Fixed point and floating-point numbers

1. FIXED-POINT AND FLOATING-POINT
NUMBERS
PRESENTED BY:
A.K.NAVEEN KUMAR
E.C.E. – THIRD YEAR
SRI SHANMUGHA COLLEGE OF
ENGINEERING & TECHNOLOGY

2. REPRESENTATIONS OF NUMBERS
 Unsigned integers
 Signed integers – 1’s and 2’s complement
representation
 Fixed-point numbers
 Floating-point numbers
2

3. BASE-10 (DECIMAL) ARITHMETIC
 Uses the ten numbers from 0 to 9
 Each column represents a power of 10
3
Thousands (103
) column
Hundreds (102
) column
Tens (101
) column
Ones (100
) column
1999.10
= 1x103
+ 9x102
+ 9x101
+ 9x100

4. STANDARD BINARY REPRESENTATION
 Uses the two numbers from 0 to 1
 Every column represents a power of 2
4
1001.2
= 1x23
+ 0x22
+ 0x21
+ 1x20
Eights (23
) column
Fours (22
) column
Twos (21
) column
Ones (20
) column

5. ADDITION
5
1. 2 5
+ 1. 5 0
2. 7 5
1. 0 1
+ 1. 1 0
1 0. 1 1
Base-10 Base-2

6. SCIENTIFIC NOTATION (1)
 One billion
= 1,000,000,000
= 1 x 109
 significand or mantissa: 1
 base or radix: 10
 exponent: 9
6

7. PRACTICE (BASE 10)
 258 = 2.58 x 102
Mantissa = 258
Radix = 10
Exponent = 2
 24.25 = 2.425 x 101
Mantissa = 2425
Radix = 10
Exponent = 1
7

8. 8-BIT FLOATING POINT FORMAT (1)
sign
1 bit
exponent
3 bits
significand
4 bits
number
base 2
number
base 10
0 001 1001 1.001x21 2.25
0 011 1100 1.1 x 23 12.0
0 111 1110 1.11 x 27 224.0
1 001 1110 1.11 x 2-1 0.875
8

9. IMPROVEMENTS
 Bias the exponent
 Always subtract a fixed amount, e.g., 3
 Allows representation of negative exponents
 Implicit one
- Leading one in a Phone number such as
1-619-556-0231 is redundant.
 Why use a bit for the leading one?
9

10. IEEE STANDARD FLOATING-POINT
 Single precision
 32 bits
 sign: 1 bit
 exponent: 8 bits
 significand: 23 bits
 Bias: 127
 Double precision
 64 bits
 sign: 1 bit
 exponent: 11 bits
 significand: 52 bits
 Bias: 511
10

11. PRACTICE( BASE 10)
 13 = 1.3 x 101
= 1.011 x 23
 1.25 = 1.25 x 100
= 1.010 x 20
11

12. 12
s exponent mantissa
3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0
s exponent mantissa
3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0

13. THANK YOU