# Famous Theorems of Mathematics/Proof style

This is an example on how to design proofs. Another one is needed for definitions and axioms.

## Irrationality of the square root of 2[edit]

This result uses the following:

**The square root of 2 is irrational**,

## Proof[edit]

This is a proof by contradiction, so we assume that and hence for some *a*, *b* that are coprime.

This implies that . Rewriting this gives .

Since the left-hand side of the equation is divisible by 2, then so must the right-hand side, i.e., . Since 2 is prime, we must have that .

So we may substitute *a* with , and we have that .

Dividing both sides with 2 yields , and using similar arguments as above, we conclude that .

Here we have a contradiction; we assumed that *a* and *b* were coprime, but we have that and .

Hence, the assumption were false, and cannot be written as a rational number. Hence, it is irrational.

## Notes[edit]

- As a generalization one can show that the square root of every prime number is irrational.
- Another way to prove the same result is to show that is an irreducible polynomial in the field of rationals using Eisenstein's criterion.