For thousands of years, mathematicians have attempted to extend their understanding ofπ, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including theEgyptians andBabylonians, required fairly accurate approximations ofπ for practical computations. Around 250BC, theGreek mathematicianArchimedes created an algorithm to approximateπ with arbitrary accuracy. In the 5th century AD,Chinese mathematicians approximatedπ to seven digits, whileIndian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula forπ, based oninfinite series, was discovered a millennium later. The earliest known use of the Greek letterπ to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematicianWilliam Jones in 1706. The invention ofcalculus soon led to the calculation of hundreds of digits ofπ, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians andcomputer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation ofπ to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test the correctness of new computer processors.
Because it relates to a circle,π is found in many formulae intrigonometry andgeometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such ascosmology,fractals,thermodynamics,mechanics, andelectromagnetism. It also appears in areas having little to do with geometry, such asnumber theory andstatistics, and in modernmathematical analysis can be defined without any reference to geometry. The ubiquity ofπ makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted toπ have been published, and record-setting calculations of the digits ofπ often result in news headlines.
Fundamentals
Name
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercaseGreek letterπ, sometimes spelled out aspi.[1] In English,π ispronounced as "pie" (/paɪ/PY).[2] In mathematical use, the lowercase letterπ is distinguished from its capitalized and enlarged counterpartΠ, which denotes aproduct of a sequence, analogously to howΣ denotessummation.
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is calledπ.
π is commonly defined as theratio of acircle'scircumferenceC to itsdiameterd:[3]The ratio is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio.
In modern mathematics, this definition is not fully satisfactory for several reasons. Firstly, it lacks a rigorous definition of thelength of a curved line. Such a definition requires at least the concept of alimit,[a] or, more generally, the concepts ofderivatives andintegrals. Also, diameters, circles and circumferences can be defined inNon-Euclidean geometries, but, in such a geometry, the ratio need not to be a constant, and need not to equal toπ.[3] Also, there are many occurrences ofπ in many branches of mathematics that are completely independent from geometry, and in modern mathematics, the trend is to built geometry fromalgebra andanalysis rather than independently from the other branches of mathematics. For these reasons, the following characterizations can be taken as definitions ofπ:[4]
π is the smallest positivezero of thesine function; that is, andπ is the smallest positive number with this property.
π is the smallest positive difference between two zeros of,, and.
The sine and the cosine satisfy thedifferential equation, and all solutions of this equation are periodic. This leads to the conceptual definition:
π is half thefundamental period of each nonzero solution of the differential equation.
The digits ofπ have no apparent pattern and have passed tests forstatistical randomness, including tests fornormality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture thatπ isnormal has not been proven or disproven.[7]
Since the advent of computers, a large number of digits ofπ have been available on which to perform statistical analysis.Yasumasa Kanada has performed detailed statistical analyses on the decimal digits ofπ, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected tostatistical significance tests, and no evidence of a pattern was found.[8] Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by theinfinite monkey theorem. Thus, because the sequence ofπ's digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as asequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation ofπ.[9] This is also called the "Feynman point" inmathematical folklore, afterRichard Feynman, although no connection to Feynman is known.
The transcendence ofπ has two important consequences: First,π cannot be expressed using any finite combination of rational numbers and square roots ornth roots (such as or). Second, since no transcendental number can beconstructed withcompass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle.[11] Squaring a circle was one of the important geometry problems of theclassical antiquity.[12] Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.[13]
As an irrational number,π cannot be represented as acommon fraction. But every number, includingπ, can be represented by an infinite series of nested fractions, called asimple continued fraction:
Truncating the continued fraction at any point yields a rational approximation forπ; the first four of these are3,22/7,333/106, and355/113. These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer toπ than any other fraction with the same or a smaller denominator.[16] Becauseπ is transcendental, it is by definition notalgebraic and so cannot be aquadratic irrational. Therefore,π cannot have aperiodic continued fraction. Although the simple continued fraction forπ (with numerators all 1, shown above) also does not exhibit any other obvious pattern,[17][18] several non-simplecontinued fractions do, such as:[19][c]
Fractions: Approximate fractions include (in order of increasing accuracy)22/7,333/106,355/113,52163/16604,103993/33102,104348/33215, and245850922/78256779.[16] (List is selected terms fromOEIS: A063674 andOEIS: A063673.)
Digits: The first 50 decimal digits are3.14159265358979323846264338327950288419716939937510...[20] (seeOEIS: A000796)
Digits in other number systems
The first 48binary (base 2) digits (calledbits) are11.001001000011111101101010100010001000010110100011... (seeOEIS: A004601)
The first 36 digits internary (base 3) are10.010211012222010211002111110221222220... (seeOEIS: A004602)
wherethe constante is the base of thenatural logarithm. This formula establishes a correspondence between imaginary powers ofe and points on theunit circle centred at the origin of the complex plane. Setting in Euler's formula results inEuler's identity, celebrated in mathematics due to it containing five important mathematical constants:[24][25]
There aren different complex numbersz satisfying, and these are called the "nthroots of unity"[26] and are given by the formula:
Surviving approximations ofπ prior to the 2nd century AD are accurate to one or two decimal places at best. The earliest written approximations are found inBabylon and Egypt, both within one percent of the true value. In Babylon, aclay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treatsπ as25/8 = 3.125.[27] In Egypt, theRhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treatsπ as.[18][27] Although somepyramidologists have theorized that theGreat Pyramid of Giza was built with proportions related toπ, this theory is not widely accepted by scholars.[28]In theShulba Sutras ofIndian mathematics, dating to an oral tradition from the 1st or 2nd millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.[29]
Polygon approximation era
Archimedes developed the polygonal approach to approximatingπ.π can be estimated by computing the perimeters of circumscribed and inscribed polygons.
