GATE CS 2012 Q7 — Number Representations (Fixed and Floating Point)

To find the IEEE 754 single-precision representation of the decimal number $0.5$:

1. Convert to binary: $$0.5_{10} = 0.1_2$$
2. Normalize the binary representation:

   Any normalized non-zero floating-point number is represented in the form:

   $$(-1)^s \times (1.M) \times 2^E$$

   Expressing $0.1_2$ in this format:

   $$0.1_2 = 1.0_2 \times 2^{-1}$$
3. Identify the components:
   • Sign ($s$): $0$ (since the number is positive).
   • Fraction/Mantissa ($M$): The bits after the binary point in $1.0_2$, which are all zeros: $000\dots000$ (since the leading $1$ is implicit and not stored).
   • Exponent ($E$): The actual exponent value is $-1$.

Therefore, the representation has fraction bits of $000\dots000$ and an exponent value of $-1$.