Paul Erdos believed that there is a book wherein god keeps the most elegant proof of every mathematical theorem. He used to say "It is not necessary to believe in god, but it is necessary to believe in The Book".
I'd like to share with you once such proof. Even if you're not the sort of person that studies proofs, this one is of sufficient elegance that it will give you, I hope, a glimpse of beauty.
A definition by way of background: a rational number is any number that can be expressed as the fraction of two integers. An irrational number is any number that can't. Some examples:
1.5 is rational (it can be expressed as 3/2)
2.2 is rational (it can be expressed as 22/10)
1.234 is rational. (it can be expressed as 1234/1000)
pi is irrational (there is no two integers that form a fraction that is equal to pi)
the square root of two is irrational (there are no two integers that form a fraction equal to the square-root of two)
Understand what an irrational number is? Good. One more bit of mathematical background:
Let's assume we have some even integer N. (It's important here to keep in mind that N is an (even) number). Since every even number is evenly divisible by 2 (or stated another way, every even integer is 2 times some other integer), that means there is some integer 'X' for which this is true:
N = 2X
If we square each side of that (that is, if we multiply each side of that equation by itself), we see that:
Since N is an even number, a way to translate that line into English is:
"The square of every even integer is 4 times some other integer".
Another, even shorter way to translate it is:
"The square of every even integer is evenly divisible by 4"
Now, with that background, on with our proof!
Pythagoras (of right-triangle fame) gave us an elegant proof that the square-root of two is irrational, and he did it by using a technique that has come to be known as "reductio ad absurdum" or "reduced to an absurdity". To use this technique to prove an assertion, we assume the (opposite) of the assertion is true and then see if that leads to any logical contradictions.
We are going to use this technique to prove that the square-root of 2 is irrational, that there are no two integers whose ratio is the square-root of two. Since we are trying to prove that there are no two integers whose square-root is two, the "reductio ad absurdum" technique means we are going to assume that there ARE two such integers, and we're going to see if that leads us to a contradiction, or an "absurdity".
We begin:
Let us suppose there are two integers A and B such that A/B = sqrt(2).
It's obvious that A and B can't both be even numbers. If they were, we could divide them both by 2, obtain a new pair of integers, and continue along with this proof. Let us take as a given that A and B can't both be even, and see where that leads. (Hint: this paragraph is going to be important)
Going back to the equation above (A/B = sqrt(2)), let's square both sides:
Now, multiply both sides by B² and we get:
A² = 2B²
Hey, now we know that A² is a multiple of two, making it an even number
The only way the square of an integer can be even is if the original integer itself is even.... so now we know that A must be an even number.
Wait a second... if A is an even number, then A² is divisible by four. Let's do that... let's create a new integer variable called "X" such that A² = 4X, and substitute it into our previous equation:
4X = 2B²
Divide both sides by 2, and we're left with:
2X = B²
Hmmm... that says B² is even... and the only way that can be true is if B is even.
We have just shown that A and B are both even.... which conflicts with our starting definition that A and B can't both be even.
By starting with the assumption that there are two integers A and B such that their ratio is the square-root of 2, we have arrived at a contradiction, a clearly absurd conclusion.
Which means... there can be no two integers that form a fraction equal to the square-root of two. The square-root of 2 is irrational.
And that is the beautiful proof that Pythagoras left us, the proof that god has written down in "The Book". I hope I've done it justice.