Question

The 32-bit IEEE 754 single precision representation of a number is \(0xC2710000\). The number in decimal representation is ______. (rounded off to two decimal places)

Hint:

Split the 32 bits into 1 sign bit, 8 exponent bits, and 23 mantissa bits, then apply the IEEE 754 formula (-1)^S x 1.mantissa x 2^(exponent-127).

Answer 1

Correct Answer: -60.25

Step 1: Write the hexadecimal number in binary.
The given hexadecimal value is \(0xC2710000\). Each hex digit expands to 4 bits:
\[ C=1100,\ 2=0010,\ 7=0111,\ 1=0001,\ 0=0000,\ 0=0000,\ 0=0000,\ 0=0000 \]
Putting these together gives the 32-bit pattern:
\[ 1100\ 0010\ 0111\ 0001\ 0000\ 0000\ 0000\ 0000 \]

Step 2: Split the pattern into sign, exponent, and mantissa.
IEEE 754 single precision uses 1 sign bit, 8 exponent bits, and 23 mantissa bits. Reading the 32 bits in that order:
Sign bit \(=1\)
Exponent bits \(=10000100\)
Mantissa bits \(=11100010000000000000000\)

Step 3: Interpret the sign bit.
A sign bit of \(1\) means the number is negative.

Step 4: Find the true exponent.
The exponent field \(10000100\) in decimal is
\[ 10000100_2 = 128+4 = 132 \]
Single precision uses a bias of \(127\), so the true exponent is
\[ 132-127=5 \]

Step 5: Find the mantissa value.
The mantissa bits \(11100010000000000000000\) represent a fraction added after an implied leading \(1\). Only bits at positions 1, 2, 3 and 7 after the point are \(1\):
\[ 2^{-1}+2^{-2}+2^{-3}+2^{-7} = 0.5+0.25+0.125+0.0078125 = 0.8828125 \]
So the full significand is
\[ 1.8828125 \]

Step 6: Combine sign, exponent, and mantissa.
The value represented is
\[ (-1)^1 \times 1.8828125 \times 2^{5} \]

Step 7: Work out the final number.
\[ 1.8828125 \times 32 = 60.25 \]
Applying the negative sign gives
\[ -60.25 \]

Step 8: Final answer.
\[ \boxed{-60.25} \]