¹ An old proof of this may be outlined thus. If √2 the ratio of two whole numbers, a, b, then √2 = a/b. If a, b have a common factor greater than I, it can be cancelled from both a and b without altering the value of a/b, so we shall assume that a, b have no such factor. Square both sides of √2 = a/b. Then 2 = a ²/b ², and hence, multiplying throughout by b ², we have 2b ² = a ², where a, b are whole numbers with the greatest common divisor I. Since 2b ² is even, and 2b ² = a ², a ² must be even. Hence a is even; say a = 2c, where c is a whole number. From 2b ² = a ² we now get 2b ² = (2c)², or 2b ² = 4c ², or b ² = 2c ². Repeating the argument we get b even; say b = 2d. So, in the original 2b ² = a ², we have b = 2d, a = 2c. That is, b, a have the common divisor a, contrary to the assumption that I is their greatest common divisor.

In this sketch many subtle properties of arithmetical divisibility are presupposed. These were all proved by the Greeks and expounded by Euclid. Unless the reader is experienced in this kind of thing, some of the presuppositions will perhaps have passed unnoticed. But they are all justifiable by strict reasoning; nothing has been put over.