# 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 | edit source]This result uses the following:

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

## Proof

[edit | edit source]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 , we have that . Since 2 is prime, 2 must be one of the prime factors of , which are also the prime factors of , thus, .

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 was false, and cannot be written as a rational number. Hence, it is irrational.

## Notes

[edit | edit source]- 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.