Pi

This article is about the number π. For the Greek letter,see Pi (letter). For other uses of pi, π, and Π, see Pi(disambiguation).

The number π is a mathematical constant, the ratio of acircle's circumference to its diameter, commonly approx-imated as 3.14159. It has been represented by the Greekletter "π" since the mid-18th century, though it is alsosometimes spelled out as "pi" (/paɪ/).Being an irrational number, π cannot be expressed ex-actly as a fraction (equivalently, its decimal representa-tion never ends and never settles into a permanent re-peating pattern). Still, fractions such as 22/7 and otherrational numbers are commonly used to approximate π.The digits appear to be randomly distributed; however,to date, no proof of this has been discovered. Also, π isa transcendental number – a number that is not the rootof any non-zero polynomial having rational coefficients.This transcendence of π implies that it is impossible tosolve the ancient challenge of squaring the circle with acompass and straightedge.Although ancient civilizations needed the value of π tobe computed accurately for practical reasons, it was notcalculated to more than seven digits, using geometricaltechniques, in Chinese mathematics and to about five inIndian mathematics in the 5th century CE. The histori-cally first exact formula for π, based on infinite series, wasnot available until a millennium later, when in the 14thcentury the Madhava–Leibniz series was discovered inIndian mathematics.[1][2] In the 20th and 21st centuries,mathematicians and computer scientists discovered newapproaches that, when combined with increasing compu-tational power, extended the decimal representation of πto, as of 2015, over 13.3 trillion (1013) digits.[3] Practi-cally all scientific applications require no more than a fewhundred digits of π, and many substantially fewer, so theprimary motivation for these computations is the humandesire to break records.[4][5] However, the extensive cal-culations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is foundin many formulae in trigonometry and geometry, espe-cially those concerning circles, ellipses or spheres. It isalso found in formulae used in other branches of sciencesuch as cosmology, number theory, statistics, fractals,thermodynamics, mechanics and electromagnetism. Theubiquity of π makes it one of the most widely knownmathematical constants both inside and outside the sci-

entific community: Several books devoted to it havebeen published, the number is celebrated on Pi Day andrecord-setting calculations of the digits of π often resultin news headlines. Attempts to memorize the value ofπ with increasing precision have led to records of over67,000 digits.

1 Fundamentals

1.1 Name

The symbol used by mathematicians to represent the ratioof a circle’s circumference to its diameter is the lowercaseGreek letter π, sometimes spelled out as pi. In English,π is pronounced as “pie” ( /paɪ/, paɪ).[6] In mathematicaluse, the lowercase letter π (or π in sans-serif font) is dis-tinguished from its capital counterpart Π, which denotesa product of a sequence.The choice of the symbol π is discussed in the sectionAdoption of the symbol π.

1.2 Definition

C

d

diame

ter

The circumference of a circle is slightly more than three times aslong as its diameter. The exact ratio is called π.

π is commonly defined as the ratio of a circle'scircumference C to its diameter d:[7]

1

2 1 FUNDAMENTALS

π =C

d

The ratio C/d is constant, regardless of the circle’s size.For example, if a circle has twice the diameter of anothercircle it will also have twice the circumference, preservingthe ratio C/d. This definition of π implicitly makes use offlat (Euclidean) geometry; although the notion of a circlecan be extended to any curved (non-Euclidean) geometry,these new circles will no longer satisfy the formula π =C/d.[7]

Here, the circumference of a circle is the arc lengtharound the perimeter of the circle, a quantity whichcan be formally defined independently of geometry us-ing limits, a concept in calculus.[8] For example, one maycompute directly the arc length of the top half of the unitcircle given in Cartesian coordinates by x2 + y2 = 1 , asthe integral:[9]

π =

∫ 1

−1

dx√1− x2

.

An integral such as this was adopted as the definition of πby Karl Weierstrass, who defined it directly as an integralin 1841.[10]

Definitions of π such as these that rely on a notion ofcircumference, and hence implicitly on concepts of theintegral calculus, are no longer common in the literature.Remmert (1991) explains that this is because in manymodern treatments of calculus, differential calculus typi-cally precedes integral calculus in the university curricu-lum, so it is desirable to have a definition of π that doesnot rely on the latter. One such definition, due to RichardBaltzer,[11] and popularized by Edmund Landau,[12] isthe following: π is twice the smallest positive number atwhich the cosine function equals 0.[7][13][14] The cosinecan be defined independently of geometry as a power se-ries,[15] or as the solution of a differential equation.[13]

In a similar spirit, π can be defined instead using prop-erties of the complex exponential, exp(z), of a complexvariable z. Like the cosine, the complex exponential canbe defined in one of several ways. The set of complexnumbers at which exp(z) is equal to one is then an (imag-inary) arithmetic progression of the form:

{. . . ,−2πi, 0, 2πi, 4πi, . . . } = {2πki|k ∈ Z}

and there is a unique positive real number π with thisproperty.[14][16] A more abstract variation on the sameidea, making use of sophisticated mathematical conceptsof topology and algebra, is the following theorem:[17]there is a unique continuous isomorphism from the groupR/Z of real numbers under addition modulo integers (thecircle group) onto the multiplicative group of complex

numbers of absolute value one. The number π is thendefined as half the magnitude of the derivative of thishomomorphism.[18]

1.3 Properties

π is an irrational number, meaning that it cannot be writ-ten as the ratio of two integers (fractions such as 22/7are commonly used to approximate π; no common frac-tion (ratio of whole numbers) can be its exact value).[19]Since π is irrational, it has an infinite number of digitsin its decimal representation, and it does not settle intoan infinitely repeating pattern of digits. There are severalproofs that π is irrational; they generally require calculusand rely on the reductio ad absurdum technique. The de-gree to which π can be approximated by rational numbers(called the irrationality measure) is not precisely known;estimates have established that the irrationality measureis larger than the measure of e or ln(2) but smaller thanthe measure of Liouville numbers.[20]

√π

r=1

Because π is a transcendental number, squaring the circle is notpossible in a finite number of steps using the classical tools ofcompass and straightedge.

More strongly, π is a transcendental number, whichmeans that it is not the solution of any non-constantpolynomial with rational coefficients, such as x5/120 −x3/6 + x = 0.[21][22]

The transcendence of π has two important consequences:First, π cannot be expressed using any finite combinationof rational numbers and square roots or n-th roots suchas 3√31 or √10. Second, since no transcendental num-ber can be constructed with compass and straightedge,it is not possible to "square the circle". In other words,it is impossible to construct, using compass and straight-edge alone, a square whose area is equal to the area of agiven circle.[23] Squaring a circle was one of the impor-tant geometry problems of the classical antiquity.[24] Am-

1.5 Approximate value 3

ateur mathematicians in modern times have sometimesattempted to square the circle and sometimes claim suc-cess despite the fact that it is impossible.[25]

The digits of π have no apparent pattern and havepassed tests for statistical randomness, including tests fornormality; a number of infinite length is called normalwhen all possible sequences of digits (of any given length)appear equally often.[26] The conjecture that π is normalhas not been proven or disproven.[26] Since the advent ofcomputers, a large number of digits of π have been avail-able on which to perform statistical analysis. YasumasaKanada has performed detailed statistical analyses on thedecimal digits of π and found them consistent with nor-mality; for example, the frequency of the ten digits 0 to9 were subjected to statistical significance tests, and noevidence of a pattern was found.[27] Despite the fact thatπ's digits pass statistical tests for randomness, π containssome sequences of digits that may appear non-random tonon-mathematicians, such as the Feynman point, which isa sequence of six consecutive 9s that begins at the 762nddecimal place of the decimal representation of π.[28]

1.4 Continued fractions

The constant π is represented in this mosaic outside the Mathe-matics Building at the Technical University of Berlin.

Like all irrational numbers, π cannot be represented as acommon fraction (also known as a simple or vulgar frac-tion), by the very definition of “irrational”. But everyirrational number, including π, can be represented by aninfinite series of nested fractions, called a continued frac-tion:

π = 3 + 1

7+1

15+1

1+1

292+1

1+1

1+1

1+. . .

A001203Truncating the continued fraction at any point yields a ra-tional approximation for π; the first four of these are 3,

22/7, 333/106, and 355/113. These numbers are amongthe most well-known and widely used historical approx-imations of the constant. Each approximation generatedin this way is a best rational approximation; that is, eachis closer to π than any other fraction with the same or asmaller denominator.[29] Because π is known to be tran-scendental, it is by definition not algebraic and so can-not be a quadratic irrational. Therefore, π cannot havea periodic continued fraction. Although the simple con-tinued fraction for π (shown above) also does not exhibitany other obvious pattern,[30] mathematicians have dis-covered several generalized continued fractions that do,such as:[31]

π =4

1 + 12

2+32

2+52

2+72

2+92

2+. . .

= 3+ 12

6+32

6+52

6+72

6+92

6+. . .

=4

1 + 12

3+22

5+32

7+42

9+. . .

1.5 Approximate value

Some approximations of pi include:

• Integers: 3

• Fractions: Approximate fractions include (inorder of increasing accuracy) 22/7, 333/106,355/113, 52163/16604, 103993/33102, and245850922/78256779.[29] (List is selected termsfrom A063674 and A063673.)

• Decimal: The first 50 decimal digits are3.14159265358979323846264338327950288419716939937510...[32]A000796

• Binary: The base 2 approximation to 48 digits is11.001001000011111101101010100010001000010110100011...

• Hexadecimal: The base 16 approximation to 20digits is 3.243F6A8885A308D31319...[33]

• Sexagesimal: A base 60 approximation to five sex-agesimal digits is 3;8,29,44,0,47[34]

2 History

Main article: Approximations of πSee also: Chronology of computation of π

2.1 Antiquity

The best known approximations to π dating to before theCommon Era were accurate to two decimal places; this

4 2 HISTORY

was improved upon in Chinese mathematics in particu-lar by the mid first millennium, to an accuracy of sevendecimal places. After this, no further progress was madeuntil the late medieval period.Some Egyptologists[35] have claimed that the ancientEgyptians used an approximation of π as 22⁄7 from asearly as the Old Kingdom.[36] This claim has met withskepticism.[37][38][39][40]

The earliest written approximations of π are found inEgypt and Babylon, both within one percent of the truevalue. In Babylon, a clay tablet dated 1900–1600 BChas a geometrical statement that, by implication, treats πas 25⁄8 = 3.1250.[41] In Egypt, the Rhind Papyrus, datedaround 1650 BC but copied from a document dated to1850 BC, has a formula for the area of a circle that treatsπ as (16⁄9)2 ≈ 3.1605.[41]

Astronomical calculations in the Shatapatha Brahmana(ca. 4th century BC) use a fractional approximation of339⁄108 ≈ 3.139 (an accuracy of 9×10−4).[42] Other Indiansources by about 150 BC treat π as √10 ≈ 3.1622[43]

2.2 Polygon approximation era

π can be estimated by computing the perimeters of circumscribedand inscribed polygons.

