# 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

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.