A proof by contradiction (a.k.a. reductio ad absurdum, reduction to absurdity) is a strategy used by mathematicians to show that a mathematical statement is true by proving that the negation of that statement leads to being able to prove that two opposite statements are simultaneously true (a contradiction).
The outline of the strategy is as follows:
To illustrate the concept, we will do a simple, non rigorous reasoning. Imagine yourself in the next situation:
You are a defense lawyer. Your client is accused of stealing the cookie from the cookie jar. You want to prove her innocence. Lets say you have evidence that the jar is still sealed. Reason as follows:
Now we will work through an actual mathematical example: we will show that is not rational; that is, it cannot be expressed as the division of two natural numbers.
Proof by contradiction is one of the most useful techniques one can use to prove anything.
In particular, if you get stuck while doing a proof, resorting to proof by contradiction is a great way to keep exploring a problem from a different perspective. Even if you do not get to solve the problem, you may get a useful insight about the problem when performing the procedure of proof by contradiction.
Also, trying to do proof by contradiction may result in a counterexample, which dissolves the problem in question.