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

1. Representing <math xmlns="http://www.w3.org/1998/Math/MathML">(−55.25)10(-55.25)_{10}</math>(−55.25)10​

Step 1: Determine the Sign Bit

Since $-55.25$ is negative, the sign bit is 1.

Step 2: Convert to Binary (Magnitude Only)

• Convert 55 to binary: <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">5510=110111255_{10} = 110111_2</math>5510​=1101112​
• Convert 0.25 to binary (fraction part):
  • $0.25\times 2=0.5\to 0$
  • $0.5\times 2=1.0\to 1$
  So, <math xmlns="http://www.w3.org/1998/Math/MathML">0.2510=0.0120.25_{10} = 0.01_2</math>0.2510​=0.012​.

Thus, <math xmlns="http://www.w3.org/1998/Math/MathML">55.251055.25_{10}</math>55.2510​ is represented in binary as:

Step 3: Normalize the Binary Number

To normalize, shift the binary point to get a single 1 to the left of the decimal point:

Step 4: Calculate the Exponent

With a bias of 127, the exponent is calculated as:

So, <math xmlns="http://www.w3.org/1998/Math/MathML">13210=100001002132_{10} = 10000100_2</math>13210​=100001002​.

Step 5: Determine the Mantissa

The mantissa is the fractional part of the normalized number, $1011101$, padded to 23 bits:

Final IEEE 754 Representation of <math xmlns="http://www.w3.org/1998/Math/MathML">(−55.25)10(-55.25)_{10}</math>(−55.25)10​

Putting it all together, the IEEE 754 single-precision format for <math xmlns="http://www.w3.org/1998/Math/MathML">(−55.25)10(-55.25)_{10}</math>(−55.25)10​ is:

So, <math xmlns="http://www.w3.org/1998/Math/MathML">(−55.25)10(-55.25)_{10}</math>(−55.25)10​ in IEEE 754 single precision is:

2. Representing <math xmlns="http://www.w3.org/1998/Math/MathML">(0.03125)10(0.03125)_{10}</math>(0.03125)10​

Step 1: Determine the Sign Bit

Since $0.03125$ is positive, the sign bit is 0.

Step 2: Convert to Binary

Convert $0.03125$ to binary:

• $0.03125\times 2=0.0625\to 0$
• $0.0625\times 2=0.125\to 0$
• $0.125\times 2=0.25\to 0$
• $0.25\times 2=0.5\to 0$
• $0.5\times 2=1.0\to 1$

So, <math xmlns="http://www.w3.org/1998/Math/MathML">0.0312510=0.0000120.03125_{10} = 0.00001_2</math>0.0312510​=0.000012​.

Step 3: Normalize the Binary Number

To normalize <math xmlns="http://www.w3.org/1998/Math/MathML">0.0000120.00001_2</math>0.000012​, shift the binary point to get a single 1 to the left of the decimal:

Step 4: Calculate the Exponent

With a bias of 127, the exponent is:

So, <math xmlns="http://www.w3.org/1998/Math/MathML">12210=011110102122_{10} = 01111010_2</math>12210​=011110102​.

Step 5: Determine the Mantissa

The mantissa is the fractional part of the normalized number (which is $0$), padded to 23 bits:

Final IEEE 754 Representation of <math xmlns="http://www.w3.org/1998/Math/MathML">(0.03125)10(0.03125)_{10}</math>(0.03125)10​

Putting it all together, the IEEE 754 single-precision format for <math xmlns="http://www.w3.org/1998/Math/MathML">(0.03125)10(0.03125)_{10}</math>(0.03125)10​ is:

So, <math xmlns="http://www.w3.org/1998/Math/MathML">(0.03125)10(0.03125)_{10}</math>(0.03125)10​ in IEEE 754 single precision is: