Pi

Pi This article is about the number π. For the Greek letter, entific entific community: community: Several Several books devoted devoted to it have see Pi see  Pi (letter). (letter). For For other other uses uses of pi, π, and Π, see P see  Pii been published, the number is celebrated on  Pi Day and Day  and (disambiguation).. (disambiguation) record-setting calculations of the digits of π often result in news headline headlines. s. Attempts Attempts to memorize memorize the value of The number  π  is a  mathematical constant, constant , the ratio the  ratio of  of a π with increasing precision have led to records of over circle's circle 's circumference  circumference to  to its diameter its diameter,, commonly approx- 67,000 digits. imated as 3.14159. It has been represented by the Greek Greek letter "π" since the mid-18th century, though it is also sometimes sometimes spelled out as " pi" (/paɪ/ (/paɪ/). ). 1 Fund Fundam amen enta tals ls Being an irrational an  irrational number, number, π cannot be expressed exactly as a fraction a  fraction (equivalently,  (equivalently, its  decimal representa- 1.1 1.1 Name tion never tion  never ends and never  settles into a permanent repeating pattern). pattern). Still, Still, fraction fractionss such as 22/7 and other The symb symbol ol used used by mathe mathema matic ticia ians ns to repr repres esent ent the ratio ratio rational numbers are commonly used to  approximate  approximate π.  π. of a circl circle’s e’s circ circumf umfer eren ence ce to its diame diameter ter is the lowe lowerc rcase ase The digits appear to be randomly distributed; however, Greek letter π, letter  π, sometimes spelled out as  pi . In English, to date, no proof of this has been discovered. Also, π is π is pronounced is  pronounced as “pie” ( “pie”  ( /paɪ/  /paɪ/,, paɪ).[6] In mathematical mathematical a transcendental number – number  – a number that is not the  root use, the lowercase lowercase letter π (or π in sans-serif in  sans-serif font)  font) is disof any non-zero polynomial non-zero  polynomial having  having rational  rational coefficients. coefficients . tinguished from from its capital counterpart Π, which denotes This transcendence of π implies that it is impossible to a product of a sequence. sequence . solve the ancient challenge of  squaring the circle  with a The choice of the symbol π is discussed in the section compass and straighted straightedge ge.. Adoption of the symbol π. Although ancient civilizations needed the value of π to be computed accurately for practical reasons, it was not calculated to more than seven digits, using geometrical 1.2 1.2 Defin Definit itio ion n techniques, techniques, in Chinese in  Chinese mathematics and mathematics  and to about five in Indian mathemati mathematics cs in  in the 5th century CE. The historicall callyy first first exac exactt formula ormula for π, based based on infin infinite ite seri series es,, was C not available until a millennium later, when in the 14th century the  Madhava–Leibniz series  was discovered in Indian mathematics. mathematics.[1][2] In the 20th and 21st centuries, mathematicians mathematicians and computer and  computer scientists discovered scientists  discovered new approaches that, when combined with increasing compud tational power, extended the decimal representation of π  e  r to, as of 2015, over 13.3 trillion (10 13 ) digits.[3] Practi e  t cally all scientific applications require no more than a few  a  m   i  d hundred digits of π, and many substantially fewer, so the primary motivation for for these computations computations is the human [4][5] desire to break records. However, the extensive calculations involved have been used to test  supercomputers and high-precision multiplication algorithms multiplication  algorithms.. Because its definition relates to the circle, π is found in many formulae in  in   trigonometry and trigonometry  and  geometry  geometry,, especially cially those concerning circles, circles, ellipses or spheres. spheres. It is also found in formulae used in other branches of science circumference ence of a circle is slightly more than three times as  such as   cosmology, cosmology,   number theory, theory,   statistics, statistics,  fractals  fractals,, The circumfer long as its diameter. The exact ratio is called π. thermodynamics,, mechanics thermodynamics  mechanics and  and electromagnetism  electromagnetism.. The ubiquity of π makes it one of the most widely known common only ly defin defined ed as the the   ratio of a   circle's circle's mathematical constants both inside and outside the sci- π is comm circumference C  to  to its diameter its diameter  d :[7] 1

2

π  =

1 FUND FUNDAME AMENT NTALS  ALS 

numbers  of  absolute value on numbers of value  one. e. The number number π is then defined as half the magnitude of the derivative of this homomorphism. [18]

C  d

The ratio  C /d  is  is constant, regardless of the circle’s size. For example, if a circle has twice the diameter of another circle circleit it willalso will alsohav havee twice twice the circumf circumfere erence, nce, preserv preserving ing the ratio C /d . This definition of π implicitly makes use of flat (Euclidean) geometry; geometry; although the notion of a circle can can be exte extend nded ed to any any curved (non-Euclid (non-Euclidean) ean) geometry geometry,, these new circles will no longer satisfy the formula π = C /d .[7] Here, the circumference of a circle is the  arc length around around the perimeter perimeter of the circle, circle, a quantity quantity which which can be formally defined independently of geometry using limits ing limits,, a concept in calculus in  calculus..[8] For example, one may compute directly directly the arc length of the top half of the unit circle given in Cartesian in  Cartesian coordinates by coordinates  by x2 + y2 = 1 , as the integral the integral::[9] 1

π  =

∫  √  −1

1.3 1.3

Prop Proper erti ties es

π is an irrational an irrational number, number , meaning that it cannot be written as the ratio the  ratio of two integers (fractions integers  (fractions such as 22/7 are commonly used to approximate π; no  common fraction (ratio tion  (ratio of whole numbers) can be its exact value). [19] Since π is irrational, it has an infinite number of digits in its  its   decimal representation, representation , and it does not settle into an infinitely repeating infinitely repeating pattern of pattern of digits. There are several several proofss that π is irrational; proof irrational; they generally require calculus and rely on the reductio the  reductio ad absurdum technique. absurdum technique. The degree gree to whic whichh π can can be appr approx oxim imat ated ed by rational numbers (called the irrationality the  irrationality measure) measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of  e  or ln(2) but smaller than the measure of Liouville of  Liouville numbers. numbers .[20]

dx

1

− x2 .

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

√π

Definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, calculus , are no longer common in the literature. Remmert (1991) (1991) explains that this is because in many r   modern treatments of calculus, calculus, differential  differential calculus typicalculus  typi=   1   cally precedes precedes integral calculus calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, definition, due to  Richard Baltzer,,[11] and popularized by  Edmund Landau, Baltzer Landau ,[12] is the following: following: π is twice the smallest positive positive number at [7][13][14] which the cosine the  cosine function equals function  equals 0. The cosine can be defined independently of geometry as a  power series,,[15] or as the solution of a differential ries a  differential equation. equation .[13] Because π is a  transcendental number  , squaring  ,  squaring the circle is circle  is not   possible in a finite number of steps using the classical tools of  In a similar spirit, π can be defined instead using propstraightedge. erties of the complex the  complex exponential, exponential , exp( z), of a complex a  complex compass and straightedge. variable  z. Like the cosine, the complex exponential exponential can More strong strongly ly,, π is a   transcendental transcendental number number,, which which be defined in one of several several ways. The set of complex More numbers at which exp( z) is equal to one is then an (imag- means that it is not the   solution  of any non-constant polynomial  with rational  with  rational coefficients,  coefficients, such as x 5 /120 − inary) arithmetic progression of the form:  = 0. [21][22]  x 3 /6 +  x  =

{. . . , −2πi, 0, 2πi, 4π i , . . . } = {2πki |k ∈ Z } and there is a unique positive real number π with this property.[14][16] A more abstract variation on the same idea, making use of sophisticated mathematical concepts of topology of  topology and  and   algebra, algebra, is the following theorem: [17] there is a unique continuous unique  continuous isomorphism from isomorphism from the group the group real numbe numbers rs under under additi addition on modulo integers integers (the R/Z of real circle group) group) onto the multiplicative group of  complex

The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or  n -th roots such roots  such 3 as  √31 or √10. Second, since no transcendental transcendental number can be   constructed with constructed  with compass  compass and straightedge, straightedge , it is not possible to "square "square the circle". circle ". In other words, it is impossible to construct, construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[23] Squaring a circle was one of the important tant geome geometry try probl problem emss of the classi classical cal antiquity antiquity..[24] Am-

3

1.5 Approximate value

ateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is impossible. [25]

22/7, 333/106, and 355/113. These numbers are among the most well-known and widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator. [29] Because π is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (shown above) also does not exhibit any other obvious pattern, [30] mathematicians have discovered several generalized continued fractions that do, such as:[31]

The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. [26] The conjecture that π is normal has not been proven or disproven. [26] Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis.   Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[27] Despite the fact that π  = 1+ π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π. [28] 1.4

1.5

Continued fractions

4

= 3+

2

1

32 2+ 52 2+ 72 2+ 92 2+ 2+

..

