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 Algebraic Methods

Edexcel A level Maths: 1.1 Proof by ContradictionCurated by Kepler, not produced by us. Watch it 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, $\textbf{assume the opposite}$ ( $P$ 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.

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

Example 1Why does the √2 proof work?

1Assume

\sqrt{2} = \tfrac{a}{b}

in lowest terms.

2Deduce

a

and

b

are both even.

3That contradicts the fraction being in lowest terms.

Answer

\sqrt{2}

cannot be rational

Other classic proofs

The same structure proves there are $\textbf{infinitely many primes}$ (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

\text{Assume } \neg P \to \bot \Rightarrow P

Structure of the proof.

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

Used in the √2 proof.

\text{rational} = \tfrac{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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

To prove a statement by contradiction, you first:
- Assume it is false
- Assume it is true
- Test examples
- Differentiate
Show answer and explanation
Answer: Assume it is false
Assume the negation.
Reaching a contradiction shows the assumption was:
- False
- True
- Irrelevant
- Incomplete
Show answer and explanation
Answer: False
So the original statement is true.
In proving √2 is irrational, we assume √2 equals:
- $\tfrac{a}{b}$ in lowest terms
- an integer
- $2b$
- an even number
Show answer and explanation
Answer: $\tfrac{a}{b}$ in lowest terms
The rational assumption.
The √2 proof reaches a contradiction because a and b turn out to both be:
- Even
- Odd
- Prime
- Negative
Show answer and explanation
Answer: Even
Contradicting “lowest terms”.
If $a^{2}$ is even, then $a$ is:
- Even
- Odd
- Prime
- Zero
Show answer and explanation
Answer: Even
Key lemma in the √2 proof.
The proof that there are infinitely many primes assumes:
- There are finitely many primes
- There are no primes
- 2 is the only prime
- All numbers are prime
Show answer and explanation
Answer: There are finitely many primes
Then constructs a number contradicting that.

Next step

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