We have to state something slightly stronger than is rational, we have to state that it has been reduced to lowest terms. We may freely make this assumption. The point of the proof is to show that if then the fraction cannot possibly be in reduced form.

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.

Thus

which can be reduced, contrary to assumption.

Thus is not rational.

-Dan