Question

A number having non terminating and recurring decimal expansion is.
A. An integer
B. A rational number
C. An irrational number
D. A whole number

Answer 1

Hint: In general we can write a non terminating and recurring decimal expansion (for example $1.5555............$) in the form of a fraction.

Complete step-by-step answer:
A number having non terminating and recurring decimal expansion is a rational number because we can write it as $\dfrac{p}{q}$ form where p and q can be any integer but q can never be equal to zero.

For example let a non- terminating and recurring decimal number is $1.3333...........$
Let it be equal to x.
$\Rightarrow x=\,1.3333.................$ …………………..(i)
On multiplying both side by 10
$\Rightarrow 10x=\,13.3333.............$ ………………………..(ii)
On subtracting equation (ii) and (i)
$\Rightarrow 10x-x=13.3333.........-1.3333.........$
$\Rightarrow 10x-x=\,12$
$\Rightarrow 9x=12$
On dividing both side by 12
$\Rightarrow \dfrac{9x}{9}=\dfrac{12}{9}$
$\Rightarrow x=\dfrac{12}{9}$
We can simplify it by dividing numerator and denominator by 3
$\Rightarrow x=\dfrac{4}{3}$
$\dfrac{4}{3}$ is not an integer so option (a) is not correct.
$\dfrac{4}{3}$ is in the form of $\dfrac{p}{q}$ form where $q=3 \ne 0$. So it is a rational number.
$\dfrac{4}{3}$ is not an irrational number because it is in the form of a rational number.
Also when 4 is divided by 3, we get a remainder =1, so it can not be a whole number.

Hence option B is correct.

Note: In general, non terminating repeating decimal numbers are rational numbers in which a digit or a sequence of digits in the decimal part keeps repeating itself upto infinity. Read the statement very carefully because non terminating and non recurring decimal expansion is an irrational number.