this article is about the mathematical constant

8/14/2019 This Article is About the Mathematical Constant

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This article is about the mathematical constant. For the Greek letter, seepi (letter). Forother uses, see Pi (disambiguation).

When a circle'sdiameteris 1, its circumference is pi.List of numbersIrrational numbers

(3) 2 3 5 e Binary 11.00100100001111110110Decimal 3.1415926535897932384626433832795Hexadecimal 3.243F6A8885A308D31319

Continued fraction

Note that this continued fraction is notperiodic.

Pi or is amathematical constantwhose value is theratio of any circle's circumferenceto its diameter in Euclidean space; this is the same value as the ratio of a circle's area tothe square of its radius. It is approximately equal to 3.14159 in the usual decimal notation(see the table for its representation in some other bases). is one of the most importantmathematical and physical constants: many formulae from mathematics, science, andengineering involve .[1]

is anirrational number, which means that its value cannot be expressed exactly as afractionm/n, where m and n are integers. Consequently, its decimal representation neverends or repeats. It is also a transcendental number, which means that no finite sequenceof algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value;proving this was a late achievement in mathematical history and a significant result of19th century German mathematics. Throughout the history of mathematics, there hasbeen much effort to determine more accurately and to understand its nature; fascinationwith the number has even carried over into non-mathematical culture.

The Greek letter , often spelled outpi in text, was adopted for the number from theGreek word forperimeter"", first by William Jones in 1707, and popularizedby Leonhard Eulerin 1737.[2] The constant is occasionally also referred to as the circularconstant,Archimedes' constant (not to be confused with an Archimedes number), or
http://en.wikipedia.org/wiki/Pi_(letter)http://en.wikipedia.org/wiki/Pi_(letter)http://en.wikipedia.org/wiki/Pi_(disambiguation)http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Ap?ry%27s_constanthttp://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_5http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Decimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Mathematical_constanthttp://en.wikipedia.org/wiki/Mathematical_constanthttp://en.wikipedia.org/wiki/Mathematical_constanthttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Pi#cite_note-0http://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Fractionhttp://en.wikipedia.org/wiki/Fractionhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Decimal_representationhttp://en.wikipedia.org/wiki/Transcendental_numberhttp://en.wikipedia.org/wiki/William_Jones_(mathematician)http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Pi#cite_note-collier-1http://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedes_numberhttp://en.wikipedia.org/wiki/File:Pi-unrolled-720.gifhttp://en.wikipedia.org/wiki/File:Pi-unrolled-720.gifhttp://en.wikipedia.org/wiki/Pi_(letter)http://en.wikipedia.org/wiki/Pi_(disambiguation)http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/List_of_numbershttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Ap?ry%27s_constanthttp://en.wikipedia.org/wiki/Square_root_of_2http://en.wikipedia.org/wiki/Square_root_of_3http://en.wikipedia.org/wiki/Square_root_of_5http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Feigenbaum_constantshttp://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Decimalhttp://en.wikipedia.org/wiki/Hexadecimalhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Periodic_functionhttp://en.wikipedia.org/wiki/Mathematical_constanthttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Pi#cite_note-0http://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Fractionhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Decimal_representationhttp://en.wikipedia.org/wiki/Transcendental_numberhttp://en.wikipedia.org/wiki/William_Jones_(mathematician)http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Pi#cite_note-collier-1http://en.wikipedia.org/wiki/Archimedeshttp://en.wikipedia.org/wiki/Archimedes_number


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Ludolph's number (from a German mathematician whose efforts to calculate more of itsdigits became famous).

Contents

[hide]

1 Fundamentalso 1.1 The letter o 1.2 Definitiono 1.3 Irrationality and transcendenceo 1.4 Numerical valueo 1.5 Calculating

2 Historyo 2.1 Geometrical periodo 2.2 Classical periodo 2.3 Computation in the computer ageo 2.4 Memorizing digits

3 Advanced propertieso 3.1 Numerical approximationso 3.2 Open questions

4 Use in mathematics and scienceo 4.1 Geometry and trigonometryo 4.2 Complex numbers and calculuso 4.3 Physicso 4.4 Probability and statistics

