Question

A number is an irrational if and only if its decimal representation is :
A. non terminating.
B. non terminating and repeating .
C. non terminating and non repeating .
D. terminating .

Answer 1

- Hint : In this question we will use the basic concept of the number system . we know that a number ‘s’ is called a irrational number if it cannot be written in the form of $\dfrac{p}{q}$, where p and q are integers and $q \ne 0$ and if that number can be written in the form of $\dfrac{p}{q}$ then it is a rational number.


Complete step-by-step solution -

Let us take an example , x= $0.2\overline {35} $ .over here ,note that 2 does not repeat, but the block 35 repeats. Since two digits are repeating , we multiply x by 100 to get
100x = 23.53535…..
100x = 23.3 + 0.23535….. = 23.3 + x
99x = $\dfrac{{233}}{{10}}$, which gives x = $\dfrac{{233}}{{990}}$ .
You can also check the reverse that $\dfrac{{233}}{{990}}$ = $0.2\overline {35} $ .
So we can say that every number with a non terminating and repeating decimal expansion can be expressed in the form of $\dfrac{p}{q}$ ($q \ne 0$) . Hence ,
The decimal expansion of rational numbers is either terminating or non terminating repeating .
So , the property for irrational numbers, similar to the property stated above for rational numbers , is
The decimal expansion of an irrational number is non terminating and non repeating. Moreover ,a number whose decimal expansion is non terminating and non repeating is irrational .
Recall s = 0.10110111011110…. , it is non terminating and non repeating .
Hence we can say that a number is irrational if and only if its decimal representation is non terminating and non repeating .
Therefore , the correct answer is option (C) .

Note : In such a type of question first we have to remember the basic concept of the number system , rational numbers , irrational numbers etc. and then we have to take an example and by using that example we have to prove the statement given in the question. Through this we will get the correct answer.