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| {{Two other uses|the mathematical constant|the Greek letter|pi (letter)}} | |||
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| IMPORTANT NOTICE: Please note that Misplaced Pages is not a database to store the millions of digits of π; please refrain from adding those to Misplaced Pages, 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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| {| class="infobox" style ="width: 370px;" | |||
| | colspan="2" align="center" | ] – ]s <br> ] – ] – ] – ] – ] – ] – ] – ] – ] | |||
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| | 11.00100100001111110110… | |||
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| | 3.14159265358979323846… | |||
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| | 3.243F6A8885A308D31319… | |||
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| | <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 ] which represents the ratio of any ]'s circumference to its diameter in ], 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, ], and ] 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 ], which means that it cannot be expressed as a ] ''m''/''n'', where ''m'' and ''n'' are ]s. Consequently its ] never ends or repeats. Beyond being ], it is a ], which means that no finite sequence of algebraic operations on ]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 ] in 1706, and popularized by ] some years later. The constant is occasionally also referred to as the '''circular constant''', ''']' constant''' (not to be confused with an ]), or ''']'s number'''. | |||
| ==Fundamentals== | |||
| === The letter π === | |||
| ] | |||
| {{main|pi (letter)}} | |||
| The name of the ] is ''pi'', and this spelling is used in ] 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 ], 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 ] {{IPA|/pi/}}. | |||
| The ] is named "π" because "π" is the first letter of the ] 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 ] ] U+03C0 ("]").<ref>{{cite web|url=http://www.w3.org/TR/MathML2/bycodes.html|title=Characters Ordered by Unicode|publisher=]|accessdate=2007-10-25}}</ref> | |||
| ===Definition=== | |||
| ] | |||
| In ], π is defined as the ] of a ]'s ] to its ]:<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 ] of all circles. | |||
| ] | |||
| Alternatively π can be also defined as the ratio of a circle's ] (A) to the area of a square whose side is equal to the ]:<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=]|date=]|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 ] length and area, for example, as twice the smallest positive ''x'' for which ](''x'') = 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 ]; that is, it cannot be written as the ratio of two ]s. This was proven in ] by ].<ref name="adm"/> In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to ], is widely known.<ref>{{cite journal|title=A simple proof that π is irrational|first=Ivan|last=Niven|authorlink=Ivan Niven|journal=]|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=]|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=]|publisher=Leibniz Rechenzentrum|accessdate=2007-11-04}}</ref> A somewhat earlier similar proof is by ].<ref>{{cite book|first=Harold|last=Jeffreys|authorlink=Harold Jeffreys|title=Scientific Inference|edition=3rd|publisher=]|year=1973}}</ref> | |||
| Furthermore, π is also ], as was proven by ] in ]. This means that there is no ] with ] coefficients of which π is a ].<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 π|accessdate=2007-11-04}}</ref> An important consequence of the transcendence of π is the fact that it is not ]. Because the coordinates of all points that can be constructed with ] are constructible numbers, it is impossible to ]: 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=]|accessdate=2007-11-04}}</ref> | |||
| ===Numerical value=== | |||
| {{seealso|numerical approximations of π}} | |||
| <!-- IMPORTANT NOTICE: Please note that Misplaced Pages is not a database to store millions of digits of π; please refrain from adding those to Misplaced Pages, 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 ] to 53 ] is:<ref>{{cite web|url=http://www.research.att.com/~njas/sequences/A000796|title=A000796: Decimal expansion of Pi|publisher=]|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 582 | |||
| :''See ] and those at sequence ] in ] for more digits.'' | |||
| While the value of pi has been computed to more than a ] (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 ] to a precision comparable to the size of a ].<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 ], its decimal expansion never ends and does not ]. 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 ] calculations that have determined over 1 ] 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=]|date=]|accessdate=2007-11-04}}</ref> Digits of π are available on many web pages, and there is ] to billions of digits on any ]. | |||
| ===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 ]<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 ], ''P<sub>n</sub> ,'' of a ] with ''n'' sides ]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 ] 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 ] and ]. However, some are quite simple, such as this form of the ]:<ref>{{cite book |first=Pierre |last=Eymard |coauthors=Jean-Pierre Lafon |others=Stephen S. Wilson (translator)|title=The Number π|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=]|title=Even from Gregory-Leibniz series π 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 ].<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=] |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 π|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 '']'' gives π as 339/108 ≈ 3.139. The ] appears to suggest, in the Book of ], 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: ]). | |||
