How Do You Write a Proof by Contradiction?

To write a proof by contradiction, assume that the statement you want to prove is false. Then use valid logical steps until this assumption produces a contradiction, so the original statement must be true.

This method is part of IB Mathematics: Analysis and Approaches HL, sub-topic AHL 1.15: Proof by induction, contradiction and counterexample, and is particularly suitable for Paper 1 reasoning questions.

The Method

This approach is especially effective when the negation gives a concrete algebraic condition, as in irrationality, divisibility or statements about infinitely many objects. Keep each implication one-way and justified; do not assume the result you are trying to prove.

Worked Example: Prove That $\sqrt{2}$ Is Irrational

Suppose, for contradiction, that $\sqrt{2}$ is rational. Then

$$\sqrt{2}=\frac{p}{q},$$

where $p,q\in\mathbb{Z}$, $q\neq 0$, and $p$ and $q$ have no common factor.

Squaring gives

$$2=\frac{p^2}{q^2}\quad\Rightarrow\quad p^2=2q^2.$$

Therefore, $p^2$ is even, so $p$ is even. This uses the fact that an odd integer has an odd square. Let $p=2k$. Substitution gives

$$4k^2=2q^2\quad\Rightarrow\quad q^2=2k^2,$$

so $q$ is also even. Thus, $p$ and $q$ share a factor of $2$, contradicting the assumption that (p/q) is in lowest terms. Therefore, $\sqrt{2}$ is irrational.

A common misconception is that finding an unexpected result is enough. The result must contradict a clearly stated assumption or established fact.

Exam Technique

For a prove command term, present a complete logical sequence. Examiners expect the contradictory assumption, justified deductions, an explicit contradiction and a final conclusion rejecting the assumption.