Number Representation

1
Number Representation

• Lecture 20
• 4.3.2001.

2
Topics

• How are numeric data items actually stored in
  computer memory?
• How much space (memory locations) is allocated
  for each type of data?
• int, float, char, etc.
• How are characters and strings stored in memory?

3
Decimal number system
Binary number system

• Ten digits 0,1,2,3,4,5,6,7,8,9
• Every digit position has a weight which is a
  power of 10.
• Base or radix is 10.

• Two digits 0,1
• Every digit position has a weight which is a
  power of 2.
• Base or radix is 2.

4
Decimal Number

• 136.25 What does this number actually mean ?
• 102 1 100.0
• 101 3 30.0
• 100 6 6.0
• 10-1 2 0.2
• 10-2 5 0.05

5
Binary Number

• 1101.01 What does this number mean?
• 23 1 1000.0 (8 in decimal)
• 22 1 100.0 (4 in decimal)
• 21 3 0.0 (0 in decimal)
• 20 6 1.0 (1 in decimal)
• 2-1 2 0.0 (0.0 in decimal)
• 2-2 5 0.01 (0.25 in decimal)

6
First integers and their binary equivalent

• decimal binary
• 0 0000 (023 022 021
  020)
• 1 0001 (023 022 021
  120)
• 2 0010 (023 022 121
  020)
• 3 0011 (023 022 121
  120)
• 4 0100 (023 122 021
  020)
• 5 0101 (023 122 021
  120)
• 6 0110 (023 122 121
  020)
• 7 0111 (023 122 121
  120)
• 8 1000 (123 022 021
  020)
• 9 1001 (123 022 021
  120)

7
Adding Binary Numbers

• Basic Rules
• 000
• 011
• 101
• 110 (carry 1)

• Example
• 01101001
• 00110100
• -------------
• 10011101

8
Weighted number systems

• N ?Mj0 bj Bj
• N the number
• M Number of digits
• b The digit
• B Systems radix

9
Examples

• 1. 101011 ? 1x25 0x24 1x23 0x22 1x21
  1x20
• 43
• (101011)2 (43)10
• 2. .0101 ? 0x2-1 1x2-2 0x2-3 1x2-4
• .3125
• (.0101)2 (.3125)10
• 3. 101.11 ? 1x22 0x21 1x20 1x2-1
  1x2-2
• 5.75
• (101.11)2 (5.75)10

10
Decimal-to-Binary Conversion

• Consider the integer and fractional parts
  separately.
• For the integer part,
• Repeatedly divide the given number by 2, and go
  on accumulating the remainders, until the number
  becomes zero.
• Arrange the remainders in reverse order.
• For the fractional part,
• Repeatedly multiply the given fraction by 2.
• Accumulate the integer part (0 or 1).
• If the integer part is 1, chop it off.
• Arrange the integer parts in the order they are
  obtained.

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Example 1 239

• 239
• 2 119 --- 1
• 59 --- 1
• 2 29 --- 1
• 14 --- 1
• 2 7 --- 0
• 3 --- 1
• 2 1 --- 1
• 2 0 --- 1

(239)10 (11101111)2
12
Example 2 64

• 64
• 2 32 --- 0
• 16 --- 0
• 2 8 --- 0
• 4 --- 0
• 2 2 --- 0
• 1 --- 0
• 2 0 --- 1

(64)10 (1000000)2
13
Example 3 .634
.634 x 2 1.268 .268 x 2
0.536 .536 x 2 1.072 .072 x 2
0.144 .144 x 2 0.288
(.634)10 (.10100)2
14
Example 4 37.0625

• (37)10 (100101)2
• (.0625)10 (.0001)2
• ?(37.0625)10 (100101.0001)2

15
Hexadecimal Numbers

• Base16
• Decimal Binary Hex
• 0 00000 0
• 1 00001 1
• 2 00010 2
• 3 00011 3
• 4 00100 4
• 5 00101 5
• 6 00110 6
• 7 00111 7
• 8 01000 8
• 9 01001 9

Decimal Binary Hex 10 01010 10
11 01011 11 12 01100 12
13 01101 13 14 01110 14
15 01111 15 16 00110 16
17 00111 17 18 01000 18
19 01001 19
16
Integers Representation