Pi

For other uses, see pi.

The minuscule, or lower-case, pi

The mathematical constant π represents the ratio of a circle's circumference to its diameter and is commonly used in mathematics, physics, and engineering. π is the lowercase Greek letter equivalent to "p" in the Roman alphabet; its name is "pi" (pronounced pie in English, but pee in Greek), and this spelling can be used in typographical contexts where the Greek letter is not available. π is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number.

In Euclidean plane geometry, π may be defined either as the ratio of a circle's circumference to its diameter, or as the ratio of a circle's area to the area of a square whose side is the radius. Advanced textbooks define π analytically using trigonometric functions, for example as the smallest positive x for which sin(x) = 0, or as twice the smallest positive x for which cos(x) = 0. All these definitions are equivalent.

The numerical value of π approximated to 70 decimal places (sequence A000796 in the OEIS) is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 

Although this precision is more than sufficient for use in engineering and science, much effort over the last few centuries has been put into computing as many digits as possible, for prestige as well as mathematical inquiry. Digits of π are available from multiple resources on the Internet, and a regular personal computer can be used to compute billions of digits. Recent calculations with supercomputers have found over 1 trillion digits of π, but despite this, no pattern in the digits has ever been found.

Properties

π is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert.

π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational (equivalently, integer) coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to square the circle, that is, it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle.

Formulae involving π

Geometry

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

Geometrical shape	Formula
Circumference of circle of radius r and diameter d	$C=\pi d=2\pi r\,\!$
Area of circle of radius r	$A=\pi r^{2}\,\!$
Area of ellipse with semiaxes a and b	$A=\pi ab\,\!$
Volume of sphere of radius r and diameter d	$V={\frac {4}{3}}\pi r^{3}={\frac {1}{6}}\pi d^{3}\,\!$
Surface area of sphere of radius r	$A=4\pi r^{2}\,\!$
Volume of cylinder of height h and radius r	$V=\pi r^{2}h\,\!$
Surface area of cylinder of height h and radius r	$A=2(\pi r^{2})+(2\pi r)h=2\pi r(r+h)\,\!$
Volume of cone of height h and radius r	$V={\frac {1}{3}}\pi r^{2}h\,\!$
Surface area of cone of height h and radius r	$A=\pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!$

(All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.)

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

Analysis

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

• François Viète, 1593 (proof):

${\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}{\frac {\sqrt {2+{\sqrt {2}}}}{2}}{\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\ldots$

• Leibniz' formula (proof):

${\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}$

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

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {\pi }{4}}$

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

${\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {\pi }{2}}$

$\prod _{n=1}^{\infty }{\frac {(2n)^{2}}{(2n)^{2}-1}}=\prod _{n=1}^{\infty }{\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}={\frac {\pi }{2}}$

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

$\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}$

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

$\zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}$

$\zeta (4)={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}$

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

• Gamma function evaluated at 1/2:

$\Gamma \left({1 \over 2}\right)={\sqrt {\pi }}$

• Stirling's approximation:

$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$

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

$e^{i\pi }+1=0\;$

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

$\sum _{k=1}^{n}\phi (k)\sim 3n^{2}/\pi ^{2}$

• Area of one quarter of the unit circle:

$\int _{0}^{1}{\sqrt {1-x^{2}}}\,dx={\pi \over 4}$

• An application of the residue theorem

$\oint {\frac {dz}{z}}=2\pi i,$

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

Continued fractions

π has many continued fractions representations, including:

${\frac {4}{\pi }}=1+{\frac {1}{3+{\frac {4}{5+{\frac {9}{7+{\frac {16}{9+{\frac {25}{11+{\frac {36}{13+...}}}}}}}}}}}}$

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

Number theory

Some results from number theory:

• 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.$\prod _{p\in \mathbb {P} }\left(1-{\frac {1}{p^{2}}}\right)={\frac {6}{\pi ^{2}}}$

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.

The remarkable fact that

$e^{\pi {\sqrt {163}}}=262537412640768743.99999999999925007...$ or equivalently,

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

can be explained by the theory of complex multiplication.

Dynamical systems / ergodic theory

In dynamical systems theory (see also ergodic theory), for almost every real-valued x₀ in the interval [0,1],

$\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}{\sqrt {x_{i}}}={\frac {2}{\pi }}\,,$

where the x_i are iterates of the logistic map for r = 4.

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 $\hbar ={\frac {h}{2\pi }}$ 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.

• Heisenberg's uncertainty principle:

$\Delta x\Delta p\geq {\frac {h}{4\pi }}$

• Einstein's field equation of general relativity:

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

• Coulomb's law for the electric force:

$F={\frac {\left|q_{1}q_{2}\right|}{4\pi \epsilon _{0}r^{2}}}$

• Magnetic permeability of free space:

$\mu _{0}=4\pi \times 10^{-7}\,\mathrm {H/m} \,$

Probability and statistics

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

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

$f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}$

• pdf for the (standard) Cauchy distribution:

$f(x)={\frac {1}{\pi (1+x^{2})}}$

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.