| ] (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 ]s and calculating the outer and inner polygons' respective perimeters:<ref name="ahop"/> | |||
| ] | |||
| ] | |||
| By using the equivalent of 96-sided polygons, he proved that 223/71 < π < 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 ] mathematician ] provided a simple and rigorous ] 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 ] 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 ] 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 ], π was known to fewer than 10 decimal digits. The next major advancement in the study of π came with the development of ], and in particular the discovery of ] which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, ] 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 ]<ref>{{citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=]|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 ] and ] 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> | |||
| ] was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the ], ], who determined 16 decimals of π. | |||
| The first major European contribution since Archimedes was made by the German mathematician ] (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.<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 ], | |||
| :<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 ] in 1593. Another famous result is ], | |||
| :<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 ] in 1655. ] 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></ref> | |||
| In 1706 ] 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 ]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 ], 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 ], 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. ] proved the irrationality of π in 1761, and ] proved in 1794 that also π<sup>2</sup> is irrational. When ] in 1735 solved the famous ] – 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 ]s. Both Legendre and Leonhard Euler speculated that π might be ], a fact that was proved in 1882 by ]. | |||
| ]' book ''A New Introduction to Mathematics'' from ] is cited as the first text where the ] was used for this constant, but this notation became particularly popular after ] 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) + ... = 3.14159... = π</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. ] used ] 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 ] (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 ] 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 π: 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 ]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 ] and ] independently discovered the ], 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 ] and ].<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 ] 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 ] of main memory, capable of carrying out 2 trillion operations per second. | |||
| An important recent development was the ] (BBP formula), discovered by ] and named after the authors of the paper in which the formula was first published, ], ], and Plouffe.<ref name="bbpf">{{cite journal | |||
| | author = ], ], and ] | |||
| | 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 = ] | |||
| | 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 ] or ] digit of π without calculating all the preceding ones.<ref name="bbpf"/> Between 1998 and 2000, the ] project ] used a modification of the BBP formula due to ] to compute the ] (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}} | |||
| ] | |||
| Even long before computers have calculated ''π'', memorizing a ''record'' number of digits became an obsession for some people. | |||
| In 2006, ], 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=]|date=]|accessdate=2007-10-27}}</ref> This, however, has yet to be verified by ]. The Guinness-recognized record for remembered digits of ''π'' is 67,890 digits, held by ], a 24-year-old graduate student from ].<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=]|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=]|title=The Life of Pi: From Archimedes to Eniac and Beyond|publisher=] Computer Science|date=]|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 '']'' 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 ] to remember the digits of ''π'', known as ]. See ] 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=]|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 ] or ] (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=]|date=]|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 ]s of ''π''. For many purposes, 3.14 or ] is close enough, although engineers often use 3.1416 (5 ]) 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 ] expansion of ''π''. The approximation ] (3.1415929…) is the best one that may be expressed with a three-digit or four-digit ].<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=]|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 π|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 ] of an ] ] ] to the ] of the ]. | |||
| ===Open questions=== | |||
| The most pressing open question about ''π'' is whether it is a ] — 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=]|date=]|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=]|date=]|accessdate=2007-11-10}}</ref> | |||
| Bailey and Crandall showed in ] that the existence of the above mentioned ] and similar formulas imply that the normality in base 2 of ''π'' and various other constants can be reduced to a plausible ] of ].<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=]|accessdate=2007-11-10}}</ref> | |||
