The number (; spelled out as "pi") is a
mathematical constant that is the
ratio of a
circle's
circumference to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
, approximately equal to 3.14159. The number appears in many formulas across
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics. It is an
irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as
are commonly
used to approximate it. Consequently, its
decimal representation never ends, nor
enters a permanently repeating pattern. It is a
transcendental number, meaning that it cannot be a solution of an
equation involving only sums, products, powers, and integers. The transcendence of implies that it is impossible to solve the ancient challenge of
squaring the circle with a
compass and straightedge. The decimal digits of appear to be
randomly distributed, but no proof of this conjecture has been found.
For thousands of years, mathematicians have attempted to extend their understanding of , sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the
Egyptians and
Babylonians, required fairly accurate approximations of for practical computations. Around 250BC, the
Greek mathematician
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
Archimedes created an algorithm to approximate with arbitrary accuracy. In the 5th century AD,
Chinese mathematicians approximated to seven digits, while
Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula for , based on
infinite series, was discovered a millennium later. The earliest known use of the Greek letter
π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician
William Jones in 1706.
The invention of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
soon led to the calculation of hundreds of digits of , enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and
computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test
supercomputers.
Because its definition relates to the circle, is found in many formulae in
trigonometry and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as
cosmology,
fractals,
thermodynamics,
mechanics, and
electromagnetism. In modern
mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as
number theory and
statistics. The ubiquity of makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to have been published, and record-setting calculations of the digits of often result in news headlines.
Fundamentals
Name
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase
Greek letter , sometimes spelled out as ''pi.''
[ In English, is pronounced as "pie" ( ). In mathematical use, the lowercase letter is distinguished from its capitalized and enlarged counterpart , which denotes a product of a sequence, analogous to how denotes summation.
The choice of the symbol is discussed in the section ''Adoption of the symbol ''.
]
Definition
is commonly defined as the ratio of a circle's circumference to its diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
:
The ratio is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio . This definition of implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula .
Here, the circumference of a circle is the arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits—a concept in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation , as the integral:
An integral such as this was adopted as the definition of by Karl Weierstrass, who defined it directly as an integral in 1841.
Integration is no longer commonly used in a first analytical definition because, as explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: is twice the smallest positive number at which the cosine function equals 0. is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential equation.
In a similar spirit, can be defined using properties of the complex exponential, , of a complex variable . Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which is equal to one is then an (imaginary) arithmetic progression of the form:
and there is a unique positive real number with this property.
A variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique ( up to automorphism) continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
isomorphism from the group R/Z of real numbers under addition modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
integers (the circle group), onto the multiplicative group of complex numbers of absolute value one. The number is then defined as half the magnitude of the derivative of this homomorphism.
Irrationality and normality
is an irrational number, meaning that it cannot be written as the ratio of two integers. Fractions such as and are commonly used to approximate , but no common fraction (ratio of whole numbers) can be its exact value. Because is irrational, it has an infinite number of digits in its decimal representation, and does not settle into an infinitely repeating pattern of digits. There are several proofs that is irrational; they generally require calculus and rely on the '' reductio ad absurdum'' technique. The degree to which can be approximated by rational numbers (called the irrationality measure
In number theory, a Liouville number is a real number ''x'' with the property that, for every positive integer ''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that
:0 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists ...
) is not precisely known; estimates have established that the irrationality measure is larger than the measure of or but smaller than the measure of Liouville numbers.
The digits of 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. The conjecture that is normal has not been proven or disproven.
Since the advent of computers, a large number of digits of have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of , 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. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would ...
. Thus, because the sequence of '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 . This is also called the "Feynman point" in mathematical folklore, after Richard Feynman, although no connection to Feynman is known.
Transcendence
In addition to being irrational, is also a transcendental number, which means that it is not the solution
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Solutio ...
of any non-constant polynomial equation with rational coefficients, such as .
