Represent (-55.25)10 and (0.03125)10 in IEEE 754 single precision format

To represent decimal numbers in IEEE 754 single precision format, we need to convert them into binary and then apply the format rules.

The IEEE 754 single precision format uses 32 bits, with 1 bit for the sign, 8 bits for the exponent, and 23 bits for the fraction (or mantissa).

Let’s represent (-55.25) and (0.03125) in IEEE 754 single precision format:

For (-55.25)10:

1. Sign Bit: Since the number is negative, the sign bit is 1.
2. Integer Part: Convert the integer part (55) to binary:

• 55 in binary: 110111

1. Fraction Part: Convert the fraction part (0.25) to binary:

• 0.25 * 2 = 0.5 –> 0
• 0.5 * 2 = 1.0 –> 1
• So, 0.25 in binary: 0.01

1. Combine Integer and Fraction Parts: The combined binary representation is -110111.01.
2. Normalize the Number: Shift the binary point to the left until there’s only one bit before the binary point:

• -110111.01 becomes -1.1011101 * 2^5

1. Exponent (8 bits): The exponent bias is 127. Since the binary point was shifted 5 positions to the left, the exponent is 127 + 5 = 132, which in binary is 10000100.
2. Mantissa (Fraction Part): The fraction part without the hidden bit is 10111010000000000000000.

Putting it all together:

• Sign bit: 1
• Exponent: 10000100
• Mantissa: 10111010000000000000000

So, (-55.25) in IEEE 754 single precision format is:

1 10000100 10111010000000000000000

For (0.03125)10:

1. Sign Bit: The number is positive, so the sign bit is 0.
2. Integer Part: There is no integer part, so it’s 0.
3. Fraction Part: Convert the fraction part (0.03125) to binary:

• 0.03125 * 2 = 0.0625 –> 0
• 0.0625 * 2 = 0.125 –> 0
• 0.125 * 2 = 0.25 –> 0
• 0.25 * 2 = 0.5 –> 0
• 0.5 * 2 = 1.0 –> 1
• So, 0.03125 in binary: 0.00001

1. Normalize the Number: Shift the binary point to the right until there’s only one bit before the binary point:

• 0.00001 becomes 1.0 * 2^(-5)

1. Exponent (8 bits): The exponent bias is 127. Since the binary point was shifted 5 positions to the right, the exponent is 127 – 5 = 122, which in binary is 01111010.
2. Mantissa (Fraction Part): The fraction part without the hidden bit is 0000000000000000000001.

Putting it all together:

• Sign bit: 0
• Exponent: 01111010
• Mantissa: 0000000000000000000001

So, (0.03125) in IEEE 754 single precision format is:

0 01111010 0000000000000000000001