| It is also unknown whether ''π'' and ] are ], although ] proved the algebraic independence of {π, ], ](1/4)} in 1996.<ref>{{cite journal|author=Nesterenko, Yuri V|authorlink=Yuri Valentinovich Nesterenko|title=Modular Functions and Transcendence Problems|journal=] 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 ] (see ]).<!-- 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=]|date=]|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 ]s, ]s, ]s, and ].<ref>{{cite web|url=http://www.math.psu.edu/courses/maserick/circle/circleapplet.html|title=Area and Circumference of a Circle by Archimedes|publisher=]|accessdate=2007-11-08}}</ref> Accordingly, π appears in ] that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the ] is given by:<ref name="udi">{{cite web|url=http://mathworld.wolfram.com/UnitDiskIntegral.html|title=Unit Disk Integral|publisher=]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=]|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 ].<ref>{{cite web|url=http://www.math.psu.edu/courses/maserick/circle/circleapplet.html|title=Area and Circumference of a Circle by Archimedes|publisher=]|accessdate=2007-11-08}}</ref> More complicated shapes can be integrated as ].<ref>{{cite web|url=http://mathworld.wolfram.com/SolidofRevolution.html|title=Solid of Revolution|publisher=]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=]|accessdate=2007-11-08}}</ref> | |||
| From the unit-circle definition of the ]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 ]s, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as ] is the easiest way to derive infinite series for π. | |||
| ===Higher analysis and number theory=== | |||
| ] | |||
| The frequent appearance of π in ] can be related to the behavior of the ] of a complex variable, described by ] | |||
| :<math>e^{i\varphi} = \cos \varphi + i\sin \varphi \!</math> | |||
| where ''i'' is the ] satisfying ''i''<sup>2</sup> = −1 and ''e'' ≈ 2.71828 is ]. This formula implies that imaginary powers of ''e'' describe rotations on the ] in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation ''φ'' = π results in the remarkable ] | |||
| :<math>e^{i \pi} = -1.\!</math> | |||
| There are ''n'' different ''n''-th ] | |||
| :<math>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).</math> | |||
| The ] | |||
| :<math>\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.</math> | |||
| A consequence is that the ] of a half-integer is a rational multiple of √π. | |||
| <!-- need some prose here on the zeta function and primes --> | |||
| ===Physics=== | |||
| Although not a ], ''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, ]s. Using units such as ] can sometimes eliminate ''π'' from formulae. | |||
| *The ]:<ref>{{cite web|first=Cole|last=Miller|url=http://www.astro.umd.edu/~miller/teaching/astr422/lecture12.pdf|format=]|title=The Cosmological Constant|publisher=]|accessdate=2007-11-08}}</ref> | |||
| ::<math>\Lambda = {{8\pi G} \over {3c^2}} \rho</math> | |||
| *], which shows that the uncertainty in the measurement of a particle's position (Δ''x'') and ] (Δ''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=]|date=]|accessdate=2007-11-09}}</ref> | |||
| ::<math> \Delta x\, \Delta p \ge \frac{h}{4\pi} </math> | |||
| *] of ]:<ref name = ein>{{cite journal| last = Einstein| first = Albert| authorlink = Albert Einstein | title = The Foundation of the General Theory of Relativity| journal = ] |date=1916| url = http://www.alberteinstein.info/gallery/gtext3.html| format = ] | 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> | |||
| *] for the ], describing the force between two ]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=]|publisher=]|date=]|accessdate=2007-11-09}}</ref> | |||
| ::<math> F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}</math> | |||
| *]:<ref>{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0 |title=Magnetic constant |accessdate=2007-11-09 |date=2006 ] recommended values |publisher=] }}</ref> | |||
| ::<math> \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,</math> | |||
| *], relating the ] (''P'') and the ] (''a'') to the ]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 ] and ], there are many ] whose formulas contain ''π'', including: | |||
| *the ] for the ] with ] μ and ] σ, due to the ]:<ref>{{cite web|url=http://mathworld.wolfram.com/GaussianIntegral.html|title=Gaussian Integral|publisher=]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=]|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) ]:<ref>{{cite web|url=http://mathworld.wolfram.com/CauchyDistribution.html|title=Cauchy Distribution|publisher=]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=]|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=]|first=Eric W|last=Weisstein|authorlink=Eric W. Weisstein|date=]|accessdate=2007-11-08}}</ref> | |||
| ] 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 ]:<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=]|date=]|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=]|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=]|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 ] 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 ] 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== | |||
| *] | |||
| *] | |||
| *] – comprising the 762nd through 767th decimal places of π, consisting of the digit 9 repeated six times. | |||
| *]. | |||
| *]. | |||
| *] on personal computers. | |||
| *]s: ] and ] | |||
| * ] resource . | |||
| == References == | |||
| {{reflist|3}} | |||
| ==External links== | |||
| {{commonscat}} | |||
| * | |||
| * at the ] | |||
| * | |||
| * at ] | |||
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| * | |||
| * at ] | |||
| * | |||
| * based on 1.2 trillion digits of PI | |||
| * | |||
| * - ''Warning'' - Roughly 2 ]s will be transferred. | |||
| * | |||
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| * (18 mb .txt file) | |||
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Revision as of 19:52, 10 October 2008
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