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:

$\pi \approx {\frac {2nL}{xS}}$

History of π

Main article: History of Pi.

π has been known in some form since antiquity. References to measurements of a circular basin in the Bible give a corresponding value of 3 for π: "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." — 1 Kings 7:23; KJV. (Nehemiah, a late antique Jewish rabbi and mathematician explained this apparent lack of precision in π, by considering the thickness of the basin, and assuming that the thirty cubits was the inner circumference, while the ten cubits was the diameter of the outside of the basin.)

Numerical approximations of π

Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. Therefore numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π.

An Egyptian scribe named Ahmes wrote the oldest known text to give an approximate value for π. The Rhind Mathematical Papyrus dates from the Egyptian Second Intermediate Period—though Ahmes states that he copied a Middle Kingdom papyrus—and describes the value in such a way that the result obtained comes out to 256 divided by 81 or 3.160.

The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.

The Indian mathematician and astronomer Aryabhata gave an accurate approximation for π. He wrote "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words (4+100)×8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.

The Chinese mathematician and astronomer Zu Chongzhi computed π to 3.1415926 to 3.1415927 and gave two approximations of π 355/113 and 22/7 in the 5th century.

The Iranian mathematician and astronomer, Ghyath ad-din Jamshid Kashani, 1350-1439, computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

2 π = 6.2831853071795865

The German mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimals. He was so proud of this accomplishment that he had them inscribed on his tombstone.

The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today.

None of the formulas given above can serve as an efficient way of approximating π. For fast calculations, one may use formulas such as Machin's:

${\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}$

together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with

$(5+i)^{4}\cdot (-239+i)=-114244-114244i.$

Formulas of this kind are known as Machin-like formulas.

Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used in the past.

The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulas were used for this:

${\frac {\pi }{4}}=12\arctan {\frac {1}{49}}+32\arctan {\frac {1}{57}}-5\arctan {\frac {1}{239}}+12\arctan {\frac {1}{110443}}$

K. Takano (1982).

${\frac {\pi }{4}}=44\arctan {\frac {1}{57}}+7\arctan {\frac {1}{239}}-12\arctan {\frac {1}{682}}+24\arctan {\frac {1}{12943}}$

F. C. W. Störmer (1896).

These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers and (obviously) for establishing new π calculation records.

In 1996 David H. Bailey, together with Peter Borwein and Simon Plouffe, discovered a new formula for π as an infinite series:

$\pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)$

This formula permits one to easily compute the k^th binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0).

Other formulas that have been used to compute estimates of π include:

${\frac {\pi }{2}}=\sum _{k=0}^{\infty }{\frac {k!}{(2k+1)!!}}=1+{\frac {1}{3}}\left(1+{\frac {2}{5}}\left(1+{\frac {3}{7}}\left(1+{\frac {4}{9}}(1+...)\right)\right)\right)$

Newton.

${\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}$

Ramanujan.

This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π.

${\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}$

David Chudnovsky and Gregory Chudnovsky.

${\pi }=20\arctan {\frac {1}{7}}+8\arctan {\frac {3}{79}}$

Euler.

On computers running Microsoft Windows OS, the program PiFast can be used to quickly calculate a large amount of digits. The largest number of digits of π calculated on a home computer, 25,000,000,000, was calculated with PiFast in 17 days.

Miscellaneous formulas

In base 60, π can be approximated to eight significant figures as

$3+{\frac {8}{60}}+{\frac {29}{60^{2}}}+{\frac {44}{60^{3}}}$

In addition, the following numerical formula will give an accurate approximation of π to 9 digits:

$(63/25)((17+15{\sqrt {5}})/(7+15{\sqrt {5}}))$

The following fraction will give an approximation of π to 17 digits:

$3+{\frac {48178703}{340262731}}$

The following sum will give an approximation of π to 3 digits:

${\sqrt {2}}+{\sqrt {3}}$

Karl Popper conjectured that Plato knew this expression, and that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry, and his repeated discussion of right triangles which are either isosceles or halves of equilateral triangles.

Least accurate approximations

In 1897, a physician and amateur mathematician from Indiana named Edward J. Goodwin found the transcendental value of π wrong. He proposed a bill to Indiana Representative T. I. Record which expresses the "new mathematical truth" in several ways:

The ratio of the diameter of a circle to its circumference is 5/4 to 4. (π= 3.2)

The ratio of the length of a 90 degree arc to the length of a segment connecting the arc's two endpoints is 8 to 7. (π ~ 3.23...)

The area of a circle equals the area of a square whose side is 1/4 the circumference of the circle. (π= 4)

It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side. (π= 9.24)

The bill also recites Goodwin's previous accomplishments: "his solutions of the trisection of the angle, doubling the cube [and the value of π] having been already accepted as contributions to science by the American Mathematical Monthly....And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man's ability to comprehend."

