Pi

Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
Area of a circle Circumference Use in other formulae
Properties
Irrationality Transcendence
Value
Less than 22/7 Approximations Madhava's correction term Memorization
People
Archimedes Liu Hui Zu Chongzhi Aryabhata Madhava Jamshīd al-Kāshī Ludolph van Ceulen François Viète Seki Takakazu Takebe Kenko William Jones John Machin William Shanks Srinivasa Ramanujan John Wrench Chudnovsky brothers Yasumasa Kanada
History
Chronology A History of Pi
In culture
Indiana pi bill Pi Day
Related topics
Squaring the circle Basel problem Six nines in π Other topics related to π
v t e

The number π ( /paɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. It is a transcendental number – a number that cannot be produced with a finite sequence of algebraic operations (sums, products, powers, and roots). The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and ruler. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered.

For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Prior to the 16th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of π. Starting in the 16th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians including Mādhava of Sañgamāgrama, Isaac Newton, Leonhard Euler, Srinivasa Ramanujan, and Carl Friedrich Gauss.

In the 20th century, mathematicians and computer scientists discovered new approaches that – when combined with increasing computer speeds – extended the decimal representation of π to over 10 trillion (10¹³) digits. 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; but the extensive calculations involved have been used to test supercomputers and high-precision 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 from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature 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 news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.

π is commonly defined as the ratio of a circle's circumference C to its diameter d:^[1]

${\displaystyle \pi ={\frac {C}{d}}.}$

The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.^[1] For this reason, some mathematicians prefer definitions of π based on calculus or trigonometry that do not rely on the circle. One such definition is: π is twice the smallest positive x for which cosine(x) equals 0.^[1][2]

The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letter π. That letter can be denoted by the Latin word pi, which is also used to represent the same ratio.^[3] In English, π is pronounced as "pie" ( /paɪ/, /ˈpaɪ/).^[4] The lower-case letter π (or π in sans-serif font) is not to be confused with the capital letter Template:PI, which denotes a product of a sequence.

The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who utilized it in 1706 in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics.^[5] Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the word periphery.^[6] Jones 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.^[7] The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.^[7]

After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Prior to 1736, mathematicians sometimes used letters such as c or p to represent the ratio of the circumference to diameter.^[7] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.^[7] 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 thereafter the Greek letter was universally adopted in the Western world.^[7]

π is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.^[8] Since π is irrational, it has an infinite number of digits in its decimal representation, and it does not end with 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.^[9]

π is a transcendental number, which means that it is not the solution of any non-constant polynomial with rational coefficients, such as ${\displaystyle \scriptstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0.}$^[10][11] The transcendence of π has two important consequences: First, π cannot be expressed using any combination of rational numbers and square roots or n-th roots such as ${\displaystyle \scriptstyle {\sqrt[{3}]{31}}}$ or ${\displaystyle \scriptstyle {\sqrt[{2}]{10}}.}$ 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.^[12] Squaring a circle was one of the important geometry problems of the classical antiquity.^[13] Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.^[14]

The digits of π have no apparent pattern and pass 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.^[15] The hypothesis that π is normal has not been proven or disproven.^[15] 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.^[16] Despite the fact that 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.^[17]

Like all irrational numbers, π cannot be represented as a simple fraction. But irrational numbers, including π, can be represented by an infinite series of nested fractions, called a continued fraction:

${\displaystyle \pi =3+\textstyle {\frac {1}{7+\textstyle {\frac {1}{15+\textstyle {\frac {1}{1+\textstyle {\frac {1}{292+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\textstyle {\frac {1}{1+\ddots }}}}}}}}}}}}}}.}$

Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate 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.^[18] Although the simple continued fraction for π does not exhibit a pattern^[19] mathematicians have discovered several generalized continued fractions which do, such as:^[20]

