GATE CS 2019 Q4 — Number Representations (Fixed and Floating Point)

To find the $16$-bit $2$’s complement representation of the decimal number $-28$, we can follow these steps:

1. Represent $+28$ in binary:

   The binary representation of $28$ is $16 + 8 + 4 = (11100)_2$.

   Extending this to $16$ bits by padding with leading zeros:

   $$\text{+28} = 0000 \: 0000 \: 0001 \: 1100$$
2. Find the $1$’s complement:

   Invert all the bits (change $0$ to $1$ and $1$ to $0$):

   $$\text{1's complement} = 1111 \: 1111 \: 1110 \: 0011$$
3. Find the $2$’s complement:

   Add $1$ to the $1$’s complement:

   $$\begin{array}{r}1111 \: 1111 \: 1110 \: 0011 \\+ \quad \quad \quad \quad \quad \quad \quad \quad 1 \\ \hline 1111 \: 1111 \: 1110 \: 0100\end{array}$$

Thus, the $16$-bit $2$’s complement representation of $-28$ is $1111 \: 1111 \: 1110 \: 0100$, which corresponds to Option C.