Question

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. \( A + B \)
• B. \( A - B \)
• C. \( B - A \)
• D. \( 2 * B \)

Hint:

In a 2's complement system, arithmetic overflow occurs if the result exceeds the range ([-2^{n-1}, 2^{n-1} - 1]), where ( n ) is the number of bits.

Answer 1

The Correct Option is B

Step 1: Understand 5-bit signed integer representation. The range of a 5-bit signed integer in 2's complement is: \[ [-16, 15]. \]
Convert \( A \) and \( B \) to decimal:
\( A = 01010 \) (positive): \( A = 10 \).
\( B = 11010 \) (negative): \( B = -6 \). Step 2: Perform the operations. 1. \( A + B \): \[ 10 + (-6) = 4 \quad (\text{within range, no overflow/underflow}). \]
2. \( A - B \): \[ 10 - (-6) = 10 + 6 = 16 \quad (\text{exceeds the range, causes overflow}). \]
3. \( B - A \): \[ -6 - 10 = -16 \quad (\text{within range, no overflow/underflow}). \]
4. \( 2 * B \): \[ 2 \times (-6) = -12 \quad (\text{within range, no overflow/underflow}). \] Step 3: Identify the problematic operation. The operation \( A - B \) causes an overflow as the result exceeds the maximum representable value \( 15 \). Final Answer: \[ \boxed{A - B} \]