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 [Pg 66] 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 be the square root of 2, so that , i.e. . Thus is an even number, and therefore must be an even number, because the square of an odd number is odd. Now if is even, must divide by 4, for if , then . Thus we shall have , where is half of . Hence , and therefore will also be the square root of 2. But then we can repeat the argument: if , will also be the square root of 2, and so on, through an unending series of numbers that are each half of its predecessor. But this is impossible; if we divide a number by 2, and then halve the half, and so on, we must reach an odd number after a finite number of steps. Or we may put the argument even more simply by assuming that the we start with is in its lowest terms; in that case, and cannot both be even; yet we have seen that, if , they must be. Thus there cannot be any fraction whose square is 2. Thus no fraction will express exactly the length of the diagonal of a square whose side is one inch long. This seems like a challenge thrown out by nature to arithmetic. However the arithmetician may boast (as Pythagoras did) about the power of numbers, nature seems able to baffle him by exhibiting lengths which no numbers can estimate in terms of the unit. But the problem did not remain in this geometrical form. As soon as algebra was invented, the same problem arose as regards the solution of equations, though here it took on a wider form, since it also involved complex numbers.