The transcendence of has two important consequences: First, cannot be expressed using any finite combination of rational numbers and square roots or ''n''-th roots (such as or ). Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to "square the circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficult ...
". 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. Squaring a circle was one of the important geometry problems of the classical antiquity
Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...
. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.
Continued fractions
Like all irrational numbers, cannot be represented as a common fraction (also known as a simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
or vulgar fraction), by the very definition of irrational number (i.e., not a rational number). But every irrational number, including , 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 ; the first four of these are , , , and . These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to than any other fraction with the same or a smaller denominator. Because is known to be transcendental, it is by definition not algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
and so cannot be a quadratic irrational. Therefore, cannot have a periodic continued fraction. Although the simple continued fraction for (shown above) also does not exhibit any other obvious pattern, mathematicians have discovered several generalized continued fractions that do, such as:
Approximate value and digits
Some approximations of ''pi'' include:
* Integers: 3
* Fractions: Approximate fractions include (in order of increasing accuracy) , , , , , , and . (List is selected terms from and .)
* Digits: The first 50 decimal digits are (see )
Digits in other number systems
* The first 48 binary ( base 2) digits (called bits) are (see )
* The first 20 digits in hexadecimal (base 16) are (see )
* The first five sexagesimal (base 60) digits are 3;8,29,44,0,47 (see )
* The first 38 digits in the ternary numeral system are (see )
Complex numbers and Euler's identity
Any complex number, say , can be expressed using a pair of real numbers. In the polar coordinate system, one number ( radius or ''r'') is used to represent 's distance from the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
of the complex plane, and the other (angle or ) the counter-clockwise rotation from the positive real line:
where is the imaginary unit satisfying = −1. The frequent appearance of in complex analysis can be related to the behaviour of the exponential function of a complex variable, described by Euler's formula:
where the constant is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of and points on the unit circle centred at the origin of the complex plane. Setting = in Euler's formula results in Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
, celebrated in mathematics due to it containing five important mathematical constants:
There are different complex numbers satisfying , and these are called the "-th roots of unity" and are given by the formula:
History
Antiquity
The best-known approximations to 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.
The earliest written approximations of are found in Babylon and Egypt, 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 as = 3.125. 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 as 3.16.[ Although some pyramidologists such as Flinders Petrie have theorized that the Great Pyramid of Giza was built with proportions related to , this theory is not widely accepted by scholars.
In the ]Shulba Sutras The ''Shulva Sutras'' or ''Śulbasūtras'' ( Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction.
Purpose and origins
T ...
of Indian mathematics, dating to an oral tradition from the first or second millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.
Polygon approximation era
The first recorded algorithm for rigorously calculating the value of was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result is sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of 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 (that is ). Archimedes' upper bound of may have led to a widespread popular belief that is equal to . Around 150 AD, Greek-Roman scientist Ptolemy, in his '' Almagest'', gave a value for of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.]
In ancient China, values for included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and (3rd century, approximately 3.1556). 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 of 3.1416. Liu later invented a faster method of calculating 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. The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that and suggested the approximations = 3.14159292035... and = 3.142857142857..., which he termed the '' Milü'' (''close ratio") and ''Yuelü'' ("approximate ratio"), respectively, using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value remained the most accurate approximation of available for the next 800 years.
The Indian astronomer Aryabhata used a value of 3.1416 in his '' Āryabhaṭīya'' (499 AD). Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value .
The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×228 sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen
Ludolph van Ceulen (, ; 28 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim. He emigrated to the Netherlands.
Biography
Van Ceulen moved to Delft most likely in 1576 to teach fencing and mathematics and in 159 ...
reached 20 digits, a record he later increased to 35 digits (as a result, was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius
Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, Snell. His name is usually associated with the law of refraction of light known as Snell's law.
The lunar crater ...
reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger
Christoph (Christophorus) Grienberger (also variously spelled Gruemberger, Bamberga, Bamberger, Banbergiera, Gamberger, Ghambergier, Granberger, Panberger) (2 July 1561 – 11 March 1636) was an Austrian Jesuit astronomer, after whom the crater ...
arrived at 38 digits in 1630 using 1040 sides. Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, ...
