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