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

Pi or π is the ratio of a circle's circumference to its diameter in Euclidean geometry, approximately 3.14159. Pi is a mathematical constant and a transcendental (and therefore irrational) real number, with many uses in mathematics, physics, and engineering. It is also known as Archimedes' constant (not to be confused with an Archimedes number) and as Ludolph's number. Image File history File links Pi-unrolled-720. ... Image File history File links Pi-unrolled-720. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... Circle illustration This article is about the shape and mathematical concept of circle. ... The circumference is the distance around a closed curve. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... 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 m/n, where m and n are integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article needs additional references or sources for verification. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... Archimedes of Syracuse (Greek: c. ... An Archimedes number, named after the ancient Greek scientist Archimedes, to determine the motion of fluids due to density differences, is a dimensionless number in the form where: g = gravitational acceleration (9. ... Ludolph van Ceulen (28 January 1540 – 31 December 1610) was a German mathematician. ...

Contents

The letter π

Main article: pi (letter)
Lower-case π (the lower case letter is used for the constant)

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. When referring to this constant, the symbol π is always pronounced like "pie" in English, the conventional English pronunciation of the letter. For other uses, see Pi (disambiguation) Pi (upper case Π, lower case Ï€ or Ï–) is the sixteenth letter of the Greek alphabet. ... Image File history File links Pi-symbol. ... For other uses, see Pi (disambiguation) Pi (upper case Π, lower case Ï€ or Ï–) is the sixteenth letter of the Greek alphabet. ... This article does not cite any references or sources. ... The English language is a West Germanic language that originates in England. ...


The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery'[1] and περίμετρος 'perimeter', i.e. 'circumference'.


π is Unicode character U+03C0 ("Greek small letter pi"). Unicode is an industry standard allowing computers to consistently represent and manipulate text expressed in any of the worlds writing systems. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ...

Circumference = π × diameter
Circumference = π × diameter

Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

Definition

In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter: Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... Circle illustration This article is about the shape and mathematical concept of circle. ... The circumference is the distance around a closed curve. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ...

 pi = frac{c}{d}

Note that the ratio c/d does not depend on the size of the circle. 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 can also be stated as saying that all circles are similar. // 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
Area of the circle = π × area of the shaded square

Alternatively π can be also defined as the ratio of a circle's area to the area of a square whose side is the radius: Image File history File links Circle_Area. ... Image File history File links Circle_Area. ... Area is a physical quantity expressing the size of a part of a surface. ... Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...

 pi = frac{A}{r^2}

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.[2] The formulæ below illustrate other (equivalent) definitions. In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...


Numerical value

The numerical value of π truncated to 50 decimal places is: In mathematics, truncation is the term used for reducing 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. ...

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
See the links below and those at sequence A000796 in OEIS for more digits.

While the value of pi has been computed to billions of digits, practical science and engineering will rarely require more than 10 decimal places. As an example, computing the circumference of the Earth's equator from its radius using only 10 decimal places of pi yields an error of less than 0.2 millimeters. A value truncated to 39 decimal places is sufficient to compute the circumference of the visible universe to a precision comparable to the size of a hydrogen atom.[3] The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... This article is about Earth as a planet. ... World map showing the equator in red In tourist areas, the equator is often marked on the sides of roads The equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe. ... A millimetre (American spelling: millimeter), symbol mm is an SI unit of length that is equal to one thousandth of a metre. ... The deepest visible-light image of the cosmos. ... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ...


Most circular objects worthy of physical study, particularly on the scale of planetary radii, have imperfections and eccentricities which account for a greater error in calculation than would be yielded by calculations using approximations of pi. The exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number). 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. 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. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. See history of numerical approximations of π. The infinity symbol ∞ in several typefaces. ... A numeral is a symbol or group of symbols that represents a number. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... 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 m/n, where m and n are integers. ... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... A supercomputer is a computer that led the world (or was close to doing so) in terms 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 numbers and probabilities. ... Over the years, several programs have been written for calculating Ï€ to many digits on personal computers. ... Best known estimates of the value of pi over the centuries This page is about the history of numerical approximations of the mathematical constant Ï€. There is a summarizing table at chronology of computation of Ï€. See also history of Ï€ for other apects of the evolution of our knowledge about mathematical properties...


Calculating π

Main article: Computing π

π can be empirically measured by drawing a large circle, then measuring its diameter and circumference, since the circumference of a circle is always π times its diameter. This article or section is in need of attention from an expert on the subject. ...


