1. Tasanawan Soonklang Department of Computing, Faculty of Science Data Representation

2. Introduction Switching circuit : On, Off  Number representation Number system Number conversion  Integer Arithmetic Addition Subtraction Negation

3. Introduction  Signed integer representation Sign magnitude Ones complement Twos complement  Character representation ASCII EBCDIC Unicode

4. Number representation  Number system N = b n b n-1 b n-2 …b 2 b 1 b 0.b -1 b -2 …b -m  r number (base-r) : 0,1,2,…,r-1 Binary (base2) : 0,1 Octal (base8) : 0,1,2,3,4,5,6,7 Decimal (base10) : 0,1,2,3,4,5,6,7,8,9 Hexadecimal (base16) : 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F

5. Number conversion Integer part N = b n 2 n + b n-1 2 n-1 + … + b 1 2 + b 0 1 (123) 10 = (1*10 2 ) + (2*10 1 ) + (3*10 0 ) Position bnbn b n-1 … b2b2 b1b1 b0b0 Base2 Base8 Base10 Base16 2 n 8 n 10 n 16 n 2 n-1 8 n-1 10 n- 1 16 n- 1 …………………………2 8 2 10 2 16 2 2 1 8 1 10 1 16 1 2 0 8 0 10 0 16 0

6. Number conversion Base-r to decimal  (11011) 2 = 1*2 4 + 1*2 3 + 0*2 2 + 1*2 1 + 1*2 0 = 1*16 + 1*8 + 0*4 + 1*2 + 1*1 = 16 + 8 + 0 + 2 + 1 = 27  (1276) 8 = 1*8 3 + 2*8 2 + 7*8 1 + 6*8 0 = 512 + 128 + 56 + 6  (2EA7) 16 = 2*16 3 + A*16 2 + E*16 1 + 7*16 0 = 2*4096 + 10*256 + 14*16 + 7

7. Number conversion Decimal to base-r (702) 10 = (1276) 8 8 ) 702 8 ) 87 6 8 ) 10 7 8 ) 1 2 0 1 (1632) 10 = (660) 16 16 ) 702 16 ) 87 0 16 ) 10 6 0 6 (19) 10 = (10011) 2 2 ) 19 mod 2 ) 9 1 2 ) 4 1 2 ) 2 0 2 ) 1 0 0 1

8. Number conversion fraction part N = b -1 2 -1 + b -2 2 -2 +…+ b -(m-1) 2 -(m-1) + b -m 2 - m position B -1 b -2 B -3 … B -(m-1) bmbm Base2 Base8 Base10 Base16 2 -1 8 -1 10 -1 16 -1 2 -2 8 -2 10 -2 16 -2 2 -3 8 -3 10 -3 16 -3 ………………………… 2 -(m-1) 8 -(m-1) 10 -(m-1) 16 -(m-1) 2 -m 8 -m 10 -m 16 -m

9. Number conversion Base-r to decimal  (.011) 2 = 0*2 -1 + 1*2 -2 + 1*2 -3 = 0 + 1/4 + 1/8 = 0 + 0.25 + 0.125  (.1142) 8 = 1*8 -1 + 1*8 -2 + 4*8 -3 + 2*8 -4 = 1/8 + 1/64 + 4/512 + 2/4098  (.1A) 16 = 1*16 -1 + A*16 -2 = 1/16 + 10/256

10. Number conversion Decimal to base-r (.375) 10 = (.011) 2.375 * 2 0.750 * 2 1.500 * 2 1.000 *

11. Integer arithmetic  Addition aba+b carry 11 0 1 10 1 0 01 1 0 00 0 0  Subtraction aba-b carry 11 0 0 10 1 0 01 1 1 00 0 0 1 1 1 1 0 0 + 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 - 1 0 0 1 1 0 0 0 1 0

12. Integer arithmetic  Bitwise complement Take the Boolean complement of each bit That is, set each 1 to 0 and each 0 to 1  Negation Add 1 to the result  Twos complement operation Two-step process Bitwise + negation 1 1 1 0 0 bitwise 0 0 0 1 1 + 1 0 0 1 0 0

13. Signed integer representation  Sign magnitude  Ones complement  Twos complement

14. Sign magnitude  Assign the high-order (leftmost) bit to the sign bit 0 -> positive (+) 1 -> negative (-)  The remaining (m-1) bits represent the magnitude of the number  Adv : familiarity  Problem : positive & negative zero

