Pi

When a circle's diameter is 1, its circumference is π.

Pi or π is a mathematical constant and a transcendental (and therefore irrational) real number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with an Archimedes number) and as Ludolph's number.

The letter π

Main article: pi (letter)

Lower-case π (the lower case letter is used for the constant)

The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter.

The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery' and περίμετρος 'perimeter', i.e. 'circumference'.

π is Unicode character U+03C0 ("Greek small letter pi").

Circumference = π × diameter

Definition

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:

$\pi ={\frac {c}{d}}$

Note that the ratio ^c/_d does not depend on the size of the circle. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference c, preserving the ratio ^c/_d. This fact can also be stated as saying that all circles are similar.

Area of the circle = π × area of the shaded square

Alternatively π can be also defined as the ratio of a circle's area to the area of a square whose side is the radius:

$\pi ={\frac {A}{r^{2}}}$

The constant π may be defined in other ways that avoid the concepts of arc length and area, for example, as twice the smallest positive x for which cos(x) = 0. The formulæ below illustrate other (equivalent) definitions. :)

Numerical value

The numerical value of π truncated to 50 decimal places is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

See the links below and those at sequence A000796 in OEIS for more digits.

While the value of pi has been computed to billions of digits, practical science and engineering will rarely require more than 10 decimal places. As an example, computing the circumference of the Earth's equator from its radius using only 10 decimal places of pi yields an error of less than 0.2 millimeters. A value truncated to 39 decimal places is sufficient to compute the circumference of the visible universe to a precision comparable to the size of a hydrogen atom.

Most circular objects worthy of physical study, particularly on the scale of planetary radii, have imperfections and eccentricities which account for a greater error in calculation than would be yielded by pi's approximate nature. The exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. See history of numerical approximations of π.

Calculating π

Main article: Computing π

π can be empirically measured by drawing a large circle, then measuring its diameter and circumference, since the circumference of a circle is always π times its diameter.

π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:

$\pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}\cdots$

While that series is easy to write and calculate, it is not immediately obvious why it yields π. A more intuitive approach is to draw an imaginary circle of radius r centered at the origin. Then any point (x,y) whose distance d from the origin is less than r, as given by the pythagorean theorem, will be inside the circle:

$d={\sqrt {x^{2}+y^{2}}}$

Finding a collection of points inside the circle allows the circle's area A to be approximated. For example, by using integer coordinate points for a big r. Since the area A of a circle is π times the radius squared, π can be approximated by using:

$\pi ={\frac {A}{r^{2}}}$

Properties

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

Furthermore, 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 compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.

History

Main article: History of π

Use of the symbol π

Often William Jones' book A New Introduction to Mathematics from 1706 is cited as the first text where the Greek letter π was used for this constant, but this notation became particularly popular after Leonhard Euler adopted it in 1737 (cf History of π).

Early approximations

Main article: History of numerical approximations of π

The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = ²⁵⁄₈, which is within 0.5% of the true value.

The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.

It is sometimes claimed that the Bible states that π = 3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. The discrepancy has been explained in various ways by different exegetes. Rabbi Nehemiah explained it by the diameter being measured from outside rim to outside rim while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers.

Principle of Archimedes' method to approximate π

Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between ²²³⁄₇₁ and ²²⁄₇. The average of these two values is roughly 3.1419.

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

The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = ⁶²⁸³²⁄₂₀₀₀₀ = 3.1416, correct when rounded off to four decimal places. He also said that this was a value that "approached" the correct number, which was interpreted in the 15th c. as meaning that $pi$ is irrational, a concept which would not be known in Europe till the 18th c.

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

The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series of arctan(1)=^π⁄₄ into the form

$\pi ={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right)$

and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of ^π⁄₄, he was able to compute π to an accuracy of 13 decimal places.

The Persian astronomer Ghyath ad-din Jamshid Kashani (1350–1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

2π = 6.2831853071795865

By 1610, the German mathematician Ludolph van Ceulen had finished computing the first 35 decimal places of π. It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone.

In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π, of which the first 126 were correct [1], and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). He published his value of pi in a book, which was promptly dubbed "the world's most boring book"! In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator).

Numerical approximations

Main article: History of numerical approximations of π

Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions. Formulae for calculating π using elementary arithmetic invariably include notation such as "...", which indicates that the formula is really a formula for an infinite sequence of approximations to π. The more terms included in a calculation, the closer to π the result will get, but none of the results will be π exactly.

Consequently, 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 precision. The approximations ²²/₇ and ³⁵⁵/₁₁₃, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. The approximation ³⁵⁵⁄₁₁₃ (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator.