5 Pi in popular culture 6 See also 7 References

8 External links

Fundamentals

The letter

Main article:pi (letter)
http://en.wikipedia.org/wiki/Ludolph_van_Ceulenhttp://toggletoc%28%29/http://en.wikipedia.org/wiki/Pi#Fundamentalshttp://en.wikipedia.org/wiki/Pi#The_letter_.CF.80http://en.wikipedia.org/wiki/Pi#Definitionhttp://en.wikipedia.org/wiki/Pi#Irrationality_and_transcendencehttp://en.wikipedia.org/wiki/Pi#Numerical_valuehttp://en.wikipedia.org/wiki/Pi#Calculating_.CF.80http://en.wikipedia.org/wiki/Pi#Historyhttp://en.wikipedia.org/wiki/Pi#Geometrical_periodhttp://en.wikipedia.org/wiki/Pi#Classical_periodhttp://en.wikipedia.org/wiki/Pi#Computation_in_the_computer_agehttp://en.wikipedia.org/wiki/Pi#Memorizing_digitshttp://en.wikipedia.org/wiki/Pi#Advanced_propertieshttp://en.wikipedia.org/wiki/Pi#Numerical_approximationshttp://en.wikipedia.org/wiki/Pi#Open_questionshttp://en.wikipedia.org/wiki/Pi#Use_in_mathematics_and_sciencehttp://en.wikipedia.org/wiki/Pi#Geometry_and_trigonometryhttp://en.wikipedia.org/wiki/Pi#Complex_numbers_and_calculushttp://en.wikipedia.org/wiki/Pi#Physicshttp://en.wikipedia.org/wiki/Pi#Probability_and_statisticshttp://en.wikipedia.org/wiki/Pi#Pi_in_popular_culturehttp://en.wikipedia.org/wiki/Pi#See_alsohttp://en.wikipedia.org/wiki/Pi#Referenceshttp://en.wikipedia.org/wiki/Pi#External_linkshttp://en.wikipedia.org/wiki/Pi_(letter)http://en.wikipedia.org/wiki/File:Pi-symbol.svghttp://en.wikipedia.org/wiki/Ludolph_van_Ceulenhttp://toggletoc%28%29/http://en.wikipedia.org/wiki/Pi#Fundamentalshttp://en.wikipedia.org/wiki/Pi#The_letter_.CF.80http://en.wikipedia.org/wiki/Pi#Definitionhttp://en.wikipedia.org/wiki/Pi#Irrationality_and_transcendencehttp://en.wikipedia.org/wiki/Pi#Numerical_valuehttp://en.wikipedia.org/wiki/Pi#Calculating_.CF.80http://en.wikipedia.org/wiki/Pi#Historyhttp://en.wikipedia.org/wiki/Pi#Geometrical_periodhttp://en.wikipedia.org/wiki/Pi#Classical_periodhttp://en.wikipedia.org/wiki/Pi#Computation_in_the_computer_agehttp://en.wikipedia.org/wiki/Pi#Memorizing_digitshttp://en.wikipedia.org/wiki/Pi#Advanced_propertieshttp://en.wikipedia.org/wiki/Pi#Numerical_approximationshttp://en.wikipedia.org/wiki/Pi#Open_questionshttp://en.wikipedia.org/wiki/Pi#Use_in_mathematics_and_sciencehttp://en.wikipedia.org/wiki/Pi#Geometry_and_trigonometryhttp://en.wikipedia.org/wiki/Pi#Complex_numbers_and_calculushttp://en.wikipedia.org/wiki/Pi#Physicshttp://en.wikipedia.org/wiki/Pi#Probability_and_statisticshttp://en.wikipedia.org/wiki/Pi#Pi_in_popular_culturehttp://en.wikipedia.org/wiki/Pi#See_alsohttp://en.wikipedia.org/wiki/Pi#Referenceshttp://en.wikipedia.org/wiki/Pi#External_linkshttp://en.wikipedia.org/wiki/Pi_(letter)


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Lower-case is used to symbolize the constant.

The name of the Greek letter ispi, and this spelling is commonly used in typographicalcontexts when the Greek letter is not available, or its usage could be problematic. It is notnormally capitalised () even at the beginning of a sentence. When referring to this

constant, the symbol is always pronounced like "pie" in English, which is theconventional English pronunciation of the Greek letter. In Greek, the name of this letter ispronounced /pi/.

The constant is named "" because "" is the first letter of the Greekwords (periphery) and (perimeter), probably referring to its use in the formula tofind the circumference, or perimeter, of a circle.[3] is UnicodecharacterU+03C0("Greek small letter pi").[4]

Definition

Circumference = diameterIn Euclidean plane geometry, is defined as the ratio of a circle's circumference to itsdiameter:[3]

The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice thediameterdof another circle it will also have twice the circumference C, preserving theratio C/d.
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Area of the circle = area of the shaded square

Alternatively can be also defined as the ratio of a circle's area (A) to the area of asquare whose side is equal to the radius:[3][5]

These definitions depend on results of Euclidean geometry, such as the fact that all circlesare similar. This can be considered a problem when occurs in areas of mathematics thatotherwise do not involve geometry. For this reason, mathematicians often prefer to define without reference to geometry, instead selecting one of itsanalytic properties as adefinition. A common choice is to define as twice the smallest positivex for whichcos(x) = 0.[6]The formulas below illustrate other (equivalent) definitions.

