# Prove that √5 is irrational.

**Solution:**

Let us prove that √5 is an irrational number.

This question can be proved with the help of the contradiction method. Let's assume that √5 is a rational number. If √5 is rational, that means it can be written in the form of a/b, where a and b integers that have no common factor other than 1 and b ≠ 0.

√5/1 = a/b

√5b = a

Squaring both sides,

5b² = a²

b² = a²/5 --- (1)

This means 5 divides a².

That means it also divides a.

a/5 = c

a = 5c

On squaring, we get

a² = 25c²

Put the value of a² in equation (1).

5b² = 25c²

b² = 5c²

b²/5 = c²

This means b² is divisible by 5 and so b is also divisible by 5. Therefore, a and b have 5 as a common factor. But this contradicts the fact that a and b are coprime. This contradiction has arisen because of our incorrect assumption that √5 is a rational number. So, we conclude that √5 is irrational.

**Video Solution:**

## Prove that √5 is irrational.

### NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.3 Question 1 - Chapter 1 Exercise 1.3 Question 1:

**Summary:**

We have proved that √5 is irrational.