π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series: Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In mathematics, Leibniz formula for Ï€, due to Gottfried Leibniz, states that Proof Consider the infinite geometric series It is the limit of the truncated geometric series Splitting the integrand as and integrating both sides from 0 to 1, we have Integrating the first integral (over the truncated geometric series ) termwise...

 pi = frac{4}{1}-frac{4}{3}+frac{4}{5}-frac{4}{7}+frac{4}{9}-frac{4}{11}cdots

While that series is easy to write and calculate, it is not immediately obvious why it yields π. A more intuitive approach is to draw an imaginary circle of radius r centered at the origin. Then any point (x,y) whose distance d from the origin is less than r, as given by the pythagorean theorem, will be inside the circle: In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

 d = sqrt{x^2 + y^2}

Finding a collection of points inside the circle allows the circle's area A to be approximated. For example, by using integer coordinate points for a big r. Since the area A of a circle is π times the radius squared, π can be approximated by using:

 pi = frac{A}{r^2}

Properties

The constant π is an irrational number; that is, it cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. See Proof that π is irrational for an elementary proof due to Ivan Niven. 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 m/n, where m and n are integers. ... The integers are commonly denoted by the above symbol. ... 1761 was a common year starting on Thursday (see link for calendar). ... Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 – September 25, 1777), was a mathematician, physicist and astronomer. ... Although the mathematical constant known as Ï€ (pi) has been studied since ancient times, as has the concept of irrational number, it was not until the 18th century that Ï€ was proved to be irrational. ... Ivan Morton Niven (October 25, 1915 – May 9, 1999) was a Canadian-American mathematician. ...


Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. 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. In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i. ... Year 1882 (MDCCCLXXXII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday of the 12-day slower Julian calendar). ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ... Squaring the circle: the areas of this square and this circle are equal. ...


History

Main article: History of π

The mathematical constant π = 3. ...

Use of the symbol π

Often 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 (cf History of π). Sir William Jones (1675 - 3 July 1749) was a mathematician. ... Events March 27 - Concluding that Emperor Iyasus I of Ethiopia had abdicated by retiring to a monastery, a council of high officials appoint Tekle Haymanot I Emperor of Ethiopia May 23 - Battle of Ramillies September 7 - The Battle of Turin in the War of Spanish Succession - forces of Austria and... For other uses, see Pi (disambiguation) Pi (upper case Π, lower case Ï€ or Ï–) is the sixteenth letter of the Greek alphabet. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... // Cf or CF may stand for: cf. ... The mathematical constant Ï€ = 3. ...


Early approximations

Main article: History of numerical approximations of π

The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π = 258, which is within 0.5% of the true value. Best known estimates of the value of pi over the centuries This page is about the history of numerical approximations of the mathematical constant Ï€. There is a summarizing table at chronology of computation of Ï€. See also history of Ï€ for other apects of the evolution of our knowledge about mathematical properties... EGGS! ... Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since...


The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160. Ahmes (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. ... The Middle Kingdom is a period in the history of ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty, roughly between 2030 BC and 1640 BC. The period comprises of 2 phases, the 11th Dynasty, which ruled from Thebes and the 12th... Papyrus plant Cyperus papyrus at Kew Gardens, London Papyrus is an early form of paper produced from the pith of the papyrus plant, Cyperus papyrus, a wetland sedge that was once abundant in the Nile Delta of Egypt. ...


It is sometimes claimed that the Bible states that π = 3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. The discrepancy has been explained in various ways by different exegetes. Rabbi Nehemiah explained it by the diameter being measured from outside rim to outside rim while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. This Gutenberg Bible is displayed by the United States Library. ... The Books of Kings (Hebrew: Sefer Melachim ספר מלכים) is a part of Judaisms Tanakh, the Hebrew Bible. ... Cubit is the name for any one of many units of measure used by various ancient peoples. ... Rabbi Nehemiah was a priest in 1 Kings of the Bible. ...


Bryson of Heraclea and Antiphon were to first to place an upper and lower bound on pi in a manner similar that used by Archimedes, but considering area instead of perimeter. Bryson of Heraclea (ca. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Principle of Archimedes' method to approximate π
Principle of Archimedes' method to approximate π

Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between 22371 and 227. The average of these two values is roughly 3.1419. Image File history File links Archimedes_pi. ... Image File history File links Archimedes_pi. ... Archimedes of Syracuse (Greek: c. ... Look up polygon in Wiktionary, the free dictionary. ...


The Chinese mathematician Liu Hui computed π to 3.141014 in AD 263 and suggested that 3.14 was a good approximation. Knowledge of Chinese mathematics before 100 BC is somewhat fragmentary, but there are elements that seem consistent. ... A possible likeness of Liu Hui on japanpostage stamp This is a Chinese name; the family name is 劉 (Liu) Liu Hui 劉徽 was a Chinese mathematician who lived in the 200s in the Wei Kingdom. ... Events The Wei Kingdom conquered the kingdom of Shu Han, one of the Chinese Three Kingdoms. ...


The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 6283220000 = 3.1416, correct when rounded off to four decimal places. He also said that this was a value that "approached" the correct number, which was interpreted in the 15th c. as meaning that pi is irrational, a concept which would not be known in Europe till the 18th c. This article is under construction. ... Hindu Astronomy is one of the ancient astronomical systems of the world. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Europe in 450 The 5th century is the period from 401 to 500 in accordance with the Julian calendar in the Christian Era. ... In philosophy: Irrationality In music: Irrational rhythm In economics: Irrational exuberance In mathematics: Irrational number Proof that e is irrational Quadratic irrational List of integrals of irrational functions See also: rational This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same...


