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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  1. 1.45 × 101
  2. 1.45 × 10-1
  3. 2.27 × 10-1
  4. 2.27 × 101

Answer (Detailed Solution Below)

Option 3 : 2.27 × 10-1
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Detailed Solution

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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 × 2E – 127

= (-1)0 × 1.1101101 × 2124 – 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
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