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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…