Irrationality and transcendence

Main article: Proof that is irrational

Being an irrational number, cannot be written as the ratio of twointegers. This wasproven in 1768 by Johann Heinrich Lambert.[7] In the 20th century, proofs were foundthat require no prerequisite knowledge beyond integral calculus. One of those, due toIvan Niven, is widely known.[8][9]A somewhat earlier similar proof is by MaryCartwright.[10]

Furthermore, is also transcendental, as was proven byFerdinand von Lindemann in1882. This means that there is nopolynomial with rational coefficients of which is aroot.[11] An important consequence of the transcendence of is the fact that it is notconstructible. Because the coordinates of all points that can be constructed with compassand straightedge are constructible numbers, it is impossible to square the circle: that is, itis impossible to construct, using compass and straightedge alone, a square whose area isequal to the area of a given circle.[12]This is historically significant, for squaring a circleis one of the easily understood elementary geometry problems left to us from antiquity;many amateurs in modern times have attempted to solve each of these problems, andtheir efforts are sometimes ingenious, but in this case, doomed to failure: a fact notalways understood by the amateur involved.
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Numerical value

See also:Numerical approximations of

The numerical value of truncatedto 50 decimal places is:[13]

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510See the links below and those at sequenceA000796in OEISfor more digits.

While the value of has been computed to more than a trillion (1012) digits,[14] elementaryapplications, such as calculating the circumference of a circle, will rarely require morethan a dozen decimal places. For example, a value truncated to 11 decimal places isaccurate enough to calculate the circumference of a circle the size of the earth with aprecision of a millimeter, and one truncated to 39 decimal places is sufficient to computethe circumference of any circle that fits in theobservable universeto a precisioncomparable to the size of a hydrogen atom.[15][16]

Because is anirrational number, its decimal expansion never ends and does not repeat.This infinite sequence of digits has fascinated mathematicians and laymen alike, andmuch effort over the last few centuries has been put into computing more digits andinvestigating the number's properties.[17] Despite much analytical work, andsupercomputercalculations that have determined over 1trilliondigits of , no simplebase-10 pattern in the digits has ever been found.[18]Digits of are available on manyweb pages, and there issoftware for calculating to billions of digits on anypersonalcomputer.

Calculating

Main article: Computing

can be empirically estimated by drawing a large circle, then measuring its diameter andcircumference and dividing the circumference by the diameter. Another geometry-basedapproach, due to Archimedes,[19]is to calculate theperimeter,Pn , of a regular polygonwith n sidescircumscribedaround a circle with diameterd. Then

That is, the more sides the polygon has, the closer the approximation approaches .Archimedes determined the accuracy of this approach by comparing the perimeter of thecircumscribed polygon with the perimeter of a regular polygon with the same number ofsides inscribed inside the circle. Using a polygon with 96 sides, he computed thefractional range: .[20]

can also be calculated using purely mathematical methods. Most formulae used forcalculating the value of have desirable mathematical properties, but are difficult to
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understand without a background intrigonometryandcalculus. However, some are quitesimple, such as this form of the Gregory-Leibniz series:[21]

While that series is easy to write and calculate, it is not immediately obvious why ityields . In addition, this series converges so slowly that 300 terms are not sufficient tocalculate correctly to 2 decimal places.[22]However, by computing this series in asomewhat more clever way by taking the midpoints of partial sums, it can be made toconverge much faster. Let

and then define

then computing 10,10 will take similar computation time to computing 150 terms of the

original series in a brute-force manner, and , correct to 9decimal places. This computation is an example of the van Wijngaardentransformation.[23]

History

See also: Chronology of computation of andNumerical approximations of

The history of parallels the development of mathematics as a whole.[24]Some authorsdivide progress into three periods: the ancient period during which was studiedgeometrically, the classical era following the development of calculus in Europe aroundthe 17th century, and the age of digital computers.[25]

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, Indianand 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.[3] TheIndian text Shatapatha Brahmana gives as 339/108 3.139. The Hebrew Bible appearsto suggest, in the Book ofKings, that = 3, which is notably worse than other estimatesavailable at the time of writing (600 BC). The interpretation of the passage isdisputed,[26][27] as some believe the ratio of 3:1 is of an interior circumference to an
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exterior diameter of a thinly walled basin, which could indeed be an accurate ratio,depending on the thickness of the walls (See: Biblical value of ).

