:= 6 &times; (6 &times; (66 − 6) + 6) − (6 + 6)/6
:= 6 + (((6 &times; 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 &times; (888/8 + 88)
:= 8 + (((88/8 − 8)8−8/8 ) − 8/8)
:= 88 + ((88 &times; (8 + 8 + 8)) − 8/8)
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) − ((9 + 9)/9))
:= ((99 + 9)/9) + 9 &times; 9 &times; (9 + 9 + 9)
:= (9 + 9)/9 + 9 &times; 9 &times; (9 + 9 + 9)
&Iuml; 2190 := (11 − 1) &times; (((1 + 1) &times; (111 − 1)) − 1)
:= 2222 − (2 &times; 2
:= 3 + 3 &times; 3
2+2
&Iuml; 2195 := ((1 + 1 + 11)1+1+1 ) − 1 − 1
:= ((22/2 + 2)
)
3+3
:= (3 &times; (3
3+3
2/2+2
)−2
+ 3)) − 3/3
&Iuml; 2200 := (1 + 1) &times; (1111 − 11)
:= 2222 − 22
:= 3 + (((3 &times; 3 + 3/3) + 3)3 )
:= 4 + (((4 − 4/4)4+4−4/4 ) + 4)
:= (4/4 + 4) &times; (444 − 4)
:= 5 &times; ((5 + 5)/(5 + 5) + 5 &times; 5 &times; 5)
:= (5 &times; (5 + 5) &times; 55) − 555
:= (5 + 5) &times; (55 &times; (5 − 5/5))
:= 6 &times; (6 &times; (66 − 6) + 6) − 6
:= 6 &times; (6 &times; (66 − 6) + 6) − 6/6
:= 6 + 6 &times; (6 &times; (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) &times; (8888/8 − (8 + 8))
:= 8 + ((88/8 − 8)8−8/8 )
:= 88 + (88 &times; (8 + 8 + 8))
:= ((9 + 9 + 9)/9) + 9 &times; 9 &times; (9 + 9 + 9)
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) − 9/9)
:= (99/9 + 9) &times; (99/9 + 99)
:= 44 + (44 &times; 44 − (4 + 4)/4)
5
&Iuml; 2191 := (1 + 1)11 + (11 &times; (1 + 1 + 11))
:= 2 + (((2/2 + 2)2/2+2+2+2 ) + 2)
&Iuml; 2196 := ((1 + 1 + 11)1+1+1 ) − 1
:= 2/2 + (2222 − 22)
:= 2222 − 22 − 2 − 2
3+3
&Iuml; 2201 := 1 + ((1 + 1) &times; (1111 − 11))
:= 3 + (3 &times; 33+3 + 33/3)
:= 3 + (3 &times; 33+3 + 3/3)
:= 3 &times; (3
:= 4 + ((4 − 4/4)4+4−4/4 )
:= 4 + (44 &times; 44 + 44 )
:= 4 + (((4/4 + 4 + 4) + 4)4−4/4 )
:= ((5 + 5)/5 + 5) &times; (55 + 5)/(5 + 5)
:= (5 − 5/5) &times; (555 − (5/5 + 5))
:= 5/5 + ((5 + 5) &times; (55 &times; (5 − 5/5)))
:= 6 &times; (6 &times; (66 − 6) + 6) − 6 + 6/6
:= 6 &times; (6 &times; (66 − 6) + 6)
:= 6 + 6 &times; (6 &times; (66 − 6) + 6) − 6/6
:= 7 + (7 &times; (7 &times; (7 &times; 7 − 7) + 7) + 77)
:= 7 + (((7 + 7 + 7)/7)7 + ((7 + 7)/7))
:= 7 + (((7 + 7 + 7)/7)7 + 7)
:= 8 + ((((8 + 8) &times; (8 &times; (8 + 8) + 8)) − 8/8) + 8)
:= 8 + (((88/8 − 8)8−8/8 ) + 8/8)
:= 8/8 + ((88 &times; (8 + 8 + 8)) + 88)
:= 9 &times; 9 + (9999/9 + 999)
:= 9 + 9 &times; 9 &times; (9 + 9 + 9)
:= 9 &times; (9 + 9) + ((((9 + 9)/9)99/9 ) − 9)
&Iuml; 2192 := (1 + 1)11 + (1 + 11)1+1
2
:= (2 &times; 22) + 2
:= 3 + ((3 &times; 3
2&times;(2+2)
3+3
− 3/3) + 3)
+ 3)
&Iuml; 2197 := (1 + 1 + 11)1+1+1
:= (22/2 + 2)
&Iuml; 2202 := (1 + 1) &times; (1 + (1111 − 11))
2/2+2
:= 2 + (2222 − 22)