The first recorded algorithm for rigorously calculatingthe value of π was a geometrical approach using poly-gons, devised around 250 BC by the Greek mathe-matician Archimedes.[44] This polygonal algorithm dom-inated for over 1,000 years, and as a result π is sometimesreferred to as “Archimedes’ constant”.[45] Archimedescomputed upper and lower bounds of π by drawing aregular hexagon inside and outside a circle, and succes-sively doubling the number of sides until he reached a96-sided regular polygon. By calculating the perimetersof these polygons, he proved that 223/71 < π < 22/7 (thatis 3.1408 < π < 3.1429).[46] Archimedes’ upper bound of22/7 may have led to a widespread popular belief that π isequal to 22/7.[47] Around 150 AD, Greek-Roman scien-tist Ptolemy, in hisAlmagest, gave a value for π of 3.1416,which he may have obtained from Archimedes or fromApollonius of Perga.[48] Mathematicians using polygonalalgorithms reached 39 digits of π in 1630, a record onlybroken in 1699 when infinite series were used to reach 71digits.[49]

In ancient China, values for π included 3.1547 (around 1AD), √10 (100 AD, approximately 3.1623), and 142/45(3rd century, approximately 3.1556).[50] Around 265

Archimedes developed the polygonal approach to approximatingπ.

AD, the Wei Kingdom mathematician Liu Hui cre-ated a polygon-based iterative algorithm and used itwith a 3,072-sided polygon to obtain a value of π of3.1416.[51][52] Liu later invented a faster method of cal-culating π and obtained a value of 3.14 with a 96-sidedpolygon, by taking advantage of the fact that the differ-ences in area of successive polygons form a geometric se-ries with a factor of 4.[51] The Chinese mathematician ZuChongzhi, around 480 AD, calculated that π ≈ 355/113(a fraction that goes by the nameMilü in Chinese), usingLiu Hui’s algorithm applied to a 12,288-sided polygon.With a correct value for its seven first decimal digits, thisvalue of 3.141592920... remained the most accurate ap-proximation of π available for the next 800 years.[53]

The Indian astronomer Aryabhata used a value of 3.1416in his Āryabhaṭīya (499 AD).[54] Fibonacci in c. 1220computed 3.1418 using a polygonal method, independentof Archimedes.[55] Italian author Dante apparently em-ployed the value 3+√2/10 ≈ 3.14142.[55]

The Persian astronomer Jamshīd al-Kāshī produced 9sexagesimal digits, roughly the equivalent of 16 decimaldigits, in 1424 using a polygon with 3×228 sides,[56][57]which stood as the world record for about 180 years.[58]French mathematician François Viète in 1579 achieved9 digits with a polygon of 3×217 sides.[58] Flemishmathematician Adriaan van Roomen arrived at 15 dec-imal places in 1593.[58] In 1596, Dutch mathematicianLudolph van Ceulen reached 20 digits, a record he laterincreased to 35 digits (as a result, π was called the

2.3 Infinite series 5

“Ludolphian number” in Germany until the early 20thcentury).[59] Dutch scientist Willebrord Snellius reached34 digits in 1621,[60] and Austrian astronomer ChristophGrienberger arrived at 38 digits in 1630 using 1040sides,[61] which remains the most accurate approximationmanually achieved using polygonal algorithms.[60]

2.3 Infinite series

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0Sn

2

4

6

8

10

12

14

16

18

20

n

3.13 3.15 3.17Sn

2

4

6

8

10

12

14

16

18

20

n

3.140 3.143Sn

2

4

6

8

10

12

14

16

18

20

n

3.1415 3.1418Sn

2

4

6

8

10

12

14

16

18

20

n

Viète

Wallis

Madhava-Gregory-Leibniz

Madhava

Newton

Nilakantha

Comparison of the convergence of several historical infinite se-ries for π. S is the approximation after taking n terms. Eachsubsequent subplot magnifies the shaded area horizontally by 10times. (click for detail)

The calculation of π was revolutionized by the devel-opment of infinite series techniques in the 16th and17th centuries. An infinite series is the sum of theterms of an infinite sequence.[62] Infinite series allowedmathematicians to compute π with much greater pre-cision than Archimedes and others who used geometri-cal techniques.[62] Although infinite series were exploitedfor π most notably by European mathematicians such asJames Gregory and Gottfried Wilhelm Leibniz, the ap-proach was first discovered in India sometime between1400 and 1500 AD.[63] The first written description of aninfinite series that could be used to compute π was laidout in Sanskrit verse by Indian astronomer Nilakantha So-mayaji in his Tantrasamgraha, around 1500 AD.[64] Theseries are presented without proof, but proofs are pre-sented in a later Indian work, Yuktibhāṣā, from around1530 AD. Nilakantha attributes the series to an ear-lier Indian mathematician, Madhava of Sangamagrama,who lived c. 1350 – c. 1425.[64] Several infinite seriesare described, including series for sine, tangent, and co-sine, which are now referred to as the Madhava series orGregory–Leibniz series.[64] Madhava used infinite seriesto estimate π to 11 digits around 1400, but that value wasimproved on around 1430 by the Persian mathematicianJamshīd al-Kāshī, using a polygonal algorithm.[65]

The first infinite sequence discovered in Europe was aninfinite product (rather than an infinite sum, which aremore typically used in π calculations) found by Frenchmathematician François Viète in 1593:[67]

2π =

√22 ·

√2+

√2

2 ·√

2+√

2+√2

2 · · ·A060294

Isaac Newton used infinite series to compute π to 15 digits, laterwriting “I am ashamed to tell you to how many figures I carriedthese computations”.[66]

The second infinite sequence found in Europe, by JohnWallis in 1655, was also an infinite product.[67] The dis-covery of calculus, by English scientist Isaac Newton andGerman mathematician GottfriedWilhelm Leibniz in the1660s, led to the development of many infinite series forapproximating π. Newton himself used an arcsin series tocompute a 15 digit approximation of π in 1665 or 1666,later writing “I am ashamed to tell you to how many fig-ures I carried these computations, having no other busi-ness at the time.”[66]

In Europe, Madhava’s formula was rediscovered by Scot-tish mathematician James Gregory in 1671, and by Leib-niz in 1674:[68][69]

arctan z = z − z3

3+

z5

5− z7

7+ · · ·

This formula, the Gregory–Leibniz series, equals π/4when evaluated with z = 1.[69] In 1699, English mathe-matician Abraham Sharp used the Gregory–Leibniz se-ries to compute π to 71 digits, breaking the previousrecord of 39 digits, which was set with a polygonalalgorithm.[70] The Gregory–Leibniz series is simple, butconverges very slowly (that is, approaches the answergradually), so it is not used in modern π calculations.[71]

In 1706 John Machin used the Gregory–Leibniz series toproduce an algorithm that converged much faster:[72]

6 2 HISTORY

π

4= 4 arctan 1

5− arctan 1

239

Machin reached 100 digits of π with this formula.[73]Other mathematicians created variants, now known asMachin-like formulae, that were used to set several suc-cessive records for calculating digits of π.[73] Machin-likeformulae remained the best-known method for calculat-ing π well into the age of computers, and were used to setrecords for 250 years, culminating in a 620-digit approxi-mation in 1946 by Daniel Ferguson – the best approxima-tion achieved without the aid of a calculating device.[74]

A record was set by the calculating prodigy ZachariasDase, who in 1844 employed a Machin-like formula tocalculate 200 decimals of π in his head at the behest ofGerman mathematician Carl Friedrich Gauss.[75] Britishmathematician William Shanks famously took 15 yearsto calculate π to 707 digits, but made a mistake in the528th digit, rendering all subsequent digits incorrect.[75]

2.3.1 Rate of convergence

Some infinite series for π converge faster than others.Given the choice of two infinite series for π, math-ematicians will generally use the one that convergesmore rapidly because faster convergence reduces theamount of computation needed to calculate π to any givenaccuracy.[76] A simple infinite series for π is the Gregory–Leibniz series:[77]

π =4

1− 4

3+

4

5− 4

7+

4

9− 4

11+

4

13− · · ·

As individual terms of this infinite series are added to thesum, the total gradually gets closer to π, and – with a suf-ficient number of terms – can get as close to π as desired.It converges quite slowly, though – after 500,000 terms,it produces only five correct decimal digits of π.[78]

An infinite series for π (published by Nilakantha inthe 15th century) that converges more rapidly than theGregory–Leibniz series is:[79]

π = 3+4

2× 3× 4− 4

4× 5× 6+

4

6× 7× 8− 4

8× 9× 10+· · ·

The following table compares the convergence rates ofthese two series:

After five terms, the sum of the Gregory–Leibniz seriesis within 0.2 of the correct value of π, whereas the sum ofNilakantha’s series is within 0.002 of the correct value ofπ. Nilakantha’s series converges faster and is more usefulfor computing digits of π. Series that converge even fasterinclude Machin’s series and Chudnovsky’s series, the lat-ter producing 14 correct decimal digits per term.[76]

2.4 Irrationality and transcendence

See also: Proof that π is irrational and Proof that π istranscendental

Not all mathematical advances relating to π were aimedat increasing the accuracy of approximations. When Eu-ler solved the Basel problem in 1735, finding the exactvalue of the sum of the reciprocal squares, he establisheda connection between π and the prime numbers that latercontributed to the development and study of the Riemannzeta function:[80]

π2

6=

1

12+

1

22+

1

32+

1

42+ · · ·

Swiss scientist Johann Heinrich Lambert in 1761 provedthat π is irrational, meaning it is not equal to the quo-tient of any two whole numbers.[19] Lambert’s proof ex-ploited a continued-fraction representation of the tangentfunction.[81] French mathematician Adrien-Marie Legen-dre proved in 1794 that π2 is also irrational. In 1882, Ger-man mathematician Ferdinand von Lindemann provedthat π is transcendental, confirming a conjecture madeby both Legendre and Euler.[82]

2.5 Adoption of the symbol π

Leonhard Euler popularized the use of the Greek letter π in workshe published in 1736 and 1748.

