Capsules

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…

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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…

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…

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…

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…

Fermat's Last Theorem

Fermat's Last Theorem states that no three positive whole numbers can satisfy the equation a to the power n, plus b to the power n, equals c to the…

The Four Color Theorem

The four color theorem states that any map drawn on a flat surface can be colored using only four colors in such a way that no two regions sharing a…

The Pythagorean Theorem

The Pythagorean theorem is one of the most famous results in all of mathematics. It states that in a right angled triangle, the square of the longest…