Given the following binary number in 32-bit (single precision) IEEE-754 format: 00111110011011010000000000000000 The decimal value closest to this floating-point number is

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Solution

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Official answer: C) 2.27 × 10-1

Number Representation and Arithmetic

In IEEE-754 single-precision (32-bit) format, the representation is structured as:

Sign bit ( $S$ ): $1$ bit
Biased Exponent ( $E$ ): $8$ bits
Mantissa/Fraction ( $F$ ): $23$ bits

Dividing the 32-bit string 0 01111100 11011010000000000000000:

$S = 0$ (positive number)
$E = 01111100_2 = 64 + 32 + 16 + 8 + 4 = 124_{10}$
$F = 0.1101101_2$

The actual unbiased exponent $e$ is:

e = E - 127 = 124 - 127 = -3

The significand $M = 1 + F$ is:

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

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

The decimal value is:

\text{Value} = (-1)^S \times M \times 2^e = 1.8515625 \times 2^{-3} = \frac{1.8515625}{8} \approx 0.231445

Among the given options, $0.231445$ is closest to $2.27 \times 10^{-1}$ ( $0.227$ ), which corresponds to (C).

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This Number Representation and Arithmetic previous year question is Q12 of the GATE CSE 2017 Set 2 paper. Find more GATE CSE PYQs on Number Representation and Arithmetic in Digital Logic, or browse all GATE CSE PYQs.

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Number Representation and Arithmetic in GATE CSE

Number Representation and Arithmetic has 33 GATE CSE PYQs across 20 papers from 2012 to 2026. 13 of them are 1-mark MCQs like this one, and GATE CSE 2017 Set 2 has 3.

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Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.