GATE CS 2016 Set 2 Q09 — Number Representations (Fixed and Floating Point)

To find the value of $X - Y$, we analyze the number of distinct integers representable in both systems using $16$ bits ($n = 16$):

• 2's Complement Representation ($X$):
  In $n$-bit 2's complement, every unique bit pattern represents a unique integer because there is only a single representation for zero ($0$).
  Thus, the number of distinct integers is:
  $$X = 2^{16} = 65536$$
• Sign-Magnitude Representation ($Y$):
  In $n$-bit sign-magnitude, there are two distinct representations for zero: $+0$ and $-0$.
  Because of this redundancy, the number of distinct integers is reduced by $1$:
  $$Y = 2^{16} - 1 = 65535$$

Therefore, the difference is: