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“π” and “ϖ” redirect here. For the Greek letter, see Pi (letter). For other uses, see Pi (disambiguation).

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 mid-18th century, though it is also sometimes spelled out as “pi” (/paɪ/).

Being an irrational number, π cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date no proof of this has been discovered. Also, π is a transcendental number – a number that is not the root of any non-zero 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.

Ancient civilizations needed the value of π to be computed accurately for practical reasons. It was calculated to seven digits, using geometrical techniques, in Chinese mathematics and to about five in Indian mathematics in the 5th century AD. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–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 2015, over 13.3 trillion (10 13 ) digits. [3] Practically all scientific applications require no more than a few hundred digits of π, and many substantially fewer, so the primary motivation for these computations is the human desire to break records. [4][5]

However, 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. Because of its special role as an eigenvalue, π appears in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics. It is also found in cosmology, 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 record-setting 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 70,000 digits.

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, and derived from the first letter of the Greek word perimetros, meaning circumference. [6] In English, π is pronounced as “pie” ( /paɪ/, paɪ). [7] In mathematical use, the lowercase letter π (or π in sans-serif 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 π.

Definition

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

π is commonly defined as the ratio of a circle’s circumference C to its diameter d: [8]

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 (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C/d. [8]

Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits, a concept in calculus. [9] For example, one may compute directly the arc length of the top half of the unit circle given in Cartesian coordinates by x 2 + y 2 = 1, as the integral: [10]

An integral such as this was adopted as the definition of π by Karl Weierstrass, who defined it directly as an integral in 1841. [11]

Definitions of π such as these that rely on a notion of circumference, and hence implicitly on concepts of the integral calculus, are no longer common in the literature. Remmert (1991) explains that this is because in many modern treatments of calculus, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer, [12] and popularized by Edmund Landau, [13] is the following: π is twice the smallest positive number at which the cosine function equals 0. [8][10][14]

The cosine can be defined independently of geometry as a power series, [15] or as the solution of a differential equation. [14]

In a similar spirit, π can be defined instead using properties of the complex exponential, exp(z), of a complex variable z. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which exp(z) is equal to one is then an (imaginary) arithmetic progression of the form:

and there is a unique positive real number π with this property. [10][16] A more abstract variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: [17]

there is a unique continuous isomorphism from the group R/Z of real numbers under addition modulo integers (the circle group) onto the multiplicative group of complex numbers of absolute value one. The number π is then defined as half the magnitude of the derivative of this homomorphism. [18]

A circle encloses the largest area that can be attained within a given perimeter. Thus the number π is also characterized as the best constant in the isoperimetric inequality (times one-fourth). There are many other, closely related, ways in which π appears as an eigenvalue of some geometrical or physical process; see below.

Irrationality and normality

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, but no common fraction (ratio of whole numbers) can be its exact value). [19]

Because 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. [20]

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. [21] The conjecture that π is normal has not been proven or disproven. [21]

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 frequencies of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found. [22] Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem. Thus, because the sequence of π’s digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π. [23]

Transcendence

Because π is a transcendental number, squaring the circle is not possible in a finite number of steps using the classical tools of compass and straightedge.

In addition to being irrational, more strongly π is a transcendental number, which means that it is not the solution of any non-constant polynomial with rational coefficients, such as x 5 120 − x 3 6 + x = 0. [24][25]

The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or n-th roots such 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 exactly equal to the area of a given circle. [26] Squaring a circle was one of the important geometry problems of the classical antiquity. [27] Amateur mathematicians in modern times have sometimes attempted to square the circle and sometimes claim success despite the fact that it is mathematically impossible. [28]

Continued fractions

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

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 number” (i.e., “not a rational number”). But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:

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 well-known and widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator. [29] Because π is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (shown above) also does not exhibit any other obvious pattern, [30]

mathematicians have discovered several generalized continued fractions that do, such as: [31]

Approximate value

Some approximations of pi include:

Integers: 3

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

Decimal: The first 50 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510… [32]

(see A000796)

Binary: The base 2 approximation to 48 digits is 11.0010 0100 0011 1111 0110 1010 1000 1000 1000 0101 1010 0011… (see A004601)

Hexadecimal: The base 16 approximation to 20 digits is 3.243F 6A88 85A3 08D3 1319… [33] (see A062964)

