February 11th, 2020

Abstract
It’s been a while since I wanted to install some math packages on my website. Finally, I managed to do it by getting asciidoctor-latex and running some quick Makefile magic. It looks like it’s working fine! Here is a simple proof that I’ve come to like over the years.
$\underline{Theorem}$

$$\sqrt{2}$$ is irrational.

$\underline{Proof}$

Let $$a,b \in \mathbb{Z} \quad, \frac{a}{b}=\sqrt{2},\quad b \neq 0, \quad (a,b)=1$$

Then $$a=\sqrt{2}b$$

$$\implies a^2=2 \times b^2 \quad (\Xi)$$

$$\implies 2|a^2$$

$$\implies 2|a$$

Then if $$a$$ is even, $$\exists k \in \mathbb{Z} \ni a = 2k$$

Then substitute into $$(\Xi)$$, we get $$(2 \times k)^2=2\times b^2$$

$$\implies 4 \times k^2 = 2 \times b^2$$

$$\implies 2 \times k^2 = b^2$$

$$\implies 2|b^2$$

$$\implies 2|b$$

If $$a$$ and $$b$$ are both even, it contradicts the initial condition $$(a,b)=1$$.

$$\therefore$$ By Law of Contradiction, $$\sqrt{2}$$ is irrational