The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355113 and 227, in the 5th century. This article or section is in need of attention from an expert on the subject. ... Zu Chongzhi (Traditional Chinese: ; Simplified Chinese: ; Hanyu Pinyin: ZÇ” ChōngzhÄ«; Wade-Giles: Tsu Chung-chih, 429–500), courtesy name Wenyuan (文遠), was a prominent Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. ... Milü (355⁄113) (Chinese: ; pinyin: mì lÇœ; literally detailed (approximation) ratio), also known as Zulü (Zus ratio),[1] roughly 3. ...


The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series of arctan(1)=π4 into the form Madhavan (മാധവന്) of Sangamagramam (1350–1425) was a prominent mathematician-astronomer from Kerala, India. ... This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

pi = sqrt{12}left(1-{1over 3cdot3}+{1over5cdot 3^2}-{1over7cdot 3^3}+cdotsright)

and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of π4, he was able to compute π to an accuracy of 13 decimal places.


The Persian astronomer Ghyath ad-din Jamshid Kashani (1350–1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as: The Persian Empire was a series of historical empires that ruled over the Iranian plateau, the old Persian homeland, and beyond in Western Asia, Central Asia and the Caucasus. ... Kashani, dubbed, the Second Ptolemy, was an outstanding Persian mathematician of the middle ages. ...

2π = 6.2831853071795865

By 1610, the German mathematician Ludolph van Ceulen had finished computing the first 35 decimal places of π. It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone. // Events January 7 - Galileo Galilei discovers the Galilean moons of Jupiter. ... Ludolph van Ceulen (28 January 1540 – 31 December 1610) was a German mathematician. ... “Tombstone” redirects here. ...


In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π, of which the first 126 were correct [1], and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct. Year 1789 (MDCCLXXXIX) was a common year starting on Thursday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Monday of the 11-day slower Julian calendar). ... Baron Jurij Bartolomej Vega (also correct Veha; official Latin Georgius Bartholomaei Vecha; German Georg Freiherr von Vega) (March 23, 1754 – September 26, 1802) was a Slovenian mathematician, physicist and artillery officer. ... John Machin, (1680—June 9, 1751), a professor of astronomy in London, is best known for developing a quickly converging series for Ï€ in 1706 and using it to compute Ï€ to 100 decimal places. ... Events March 27 - Concluding that Emperor Iyasus I of Ethiopia had abdicated by retiring to a monastery, a council of high officials appoint Tekle Haymanot I Emperor of Ethiopia May 23 - Battle of Ramillies September 7 - The Battle of Turin in the War of Spanish Succession - forces of Austria and... 1841 is a common year starting on Friday (link will take you to calendar). ... William Rutherford (1798–1871) was an English mathematician. ...


The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). He published his value of pi in a book, which was promptly dubbed "the world's most boring book"! In 1944, D. F. Ferguson found that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. By 1947, Ferguson had recalculated pi to 808 decimal places (with the aid of a mechanical desk calculator). William Shanks (January 25, 1812 -- 1882 in Houghton-le-Spring, Durham, England) was a British amateur mathematician. ... 1873 (MDCCCLXXIII) was a common year starting on Wednesday (see link for calendar). ... 1944 (MCMXLIV) was a leap year starting on Saturday. ...


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. Formulae for calculating π using elementary arithmetic invariably include notation such as "...", which indicates that the formula is really a formula for an infinite sequence of approximations to π. The more terms included in a calculation, the closer to π the result will get, but none of the results will be π exactly. Best known estimates of the value of pi over the centuries This page is about the history of numerical approximations of the mathematical constant Ï€. There is a summarizing table at chronology of computation of Ï€. See also history of Ï€ for other apects of the evolution of our knowledge about mathematical properties...


Consequently, numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (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. The approximation 355113 (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator. It has been suggested that this article or section be merged with estimation. ... Proofs of the famous mathematical result that the rational number 22⁄7 is greater than Ï€ date back to antiquity. ... Rounding to n significant figures is a form of rounding. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Milü (355⁄113) (Chinese: ; pinyin: mì lÇœ; literally detailed (approximation) ratio), also known as Zulü (Zus ratio),[1] roughly 3. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ...


The earliest numerical approximation of π is almost certainly the value 3. 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. Look up three in Wiktionary, the free dictionary. ... 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. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... A regular hexagon. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... Circle illustration This article is about the shape and mathematical concept of circle. ...


Formulae

Geometry

The constant π appears in many formulæ in geometry involving circles and spheres. Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Circle illustration This article is about the shape and mathematical concept of circle. ... A sphere is a perfectly symmetrical geometrical object. ...

