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Part of a series of articles on the 
mathematical constant π 

File:Piunrolled720.gif 
Uses 
Properties 
Value 
People 
History 
In culture 
Related topics 
The number π is a mathematical constant, the ratio of a circle's circumference to its diameter, commonly approximated as 3.14159. It has been represented by the Greek letter "π" since the mid18th century, though it is also sometimes spelled out as "pi" (/paÉª/).
Being an irrational number, π cannot be expressed exactly as a common fraction, although fractions such as 22/7 and other rational numbers are commonly used to approximate π. Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed; however, to date, no proof of this has been discovered. Also, π is a transcendental number â€“ a number that is not the root of any nonzero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
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 techniques, in Chinese mathematics and to about five in Indian mathematics in the 5th century CE. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhavaâ€“Leibniz series was discovered in Indian mathematics.^{[1]}^{[2]} In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to, as of late 2013, over 13.3 trillion (10^{13}) digits.^{[3]} Scientific applications generally require no more than 40 digits of π so the primary motivation for these computations is the human desire to break records. However, the extensive calculations involved have been used to test supercomputers and highprecision multiplication algorithms.
Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses or spheres. It is also found in formulae used in other branches of science such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics and electromagnetism. The ubiquity of π makes it one of the most widely known mathematical constants both inside and outside the scientific community: Several books devoted to it have been published, the number is celebrated on Pi Day and recordsetting calculations of the digits of π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 67,000 digits.
Fundamentals
Name
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase Greek letter π, sometimes spelled out as pi. In English, π is pronounced as "pie" ( /paÉª/, paÉª).^{[4]} In mathematical use, the lowercase letter π (or Ï€ in sansserif font) is distinguished from its capital counterpart Π, which denotes a product of a sequence.
The choice of the symbol π is discussed in the section Adoption of the symbol Ï€.
Definition
π is commonly defined as the ratio of a circle's circumference C to its diameter d:^{[5]}
 <math> \pi = \frac{C}{d}</math>
The ratio C/d 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 C/d. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curved (nonEuclidean) geometry, these new circles will no longer satisfy the formula π = C/d.^{[5]} There are also other definitions of π that do not immediately involve circles at all. For example, π is twice the smallest positive x for which cos(x) equals 0.^{[5]}^{[6]}
Properties
π is an irrational number, meaning that it cannot be written as the ratio of two integers (fractions such as 22/7 are commonly used to approximate π; no common fraction (ratio of whole numbers) can be its exact value).^{[7]} Since π is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality 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 numbers.^{[8]}
More strongly, π is a transcendental number, which means that it is not the solution of any nonconstant polynomial with rational coefficients, such as x^{5}/120 − x^{3}/6 + x = 0.^{[9]}^{[10]} The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or nth roots such as ^{3}√31 or √10. Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.^{[11]} Squaring a circle was one of the important geometry problems of the classical antiquity.^{[12]} Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is impossible.^{[13]}
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.^{[14]} The conjecture that π is normal has not been proven or disproven.^{[14]} 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.^{[15]} Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear nonrandom to nonmathematicians, 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.^{[16]}
Continued fractions
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:
 <math>
\pi=3+\textstyle \frac{1}{7+\textstyle \frac{1}{15+\textstyle \frac{1}{1+\textstyle \frac{1}{292+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\textstyle \frac{1}{1+\ddots}}}}}}}</math> OEIS A001203
Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22/7, 333/106, and 355/113. These numbers are among the most wellknown 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.^{[17]} 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,^{[18]} mathematicians have discovered several generalized continued fractions that do, such as:^{[19]}
 <math>\pi=\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}}
=3+\textstyle \frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}} =\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{3+\textstyle \frac{2^2}{5+\textstyle \frac{3^2}{7+\textstyle \frac{4^2}{9+\ddots}}}}}</math>
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.^{[17]} (List is selected terms from OEIS A063674 and OEIS A063673.)
 Decimal: The first 50 decimal digits are Script error: No such module "Gaps".^{[20]} OEIS A000796
 Binary: The base 2 approximation to 48 digits is Script error: No such module "Gaps".
 Hexadecimal: The base 16 approximation to 20 digits is Script error: No such module "Gaps".^{[21]}
 Sexagesimal: A base 60 approximation to five sexagesimal digits is 3;8,29,44,0,47^{[22]}
History
Antiquity
The best known approximations to π dating to before the Common Era were accurate to two decimal places; this 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^{[23]} have claimed that the ancient Egyptians used an approximation of π as ^{22}⁄_{7} from as early as the Old Kingdom.^{[24]} This claim has met with skepticism.^{[25]}^{[26]}^{[27]}^{[28]}
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 1900â€“1600 BC has a geometrical statement that, by implication, treats π as ^{28}⁄_{5} = 3.1250.^{[29]} 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} â‰ˆ 3.1605.^{[29]}
Astronomical calculations in the Shatapatha Brahmana (ca. 4th century BC) use a fractional approximation of ^{339}⁄_{108} â‰ˆ 3.139 (an accuracy of 9Ã—10^{âˆ’4}).^{[30]} Other Indian sources by about 150 BC treat π as √10 â‰ˆ 3.1622^{[31]}
Polygon approximation era
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.^{[32]} This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant".^{[33]} 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 96sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (that is 3.1408 < π < 3.1429).^{[34]} Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.^{[35]} Around 150 AD, GreekRoman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.^{[36]} 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.^{[37]}
In ancient China, values for π included 3.1547 (around 1 AD), √10 (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).^{[38]} Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygonbased iterative algorithm and used it with a 3,072sided polygon to obtain a value of π of 3.1416.^{[39]}^{[40]} Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.^{[39]} The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π â‰ˆ 355/113 (a fraction that goes by the name MilÃ¼ in Chinese), using Liu Hui's algorithm applied to a 12,288sided 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.^{[41]}
The Indian astronomer Aryabhata used a value of 3.1416 in his Ä€ryabhaá¹Ä«ya (499 AD).^{[42]} Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.^{[43]} Italian author Dante apparently employed the value 3+√2/10 â‰ˆ 3.14142.^{[43]}
The Persian astronomer JamshÄ«d alKÄshÄ« produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3Ã—2^{28} sides,^{[44]}^{[45]} which stood as the world record for about 180 years.^{[46]} French mathematician FranÃ§ois ViÃ¨te in 1579 achieved 9 digits with a polygon of 3Ã—2^{17} sides.^{[46]} Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.^{[46]} In 1596, Dutch mathematician Ludolph 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).^{[47]} Dutch scientist Willebrord Snellius reached 34 digits in 1621,^{[48]} and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10^{40} sides,^{[49]} which remains the most accurate approximation manually achieved using polygonal algorithms.^{[48]}
