Hassle-free LaTeX with Overleaf

There is something delightful about LaTeX. However, the last time I bothered with it was in college, since I don’t have much call for PDFs in day-to-day life. I recently came across Overleaf, which is an online LaTeX editor. The nice part is that it live-renders your work and you can right-click->Save as an PNG. Thus, you can suddenly embed gorgeously formatted math anywhere. For example, here’s one of my favorite proofs, that the square root of two is not a rational number:

Proof by contradiction.

Source code:

\documentclass[varwidth=true, border=10pt]{standalone}


Suppose $\sqrt{2}$ was rational. Then we could write:

\[ \sqrt{2} = \frac{a}{b} \]

...where $a/b$ is in lowest terms. Squaring both sides yields:

\[ 2 = \frac{a^{2}}{b^{2}} \]

Now multiply both sides by $b^{2}$:

\[ 2b^{2} = a^{2} \]

$a^{2}$ must be even, since $b^{2}$ is multiplied by 2. For $a^{2}$ to be even, $a$ must be even, so we can say that $a = 2c$ for some $c$. 

Thus, we can write this equation as:

\[ 2b^{2} = (2c)^{2} \]


\[ 2b^{2} = 4c^{2} \]

Now we can divide both sides by 2... but we end up with $b^{2} = 2c^{2}$, which is shaped the same as $2b^{2} = a^{2}$ above!

We can continue expanding this equation out forever, so there are no whole numbers that $a$ and $b$ can resolve to.

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



Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: