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pie is something you eat and it comes in many different flavors
{{Two other uses|the mathematical constant|the Greek letter|pi (letter)}}
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IMPORTANT NOTICE: Please note that Wikipedia is not a database to store the millions of digits of π; please refrain from adding those to Wikipedia, as it could cause technical problems (and it makes the page unreadable, or at least unattractive, in the opinion of most readers). Instead, you could add links in the "External links" section, to other web sites containing information regarding digits of π.

This has been established by a very clear consensus and any editor adding lists of digits of pi is liable to be blocked from editing without further warning.

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[[Image:Pi-unrolled-720.gif|thumb|360px|right|When a circle's diameter is 1, its circumference is π.]]
{| class="infobox" style ="width: 370px;"
| colspan="2" align="center" | [[List of numbers]] – [[Irrational number]]s <br> [[Apéry's constant|&zeta;(3)]] – [[Square root of 2|√2]] – [[Square root of 3|√3]] – [[Square root of 5|√5]] – [[Golden ratio|&phi;]] – [[Feigenbaum constants|&alpha;]] – [[E (mathematical constant)|e]] – [[Pi|&pi;]] – [[Feigenbaum constants|&delta;]]
|-
|[[Binary numeral system|Binary]]
| 11.00100100001111110110…
|-
| [[Decimal]]
| 3.14159265358979323846…
|-
| [[Hexadecimal]]
| 3.243F6A8885A308D31319…
|-
| [[Continued fraction]]
| <math>3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \ddots}}}}</math><br><small>Note that this continued fraction is not periodic.</small>
|}

'''Pi''' or '''π''' is a [[mathematical constant]] which represents the ratio of any [[circle]]'s circumference to its diameter in [[Euclidean geometry]], which is the same as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159. Pi is one of the most important mathematical constants: many formulae from mathematics, [[science]], and [[engineering]] involve π.<ref>{{cite book | title = An Introduction to the History of Mathematics | author = Howard Whitley Eves | year = 1969 | publisher = Holt, Rinehart & Winston | url = http://books.google.com/books?id=LIsuAAAAIAAJ&q=%22important+numbers+in+mathematics%22&dq=%22important+numbers+in+mathematics%22&pgis=1 }}</ref>

Pi is an [[irrational number]], which means that it cannot be expressed as a [[fraction]] ''m''/''n'', where ''m'' and ''n'' are [[integer]]s. Consequently its [[decimal representation]] never ends or repeats. Beyond being [[irrational number|irrational]], it is a [[transcendental number]], which means that no finite sequence of algebraic operations on [[integer]]s (powers, roots, sums, etc.) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.

The Greek letter π, often spelled out ''pi'' in text, was adopted for the number from the Greek word for ''perimeter'' "περίμετρος", probably by [[William Jones (mathematician)|William Jones]] in 1706, and popularized by [[Leonhard Euler]] some years later. The constant is occasionally also referred to as the '''circular constant''', '''[[Archimedes]]' constant''' (not to be confused with an [[Archimedes number]]), or '''[[Ludolph van Ceulen|Ludolph]]'s number'''.

==Fundamentals==
=== The letter π ===
[[Image:Pi-symbol.svg|thumb|140px|right|Lower-case ''π'' is used for the constant.]]
{{main|pi (letter)}}
The name of the [[pi (letter)|Greek letter π]] is ''pi'', and this spelling is used in [[typography|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 language|English]], the conventional ''English'' pronunciation of the letter.<!--only state this fact, try not to justify here: see Talk page --> In Greek, the name of this letter is [[Help:IPA|pronounced]] {{IPA|/pi/}}.

The [[constant]] is named "π" because "π" is the first letter of the [[Greek language|Greek]] words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.<ref name="adm">{{cite web|url=http://mathforum.org/dr.math/faq/faq.pi.html|title=About Pi|work=Ask Dr. Math FAQ|accessdate=2007-10-29}}</ref> π is [[Unicode]] [[character (computing)|character]] U+03C0 ("[[Greek alphabet|Greek small letter pi]]").<ref>{{cite web|url=http://www.w3.org/TR/MathML2/bycodes.html|title=Characters Ordered by Unicode|publisher=[[World Wide Web Consortium|W3C]]|accessdate=2007-10-25}}</ref>

===Definition===
[[Image:Pi eq C over d.svg|thumb|left|Circumference = π × diameter]]
In [[Euclidean geometry|Euclidean plane geometry]], π is defined as the [[ratio]] of a [[circle]]'s [[circumference]] to its [[diameter]]:<ref name="adm"/>

:<math> \pi = \frac{c}{d}. </math>

Note that the ratio <sup>''c''</sup>/<sub>''d''</sub> 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 <sup>''c''</sup>/<sub>''d''</sub>. This fact is a consequence of the [[similarity (geometry)|similarity]] of all circles.

[[Image:Circle Area.svg|right|thumb|Area of the circle = π × area of the shaded square]]
Alternatively π can be also defined as the ratio of a circle's [[area]] (A) to the area of a square whose side is equal to the [[radius]]:<ref name="adm"/><ref>{{cite web|url=http://www.wku.edu/~tom.richmond/Pir2.html|title=Area of a Circle|first=Bettina|last=Richmond|publisher=[[Western Kentucky University]]|date=[[1999-01-12]]|accessdate=2007-11-04}}</ref>

:<math> \pi = \frac{A}{r^2}. </math>

The constant π may be defined in other ways that avoid the concepts of [[arc (geometry)|arc]] length and area, for example, as twice the smallest positive ''x'' for which [[trigonometric function|cos]](''x'')&nbsp;=&nbsp;0.<ref>{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X | pages = 183}}</ref> The formulas below illustrate other (equivalent) definitions.

===Irrationality and transcendence===
{{main|Proof that π is irrational}}
The constant π is an [[irrational number]]; that is, it cannot be written as the ratio of two [[integer]]s. This was proven in [[1761]] by [[Johann Heinrich Lambert]].<ref name="adm"/> In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to [[Ivan M. Niven|Ivan Niven]], is widely known.<ref>{{cite journal|title=A simple proof that &pi; is irrational|first=Ivan|last=Niven|authorlink=Ivan Niven|journal=[[Bulletin of the American Mathematical Society]]|volume=53|number=6|pages=509|year=1947|url=http://www.ams.org/bull/1947-53-06/S0002-9904-1947-08821-2/S0002-9904-1947-08821-2.pdf|format=[[Portable Document Format|PDF]]|accessdate=2007-11-04|doi=10.1090/S0002-9904-1947-08821-2}}</ref><ref>{{cite web|first=Helmut|last=Richter|url=http://www.lrz-muenchen.de/~hr/numb/pi-irr.html|title=Pi Is Irrational|date=[[1999-07-28]]|publisher=Leibniz Rechenzentrum|accessdate=2007-11-04}}</ref> A somewhat earlier similar proof is by [[Mary Cartwright]].<ref>{{cite book|first=Harold|last=Jeffreys|authorlink=Harold Jeffreys|title=Scientific Inference|edition=3rd|publisher=[[Cambridge University Press]]|year=1973}}</ref>

Furthermore, 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 (mathematics)|root]].<ref name="ttop">{{cite web|first=Steve|last=Mayer|url=http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html|title=The Transcendence of &pi;|accessdate=2007-11-04}}</ref> 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 [[compass and straightedge constructions|compass and straightedge]] are constructible numbers, it is impossible to [[squaring the circle|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.<ref>{{cite web|url=http://www.cut-the-knot.org/impossible/sq_circle.shtml|title=Squaring the Circle|publisher=[[cut-the-knot]]|accessdate=2007-11-04}}</ref>

===Numerical value===
{{seealso|numerical approximations of π}}
<!-- IMPORTANT NOTICE: Please note that Wikipedia is not a database to store millions of digits of π; please refrain from adding those to Wikipedia, as it could cause technical problems (and it makes the page unreadable or at least unattractive in the opinion of most readers). Instead, you could add links in the "External links" section, to other web sites containing information regarding digits of π.-->
The numerical value of π [[truncation|truncated]] to 50 [[decimal|decimal places]] is:<ref>{{cite web|url=http://www.research.att.com/~njas/sequences/A000796|title=A000796: Decimal expansion of Pi|publisher=[[On-Line Encyclopedia of Integer Sequences]]|accessdate=2007-11-04}}</ref>

:<!--Please discuss any changes to this on the Talk page.-->3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
:''See [[#External links|the links below]] and those at sequence [[oeis:A000796|A000796]] in [[On-Line Encyclopedia of Integer Sequences|OEIS]] for more digits.''

While the value of pi has been computed to more than a [[orders of magnitude (numbers)#1012|trillion]] (10<sup>12</sup>) digits,<ref>{{cite web |url=http://www.super-computing.org/pi_current.html |title=Current publicized world record of pi |accessdate=2007-10-14}}</ref> elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the [[observable universe]] to a precision comparable to the size of a [[hydrogen atom]].<ref>{{cite book |title=Excursions in Calculus |last=Young |first=Robert M. |year=1992 |publisher=Mathematical Association of America (MAA)|location=Washington |isbn=0883853175 |pages=417 | url = http://books.google.com/books?id=iEMmV9RWZ4MC&pg=PA238&dq=intitle:Excursions+intitle:in+intitle:Calculus+39+digits&lr=&as_brr=0&ei=AeLrSNKJOYWQtAPdt5DeDQ&sig=ACfU3U0NSYsF9kVp6om4Zyw3a7F82QCofQ }}</ref><ref>{{cite web |url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000004000298000001&idtype=cvips&gifs=yes |title=Statistical estimation of pi using random vectors |accessdate=2007-08-12 |format= |work=}}</ref>

Because π is an [[irrational number]], its decimal expansion never ends and does not [[Repeating decimal|repeat]]. 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.<ref>{{MathWorld|urlname=PiDigits|title=Pi Digits}}</ref> Despite much analytical work, and [[supercomputer]] calculations that have determined over 1 [[orders of magnitude (numbers)#1012|trillion]] digits of π, no simple pattern in the digits has ever been found.<ref>{{cite news|first=Chad|last=Boutin|url=http://www.purdue.edu/UNS/html4ever/2005/050426.Fischbach.pi.html|title=Pi seems a good random number generator - but not always the best|publisher=[[Purdue University]]|date=[[2005-04-26]]|accessdate=2007-11-04}}</ref> Digits of π are available on many web pages, and there is [[software for calculating π]] to billions of digits on any [[personal computer]].

===Calculating π===
{{main|Computing π}}

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to [[Archimedes]]<ref name="NOVA">{{cite web|first=Rick|last=Groleau|url=http://www.pbs.org/wgbh/nova/archimedes/pi.html|title=Infinite Secrets: Approximating Pi|publisher=NOVA|date=09-2003|accessdate=2007-11-04}}</ref>, is to calculate the [[perimeter]], ''P<sub>n</sub> ,'' of a [[regular polygon]] with ''n'' sides [[circumscribe]]d around a circle with diameter ''d.'' Then

:<math>\pi = \lim_{n \to \infty}\frac{P_{n}}{d}</math>

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides [[Inscribed figure|inscribed]] inside the circle. Using a polygon with 96 sides, he computed the fractional range: <math>\begin{smallmatrix}3\frac{10}{71}\ <\ \pi\ <\ 3\frac{1}{7}\end{smallmatrix}</math>.<ref>{{cite book
| first=Petr | last=Beckmann
| year=1989
| title=A History of Pi
| publisher=Barnes & Noble Publishing
| isbn=0880294183 }}</ref>

π can also be calculated using purely mathematical methods. Most formulas 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 [[Leibniz formula for pi|Gregory-Leibniz series]]:<ref>{{cite book |first=Pierre |last=Eymard |coauthors=Jean-Pierre Lafon |others=Stephen S. Wilson (translator)|title=The Number &pi;|url=http://books.google.com/books?id=qZcCSskdtwcC&pg=PA53&dq=leibniz+pi&ei=uFsuR5fOAZTY7QLqouDpCQ&sig=k8VlN5VTxcX9a6Ewc71OCGe_5jk |accessdate=2007-11-04 |year=2004 |month=02 |publisher=American Mathematical Society |isbn=0821832468 |pages=53 |chapter=2.6 }}</ref>

:<math>\pi = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots\! </math>.

