GATE CS 2024 Set 1 Q3 — Number Representations (Fixed and Floating Point)

In a $5$-bit $2$'s complement representation, the range of representable signed integers is:

First, let's determine the decimal values of the given binary representations $A$ and $B$:

• $A = 01010_2$: Since the MSB (sign bit) is $0$, it is a positive number. $$A = (0 \times -16) + (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) = 10$$
• $B = 11010_2$: Since the MSB is $1$, it is a negative number. $$B = -16 + (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) = -6$$

Now, let's evaluate each option to see if the result falls outside the representable range $[-16, 15]$:

1. $A + B$: $$10 + (-6) = 4$$ Since $4 \in [-16, 15]$, no overflow or underflow occurs.
2. $A - B$: $$10 - (-6) = 16$$ Since $16 \notin [-16, 15]$, this operation results in an arithmetic overflow (positive overflow).
3. $B - A$: $$-6 - 10 = -16$$ Since $-16 \in [-16, 15]$, no overflow or underflow occurs.
4. $2 * B$: $$2 \times (-6) = -12$$ Since $-12 \in [-16, 15]$, no overflow or underflow occurs.

Thus, only the operation $A - B$ results in an arithmetic overflow.