12 32 6+ 52 6+ 72 6+ 92 6+

.

6+

..

=

4 1

1+ 3+ 5+

7

.

Approximate value

Some approximations of  pi  include:

•   Integers: 3 •   Fractions:

Approximate fractions include (in order of increasing accuracy) 22/7, 333/106, 355/113, 52163/16604, 103993/33102, and 245850922/78256779.[29] (List is selected terms from   A063674 and   A063673.)

•   Decimal:

The first 50 decimal digits are 3.14159265358979323846264338327950288419716939937510...[32 A000796

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

•   Binary: The base 2 approximation to 48 digits is

Like all irrational numbers, π cannot be represented as a common fraction (also known as a simple or vulgar fraction), by the very definition of “irrational”. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a  continued fraction:

•   Hexadecimal:

11.001001000011111101101010100010001000010110100011... The base 16 approximation to 20 digits is 3.243F6A8885A308D31319... [33]

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

2 π  = 3 +

1 7+

15+

1+

292+

1

1

1+

1

1

1+

History

Main article: Approximations of π See also: Chronology of computation of π 1

1

1+

..

.

A001203 Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3,

2.1

Antiquity

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

4

2 HISTORY 

was improved upon in Chinese mathematics in particular by the mid first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period. Some Egyptologists[35] have claimed that the   ancient Egyptians used an approximation of π as 22 ⁄ 7  from as early as the Old Kingdom.[36] This claim has met with skepticism.[37][38][39][40] The earliest written approximations of π are found in Egypt and Babylon, both within one percent of the true value. In Babylon, a clay tablet  dated 1900–1600 BC has a geometrical statement that, by implication, treats π as  25 ⁄ 8  = 3.1250. [41] In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats π as (16 ⁄ 9 )2 ≈ 3.1605.[41] Astronomical calculations in the  Shatapatha Brahmana (ca. 4th century BC) use a fractional approximation of 339 ⁄   ≈ 3.139 (an accuracy of 9×10 −4 ).[42] Other Indian 108 sources by about 150 BC treat π as √10 ≈ 3.1622 [43] 2.2

Polygon approximation era Archimedes  developed the polygonal approach to approximating  π.

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

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes.[44] This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as “Archimedes’ constant”. [45] Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (that is 3.1408 < π < 3.1429).[46] Archimedes’ upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7. [47] Around 150 AD, Greek-Roman scientist Ptolemy,inhis Almagest , gavea value forπ of3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[48] Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[49]

AD, the   Wei Kingdom   mathematician   Liu Hui   created a   polygon-based iterative algorithm   and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[51][52] Liu later invented a faster method of calculating π 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.[51] The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name  Milü in Chinese), using Liu Hui’s algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of π available for the next 800 years. [53] The Indian astronomer Aryabhata used a value of 3.1416 in his  Āryabhaṭīya  (499 AD).[54] Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[55] Italian author Dante apparently employed the value 3+√2/10 ≈ 3.14142. [55]

The Persian astronomer   Jamshīd al-Kāshī  produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×2 28 sides,[56][57] which stood as the world record for about 180 years. [58] French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×2 17 sides.[58] Flemish mathematician  Adriaan van Roomen arrived at 15 decIn ancient China, values for π included 3.1547 (around 1 imal places in 1593. [58] In 1596, Dutch mathematician AD), √10 (100 AD, approximately 3.1623), and 142/45 Ludolph van Ceulen reached 20 digits, a record he later (3rd century, approximately 3.1556).[50] Around 265 increased to 35 digits (as a result, π was called the

5

2.3 Infinite series 

“Ludolphian number” in Germany until the early 20th century).[59] Dutch scientist Willebrord Snellius reached 34 digits in 1621, [60] and Austrian astronomer Christoph Grienberger   arrived at 38 digits in 1630 using 10 40 sides,[61] which remains the most accurate approximation manually achieved using polygonal algorithms. [60] 2.3

Infinite series Sn 2.6 2.8   3 . 0 3 . 2 3 . 4 3 .6 3 .8 4.0

Sn 3.140 3.143

Sn 3.1415 3.1418

2

2

2

2

4

4

4

4

6

6

6

8

8

8

10

10

10

6

Viète

8 n

Sn 3.13 3.15 3.17

Wallis

10

MadhavaGregoryLeibniz

12 14

n

Madhava

16

Newton

18

Nilakantha

20

n

n

12

12

12

14

14

14

16

16

16

18

18

18

20

20

20

Comparison of the convergence of several historical infinite series for π. S  is the approximation after taking  n   terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)

The calculation of π was revolutionized by the development of   infinite series   techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[62] Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques. [62] Although infinite series were exploited for π most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach was first discovered in  India sometime between 1400 and 1500 AD. [63] The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indianastronomer Nilakantha Somayaji in his  Tantrasamgraha, around 1500 AD.[64] The series are presented without proof, but proofs are presented in a later Indian work,  Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 – c. 1425.[64] Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory–Leibniz series.[64] Madhava used infinite series to estimate π to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī , using a polygonal algorithm.[65]

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

The second infinite sequence found in Europe , by John Wallis in 1655, was also an infinite product. [67] The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing “I am ashamed to tell you to how many figures I carried these computations, having no other business at the time.” [66] In Europe, Madhava’s formula was rediscovered by Scottish mathematician James Gregory in 1671, and by Leibniz in 1674:[68][69]

arctan z  =  z −

z3

3

+

z5

5

−

z7

7

+

···

This formula, the Gregory–Leibniz series, equals π/4 when evaluated with z  = 1.[69] In 1699, English matheThe first infinite sequence discovered in Europe  was an matician Abraham Sharp used the Gregory–Leibniz seinfinite product (rather than an  infinite sum, which are ries to compute π to 71 digits, breaking the previous 39 digits, which was set with a polygonal more typically used in π calculations) found by French record of [70] algorithm. The Gregory–Leibniz series is simple, but mathematician François Viète in 1593: [67] converges very slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[71] √  √  √  √  2

π

=

√ 

A060294

2 2

·

√ 

2+ 2 2

·

2+

2+ 2

2

···

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

6

2 HISTORY 

2.4 π

4

= 4  arctan

 1 5

1  − arctan 239

Machin reached 100 digits of π with this formula. [73] Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of π.[73] Machin-like formulae remained the best-known method for calculating π 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. [74]

Irrationality and transcendence

See also:  Proof that π is irrational and  Proof that π is transcendental Not all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[80]

A record was set by the calculating prodigy   Zacharias Dase, who in 1844 employed a Machin-like formula to π2 1 1 1 1 calculate 200 decimals of π in his head at the behest of 6 = 12 + 22 + 32 + 42 + ··· German mathematician Carl Friedrich Gauss.[75] British mathematician  William Shanks  famously took 15 years Swiss scientist Johann Heinrich Lambert in 1761 proved to calculate π to 707 digits, but made a mistake in the that π is irrational, meaning it is[19]not equal to the quo528th digit, rendering all subsequent digits incorrect. [75] tient of any two whole numbers. Lambert’s proof exploited a continued-fraction representation of the tangent function.[81] French mathematician Adrien-Marie Legen2.3.1 Rate of convergence dre provedin1794thatπ2 is also irrational. In 1882, German mathematician  Ferdinand von Lindemann  proved Some infinite series for π  converge faster than others. that π is  transcendental, confirming a conjecture made Given the choice of two infinite series for π, math- by both Legendre and Euler.[82] ematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given 2.5 Adoption of the symbol π accuracy.[76] A simple infinite series for π is the Gregory– Leibniz series:[77] π  =

