|
=== [[Korean War]] ===
{{otheruses|pie}}
I have been editing this article for a few days. I ran down through the article and fixed any grammer mistakes I could find. I would appreciate any suggestions on what to do and how I can really improve it further. Thanks a lot. [[User:Mr. Killigan|Mr. Killigan]] 06:17, 12 July 2007 (UTC)
==== Kirill Lokshin ====
[[Image:Greek pi.png|right|thumb|110px|Lower-case <b>pi</b>]]
There are a number of areas to work on, at this point; keep in mind, though, that this is a very high-profile article, so you should be careful to move slowly and carefully to avoid getting entangled in any editorial conflicts here.
The [[mathematical constant]] '''π''' is a [[real number]] which may be defined as the [[ratio]] of a [[circle]]'s [[circumference]] ([[Greek language|Greek]]<!-- It would be nice if <nobr> worked: --> <u>'''π'''</u>εριφέρεια, periphery) to its [[diameter]] in [[Euclidean geometry]], and which is in common use in [[mathematics]], [[physics]], and [[engineering]]. The name of the [[Greek alphabet|Greek letter]] [[Pi (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|Archimedes' number]]) and '''[[Ludolph van Ceulen|Ludolph]]'s number'''.
* <s>The article is, in my opinion, simply too long; we're looking at 72K (~12,000 words) of prose. The "Legacy" section, in particular, is ripe for splitting out into a separate [[Legacy of the Korean War]] article, with a much shorter summary left in the main one.</s>
* <s>"Korean War (1950 – 1953)" should really be something like "Course of the war"; you probably don't want to repeat the article title as a section heading.</s>
* The citations need cleanup; at a minimum, all of the embedded external links should be converted to footnotes. There are also a number of "citation needed" tags floating around. Beyond that, more thorough citation would be appropriate throughout the article; see [[WP:MILHIST#CITE]] for some guidelines.
* <s>The "Depictions" section should be turned into prose, rather than a laundry list of films; see also [[WP:MILHIST#POP]].</s>
* <s>The "Names" section, as it's presently constituted, would work much better as a narrow sidebar; it's of some interest, but I doubt there's enough material to sustain a separate section.</s>
* <s>The "See also" section should be eliminated. If something isn't worth linking from the text, it's generally not worth linking at all. </s>
* <s>The rump "Bibliography" section should be removed as well.</s>
* The "External links" section could use trimming.
Hope that helps! [[User:Kirill Lokshin|Kirill]] 04:08, 13 July 2007 (UTC)<s>cool</s>
::Thank you very much for offering your opinion! [[User:Mr. Killigan|Mr. Killigan]] 00:57, 14 July 2007 (UTC)
In [[Euclidean geometry|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 [[mathematical analysis|analytically]] using [[trigonometric function]]s, for example as the smallest positive ''x'' for which [[trigonometric function|sin]](''x'') = 0, or as twice the smallest positive ''x'' for which [[trigonometric function|cos]](''x'') = 0.
All these definitions are equivalent.
The numerical value of π truncated to 50 [[decimal|decimal places]] {{OEIS|id=A000796}} is:
:<!--Discussion on the Talk page has determined that 50 digits are appropriate here.
Please discuss any changes to this on the Talk page.-->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 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 [[Software for calculating π|available software]].
== Properties ==
π is an [[irrational number]]; that is, it cannot be written as the ratio of two [[integer]]s, as was proven in [[1761]] by [[Johann Heinrich Lambert]].
π is also [[transcendental number|transcendental]], as was proven by [[Ferdinand von Lindemann]] in [[1882]]. This means that there is no [[polynomial]] with [[rational number|rational]] coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not [[constructible number|constructible]]. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to [[squaring the circle|square the circle]], that is, it is impossible to construct, using [[ruler-and-compass construction|ruler and compass]] alone, a square whose area is equal to the area of a given circle.
== Formulæ involving π ==
===Geometry===
<math>\pi</math> appears in many formulæ in [[geometry]] involving [[circle]]s and [[sphere]]s.
