When a circle's diameter is 1, its circumference is π.

List of numbers – Irrational numbers ζ(3) – √2 – √3 – √5 – φ – α – e – π – δ
Binary	11. This is a list of articles about Numbers ( not about Numerals. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, Apéry's Constant is a curious number that occurs in a variety of situations The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2}   or   √2 Geometry If an equilateral triangle ( Equilateral polygon with three sides with sides of length 1 is cut into two equal halves by bisecting an internal angle across Continued fraction It can be expressed as the Continued fraction 4 4 4 4 4 In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. 00100100001111110110…
Decimal	3. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 14159265358979323846…
Hexadecimal	3. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a 243F6A8885A308D31319…
Continued fraction	Note that this continued fraction is not periodic. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}}

Pi or π is a mathematical constant which represents the ratio of any circle's circumference to its diameter in Euclidean geometry, which is the same as the ratio of a circle's area to the square of its radius. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. It is approximately equal to 3. 14159. Pi is one of the most important mathematical constants: many formulae from mathematics, science, and engineering involve π. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Engineering is the Discipline and Profession of applying technical and scientific Knowledge and ^[1]

Pi is an irrational number, which means that it cannot be expressed as a fraction m/n, where m and n are integers. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Consequently its decimal representation never ends or repeats. This article gives a mathematical definition For a more accessible article see Decimal. Beyond being irrational, it is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French ) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", probably by William Jones in 1706, and popularized by Leonhard Euler some years later. William Jones (1675 &ndash 3 July 1749) was a Welsh Mathematician, born in the village of Llanfihangel Tre'r Beirdd, on the The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer An Archimedes number (not to be confused with Archimedes' constant, π) named after the ancient Greek scientist Archimedes —used to determine the motion Ludolph van Ceulen ( 28 January 1540 &ndash 31 December 1610) was a German Mathematician from Hildesheim.

Fundamentals

The letter π

Lower-case π is used for the constant.

Main article: pi (letter)

The name of the Greek letter π is pi, and this spelling is used in typographical contexts where the Greek letter is not available or where its usage could be problematic. Pi (uppercase &Pi, lower case &pi) is the sixteenth letter of the Greek alphabet. Pi (uppercase &Pi, lower case &pi) is the sixteenth letter of the Greek alphabet. Typography is the art and techniques of arranging type, Type design, and modifying type Glyphs Type glyphs are created and modified using a variety When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States In Greek, the name of this letter is pronounced /pi/.

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly ^[2] π is Unicode character U+03C0 ("Greek small letter pi"). In Computing, Unicode is an Industry standard allowing Computers to consistently represent and manipulate text expressed in most of the world's For other uses see Character. In Computer and machine-based Telecommunications terminology a character is a unit of The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early ^[3]

Definition

Circumference = π × diameter

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:^[2]

Note that the ratio ^c/_d does not depend on the size of the circle. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. A ratio is an expression which compares quantities relative to each other Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the For example, if a circle has twice the diameter d of another circle it will also have twice the circumference c, preserving the ratio ^c/_d. This fact is a consequence of the similarity of all circles. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other

Area of the circle = π × area of the shaded square

Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius:^[2][4]

The constant π may be defined in other ways that avoid the concepts of arc length and area, for example, as twice the smallest positive x for which cos(x) = 0. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers In Geometry, an arc is a closed segment of a Differentiable Curve in the two-dimensional plane; for example a circular ^[5] The formulas below illustrate other (equivalent) definitions.

Irrationality and transcendence

Main article: Proof that π is irrational

The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. Although the Mathematical constant known as &pi (pi has been studied since ancient times and so has the concept of Irrational number, it was not until the 18th In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This was proven in 1761 by Johann Heinrich Lambert. Year 1761 ( MDCCLXI) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and ^[2] In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known. Ivan Morton Niven ( October 25 1915 &ndash May 9 1999) was a Canadian - American Mathematician, specializing in ^[6][7] A somewhat earlier similar proof is by Mary Cartwright. Dame Mary Lucy Cartwright DBE ( December 17, 1900 &ndash April 3, 1998) was a leading 20th-century British Mathematician ^[8]

Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation Carl Louis Ferdinand von Lindemann ( April 12, 1852 &ndash March 6 1939) was a German Mathematician, noted for his proof Year 1882 ( MDCCCLXXXII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common This means that there is no polynomial with rational coefficients of which π is a root. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions This article is about the zeros of a function which should not be confused with the value at zero. ^[9] An important consequence of the transcendence of π is the fact that it is not constructible. A point in the Euclidean plane is a constructible point if given a fixed Coordinate system (or a fixed Line segment of unit Length Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles Squaring the circle is a problem proposed by ancient Geometers. ^[10]

Numerical value

See also: numerical approximations of π

The numerical value of π truncated to 50 decimal places is:^[11]

3. This page is about the history of numerical approximations of the Mathematical constant &pi. In Mathematics, truncation is the term for limiting the number of digits right of the Decimal point, by discarding the least significant ones The decimal ( base ten or occasionally denary) Numeral system has ten as its base. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

See the links below and those at sequence A000796 in OEIS for more digits. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

While the value of pi has been computed to more than a trillion (10¹²) digits,^[12] elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. This list compares various sizes of positive Numbers including counts of things Dimensionless quantity and probabilities. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom. In Big Bang Cosmology, the observable universe is the region of space bounded by a Sphere, centered on the observer that is small enough that A hydrogen atom is an atom of the chemical element Hydrogen. The electrically neutral ^[13][14]

Because π is an irrational number, its decimal expansion never ends and does not repeat. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. ^[15] Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found. A supercomputer is a Computer that is at the frontline of processing capacity particularly speed of calculation (at the time of its introduction This list compares various sizes of positive Numbers including counts of things Dimensionless quantity and probabilities. ^[16] Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. Over the years several programs have been written for calculating Pi (π to many digits on Personal computers General-purpose Most Computer A personal computer ( PC) is any Computer whose original sales price size and capabilities make it useful for individuals and which is intended to be operated

Calculating π

Main article: Computing π

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Standard methods Circles π is defined as the ratio of the circumference of the circle to its diameter Another geometry-based approach, due to Archimedes^[17], is to calculate the perimeter, P_n , of a regular polygon with n sides circumscribed around a circle with diameter d. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer The perimeter is the distance around a given two-dimensional object General properties These properties apply to both convex and star regular polygons Then

i. e. , the more sides the polygon has, the closer the approximation.

Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. In Geometry, an inscribed Planar Shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid

π can also be calculated using purely mathematical methods. Most formulas used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives However, some are quite simple, such as this form of the Gregory-Leibniz series:^[18]

. See Leibniz formula for other formulas known under the same name

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that 300 terms are not sufficient to calculate π correctly to 2 decimal places. ^[19] However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

and then define

for all

then computing π_10,10 will take similar computation time to computing 150 terms of the original series in a brute force manner, and , correct to 9 decimal places. This computation is an example of the Van Wijngaarden transformation^[20]. In Mathematics and Numerical analysis, in order to accelerate convergence Euler's transform can be implemented as follows compute the partial sums of an alternating

History

See also: Chronology of computation of π and Numerical approximations of π

The history of π parallels the development of mathematics as a whole. The table below is a brief chronology of computed numerical values of or bounds on the mathematical constant &pi. This page is about the history of numerical approximations of the Mathematical constant &pi. ^[21] Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers. ^[22]

Geometrical period

That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value. ^[2] The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3. The Shatapatha Brahmana (sa शतपथ ब्राह्मण śatapatha brāhmaṇa, " Brahmana of one-hundred paths" abbreviated ŚB 139. The Tanakh appears to suggest, in the Book of Kings, that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The term Hebrew Bible is a generic reference to those books of the Bible originally written in Biblical Hebrew (and the related Biblical Aramaic The interpretation of the passage is disputed,^[23][24] as some believe the ratio of 3:1 is of an exterior circumference to an interior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls. See: Biblical value of Pi. This page is about the history of numerical approximations of the Mathematical constant &pi.

