GATE CS 2026 Set 2 Q18 — Number Representations (Fixed and Floating Point)

In a $4$-bit two's complement representation, the range of representable integers is:

An arithmetic overflow occurs if the result of an operation falls outside this range $[-8, +7]$.

First, let's convert the given $4$-bit two's complement numbers to their decimal equivalents:

• $N1 = 1011_2 \rightarrow -8 + 2 + 1 = -5$
• $N2 = 1101_2 \rightarrow -8 + 4 + 1 = -3$
• $N3 = 1010_2 \rightarrow -8 + 2 = -6$
• $N4 = 1001_2 \rightarrow -8 + 1 = -7$

Now, let's evaluate each option:

1. $N1 + N2$:
   $-5 + (-3) = -8$
   Since $-8$ lies within the range $[-8, +7]$, no overflow occurs.
2. $N2 + N3$:
   $-3 + (-6) = -9$
   Since $-9$ is outside the range $[-8, +7]$, an overflow occurs.
   Binary verification: $1101_2 + 1010_2 = 10111_2 \rightarrow 0111_2$ (positive result $+7$ from adding two negative numbers).
3. $N3 - N4$:
   $-6 - (-7) = -6 + 7 = +1$
   Since $+1$ lies within the range $[-8, +7]$, no overflow occurs.
4. $N1 + N4$:
   $-5 + (-7) = -12$
   Since $-12$ is outside the range $[-8, +7]$, an overflow occurs.
   Binary verification: $1011_2 + 1001_2 = 10100_2 \rightarrow 0100_2$ (positive result $+4$ from adding two negative numbers).

Thus, the operations that result in arithmetic overflow are B and D.