:= ((3 &times; 3 + 3/3) + 3)
3
:= 3 + ((3 &times; (33+3 + 3)) + 3)
:= 44 + 44 &times; 44
:= ((4/4 + 4 + 4) + 4)4−4/4
:= 44 + (44 &times; 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 &times; 6/(6 + 6))6/6+6) − 6/6)
:= 6 &times; (6 &times; (66 − 6) + 6) + 6/6
:= 6 + 6 &times; (6 &times; (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) &times; (8 &times; (8 + 8) + 8)) + 8)
:= ((88 + 8 + 8)/8)88/8−8
:= 88 + ((88 &times; (8 + 8 + 8)) + ((8 + 8)/8))
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) + 9/9)
:= ((9 + 9)/9) &times; (((9999 − 9)/9) − 9)
:= 9 &times; 9 &times; (9 + 9 + 9) + ((9 &times; 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
&Iuml; 2203 := 1 + ((1 + 1) &times; (1 + (1111 − 11)))
:= 2 + ((2222 − 22) + 2/2)
&Iuml; 2208 := 11 + ((1 + 1 + 11)1+1+1 )
:= 2 + (2222 − 2
3
2+2
&Iuml; 2213 := ((1 + 1) &times; (1 + 1111)) − 11
:= 2 + (2222 − 22/2)
)
:= 3 + ((((3 &times; 3 + 3/3) + 3) ) + 3)
:= 3 + (3 &times; ((33+3 + 3) + 3))
:= 33 + (3 &times; 33+3 − 3/3)
:= 4 &times; 4 + ((4 − 4/4)4+4−4/4 )
:= (4 + 4) &times; ((4 &times; 4 + 44 ) + 4)
:= 4 + ((44 − 4/4 + 4)(4+4)/4 )
:= 5 + ((((5 + 5)/5)55/5 ) + 5 &times; (5 &times; 5 + 5))
:= (5 − 5/5) &times; (((5 + 5)/5 − 5) + 555)
:= ((5 − 5/5) &times; 555) − ((5 + 5)/5 + 5)
:= 6 + 6 &times; (6 &times; (66 − 6) + 6) + 6/6
:= 6 &times; (6 &times; (66 − 6) + 6) + 6 + 6
:= 6 + (6 &times; 6 &times; (66 − 6 + 6) + (66/6))
:= 7 + ((((7 + 7 + 7)/7) + ((7 + 7)/7)) + 7)
:= 7 + ((7 + 7 + 7)/7) + 7 + 7
:= 7 + ((7 &times; (7 &times; (7 &times; 7 + 7) − 77)) + 7/7)
:= 8 + (((88/8 − 8)8−8/8 ) + 8)
:= 8 + ((88 &times; (8 + 8 + 8)) + 88)
:= ((88 + 8 + 8)/8)88/8−8 + 8 + 8
:= 9 + ((9 &times; 9 &times; (9 + 9 + 9) − ((9 + 9)/9)) + 9)
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) + ((99 + 9)/9))
:= (9 + 9 + 9) &times; (9/9 + 9 &times; 9) − 9/9
7
&Iuml; 2204 := (1 + 1) &times; (1 + (1 + (1111 − 11)))
7
&Iuml; 2209 := ((1 + 1) &times; (1111 − 1)) − 11
:= ((2 &times; 22 + 2/2) + 2)
:= 2 + ((2222 − 22) + 2)
3+3
2
:= (3 &times; ((33+3 + 3) + 3)) − 3/3
:= 3 + ((3 &times; ((3
:= 4 + ((4/4 + 4) &times; (444 − 4))
:= (44 − 4/4 + 4)(4+4)/4
:= (5 − 5/5) &times; ((5 + 5) &times; 55 + 5/5)
:= ((5 − 5/5) &times; 555) − 55/5
:= 6 + (6 &times; 6 &times; (66 − 6 + 6) + ((6 + 6)/6))
:= (66/6 + 6 &times; 6)
+ 3) + 3)) + 3/3)
(6+6)/6
&Iuml; 2214 := 1 + (((1 + 1) &times; (1 + 1111)) − 11)
:= 2222 − 2 &times; (2 + 2)
:= 3 &times; (33+3 + 3 &times; 3)
:= (4 + 4)/4 &times; (4444/4 − 4)
:= ((5 − 5/5) &times; 555) − (5/5 + 5)
:= 6 &times; (66 &times; 66/(6 + 6) + 6)
:= (7 &times; 7 − (7 + 7)/7)(7+7)/7
:= 7 + ((((((7 + 7 + 7)/7)7 − 7/7) + 7) + 7) + 7)
