GATE CS 2012 Q.7 (CS Q.7): Number Representations and Computer Arithmetic

The decimal value 0.5 in IEEE single precision floating point representation has

fraction bits of $000\dots000$ and exponent value of 0

fraction bits of $000\dots000$ and exponent value of $-1$

fraction bits of $100\dots000$ and exponent value of 0

no exact representation

Official answer and solution

Solution

Convert the decimal number 0.5 to normalized binary scientific notation and compute the IEEE 754 single precision fields.

We represent $0.5_{10}$ in binary scientific notation:

0.5_{10} = 2^{-1} = 1.0_2 \times 2^{-1}

In IEEE 754 single precision (32-bit) floating point format:

Sign bit $S = 0$ (positive).
The normalized form is $(-1)^S \times (1.M) \times 2^{E - 127}$ , where $M$ is the 23-bit fractional mantissa and $E$ is the 8-bit biased exponent.
The unbiased exponent value is $e = -1$ .
The biased exponent stored is $E = e + 127 = -1 + 127 = 126$ .
The significand is $1.0_2$ , so the fraction bits (stored mantissa $M$ ) are all zeros: $000\dots000$ .

Therefore, the representation has fraction bits of $000\dots000$ and an exponent value of $-1$ .
This corresponds to option (B).

Final answer

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