The earliest numerical approximation of π is almost certainly the value 3. In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.

Formulae

Geometry

The constant π appears in many formulæ in geometry involving circles and spheres.

Geometrical shape	Formula
Circumference of circle of radius r and diameter d	$C=2\pi r=\pi d\,\!$
Area of circle of radius r	$A=\pi r^{2}={\frac {1}{4}}\pi d^{2}\,\!$
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 and diameter d	$A=4\pi r^{2}=\pi d^{2}\,\!$
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	$A=\pi r^{2}+\pi r{\sqrt {r^{2}+h^{2}}}=\pi r(r+{\sqrt {r^{2}+h^{2}}})\,\!$

All of these formulae are a consequence of the formula for circumference. For example, the area of a circle of radius R can be accumulated by summing annuli of infinitesimal width using the integral $A=\int _{0}^{R}2\pi rdr=\pi R^{2}.$. The others concern a surface or solid of revolution.

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

Analysis

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

• The area of the unit disc

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

• Half the circumference of the unit circle

$\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi$

• François Viète, 1593 (proof)

${\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots ={\frac {2}{\pi }}$

• Leibniz' formula (proof)

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}$

• Wallis product, 1655 (proof)

$\prod _{n=1}^{\infty }\left({\frac {n+1}{n}}\right)^{(-1)^{n-1}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots ={\frac {\pi }{2}}$

• Ramanujan formula, 1914

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

• Chebyshev series Y. Luke, Math. Tabl. Aids Comp. 11 (1957) 16

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}({\sqrt {2}}-1)^{2n+1}}{2n+1}}={\frac {\pi }{8}}.$

$\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2-{\sqrt {3}})^{2n+1}}{2n+1}}={\frac {\pi }{12}}.$

• Chudnovsky formula, 1989

${\frac {426880{\sqrt {10005}}}{\sum _{n=0}^{\infty }{\frac {(6n)!(545140134n+13591409)}{(n!)^{3}(3n)!(-640320)^{3n}}}}}=\pi$

• Symmetric formula (see Sondow, 1997)

${\frac {\displaystyle \prod _{n=1}^{\infty }\left(1+{\frac {1}{4n^{2}-1}}\right)}{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}}}={\frac {\displaystyle \left(1+{\frac {1}{3}}\right)\left(1+{\frac {1}{15}}\right)\left(1+{\frac {1}{35}}\right)\cdots }{\displaystyle {\frac {1}{3}}+{\frac {1}{15}}+{\frac {1}{35}}+\cdots }}=\pi$

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

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

• Faster product (see Sondow, 2005 and Sondow web page)

$\left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/4}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/8}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/16}\cdots ={\frac {\pi }{2}}$

where the nth factor is the 2ⁿth root of the product

$\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}.$

• 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

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

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

• 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")

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

• A property of Euler's totient function (see also Farey sequence)

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

• An application of the residue theorem

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

where the path of integration is a closed curve around the origin, traversed in the standard anticlockwise direction.

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 but not sign) is ^π/₄.

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

• The product of (1 − ¹/_p²) over the primes, p, is ⁶/_π².

The theory of elliptic curves and complex multiplication derives the approximation

$\pi \approx {\ln(640320^{3}+744) \over {\sqrt {163}}}$

which is valid to about 30 digits.

Dynamical systems and ergodic theory

Consider the recurrence relation

$x_{i+1}=4x_{i}(1-x_{i})\,$

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

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

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:

$\Lambda ={{8\pi G} \over {3c^{2}}}\rho$

• 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:

$\mu _{0}=4\pi \cdot 10^{-7}\,\mathrm {N/A^{2}} \,$

• Kepler's third law constant:

${\frac {P^{2}}{a^{3}}}={(2\pi )^{2} \over G(M+m)}$

Probability and statistics

In probability and statistics, there are many distributions whose formulæ 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 formulæ can be used to produce other integral formulae for π.

A semi-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:

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

[As a practical matter, this approximation is poor and converges very slowly.]

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. π, the area of a unit circle, is then approximated as 4×(points in the quarter circle) ÷ (total points).

Efficient methods

In the early years of the computer, the first expansion of π to 100,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in 1961.

Daniel Shanks and his team used two different power series for calculating the digits of π. For one it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,000 digits of π were published by the Naval Research Laboratory.

None of the formulæ given above can serve as an efficient way of approximating π. For fast calculations, one may use a formula 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

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

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

Many other expressions for π were developed and published by Indian mathematician Srinivasa Ramanujan. He worked with mathematician Godfrey Harold Hardy in England for a number of years.

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.

