, where
is any ratio; and
is of course the converse of
. This is not the only possible way of defining positive and negative ratios, but it is a way which, for our purpose, has the merit of being an obvious adaptation of the way we adopted in the case of integers.
We come now to a more interesting extension of the idea of number, i.e. the extension to what are called "real" numbers, which are the kind that embrace irrationals. In Chapter I. we had occasion to mention "incommensurables" and their discovery by Pythagoras. It was through them, i.e. through geometry, that irrational numbers were first thought of. A square of which the side is one inch long will have a diagonal of which the length is the square root of 2 inches. But, as the ancients discovered, there is no fraction of which the square is 2. This proposition is proved in the tenth book of Euclid, which is one of those books that schoolboys supposed to be fortunately lost in the days when Euclid was still used as a text-book. The proof is extraordinarily simple. If possible, let