Systèmes de numération

10
10
10 0,1,2,3,4,5,6,7,8 9
2 0 1
8 0,1,2,3,4,5,6,7
16 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E, F
60 2000
60
2
(0)16 (F F )16
(23)12 12
10
(23)douze
b N
b
anan−1. . . a2a1a0
(anan−1. . . a2a1a0)b

an×bn+an−1×bn−1+· · · +a2×b2+a1×b1+a0×b0
b1=b b0= 1 b
n b > 2n b
1999
MCM XCIX =MCMXCV IIII
(201)3= 19
(201)7= 99
(201)8= 129
(201)10 = 201
(201)16 = 513
n
b n b
b
b
2 3317
3317 = 2048 + 1269 = 211 + 1269
1269 = 1024 + 245 = 210 + 245
245 = 128 + 117 = 27+ 117
117 = 64 + 53 = 26+ 53
53 = 32 + 21 = 25+ 21
21 = 16 + 5 = 24+ 5
5 = 4 + 1 = 22+ 20
3317
211 + 210 + 27+ 26+ 25+ 24+ 22+ 20= (110011110101)2
b

n b > 2n
n= (apap−1. . . a2a1a0)b
n b
n=q×b+r
r n b
r=a0
q
q= (apap−1. . . a1)b
(543)10
(543)10 10 (54)10 3
(54)10 10 (5)10 4
(5)10 10 (0)10 5
n
b
n b
n=q×b+r
r n
q
317
402
b
b
10 b10 = (13)7
b
7

+ 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6
1 1 2 3 4 5 6 10
2 2 3 4 5 6 10 11
3 3 4 5 6 10 11 12
4 4 5 6 10 11 12 13
5 5 6 10 11 12 13 14
6 6 10 11 12 13 14 15
×0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 11 13 15
3 0 3 6 12 15 21 24
4 0 4 11 15 22 26 33
5 0 5 13 21 26 34 42
6 0 6 15 24 33 42 51
(1363)7
7
1363 = 104 ∗13 + 5
104 = 5 ∗13 + 3
5=0∗13 + 5
(1363)7= 535
n= (anan−1. . . a0)b
n=an×bn+· · · +a1×b1+a0×b0
n
b
an×bn+· · · +a1×b+a0
(an×bn−1+· · · +a1)×b+a0

((. . . (((an)×b+an−1)×b+an−2)×b+· · · +a2)×b+a1)×b+a0
(30624)8
(30624)8= ((((3) ×8 + 0) ×8 + 6) ×8 + 2) ×8 + 4 = 12692
2,8 16
m/n m n
b
b
(0, a1a2. . . an)b=a1×b−1+a2×b−2+· · · +an×b−n
(301,23)4= 3 ×42+ 0 ×41+ 1 ×40+ 2 ×4−1+ 3 ×4−2
= 3 ×16 + 1 + 2 ×0,25 + 3 ×0,0625
= 49,3875
(0,36207)8= 3 ×8−1+ 6 ×8−2+ 2 ×8−3+ 0 ×8−4+ 7 ×8−5
= 8−5(7 + 8(0 + 8(2 + 8(6 + 8(3)))))
= 15495/32768
= 0,47286987 . . .
n b