Categories: Transcendental numbers | Mathematical constants | Famous numbers | Pi

Pi

The mathematical constant π (written as "pi" when the Greek letter is not available) is commonly used in mathematics and physics. In Euclidean plane geometry, π may be defined as either the ratio of a circle's circumference to its diameter, or as the area of a circle of radius 1 (the unit circle). Most modern 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.

Pi is also known as Archimedes' constant (not to be confused with Archimedes' number) and Ludolph's number.

The numerical value of π approximated to 64 decimal places is:

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 5923

More digits of π are also available. See pi to 1,000 places , 10,000 places , 100,000 places , and 1,000,000 places . This page is also good for up to 200 million places: pi search page

Contents

1 Properties

2 Formulas involving π

2.1 Geometry
2.2 Analysis
2.3 Complex analysis
2.4 Continued fractions
2.5 Number theory
2.6 Dynamical systems / ergodic theory
2.7 Physics
2.8 Probability and statistics

3 History

4 Numerical approximations of π

5 Open questions

6 The nature of π

7 π culture

8 Related articles

9 External links

9.1 Digit resources
9.2 Calculation
9.3 General
9.4 Mnemonics

Properties

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. In fact, the number is 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. This means that it is impossible to express π using only a finite number of integers, fractions and their square roots. This result establishes the impossibility of squaring the circle: it is impossible to construct, using ruler and compass alone, a square whose area is equal to the area of a given circle. The reason is that the coordinates of all points that can be constructed with ruler and compass are constructible numbers.

While the original Greek letter π was phonetically equivalent to the English letter p, it has now evolved to be pronounced like the word pie in most circles.

Formulas involving π

Geometry

π appears in many formulas 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	$A = \pi a b \,\!$
Volume of sphere of radius r	$V = \frac{4}{3} \pi r^3 \,\!$
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	$A = \pi r \sqrt{r^2 + h^2} + \pi r^2 = \pi r (r + \sqrt{r^2 + h^2}) \,\!$

Also, the angle measurement 180° (in 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.

François Viète, 1593 (proof):

$\frac2\pi= \frac{\sqrt2}2 \frac{\sqrt{2+\sqrt2}}2 \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\ldots$

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:

$\sum_{n=0}^{\infty} (-1)^{n} \left (\frac{1}{2n+1}\right ) = \frac{\pi}{4}$

Wallis's product (proof):

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

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,

ζ(2 n)

is a rational multiple of

π 2 n

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):

$\sum_{k=0}^{n} \phi (k) \sim 3 n^2 / \pi^2$

Area of one quarter of the unit circle:

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

Complex analysis

A special case of Euler's formula

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

An application of the residue theorem

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

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.

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.

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 formulas 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 formulas 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:

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

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:

$F = \frac{\left|q_1q_2\right|}{4 \pi \epsilon_0 r^2}$

Magnetic permeability of free space:

$\mu_0 = 4 \pi \times 10^{-7} \,$

Probability and statistics

In probability and statistics, there are many distributions whose formulas 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 formulas 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

The symbol "π" for Archimedes' constant was first introduced in 1706 by William Jones when he published A New Introduction to Mathematics, although the same symbol had been used earlier to indicate the circumference of a circle. The notation became standard after it was adopted by Leonhard Euler. In either case, π is the first letter of περιμετρος (perimetros), meaning 'measure around' in Greek.

Here is a brief chronology of π:

