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COE 205. Lecture 5: Data Representation. 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 - PowerPoint PPT Presentation
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COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 1© 2005 Dr. Abdelhafid Bouhraoua
COE 205
Lecture 5:
Data Representation
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 2© 2005 Dr. Abdelhafid Bouhraoua
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)
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 3© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 4© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 5© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 6© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 7© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 8© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 9© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 10© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 11© 2005 Dr. Abdelhafid Bouhraoua
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
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 12© 2005 Dr. Abdelhafid Bouhraoua
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)
COE 205COE 205
King Fahd University of Petroleum and MineralsKing Fahd University of Petroleum and Minerals
Computer Engineering Department
Computer Engineering Department
College of ComputerScience And Engineering
College of ComputerScience And Engineering
Lecture 5 – Data Representation 13© 2005 Dr. Abdelhafid Bouhraoua
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