The number (; spelled out as "pi") is a
mathematical constant that is the
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
's
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
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 fo ...
, approximately equal to 3.14159. The number appears in many formulas across
mathematics and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. It is an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, 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
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
never ends, nor
enters a permanently repeating pattern. It is a
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
, 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
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. 1 ...
, required fairly accurate approximations of for practical computations. Around 250BC, the
Greek mathematician Archimedes created an algorithm to approximate with arbitrary accuracy. In the 5th century AD,
Chinese mathematicians
Mathematics in China emerged independently by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system ( base 2 and base 10), algebra, geomet ...
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
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, 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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
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
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
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 ...
, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
,
fractals,
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
,
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
, and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
. In modern
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, 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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
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
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
, 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
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of a circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
's circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
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 fo ...
:
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
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
of the circle, a quantity which can be formally defined independently of geometry using limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
—a concept in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. 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
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
:
An integral such as this was adopted as the definition of by Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, 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
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopol ...
, 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 mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
.
In a similar spirit, can be defined using properties of the complex exponential
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
, , of a complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, is the following theorem: there is a unique ( up to automorphism) continuous isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
from the group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
R/Z of real numbers under addition modulo integers (the circle group), onto the multiplicative group of complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
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
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, 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
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
(ratio of whole numbers) can be its exact value. Because is irrational, it has an infinite number of digits in its decimal representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, i ...
, 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
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
'' technique. The degree to which can be approximated by rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s (called the irrationality measure) 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 number
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 ...
s.
The digits of have no apparent pattern and have passed tests for statistical randomness A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll or the digits of π exhibit statistical randomness.
Statistical randomness does ...
, 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
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
was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of . He set the record 11 of the past 21 times.
Kanada was a professor in the Department of Information Science at ...
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
In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics, or ...
, after Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
, although no connection to Feynman is known.
Transcendence
In addition to being irrational, is also a transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
, which means that it is not the solution of any non-constant polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
with rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
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". 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
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
(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
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
), 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
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
:
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 and so cannot be a quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...
. Therefore, cannot have a periodic continued fraction
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
:
x = a_0 + \cfrac
where the initial block of ''k'' + 1 partial denominators is followed by a block 'a'k''+1, ''a'k''+2,.. ...
. Although the simple continued fraction for (shown above) also does not exhibit any other obvious pattern, mathematicians have discovered several generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.
A ge ...
s 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
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
( base 2) digits (called bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s) are (see )
* The first 20 digits in hexadecimal (base 16) are (see )
* The first five sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
(base 60) digits are 3;8,29,44,0,47 (see )
* The first 38 digits in the ternary numeral system
A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information.
Although ''ternary'' m ...
are (see )
Complex numbers and Euler's identity
Any complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
, say , can be expressed using a pair of real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. In the polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, one number (radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
or ''r'') is used to represent 's distance from the origin of the complex plane, and the other (angle or ) the counter-clockwise rotation from the positive real line:
where is the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
satisfying = −1. The frequent appearance of in complex analysis can be related to the behaviour of the exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
of a complex variable, described by Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
:
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
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
centred at the origin of the complex plane. Setting = in Euler's formula results in Euler's identity, celebrated in mathematics due to it containing five important mathematical constants:
There are different complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s satisfying , and these are called the "-th roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
" and are given by the formula:
History
Antiquity
The best-known approximations to dating before the Common Era
Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
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
In the Ancient Near East, clay tablets ( Akkadian ) were used as a writing medium, especially for writing in cuneiform, throughout the Bronze Age and well into the Iron Age.