The earliest known use of the Greek letter π to repre-sent the ratio of a circle’s circumference to its diameter

3.1 Computer era and iterative algorithms 7

was by Welsh mathematician William Jones in his 1706work Synopsis Palmariorum Matheseos; or, a New Intro-duction to the Mathematics.[83] The Greek letter first ap-pears there in the phrase “1/2 Periphery (π)" in the dis-cussion of a circle with radius one. Jones may have cho-sen π because it was the first letter in the Greek spelling ofthe word periphery.[84] However, he writes that his equa-tions for π are from the “ready pen of the truly ingeniousMr. John Machin”, leading to speculation that Machinmay have employed the Greek letter before Jones.[85] Ithad indeed been used earlier for geometric concepts.[85]William Oughtred used π and δ, the Greek letter equiva-lents of p and d, to express ratios of periphery and diame-ter in the 1647 and later editions of Clavis Mathematicae.After Jones introduced the Greek letter in 1706, it wasnot adopted by other mathematicians until Euler startedusing it, beginning with his 1736 work Mechanica. Be-fore then, mathematicians sometimes used letters suchas c or p instead.[85] Because Euler corresponded heav-ily with other mathematicians in Europe, the use of theGreek letter spread rapidly.[85] In 1748, Euler used π inhis widely read work Introductio in analysin infinitorum(he wrote: “for the sake of brevity we will write this num-ber as π; thus π is equal to half the circumference of a cir-cle of radius 1”) and the practice was universally adoptedthereafter in the Western world.[85]

3 Modern quest for more digits

3.1 Computer era and iterative algorithms

The Gauss–Legendre iterative algorithm:Initialize

a0=1 b0=1√2

t0=14 p0=1

Iterate

an+1=an+bn

2 bn+1=√anbn

tn+1=tn−pn(an−an+1)2 pn+1=2pn

Then an estimate for π is given by

π≈ (an+bn)2

4tn

The development of computers in the mid-20th centuryagain revolutionized the hunt for digits of π. Americanmathematicians John Wrench and Levi Smith reached1,120 digits in 1949 using a desk calculator.[86] Usingan inverse tangent (arctan) infinite series, a team led byGeorge Reitwiesner and John von Neumann that sameyear achieved 2,037 digits with a calculation that took70 hours of computer time on the ENIAC computer.[87]

John von Neumann was part of the team that first used a digitalcomputer, ENIAC, to compute π.

The record, always relying on an arctan series, was brokenrepeatedly (7,480 digits in 1957; 10,000 digits in 1958;100,000 digits in 1961) until 1million digits were reachedin 1973.[88]

Two additional developments around 1980 once againaccelerated the ability to compute π. First, the dis-covery of new iterative algorithms for computing π,which were much faster than the infinite series; and sec-ond, the invention of fast multiplication algorithms thatcould multiply large numbers very rapidly.[89] Such algo-rithms are particularly important in modern π computa-tions, because most of the computer’s time is devoted tomultiplication.[90] They include the Karatsuba algorithm,Toom–Cookmultiplication, and Fourier transform-basedmethods.[91]

The iterative algorithms were independently published in1975–1976 by American physicist Eugene Salamin andAustralian scientist Richard Brent.[92] These avoid re-liance on infinite series. An iterative algorithm repeats aspecific calculation, each iteration using the outputs fromprior steps as its inputs, and produces a result in each stepthat converges to the desired value. The approach wasactually invented over 160 years earlier by Carl FriedrichGauss, in what is now termed the arithmetic–geometricmean method (AGM method) or Gauss–Legendre algo-rithm.[92] As modified by Salamin and Brent, it is alsoreferred to as the Brent–Salamin algorithm.The iterative algorithms were widely used after 1980because they are faster than infinite series algorithms:

8 3 MODERN QUEST FOR MORE DIGITS

whereas infinite series typically increase the number ofcorrect digits additively in successive terms, iterative al-gorithms generally multiply the number of correct digitsat each step. For example, the Brent-Salamin algorithmdoubles the number of digits in each iteration. In 1984,the Canadian brothers John and Peter Borwein producedan iterative algorithm that quadruples the number of dig-its in each step; and in 1987, one that increases the num-ber of digits five times in each step.[93] Iterative methodswere used by Japanese mathematician Yasumasa Kanadato set several records for computing π between 1995 and2002.[94] This rapid convergence comes at a price: theiterative algorithms require significantly more memorythan infinite series.[94]

3.2 Motivations for computing π

1

100

104

106

108

1010

1012

1014

2000BCE

250 BCE

480 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000

Nu

mb

er

of

decim

al d

igit

s

Year

Record approximations of pi

As mathematicians discovered new algorithms, and computersbecame available, the number of known decimal digits of π in-creased dramatically. Note that the vertical scale is logarithmic.

For most numerical calculations involving π, a handfulof digits provide sufficient precision. According to JörgArndt and Christoph Haenel, thirty-nine digits are suffi-cient to perform most cosmological calculations, becausethat is the accuracy necessary to calculate the circumfer-ence of the observable universe with a precision of oneatom.[95] Despite this, people have worked strenuously tocompute π to thousands and millions of digits.[96] Thiseffort may be partly ascribed to the human compulsionto break records, and such achievements with π oftenmake headlines around the world.[97][98] They also havepractical benefits, such as testing supercomputers, testingnumerical analysis algorithms (including high-precisionmultiplication algorithms); and within pure mathematicsitself, providing data for evaluating the randomness of thedigits of π.[99]

3.3 Rapidly convergent series

Modern π calculators do not use iterative algorithmsexclusively. New infinite series were discovered inthe 1980s and 1990s that are as fast as iterative algo-rithms, yet are simpler and less memory intensive.[94]The fast iterative algorithms were anticipated in 1914,when the Indian mathematician Srinivasa Ramanujan

Srinivasa Ramanujan, working in isolation in India, producedmany innovative series for computing π.

published dozens of innovative new formulae for π, re-markable for their elegance, mathematical depth, andrapid convergence.[100] One of his formulae, based onmodular equations, is

1

π=

2√2

9801

∞∑k=0

(4k)!(1103 + 26390k)

k!4(3964k).

This series converges much more rapidly than most arc-tan series, including Machin’s formula.[101] Bill Gosperwas the first to use it for advances in the calculation of π,setting a record of 17million digits in 1985.[102] Ramanu-jan’s formulae anticipated the modern algorithms devel-oped by the Borwein brothers and the Chudnovsky broth-ers.[103] The Chudnovsky formula developed in 1987 is

1

π=

12

6403203/2

∞∑k=0

(6k)!(13591409 + 545140134k)

(3k)!(k!)3(−640320)3k.

It produces about 14 digits of π per term,[104] and hasbeen used for several record-setting π calculations, in-cluding the first to surpass 1 billion (109) digits in 1989by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digitsby Fabrice Bellard in 2009, and 10 trillion (1013) digits in2011 by Alexander Yee and Shigeru Kondo.[105][106] Forsimilar formulas, see also the Ramanujan–Sato series.In 2006, Canadianmathematician Simon Plouffe used thePSLQ integer relation algorithm[107] to generate several

9

new formulas for π, conforming to the following tem-plate:

πk =∞∑

n=1

1

nk

(a

qn − 1+

b

q2n − 1+

c

q4n − 1

),

where q is eπ (Gelfond’s constant), k is an odd num-ber, and a, b, c are certain rational numbers that Plouffecomputed.[108]

3.4 Spigot algorithms

Two algorithms were discovered in 1995 that opened upnew avenues of research into π. They are called spigot al-gorithms because, like water dripping from a spigot, theyproduce single digits of π that are not reused after theyare calculated.[109][110] This is in contrast to infinite seriesor iterative algorithms, which retain and use all interme-diate digits until the final result is produced.[109]

American mathematicians Stan Wagon and StanleyRabinowitz produced a simple spigot algorithm in1995.[110][111][112] Its speed is comparable to arctan al-gorithms, but not as fast as iterative algorithms.[111]

Another spigot algorithm, the BBP digit extraction algo-rithm, was discovered in 1995 by Simon Plouffe:[113][114]

π =

∞∑k=0

1

16k

(4

8k + 1− 2

8k + 4− 1

8k + 5− 1

8k + 6

)

This formula, unlike others before it, can produce any in-dividual hexadecimal digit of π without calculating all thepreceding digits.[113] Individual binary digits may be ex-tracted from individual hexadecimal digits, and octal dig-its can be extracted from one or two hexadecimal digits.Variations of the algorithm have been discovered, but nodigit extraction algorithm has yet been found that rapidlyproduces decimal digits.[115] An important application ofdigit extraction algorithms is to validate new claims ofrecord π computations: After a new record is claimed,the decimal result is converted to hexadecimal, and then adigit extraction algorithm is used to calculate several ran-dom hexadecimal digits near the end; if they match, thisprovides a measure of confidence that the entire compu-tation is correct.[106]

Between 1998 and 2000, the distributed computingproject PiHex used Bellard’s formula (a modification ofthe BBP algorithm) to compute the quadrillionth (1015th)bit of π, which turned out to be 0.[116] In September2010, a Yahoo! employee used the company’s Hadoopapplication on one thousand computers over a 23-day pe-riod to compute 256 bits of π at the two-quadrillionth(2×1015th) bit, which also happens to be zero.[117]

4 Use

Main article: List of formulae involving π

Because π is closely related to the circle, it is found inmany formulae from the fields of geometry and trigonom-etry, particularly those concerning circles, spheres, or el-lipses. Formulae from other branches of science alsoinclude π in some of their important formulae, includ-ing sciences such as statistics, fractals, thermodynamics,mechanics, cosmology, number theory, and electromag-netism.

4.1 Geometry and trigonometry

Area =

Circle Area =π× r2

r2

The area of the circle equals π times the shaded area.

π appears in formulae for areas and volumes of geomet-rical shapes based on circles, such as ellipses, spheres,cones, and tori. Below are some of the more commonformulae that involve π.[118]

• The circumference of a circle with radius r is 2πr.

• The area of a circle with radius r is πr2.

• The volume of a sphere with radius r is 4/3πr3.

• The surface area of a sphere with radius r is 4πr2.

