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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GATE CS 2017 Official Paper: Shift 2

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Answer 1

Concept:

32-bit floating-point representation of a binary number in IEEE- 754 is

Sign (1 bit)

Exponent (8 bit)

Mantissa bit (23 bits)

Calculation:

Given binary number is

00111110011011010000000000000000

Here, sign bit is 0. So, number is positive.

0

01111100

11011010000000000000000

Exponent bits = E = 01111100 = 124 (in decimal)

Mantissa bits M = 11011010000000000000000

In IEEE-754 format, 32-bit (single precision)

(-1)^s × 1.M × 2^{E – 127}

= (-1)⁰ × 1.1101101 × 2^{124 – 127}

= 1.1101101 × 2^-3

= (1 + 2^-1 + 2^-2 + 2^-4 + 2^-5 + 2^-7) × 2-3

= 0.231 = 2.31 × 10^-1 ≈ 2.27 × 10^-1