:= ((8 + 8)/8) &times; ((8888 − 8)/8 − 8)
:= (8 &times; 8 − (8/8 + 8 + 8))(8+8)/8
:= (((8 + 8)/8) &times; 8888/8) − 8
:= 9 + ((9 &times; 9 &times; (9 + 9 + 9) − 9/9) + 9)
:= 9 + ((99/9 + 9) &times; (99/9 + 99))
:= 7 + ((7 − 7/7 + 7)
(7+7+7)/7
)
&Iuml; 2205 := 1 + ((1 + 1) &times; (1 + (1 + (1111 − 11))))
2
:= (2/2 + 2 + 2) &times; ((22 − 2/2) )
&Iuml; 2210 := (11 − 1) &times; ((1 + 1) &times; 111 − 1)
:= 2222 − (2 &times; (2 + 2 + 2))
:= (9 + 9 + 9) &times; (9/9 + 9 &times; 9)
&Iuml; 2215 := ((1 + 1) &times; (1 + (1 + 1111))) − 11
:= 2 + ((2222 − 22/2) + 2)
:= 3 + ((3 &times; (33+3 + 3)) + 33/3)
:= 33 + (3 &times; 33+3 + 3/3)
:= (4/4 + 4) &times; ((444 − 4) + 4/4)
:= (4/4 + 4) &times; (444 − (4 + 4)/4)
:= (4/4 + 4) &times; (444 − 4/4)
:= 5 + ((5 + 5) &times; (55 &times; (5 − 5/5)))
:= ((5 − 5/5) &times; 555) − 5 − 5
:= ((5 − 5/5) &times; 555) − 5
:= 6 + ((((6 &times; 6/(6 + 6))6/6+6 ) + 6) + 6)
:= ((6 + 6)/6) &times; ((6666/6) − 6)
:= 6 + ((66/6 + 6 &times; 6)(6+6)/6 )
:= 7 &times; (7 &times; (7 &times; 7 + 7) − 77)
:= 7/7 + (7 &times; 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) &times; (8 &times; (8 + 8) + ((8 + 8)/8))
:= 888/8 + ((88 &times; (8 + 8 + 8)) − 8)
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) + 9)
:= 9 &times; (9 + 9) + (((9 + 9)/9)99/9 )
:= 9/9 + (9 + 9 + 9) &times; (9/9 + 9 &times; 9)
:= 3 &times; ((3
3+3
+ 3) + 3)
&Iuml; 2206 := 11 + (((1 + 1 + 11)1+1+1 ) − (1 + 1))
&Iuml; 2211 := ((1 + 1) &times; 1111) − 11
&Iuml; 2216 := (1 + 1) &times; (1111 − (1 + 1 + 1))
:= 2222 − 22/2
:= 2222 − (2 + 2 + 2)
:= 3 &times; 3 + (((3 &times; 3 + 3/3) + 3)3 )
:= 33 + (3 &times; 33+3 − 3)
:= 3 + ((3 &times; 33+3 − 3/3) + 33 )
:= 4 &times; 4444/(4 + 4) − 4 &times; 4
:= ((4/4 + 4) &times; (444 − 4/4)) − 4
:= (4/4 + 4) &times; 444 − 4
:= 5 + (((5 + 5) &times; (55 &times; (5 − 5/5))) + 5/5)
:= ((5 − 5/5) &times; (555 − 5/5)) − 5
:= (5 − 5/5) &times; (555 − 5/5)
:= 6 &times; (6 &times; (66 − 6) + 6) + (66 − 6)/6
:= (66 + 6/6) &times; (66 &times; 6/(6 + 6))
:= (((6 + 6)/6) &times; (6666/6)) − 6
:= 7/7 + (7 &times; (7 &times; (7 &times; 7 + 7) − 77))
:= 7 + (((7 − 7/7 + 7)(7+7+7)/7 ) + 7)
:= 7 + ((7 &times; 7 − (7 + 7)/7)(7+7)/7 )
:= ((8 + 8)/8) &times; (8888/8 − 8)
:= 8 + ((((88/8 − 8)8−8/8 ) + 8) + 8)
:= ((88 + 8) &times; (8 + 8 + 8)) − 88
:= 9 + ((9 &times; 9 &times; (9 + 9 + 9) + 9/9) + 9)
:= ((99/9 + 9) &times; 999/9) − 9
:= 9 + ((9 &times; 9 &times; (9 + 9 + 9) + (99/9)) + 9)
:= 2222 − 2
2+2
&Iuml; 2207 := 11 + (((1 + 1 + 11)1+1+1 ) − 1)
2
&Iuml; 2212 := 1 + (((1 + 1) &times; 1111) − 11)
&Iuml; 2217 := ((1 + 1) &times; (1111 − (1 + 1))) − 1
:= ((2 − 22)/2) + 2222
