GATE CS 2026 Set 1 — Question 22
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Digital Logic → Number Systems & Arithmetic → Binary Arithmetic & Overflow
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Question
Consider the 8-bit signed integers and represented using the sign-magnitude form. The binary representations of and are as follows:Which of the following operations to compute result(s) in an arithmetic overflow?
Correct answer
(B) Z = X - Y; (C) Z = -X + Y
Solution
In 8-bit sign-magnitude representation, the Most Significant Bit (MSB) is the sign bit ( for positive, for negative), and the remaining 7 bits represent the magnitude. The range of representable values is to , i.e., .Given:
. Sign bit is (negative). Magnitude is . So, .
. Sign bit is (positive). Magnitude is . So, .Let's evaluate the options:
(A) . This is within . No overflow.
(B) . The minimum representable value is . Since , this causes an overflow.
(C) . The maximum representable value is . Since , this causes an overflow.
(D) . This is within . No overflow.Thus, operations (B) and (C) result in arithmetic overflow.
. Sign bit is (negative). Magnitude is . So, .
. Sign bit is (positive). Magnitude is . So, .Let's evaluate the options:
(A) . This is within . No overflow.
(B) . The minimum representable value is . Since , this causes an overflow.
(C) . The maximum representable value is . Since , this causes an overflow.
(D) . This is within . No overflow.Thus, operations (B) and (C) result in arithmetic overflow.
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