GATE CS 2018 Q.43 (CS Q.33): Number Representations and Computer Arithmetic

Consider the unsigned 8-bit fixed point binary number representation below,

b_7 \; b_6 \; b_5 \; b_4 \; b_3 \, . \, b_2 \; b_1 \; b_0

where the position of the binary point is between $b_3$ and $b_2$ . Assume $b_7$ is the most significant bit. Some of the decimal numbers listed below cannot be represented exactly in the above representation:
(i) 31.500
(ii) 0.875
(iii) 12.100
(iv) 3.001

Which one of the following statements is true?

None of (i), (ii), (iii), (iv) can be exactly represented

Only (ii) cannot be exactly represented

Only (iii) and (iv) cannot be exactly represented

Only (i) and (ii) cannot be exactly represented

Official answer and solution

Official answer: C
From: Official GATE 2018 answer key

Solution

Examine whether the integer and fractional parts can be represented exactly using 5 integer bits and 3 fractional bits.

In the format $b_7 b_6 b_5 b_4 b_3 . b_2 b_1 b_0$ :

The integer part has 5 bits ( $b_7$ to $b_3$ ), representing integers from 0 to $2^5 - 1 = 31$ .
The fractional part has 3 bits ( $b_2, b_1, b_0$ ), representing fractions of the form $\frac{k}{8}$ where $k \in \{0, 1, \dots, 7\}$ , i.e., multiples of $0.125$ .

Evaluate each number:

(i) $31.500$ : Integer 31 fits in 5 bits ( $11111_2$ ). Fractional part $0.5 = 4/8 = 0.100_2$ fits in 3 bits. Represented exactly.
(ii) $0.875$ : Integer 0 fits. Fractional part $0.875 = 7/8 = 0.111_2$ fits in 3 bits. Represented exactly.
(iii) $12.100$ : Integer 12 fits. Fractional part $0.1 = 1/10$ . Since 10 has prime factor 5, $0.1$ has an infinite repeating binary expansion ( $0.000110011\dots_2$ ) and cannot be represented exactly in any finite number of binary fractional bits.
(iv) $3.001$ : Integer 3 fits. Fractional part $0.001 = 1/1000$ cannot be written as $k/8$ . Cannot be represented exactly.

Thus, only (iii) and (iv) cannot be represented exactly, matching option C.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2018 Computer Science and Information Technology (CS)
Question: 43 of 65, printed as CS Q.33
Section: Computer Science and Information Technology

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