Question

Represent the following in IEEE format. Express your answers in Hexadecimal Format (Base 16)

1. -10.5₁₀
2. 9.999 x 10¹ + 1.610 x 10^-1
3. 0.5₁₀ x (-0.4375)₁₀
4. (1.10 x 10¹⁰) x (9.2 x 10^-5)

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

a) -10.5

Convert the number into binary form:

(10)₁₀ =(1010)₂

(0.5)₁₀ = (1)₂

     10.5 = 1010.1

               = 1.0101 * 2³

Sign = 1 (as the number is negative)

Single Precision Representation:

Biased exponent = 127+(3) = 130

130 = 10000010

Normalised matissa = 0101

The IEEE 754 Single Precision : 1 10000010 01010000000000000000000

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

Hexadecimal Representation = C1280000

Double Precision Representation:

Biased exponent = 1023+(3) = 1026

1026 = 10000000010

Normalised Mantissa = 0101

The IEEE 754 Double Precision : 1 10000000010 0101000000000000000000000000000000000000000000000000

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

Hexadecimal Representation: C025000000000000

b) 9.999 x 10¹ + 1.610 x 10^-1 = 99.99+0.1610 = 100.151

Convert the number into binary form:

(100)₁₀ =(1100100)₂

(0.151)₁₀ = (00100110101001111111)₂

     100.151 = 1100100.0010011010100

               = 1.100100001001101010100 * 2⁶

Sign = 0 (as the number is negative)

Single Precision Representation:

Biased exponent = 127+(6) = 133

130 = 10000101

Normalised matissa = 1001000010011010101

The IEEE 754 Single Precision : 0 10000101 10010000100110101010000

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

Hexadecimal Representation = 42C84D50

Double Precision Representation:

Biased exponent = 1023+(6) = 1029

1029 = 10000000101

Normalised Mantissa = 1001000010011010100111111011111001110110110010001011

The IEEE 754 Double Precision : 0 10000000101 11001000010011010100111111011111001110110110010001011

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

Hexadecimal Representation: 405909A9FBE76C8B

c) 0.5 x (-0.4375) = - 0.21875

Convert the number into binary form:

(0.21875)₁₀ = (0.00111)₂

     0.21875 = 0.00111

               = 1.11* 2^-3

Sign = 1 (as the number is negative)

Single Precision Representation:

Biased exponent = 127+(-3) = 124

124 = 01111100

Normalised matissa = 11

The IEEE 754 Single Precision :1 01111100 11000000000000000000000

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

Hexadecimal Representation = BE600000

Double Precision Representation:

Biased exponent = 1023+(-3) = 1020

1020 = 01111111100

Normalised Mantissa = 11

The IEEE 754 Double Precision : 1 01111111100 1100000000000000000000000000000000000000000000000000

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

Hexadecimal Representation: BFCC000000000000

d) (1.10 x 10¹⁰) x (9.2 x 10^-5) = 11000000000 x 0.000092 = 1012000

Convert the number into binary form:

(1012000)₁₀ = (11110111000100100000)₂

     1012000 = 11110111000100100000

               = 1.1110111000100100000 * 2¹⁹

Sign = 0 (as the number is negative)

Single Precision Representation:

Biased exponent = 127+(19) = 146

146 = 10010010

Normalised matissa = 1110111000100100000

The IEEE 754 Single Precision :0 10010010 11101110001001000000000

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

Hexadecimal Representation = 49771200

Double Precision Representation:

Biased exponent = 1023+(19) = 1042

1042 = 49771200

Normalised Mantissa = 1110111000100100000

The IEEE 754 Double Precision : 0 10000010010 1110111000100100000000000000000000000000000000000000

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

Hexadecimal Representation: 412EE24000000000