# Real Numbers - Irrational Numbers

Ex 1.3 Q1. CBSE 10th Maths NCERT exercise solution

#### Question: Prove that $$sqrt5$ is irrational. #### Explanatory Answer Let us assume the contrary. i.e., $\sqrt5$ is rational. So, $\sqrt5$ = $\frac{a}{b}$ where a and b are co-prime integers and b ≠ 0 ∴ a = b$\sqrt5$ Squaring both sides, a2 = 5b2 -----------$1)

Inference 1: So, we can infer that 5 divides a2
Because ‘a’ is a positive integer, if 5 divides a2, 5 will also divide a.
∴ ‘a’ can be written as a multiple of 5 i.e., a = 5c for some integer c
Substitute a = 5c in (1)
25c2 = 5b2 or b2 = 5c2
Inference 2: 5 divides b2.
Because ‘b’ is a positive integer, if 5 divides b2, 5 will also divide b.
5 divides a. 5 divides b
a and b have 5 as common factor.
But we picked ‘a’ and ‘b’ as co-prime integers. So, 'a' and 'b' cannot have 5 as a common factor.
The contradiction is because of our incorrect assumption that $\sqrt5$ is rational.
So, we can conclude that $\sqrt5$ is irrational.