GATE CS 1999 Q2.17 — Number Representations (Fixed and Floating Point)

To determine which number systems have two representations for zero, let us analyze an $n$-bit binary representation for each case:

1. Sign-Magnitude Representation:
   • The most significant bit (MSB) represents the sign ($0$ for positive, $1$ for negative), and the remaining $n-1$ bits represent the magnitude.
   • Positive zero ($+0$) is represented as: $000\dots0$
   • Negative zero ($-0$) is represented as: $100\dots0$
   • Thus, there are two distinct representations for zero.
2. 1's Complement Representation:
   • Negative numbers are obtained by bitwise inverting the positive representation.
   • Positive zero ($+0$) is represented as: $000\dots0$
   • Negative zero ($-0$) is represented as: $111\dots1$
   • Thus, there are two distinct representations for zero.
3. 2's Complement Representation:
   • Negative numbers are obtained by taking the 1's complement and adding $1$.
   • Positive zero ($+0$) is represented as: $000\dots0$
   • Taking the 2's complement of $000\dots0$ yields $111\dots1 + 1 = 1000\dots0$. Discarding the carry bit beyond $n$ bits leaves $000\dots0$.
   • Thus, there is only one unique representation for zero.

Therefore, both Sign-magnitude (A) and 1's complement (C) have two representations for zero.