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1 ( )

2 Page 2

3 Page 3

4 0 = 1 = Page 4

5 = = = 1101 ( ) b n 1 b n 2 b 0 (b n 1,,b 0 {0,1}) = 2 n 1 b n n 2 b n b 0 n n bit( ) Page 5

6 n n n {}}{{}}{ 00 0 = = 2 n k 1/2 k 2 k b n 1 b 0 b n 1 b 0 1/2 k b n 1 b k b k 1 b 0 k {}}{ 0 0 k {}}{ 0 0b n 1 b k Page 6

7 Page 7

8 8bit Page 8

9 0 + 0 = = = = 10 Page 9

10 Page 10

11 Page 11

12 34 34 = = 34 = 34 Page 12

13 =? = ( 34) = 191 Page 13

14 b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 b 7 b 6 b 5 b 4 b 3 b 2 b 1 b (b bit b ) b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 + (b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 + 1) = (b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 + b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 ) + 1 = = bit 1 ( )+( )=0 Page 14

15 ( : n bit 0 2 n 1 ) a b modulo m a bmodm : k Z a b = km (a m b m ) a mod m a + mz = {b a bmodm} (= {a + mk k Z}) : a bmodm c d modm a + c b + d mod m, a c b d mod m Page 15

16 = : m = 2 8 (= 256) ( 1) Z = {, 513, 257, 1,255,511, }, Z = {, 512, 256,0,256,512, }, Z = {, 511, 255,1,257,513, }, Z = {, 510, 254,2,258,514, }, ( 1) (mod 2 8 ) Page 16

17 = bit 1 = 2 8 (= 256) bit n n Z {, 256,0,256, } {, 255,1,257, } {, 129,127,383, } {, 128,128,384, } {, 127,129,385, } {, 1,255,511, } 1 bit Page 17

18 nbit n 1 n 1 {}}{{}}{ 10 0 = 2 n = 2 n 1 1 bit Page 18

19 (unsigned) (signed) 0 2 n 1 2 n 1 2 n 1 1 bit bit Page 19

20 Page 20

21 1bit = = = = = = = = 11 1bit : 1bit (carry in) : 1bit (carry out) Page 21

22 1bit a b cin cout sum Page 22

23 1bit bit (cout) (sum) 1bit Page 23

24 (cout=1 ) (a,b,cin) 1 (cout=0) cout = (a b cin) (a b) (a cin) (b cin) = (a b) (a cin) (b cin) Page 24

25 cout = (a b) (a cin) (b cin) Page 25

26 1(sum=1) (a,b,cin) 1 0(sum=0) sum = (a b cin) (a b cin) ( a b cin) ( a b cin) Exercise. Page 26

27 1bit Page 27

28 n bit n = 4 : a n a 1, b n b 1 (n bit ) : d n d 1 ( ) cout ( 1bit) Page 28

29 n bit n = 4 : a n a 1, b n b 1 (n bit ) : d n d 1 ( b n b 1 a n a 1 ) Page 29

30 (= ) = 0001(= 1) (= 6 + 5) = 1011(= 5) (= 3 ( 5)) = 1000(= 8) (= ( 5) 6) = 0101(= 5) ( 5) 6 5 mod 2 4, etc. Page 30

31 4bit a 4 a 3 a 2 a 1 7 7bit 2 d 7 d 6 d 5 d 4 d 3 d 2 d (= 13) (= 91) n bit (bit ) ( ) 2 Page 31

32 7 = (1) 8 a 4 a 3 a 2 a 1 a 4 a 3 a 2 a (3bit ) (2) a 4 a 3 a 2 a (2 ) 8 a 4 a 3 a 2 a a 4 a 3 a 2 a a 4 a 3 a 2 a a 4 a 3 a 2 a 1 0a 4 a 3 a 2 a a 4 a 3 a 2 a 1 2 (3) 8bit 7bit 0 105(= ) 8 2 ( 0 ) Page 32

33 Page 33

34 susumu/lecture/digital11.pdf Page 34

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