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that 300 terms are not sufficient to calculate '''π''' correctly to 2 decimal places.<ref>{{cite journal|url=http://www.scm.org.co/Articulos/832.pdf|format=[[PDF]]|title=Even from Gregory-Leibniz series &pi; could be computed: an example of how convergence of series can be accelerated|journal=Lecturas Mathematicas|volume=27|year=2006|pages=21–25|first=Vito|last=Lampret, Spanish|accessdate=2007-11-04}}</ref> However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

<math>\pi_{0,1} = \frac{4}{1}, \pi_{0,2} =\frac{4}{1}-\frac{4}{3}, \pi_{0,3} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}, \pi_{0,4} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}, \cdots\! </math>

and then define

<math>\pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2}</math> for all <math>i,j\ge 1</math>

then computing <math>\pi_{10,10}</math> will take similar computation time to computing 150 terms of the original series in a brute force manner, and <math>\pi_{10,10}=3.141592653\cdots</math>, correct to 9 decimal places. This computation is an example of the [[Van Wijngaarden transformation]].<ref>A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Process Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp. 51-60.</ref>

==History==
{{seealso|Chronology of computation of π|Numerical approximations of π}}
The history of π parallels the development of mathematics as a whole.<ref>{{cite book |last=Beckmann |first=Petr |authorlink=Petr Beckmann |title=A History of π |year=1976 |publisher=[[St. Martin's Press|St. Martin's Griffin]] |id=ISBN 0-312-38185-9}}</ref> Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.<ref>{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/pi.html|title=Archimedes' constant &pi;|accessdate=2007-11-04}}</ref>

===Geometrical period===

That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.<ref name="adm"/> The Indian text ''[[Shatapatha Brahmana]]'' gives π as 339/108 ≈ 3.139. The [[Hebrew Bible|Tanakh]] appears to suggest, in the Book of [[Book of Kings|Kings]], that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The interpretation of the passage is disputed,<ref>{{cite web|first=H. Peter|last=Aleff|url=http://www.recoveredscience.com/const303solomonpi.htm|title=Ancient Creation Stories told by the Numbers: Solomon's Pi|publisher=recoveredscience.com|accessdate=2007-10-30}}</ref><ref name="ahop">{{cite web|first=J J|last=O'Connor|coauthors=E F Robertson|url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html|title=A history of Pi|date=2001-08|accessdate=2007-10-30}}</ref> as some believe the ratio of 3:1 is of an exterior circumference to an interior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls (See: [[History_of_numerical_approximations_of_%CF%80#Biblical_value|Biblical value of π]]).

[[Archimedes]] (287-212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in [[regular polygon]]s and calculating the outer and inner polygons' respective perimeters:<ref name="ahop"/>

[[Image:Archimedes pi.svg|350px|center|]]
[[Image:Cutcircle2.svg|thumb|right|250px|Liu Hui's Pi algorithm]]
By using the equivalent of 96-sided polygons, he proved that 223/71 &lt; π &lt; 22/7.<ref name="ahop"/> Taking the average of these values yields 3.1419.

In the following centuries further development took place in India and China. Around 265, the [[Wei Kingdom]] mathematician [[Liu Hui]] provided a simple and rigorous [[Liu Hui's π algorithm|iterative algorithm]] to calculate π to any degree of accuracy. He himself carried through the calculation to 3072-gon and obtained an approximate value for π of 3.1416.
: <math>
\begin{align}
\pi \approx A_{3072} & {} = 768 \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2+1}}}}}}}}} \\
& {} \approx 3.14159.
\end{align}
</math>

Later, Liu Hui invented a [[Liu Hui's π algorithm#Quick method|quick method of calculating π]] and obtained an approximate value of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician [[Zu Chongzhi]] demonstrated that π ≈ 355/113, and showed that 3.1415926 &lt; π &lt; 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would stand as the most accurate approximation of π over the next 900 years.

===Classical period===

Until the [[2nd millennium|second millennium]], π was known to fewer than 10 decimal digits. The next major advancement in the study of π came with the development of [[calculus]], and in particular the discovery of [[Series (mathematics)|infinite series]] which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, [[Madhava of Sangamagrama]] found the first known such series:

:<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\!</math>

This is now known as the [[Leibniz formula for pi|Madhava-Leibniz series]]<ref>{{citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=[[Cambridge University Press]]|year=1999|isbn=0521789885|page=58}}</ref><ref>{{citation|first=R. C.|last=Gupta|title=On the remainder term in the Madhava-Leibniz's series|journal=Ganita Bharati|volume=14|issue=1-4|year=1992|pages=68-71}}</ref> or Gregory-Leibniz series since it was rediscovered by [[James Gregory (astronomer and mathematician)|James Gregory]] and [[Gottfried Leibniz]] in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

:<math>\pi = \sqrt{12} \, \left(1-\frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} - \frac{1}{7 \cdot 3^3} + \cdots\right)\!</math>

[[Madhava of Sangamagrama| Madhava]] was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the [[Islamic mathematics|Persian mathematician]], [[Jamshīd al-Kāshī]], who determined 16 decimals of π.

The first major European contribution since Archimedes was made by the German mathematician [[Ludolph van Ceulen]] (1540&ndash;1610), who used a geometrical method to compute 35 decimals of π. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.<ref>{{cite book | title = Mathematical Tables; Containing the Common, Hyperbolic, and Logistic Logarithms... | author = Charles Hutton | publisher = London: Rivington | year = 1811 | pages = p.13 | url = http://books.google.com/books?id=zDMAAAAAQAAJ&pg=PA13&dq=snell+descartes+date:0-1837&lr=&as_brr=1&ei=rqPgR7yeNqiwtAPDvNEV }}</ref>

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the [[Viète's formula]],
:<math>\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots\!</math>

found by [[François Viète]] in 1593. Another famous result is [[Wallis product|Wallis' product]],

:<math>\frac{\pi}{2} = \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\!</math>

written down by [[John Wallis]] in 1655. [[Isaac Newton]] himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." <ref>[http://query.nytimes.com/gst/fullpage.html?res=9B0DE0DB143FF93BA35750C0A961948260 The New York Times: Even Mathematicians Can Get Carried Away]</ref>

In 1706 [[John Machin]] was the first to compute 100 decimals of π, using the formula

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

with

:<math>\arctan \, x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\!</math>

Formulas of this type, now known as [[Machin-like formula]]s, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy [[Zacharias Dase]], who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head. The best value at the end of the 19th century was due to [[William Shanks]], who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. [[Johann Heinrich Lambert]] proved the irrationality of π in 1761, and [[Adrien-Marie Legendre]] proved in 1794 that also π<sup>2</sup> is irrational. When [[Leonhard Euler]] in 1735 solved the famous [[Basel problem]] &ndash; finding the exact value of

:<math>\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots\!</math>

which is π<sup>2</sup>/6, he established a deep connection between π and the [[prime number]]s. Both Legendre and Leonhard Euler speculated that π might be [[transcendental number|transcendental]], a fact that was proved in 1882 by [[Ferdinand von Lindemann]].