4 1

 −  43  + 54 − 74  + 94 − 114  + 134  − · · ·

As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and – with a sufficient 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 in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:[79] π  = 3+

4 3

4 4 4 − − + +··· 2 × × 4 4 × 5 × 6 6 × 7 × 8 8 × 9 × 10

The following table compares the convergence rates of these two series:

After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha’s series is within 0.002 of the correct value of π. Nilakantha’s series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin’s series and Chudnovsky’s series, the latter producing 14 correct decimal digits per term. [76]

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

The earliest known use of the Greek letter π to represent 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 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics .[83] The Greek letter first appears there in the phrase “1/2 Periphery (π)" in the discussion of a circle with radius one. Jones may have chosen π because it was the first letter in the Greek spelling of the word periphery.[84] However, he writes that his equations for π 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. [85] It had indeed been used earlier for geometric concepts. [85] William Oughtred used π and δ, the Greek letter equivalents of p and d, to express ratios of periphery and diameter in the 1647 and later editions of  Clavis Mathematicae. After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler started using it, beginning with his 1736 work  Mechanica. Before then, mathematicians sometimes used letters such as  c  or  p  instead.[85] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly. [85] In 1748, Euler used π in his widely read work  Introductio in analysin infinitorum (he wrote: “for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a cir- John von Neumann was part of the team that first used a digital  cle of radius 1”) and the practice was universally adopted computer, ENIAC  , to compute π. thereafter in the Western world.[85]

3

Therecord, always relying on an arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached Computer era and iterative algorithms in 1973.[88]

Modern quest for more digits

3.1

The Gauss–Legendre iterative algorithm: Initialize

a0 =1

1 b0 = √ 

2

t0 = 1 4

p0 =1

Iterate

+

an+1 = an 2 bn tn+1 =tn  pn (an

−

√ 

bn+1 = an bn

−a

n+1

)2

 pn+1 =2 pn

Then an estimate for π is given by

π

≈(

2

an +bn ) 4tn

Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new   iterative algorithms   for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly. [89] Such algorithms are particularly important in modern π computations, because most of the computer’s time is devoted to multiplication. [90] They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods.[91] The iterative algorithms were independently published in 1975–1976 by American physicist  Eugene Salamin and Australian scientist  Richard Brent.[92] 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 by  Carl Friedrich Gauss, in what is now termed the  arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm.[92] As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians  John Wrench  and Levi Smith reached 1,120 digits in 1949 using a desk calculator. [86] Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and  John von Neumann  that same year achieved 2,037 digits with a calculation that took The iterative algorithms were widely used after 1980 70 hours of computer time on the  ENIAC computer.[87] because they are faster than infinite series algorithms:

8

3 MODERN QUEST FOR MORE DIGITS 

whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally  multiply the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter 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. [93] Iterative methods were used by Japanese mathematician  Yasumasa Kanada to set several records for computing π between 1995 and 2002.[94] This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series. [94] 3.2

Motivations for computing π Record approximations of pi 14

10

12

10

  s    t    i   g 10    i    d10    l   a   m108    i   c   e    d 6    f 10   o   r   e    b 104   m   u    N 100 1 2000 BCE

250 BCE

480

1400

1450

1500

1550

1600

1650

1700

1750

1800

1850

1900

1950

2000

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

 Year

As mathematicians discovered new algorithms, and computers  became available, the number of known decimal digits of π increased dramatically. Note that the vertical scale is  logarithmic .

For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom.[95] Despite this, people have worked strenuously to compute π to thousands and millions of digits. [96] This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world. [97][98] They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including  high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[99]

published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence. [100] One of his formulae, based on modular equations, is

√ 

∞ 2 2 (4k)!(1103 + 26390k) = . 9801 π k!4 (3964k ) k 1

∑

=0

This series converges much more rapidly than most arctan series, including Machin’s formula. [101] Bill Gosper was the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985. [102] Ramanujan’s formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[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 has been used for several record-setting π calculations, in3.3 Rapidly convergent series cluding the first to surpass 1 billion (10 9 ) digits in 1989 12 ) digits Modern π calculators do not use iterative algorithms by the Chudnovsky brothers, 2.7 trillion (2.7×10 13 ) digits in exclusively. New infinite series were discovered in by Fabrice Bellard in 2009, and 10 trillion (10[105][106] For the 1980s and 1990s that are as fast as iterative algo- 2011 by Alexander Yee and Shigeru Kondo. similar formulas, see also the Ramanujan–Sato series. rithms, yet are simpler and less memory intensive. [94] The fast iterative algorithms were anticipated in 1914, In 2006, Canadian mathematician Simon Plouffeused the when the Indian mathematician   Srinivasa Ramanujan PSLQ integer relation algorithm [107] to generate several

9 new formulas for π, conforming to the following template:

4

Use

Main article: List of formulae involving π ∞

∑ � 1

a

b

c

�

Because π is closely related to the circle, it is found in many formulae from the fields of geometry and trigonomn=1 etry, particularly those concerning circles, spheres, or elπ where q is e (Gelfond’s constant), k   is an  odd num- lipses. Formulae from other branches of science also ber, and  a ,  b,  c  are certain rational numbers that Plouffe include π in some of their important formulae, including sciences such as statistics, fractals, thermodynamics, computed.[108] mechanics, cosmology, number theory, and electromagnetism. k

π =

3.4

nk

q n

− 1 + q 2 − 1 + q 4 − 1 n

n

,

Spigot algorithms

Two algorithms were discovered in 1995 that opened up new avenues of research into π. They are called  spigot algorithms because, like water dripping from a  spigot, they produce single digits of π that are not reused after they are calculated. [109][110] This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced. [109]

4.1

Geometry and trigonometry

American mathematicians   Stan Wagon   and Stanley Rabinowitz produced a simple spigot algorithm in 1995.[110][111][112] Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[111]

Area =

2

r 

Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe: [113][114] Circle Area = π  =

∞

2

∑ �

k =0

1 16k

4 8k + 1

 −

2 8k + 4

 −

1 8k + 5

 −

1 8k + 6

 π × r 

�

This formula, unlike others before it, can produce any individual hexadecimal digit of π without calculating all the preceding digits.[113] Individual binary digits may be extracted from individual hexadecimal digits, and  octal digits can be extracted from one or two hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits. [115] An important application of digit extraction algorithms is to validate new claims of record π 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 random hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is correct.[106] Between 1998 and 2000, the   distributed computing project PiHex used Bellard’s formula (a modification of theBBP algorithm) to compute the quadrillionth (1015 th) bit of π, which turned out to be 0. [116] In September 2010, a Yahoo!  employee used the company’s  Hadoop application on one thousand computers over a 23-day period to compute 256 bits of π at the two-quadrillionth (2×1015 th) bit, which also happens to be zero. [117]

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

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

•  The circumference of a circle with radius  r  is 2πr . • The area of a circle with radius r  is πr 2. • The volume of a sphere with radius  r  is 4/3πr 3. •  The surface area of a sphere with radius  r  is 4πr 2. The formulae above are special cases of the surface area Sn(r ) and volume  Vn(r ) of an  n-dimensional sphere. nπn/2

S n (r) =

r Γ( n 2  +1)

V  n (r) =

r Γ( n 2 +1)

π n/2

n 1

−

n

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

10

4 USE  

example, an integral that specifies half the area of a circle of radius one is given by: [119] 1

∫  √  − 1

−1

x2 dx =

π

2

.