The first recorded algorithm for rigorously calculating the value ofπ was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes, implementing themethod of exhaustion.[30] This polygonal algorithm dominated for over 1,000 years, and as a resultπ is sometimes referred to as Archimedes's constant.[31] Archimedes computed upper and lower bounds ofπ by drawing a regular hexagon inside and outside a circle, andsuccessively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that223/71 <π <22/7 (that is,3.1408 <π < 3.1429).[32] Archimedes' upper bound of22/7 may have led to a widespread popular belief thatπ is equal to22/7.[33] Around 150 AD, Greco-Roman scientistPtolemy, in hisAlmagest, gave a value forπ of 3.1416, which he may have obtained from Archimedes or fromApollonius of Perga.[34][35] Mathematicians using polygonal algorithms reached 39 digits ofπ in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]
Inancient China, values forπ included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and142/45 (3rd century, approximately 3.1556).[37] Around 265 AD, theCao Wei mathematicianLiu Hui created apolygon-based iterative algorithm, with which he constructed a 3,072-sided polygon to approximateπ as 3.1416.[38][39] Liu later invented a faster method of calculatingπ and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38] Around 480 AD,Zu Chongzhi calculated that and suggested the approximations and, which he termed themilü ('close ratio') andyuelü ('approximate ratio') respectively, iterating with Liu Hui's algorithm up to a 12,288-sided polygon. With a correct value for its seven first decimal digits, Zu's result remained the most accurate approximation ofπ for the next 800 years.[40]
The Indian astronomerAryabhata used a value of 3.1416 in hisĀryabhaṭīya (499 AD).[41] Around 1220,Fibonacci computed 3.1418 using a polygonal method devised independently of Archimedes.[42] Italian authorDante apparently employed the value.[42]
The Persian astronomerJamshīd al-Kāshī produced ninesexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with sides,[43] which stood as the world record for about 180 years.[44] French mathematicianFrançois Viète in 1579 achieved nine digits with a polygon of sides.[44] Flemish mathematicianAdriaan van Roomen arrived at 15 decimal places in 1593.[44] In 1596, Dutch mathematicianLudolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result,π was called the "Ludolphian number" in Germany until the early 20th century).[45] Dutch scientistWillebrord Snellius reached 34 digits in 1621,[46] and Austrian astronomerChristoph Grienberger arrived at 38 digits in 1630 using 1040 sides.[47]Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent toRichardson extrapolation.[48]
Infinite series
Comparison of the convergence of several historical infinite series forπ.Sn is the approximation after takingn terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.(click for detail)
The calculation ofπ was revolutionized by the development ofinfinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinitesequence. Infinite series allowed mathematicians to computeπ with much greater precision thanArchimedes and others who used geometrical techniques.[49] Although infinite series were exploited forπ most notably by European mathematicians such asJames Gregory andGottfried Wilhelm Leibniz, the approach also appeared in theKerala school sometime in the 14th or 15th century.[50][51] Around 1500, an infinite series that could be used to computeπ, written in the form ofSanskrit verse, was presented inTantrasamgraha byNilakantha Somayaji.[50] The series are presented without proof, but proofs are presented in the later workYuktibhāṣā, published around 1530. Several infinite series are described, including series for sine (which Nilakantha attributes toMadhava of Sangamagrama), cosine, and arctangent which are now sometimes referred to asMadhava series. The series for arctangent is sometimes calledGregory's series or the Gregory–Leibniz series.[50] Madhava used infinite series to estimateπ to 11 digits around 1400.[52][53][54]
Isaac Newton usedinfinite series to computeπ to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[57]
In the 1660s, the English scientistIsaac Newton and German mathematicianGottfried Wilhelm Leibniz discoveredcalculus, which led to the development of many infinite series for approximatingπ. Newton himself used an arcsine series to compute a 15-digit approximation ofπ in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[57]
This series, sometimes called theGregory–Leibniz series, equals when evaluated with.[59] But for,it converges impractically slowly (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.[60]
In 1699, English mathematicianAbraham Sharp used the Gregory–Leibniz series for to computeπ to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[61]
In 1706,John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster:[62][63][64]
Machin reached 100 digits ofπ with this formula.[65] Other mathematicians created variants, now known asMachin-like formulae, that were used to set several successive records for calculating digits ofπ.[66][65]
Isaac Newtonaccelerated the convergence of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):[67]
Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including with which he computed 20 digits ofπ in one hour.[68]
Machin-like formulae remained the best-known method for calculatingπ well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device.[69]
In 1844, a record was set byZacharias Dase, who employed a Machin-like formula to calculate 200 decimals ofπ in his head at the behest of German mathematicianCarl Friedrich Gauss.[70]
In 1853, British mathematicianWilliam Shanks calculatedπ to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.[71]
Rate of convergence
Some infinite series forπconverge faster than others. Given the choice of two infinite series forπ, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculateπ to any given accuracy.[72] A simple infinite series forπ is theGregory–Leibniz series:[73]
As individual terms of this infinite series are added to the sum, the total gradually gets closer toπ, and – with a sufficient number of terms – can get as close toπ as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits ofπ.[74]
An infinite series forπ (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:[75][76]
The following table compares the convergence rates of these two series:
Infinite series forπ
After 1st term
After 2nd term
After 3rd term
After 4th term
After 5th term
Converges to:
4.0000
2.6666 ...