Geometrical shape Formula
Circumference of circle of radius r and diameter d C = 2 pi r = pi d ,!
Area of circle of radius r A = pi r^2 = frac{1}{4} pi d^2 ,!
Area of ellipse with semiaxes a and b A = pi a b ,!
Volume of sphere of radius r and diameter d V = frac{4}{3} pi r^3 = frac{1}{6} pi d^3 ,!
Surface area of sphere of radius r and diameter d A = 4 pi r^2 = pi d^2 ,!
Volume of cylinder of height h and radius r V = pi r^2 h ,!
Surface area of cylinder of height h and radius r A = 2 (pi r^2) + ( 2 pi r)h = 2 pi r (r+h) ,!
Volume of cone of height h and radius r V = frac{1}{3} pi r^2 h ,!
Surface area of cone of height h and radius r A = pi r^2 + pi r sqrt{r^2 + h^2} = pi r (r + sqrt{r^2 + h^2}) ,!

All of these formulae are a consequence of the formula for circumference. For example, the area of a circle of radius R can be accumulated by summing annuli of infinitesimal width using the integral A = int_0^R 2pi r dr = pi R^2.. The others concern a surface or solid of revolution. The circumference is the distance around a closed curve. ... Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... For other uses, see Ellipse (disambiguation). ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Area is the measure of how much exposed area any two dimensional object has. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... This article is about the geometric object, for other uses see Cone. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ... In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane. ...


Also, the angle measure of 180° (degrees) is equal to π radians. ∠, the angle symbol. ... This article describes the unit of angle. ... Some common angles, measured in radians. ...


Analysis

Many formulas in analysis contain π, including infinite series (and infinite product) representations, integrals, and so-called special functions. Analysis has its beginnings in the rigorous formulation of calculus. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... In mathematics, several functions or groups of functions are important enough to deserve their own names. ...

2int_{-1}^1 sqrt{1-x^2},dx = pi
int_{-1}^1frac{dx}{sqrt{1-x^2}} = pi
frac{sqrt2}2 cdot frac{sqrt{2+sqrt2}}2 cdot frac{sqrt{2+sqrt{2+sqrt2}}}2 cdot cdots = frac2pi
sum_{n=0}^{infty} frac{(-1)^{n}}{2n+1} = frac{1}{1} - frac{1}{3} + frac{1}{5} - frac{1}{7} + frac{1}{9} - cdots = frac{pi}{4}
 prod_{n=1}^{infty} left ( frac{n+1}{n} right )^{(-1)^{n-1}} = frac{2}{1} cdot frac{2}{3} cdot frac{4}{3} cdot frac{4}{5} cdot frac{6}{5} cdot frac{6}{7} cdot frac{8}{7} cdot frac{8}{9} cdots = frac{pi}{2}
frac{2sqrt{2}}{9801}sum_{n=0}^{infty}frac{(1103+26390n)cdot(4n)!}{396^{4n}cdot(n!)^4}=frac{1}{pi}
  • Chebyshev series Y. Luke, Math. Tabl. Aids Comp. 11 (1957) 16
sum_{n=0}^inftyfrac{(-1)^n(sqrt{2}-1)^{2n+1}}{2n+1} = frac{pi}{8}.
sum_{n=0}^inftyfrac{(-1)^n(2-sqrt{3})^{2n+1}}{2n+1}=frac{pi}{12}.
frac{426880sqrt{10005}}{sum_{n=0}^{infty}frac{(6n)!(545140134n+13591409)}{(n!)^3(3n)!(-640320)^{3n}}}=pi
  • Symmetric formula (see Sondow, 1997)
 frac {displaystyle prod_{n=1}^{infty} left (1 + frac{1}{4n^2-1} right )}{displaystylesum_{n=1}^{infty} frac {1}{4n^2-1}} = frac {displaystyleleft (1 + frac{1}{3} right ) left (1 + frac{1}{15} right ) left (1 + frac{1}{35} right ) cdots} {displaystyle frac{1}{3} + frac{1}{15} + frac{1}{35} + cdots} = pi
sum_{n=0}^inftyfrac{1}{16^n}left(frac {4}{8n+1} - frac {2}{8n+4} - frac {1}{8n+5} - frac {1}{8n+6}right) = pi
 left ( frac{2}{1} right )^{1/2} left (frac{2^2}{1 cdot 3} right )^{1/4} left (frac{2^3 cdot 4}{1 cdot 3^3} right )^{1/8} left (frac{2^4 cdot 4^4}{1 cdot 3^6 cdot 5} right )^{1/16} cdots = frac{pi}{2}
where the nth factor is the 2nth root of the product
prod_{k=0}^n (k+1)^{(-1)^{k+1}{n choose k}}.
zeta(2)= frac{1}{1^2} + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + cdots = frac{pi^2}{6}
zeta(4)= frac{1}{1^4} + frac{1}{2^4} + frac{1}{3^4} + frac{1}{4^4} + cdots = frac{pi^4}{90}
and generally, ζ(2n) is a rational multiple of π2n for positive integer n
int_{-infty}^{infty} e^{-x^2},dx = sqrt{pi}
Gammaleft({1 over 2}right)=sqrt{pi}
n! sim sqrt{2 pi n} left(frac{n}{e}right)^n
e^{i pi} + 1 = 0;
sum_{k=1}^{n} phi (k) sim frac{3n^2}{pi^2}
ointfrac{dz}{z}=2pi i ,
where the path of integration is a closed curve around the origin, traversed in the standard anticlockwise direction.

