GATE CS 2008 Q4 — Number Representations (Fixed and Floating Point)

In the standard IEEE 754 single-precision (32-bit) floating-point representation, the bits are distributed as follows:

• Sign bit ($S$): 1 bit (MSB)
• Biased Exponent ($E$): 8 bits
• Fraction/Mantissa ($F$): 23 bits

The hexadecimal value $0\text{x}00000000$ in binary is represented by 32 zeros:

$$\underbrace{0}_{S} \quad \underbrace{00000000}_{E} \quad \underbrace{00000000000000000000000}_{F}$$

Here, we have:

• $S = 0$ (indicates a positive number)
• $E = 0$
• $F = 0$

According to the IEEE 754 standard:

• When the biased exponent $E = 0$ and the fraction $F = 0$, the representation corresponds to the special value $0$ (specifically $+0$ if $S = 0$, and $-0$ if $S = 1$).
• Normalized numbers must have a biased exponent in the range $1 \le E \le 254$. Since $E = 0$, this is not a normalized value.

Thus, $0\text{x}00000000$ represents the special value $+0$.