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

In that integral the function √1- x 2 represents the top π ≈ 2nℓ half of a circle (the  square root is a consequence of the xt Pythagorean theorem), and the integral ∫1 Monte Carlo method for computing π is to draw −1 computes the area between that half of a circle and Another a circle inscribed in a square, and randomly place dots in the x  axis. the square. The ratio of dots inside the circle to the total number of dots will approximately equal π/4. [124] Monte Carlo methods for approximating π are very slow compared to other methods, and are never used to approximate π when speed or accuracy are desired. [125] 4.2

Complex numbers and analysis

Sine and  cosine functions repeat with period 2π.

The trigonometric functions rely on angles, and mathematicians 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 angle of 2π radians.[120] The angle measure of 180° is equal to π radians, and 1° = π/180 radians. [120] Common trigonometric functions have periods that are multiples of π; for example, sine and cosine have period 2π,[121] so for any angle  θ and any integer  k , sin θ  = sin (θ + 2 πk ) and cos θ  =  cos (θ + 2 πk ) .  [121] 4.1.1

Monte Carlo methods

t  a

b l

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

The association between imaginary powers of the number  e  and   points on the unit circle centered at the origin in the complex plane  given by Euler’s formula.

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

Random dots are placed on the quadrant of a square with a circle inscribed in it. z  =  r · (cos ϕ + i sin ϕ), Monte Carlo methods, based on random trials, can be used to approximate π. where  i  is the imaginary unit satisfying  i 2 = −1. The frequent appearance of π in complex analysis can be related Monte Carlo methods, which evaluate the results of mul- to the behavior of the exponential function of a complex tiple random trials, can be used to create approximations variable, described by Euler’s formula:[127]

11

4.3 Number theory and Riemann zeta function

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

eiϕ = cos ϕ + i sin ϕ,

where the constant  e is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of  e and points on the unit circle centered at the origin of the complex plane. Setting  φ  = π in Euler’s formula results in Euler’s identity, celebrated by mathematicians because it contains the five most important mathematical constants: [127][128] eiπ + 1 = 0.

The gamma function  extends the concept of  factorial (normally defined only for non-negative integers) to all complex 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 4 π .[134] The gamma function can be used to create a simple approximation to n! for large n: √  n which is known as Stirling’s approxin! ∼ 2πn ne [135] mation.

��

4.3

Number theory and Riemann zeta function

There are  n  different complex numbers  z  satisfying  z n = 1, and these are called the " n-th roots of unity".[129] They The Riemann zeta function  ζ (s ) is used in many areas of are given by this formula: mathematics. When evaluated at  s   = 2 it can be written as e2πik /n

(k  = 0, 1, 2, . . . , n

− 1).

Cauchy’s integral formula   governs   complex analytic functions and establishes an important relationship between integration and differentiation, including the fact that the values of a complex function within a closed boundary are entirely determined by the values on the boundary:[130][131]

ζ (2) =

1 1 1 + 2+ 2+ 2 1 2 3

···

Finding a simple solution for this infinite series was a famous problem in mathematics called the  Basel problem. Leonhard Euler solved it in 1735 when he showed it was equal to π2 /6.[80] Euler’s result leads to the number theory result that the probability of two random numbers being relatively prime (that is, having no shared factors) is equal 1 f (z ) to 6/π2.[136][137] This probability is based on the observaf (z0 ) = dz 2πi γ  z − z0 tion that the probability that any number is divisible by a prime p is 1/ p (for example, every 7th integer is divisible An occurrence of π in the Mandelbrot set fractal was disby 7.) Hence the probability that two numbers are both divisible by this prime is 1/ p2 , and the probability that at least one of them is not is 1-1/ 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: [138]

 

∏ � − � �∏ ∞  p

1

1

 p2

=

∞

1

 p

1  p−2

−

�

−1 =

1+

1 22

1 + 312 +

This probability canbe used in conjunction with a random number generator to approximate π using a Monte Carlo approach. [139]  π can be computed fromthe Mandelbrot set  , by counting the number of iterations required before point (−0.75, ε) diverges.

4.4

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

The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution.[140] π is found in the Gaussian function (which is the probability density function  of the normal distribution) with mean μ and standard deviation σ:[141]

Probability and statistics

···

=

1 6 = ζ (2) π

12

5 OUTSIDE MATHEMATICS 

(Δ x ) and momentum (Δ p) cannotbothbe arbitrarilysmall at the same time (where h  is Planck’s constant):[143]

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

 ≥ 4hπ .

1.5

∆x ∆ p

In the domain of   cosmology, π appears in   Einstein’s field equation, a fundamental formula which forms the basis of the   general theory of relativity  and describes the  fundamental interaction of gravitation as a result of spacetime being curved by matter and energy:[144]

1

0.5

0

-0.5 -2

-1

0

1

 − g 2R + Λg

2

ik

Rik A graph of the Gaussian function 2 ƒ(x) =  e −x . The √ colored region between the function and the x-axis has area π .

ik

=

8πG c4

T ik ,

where  Rik  is the Ricci curvature tensor, R  is the scalar curvature,  gik  is the metric tensor, Λ is the cosmological constant, G  is Newton’s gravitational constant, c   is the speed of light in vacuum, and Tik  is the  stress–energy tensor.

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 representing The area under the graph of the normal distribution curve the vacuum permittivity of free space): [145] is given by the Gaussian integral:[141] f (x) =

∞

∫ 

1 √  e−( − ) /(2 σ 2π

2 e−x dx =

x µ

2

σ2 )

√ π,

−∞

while the related integral for the Cauchy distribution is

F  =

|q 1q 2| .

4πε0 r 2

The fact that π is approximately equal to 3 plays a role in the relatively long lifetime of   orthopositronium. The inverse lifetime to lowest order in the fine structure constant α is[146]

∞

1  dx  =  π. −∞ x2 + 1

∫ 

1 τ 

5

Outside mathematics

=2

π2

− 9 mα6,

9π

where  m is the mass of the electron.

π is present in some structural engineering formulae, such 5.1 Describing physical phenomena as the buckling formula derived by Euler, which gives the maximum axial load F   that a long, slender column Although not a physical constant, π appears routinely in of length  L, modulus of elasticity  E , and area moment of equations describing fundamental principles of the uni- inertia  I  can carry without buckling:[147] verse, often because of π's relationship to the circle and to spherical coordinate systems. A simple formula from π 2 EI  the field of classical mechanics gives the approximate peF  = . riod T  of a simple pendulum of length  L, swinging with L2 a small amplitude ( g is the earth’s gravitational accelera- The field of  fluid dynamics  contains π in Stokes’ law, tion):[142] which approximates the   frictional force F   exerted on

 ≈  2π

T 

� 

small, spherical objects of radius  R, moving with velocity v in a fluid with dynamic viscosity  η:[148]

L  . g F  = 6 π η R v.

One of the key formulae of   quantum mechanics is Heisenberg’s uncertainty principle, which shows that the The Fourier transform, defined below, is a mathematical uncertainty in the measurement of a particle’s position operation that expresses time as a function of frequency,

13

5.3 In popular culture

known as its frequency spectrum. It has many applica- 5.3 tions in  physics and engineering, particularly in  signal processing.[149]

f ˆ(ξ ) =

∞

∫ 

In popular culture

f (x)  e−2πixξ dx

−∞

Under ideal conditions (uniform gentle slope on an homogeneously erodible substrate), the   sinuosity of a meandering river approaches π. The sinuosity is the ratio between the actual length and the straight-line distance from source to mouth. Faster currents along the outside edges of a river’s bends cause more erosion than along the inside edges, thus pushing the bends even farther out, and increasing the overall loopiness of the river. However, that loopiness eventually causes the river to double back on itself in places and “short-circuit”, creating an  ox-bow lake in the process. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth.[150][151] 5.2

Memorizing digits

Main article: Piphilology Many persons have memorized large numbers of digits of π, a practice called piphilology.[152] One common technique is to memorize a story or poem in which the word lengths represent the digits of π: 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. An early example of a memorization aid, originally devised by English scientist James Jeans, is “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.” [152] When a poem is used, it is sometimes referred to as a  piem. Poems for memorizing π have been composed in several languages in addition to English. [152]