The formulae above are special cases of the surface areaSn(r) and volume Vn(r) of an n-dimensional sphere.

Sn(r) =nπn/2

Γ(n2 +1)r

n−1

Vn(r) =πn/2

Γ(n2 +1)r

n

π appears in definite integrals that describe circumfer-ence, area, or volume of shapes generated by circles. For

10 4 USE

example, an integral that specifies half the area of a circleof radius one is given by:[119]

∫ 1

−1

√1− x2 dx =

π

2.

In that integral the function √1-x2 represents the tophalf of a circle (the square root is a consequence of thePythagorean theorem), and the integral ∫1−1 computes the area between that half of a circle andthe x axis.

Sine and cosine functions repeat with period 2π.

The trigonometric functions rely on angles, and mathe-maticians generally use radians as units of measurement.π plays an important role in angles measured in radians,which are defined so that a complete circle spans an angleof 2π radians.[120] The angle measure of 180° is equal toπ radians, and 1° = π/180 radians.[120]

Common trigonometric functions have periods that aremultiples of π; for example, sine and cosine have pe-riod 2π,[121] so for any angle θ and any integer k, sin θ =sin (θ + 2πk) and cos θ = cos (θ + 2πk) . [121]

4.1.1 Monte Carlo methods

al

b

t

Buffon’s needle. Needles a and b are dropped randomly.

Random dots are placed on the quadrant of a square witha circle inscribed in it.Monte Carlo methods, based on random trials, can beused to approximate π.

Monte Carlo methods, which evaluate the results of mul-tiple random trials, can be used to create approximations

of π.[122] Buffon’s needle is one such technique: If a nee-dle of length ℓ is dropped n times on a surface on whichparallel lines are drawn t units apart, and if x of thosetimes it comes to rest crossing a line (x > 0), then onemay approximate π based on the counts:[123]

π ≈ 2nℓ

xt

Another Monte Carlo method for computing π is to drawa circle inscribed in a square, and randomly place dots inthe square. The ratio of dots inside the circle to the totalnumber of dots will approximately equal π/4.[124]

Monte Carlo methods for approximating π are very slowcompared to other methods, and are never used to ap-proximate π when speed or accuracy are desired.[125]

4.2 Complex numbers and analysis

The association between imaginary powers of the number e andpoints on the unit circle centered at the origin in the complex planegiven by Euler’s formula.

Any complex number, say z, can be expressed using apair of real numbers. In the polar coordinate system, onenumber (radius or r) is used to represent z's distance fromthe origin of the complex plane and the other (angle or φ)to represent a counter-clockwise rotation from the posi-tive real line as follows:[126]

z = r · (cosφ+ i sinφ),

where i is the imaginary unit satisfying i2 = −1. The fre-quent appearance of π in complex analysis can be relatedto the behavior of the exponential function of a complexvariable, described by Euler’s formula:[127]

4.3 Number theory and Riemann zeta function 11

eiφ = cosφ+ i sinφ,

where the constant e is the base of the natural logarithm.This formula establishes a correspondence between imag-inary powers of e and points on the unit circle centered atthe origin of the complex plane. Setting φ = π in Euler’sformula results in Euler’s identity, celebrated by math-ematicians because it contains the five most importantmathematical constants:[127][128]

eiπ + 1 = 0.

There are n different complex numbers z satisfying zn =1, and these are called the "n-th roots of unity".[129] Theyare given by this formula:

e2πik/n (k = 0, 1, 2, . . . , n− 1).

Cauchy’s integral formula governs complex analyticfunctions and establishes an important relationship be-tween integration and differentiation, including the factthat the values of a complex function within a closedboundary are entirely determined by the values on theboundary:[130][131]

f(z0) =1

2πi

∮γ

f(z)

z − z0dz

An occurrence of π in theMandelbrot set fractal was dis-

π can be computed from theMandelbrot set, by counting the num-ber of iterations required before point (−0.75, ε) diverges.

covered by American David Boll in 1991.[132] He exam-ined the behavior of the Mandelbrot set near the “neck”at (−0.75, 0). If points with coordinates (−0.75, ε) areconsidered, as ε tends to zero, the number of iterationsuntil divergence for the point multiplied by ε convergesto π. The point (0.25, ε) at the cusp of the large “valley”on the right side of the Mandelbrot set behaves similarly:

the number of iterations until divergence multiplied bythe square root of ε tends to π.[132][133]

The gamma function extends the concept of factorial(normally defined only for non-negative integers) to allcomplex numbers, except the negative real integers.When the gamma function is evaluated at half-integers,the result contains π; for example Γ(1/2) =

√π and

Γ(5/2) = 3√π

4 .[134] The gamma function can be usedto create a simple approximation to n! for large n:n! ∼

√2πn

(ne

)n which is known as Stirling’s approxi-mation.[135]

4.3 Number theory and Riemann zetafunction

The Riemann zeta function ζ(s) is used in many areas ofmathematics. When evaluated at s = 2 it can be writtenas

ζ(2) =1

12+

1

22+

1

32+ · · ·

Finding a simple solution for this infinite series was a fa-mous problem in mathematics called the Basel problem.Leonhard Euler solved it in 1735 when he showed it wasequal to π2/6.[80] Euler’s result leads to the number theoryresult that the probability of two random numbers beingrelatively prime (that is, having no shared factors) is equalto 6/π2.[136][137] This probability is based on the observa-tion that the probability that any number is divisible by aprime p is 1/p (for example, every 7th integer is divisibleby 7.) Hence the probability that two numbers are bothdivisible by this prime is 1/p2, and the probability that atleast one of them is not is 1-1/p2. For distinct primes,these divisibility events are mutually independent; so theprobability that two numbers are relatively prime is givenby a product over all primes:[138]

∞∏p

(1− 1

p2

)=

( ∞∏p

1

1− p−2

)−1

=1

1 + 122 + 1

32 + · · ·=

1

ζ(2)=

6

π2≈ 61%

This probability can be used in conjunction with a randomnumber generator to approximate π using a Monte Carloapproach.[139]

4.4 Probability and statistics

The fields of probability and statistics frequently use thenormal distribution as a simple model for complex phe-nomena; for example, scientists generally assume that theobservational error in most experiments follows a nor-mal distribution.[140] π is found in the Gaussian function(which is the probability density function of the normaldistribution) with mean μ and standard deviation σ:[141]

12 5 OUTSIDE MATHEMATICS

-0.5

0

0.5

1

1.5

2

2.5

-2 -1 0 1 2

Area=sqrt(pi)e^(-x^2)

A graph of the Gaussian functionƒ(x) = e−x2 . The colored region between the function and thex-axis has area√π .

f(x) =1

σ√2π

e−(x−µ)2/(2σ2)

The area under the graph of the normal distribution curveis given by the Gaussian integral:[141]

∫ ∞

−∞e−x2

dx =√π,

while the related integral for the Cauchy distribution is

∫ ∞

−∞

1

x2 + 1dx = π.

5 Outside mathematics

5.1 Describing physical phenomena

Although not a physical constant, π appears routinely inequations describing fundamental principles of the uni-verse, often because of π's relationship to the circle andto spherical coordinate systems. A simple formula fromthe field of classical mechanics gives the approximate pe-riod T of a simple pendulum of length L, swinging witha small amplitude (g is the earth’s gravitational accelera-tion):[142]

T ≈ 2π

√L

g.

One of the key formulae of quantum mechanics isHeisenberg’s uncertainty principle, which shows that theuncertainty in the measurement of a particle’s position

(Δx) andmomentum (Δp) cannot both be arbitrarily smallat the same time (where h is Planck’s constant):[143]

∆x∆p ≥ h

4π.

In the domain of cosmology, π appears in Einstein’sfield equation, a fundamental formula which forms thebasis of the general theory of relativity and describesthe fundamental interaction of gravitation as a result ofspacetime being curved by matter and energy:[144]

Rik − gikR

2+ Λgik =

8πG

c4Tik,

where Rik is the Ricci curvature tensor, R is the scalarcurvature, gik is the metric tensor, Λ is the cosmologicalconstant, G is Newton’s gravitational constant, c is thespeed of light in vacuum, and Tik is the stress–energytensor.Coulomb’s law, from the discipline of electromagnetism,describes the electric field between two electric charges(q1 and q2) separated by distance r (with ε0 representingthe vacuum permittivity of free space):[145]

F =|q1q2|4πε0r2

.

The fact that π is approximately equal to 3 plays a role inthe relatively long lifetime of orthopositronium. The in-verse lifetime to lowest order in the fine structure constantα is[146]

1

τ= 2

π2 − 9

9πmα6,

where m is the mass of the electron.π is present in some structural engineering formulae, suchas the buckling formula derived by Euler, which givesthe maximum axial load F that a long, slender columnof length L, modulus of elasticity E, and area moment ofinertia I can carry without buckling:[147]

F =π2EI

L2.

The field of fluid dynamics contains π in Stokes’ law,which approximates the frictional force F exerted onsmall, spherical objects of radius R, moving with velocityv in a fluid with dynamic viscosity η:[148]

F = 6π η R v.

The Fourier transform, defined below, is a mathematicaloperation that expresses time as a function of frequency,

5.3 In popular culture 13

known as its frequency spectrum. It has many applica-tions in physics and engineering, particularly in signalprocessing.[149]

f̂(ξ) =

∫ ∞

−∞f(x) e−2πixξ dx

Under ideal conditions (uniform gentle slope on anhomogeneously erodible substrate), the sinuosity of ameandering river approaches π. The sinuosity is the ratiobetween the actual length and the straight-line distancefrom source to mouth. Faster currents along the outsideedges of a river’s bends cause more erosion than along theinside edges, thus pushing the bends even farther out, andincreasing the overall loopiness of the river. However,that loopiness eventually causes the river to double backon itself in places and “short-circuit”, creating an ox-bowlake in the process. The balance between these two op-posing factors leads to an average ratio of π between theactual length and the direct distance between source andmouth.[150][151]

5.2 Memorizing digits

Main article: Piphilology

Many persons have memorized large numbers of digits ofπ, a practice called piphilology.[152] One common tech-nique is to memorize a story or poem in which the wordlengths represent the digits of π: The first word has threeletters, the second word has one, the third has four, thefourth has one, the fifth has five, and so on. An early ex-ample of a memorization aid, originally devised by En-glish scientist James Jeans, is “How I want a drink, alco-holic of course, after the heavy lectures involving quan-tum mechanics.”[152] When a poem is used, it is some-times referred to as a piem. Poems for memorizing πhave been composed in several languages in addition toEnglish.[152]

The record for memorizing digits of π, certified byGuinness World Records, is 70,000 digits, recited in In-dia by Rajveer Meena in 9 hours and 27 minutes on 21March 2015.[153] In 2006, Akira Haraguchi, a retiredJapanese engineer, claimed to have recited 100,000 dec-imal places, but the claim was not verified by GuinnessWorld Records.[154] Record-setting π memorizers typ-ically do not rely on poems, but instead use methodssuch as remembering number patterns and the methodof loci.[155]

A few authors have used the digits of π to establisha new form of constrained writing, where the wordlengths are required to represent the digits of π. TheCadaeic Cadenza contains the first 3835 digits of π in thismanner,[156] and the full-length bookNot aWake contains10,000 words, each representing one digit of π.[157]

5.3 In popular culture

A pi pie. The circular shape of pie makes it a frequent subject ofpi puns.