[[William Jones (mathematician)|William Jones]]' book ''A New Introduction to Mathematics'' from [[1706]] is cited as the first text where the [[pi (letter)|Greek letter π]] was used for this constant, but this notation became particularly popular after [[Leonhard Euler]] adopted it in 1737.<ref>{{cite web|url=http://www.famousWelsh.com/cgibin/getmoreinf.cgi?pers_id=737|title=About: William Jones|work=Famous Welsh|accessdate=2007-10-27}}</ref> He wrote:
{{cquote|<nowiki>There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) +&nbsp;...&nbsp;=&nbsp;3.14159...&nbsp;=&nbsp;&pi;</nowiki><ref name="adm"/>}}
{{seealso|history of mathematical notation}}

===Computation in the computer age===

The advent of digital computers in the 20th century led to an increased rate of new π calculation records. [[John von Neumann]] used [[ENIAC]] to compute 2037 digits of π in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the [[fast Fourier transform]] (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician [[Srinivasa Ramanujan]] found many new formulas for π, some remarkable for their elegance and mathematical depth.<ref name="rad">{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/piramanujan.html|title=The constant &pi;: Ramanujan type formulas|accessdate=2007-11-04}}</ref> Two of his most famous formulas are the series

:<math>\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!</math>
and
:<math>\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!</math>

which deliver 14 digits per term.<ref name="rad"/> The Chudnovsky brothers used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the [[supercomputer]]s used to set modern records.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that ''multiply'' the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when [[Richard Brent (scientist)|Richard Brent]] and [[Eugene Salamin]] independently discovered the [[Gauss–Legendre algorithm|Brent–Salamin algorithm]], which uses only arithmetic to double the number of correct digits at each step.<ref name="brent">{{Citation | last=Brent | first=Richard | author-link=Richard Brent (scientist) | year=1975 | title=Multiple-precision zero-finding methods and the complexity of elementary function evaluation | periodical=Analytic Computational Complexity | publication-place=New York | publisher=Academic Press | editor-last=Traub | editor-first=J F | pages=151–176 | url=http://wwwmaths.anu.edu.au/~brent/pub/pub028.html | accessdate=2007-09-08}}</ref> The algorithm consists of setting

:<math>a_0 = 1 \quad \quad \quad b_0 = \frac{1}{\sqrt 2} \quad \quad \quad t_0 = \frac{1}{4} \quad \quad \quad p_0 = 1\!</math>

and iterating

:<math>a_{n+1} = \frac{a_n+b_n}{2} \quad \quad \quad b_{n+1} = \sqrt{a_n b_n}\!</math>
:<math>t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \quad \quad \quad p_{n+1} = 2 p_n\!</math>

until ''a<sub>n</sub>'' and ''b<sub>n</sub>'' are close enough. Then the estimate for π is given by

:<math>\pi \approx \frac{(a_n + b_n)^2}{4 t_n}\!</math>.

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by [[Jonathan Borwein|Jonathan]] and [[Peter Borwein]].<ref>{{cite book|first=Jonathan M|last=Borwein|authorlink=Jonathan Borwein|coauthors=Borwein, Peter, Berggren, Lennart|date=2004|title=Pi: A Source Book|publisher=Springer|isbn=0387205713}}</ref> The methods have been used by [[Yasumasa Kanada]] and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is 1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-like formulas that were slower but crucially reduced memory consumption. The calculation was performed on a 64-node Hitachi supercomputer with 1 [[terabyte]] of main memory, capable of carrying out 2 trillion operations per second.

An important recent development was the [[Bailey–Borwein–Plouffe formula]] (BBP formula), discovered by [[Simon Plouffe]] and named after the authors of the paper in which the formula was first published, [[David H. Bailey]], [[Peter Borwein]], and Plouffe.<ref name="bbpf">{{cite journal
| 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
| format = [[PDF]]
| doi = 10.1090/S0025-5718-97-00856-9
}}</ref> The formula,

:<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>

is remarkable because it allows extracting any individual [[hexadecimal]] or [[Binary numeral system|binary]] digit of π without calculating all the preceding ones.<ref name="bbpf"/> Between 1998 and 2000, the [[distributed computing]] project [[PiHex]] used a modification of the BBP formula due to [[Fabrice Bellard]] to compute the [[Orders of magnitude (numbers) #1015|quadrillionth]] (1,000,000,000,000,000:th) bit of π, which turned out to be 0.<ref>{{cite web|url=http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html|title=A new formula to compute the n<sup>th</sup> binary digit of pi|first=Fabrice|last=Bellard|authorlink=Fabrice Bellard|accessdate=2007-10-27}}</ref>

===Memorizing digits===
{{main|Piphilology}}
[[Image:PiDigits.svg|right|thumb|300px|right|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.
In 2006, [[Akira Haraguchi]], a retired Japanese engineer, claimed to have recited 100,000 decimal places.<ref name="japantimes">{{cite news|first=Tomoko|last=Otake|url=http://search.japantimes.co.jp/print/fl20061217x1.html|title=How can anyone remember 100,000 numbers?|work=[[The Japan Times]]|date=[[2006-12-17]]|accessdate=2007-10-27}}</ref> 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]].<ref>{{cite web|url=http://www.pi-world-ranking-list.com/news/index.htm|title=Pi World Ranking List|accessdate=2007-10-27}}</ref> It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of ''π'' without an error.<ref>{{cite news|url=http://www.newsgd.com/culture/peopleandlife/200611280032.htm|title=Chinese student breaks Guiness record by reciting 67,890 digits of pi|work=News Guangdong|date=[[2006-11-28]]|accessdate=2007-10-27}}</ref>

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.''<ref>{{cite web|first=Jonathan M|last=Borwein|authorlink=Jonathan Borwein|url=http://users.cs.dal.ca/~jborwein/pi-culture.pdf|format=[[PDF]]|title=The Life of Pi: From Archimedes to Eniac and Beyond|publisher=[[Dalhousie University]] Computer Science|date=[[2005-09-25]]|accessdate=2007-10-29}}</ref> 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.<ref>{{cite web|first=Mike|last=Keith|authorlink=Mike Keith (mathematician)|url=http://users.aol.com/s6sj7gt/solution.htm|title=Cadaeic Cadenza: Solution & Commentary|date=1996|accessdate=2007-10-30}}</ref> Piems are related to the entire field of humorous yet serious study that involves the use of [[Mnemonic|mnemonic techniques]] to remember the digits of ''π'', known as [[piphilology]]. See [[:q:English mathematics mnemonics#Pi|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.<ref>{{cite web|first=Yicong|last=Liu|url=http://silverchips.mbhs.edu/inside.php?sid=3577|title=Oh my, memorizing so many digits of pi.|publisher=Silver Chips Online|date=[[2004-05-19]]|accessdate=2007-11-04}}</ref>

==Advanced properties==
===Numerical approximations===
{{main|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.<ref name="ttop"/> Formulas for calculating ''π'' using elementary arithmetic typically include [[series (mathematics)|series]] or [[Summation#Capital-sigma notation|summation notation]] (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to ''π''.<ref>{{cite web|url=http://mathworld.wolfram.com/PiFormulas.html|title=Pi Formulas|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|publisher=[[MathWorld]]|date=[[2007-09-27]]|accessdate=2007-11-10}}</ref> The more terms included in a calculation, the closer to ''π'' the result will get.

