Root 2 is irrational:

Suppose root 2 is rational. that means it can be written as the ratio of two intergers p and q
external image gif.latex?%5Cinline%20%5E%5Csqrt%7B2%7D=%5Cfrac%7Bp%7D%7Bq%7D
Where we may assume that p and q have NO common factors. Squaring in on both sides implies:
external image gif.latex?%5Cinline%202=%5Ctfrac%7Bp%5E%7B2%7D%7D%7Bq%5E%7B2%7D%7D
Thus p² is even. The only way that this can be true is if that p itself is even. But then p² is actually divisible by 4. hence q² and q must also be even. So p and q are both even which means they have common factors, which contradicts what was said before, so root 2 MUST be irrational as it cannot be rational.

The area of a (perfect) circle and the irrationality of pi:


The area of a circle must be:
external image gif.latex?%5C200dpi%20%5CLARGE%20%5Cpi%20r%20%5E%7B2%7D
external image 180px-PI.svg.png
The real thing here is the 'magic' number 'pi', which is 3.14... It is irrational, it has been proved by some really smart mathematicians so it's too difficult to explain on this wiki without me having to do a degree, and you having to do alot of reading! But anyway: http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational.

Ed Walden 8J
Add anything you want to do with theorems!