Proof by contradiction (reductio ad absurdum) assumes the negation of what you want to prove, then derives a logical contradiction, thereby establishing that the original statement must be true. Proof by contradiction is essential when direct proof is difficult — it is used to prove irrationality of \sqrt{2}, infinitude of primes, and countless other results.
Definition
Proof by contradiction (reductio ad absurdum) assumes the negation of what you want to prove, then derives a logical contradiction, thereby establishing that the original statement must be true.
💡 Intuition
Assume the opposite of what you want to prove, then follow the logic to a statement that is impossibly false — proving your assumption must have been wrong.
🎯 Core Idea
A contradiction in assumptions validates the original statement.
Example
Notation
eg P, derive contradiction, conclude P$.
🌟 Why It Matters
Proof by contradiction is essential when direct proof is difficult — it is used to prove irrationality of \sqrt{2}, infinitude of primes, and countless other results.
💭 Hint When Stuck
Write 'Assume for contradiction that [negation of what you want to prove].' Then reason forward from that assumption until you reach a statement that contradicts a known fact or your own assumption.
Formal View
Related Concepts
🚧 Common Stuck Point
Students derive a surprising result, but not an actual contradiction.
⚠️ Common Mistakes
- Forgetting to clearly state the assumption you are contradicting at the start
- Reaching a 'contradiction' that is not actually contradictory — make sure the two statements genuinely conflict
- Confusing proof by contradiction with proof by contrapositive — contradiction assumes \neg Q AND P, contrapositive assumes only \neg Q
Frequently Asked Questions
What is Proof by Contradiction in Math?
Proof by contradiction (reductio ad absurdum) assumes the negation of what you want to prove, then derives a logical contradiction, thereby establishing that the original statement must be true.
When do you use Proof by Contradiction?
Write 'Assume for contradiction that [negation of what you want to prove].' Then reason forward from that assumption until you reach a statement that contradicts a known fact or your own assumption.
What do students usually get wrong about Proof by Contradiction?
Students derive a surprising result, but not an actual contradiction.
Prerequisites
Cross-Subject Connections
How Proof by Contradiction Connects to Other Ideas
To understand proof by contradiction, you should first be comfortable with contradiction, logical statement and direct proof.