15. Sign magnitude  8 bits : 1 sign bit, 7 magnitude base10base2 - 1271111 1111 - 1261111 1110 - 11000 0001 - 01000 0000 + 00000 0000 + 10000 0001 + 1260111 1110 + 1270111 1111 2 8 = 256 numbers - 01000 0000 + 00000 0000

16. Ones complement  Similar to sign magnitude  Assign the high-order bit to the sign bit 0 -> positive (+) 1 -> negative (-)  Take the bitwise complement of the remaining bits to represent the magnitude  Problem : positive & negative zero

17. Ones complement  8 bits : 1 sign bit, 7 magnitude base10base2 - 1271000 0000 - 1261000 0001 - 11111 1110 - 01000 0000 + 00000 0000 + 10000 0001 + 1260111 1110 + 1270111 1111 2 8 = 256 numbers - 01000 0000 + 00000 0000

18. Twos complement  Similar to ones complement  Assign the high-order bit to the sign bit 0 -> positive (+) 1 -> negative (-)  Take the twos complement operation of the remaining bits to represent the magnitude  Solution for positive & negative zero

19. Twos complement  8 bits : 1 sign bit, 7 magnitude base10base2 - 1281000 0000 - 1271000 0001 - 11111 1111 00000 0000 + 10000 0001 + 1260111 1110 + 1270111 1111 00000 0000 0 = 0 0 0 0 bitwise 1 1 1 1 + 1 1 0 0 0 0 - 0 = 0 ignore overflow -8 = 1 0 0 0 bitwise 0 1 1 1 + 1 1 0 0 0 -(-8) = -8 monitor sign bit

20. Addition & Subtraction  Normal binary addition  Monitor sign for overflow  Overflow : the result is larger than can be held in the word size  Take twos compliment of subtrahend and add to minuend  A – B = A + (-B)

21. Twos complement  8 bits : 1 sign bit, 7 magnitude  The negative of the negative of that number is itself. +18 = 0 0 0 1 0 0 1 0 bitwise = 1 1 1 0 1 1 0 1 + 1 1 1 1 0 1 1 1 0 = -18 -18 = 1 1 1 0 1 1 1 0 bitwise = 1 1 1 0 1 1 0 1 - 1 0 0 0 1 0 0 1 0 = +18

22. Twos complement

25. Character representation  BCD – Binary code decimal  EBCDIC – Extended binary code decimal interchange code  ASCII – American standard code for information interchange  Unicode

26. BCD  Binary system  6 bits for representing 1 character  2 6 = 64 codes  2 parts : Zone Bit (first 2 bits) and Numeric Bit (last 4 bits)

27. EBCDIC  Binary system  8 bits for representing 1 character  2 8 = 256 codes  2 parts : Zone Bit (first 4 bits) and Numeric Bit (last 4 bits)  Developed by IBM

28. EBCDIC CharacterZoneDigit A,B,C,…,I11000001-1001 J,K,L,…,R11010001-1001 S,T,U,…,Z11100001-1001 0,1,2,…,911110000-1001 a,b,c,…,i10000001-1001 j,k,l,…,r10010001-1001 s,t,u,…,z10100001-1001 blank,$,.,,(,+10110001-1001 &,!,*,),;01000001-1001 -,/,’,_,?01010001-1001 :,#,@,=,”01110001-1001

29. ASCII  8 bits for representing 1 character  2 8 = 256 codes  3 parts consist of 0-32 : control character 32-127 (lower ASCII): English alphabets, numbers, symbols 128-256 (higher ASCII) : other language alphabets (e.g. Thai)  Developed by ANSI (American national standard institute)

30. ASCII Lower ASCII Higher ASCII

31. Unicode  16 bits for representing 1 character  2 16 = 65,536 codes  Enough for alphabets in other language such as Chinese, Japanese special symbols such as mathematic symbols  Widely use in many operating systems, applications and programming languages  Developed by Unicode consortium

32. Unicode  16 bits for representing 1 character  2 16 = 65,536 codes  Enough for alphabets in other language such as Chinese, Japanese special symbols such as mathematic symbols  Widely use in many operating systems, applications and programming languages  Developed by Unicode consortium

33. Unit NameAbbr.SizeByte KiloK2 10 1,024 MegaM2 20 1,048,576 GigaG2 30 1,073,741,824 TeraT2 40 1,099,511,627,776 PetaP2 50 1,125,899,906,842,624 ExaE2 60 1,152,921,504,606,846,976 ZettaZ2 70 1,180,591,620,717,411,303,424