GATE CS 1991 Q01-iii — Number Representations (Fixed and Floating Point)

The given decimal expression is:

$$N = 9 \times 16^3 + 7 \times 16^2 + 5 \times 16^1 + 3 \times 16^0$$

This expression represents a number in base-16 (hexadecimal) notation. Writing it directly in hexadecimal form:

$$N = (9753)_{16}$$

To find the number of $1$s in its unsigned binary representation, we can convert each hexadecimal digit into its equivalent $4$-bit binary representation:

• $9_{16} = 1001_2$ (contains $2$ ones)
• $7_{16} = 0111_2$ (contains $3$ ones)
• $5_{16} = 0101_2$ (contains $2$ ones)
• $3_{16} = 0011_2$ (contains $2$ ones)

Combining these, the binary representation of the number is:

$$N = (1001\ 0111\ 0101\ 0011)_2$$

Now, we sum the number of $1$s in each digit:

$$\text{Total number of } 1\text{'s} = 2 + 3 + 2 + 2 = 9$$