Perhaps because of the simplicity of its definition andits ubiquitous presence in formulae, π has been repre-sented in popular culture more than other mathematicalconstructs.[158]

In the 2008 Open University and BBC documentary co-production, The Story of Maths, aired in October 2008on BBC Four, British mathematician Marcus du Sautoyshows a visualization of the - historically first exact -formula for calculating π when visiting India and explor-ing its contributions to trigonometry.[159]

In the Palais de la Découverte (a science museum inParis) there is a circular room known as the pi room. Onits wall are inscribed 707 digits of π. The digits are largewooden characters attached to the dome-like ceiling. Thedigits were based on an 1853 calculation by Englishmath-ematician William Shanks, which included an error be-ginning at the 528th digit. The error was detected in 1946and corrected in 1949.[160]

In Carl Sagan's novel Contact it is suggested that the cre-ator of the universe buried a message deep within the dig-its of π.[161] The digits of π have also been incorporatedinto the lyrics of the song “Pi” from the album Aerial byKate Bush,[162] and a song by Hard 'n Phirm.[163]

Many schools in the United States observe Pi Day on14 March (written 3/14 in the US style).[164] π and itsdigital representation are often used by self-described“math geeks" for inside jokes among mathematically andtechnologically minded groups. Several college cheersat the Massachusetts Institute of Technology include“3.14159”.[165] Pi Day in 2015 was particularly signifi-cant because the date and time 3/14/15 9:26:53 reflectedmany more digits of pi.[166]

14 7 NOTES

During the 2011 auction for Nortel's portfolio of valuabletechnology patents, Google made a series of unusuallyspecific bids based on mathematical and scientific con-stants, including π.[167]

Some formulas using the 2π definition of τ.

In 1958 Albert Eagle proposed replacing π by τ = π/2to simplify formulas.[168] However, no other authors areknown to use τ in this way. Some people use a dif-ferent value, τ = 6.283185... = 2π,[169] arguing thatτ, as the number of radians in one turn or as the ra-tio of a circle’s circumference to its radius rather thanits diameter, is more natural than π and simplifies manyformulas.[170][171] Celebrations of this number, because itapproximately equals 6.28, by making 28 June “Tau Day”and eating “twice the pie”,[172] have been reported in themedia. However this use of τ has not made its way intomainstream mathematics.[173]

In 1897, an amateur American mathematician attemptedto persuade the Indiana legislature to pass the Indiana PiBill, which described a method to square the circle andcontained text that implied various incorrect values for π,including 3.2. The bill is notorious as an attempt to estab-lish a value of scientific constant by legislative fiat. Thebill was passed by the Indiana House of Representatives,but rejected by the Senate.[174]

6 See also

• Chronology of computation of π

• Proof that π is irrational

• Proof that π is transcendental

• Mathematical constants and functions

• Approximations of π

7 Notes

Footnotes

[1] George E. Andrews, Richard Askey, Ranjan Roy (1999).Special Functions. Cambridge University Press. p. 58.ISBN 0-521-78988-5.

[2] Gupta, R. C. (1992). “On the remainder term in theMadhava–Leibniz’s series”. Ganita Bharati 14 (1-4): 68–71.

[3] http://www.numberworld.org/y-cruncher/

[4] Arndt & Haenel 2006, p. 17

[5] David Bailey; Jonathan Borwein; Peter Borwein; SimonPlouffe (1997), “The Quest for Pi”, The Mathematical In-telligencer 19 (1): 50–56

[6] “pi”. Dictionary.reference.com. 2 March 1993. Re-trieved 18 June 2012.

[7] Arndt & Haenel 2006, p. 8

[8] Tom Apostol (1967), Calculus, volume 1 (2nd ed.), Wi-ley. Page 102: “From a logical point of view, this is un-satisfactory at the present stage because we have not yetdiscussed the concept of arc length.” Arc length is intro-duced on page 529.

[9] Reinhold Remmert (1991), “What is π?", Numbers,Springer, p. 129

[10] Reinhold Remmert (1991), “What is π?", Numbers,Springer, p. 129. The precise integral that Weierstrassused was π =

∫∞−∞

dx1+x2 .

[11] Richard Baltzer (1870), Die Elemente der Mathematik,Hirzel, p. 195

[12] Edmund Landau (1934), Einführung in die Differential-rechnung und Integralrechnung, Noordoff, p. 193

[13] Rudin, Walter (1976). Principles of Mathematical Analy-sis. McGraw-Hill. ISBN 0-07-054235-X., p 183.

[14] Reinhold Remmert (1991), “What is π?", Numbers,Springer, p. 129

[15] Rudin, Walter (1986). Real and complex analysis.McGraw-Hill., p 2.

[16] Lars Ahlfors (1966), Complex analysis, McGraw-Hill, p.46

[17] Nicolas Bourbaki (1981), Topologie generale, Springer,§VIII.2

[18] Nicolas Bourbaki (1979), Fonctions d'une variable réelle,Springer, §II.3.

[19] Arndt & Haenel 2006, p. 5

[20] Salikhov, V. (2008). “On the Irrationality Mea-sure of pi”. Russian Mathematical Survey 53(3): 570–572. Bibcode:2008RuMaS..63..570S.doi:10.1070/RM2008v063n03ABEH004543.

[21] Mayer, Steve. “The Transcendence of π". Archived fromthe original on 2000-09-29. Retrieved 4 November 2007.

[22] The polynomial shown is the first few terms of the Taylorseries expansion of the sine function.

15

[23] Posamentier & Lehmann 2004, p. 25

[24] Eymard & Lafon 1999, p. 129

[25] Beckmann 1989, p. 37Schlager, Neil; Lauer, Josh (2001). Science and Its Times:Understanding the Social Significance of Scientific Discov-ery. Gale Group. ISBN 0-7876-3933-8., p 185.

[26] Arndt & Haenel 2006, pp. 22–23Preuss, Paul (23 July 2001). “Are The Digits of Pi Ran-dom? Lab Researcher May Hold The Key”. LawrenceBerkeley National Laboratory. Retrieved 10 November2007.

[27] Arndt & Haenel 2006, pp. 22, 28–30

[28] Arndt & Haenel 2006, p. 3

[29] Eymard & Lafon 1999, p. 78

[30] "Sloane’s A001203 : Continued fraction for Pi", The On-Line Encyclopedia of Integer Sequences. OEIS Founda-tion. Retrieved 12 April 2012.

[31] Lange, L. J. (May 1999). “An Elegant Continued Frac-tion for π". The American Mathematical Monthly 106 (5):456–458. doi:10.2307/2589152. JSTOR 2589152.

[32] Arndt & Haenel 2006, p. 240

[33] Arndt & Haenel 2006, p. 242

[34] Kennedy, E. S., “Abu-r-Raihan al-Biruni, 973-1048”, Journal for the History of Astronomy 9: 65,Bibcode:1978JHA.....9...65K. Ptolemy used a three-sexagesimal-digit approximation, and Jamshīd al-Kāshīexpanded this to nine digits; see Aaboe, Asger (1964),Episodes from the Early History of Mathematics, NewMathematical Library 13, New York: Random House, p.125.

[35] Petrie, W.M.F.Wisdom of the Egyptians (1940)

[36] Based on the Great Pyramid of Giza, supposedly built sothat the circle whose radius is equal to the height of thepyramid has a circumference equal to the perimeter of thebase (it is 1760 cubits around and 280 cubits in height).Verner, Miroslav. The Pyramids: The Mystery, Culture,and Science of Egypt’s Great Monuments. Grove Press.2001 (1997). ISBN 0-8021-3935-3

[37] Rossi, Corinna Architecture and Mathematics in AncientEgypt, Cambridge University Press. 2007. ISBN 978-0-521-69053-9.

[38] Legon, J. A. R. On Pyramid Dimensions and Proportions(1991) Discussions in Egyptology (20) 25-34

[39] “We can conclude that although the ancient Egyptianscould not precisely define the value of π, in practice theyused it”. Verner, M. (2003). “The Pyramids: Their Ar-chaeology and History”., p. 70.Petrie (1940). “Wisdom of the Egyptians”., p. 30.See also Legon, J. A. R. (1991). “On PyramidDimensionsand Proportions”. Discussions in Egyptology 20: 25–34..See also Petrie, W. M. F. (1925). “Surveys ofthe Great Pyramids”. Nature 116 (2930): 942–942.Bibcode:1925Natur.116..942P. doi:10.1038/116942a0.

[40] Egyptologist: Rossi, Corinna,Architecture andMathemat-ics in Ancient Egypt, Cambridge University Press, 2004,pp 60–70, 200, ISBN 9780521829540.Skeptics: Shermer, Michael, The Skeptic Encyclopediaof Pseudoscience, ABC-CLIO, 2002, pp 407–408, ISBN9781576076538.See also Fagan, Garrett G., Archaeological Fantasies:How Pseudoarchaeology Misrepresents The Past and Mis-leads the Public, Routledge, 2006, ISBN 9780415305938.For a list of explanations for the shape that do not in-volve π, see Roger Herz-Fischler (2000). The Shape of theGreat Pyramid. Wilfrid Laurier University Press. pp. 67–77, 165–166. ISBN 9780889203242. Retrieved 2013-06-05.

[41] Arndt & Haenel 2006, p. 167

[42] Chaitanya, Krishna. A profile of Indian culture. IndianBook Company (1975). p.133.