The first one million digits of π and ¹/_π are available from Project Gutenberg (see external links below). The record as of 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 formulæ 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. (Normality of π will always depend on the infinite string of digits on the end, not on any finite computation.)

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:

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

This formula permits one to fairly readily 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).

Fabrice Bellard claims to have beaten the efficiency record set by Bailey, Borwein, and Plouffe with his formula to calculate binary digits of π [2]:

$\pi ={\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {{(-1)}^{n}}{2^{10n}}}\left(-{\frac {2^{5}}{4n+1}}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)$

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

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

Newton.

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

Srinivasa 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.

Miscellaneous formulæ

The base 60 representation of π, correct to eight significant figures (in base 10) is:

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

In addition, the following expressions approximate π:

• accurate to 9 decimal places:

${\frac {63}{25}}\times {\frac {17+15{\sqrt {5}}}{7+15{\sqrt {5}}}}$

• accurate to 9 places:

${\sqrt[{4}]{\frac {2143}{22}}}$

Ramanujan claimed he had a dream in which the goddess Namagiri appeared and told him the true value of π.

• accurate to 3 decimal places:

${\sqrt[{3}]{31}}$

• accurate to 2 decimal places:

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

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 special right triangles that are either isosceles or halves of equilateral triangles.

• The continued fraction representation of π can be used to generate successively better rational approximations, which start off: ²²/₇, ³³³/₁₀₆, ³⁵⁵/₁₁₃…. These approximations are the best possible rational approximations of π relative to the size of their denominators.

Memorizing digits

Main article: Piphilology

Recent decades have seen a surge in the record number of digits memorized.

Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. A Japanese man named Akira Haraguchi claims to have memorized 100,000 decimal places. This, however, has yet to be verified by Guinness World Records. The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China. It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.

There are many ways to memorize π, including the use of piems, which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: How I need a drink, alcoholic in nature (or: of course), after the heavy lectures involving quantum mechanics. Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this manner. Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. See Pi mnemonics for examples. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of pi. Other methods include remembering patterns in the numbers (for instance, the year 1971 appears in the first fifty digits of pi).

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", and that this is true in every base, not just base 10. Current knowledge on this point 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 (see Lindemann–Weierstrass theorem).

Naturality

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 formulæ 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 (SI units)

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

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.

See also

• List of topics related to pi
• Calculus
• Geometry
• Trigonometric function
• Pi through experiment
• A simple proof that 22/7 exceeds π
• Feynman point comprising the 762nd through 767th decimal places of π, consisting of the digit 9 repeated six times.
• Indiana Pi Bill the attempt to legislate π = 3.2
• Pi Day
• Lucy Tuning
• Cadaeic Cadenza
• Software for calculating π on personal computers
• Mathematical constants: e and φ

References

Additional

• Bailey, David H., Borwein, Peter B., and Plouffe, Simon (1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: multiple names: authors list (link)
• Rudin, Walter (1976) [1953]. Principles of mathematical analysis (3e ed.). McGraw-Hill. ISBN 0-07-054235-X.
• A new formula to compute the n'th binary digit of pi by Fabrice Bellard, retrieved March 22, 2006
• Beckmann, Petr (1976). A History of π. St. Martin's Griffin. ISBN 0-312-38185-9.
• Jonathan Sondow, "A faster product for pi and a new integral for ln ^pi/₂," Amer. Math. Monthly 112 (2005) 729-734.
• Jonathan Sondow, Problem 88, Math Horizons 5 (Sept., 1997) 32, 34
• Borwein, Jonathan M.; Borwein, Peter; and Berggren, Lennart (2004). Pi: A Source Book, Springer. ISBN 0-387-20571-3.

External links

Digits

• The First Billion Digits of Pi (π)
• First 4 Million Digits of π - Warning - Roughly 2 megabytes will be transferred.
• One million digits of pi at piday.org
• Project Gutenberg E-Text containing a million digits of π
• Search the first 200 million digits of π for arbitrary strings of numbers
• Source code for calculating the digits of π
• Pi World ranking list - List of many people who have memorized large numbers of digits of π (not up-to-date).

General

• The Joy of Pi by David Blatner
• J J O'Connor and E F Robertson: A history of pi. Mac Tutor project
• A proof that π Is Irrational
• Lots of formulæ for π at MathWorld
• PlanetMath: Pi
• Finding the value of π
• Determination of π at cut-the-knot
• The Life of Pi by Jonathan Borwein
• BBC Radio Program about π
• Decimal expansions of Pi and related links at the On-Line Encyclopedia of Integer Sequences
• Statistical Distribution Information on PI based on 1.2 trillion digits of PI
• How to calculate Pi using the Monte Carlo method, explanation and source code in C++

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