Archimedes (287-212 BC) was the first to estimate π rigorously. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters:^[24]

Liu Hui's Pi algorithm

By using the equivalent of 96-sided polygons, he proved that 223/71 < π < 22/7. General properties These properties apply to both convex and star regular polygons ^[24] Taking the average of these values yields 3. 1419.

In the following centuries, most significant development took place in India and China. Around 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate Pi to any degree of accuracy. Cao Wei ( was one of the empires that competed for control of China during the Three Kingdoms period Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom. He himself carried through the calculation to 3072-gon and obtained Pi=3. 1416.

Later, Liu Hui invented a quick method of calculating Pi and obtained Pi=3. Liu Hui's π algorithm is a mathematical algorithm invented by Liu Hui (fl 1416 with only 96-gon, by taking advantage of the fact that the difference in area of successive polygons formed a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi gave the approximation π = 355/113, and showed that 3. Zu Chongzhi ( 429–500 Courtesy name Wenyuan (文遠 was a prominent Chinese mathematician and astronomer during the Liu 1415926 < π < 3. 1415927, calculated with Liu Hui's π algorithm to 12288-gon; this value of Pi would stand as the most accurate value for π over the next 900 years. Liu Hui's π algorithm is a mathematical algorithm invented by Liu Hui (fl

Classical period

Until the second millennium, π was known to fewer than 10 decimal digits. The second millennium is a period of time that commenced on January 1, 1001, and ended on December 31, 2000. The next major advancement in the study of π came with the development of calculus, and in particular the discovery of infinite series which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Around 1400, Madhava of Sangamagrama found the first known such series:

(now known as the Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century). Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c See Leibniz formula for other formulas known under the same name James Gregory (November 1638 &ndash October 1675 was a Scottish Mathematician and Astronomer. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

Madhava was able to calculate π as 3. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c 14159265359, correct to 11 decimal places. The record was beaten in 1424 by the Persian astronomer Jamshīd al-Kāshī, who determined 16 decimals of π. (or, Persian: غیاث‌الدین جمشید کاشانی (c

The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometrical method to compute 35 decimals of π. Ludolph van Ceulen ( 28 January 1540 &ndash 31 December 1610) was a German Mathematician from Hildesheim. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. ^[25]

Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète in 1593. This article is not about Viète's formulas for symmetric polynomials François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta, Another famous result is Wallis' product,

written down by John Wallis in 1655. In Mathematics, Wallis' product for &pi, written down in 1655 by John Wallis, states that \prod_{n=1}^{\infty} John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements " ^[26]

In 1706 John Machin was the first to compute 100 decimals of π, using the formula

with

Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. John Machin, ( bapt 1686?&mdash June 9, 1751) a professor of Astronomy at Gresham College, London is best known for developing In Mathematics, Machin-like formulas are a class of identities involving &pi = 3 A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head. Johann Martin Zacharias Dase ( June 23, 1824, Hamburg - September 11, 1861, Hamburg was a German Mental calculator The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. William Shanks ( January 25, 1812 &ndash summer 1882 Houghton-le-Spring, Durham, England) was a British Amateur (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct. )

Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien-Marie Legendre proved in 1794 that also π² is irrational. Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of

which is π²/6, he established a deep connection between π and the prime numbers. The Basel problem is a famous problem in Number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735 In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Both Legendre and Leonhard Euler speculated that π might be transcendental, a fact that was proved in 1882 by Ferdinand von Lindemann. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation Carl Louis Ferdinand von Lindemann ( April 12, 1852 &ndash March 6 1939) was a German Mathematician, noted for his proof

William Jones' book A New Introduction to Mathematics from 1706 is cited as the first text where the Greek letter π was used for this constant, but this notation became particularly popular after Leonhard Euler adopted it in 1737. William Jones (1675 &ndash 3 July 1749) was a Welsh Mathematician, born in the village of Llanfihangel Tre'r Beirdd, on the Year 1706 ( MDCCVI) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar (or a Pi (uppercase &Pi, lower case &pi) is the sixteenth letter of the Greek alphabet. ^[27] He wrote:

See also: history of mathematical notation

Computation in the computer age

The advent of digital computers in the 20th century led to an increased rate of new π calculation records. Mathematical notation comprises the Symbols used to write mathematical Equations and Formulas It includes Arabic numerals, letters from the John von Neumann used ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours. ENIAC, short for Electronic Numerical Integrator And Computer, was the first general-purpose electronic Computer. Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.

