TIFR CSE 2019 | Part B | Question: 1

Question

Which of the following decimal numbers can be exactly represented in binary notation with a finite number of bits ?

• A. $0.1$
• B. $0.2$
• C. $0.4$
• D. $0.5$
• E. All the above

Comments

0.5

Answer 1

Binary representation of $0.5$ is $0.1_2$ and it is the only fraction among the given options that is terminating. So it is the only fraction that can be represented exactly.

All the other options have non-terminating fraction part and cannot be represented exactly with any finite number of bits.

Comments

how 0.1 is terminating? I can only see 0.5 is terminating fraction, so the answer should be 0.5

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Option D is Correct.

Only 0.5 is terminating, not 0.1.

For 0.5 – 

0.5 + 0.5 = 1.0 →1

0.5 In Binary = 0.1

For 0.1 – 

0.1 + 0.1 = 0.2 →0

0.2 + 0.2 = 0.4 →0

0.4 + 0.4 = 0.8 →0

0.8 + 0.8 = 1.6 →1

0.6 + 0.6 = 1.2 →1

0.2 + 0.2 = 0.4 →0 (Repeating Again)

All other options have non-terminating fraction part and cannot be represented exactly with finite number of bits.

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Option D is Correct.

Only 0.5 is terminating, not 0.1.

For 0.5 – 

0.5 + 0.5 = 1.0 →1

0.5 In Binary = 0.1

For 0.1 – 

0.1 + 0.1 = 0.2 →0

0.2 + 0.2 = 0.4 →0

0.4 + 0.4 = 0.8 →0

0.8 + 0.8 = 1.6 →1

0.6 + 0.6 = 1.2 →1

0.2 + 0.2 = 0.4 →0 (Repeating Again)

All the others options are non terminating so we can not express them exactly with finite number of bits

Answer 2

The rational numbers that can be represented exactly in binary with a finite sequence of digits are precisely those of the form x/2^y for x,y be some integers.
Option D only satisfies this.

This is how i arrived –
9/8= 1.001 in binary
9/2^3 = 9*2^-3= 1001* 2^-3= 1.001