Q1 Prove that is irrational.
Let us assume is rational.
It means can be written in the form where p and q are co-primes and
Squaring both sides we obtain
From the above equation, we can see that p2 is divisible by 5, Therefore p will also be divisible by 5 as 5 is a prime number.
Therefore p can be written as 5r
p = 5r
p2 = (5r)2
5q2 = 25r2
q2 = 5r2
From the above equation, we can see that q2 is divisible by 5, Therefore q will also be divisible by 5 as 5 is a prime number.
From (i) and (ii) we can see that both p and q are divisible by 5. This implies that p and q are not co-primes. This contradiction arises because our initial assumption that is rational was wrong. Hence proved that is irrational.