3.4666 ...
2.8952 ...
3.3396 ...
π = 3.1415 ...
3.0000
3.1666 ...
3.1333 ...
3.1452 ...
3.1396 ...
After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value ofπ, whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits ofπ. Series that converge even faster includeMachin's series andChudnovsky's series, the latter producing 14 correct decimal digits per term.[72]
Not all mathematical advances relating toπ were aimed at increasing the accuracy of approximations. When Euler solved theBasel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection betweenπ and theprime numbers that later contributed to the development and study of theRiemann zeta function:[77]
Swiss scientistJohann Heinrich Lambert in 1768 proved thatπ isirrational, meaning it is not equal to the quotient of any two integers.[5]Lambert's proof exploited a continued-fraction representation of the tangent function.[78] French mathematicianAdrien-Marie Legendre proved in 1794 thatπ2 is also irrational. In 1882, German mathematicianFerdinand von Lindemann proved thatπ istranscendental,[79] confirming a conjecture made by bothLegendre and Euler.[80][81] Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".[82]
Adoption of the symbolπ
The earliest known use of the Greek letterπ to represent the ratio of a circle's circumference to its diameter was by Welsh mathematicianWilliam Jones in 1706.
Leonhard Euler popularized the use of the Greek letterπ in works he published in 1736 and 1748.
The first recorded use of the symbolπ in circle geometry is inOughtred'sClavis Mathematicae (1648),[83][84]where theGreek lettersπ andδ were combined into the fraction for denoting the ratiossemiperimeter tosemidiameter and perimeter to diameter, that is, what is presently denoted asπ.[1][85][86][87] (Before then, mathematicians sometimes used letters such asc orp instead.[84])Barrow likewise used the same notation,[88] whileGregory instead used to represent6.28... .[89][86]
The earliest known use of the Greek letterπ alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematicianWilliam Jones in his 1706 workSynopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.[62][90] The Greek letter appears on p. 243 in the phrase " Periphery (π)", calculated for a circle with radius one. However, Jones writes that his equations forπ are from the "ready pen of the truly ingenious Mr.John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[84] Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.[85][91]
Euler started using the single-letter form beginning with his 1727Essay Explaining the Properties of Air, though he usedπ = 6.28..., the ratio of periphery to radius, in this and some later writing.[92] Euler first usedπ = 3.14... in his 1736 workMechanica,[93] and continued in his widely read 1748 workIntroductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number asπ; thusπ is equal to half the circumference of a circle of radius1").[94] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in theWestern world,[84] though the definition still varied between3.14... and6.28... as late as 1761.[95]
Modern quest for more digits
Motives for computingπ
As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits ofπ increased dramatically. The vertical scale islogarithmic.
For most numerical calculations involvingπ, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform mostcosmological calculations, because that is the accuracy necessary to calculate the circumference of theobservable universe with a precision of one atom. Accounting for additional digits needed to compensate for computationalround-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to computeπ to thousands and millions of digits.[96] This effort may be partly ascribed to the human compulsion to break records, and such achievements withπ often make headlines around the world.[97] They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (includinghigh-precision multiplication algorithms) –and within pure mathematics itself, providing data for evaluating the randomness of the digits ofπ.[98][99]
Computer era and iterative algorithms
The development of computers in the mid-20th century again revolutionized the hunt for digits ofπ. MathematiciansJohn Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator.[100] Using aninverse tangent (arctan) infinite series, a team led by George Reitwiesner andJohn von Neumann that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on theENIAC computer.[101][102] The record, always relying on an arctan series, was broken repeatedly (3089 digits in 1955,[103] 7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[101]
The iterative algorithms were independently published in 1975–1976 by physicistEugene Salamin and scientistRichard Brent.[107] These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier byCarl Friedrich Gauss, in what is now termed thearithmetic–geometric mean method (AGM method) orGauss–Legendre algorithm.[107] As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.
The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generallymultiply the number of correct digits at each step. For example, the Brent–Salamin algorithm doubles the number of digits in each iteration. In 1984, brothersJohn andPeter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[108] Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computingπ between 1995 and 2002.[109] This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[109]
Rapidly convergent series
Srinivasa Ramanujan, working in isolation in India, produced many innovative series for computingπ.
Modernπ calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[109] The fast iterative algorithms were anticipated in 1914, when Indian mathematicianSrinivasa Ramanujan published dozens of innovative new formulae forπ, remarkable for their elegance, mathematical depth and rapid convergence.[110] One of his formulae, based onmodular equations, is
This series converges much more rapidly than most arctan series, including Machin's formula.[111]Bill Gosper was the first to use it for advances in the calculation ofπ, setting a record of 17 million digits in 1985.[112] Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (Jonathan andPeter) and theChudnovsky brothers.[113] TheChudnovsky formula developed in 1987 is
It produces about 14 digits ofπ per term[114] and has been used for several record-settingπ calculations, including the first to surpass 1 billion (109) digits in 1989 by the Chudnovsky brothers, 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo,[115] and 100 trillion digits byEmma Haruka Iwao in 2022.[116][117] For similar formulae, see also theRamanujan–Sato series.
