GATE CSE 2003 | Question: 43

Question

The following is a scheme for floating point number representation using $16$ bits.

Let $s, e,$ and $m$ be the numbers represented in binary in the sign, exponent, and mantissa fields respectively. Then the floating point number represented is:

$$\begin{cases}(-1)^s \left(1+m \times 2^{-9}\right) 2^{e-31}, & \text{ if the exponent } \neq 111111 \\  0, & \text{ otherwise} \end{cases}$$

What is the maximum difference between two successive real numbers representable in this system?

• A. $2^{-40}$
• B. $2^{-9}$
• C. $2^{22}$
• D. $2^{31}$

Comments

First of all this is one of the best question.

Let us do our analysis in decimal system -

Let we us have 1-Digit Mantissa and 2-Digit Exponent then Mantissa range $-9$ to $9$, and 2-Digit Exponent range $0$ to $100$.

Now possible numbers are $-900, -800 ...... -1, 0, 1, 2 ....10, 20, 30,...800, 900 $

It could be observed clearly that max difference between two consecutive number will be max(100) when exponent value is max. Similarly min difference between two consecutive number will be minimum(1) when exponent value is least.

Same principle could be used in the mentioned question.

ping @Puja Mishra, @Shivam Chauhan, @Manu Thakur, @Anu007,  @Hemant Parihar, @ sushmita,  @VS  @Shweta Nair @Krish__,  @Ashwin Kulkarni @reena_kandari  and @srestha ji.

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Gate 2021 question:

https://gateoverflow.in/357536/gate-cse-2021-set-2-question-4

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in floating point representation as we move AWAY from zero, numbers are sparsely spaced. This means the largest difference between 2 successive numbers will be at the VERY ends. 
For better explanation →https://gateoverflow.in/934/gate-cse-2003-question-43?show=410241#a410241

Answer 1

for same exponent any two successive real numbers can differ maximum by magnitude of the least significant matissa. Here the least representable matissa would be 2^-9. the max difference will hence occur for the highest exponent cases. Here 31 (62-31 as explained in other answers)... so the magnitude of max difference will be 2^-9 *2^31 = 2^22