Rosio Pavoris

I like this proof. It’s simple and easy to follow, but it still gives an idea of the power of mathematics. It’s the sort of thing most people probably saw in high school, but most people forget.

What we’re trying to prove is that the square root of 2 (1.4142135623&c.) is an irrational number. A number is irrational if it cannot be expressed as a fraction; that is, the ratio of two whole numbers.
To prove this, we’ll use a reductio ad absurdum—we’ll assume √2 is rational, and we’ll see if this leads us to a contradiction.

If it is rational, the expression

,

where A and B are integers without common divisors, must be true.
Squaring both sides, we get

.

Let’s also multiply both sides by B².

So, we clearly see A² is even, and since the square of an odd number is always odd itself (right?), it follows that A itself is even as well.
So A is the double of a whole number, let’s say C. Let’s substitute C in the equation.

Dividing both sides by 2, we get

So B is an even number too!
We’ve said that A and B didn’t have any common divisors, and if they’re both even they have 2 as a common divisor, so this is clearly a contradiction. It follows that √2 cannot be expressed as the ratio of two whole number.
As such, √2 is irrational. Quantum electrodynamics Quod erat demonstrandum.

By posting this, I’ve probably done someone’s homework for them. Oh well.
Another interesting thing about √2, and really the square root of any positive integer, is that it can be expressed as a periodic continued fraction.

In a sense, they are the simplest of the irrational numbers. This is rather beyond my ability to prove at the moment, though.