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

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

In the  Palais de la Découverte  (a science museum in Paris) there is a circular room known as the  pi room. On its wall are inscribed 707 digits of π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1853 calculation by English mathematician William Shanks, which included an error beThe error was detected in 1946 The record for memorizing digits of π, certified by ginning at the 528th digit. [160] and corrected in 1949. Guinness World Records , is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 In Carl Sagan's novel Contact  it is suggested that the creMarch 2015. [153] In 2006,   Akira Haraguchi, a retired ator of the universe buried a message deep within the digJapanese engineer, claimed to have recited 100,000 dec- its of π.[161] The digits of π have also been incorporated imal places, but the claim was not verified by Guinness into the lyrics of the song “Pi” from the album  Aerial  by World Records. [154] Record-setting π memorizers typ- Kate Bush,[162] and a song by Hard 'n Phirm.[163] ically do not rely on poems, but instead use methods Many schools in the United States observe  Pi Day on such as remembering number patterns and the  method 14 March (written 3/14 in the US style). [164] π and its of loci.[155] digital representation are often used by self-described A few authors have used the digits of π to establish a new form of   constrained writing, where the word lengths are required to represent the digits of π. The Cadaeic Cadenza contains the first 3835 digits of π in this manner,[156] and the full-lengthbook Not a Wake contains 10,000 words, each representing one digit of π. [157]

“math geeks" for inside jokes among mathematically and technologically minded groups. Several college  cheers at the   Massachusetts Institute of Technology   include “3.14159”.[165] Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi.[166]

14

7 NOTES 

During the 2011 auction for Nortel's portfolio of valuable technology patents,   Google made a series of unusually specific bids based on mathematical and scientific constants, including π. [167]

[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 the Madhava–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; Simon Plouffe (1997), “The Quest for Pi”, The Mathematical Intelligencer  19  (1): 50–56 [6]   “pi”. Dictionary.reference.com. 2 March 1993. Retrieved 18 June 2012. [7]  Arndt & Haenel 2006, p. 8

Some formulas using the 2π definition of τ.

[8]  Tom Apostol (1967),  Calculus, volume 1  (2nd ed.), Wiley. Page 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.” Arc length is introduced on page 529.

In 1958 Albert Eagle proposed replacing π by τ = π/2 to simplify formulas. [168] However, no other authors are known to use τ in this way. Some people use a different value, τ = 6.283185... = 2π, [169] arguing that [9] Reinhold Remmert (1991), “What is π?",   Numbers , Springer, p. 129 τ, as the number of radians in one  turn or as the ratio of a circle’s circumference to its radius rather than [10] Reinhold Remmert (1991), “What is π?",   Numbers , its diameter, is more natural than π and simplifies many Springer, p. 129. ∫  The precise integral that Weierstrass used was formulas.[170][171] Celebrations of this number, because it 1+ 2 approximately equals 6.28, by making 28 June “Tau Day” [11] Richard Baltzer (1870),  Die Elemente der Mathematik , and eating “twice the pie”, [172] have been reported in the Hirzel, p. 195 media. However this use of τ has not made its way into mainstream mathematics. [173] [12] Edmund Landau (1934),  Einführung in die Differential∞

π

=

−∞

dx

x

 .

rechnung und Integralrechnung, Noordoff, p. 193

In 1897, an amateur American mathematician attempted to persuade the Indiana legislature to pass the Indiana Pi [13] Rudin, Walter (1976). Principles of Mathematical Analysis . McGraw-Hill. ISBN 0-07-054235-X., p 183. Bill, which described a method to  square the circle and contained text that implied various incorrect values for π, [14] Reinhold Remmert (1991), “What is π?",   Numbers , including 3.2. The bill is notorious as an attempt to estabSpringer, p. 129 lish a value of scientific constant by legislative fiat. The bill was passed by the Indiana House of Representatives, [15] Rudin, Walter (1986). Real and complex analysis . McGraw-Hill., p 2. 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

[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 MeaRussian Mathematical Survey 53 sure of pi”. (3): 570–572.   Bibcode:2008RuMaS..63..570S. doi:10.1070/RM2008v063n03ABEH004543. [21] Mayer, Steve. “The Transcendence of π". Archived from the original on 2000-09-29. Retrieved 4 November 2007. [22] The polynomial shown is the first few terms of the Taylor series expansion of the sine function.

15 [23]  Posamentier & Lehmann 2004, p. 25

[40] Egyptologist: Rossi, Corinna, Architecture and Mathematics in Ancient Egypt , Cambridge University Press, 2004, [24]  Eymard & Lafon 1999, p. 129 pp 60–70, 200, ISBN 9780521829540. Skeptics:   Shermer, Michael,   The Skeptic Encyclopedia [25]  Beckmann 1989, p. 37 of Pseudoscience , ABC-CLIO, 2002, pp 407–408, ISBN Schlager, Neil; Lauer, Josh (2001).  Science and Its Times:  9781576076538. Understanding the Social Significance of Scientific DiscovSee also Fagan, Garrett G.,   Archaeological Fantasies:  ery. Gale Group. ISBN 0-7876-3933-8., p 185. [26]  Arndt & Haenel 2006, pp. 22–23 Preuss, Paul (23 July 2001). “Are The Digits of Pi Random? Lab Researcher May Hold The Key”.   Lawrence Berkeley National Laboratory. Retrieved 10 November 2007. [27]  Arndt & Haenel 2006, pp. 22, 28–30 [28]  Arndt & Haenel 2006, p. 3 [29]  Eymard & Lafon 1999, p. 78

How Pseudoarchaeology Misrepresents The Past and Misleads the Public , Routledge, 2006, ISBN 9780415305938.

For a list of explanations for the shape that do not involve π, see Roger Herz-Fischler(2000). The Shape of the Great Pyramid . Wilfrid Laurier University Press. pp. 67– 77, 165–166.   ISBN 9780889203242. Retrieved 201306-05. [41]  Arndt & Haenel 2006, p. 167 [42] Chaitanya, Krishna.  A profile of Indian culture.   Indian Book Company (1975). p.133.

[30] "Sloane’s A001203 : Continued fraction for Pi", The On- [43]  Arndt & Haenel 2006, p. 169 Line Encyclopedia of Integer Sequences . OEIS Founda[44]  Arndt & Haenel 2006, p. 170 tion. Retrieved 12 April 2012. [31] Lange, L. J. (May 1999). “An Elegant Continued Frac- [45]  Arndt & Haenel 2006, pp. 175, 205 tion for π". The American Mathematical Monthly 106 (5): [46]  “The Computation of Pi by Archimedes: The Computa456–458. doi:10.2307/2589152. JSTOR 2589152. tion of Pi by Archimedes – File Exchange – MATLAB Central”. Mathworks.com. Retrieved 2013-03-12. [32]  Arndt & Haenel 2006, p. 240 [33]  Arndt & Haenel 2006, p. 242