{| border="1" cellspacing="4" cellpadding="4" style="border-collapse: collapse;"
!Geometrical shape
!Formula
|-
|[[Circumference]] of circle of [[radius]] ''r'' and [[diameter]] ''d''
|<math>C = \pi d = 2 \pi r \,\!</math>
|-
|[[area (geometry)|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 (geometry)|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 (solid)|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'' = ∫(2''πr'')d''r'' ("sum of [[annulus|annuli]] of infinitesimal width"), and others concern a surface or [[solid of revolution]].)
Also, the [[angle]] measure of 180° ([[Degree (angle)|degrees]]) is equal to π [[radian]]s.
===Analysis===
Many formulæ in [[Mathematical analysis|analysis]] contain π, including [[infinite series]] (and [[infinite product]]) representations, [[integral]]s, and so-called [[List of mathematical functions|special functions]].
*[[François Viète]], [[1593]] ([[Viète formula|proof]]):
:<math>\frac2\pi=
\frac{\sqrt2}2
\frac{\sqrt{2+\sqrt2}}2
\frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\ldots</math>
*[[Gottfried Leibniz|Leibniz]]' formula ([[Leibniz formula for pi|proof]]):
:<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>
*[[John Wallis|Wallis]] product ([[Wallis product|proof]]):
:<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>
* Bailey-Borwein-Plouffe algorithm (See Bailey, 1997 and [http://www.nersc.gov/~dhbailey/ Bailey web page])
:<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>
*An [[integral]] formula from [[calculus]] (see also [[Error function]] and [[Normal distribution]]):
:<math>\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}</math>
*[[Basel problem]], first solved by [[Leonhard Euler|Euler]] (see also [[Riemann zeta function]]):
:<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
*[[Gamma function]] evaluated at 1/2:
:<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}</math>
*[[Stirling's approximation]]:
:<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math>
*[[Euler's identity]] (called by [[Richard Feynman]] "the most remarkable formula in mathematics"):
:<math>e^{i \pi} + 1 = 0\;</math>
*Property of [[Euler's totient function]] (see also [[Farey sequence]]):
:<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>
*An application of the [[residue theorem]]
:<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 fraction]]s 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 [http://functions.wolfram.com/Constants/Pi/10/ The Wolfram Functions Site].)
===Number theory===
Some results from [[number theory]]:
*The [[probability]] that two [[random]]ly chosen integers are [[coprime]] is 6/π<sup>2</sup>.
*The probability that a randomly chosen integer is [[square-free]] is 6/π<sup>2</sup>.
*The [[mean|average]] number of ways to write a positive integer as the sum of two [[perfect square]]s (order matters) is π/4.
* The [[Product (mathematics)|product]] of (1-1/p<sup>2</sup>) over the primes, ''p'', is 6/π<sup>2</sup>.<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 (mathematics)|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 everywhere|almost every]] initial value ''x''<sub>0</sub> 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 system]]s 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]]:
:<math>\Lambda = {{8\pi G} \over {3c^2}} \rho</math>
*[[Uncertainty principle|Heisenberg's uncertainty principle]]:
:<math> \Delta x \Delta p \ge \frac{h}{4\pi} </math>
*[[Einstein's field equation]] of [[general relativity]]:
:<math> R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik} </math>
*[[Coulomb's law]] for the [[electric force]]:
:<math> F = \frac{\left|q_1q_2\right|}{4 \pi \epsilon_0 r^2} </math>
*[[Permeability (electromagnetism)|Magnetic permeability of free space]]:
:<math> \mu_0 = 4 \pi \times 10^{-7}\,\mathrm{H/m}\,</math>
===Probability and statistics===
In [[probability]] and [[statistics]], there are many [[probability distribution|distributions]] whose formulæ contain π, including:
*[[probability density function]] (pdf) for the [[normal distribution]] with [[mean]] μ and [[standard deviation]] σ:
:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math>
*pdf for the (standard) [[Cauchy distribution]]:
:<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 formulæ can be used to produce other integral formulæ 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 [http://www.statisticool.com/pi.htm 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 Antiquity|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 [[approximation]]s 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 [[Moscow and Rhind Mathematical Papyri|Rhind Mathematical Papyrus]] dates from the [[Ancient Egypt|Egyptian]] [[Second Intermediate Period]]—though Ahmes stated that he copied a [[Middle Kingdom of Egypt|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 [[tomb stone|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 [[John_Machin|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 number]]s, starting with
:<math>(5+i)^4\cdot(-239+i)=-114244-114244i.</math>
Formulae of this kind are known as ''[[Machin-like formula]]e''.
