File: images (1).jpg (42 KB, 370x450)

42 KB JPG

Definition: Two magnitudes are commensurable if and only if their respective lengths can be compared without remainder.

Lemma: If the square of a positive whole number is even then the whole number itself is even.

Assume the square root of two is commensurable with some positive whole number magnitude i.e. it is equivalent to the ratio of two commensurable magnitudes with positive whole number lengths that share no common factor that we will call: Length One and Length Two.

This implies that twice the square of Length Two is equal to the square of Length One, implying that Length One is even which means that twice the square of the Length Two is equal to the square of an even number.

But since twice the square of Length Two is the square of an even number, then the square of Length Two is an even number, and hence Length Two is an even number.

But then Length One and Length Two share a common factor, which contradicts our assumption that Length One and Length Two share no common factors.

Thus by reductio ad absurdum we must conclude that the square root of two cannot be expressed by the ratio of two commensurable magnitudes and thus there does not exist any positive whole number magnitude which can be evenly compared with the square root of two.