Infinite series
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.^{[50]} Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques.^{[50]} 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.^{[51]} The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.^{[52]} The series are presented without proof, but proofs are presented in a later Indian work, YuktibhÄá¹£Ä, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 â€“ c. 1425.^{[52]} Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregoryâ€“Leibniz series.^{[52]} Madhava used infinite series to estimate π to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician JamshÄ«d alKÄshÄ«, using a polygonal algorithm.^{[53]}
The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician FranÃ§ois ViÃ¨te in 1593:^{[55]}
 <math> \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots</math> OEIS A060294
The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.^{[55]} 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."^{[54]}
In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregory in 1671, and by Leibniz in 1674:^{[56]}^{[57]}
 <math>
\arctan z = z  \frac {z^3} {3} +\frac {z^5} {5} \frac {z^7} {7} +\cdots </math>
This formula, the Gregoryâ€“Leibniz series, equals Ï€/4 when evaluated with z = 1.^{[57]} In 1699, English mathematician Abraham Sharp used the Gregoryâ€“Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.^{[58]} The Gregoryâ€“Leibniz series is simple, but converges very slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.^{[59]}
In 1706 John Machin used the Gregoryâ€“Leibniz series to produce an algorithm that converged much faster:^{[60]}
 <math> \frac{\pi}{4} = 4 \, \arctan \frac{1}{5}  \arctan \frac{1}{239}</math>
Machin reached 100 digits of π with this formula.^{[61]} Other mathematicians created variants, now known as Machinlike formulae, that were used to set several successive records for calculating digits of π.^{[61]} Machinlike formulae remained the bestknown method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620digit approximation in 1946 by Daniel Ferguson â€“ the best approximation achieved without the aid of a calculating device.^{[62]}
A record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machinlike formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss.^{[63]} British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.^{[63]}
Rate of convergence
Some infinite series for π converge faster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.^{[64]} A simple infinite series for π is the Gregoryâ€“Leibniz series:^{[65]}
 <math> \pi = \frac{4}{1}  \frac{4}{3} + \frac{4}{5}  \frac{4}{7} + \frac{4}{9}  \frac{4}{11} + \frac{4}{13}  \cdots</math>
As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and â€“ with a sufficient number of terms â€“ can get as close to π as desired. It converges quite slowly, though â€“ after 500,000 terms, it produces only five correct decimal digits of π.^{[66]}
An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregoryâ€“Leibniz series is:^{[67]}
 <math> \pi = 3 + \frac{4}{2\times3\times4}  \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8}  \frac{4}{8\times9\times10} + \cdots</math>
The following table compares the convergence rates of these two series:
Infinite series for π  After 1st term  After 2nd term  After 3rd term  After 4th term  After 5th term  Converges to: 

<math>\pi = \frac{4}{1}  \frac{4}{3} + \frac{4}{5}  \frac{4}{7} + \frac{4}{9}  \frac{4}{11} + \frac{4}{13} \cdots.</math>  4.0000  2.6666...  3.4666...  2.8952...  3.3396...  π = 3.1415... 
<math>\pi = Unexpected use of template {{3}}  see Template:3 for details. + \frac Unexpected use of template {{4}}  see Template:4 for details.{2\times3\times4}  \frac Unexpected use of template {{4}}  see Template:4 for details.{4\times5\times6} + \frac Unexpected use of template {{4}}  see Template:4 for details.{6\times7\times8} \cdots. </math> 
3.0000  3.1666...  3.1333...  3.1452...  3.1396... 
After five terms, the sum of the Gregoryâ€“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.^{[64]}
Irrationality and transcendence
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:^{[68]}
 <math> \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots</math>
Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.^{[7]} Lambert's proof exploited a continuedfraction representation of the tangent function.^{[69]} French mathematician AdrienMarie Legendre proved in 1794 that π^{2} is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.^{[70]}
Adoption of the symbol π
The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.^{[71]} 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.^{[72]} 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.^{[73]} It had indeed been used earlier for geometric concepts.^{[73]} William Oughtred used π and Î´, 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.^{[73]} Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.^{[73]} 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 circle of radius 1") and the practice was universally adopted thereafter in the Western world.^{[73]}
Modern quest for more digits
Computer era and iterative algorithms
The Gaussâ€“Legendre iterative algorithm:
Initialize
 <math>\scriptstyle a_0 = 1 \quad b_0 = \frac{1}{\sqrt 2} \quad t_0 = \frac{1}{4} \quad p_0 = 1</math>
Iterate
 <math>\scriptstyle a_{n+1} = \frac{a_n+b_n}{2} \quad \quad b_{n+1} = \sqrt{a_n b_n}</math>
 <math>\scriptstyle t_{n+1} = t_n  p_n (a_na_{n+1})^2 \quad \quad p_{n+1} = 2 p_n</math>
Then an estimate for π is given by
 <math>\scriptstyle \pi \approx \frac{(a_n + b_n)^2}{4 t_n}</math>
The development of computers in the mid20th 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.^{[74]} 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 70 hours of computer time on the ENIAC computer.^{[75]} The record, 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 in 1973.^{[76]}
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.^{[77]} Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.^{[78]} They include the Karatsuba algorithm, Toomâ€“Cook multiplication, and Fourier transformbased methods.^{[79]}
The iterative algorithms were independently published in 1975â€“1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.^{[80]} 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 arithmeticâ€“geometric mean method (AGM method) or Gaussâ€“Legendre algorithm.^{[80]} As modified by Salamin and Brent, it is also referred to as the Brentâ€“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 generally multiply the number of correct digits at each step. For example, the BrentSalamin 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.^{[81]} Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing π between 1995 and 2002.^{[82]} This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.^{[82]}
Motivations for computing π
For most numerical calculations involving π, a handful of digits provide sufficient precision. According to JÃ¶rg Arndt and Christoph Haenel, thirtynine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.^{[83]} Despite this, people have worked strenuously to compute π to thousands and millions of digits.^{[84]} This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.^{[85]}^{[86]} They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including highprecision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.^{[87]}
Rapidly convergent series
Modern π 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.^{[82]} The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.^{[88]} One of his formulae, based on modular equations, is
 <math>\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{k!^4(396^{4k})}.</math>
This series converges much more rapidly than most arctan series, including Machin's formula.^{[89]} Bill Gosper was the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.^{[90]} Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.^{[91]} The Chudnovsky formula developed in 1987 is
 <math>\frac{1}{\pi} = \frac{12}{640320^{3/2}} \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (640320)^{3k}}.</math>
It produces about 14 digits of π per term,^{[92]} and has been used for several recordsetting π calculations, including the first to surpass 1 billion (10^{9}) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7Ã—10^{12}) digits by Fabrice Bellard in 2009, and 10 trillion (10^{13}) digits in 2011 by Alexander Yee and Shigeru Kondo.^{[93]}^{[3]} For similar formulas, see also the Ramanujanâ€“Sato series.