A disc of unit radius on a plane is called a unit disc. ... The circumference is the distance around a closed curve. ... Illustration of a unit circle. ... François Viète. ... In mathematics, the Viète formula, named after François Viète, is the following infinite product type representation of the mathematical constant Ï€: The expression on the right hand side has to be understood as a limit expression where with initial condition Do some simplification , a pretty formula for Ï€ is... It has been suggested that this article be split into multiple articles. ... In mathematics, Leibniz formula for Ï€, due to Gottfried Leibniz, states that Proof Consider the infinite geometric series It is the limit of the truncated geometric series Splitting the integrand as and integrating both sides from 0 to 1, we have Integrating the first integral (over the truncated geometric series ) termwise... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... In mathematics, Wallis product for π, written down in 1655 by John Wallis, states that Proof First of all, consider the root of sin(x)/x is ±nπ, where n = 1, 2, 3, ... Then, we can express sine as an infinite product of linear factors given by its roots... Srinivasa Ramanujan Iyengar (Tamil: ) (22 December 1887 – 26 April 1920) was an Indian mathematician who is widely regarded as one of the greatest mathematical minds in recent history. ... Pafnuty Lvovich Chebyshev (Russian: ) ( May 26 [O.S. May 14] 1821 – December 8 [O.S. November 26] 1894) was a Russian mathematician. ... The Chudnovsky Brothers are mathematicians known for their wide-ranging mathematical abilities, their home-built supercomputers, and their close working relationship. ... The Bailey-Borwein-Plouffe formula (BBP formula) permits the computation of the nth binary digit of Ï€. It is a Ï€ summation formula discovered in 1995 by Simon Plouffe. ... The Basel problem is a famous problem in number theory, first posed by Pietro Mengoli in 1644, and solved by Leonhard Euler in 1735. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ... The relative difference between (ln x!) and (x ln x - x) approaches zero as x increases. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... Richard Phillips Feynman (May 11, 1918 – February 15, 1988; IPA: ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ... The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. ... The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...

Number theory

Some results from number theory: Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

  • The probability that a randomly chosen integer is square-free is 6/π2.
  • The average number of ways to write a positive integer as the sum of two perfect squares (order matters but not sign) is π/4.

In the above three statements, "probability", "average", and "random" are taken in a limiting sense, i.e. we consider the probability for the set of integers {1, 2, 3,…, N}, and then take the limit as N approaches infinity. Probability is the likelihood that something is the case or will happen. ... The word random is used to express lack of order, purpose, cause, or predictability in non-scientific parlance. ... Coprime - Wikipedia /**/ @import /skins-1. ... In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. ... In statistics, mean has two related meanings: the arithmetic mean (and is distinguished from the geometric mean or harmonic mean). ... The term perfect square is used in mathematics in two meanings: an integer which is the square of some other integer, i. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

The theory of elliptic curves and complex multiplication derives the approximation In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at...

pi approx {ln(640320^3+744)oversqrt{163}}

which is valid to about 30 digits.


Dynamical systems and ergodic theory

Consider the recurrence relation In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

x_{i+1} = 4 x_i (1 - x_i) ,

Then for almost every initial value x0 in the unit interval [0,1], In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

 lim_{n to infty} frac{1}{n} sum_{i = 1}^{n} sqrt{x_i} = frac{2}{pi}

This recurrence relation is the logistic map with parameter r = 4, known from dynamical systems theory. See also: ergodic theory. The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ...


Physics

The number 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.

Lambda = {{8pi G} over {3c^2}} rho
 Delta x Delta p ge frac{h}{4pi}
 R_{ik} - {g_{ik} R over 2} + Lambda g_{ik} = {8 pi G over c^4} T_{ik}
 F = frac{left|q_1q_2right|}{4 pi epsilon_0 r^2}
 mu_0 = 4 pi cdot 10^{-7},mathrm{N/A^2},
frac{P^2}{a^3}={(2pi)^2 over G (M+m)}

In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. ... Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ...

Probability and statistics

In probability and statistics, there are many distributions whose formulæ contain π, including: Probability is the likelihood that something is the case or will happen. ... A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...

f(x) = {1 over sigmasqrt{2pi} },e^{-(x-mu )^2/(2sigma^2)}
f(x) = frac{1}{pi (1 + x^2)}

Note that since int_{-infty}^{infty} f(x),dx = 1, for any pdf f(x), the above formulæ can be used to produce other integral formulae for π. In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... 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 of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ...


A semi-interesting empirical approximation of π is based on Buffon's needle problem. 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: In mathematics, Buffons 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. ...

pi approx frac{2nL}{xS}

[As a practical matter, this approximation is poor and converges very slowly.] In numerical analysis (a branch of mathematics), the speed at which a convergent sequence approaches its limit is called the rate of convergence. ...


