GATE CS 2001 Q2.10 — Number Representations (Fixed and Floating Point)

To find the $2$'s complement representation of $(-539)_{10}$ in hexadecimal, we can use either the binary method or the direct hexadecimal subtraction method.

Method 1: Direct Hexadecimal Subtraction

First, convert the magnitude $539_{10}$ to hexadecimal:

Since the options contain $3$ hexadecimal digits (representing $12$ bits), we find the $2$'s complement by subtracting $21\text{B}_{16}$ from $16^3 = 1000_{16}$:

Thus, the representation is $\text{DE5}_{16}$.

Method 2: Binary Conversion

1. Convert $+539_{10}$ to $12$-bit binary: $$539_{10} = 512 + 16 + 8 + 2 + 1 = 0010\ 0001\ 1011_2$$
2. Find the $1$'s complement: $$1101\ 1110\ 0100_2$$
3. Add $1$ to get the $2$'s complement: $$1101\ 1110\ 0101_2$$
4. Convert groups of $4$ bits to hexadecimal:
   • $1101_2 = \text{D}_{16}$
   • $1110_2 = \text{E}_{16}$
   • $0101_2 = 5_{16}$
   This gives $\text{DE5}_{16}$.