Question

Find two irrational number between $ 1 $ and $ 2 $

Answer 1

Hint: We are asked to find the irrational number between the numbers. Here first we will find the number between $ 1 $ and $ 2 $ . And then by using the definition of irrational number will find the non-terminating and non-repeating number between the two.

Complete step-by-step answer:
To know the irrational number between $ 1 $ and $ 2 $
First find the middle point –
 $ \Rightarrow 1 < \dfrac{{1 + 2}}{2} < 2 $
Simplify the above expressions –
 $ \Rightarrow 1 < \dfrac{3}{2} < 2 $
As we know that the irrational numbers are all the non-terminating and the non-repeating numbers.
Irrational numbers between the two given fraction are –
 $ 1.71000002,{\text{ 1}}{\text{.500000008,}}.... $
It can be any number with any digits after decimal points.
It can be any number between the two with any number of digits. Hence we have an infinite number of irrational numbers between the two numbers.
Hence, two irrational numbers are - $ 1.71000002{\text{ and 1}}{\text{.500000008}} $
So, the correct answer is “ $ 1.71000002{\text{ and 1}}{\text{.500000008}} $ ”.

Note: Always remember that between any two given numbers there are infinite rational and irrational numbers irrespective of how small or large the difference between the two may be. In irrational numbers the decimal form is in the non-repeating and non-terminating numbers.