Just out of interest...

[marnitzg]

How many people here can prove that Sqrt(2) is irrational?

[[Digital-X-Treme]]

I can... after quickly flicking through my A Level Pure Maths text book to the 'Proof By Contradiction' chapter...

Later

[Guv]

The proof goes roughly as follows.

Assume that SquareRoot(2) = P / Q, where P and Q are integers with no common factor. Then we have the following.

P = Q * Squareroot(2)
P^2 = 2 * Q^2
Then 2 is a factor of P^2
But then 2 is also a factor of P
Then P^2 = 4 * X^2
Then 4 * X^2 = 2 * Q^2
or 2 * X^2 = Q^2
Now 2 must be a factor of Q^2
Hence 2 must be a factor of Q
But P and Q are not supposed to have factors.

Note that we could keep going along the same lines and prove that P and Q have 2 as a factor many times.

2 * X*2 = 4 * Y^2, where Y = 2 * Q.
X^2 = 2 * Y^2
Now X has a factor of 2, and hence P has 4 as a factor.