Archimedes (287212 BC) was the first to estimate rigorously. He realized that itsmagnitude can be bounded from below and above by inscribing circles in regular

polygons and calculating the outer and inner polygons' respective perimeters:[27]

Liu Hui's algorithm

By using the equivalent of 96-sided polygons, he proved that 223/71 < < 22/7.[27]

Taking the average of these values yields 3.1419.

In the following centuries further development took place in India and China. Around AD265, the Wei Kingdom mathematicianLiu Huiprovided a simple and rigorousiterativealgorithm to calculate to any degree of accuracy. He himself carried through thecalculation to a 3072-gon and obtained an approximate value for of 3.1416.

Later, Liu Hui invented a quick method of calculating and obtained an approximatevalue of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference inarea of successive polygons forms a geometric series with a factor of 4.
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Around 480, the Chinese mathematicianZu Chongzhi demonstrated that 355/113,and showed that 3.1415926 < < 3.1415927 using Liu Hui's algorithm applied to a12288-gon. This value was the most accurate approximation of available for the next900 years.

Classical period

Until thesecond millennium, was known to fewer than 10 decimal digits. The nextmajor advance in studies came with the development ofcalculus, and in particular thediscovery ofinfinite series which in principle permit calculating to any desiredaccuracy by adding sufficiently many terms. Around 1400, Madhava of Sangamagramafound the first known such series:

This is now known as the MadhavaLeibniz series[28] [29] or Gregory-Leibniz series since itwas rediscovered by James GregoryandGottfried Leibnizin 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, bytransforming the series into

Madhava was able to calculate as 3.14159265359, correct to 11 decimal places. Therecord was beaten in 1424 by thePersian mathematician,Jamshd al-Ksh, whodetermined 16 decimals of .

The first major European contribution since Archimedes was made by the GermanmathematicianLudolph van Ceulen (15401610), who used a geometric method tocompute 35 decimals of . He was so proud of the calculation, which required the greaterpart of his life, that he had the digits engraved into his tombstone.[30]

Around the same time, the methods of calculus and determination of infinite series andproducts for geometrical quantities began to emerge in Europe. The first suchrepresentation was the Vite's formula,

found by Franois Vite in 1593. Another famous result is Wallis' product,
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by John Wallis in 1655. Isaac Newton himself derived a series for and calculated 15digits, 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."[31]

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

with

Formulas of this type, now known as Machin-like formulas, were used to set severalsuccessive records and remained the best known method for calculating well into theage of computers. A remarkable record was set by the calculating prodigy ZachariasDase, who in 1844 employed a Machin-like formula to calculate 200 decimals of in hishead at the behest ofGauss. The best value at the end of the 19th century was due toWilliam Shanks, who took 15 years to calculate with 707 digits, although due to amistake only the first 527 were correct. (To avoid such errors, modern record calculationsof any kind are often performed twice, with two different formulas. If the results are thesame, they are likely to be correct.)

Theoretical advances in the 18th century led to insights about 's nature that could not beachieved through numerical calculation alone.Johann Heinrich Lambertproved theirrationality of in 1761, andAdrien-Marie Legendre also proved in 1794 2 to beirrational. When Leonhard Eulerin 1735 solved the famous Basel problem finding theexact value of

which is 2/6, he established a deep connection between and theprime numbers. Both

Legendre and Leonhard Euler speculated that might betranscendental, which wasfinally proved in 1882 by Ferdinand von Lindemann.

William Jones' bookA New Introduction to Mathematics from 1706 is said to be the firstuse of the Greek letter for this constant, but the notation became particularly popularafterLeonhard Euleradopted it in 1737.[32] He wrote:
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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, theDiameter is to the Circumference as 1 to(16/5 4/239) 1/3(16/53 4/2393) + ... = 3.14159... = [3]}}

