let root 5 be rational then it must in the form of p/q [q is not equal to 0][p and q are co-prime] root 5=p/q => root 5 * q = p squaring on both sides => 5*q*q = p*p ——> 1 p*p is divisible by 5 p is divisible by 5 p = 5c [c is a positive integer] [squaring on both sides ] p*p = 25c*c ——— > 2 sub p*p in 1 5*q*q = 25*c*c q*q = 5*c*c => q is divisble by 5 thus q and p have a common factor 5 there is a contradiction as our assumsion p &q are co prime but it has a common factor so √5 is an irrational