Sexagesimal: A base 60 approximation to five sexagesimal digits is 3;8,29,44,0,47 [34] (see A060707)

Complex numbers and Euler’s identity

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

Any complex number, say z, can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or r) is used to represent z’s distance from the origin of the complex plane and the other (angle or φ) to represent a counter-clockwise rotation from the positive real line as follows: [35]

where i is the imaginary unit satisfying i 2 = −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: [36]

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: [36][37]

There are n different complex numbers z satisfying z n = 1, and these are called the “n-th roots of unity”. [38] They are given by this formula:

Spectral characterizations

The overtones of a vibrating string are eigenfunctions of the second derivative, and form a harmonic progression. The associated eigenvalues form the arithmetic progression of integer multiples of π.

Many of the appearances of π in the formulas of mathematics and the sciences have to do with its close relationship with geometry. However, π also appears in many natural situations having apparently nothing to do with geometry.

In many applications it plays a distinguished role as an eigenvalue. For example, an idealized vibrating string can be modelled as the graph of a function f on the unit interval [0,1], with fixed ends f(0) = f(1) = 0. The modes of vibration of the string are solutions of the differential equation f “(x) + λ 2 f(x) = 0. Here λ is an associated eigenvalue, which is constrained by Sturm–Liouville theory to take on only certain specific values. The value λ = π is one such eigenvalue, as the function f(x) = sin(π x) satisfies the boundary conditions and the differential equation with λ = π. [39]

The ancient city of Carthage was the solution to an isoperimetric problem, according to a legend recounted by Lord Kelvin (Thompson 1894): those lands bordering the sea that Queen Dido could enclose on all other sides within a single given oxhide, cut into strips.

The value π is in fact the least such eigenvalue, and is associated with the fundamental mode of vibration of the string. One way to obtain this is by estimating the energy. The energy satisfies an inequality, Wirtinger’s inequality for functions, [40] which states that if a function f : [0, 1] → is given such that f(0) = f(1) = 0 and f and f ‘ are both square integrable, then the inequality holds:

and the case of equality holds precisely when f is a multiple of sin(π x). So π appears as an optimal constant in Wirtinger’s inequality, and from this it follows that it is the smallest such eigenvalue (by Rayleigh quotient methods).

The number π serves a similar role in higher-dimensional analysis, appearing as eigenvalues for other similar kinds of problems. As mentioned above, it can be characterized via its role as the best constant in the isoperimetric inequality: the area A enclosed by a plane Jordan curve of perimeter P satisfies the inequality

and equality is clearly achieved for the circle, since in that case A = πr 2 and P = 2πr. [41]

An animation of a geodesic in the Heisenberg group, showing the close connection between the Heisenberg group, isoperimetry, and the constant π. The cumulative height of the geodesic is equal to the area of the shaded portion of the unit circle, while the arc length (in the Carnot–Carathéodory metric) is equal to the circumference.

Ultimately as a consequence of the isoperimetric inequality, the constant π is associated with best constants of the Poincaré inequality. [42] As a special case, π appears as the optimal smallest eigenvalue of the Dirichlet energy, in dimensions 1 and 2, which thus characterizes the role of π in many physical phenomena as well, for example those of classical potential theory. [43][44][45] The one-dimensional case is just Wirtinger’s inequality.

The constant π also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as:

There are several different conventions for the Fourier transform, all of which involve a factor of π that is placed somewhere. The appearance of π is essential in these formulas, as there is there is no possibility to remove π altogether from the Fourier transform and its inverse transform. The definition given above is the most canonical however, because it describes the unique unitary operator on L 2 that is also an algebra homomorphism of L 1 to L ∞ . [46]

The Heisenberg uncertainty principle also contains the number π. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,

The physical consequence, about the uncertainty in simultaneous position and momentum observations of a quantum mechanical system, is discussed below. The appearance of π in the formulae of Fourier analysis is ultimately a consequence of the Stone–von Neumann theorem, asserting the uniqueness of the Schrödinger representation of the Heisenberg group. [47]

Gaussian integrals

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

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. [48] The Gaussian function, which is the probability density function of the normal distribution with mean μ and standard deviation σ, naturally contains π: [49]

For this to be a probability density, the area under the graph of f needs to be equal to one. This follows from a change of variables in the Gaussian integral: [49]

which says that the area under the basic Bell curve in the figure is equal to the square root of π.