${\displaystyle \pi =\textstyle {\cfrac {4}{1+\textstyle {\frac {1^{2}}{2+\textstyle {\frac {3^{2}}{2+\textstyle {\frac {5^{2}}{2+\textstyle {\frac {7^{2}}{2+\textstyle {\frac {9^{2}}{2+\ddots }}}}}}}}}}}}=3+\textstyle {\frac {1^{2}}{6+\textstyle {\frac {3^{2}}{6+\textstyle {\frac {5^{2}}{6+\textstyle {\frac {7^{2}}{6+\textstyle {\frac {9^{2}}{6+\ddots }}}}}}}}}}=\textstyle {\cfrac {4}{1+\textstyle {\frac {1^{2}}{3+\textstyle {\frac {2^{2}}{5+\textstyle {\frac {3^{2}}{7+\textstyle {\frac {4^{2}}{9+\ddots }}}}}}}}}}.}$

Some approximations of π include:

• Decimal – The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....^[21]
• Binary – 11.00100100001111 ....
• Hexadecimal – The base 16 approximation to 20 digits is ^[22]
• Sexagesimal – A base 60 approximation is 3:8:30.^[23]
• Fractions – Approximate fractions include (in order of increasing accuracy) ⁠22/7⁠, ⁠333/106⁠, ⁠355/113⁠, ⁠52163/16604⁠, and ⁠103993/33102⁠.^[24]

The Great Pyramid at Giza, constructed c.2589–2566 BC, was built with a perimeter of approximately 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 6.2857 is about equal to 2π ≈ 6.2832. A few pyramidologists conclude from this value that the pyramid builders had knowledge of π and deliberately designed the pyramid to incorporate the value.^[25] However, mainstream historians believe that ancient Egyptians had no concept of π and that it is merely a coincidence that the ratio of perimeter to height is about 2π.^[26]

The earliest written approximations of π are found in Babylon and Egypt, both within 1 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.^[27] In Egypt, the Rhind Papyrus, dated around 1650 BC, has a formula for the area of a circle that treats π as (16/9)² ≈ 3.1605.^[27]

In India, around 600 BC, the ancient Indian math texts Shulba Sutras, written in Sanskrit, treat π as (9785/5568)² ≈ 3.088.^[28] In 150 BC, or perhaps earlier, Indian sources treat π as ${\displaystyle \scriptstyle {\sqrt {10}}}$ ≈ 3.1622.^[29]

The Hebrew Bible, written between 8th and 3rd centuries BC, contains two verses which suggest that π has a value of three. The two verses, 1 Kings 7:23 and 2 Chronicles 4:2, discuss a ceremonial pool in the temple of King Solomon with a diameter of ten cubits and a circumference of thirty cubits.^[30][31]

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach utilizing polygons that was used around 250 BC by Greek mathematician Archimedes.^[32] This polygonal algorithm remained the primary approach for computing π 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 regular polygons inside and outside a circle, and calculating the perimeters of the outer and inner polygons.^[34] By using the equivalent of 96-sided 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 widespread belief that π was equal to 22/7.^[35] Around 150 AD, Greek-Roman 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 0 AD), ${\displaystyle \scriptstyle {\sqrt {10}}}$ (100 AD, approximately 3.1623), and 142/45 (third century, approximately 3.1556).^[38] Around 265 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.^[39][40] Hui 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 difference in area of successive polygons forms a geometric series with a factor of 4.^[39] The Chinese mathematician Zu Chongzhi, around 480, calculated that π ≃ 355/113 (a fraction which 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... would remain the most accurate approximation of π available for the next 800 years.^[41]

In India, astronomer Aryabhata used a value of 3.1416 in his 499 AD work Āryabhaṭīya.^[42] Fibonacci in circa 1220 computed 3.1418 using a polygonal method, independent from Archimedes.^[43] Italian author Dante apparently employed the value ${\displaystyle \scriptstyle 3+{\sqrt {2}}/10}$ ≃ 3.14142.^[43]

Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1430 using a polygon with 3 x 2²⁸ sides, which stood as the world record for about 180 years.^[44] French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3 x 2¹⁷ sides.^[44] Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.^[44] 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).^[45] Dutch scientist Willebrord Snellius reached 34 digits in 1621,^[46] and Austrian astronomer Christoph Grienberger arrived at 39 digits in 1630, which would remain the most accurate approximation manually achieved using polygonal algorithms.^[46]

The calculation of π was revolutionized in the 16th and 17th centuries by the discovery of infinite series, which are sums containing an infinite number of terms.^[47] Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques.^[47] Although infinite series were exploited for π most notably by European mathematicians such as James Gregory and Gottfried Leibniz, the approach was first discovered in India sometime between 1400 and 1500 AD.^[48] The first written description of an infinite series which could be used to compute π was written in Sanskrit verse by Indian astronomer Nilakantha Somayaji in the work Tantrasamgraha, dating from around 1500 AD.^[49] 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.^[49] 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.^[49] Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician, Jamshīd al-Kāshī using a polygonal algorithm.^[50]

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 François Viète in 1593:^[52]

${\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots }$

The second infinite sequence found in Europe, by John Wallis in 1655, also was an infinite product.^[52] The discovery of calculus by English scientist Isaac Newton and German mathematician Leibniz in the 1660s, created the foundation for a large number of infinite series which could be exploited by people who calculated digits of 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".^[51]

In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregory in 1671 and Leibniz in 1674:^[53][54]

${\displaystyle {\begin{aligned}\arctan z&{}=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdot \cdots \end{aligned}}}$

This formula, the Gregory–Leibniz series, equals ${\displaystyle \scriptstyle \pi /4}$ when evaluated with z=1.^[54] In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.^[55] The Gregory–Leibniz series is simple, but converges very slowly (that is, approaches the answer gradually) so is not used in modern π calculations.^[56]

In 1706 John Machin utilized the Gregory–Leibniz series to produce an algorithm which converged much faster:^[57]

${\displaystyle {\frac {\pi }{4}}=4\,\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}$

Machin reached 100 digits of π with this formula.^[58] Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.^[58] 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 calculated without the aid of a calculating device.^[59]

A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss.^[60] 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.^[60]

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.^[61] A simple infinite series for π is the Gregory–Leibniz series:^[62]

${\displaystyle {\pi }={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdot \cdots }$

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. The Gregory–Leibniz series converges slowly: After 500,000 terms, it produces only five correct decimal digits of π.^[63] An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:^[64]

${\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdot \cdots }$

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:
${\displaystyle \scriptstyle {\pi }={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}\cdot \cdots }$	4.0000	2.6666...	3.4666...	2.8952...	3.3396...	π = 3.1415...
${\displaystyle \scriptstyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}\cdot \cdots }$	3.0000	3.1666...	3.1333...	3.1452...	3.1396...	π = 3.1415...

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.^[61]

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:^[65]

${\displaystyle {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdot \cdots }$

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.^[8] Lambert's proof utilized a continued fraction representation of the tangent function.^[66] French mathematician Adrien-Marie Legendre proved in 1794 that π² is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π was transcendental, confirming a conjecture made by both Legendre and Euler.^[67]

The development of computers in the mid twentieth 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.^[68] Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann used the ENIAC computer to compute 2,037 digits of π in 1949, a calculation that took 70 hours of computer time.^[69] The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.^[70]

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 together very rapidly.^[71] The fast multiplication algorithms are particularly important in computer-based π computations, because the majority of the computer's time is typically spent performing multiplications.^[72] Fast multiplication algorithms include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform based methods.^[73]

The iterative algorithms were independently published in 1975–1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.^[74] These algorithms were unique because they utilized an iterative approach rather than an 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. Salamin and Brent were not the first to discover the iterative approach for π: It 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.^[74] The algorithm, as modified by Salamin and Brent, is also referred to as the Brent–Salamin algorithm.