Cuneiform characters were imprinted on a wet clay tablet with a sty ...
dated 1900–1600 BC has a geometrical statement that, by implication, treats as = 3.125. In Egypt, the Rhind Papyrus
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
, 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
Sir William Matthew Flinders Petrie ( – ), commonly known as simply Flinders Petrie, was a British Egyptologist and a pioneer of systematic methodology in archaeology and the preservation of artefacts. He held the first chair of Egyp ...
have theorized that the Great Pyramid of Giza
The Great Pyramid of Giza is the biggest Egyptian pyramid and the tomb of Fourth Dynasty pharaoh Khufu. Built in the early 26th century BC during a period of around 27 years, the pyramid is the oldest of the Seven Wonders of the Ancient Worl ...
was built with proportions related to , this theory is not widely accepted by scholars.
In the Shulba Sutras 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
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
, 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
Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
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
Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...
, 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
Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which ...
used a value of 3.1416 in his ''Āryabhaṭīya
''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Indian astronomy, Sanskrit astronomical treatise, is the ''Masterpiece, magnum opus'' and only known surviving work of the 5th century Indian mathematics, Indian mathematician Aryabhata. Philos ...
'' (499 AD). Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante
Dante Alighieri (; – 14 September 1321), probably baptized Durante di Alighiero degli Alighieri and often referred to as Dante (, ), was an Italian people, Italian Italian poetry, poet, writer and philosopher. His ''Divine Comedy'', origin ...
apparently employed the value .
The Persian astronomer Jamshīd al-Kāshī
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer ...
produced 9 sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
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
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
in 1579 achieved 9 digits with a polygon of 3×217 sides. Flemish mathematician Adriaan van Roomen
Adriaan van Roomen (29 September 1561 – 4 May 1615), also known as Adrianus Romanus, was a mathematician, professor of medicine and medical astronomer from the Duchy of Brabant in the Habsburg Netherlands who was active throughout Central Europ ...
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 reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger 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.
Infinite series
The calculation of was revolutionized by the development of infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. 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
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
, 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
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
verse in ''Tantrasamgraha
Tantrasamgraha, or Tantrasangraha, (literally, ''A Compilation of the System'') is an important astronomical treatise written by Nilakantha Somayaji, an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics.
The ...
'' by Nilakantha Somayaji
Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...
. 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
Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He ...
, 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 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
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer ...
, using a polygonal algorithm.
In 1593, François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
published what is now known as Viète's formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant :
\frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots
It can also be represented as:
\frac2\pi ...
, an infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
(rather than an infinite sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, 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
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
and German mathematician Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
discovered calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, 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
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
in 1674, published the series:
This series, sometimes called the Gregory–Leibniz series, equals when evaluated with = 1.
In 1699, English mathematician Abraham Sharp 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 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
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
.
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
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
* Converge ICT, internet service provider in the Philippines
*CONVERGE CFD s ...
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
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
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 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 because the only ways ...
s that later contributed to the development and study of the Riemann zeta function:
Swiss scientist Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
in 1768 proved that is irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, 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
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
proved in 1794 that 2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
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 na ...
(''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 fo ...
or semidiameter) or ρ (for radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
) 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", 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
''Mechanica'' ( la, Mechanica sive motus scientia analytice exposita; 1736) is a two-volume work published by mathematician Leonhard Euler which describes analytically the mathematics governing movement.
Euler both developed the techniques of ...
'', 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
The Western world, also known as the West, primarily refers to the various nations and states in the regions of Europe, North America, and Oceania. , 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 (arctan) infinite series, a team led by George Reitwiesner and John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC
ENIAC (; Electronic Numerical Integrator and Computer) was the first programmable, electronic, general-purpose digital computer, completed in 1945. There were other computers that had these features, but the ENIAC had all of them in one pac ...
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 algorithm
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
s 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
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962.
Knuth D.E. (1969) ''The Art of Computer Programming. v.2.'' Addison-Wesley Publ.Co., 724 pp.
It is a div ...
, Toom–Cook multiplication Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers.
Given two large integ ...
, 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
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, in what is now termed the arithmetic–geometric mean method (AGM method) or Gauss–Legendre algorithm. 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
John is a common English name and surname:
* John (given name)
* John (surname)
John may also refer to:
New Testament
Works
* Gospel of John, a title often shortened to John
* First Epistle of John, often shortened to 1 John
* Secon ...
