Mathematics is the one field that settles its questions for good. A theorem, once properly proved, does not get overturned by the next experiment; it stays true. That permanence is what makes the subject both beautiful and forbidding, and it is why a good explanation matters so much. The aim of these pages is to show you why a famous result is true, not just to tell you that it is.
The explainers here fall into a few families. One is the deep world of whole numbers, where the simplest questions turn out to be the hardest: whether the primes thin out in a predictable way (the Riemann hypothesis), whether they keep pairing up forever (the twin prime conjecture), whether every even number splits into two of them (Goldbach's conjecture), and the margin note that took 358 years to settle (Fermat's Last Theorem).
A second family is about the infinite and the limits of logic itself. Cantor's diagonal argument shows that some infinities are strictly larger than others; Gödel's incompleteness theorems show that any rich enough system of mathematics must contain true statements it cannot prove; and Turing machines mark the boundary of what any computer can ever decide. Those three results, from the 1870s to the 1930s, redrew the edges of the knowable.
A third is chance and strategy, where intuition is famously unreliable: Bayes' theorem, the birthday paradox, the Monty Hall problem, and the Nash equilibrium that underpins game theory. A fourth is change and pattern, the machinery for describing a moving world: calculus, Fourier transforms, the Fibonacci sequence, and the endlessly detailed edge of the Mandelbrot set. Threaded through all of it are the questions of what a computer can and cannot do, from public-key cryptography to the Millennium Prize Problems.
Every explainer below is written at three depths on the same page. Level 1 is for a curious twelve-year-old, Level 2 for someone who has just finished school, and Level 3 for an undergraduate. Start wherever you like and change depth whenever the ground feels too soft or too familiar.
All mathematics topics
- The Seven Millennium Prize Problems
- Fermat's Last Theorem
- The Twin Prime Conjecture and Prime Gaps
- Goldbach's Conjecture: Every Even Number, Two Primes
- Pythagoras' Theorem
- The Collatz Conjecture: The 3n+1 Problem
- Gödel's Incompleteness Theorems: The Limits of Proof
- The Birthday Paradox
- Nash Equilibrium
- Calculus
- The Fibonacci Sequence
- Fourier Transforms
- Bayes' Theorem
- Graph Theory & the Bridges of Königsberg
- The Riemann Hypothesis
- The Monty Hall Problem
- Infinity and Cantor's Diagonal
- Turing Machines and Computability
- Public-Key Cryptography
- The Mandelbrot Set and Fractals