π can be computed from the distribution of zeros of a one-dimensional Wiener process

The central limit theorem explains the central role of normal distributions, and thus of π, in probability and statistics. This theorem is ultimately connected with the spectral characterization of π as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function. [50] Equivalently, π is the unique constant making the Gaussian normal distribution e -πx 2 equal to its own Fourier transform. [51] Indeed, according to Howe (1980), the “whole business” of establishing the fundamental theorems Fourier analysis reduces to the Gaussian integral.

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

Antiquity

The best known approximations to π dating 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 [52] have claimed that the ancient Egyptians used an approximation of π as 22 7 from as early as the Old Kingdom. [53] This claim has met with skepticism. [54][55][56][57]

The earliest written approximations of π are found in Egypt and Babylon, both within one percent of the true value. In Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25 8 = 3.125. [58] In Egypt, the Rhind Papyrus, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats π as ( 16 9 ) 2 ≈ 3.1605. [58]

Astronomical calculations in the Shatapatha Brahmana (ca. 4th century BC) use a fractional approximation of 339 108 ≈ 3.139 (an accuracy of 9×10 −4 ). [59]

Other Indian sources by about 150 BC treat π as √10 ≈ 3.1622. [60]

Polygon approximation era

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

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. [61]

This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as “Archimedes’ constant”. [62] Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223 71 < π < 22 7 (that is 3.1408 < π < 3.1429). [63]

Archimedes' upper bound of 22 7 may have led to a widespread popular belief that π is equal to 22 7 . [64] 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. [65] 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. [66]

Archimedes developed the polygonal approach to approximating π.

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). [67]

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. [68][69] Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4. [68] The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355 113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920… remained the most accurate approximation of π available for the next 800 years. [70]

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabha īya (499 AD). [71] Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. [72]

Italian author Dante apparently employed the value 3+ √2 10 ≈ 3.14142. [72]

The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×2 28 sides, [73][74] which stood as the world record for about 180 years. [75]

French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×2 17 sides. [75] Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. [75] 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). [76] Dutch scientist Willebrord Snellius reached 34 digits in 1621, [77]

and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10 40 sides, [78] which remains the most accurate approximation manually achieved using polygonal algorithms. [77]

Infinite series

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

The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. [79]

Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques. [79]

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. [80] 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. [81] The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhā ā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 –c. 1425. [81] 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. [81] Madhava used infinite series to estimate π to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm. [82]

Isaac Newton used infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations". [83]

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: [84][85]

The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product: [84]

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." [83]

In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregory in 1671, and by Leibniz in 1674: [86][87]

This formula, the Gregory–Leibniz series, equals π/4 when evaluated with z = 1. [87] In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. [88] The Gregory–Leibniz for series is simple, but converges very slowly (that is, approaches the answer gradually), so it is not used in modern π calculations. [89]

In 1706 John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster: [90]

Machin reached 100 digits of π with this formula. [91] Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of π. [91] Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device. [92]

A 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. [93] 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. [93]

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. [94] A simple infinite series for π is the Gregory–Leibniz series: [95]

As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and – with a sufficient number of terms – can get as close to π as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of π. [96]

An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is: [97]

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

After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term. [94]

Irrationality and transcendence

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

Not all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function: [98]

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. [19]

Lambert's proof exploited a continued-fraction representation of the tangent function. [99] French mathematician Adrien-Marie 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. [100][101] Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers". [102]

Adoption of the symbol π

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

The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics. [103]

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. [104] 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. [105] It had indeed been used earlier for geometric concepts. [105] William Oughtred used π and δ, the Greek letter equivalents of p and d, to express ratios of periphery and diameter in the 1647 and later editions of Clavis Mathematicae. [106]

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. [105] Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly. [105] 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. [105]

Computer era and iterative algorithms

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

The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. [107] 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. [108] 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. [109]

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. [110] Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication. [111] They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods. [112]

The iterative algorithms were independently published in 1975–1976 by American physicist Eugene Salamin and Australian scientist Richard Brent. [113] These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm. [113] As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.

The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiply the number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borwein produced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step. [114] Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing π between 1995 and 2002. [115] This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series. [115]

Motivations for computing π

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

For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the circumference of the observable universe with a precision of one atom. [116] Despite this, people have worked strenuously to compute π to thousands and millions of digits. [117]

This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world. [118][119] They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π. [120]

Rapidly convergent series

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

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. [115] 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,

Aliearia