Pi

The mathematical constant ? is the ratio of a circle's circumference (Greek ??????????, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. The name of the Greek letter ? is pi (pronounced pie), and this spelling can be used in typographical contexts where the Greek letter is not available. ? is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number.

Formulae involving π

Geometry

pi appears in many formulae in geometry involving circles and spheres.

Related Topics:
Geometry - Circle - Sphere

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(All of these are a consequence of the first one, as the area of a circle can be written as

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A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.)

Related Topics:
Annuli - Solid of revolution

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Also, the angle measure of 180° (degrees) is equal to π radians.

Related Topics:
Angle - Degrees - Radian

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Analysis

Many formulae in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions.

Related Topics:
Analysis - Infinite series - Infinite product - Integral - Special functions

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• François Viète, 1593 (proof):

:rac2pi=

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rac{sqrt2}2

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rac{sqrt{2+sqrt2}}2

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rac{sqrt{2+sqrt{2+sqrt2}}}2ldots

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• Leibniz' formula (proof):

:rac{1}{1} - rac{1}{3} + rac{1}{5} - rac{1}{7} + rac{1}{9} - cdots = rac{pi}{4}

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:This commonly cited infinite series is usually written as above, but is more technically expressed as:

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:sum_{n=0}^{infty} rac{(-1)^{n}}{2n+1} = rac{pi}{4}

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• Wallis's product (see that article for a proof):

: rac{2}{1} cdot rac{2}{3} cdot rac{4}{3} cdot rac{4}{5} cdot rac{6}{5} cdot rac{6}{7} cdot rac{8}{7} cdot rac{8}{9} cdots = rac{pi}{2}

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: prod_{n=1}^{infty} rac{(2n)^2}{(2n)^2-1} = prod_{n=1}^{infty} rac{2n}{2n-1} cdot rac{2n}{2n+1} = rac{pi}{2}

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• 1995 Bailey-Borwein-Plouffe algorithm

:pi=sum_{k=0}^inftyrac{1}{16^k}left

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• An integral formula from calculus (see also Error function and Normal distribution):

:int_{-infty}^{infty} e^{-x^2},dx = sqrt{pi}

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• Basel problem, first solved by Euler (see also Riemann zeta function):

:zeta(2) = rac{1}{1^2} + rac{1}{2^2} + rac{1}{3^2} + rac{1}{4^2} + cdots = rac{pi^2}{6}

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:zeta(4)= rac{1}{1^4} + rac{1}{2^4} + rac{1}{3^4} + rac{1}{4^4} + cdots = rac{pi^4}{90}

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:and generally, zeta(2n) is a rational multiple of pi^{2n} for positive integer n

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• Gamma function evaluated at 1/2:

:Gammaleft({1 over 2} ight)=sqrt{pi}

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• Stirling's approximation:

:n! sim sqrt{2 pi n} left(rac{n}{e} ight)^n

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• Euler's identity (called by Richard Feynman "the most remarkable formula in mathematics"):

:e^{i pi} + 1 = 0;

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• Property of Euler's totient function (see also Farey sequence):

:sum_{k=1}^{n} phi (k) sim 3 n^2 / pi^2

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• Area of one quarter of the unit circle:

:int_0^1 sqrt{1-x^2},dx = {pi over 4}

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• An application of the residue theorem

:ointrac{dz}{z}=2pi i ,

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:where the path of integration is a circle around the origin, traversed in the standard (anti-clockwise) direction.

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Continued fractions

π has many continued fractions representations, including:

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: rac{4}{pi} = 1 + rac{1}{3 + rac{4}{5 + rac{9}{7 + rac{16}{9 + rac{25}{11 + rac{36}{13 + ...}}}}}}

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(Other representations are available at The Wolfram Functions Site.)

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Number theory

Some results from number theory:

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• The probability that two randomly chosen integers are coprime is 6/π².
• The probability that a randomly chosen integer is square-free is 6/π².
• The average number of ways to write a positive integer as the sum of two perfect squares (order matters) is π/4.
• The product of (1-1/p²) over the primes, p, is 6/π².

Here, "probability", "average", and "random" are taken in a limiting sense, e.g. we consider the probability for the set of integers {1, 2, 3,..., N}, and then take the limit as N approaches infinity.

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The remarkable fact that

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: e^{pi sqrt{163}} = 262537412640768743.99999999999925007...

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or equivalently,

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: e^{pi sqrt{163}} = 640320^3+743.99999999999925007...

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can be explained by the theory of complex multiplication.

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Dynamical systems and ergodic theory

Consider the recurrence relation

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:x_{i+1} = 4 x_i (1 - x_i) ,

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Then for almost every initial value x0 in the unit interval ,

Related Topics:
Almost every - Unit interval

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: lim_{n o infty} rac{1}{n} sum_{i = 1}^{n} sqrt{x_i} = rac{2}{pi}

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This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory.

Related Topics:
Logistic map - Dynamical system - Ergodic theory

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Physics

In physics, appearance of π in formulae is usually only a matter of convention and normalization. For example, by using the reduced Planck's constant hbar = rac{h}{2pi} one can avoid writing π explicitly in many formulae of quantum mechanics. In fact, the reduced version is the more fundamental, and presence of factor 1/2π in formulas using h can be considered an artifact of the conventional definition of Planck's constant.

Related Topics:
Physics - Planck's constant

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• Heisenberg's uncertainty principle:

: Delta x Delta p ge rac{h}{4pi}

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• Einstein's field equation of general relativity:

: R_{ik} - {g_{ik} R over 2} + Lambda g_{ik} = {8 pi G over c^4} T_{ik}

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• Coulomb's law for the electric force:

: F = rac{left|q_1q_2 ight|}{4 pi epsilon_0 r^2}

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• Magnetic permeability of free space:

: mu_0 = 4 pi imes 10^{-7},mathrm{H/m},

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Probability and statistics

In probability and statistics, there are many distributions whose formulae contain π, including:

Related Topics:
Probability - Statistics - Distributions

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• probability density function (pdf) for the normal distribution with mean μ and standard deviation σ:

:f(x) = {1 over sigmasqrt{2pi} },e^{-(x-mu )^2/(2sigma^2)}

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• pdf for the (standard) Cauchy distribution:

:f(x) = rac{1}{pi (1 + x^2)}

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Note that since int_{-infty}^{infty} f(x),dx = 1, for any pdf f(x), the above formulae can be used to produce other integral formulas for π.

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An interesting empirical approximation of π is based on Buffon's needle problem. Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using:

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:pi pprox rac{2nL}{xS}

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