In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm^{[94]} to generate several new formulas for π, conforming to the following template:
 <math>\pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{a}{q^n1} + \frac{b}{q^{2n}1} + \frac{c}{q^{4n}1}\right),</math>
where q is e^{π} (Gelfond's constant), k is an odd number, and a, b, c are certain rational numbers that Plouffe computed.^{[95]}
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.^{[96]}^{[97]} This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.^{[96]}
American mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995.^{[97]}^{[98]}^{[99]} Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.^{[98]}
Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe:^{[100]}^{[101]}
 <math> \pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1}  \frac{2}{8k + 4}  \frac{1}{8k + 5}  \frac{1}{8k + 6}\right)</math>
This formula, unlike others before it, can produce any individual hexadecimal digit of π without calculating all the preceding digits.^{[100]} 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.^{[102]} 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.^{[3]}
Between 1998 and 2000, the distributed computing project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (10^{15}th) bit of π, which turned out to be 0.^{[103]} In September 2010, a Yahoo! employee used the company's Hadoop application on one thousand computers over a 23day period to compute 256 bits of π at the twoquadrillionth (2Ã—10^{15}th) bit, which also happens to be zero.^{[104]}
Use
Because π is closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Formulae from other branches of science also include π in some of their important formulae, including sciences such as statistics, fractals, thermodynamics, mechanics, cosmology, number theory, and electromagnetism.
Geometry and trigonometry
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.^{[105]}
 The circumference of a circle with radius r is 2Ï€r.
 The area of a circle with radius r is Ï€r^{2}.
 The volume of a sphere with radius r is 4/3Ï€r^{3}.
 The surface area of a sphere with radius r is 4Ï€r^{2}.
The formulae above are special cases of the surface area S_{n}(r) and volume V_{n}(r) of an ndimensional sphere.
<math>S_n(r) = \frac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^{n1}</math>
<math>V_n(r) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^n</math>
π appears in definite integrals that describe circumference, area, or volume of shapes generated by circles. For example, an integral that specifies half the area of a circle of radius one is given by:^{[106]}
 <math>\int_{1}^1 \sqrt{1x^2}\,dx = \frac{\pi}{2}.</math>
In that integral the function √1x^{2} represents the top half of a circle (the square root is a consequence of the Pythagorean theorem), and the integral Template:Intmath computes the area between that half of a circle and the x axis.
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.^{[107]} The angle measure of 180Â° is equal to π radians, and 1Â° = π/180 radians.^{[107]}
Common trigonometric functions have periods that are multiples of π; for example, sine and cosine have period 2π,^{[108]} so for any angle Î¸ and any integer k, <math> \sin\theta = \sin\left(\theta + 2\pi k \right)</math> and <math> \cos\theta = \cos\left(\theta + 2\pi k \right).</math>^{[108]}
Monte Carlo methods
Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of π.^{[109]} Buffon's needle is one such technique: If a needle of length â„“ 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:^{[110]}
 <math>\pi \approx \frac{2n\ell}{xt}</math>
Another Monte Carlo method for computing π 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 equal Ï€/4.^{[111]}
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.^{[112]}
Complex numbers and analysis
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 Ï†) to represent a counterclockwise rotation from the positive real line as follows:^{[113]}
 <math>z = r\cdot(\cos\varphi + i\sin\varphi),</math>
where i is the imaginary unit satisfying i^{2} = âˆ’1. The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula:^{[114]}
 <math>e^{i\varphi} = \cos \varphi + i\sin \varphi,</math>
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 Ï† = π in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:^{[114]}^{[115]}
 <math>e^{i \pi} + 1 = 0.</math>
There are n different complex numbers z satisfying z^{n} = 1, and these are called the "nth roots of unity".^{[116]} They are given by this formula:
 <math>e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n  1).</math>
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:^{[117]}^{[118]}
 <math>f (z_{0}) = \frac{1}{ 2\pi i } \oint_\gamma { f(z) \over zz_0 }\,dz</math>
An occurrence of π in the Mandelbrot set fractal was discovered by American David Boll in 1991.^{[119]} He examined the behavior of the Mandelbrot set near the "neck" at (âˆ’0.75, 0). If points with coordinates (âˆ’0.75, Îµ) are considered, as Îµ tends to zero, the number of iterations until divergence for the point multiplied by Îµ converges to π. The point (0.25, Îµ) 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 of Îµ tends to π.^{[119]}^{[120]}
The gamma function extends the concept of factorial (normally defined only for nonnegative integers) to all complex numbers, except the negative real integers. When the gamma function is evaluated at halfintegers, the result contains π; for example <math> \Gamma(1/2) = \sqrt{\pi} </math> and <math>\Gamma(5/2) = \frac {3 \sqrt{\pi}} {4} </math>.^{[121]} The gamma function can be used to create a simple approximation to n! for large n: <math> n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n</math> which is known as Stirling's approximation.^{[122]}
Number theory and Riemann zeta function
The Riemann zeta function Î¶(s) is used in many areas of mathematics. When evaluated at s = 2 it can be written as
 <math> \zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots</math>
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 Ï€^{2}/6.^{[68]} 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 to 6/Ï€^{2}.^{[123]}^{[124]} This probability is based on the observation that the probability that any number is divisible by a prime p is 1/p (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is 1/p^{2}, and the probability that at least one of them is not is 11/p^{2}. 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:^{[125]}
 <math>\prod_p^{\infty} \left(1\frac{1}{p^2}\right) = \left( \prod_p^{\infty} \frac{1}{1p^{2}} \right)^{1} = \frac{1}{1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots } = \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 61\% </math>
This probability can be used in conjunction with a random number generator to approximate π using a Monte Carlo approach.^{[126]}
Probability and statistics
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.^{[127]} π is found in the Gaussian function (which is the probability density function of the normal distribution) with mean Î¼ and standard deviation Ïƒ:^{[128]}
 <math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{(x\mu )^2/(2\sigma^2)}</math>
The area under the graph of the normal distribution curve is given by the Gaussian integral:^{[128]}
 <math>\int_{\infty}^\infty e^{x^2} \, dx=\sqrt{\pi},</math>
while the related integral for the Cauchy distribution is
 <math>\int_{\infty }^{\infty } \frac{1}{x^2+1} \, dx = \pi.</math>
Outside mathematics
Describing physical phenomena
Although not a physical constant, π appears routinely in equations describing fundamental principles of the universe, often because of π's relationship to the circle and to spherical coordinate systems. A simple formula from the field of classical mechanics gives the approximate period T of a simple pendulum of length L, swinging with a small amplitude (g is the earth's gravitational acceleration):^{[129]}
 <math>T \approx 2\pi \sqrt\frac{L}{g}.</math>
One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Î”x) and momentum (Î”p) cannot both be arbitrarily small at the same time (where h is Planck's constant):^{[130]}
 <math> \Delta x\, \Delta p \ge \frac{h}{4\pi}.</math>
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:^{[131]}
 <math> R_{ik}  {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik},</math>
where R_{ik} is the Ricci curvature tensor, R is the scalar curvature, g_{ik} is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and T_{ik} is the stressâ€“energy tensor.
