Question

Proved that √5 is irrational?

Answer 1

Given : √5

We need to prove that √5 is irrational.

Proof :

Let's assume that √5 is a rational number.

So, it can be expressed in the form of p/q where p and q are integers and q ≠ 0.

=> √5 = p/q

On squaring both sides, we get :

=> 5 = p²/q²

=> 5q² = p² ...i)

p²/5 = q²

So, 5 divides p

p is a multiple of 5.

Let p = 5r for some integer r.

=> p = 5r

=> p² = 25r² ...ii)

From equations i) and ii) , we get :

=> 5q² = 25r²

=> q² = 5r²

=> q² is a multiple of 5

=> q is a multiple of 5

Hence, p and q have common factor 5. This contradicts our assumption that they are co primes.

Therefore, √5 is an irrational number.