Consequently, numerical calculations must use [[approximation]]s of ''π''. For many purposes, 3.14 or [[Proof that 22/7 exceeds π|<sup>22</sup>/<sub>7</sub>]] is close enough, although engineers often use 3.1416 (5 [[significant figures]]) or 3.14159 (6 significant figures) for more precision. The approximations <sup>22</sup>/<sub>7</sub> and <sup>355</sup>/<sub>113</sub>, with 3 and 7 significant figures respectively, are obtained from the simple [[continued fraction]] expansion of ''π''. The approximation [[Milü|<sup>355</sup>⁄<sub>113</sub>]] (3.1415929…) is the best one that may be expressed with a three-digit or four-digit [[fraction (mathematics)|numerator and denominator]].<ref>{{cite news|language=Chinese|author=韩雪涛|title=数学科普:常识性谬误流传令人忧|publisher=中华读书报|date=[[2001-08-29]]|url=http://www.xys.org/~xys/xys/ebooks/others/science/dajia/shuxuekepu.txt|accessdate=2006-10-06}}</ref><ref>{{cite web|url=http://www.kaidy.com/PiReward.htm|title=Magic of 355 ÷ 113|publisher=Kaidy Educational Resources|accessdate=2007-11-08}}</ref><ref>{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/piApprox.html|title=Collection of approximations for &pi;|publisher=Numbers, constants and computation|first=Xavier|last=Gourdon|coauthors=Pascal Sebah|accessdate=2007-11-08}}</ref>

The earliest numerical approximation of ''π'' is almost certainly the value {{num|3}}.<ref name="ahop"/> 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 figure|inscribed]] [[regular polygon|regular]] [[hexagon]] to the [[diameter]] of the [[circle]].

===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.<ref>{{cite web|url=http://mathworld.wolfram.com/NormalNumber.html|title=Normal Number|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|publisher=[[MathWorld]]|date=[[2005-12-22]]|accessdate=2007-11-10}}</ref> 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''.<ref>{{cite news|url=http://www.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are The Digits of Pi Random? Lab Researcher May Hold The Key|first=Paul|last=Preuss|authorlink=Paul Preuss|publisher=[[Lawrence Berkeley National Laboratory]]|date=[[2001-07-23]]|accessdate=2007-11-10}}</ref>

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]].<ref>{{cite news|url=http://www.sciencenews.org/articles/20010901/bob9.asp|title=Pi à la Mode: Mathematicians tackle the seeming randomness of pi's digits|first=Ivars|last=Peterson|authorlink=Ivars Peterson|work=Science News Online|date=[[2001-09-01]]|accessdate=2007-11-10}}</ref>

It is also unknown whether ''π'' and [[E (mathematical constant)|''e'']] are [[Algebraic independence|algebraically independent]], although [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]] proved the algebraic independence of {π, [[Gelfond's constant|''e''<sup>&pi;</sup>]], [[Gamma function|&Gamma;]](1/4)} in 1996.<ref>{{cite journal|author=Nesterenko, Yuri V|authorlink=Yuri Valentinovich Nesterenko|title=Modular Functions and Transcendence Problems|journal=[[Comptes rendus de l'Académie des sciences]] Série 1|volume=322|number=10|pages=909–914|year=1996}}</ref> However it is known that at least one of ''πe'' and ''π'' + ''e'' is [[transcendental number|transcendental]] (see [[Lindemann–Weierstrass theorem]]).<!-- redundant wikilink intentional: specifically relevant to this section-->

==Use in mathematics and science==
{{main|List of formulas involving π}}
π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.<ref>{{cite web|url=http://news.bbc.co.uk/1/hi/world/asia-pacific/4644103.stm|title=Japanese breaks pi memory record|work=[[BBC News]]|date=[[2005-07-02]]|accessdate=2007-10-30}}</ref>

===Geometry and trigonometry===
{{seealso|Area of a disk}}
For any circle with radius ''r'' and diameter ''d'' = 2''r'', the circumference is π''d'' and the area is π''r''<sup>2</sup>. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as [[ellipse]]s, [[sphere]]s, [[Cone (geometry)|cone]]s, and [[torus|tori]].<ref>{{cite web|url=http://www.math.psu.edu/courses/maserick/circle/circleapplet.html|title=Area and Circumference of a Circle by Archimedes|publisher=[[Pennsylvania State University|Penn State]]|accessdate=2007-11-08}}</ref> Accordingly, π appears in [[Integral|definite integrals]] that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the [[unit disk]] is given by:<ref name="udi">{{cite web|url=http://mathworld.wolfram.com/UnitDiskIntegral.html|title=Unit Disk Integral|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2006-01-28]]|accessdate=2007-11-08}}</ref>
:<math>\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}</math>
and
:<math>\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\,dx = \pi</math>
gives half the circumference of the [[unit circle]].<ref>{{cite web|url=http://www.math.psu.edu/courses/maserick/circle/circleapplet.html|title=Area and Circumference of a Circle by Archimedes|publisher=[[Pennsylvania State University|Penn State]]|accessdate=2007-11-08}}</ref> More complicated shapes can be integrated as [[solid of revolution|solids of revolution]].<ref>{{cite web|url=http://mathworld.wolfram.com/SolidofRevolution.html|title=Solid of Revolution|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2006-05-04]]|accessdate=2007-11-08}}</ref>

From the unit-circle definition of the [[trigonometric function]]s also follows that the sine and cosine have period 2π. That is, for all ''x'' and integers ''n'', sin(''x'') = sin(''x'' + 2π''n'') and cos(''x'') = cos(''x'' + 2π''n''). Because sin(0) = 0, sin(2π''n'') = 0 for all integers ''n''. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often ''defined'' using trigonometric functions, for example as the smallest positive ''x'' for which sin ''x'' = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the [[inverse trigonometric function]]s, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as [[power series]] is the easiest way to derive infinite series for π.

