Pi

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Lower-case pi

The mathematical constant π is a real number which may be defined as the ratio of a circle's circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. The name of the Greek letter π is pi (pronounced pie in English), 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 Template: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, the exact value of π has decimal places that never end. Much effort over 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

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.

Formulae involving π

Geometry

<math>\pi</math> appears in many formulae in geometry involving circles and spheres.

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

(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.

<math>\frac2\pi=

\frac{\sqrt2}2 \frac{\sqrt{2+\sqrt2}}2 \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\ldots</math>

<math>\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4}</math>
This commonly cited infinite series is usually written as above, but is more technically expressed as:
<math>\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} = \frac{\pi}{4}</math>
<math> \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} </math>
<math>\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}</math>
<math>\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 ]</math>
<math>\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}</math>
<math>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}</math>
<math>\zeta(4)= \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots = \frac{\pi^4}{90}</math>
and generally, <math>\zeta(2n)</math> is a rational multiple of <math>\pi^{2n}</math> for positive integer n
<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}</math>
<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math>
<math>e^{i \pi} + 1 = 0\;</math>
<math>\sum_{k=1}^{n} \phi (k) \sim 3 n^2 / \pi^2</math>
  • Area of one quarter of the unit circle:
<math>\int_0^1 \sqrt{1-x^2}\,dx = {\pi \over 4}</math>
<math>\oint\frac{dz}{z}=2\pi i ,</math>
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:

<math> \frac{4}{\pi} = 1 + \frac{1}{3 + \frac{4}{5 + \frac{9}{7 + \frac{16}{9 + \frac{25}{11 + \frac{36}{13 + ...}}}}}} </math>

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

Number theory

Some results from number theory:

  • 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.<math> \prod_{p\in\mathbb{P}} \left(1-\frac {1} {p^2} \right) = \frac {6} {\pi^2} </math>

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 fact (note the order to which the number approaches an integer) that

<math>e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007...</math>

or equivalently,

<math>e^{\pi \sqrt{163}} = 640320^3+743.99999999999925007...</math>

can be explained by the theory of complex multiplication.

Dynamical systems and ergodic theory

Consider the recurrence relation

<math>x_{i+1} = 4 x_i (1 - x_i) \,</math>

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

<math> \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{n} \sqrt{x_i} = \frac{2}{\pi} </math>

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.

<math>\Lambda = {{8\pi G} \over {3c^2}} \rho</math>
<math> \Delta x \Delta p \ge \frac{h}{4\pi} </math>
<math> R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik} </math>
<math> F = \frac{\left|q_1q_2\right|}{4 \pi \epsilon_0 r^2} </math>
<math> \mu_0 = 4 \pi \times 10^{-7}\,\mathrm{H/m}\,</math>

Probability and statistics

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

<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math>
<math>f(x) = \frac{1}{\pi (1 + x^2)}</math>

Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1</math>, 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:

<math>\pi \approx \frac{2nL}{xS}</math>

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 appears to 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. The thickness of the basin was recorded as a "hand-breadth." If one considers a hand's breadth to be four inches, and uses a figure of 17.75 inches for a cubit, the true Biblical value of π comes out to 355/113, a suprisingly accurate approximation.

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 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.

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:

<math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} </math>

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

<math>(5+i)^4\cdot(-239+i)=-114244-114244i.</math>

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:

<math> \frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}</math>
K. Takano (1982).
<math> \frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}</math>
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 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series:

<math>\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)</math>

This formula permits one to easily compute the kth 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:

<math>

\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) </math>

Newton.
<math> \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} </math>
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.

<math> \frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}} </math>
David Chudnovsky and Gregory Chudnovsky.
<math>{\pi} = 20 \arctan\frac{1}{7} + 8 \arctan\frac{3}{79} </math>
Euler.

Miscellaneous formulae

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

<math> 3 + \frac{8}{60} + \frac{29}{60^2} + \frac{44}{60^3}</math>

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

  • accurate to 9 digits:
<math>(63/25)((17+15\sqrt 5)/(7+15\sqrt5))</math>
  • accurate to 17 digits:
<math>3 + \frac{48178703}{340262731}</math>
  • accurate to 3 digits:
<math>\sqrt{2} + \sqrt{3}</math>
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.
  • The continued fraction representation of π can be used to generate successively better rational approximations of pi, 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.

Less accurate approximations

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

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, now called Lucy Tuning.

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

<math> F = \frac{1}{ 4 \pi \epsilon_0} \frac{\left|q_1 q_2\right|}{r^2} </math>.

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

<math> F = \frac{q_1 q_2}{r^2} </math>

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

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!"

Singer Kate Bush's recently released album "Aerial" contains a song titled "π," in which she sings π to over one hundred decimal places. Fans have discovered that she got some of them wrong[2], however, and actually misses twenty-two numbers. Fans are calling Bush's version "Kate's π."

See also

References

External links

Digit resources

Calculation

General

Mnemonics

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