GATE CS 2005 Q16 — Number Representations (Fixed and Floating Point)

In an $n$-bit 2's complement representation, the most significant bit (MSB) is the sign bit, which has a negative weight of $-2^{n-1}$. The remaining $n-1$ bits have positive weights.

To find the range of representable integers:

• Minimum Value: This occurs when the MSB is $1$ and all other $n-1$ bits are $0$.
  $$\text{Minimum} = -1 \cdot 2^{n-1} + 0 \cdot 2^{n-2} + \dots + 0 \cdot 2^0 = -2^{n-1}$$
• Maximum Value: This occurs when the MSB is $0$ and all other $n-1$ bits are $1$.
  $$\text{Maximum} = 0 \cdot 2^{n-1} + 1 \cdot 2^{n-2} + \dots + 1 \cdot 2^0 = 2^{n-1} - 1$$

Therefore, the range of integers is from $-2^{n-1}$ to $(2^{n-1} - 1)$.

Example (for $n = 4$):

• Minimum value: $-2^{4-1} = -8$ (represented as $1000_2$)
• Maximum value: $2^{4-1} - 1 = 7$ (represented as $0111_2$)
• Range: $-8$ to $+7$

Hence, the correct option is A.