.
Infinite series
The calculation of 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. Infinite series allowed mathematicians to compute with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach also appeared in the Kerala school sometime between 1400 and 1500 AD. Around 1500 AD, a written description of an infinite series that could be used to compute was laid out in Sanskrit verse in '' Tantrasamgraha'' by Nilakantha Somayaji. The series are presented without proof, but proofs are presented in a later work, '' Yuktibhāṣā'', from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 – c. 1425. Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series
In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by an Indian Mathematician and Astronomer Madhava of Sangamagrama (c.&nbs ...
or Gregory–Leibniz series. Madhava used infinite series to estimate to 11 digits around 1400, but that value was improved on around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.
In 1593, François Viète published what is now known as Viète's formula, an infinite product (rather than an infinite sum, which is more typically used in calculations):
In 1655, John Wallis published what is now known as Wallis product, also an infinite product:
In the 1660s, the English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz discovered calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, which led to the development of many infinite series for approximating . Newton himself used an arcsin series to compute a 15-digit approximation of in 1665 or 1666, writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[. Newton quoted by Arndt.]
In 1671, James Gregory, and independently, Leibniz in 1674, published the series:
This series, sometimes called the Gregory–Leibniz series, equals when evaluated with = 1.
In 1699, English mathematician Abraham Sharp
Abraham Sharp (1653 – 18 July 1742) was an English mathematician and astronomer.
Life
Sharp was born in Horton Hall in Little Horton, Bradford, the son of well-to-do merchant John Sharp and Mary (née Clarkson) Sharp and was educated at Bradf ...
used the Gregory–Leibniz series for to compute to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm. The Gregory–Leibniz series for is simple, but converges very slowly (that is, approaches the answer gradually), so it is not used in modern calculations.
In 1706, John Machin
John Machin (bapt. c. 1686 – June 9, 1751) was a professor of astronomy at Gresham College, London. He is best known for developing a quickly converging series for pi in 1706 and using it to compute pi to 100 decimal places.
History ...
used the Gregory–Leibniz series to produce an algorithm that converged much faster:
Machin reached 100 digits of with this formula. Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of . Machin-like formulae remained the best-known method for calculating 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.
In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of in his head at the behest of German mathematician Carl Friedrich Gauss.
In 1853, British mathematician William Shanks calculated to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.
Rate of convergence
Some infinite series for converge faster than others. Given the choice of two infinite series for , mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate to any given accuracy.
A simple infinite series for is the Gregory–Leibniz series:
As individual terms of this infinite series are added to the sum, the total gradually gets closer to , and – with a sufficient number of terms – can get as close to as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of .
An infinite series for (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:
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 , whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of . Series that converge even faster include Machin's series and Chudnovsky's series, the latter producing 14 correct decimal digits per term.
Irrationality and transcendence
Not all mathematical advances relating to 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 and the prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s that later contributed to the development and study of the Riemann zeta function:
Swiss scientist Johann Heinrich Lambert in 1768 proved that is irrational, meaning it is not equal to the quotient of any two integers. Lambert's proof exploited a continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that 2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that is transcendental, confirming a conjecture made by both Legendre and Euler. Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".
Adoption of the symbol
In the earliest usages, the Greek letter was used to denote the semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate ...
(''semiperipheria'' in Latin) of a circle. and was combined in ratios with δ (for diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
or semidiameter) or ρ (for radius) to form circle constants. (Before then, mathematicians sometimes used letters such as ''c'' or ''p'' instead.) The first recorded use is Oughtred's , to express the ratio of periphery and diameter in the 1647 and later editions of . Barrow likewise used "" to represent the constant 3.14..., while Gregory instead used "" to represent 6.28... .
