[PDF] Version numérique pour la préparation des cours dinformatique en

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.,'/,'01 &223332,22"####### ##4567787857 )"&&2239"2"2+"::::::: : n O n3 R 0 R 1 R n (R1,R2) R1 R2

R2 R1

PC PC IR PC PC IR IR

PC n

n+ 1 Ċ PC PC

PC 4 32

32 4 8 64

(n,i)n i

1248124816

32

12412816 32

555
n n

0123456789 2359

953 2

24= 16

28= 256

11012
1101
k k kĊ n k kk ai...a1a0 n k nk i←0 n?= 0 a i← nk n← nk i←i+ 1 ai...a1a0 58

58 = 11×5 +3= (2×5 +1)×5 +3= (2×52) + (1×51) + (3×50)

58 = 2135

872

872 = 114425

1207

16 0123456789 abc

de f

1207 = 4b716

k k k Ċ k k×k=k2 k×k×k=k3 k ai...a1a0 n k n=a0+a1k+a2k2+...+aiki n= ((...(aik+ai-1)k+...)k+a1)k+a0

2024135

202413

5= (3×50) + (1×51) + (4×52) + (2×53) + (0×54) + (2×55) = 6608

5ad916

5ad916= ((5×16 + 10)×16 + 13)×16 + 9 = 23257

1101
101 1
3,2

13 = 1101

8163264 0

11111111

= 255 01111111111111111 = 65535

011111111111111111111111111111111

= 4294967295 0

18446744073709551615

14

14 = 1110

10101010

10101010

= (0×20)+(1×21)+(0×22)+(1×23)+(0×24)+(1×25)+(0×26)+(1×27) = 170

886271451

441352125

10000

137153163

371854920482919

11011011

11110110001

10 fedc166a5c116

209 46 m

n mm? m n? m? nn? 2 n+n?= 255

255-n n

n 02n-1 n 32
0

12×10 = 120

n nm n×m n mi...m1m0 m 16 1 15 -111111111111111 =-(215-1) =-32767111111111111111 = 2

15-1 = 32767

16 -3276832767 x x x x+216=x+65536 3276865535

032767

3276865535

-1 65535 0 1 n -2n-12n-1-1 x x 02n-1-1 x x+ 2n 2n-12n-1 32
-21474836482147483647

64 64

-92233720368547758089223372036854775807 n x x x+ 2n 0-128

0 0 -128

-128 + 256 = 128 n m p p 2n-1

2n-1 p-2n

23 23

140 140-256 =-116

p x p? x 0127 p=x -x p ?=-x+ 256 = 256-p x -127-1 p=x+256 -x p?=-x= 256-p x=-128 x p -x p?= 256-p= (255-p) + 1

255-p= 11111111

-p ċ p

ċ 1

11

11 11 = 00001011

1 -11 -11 + 256 = 245 32
32
2

32 -294967296

4 3264

32 x

x 15

16 Ċ x

2 15

Ċ x

215 x

100010000011000010000000100000000

33 100010000011000010000000100000000

000000100000000

= 256

010000011000010

= 8386 100
= 4 Ċ0

64 x

30 Ċ 230

127-127 101

-94 8

1632 64

8 32
a←x+y a←a-z b←x-z b←b+y xy ab 4

16 0,1,2,...,7,-1,-2,...,-8

1-12-2

-8

9957 8

n

32 64

16 16 n n 10

13 + 28

xy n mp xy 2 xy mp x+y 2 1 xy x < y x-y N x-y N= 1 V x-y

V N V

x < y 1,01 sm2ns nm 1 2

64 1

11 52

n-1022 n+ 1023 12046 0

2047 +∞-∞

m 1 2

52Ċ 12

Ċ1

52 52Ċ

53
s m n sm2n s e1...e11 m1...m52 (-1)s×2e1...e11 -1023×?

1 +52?

i=1m i1 2 i? - 10001000110 1094
n= 1094-1023 = 71 m= 1,1001001111000011100000000000000000000000000000000000 = 1 + 1/2 + 1/24+ 1/27+ 1/28+ 1/29+ 1/210+ 1/215+ 1/216+ 1/217 = (2

17+ 216+ 213+ 210+ 29+ 28+ 27+ 22+ 2 + 1)/217

206727

131072

-206 727

131 072

×271? -3,724...×1021

1

Ċ 0 -1022

64
52
64
11

52 1 52 Ċ1

0 -1022 11

64 Ċ

0,4 2

1 2 2+1 2 3+1 2 6+1 2 7+1 2 10+1 2 11+1 2 14+1 2

15+...= 0,011001100110011

64
0,4

7205759403792794

18014398509481984

?0,400000000000000022204460492503 k 2 n 1 + 2 -54= 1,000000000000000000000000000000000000000000000000000001

54 1

Ċ 1 Ċ

0 (1 + (-1)) + 2-54?= 1 +?(-1) + 2-54? ?1 + 2-50?2

2-1 022 2,225×10-308

64
13 13,0 52
64
64
64

32 1 8 23

64 1

1 k 12 1 2 x1 64 x2 x1 x1x2 x2-x1 x 1-0 y1y2 y2-y1 x 2-x1 -1022 -1023 64

52 2-52

0,1 64
64
64
0 64
nm n,mm?= 0 (q,r)?Z2 n=qm+r r 0 m qr nm r m

8 = 1×5 + 3

1 = 0×2 + 1

n,mm?= 0 (q,r)?Z×N n=qm+r0⩽r <|m| 3-2

3 =-2× -2 + (-1) =-1× -2 + 1

(-2,-1) -2<-1⩽0 (-1,1)0⩽1<| -2|= 2

2 0132 013

5 m10nm n 0,0012 = 12×10-4-1 230 = -123×101

7 10100

2m

2,7×1016

6,022 141 29×1023±0,000 000 27×1023 5×10-8

100
1017
10-4 10-4 42
x |x|

Ċ x

(x,y) (z,t) r (x,y)(z,t) (x,y)(z,t) abc n 5 mn mn mnp n 7 10100 mnp x 42 y z E D V

V × D

{(x,1),(y,2),(z,3)} S :E × S ?→ D (e1+e2,s) =(e1,s) +(e2,s) 2 1 0 1 n n n ċ n n 0 1 n n Ċ n Ċ n= 01 0 1 1 n p (n+p) n 1 n n n n n n n n n n j⩽i j n n n n 10 0255
n n s n i(e) i e i1i2 i2(i1(e)) e i1 i2 i3 i1 i2 i3 e? 1 1 2 1 ax+by+c= 0 abc 01 0 1 0 1 [a;b] a b

Ja;bK ab

n

2n 0

n 02 0 1 1

0 n

03 ċ

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