Question

Which of the following is a rational number?
(a) \[\sqrt{5}\]
(b) \[\pi \]
(c) \[0.101001000100001....\]
(d) \[0.853853853...\]

Answer 1

Hint: We solve this problem by using the definition of a rational number.
A number that can be represented in the form of \[\dfrac{p}{q}\] where \[q\ne 0\] is called a rational number.
We have the condition that the rational number may have either finite digits after the decimal point or there will be repeating digits after the decimal point. If a number is not in this form then the number is not a rational number.

Complete step by step answer:
We are asked to check the numbers given are rational or not.
Let us check each and every option one by one.
(a) \[\sqrt{5}\]
Here, we can see that the number 5 is not a perfect square.
We know that the square root of a non – perfect square is always an irrational number.
We also know that the square root of a prime number is an irrational number.
Therefore, we can conclude that the number \[\sqrt{5}\] is not a rational number.
(b) \[\pi \]
We know that the value of \[\pi \] is a universal irrational number.
Therefore, we can conclude that the number \[\pi \] is not a rational number.

(c) \[0.101001000100001....\]
We know that the condition that the rational number may have either finite digits after the decimal point or there will be repeating digits after the decimal point.
Here, we can see that the given number is not at all repeating and also not a finite number
Therefore we can conclude that the number \[0.101001000100001....\] is not a rational number.
(d) \[0.853853853...\]
We know that the condition that the rational number may have either finite digits after the decimal point or there will be repeating digits after the decimal point.
Here, we can see that in the given number there are repeating digits that are 853
Therefore, we can conclude that the number \[0.853853853...\] is a rational number.
So, we can conclude that the rational number of all given numbers is \[0.853853853...\]

Therefore, option (d) is the correct answer.


Note:
 Students may do mistake in considering the rational numbers statement to option (c)
We are given that the number in option (c) as \[0.101001000100001....\]
We have the condition that the rational number may have either finite digits after the decimal point or there will be repeating digits after the decimal point
Here, the repeating digits indicate that fixed-term will repeat.
But students may misunderstand that repeating digits as repeating in a certain order. This is not correct that is not repeating digits in a certain order but repeating of the fixed term.