[43] Arndt & Haenel 2006, p. 169

[44] Arndt & Haenel 2006, p. 170

[45] Arndt & Haenel 2006, pp. 175, 205

[46] “The Computation of Pi by Archimedes: The Computa-tion of Pi by Archimedes – File Exchange – MATLABCentral”. Mathworks.com. Retrieved 2013-03-12.

[47] Arndt & Haenel 2006, p. 171

[48] Arndt & Haenel 2006, p. 176Boyer & Merzbach 1991, p. 168

[49] Arndt & Haenel 2006, pp. 15–16, 175, 184–186, 205.Grienberger achieved 39 digits in 1630; Sharp 71 digits in1699.

[50] Arndt & Haenel 2006, pp. 176–177

[51] Boyer & Merzbach 1991, p. 202

[52] Arndt & Haenel 2006, p. 177

[53] Arndt & Haenel 2006, p. 178

[54] Arndt & Haenel 2006, pp. 179

[55] Arndt & Haenel 2006, pp. 180

[56] Azarian, Mohammad K. (2010). “al-Risāla al-muhītīyya:A Summary”. Missouri Journal of Mathematical Sciences22 (2): 64–85.

[57] O'Connor, John J.; Robertson, Edmund F. (1999).“Ghiyath al-Din Jamshid Mas’ud al-Kashi”. MacTutorHistory of Mathematics archive. Retrieved August 11,2012.

[58] Arndt & Haenel 2006, p. 182

[59] Arndt & Haenel 2006, pp. 182–183

[60] Arndt & Haenel 2006, p. 183

[61] Grienbergerus, Christophorus (1630). Elementa Trigono-metrica (PDF) (in Latin). Archived from the original(PDF) on 2014-02-01. His evaluation was 3.14159 2653589793 23846 26433 83279 50288 4196 < π < 3.1415926535 89793 23846 26433 83279 50288 4199.

16 7 NOTES

[62] Arndt & Haenel 2006, pp. 185–191

[63] Roy 1990, pp. 101–102Arndt & Haenel 2006, pp. 185–186

[64] Roy 1990, pp. 101–102

[65] Joseph 1991, p. 264

[66] Arndt & Haenel 2006, p. 188. Newton quoted by Arndt.

[67] Arndt & Haenel 2006, p. 187

[68] Arndt & Haenel 2006, pp. 188–189

[69] Eymard & Lafon 1999, pp. 53–54

[70] Arndt & Haenel 2006, p. 189

[71] Arndt & Haenel 2006, p. 156

[72] Arndt & Haenel 2006, pp. 192–193

[73] Arndt & Haenel 2006, pp. 72–74

[74] Arndt & Haenel 2006, pp. 192–196, 205

[75] Arndt & Haenel 2006, pp. 194–196

[76] Borwein, J. M.; Borwein, P. B. (1988). “Ra-manujan and Pi”. Scientific American 256(2): 112–117. Bibcode:1988SciAm.258b.112B.doi:10.1038/scientificamerican0288-112.Arndt & Haenel 2006, pp. 15–17, 70–72, 104, 156,192–197, 201–202

[77] Arndt & Haenel 2006, pp. 69–72

[78] Borwein, J. M.; Borwein, P. B.; Dilcher, K. (1989).“Pi, Euler Numbers, and Asymptotic Expansions”.American Mathematical Monthly 96 (8): 681–687.doi:10.2307/2324715.

[79] Arndt & Haenel 2006, p. 223, (formula 16.10). Note that(n − 1)n(n + 1) = n3 − n.Wells, David (1997). The Penguin Dictionary of Curiousand Interesting Numbers (revised ed.). Penguin. p. 35.ISBN 978-0-140-26149-3.

[80] Posamentier & Lehmann 2004, pp. 284

[81] Lambert, Johann, “Mémoire sur quelques propriétés re-marquables des quantités transcendantes circulaires et log-arithmiques”, reprinted in Berggren, Borwein & Borwein1997, pp. 129–140

[82] Arndt & Haenel 2006, p. 196

[83] Arndt & Haenel 2006, p. 165. A facsimile of Jones’ textis in Berggren, Borwein & Borwein 1997, pp. 108–109

[84] See Schepler 1950, p. 220: William Oughtred used theletter π to represent the periphery (i.e., circumference) ofa circle.

[85] Arndt & Haenel 2006, p. 166

[86] Arndt & Haenel 2006, pp. 205

[87] Arndt &Haenel 2006, p. 197. See also Reitwiesner 1950.

[88] Arndt & Haenel 2006, p. 197

[89] Arndt & Haenel 2006, pp. 15–17

[90] Arndt & Haenel 2006, pp. 131

[91] Arndt & Haenel 2006, pp. 132, 140

[92] Arndt & Haenel 2006, p. 87

[93] Arndt & Haenel 2006, pp. 111 (5 times); pp. 113–114 (4times).See Borwein & Borwein 1987 for details of algorithms.

[94] Bailey, David H. (16 May 2003). “Some Backgroundon Kanada’s Recent Pi Calculation” (PDF). Retrieved 12April 2012.

[95] Arndt & Haenel 2006, p. 17. “39 digits of π are suffi-cient to calculate the volume of the universe to the nearestatom.”Accounting for additional digits needed to compensate forcomputational round-off errors, Arndt concludes that afew hundred digits would suffice for any scientific appli-cation.

[96] Arndt & Haenel 2006, pp. 17–19

[97] Schudel, Matt (25 March 2009). “John W. Wrench, Jr.:Mathematician Had a Taste for Pi”. The Washington Post.p. B5.

[98] Connor, Steve (8 January 2010). “The Big Question: Howclose have we come to knowing the precise value of pi?".The Independent (London). Retrieved 14 April 2012.

[99] Arndt & Haenel 2006, p. 18

[100] Arndt & Haenel 2006, pp. 103–104

[101] Arndt & Haenel 2006, p. 104

[102] Arndt & Haenel 2006, pp. 104, 206

[103] Arndt & Haenel 2006, pp. 110–111

[104] Eymard & Lafon 1999, p. 254

[105] Arndt & Haenel 2006, pp. 110–111, 206Bellard, Fabrice, “Computation of 2700 billion decimaldigits of Pi using a Desktop Computer”, 11 Feb 2010.

[106] “Round 2... 10 Trillion Digits of Pi”, NumberWorld.org,17 Oct 2011. Retrieved 30 May 2012.

[107] PSLQ means Partial Sum of Least Squares.

[108] Plouffe, Simon (April 2006). “Identities inspired by Ra-manujan’s Notebooks (part 2)" (PDF). Retrieved 10 April2009.

[109] Arndt & Haenel 2006, pp. 77–84

[110] Gibbons, Jeremy, “Unbounded Spigot Algorithms for theDigits of Pi”, 2005. Gibbons produced an improved ver-sion of Wagon’s algorithm.

[111] Arndt & Haenel 2006, p. 77

17

[112] Rabinowitz, Stanley; Wagon, Stan (March 1995).“A spigot algorithm for the digits of Pi”. Amer-ican Mathematical Monthly 102 (3): 195–203.doi:10.2307/2975006. A computer program hasbeen created that implements Wagon’s spigot algorithmin only 120 characters of software.

[113] Arndt & Haenel 2006, pp. 117, 126–128

[114] Bailey, David H.; Borwein, Peter B.; and Plouffe, Simon(April 1997). “On the Rapid Computation of VariousPolylogarithmic Constants” (PDF). Mathematics of Com-putation 66 (218): 903–913. doi:10.1090/S0025-5718-97-00856-9.

[115] Arndt & Haenel 2006, p. 128. Plouffe did create a dec-imal digit extraction algorithm, but it is slower than full,direct computation of all preceding digits.

[116] Arndt & Haenel 2006, p. 20Bellards formula in: Bellard, Fabrice. “A new formula tocompute the nth binary digit of pi”. Archived from theoriginal on 12 September 2007. Retrieved 27 October2007.

[117] Palmer, Jason (16 September 2010). “Pi record smashedas team finds two-quadrillionth digit”. BBC News. Re-trieved 26 March 2011.

[118] Bronshteĭn & Semendiaev 1971, pp. 200, 209

[119] Weisstein, Eric W., “Semicircle”, MathWorld.

[120] Ayers 1964, p. 60

[121] Bronshteĭn & Semendiaev 1971, pp. 210–211

[122] Arndt & Haenel 2006, p. 39

[123] Ramaley, J. F. (October 1969). “Buffon’s Noodle Prob-lem”. The American Mathematical Monthly 76 (8): 916–918. doi:10.2307/2317945. JSTOR 2317945.

[124] Arndt & Haenel 2006, pp. 39–40Posamentier & Lehmann 2004, p. 105

[125] Arndt & Haenel 2006, pp. 43Posamentier & Lehmann 2004, pp. 105–108

[126] Ayers 1964, p. 100

[127] Bronshteĭn & Semendiaev 1971, p. 592

[128] Maor, Eli, E: The Story of a Number, Princeton UniversityPress, 2009, p 160, ISBN 978-0-691-14134-3 (“five mostimportant” constants).

[129] Weisstein, Eric W., “Roots of Unity”, MathWorld.

[130] Weisstein, Eric W., “Cauchy Integral Formula”,MathWorld.

[131] Joglekar, S. D., Mathematical Physics, Universities Press,2005, p 166, ISBN 978-81-7371-422-1.

[132] Klebanoff, Aaron (2001). “Pi in the Man-delbrot set” (PDF). Fractals 9 (4): 393–402.doi:10.1142/S0218348X01000828. Retrieved 14April 2012.

[133] Peitgen, Heinz-Otto, Chaos and fractals: new frontiers ofscience, Springer, 2004, pp. 801–803, ISBN 978-0-387-20229-7.

[134] Bronshteĭn & Semendiaev 1971, pp. 191–192

[135] Bronshteĭn & Semendiaev 1971, p. 190

[136] Arndt & Haenel 2006, pp. 41–43

[137] This theorem was proved by Ernesto Cesàro in 1881. Fora more rigorous proof than the intuitive and informal onegiven here, seeHardy, G. H.,An Introduction to the Theoryof Numbers, Oxford University Press, 2008, ISBN 978-0-19-921986-5, theorem 332.

[138] Ogilvy, C. S.; Anderson, J. T., Excursions in Number The-ory, Dover Publications Inc., 1988, pp. 29–35, ISBN 0-486-25778-9.