See also: 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 calculation records. John von Neumannet. al. usedENIACto compute 2037 digits of in1949, a calculation that took 70 hours. [33][34]Additional thousands of decimal places wereobtained 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 mostsignificant developments was the discovery of the fast Fourier transform(FFT) in the1960s, which allows computers to perform arithmetic on extremely large numbers

quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujanfound many new formulas for , some remarkable for their elegance and mathematicaldepth.[35]One of his formulas is the series,

and the related one found by theChudnovsky brothers in 1987,

which deliver 14 digits per term.[35]The Chudnovskys used this formula to set several computing records in the end of the 1980s, including the first calculation of over onebillion (1,011,196,691) decimals in 1989. It remains the formula of choice for calculating software that runs on personal computers, as opposed to the supercomputersused 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. Abreakthrough was made in 1975, when Richard Brent and Eugene Salaminindependentlydiscovered the BrentSalamin algorithm, which uses only arithmetic to double thenumber of correct digits at each step.[36]The algorithm consists of setting
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and iterating

until an and bn are close enough. Then the estimate for 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 PeterBorwein.[37] The methods have been used by Yasumasa Kanada and team to set most ofthe calculation records since 1980, up to a calculation of 206,158,430,000 decimals of in 1999. The current record is 1,241,100,000,000 decimals, set by Kanada and team in2002. Although most of Kanada's previous records were set using the Brent-Salaminalgorithm, the 2002 calculation made use of two Machin-like formulas that were slowerbut crucially reduced memory consumption. The calculation was performed on a 64-nodeHitachi supercomputer with 1 terabyte of main memory, capable of carrying out 2 trillionoperations per second.

An important recent development was the BaileyBorweinPlouffe formula (BBP

formula), discovered by Simon Plouffe and named after the authors of the paper in whichthe formula was first published, David H. Bailey, Peter Borwein, and Plouffe.[38] Theformula,

is remarkable because it allows extracting any individual hexadecimal orbinarydigit of without calculating all the preceding ones.[38] Between 1998 and 2000, the distributedcomputingproject 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 , which turned out to be0.[39]

In 2006, Simon Plouffe, using the integer relation algorithm PSLQ, found a series ofbeautiful formulas.[40] Let q = e, then
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and others of form,

where q = e, kis anodd number, and a, b, c arerational numbers. Ifkis of the form4m + 3, then the formula has the particularly simple form,

for some rational numberp where thedenominatoris a highly factorable number, thoughno rigorous proof has yet been given.

Memorizing digits

Main article: Piphilology

Recent decades have seen a surge in the record for number of digits memorized.

Even long before computers have calculated , memorizing a recordnumber of digitsbecame an obsession for some people. In 2006, Akira Haraguchi, a retired Japaneseengineer, claimed to have recited 100,000 decimal places.[41] This, however, has yet to beverified by Guinness World Records. The Guinness-recognized record for remembereddigits of is 67,890 digits, held by Lu Chao, a 24-year-old graduate student fromChina.[42]It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of without an error.[43]
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There are many ways to memorize , including the use of "piems", which are poems thatrepresent in a way such that the length of each word (in letters) represents a digit. Hereis an example of a piem, originally devised bySir James Jeans:How I need(or: want) adrink, alcoholic in nature (or: of course), after the heavy lectures (or: chapters) involvingquantum mechanics.[44][45] 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 containsthe first 3834 digits of in this manner.[46]Piems are related to the entire field ofhumorous yet serious study that involves the use ofmnemonic techniques to rememberthe digits of , known aspiphilology. In other languages there are similar methods ofmemorization. However, this method proves inefficient for large memorizations of .Other methods include remembering patterns in the numbers.[47]

Advanced properties

Numerical approximations

Main article: History of numerical approximations of

Due to the transcendental nature of , there are no closed form expressions for thenumber in terms of algebraic numbers and functions.[11]Formulas for calculating usingelementary arithmetic typically include series orsummation notation (such as "..."),which indicates that the formula is really a formula for an infinite sequence ofapproximations to .[48] The more terms included in a calculation, the closer to the resultwill get.

Consequently, numerical calculations must use approximations of . For many purposes,3.14 or22/7 is close enough, although engineers often use 3.1416 (5significant figures) or

3.14159 (6 significant figures) for more precision. The approximations 22/7 and 355/113, with3 and 7 significant figures respectively, are obtained from the simplecontinued fractionexpansion of . The approximation 355113 (3.1415929) is the best one that may beexpressed with a three-digit or four-digit numerator and denominator.[49][50][51]

The earliest numerical approximation of is almost certainly the value 3.[27] In caseswhere little precision is required, it may be an acceptable substitute. That 3 is anunderestimate follows from the fact that it is the ratio of theperimeterof aninscribedregularhexagon to thediameterof the circle.