The iterative algorithms were widely used following 1980 because the iterative algorithms have the potential to be faster than infinite series algorithms: Whereas infinite series typically increase the number of correct digits by a fixed amount for each added term, 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 they discovered an iterative algorithm that increases the number of digits five times each iteration.^[75] Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing π between 1995 and 2002.^[76] The rapid convergence of iterative algorithms comes at a price: The iterative algorithms require significantly more memory usage than infinite series.^[76]

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 which is necessary to calculate the diameter of the observable universe with a precision of one atom.^[77] Despite this, people have worked strenuously to compute π to thousands and millions of digits.^[78] The desire for large number of digits may be partly ascribed to the human compulsion to break records, as new records for computing π often make news headlines around the world.^[79][80] Computing a large number of digits of π has practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms), and providing raw data to evaluate the randomness of the digits of π.^[81]

Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s which are as fast as iterative algorithms, yet simpler and less memory intensive.^[76] 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.^[82] One of his formulae, based on modular equations, was:

${\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.}$

This series converges much more rapidly than most arctan series, including Machin's formula.^[83] Ramanujan's formula was not used for calculating digits of π until Bill Gosper used it in 1985 to set a record of 17 million digits.^[84] Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.^[85] The Chudnovsky formula developed in 1987 is:

${\displaystyle {\frac {426880{\sqrt {10005}}}{\pi }}=\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}(-640320)^{3k}}}.}$

which produces 14 digits of π per term.^[86] The Chudnovsky formula has been used for several record-setting π calculations including the first calculation of over one billion (10⁹) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×10¹²) digits by Fabrice Bellard in 2009, and 10 trillion (10¹³) digits in 2011 by Alexander Yee and Shigeru Kondo.^[87][88]

In 2006, Canadian mathematician Simon Plouffe, using the PSLQ integer relation algorithm^[89] found several formulae for π, which conformed to the following template:

${\displaystyle \pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right)}$

where ${\displaystyle {\mathit {q}}}$ is e^π (Gelfond's constant), ${\displaystyle {\mathit {k}}}$ is an odd number, and ${\displaystyle {\mathit {a,b,c}}}$ are certain rational numbers that Plouffe computed.^[90]

Two algorithms were discovered in 1995 that opened up new avenues of research into π. The algorithms 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.^[91][92] This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.^[91]

American mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995.^[92][93][94] The algorithm's speed is comparable to arctan algorithms but not as fast as iterative algorithms.^[93]

Another spigot algorithm that originated in 1995 is the BBP digit extraction algorithm discovered by Simon Plouffe:^[95][96]

${\displaystyle \pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right).}$

This formula was a breakthrough in calculating π because it can produce any individual hexadecimal digit of π without calculating all the preceding digits.^[95] From the hexadecimal digit, octal or binary digits may be readily extracted. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found which rapidly produces decimal digits.^[97] 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, and if they match, it provides a measure of confidence that the entire computation is correct.^[88]

Between 1998 and 2000, the distributed computing project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (10¹⁵th) bit of π, which turned out to be 0.^[98] In September 2010, a Yahoo! employee used the company's Hadoop application on 1,000 computers over a 23-day period to compute 256 bits of π at the two-quadrillion^th (2×10¹⁵th) bit.^[99]

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.

π appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Some of the more common formulae that involve π are:^[100]

• The circumference of a circle with radius r is ${\displaystyle 2\pi r}$
• The area of a circle with radius r is ${\displaystyle \pi r^{2}}$
• The volume of a sphere with radius r is ${\displaystyle {\tfrac {4}{3}}\pi r^{3}}$
• The surface area of a sphere with radius r is ${\displaystyle 4\pi r^{2}}$

π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. For example, an integral which specifies half the area of a circle of radius one is given by:^[101]

${\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}.}$

In the above integral, the function ${\displaystyle \scriptstyle {\sqrt {1-x^{2}}}}$ represents the top half of a circle (the square root is a consequence of the Pythagorean theorem), and the integral ${\displaystyle \scriptstyle \int _{-1}^{1}}$ is an operation which computes the area between that half 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: A complete circle spans an angle of 2π radians.^[102] The angle measure of 180° is equal to π radians, and 1° = (π/180) radians.^[102]

Common trigonometric functions have periods that are multiples of π, for example, sine and cosine have period 2π.^[103] Thus, for any angle θ and any integer k, ${\displaystyle \scriptstyle \sin \theta =\sin \left(\theta +2\pi k\right)}$ and ${\displaystyle \scriptstyle \cos \theta =\cos \left(\theta +2\pi k\right).}$^[103]

Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of π.^[104] Buffon's needle is one such technique: If a needle of length l is dropped n times on a surface containing parallel lines 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:^[105]

${\displaystyle \pi \approx {\frac {2nl}{xt}}.}$

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 ${\displaystyle \scriptstyle \pi /4}$.^[106]

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.^[107]

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:^[108]

${\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi ).}$

where i is the imaginary unit satisfying i² = −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:^[109]

${\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi }$

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:^[109][110]

${\displaystyle e^{i\pi }+1=0.}$

There are n different complex numbers z satisfying ${\displaystyle z^{n}=1}$ and these numbers are called the "n-th roots of unity".^[111] The n-th roots of unity are

${\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}$

Cauchy's integral formula governs complex analytic functions and establishes an important relationship between integration and differentiation, including the remarkable fact that the values of a complex function within a closed boundary are entirely determined by the values on the boundary:^[112][113]

${\displaystyle f(z_{0})={\frac {1}{2\pi i}}\oint _{\gamma }{f(z) \over z-z_{0}}\,dz.}$

An occurrence of π in the Mandelbrot set fractal was discovered by American David Boll in 1991.^[114] He examined the behavior of the Mandelbrot set near the "neck" at (−.75, 0). If points with coordinates (−.75, ε) are considered then, as ε tends to zero, the number of iterations until divergence for the point multiplied by ε converges to π. The point (.25, ε) at the cusp of the large "valley" on the right side of the Mandelbrot set behaves similarly: the number of iterations until divergence, times the square root of ε, tends to π.^[114][115]

The Gamma function extends the concept of factorial – which is normally defined only for whole numbers – to all real numbers. When the Gamma function is evaluated at half-integers, the result contains π; for example ${\displaystyle \scriptstyle \Gamma (1/2)={\sqrt {\pi }}}$ and ${\displaystyle \scriptstyle \Gamma (5/2)={\frac {3{\sqrt {\pi }}}{4}}}$.^[116] The Gamma function can be used to create a simple approximation to ${\displaystyle \scriptstyle n!}$ for large ${\displaystyle \scriptstyle n}$: ${\displaystyle \scriptstyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}$ which is known as Stirling's approximation.^[117]

The Riemann zeta function ζ(s) is a function which is utilized in many areas of mathematics. When evaluated at ${\displaystyle \scriptstyle s=2}$ it can be written as

${\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdot \cdots }$

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 ${\displaystyle \scriptstyle {\frac {\pi ^{2}}{6}}}$.^[65] 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 ${\displaystyle \scriptstyle 6/\pi ^{2}}$.^[118][119] This probability is based on the observation that the probability that any number is divisible by a prime ${\displaystyle \scriptstyle p}$ is ${\displaystyle \scriptstyle 1/p}$ (for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is ${\displaystyle \scriptstyle 1/p^{2}}$, and the probability that at least one of them is not is ${\displaystyle \scriptstyle 1-1/p^{2}}$. For distinct primes, these divisibility events are mutually independent, thus the probability that two numbers are relatively prime is given by a product over all primes:^[120]

