reductio ad absurdum

Theorem proving technique that applies the modus tollens inference method. In practice,

1. you want to prove thesis A
2. you assume that A is false
3. you prove that the logical consequence of A being false is absurdity
4. you deduce that A is true

two favorite theorems proved by reductio ad absurdum (the latin phrase means "reduction to absurd") are:

• the square root of 2 is irrational, proved by assuming that square root of 2 is rational, and showing that that requires a certain number to be both even and odd at the same time.
• There is no largest prime number, proved by assuming that there is in fact such a number, and showing that you can always construct a bigger prime number. Contradiction results, although not as elegant as the previous one.

This is part of 10998521's Mathematics for the Layman project, which you can read more about on his homenode.

Reductio ad absurdum sounds like fancy Latin, but it's actually a very widely-used method of proof. Say you start with some idea that you want to prove: this is the postulate or conjecture. To prove it, we first of all assume the exact opposite. Then, we have to show that this assumption leads to some contradiction: this could be something obvious like 1 = 3, or more subtle. Then, if the opposite is false, that implies that your original postulate was true. Beautifully logical!

As baffo mentions, among the most famous applications of this is the proof that the square root of two is irrational: it can't be written as one integer divided by another. There is a good explanation of this here, and you can see the steps I described above:

• Assume the opposite: That the square root of 2 is rational
• Show that this leads to a contradiction
• Conclude that it must therefore be irrational.