Date	Person	Value of π (world records in bold)
20th century BC	Babylonians	25/8 = 3.125
20th century BC	Egyptian Rhind Mathematical Papyrus	(16/9)² = 3.160493...
12th century BC	Chinese	3
mid 6th century BC	1 Kings 7:23	3
434 BC	Anaxagoras tried to square the circle with straightedge and compass
3rd century BC	Archimedes	223/71 < π < 22/7 (3.140845... < π < 3.142857...) 211875/67441 = 3.14163...
20 BC	Vitruvius	25/8 = 3.125
130	Chang Hong	√10 = 3.162277...
150	Ptolemy	377/120 = 3.141666...
250	Wang Fau	142/45 = 3.155555...
263	Liu Hui	3.14159
480	Zu Chongzhi	3.1415926 < π < 3.1415927
499	Aryabhatta	62832/20000 = 3.1416
598	Brahmagupta	√10 = 3.162277...
800	Al Khwarizmi	3.1416
12th Century	Bhaskara	3.14156
1220	Fibonacci	3.141818
1400	Madhava	3.14159265359
All records from 1424 are given as the number of correct decimal places (dps).
1424	Jamshid Masud Al Kashi	16 dps
1573	Valenthus Otho	6 dps
1593	François Viète	9 dps
1593	Adriaen van Roomen	15 dps
1596	Ludolph van Ceulen	20 dps
1615	Ludolph van Ceulen	32 dps
1621	Willebrord Snell (Snellius), a pupil of Van Ceulen	35 dps
1665	Isaac Newton	16 dps
1699	Abraham Sharp	71 dps
1700	Seki Kowa	10 dps
1706	John Machin	100 dps
1706	William Jones introduced the Greek letter π
1730	Kamata	25 dps
1719	De Lagny calculated 127 decimal places, but not all were correct	112 dps
1723	Takebe	41 dps
1734	Leonhard Euler adopted the Greek letter π and assured its popularity
1739	Matsunaga	50 dps
1761	Johann Heinrich Lambert proved that π is irrational
1775	Euler pointed out the possibility that π might be transcendental
1789	Jurij Vega calculated 140 decimal places, but not all are correct	137 dps
1794	Adrien-Marie Legendre showed that π² (and hence π) is irrational, and mentioned the possibility that π might be transcendental.
1841	Rutherford calculated 208 decimal places, but not all were correct	152 dps
1844	Zacharias Dase and Strassnitzky	200 dps
1847	Thomas Clausen	248 dps
1853	Lehmann	261 dps
1853	Rutherford	440 dps
1853	William Shanks	527 dps
1855	Richter	500 dps
1874	William Shanks took 15 years to calculate 707 decimal places, but not all were correct (the error was found by D. F. Ferguson in 1946)	527 dps
1882	Lindemann proved that π is transcendental (the Lindemann-Weierstrass theorem)
1946	D. F. Ferguson used a desk calculator	620 dps
1947		710 dps
1947		808 dps
All records from 1949 onwards were calculated with electronic computers.
1949	J. W. Wrench, Jr, and L. R. Smith were the first to use an electronic computer (the Eniac) to calculate π	2,037 dps
1953	Mahler showed that π is not a Liouville number
1955	J. W. Wrench, Jr, and L. R. Smith	3,089 dps
1961		100,000 dps
1966		250,000 dps
1967		500,000 dps
1974		1,000,000 dps
1992		2,180,000,000 dps
1995	Yasumasa Kanada	> 6,000,000,000 dps
1997	Kanada and Takahashi	> 51,500,000,000 dps
1999	Kanada and Takahashi	> 206,000,000,000 dps
2002	Kanada and team	> 1,240,000,000,000 dps
2003	Kanada and team	> 1,241,100,000,000 dps

Numerical approximations of π

Due to the transcendental nature of π, there are no nice closed expressions for π. Therefore numerical calculations must use approximations to the number. 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 is the source of 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 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 radius 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 digit 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) 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.

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

Ramanujan.

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

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.

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 Coulomb's law $F = \frac{1}{ 4 \pi \epsilon_0} \frac{\left|q_1 q_2\right|}{r^2}$ , here $4π r 2$ is just the surface area of sphere of radius r, which 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.

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) marks Pi Day which is celebrated by many lovers of π. On July 22, Pi Approximation Day is celebrated (22/7 is a popular approximation of π).

Furthermore, many talk of "pi o clock" [fifteen seconds past fourteen minutes past 3 (3:14:15) is slightly less than pi o clock; 3:08:30 would be closest to π hours past noon or midnight in whole seconds].

Another example of math-humor is this approximation of π: Take the number "1234", transpose the first two digits and the last two digits, so the number becomes "2143". Divide that number by "two-two" (22, so 2143/22 = 97.40909...). Take the two-squaredth root (4th root) of this number. The final outcome is remarkably close to π: 3.14159265.

External links

Digit resources

Wikisource - Pi to 1,000 Places | 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

General

Mnemonics

Categories: Transcendental numbers | Mathematical constants | Famous numbers | Pi

Last updated: 10-19-2005 23:29:19

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