Random dots are placed on a square and a circle inscribed inside.
Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations ofπ.[120]Buffon's needle is one such technique: If a needle of lengthℓ is droppedn times on a surface on which parallel lines are drawnt units apart, and ifx of those times it comes to rest crossing a line (x > 0), then one may approximateπ based on the counts:[121]
Another Monte Carlo method for computingπ is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equalπ/4.[122]
Five random walks with 200 steps. The sample mean of|W200| isμ = 56/5, and so2(200)μ−2 ≈ 3.19 is within0.05 of π.
Another way to calculateπ using probability is to start with arandom walk, generated by a sequence of (fair) coin tosses: independentrandom variablesXk such thatXk ∈ {−1,1} with equal probabilities. The associated random walk is
This Monte Carlo method is independent of any relation to circles, and is a consequence of thecentral limit theorem, discussedbelow.
These Monte Carlo methods for approximatingπ are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximateπ when speed or accuracy is desired.[124]
Spigot algorithms
Two algorithms were discovered in 1995 that opened up new avenues of research intoπ. They are calledspigot algorithms because, like water dripping from aspigot, they produce single digits ofπ that are not reused after they are calculated.[125][126] This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[125]
MathematiciansStan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995.[126][127][128] Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[127]
This formula, unlike others before it, can produce any individualhexadecimal digit ofπ without calculating all the preceding digits.[129] Individual binary digits may be extracted from individual hexadecimal digits, andoctal digits can be extracted from one or two hexadecimal digits. An important application of digit extraction algorithms is to validate new claims of recordπ computations: After a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several randomly selected hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is correct.[115]
Between 1998 and 2000, thedistributed computing projectPiHex usedBellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (1015th) bit ofπ, which turned out to be 0.[131] In September 2010, aYahoo employee used the company'sHadoop application on one thousand computers over a 23-day period to compute 256bits ofπ at the two-quadrillionth (2×1015th) bit, which also happens to be zero.[132]
In 2022, Plouffe found a base-10 algorithm for calculating digits ofπ.[133]
Role and characterizations in mathematics
Becauseπ is closely related to the circle, it is found inmany formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics,Fourier analysis, and number theory, also includeπ in some of their important formulae.
Geometry and trigonometry
The area of the circle equalsπ times the shaded area. The area of theunit circle isπ.
π appears in formulae for areas and volumes of geometrical shapes based on circles, such asellipses,spheres,cones, andtori.[134] Below are some of the more common formulae that involveπ.[135]
The circumference of a circle with radiusr is2πr.[136]
Apart from circles, there are othercurves of constant width. ByBarbier's theorem, every curve of constant width has perimeterπ times its width. TheReuleaux triangle (formed by the intersection of three circles with the sides of anequilateral triangle as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circularsmooth and evenalgebraic curves of constant width.[138]
Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involveπ. For example, an integral that specifies half the area of a circle of radius one is given by:[139]
In that integral, the function represents the height over the-axis of asemicircle (thesquare root is a consequence of thePythagorean theorem), and the integral computes the area below the semicircle. The existence of such integrals makesπ analgebraic period.[140]
Thetrigonometric functions rely on angles, and mathematicians generally use theradian as a unit of measurement.π plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2π radians. The angle measure of 180° is equal toπ radians, and1° =π/180 radians.[141]
Common trigonometric functions have periods that are multiples ofπ; for example, sine and cosine have period 2π, so for any angleθ and any integerk,[142]
Many of the appearances ofπ in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However,π also appears in many natural situations having apparently nothing to do with geometry.
In many applications, it plays a distinguished role as aneigenvalue. For example, an idealizedvibrating string can be modelled as the graph of a functionf on the unit interval[0, 1], withfixed endsf(0) =f(1) = 0. The modes of vibration of the string are solutions of thedifferential equation, or. Thusλ is an eigenvalue of the second derivativeoperator, and is constrained bySturm–Liouville theory to take on only certain specific values. It must be positive, since the operator isnegative definite, so it is convenient to writeλ =ν2, whereν > 0 is called thewavenumber. Thenf(x) = sin(πx) satisfies the boundary conditions and the differential equation withν =π.[143]
The valueπ is, in fact, theleast such value of the wavenumber, and is associated with thefundamental mode of vibration of the string. One way to show this is by estimating theenergy, which satisfiesWirtinger's inequality:[144] for a function withf(0) =f(1) = 0 andf,f′ bothsquare integrable, we have:
with equality precisely whenf is a multiple ofsin(πx). Hereπ appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using thevariational characterization of the eigenvalue. As a consequence,π is the smallestsingular value of the derivative operator on the space of functions on[0, 1] vanishing at both endpoints (theSobolev space).
