Prove that √5 is irrational.

Solution:

Let us assume, that √5 is a rational number.

i.e. √5 = x/y (where, x and y are co-primes)

y√5= x

Squaring on both sides, we get,

(y√5)² = x²

⇒ 5y² = x²…. (1)

Thus, x² is divisible by 5, so x is also divisible by 5.

Let us say, x = 5k, for some value of k and substituting the value of x in equation (1), we get,

5y² = (5k)²

⇒ y² = 5k²

is divisible by 5 it means y is divisible by 5.

Clearly, x and y are not co-primes. Thus, our assumption about √5 is rational is incorrect.

Hence, √5 is an irrational number.