GATE CS 2025 Set 1 Q.25: Number Representations and Computer Arithmetic

The number $-6$ can be represented as $1010$ in 4-bit 2's complement representation. Which of the following is/are CORRECT 2's complement representation(s) of $-6$ ?

$1000\; 1010$ in 8-bits

$1111\; 1010$ in 8-bits

$1000\; 0000\; 0000\; 1010$ in 16-bits

$1111\; 1111\; 1111\; 1010$ in 16-bits

Official answer and solution

Official answer: B and D
From: Official GATE 2025 answer key

Solution

Apply sign extension by replicating the most significant (sign) bit 1 into all higher-order bit positions.

In 2's complement representation, extending a negative number (1010 for $-6$ ) to a wider bit width requires sign extension (replicating the sign bit 1 in all new left positions):

8-bit representation: 1111 1010 ( $-128 + 64 + 32 + 16 + 8 + 2 = -6$ ) $\implies$ (B) is CORRECT.
16-bit representation: 1111 1111 1111 1010 $\implies$ (D) is CORRECT.

Final answer

Options (B), (D)

Common trap

Confusing 2's complement sign extension with sign-magnitude representation in (A) and (C).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 25 of 65
Section: Computer Science and Information Technology

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