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.