Prove that between any two rational numbers, there is an irrational number.

We will find an irrational number between two arbitrary rational numbers.

Answer: Between any two rational numbers, there is an irrational number is proved.

Follow the explanation given below.

Explanation:

Assume a and b to be two arbitrary rational numbers such that b > a. We claim c = a + (b - a)/√2 is an irrational number that lies between a and b.

1/√2 is an irrational number that lies between 0 and 1. This means c is also an irrational number.

0 < 1/√2 < 1

0 < (1/√2) (b - a) < (b - a)     [Since b > a, b - a > 0]

a + 0 < a + 1/√2 (b - a) < a + (b - a)

a < a + (b - a)/√2 < b

This means c = a + (b - a)/√2 lies between a and b.

So, we found an irrational number between two arbitrary rational numbers.