Analysis and topology
Above,π was defined as the ratio of a circle's circumference to its diameter. The circumference of a circle is thearc length around theperimeter of the circle, a quantity which can be formally defined independently of geometry usinglimits—a concept incalculus.[145] For example, one may directly compute the arc length of the top half of the unit circle, given inCartesian coordinates by the equation, as theintegral:[146]
An integral such as this was proposed as a definition ofπ byKarl Weierstrass, who defined it directly as an integral in 1841.[d]
Integration is no longer commonly used in a first analytical definition because, asRemmert 2012 explains,differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition ofπ that does not rely on the latter. One such definition, due to Richard Baltzer[148] and popularized byEdmund Landau,[149] is the following:π is twice the smallest positive number at which thecosine function equals 0.[3][146][4]π is also the smallest positive number at which thesine function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as apower series,[150] or as the solution of adifferential equation.[4]
In a similar spirit,π can be defined using properties of thecomplex exponential,expz, of acomplex variablez. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at whichexpz is equal to one is then an (imaginary) arithmetic progression of the form:
and there is a unique positive real numberπ with this property.[146][151]
Theancient city of Carthage was the solution to an isoperimetric problem, according to a legend recounted byLord Kelvin:[154] those lands bordering the sea thatQueen Dido could enclose on all other sides within a single given oxhide, cut into strips.
The numberπ serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentionedabove, it can be characterized via its role as the best constant in theisoperimetric inequality: the areaA enclosed by a planeJordan curve of perimeterP satisfies the inequality
and equality is clearly achieved for the circle, since in that caseA = πr2 andP = 2πr.[155]
Ultimately, as a consequence of the isoperimetric inequality,π appears in the optimal constant for the criticalSobolev inequality inn dimensions, which thus characterizes the role ofπ in many physical phenomena as well, for example those of classicalpotential theory.[156][157][158] In two dimensions, the critical Sobolev inequality is
forf a smooth function with compact support inR2, is thegradient off, and and refer respectively to theL2 andL1-norm. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.
Wirtinger's inequality also generalizes to higher-dimensionalPoincaré inequalities that provide best constants for theDirichlet energy of ann-dimensional membrane. Specifically,π is the greatest constant such that
for allconvex subsetsG ofRn of diameter 1, and square-integrable functionsu onG of mean zero.[159] Just as Wirtinger's inequality is thevariational form of theDirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of theNeumann eigenvalue problem, in any dimension.
Fourier transform and Heisenberg uncertainty principle
The constantπ also appears as a critical spectral parameter in theFourier transform. This is theintegral transform, that takes a complex-valued integrable functionf on the real line to the function defined as:
Although there are several different conventions for the Fourier transform and its inverse, any such convention must involveπsomewhere. The above is the most canonical definition, however, giving the unique unitary operator onL2 that is also an algebra homomorphism ofL1 toL∞.[160]
TheHeisenberg uncertainty principle also contains the numberπ. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,
The factor of makes the area under the graph off equal to one, as is required for a probability distribution. This follows from achange of variables in theGaussian integral:[163]
which says that the area under the basicbell curve in the figure is equal to the square root ofπ.
Thecentral limit theorem explains the central role of normal distributions, and thus ofπ, in probability and statistics. This theorem is ultimately connected with thespectral characterization ofπ as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function.[164] Equivalently,π is the unique constant making the Gaussian normal distributione−πx2 equal to its own Fourier transform.[165] Indeed, according toHowe (1980), the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.[161]
whereχ(Σ) is theEuler characteristic, which is an integer.[166] An example is the surface area of a sphereS of curvature 1 (so that itsradius of curvature, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from itshomology groups and is found to be equal to two. Thus we have
reproducing the formula for the surface area of a sphere of radius 1.
Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles. Here illustrated is the complex logarithm of the Gamma function.
Although the curveγ is not a circle, and hence does not have any obvious connection to the constantπ, a standard proof of this result usesMorera's theorem, which implies that the integral is invariant underhomotopy of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates. More generally, it is true that if a rectifiable closed curveγ does not containz0, then the above integral is2πi times thewinding number of the curve.
The general form of Cauchy's integral formula establishes the relationship between the values of acomplex analytic functionf(z) on the Jordan curveγ and the value off(z) at any interior pointz0 ofγ:[169]
providedf(z) is analytic in the region enclosed byγ and extends continuously toγ. Cauchy's integral formula is a special case of theresidue theorem, that ifg(z) is ameromorphic function the region enclosed byγ and is continuous in a neighbourhood ofγ, then
In higher dimensions, factors ofπ are present because of a normalization by the n-dimensional volume of the unitn sphere. For example, in three dimensions, the Newtonian potential is:[173]
which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator.
Total curvature
This curve has total curvature6π and turning number3; it haswinding number2 aboutp and an additional loop which does not containp.
In thedifferential geometry of curves, thetotal curvature of a smooth plane curve is the amount it turns anticlockwise, in radians, from start to finish, computed as the integral of signedcurvature with respect to arc length:
For a closed curve, this quantity is equal to2πN for an integerN called theturning number orindex of the curve.N is thewinding number about the origin of thehodograph of the curve parametrized by arclength, a new curve lying on the unit circle, described by the normalizedtangent vector at each point on the original curve. Equivalently,N is thedegree of the map taking each point on the curve to the corresponding point on the hodograph, analogous to theGauss map for surfaces.
