Beautiful Proof

Okay guys, if you're not a mathematician, bear with me. It's sad I know, but to try and convince myself that I was working earlier, I produced a lovely typed copy of the proof that the square root of two is irrational (cannot be written as a fraction). This is something that I know off by heart and really had no need to type up, but I really think it's rather elegant.If you have any interest in logical thinking/reasoning etc. (even without knowing/caring about maths itself), you will find that this does flow very logically, and is about as beautiful as this area of maths can get, I reckon. Anyway, I hope this will if nothing else give people who know my character an understanding of why I love maths so much, and even possibly get a few people a little bit excited by, or at least less hateful of, mathematics.

Assume: √2 = a/b such that a,b are two integers (whole numbers) which share no common factors.

Therefore: a = √2.b (multiplication by b)

Squaring: a² = 2b²

Because a² is a multiple of 2; a² is an even number.

[Note at this point if x is an odd integer; x² is also an odd integer:

If x is an odd integer, there exists some integer, n, such that x = 2n+1.

Therefore: x² = (2n+1)²

= (2n+1)(2n+1)

= 4n²+4n+1

Taking a factor of 2 from the n components: x² = 2(2n²+2n)+1.

If n is an integer, n² is an integer. Likewise 2n is an integer.

Therefore: 2n² is also an integer.

Therefore: 2n²+2n as a sum of integers is also an integer.

Thus: x² = 2(some integer)+1

which is equivalent to 2n+1, as n is also 'some integer'.

Thus, x² conforms with the initial condition defining odd numbers;

x² is shown also to be an odd number.]

As: a² is even, a must also be even (as an odd number squared is odd; see above).

Therefore: a = 2c (where c is some integer)

Squaring: a² = 4c²

Using (a² = 2b²):

4c² = 2b²

Therefore: 2c² = b² (dividing by 2)

Because b² is a multiple of 2; b² is an even number.

As: b² is even, b must also be even (as odd squares are odd; see above).

Therefore: a and b share at least one common factor, 2, thus contradicting the initial assumption such that a,b share no common factors.

Therefore: √2 cannot be expressed as a ratio of two factors a,b and is irrational.

QED