Coulomb's law, from the discipline of electromagnetism, describes the electric field between two electric charges (q_{1} and q_{2}) separated by distance r (with Îµ_{0} representing the vacuum permittivity of free space):^{[132]}
 <math> F = \frac{\leftq_1q_2\right}{4 \pi \varepsilon_0 r^2}.</math>
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 Î± is^{[133]}
 <math>\frac{1}{\tau} = 2\frac{\pi^2  9}{9\pi}m\alpha^{6},</math>
where m is the mass of the electron.
π is present in some structural engineering formulae, such as the buckling formula derived by Euler, which gives the maximum axial load F that a long, slender column of length L, modulus of elasticity E, and area moment of inertia I can carry without buckling:^{[134]}
 <math>F =\frac{\pi^2EI}{L^2}.</math>
The field of fluid dynamics contains π in Stokes' law, which approximates the frictional force F exerted on small, spherical objects of radius R, moving with velocity v in a fluid with dynamic viscosity Î·:^{[135]}
 <math>F =6 \, \pi \, \eta \, R \, v .</math>
The Fourier transform, defined below, is a mathematical operation that expresses time as a function of frequency, known as its frequency spectrum. It has many applications in physics and engineering, particularly in signal processing.^{[136]}
 <math>\hat{f}(\xi) = \int_{\infty}^{\infty} f(x)\ e^{ 2\pi i x \xi}\,dx</math>
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 straightline 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 "shortcircuit", creating an oxbow 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.^{[137]}^{[138]}
Memorizing digits
Many persons have memorized large numbers of digits of π, a practice called piphilology.^{[139]} 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."^{[139]} 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.^{[139]}
The record for memorizing digits of π, certified by Guinness World Records, is 67,890 digits, recited in China by Lu Chao in 24 hours and 4 minutes on 20 November 2005.^{[140]}^{[141]} 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.^{[142]} Recordsetting π memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the method of loci.^{[143]}
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,^{[144]} and the fulllength book Not a Wake contains 10,000 words, each representing one digit of π.^{[145]}
In popular culture
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.^{[146]}
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 the Ï€ when visiting India and exploring its contributions to trigonometry.^{[147]}
In the Palais de la DÃ©couverte (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 domelike ceiling. The digits were based on an 1853 calculation by English mathematician William Shanks, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.^{[148]}
In Carl Sagan's novel Contact it is suggested that the creator of the universe buried a message deep within the digits of π.^{[149]} The digits of π have also been incorporated into the lyrics of the song "Pi" from the album Aerial by Kate Bush,^{[150]} and a song by Hard 'n Phirm.^{[151]}
Many schools in the United States observe Pi Day on 14 March (written 3/14 in the US style).^{[152]} π and its digital representation are often used by selfdescribed "math geeks" for inside jokes among mathematically and technologically minded groups. Several college cheers at the Massachusetts Institute of Technology include "3.14159".^{[153]} Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi.^{[154]}
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.^{[155]}
In 1958 Albert Eagle proposed replacing π by τ = π/2 to simplify formulas.^{[156]} However, no other authors are known to use tau in this way. Some people use a different value for tau, τ = 6.283185... = 2π,^{[157]} arguing that τ, as the number of radians in one turn or as the ratio of a circle's circumference to its radius rather than its diameter, is more natural than π and simplifies many formulas.^{[158]}^{[159]} Celebrations of this number, because it approximately equals 6.28, by making 28 June "Tau Day" and eating "twice the pie",^{[160]} have been reported in the media. However this use of Ï„ has not made its way into mainstream mathematics.^{[161]}
In 1897, an amateur American mathematician attempted to persuade the Indiana legislature to pass the Indiana Pi Bill, which described a method to square the circle and contained text that implied various incorrect values for π, including 3.2. The bill is notorious as an attempt to establish a value of scientific constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate.^{[162]}
See also
 Chronology of computation of Ï€
 Proof that Ï€ is irrational
 Proof that Ï€ is transcendental
 Mathematical constants and functions
 Approximations of Ï€
Notes
 Footnotes
 ^ George E. Andrews, Richard Askey, Ranjan Roy (1999). Special Functions. Cambridge University Press. p. 58. ISBN 0521789885.
 ^ Gupta, R. C. (1992). "On the remainder term in the Madhavaâ€“Leibniz's series". Ganita Bharati 14 (14): 68â€“71.
 ^ ^{a} ^{b} ^{c} "Round 2... 10 Trillion Digits of Pi", NumberWorld.org, 17 Oct 2011. Retrieved 30 May 2012.