In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance and mathematical depth. ^[28] Two of his most famous formulas are the series

and

which deliver 14 digits per term. ^[28] The Chudnovsky brothers used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records. A supercomputer is a Computer that is at the frontline of processing capacity particularly speed of calculation (at the time of its introduction

Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent-Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. Richard Peirce Brent is an Australian Mathematician and Computer scientist, born in 1946 Eugene Salamin is a Mathematician who discovered (independently with Richard Brent) the Salamin-Brent algorithm, used in high-precision calculation of ^[29] The algorithm consists of setting

and iterating

until a_n and b_n are close enough. Then the estimate for π is given by

.

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein. Jonathan Michael Borwein (born 1951 is a Canadian mathematician noted for his prolific and creative work throughout the international mathematical community Peter Benjamin Borwein ( St Andrews, Scotland, 1953 is a Canadian Mathematician, co-developer of an algorithm for calculating π ^[30] The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. is a Japanese Mathematician most known for his numerous world records over the past two decades for calculating digits of π. The current record is 1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-like formulas that were slower but crucially reduced memory consumption. The calculation was performed on a 64-node Hitachi supercomputer with 1 terabyte of main memory, capable of carrying out 2 trillion operations per second. A terabyte (derived from the prefix Tera- and commonly abbreviated TB) is a measurement term for data storage capacity.

An important recent development was the Bailey-Borwein-Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Plouffe. Simon Plouffe is a Quebec Mathematician born on June 11 1956 in, Quebec. David Harold Bailey is a Mathematician and Computer scientist. Peter Benjamin Borwein ( St Andrews, Scotland, 1953 is a Canadian Mathematician, co-developer of an algorithm for calculating π ^[31] The formula,

is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. ^[31] Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0. Distributed computing deals with Hardware and Software Systems containing more than one processing element or Storage element concurrent PiHex was a Distributed computing Project to calculate specific Bits of Pi, the greatest calculation of Pi ever successfully attempted Fabrice Bellard is a Computer programmer who is best known as the founder of FFmpeg and project leader for QEMU. This list compares various sizes of positive Numbers including counts of things Dimensionless quantity and probabilities. ^[32]

Memorizing digits

Main article: Piphilology

Recent decades have seen a surge in the record number of digits memorized. Piphilology comprises the creation and use of Mnemonic techniques to remember a span of digits of the mathematical constant &pi. A decade is a period of 10 Years (since 1594 a factor of 10 difference between two numbers, or sometimes a set or a group of ten (since 1451

Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places. Akira Haraguchi (原口 證 (born 1946 a retired Japanese engineer currently working as a mental health counsellor and business consultant in Mobara City, is known ^[33] This, however, has yet to be verified by Guinness World Records. Guinness World Records, known until 2000 as The Guinness Book of Records (and in previous U The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National ^[34] It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error. ^[35]

There are many ways to memorize π, including the use of "piems", which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem: How I need a drink, alcoholic in nature (or: of course), after the heavy lectures involving quantum mechanics. ^[36] Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The Cadaeic Cadenza contains the first 3834 digits of π in this manner. Cadaeic Cadenza is a 1996 Short story by Mike Keith. It is an example of Constrained writing, a book with restrictions on how it can be written ^[37] Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. A mnemonic device (nəˈmɒnɪk is a Memory aid Commonly met mnemonics are often verbal something such as a very short poem or a special word used to help a person remember Piphilology comprises the creation and use of Mnemonic techniques to remember a span of digits of the mathematical constant &pi. See Pi mnemonics for examples. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of pi. Other methods include remembering patterns in the numbers. ^[38]

Advanced properties

Numerical approximations

Main article: History of numerical approximations of π

Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions. This page is about the history of numerical approximations of the Mathematical constant &pi. ^[9] Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as ". In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with . . "), which indicates that the formula is really a formula for an infinite sequence of approximations to π. ^[39] The more terms included in a calculation, the closer to π the result will get.

