# User:Paxinum/Proof styles

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## The square rooth of 2 is irrational theorem[edit]

This result uses the following: | [hide] |
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Definition of rational number. | |

Definition of prime and coprime. | |

Definition of square rooth. | |

Gödels incompleteness theorem =) |

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

## Proof[edit]

This is a proof by contradiction, so we assumes 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.

## History[edit]

Some nice history about the one that first proved this theorem.