===Higher analysis and number theory===

[[Image:Euler's formula.svg|thumb|250px]]

The frequent appearance of π in [[complex analysis]] can be related to the behavior of the [[exponential function]] of a complex variable, described by [[Euler's formula]]

:<math>e^{i\varphi} = \cos \varphi + i\sin \varphi \!</math>

where ''i'' is the [[imaginary unit]] satisfying ''i''<sup>2</sup> = &minus;1 and ''e'' ≈ 2.71828 is [[E (mathematical constant)|Euler's number]]. This formula implies that imaginary powers of ''e'' describe rotations on the [[unit circle]] in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation ''φ'' = π results in the remarkable [[Euler's identity]]

:<math>e^{i \pi} = -1.\!</math>

There are ''n'' different ''n''-th [[Root of unity|roots of unity]]
:<math>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).</math>

The [[Gaussian integral]]

:<math>\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.</math>

A consequence is that the [[gamma function]] of a half-integer is a rational multiple of √π.
<!-- need some prose here on the zeta function and primes -->

===Physics===
Although not a [[physical constant]], ''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 system]]s. Using units such as [[Planck units]] can sometimes eliminate ''π'' from formulae.

*The [[cosmological constant]]:<ref>{{cite web|first=Cole|last=Miller|url=http://www.astro.umd.edu/~miller/teaching/astr422/lecture12.pdf|format=[[PDF]]|title=The Cosmological Constant|publisher=[[University of Maryland, College Park|University of Maryland]]|accessdate=2007-11-08}}</ref>
::<math>\Lambda = {{8\pi G} \over {3c^2}} \rho</math>
*[[Uncertainty principle|Heisenberg's uncertainty principle]], which shows that the uncertainty in the measurement of a particle's position (&Delta;''x'') and [[momentum]] (&Delta;''p'') can not both be arbitrarily small at the same time:<ref>{{cite web|first=James M|last=Imamura|url=http://zebu.uoregon.edu/~imamura/208/jan27/hup.html|title=Heisenberg Uncertainty Principle|publisher=[[University of Oregon]]|date=[[2005-08-17]]|accessdate=2007-11-09}}</ref>
::<math> \Delta x\, \Delta p \ge \frac{h}{4\pi} </math>
*[[Einstein field equations|Einstein's field equation]] of [[general relativity]]:<ref name = ein>{{cite journal| last = Einstein| first = Albert| authorlink = Albert Einstein | title = The Foundation of the General Theory of Relativity| journal = [[Annalen der Physik]] |date=1916| url = http://www.alberteinstein.info/gallery/gtext3.html| format = [[PDF]] | id = | accessdate = 2007-11-09 }}</ref>
::<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 field|electric force]], describing the force between two [[electric charge]]s (''q<sub>1</sub>'' and ''q<sub>2</sub>'') separated by distance ''r'':<ref>
{{cite web|first=C. Rod|last=Nave|url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c3|title=Coulomb's Constant|work=[[HyperPhysics]]|publisher=[[Georgia State University]]|date=[[2005-06-28]]|accessdate=2007-11-09}}</ref>
::<math> F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}</math>
*[[Magnetic constant|Magnetic permeability of free space]]:<ref>{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0 |title=Magnetic constant |accessdate=2007-11-09 |date=2006 [[Committee on Data for Science and Technology|CODATA]] recommended values |publisher=[[National Institute of Standards and Technology|NIST]] }}</ref>
::<math> \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,</math>
*[[Kepler's laws of planetary motion#Kepler's third law|Kepler's third law constant]], relating the [[orbital period]] (''P'') and the [[semimajor axis]] (''a'') to the [[mass]]es (''M'' and ''m'') of two co-orbiting bodies:
::<math>\frac{P^2}{a^3}={(2\pi)^2 \over G (M+m)} </math>

===Probability and statistics===
In [[probability]] and [[statistics]], there are many [[probability distribution|distributions]] whose formulas contain ''π'', including:
*the [[probability density function]] for the [[normal distribution]] with [[mean]] μ and [[standard deviation]] σ, due to the [[Gaussian integral]]:<ref>{{cite web|url=http://mathworld.wolfram.com/GaussianIntegral.html|title=Gaussian Integral|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2004-10-07]]|accessdate=2007-11-08}}</ref>

:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}</math>
*the probability density function for the (standard) [[Cauchy distribution]]:<ref>{{cite web|url=http://mathworld.wolfram.com/CauchyDistribution.html|title=Cauchy Distribution|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2005-10-11]]|accessdate=2007-11-08}}</ref>

:<math>f(x) = \frac{1}{\pi (1 + x^2)}.</math>

Note that since <math>\int_{-\infty}^{\infty} f(x)\,dx = 1</math> for any probability density function ''f''(''x''), the above formulas can be used to produce other integral formulas for ''π''.<ref>{{cite web|url=http://mathworld.wolfram.com/ProbabilityFunction.html|title=Probability Function|publisher=[[MathWorld]]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=[[2003-07-02]]|accessdate=2007-11-08}}</ref>

[[Buffon's needle]] problem is sometimes quoted as a empirical approximation of ''π'' in "popular mathematics" works. Consider dropping a needle of length ''L'' repeatedly on a surface containing parallel lines drawn ''S'' units apart (with ''S''&nbsp;>&nbsp;''L''). If the needle is dropped ''n'' times and ''x'' of those times it comes to rest crossing a line (''x''&nbsp;>&nbsp;0), then one may approximate ''π'' using the [[Monte Carlo method]]:<ref name="bn">{{cite web|url=http://mathworld.wolfram.com/BuffonsNeedleProblem.html|title=Buffon's Needle Problem|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|publisher=[[MathWorld]]|date=[[2005-12-12]]|accessdate=2007-11-10}}</ref><ref>{{cite web|first=Alex|last=Bogomolny|url=http://www.cut-the-knot.org/ctk/August2001.shtml|title=Math Surprises: An Example|work=[[cut-the-knot]]|date=2001-08|accessdate=2007-10-28}}</ref><ref>{{cite journal|last = Ramaley|first = J. F.|title = Buffon's Noodle Problem|journal = The American Mathematical Monthly|volume = 76|issue = 8|date=Oct 1969|pages = 916–918|doi = 10.2307/2317945}}</ref><ref>{{cite web|url=http://www.datastructures.info/the-monte-carlo-algorithmmethod/|title=The Monte Carlo algorithm/method|work=datastructures|date=[[2007-01-09]]|accessdate=2007-11-07}}</ref>
:<math>\pi \approx \frac{2nL}{xS}.</math>
Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of ''π'' ''by experiment''. Reliably getting just three digits (including the initial "3") right requires millions of throws,<ref name="bn"/> and the number of throws grows [[exponential growth|exponentially]] with the number of digits desired. Furthermore, any error in the measurement of the lengths ''L'' and ''S'' will transfer directly to an error in the approximated ''π''. For example, a difference of a single [[atom]] in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.