:= 2222 − (2/2 + 2 + 2)
:= 33 + ((3 &times; 33+3 − 3) + 3/3)
:= 3 + (3 &times; 33+3 + 33 )
:= ((4 + 4) &times; ((4 &times; 4 + 44 ) + 4)) − 4/4
:= 4 + ((4 + 4) &times; ((4 &times; 4 + 44 ) + 4))
:= 4/4 + ((4/4 + 4) &times; 444 − 4)
:= ((5 + 5)/5 &times; (5555/5 − 5)) − 5
:= (5 + 5)/5 &times; (5555/5 − 5)
:= ((5 + 5)/5 &times; 5555/5) − 5
:= 6 &times; (6 &times; (66 − 6) + 6) + 66/6
:= ((6 + 6)/6) &times; ((6666 + 6)/6 − 6)
:= 6 + ((66 + 6/6) &times; (66 &times; 6/(6 + 6)))
:= 7 + (((((7 + 7 + 7)/7)7 − 7/7) + 7) + 7)
:= 7 + (7 &times; (7 &times; (7 &times; 7 + 7) − 77))
:= 7 &times; (7 &times; 7 &times; 7 − 7) − (((7 + 7)/7)7 + 7)
:= 8 + (((88 &times; (8 + 8 + 8)) − 8/8) + 88)
:= (((8 + 8)/8) &times; (8888 − 8)/8) − 8
:= 8 + ((8 &times; 8 − (8/8 + 8 + 8))(8+8)/8 )
:= 9 + (9 &times; 9 &times; (9 + 9 + 9) + (99/9))
:= ((9/9 + 9 + 9) + 9) &times; (9 &times; 9 − ((9 + 9)/9))
:= 999/9 + (9 &times; (9 &times; (9 + 9 + 9) − 9))
:= (((2 &times; 22 + 2/2) + 2) ) − 2
:= 33/3 + (3 &times; (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
&Iuml; 2218 := (1 + 1) &times; (1111 − (1 + 1))
:= 3 + ((3 &times; 3
&Iuml; 2228 := (1 + 1) &times; (1 + (1 + (1 + 1111)))
&Iuml; 2223 := 1 + ((1 + 1) &times; 1111)
:= 2 + ((2222 + 2) + 2)
:= 2/2 + 2222
:= 2222 − 2 − 2
3+3
3
+ 3 ) + 3/3)
:= 3 + (3 &times; 3
3+3
:= 33 + ((3 &times; (33+3 + 3)) − 3/3)
+ 33)
:= 4 &times; 4444/(4 + 4) − 4
:= 4 + ((4/4 + 4) &times; 444 − 4/4)
:= 4 + ((4/4 + 4) &times; 444 + 4)
:= ((5 − 5/5) &times; 555) − (5 + 5)/5
:= 5 + (((5 − 5/5) &times; 555) − ((5 + 5)/5))
:= (5 − 5/5) &times; (555 + (5 + 5)/5)
:= ((6 + 6)/6) &times; ((6666 − (6 + 6))/6)
:= 6 &times; 6 + ((6 &times; 6/(6 + 6))
6/6+6)
:= 6 + (((6 + 6)/6) &times; (6666/6))
:= 7 &times; 7 + (((7 + 7 + 7)/7)7 − (7/7 + 7))
:= (7 + 7)/7 &times; (7777 − 7 − 7)/7
:= 7 + (((7 &times; 7 − (7 + 7)/7)
:= ((8 + 8)/8) &times; ((8888 − (8 + 8))/8)
:= 888/8 + (88 &times; (8 + 8 + 8))
:= 8 + (((8 + 8)/8) &times; (8888 − 8)/8)
:= ((9 + 9)/9) &times; ((9999 − (9 + 9))/9)
:= 9 + (9 + 9 + 9) &times; (9/9 + 9 &times; 9)
:= 99 + ((((9 + 9)/9)99/9 ) + 9 &times; 9)
&Iuml; 2219 := ((1 + 1) &times; (1111 − 1)) − 1
:= 2222 − 2/2 − 2
(7+7)/7
) + 7)
&Iuml; 2229 := 1 + ((1 + 1) &times; (1 + (1 + (1 + 1111))))
&Iuml; 2224 := (1 + 1) &times; (1 + 1111)
:= 2 + 2222
:= 2 + (((2222 + 2/2) + 2) + 2)
:= 33 + (((3 &times; 3 + 3/3) + 3)3 )
:= 33 + (3 &times; (33+3 + 3))
:= (4/4 + 4) &times; 444 − 4/4
:= 4 + (4/4 + 4) &times; 444
:= 4 + ((4/4 + 4) &times; (444 + 4/4))
:= 33 + (3 &times; 3
3+3
− 3/3)
:= ((5 − 5/5) &times; 555) − 5/5
:= (5 − 5/5) &times; (555 + 5/5)
:= 5 + ((5 − 5/5) &times; (555 + 5/5))
