GATE CS 2017 Set 2 Q12 — Number Representations (Fixed and Floating Point)

To find the decimal value of the given $32$-bit IEEE-754 single-precision floating-point number, we divide the binary representation into three fields:

1. Sign ($S$): $0 \implies$ Positive number.
2. Biased Exponent ($E$): $$E = 01111100_2 = 64 + 32 + 16 + 8 + 4 = 124_{10}$$ The actual exponent $e$ is: $$e = E - \text{Bias} = 124 - 127 = -3$$
3. Mantissa ($M$): The fraction part is $0.1101101_2$. $$\text{Significand} = 1 + M = 1 + 2^{-1} + 2^{-2} + 2^{-4} + 2^{-5} + 2^{-7}$$ $$1 + 0.5 + 0.25 + 0.0625 + 0.03125 + 0.0078125 = 1.8515625$$

Now, calculate the final value:

Comparing this with the given options:

• Option B: $1.45 \times 10^{-1} = 0.145$
• Option C: $2.27 \times 10^{-1} = 0.227$

The value $0.2314$ is closest to $0.227$, which is $2.27 \times 10^{-1}$.