GATE CS 2026 Set 2 Q24 — Number Representations (Fixed and Floating Point)

To find the decimal representation of the given 32-bit IEEE 754 single-precision floating-point number, we first convert the hexadecimal representation $0\text{xC}2710000$ into binary:

Now, we extract the three components of the IEEE 754 representation:

1. Sign Bit ($S$): $$S = 1 \implies \text{The number is negative.}$$
2. Biased Exponent ($E$): $$E = 10000100_2 = 128 + 4 = 132_{10}$$ The unbiased exponent ($e$) is calculated as: $$e = E - 127 = 132 - 127 = 5$$
3. Mantissa ($F$): $$F = 0.1110001_2$$ The significand is $(1 + F) = 1.1110001_2$.

The value of the floating-point number is given by:

Multiplying by $2^5$ shifts the binary point $5$ places to the right:

Converting this binary value to decimal:

• Integer part: $111100_2 = 32 + 16 + 8 + 4 = 60_{10}$
• Fractional part: $0.01_2 = 0 \times 2^{-1} + 1 \times 2^{-2} = 0.25_{10}$

Thus, the decimal representation of the number is $-60.25$.