Question

Represent the following decimal numbers using IEEE-754 floating point representation. please show all work

i. -0.75

ii. 0

iii. - infinity

iv. 23

v. 10.25

Answer 1

In Single Point IEEE-754 Floating Point Representation:

Sign = 1 bit

Exponent = 8 bits

Mantissa = 23 bits

In Single Point IEEE-754 Floating Point Representation:

Sign = 1 bit

Exponent = 11 bits

Mantissa = 52 bits

i) -0.75

Convert the number into binary form:

(0.75)₁₀ =(0.11)₂

                = 1.1 * 2^-1

Sign = 1 (as the number is negative)

Single Precision Representation:

Biased exponent = 127+(-1) = 126

126 = 01111110

Normalised matissa = 1

The IEEE 754 Single Precision : 1 01111110 100000000000000000000000

(add 0's to normalised matissa to make it 23 bits)

Hexadecimal Representation = BF400000

Double Precision Representation:

Biased exponent = 1023+(-1) = 1022

1022 = 1111111110

Normalised Mantissa = 1

The IEEE 754 Double Precision : 1 01111111110 1000000000000000000000000000000000000000000000000000

(add 0's to normalised matissa to make it 52 bits)

Hexadecimal Representation: BFE8000000000000

ii) 0

Convert the number into binary form:

(0)₁₀ =(0)₂

                = 0

Sign = 0 (as the number is positive)

Single Precision Representation:

The IEEE 754 Single Precision : 0 00000000 000000000000000000000000

(add 0's to normalised matissa to make it 23 bits)

Hexadecimal Representation = 00000000

Double Precision Representation:

The IEEE 754 Double Precision : 0 00000000000 0000000000000000000000000000000000000000000000000000

(add 0's to normalised matissa to make it 52 bits)

Hexadecimal Representation: 0000000000000000

iii) - infinity

It cannot be converted to any form as it is boundless and is something that is unknown and is something that can be larger or smaller than any number that is known.

iv) 23

Convert the number into binary form:

(23)₁₀ =(10111)₂

                = 1.0111 * 2⁴

Sign = 0 (as the number is positive)

Single Precision Representation:

Biased exponent = 127+(4) = 131

131 = 10000011

Normalised matissa = 0111

The IEEE 754 Single Precision : 0 10000011 011100000000000000000000

(add 0's to normalised matissa to make it 23 bits)

Hexadecimal Representation = 41B80000

Double Precision Representation:

Biased exponent = 1023+(4) = 1027

1027 = 10000000011

Normalised Mantissa = 0111

The IEEE 754 Double Precision : 0 10000000011 0111000000000000000000000000000000000000000000000000

(add 0's to normalised matissa to make it 52 bits)

Hexadecimal Representation: 4037000000000000

v) 10.25

Convert the number into binary form:

(10)₁₀ =(1010)₂

(0.25)₁₀ = (01)₂

     10.25 = 1010.01

               = 1.01001 * 2³

Sign = 0 (as the number is positive)

Single Precision Representation:

Biased exponent = 127+(3) = 130

130 = 10000010

Normalised matissa = 01001

The IEEE 754 Single Precision : 0 10000010 01001000000000000000000

(add 0's to normalised matissa to make it 23 bits)

Hexadecimal Representation = 41240000

Double Precision Representation:

Biased exponent = 1023+(3) = 1026

1026 = 10000000010

Normalised Mantissa = 01001

The IEEE 754 Double Precision : 0 10000000010 0100100000000000000000000000000000000000000000000000

(add 0's to normalised matissa to make it 52 bits)

Hexadecimal Representation: 4024800000000000