Prove that √5 is irrational.


Let us assume, that 5 is a rational number.

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

y5= x

Squaring on both sides, we get,

(y5)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.