The Indiana Assembly referred the bill to the Committee on Swamp Lands, which Petr Beckmann has seen as symbolic. It was transferred to the Committee on Education, which reported favorably, and the bill passed unanimously. One argument used was that Goodwin had copyrighted his discovery, and proposed to let the State use it in the public schools for free. As this debate concluded, Professor C. A. Waldo arrived in Indianapolis to secure the annual appropriation for the Indiana Academy of Sciences. An assemblyman handed him the bill, offering to introduce him to the genius who wrote it. He declined, saying that he already knew as many crazy people as he cared to.

The Indiana Senate had not yet finally passed the bill (which they had referred to the Committee on Temperance), and Professor Waldo coached enough Senators overnight that they postponed the bill indefinitely. source

Open questions

The most pressing open question about π is whether it is a normal number -- whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly". This must be true in any base, not just in base 10. Current knowledge in this direction is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of π.

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details.

It is also unknown whether π and e are algebraically independent, i.e. whether there is a polynomial relation between π and e with rational coefficients.

John Harrison, (1693-1776) (of Longitude fame), devised a meantone temperament musical tuning system derived from π. This Lucy Tuning system (due to the unique mathematical properties of π), can map all musical intervals, harmony and harmonics. This suggests that musical harmonics beat, and that using π could provide a more precise model for the analysis of both musical and other harmonics in vibrating systems.

The nature of π

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulae in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. The reason it occurs so often in physics is simply because it's convenient in many physical models.

For example, consider Coulomb's law

$F={\frac {1}{4\pi \epsilon _{0}}}{\frac {\left|q_{1}q_{2}\right|}{r^{2}}}$.

Here, 4πr² is just the surface area of sphere of radius r. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance r from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If Planck charge is used, it can be written as

$F={\frac {q_{1}q_{2}}{r^{2}}}$

and thus eliminate the need for π.

Fictional references

• Contact -- Carl Sagan's science fiction work. Sagan contemplates the possibility of finding a signature embedded in the base-11 expansion of Pi by the creators of the universe.
• π (movie) -- On the relationship between numbers and nature: finding one without being a numerologist.
• Time's Eye -- science fiction by Arthur C. Clarke and Stephen Baxter. In a world restructured by alien forces, a spherical device is observed whose circumference to diameter ratio appears to be an exact integer 3 across all planes.

π culture

There is an entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, which is known as piphilology. See Pi mnemonics for examples.

March 14 (3/14 in US date format) marks Pi Day which is celebrated by many lovers of π.

On July 22, Pi Approximation Day is celebrated (22/7 - in European date format - is a popular approximation of π).

In the early hours of Saturday 2 July, 2005, a Japanese mental health counsellor, Akira Haraguchi, 59, managed to recite Pi's first 83,431 decimal places from memory, thus breaking the standing world record [1].

Related articles

• List of topics related to pi
• Greek letter pi
• Calculus
• Geometry
• Trigonometric function
• Pi through experiment
• Proof that π is transcendental
• A simple proof that 22/7 exceeds π
• Feynman point
• Petr Beckmann, A History of Pi
• Pi Day
• Lucy Tuning

External links

Digit resources

10,000 Places | 100,000 Places | 1,000,000 Places

• Project Gutenberg E-Text containing a million digits of Pi
• Archives of Pi calculated to 1,000,000 or 10,000,000 places.
• Search π – search and print π's digits (400 million places)
• Statistics about the first 1.2 trillion digits of Pi
• A banner of approximately 220 million digits of pi

Calculation

• Calculating Pi: The open source project for calculating Pi.
• Background Pi: An open source project for calculating Pi over many computers. (Inspired by "Calulating Pi", Above)
• PiFast: a fast program for calculating Pi with a large number of digits
• PiHex Project
• Super Pi: Another program to calculate Pi to the 33.55 millionth digit. Also used a benchmark
• PiSlice: A distributed computing project to calculate Pi
• Calculating the digits of π using generalised continued fractions - open source Python code

General

• J J O'Connor and E F Robertson: A history of Pi. Mac Tutor project
• A collection of Machin-type formulas for Pi
• A proof that Pi Is Irrational
• PiFacts-Record Broken
• The Joy of Pi-About the Book
• From the Wolfram Mathematics site lots of formulae for π
• PlanetMath: Pi
• The pi-hacks Yahoo! Group
• Finding the value of Pi
• Proof that Pi exists
• Friends of Pi Club (German and English)
• Determination of Pi
• LucyTuning - musical tuning derived from Pi

Mnemonics

• One of the more popular mnemonic devices for remembering pi
• Andreas P. Hatzipolakis: PiPhilology. A site with hundreds of examples of π mnemonics
• Pi memorised as poetry