February 11th, 2020

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.

\(\sqrt{2}\) is irrational.


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