${\displaystyle \prod _{p}^{\infty }\left(1-{\frac {1}{p^{2}}}\right)=\left(\prod _{p}^{\infty }{\frac {1}{1-p^{-2}}}\right)^{-1}={\frac {1}{1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }}={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\%.}$

This probability can be used in conjunction with a random number generator to approximate π using a Monte Carlo approach.^[121]

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): ^[122]

${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}.}$

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 (Δx) and momentum (Δp) can not both be arbitrarily small at the same time (where h is Planck's constant):^[123]

${\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}.}$

In the domain of cosmology, π appears in one of the fundamental formulae, Einstein's field equation, 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:^[124]

${\displaystyle R_{ik}-{g_{ik}R \over 2}+\Lambda g_{ik}={8\pi G \over c^{4}}T_{ik}}$

where ${\displaystyle R_{ik}\,}$ is the Ricci curvature tensor, ${\displaystyle R\,}$ the scalar curvature, ${\displaystyle g_{ik}\,}$ the metric tensor, ${\displaystyle \Lambda \,}$ is the cosmological constant, ${\displaystyle G\,}$ is Newton's gravitational constant, ${\displaystyle c\,}$ the speed of light in vacuum, and ${\displaystyle T_{ik}\,}$ 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 vacuum permittivity of free space):^[125]

${\displaystyle F={\frac {\left|q_{1}q_{2}\right|}{4\pi \varepsilon _{0}r^{2}}}.}$

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.^[126] π is found in the Gaussian function (which is the probability density function of the normal distribution) with mean μ and standard deviation σ:^[127]

${\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}$

The area under the graph of the normal distribution curve is given by the Gaussian integral:^[127]

${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}dx={\sqrt {\pi }}.}$

π is present in some structural engineering formulae, such as the buckling formula, derived by Euler, that 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:^[128]

${\displaystyle F={\frac {\pi ^{2}EI}{s^{2}}}.}$

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 η:^[129]

${\displaystyle F=6\,\pi \,\eta \,R\,v.}$

The Fourier transform is a mathematical operation that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum. It has many applications in physics and engineering, particularly in signal processing:^[130]

${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx.}$

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.^[131][132]

Many persons have memorized large numbers of digits of π, a practice called piphilology.^[133] 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."^[133] When a poem is utilized, it is sometimes referred to as a "piem". Poems for memorizing π have been composed in several languages in addition to English.^[133]

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 November 20, 2005.^[134][135] 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.^[136] Record-setting π memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the method of loci.^[137]

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,^[138] and a full-length novel has been published which contains 10,000 words, each representing one digit of π.^[139]

Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae, π has been represented in popular culture more than most other mathematical constructs. Palais de la Découverte, a science museum in Paris, contains a circular room known as the "pi room". On its wall is inscribed 707 digits of π. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on a 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.^[140]

Many schools in the United States observe Pi Day on March 14 (March is the third month, hence the date is 3/14).^[142] π and its digital representation are often used by self-described "math geeks" for inside jokes among mathematically and technologically minded groups. Several college cheers at the Massachusetts Institute of Technology include "3.14159".^[141] 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.^[143]

Some individuals have proposed a new mathematical constant tau (τ), which equals two times π. Its proponents have argued that a constant based on the ratio of a circle's circumference to its radius rather than its diameter would be a more natural choice than π and would simplify many formulae.^[144][145] While their proposals, which include celebrating June 28 as "Tau Day", have been reported in the media, they have not been reflected in the scientific literature.^[146][147]

In Carl Sagan's novel Contact, π played a key role in the story. The novel suggested that there was a message buried deep within the digits of π placed there by the creator of the universe.^[148]

In 1897, an amateur mathematician attempted to persuade the Indiana legislature to pass the Indiana Pi Bill, which described a method to square the circle, and contained text which assumes various incorrect values of π, including 3.2. The bill is notorious as an attempt to establish scientific truth by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate.^[149]

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