[139] Arndt & Haenel 2006, p. 43

[140] Feller, W. An Introduction to Probability Theory and ItsApplications, Vol. 1, Wiley, 1968, pp 174–190.

[141] Bronshteĭn & Semendiaev 1971, pp. 106–107, 744, 748

[142] Halliday, David; Resnick, Robert; Walker, Jearl, Funda-mentals of Physics, 5th Ed., John Wiley & Sons, 1997, p381, ISBN 0-471-14854-7.

[143] Imamura, James M (17 August 2005). “Heisenberg Un-certainty Principle”. University of Oregon. Archivedfrom the original on 12 October 2007. Retrieved 9September 2007.

[144] Yeo, Adrian, The pleasures of pi, e and other interestingnumbers, World Scientific Pub., 2006, p 21, ISBN 978-981-270-078-0.Ehlers, Jürgen, Einstein’s Field Equations and Their Phys-ical Implications, Springer, 2000, p 7, ISBN 978-3-540-67073-5.

[145] Nave, C. Rod (28 June 2005). “Coulomb’s Constant”.HyperPhysics. Georgia State University. Retrieved 9November 2007.

[146] C. Itzykson, J-B. Zuber, Quantum Field Theory,McGraw-Hill, 1980.

[147] Low, Peter, Classical Theory of Structures Based on theDifferential Equation, CUP Archive, 1971, pp 116–118,ISBN 978-0-521-08089-7.

[148] Batchelor, G. K., An Introduction to Fluid Dynamics,Cambridge University Press, 1967, p 233, ISBN 0-521-66396-2.

[149] Bracewell, R. N., The Fourier Transform and Its Applica-tions, McGraw-Hill, 2000, ISBN 0-07-116043-4.

[150] Hans-Henrik Stølum (22 March 1996). “River Mean-dering as a Self-Organization Process”. Science 271(5256): 1710–1713. Bibcode:1996Sci...271.1710S.doi:10.1126/science.271.5256.1710.

[151] Posamentier & Lehmann 2004, pp. 140–141

[152] Arndt & Haenel 2006, pp. 44–45

18 7 NOTES

[153] “Most Pi Places Memorized”, Guinness World Records.

[154] Otake, Tomoko (17 December 2006). “How can anyoneremember 100,000 numbers?". The Japan Times. Re-trieved 27 October 2007.

[155] Raz, A.; Packard, M. G. (2009). “A slice of pi: An ex-ploratory neuroimaging study of digit encoding and re-trieval in a superior memorist”. Neurocase 15: 361–372.doi:10.1080/13554790902776896. PMID 19585350.

[156] Keith, Mike. “Cadaeic Cadenza Notes & Commentary”.Retrieved 29 July 2009.

[157] Keith, Michael; Diana Keith (February 17, 2010). NotA Wake: A dream embodying (pi)'s digits fully for 10000decimals. Vinculum Press. ISBN 978-0963009715.

[158] For instance, Pickover calls π “the most famous mathe-matical constant of all time”, and Peterson writes, “Of allknown mathematical constants, however, pi continues toattract the most attention”, citing the Givenchy π perfume,Pi (film), and Pi Day as examples. See Pickover, CliffordA. (1995), Keys to Infinity, Wiley & Sons, p. 59, ISBN9780471118572; Peterson, Ivars (2002), MathematicalTreks: From Surreal Numbers to Magic Circles, MAAspectrum, Mathematical Association of America, p. 17,ISBN 9780883855379.

[159] BBC documentary “The Story of Maths”, second part,showing a visualization of the historically first exact for-mula, starting at 35 min and 20 sec into the second partof the documentary.

[160] Posamentier & Lehmann 2004, p. 118Arndt & Haenel 2006, p. 50

[161] Arndt & Haenel 2006, p. 14. This part of the story wasomitted from the film adaptation of the novel.

[162] Gill, Andy (4 November 2005). “Review of Aerial”.The Independent. the almost autistic satisfaction of theobsessive-compulsive mathematician fascinated by 'Pi'(which affords the opportunity to hear Bush slowly singvast chunks of the number in question, several dozen dig-its long)

[163] Board, Josh (1 December 2010). “PARTY CRASHER:Laughing With Hard 'N Phirm”. SanDiego.com. Therewas one song about Pi. Nothing like hearing people har-monizing over 200 digits.

[164] Pi Day activities.

[165] MIT cheers. Retrieved 12 April 2012.

[166] “Happy Pi Day! Watch these stunning videos of kidsreciting 3.14”. USAToday.com. 2015-03-14. Retrieved2015-03-14.

[167] “Google’s strange bids for Nortel patents”. Financial-Post.com. Reuters. 2011-07-05. Retrieved 16 August2011.

[168] Eagle, Albert (1958). The Elliptic Functions as TheyShould be: An Account, with Applications, of the Functionsin a New Canonical Form. Galloway and Porter, Ltd. p.ix.

[169] Sequence A019692,

[170] Abbott, Stephen (April 2012). “My Conversionto Tauism” (PDF). Math Horizons 19 (4): 34.doi:10.4169/mathhorizons.19.4.34.

[171] Palais, Robert (2001). "π Is Wrong!" (PDF).The Mathematical Intelligencer 23 (3): 7–8.doi:10.1007/BF03026846.

[172] Tau Day: Why you should eat twice the pie – Light Years– CNN.com Blogs

[173] “Life of pi in no danger – Experts cold-shoulder campaignto replace with tau”. Telegraph India. 2011-06-30.

[174] Arndt & Haenel 2006, pp. 211–212Posamentier & Lehmann 2004, pp. 36–37Hallerberg, Arthur (May 1977). “Indiana’s squaredcircle”. Mathematics Magazine 50 (3): 136–140.doi:10.2307/2689499. JSTOR 2689499.

References

• Arndt, Jörg; Haenel, Christoph (2006). Pi Un-leashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. English translation byCatriona and David Lischka.

• Ayers, Frank (1964). Calculus. McGraw-Hill.ISBN 978-0-070-02653-7.

• Berggren, Lennart; Borwein, Jonathan; Borwein,Peter (1997). Pi: a Source Book. Springer-Verlag.ISBN 978-0-387-20571-7.

• Beckmann, Peter (1989) [1974]. History of Pi. St.Martin’s Press. ISBN 978-0-88029-418-8.

• Borwein, Jonathan; Borwein, Peter (1987). Pi andthe AGM: a Study in Analytic Number Theory andComputational Complexity. Wiley. ISBN 978-0-471-31515-5.

• Boyer, Carl B.; Merzbach, Uta C. (1991). A Historyof Mathematics (2 ed.). Wiley. ISBN 978-0-471-54397-8.

• Bronshteĭn, Ilia; Semendiaev, K. A. (1971). AGuide Book toMathematics. H. Deutsch. ISBN 978-3-871-44095-3.

• Eymard, Pierre; Lafon, Jean Pierre (1999). TheNumber Pi. American Mathematical Society. ISBN978-0-8218-3246-2., English translation by StephenWilson.

• Joseph, George Gheverghese (1991). The Crestof the Peacock: Non-European Roots of Mathemat-ics. Princeton University Press. ISBN 978-0-691-13526-7. Retrieved 2013-06-05.

19

• Posamentier, Alfred S.; Lehmann, Ingmar (2004).Pi: A Biography of the World’s Most MysteriousNumber. Prometheus Books. ISBN 978-1-59102-200-8.

• Reitwiesner, George (1950). “An ENIAC Determi-nation of pi and e to 2000 Decimal Places”. Mathe-matical Tables andOther Aids to Computation 4 (29):11–15. doi:10.2307/2002695.

• Roy, Ranjan (1990). “The Discovery of the Se-ries Formula for pi by Leibniz, Gregory, and Ni-lakantha”. Mathematics Magazine 63 (5): 291–306.doi:10.2307/2690896.

• Schepler, H. C. (1950). “The Chronology ofPi”. Mathematics Magazine (Mathematical Asso-ciation of America) 23 (3): 165–170 (Jan/Feb),216–228 (Mar/Apr), and 279–283 (May/Jun).doi:10.2307/3029284.. issue 3 Jan/Feb, issue 4Mar/Apr, issue 5 May/Jun

8 Further reading• Blatner, David (1999). The Joy of Pi. Walker &Company. ISBN 978-0-8027-7562-7.

• Borwein, Jonathan; Borwein, Peter (1984). “TheArithmetic-Geometric Mean and Fast Computationof Elementary Functions”. SIAM Review 26: 351–365. doi:10.1137/1026073.

• Borwein, Jonathan; Borwein, Peter; Bailey, DavidH. (1989). “Ramanujan, Modular Equations, andApproximations to Pi or How to Compute One Bil-lion Digits of Pi”. The American MathematicalMonthly 96: 201–219. doi:10.2307/2325206.

• Chudnovsky, David V. and Chudnovsky, GregoryV., “Approximations and Complex MultiplicationAccording to Ramanujan”, in Ramanujan Revisited(G.E. Andrews et al. Eds), Academic Press, 1988,pp 375–396, 468–472

• Cox, David A., “The Arithmetic-Geometric Meanof Gauss”, L' Ensignement Mathematique, 30(1984)275–330

• Delahaye, Jean-Paul, “Le Fascinant Nombre Pi”,Paris: Bibliothèque Pour la Science (1997) ISBN2902918259

• Engels, Hermann (1977). “Quadrature of the Circlein Ancient Egypt”. Historia Mathematica 4: 137–140. doi:10.1016/0315-0860(77)90104-5.

• Euler, Leonhard, “On the Use of the DiscoveredFractions to Sum Infinite Series”, in Introduction toAnalysis of the Infinite. Book I, translated from theLatin by J. D. Blanton, Springer-Verlag, 1964, pp137–153

• Heath, T. L., TheWorks of Archimedes, Cambridge,1897; reprinted in The Works of Archimedes withThe Method of Archimedes, Dover, 1953, pp 91–98

• Huygens, Christiaan, “De Circuli Magnitudine In-venta”, Christiani Hugenii Opera Varia I, Leiden1724, pp 384–388

• Lay-Yong, Lam; Tian-Se, Ang (1986). “Cir-cle Measurements in Ancient China”. HistoriaMathematica 13: 325–340. doi:10.1016/0315-0860(86)90055-8.

• Lindemann, Ferdinand (1882). “Ueber die Zahlpi”. Mathematische Annalen 20: 213–225.doi:10.1007/bf01446522.

