根号 2 的无理性证明

A proof that sqrt 2 is irrational

is irrational.

proof by contradiction.

If is a rational number, then it must be written as the ratio of two positive integers of mutual prime (that is, the common factor of both is only) and .

Square both sides to derive:

is even.

is even.

let

is even.

Both and are even, which contradicts the assumption that they are prime, so is not rational, only irrational.

证明根号 2 是无理数

命题： 是无理数.

反证法.

假设： 是有理数.

如果 是有理数，那么它一定可以表示为两个互质的正整数（即两者的公因数只有 ） 和 的比值。

对等式两边平方可得：

是偶数.

是偶数.

令

是偶数.

和 都是偶数, 这与他们是素数的假设相矛盾, 因此 不是有理数, 是无理数.