Proof by Contradiction

Proof by contradiction is one of the most powerful proof techniques in mathematics. It works by assuming the opposite of what you want to prove and showing that this assumption leads to a logical impossibility.

---

The Method

1. Assume the negation of the statement you want to prove.
2. Use logical reasoning and known results to derive a contradiction — a statement that is impossible or contradicts a known fact.
3. Conclude that the original assumption was false, so the statement must be true.

---

Classic Proof: √2 Is Irrational

Statement

Prove that √2 is irrational.

Proof

Assume for contradiction that √2 is rational. Then √2 = a/b where a and b are integers with no common factors (the fraction is in its lowest terms).

Squaring: 2 = a²/b² → a² = 2b²

So a² is even, which means a is even. Write a = 2k.

Then (2k)² = 2b² → 4k² = 2b² → b² = 2k²

So b² is even, which means b is even.