Prove that √5 is irrational by the method of Contradiction.
Let √5 be a rational number.
then it must be in form of pq where, q≠0 ( p and q are co-prime)
√5=pq
√5×q=p
Suaring on both sides,
5q2=p2 --------------(1)
p2 is divisible by 5.
So, p is divisible by 5.
p=5c
Suaring on both sides,
p2=25c2 --------------(2)
Put p2 in eqn.(1)
5q2=25(c)2
q2=5c2
So, q is divisible by 5.
.
Thus p and q have a common factor of 5.
So, there is a contradiction as per our assumption.
We have assumed p and q are co-prime but here they a common factor of 5.
The above statement contradicts our assumption.
Therefore, √5 is an irrational number.