:= ((66 − 6) &times; (6 &times; 6 + 6/6)) − 6/6
:= ((6 + 6)/6) &times; (6666 + 6)/6
:= 6 &times; (6 &times; 66 − 6) − 666/6
:= 7 + ((7 &times; (7 &times; (7 &times; 7 + 7) − 77)) + 7)
:= ((7 + 7)/7) &times; (7777 + 7)/7
:= 7 &times; 7 + (((7 + 7 + 7)/7)7 − 7)
)
:= (8 + 8) &times; (8 &times; (8 + 8) + (88/8))
:= (8 &times; (8 &times; (8 + 8 + 8) + 88)) − 88/8
) + 9 &times; (9 + 9))
:= ((9 + 9)/9) &times; (9999 + 9)/9
:= 9 + ((99/9 + 9) &times; 999/9)
:= 888 + ((88/8)
88/8−8
99/9
:= 9 + ((((9 + 9)/9)
&Iuml; 2220 := (1 + 1) &times; (1111 − 1)
&Iuml; 2230 := (11 − 1) &times; (1 + (1 + 1) &times; 111)
:= 2 + (2222 + 2/2)
:= 2222 − 2
:= 33 + 3 &times; 3
&Iuml; 2225 := 1 + ((1 + 1) &times; (1 + 1111))
3
:= 2 &times; (2 + 2) + 2222
3
:= 33 + (((3 &times; 3 + 3/3) + 3)3 )
:= 3 + ((33 &times; 3 ) + (33/3) )
3+3
:= (4/4 + 4) &times; 444
:= (4/4 + 4) &times; (444 + 4/4)
:= 4 + (4 &times; 4444/(4 + 4) + 4)
:= (5 − 5/5) &times; 555
:= 5 + ((5 − 5/5) &times; 555)
:= 5 + (((5 − 5/5) &times; 555) + 5)
:= (66 − 6) &times; (6 &times; 6 + 6/6)
:= 66 + (6 &times; 6 &times; (66 − 6) − 6/6)
:= 6 + (6 &times; 6 &times; (66 − 6) + ((6 + 6)/6)6 )
:= ((7 + 7)/7) &times; (7777 − 7)/7
:= 7 &times; 7 + (((7 + 7 + 7)/7)7 − (77/7))
:= 7/7 + ((((7 + 7 + 7)/7)7 − 7) + 7 &times; 7)
:= ((8 + 8)/8) &times; (8888 − 8)/8
:= (8/8 + 88) &times; (8/8 + 8 + 8 + 8)
:= 8 + (((8 + 8)/8) &times; 8888/8)
:= (99/9 + 9) &times; 999/9
:= 99/9 + (9 + 9 + 9) &times; (9/9 + 9 &times; 9)
:= 9 + (((99/9 + 9) &times; 999/9) + 9/9)
&Iuml; 2221 := ((1 + 1) &times; 1111) − 1
&Iuml; 2231 := 11 + ((1 + 1) &times; (1111 − 1))
:= 2 + (2222 + 2)
:= 2222 − 2/2
:= 3/3 + (3 &times; 3
&Iuml; 2226 := (1 + 1) &times; (1 + (1 + 1111))
3+3
+ 33)
:= 3 + ((3 &times; 3
3+3
:= 22/2 + (2222 − 2)
:= 33 + (3 &times; 33+3 + 33/3)
+ 33) + 3)
:= 4/4 + (4/4 + 4) &times; 444
:= 4 + 4 &times; 4444/(4 + 4)
:= 44 + ((4 − 4/4)4+4−4/4 )
:= 5/5 + ((5 − 5/5) &times; 555)
:= 5 + (((5 − 5/5) &times; 555) + 5/5)
:= 55/5 + ((5 − 5/5) &times; 555)
:= 6/6 + ((66 − 6) &times; (6 &times; 6 + 6/6))
:= 66 + 6 &times; 6 &times; (66 − 6)
:= (6 &times; ((6 &times; (66 − 6) + 6) + 6)) − 6/6
:= (((7 + 7)/7) &times; 7777/7) − 7/7
:= 7 &times; (7 &times; 7 &times; 7 − 7 − 7) − 77
:= 7 + (((7 + 7)/7) &times; (7777 + 7)/7)
:= (((8 + 8)/8) &times; 8888/8) − 8/8
:= ((8 + 8)/8) &times; (((8888 + 8) + 8)/8)
:= 8 + ((88 &times; (8 + 8 + 8)) + 888/8)
:= 9/9 + ((99/9 + 9) &times; 999/9)
:= ((9 + 9)/9) &times; (((9999 + 9) + 9)/9)
:= 9 + (((9 + 9)/9) &times; 9999/9)
&Iuml; 2222 := (1 + 1) &times; 1111
&Iuml; 2227 := 1 + ((1 + 1) &times; (1 + (1 + 1111)))
:= 2222
:= 2 + ((2222 + 2/2) + 2)
