# Proof by contradiction

Assume the opposite of what you want to prove. Follow the logic until something impossible appears. So the opposite was wrong.

Maths (university) · Part 01: Logic, sets and proof · Card 6 of 9 · Last updated 2026-10-19

## Analogy

A detective who assumes the butler was in the kitchen, then finds the kitchen was locked all night.

## Example

**sqrt(2) is irrational**

Assume sqrt(2) = a/b in lowest terms.
Then a^2 = 2b^2, so a is even: a = 2c.
4c^2 = 2b^2 gives b^2 = 2c^2, so b is even.
Both even breaks 'lowest terms'. Contradiction.

## In practice

Useful for 'does not exist' and 'is irrational' claims, where direct steps are hard to find.

## Practice

### Flashcards

- **Which method suits 'sqrt(2) is irrational'?** Proof by contradiction: assume sqrt(2) = a/b in lowest terms and reach a clash.

### Interview questions

**Explain why the square root of 2 is irrational.**

- Assume sqrt(2) = a/b in lowest terms
- Square to get a^2 = 2b^2
- So a is even, a = 2c
- Then b^2 = 2c^2, so b is even
- Both even contradicts lowest terms

## Questions people ask

### Why does proof by contradiction work?

A statement and its opposite cannot both be true. If assuming the opposite forces something impossible, such as a number being both even and odd, then the opposite must be false. So the original claim is true. It relies on every statement being true or false.

### When should I use contradiction instead of a direct proof?

Choose it when the claim says something does not exist or cannot happen, since assuming it does gives you an object to work with. Proofs that the square root of 2 is irrational, or that there are infinitely many primes, are classic examples.

## Related concepts

- [Contrapositive](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-3)
- [Direct proof](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-5)
- [Induction](https://studyvisually.com/learn/science/maths-university/maths-university-part-1/maths-university-part-1-concept-7)
- [Epsilon-delta definition](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-9)

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