==See also==
*[[List of topics related to π]]
*[[Proof that 22/7 exceeds π]]
*[[Feynman point]] &ndash; comprising the 762nd through 767th decimal places of π, consisting of the digit 9 repeated six times.
*[[Indiana Pi Bill]].
*[[Pi Day]].
*[[Software for calculating π]] on personal computers.
*[[Mathematical constant]]s: [[E (mathematical constant)|e]] and [[Golden ratio|φ]]
* [[Statistics Online Computational Resource|SOCR]] resource [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_BuffonNeedleExperiment#Buffon.27s_needle_experiment_and_estimation_of_the_constant_.CF.80 hands-on activity for estimation of ''π'' using needle-dropping simulation].

== References ==

{{reflist|3}}

==External links==
{{commonscat}}
*[http://www.joyofpi.com The Joy of Pi by David Blatner]
*[http://www.research.att.com/~njas/sequences/A000796 Decimal expansions of Pi and related links] at the [[On-Line Encyclopedia of Integer Sequences]]
*[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://mathworld.wolfram.com/PiFormulas.html Lots of formulas for ''π''] at [[MathWorld]]
*[http://planetmath.org/encyclopedia/Pi.html PlanetMath: Pi]
*[http://mathforum.org/isaac/problems/pi1.html Finding the value of ''π'']
*[http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml Determination of ''π''] at [[cut-the-knot]]
*[http://www.bbc.co.uk/radio4/science/5numbers2.shtml BBC Radio Program about ''π'']
*[http://www.super-computing.org/pi-decimal_current.html Statistical Distribution Information on PI] based on 1.2 trillion digits of PI
*[http://www.joyofpi.com/pi.html The Digits of Pi &mdash; First ten thousand]
*[http://www.zenwerx.com/pi.php First 4 Million Digits of ''π''] - ''Warning'' - Roughly 2 [[megabyte]]s will be transferred.
*[http://www.piday.org/million.php One million digits of pi at piday.org]
*[http://www.gutenberg.net/etext/50 Project Gutenberg E-Text containing a million digits of ''π'']
*[http://www.angio.net/pi/piquery Search the first 200 million digits of ''π'' for arbitrary strings of numbers]
*[http://www.codecodex.com/wiki/index.php?title=Digits_of_pi_calculation Source code for calculating the digits of ''π'']
*[http://www.math.utah.edu/~palais/pi.pdf π is Wrong! An opinion column on why 2π is more useful in mathematics.]
*[http://ja0hxv.calico.jp/pai/estart.html 70 Billion digits of Pi(π) downloads.]
*[http://filebin.ca/nastsa/pi_data.txt The first 16 million digits of Pi] (18 mb .txt file)

[[Category:Pi| ]]
[[Category:Transcendental numbers]]
[[Category:Mathematical constants]]
[[Category:Dimensionless numbers]]

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Revision as of 19:22, 12 October 2008

Template:Two other uses

When a circle's diameter is 1, its circumference is π.
List of numbersIrrational numbers
ζ(3)√2√3√5φαeπδ
Binary 11.00100100001111110110…
Decimal 3.14159265358979323846…
Hexadecimal 3.243F6A8885A308D31319…
Continued fraction
Note that this continued fraction is not periodic.

Pi or π is a mathematical constant which represents the ratio of any circle's circumference to its diameter in Euclidean geometry, which is the same as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159. Pi is one of the most important mathematical constants: many formulae from mathematics, science, and engineering involve π.[1]

Pi is an irrational number, which means that it cannot be expressed as a fraction m/n, where m and n are integers. Consequently its decimal representation never ends or repeats. Beyond being irrational, it is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", probably by William Jones in 1706, and popularized by Leonhard Euler some years later. The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number.

Fundamentals

The letter π

Lower-case π 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. In Greek, the name of this letter is pronounced /pi/.

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.[2] π is Unicode character U+03C0 ("Greek small letter pi").[3]

Definition

Circumference = π × diameter

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:[2]

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 is a consequence of the similarity of all circles.

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

Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius:[2][4]

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.[5] The formulas below illustrate other (equivalent) definitions.

Irrationality and transcendence

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.[2] In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known.[6][7] A somewhat earlier similar proof is by Mary Cartwright.[8]

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.[9] 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.[10]

Numerical value

The numerical value of π truncated to 50 decimal places is:[11]

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 more than a trillion (1012) digits,[12] elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.[13][14]

Because π is an irrational number, its decimal expansion never ends and does not repeat. 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.[15] 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.[16] Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer.

Calculating π

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to Archimedes[17], is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range: .[18]

π can also be calculated using purely mathematical methods. Most formulas 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:[19]

.

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that 300 terms are not sufficient to calculate π correctly to 2 decimal places.[20] However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

and then define

for all

then computing will take similar computation time to computing 150 terms of the original series in a brute force manner, and , correct to 9 decimal places. This computation is an example of the Van Wijngaarden transformation.[21]

History

The history of π parallels the development of mathematics as a whole.[22] Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.[23]

Geometrical period

That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.[2] The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3.139. The Tanakh appears to suggest, in the Book of Kings, that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The interpretation of the passage is disputed,[24][25] as some believe the ratio of 3:1 is of an exterior circumference to an interior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls (See: Biblical value of π).

Archimedes (287-212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters:[25]

Liu Hui's Pi algorithm

By using the equivalent of 96-sided polygons, he proved that 223/71 < π < 22/7.[25] Taking the average of these values yields 3.1419.

In the following centuries further development took place in India and China. Around 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to 3072-gon and obtained an approximate value for π of 3.1416.

Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value would stand as the most accurate approximation of π over the next 900 years.

Classical period

Until the second millennium, π was known to fewer than 10 decimal digits. The next major advancement in the study of π came with the development of calculus, and in particular the discovery of infinite series which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, Madhava of Sangamagrama found the first known such series:

This is now known as the Madhava-Leibniz series[26][27] or Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

Madhava was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī, who determined 16 decimals of π.

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometrical method to compute 35 decimals of π. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.[28]

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète in 1593. Another famous result is Wallis' product,

written down by John Wallis in 1655. Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." [29]

In 1706 John Machin was the first to compute 100 decimals of π, using the formula

with

Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head. The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien-Marie Legendre proved in 1794 that also π2 is irrational. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of

which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre and Leonhard Euler speculated that π might be transcendental, a fact that was proved in 1882 by Ferdinand von Lindemann.

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.[30] He wrote:

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) + ... = 3.14159... = π[2]

Computation in the computer age

The advent of digital computers in the 20th century led to an increased rate of new π calculation records. John von Neumann used ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance and mathematical depth.[31] Two of his most famous formulas are the series

and

which deliver 14 digits per term.[31] The Chudnovsky brothers used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records.

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step.[32] The algorithm consists of setting

and iterating

until an and bn are close enough. Then the estimate for π is given by

.

