GATE CS 2018 Q33 — Number Representations (Fixed and Floating Point)

The given unsigned 8-bit fixed-point binary representation is:

$$\underbrace{b_7 \quad b_6 \quad b_5 \quad b_4 \quad b_3}_{\text{5 integer bits}} \quad \cdot \quad \underbrace{b_2 \quad b_1 \quad b_0}_{\text{3 fractional bits}}$$

1. Range and Resolution Analysis

• Integer Part: Represented by 5 bits ($b_7$ to $b_3$). Since it is unsigned, the range of the integer part is from $0$ to $2^5 - 1 = 31$.
• Fractional Part: Represented by 3 bits ($b_2$ to $b_0$). The resolution (step size) is $2^{-3} = 0.125$. Thus, any representable fractional part must be an integer multiple of $0.125$.

The only exactly representable fractional parts are:

$$0.000, \; 0.125, \; 0.250, \; 0.375, \; 0.500, \; 0.625, \; 0.750, \; 0.875$$

2. Evaluating the Options

1. $31.500$:
   • Integer part $31 \le 31$ (representable as $11111_2$).
   • Fractional part $0.500$ is a multiple of $0.125$ (representable as $100_2$).
   • Can be represented exactly.
2. $0.875$:
   • Integer part $0 \le 31$ (representable as $00000_2$).
   • Fractional part $0.875$ is a multiple of $0.125$ (representable as $111_2$).
   • Can be represented exactly.
3. $12.100$:
   • Fractional part $0.100$ is not a multiple of $0.125$.
   • Cannot be represented exactly.
4. $3.001$:
   • Fractional part $0.001$ is not a multiple of $0.125$.
   • Cannot be represented exactly.

Thus, only iii and iv cannot be represented exactly.