Open questions

The most pressing open question about is whether it is a normal numberwhether anydigit block occurs in the expansion of just as often as one would statistically expect ifthe digits had been produced completely "randomly", and that this is true in every base,not just base 10.[52] Current knowledge on this point is very weak; e.g., it is not evenknown which of the digits 0,,9 occur infinitely often in the decimal expansion of .[53]
http://en.wikipedia.org/wiki/James_Hopwood_Jeanshttp://en.wikipedia.org/wiki/James_Hopwood_Jeanshttp://en.wikipedia.org/wiki/Pi#cite_note-43http://en.wikipedia.org/wiki/Pi#cite_note-44http://en.wikipedia.org/wiki/Cadaeic_Cadenzahttp://en.wikipedia.org/wiki/Pi#cite_note-45http://en.wikipedia.org/wiki/Pi#cite_note-45http://en.wikipedia.org/wiki/Mnemonichttp://en.wikipedia.org/wiki/Piphilologyhttp://en.wikipedia.org/wiki/Pi#cite_note-46http://en.wikipedia.org/wiki/Pi#cite_note-46http://en.wikipedia.org/wiki/History_of_numerical_approximations_of_?http://en.wikipedia.org/wiki/Pi#cite_note-ttop-10http://en.wikipedia.org/wiki/Pi#cite_note-ttop-10http://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Summation#Capital-sigma_notationhttp://en.wikipedia.org/wiki/Summation#Capital-sigma_notationhttp://en.wikipedia.org/wiki/Pi#cite_note-47http://en.wikipedia.org/wiki/Approximationhttp://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_?http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_?http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_?http://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Mil?http://en.wikipedia.org/wiki/Mil?http://en.wikipedia.org/wiki/Mil?http://en.wikipedia.org/wiki/Mil?http://en.wikipedia.org/wiki/Fraction_(mathematics)http://en.wikipedia.org/wiki/Pi#cite_note-48http://en.wikipedia.org/wiki/Pi#cite_note-49http://en.wikipedia.org/wiki/Pi#cite_note-50http://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/Pi#cite_note-ahop-26http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Inscribed_figurehttp://en.wikipedia.org/wiki/Inscribed_figurehttp://en.wikipedia.org/wiki/Regular_polygonhttp://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Normal_numberhttp://en.wikipedia.org/wiki/Pi#cite_note-51http://en.wikipedia.org/wiki/Pi#cite_note-52http://en.wikipedia.org/wiki/James_Hopwood_Jeanshttp://en.wikipedia.org/wiki/Pi#cite_note-43http://en.wikipedia.org/wiki/Pi#cite_note-44http://en.wikipedia.org/wiki/Cadaeic_Cadenzahttp://en.wikipedia.org/wiki/Pi#cite_note-45http://en.wikipedia.org/wiki/Mnemonichttp://en.wikipedia.org/wiki/Piphilologyhttp://en.wikipedia.org/wiki/Pi#cite_note-46http://en.wikipedia.org/wiki/History_of_numerical_approximations_of_?http://en.wikipedia.org/wiki/Pi#cite_note-ttop-10http://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Summation#Capital-sigma_notationhttp://en.wikipedia.org/wiki/Pi#cite_note-47http://en.wikipedia.org/wiki/Approximationhttp://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_?http://en.wikipedia.org/wiki/Significant_figureshttp://en.wikipedia.org/wiki/Continued_fractionhttp://en.wikipedia.org/wiki/Mil?http://en.wikipedia.org/wiki/Fraction_(mathematics)http://en.wikipedia.org/wiki/Pi#cite_note-48http://en.wikipedia.org/wiki/Pi#cite_note-49http://en.wikipedia.org/wiki/Pi#cite_note-50http://en.wikipedia.org/wiki/3_(number)http://en.wikipedia.org/wiki/Pi#cite_note-ahop-26http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Inscribed_figurehttp://en.wikipedia.org/wiki/Regular_polygonhttp://en.wikipedia.org/wiki/Hexagonhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Normal_numberhttp://en.wikipedia.org/wiki/Pi#cite_note-51http://en.wikipedia.org/wiki/Pi#cite_note-52


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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 andvarious other constants can be reduced to a plausibleconjecture ofchaos theory.[54]

It is also unknown whether ande arealgebraically independent, although Yuri

Nesterenko proved the algebraic independence of {, e

,(1/4)} in 1996.[55]

Use in mathematics and science

Main article: List of formulas involving

is ubiquitous in mathematics, appearing even in places that lack an obvious connectionto the circles of Euclidean geometry.[56]

Geometry and trigonometry

See also: Area of a disk

For any circle with radius rand diameterd= 2r, the circumference is dand the area isr2. Further, appears in formulas for areas and volumes of many other geometricalshapes based on circles, such asellipses,spheres,cones, and tori.[57] Accordingly, appears in definite integrals that describe circumference, area or volume of shapesgenerated by circles. In the basic case, half the area of the unit diskis given by:[58]

and

gives half the circumference of the unit circle.[57] More complicated shapes can beintegrated as solids of revolution.[59]

From the unit-circle definition of the trigonometric functionsalso follows that the sineand cosine have period 2. That is, for allx and integers n, sin(x) = sin(x + 2n) and

cos(x) = cos(x + 2n). Because sin(0) = 0, sin(2n) = 0 for all integers n. Also, the anglemeasure of 180 is equal to radians. In other words, 1 = (/180) radians.

