**Pi**

Lower-case pi |

The mathematical constant π is the ratio of a circle's circumference (Greek
περιφέρεια, periphery) to its diameter and is commonly used in mathematics,
physics, and engineering. The name of the Greek letter π is pi (pronounced pie),
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 truncated to 50 decimal places
(sequence A000796 in OEIS) is:

3.14159 26535 89793 23846 26433 83279
50288 41971 69399 37510

Although this precision is more than sufficient
for use in engineering and science, much effort o

ver the last few centuries has been put into computing more digits and
investigating the number's properties. Despite much analytical work, in addition
to supercomputer calculations that have determined over 1 trillion digits of π,
no pattern in the digits has ever been found. Digits of π are available from
multiple resources on the Internet, and a regular personal computer can compute
billions of digits with available software.

__Contents__

1
Properties

2 Formulae involving
π

2.1
Geometry

2.2
Analysis

2.3 Continued
fractions

2.4 Number
theory

2.5 Dynamical systems and ergodic
theory

2.6
Physics

2.7 Probability and
statistics

3 History of π

4 Numerical
approximations of π

4.1 Miscellaneous
formulae

4.2 Less accurate
approximations

5 Open questions

6 The
nature of π

7 Fictional references

8 π
culture

9 See also

10
References

11 External links

11.1 Digit resources

11.2
Calculation

11.3
General

11.4 Mnemonics

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__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
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.

[edit]__Formulae involving π__

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__Geometry__

π appears in many formulae in geometry involving circles and spheres

.Geometrical shape |
Formula |

Circumference of circle of radius r and diameter d | |

Area of circle of radius r | |

Area of ellipse with semiaxes a and b | |

Volume of sphere of radius r and diameter d | |

Surface area of sphere of radius r | |

Volume of cylinder of height h and radius r | |

Surface area of cylinder of height h and radius r | |

Volume of cone of height h and radius r | |

Surface area of cone of height h and radius r |

(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.

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* François Viète, 1593 (proof):

* Leibniz' formula (proof):

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

* Wallis product (proof):

* Bailey-Borwein-Plouffe algorithm (See Bailey, 1997 and Bailey web
page)

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

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

and generally,
ζ(2n) is a rational multiple of π^{2n} for positive integer n

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* Gamma function evaluated at 1/2:

*
Stirling's approximation:

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

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

* Area of
one quarter of the unit circle:

* An
application of the residue theorem

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

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π has many continued fractions representations, including:

(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/π^{2}.

* The probability that a randomly chosen integer is
square-free is 6/π^{2}.

* 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/p^{2}) over the primes, p, is
6/π^{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 (note the order to which the number approaches an integer)
that

or
equivalently,

can be explained by the theory of complex
multiplication.__Dynamical systems and ergodic theory__

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Consider the recurrence relation

Then for almost every initial
value x0 in the unit interval [0,1],

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

The number π appears
routinely in equations describing fundamental principles of the universe, due in
no small part to its relationship to the nature of the circle and,
correspondingly, spherical coordinate systems.

* The cosmological
constant:

*
Heisenberg's uncertainty principle:

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* Einstein's field equation of general relativity:

* Coulomb's law for the electric
force:

* Magnetic permeability of free space:

__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:

*
pdf for the (standard) Cauchy distribution:

Note that since , for any pdf f(x), the above
formulae can be used to produce other integral formulae 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:

Another approximation of π is to
throw points randomly into a quarter of a circle with radius 1 that is inscribed
in a square of length 1. Pi, the area of a unit circle, is then approximated as
4*(points in the quarter circle)/(total points).

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.

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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 stated 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.

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

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

Formulae of this kind are known as Machin-like formulae.

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 formulae were used for this:

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.

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In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper
(Bailey, 1997) on a new formula for π as an infinite series:

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 formulae that have been used to
compute estimates of π include:__

Newton.

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.

David Chudnovsky and
Gregory Chudnovsky.

Euler.

__Miscellaneous formulae__

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

In addition, the following expressions can be used to
estimate π

* accurate to 9 digits:

* accurate to 17
digits:

*
accurate to 3 digits:

Karl Popper conjectured that Plato knew this expression; 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 Plato's repeated
discussion of right triangles which are either isosceles or halves of
equilateral triangles. __Less accurate approximations__

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In 1897, a physician and amateur mathematician from Indiana named Edward J.
Goodwin believed that the transcendental value of π was wrong. He proposed a
bill to Indiana Representative T. I. Record which expressed 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 if rectangle is emended to triangle; if not, as
above.)

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." These false claims are typical of a mathematical crank.
The claims trisection of an angle and the doubling of the cube are particularly
widespread in crank literature.

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

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__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 formulae 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. However it is known that at least one of πe and π + e is
transcendental (q.v.).

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. Nonetheless, it occurs often in physics.

For example,
consider Coulomb's law

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Here,
4πr^{2} 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

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.

*
Eon -- science fiction novel by Greg Bear. The protagonists measure the amount
of space curvature using a device that computes π. Only in completely flat
space/time will a circle have a circumference, diameter ratio of 3.14159...
.

* Going Postal -- fantasy novel by Terry Pratchett. Famous inventor Bloody
Stupid Johnson invents an organ/mail sorter that contains a wheel for which pi
is exactly 3. This "New Pie" starts a chain of events that leads to the failure
of the Ankh-Morpork Post Office (and possibly the destruction of the Universe
all in one go.)

* π (film) -- On the relationship between numbers and nature:
finding one without being a numerologist.

* The Simpsons -- "Pi is exactly
3!" was an announcement used by Professor Frink to gain the full attention of a
hall full of scientists.

* 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. It is the first book in The Time Odyssey
series.

π culture

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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 π's
first 83,431 decimal places from memory, thus breaking the standing world record
[1].

355/113 (~3.1415929) is sometimes jokingly referred to as "not π,
but an incredible simulation!"__See also__

* 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

*
Pi Day

* Lucy Tuning

* Cadaeic Cadenza

* Software for calculating π on
personal computers

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__References__

* Bailey, David H., Borwein, Peter B., and
Plouffe, Simon (April 1997). On the Rapid Computation of Various Polylogarithmic
Constants, Mathematics of Computation, 66(218): 903-913.

* Petr Beckmann,
A History of Pi__Digit resources__

* Project
Gutenberg E-Text containing a million digits of Pi

* Pi to a million
places

* Archives of Pi calculated to 1,000,000 or 10,000,000 places.

*
Search π – search and print π's digits (up to 3.2 billion 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

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__General__

* J J O'Connor and E F Robertson: A history of
Pi. Mac Tutor project

* A collection of Machin-type formulae 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

* Pi
Symphony : An orchestral work by Lars Erickson based on the digits of pi and
'e'.

* 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

* The Pi Is Rational Page__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

* First fifty digits of Pi, memorised as a humorous
song

* Phrase to easily remember upto 8 decimal places of the value of Pi
(See Item #3 on page)

* Free software to help memorise Pi

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