A Working Library
Applied Mathematics
The handful of plain ideas that quietly run the world — taken one at a time, from the first principle all the way out to the machine it ends up inside.
Here's a funny thing about the mathematics that actually gets used: it's really a small handful of ideas, each one wearing a dozen different costumes. So that's what we'll do here — take them one at a time. Build each one up from nothing, find the single move that makes it useful, and then chase it out into the wild, into the places it earns its keep: money, signals, learning machines, falling balls, networks straining under load. And we won't just talk about it — every section is yours to poke at. Read the idea, then grab a slider and watch the whole thing move.
Contents
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Geometric Series
Take a number and shave it by the same fraction, over and over — that's the whole idea. And yet it prices a bond that pays you forever, smooths the jitter out of a noisy sensor, discounts tomorrow's rewards, and tells you whether a viral loop will catch fire or quietly run out of breath — and, when you're not careful, exactly how it lies to you.
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Binomial Series
Raise anything to a power while it sits just above 1, and the whole curve straightens into a line you can read off in your head. That one move hides Newton's ½mv² inside Einstein's energy, tells a bond how its price bends with yield, and turns a 1% slip in a measurement into a 3% slip in the answer.
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Taylor Series
The master machine. Match a function's value, its slope, its bend, and a few more, and you've rebuilt the whole thing out of a plain polynomial — which is how a chip computes a sine, why every gentle wobble in nature is secretly a spring, and where every other series in this book quietly comes from.
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Exponential Series
When a thing grows at a rate set by how much of it there already is, you get the one function that is its own shadow. It dates the bones of the long-dead, charges the capacitor in your phone, and — fed an imaginary number — stops growing and begins to spin.
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Logarithmic Series
The geometric series turned inside out. It trades multiplication for addition and crushes the unimaginably large down to a number you can hold in your hand — the bit, the decibel, the pH, the O(log n) that makes a search through a billion things feel like twenty — and it pays for the magic by crawling.
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More ideas on the way — one costume at a time.