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, Ltd.|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 ''k''<sup>th</sup> [[Binary numeral system|binary]] or [[hexadecimal]] digit of π, without
having to compute the preceding ''k'' − 1 digits. [http://www.nersc.gov/~dhbailey/ Bailey's website] contains the derivation as well as implementations in various [[programming language|programming languages]]. The [[PiHex]] project computed 64-bits around the [[quadrillion]]th 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>
:[[Isaac Newton|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 (mathematician)|David Chudnovsky]] and [[Gregory Chudnovsky]].
: <math>{\pi} = 20 \arctan\frac{1}{7} + 8 \arctan\frac{3}{79} </math>
:[[Euler]].
===Miscellaneous formulae===
In [[radix|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 triangle]]s 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 number|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 (person)|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. [http://faqs.jmas.co.jp/FAQs/sci-math-faq/indianabill 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 (mathematical constant)|''e'']] are [[algebraically independent]]. However it is known that at least one of π''e'' and π + ''e'' is [[transcendental number|transcendental]] (''q.v.'').<!-- redundant wikilink intentional: specifically relevant to this section-->
[[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 (geometry)|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, 4''πr''<sup>2</sup> 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 (novel)|Contact]]'' -- [[Carl Sagan|Carl Sagan's]] [[science fiction]] work. Sagan contemplates the possibility of finding a signature embedded in the [[Positional notation|base-11]] expansion of Pi by the creators of the universe.
* ''[[Eon (novel)|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.)
* ''[[Pi (film)|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 technique]]s to remember the digits of π, which is known as [[piphilology]]. See [[:q:English_mnemonics#Pi|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 [[Japan|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 [http://news.bbc.co.uk/1/hi/world/asia-pacific/4644103.stm].
355/113 (~3.1415929) is sometimes jokingly referred to as "not π, but an incredible simulation!"
Singer [[Kate Bush]]'s recently released album "[[Aerial (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, however, and actually misses twenty-two numbers. Fans are calling Bush's version "Kate's π."
==See also==
*[[List of topics related to pi]]
*[[Pi (letter)|Greek letter pi]]
*[[Calculus]]
*[[Geometry]]
*[[Trigonometric function]]
*[[Pi through experiment]]
*[[Lindemann-Weierstrass theorem|Proof that π is transcendental]]
*[[Proof that 22 over 7 exceeds π|A simple proof that 22/7 exceeds π]]
*[[Feynman point]]
*[[Pi Day]]
*[[Lucy Tuning]]
*[[Cadaeic Cadenza]]
*[[Software for calculating π]] on personal computers
==References==
* {{Journal reference
| Author = [[David H. Bailey|Bailey, David H.]], [[Peter Borwein|Borwein, Peter B.]], and [[Simon Plouffe|Plouffe, Simon]] | Year = 1997
| Month = April | Title = On the Rapid Computation of Various Polylogarithmic Constants
| Journal = Mathematics of Computation | Volume = 66 | Issue = 218 | Pages = 903-913
| URL = http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf }}
*[[Petr Beckmann]], ''A History of Pi''
== External links ==
===Digit resources===
*[http://www.gutenberg.net/etext/50 Project Gutenberg E-Text containing a million digits of Pi]
*[http://3.141592653589793238462643383279502884197169399375105820974944592.com/ Pi to a million places]
*[http://www.solidz.com/pi/ Archives of Pi calculated to 1,000,000 or 10,000,000 places.]