 ^ "pi". Dictionary.reference.com. 2 March 1993. Retrieved 18 June 2012.
 ^ ^{a} ^{b} ^{c} Arndt & Haenel 2006, p. 8
 ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGrawHill. ISBN 007054235X., p 183.
 ^ ^{a} ^{b} Arndt & Haenel 2006, p. 5
 ^ Salikhov, V. (2008). "On the Irrationality Measure of pi". Russian Mathematical Survey 53 (3): 570â€“572. Bibcode:2008RuMaS..63..570S. doi:10.1070/RM2008v063n03ABEH004543.
 ^ Mayer, Steve. "The Transcendence of π". Archived from the original on 20000929. Retrieved 4 November 2007.
 ^ The polynomial shown is the first few terms of the Taylor series expansion of the sine function.
 ^ Posamentier & Lehmann 2004, p. 25
 ^ Eymard & Lafon 1999, p. 129
 ^ Beckmann 1989, p. 37
Schlager, Neil; Lauer, Josh (2001). Science and Its Times: Understanding the Social Significance of Scientific Discovery. Gale Group. ISBN 0787639338., p 185.  ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 22â€“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.  ^ Arndt & Haenel 2006, pp. 22, 28â€“30
 ^ Arndt & Haenel 2006, p. 3
 ^ ^{a} ^{b} Eymard & Lafon 1999, p. 78
 ^ "Sloane's A001203 : Continued fraction for Pi", The OnLine Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 April 2012.
 ^ Lange, L. J. (May 1999). "An Elegant Continued Fraction for π". The American Mathematical Monthly 106 (5): 456â€“458. JSTOR 2589152. doi:10.2307/2589152.
 ^ Arndt & Haenel 2006, p. 240
 ^ Arndt & Haenel 2006, p. 242
 ^ Kennedy, E. S., "AburRaihan alBiruni, 9731048", Journal for the History of Astronomy 9: 65, Bibcode:1978JHA.....9...65K. Ptolemy used a threesexagesimaldigit approximation, and JamshÄ«d alKÄshÄ« expanded this to nine digits; see Aaboe, Asger (1964), Episodes from the Early History of Mathematics, New Mathematical Library 13, New York: Random House, p. 125.
 ^ Petrie, W.M.F. Wisdom of the Egyptians (1940)
 ^ Based on the Great Pyramid of Giza, supposedly built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base (it is 1760 cubits around and 280 cubits in height). Verner, Miroslav. The Pyramids: The Mystery, Culture, and Science of Egypt's Great Monuments. Grove Press. 2001 (1997). ISBN 0802139353
 ^ Rossi, Corinna Architecture and Mathematics in Ancient Egypt, Cambridge University Press. 2007. ISBN 9780521690539.
 ^ Legon, J. A. R. On Pyramid Dimensions and Proportions (1991) Discussions in Egyptology (20) 2534 [1]
 ^ "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.
See also Legon, J. A. R. (1991). "On Pyramid Dimensions and Proportions". Discussions in Egyptology 20: 25â€“34..
See also Petrie, W. M. F. (1925). "Surveys of the Great Pyramids". Nature Journal 116 (2930): 942â€“942. Bibcode:1925Natur.116..942P. doi:10.1038/116942a0.  ^ Egyptologist: Rossi, Corinna, Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2004, pp 60â€“70, 200, ISBN 9780521829540.
Skeptics: Shermer, Michael, The Skeptic Encyclopedia of Pseudoscience, ABCCLIO, 2002, pp 407â€“408, ISBN 9781576076538.
See also Fagan, Garrett G., Archaeological Fantasies: 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 HerzFischler (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. pp. 67â€“77, 165â€“166. ISBN 9780889203242. Retrieved 20130605.  ^ ^{a} ^{b} Arndt & Haenel 2006, p. 167
 ^ Chaitanya, Krishna. A profile of Indian culture. Indian Book Company (1975). p.133.
 ^ Arndt & Haenel 2006, p. 169
 ^ Arndt & Haenel 2006, p. 170
 ^ Arndt & Haenel 2006, pp. 175, 205
 ^ "The Computation of Pi by Archimedes: The Computation of Pi by Archimedes â€“ File Exchange â€“ MATLAB Central". Mathworks.com. Retrieved 20130312.
 ^ Arndt & Haenel 2006, p. 171
 ^ Arndt & Haenel 2006, p. 176
Boyer & Merzbach 1991, p. 168  ^ Arndt & Haenel 2006, pp. 15â€“16, 175, 184â€“186, 205. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.
 ^ Arndt & Haenel 2006, pp. 176â€“177
 ^ ^{a} ^{b} Boyer & Merzbach 1991, p. 202
 ^ Arndt & Haenel 2006, p. 177
 ^ Arndt & Haenel 2006, p. 178
 ^ Arndt & Haenel 2006, pp. 179
 ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 180
 ^ Azarian, Mohammad K. (2010). "alRisÄla almuhÄ«tÄ«yya: A Summary". Missouri Journal of Mathematical Sciences 22 (2): 64â€“85.
 ^ O'Connor, John J.; Robertson, Edmund F. (1999). "Ghiyath alDin Jamshid Mas'ud alKashi". MacTutor History of Mathematics archive. Retrieved August 11, 2012.
 ^ ^{a} ^{b} ^{c} Arndt & Haenel 2006, p. 182
 ^ Arndt & Haenel 2006, pp. 182â€“183
 ^ ^{a} ^{b} Arndt & Haenel 2006, p. 183
 ^ Grienbergerus, Christophorus (1630). Elementa Trigonometrica (PDF) (in Latin). Archived from the original (PDF) on 20140201. His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < π < 3.14159 26535 89793 23846 26433 83279 50288 4199.
 ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 185â€“191
 ^ Roy 1990, pp. 101â€“102
Arndt & Haenel 2006, pp. 185â€“186  ^ ^{a} ^{b} ^{c} Roy 1990, pp. 101â€“102
 ^ Joseph 1991, p. 264
 ^ ^{a} ^{b} Arndt & Haenel 2006, p. 188. Newton quoted by Arndt.
