GATE CS 2016 Set 1 Q07 — Number Representations (Fixed and Floating Point)

To find the decimal representation of the given $16$-bit $2$'s complement number, we first observe its Most Significant Bit (MSB):

$$\text{Binary number: } \mathbf{1}111 \quad 1111 \quad 1111 \quad 0101$$

Since the MSB is $1$, the number is negative. We can find its magnitude by taking its $2$'s complement:

1. Find the $1$'s complement (flip all the bits):$$\text{1's complement} = 0000 \quad 0000 \quad 0000 \quad 1010$$
2. Add $1$ to get the $2$'s complement:$$\text{2's complement} = 0000 \quad 0000 \quad 0000 \quad 1011$$

Now, convert this binary magnitude to decimal:

$$1011_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 8 + 0 + 2 + 1 = 11$$

Since the original number was negative, the decimal representation is $-11$.