GATE CS 2017 Set 1 Q.7 (CS Q.7): Number Representations and Computer Arithmetic

The $n$ -bit fixed-point representation of an unsigned real number $X$ uses $f$ bits for the fraction part. Let $i = n - f$ . The range of decimal values for $X$ in this representation is

$2^{-f}$ to $2^i$

$2^{-f}$ to $(2^i - 2^{-f})$

$0$ to $2^i$

$0$ to $(2^i - 2^{-f})$

Official answer and solution

Official answer: D
From: Official GATE 2017 answer key

Solution

Determine the minimum and maximum representable decimal values for an $n$ -bit unsigned fixed-point representation with $i$ integer bits and $f$ fractional bits.

In an $n$ -bit unsigned fixed-point format with $i$ integer bits and $f$ fractional bits ( $n = i + f$ ), the weight of each bit position $k$ from $k = -f$ to $i - 1$ is $2^k$ .

The minimum value occurs when all $n$ bits are 0:

X_{\min} = 0

The maximum value occurs when all $n$ bits are 1:

X_{\max} = \sum_{k=-f}^{i-1} 2^k = 2^{-f} \sum_{j=0}^{i+f-1} 2^j = 2^{-f} (2^{i+f} - 1) = 2^i - 2^{-f}

Thus, the range of decimal values for $X$ is $0$ to $(2^i - 2^{-f})$ .

Final answer

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

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 1
Question: 7 of 65, printed as CS Q.7
Section: Computer Science and Information Technology

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