1. Prove that √5 is irrational.

Solutions: Let us assume, that √5 is rational number. i.e. √5 = x/y (where, x and y are co-primes) y√5= x Squaring both the 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.