[47]  Arndt & Haenel 2006, p. 171

[34] Kennedy, E. S., “Abu-r-Raihan al-Biruni, 973- [48]  Arndt & Haenel 2006, p. 176 Boyer & Merzbach 1991, p. 168 1048”,   Journal for the History of Astronomy 9: 65, Bibcode:1978JHA.....9...65K.   Ptolemy   used a three[49]  Arndt & Haenel 2006, pp. 15–16, 175, 184–186, 205. sexagesimal-digit approximation, and Jamshīd al-Kāshī  Grienberger achieved 39 digits in 1630; Sharp 71 digits in expanded this to nine digits; see Aaboe, Asger (1964), 1699. Episodes from the Early History of Mathematics , New Mathematical Library  13, New York: Random House, p. [50]  Arndt & Haenel 2006, pp. 176–177 125. [51] Boyer & Merzbach 1991, p. 202 [35] Petrie, W.M.F. Wisdom of the Egyptians  (1940) [52]  Arndt & Haenel 2006, p. 177 [36] Based on the Great Pyramid of Giza, supposedly built so that the circle whose radius is equal to the height of the [53]  Arndt & Haenel 2006, p. 178 pyramid has a circumference equal to the perimeter of the [54]  Arndt & Haenel 2006, pp. 179 base (it is 1760 cubits around and 280 cubits in height). Verner, Miroslav.   The Pyramids: The Mystery, Culture, [55]  Arndt & Haenel 2006, pp. 180 and Science of Egypt’s Great Monuments.   Grove Press. [56] Azarian, Mohammad K. (2010). “al-Risāla al-muhītīyya: 2001 (1997). ISBN 0-8021-3935-3 A Summary”.  Missouri Journal of Mathematical Sciences  [37] Rossi,  Corinna Architecture and Mathematics in Ancient  22 (2): 64–85. Egypt, Cambridge University Press. 2007. ISBN 978-0[57] O'Connor, John J.; Robertson, Edmund F. (1999). 521-69053-9. “Ghiyath al-Din Jamshid Mas’ud al-Kashi”.   MacTutor  [38] Legon, J. A. R.  On Pyramid Dimensions and Proportions  History of Mathematics archive . Retrieved August 11, (1991) Discussions in Egyptology (20) 25-34 2012. [39] “We can conclude that although the ancient Egyptians could not precisely define the value of π, in practice they used it”. Verner, M. (2003). “The Pyramids: Their Archaeology and History”., p. 70. Petrie (1940). “Wisdom of the Egyptians”., p. 30. SeealsoLegon,J.A.R.(1991). “On Pyramid Dimensions and Proportions”.  Discussions in Egyptology  20: 25–34.. See also Petrie, W. M. F. (1925). “Surveys of the Great Pyramids”.   Nature 116 (2930): 942–942. Bibcode:1925Natur.116..942P. doi:10.1038/116942a0.

[58]  Arndt & Haenel 2006, p. 182 [59]  Arndt & Haenel 2006, pp. 182–183 [60]  Arndt & Haenel 2006, p. 183 [61]   Grienbergerus, Christophorus (1630).  Elementa Trigonometrica   (PDF) (in Latin). Archived from the original (PDF)on 2014-02-01. His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < π < 3.14159 26535 89793 23846 26433 83279 50288 4199.

16

7 NOTES 

[62]  Arndt & Haenel 2006, pp. 185–191

[88]  Arndt & Haenel 2006, p. 197

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

[89]  Arndt & Haenel 2006, pp. 15–17

[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

[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 (4 times). See Borwein & Borwein 1987 for details of algorithms. [94] Bailey, David H. (16 May 2003).   “Some Background on Kanada’s Recent Pi Calculation” (PDF). Retrieved 12 April 2012. [95]  Arndt & Haenel 2006, p. 17. “39 digits of π are sufficient to calculate the volume of the universe to the nearest atom.” Accounting for additional digits needed to compensate for computational  round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. [96]  Arndt & Haenel 2006, pp. 17–19

[76] Borwein, J. M.; Borwein, P. B. (1988). “RaScientific American 256 [97] Schudel, Matt (25 March 2009). “John W. Wrench, Jr.: manujan and Pi”. Mathematician Had a Taste for Pi”.  The Washington Post . (2): 112–117.   Bibcode:1988SciAm.258b.112B. p. B5. doi:10.1038/scientificamerican0288-112. Arndt & Haenel 2006, pp. 15–17, 70–72, 104, 156, [98] Connor, Steve (8 January 2010). “The BigQuestion: How 192–197, 201–202 close have we come to knowing the precise value of pi?". The Independent  (London). Retrieved 14 April 2012. [77]  Arndt & Haenel 2006, pp. 69–72 [78] Borwein, J. M.; Borwein, P. B.; Dilcher, K. (1989). [99]  Arndt & Haenel 2006, p. 18 “Pi, Euler Numbers, and Asymptotic Expansions”. American Mathematical Monthly 96 (8): 681–687. [100]  Arndt & Haenel 2006, pp. 103–104 doi:10.2307/2324715. [101]  Arndt & Haenel 2006, p. 104 [79] Arndt & Haenel 2006, p. 223, (formula 16.10). Note that [102]  Arndt & Haenel 2006, pp. 104, 206 (n − 1)n(n + 1) = n3 − n. Wells, David (1997).  The Penguin Dictionary of Curious  [103]  Arndt & Haenel 2006, pp. 110–111 and Interesting Numbers  (revised ed.). Penguin. p. 35. ISBN 978-0-140-26149-3. [104]  Eymard & Lafon 1999, p. 254 [80]  Posamentier & Lehmann 2004, pp. 284

[105]  Arndt & Haenel 2006, pp. 110–111, 206 Bellard, Fabrice, “Computation of 2700 billion decimal [81] Lambert, Johann, “Mémoire sur quelques propriétés redigits of Pi using a Desktop Computer”, 11 Feb 2010. marquables des quantités transcendantescirculaires et logarithmiques”, reprinted in Berggren, Borwein & Borwein [106]  “Round 2... 10 Trillion Digits of Pi”, NumberWorld.org, 1997, pp. 129–140 17 Oct 2011. Retrieved 30 May 2012. [82]  Arndt & Haenel 2006, p. 196

[107] PSLQ means Partial Sum of Least Squares.

[83]  Arndt & Haenel 2006, p. 165. A facsimile of Jones’ text is in Berggren, Borwein & Borwein 1997, pp. 108–109 [108]   Plouffe, Simon (April 2006).  “Identities inspired by Ramanujan’s Notebooks (part 2)" (PDF). Retrieved 10 April 2009. [84] See Schepler 1950, p. 220: William Oughtred used the letter π to represent the periphery (i.e., circumference) of [109]  Arndt & Haenel 2006, pp. 77–84 a circle. [85]  Arndt & Haenel 2006, p. 166 [86]  Arndt & Haenel 2006, pp. 205

[110] Gibbons, Jeremy, “Unbounded Spigot Algorithms for the Digits of Pi”, 2005. Gibbons produced an improved version of Wagon’s algorithm.

[87]   Arndt & Haenel 2006, p. 197. See alsoReitwiesner 1950. [111]  Arndt & Haenel 2006, p. 77

17 [112] Rabinowitz, Stanley; Wagon, Stan (March 1995). [133] Peitgen, Heinz-Otto, Chaos and fractals: new frontiers of  science, Springer, 2004, pp. 801–803, ISBN 978-0-387“A spigot algorithm for the digits of Pi”.   Amer195–203. 20229-7. ican Mathematical Monthly 102 (3): doi:10.2307/2975006. A computer program has been created that implements Wagon’s spigot algorithm [134] Bronshteĭn & Semendiaev 1971, pp. 191–192 in only 120 characters of software. [135] Bronshteĭn & Semendiaev 1971, p. 190 [113]  Arndt & Haenel 2006, pp. 117, 126–128 [136]  Arndt & Haenel 2006, pp. 41–43 [114]  Bailey, David H.; Borwein, Peter B.; and Plouffe, Simon [137] This theorem was proved by Ernesto Cesàro in 1881. For (April 1997). “On the Rapid Computation of Various a more rigorous proof than the intuitive and informal one Polylogarithmic Constants” (PDF). Mathematics of Comgiven here, see Hardy, G. H., An Introductionto theTheory  putation 66  (218): 903–913. doi:10.1090/S0025-5718of Numbers , Oxford University Press, 2008, ISBN 978-097-00856-9. 19-921986-5, theorem 332. [115]  Arndt & Haenel 2006, p. 128. Plouffe did create a dec- [138] Ogilvy, C. S.; Anderson, J. T., Excursions in Number Theimal digit extraction algorithm, but it is slower than full, ory, Dover Publications Inc., 1988, pp. 29–35, ISBN 0direct computation of all preceding digits. 486-25778-9. [116]  Arndt & Haenel 2006, p. 20 [139]  Arndt & Haenel 2006, p. 43 Bellards formula in: Bellard, Fabrice. “A new formula to compute the nth binary digit of pi”. Archived from the [140] Feller, W. An Introduction to Probability Theory and Its  original on 12 September 2007. Retrieved 27 October Applications, Vol. 1, Wiley, 1968, pp 174–190. 2007. [141] Bronshteĭn & Semendiaev 1971, pp. 106–107, 744, 748 [117] Palmer, Jason (16 September 2010). “Pi record smashed as team finds two-quadrillionth digit”.   BBC News . Re- [142] Halliday, David; Resnick, Robert; Walker, Jearl, Fundamentals of Physics, 5th Ed. , John Wiley & Sons, 1997, p trieved 26 March 2011. 381, ISBN 0-471-14854-7. [118] Bronshteĭn & Semendiaev 1971, pp. 200, 209 [143] Imamura, James M (17 August 2005).  “Heisenberg Un MathWorld  [119]  Weisstein, Eric W., “Semicircle”, . certainty Principle”. University of Oregon. Archived from   the original   on 12 October 2007. Retrieved 9 [120]  Ayers 1964, p. 60 September 2007. [121] Bronshteĭn & Semendiaev 1971, pp. 210–211