&Iuml; 2232 := 11 + (((1 + 1) &times; 1111) − 1)
:= 2 + (2222 + 2 &times; (2 + 2))
3
3
:= (3 − 3/3) &times; 3333/3
:= 3 + ((((3 &times; 3 + 3/3) + 3) ) + 3 )
:= 3 + ((3 &times; (33+3 + 3)) + 33)
:= 4 &times; 4444/(4 + 4)
:= 4 + (((4/4 + 4) &times; 444 − 4/4) + 4)
:= (4/4 + 4 + 4) &times; (44 − 4 − 4)
:= (5 + 5)/5 &times; 5555/5
:= 5 + ((5 + 5)/5 &times; 5555/5)
:= (5 + 5)/5 &times; (5555/5 + 5)
:= ((6 + 6)/6) &times; (6666/6)
:= 66 + (6 &times; 6 &times; (66 − 6) + 6/6)
:= 6 &times; ((6 &times; (66 − 6) + 6) + 6)
:= ((7 + 7)/7) &times; 7777/7
:= ((7 &times; 7 − 7/7)(7+7)/7 ) − 77
:= 77 + (77 &times; (7 + 7 + 7 + 7) − 7/7)
:= ((8 + 8)/8) &times; 8888/8
:= 8 + (((88/8)88/8−8 ) + 888)
:= (8/8 + 8) &times; ((8 + 8) &times; (8 + 8) − 8)
:= ((9 + 9)/9) &times; 9999/9
:= 9 + (((9 + 9)/9) &times; ((9999 − (9 + 9))/9))
:= 9 + ((9 + 9 + 9) &times; (9/9 + 9 &times; 9) + 9)
158
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 2238 := (1 + 1) &times; (((11 − 1) &times; (1 + 111)) − 1)
&Iuml; 2233 := 11 + ((1 + 1) &times; 1111)
:= 22/2 + 2222
:= 2
2+2
&Iuml; 2243 := ((1 + 1) &times; (11 + 1111)) − 1
:= 22 + (2222 − 2/2)
+ 2222
:= 3 + ((((3 &times; 3 + 3/3) + 3) ) + 33)
:= 33 + (3 &times; ((33+3 + 3) + 3))
:= 3 + (((3 − 3/3)3&times;3 ) + (3 &times; 3 + 3)3 )
:= 4 + (((4/4 + 4) &times; (444 + 4/4)) + 4)
:= 4 &times; 4 + 4 &times; 4444/(4 + 4)
:= (4 &times; (4/4 + 4)4 ) − (4/4 + 44 )
:= 5 + ((5 − 5/5) &times; (555 + (5 + 5)/5))
:= ((5 − 5/5) &times; (555 + 5)) − (5 + 5)/5
:= 5 + (((5 − 5/5) &times; (555 + 5)) − ((5 + 5)/5))
:= 6/6 + (6 &times; ((6 &times; (66 − 6) + 6) + 6))
:= 6 + (6 &times; ((6 &times; (66 − 6) + 6) + 6))
:= (66 &times; (6 &times; 6 − ((6 + 6)/6))) − 6/6
3
:= 77 + 77 &times; (7 + 7 + 7 + 7)
:= 7 &times; 7 + (((7 + 7 + 7)/7) + ((7 + 7)/7))
:= 7 + (((7 + 7 + 7)/7)7 + 7 &times; 7)
:= 8 + ((8/8 + 88) &times; (8/8 + 8 + 8 + 8))
:= ((8 + 8)/8) &times; (8888/8 + 8)
:= 8 &times; 8 + (((88/8 − 8)8−8/8 ) − 8)
:= 9 + (((9 + 9)/9) &times; (9999 + 9)/9)
:= 9 + (((99/9 + 9) &times; 999/9) + 9)
:= (9 + 9)/9 + ((9 + 9 + 9) &times; (((9 + 9)/9) + 9 &times; 9))
7
&Iuml; 2234 := 1 + (11 + ((1 + 1) &times; 1111))
:= 2222 + (2 &times; (2 + 2 + 2))
&Iuml; 2239 := ((1 + 1) &times; ((11 − 1) &times; (1 + 111))) − 1
&Iuml; 2244 := (1 + 1) &times; (11 + 1111)
:= 2/2 + (2222 + 22+2 )
:= 22 + 2222
:= 3 &times; 3 + ((((3 &times; 3 + 3/3) + 3)3 ) + 33)
:= 3 + (3 &times; (33+3 + (3 &times; (3 + 3))))
:= 4 + ((4 &times; 4444/(4 + 4) + 4) + 4)
:= ((4/4 + 4) &times; (444 + 4)) − 4/4
:= (4 &times; (4/4 + 4)4 ) − 44
:= 5 + (((5 − 5/5) &times; (555 + 5/5)) + 5)
:= ((5 − 5/5) &times; (555 + 5)) − 5/5
:= (5 − 5/5) &times; ((555 + 5/5) + 5)