The earliest known use of the Greek letter alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work ''; or, a New Introduction to the Mathematics''. The Greek letter first appears there in the phrase "1/2 Periphery ()" in the discussion of a circle with radius one. However, he writes that his equations for are from the "ready pen of the truly ingenious Mr. John Machin
John Machin (bapt. c. 1686 – June 9, 1751) was a professor of astronomy at Gresham College, London. He is best known for developing a quickly converging series for pi in 1706 and using it to compute pi to 100 decimal places.
History ...
", leading to speculation that Machin may have employed the Greek letter before Jones. Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.
Euler started using the single-letter form beginning with his 1727 ''Essay Explaining the Properties of Air'', though he used , the ratio of periphery to radius, in this and some later writing. Euler first used in his 1736 work '' Mechanica'', and continued in his widely-read 1748 work (he wrote: "for the sake of brevity we will write this number as ; thus is equal to half the circumference of a circle of radius 1"). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the Western world, though the definition still varied between 3.14... and 6.28... as late as 1761.
Modern quest for more digits
Computer era and iterative algorithms
The development of computers in the mid-20th century again revolutionized the hunt for digits of . Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an inverse tangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
(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. 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.
Two additional developments around 1980 once again accelerated the ability to compute . First, the discovery of new iterative algorithms for computing , which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly. Such algorithms are particularly important in modern computations because most of the computer's time is devoted to multiplication. They include the Karatsuba algorithm, Toom–Cook multiplication, and Fourier transform-based methods.
The iterative algorithms were independently published in 1975–1976 by physicist Eugene Salamin and scientist Richard Brent. 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 The Gauss–Legendre algorithm is an algorithm to compute the digits of . It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of . However, it has some drawbacks (for example, it is computer ...
. 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, brothers John and Peter Borwein
Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician
and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–P ...
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. Iterative methods were used by Japanese mathematician Yasumasa Kanada to set several records for computing between 1995 and 2002. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.
Motives for computing
For most numerical calculations involving , 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. Accounting for additional digits needed to compensate for computational round-off errors, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with often make headlines around the world. 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 .
Rapidly convergent series
Modern 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. The fast iterative algorithms were anticipated in 1914, when Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for , remarkable for their elegance, mathematical depth and rapid convergence. One of his formulae, based on modular equation
In mathematics, a modular equation is an algebraic equation satisfied by ''moduli'', in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other word ...
s, is
This series converges much more rapidly than most arctan series, including Machin's formula. Bill Gosper was the first to use it for advances in the calculation of , setting a record of 17 million digits in 1985. Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (Jonathan
Jonathan may refer to:
*Jonathan (name), a masculine given name
Media
* ''Jonathan'' (1970 film), a German film directed by Hans W. Geißendörfer
* ''Jonathan'' (2016 film), a German film directed by Piotr J. Lewandowski
* ''Jonathan'' (2018 ...
and Peter) and the Chudnovsky brothers. The Chudnovsky formula developed in 1987 is
It produces about 14 digits of per term, and has been used for several record-setting calculations, including the first to surpass 1 billion (109) digits in 1989 by the Chudnovsky brothers, 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo, and 100 trillion digits by Emma Haruka Iwao in 2022. For similar formulas, see also the Ramanujan–Sato series In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as,
:\frac = \frac \sum_^\infty \frac \frac
to the form
:\frac = \sum_^\infty s(k) \frac
by using other well-defined sequences of integers s(k) obeying a c ...
.
In 2006, mathematician Simon Plouffe used the PSLQ integer relation algorithm to generate several new formulas for , conforming to the following template:
where is (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.
Monte Carlo methods
Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of . Buffon's needle is one such technique: If a needle of length is dropped times on a surface on which parallel lines are drawn units apart, and if of those times it comes to rest crossing a line ( > 0), then one may approximate based on the counts:
Another Monte Carlo method for computing 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 .
