A Level Maths: Proof By Contradiction Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Proof — the three other proof methods provide essential foundation
- Indices — useful for proofs involving divisibility and powers
- Quadratic Equations — algebraic manipulation is central
A Level Maths proof by contradiction is the fourth and most subtle proof technique, building on Proof. To prove a statement, you assume its OPPOSITE is true, then derive a logical contradiction. This shows the assumption must be false, so the original statement is true. It is a powerful technique for statements that resist direct proof.
The strategy is: state the result you want to prove, then assume its negation (the OPPOSITE of what you want to show). Use logical deduction to derive a contradiction — usually with a known fact, a definition, or with the assumption itself. The classic examples are the irrationality of $\sqrt{2}$ (assume $\sqrt{2} = p/q$ in lowest terms, square, deduce $p$ and $q$ are both even, contradicting 'lowest terms'), and the infinity of primes (assume there are finitely many, construct a new prime not in the list, contradicting completeness). You apply contradiction to unfamiliar statements about numbers, divisibility, and basic algebraic results.
Proof by contradiction is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: the OPPOSITE of 'all $X$ are $Y$' is 'there exists an $X$ that is not $Y$' — not 'no $X$ are $Y$'; the contradiction must be LOGICAL, not just an unexpected result — finding $\sqrt{2}$ is irrational does not contradict '$\sqrt{2}$ is rational' just because it looks weird, you need a concrete logical conflict; the assumption you make must be the FULL negation, not a weaker statement; and once the contradiction is reached, you must explicitly state that the assumption was false and therefore the original statement is true.