1.1PureStretch

Proof by Contradiction

Proof by contradiction is a powerful technique: to prove a statement true, you assume it is false and show that this forces an impossibility. It is the method behind the classic proofs that √2 is irrational and that there are infinitely many primes.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel A level Maths: 1.1 Proof by ContradictionWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand the logic of proof by contradiction
Set up the negation of a statement
Prove a statement by reaching a contradiction
Follow the standard proofs (√2 irrational, infinite primes)

The logic

To prove a statement $P$ by contradiction, $assume the opposite$ ( $P$ is false) and reason logically until you reach something impossible. Since the reasoning is valid, the only flaw must be the assumption — so $P$ must be true.

Assume \neg P \Rightarrow contradiction \Rightarrow P is true

Assume the negation; derive an impossibility.

Tip — Start every proof by clearly writing the assumption: “Assume, for contradiction, that …”.

Worked idea: √2 is irrational

Assume $2$ is rational, so $2 = \frac{a}{b}$ in lowest terms. Squaring gives $a^{2} = 2 b^{2}$ , so $a^{2}$ is even, hence $a$ is even. Writing $a = 2 k$ leads to $b^{2} = 2 k^{2}$ , so $b$ is also even — but then $a$ and $b$ share a factor of 2, contradicting “lowest terms”.

1Assume

2 = \frac{a}{b}

in lowest terms.

2Deduce

a

and

b

are both even.

3That contradicts the fraction being in lowest terms.

Answerso

2

cannot be rational

Other classic proofs

The same structure proves there are $infinitely many primes$ (assume finitely many, multiply them all and add 1, get a number with no prime factor in the list) and that there is no smallest positive rational number. Recognising the “assume the negation” opening is the key skill.

Formula recap

Assume \neg P \to ⊥ \Rightarrow P

Structure of the proof.

a even \Rightarrow a^{2} even (and converse)

Used in the √2 proof.

rational = \frac{a}{b}, b \neq = 0

Definition to negate.

Common mistakes to avoid

Assuming the statement you are trying to prove.

Assume its NEGATION — the opposite — then seek a contradiction.

Stopping before clearly stating the contradiction.

Explicitly identify the impossibility (e.g. “contradicts lowest terms”).

Key takeaways

Assume the statement is false, then derive a contradiction.
The contradiction shows the assumption was wrong, so the statement is true.
Know the standard results: √2 irrational, infinitely many primes.

Test yourself

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