[144] Yeo, Adrian, The pleasures of pi, e and other interesting numbers , World Scientific Pub., 2006, p 21, ISBN 978[122]  Arndt & Haenel 2006, p. 39 981-270-078-0. Ehlers, Jürgen, Einstein’s Field Equations and Their Phys[123] Ramaley, J. F. (October 1969). “Buffon’s Noodle Probical Implications , Springer, 2000, p 7, ISBN 978-3-540lem”.  The American Mathematical Monthly  76  (8): 916– 67073-5. 918. doi:10.2307/2317945. JSTOR 2317945. [124]  Arndt & Haenel 2006, pp. 39–40 Posamentier & Lehmann 2004, p. 105 [125]  Arndt & Haenel 2006, pp. 43 Posamentier & Lehmann 2004, pp. 105–108 [126]  Ayers 1964, p. 100 [127] Bronshteĭn & Semendiaev 1971, p. 592

[145] Nave, C. Rod (28 June 2005).   “Coulomb’s Constant”. HyperPhysics . Georgia State University. Retrieved 9 November 2007. [146] C. Itzykson, J-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980. [147] Low, Peter, Classical Theory of Structures Based on the Differential Equation, CUP Archive, 1971, pp 116–118, ISBN 978-0-521-08089-7.

[128] Maor, Eli, E: The Story of a Number , Princeton University Press, 2009, p 160, ISBN 978-0-691-14134-3 (“five most [148] Batchelor, G. K.,   An Introduction to Fluid Dynamics , important” constants). Cambridge University Press, 1967, p 233, ISBN 0-52166396-2. [129]  Weisstein, Eric W., “Roots of Unity”, MathWorld . [149] Bracewell, R. N., The Fourier Transform and Its Applica[130] Weisstein, Eric W.,   “Cauchy Integral Formula”, tions , McGraw-Hill, 2000, ISBN 0-07-116043-4. MathWorld . [150] Hans-Henrik Stølum (22 March 1996). “River Mean[131] Joglekar, S. D., Mathematical Physics , Universities Press, dering as a Self-Organization Process”.   Science 271 2005, p 166, ISBN 978-81-7371-422-1. (5256): 1710–1713.   Bibcode:1996Sci...271.1710S. doi:10.1126/science.271.5256.1710. [132] Klebanoff, Aaron (2001). “Pi in the Mandelbrot set”   (PDF).   Fractals  9 (4): 393–402. [151]  Posamentier & Lehmann 2004, pp. 140–141 doi:10.1142/S0218348X01000828. Retrieved 14 April 2012. [152]  Arndt & Haenel 2006, pp. 44–45

18

7 NOTES 

[153]  “Most Pi Places Memorized”, Guinness World Records. [169] Sequence   A019692, [154] Otake, Tomoko (17 December 2006). “How can anyone [170] Abbott, Stephen (April 2012). “My Conversion remember 100,000 numbers?".   The Japan Times . Reto Tauism”   (PDF).   Math Horizons  19 (4): 34. trieved 27 October 2007. doi:10.4169/mathhorizons.19.4.34. [155] Raz, A.; Packard, M. G. (2009). “A slice of pi: An ex[171] Palais, Robert (2001). "π Is Wrong!"   (PDF). ploratory neuroimaging study of digit encoding and reThe Mathematical Intelligencer  23 (3): 7–8. trieval in a superior memorist”.   Neurocase  15: 361–372. doi:10.1007/BF03026846. doi:10.1080/13554790902776896. PMID 19585350. [156]   Keith, Mike. “Cadaeic Cadenza Notes & Commentary”. [172]  Tau Day: Why you should eat twice the pie – Light Years – CNN.com Blogs Retrieved 29 July 2009. [157] Keith, Michael; Diana Keith (February 17, 2010). Not  [173]  “Life of pi in no danger – Experts cold-shoulder campaign to replace with tau”.  Telegraph India. 2011-06-30. A Wake: A dream embodying (pi)'s digits fully for 10000 decimals . Vinculum Press. ISBN 978-0963009715. [174]  Arndt & Haenel 2006, pp. 211–212 [158] For instance, Pickover calls π “the most famous mathePosamentier & Lehmann 2004, pp. 36–37 matical constant of all time”, and Peterson writes, “Of all Hallerberg, Arthur (May 1977). “Indiana’s squared known mathematical constants, however, pi continues to circle”. Mathematics Magazine 50 (3): 136–140. attract the most attention”, citing theGivenchy π perfume, doi:10.2307/2689499. JSTOR 2689499. Pi (film), and Pi Day as examples. See Pickover, Clifford A. (1995),  Keys to Infinity , Wiley & Sons, p. 59, ISBN 9780471118572;   Peterson, Ivars (2002),  Mathematical  References Treks: From Surreal Numbers to Magic Circles , MAA spectrum, Mathematical Association of America, p. 17,   Arndt, Jörg; Haenel, Christoph (2006).   Pi UnISBN 9780883855379. leashed . Springer-Verlag. ISBN 978-3-540-665724. Retrieved 2013-06-05. English translation by [159]  BBC documentary “The Story of Maths”, second part, Catriona and David Lischka. showing a visualization of the historically first exact formula, starting at 35 min and 20 sec into the second part   Ayers, Frank (1964).   Calculus . McGraw-Hill. of the documentary.

• •

[160]  Posamentier & Lehmann 2004, p. 118 Arndt & Haenel 2006, p. 50 [161]  Arndt & Haenel 2006, p. 14. This part of the story was omitted from the film adaptation of the novel. [162] Gill, Andy (4 November 2005). “Review of Aerial”. The Independent . 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) [163] Board, Josh (1 December 2010).  “PARTY CRASHER: Laughing With Hard 'N Phirm”. SanDiego.com. There was one song about Pi. Nothing like hearing people harmonizing over 200 digits. [164]  Pi Day activities. [165]  MIT cheers. Retrieved 12 April 2012. [166]   “Happy Pi Day! Watch these stunning videos of kids reciting 3.14”. USAToday.com. 2015-03-14. Retrieved 2015-03-14. [167]   “Google’s strange bids for Nortel patents”.   FinancialPost.com. Reuters. 2011-07-05. Retrieved 16 August 2011. [168] Eagle, Albert (1958). The Elliptic Functions as They Should be: An Account, with Applications, of the Functions  in a New Canonical Form. Galloway and Porter, Ltd. p.

ix.

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 and  the AGM: a Study in Analytic Number Theory and  Computational Complexity. Wiley.   ISBN 978-0-

471-31515-5.

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

54397-8.

•   Bronshteĭn, Ilia;

Semendiaev, K. A. (1971). A Guide Book to Mathematics . H. Deutsch. ISBN 9783-871-44095-3.

•  Eymard, Pierre; Lafon, Jean Pierre (1999).

The

Number Pi . American Mathematical Society. ISBN

978-0-8218-3246-2., English translation by Stephen Wilson.

•   Joseph,

George Gheverghese (1991).   The Crest 

of the Peacock: Non-European Roots of Mathematics . 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 Mysterious  Number . Prometheus Books.  ISBN 978-1-59102-

200-8.

• Reitwiesner, George (1950). “An ENIAC Determi-

nation of pi and e to 2000 Decimal Places”.  Mathematical Tables andOther Aids to Computation 4 (29): 11–15. doi:10.2307/2002695.

