Proof by Contradiction

A proof by contradiction turns a statement around. Instead of building it up directly, you assume the statement is false — that the opposite is true — and then reason carefully until you deduce something impossible. Since correct reasoning can never lead to an absurdity, the only thing that could have been wrong is your starting assumption. So the assumption is false, and the original statement must be true.

The most famous example is that \sqrt{2} is irrational — it cannot be written as a fraction. Assume the opposite: suppose it can, as a fraction in lowest terms,

\sqrt{2} = \frac{a}{b}, \qquad \text{with } \frac{a}{b} \text{ in lowest terms.}

Square both sides and rearrange:

2 = \frac{a^2}{b^2} \quad\Longrightarrow\quad 2b^2 = a^2.

So a^2 is even, which forces a to be even, say a = 2c. Substitute back:

2b^2 = (2c)^2 = 4c^2 \quad\Longrightarrow\quad b^2 = 2c^2.

Now b^2 is even too, so b is even. But if a and b are both even, the fraction \tfrac{a}{b} was not in lowest terms — contradicting how we chose it. The assumption collapses, so \sqrt{2} is irrational.

The pattern

Every proof by contradiction has the same shape: flip the claim, follow the logic honestly, and wait for the impossible to appear.

To prove a statement P by contradiction:

assume the opposite — suppose P is false;
reason with correct, valid steps until you reach a contradiction — something impossible;
conclude that the assumption must have been wrong, so P is true;
classic results proved this way: \sqrt{2} is irrational, and there are infinitely many primes.