Gamma function and Stirling's approximation
Plot of the gamma function on the real axis
Thefactorial function is the product of all of the positive integers throughn. Thegamma function extends the concept offactorial (normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity. When the gamma function is evaluated at half-integers, the result containsπ. For example, and.[174]
The gamma function is used to calculate the volumeVn(r) of then-dimensional ball of radiusr in Euclideann-dimensional space, and the surface areaSn−1(r) of its boundary, the(n−1)-dimensional sphere:[176]
The gamma function can be used to create a simple approximation to the factorial functionn! for largen: which is known asStirling's approximation.[177] Equivalently,
As a geometrical application of Stirling's approximation, letΔn denote thestandard simplex inn-dimensional Euclidean space, and(n + 1)Δn denote the simplex having all of its sides scaled up by a factor ofn + 1. Then
Each prime has an associatedPrüfer group, which are arithmetic localizations of the circle. TheL-functions of analytic number theory are also localized in each primep.Solution of the Basel problem using theWeil conjecture: the value ofζ(2) is thehyperbolic area of a fundamental domain of themodular group, timesπ/2.
TheRiemann zeta functionζ(s) is used in many areas of mathematics. When evaluated ats = 2 it can be written as
Finding asimple solution for this infinite series was a famous problem in mathematics called theBasel problem.Leonhard Euler solved it in 1735 when he showed it was equal toπ2/6.[77] Euler's result leads to thenumber theory result that the probability of two random numbers beingrelatively prime (that is, having no shared factors) is equal to6/π2.[179] This probability is based on the observation that the probability that any number isdivisible by a primep is1/p (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is1/p2, and the probability that at least one of them is not is1 − 1/p2. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:[180]
This probability can be used in conjunction with arandom number generator to approximateπ using a Monte Carlo approach.[181]
The solution to the Basel problem implies that the geometrically derived quantityπ is connected in a deep way to the distribution of prime numbers. This is a special case ofWeil's conjecture on Tamagawa numbers, which asserts the equality of similar such infinite products ofarithmetic quantities, localized at each primep, and ageometrical quantity: the reciprocal of the volume of a certainlocally symmetric space. In the case of the Basel problem, it is thehyperbolic 3-manifoldSL2(R)/SL2(Z).[182]
The zeta function also satisfies Riemann's functional equation, which involvesπ as well as the gamma function:
Furthermore, the derivative of the zeta function satisfies
The constantπ also appears naturally inFourier series ofperiodic functions. Periodic functions are functions on the groupT =R/Z of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued functionf onT can be written as an infinite linear superposition ofunitary characters ofT. That is, continuousgroup homomorphisms fromT to thecircle groupU(1) of unit modulus complex numbers. It is a theorem that every character ofT is one of the complex exponentials.
There is a unique character onT, up to complex conjugation, that is a group isomorphism. Using theHaar measure on the circle group, the constantπ is half the magnitude of theRadon–Nikodym derivative of this character. The other characters have derivatives whose magnitudes are positive integral multiples of 2π.[153] As a result, the constantπ is the unique number such that the groupT, equipped with its Haar measure, isPontrjagin dual to thelattice of integral multiples of 2π.[186] This is a version of the one-dimensionalPoisson summation formula.
In Fourier analysis, the numberπ rather than 2π also appears, and sometimes this difference has important consequences. The basic exponential is no longer a character of the groupT, but instead is twisted by a sign after one turn of the circle group. This is known as aprojective representation: it is a representation not ofT but of itsdouble cover. It is the most basic projective representation, being associated with the most elementary compact group, andπ (rather than 2π) often appears in projective representations requiring a double cover.Spinors, for instance, exhibit this behavior in physics, representing rotations with a twist by a sign.Certain representations of the groupSL(2,R) of real matrices of determinant one also require this extra twist, as do representations of theHeisenberg group.
Modular forms and theta functions
Theta functions transform under thelattice of periods of an elliptic curve.
which is a kind of modular form called aJacobi form.[187] This is sometimes written in terms of thenome.
The constantπ is the unique constant making the Jacobi theta function anautomorphic form, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example is
TheWitch of Agnesi, named forMaria Agnesi (1718–1799), is a geometrical construction of the graph of the Cauchy distribution.The Cauchy distribution governs the passage ofBrownian particles through a membrane.
The constantπ is the unique (positive) normalizing factor such thatH defines alinear complex structure on theHilbert space of square-integrable real-valued functions on the real line.[189] The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert spaceL2(R): up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line.[190] The constantπ is the unique normalizing factor that makes this transformation unitary.
An occurrence ofπ in thefractal called theMandelbrot set was discovered by David Boll in 1991. He examined the behaviour of the Mandelbrot set near the "neck" at(−0.75, 0). When the number of iterations until divergence for the point(−0.75,ε) is multiplied byε, the result approachesπ asε approaches zero. The point(0.25 +ε, 0) at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root ofε tends toπ.[191]
The fact thatπ is approximately equal to 3 plays a role in the relatively long lifetime oforthopositronium. The inverse lifetime to lowest order in thefine-structure constantα is[194]
whereme is the mass of the electron.
