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King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals

Computer Engineering Department


College of ComputerScience And Engineering


Lecture 5 – Data Representation 1© 2005 Dr. Abdelhafid Bouhraoua

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Lecture 5:

Data Representation

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Data in Computers• Computers understand binary numbers only:• Numbers:

– Represented in binary• Integers: +33, -37, +267• Reals: -123.45 E16, -59.6 E-12

• Characters: Alphanumeric: A, c, Ctrl, Shift– Represented by numeric codes

• Multimedia– Still Images: Many formats, all numeric. Based on numeric representation of the

RGB components of each point of the image (pixel)– Video Images: Many formats also. Based on same principal as still images but

applied on moving portions of the image only.– Sounds: Many formats. All numeric. Based on numeric representation of the

amplitude of the sound wave over time.• Records and Database Elements:

– Combination of Alphanumeric strings and numbers• Scientific Data

– Combination of Numbers, Records, Multimedia and Statistical Data (numbers)

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What need to be represented

From CPU point of view: ONLY BinaryTo Standardize: At CPU Level Need to represent:• Numbers• Characters• No need to represent other items

– No need to represent Images at CPU level– No need to represent Sounds at CPU level– No need to represent records at CPU level– Taken care by HL language constructs– Goal: Machine independent representation: STANDARD

Represent ONLY Numbers and Characters

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Numbering SystemsN = dw-1Bw-1 +…+ diBi + d1B1 +d0B0

di: digits; B: Basedi in [0, B-1]

Numbering System Base Digits Set

Binary 2 0, 1

Octal 8 0, 1, 2, 3, 4, 5, 6, 7

Decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Hexadecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

(728)10 = (1011011000)2 = (1330)8 = (2D8)16

728d = 1011011000b = 1330o = 2D8h

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Number ConversionBinary - HexadecimalHexa-

decimal

Bin. Sym.

0000 0

0001 1

0010 2

0011 3

0100 4

0101 5

0110 6

0111 7

1000 8

1001 9

1010 A

1011 B

1100 C

1101 D

1110 E

1111 F

Binary to Hexadecimal000100110100110101101100

0

134D6C

Conversion Table Lookup

Hexadecimal to Binary5D1F8

0101 1101 0001 1111 1000

Conversion Table Lookup

Octal

Bin. Sym.

000 0

001 1

010 2

011 3

100 4

101 5

110 6

111 7

Binary to Octal001110101101100

0 0

16554

Octal to Binary5361

Conv. Table Lookup

Conv. Table Lookup

101 011 110 001

Hex Octal: Go to Binary First

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Number ConversionDecimal – Binary, Octal, Hex

• Binary, Octal, Hex Decimal:– Represent the number in base 10 and compute the operations:– 110110b = 1x25 + 1x24 + 0x23 + 1x22 + 1x21 + 0x20 = 54– 236o = 2x82 + 3x81 + 6x80 = 158– 3Ch = 3x161 + Cx160 = 60

• Decimal Binary, Octal, Hex– Remainder Set of Successive Division by Target Base (2, 8 or 16)

N = Q0 x 2 + R0 First DivisionQ0 = Q1 x 2 + R1 Second DivisionQ1 = Q2 x 2 + R2 Third Division….Qn-1 = 0 x 2 + Rn Cannot divide anymoreN = (((((…(Rn x2) x 2 + Rn-1) x 2 + .) x2 + …..x2 + R1) x2 + R0

N = Rn x 2n + Rn-1 x 2n-1 + ….. + R1 x 2 + R0

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ACSII or Decimal vs Binary Numbers

• Attractive because: Human- Friendly representation• Non attractive to computers:

– Understand only binary

– Binary operations easy to perform

– Does not know how to perform operations on ASCII or Decimal number.

– Need to convert ASCII/Decimal to binary before and after operation

• Used only for I/O (User Interface)

Use Binary Representation

For Internal Computations

Convert to/from ASCII for I/O

Only

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Number Representation• Numbers in computers

– Just a representation of real numbers

– Real numbers have an infinite number of digits

153ten = …..00000…0000010011001two

= 0000 0000 1001 1001two in a 16 bits word

Add, Subtract, Multiply, Divide

Result bigger than number of

available slots (bits)

Overflow

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Signed Numbers• Subtraction of a big number from small

number is a negative numberNeed for Signed Numbers Representation

• Three Main ways:– Sign + Magnitude– 1’s Complement– 2’s Complement

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Sign and Magnitude

Intuitive Representation: Sign + Magnitude

Sign Magnitude• Range [-2n-1 -1 , +2n-1-1]

– 8 bits: [-127, +127]

– 12 bits: [-2047, +2047]

• Advantages: Straight Forward• Disadvantages

– Has the problem of double representing the 0 (–0 and +0),

– Complicates the design of the logic circuits that handle signed-numbers arithmetic

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One’s Complement• Negative numbers are bit-to-bit complements of

positive numbers– +9 on 8 bits = 00001001– -9 on 8 bits = 11110110

• Range [-2n-1 -1 , +2n-1-1]– 8 bits: [-127, +127]– 12 bits: [-2047, +2047]

• Advantages– Easy Representation– Easier arithmetic operations (add/sub) than sign + magnitude

• Disadvantages– Has the problem of double representing the 0 (–0 and +0), – Add/Sub still relatively complex

Examples: 0111 + 01110001 – 01110111 – 0001

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Two’s Complement

• Range [-2n-1, +2n-1-1]– 8 bits: [-128, +127]– 12 bits: [-2048, +2047]

• Advantages– No double representation of 0 (the 2's complement of 0 is still 0), – Simplest Add/Subtract circuit design,– Add/Subtract operations is done in one-step, – The end result is already represented in 2's complement (not when overflow).

• Negative number represented as bit to bit complement of positive number plus 1

• Coming from: x + (-x) = 0 • Given x + x = -1 (check for any number)

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Character Representation• ASCII

– Broadly Used– Standard– Limited in representing other languages– 7 bits + 1 bit (parity)

• Even Parity: put 1 in PB if # of 1s is even• Odd Parity: put 1 in PB if # of 1s is odd

• Unicode– 16 bits– Used to represent chars of all languages

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