Famous Theorems of Mathematics/Proof style

This is an example on how to design proofs. Another one is needed for definitions and axioms.

Irrationality of the square root of 2
 This result uses the following:


 * Definition of rational number.
 * Definition of prime and coprime.
 * Definition of square root.

The square root of 2 is irrational, $$ \sqrt{2} \notin \mathbb{Q} $$

Proof
This is a proof by contradiction, so we assume that $$ \sqrt{2} \in \mathbb{Q} $$ and hence $$ \sqrt{2} = \frac{a}{b} $$ for some a, b that are coprime.

This implies that $$2 = \frac{a^2}{b^2}$$. Rewriting this gives $$2b^2 = a^2 \!\,$$.

Since $$b^2 \in \mathbb{Z}$$, we have that $$2 | a^2$$. Since 2 is prime, 2 must be one of the prime factors of $$a^2$$, which are also the prime factors of $$a$$, thus, $$2 | a$$.

So we may substitute a with $$2k, k \in \mathbb{Z}$$, and we have that $$2b^2 = 4k^2 \!\,$$.

Dividing both sides with 2 yields $$b^2 = 2k^2 \!\,$$, and using similar arguments as above, we conclude that  $$2 | b $$.

Here we have a contradiction; we assumed that a and b were coprime, but we have that $$2 | a $$ and $$2 | b $$.

Hence, the assumption was false, and $$ \sqrt{2} $$ cannot be written as a rational number. Hence, it is irrational.