π is present in some structural engineering formulae, such as thebuckling formula derived by Euler, which gives the maximum axial loadF that a long, slender column of lengthL,modulus of elasticityE, andarea moment of inertiaI can carry without buckling:[195]
Piphilology is the practice of memorizing large numbers of digits ofπ,[197] and world-records are kept by theGuinness World Records. The record for memorizing digits ofπ, certified by Guinness World Records, is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015.[198] In 2006,Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.[199]
One common technique is to memorize a story or poem in which the word lengths represent the digits ofπ: The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. Such memorization aids are calledmnemonics. An early example of a mnemonic for pi, originally devised by English scientistJames Jeans, is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."[197] When a poem is used, it is sometimes referred to as apiem.[200] Poems for memorizingπ have been composed in several languages in addition to English.[197] Record-settingπ memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and themethod of loci.[201]
A few authors have used the digits ofπ to establish a new form ofconstrained writing, where the word lengths are required to represent the digits ofπ. TheCadaeic Cadenza contains the first 3835 digits ofπ in this manner,[202] and the full-length bookNot a Wake contains 10,000 words, each representing one digit ofπ.[203]
In popular culture
A pi pie. Manypies are circular, and "pie" andπ arehomophones, making pie a frequent subject of pipuns.
Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae,π has been represented in popular culture more than other mathematical constructs.[204]
In thePalais de la Découverte (a science museum in Paris) there is a circular room known as thepi room. On its wall are inscribed 707 digits ofπ. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1873 calculation by English mathematicianWilliam Shanks, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.[205]
InCarl Sagan's 1985 novelContact it is suggested that the creator of the universe buried a message deep within the digits ofπ. This part of the story was omitted from thefilm adaptation of the novel.[206] The digits ofπ have also been incorporated into the lyrics of the song "Pi" from the 2005 albumAerial byKate Bush.[207] In the 1967Star Trek episode "Wolf in the Fold", a computer possessed by a demonic entity is contained by being instructed to "Compute to the last digit the value ofπ".[32]
In the United States,Pi Day falls on 14 March (written 3/14 in the US style), and is popular among students.[32]π and its digital representation are often used by self-described "mathgeeks" forinside jokes among mathematically and technologically minded groups. Acollege cheer variously attributed to theMassachusetts Institute of Technology or theRensselaer Polytechnic Institute includes "3.14159".[208] Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi.[209] In parts of the world where dates are commonly noted in day/month/year format, 22 July represents "Pi Approximation Day", as 22/7 ≈ 3.142857.[210]
Some have proposed replacingπ byτ = 2π, arguing thatτ, as the number of radians in oneturn or the ratio of a circle's circumference to its radius, is more natural thanπ and simplifies many formulae.[211][212] This use ofτ has not made its way into mainstream mathematics,[213] but since 2010 this has led to people celebrating Two Pi Day or Tau Day on June 28.[214]
In 1897, an amateur mathematician attempted to persuade theIndiana legislature to pass theIndiana Pi Bill, which described a method tosquare the circle and contained text that implied various incorrect values forπ, including 3.2. The bill is notorious as an attempt to establish a value of mathematical constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, and thus it did not become a law.[215]
In contemporaryinternet culture, individuals and organizations frequently pay homage to the numberπ. For instance, thecomputer scientistDonald Knuth let the version numbers of his programTeX approachπ. The versions are 3, 3.1, 3.14, and so forth.[216]
^abcBorwein, Jonathan M. (2014). "The life ofπ: from Archimedes to ENIAC and beyond". In Sidoli, Nathan; Van Brummelen, Glen (eds.).From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren. Heidelberg: Springer. pp. 531–561.doi:10.1007/978-3-642-36736-6_24.ISBN978-3-642-36735-9.MR3203895.
^Horvath, Miklos (1983)."On the Leibnizian quadrature of the circle"(PDF).Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica).4:75–83. Archived fromthe original(PDF) on 7 March 2023. Retrieved21 February 2023.
^abJones, William (1706).Synopsis Palmariorum Matheseos. London: J. Wale. pp. 243,263. p. 263:There are various other ways of finding theLengths, orAreas of particularCurve Lines orPlanes, which may very much facilitate the Practice; as for instance, in theCircle, the Diameter is to Circumference as 1 to 3.14159, &c. =π. ThisSeries (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr.John Machin; and by means thereof,Van Ceulen's Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch. Reprinted inSmith, David Eugene (1929)."William Jones: The First Use ofπ for the Circle Ratio".A Source Book in Mathematics. McGraw–Hill. pp. 346–347.
^Roy, Ranjan (2021) [1st ed. 2011].Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216,219–220.Newton, Isaac (1971)."De computo serierum" [On the computation of series]. InWhiteside, Derek Thomas (ed.).The Mathematical Papers of Isaac Newton. Vol. 4,1674–1684. Cambridge University Press. "De transmutatione serierum" [On the transformation of series] §3.2.2 pp. 604–615.
^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.hdl:1959.13/1043679.JSTOR2324715.
^Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted inBerggren, Borwein & Borwein 1997, pp. 129–140.
^Hardy and Wright 1938 and 2000: 177 footnote § 11.13–14 references Lindemann's proof as appearing atMath. Ann. 20 (1882), 213–225.
^cf Hardy and Wright 1938 and 2000:177 footnote § 11.13–14. The proofs thate andπ are transcendental can be found on pp. 170–176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417–419 for full citations.
^abCajori, Florian (2007).A History of Mathematical Notations: Vol. II. Cosimo, Inc. pp. 8–13.ISBN978-1-60206-714-1.the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented3.14159... byδ:π, as did Oughtred more than a century earlier
^Archibald, R. C. (1921). "Historical Notes on the Relatione−(π/2) =ii".The American Mathematical Monthly.28 (3):116–121.doi:10.2307/2972388.JSTOR2972388.It is noticeable that these letters arenever used separately, that is,π isnot used for 'Semiperipheria'
^Barrow, Isaac (1860)."Lecture XXIV". In Whewell, William (ed.).The mathematical works of Isaac Barrow (in Latin). Harvard University. Cambridge University press. p. 381.
