Question

If a number has non-terminating and non-recurring decimal expansion, then it is.
A. A rational number
B. A natural number
C. An irrational number
D. An integer

Answer 1

Hint:First we have to write the definition of each of these numbers in the given option with some good examples and try to show that these examples explain the property of numbers and then we will try to eliminate our options to find the correct answer.

Complete step-by-step answer:
Let’s first check the definition of natural number.
Natural number: Any number which is greater than equal to 1 and it doesn’t have decimal expansion. Ex- 1, 2, 3, …….
So, natural numbers are not the correct answer.
Now let’s check the integers.
Integer: A number of set of positive whole numbers $\left\{ 1,2,3,4,....... \right\}\text{ }$, negative whole numbers$\left\{ -1,-2,-3,........ \right\}$ and zero {0}.
So, integer doesn’t contain decimal expansion hence it is also incorrect.
Now let’s check the rational numbers.
Rational number: A rational number is a number that can be expressed as the quotient or fraction $\dfrac{p}{q}$ of two integers, a numerator p and a non-zero denominator q. It has repeating or terminating decimal expansion. Ex- $\dfrac{7}{5},3,\dfrac{2}{9}$ .
So, a rational number is also an incorrect option.
Now let’s check irrational numbers.
Irrational number: A number having non-terminating and non-recurring decimal expansion is an Irrational number. And it cannot be expressed as a ratio of two integers. Ex- $\pi ,\sqrt{2},\sqrt{3},etc.$
The value of $\pi =3.1415926535897932384626433832............$
Therefore, it is clear that it is non-terminating and non-recurring decimal expansion so it is the correct answer.
Hence, option (C) is the correct answer.

Note: There is another method to solve this problem by using the definition of rational numbers. First we have to prove the contrapositive. If x has a repeating decimal expansion (this includes terminating decimal expansions), then x is rational. And another definition of irrational number that any number which is not rational is irrational.