Another way to calculate using probability is to start with a random walk, generated by a sequence of (fair) coin tosses: independent random variables such that with equal probabilities. The associated random walk is
so that, for each , is drawn from a shifted and scaled binomial distribution. As varies, defines a (discrete) stochastic process. Then can be calculated by
This Monte Carlo method is independent of any relation to circles, and is a consequence of the central limit theorem, discussed below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
These Monte Carlo methods for approximating are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximate when speed or accuracy is desired.
Spigot algorithms
Two algorithms were discovered in 1995 that opened up new avenues of research into . They are called spigot algorithm A spigot algorithm is an algorithm for computing the value of a transcendental number (such as or ''e'') that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot alg ...
s because, like water dripping from a spigot, they produce single digits of that are not reused after they are calculated. This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.
Mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995. Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.
Another spigot algorithm, the BBP digit extraction algorithm, was discovered in 1995 by Simon Plouffe:
This formula, unlike others before it, can produce any individual hexadecimal digit of without calculating all the preceding digits. Individual binary digits may be extracted from individual hexadecimal digits, and octal
The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...
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. An important application of digit extraction algorithms is to validate new claims of record 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.
Between 1998 and 2000, the distributed computing project PiHex PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of . 1,246 contributors used idle time slices on almost two thousand computers to make its calculations. The software used for the project made use o ...
used Bellard's formula
Bellard's formula is used to calculate the ''n''th digit of π in base 16.
Bellard's formula was discovered by Fabrice Bellard in 1997. It is about 43% faster than the Bailey–Borwein–Plouffe formula
The Bailey–Borwein–Plouffe formula (BB ...
(a modification of the BBP algorithm) to compute the quadrillionth (1015th) bit of , which turned out to be 0. In September 2010, a Yahoo! employee used the company's Hadoop application on one thousand computers over a 23-day period to compute 256 bits of at the two-quadrillionth (2×1015th) bit, which also happens to be zero.
Role and characterizations in mathematics
Because 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. Other branches of science, such as statistics, physics, Fourier analysis, and number theory, also include in some of their important formulae.
Geometry and trigonometry
appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s, cones, and tori. Below are some of the more common formulae that involve .
* The circumference of a circle with radius is .
* The area of a circle with radius is .
* The area of an ellipse with semi-major axis and semi-minor axis is .
* The volume of a sphere with radius is .
* The surface area of a sphere with radius is .
Some of the formulae above are special cases of the volume of the ''n''-dimensional ball and the surface area of its boundary, the (''n''−1)-dimensional sphere, given below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
Apart from circles, there are other curves of constant width. By Barbier's theorem, every curve of constant width has perimeter times its width. The Reuleaux triangle (formed by the intersection of three circles with the sides of an equilateral triangle as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular smooth and even algebraic curves of constant width.
Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involve . For example, an integral that specifies half the area of a circle of radius one is given by:
In that integral the function represents the height over the -axis of a semicircle (the square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
is a consequence of the Pythagorean theorem), and the integral computes the area below the semicircle.
Units of angle
The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2 radians. The angle measure of 180° is equal to radians, and 1° = /180 radians.
Common trigonometric functions have periods that are multiples of ; for example, sine and cosine have period 2, so for any angle and any integer ,
Eigenvalues
Many of the appearances of in the formulas of mathematics and the sciences have to do with its close relationship with geometry. However, 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 on the unit interval , with fixed ends . The modes of vibration of the string are solutions of the differential equation , or . Thus is an eigenvalue of the second derivative operator , and is constrained by Sturm–Liouville theory to take on only certain specific values. It must be positive, since the operator is negative definite, so it is convenient to write , where is called the wavenumber. Then satisfies the boundary conditions and the differential equation with .
The value is, in fact, the ''least'' such value of the wavenumber, and is associated with the fundamental mode of vibration of the string. One way to show this is by estimating the energy, which satisfies Wirtinger's inequality: for a function with and , both square integrable, we have:
with equality precisely when is a multiple of . Here appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the variational characterization of the eigenvalue. As a consequence, is the smallest singular value of the derivative operator on the space of functions on vanishing at both endpoints (the Sobolev space