 ^ ^{a} ^{b} Arndt & Haenel 2006, p. 187
 ^ Arndt & Haenel 2006, pp. 188â€“189
 ^ ^{a} ^{b} Eymard & Lafon 1999, pp. 53â€“54
 ^ Arndt & Haenel 2006, p. 189
 ^ Arndt & Haenel 2006, p. 156
 ^ Arndt & Haenel 2006, pp. 192â€“193
 ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 72â€“74
 ^ Arndt & Haenel 2006, pp. 192â€“196, 205
 ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 194â€“196
 ^ ^{a} ^{b} Borwein, J. M.; Borwein, P. B. (1988). "Ramanujan and Pi". Scientific American 256 (2): 112â€“117. Bibcode:1988SciAm.258b.112B. doi:10.1038/scientificamerican0288112.
Arndt & Haenel 2006, pp. 15â€“17, 70â€“72, 104, 156, 192â€“197, 201â€“202  ^ Arndt & Haenel 2006, pp. 69â€“72
 ^ Borwein, J. M.; Borwein, P. B.; Dilcher, K. (1989). "Pi, Euler Numbers, and Asymptotic Expansions". American Mathematical Monthly 96 (8): 681â€“687. doi:10.2307/2324715.
 ^ Arndt & Haenel 2006, p. 223, (formula 16.10). Note that (n âˆ’ 1)n(n + 1) = n^{3} âˆ’ n.
Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). Penguin. p. 35. ISBN 9780140261493.  ^ ^{a} ^{b} Posamentier & Lehmann 2004, pp. 284
 ^ Lambert, Johann, "MÃ©moire sur quelques propriÃ©tÃ©s remarquables des quantitÃ©s transcendantes circulaires et logarithmiques", reprinted in Berggren, Borwein & Borwein 1997, pp. 129â€“140
 ^ Arndt & Haenel 2006, p. 196
 ^ Arndt & Haenel 2006, p. 165. A facsimile of Jones' text is in Berggren, Borwein & Borwein 1997, pp. 108â€“109
 ^ See Schepler 1950, p. 220: William Oughtred used the letter π to represent the periphery (i.e., circumference) of a circle.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Arndt & Haenel 2006, p. 166
 ^ Arndt & Haenel 2006, pp. 205
 ^ Arndt & Haenel 2006, p. 197. See also Reitwiesner 1950.
 ^ Arndt & Haenel 2006, p. 197
 ^ Arndt & Haenel 2006, pp. 15â€“17
 ^ Arndt & Haenel 2006, pp. 131
 ^ Arndt & Haenel 2006, pp. 132, 140
 ^ ^{a} ^{b} Arndt & Haenel 2006, p. 87
 ^ Arndt & Haenel 2006, pp. 111 (5 times); pp. 113â€“114 (4 times).
See Borwein & Borwein 1987 for details of algorithms.  ^ ^{a} ^{b} ^{c} Bailey, David H. (16 May 2003). "Some Background on Kanadaâ€™s Recent Pi Calculation" (PDF). Retrieved 12 April 2012.
 ^ 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 roundoff errors, Arndt concludes that a few hundred digits would suffice for any scientific application.  ^ Arndt & Haenel 2006, pp. 17â€“19
 ^ Schudel, Matt (25 March 2009). "John W. Wrench, Jr.: Mathematician Had a Taste for Pi". The Washington Post. p. B5.
 ^ Connor, Steve (8 January 2010). "The Big Question: How close have we come to knowing the precise value of pi?". The Independent (London). Retrieved 14 April 2012.
 ^ Arndt & Haenel 2006, p. 18
 ^ Arndt & Haenel 2006, pp. 103â€“104
 ^ Arndt & Haenel 2006, p. 104
 ^ Arndt & Haenel 2006, pp. 104, 206
 ^ Arndt & Haenel 2006, pp. 110â€“111
 ^ Eymard & Lafon 1999, p. 254
 ^ Arndt & Haenel 2006, pp. 110â€“111, 206
Bellard, Fabrice, "Computation of 2700 billion decimal digits of Pi using a Desktop Computer", 11 Feb 2010.  ^ PSLQ means Partial Sum of Least Squares.
 ^ Plouffe, Simon (April 2006). "Identities inspired by Ramanujan's Notebooks (part 2)" (PDF). Retrieved 10 April 2009.
 ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 77â€“84
 ^ ^{a} ^{b} Gibbons, Jeremy, "Unbounded Spigot Algorithms for the Digits of Pi", 2005. Gibbons produced an improved version of Wagon's algorithm.
 ^ ^{a} ^{b} Arndt & Haenel 2006, p. 77
 ^ Rabinowitz, Stanley; Wagon, Stan (March 1995). "A spigot algorithm for the digits of Pi". American Mathematical Monthly 102 (3): 195â€“203. doi:10.2307/2975006. A computer program has been created that implements Wagon's spigot algorithm in only 120 characters of software.
 ^ ^{a} ^{b} Arndt & Haenel 2006, pp. 117, 126â€“128
 ^ Bailey, David H.; Borwein, Peter B.; and Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation 66 (218): 903â€“913. doi:10.1090/S0025571897008569.
 ^ Arndt & Haenel 2006, p. 128. Plouffe did create a decimal digit extraction algorithm, but it is slower than full, direct computation of all preceding digits.
 ^ Arndt & Haenel 2006, p. 20
Bellards formula in: Bellard, Fabrice. "A new formula to compute the n^{th} binary digit of pi". Archived from the original on 12 September 2007. Retrieved 27 October 2007.  ^ Palmer, Jason (16 September 2010). "Pi record smashed as team finds twoquadrillionth digit". BBC News. Retrieved 26 March 2011.
 ^ BronshteÄn & Semendiaev 1971, pp. 200, 209
 ^ Weisstein, Eric W., "Semicircle", MathWorld.
 ^ ^{a} ^{b} Ayers 1964, p. 60
 ^ ^{a} ^{b} BronshteÄn & Semendiaev 1971, pp. 210â€“211
 ^ Arndt & Haenel 2006, p. 39
 ^ Ramaley, J. F. (October 1969). "Buffon's Noodle Problem". The American Mathematical Monthly 76 (8): 916â€“918. JSTOR 2317945. doi:10.2307/2317945.
