The symbol π was first introduced by William Jones [7, 8] in 1706 to denote the ratio between the perimeterPlanetmathPlanetmath and the diameterMathworldPlanetmath on any given circle. In other words, dividing the perimeter of any circle by its diameter always gives the same answer, and this number is defined to be π. A 12-digit approximation of π is given by 3.14159265358

Over human history there were many attempts to calculate this number precisely. One of the oldest approximations appears in the Rhind Papyrus (circa 1650 B.C.) where a geometrical construction is given where (16/9)2=3.1604 is used as an approximation to π although this was not explicitly mentioned.

It wasn’t until the Greeks that there were systematical attempts to calculate π. Archimedes [1], in the third century B.C. used regular polygonsMathworldPlanetmath inscribedMathworldPlanetmath and circumscribedMathworldPlanetmath to a circle to approximate π: the more sides a polygonMathworldPlanetmathPlanetmath has, the closer to the circle it becomes and therefore the ratio between the polygon’s area between the square of the radius yields approximations to π. Using this method he showed that 223/71<π<22/7 (3.140845<π<3.142857).

Around the world there were also attempts to calculate π. Brahmagupta [1] gave the value of 10=3.16227 using a method similarMathworldPlanetmath to Archimedes’. Chinese mathematician Tsu Chung-Chih (ca. 500 A.D.) gave the approximation 355/113=3.141592920.

Later, during the renaissance, Leonardo de Pisa (Fibonacci) [1] used 96-sideed regular polygons to find the approximation 864/275=3.141818

For centuries, variations on Archimedes’ method were the only tool known, but Viète [1] gave in 1593 the formula


which was the first analytical expression for π involving infinite summations or products. Later with the advent of calculus many of these formulas were discovered. Some examples are Wallis’ [1] formula:


and Leibniz’s formula,


obtained by developing arctan(π/4) using power series, and with some more advanced techniques,


found by determining the value of the Riemann Zeta functionMathworldPlanetmath at s=2 (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2).

The Leibniz expression provides an alternate way to define π (namely 4 times the limit of the series) and it is one of the formal ways to define π when studying analysis in order to avoid the geometrical definition.

It is known that π is not a rational numberPlanetmathPlanetmathPlanetmath (quotient of two integers). Moreover, π is not algebraic over the rationals (that is, it is a transcendental numberMathworldPlanetmath). This means that no polynomialPlanetmathPlanetmath with rational coefficients can have π as a root. Its irrationality implies that its decimal expansion (or any integer base for that matter) is not finite nor periodic.


Title pi
Canonical name Pi
Date of creation 2013-03-22 11:51:41
Last modified on 2013-03-22 11:51:41
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 27
Author mathcam (2727)
Entry type Definition
Classification msc 01A40
Classification msc 01A32
Classification msc 01A25
Classification msc 01A20
Classification msc 01A16
Classification msc 51-00
Classification msc 11-00
Classification msc 22A22
Classification msc 46L05
Classification msc 82-00
Classification msc 83-00
Classification msc 81-00
Related topic RegularPolygon
Related topic Limit
Related topic Diameter
Related topic TranscedentalNumber
Related topic AlgebraicNumber
Related topic ExtensionMathbbRmathbbQIsNotFinite
Related topic WallisFormulae