• Matar, K. Mukunda; Rajagonal, C. (1944). “On theHindu Quadrature of the Circle” (Appendix by K.Balagangadharan)". Journal of the Bombay Branchof the Royal Asiatic Society 20: 77–82.

• Niven, Ivan, “A Simple Proof that pi Is Irrational”,Bulletin of the American Mathematical Society, 53:7(July 1947), 507

• Ramanujan, Srinivasa, “Modular Equations andApproximations to π", Quarterly Journal of Pureand Applied Mathematics, XLV, 1914, 350–372.Reprinted in G.H. Hardy, P.V. Seshu Aiyar, and B.M. Wilson (eds), Srinivasa Ramanujan: CollectedPapers, 1927 (reprinted 2000), pp 23–29

• Shanks, William, Contributions to MathematicsComprising Chiefly of the Rectification of the Circleto 607 Places of Decimals, 1853, pp. i–xvi, 10

• Shanks, Daniel; Wrench, John William (1962).“Calculation of pi to 100,000 Decimals”.Mathematics of Computation 16: 76–99.doi:10.1090/s0025-5718-1962-0136051-9.

• Tropfke, Johannes, Geschichte Der Elementar-Mathematik in Systematischer Darstellung (The his-tory of elementary mathematics), BiblioBazaar,2009 (reprint), ISBN 978-1-113-08573-3

• Viete, Francois, Variorum de Rebus MathematicisReponsorum Liber VII. F. Viete, Opera Mathematica(reprint), Georg Olms Verlag, 1970, pp 398–401,436–446

• Wagon, Stan, “Is Pi Normal?", The MathematicalIntelligencer, 7:3(1985) 65–67

• Wallis, John, Arithmetica Infinitorum, sive NovaMethodus Inquirendi in Curvilineorum Quadratum,aliaque difficiliora Matheseos Problemata, Oxford1655–6. Reprinted in vol. 1 (pp 357–478) of OperaMathematica, Oxford 1693

• Zebrowski, Ernest, A History of the Circle: Mathe-matical Reasoning and the Physical Universe, Rut-gers University Press, 1999, ISBN 978-0-8135-2898-4

20 9 EXTERNAL LINKS

9 External links• Digits of Pi at DMOZ

• “Pi” at Wolfram Mathworld

• Representations of Pi at Wolfram Alpha

• Pi Search Engine: 2 billion searchable digits of π,√2, and e

• Eaves, Laurence (2009). "π – Pi”. Sixty Symbols.Brady Haran for the University of Nottingham.

• Grime, Dr. James (2014). “Pi is Beautiful – Num-berphile”. Numberphile. Brady Haran.

• Demonstration by Lambert (1761) of irrationality ofπ, online and analyzed BibNum (PDF).

21

10 Text and image sources, contributors, and licenses

10.1 Text• Pi Source: https://en.wikipedia.org/wiki/Pi?oldid=684386320 Contributors: AxelBoldt, Chuck Smith, Brion VIBBER, Mav, Wesley, Zun-

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22 10 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

John Reid, Insanity13, Ohconfucius, Sesquialtera II, The undertow, SashatoBot, Lambiam, Esrever, -Ilhador-, Nishkid64, Maddogmike,Akubra, Gmvgs, Saccerzd, Turbothy, Doug Bell, Johncatsoulis, BorisFromStockdale, Attys, JzG, Dbtfz, Kuru, Titus III, Richard L. 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10.2 Images 23

Rayizcoolio, Utrechtse, Citation bot 1, Tkuvho, Theory2reality, Jaymie94, ChrisJBenson, Kiefer.Wolfowitz, PrincessofLlyr, Jonesey95,Grscomet7, DAVilla, Allthingstoallpeople, 1to0to-1, Σ, Foobarnix, Spiro Liacos, Electricmaster, Designate, FoxBot, Delorme, Tobe-Bot, Trappist the monk, DixonDBot, Jocme, Lotje, Red Denim, Efficiencyjacky154, Aoidh, Leopoldwilson, Pilot850, Brian the Editor,N4m3, Suffusion of Yellow, Skullpatrol, Tbhotch, RjwilmsiBot, Jowa fan, Balph Eubank, Mandolinface, Afteread, DarkLightA, Dfs72,Matsgranvik, Chip McShoulder, EmausBot, John of Reading, Surlyduff50, Timtempleton, Jurvetson2, Fly by Night, ZxxZxxZ, TeleCom-NasSprVen, Slawekb, Cegalegolog99, Werieth, Mirificium, AvicBot, Misty MH, Knight1993, The Nut, Alpha Quadrant, Fixblor, FredGandt, LordSuryaofShropshire, SPARTAN T-82, Bamyers99, H3llBot, Quondum, D.Lazard, Ahann, Brandmeister, Vanished user fi-jtji34toksdcknqrjn54yoimascj, Stephenj642, Maschen, Chris857, Pyrospark, ChuispastonBot, Albert Nestar, Teapeat, Llightex, Mjbmrbot,Liuthar, L1ght5h0w, Rememberway, Zytigon, JimsMaher, Bulldog73, Nohwave, Frietjes, Delusion23, Muon, Braincricket, Boobietime,Joel B. Lewis, Bradrangers, Lord Nordeck, Danim, Nobletripe, MerlIwBot, Be..anyone, Helpful Pixie Bot, Sceptic1954, Zibart, Ignacitum,Bibcode Bot, 2001:db8, Doorknob747, Picklebobdogflog, BG19bot, Max Longint, Consorveyapaaj2048394, Northamerica1000, Leonxlin,PhnomPencil, Portlandium, Ke5skw, Pitzik4, Joseph Lindenberg, Piisawesome, Snake of Intelligent Ignorance, Ebbillings, Max Ijzersteen,Chmarkine, MisterCSharp, NotWith, 23haveblue, Brad7777, Unknownkarma, Jawadreventon, Neotarf, Minsbot, Mike Agricola, Life421,ArrakisFrance, Kiewbra, Bismarck rules the sea, Hyuganatsu, JordanKyser22, Shwangtianyuan, CrunchySkies, SD5bot, JYBot, Dexbot,Joy if, Br'er Rabbit, Mogism, RazrRekr201, Numbermaniac, Czech is Cyrillized, Zstk, Chrisrox50267899, Pokajanje, Jochen Burghardt,Kevin12xd, Vahid alpha, Leijurv, JustAMuggle, Junvfr, Faizan, Epicgenius, Djkauffman, Light Peak, Magnolia677, JPaestpreornJeolhlna,Pdecalculus, Escspeed, XndrK, Mathmensch, Ram Zaltsman, Johndric Valdez, Hotchotmin, Meteor sandwich yum, Themessengerof-knowledge, ARUNEEK, Wyattbergeron1, HectorCabreraJr, Mahusha, 22merlin, Monkbot, GinAndChronically, Owais Khursheed, Qwer-tyxp2000, Gamemaster eleven, Awesome5860, Samuelrowland, NCCL2310, Whikie, Loraof, Orduin, SoSivr, Dandtiks69, Helloholabon-journihaonamastegutentag, KasparBot and Anonymous: 1081

10.2 Images• File:Archimedes_pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Archimedes_pi.svg License: CC-BY-SA-3.0Contributors: Own work Original artist: Leszek Krupinski (disputed, see File talk:Archimedes pi.svg)

• File:Buffon_needle.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/58/Buffon_needle.svg License: CCBY 2.5 Contrib-utors:

• Buffon_needle.gif Original artist: Buffon_needle.gif: Claudio Rocchini• File:Circle_Area.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/ce/Circle_Area.svg License: Public domain Contribu-tors: ? Original artist: ?

• File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Originalartist: ?

• File:Comparison_pi_infinite_series.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f5/Comparison_pi_infinite_series.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Cmglee

• File:Domenico-Fetti_Archimedes_1620.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/e7/Domenico-Fetti_Archimedes_1620.jpg License: Public domain Contributors: http://archimedes2.mpiwg-berlin.mpg.de/archimedes_templates/popup.htmOriginal artist: Domenico Fetti

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• File:Euler’{}s_formula.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/71/Euler%27s_formula.svg License: CC-BY-SA-3.0 Contributors: Drawn by en User:Gunther, modified by others. Original artist: Originally created by gunther using xfig, recreated inInkscape by Wereon, italics fixed by lasindi.

• File:GodfreyKneller-IsaacNewton-1689.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg License: Public domain Contributors: http://www.newton.cam.ac.uk/art/portrait.html Orig-inal artist: Sir Godfrey Kneller

• File:JohnvonNeumann-LosAlamos.gif Source: https://upload.wikimedia.org/wikipedia/commons/5/5e/JohnvonNeumann-LosAlamos.gif License: Public domain Contributors: http://www.lanl.gov/history/atomicbomb/images/NeumannL.GIF(Archive copy at the Wayback Machine (archived 11 March 2010)) Original artist: LANL

• File:Leonhard_Euler.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg License: Public domainContributors:2. Kunstmuseum BaselOriginal artist: Jakob Emanuel Handmann

• File:Mandel_zoom_00_mandelbrot_set.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_mandelbrot_set.jpg License: CC-BY-SA-3.0 Contributors: ? Original artist: ?

• File:Matheon2.jpg Source: https://upload.wikimedia.org/wikipedia/commons/8/8c/Matheon2.jpg License: CC-BY-SA-3.0 Contributors:Own work (own photo) Original artist: Holger Motzkau

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• File:OEISicon_light.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/OEISicon_light.svg License: Public domainContributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)

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24 10 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

• File:Pi_pie2.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d4/Pi_pie2.jpg License: Public domain Contributors:Pi_pie2.jpg Original artist: GJ

• File:Record_pi_approximations.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/25/Record_pi_approximations.svgLicense: CC BY-SA 3.0 Contributors: Own work Original artist: Nageh

• File:Sine_cosine_one_period.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/71/Sine_cosine_one_period.svg License:CC BY 3.0 Contributors: Own work Original artist: Geek3

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• File:Srinivasa_Ramanujan_-_OPC_-_1.jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/c1/Srinivasa_Ramanujan_-_OPC_-_1.jpg License: Public domain Contributors: link Original artist: Unknown

• File:Tau_uses.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/96/Tau_uses.jpg License: CC0 Contributors: Own workOriginal artist: Helloholabonjournihaonamastegutentag

10.3 Content license• Creative Commons Attribution-Share Alike 3.0