Another approximation of π is to throw points randomly into a quarter of a circle with radius 1 that is inscribed in a square of length 1. π, the area of a unit circle, is then approximated as 4×(points in the quarter circle) ÷ (total points).


Efficient methods

In the early years of the computer, the first expansion of π to 100,000 decimal places was computed by Maryland mathematician Dr. Daniel Shanks and his team at the United States Naval Research Laboratory (N.R.L.) in 1961. Wikipedia does not have an article with this exact name. ... This article or section is in need of attention from an expert on the subject. ... Bust of Thomas Edison at the front gate of the Naval Research Laboratory. ...


Daniel Shanks and his team used two different power series for calculating the digits of π. For one it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,000 digits of π were published by the Naval Research Laboratory.


None of the formulæ given above can serve as an efficient way of approximating π. For fast calculations, one may use a formula such as Machin's: John Machin, (1680—June 9, 1751), a professor of astronomy in London, is best known for developing a quickly converging series for Ï€ in 1706 and using it to compute Ï€ to 100 decimal places. ...

frac{pi}{4} = 4 arctanfrac{1}{5} - arctanfrac{1}{239}

together with the Taylor series expansion of the function arctan(x). This formula is most easily verified using polar coordinates of complex numbers, starting with As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

(5+i)^4cdot(-239+i)=-114244-114244i.

Formulæ of this kind are known as Machin-like formulae. In mathematics, Machin-like formulas are a class of identities involving Ï€ = 3. ...


Many other expressions for π were developed and published by Indian mathematician Srinivasa Ramanujan. He worked with mathematician Godfrey Harold Hardy in England for a number of years. Srinivasa Ramanujan Iyengar (Tamil: ) (22 December 1887 – 26 April 1920) was an Indian mathematician who is widely regarded as one of the greatest mathematical minds in recent history. ... G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...


Extremely long decimal expansions of π are typically computed with the Gauss-Legendre algorithm and Borwein's algorithm; the Salamin-Brent algorithm which was invented in 1976 has also been used. The Gauss-Legendre algorithm is an algorithm to compute the digits of Ï€. The method is based on the individual work of Carl Friedrich Gauss (1777-1855) and Adrien-Marie Legendre (1752-1833) combined with modern algorithms for multiplication and square roots. ... Borweins algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. It works as follows: Start out by setting Then iterate Then ak converges quartically against 1/π; that is, each iteration approximately quadruples the number of correct digits. ... The Gauss-Legendre algorithm is an algorithm to compute the digits of π. The method is based on the individual work of Carl Friedrich Gauss (1777 - 1855) and Adrien-Marie Legendre (1752-1833) combined with modern algorithms for multiplication and square roots. ... Year 1976 Pick up sticks(MCMLXXVI) was a leap year starting on Thursday (link will display full calendar) of the Gregorian calendar. ...


The first one million digits of π and 1/π are available from Project Gutenberg (see external links below). The record as of December 2002 by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulæ were used for this: Project Gutenberg, abbreviated as PG, is a volunteer effort to digitize, archive, and distribute cultural works. ... Also see: 2002 (number). ... Yasumasa Kanada (金田 康正) is a Japanese mathematician most known for his numerous world records over the past two decades for calculating digits of Ï€. Kanada is a professor in the Department of Information Science at the University of Tokyo in Tokyo, Japan. ... The Yasuda Auditorium on the University of Tokyos Hongo Campus. ... It has been suggested that Hitachi Works be merged into this article or section. ... A supercomputer is a computer that led the world (or was close to doing so) in terms of processing capacity, particularly speed of calculation, at the time of its introduction. ...

 frac{pi}{4} = 12 arctanfrac{1}{49} + 32 arctanfrac{1}{57} - 5 arctanfrac{1}{239} + 12 arctanfrac{1}{110443}
K. Takano (1982).
 frac{pi}{4} = 44 arctanfrac{1}{57} + 7 arctanfrac{1}{239} - 12 arctanfrac{1}{682} + 24 arctanfrac{1}{12943}
F. C. W. Störmer (1896).

These approximations have so many digits that they are no longer of any practical use, except for testing new supercomputers. (Normality of π will always depend on the infinite string of digits on the end, not on any finite computation.) Year 1982 (MCMLXXXII) was a common year starting on Friday (link displays the 1982 Gregorian calendar). ... Year 1896 (MDCCCXCVI) was a leap year starting on Wednesday (link will display calendar). ...


