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# Beautiful Math

Three of the most beautiful equations in mathematics, and why they’re beautiful.

*These are my own personal favourites that I’ve come across in my studies, and not based on any rigorous definition of beauty. Because there isn’t one. - Adriano*

## Euler’s Identity

$$ e^{i\pi} + 1 = 0 $$

### In words

$1$ and $0$ are whole numbers.

$e$ and $\pi$ are also just numbers, ($2.71828...$ and $3.14159...$), but whose digits go on forever and don’t repeat.

$i$ is the only “weird” one here, but really it just means “the number that when multiplied by itself, equals $-1$“.

And somehow, if you take $e$ and multiply it by itself “$i\pi$ times” (whatever that means), and add one, you get $0$ *exactly*.

To see how incredible that is, look at it this way:

$$(2.7182818284590452…)^{\sqrt{-1}\times 3.141592653589793238462643…}+1=0$$

## Cauchy Residue Theorem

$$\oint_\gamma f(z)\, dz = 2\pi i \sum \operatorname{Res}( f, a_k ) $$

### In words

$f(z)$ is some function, meaning you give it an $z$ and it returns a plain old number.

In this case, $z$ can be “complex”, in the sense that it’s a two-part number: $z_1+iz_2$.

Now if $f(z)$ is *infinite* at some point $a_k$, the part called $\operatorname{Res}( f, a_k )$ is way of asking: *how infinite* is $f$ at $a_k$?

Just a little infinite, or a lot infinite?

This formula says that: if you take a circle around a function with a bunch of infinities, and you add up the value of the function *along the edge* of that circle, it’ll be the same as adding up the infinities *trapped inside* the circle.

This is true no matter how big the circle is (yes, you guessed it, even if it’s infinite).

Let me say this another way: the few infinity spikes in that function somehow contain within them information about the whole function all the way out to forever.

**Mind-blowing**.

## Euler-Lagrange Equation

$$\frac{\partial L}{\partial f}=\frac{d}{dx} \frac{\partial L}{\partial f’}$$

### In words

This is the foundation of the field of “Variational Calculus”, and is related to somehow magically choosing the right path along a surface, out of a massive infinity of possible paths…