 ^ Arndt & Haenel 2006, pp. 39â€“40
Posamentier & Lehmann 2004, p. 105  ^ Arndt & Haenel 2006, pp. 43
Posamentier & Lehmann 2004, pp. 105â€“108  ^ Ayers 1964, p. 100
 ^ ^{a} ^{b} BronshteÄn & Semendiaev 1971, p. 592
 ^ Maor, Eli, E: The Story of a Number, Princeton University Press, 2009, p 160, ISBN 9780691141343 ("five most important" constants).
 ^ Weisstein, Eric W., "Roots of Unity", MathWorld.
 ^ Weisstein, Eric W., "Cauchy Integral Formula", MathWorld.
 ^ Joglekar, S. D., Mathematical Physics, Universities Press, 2005, p 166, ISBN 9788173714221.
 ^ ^{a} ^{b} Klebanoff, Aaron (2001). "Pi in the Mandelbrot set" (PDF). Fractals 9 (4): 393â€“402. doi:10.1142/S0218348X01000828. Retrieved 14 April 2012.
 ^ Peitgen, HeinzOtto, Chaos and fractals: new frontiers of science, Springer, 2004, pp. 801â€“803, ISBN 9780387202297.
 ^ BronshteÄn & Semendiaev 1971, pp. 191â€“192
 ^ BronshteÄn & Semendiaev 1971, p. 190
 ^ Arndt & Haenel 2006, pp. 41â€“43
 ^ This theorem was proved by Ernesto CesÃ ro in 1881. For a more rigorous proof than the intuitive and informal one given here, see Hardy, G. H., An Introduction to the Theory of Numbers, Oxford University Press, 2008, ISBN 9780199219865, theorem 332.
 ^ Ogilvy, C. S.; Anderson, J. T., Excursions in Number Theory, Dover Publications Inc., 1988, pp. 29â€“35, ISBN 0486257789.
 ^ Arndt & Haenel 2006, p. 43
 ^ Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968, pp 174â€“190.
 ^ ^{a} ^{b} BronshteÄn & Semendiaev 1971, pp. 106â€“107, 744, 748
 ^ Halliday, David; Resnick, Robert; Walker, Jearl, Fundamentals of Physics, 5th Ed., John Wiley & Sons, 1997, p 381, ISBN 0471148547.
 ^ Imamura, James M (17 August 2005). "Heisenberg Uncertainty Principle". University of Oregon. Archived from the original on 12 October 2007. Retrieved 9 September 2007.
 ^ Yeo, Adrian, The pleasures of pi, e and other interesting numbers, World Scientific Pub., 2006, p 21, ISBN 9789812700780.
Ehlers, JÃ¼rgen, Einstein's Field Equations and Their Physical Implications, Springer, 2000, p 7, ISBN 9783540670735.  ^ Nave, C. Rod (28 June 2005). "Coulomb's Constant". HyperPhysics. Georgia State University. Retrieved 9 November 2007.
 ^ C. Itzykson, JB. Zuber, Quantum Field Theory, McGrawHill, 1980.
 ^ Low, Peter, Classical Theory of Structures Based on the Differential Equation, CUP Archive, 1971, pp 116â€“118, ISBN 9780521080897.
 ^ Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, 1967, p 233, ISBN 0521663962.
 ^ Bracewell, R. N., The Fourier Transform and Its Applications, McGrawHill, 2000, ISBN 0071160434.
 ^ HansHenrik StÃ¸lum (22 March 1996). "River Meandering as a SelfOrganization Process". Science 271 (5256): 1710â€“1713. Bibcode:1996Sci...271.1710S. doi:10.1126/science.271.5256.1710.
 ^ Posamentier & Lehmann 2004, pp. 140â€“141
 ^ ^{a} ^{b} ^{c} Arndt & Haenel 2006, pp. 44â€“45
 ^ "Chinese student breaks Guinness record by reciting 67,890 digits of pi". News Guangdong. 28 November 2006. Retrieved 27 October 2007.
 ^ "Most Pi Places Memorized", Guinness World Records. Retrieved 3 April 2012.
 ^ Otake, Tomoko (17 December 2006). "How can anyone remember 100,000 numbers?". The Japan Times. Retrieved 27 October 2007.
 ^ Raz, A.; Packard, M. G. (2009). "A slice of pi: An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist". Neurocase 15: 361â€“372. PMID 19585350. doi:10.1080/13554790902776896.
 ^ Keith, Mike. "Cadaeic Cadenza Notes & Commentary". Retrieved 29 July 2009.
 ^ Keith, Michael; Diana Keith (February 17, 2010). Not A Wake: A dream embodying (pi)'s digits fully for 10000 decimals. Vinculum Press. ISBN 9780963009715.
 ^ For instance, Pickover calls Ï€ "the most famous mathematical constant of all time", and Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the Givenchy Ï€ perfume, 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 Treks: From Surreal Numbers to Magic Circles, MAA spectrum, Mathematical Association of America, p. 17, ISBN 9780883855379.
 ^ BBC documentary "The Story of Maths", second part, showing a visualization of the historically first exact formula, starting at 35 min and 20 sec into the second part of the documentary.
 ^ Posamentier & Lehmann 2004, p. 118
Arndt & Haenel 2006, p. 50  ^ Arndt & Haenel 2006, p. 14. This part of the story was omitted from the film adaptation of the novel.
 ^ Gill, Andy (4 November 2005). "Review of Aerial". The Independent.
the almost autistic satisfaction of the obsessivecompulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits long)
 ^ 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.
 ^ Pi Day activities.
 ^ MIT cheers. Retrieved 12 April 2012.
 ^ "Happy Pi Day! Watch these stunning videos of kids reciting 3.14". USAToday.com. 20150314. Retrieved 20150314.
 ^ "Google's strange bids for Nortel patents". FinancialPost.com. Reuters. 20110705. Retrieved 16 August 2011.
 ^ 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.
 ^ Sequence OEIS A019692,
 ^ Abbott, Stephen (April 2012). "My Conversion to Tauism" (PDF). Math Horizons 19 (4): 34. doi:10.4169/mathhorizons.19.4.34.
 ^ Palais, Robert (2001). "π Is Wrong!" (PDF). The Mathematical Intelligencer 23 (3): 7â€“8. doi:10.1007/BF03026846.
 ^ Tau Day: Why you should eat twice the pie â€“ Light Years â€“ CNN.com Blogs
 ^ "Life of pi in no danger â€“ Experts coldshoulder campaign to replace with tau". Telegraph India. 20110630.