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein.[33] The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is 1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-like formulas that were slower but crucially reduced memory consumption. The calculation was performed on a 64-node Hitachi supercomputer with 1 terabyte of main memory, capable of carrying out 2 trillion operations per second.

An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Plouffe.[34] The formula,

is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones.[34] Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0.[35]

Memorizing digits

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. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places.[36] 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.[37] It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.[38]

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.[39] 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.[40] 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.[41]

Advanced properties

Numerical approximations

Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions.[9] Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π.[42] The more terms included in a calculation, the closer to π the result will get.

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 355113 (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator.[43][44][45]

The earliest numerical approximation of π is almost certainly the value 3.[25] 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.

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.[46] 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.[47]

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.[48]

It is also unknown whether π and e are algebraically independent, although Yuri Nesterenko proved the algebraic independence of {π, eπ, Γ(1/4)} in 1996.[49] However it is known that at least one of πe and π + e is transcendental (see Lindemann–Weierstrass theorem).

Use in mathematics and science

π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.[50]

Geometry and trigonometry

For any circle with radius r and diameter d = 2r, the circumference is πd and the area is πr2. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipses, spheres, cones, and tori.[51] Accordingly, π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the unit disk is given by:[52]

and

gives half the circumference of the unit circle.[53] More complicated shapes can be integrated as solids of revolution.[54]

From the unit-circle definition of the trigonometric functions also follows that the sine and cosine have period 2π. That is, for all x and integers n, sin(x) = sin(x + 2πn) and cos(x) = cos(x + 2πn). Because sin(0) = 0, sin(2πn) = 0 for all integers n. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often defined using trigonometric functions, for example as the smallest positive x for which sin x = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the inverse trigonometric functions, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as power series is the easiest way to derive infinite series for π.

Higher analysis and number theory

The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula

where i is the imaginary unit satisfying i2 = −1 and e ≈ 2.71828 is Euler's number. This formula implies that imaginary powers of e describe rotations on the unit circle in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation φ = π results in the remarkable Euler's identity

There are n different n-th roots of unity

The Gaussian integral

A consequence is that the gamma function of a half-integer is a rational multiple of √π.

Physics

Although not a physical constant, 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. Using units such as Planck units can sometimes eliminate π from formulae.

Probability and statistics

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

Note that since for any probability density function f(x), the above formulas can be used to produce other integral formulas for π.[62]

Buffon's needle problem is sometimes quoted as a empirical approximation of π in "popular mathematics" works. 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 the Monte Carlo method:[63][64][65][66]

Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of π by experiment. Reliably getting just three digits (including the initial "3") right requires millions of throws,[63] and the number of throws grows exponentially with the number of digits desired. Furthermore, any error in the measurement of the lengths L and S will transfer directly to an error in the approximated π. For example, a difference of a single atom in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.

See also

References

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  2. ^ a b c d e f "About Pi". Ask Dr. Math FAQ. Retrieved 2007-10-29.
  3. ^ "Characters Ordered by Unicode". W3C. Retrieved 2007-10-25.
  4. ^ Richmond, Bettina (1999-01-12). "Area of a Circle". Western Kentucky University. Retrieved 2007-11-04. {{cite web}}: Check date values in: |date= (help)
  5. ^ Rudin, Walter (1976) [1953]. Principles of mathematical analysis (3e ed.). McGraw-Hill. p. 183. ISBN 0-07-054235-X.
  6. ^ Niven, Ivan (1947). "A simple proof that π is irrational" (PDF). Bulletin of the American Mathematical Society. 53 (6): 509. doi:10.1090/S0002-9904-1947-08821-2. Retrieved 2007-11-04.
  7. ^ Richter, Helmut (1999-07-28). "Pi Is Irrational". Leibniz Rechenzentrum. Retrieved 2007-11-04. {{cite web}}: Check date values in: |date= (help)
  8. ^ Jeffreys, Harold (1973). Scientific Inference (3rd ed.). Cambridge University Press.
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  17. ^ Groleau, Rick (09-2003). "Infinite Secrets: Approximating Pi". NOVA. Retrieved 2007-11-04. {{cite web}}: Check date values in: |date= (help)
  18. ^ Beckmann, Petr (1989). A History of Pi. Barnes & Noble Publishing. ISBN 0880294183.
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  20. ^ Lampret, Spanish, Vito (2006). "Even from Gregory-Leibniz series π could be computed: an example of how convergence of series can be accelerated" (PDF). Lecturas Mathematicas. 27: 21–25. Retrieved 2007-11-04.
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  24. ^ Aleff, H. Peter. "Ancient Creation Stories told by the Numbers: Solomon's Pi". recoveredscience.com. Retrieved 2007-10-30.
  25. ^ a b c d O'Connor, J J (2001-08). "A history of Pi". Retrieved 2007-10-30. {{cite web}}: Check date values in: |date= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  26. ^ George E. Andrews, Richard Askey, Ranjan Roy (1999), Special Functions, Cambridge University Press, p. 58, ISBN 0521789885
  27. ^ Gupta, R. C. (1992), "On the remainder term in the Madhava-Leibniz's series", Ganita Bharati, 14 (1–4): 68–71
  28. ^ Charles Hutton (1811). Mathematical Tables; Containing the Common, Hyperbolic, and Logistic Logarithms... London: Rivington. pp. p.13. {{cite book}}: |pages= has extra text (help)
  29. ^ The New York Times: Even Mathematicians Can Get Carried Away
  30. ^ "About: William Jones". Famous Welsh. Retrieved 2007-10-27.
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  32. ^ Brent, Richard (1975), Traub, J F (ed.), "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", Analytic Computational Complexity, New York: Academic Press, pp. 151–176, retrieved 2007-09-08
  33. ^ Borwein, Jonathan M (2004). Pi: A Source Book. Springer. ISBN 0387205713. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  34. ^ a b 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. doi:10.1090/S0025-5718-97-00856-9. {{cite journal}}: Unknown parameter |month= ignored (help)CS1 maint: multiple names: authors list (link)
  35. ^ Bellard, Fabrice. "A new formula to compute the nth binary digit of pi". Retrieved 2007-10-27.
  36. ^ Otake, Tomoko (2006-12-17). "How can anyone remember 100,000 numbers?". The Japan Times. Retrieved 2007-10-27. {{cite news}}: Check date values in: |date= (help)
  37. ^ "Pi World Ranking List". Retrieved 2007-10-27.
  38. ^ "Chinese student breaks Guiness record by reciting 67,890 digits of pi". News Guangdong. 2006-11-28. Retrieved 2007-10-27. {{cite news}}: Check date values in: |date= (help)
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