Prove that is an irrational number.
Lets assume is a rational number.
For rational number,
it should be in a form of p/q where and HCF of P and q should be 1 in their least terms.
So,
= p/q.
After squaring both sides
---(1)
From the above eqn, 2 is the divisor of
Here 2 is a factor of
and atlast we can say 2 is a factor of p.
Let p=2a for all a (where a is a positive integer)
Squaring both sides,
---(2)
From eqn(1) and eqn(2)
From this, 2 is a divisor of
So 2 is a divisor of
We can conclude 2 is a factor of q,
We have seen that 2 is a factor of p as well as q which is against the condition HCF(p,q) =1.
Therefore there is a contradiction because our assumption is wrong.
Hence is an irrational number.