•   Roy, Ranjan (1990).

“The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha”.  Mathematics Magazine  63  (5): 291–306. doi:10.2307/2690896.

•   Schepler,

H. C. (1950).

“The Chronology of Pi”.   Mathematics Magazine  (Mathematical Association 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 4 Mar/Apr, issue 5 May/Jun

•  Heath, T. L., The Works of Archimedes , Cambridge,

1897; reprinted in  The Works of Archimedes with The Method of Archimedes , Dover, 1953, pp 91–98

•  Huygens, Christiaan, “De Circuli Magnitudine In-

venta”,   Christiani Hugenii Opera Varia I , Leiden 1724, pp 384–388

•   Lay-Yong,

Lam; Tian-Se, Ang (1986). “Circle Measurements in Ancient China”.   Historia Mathematica 13: 325–340. doi:10.1016/03150860(86)90055-8.

•   Lindemann, Ferdinand  (1882).

“Ueber die Zahl pi”. Mathematische Annalen 20: 213–225. doi:10.1007/bf01446522.

• Matar, K. Mukunda; Rajagonal, C. (1944). “On the

Hindu Quadrature of the Circle” (Appendix by K. Balagangadharan)".  Journal of the Bombay Branch of the Royal Asiatic Society 20: 77–82.

•  Niven, Ivan, “A Simple Proof that pi Is Irrational”,

Bulletin of the American Mathematical Society,  53:7

8

(July 1947), 507

Further reading

•  Blatner, David (1999).   The Joy of Pi .

Walker &

Company.  ISBN 978-0-8027-7562-7.

•  Borwein, Jonathan;  Borwein, Peter   (1984).

“The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions”.  SIAM Review  26 : 351– 365. doi:10.1137/1026073.

•  Borwein, Jonathan; Borwein, Peter;  Bailey, David

H. (1989). “Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi”. The American Mathematical  Monthly 96: 201–219. doi:10.2307/2325206.

•   Ramanujan,

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

• Shanks,

William,   Contributions to Mathematics 

Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals , 1853, pp. i–xvi, 10

•   Shanks,

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

•  Chudnovsky, David V.  and  Chudnovsky, Gregory

•   Tropfke,

•  Cox, David A., “The Arithmetic-Geometric Mean

•   Viete, Francois,   Variorum de Rebus Mathematicis 

•  Delahaye, Jean-Paul, “Le Fascinant Nombre Pi”,

•  Wagon, Stan, “Is Pi Normal?",  The Mathematical 

V., “Approximations and Complex Multiplication According to Ramanujan”, in  Ramanujan Revisited  (G.E. Andrews et al. Eds), Academic Press, 1988, pp 375–396, 468–472 of Gauss”,  L' Ensignement Mathematique, 30(1984) 275–330 Paris: Bibliothèque Pour la Science (1997)  ISBN 2902918259

• Engels, Hermann (1977). “Quadrature of the Circle

in Ancient Egypt”.   Historia Mathematica  4 : 137– 140. doi:10.1016/0315-0860(77)90104-5.

•   Euler, Leonhard, “On the Use of the Discovered

Fractions to Sum Infinite Series”, in  Introduction to Analysis of the Infinite. Book I , translated from the Latin by J. D. Blanton, Springer-Verlag, 1964, pp 137–153

Johannes,   Geschichte Der ElementarMathematik in Systematischer Darstellung  ( The history of elementary mathematics ), BiblioBazaar, 2009 (reprint), ISBN 978-1-113-08573-3 Reponsorum Liber VII. F. Viete, Opera Mathematica

(reprint), Georg Olms Verlag, 1970, pp 398–401, 436–446 Intelligencer , 7:3(1985) 65–67

•   Wallis,

John,   Arithmetica Infinitorum, sive Nova

Methodus Inquirendi in Curvilineorum Quadratum, aliaque difficiliora Matheseos Problemata, Oxford 1655–6. Reprinted in vol. 1 (pp 357–478) of Opera Mathematica , Oxford 1693

•  Zebrowski, Ernest,  A History of the Circle: Mathematical Reasoning and the Physical Universe , Rut-

gers University Press, 1999,   ISBN 978-0-81352898-4

20

9

9 EXTERNAL LINKS 

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 – Numberphile”.  Numberphile. Brady Haran.

• Demonstrationby Lambert (1761) of irrationalityof π, 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, J ohncatsoulis, BorisFromStockdale, Attys, JzG, Dbtfz, Kuru, Titus III, Richard L. 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10.2 Images 

23

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10.2 •

Images

  File:Archimedes_pi.svg  Source:   https://upload.wikimedia.org/wikipedia/commons/c/c9/Archimedes_pi.svg  License:    CC-BY-SA-3.0

Contributors:  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-

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  Buffon_needle.gif  Original artist:   Buffon_needle.gif: Claudio Rocchini

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  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:  ?  Original 

artist:  ?

  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

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  File:Comparison_pi_infinite_series.svg   Source: 

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  File:Domenico-Fetti_Archimedes_1620.jpg   Source: 

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  File:E\char"005E\relax{}(-x\char"005E\relax{}2).svg Source:   https://upload.wikimedia.org/wikipedia/commons/2/2f/E%5E%28-x%

  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.htm Original artist:   Domenico Fetti 5E2%29.svg License:  CC BY-SA 3.0  Contributors:  Own work  Original artist:   Autopilot

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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 in Inkscape by Wereon, italics fixed by lasindi.   https://upload.wikimedia.org/wikipedia/commons/3/39/   Source:  GodfreyKneller-IsaacNewton-1689.jpg   License:    Public domain   Contributors:    http://www.newton.cam.ac.uk/art/portrait.html   Original artist:   Sir Godfrey Kneller

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  File:GodfreyKneller-IsaacNewton-1689.jpg

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  File:JohnvonNeumann-LosAlamos.gif

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  File:Leonhard_Euler.jpg  Source:  https://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg  License:  Public domain

  https://upload.wikimedia.org/wikipedia/commons/5/5e/   Source:  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 Contributors: 

2. Kunstmuseum Basel Original artist:  Jakob Emanuel Handmann   https://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_ mandelbrot_set.jpg License:  CC-BY-SA-3.0  Contributors:  ?  Original artist:  ?

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  File:Mandel_zoom_00_mandelbrot_set.jpg   Source: 

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  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 •

  File:Nuvola_apps_edu_mathematics_blue-p.svg  Source:    https://upload.wikimedia.org/wikipedia/commons/3/3e/Nuvola_apps_edu_

mathematics_blue-p.svg  License:  GPL  Contributors:  Derivative work from Image:Nuvola apps edu mathematics.png  and Image:Nuvola apps edu mathematics-p.svg Original artist:  David Vignoni (original icon); Flamurai (SVG convertion); bayo (color) •

  File:OEISicon_light.svg  Source:   https://upload.wikimedia.org/wikipedia/commons/d/d8/OEISicon_light.svg  License:    Public

domain

Contributors:  Own work  Original artist:   Watchduck  (a.k.a. Tilman Piesk) •

  File:Pi_30K.gif Source:  https://upload.wikimedia.org/wikipedia/commons/8/84/Pi_30K.gif License:  CCBY 3.0 Contributors:   This math-

ematical image was created with Mathematica Original artist:  CaitlinJo •

  File:Pi_eq_C_over_d.svg Source:   https://upload.wikimedia.org/wikipedia/commons/3/36/Pi_eq_C_over_d.svg  License:  Public domain

Contributors:  Own work  Original artist:   Kjoonlee, based on previous work by w:User:Papeschr

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10 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

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  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.svg License:  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   File:Squaring_the_circle.svg  Source:   https://upload.wikimedia.org/wikipedia/commons/a/a7/Squaring_the_circle.svg  License:   Public domain  Contributors:  Pd-self image by Plynn9  Original artist:  Original PNG by Plynn9; SVG by Alexei Kouprianov   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 work Original artist:   Helloholabonjournihaonamastegutentag

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