Pi

It is approximately equal to: 3.141592653589793238462643383279...

It appears in many formulae across mathematics and physics.

Consequently, its decimal representation never ends, nor does it enter a permanently repeating pattern. It is also a transcendental number, meaning it is not a root of any non-zero polynomial with rational coefficients.

Fundamentals

Name

Definition

Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined using limits – a concept in calculus.

The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential equation.

Like the cosine, the complex exponential can be defined in one of several ways.

Irrationality and normality

Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem.

Transcendence

Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success – despite the fact that it is mathematically impossible.

This would be resolved by Schanuel's conjecture – a currently unproven generalization of the Lindemann–Weierstrass theorem.

Continued fractions

These numbers are among the best-known and most widely used historical approximations of the constant.

Complex numbers and Euler's identity

History

The earliest written approximations are found in Babylon and Egypt, both within one percent of the true value. In the Shulba Sutras of Indian mathematics, dating to an oral tradition from the 1st or 2nd millennium BCE, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.

Polygon approximation era

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD). Around 1220, Fibonacci computed 3.1418 using a polygonal method devised independently of Archimedes.

Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10^40 sides. Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation.

Infinite series

An infinite series is the sum of the terms of an infinite sequence. The series are presented without proof, but proofs are presented in the later work Yuktibhāṣā, published around 1530. Several infinite series are described, including series for sine (which Nilakantha attributes to Madhava of Sangamagrama), cosine, and arctangent which are now sometimes referred to as Madhava series. The series for arctangent is sometimes called Gregory's series or the Gregory–Leibniz series.

In 1655, John Wallis published what is now known as the Wallis product, also an infinite product:

"I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."

In 1671, James Gregory, and independently, Leibniz in 1673, discovered the Taylor series expansion for arctangent:

In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:

Isaac Newton accelerated the convergence of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):

Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.

The following table compares the convergence rates of these two series:

Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.

Irrationality and transcendence

Lambert's proof exploited a continued-fraction representation of the tangent function.

Adoption of the symbol π

(Before then, mathematicians sometimes used letters such as c or p instead.)

Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.

).

Modern quest for more digits

Motives for computing π

According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom. Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application.

Computer era and iterative algorithms

Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and John von Neumann that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC computer. The record, always relying on an arctan series, was broken repeatedly (3,089 digits in 1955, 7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.

They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods.

The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent. These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm. As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiply the number of correct digits at each step. For example, the Brent–Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.

Rapidly convergent series

New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive. One of his formulae, based on modular equations, is

This series converges much more rapidly than most arctan series, including Machin's formula. Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (Jonathan and Peter) and the Chudnovsky brothers. The Chudnovsky formula developed in 1987 is

For similar formulae, see also the Ramanujan–Sato series.

Monte Carlo methods

The associated random walk is

This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below.

Spigot algorithms

This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.

Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995. Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.

Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe:

Individual binary digits may be extracted from individual hexadecimal digits, and octal digits can be extracted from one or two hexadecimal digits.

Role and characterizations in mathematics

Geometry and trigonometry

Some of the formulae above are special cases of the volume of the n-dimensional ball and the surface area of its boundary, the (n−1)-dimensional sphere, given below.

Apart from circles, there are other curves of constant width. The Reuleaux triangle (formed by the intersection of three circles with the sides of an equilateral triangle as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular smooth and even algebraic curves of constant width.

For example, an integral that specifies half the area of a circle of radius one is given by:

Unit of angle

Eigenvalues

In many applications, it plays a distinguished role as an eigenvalue.

Inequalities

In two dimensions, the critical Sobolev inequality is

The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.

Wirtinger's inequality also generalizes to higher-dimensional Poincaré inequalities that provide best constants for the Dirichlet energy of an n-dimensional membrane.

Just as Wirtinger's inequality is the variational form of the Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any dimension.

Fourier transform and Heisenberg uncertainty principle

The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,

The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below.

Gaussian integrals

The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in experiments follows a normal distribution.

This follows from a change of variables in the Gaussian integral:

Indeed, according to Howe (1980), the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.

Topology

An example is the surface area of a sphere S of curvature 1 (so that its radius of curvature, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from its homology groups and is found to be equal to two. Thus we have

The constant appears in many other integral formulae in topology, in particular, those involving characteristic classes via the Chern–Weil homomorphism.

Cauchy's integral formula

Vector calculus and physics

Perhaps the simplest example of this is the two-dimensional Newtonian potential, representing the potential of a point source at the origin, whose associated field has unit outward flux through any smooth and oriented closed surface enclosing the source:

For example, in three dimensions, the Newtonian potential is:

Total curvature

In the differential geometry of curves, the total curvature of a smooth plane curve is the amount it turns anticlockwise, in radians, from start to finish, computed as the integral of signed curvature with respect to arc length:

Gamma function and Stirling's approximation

The gamma function is defined by its Weierstrass product development:

Further, it follows from the functional equation that

Equivalently,

Then

Ehrhart's volume conjecture predicts that this is the (optimal) upper bound on the volume of a convex body containing only one integer lattice point.

Number theory and Riemann zeta function

Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:

This is a special case of Weil's conjecture on Tamagawa numbers, which asserts the equality of similar such infinite products of arithmetic quantities, localized at each prime p, and a geometrical quantity: the reciprocal of the volume of a certain locally symmetric space.

Furthermore, the derivative of the zeta function satisfies

This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula. The calculation can be recast in quantum mechanics, specifically the variational approach to the spectrum of the hydrogen atom.

Fourier series

This is a version of the one-dimensional Poisson summation formula.

This is the simplest example of a projective representation, and the same phenomenon recurs more broadly in harmonic analysis. The same ideas also connect Fourier analysis with arithmetic, since theta functions and modular forms of half-integral weight are naturally governed by these metaplectic and Heisenberg-theoretic constructions.

Modular forms and theta functions

An example is the Jacobi theta function

Certain identities hold for all automorphic forms. An example is

Cauchy distribution and potential theory

The Cauchy distribution

The total probability is equal to one, owing to the integral:

The Cauchy distribution plays an important role in potential theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane. Conjugate harmonic functions and hence the Hilbert transform are associated with the asymptotics of the Poisson kernel. The Hilbert transform H is the integral transform given by the Cauchy principal value of the singular integral

In the Mandelbrot set

Outside mathematics

Describing physical phenomena

Before 20 May 2019, it was defined as exactly

Memorizing digits

In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.

Such memorization aids are called mnemonics. An early example of a mnemonic for pi, originally devised by English scientist James Jeans, is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." When a poem is used, it is sometimes referred to as a piem.

In popular culture

In the Palais de la Découverte (a science museum in Paris) there is a circular room known as the pi room. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1873 calculation by English mathematician William Shanks, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.

This part of the story was omitted from the film adaptation of the novel.

A college cheer variously attributed to the Massachusetts Institute of Technology or the Rensselaer Polytechnic Institute includes "3.14159". Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi. In parts of the world where dates are commonly noted in day/month/year format, 22 July represents "Pi Approximation Day", as 22/7 ≈ 3.142857.

The bill is notorious as an attempt to establish a value of mathematical constant by legislative fiat. While the bill was passed by the Indiana House of Representatives, it was rejected by the Indiana Senate, and thus did not become a law.