Tasanawan Soonklang Department of Computing, Faculty of Science Data Representation

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

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

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

Number conversion Integer part N = b n 2 n + b n-1 2 n-1 + … + b 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 n- 1 …………………………

Number conversion Base-r to decimal  (11011) 2 = 1* * * * *2 0 = 1*16 + 1*8 + 0*4 + 1*2 + 1*1 = = 27  (1276) 8 = 1* * * *8 0 =  (2EA7) 16 = 2* A* E* *16 0 = 2* * *16 + 7

Number conversion Decimal to base-r (702) 10 = (1276) 8 8 ) ) ) ) (1632) 10 = (660) ) ) ) (19) 10 = (10011) 2 2 ) 19 mod 2 ) ) ) )

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

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

Number conversion Decimal to base-r (.375) 10 = (.011) * * * *

Integer arithmetic  Addition aba+b carry  Subtraction aba-b carry

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 bitwise

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

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

Sign magnitude  8 bits : 1 sign bit, 7 magnitude base10base = 256 numbers

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

Ones complement  8 bits : 1 sign bit, 7 magnitude base10base = 256 numbers

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

Twos complement  8 bits : 1 sign bit, 7 magnitude base10base = bitwise = 0 ignore overflow -8 = bitwise (-8) = -8 monitor sign bit

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)

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

Twos complement

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

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)

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

EBCDIC CharacterZoneDigit A,B,C,…,I J,K,L,…,R S,T,U,…,Z ,1,2,…, a,b,c,…,i j,k,l,…,r s,t,u,…,z blank,$,.,,(, &,!,*,),; ,/,’,_,?

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

ASCII Lower ASCII Higher ASCII

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

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

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