So assume we can write
where x and y are integers ( ) and the fraction is in reduced form. Then by squaring both sides we see that:
Thus is an even number.
So if x is an even number it is of the form where n is an integer. Thus and so
Thus y^2 is also an even number.
Thus y is of the form where m is a (non-zero) integer.
which can be reduced, contrary to assumption.
Thus is not rational.