GATE CS 2002 Q1.15 — Number Representations (Fixed and Floating Point)

To find the $2$'s complement representation of the decimal value $-15$, we first determine the minimum number of bits required.

The range of numbers that can be represented in $n$-bit $2$'s complement form is:

• For $n = 4$ bits, the range is $[-8, 7]$. Since $-15$ lies outside this range, it cannot be represented in $4$ bits. Thus, option A is incorrect.
• For $n = 5$ bits, the range is $[-16, 15]$. Since $-15$ lies within this range, we can represent it using $5$ bits.

Now, let's find the $5$-bit $2$'s complement representation of $-15$:

1. Represent the positive value $+15$ in $5$-bit binary: $$+15_{10} = 01111_2$$
2. Find the $1$'s complement by inverting all the bits: $$10000_2$$
3. Add $1$ to the Least Significant Bit (LSB) to get the $2$'s complement: $$10000_2 + 1 = 10001_2$$

Thus, the $2$'s complement representation of $-15$ is $10001$.