GATE CS 2024 Set 1 — Question 13
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Digital Logic → Number Systems & Arithmetic → Binary Arithmetic & Overflow
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Question
Consider a system that uses 5 bits for representing signed integers in 2’s complement format. In this system, two integers and are represented as and . Which one of the following operations will result in either an arithmetic overflow or an arithmetic underflow?
Correct answer
(B) A - B
Solution
In a 5-bit 2's complement system, the range of representable integers is from to , i.e., from to .Given:
. Since the MSB is 0, it is positive. Value .
. Since the MSB is 1, it is negative. To find the magnitude, we take the 2's complement: invert bits () and add 1 (), which is 6. So, .Let's evaluate the options:(A) . This is within the range . No overflow.(B) . The maximum positive value representable is 15. Since , an arithmetic overflow occurs.(C) . This is within the range . No overflow ( is represented as ).(D) . This is within the range . No overflow.Thus, the operation results in an overflow.
. Since the MSB is 0, it is positive. Value .
. Since the MSB is 1, it is negative. To find the magnitude, we take the 2's complement: invert bits () and add 1 (), which is 6. So, .Let's evaluate the options:(A) . This is within the range . No overflow.(B) . The maximum positive value representable is 15. Since , an arithmetic overflow occurs.(C) . This is within the range . No overflow ( is represented as ).(D) . This is within the range . No overflow.Thus, the operation results in an overflow.
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