In 1997, David H. Bailey, Peter Borwein and Simon Plouffe published a paper (Bailey, 1997) on a new formula for π as an infinite series: Year 1997 (MCMXCVII) was a common year starting on Wednesday (link will display full 1997 Gregorian calendar). ... David H. Bailey is a mathematician who, together with Peter Borwein and Simon Plouffe, found a formula for Ï€ in 1996 that permits one to calculate binary or hexadecimal digits of Ï€ beginning at an arbitrary position. ... Peter B. Borwein is a Canadian mathematician, co-developer of an algorithm for calculating Ï€ to the nth digit, co-discoverer of the billionth, four billionth, 40th billionth, and quadrillionth digits of Ï€, and professor at Simon Fraser University. ... Simon Plouffe is a Quebec mathematician born on June 11, 1956 in St-Jovite. ... In mathematics, a series is a sum of a sequence of terms. ...

pi = sum_{k = 0}^{infty} frac{1}{16^k} left( frac{4}{8k + 1} - frac{2}{8k + 4} - frac{1}{8k + 5} - frac{1}{8k + 6}right).

This formula permits one to fairly readily compute the kth binary or hexadecimal digit of π, without having to compute the preceding k − 1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. The PiHex project computed 64-bits around the quadrillionth bit of π (which turns out to be 0). The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... PiHex was a distributed computing project to calculate specific bits of Pi, the greatest calculations of Pi ever successfully attempted. ... The quadrillion is a large number which has one of two values depending on how or where it is being used. ...


Fabrice Bellard claims to have beaten the efficiency record set by Bailey, Borwein, and Plouffe with his formula to calculate binary digits of π [2]: Fabrice Bellard is a computer programmer who is best known as the founder of FFmpeg and project leader for Qemu. ...

pi = frac{1}{2^6} sum_{n=0}^{infty} frac{{(-1)}^n}{2^{10n}} left( - frac{2^5}{4n+1} - frac{1}{4n+3} + frac{2^8}{10n+1} - frac{2^6}{10n+3} - frac{2^2}{10n+5} - frac{2^2}{10n+7} + frac{1}{10n+9} right)

Other formulæ that have been used to compute estimates of π include:

 frac{pi}{2}= sum_{k=0}^inftyfrac{k!}{(2k+1)!!}= 1+frac{1}{3}left(1+frac{2}{5}left(1+frac{3}{7}left(1+frac{4}{9}(1+cdots)right)right)right)
Newton.
 frac{1}{pi} = frac{2sqrt{2}}{9801} sum^infty_{k=0} frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}
Srinivasa Ramanujan.

This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π. Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Srinivasa Ramanujan Iyengar (Tamil: ) (22 December 1887 – 26 April 1920) was an Indian mathematician who is widely regarded as one of the greatest mathematical minds in recent history. ...

 frac{1}{pi} = 12 sum^infty_{k=0} frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}
David Chudnovsky and Gregory Chudnovsky.

David Chudnovsky is a mathematician, and is the brother of Gregory Chudnovsky. ... Gregory Chudnovsky is a mathematician with a particular interest in number theory. ...

Miscellaneous formulæ

The base 60 representation of π, correct to eight significant figures (in base 10) is: Wikipedia does not have an article with this exact name. ... This article or section is in need of attention from an expert on the subject. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...

 3 + frac{8}{60} + frac{29}{60^2} + frac{44}{60^3}

In addition, the following expressions approximate π:

  • accurate to 9 decimal places: [4]
frac{63}{25} times frac{17 + 15 sqrt{5}}{7 + 15 sqrt{5}}
  • accurate to 9 places:
sqrt[4]{frac{2143}{22}}
Ramanujan claimed he had a dream in which the goddess Namagiri appeared and told him the true value of π. [5]
  • accurate to 3 decimal places: [4]
sqrt[3]{31}
  • accurate to 2 decimal places:
sqrt{2} + sqrt{3}
Karl Popper conjectured that Plato knew this expression; that he believed it to be exactly π; and that this is responsible for some of Plato's confidence in the omnicompetence of mathematical geometry — and Plato's repeated discussion of special right triangles that are either isosceles or halves of equilateral triangles.
  • The continued fraction representation of π can be used to generate successively better rational approximations, which start off: 22/7, 333/106, 355/113…. These approximations are the best possible rational approximations of π relative to the size of their denominators.

Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ... It has been suggested that this article or section be merged into Vishnu. ... Sir Karl Raimund Popper, CH, FRS, FBA, (July 28, 1902 – September 17, 1994), was an Austrian and British[1] philosopher and a professor at the London School of Economics. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A triangle. ... A triangle. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

Memorizing digits

Main article: Piphilology
Recent decades have seen a surge in the record number of digits memorized.
Recent decades have seen a surge in the record number of digits memorized.

Even long before computers have calculated π, memorizing a record number of digits became an obsession for some people. A Japanese man named Akira Haraguchi claims to have memorized 100,000 decimal places. This, however, has yet to be verified by Guinness World Records. 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.[6] It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.[7] Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant Ï€. The word is a play on Pi itself and the linguistic field of philology. ... Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... A decade is a set or a group of ten, commonly a period of 10 years in contemporary English, or a period of 10 days in the French revolutionary calendar. ... Akira Haraguchi (原口證) (born 1946) is a Japanese mental health counsellor best known for memorizing and reciting digits of Pi. ... Guinness World Records 2007 edition. ...


