GATE CS 1998 Q2.20 — Number Representations (Fixed and Floating Point)

To find the percentage reduction in storage space, we compare the storage requirements of the attribute in character form versus integer form.

1. Storage in Character Form

A signed four-digit number (e.g., -9999 or +1234) consists of:

• $1$ sign character (+ or -)
• $4$ digit characters

This requires a total of $5$ characters. Since each character typically takes $1$ byte of storage, the total space required in character form is:

$$\text{Storage}_{\text{char}} = 5 \text{ bytes}$$

2. Storage in Integer Form

The range of signed four-digit numbers is from $-9999$ to $+9999$. The total number of distinct values in this range is:

$$9999 - (-9999) + 1 = 19999 \text{ values}$$

To represent $19999$ distinct values, the minimum number of bits required is:

$$\lceil \log_2(19999) \rceil = 15 \text{ bits}$$

In standard computer systems, memory is allocated in bytes. The smallest standard integer size that can accommodate $15$ bits is a $2$-byte ($16$-bit) short integer (which can store values from $-32768$ to $+32767$). Thus, the space required in integer form is:

$$\text{Storage}_{\text{int}} = 2 \text{ bytes}$$

3. Percentage Reduction

The reduction in storage space is:

$$\text{Percentage Reduction} = \frac{\text{Storage}_{\text{char}} - \text{Storage}_{\text{int}}}{\text{Storage}_{\text{char}}} \times 100\%$$

$$\text{Percentage Reduction} = \frac{5 - 2}{5} \times 100\% = 60\%$$

Hence, the percentage of reduction in storage space is 60%.