*[http://www.pisearch.de.vu Search π] – search and print π's digits (up to 3.2 billion places)
*[http://www.super-computing.org/pi-decimal_current.html Statistics about the first 1.2 trillion digits of Pi]
*[http://3.14.maxg.org/ A banner of approximately 220 million digits of pi]
*[http://3.141592653589793238462643383279502884197169399375105820974944592.com/ Pi to 1 million decimal places]
===Calculation===
*[http://projectpi.sourceforge.net/ Calculating Pi: The open source project for calculating Pi.]
*[http://backpi.sourceforge.net Background Pi: An open source project for calculating Pi over many computers. (Inspired by "Calulating Pi", Above)]
*[http://numbers.computation.free.fr/Constants/PiProgram/pifast.html PiFast: a fast program for calculating Pi with a large number of digits]
*[http://oldweb.cecm.sfu.ca/projects/pihex/index.html PiHex Project]
*[http://files.extremeoverclocking.com/file.php?f=36 Super Pi: Another program to calculate Pi to the 33.55 millionth digit. Also used a benchmark]
*[http://www.pislice.com/ PiSlice: A distributed computing project to calculate Pi]
*[[wikisource:Calculating the digits of pi|Calculating the digits of π using generalised continued fractions]] - open source [[Python programming language|Python]] code
===General===
*[http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Pi_through_the_ages.html J J O'Connor and E F Robertson: ''A history of Pi''. Mac Tutor project]
*[http://machination.mysite.freeserve.com/ A collection of Machin-type formulae for Pi]
*[http://www.lrz-muenchen.de/~hr/numb/pi-irr.html A proof that Pi Is Irrational]
*[http://www.joyofpi.com/pifacts.html PiFacts-Record Broken]
*[http://www.joyofpi.com/thebook.html The Joy of Pi-About the Book]
*[http://mathworld.wolfram.com/PiFormulas.html From the Wolfram Mathematics site lots of formulae for π]
*[http://www.pisymphony.com/gpage.html Pi Symphony : An orchestral work by Lars Erickson based on the digits of pi and 'e'.]
*[http://planetmath.org/encyclopedia/Pi.html PlanetMath: Pi]
*[http://groups.yahoo.com/group/pi-hacks The pi-hacks Yahoo! Group]
*[http://mathforum.org/isaac/problems/pi1.html Finding the value of Pi]
*[http://cf.geocities.com/ilanpi/pi-exists.html Proof that Pi exists]
*[http://pi314.at/ Friends of Pi Club ''(German and English)'']
* [http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml Determination of Pi] at [[cut-the-knot]]
*[http://www.lucytune.co.uk LucyTuning - musical tuning derived from Pi]
*[http://dse.webonastick.com/pi/ The Pi Is Rational Page]
===Mnemonics===
*[http://users.aol.com/s6sj7gt/mikerav.htm One of the more popular mnemonic devices for remembering pi]
*[http://www.cilea.it/~bottoni/www-cilea/F90/piph.htm Andreas P. Hatzipolakis: ''PiPhilology''. A site with hundreds of examples of π mnemonics]
*[http://www.startfromhere.freeserve.co.uk/nudesci/abc/pi.htm Pi memorised as poetry]
*[http://www.archivestowearpantsto.com/tracks/0052_i_am_the_first_fifty_digits_of_pi.mp3 First fifty digits of Pi, memorised as a humorous song]
*[http://www.geocities.com/sviveknayak/riddles.htm Phrase to easily remember upto 8 decimal places of the value of Pi (See Item #3 on page)]
*[http://brianbondy.com/other/pi.aspx Free software to help memorise Pi]
[[Category:Pi| ]]
[[Category:Famous numbers|3.1416]]
[[Category:Transcendental numbers]]
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