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 sign, 8-bit biased exponent, and 23-bit mantissa; subtract 127 from the exponent and add the implicit leading 1 to the mantissa before scaling.

Answer 1

Correct Answer: -60.25

Step 1: Convert the hexadecimal value \(0xC2710000\) to binary: \(1100\ 0010\ 0111\ 0001\ 0000\ 0000\ 0000\ 0000\).
Step 2: In IEEE 754 single precision, the first bit is the sign, the next 8 bits are the exponent, and the remaining 23 bits are the mantissa. Here sign \(= 1\) (negative), exponent bits \(= 10000100\), mantissa bits \(= 11100010000000000000000\).
Step 3: Convert the exponent bits \(10000100\) to decimal: \(128 + 4 = 132\).
Step 4: Subtract the bias of \(127\) to get the actual exponent: \(132 - 127 = 5\).
Step 5: The mantissa bits \(111\ 0001\ 0\ldots0\) give a fractional value of \(0.5 + 0.25 + 0.125 + 0.0078125 = 0.8828125\). With the implicit leading \(1\), the significand is \(1.8828125\).
Step 6: The value is \((-1)^1 \times 1.8828125 \times 2^5 = -1.8828125 \times 32 = -60.25\).
Final Answer: \[\boxed{-60.25}\]