Reductio ad Absurdum — Definition, Formula & Examples

Reductio ad absurdum is a proof technique where you assume the opposite of what you want to prove, then show that assumption leads to a logical contradiction. Since the assumption produces something impossible, the original statement must be true.

To prove a proposition $P$ , assume $\lnot P$ (the negation of $P$ ). If $\lnot P$ , together with accepted axioms and previously established results, logically implies a contradiction $Q \land \lnot Q$ , then $\lnot P$ is false, and therefore $P$ is true.

How It Works

Start by clearly stating what you want to prove. Then assume the exact opposite is true. Using that assumption along with known facts, reason step by step until you arrive at two statements that contradict each other. The contradiction means your assumption was wrong, so the original statement must hold. This technique is also called "proof by contradiction" and is especially powerful when a direct proof is hard to construct.

Example

Problem: Prove that the square root of 2 is irrational.

Assume the opposite: Assume √2 is rational. Then it can be written as a fraction in lowest terms:

\sqrt{2} = \frac{a}{b}, \quad \gcd(a, b) = 1

Square both sides: Squaring gives 2b² = a², which means a² is even, so a must be even. Write a = 2k.

2b^2 = a^2 = (2k)^2 = 4k^2 \implies b^2 = 2k^2

Identify the contradiction: Now b² is even, so b is also even. But if both a and b are even, they share a factor of 2, contradicting gcd(a, b) = 1.

Answer: The assumption that √2 is rational leads to a contradiction, so √2 must be irrational.

Why It Matters

Reductio ad absurdum is used throughout geometry, number theory, and analysis to prove results that resist direct argument. Many foundational results — the infinitude of primes, the irrationality of √2 — rely on it. You will encounter this technique repeatedly in proof-based courses from high school geometry through college-level real analysis.

Common Mistakes

Mistake: Confusing proof by contradiction with proof by contrapositive.

Correction: A contrapositive proof rewrites "if P then Q" as "if not Q then not P" and proves that directly — no contradiction is needed. Reductio ad absurdum assumes the negation of the entire conclusion and derives an impossible statement.