The Mathematics of π
Exploring the formulas that define and approximate pi
Pi (π) appears in countless formulas across mathematics, physics, and engineering. Below are some of the most important and beautiful formulas involving this fundamental constant.
Fundamental Definitions
Circle Circumference
DefinitionThe most fundamental definition: π is the ratio of a circle's circumference to its diameter.
Circle Area
DefinitionThe area of a circle is proportional to the square of its radius, with π as the constant of proportionality.
Infinite Series
Leibniz Formula
1674One of the simplest infinite series for π, discovered by Gottfried Wilhelm Leibniz. While elegant, it converges very slowly (after 500,000 terms, it's only accurate to 5 decimal places).
Nilakantha Series
15th CenturyDiscovered by the Indian mathematician Nilakantha Somayaji, this series converges much faster than the Leibniz formula.
Ramanujan's Formula
1910An extremely rapidly converging series discovered by Srinivasa Ramanujan. Each term adds about 8 decimal digits of precision.
Infinite Products
Wallis Product
1656Discovered by John Wallis, this was the first infinite sequence ever found for π.
Viète's Formula
1593The first infinite product in mathematics, discovered by François Viète.
Continued Fractions
Simple Continued Fraction
GeneralThe simple continued fraction representation of π has no apparent pattern.
Generalized Continued Fraction
1706Discovered by William Brouncker, this was the first continued fraction representation of π.
Integrals and Calculus
Gaussian Integral
ProbabilityThis integral is fundamental in probability theory and statistics.
Basel Problem
1734Solved by Euler, this showed a deep connection between π and the integers.
Modern Computation Formulas
Chudnovsky Algorithm
1989Used to set several π calculation records, including the first computation of over 1 billion digits.
BBP Formula
1995The Bailey-Borwein-Plouffe formula allows calculating specific binary or hexadecimal digits of π without calculating preceding digits.