Prove that is irrational, where p, q are primes.
We will do it by method of contradiction: We will assume is a rational number. If it leads to some absurd outcome then it is wrong assumption.
Let is a rational number
Squaring both sides
(Using (a – b)2 = a2 + b2 – 2ab)
RHS is a rational number but LHS is an irrational number.
This is not possible, hence our assumption was wrong.
Here has to be irrational.