:= ((6 + 6)/6) &times; ((6666/6) + 6)
:= 6 + ((6 &times; ((6 &times; (66 − 6) + 6) + 6)) + 6/6)
:= 66 &times; (6 &times; 6 − ((6 + 6)/6))
:= ((7 + 7) &times; (777/7 + 7 &times; 7)) − 7/7
:= 7 + ((((7 + 7 + 7)/7)7 + 7 &times; 7) + 7/7)
:= 8 + (((8 + 8)/8) &times; (((8888 + 8) + 8)/8))
:= (8 &times; (8 &times; (8 + 8 + 8) + 88)) − 8/8
:= ((8 + 8)/8) &times; ((8888 + 88)/8)
:= 9 + ((9 + 9 + 9) &times; (9/9 + 9 &times; 9) + (99/9))
:= 99 + ((99/9 + 9) &times; ((99 − 9/9) + 9))
:= ((9 + 9)/9) &times; ((9999 + 99)/9)
:= 3 + ((3 &times; 3
3+3
+ 33/3) + 33)
:= 7 + (((7 &times; 7 − 7/7)(7+7)/7 ) − 77)
&Iuml; 2235 := 11 + ((1 + 1) &times; (1 + 1111))
:= 2 + (2222 + 22/2)
&Iuml; 2240 := (1 + 1) &times; ((11 − 1) &times; (1 + 111))
:= 2 + (2222 + 2
2+2
:= 22 + (2222 + 2/2)
)
:= (3 &times; (33+3 + 33 )) − 33
:= (3 &times; 3 + 3) + ((3 − 3/3)
:= (4/4 + 4) &times; ((444 − 4/4) + 4)
:= (4/4 + 4) &times; (444 + 4)
:= ((5 − 5/5) &times; (555 + 5)) − 5
:= (5 − 5/5) &times; (555 + 5)
3
:= 6 + (6 &times; (6 &times; 66 − 6) − 666/6)
&Iuml; 2245 := 1 + ((1 + 1) &times; (11 + 1111))
3&times;3
:= 3 + ((3 &times; (33+3 + (3 &times; (3 + 3)))) + 3/3)
)
:= 4/4 + ((4 &times; (4/4 + 4)4 ) − 44 )
:= 5 + ((5 − 5/5) &times; (555 + 5))
6
:= (6 &times; 6 − 6/6) &times; ((6 + 6)/6)
:= 6/6 + (66 &times; (6 &times; 6 − ((6 + 6)/6)))
:= 7 + ((((7 + 7 + 7)/7)7 + ((7 + 7)/7)) + 7 &times; 7)
:= 8 &times; 8 + (((88/8 − 8)8−8/8 ) − (8 + 8))
:= (7 + 7) &times; (777/7 + 7 &times; 7)
:= 8 &times; (8 &times; (8 + 8 + 8) + 88)
:= 8 &times; 8 &times; (8 + 8) + 88/8 &times; 888/8
:= 9 + (((9 + 9)/9) &times; (((9999 + 9) + 9)/9))
:= ((9 + 9)/9) &times; (9999/9 + 9)
:= 9999/9 + (9 &times; ((99 + 9 + 9) + 9))
:= 7 &times; 7 + (((7 + 7 + 7)/7)7 − 7/7)
&Iuml; 2236 := 1 + (11 + ((1 + 1) &times; (1 + 1111)))
2+2
&Iuml; 2241 := 1 + ((1 + 1) &times; ((11 − 1) &times; (1 + 111)))
&Iuml; 2246 := (1 + 1) &times; (1 + (11 + 1111))
:= 22 + (2222 − (2/2 + 2))
:= 2 + (2222 + 22)
:= 3 + (((((3 &times; 3 + 3/3) + 3) ) + 33) + 3)
:= 3 &times; (33+3 + (3 &times; (3 + 3)))
:= 33 + ((3 &times; 33+3 − 3/3) + 33)
:= 4 &times; 4 + (4/4 + 4) &times; 444
:= 4/4 + ((4/4 + 4) &times; (444 + 4))
:= (4 + 4)/4 + ((4 &times; (4/4 + 4)4 ) − 44 )
:= (5 − 5/5) &times; (555 − 5/5 + 5)
:= 5/5 + ((5 − 5/5) &times; (555 + 5))
:= 5 + (((5 − 5/5) &times; (555 + 5)) + 5/5)
:= ((6 + 6)/6) &times; ((6666 + 6)/6 + 6)
:= 66 + ((66 &times; 66 − 6)/((6 + 6)/6))
:= 6 + ((6 &times; 6 − 6/6) &times; ((6 + 6)/6)6 )
:= 7 &times; 7 + ((7 + 7 + 7)/7)7
:= 7 &times; (7 &times; 7 &times; 7 − 7) − 777/7
:= 7 &times; 7 + ((7 − 7/7 + 7)(7+7+7)/7 )
:= ((8 + 8)/8) &times; ((8888 − 8)/8 + 8)