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. 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. 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. 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 (for instance, the year 1971 appears in the first fifty digits of pi). This article needs to be cleaned up to conform to a higher standard of quality. ... A mnemonic (AmE [] or BrE []) is a memory aid. ... Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant Ï€. The word is a play on Pi itself and the linguistic field of philology. ...


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. 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. In mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely. ...


Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulae imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory. See Bailey's above mentioned web site for details. 2000 (MM) was a leap year starting on Saturday of the Gregorian calendar. ... The Bailey-Borwein-Plouffe formula (BBP formula) permits the computation of the nth binary digit of Ï€. It is a Ï€ summation formula discovered in 1995 by Simon Plouffe. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). ...


It is also unknown whether π and e are algebraically independent. However it is known that at least one of πe and π + e is transcendental (see Lindemann–Weierstrass theorem). e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. This means that for every finite sequence α1, ..., αn of elements of S, no two the... In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, the Lindemann–Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ...


Naturality

In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π. This does not change the definition of π, but it does affect many formulæ in which π appears. So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements. Nonetheless, it occurs often in physics. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Some common angles, measured in radians. ... The shape of the Universe is an informal name for a subject of investigation within physical cosmology. ... In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ...


For example, consider Coulomb's law (SI units) Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ...

 F = frac{1}{ 4 pi epsilon_0} frac{left|q_1 q_2right|}{r^2} .

Here, 4πr2 is just the surface area of sphere of radius r. In this form, it is a convenient way of describing the inverse square relationship of the force at a distance r from a point source. It would of course be possible to describe this law in other, but less convenient ways, or in some cases more convenient. If Planck charge is used, it can be written as In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ...

 F = frac{q_1 q_2}{r^2}

and thus eliminate the need for π.


See also

This is a list of topics related to π (pi), the fundamental mathematical constant. ... Although the mathematical constant known as Ï€ (pi) has been studied since ancient times, as has the concept of irrational number, it was not until the 18th century that Ï€ was proved to be irrational. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... In mathematics, Buffons 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. ... Proofs of the famous mathematical result that the rational number 22⁄7 is greater than Ï€ date back to antiquity. ... The Feynman point comprises the 762nd through 767th decimal places of π, consisting of the digit 9 repeated six times. ... The Indiana Pi Bill is the popular name for Indiana House of Representatives bill #246 of 1897, which is one of the most famous historical attempts to (erroneously) define scientific truth by legislative fiat. ... Pies for a celebration at the Massachusetts Institute of Technology Free pie being handed out at the University of Waterloo Larry Shaw, the founder of Pi Day at the Exploratorium Pi Day and Pi Approximation Day are two unofficial holidays held to celebrate the mathematical constant Ï€ (Pi). ... LucyTuning is a form of meantone temperament, in which the fifth is of size 600+300/π (= approximately 695. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Over the years, several programs have been written for calculating Ï€ to many digits on personal computers. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... The golden section is a line segment sectioned into two according to the golden ratio. ...

References

Footnotes

  1. ^ OED: probably περιφέρεια or periphery
  2. ^ Rudin p.183
  3. ^ Statistical estimation of pi using random vectors. Retrieved on 2007-08-12.
  4. ^ a b Eric W. Weisstein, Pi Approximations at MathWorld.
  5. ^ Robert Kanigel (1991), The Man Who Knew Infinity: a life of the genius Ramanujan ISBN 0-671-75061-5
  6. ^ http://english.people.com.cn/200611/27/eng20061127_325612.htm
  7. ^ http://www.newsgd.com/culture/peopleandlife/200611280032.htm

Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... is the 224th day of the year (225th in leap years) in the Gregorian calendar. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Additional

David H. Bailey is a mathematician who, together with Peter Borwein and Simon Plouffe, found a formula for π in 1996 that permits one to calculate binary or hexadecimal digits of π beginning at an arbitrary position. ... Peter B. Borwein is a Canadian mathematician, co-developer of an algorithm for calculating π to the nth digit, co-discoverer of the billionth, four billionth, 40th billionth, and quadrillionth digits of π, and professor at Simon Fraser University. ... Simon Plouffe is a Quebec mathematician born on June 11, 1956 in St-Jovite. ... Walter Rudin Walter Rudin is an American mathematician, formerly a professor of mathematics at the University of Wisconsin, Madison. ... Petr Beckmann (1924-1993) was a physicist who defected to the United States from Czechoslovakia in 1963 and became a Professor of electrical engineering at the University of Colorado. ... Headquartered in the legendary Flatiron Building in New York City, St. ...

External links

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Image File history File links Commons-logo. ... This article is about a unit of data measurement. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...


  Results from FactBites:
 
Pi Day - Wikipedia, the free encyclopedia (487 words)
Pi Day and Pi Approximation Day are two unofficial holidays held to celebrate the mathematical constant π (Pi).
Pi Day is observed on March 14; Pi Approximation Day may be observed on any of several dates.
Pi Day is often celebrated at 1:59 p.m.
  More results at FactBites »

 
 

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