Data Representation

1
Data Representation

• Reading
• Chapter 3 of Goodman and Millar - Number Systems
• Chapter 4 of Goodman and Millar - Data
  Representation
• Goal
• Describe the methods by which data can be
  represented and transmitted in a computer.

2
Introduction

• Inside a computer all data must be coded in a
  binary format
• Coding format used will depend on the type of
  data
• e.g. consider the number 9
• 9 may be represented as 1001
• -9 may be represented as 10111
• 9.0 may be represented as 0000010010000001
• the character 9 may be represented as 0111001

3
Introduction

• When a coding format is being devised a number of
  considerations need to be made
• Ease of manipulation
• Conciseness and machine efficiency
• Accuracy
• Standardised data communication

4
Binary Data Representation

• Some things that a computer may represent are
• A machine instruction
• A numeric quantity
• A character
• A graphic
• A sound
• Machine Instructions
• Relationship between an instruction and its bit
  pattern is determined by the computer designer
  (e.g. the same bit pattern on the 68000 will have
  a different effect on an 8086)

5
Numbers

• Need to represent ve and -ve numbers as well as
  very small and very large numbers
• Number formats include BCD, signed magnitude,
  excess notation, two compliment, floating point.
• Both decimal and Binary number systems are forms
  of positional number systems
• e.g. 372910 3 x103 7 x 102 2 x 101 9 x
  100 101102 1 x 24 0 x 23 1 x 22 1 x 21
  0 x 20 2210

20
2-1
2-2
2-3
2-4
21
22
23
24
25
32
16
8
4
2
1
.
1/2
1/4
1/8
1/16
6
Numbers

• representation of binary numbers pose problems
  for people
• hard to read
• easy to make errors
• e.g. consider 1010011101011100
• To minimise these errors use a different number
  system
• Hexadecimal
• Base 16
• Uses the digit values 0, 1, 2, 3, 4, 5, 6, 7, 8,
  9, A, B, C, D, E, F
• Each hex digit represents 4 binary digits
• e.g. 10100111010111002 A75C
• require 2 hex digits per byte

7
Numbers

• Octal
• Base 8
• Uses the digit values 0, 1, 2, 3, 4, 5, 6, 7
• e.g. 10100111010111002 1235348

8
Numeric Representation

• BCD (binary coded decimal)
• Each Decimal digit is coded as a 4 bit binary
  code
• Developed for early calculators
• e.g. 35910 0011 0101 1001bcd
• Easy for people to understand, Hard for computers
  to work with
• Signed Magnitude
• Extra bit added to the code to represent the sign
• In most cases
• a 0 represents a ve
• a 1 represents a ve

9
Excess Notation

• fix number of bit positions used
• smallest number with 1 in the MSB represents zero
  (e.g. 1000)
• All bit strings greater than this represent ve
  numbers
• All bit strings less than this represent -ve
  numbers
• Example is known as excess eight notation because
  8 (100) represents zero

10
Twos Complement

• Can represent numbers in the range -2n-1 to
  (2n-1-1)
• One representation of zero
• Can be sign extended
• 510 00000101 0000000000000101
• -510 11111011 1111111111111011

Binary
Negative
Compliment
Positive
0 1
1 0
1
Result
11
Twos Complement
0
-1
1
-2
0000
1111
1110
0001
-3
2
1101
0010
-4
3
0011
1100
-5
1011
4
0100
1010
0101
-6
5
1001
0110
6
-7
0111
1000
7
-8