Question 19

1. 16-bit 2's complement representation:

• The range of representable integers is $[-2^{15}, 2^{15} - 1] = [-32768, 32767]$.
• In 2's complement, every bit pattern represents a unique integer (zero has a single representation $0000\dots0$).
• Number of distinct integers: $X = 2^{16} = 65536$.

2. 16-bit sign-magnitude representation:

• The range of representable integers is $[-(2^{15} - 1), 2^{15} - 1] = [-32767, 32767]$.
• Zero has two representations: $+0$ ($0000\dots0$) and $-0$ ($1000\dots0$).
• Number of distinct integers: $Y = 2^{16} - 1 = 65535$.

3. Calculate the difference:

Thus, $X - Y = 1$.