Question:hard
Consider the 8-bit signed integers \( X, Y \) and \( Z \) represented using the sign-magnitude form. The binary representations of \( X \) and \( Y \) are as follows:\[ X: 10110100 \qquad Y: 01001100 \]Which of the following operations to compute \( Z \) result(s) in an arithmetic overflow?
Consider the 8-bit signed integers \( X, Y \) and \( Z \) represented using the sign-magnitude form. The binary representations of \( X \) and \( Y \) are as follows:
\[ X: 10110100 \qquad Y: 01001100 \]
Which of the following operations to compute \( Z \) result(s) in an arithmetic overflow?
Show Hint
Overflow in sign-magnitude addition happens only when two same-signed operands' magnitudes add up to more than 127 (7-bit limit); opposite-signed operands only ever subtract magnitudes and can never overflow.
Updated On: Jul 22, 2026
- \( Z = X + Y \)
- \( Z = X - Y \)
- \( Z = -X + Y \)
- \( Z = -X - Y \)
Show Solution
The Correct Option is B, C
Solution and Explanation
Method: Perform each operation by literally adding or subtracting the 7-bit magnitudes according to the signs, the way a sign-magnitude ALU would, and check whether the resulting magnitude fits in 7 bits (max 127).
Decoded values: X = -52 (magnitude 0110100), Y = +76 (magnitude 1001100).
(A) X+Y: opposite signs, subtract smaller magnitude from larger: 76-52=24, sign follows the larger-magnitude operand (positive). Result +24, fits in 7 bits. No overflow.
(B) X-Y = X+(-Y): both operands negative (X=-52, -Y=-76), same-sign addition: 52+76=128, sign negative. Result -128; magnitude 128 needs 8 bits, cannot be stored in 7 bits: overflow.
(C) -X+Y: -X=+52 and Y=+76, both positive, same-sign addition: 52+76=128, sign positive. Result +128, again needs 8 magnitude bits: overflow.
(D) -X-Y = (-X)+(-Y): -X=+52 and -Y=-76, opposite signs, subtract magnitudes: 76-52=24, sign follows the larger magnitude operand -Y (negative). Result -24, fits comfortably in 7 bits. No overflow.
So overflow happens exactly for (B) and (C), where two same-signed operands' magnitudes sum past 127.
\[ \boxed{\text{Options (B) and (C)}} \]Was this answer helpful?
0
Top Questions on Number Representation - Sign-Magnitude Arithmetic and Overflow
Questions Asked in GATE CS exam
A database has 25,000 fixed-length records of 100 bytes (primary key 15 bytes). The data file is block-aligned so each record is fully contained within a block. The file is indexed by a primary index file, also block-aligned and ordered. Block size is 1024 bytes and a block pointer is 5 bytes. Each index entry stores the block’s anchor key (15 bytes) and a pointer (5 bytes). Binary search on an index file of \(b\) blocks needs \(\lceil \log_2 b \rceil\) block accesses in the worst case. Given a key, the number of block accesses required to identify the data file block that may contain the record, in the worst case, is ____________.
- GATE CS - 2023
- Database Management Systems