[
{
"content": "# Beautiful Math\nThree of the most beautiful equations in mathematics, and why they're beautiful.\n\n*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*",
"children": [
{
"content": "## Euler's Identity\n$$ e^{i\\pi} + 1 = 0 $$",
"children": [
{
"content": "### In words\n$1$ and $0$ are whole numbers.\n\n$e$ and $\\pi$ are also just numbers, ($2.71828...$ and $3.14159...$), but whose digits go on forever and don't repeat.\n\n$i$ is the only \"weird\" one here, but really it just means \"the number that when multiplied by itself, equals $-1$\".\n\nAnd somehow, if you take $e$ and multiply it by itself \"$i\\pi$ times\" (whatever that means), and add one, you get $0$ *exactly*.\n\nTo see how incredible that is, look at it this way:\n$$(2.7182818284590452...)^{\\sqrt{-1}\\times 3.141592653589793238462643...}+1=0$$"
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]
},
{
"content": "## Cauchy Residue Theorem\n$$\\oint_\\gamma f(z)\\, dz = 2\\pi i \\sum \\operatorname{Res}( f, a_k ) $$",
"children": [
{
"content": "### In words\n$f(z)$ is some function, meaning you give it an $z$ and it returns a plain old number.\n\nIn this case, $z$ can be \"complex\", in the sense that it's a two-part number: $z_1+iz_2$.\n\nNow 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 $a_k$?\nJust a little infinite, or a lot infinite?\n\nThis 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.\n\nThis is true no matter how big the circle is (yes, you guessed it, even if it's infinite).\n\nLet 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.\n\n**Mind-blowing**."
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]
},
{
"content": "## Euler-Lagrange Equation\n$$\\frac{\\partial L}{\\partial f}=\\frac{d}{dx} \\frac{\\partial L}{\\partial f'}$$",
"children": [
{
"content": "### In words\n\nThis 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..."
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]
}
]
}
]