Question

Show that root 5 is an irrational number

Answer 1

firstly let us assume to the contrary that root 5is rational
so we can find integer p and q such that p/q=√5......(p and q are co primes)
now squaring both side
(p/q)2 =(5)2
p2 = 5 q2.......1
p2 is divisible by 5
p is also divisible by 5
p= 5a for any integer a
now again squaring both side
p2 = (5a)2
p2= 25a2
5q2 = 25a2 ...........from 1
q2 = 5 a2
therefore . q2 is divisible by 5
so q is also divisible by 5
therefore p and q both are divisible by 5
p and q hv common factor 5 which contradicts the fact that p and q no other common factor than 1
thus contradiction has arisen coz of our incorrect assumption....
hence we conclude.that √5 is irrational