GATE CS 2022 Q8 — Number Representations (Fixed and Floating Point)

For a $4$-bit register storing numbers in $2$’s complement form, the range of representable integers is:

An arithmetic overflow occurs during addition if the mathematical sum of the two numbers falls outside this representable range $[-8, 7]$. Alternatively, overflow occurs when adding two numbers of the same sign yields a result with the opposite sign.

Let us evaluate each option by converting the $2$’s complement binary numbers to their decimal equivalents:

• Option A:
  • $R1 = 1011_2 = -5$
  • $R2 = 1110_2 = -2$
  • $\text{Sum} = -5 + (-2) = -7$ (within range $[-8, 7]$, no overflow)
• Option B:
  • $R1 = 1100_2 = -4$
  • $R2 = 1010_2 = -6$
  • $\text{Sum} = -4 + (-6) = -10$ (outside range $[-8, 7]$, overflow occurs)
• Option C:
  • $R1 = 0011_2 = +3$
  • $R2 = 0100_2 = +4$
  • $\text{Sum} = 3 + 4 = 7$ (within range $[-8, 7]$, no overflow)
• Option D:
  • $R1 = 1001_2 = -7$
  • $R2 = 1111_2 = -1$
  • $\text{Sum} = -7 + (-1) = -8$ (within range $[-8, 7]$, no overflow)

Thus, only the values in Option B result in an arithmetic overflow.