Tasanawan SoonklangTasanawan SoonklangDepartment of Computing, Faculty of ScienceDepartment of Computing, Faculty of Science

Data RepresentationData Representation

IntroductionIntroduction

Switching circuit : On, Off Number representation

Number system Number conversion

Integer Arithmetic Addition Subtraction Negation

IntroductionIntroduction

Signed integer representation Sign magnitude Ones complement Twos complement

Character representation ASCII EBCDIC Unicode

Number representation

Number system

N = bnbn-1bn-2…b2b1b0.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 conversionNumber conversion

Integer part

N = bn2n + bn-12n-1

+ … + b12 + b01

(123)10 = (1*102) + (2*101) + (3*100)

Position bn bn-1 … b2 b1 b0

Base2

Base8

Base10

Base16

2n

8n

10n

16n

2n-1

8n-1

10n-1

16n-1

…

…

…

…

…

22

82

102

162

21

81

101

161

20

80

100

160

Number conversionNumber conversion

Base-r to decimal (11011)2 = 1*24 + 1*23 + 0*22 + 1*21 + 1*20

= 1*16 + 1*8 + 0*4 + 1*2 + 1*1

= 16 + 8 + 0 + 2 + 1

= 27 (1276)8 = 1*83 + 2*82 + 7*81 + 6*80

= 512 + 128 + 56 + 6

(2EA7)16 = 2*163 + A*162 + E*161 + 7*160

= 2*4096 + 10*256 + 14*16 + 7

Number conversionNumber 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

Number conversionNumber conversion

fraction part

N = b-12-1 + b-22-2

+…+ b-(m-1)2-(m-1) + b-m2-m

position B-1 b-2 B-3 … B-(m-1) bm

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

Number conversionNumber 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

Number conversionNumber conversion

Decimal to base-r

(.375)10 = (.011)2

.375 * 2 0 .750 *

2 1 .500 *

2 1 .000 *

Integer arithmeticInteger arithmetic

Additiona b a+b carry1 1 0 1

1 0 1 0

0 1 1 0

0 0 0 0

Subtractiona b a-b carry1 1 0 0

1 0 1 0

0 1 1 1

0 0 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

Integer arithmeticInteger arithmetic

Bitwise complementTake the Boolean complement of each bitThat is, set each 1 to 0 and each 0 to 1

NegationAdd 1 to the result

Twos complement operationTwo-step process Bitwise + negation

1 1 1 0 0 bitwise 0 0 0 1 1 + 1

0 0 1 0 0

Signed integer representationSigned integer representation

Sign magnitude Ones complement Twos complement

Sign magnitudeSign magnitude

Assign the high-order (leftmost) bit to the sign bit0 -> positive (+)1 -> negative (-)

The remaining (m-1) bits represent the magnitude of the number

Adv : familiarity Problem : positive & negative zero

Sign magnitudeSign magnitude

8 bits : 1 sign bit, 7 magnitudebase10 base2 - 127 1111 1111

- 126 1111 1110

- 1 1000 0001

- 0 1000 0000

+ 0 0000 0000

+ 1 0000 0001

+ 126 0111 1110

+ 127 0111 1111

28 = 256 numbers - 0 1000 0000+ 0 0000 0000

Ones complementOnes 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 complementOnes complement

8 bits : 1 sign bit, 7 magnitudebase10 base2 - 127 1000 0000

- 126 1000 0001

- 1 1111 1110

- 0 1000 0000

+ 0 0000 0000

+ 1 0000 0001

+ 126 0111 1110

+ 127 0111 1111

28 = 256 numbers - 0 1000 0000+ 0 0000 0000

Twos complementTwos 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 complementTwos complement

8 bits : 1 sign bit, 7 magnitudebase10 base2 - 128 1000 0000

- 127 1000 0001

- 1 1111 1111

0 0000 0000

+ 1 0000 0001

+ 126 0111 1110

+ 127 0111 1111

0 0000 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 0bitwise 0 1 1 1

+ 1 1 0 0 0

-(-8) = -8 monitor sign bit

Addition & SubtractionAddition & 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 complementTwos 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 0bitwise = 1 1 1 0 1 1 0 1 - 1

0 0 0 1 0 0 1 0 = +18

Twos complementTwos complement

Twos complementTwos complement

CharacterCharacter representationrepresentation

BCD – Binary code decimal EBCDIC – Extended binary code

decimal interchange code ASCII – American standard code for

information interchange Unicode

BCDBCD

Binary system 6 bits for representing 1 character 26 = 64 codes 2 parts : Zone Bit (first 2 bits) and

Numeric Bit (last 4 bits)

EBCDICEBCDIC

Binary system 8 bits for representing 1 character 28 = 256 codes 2 parts : Zone Bit (first 4 bits) and

Numeric Bit (last 4 bits) Developed by IBM

EBCDICEBCDIC

Character Zone Digit

A,B,C,…,I 1100 0001-1001

J,K,L,…,R 1101 0001-1001

S,T,U,…,Z 1110 0001-1001

0,1,2,…,9 1111 0000-1001

a,b,c,…,i 1000 0001-1001

j,k,l,…,r 1001 0001-1001

s,t,u,…,z 1010 0001-1001

blank,$,.,<,>,(,+ 1011 0001-1001

&,!,*,),; 0100 0001-1001

-,/,’,_,? 0101 0001-1001

:,#,@,=,” 0111 0001-1001

ASCIIASCII

8 bits for representing 1 character 28 = 256 codes 3 parts consist of

0-32 : control character32-127 (lower ASCII): English alphabets,

numbers, symbols128-256 (higher ASCII) : other language

alphabets (e.g. Thai)

Developed by ANSI (American national standard institute)

ASCIIASCII

Lower ASCII Higher ASCII

UnicodeUnicode

16 bits for representing 1 character 216 = 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

UnicodeUnicode

16 bits for representing 1 character 216 = 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

UnitUnit

Name Abbr. Size Byte

Kilo K 210 1,024

Mega M 220 1,048,576

Giga G 230 1,073,741,824

Tera T 240 1,099,511,627,776

Peta P 250 1,125,899,906,842,624

Exa E 260 1,152,921,504,606,846,976

Zetta Z 270 1,180,591,620,717,411,303,424