In modern mathematics, is often definedusing trigonometric functions, for example asthe smallest positivex for which sinx = 0, to avoid unnecessary dependence on thesubtleties of Euclidean geometry and integration. Equivalently, can be defined using theinverse trigonometric functions, for example as = 2 arccos(0) or = 4 arctan(1).
http://en.wikipedia.org/wiki/Richard_Crandallhttp://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formulahttp://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formulahttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Chaos_theoryhttp://en.wikipedia.org/wiki/Chaos_theoryhttp://en.wikipedia.org/wiki/Pi#cite_note-53http://en.wikipedia.org/wiki/Pi#cite_note-53http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Algebraic_independencehttp://en.wikipedia.org/wiki/Algebraic_independencehttp://en.wikipedia.org/wiki/Yuri_Valentinovich_Nesterenkohttp://en.wikipedia.org/wiki/Yuri_Valentinovich_Nesterenkohttp://en.wikipedia.org/wiki/Gelfond's_constanthttp://en.wikipedia.org/wiki/Gelfond's_constanthttp://en.wikipedia.org/wiki/Gelfond's_constanthttp://en.wikipedia.org/wiki/Gelfond's_constanthttp://en.wikipedia.org/wiki/Gamma_functionhttp://en.wikipedia.org/wiki/Pi#cite_note-54http://en.wikipedia.org/wiki/List_of_formulas_involving_?http://en.wikipedia.org/wiki/Pi#cite_note-55http://en.wikipedia.org/wiki/Area_of_a_diskhttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Cone_(geometry)http://en.wikipedia.org/wiki/Cone_(geometry)http://en.wikipedia.org/wiki/Torushttp://en.wikipedia.org/wiki/Pi#cite_note-psu1-56http://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Unit_diskhttp://en.wikipedia.org/wiki/Pi#cite_note-udi-57http://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Pi#cite_note-psu1-56http://en.wikipedia.org/wiki/Solid_of_revolutionhttp://en.wikipedia.org/wiki/Solid_of_revolutionhttp://en.wikipedia.org/wiki/Pi#cite_note-58http://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionhttp://en.wikipedia.org/wiki/Richard_Crandallhttp://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formulahttp://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formulahttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Chaos_theoryhttp://en.wikipedia.org/wiki/Pi#cite_note-53http://en.wikipedia.org/wiki/E_(mathematical_constant)http://en.wikipedia.org/wiki/Algebraic_independencehttp://en.wikipedia.org/wiki/Yuri_Valentinovich_Nesterenkohttp://en.wikipedia.org/wiki/Yuri_Valentinovich_Nesterenkohttp://en.wikipedia.org/wiki/Gelfond's_constanthttp://en.wikipedia.org/wiki/Gamma_functionhttp://en.wikipedia.org/wiki/Pi#cite_note-54http://en.wikipedia.org/wiki/List_of_formulas_involving_?http://en.wikipedia.org/wiki/Pi#cite_note-55http://en.wikipedia.org/wiki/Area_of_a_diskhttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Cone_(geometry)http://en.wikipedia.org/wiki/Torushttp://en.wikipedia.org/wiki/Pi#cite_note-psu1-56http://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Unit_diskhttp://en.wikipedia.org/wiki/Pi#cite_note-udi-57http://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Pi#cite_note-psu1-56http://en.wikipedia.org/wiki/Solid_of_revolutionhttp://en.wikipedia.org/wiki/Pi#cite_note-58http://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Inverse_trigonometric_function


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Expanding inverse trigonometric functions aspower series is the easiest way to deriveinfinite series for .

Complex numbers and calculus

The frequent appearance of in complex analysis can be related to the behavior of theexponential 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. Thisformula implies that imaginary powers ofe describe rotations on theunit circle in the

complex plane; these rotations have a period of 360 = 2. In particular, the 180 rotation = 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 functionof a half-integer is a rational multiple of .

Physics
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Although not aphysical constant, appears routinely in equations describingfundamental principles of the Universe, due in no small part to its relationship to thenature of the circle and, correspondingly, spherical coordinate systems. Using units suchas Planck units can sometimes eliminate from formulae.

