GATE CS 2026 Set 1 Q12 — Number Representations (Fixed and Floating Point)

In an $8$-bit signed sign-magnitude representation:

• The most significant bit (MSB) represents the sign ($0$ for positive, $1$ for negative).
• The remaining $7$ bits represent the magnitude.
• The range of representable integers is from $-(2^7 - 1)$ to $+(2^7 - 1)$, which is $[-127, +127]$.

Let us determine the decimal values of $X$ and $Y$:

• For $X = 10110100_2$:
  • Sign bit is $1$ (negative).
  • Magnitude bits: $0110100_2 = 32 + 16 + 4 = 52$.
  • Thus, $X = -52$.
• For $Y = 01001100_2$:
  • Sign bit is $0$ (positive).
  • Magnitude bits: $1001100_2 = 64 + 8 + 4 = 76$.
  • Thus, $Y = +76$.

Now, we evaluate each operation to check if the result falls outside the representable range $[-127, +127]$:

1. $Z = X + Y$:
   $Z = -52 + 76 = +24$
   Since $+24 \in [-127, +127]$, there is no overflow.
2. $Z = X - Y$:
   $Z = -52 - 76 = -128$
   Since $-128 \notin [-127, +127]$, this results in an arithmetic overflow.
3. $Z = -X + Y$:
   $Z = -(-52) + 76 = 52 + 76 = +128$
   Since $+128 \notin [-127, +127]$, this results in an arithmetic overflow.
4. $Z = -X - Y$:
   $Z = -(-52) - 76 = 52 - 76 = -24$
   Since $-24 \in [-127, +127]$, there is no overflow.

Thus, operations B and C result in an arithmetic overflow.