Consequently, numerical calculations must use approximations of π. An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful For many purposes, 3. 14 or ²²/₇ is close enough, although engineers often use 3. Proofs of the famous mathematical result that the Rational number 22⁄7 is greater than π date back to antiquity 1416 (5 significant figures) or 3. The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy 14159 (6 significant figures) for more precision. The approximations ²²/₇ and ³⁵⁵/₁₁₃, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} The approximation ³⁵⁵⁄₁₁₃ (3. \pi \approx 3141\ 592\ 653\dots\\frac{355}{113} \approx 3141\ 592\ 920\dots\The best rational 1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object ^[40][41][42]

The earliest numerical approximation of π is almost certainly the value 3. ---- In mathematics Three is the first odd Prime number, and the second smallest prime ^[24] In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle. The perimeter is the distance around a given two-dimensional object In Geometry, an inscribed Planar Shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid General properties These properties apply to both convex and star regular polygons Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees Geometry, a diameter of a Circle is any straight Line segment that passes through the center of the circle and whose Endpoints are on the Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

Open questions

The most pressing open question about π is whether it is a normal number — whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every base, not just base 10. A different topic is treated in the article titled Normal number (computing. ^[43] Current knowledge on this point is very weak; e. g. , it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π. ^[44]

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. 2000 ( MM) was a Leap year that started on Saturday of the Common Era, in accordance with the Gregorian calendar. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that ^[45]

It is also unknown whether π and e are algebraically independent, although Yuri Nesterenko proved the algebraic independence of {π, e^π, Γ(1/4)} in 1996. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In Abstract algebra, a Subset S of a field L is algebraically independent over a subfield K if the elements Yuri Valentinovich Nesterenko (Юрий Валентинович Нестеренко born December 5, 1946) is a mathematician who has written papers in Algebraic In Mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e^\pi \ that is e to the In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function ^[46] However it is known that at least one of πe and π + e is transcendental (see Lindemann–Weierstrass theorem). In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers

Use in mathematics and science

Main article: List of formulas involving π

π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry. The following is a list of significant formulae involving the Mathematical constant π. ^[47]

Geometry and trigonometry

See also: Area of a disk

For any circle with radius r and diameter d = 2r, the circumference is πd and the area is πr². The area of a disk (the region inside a Circle) is &pi r 2 when the circle has Radius r. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar ^[48] Accordingly, π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In the basic case, half the area of the unit disk is given by:^[49]

and

gives half the circumference of the unit circle. In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is In Mathematics, a unit circle is ^[50] More complicated shapes can be integrated as solids of revolution. In Mathematics, Engineering, and Manufacturing, a solid of revolution is a solid figure obtained by rotating a Plane curve around ^[51]

From the unit-circle definition of the trigonometric functions also follows that the sine and cosine have period 2π. That is, for all x and integers n, sin(x) = sin(x + 2πn) and cos(x) = cos(x + 2πn). Because sin(0) = 0, sin(2πn) = 0 for all integers n. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.

In modern mathematics, π is often defined using trigonometric functions, for example as the smallest positive x for which sin x = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the inverse trigonometric functions, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as power series is the easiest way to derive infinite series for π. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 +

Higher analysis and number theory

The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula

where i is the imaginary unit satisfying i² = −1 and e ≈ 2. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation 71828 is Euler's number. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line This formula implies that imaginary powers of e describe rotations on the unit circle in the complex plane; these rotations have a period of 360° = 2π. In Mathematics, a unit circle is In particular, the 180° rotation φ = π results in the remarkable Euler's identity

There are n different n-th roots of unity

The Gaussian integral

A consequence is that the gamma function of a half-integer is a rational multiple of √π. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power The Gaussian integral, or probability integral, is the Improper integral of the Gaussian function e^ over the entire real line In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function

Physics

Although not a physical constant, π appears routinely in equations describing fundamental principles of the Universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial Using units such as Planck units can sometimes eliminate π from formulae. Planck units are Units of measurement named after the German physicist Max Planck, who first proposed them in 1899

• The cosmological constant:^[52]

• Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δx) and momentum (Δp) can not both be arbitrarily small at the same time:^[53]

