GATE CS 2017 Set 2 Q.12 (CS Q.12): Number Representations and Computer Arithmetic

Given the following binary number in 32-bit (single precision) IEEE-754 format:

00111110011011010000000000000000

The decimal value closest to this floating-point number is

$1.45 \times 10^1$

$1.45 \times 10^{-1}$

$2.27 \times 10^{-1}$

$2.27 \times 10^1$

Official answer and solution

Solution

Decode the 32-bit IEEE-754 single-precision floating-point bit pattern into sign, exponent, and mantissa components.

Partition the 32-bit pattern 0011 1110 0110 1101 0000 0000 0000 0000:

Sign bit ( $S$ ): 0 (positive)
Biased exponent ( $E$ , 8 bits): 01111100 in binary: $E = 64 + 32 + 16 + 8 + 4 = 124_{10}$ Unbiased exponent $e = E - 127 = 124 - 127 = -3$ .
Mantissa ( $M$ , 23 bits): 11011010000000000000000

Compute the normalized significand $(1.M)_2$ :

1 + 2^{-1} + 2^{-2} + 2^{-4} + 2^{-5} + 2^{-7}

= 1 + 0.5 + 0.25 + 0.0625 + 0.03125 + 0.0078125 = 1.8515625

Now multiply by $2^e = 2^{-3} = 0.125$ :

\text{Value} = 1.8515625 \times 2^{-3} = \frac{1.8515625}{8} \approx 0.2314453125 = 2.31 \times 10^{-1}

Among the given choices, $2.27 \times 10^{-1}$ is the closest decimal value.

Final answer

Common trap

Forgetting the implicit leading 1 in the normalized significand for non-zero biased exponents.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.