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. Iterative methods were used by Japanese mathematician Yasumasa Kanada
was a Japanese computer scientist most known for his numerous world records over the past three decades for calculating digits of . He set the record 11 of the past 21 times.
Kanada was a professor in the Department of Information Science at ...
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
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
calculations, because that is the accuracy necessary to calculate the circumference of the observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
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 equations, is
This series converges much more rapidly than most arctan series, including Machin's formula. Bill Gosper
Ralph William Gosper Jr. (born April 26, 1943), known as Bill Gosper, is an American mathematician and programmer. Along with Richard Greenblatt, he may be considered to have founded the hacker community, and he holds a place of pride in the ...
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 and Peter
Peter may refer to:
People
* List of people named Peter, a list of people and fictional characters with the given name
* Peter (given name)
** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church
* Peter (surname), a sur ...
) and the Chudnovsky brothers
David Volfovich Chudnovsky (born January 22, 1947 in Kyiv) and Gregory Volfovich Chudnovsky (born April 17, 1952 in Kyiv) are Ukrainian-born American mathematicians and engineers known for their world-record mathematical calculations and developing ...
. 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 2006, mathematician Simon Plouffe
Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the '' ...
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
In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:
:Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. ...
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
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
, 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
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
, discussed 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 algorithms 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
Stanley Wagon is a Canadian-American mathematician, a professor of mathematics at Macalester College in Minnesota. He is the author of multiple books on number theory, geometry, and computational mathematics, and is also known for his snow sculp ...
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 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 ...
, 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 base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 number ...
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
A distributed system is a system whose components are located on different networked computers, which communicate and coordinate their actions by passing messages to one another from any system. Distributed computing is a field of computer sci ...
project PiHex used Bellard's formula (a modification of the BBP algorithm) to compute the quadrillionth (1015th) bit of , which turned out to be 0. In September 2010, a Yahoo!
Yahoo! (, styled yahoo''!'' in its logo) is an American web services provider. It is headquartered in Sunnyvale, California and operated by the namesake company Yahoo Inc., which is 90% owned by investment funds managed by Apollo Global Manage ...
employee used the company's Hadoop
Apache Hadoop () is a collection of open-source software utilities that facilitates using a network of many computers to solve problems involving massive amounts of data and computation. It provides a software framework for distributed storage an ...
application on one thousand computers over a 23-day period to compute 256 bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s 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 th ...
s, cones
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines conn ...
, 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
In geometry, the area enclosed by a circle of radius is . Here the Greek letter represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.
One method of deriving this formula, which origi ...
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.
Apart from circles, there are other curves of constant width. By Barbier's theorem
In geometry, Barbier's theorem states that every curve of constant width has perimeter times its width, regardless of its precise shape. This theorem was first published by Joseph-Émile Barbier in 1860.
Examples
The most familiar examples of ...
, 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
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
and even algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s 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
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
(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 function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s rely on angles, and mathematicians generally use radians as units of measurement. plays an important role in angles measured in radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s, 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
. For example, an idealized vibrating string
A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or tension of the string is correctly adjusted, the sound produced is a musical tone. Vibrating ...
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
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, 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
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
. 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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
, which satisfies Wirtinger's inequality: for a function with and , both square integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
, 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 In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self ...
of the derivative operator on the space of functions on vanishing at both endpoints (the Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
when visiting India and exploring its contributions to trigonometry.
(a science museum in Paris) there is a circular room known as the ''pi room''. On its wall are inscribed 707 digits of . The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1873 calculation by English mathematician
, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.
In
'' it is suggested that the creator of the universe buried a message deep within the digits of . The digits of have also been incorporated into the lyrics of the song "Pi" from the 2005 album ''
. In the 1967 ''
", an out-of-control computer is contained by being instructed to "Compute to the last digit the value of ".