GATE CS 2010 Q8 — Number Representations (Fixed and Floating Point)

The $16$-bit signed integer $P$ is represented in $2$'s complement form as $(F87B)_{16}$.

Converting $(F87B)_{16}$ to its binary representation:

Multiplying a signed integer by $8$ ($2^3$) is equivalent to shifting its binary representation to the left by $3$ bits, filling the least significant bits with $0$s:

Since the discarded MSBs are all $1$s and the new sign bit is also $1$, no overflow occurs. Converting this binary result back to hexadecimal:

• $1100_2 = C_{16}$
• $0011_2 = 3_{16}$
• $1101_2 = D_{16}$
• $1000_2 = 8_{16}$

Thus, the $2$'s complement representation of $8 \times P$ is $(C3D8)_{16}$.