• Einstein's field equation of general relativity:^[54]

• Coulomb's law for the electric force, describing the force between two electric charges (q₁ and q₂) separated by distance r:^[55]

• Magnetic permeability of free space:^[56]

• Kepler's third law constant, relating the orbital period (P) and the semimajor axis (a) to the masses (M and m) of two co-orbiting bodies:

Probability and statistics

In probability and statistics, there are many distributions whose formulas contain π, including:

• the probability density function for the normal distribution with mean μ and standard deviation σ, due to the Gaussian integral:^[57]

• the probability density function for the (standard) Cauchy distribution:^[58]

Note that since for any probability density function f(x), the above formulas can be used to produce other integral formulas for π. In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. The orbital period is the time taken for a given object to make one complete Orbit about another object In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Probability is the likelihood or chance that something is the case or will happen Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values The Gaussian integral, or probability integral, is the Improper integral of the Gaussian function e^ over the entire real line The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. ^[59]

Buffon's needle problem is sometimes quoted as a empirical approximation of π in "popular mathematics" works. 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 Consider dropping a needle of length L repeatedly on a surface containing parallel lines drawn S units apart (with S > L). If the needle is dropped n times and x of those times it comes to rest crossing a line (x > 0), then one may approximate π using the Monte Carlo method:^{[60][61][62][63]}

Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of π by experiment. Monte Carlo methods are a class of Computational Algorithms that rely on repeated Random sampling to compute their results Reliably getting just three digits (including the initial "3") right requires millions of throws,^[60] and the number of throws grows exponentially with the number of digits desired. Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value Furthermore, any error in the measurement of the lengths L and S will transfer directly to an error in the approximated π. For example, a difference of a single atom in the length of a 10-centimeter needle would show up around the 9th digit of the result. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.

See also

• List of topics related to π
• Proof that 22/7 exceeds π
• Feynman point – comprising the 762nd through 767th decimal places of π, consisting of the digit 9 repeated six times. This is a list of topics related to π (pi the fundamental Mathematical constant. Proofs of the famous mathematical result that the Rational number 22⁄7 is greater than π date back to antiquity The Feynman point is the sequence of six 9s which begins at the 762nd decimal place of π.
• Indiana Pi Bill. The Indiana Pi Bill is the popular name for bill #246 of the 1897 sitting of the Indiana General Assembly, which is one of the most famous attempts to establish scientific truth
• Pi Day. Pi Day and Pi Approximation Day are two Holidays held to celebrate the Mathematical constant π (pi
• Software for calculating π on personal computers. Over the years several programs have been written for calculating Pi (π to many digits on Personal computers General-purpose Most Computer
• Mathematical constants: e and φ
• SOCR resource hands-on activity for estimation of π using needle-dropping simulation. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In Mathematics and the Arts two quantities are in the Golden ratio if the Ratio between the sum of those quantities and the larger one is the The Statistics Online Computational Resource (SOCR is a suite of online tools and interactive aids for hands-on learning and teaching concepts in statistical analyses and

References

External links

• The Joy of Pi by David Blatner
• Decimal expansions of Pi and related links at the On-Line Encyclopedia of Integer Sequences
• J J O'Connor and E F Robertson: A history of pi. Mac Tutor project
• Lots of formulas for π at MathWorld
• PlanetMath: Pi
• Finding the value of π
• Determination of π at cut-the-knot
• BBC Radio Program about π
• Statistical Distribution Information on PI based on 1. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences MathWorld is an online Mathematics reference work created and largely written by Eric W Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. 2 trillion digits of PI
• First 4 Million Digits of π - Warning - Roughly 2 megabytes will be transferred. A megabyte is a unit of Information or Computer storage equal to either 106 (1000000 Bytes or 220 (1048576 bytes depending on
• One million digits of pi at piday.org
• Project Gutenberg E-Text containing a million digits of π
• Search the first 200 million digits of π for arbitrary strings of numbers
• Source code for calculating the digits of π
• π is Wrong! An opinion column on why 2π is more useful in mathematics.
• 70 Billion digits of Pi(π) downloads.
• The first 16 million digits of Pi (18 mb . txt file)