GATE CS 2024 Set 1 Q.13: Number Representations and Computer Arithmetic

Consider a system that uses 5 bits for representing signed integers in 2's complement format. In this system, two integers $A$ and $B$ are represented as $A=01010$ and $B=11010$ . Which one of the following operations will result in either an arithmetic overflow or an arithmetic underflow?

$A + B$

$A - B$

$B - A$

$2 * B$

Official answer and solution

Official answer: B
From: Official GATE 2024 answer key

Solution

Convert the 5-bit 2's complement numbers to decimal and check which result falls outside the representable range $[-16, +15]$ .

In a 5-bit 2's complement system, the representable signed integer range is $[-2^{5-1},\; 2^{5-1}-1] = [-16, +15]$ .

$A = 01010_2 = +10$
$B = 11010_2 = -16 + 8 + 2 = -6$

Evaluating each operation:

(A) $A + B = 10 + (-6) = +4 \in [-16, +15]$ (no overflow)
(B) $A - B = 10 - (-6) = +16 > +15$ , which exceeds the maximum positive value $+15$ and causes an arithmetic overflow.
(C) $B - A = -6 - 10 = -16 \in [-16, +15]$ (representable as $10000_2$ , no underflow)
(D) $2 * B = 2 \times (-6) = -12 \in [-16, +15]$ (no underflow)

Final answer

Option (B)

Common trap

Assuming $-16$ overflows in 5-bit 2's complement and choosing $B - A$ , forgetting that the asymmetric range $[-16, +15]$ includes $-16$ .

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

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 13 of 65
Section: Computer Science and Information Technology

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