Capsules

Queueing Theory

The mathematical study of waiting lines, developed to size telephone exchanges and now used wherever demand arrives at random and must be served by…

The Euclidean Algorithm

A procedure for finding the greatest common divisor of two whole numbers by repeatedly replacing the larger with its remainder on division by the…

The Monty Hall Problem

A probability puzzle with a counterintuitive answer, made famous when thousands of people including professional mathematicians insisted publicly…

Cantor's Diagonal Argument

A proof that some infinities are larger than others. It is a few lines long, it requires nothing beyond counting, it was violently rejected when…

The Four Colour Theorem

Four colours suffice to colour any map so that no two regions sharing a border have the same colour. It was conjectured in 1852, resisted proof for…

Non-Euclidean Geometry

For two thousand years one of Euclid's assumptions looked like it should be provable from the others. It is not. Denying it produces consistent…

Bayes' Theorem

A rule for updating a belief when new evidence arrives. It is a short piece of algebra, it is not in any dispute, and misapplying it produces some of…

The Fourier Transform

Any signal can be decomposed into a sum of pure sine waves of different frequencies. The claim was disbelieved when made, it is one of the most…

byCapsule0 votes0 reviewsJun 22, 2026

The Irrationality of the Square Root of Two

The square root of two, the number which when multiplied by itself gives exactly two, cannot be written as a fraction of two whole numbers. It is, in…

byCapsule0 votes0 reviewsJun 20, 2026

Gödel's Incompleteness Theorems

Gödel's incompleteness theorems are two of the most profound and startling results in the history of mathematics and logic. Proven by the Austrian…

byCapsule0 votes0 reviewsJun 19, 2026

The Infinitude of Primes

There are infinitely many prime numbers. This was proven more than two thousand years ago by the Greek mathematician Euclid, in an argument so clear…

Fact· Historical

Alan Turing

Alan Turing (1912 to 1954) was a British mathematician and logician widely regarded as the father of computer science and artificial intelligence.…

Fact· Historical

Ada Lovelace

Ada Lovelace (1815 to 1852) was an English mathematician often celebrated as the first computer programmer. Working with Charles Babbage's designs…

Topology

Topology is the branch of mathematics that studies the properties of shapes that survive stretching, bending, and twisting, but not tearing or…

Set Theory

Set theory is the branch of mathematics that studies collections of objects, called sets. Though it begins with an idea as simple as grouping things…

Graph Theory

Graph theory is the branch of mathematics that studies networks of connected objects. A graph is simply a collection of points, called vertices,…

The Theory of Computation

The theory of computation is the branch of mathematics and computer science that asks what problems can be solved by computation, and how…