The cosmological constant:[60]

Heisenberg's uncertainty principle, which shows that the uncertainty in themeasurement of a particle's position (x) and momentum (p) can not both bearbitrarily small at the same time:[61]

Einstein's field equation ofgeneral relativity:[62]

Coulomb's law for theelectric force, describing the force between two electriccharges (q1 and q2) separated by distance r:[63]

Magnetic permeability of free space:[64]

Kepler's third law constant, relating the orbital period (P) and the semimajor axis(a) to the masses (Mand m) of two co-orbiting bodies:

Probability and statistics

Inprobability and statistics, there are manydistributions whose formulas contain ,including:
http://en.wikipedia.org/wiki/Physical_constanthttp://en.wikipedia.org/wiki/Spherical_coordinate_systemhttp://en.wikipedia.org/wiki/Planck_unitshttp://en.wikipedia.org/wiki/Cosmological_constanthttp://en.wikipedia.org/wiki/Pi#cite_note-59http://en.wikipedia.org/wiki/Uncertainty_principlehttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Pi#cite_note-60http://en.wikipedia.org/wiki/Pi#cite_note-60http://en.wikipedia.org/wiki/Einstein_field_equationshttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Pi#cite_note-ein-61http://en.wikipedia.org/wiki/Coulomb's_lawhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Pi#cite_note-62http://en.wikipedia.org/wiki/Magnetic_constanthttp://en.wikipedia.org/wiki/Pi#cite_note-63http://en.wikipedia.org/wiki/Pi#cite_note-63http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Kepler.27s_third_lawhttp://en.wikipedia.org/wiki/Orbital_periodhttp://en.wikipedia.org/wiki/Semimajor_axishttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Probability_distributionhttp://en.wikipedia.org/wiki/Physical_constanthttp://en.wikipedia.org/wiki/Spherical_coordinate_systemhttp://en.wikipedia.org/wiki/Planck_unitshttp://en.wikipedia.org/wiki/Cosmological_constanthttp://en.wikipedia.org/wiki/Pi#cite_note-59http://en.wikipedia.org/wiki/Uncertainty_principlehttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Pi#cite_note-60http://en.wikipedia.org/wiki/Einstein_field_equationshttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Pi#cite_note-ein-61http://en.wikipedia.org/wiki/Coulomb's_lawhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Pi#cite_note-62http://en.wikipedia.org/wiki/Magnetic_constanthttp://en.wikipedia.org/wiki/Pi#cite_note-63http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Kepler.27s_third_lawhttp://en.wikipedia.org/wiki/Orbital_periodhttp://en.wikipedia.org/wiki/Semimajor_axishttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Probability_distribution


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theprobability density function for thenormal distribution with mean andstandard deviation, due to the Gaussian integral:[65]

the probability density function for the (standard) Cauchy distribution:[66]

Note that since for any probability density functionf(x), the aboveformulas can be used to produce other integral formulas for .[67]

Buffon's needle problem is sometimes quoted as a empirical approximation of in"popular mathematics" works. Consider dropping a needle of lengthL repeatedly on asurface containing parallel lines drawn Sunits apart (with S>L). If the needle is droppedn times andx of those times it comes to rest crossing a line (x > 0), then one mayapproximate using the Monte Carlo method:[68][69][70][71]

Though this result is mathematically impeccable, it cannot be used to determine morethan very few digits of by experiment. Reliably getting just three digits (including the

initial "3") right requires millions of throws,[68] and the number of throws growsexponentially with the number of digits desired. Furthermore, any error in themeasurement of the lengthsL and Swill transfer directly to an error in the approximated. For example, a difference of a single atom in the length of a 10-centimeter needlewould show up around the 9th digit of the result. In practice, uncertainties in determiningwhether the needle actually crosses a line when it appears to exactly touch it will limit theattainable accuracy to much less than 9 digits.

Pi in popular culture
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A whimsical "Pi plate".

Probably because of the simplicity of its definition, the concept of pi and, especially itsdecimal expression, have become entrenched in popular culture to a degree far greaterthan almost any other mathematical construct.[72] It is, perhaps, the most common ground

between mathematicians and non-mathematicians.[73] Reports on the latest, most-precisecalculation of (and related stunts) are common news items.[74]Pi Day (March 14, from3.14) is observed in many schools.[75] At least one cheer at the Massachusetts Institute ofTechnology includes "3.14159!"[76] One can buy a "Pi plate": a pie dish with both "" anda decimal expression of it appearing on it.[77
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this article is about the mathematical constant

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