Proof that is irrational
Let be . Then
If were a rational number it would be expressible in the form
Where and are rational numbers with no common divisor (other than 1). It follows that
So is even. This implies that is even. So we can write
Thus is even. We have therefore shown that and are divisible by 2. This is a contradiction. It therefore follows that cannot be rational, i.e. must be an irrational real number.