GATE CS 2026 Set 2 Q.28: Number Representations and Computer Arithmetic

In a system, numbers are represented using 4-bit two's complement form. Consider four numbers $N_1 = 1011$ , $N_2 = 1101$ , $N_3 = 1010$ and $N_4 = 1001$ in the system. Which of the following operations will result in arithmetic overflow?

$N_1 + N_2$

$N_2 + N_3$

$N_3 - N_4$

$N_1 + N_4$

Official answer and solution

Official answer: B and D
From: Official GATE 2026 answer key

Solution

Convert each 4-bit two's complement number to decimal and check which results fall outside the representable range $[-8, +7]$ .

The representable range of 4-bit two's complement integers is $[-8, +7]$ . Converting the four operands to decimal:

$N_1 = 1011_2 = -8 + 3 = -5$
$N_2 = 1101_2 = -8 + 5 = -3$
$N_3 = 1010_2 = -8 + 2 = -6$
$N_4 = 1001_2 = -8 + 1 = -7$

Checking each operation:

(A) $N_1 + N_2 = -5 + (-3) = -8 \in [-8, 7]$ (no overflow).
(B) $N_2 + N_3 = -3 + (-6) = -9 < -8$ (overflow).
(C) $N_3 - N_4 = -6 - (-7) = +1 \in [-8, 7]$ (no overflow).
(D) $N_1 + N_4 = -5 + (-7) = -12 < -8$ (overflow).

Final answer

Options (B), (D)

Common trap

Treating $-8$ ( $1000_2$ ) in option (A) as an overflow when $-8$ is valid in 4-bit two's complement.

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

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 2 (CS2)
Question: 28 of 65
Section: Computer Science and Information Technology

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