Contradiction and counterexamples

Example

To prove \(\sqrt2\) irrational, suppose \(\sqrt2=\dfrac pq\), with integers \(p,q\), \(q\ne0\), in lowest terms. Then \(p^2=2q^2\), so \(p\) is even. Write \(p=2k\).

\(\begin{aligned}4k^2&=2q^2\\q^2&=2k^2.\end{aligned}\)

Then \(q\) is also even, contradicting lowest terms. Therefore \(\sqrt2\) is irrational. Here we used the fact that an odd integer has an odd square.

Example

The statement “the sum of two irrational numbers is always irrational” is false. Both \(\sqrt2\) and \(-\sqrt2\) are irrational, but their sum is zero, which is rational.

A counterexample must show both that the example is admissible and that the conclusion fails. A long list of successful examples cannot prove a universal statement.