Field
Mathematics
From the derivative to the Fourier transform — the structures and patterns that underlie everything else.
19 articles · 38 widgets
Logarithms: Taming Huge Numbers
Logarithms are the inverse of exponents: they answer what power produces a number, turn multiplication into addition, and — despite the name — grow astonishingly slowly.
Matrices: Functions That Move Space
A matrix is a linear transformation of space in disguise. Its columns tell you where the basis vectors land, and multiplication is just composition.
Infinity: Not All Infinities Are Equal
Infinity is not a number. Learn how bijections measure the size of infinite sets, why the naturals and the reals differ, and Cantor's diagonal proof.
The Integral: Adding Up the Infinitely Small
The integral is not just a reversed derivative. It is the limit of a sum that measures accumulated area, and the Fundamental Theorem is what ties those two ideas together.
Information Theory
Shannon entropy counts the yes/no questions you need to pin down an unknown. That count turns out to be the exact limit of lossless compression, and the reason error correction has a price.
Markov Chains
How a process that remembers only where it is right now can settle into a stable long-run distribution — the idea behind PageRank, MCMC, and text generation.
Eigenvectors and Eigenvalues
Almost every vector gets knocked off its line by a matrix. The rare few that don't reveal what the transformation is really doing.
Game Theory and the Nash Equilibrium
How a single fixed-point idea — nobody can improve by changing their mind alone — explains cartels, auctions, arms races, and why cooperation still manages to survive.
Chaos and Strange Attractors
How a rounding error in a weather simulation revealed that deterministic systems can be unpredictable in principle — and why that caps forecasting at about two weeks.
Gradient Descent
How repeatedly stepping in the direction of steepest descent minimizes functions in millions of dimensions — and trains essentially every modern neural network.
The Derivative
Speed is distance over time, yet a speedometer reads a speed right now. Resolving that paradox is the derivative — the foundation under optimization, physics, and machine learning.
Complex Numbers
Nobody invented i to solve x² = −1 — the cubic formula forced it on them. How a bookkeeping trick became a geometric plane where multiplication is rotation.
Bayes' Theorem
The rule for updating belief in the face of evidence, and why ignoring the base rate makes accurate tests wildly misleading.
Euler's Formula
Why eˣ has anything to do with circles — and what rotating arrows have to do with waves.
Taylor Series
Turn sine, cosine, and eˣ into polynomials — and discover why this works everywhere.
The Central Limit Theorem
The theorem that explains why the bell curve appears everywhere from exam scores to measurement errors.
Newton's Method
The algorithm that squares the number of correct decimal places with every iteration.
Sieve of Eratosthenes
Start with every number, then cross out the non-primes. What's left is beautiful.
The Fourier Transform
Decompose any wave into its frequency components — and see why this changes everything.