Converting IEEE 754 Single Precision Floating Point Numbers

Question:

Convert the following binary format IEEE 754 single precision floating point numbers into decimal numbers. Show every step in your calculation to get full marks.

• a. Ob 0 0111 1100 100 1100 1100 1100 1100 1101
• b. Ob 1 0111 1110 001 1001 1001 1001 1001 1010

Answer:

To convert the given binary format IEEE 754 single precision floating point numbers into decimal numbers, we need to follow certain steps. First, we need to determine the sign, exponent, and significant of the number. Let's start with number a:

For number a:

1. Sign bit: 0 (which indicates a positive number)
2. Exponent: 0111 1100 (which equals 124 in decimal after subtracting the bias value)
3. Significant: 1.1001100110011001101 (note that there is an implicit leading 1)

By combining the sign, exponent, and significant, we get the number 1.1001100110011001101 x 2^(124).

Converting this binary number to decimal, we get 1.1001100110011001101 x 2^(124) ≈ 3198.79

Therefore, the decimal representation of number a is approximately 3198.79.

For number b:

1. Sign bit: 1 (which indicates a negative number)
2. Exponent: 0111 1110 (which equals 126 in decimal after subtracting the bias value)
3. Significant: 1.10011001100110011010 (again, note the implicit leading 1)

Combining the sign, exponent, and significant, we get the number -1.10011001100110011010 x 2^(126).

Converting this binary number to decimal, we get -1.10011001100110011010 x 2^(126) ≈ -342.65

Therefore, the decimal representation of number b is approximately -342.65.

Explanation:

IEEE 754 Single Precision Floating Point Format: IEEE 754 is a widely used standard for representing floating-point numbers in computers. It defines the format for 32-bit single-precision floating-point numbers, which includes a sign bit, exponent, and significant.

Steps to Convert Binary to Decimal:

1. Determine the sign bit: 0 for positive, 1 for negative.

2. Calculate the exponent: Convert the exponent bits to decimal and subtract the bias value (127 for single-precision).

3. Determine the significant: Add an implicit leading 1 to the fraction bits.

4. Combine the sign, exponent, and significant to get the binary number.

5. Convert the binary number to decimal using the formula: (-1)^S x 1.M x 2^(E - bias).

Decimal Representation: The decimal representation of a floating-point number gives its value in base 10, which is easier to understand and work with for most applications. By converting the binary IEEE 754 single precision floating point numbers into decimal, we can interpret and use these numbers effectively in various computations and analyses.