A-Level Mathematics · Proof · A5

Proof by contradiction.

Assume the opposite of a claim and show that this produces an impossible conclusion.

A5 Proof11 MinutesAnimated lesson

WATCH THE IDEA MOVE

Animated
explanation.

Pause, replay and follow every logical step at your own pace.

01 / KEY IDEA

If denying a statement forces a contradiction, the original statement must be true.

02 / LEARNING OUTCOMES
  • Negate a statement
  • Derive a contradiction
  • Conclude the original claim

WORKED EXAMPLE

Follow every
reason.

Prove that there are infinitely many prime numbers.

01

Assume there are only finitely many primes p₁, …, pₙ.

02

Form N = p₁p₂…pₙ + 1. None of the listed primes divides N.

03

N has a prime factor not in the list, contradicting completeness. Hence infinitely many primes exist.

QUICK CHECK

Test the
logic.

Choose the statement that describes a valid mathematical proof.

What makes a mathematical proof valid?