Theorem. There is no rational number whose square is 2.

Theorem. There is no rational number whose square is 2. Proof. We use indirect reasoning. Suppose x is a rational number whose square is 2. Then x can be written in lowest terms as a b, where a is an integer and b is a positive integer. Since x2 = 2, ⇣a b ⌘ 2 = 2, so a2 b2 = 2. Then a2 = 2b2, so a2 is even. But then a is even, so a = 2n for ... ................
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