:= 8/8 + (8 &times; (8 &times; (8 + 8 + 8) + 88))
:= 8 + (((8 + 8)/8) &times; (8888/8 + 8))
:= ((9 + 9)/9) &times; (((9999 − (9 + 9))/9) + 9)
:= (9 + 9 + 9) &times; (((9 + 9)/9) + 9 &times; 9)
:= 99 + ((((9 + 9)/9)99/9 ) + 99)
:= 2
+ (2222 − 2)
3
&Iuml; 2237 := 11 + ((1 + 1) &times; (1 + (1 + 1111)))
:= 2 + ((2222 + 22/2) + 2)
:= 33 + ((3 &times; ((3
3+3
+ 3) + 3)) − 3/3)
:= 4 &times; 4 + ((4/4 + 4) &times; 444 + 4/4)
&Iuml; 2242 := (1 + 1) &times; (11 + (1111 − 1))
&Iuml; 2247 := 1 + ((1 + 1) &times; (1 + (11 + 1111)))
:= 22 + (2222 − 2)
:= 3/3 + (3 &times; (3
3+3
:= 2 + ((2222 + 22) + 2/2)
+ (3 &times; (3 + 3))))
:= 4 + (4 &times; 4444/(4 + 4) + 4 &times; 4)
(5+5)/5+5
:= 5 + ((5 + 5)/5 &times; (5555/5 + 5))
:= 55 + ((5 − (5 + 5)/5)
:= 6 + ((6 &times; ((6 &times; (66 − 6) + 6) + 6)) − 6/6)
:= ((6 + 6)/6 + 6 &times; 6) &times; (66 − (6/6 + 6))
)
:= 33 + (3 &times; 33+3 + 33)
:= 4 + ((4 &times; (4/4 + 4)4 ) − (4/4 + 44 ))
:= 5 &times; 5 + ((5 + 5)/5 &times; 5555/5)
:= 66 + ((66 &times; 66 + 6)/((6 + 6)/6))
:= 7/7 + (((7 + 7 + 7)/7)7 + 7 &times; 7)
:= 7 + ((((7 + 7 + 7)/7) − 7/7) + 7 &times; 7)
:= 8 + ((8 &times; (8 &times; (8 + 8 + 8) + 88)) − (88/8))
:= (8 + 8)/8 + (8 &times; (8 &times; (8 + 8 + 8) + 88))
:= 8 + ((8 &times; (8 &times; (8 + 8 + 8) + 88)) − 8/8)
:= 9 + (((((9 + 9)/9)99/9 ) + 99) + 9 &times; 9)
:= 9/9 + ((9 + 9 + 9) &times; (((9 + 9)/9) + 9 &times; 9))
:= (((99 + 9)/9) + 9) &times; ((99 − 9/9) + 9)
7
159
:= 7 &times; 7 &times; 7 &times; 7 − 77 − 77
Inder J. Taneja
Single Digit Representations of Natural Numbers From 1 to 5000, Zenodo, Open Access, pp. 1-345, Jan. 2019
&Iuml; 2248 := (1 + 1) &times; (1 + (1 + (11 + 1111)))
3
&Iuml; 2253 := 11 + ((1 + 1) &times; (11 + (1111 − 1)))
3+3
3+3
&Iuml; 2258 := (1 + 1) &times; (((11 − 1) &times; (1 + 1 + 111)) − 1)
:= (2 + 2 + 2)2 + 2222
:= 22 + ((2222 − 2) + 22/2)
:= 2 + ((2222 + 22) + 2)
:= 3 &times; 33+3 + (((3 + 3)3 − 3)/3)
− 3)
:= 33 + (3 &times; 3
4
:= 4 + ((4 &times; (4/4 + 4) ) − 4 )
:= 44 + ((44 − 4/4 + 4)(4+4)/4 )
:= 4 + (4 &times; 4444/(4 + 4) + 4 &times; (4 + 4))
:= (5 − 5/5) &times; ((555 + (5 + 5)/5) + 5)
:= 5 + ((5 − 5/5) &times; ((555 + (5 + 5)/5) + 5))
:= 55 + (((5 − 55 )/(5 + 5)) − 555)
:= ((3/3 + 3) ) + (3 &times; 3
4
6/6+6)
+ 33)
6/6+6)
:= 6 &times; 6 + (((6 + 6)/6) &times; (6666/6))
:= 6 &times; 6 &times; 66 − (((6 + 6)/6)
:= 66 + ((6 &times; 6/(6 + 6))
:= 7 + (7 &times; (7 &times; 7 &times; 7 − 7) − 777/7)
:= (7 &times; (7 &times; 7 &times; 7 − 7 − 7 − 7)) − 7/7
:= 7/7 + ((((7 + 7 + 7)/7)7 − 7) + 77)
:= 8 + (8 &times; (8 &times; (8 + 8 + 8) + 88))
:= ((88/8 + 8) &times; (888/8 + 8)) − 8
:=```