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

An alternative way to decode this IEEE 754 pattern is to break the hex digits directly into the three fields. Writing \(0xC2710000\) in binary as \(1100\,0010\,0111\,0001\,0000\,0000\,0000\,0000\), the leading bit is \(1\), telling us the number is negative. The next eight bits, \(10000100\), form the biased exponent; converting to decimal gives \(132\), and after removing the bias of \(127\) used in single precision, the true exponent is \(5\), meaning the binary point shifts right by \(5\) places. The remaining 23 bits, \(11100010000000000000000\), are the fraction, and prefixing the implicit leading \(1\) used for normalized numbers gives the significand \(1.111000100000000000000_2\). Converting this binary fraction to decimal: \(1 + 2^{-1} + 2^{-2} + 2^{-3} + 2^{-7} = 1 + 0.5 + 0.25 + 0.125 + 0.0078125 = 1.8828125\). Multiplying by \(2^5 = 32\) gives \(60.25\), and applying the negative sign from step one gives the final decimal value \(-60.25\).