**Method** | **Opening move** | **What finishes the argument** Contradiction | Assume the negation of the required claim. | Derive an impossibility, then reject that assumption. Counterexample | Choose one value satisfying every hypothesis. | Calculate why the claimed conclusion fails. Example To prove 2 irrational, suppose 2= pq, with integers p,q, q 0, in lowest terms. Then p2=2q2, so p is even. Write p=2k. 4k2=2q2 q2=2k2. Then q is also even, contradicting lowest terms. Therefore 2 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 2 and - 2 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.

Assume the negation of the required claim.

Derive an impossibility, then reject that assumption.

Choose one value satisfying every hypothesis.

Calculate why the claimed conclusion fails.

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.

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.

Mathematical Induction

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Contradiction and counterexamples

Contradiction and counterexamples