^Segner, Joannes Andreas (1756).Cursus Mathematicus (in Latin). Halae Magdeburgicae. p. 282.Archived from the original on 15 October 2017. Retrieved15 October 2017.
^Euler, Leonhard (1727)."Tentamen explicationis phaenomenorum aeris"(PDF).Commentarii Academiae Scientiarum Imperialis Petropolitana (in Latin).2: 351.E007.Archived(PDF) from the original on 1 April 2016. Retrieved15 October 2017.Sumatur pro ratione radii ad peripheriem,I : πEnglish translation by Ian BruceArchived 10 June 2016 at theWayback Machine: "π is taken for the ratio of the radius to the periphery [note that in this work, Euler'sπ is double ourπ.]"Euler, Leonhard (1747). Henry, Charles (ed.).Lettres inédites d'Euler à d'Alembert. Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche (in French). Vol. 19 (published 1886). p. 139.E858.Car, soit π la circonference d'un cercle, dout le rayon est= 1 English translation inCajori, Florian (1913). "History of the Exponential and Logarithmic Concepts".The American Mathematical Monthly.20 (3):75–84.doi:10.2307/2973441.JSTOR2973441.Lettingπ be the circumference (!) of a circle of unit radius
^Reitwiesner, George (1950). "An ENIAC Determination of pi and e to 2000 Decimal Places".Mathematical Tables and Other Aids to Computation.4 (29):11–15.doi:10.2307/2002695.JSTOR2002695.
^Nicholson, J. C.; Jeenel, J. (1955). "Some comments on a NORC Computation of π".Math. Tabl. Aids. Comp.9 (52):162–164.doi:10.2307/2002052.JSTOR2002052.
^Arndt & Haenel 2006, p. 111 (5 times); pp. 113–114 (4 times). For details of algorithms, seeBorwein, Jonathan; Borwein, Peter (1987).Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity. Wiley.ISBN978-0-471-31515-5.
^Rabinowitz, Stanley; Wagon, Stan (March 1995). "A spigot algorithm for the digits of Pi".American Mathematical Monthly.102 (3):195–203.doi:10.2307/2975006.JSTOR2975006.
See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, § 5.3.3, pp. 111–112.
^Apostol, Tom (1967).Calculus. Vol. 1 (2nd ed.). Wiley. p. 102.From a logical point of view, this is unsatisfactory at the present stage because we have not yet discussed the concept of arc length
^Baltzer, Richard (1870).Die Elemente der Mathematik [The Elements of Mathematics] (in German). Hirzel. p. 195.Archived from the original on 14 September 2016.
^Landau, Edmund (1934).Einführung in die Differentialrechnung und Integralrechnung (in German). Noordoff. p. 193.
^Rudin, Walter (1986).Real and complex analysis. McGraw-Hill. p. 2.
^Ahlfors, Lars (1966).Complex analysis. McGraw-Hill. p. 46.
^Arndt & Haenel 2006, pp. 41–43. This theorem was proved byErnesto Cesàro in 1881. For a more rigorous proof than the intuitive and informal one given here, seeHardy, G. H. (2008).An Introduction to the Theory of Numbers. Oxford University Press. Theorem 332.ISBN978-0-19-921986-5.
^Tate, John T. (1967). "Fourier analysis in number fields, and Hecke's zeta-functions". In Cassels, J. W. S.; Fröhlich, A. (eds.).Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965). Thompson, Washington, DC. pp. 305–347.ISBN978-0-9502734-2-6.MR0217026.
^Port, Sidney; Stone, Charles (1978).Brownian motion and classical potential theory. Academic Press. p. 29.
^Titchmarsh, E. (1948).Introduction to the Theory of Fourier Integrals (2nd ed.). Oxford University: Clarendon Press (published 1986).ISBN978-0-8284-0324-5.{{cite book}}:ISBN / Date incompatibility (help)
^Stein, Elias (1970).Singular Integrals and Differentiability Properties of Functions. Princeton University Press.; Chapter II.
^Keith, Michael; Diana Keith (17 February 2010).Not A Wake: A dream embodying (pi)'s digits fully for 10,000 decimals. Vinculum Press.ISBN978-0-9630097-1-5.
^Gill, Andy (4 November 2005)."Review of Aerial".The Independent.Archived from the original on 15 October 2006.the almost autistic satisfaction of the obsessive-compulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits long)
Arndt, Jörg; Haenel, Christoph (2006).Pi Unleashed. Springer-Verlag.ISBN978-3-540-66572-4. Retrieved5 June 2013. English translation by Catriona and David Lischka.
Dym, H.; McKean, H. P. (1972).Fourier series and integrals. Academic Press.
Eymard, Pierre; Lafon, Jean Pierre (2004).The Numberπ. Translated by Wilson, Stephen. American Mathematical Society.ISBN978-0-8218-3246-2. English translation ofAutour du nombreπ (in French). Hermann. 1999.
Remmert, Reinhold (2012)."Ch. 5 What is π?". In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.).Numbers. Springer.ISBN978-1-4612-1005-4.
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