 ^ Arndt & Haenel 2006, pp. 211â€“212
Posamentier & Lehmann 2004, pp. 36â€“37
Hallerberg, Arthur (May 1977). "Indiana's squared circle". Mathematics Magazine 50 (3): 136â€“140. JSTOR 2689499. doi:10.2307/2689499.
 References
 Arndt, JÃ¶rg; Haenel, Christoph (2006). Pi Unleashed. SpringerVerlag. ISBN 9783540665724. Retrieved 20130605. English translation by Catriona and David Lischka.
 Ayers, Frank (1964). Calculus. McGrawHill. ISBN 9780070026537.
 Berggren, Lennart; Borwein, Jonathan; Borwein, Peter (1997). Pi: a Source Book. SpringerVerlag. ISBN 9780387205717.
 Beckmann, Peter (1989) [1974]. History of Pi. St. Martin's Press. ISBN 9780880294188.
 Borwein, Jonathan; Borwein, Peter (1987). Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity. Wiley. ISBN 9780471315155.
 Boyer, Carl B.; Merzbach, Uta C. (1991). A History of Mathematics (2 ed.). Wiley. ISBN 9780471543978.
 BronshteÄn, Ilia; Semendiaev, K. A. (1971). A Guide Book to Mathematics. H. Deutsch. ISBN 9783871440953.
 Eymard, Pierre; Lafon, Jean Pierre (1999). The Number Pi. American Mathematical Society. ISBN 9780821832462., English translation by Stephen Wilson.
 Joseph, George Gheverghese (1991). The Crest of the Peacock: NonEuropean Roots of Mathematics. Princeton University Press. ISBN 9780691135267. Retrieved 20130605.
 Posamentier, Alfred S.; Lehmann, Ingmar (2004). Pi: A Biography of the World's Most Mysterious Number. Prometheus Books. ISBN 9781591022008.
 Reitwiesner, George (1950). "An ENIAC Determination of pi and e to 2000 Decimal Places". Mathematical Tables and Other Aids to Computation 4 (29): 11â€“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): 291â€“306. doi:10.2307/2690896.
 Schepler, H. C. (1950). "The Chronology of Pi". Mathematics Magazine (Mathematical Association of America) 23 (3): 165â€“170 (Jan/Feb), 216â€“228 (Mar/Apr), and 279â€“283 (May/Jun). doi:10.2307/3029284.. issue 3 Jan/Feb, issue 4 Mar/Apr, issue 5 May/Jun
Further reading
 Blatner, David (1999). The Joy of Pi. Walker & Company. ISBN 9780802775627.
 Borwein, Jonathan; Borwein, Peter (1984). "The ArithmeticGeometric Mean and Fast Computation of Elementary Functions". SIAM Review 26: 351â€“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: 201â€“219. doi:10.2307/2325206.
 Chudnovsky, David V. and Chudnovsky, Gregory V., "Approximations and Complex Multiplication According to Ramanujan", in Ramanujan Revisited (G.E. Andrews et al. Eds), Academic Press, 1988, pp 375â€“396, 468â€“472
 Cox, David A., "The ArithmeticGeometric Mean of Gauss", L' Ensignement Mathematique, 30(1984) 275â€“330
 Delahaye, JeanPaul, "Le Fascinant Nombre Pi", Paris: BibliothÃ¨que Pour la Science (1997) ISBN 2902918259
 Engels, Hermann (1977). "Quadrature of the Circle in Ancient Egypt". Historia Mathematica 4: 137â€“140. doi:10.1016/03150860(77)901045.
 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, SpringerVerlag, 1964, pp 137â€“153
 Heath, T. L., The Works of Archimedes, Cambridge, 1897; reprinted in The Works of Archimedes with The Method of Archimedes, Dover, 1953, pp 91â€“98
 Huygens, Christiaan, "De Circuli Magnitudine Inventa", Christiani Hugenii Opera Varia I, Leiden 1724, pp 384â€“388
 LayYong, Lam; TianSe, Ang (1986). "Circle Measurements in Ancient China". Historia Mathematica 13: 325â€“340. doi:10.1016/03150860(86)900558.
 Lindemann, Ferdinand (1882). "Ueber die Zahl pi". Mathematische Annalen 20: 213â€“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: 77â€“82.
 Niven, Ivan, "A Simple Proof that pi Is Irrational", Bulletin of the American Mathematical Society, 53:7 (July 1947), 507
 Ramanujan, Srinivasa, "Modular Equations and Approximations to Ï€", Quarterly Journal of Pure and Applied Mathematics, XLV, 1914, 350â€“372. Reprinted in G.H. Hardy, P.V. Seshu Aiyar, and B. M. Wilson (eds), Srinivasa Ramanujan: Collected Papers, 1927 (reprinted 2000), pp 23â€“29
 Shanks, William, Contributions to Mathematics Comprising Chiefly of the Rectification of the Circle to 607 Places of Decimals, 1853, pp. iâ€“xvi, 10
 Shanks, Daniel; Wrench, John William (1962). "Calculation of pi to 100,000 Decimals". Mathematics of Computation 16: 76â€“99. doi:10.1090/s00255718196201360519.
 Tropfke, Johannes, Geschichte Der ElementarMathematik in Systematischer Darstellung (The history of elementary mathematics), BiblioBazaar, 2009 (reprint), ISBN 9781113085733
 Viete, Francois, Variorum de Rebus Mathematicis Reponsorum Liber VII. F. Viete, Opera Mathematica (reprint), Georg Olms Verlag, 1970, pp 398â€“401, 436â€“446
 Wagon, Stan, "Is Pi Normal?", The Mathematical Intelligencer, 7:3(1985) 65â€“67
 Wallis, John, Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilineorum Quadratum, aliaque difficiliora Matheseos Problemata, Oxford 1655â€“6. Reprinted in vol. 1 (pp 357â€“478) of Opera Mathematica, Oxford 1693
 Zebrowski, Ernest, A History of the Circle: Mathematical Reasoning and the Physical Universe, Rutgers University Press, 1999, ISBN 9780813528984
External links
40x40px  Wikimedia Commons has media related to Pi. 
 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 â€“ Pi". Sixty Symbols. Brady Haran for the University of Nottingham.
 Grime, Dr. James (2014). "Pi is Beautiful â€“ Numberphile". Numberphile. Brady Haran.
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