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What dog ran 100 meters in 20 seconds. What was the dog's speed (Use the equation s d and Atilde and middot t.)?
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What is 2(53) equal?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and Natural Resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual
What is (x,20) and (20,-13)slope;0?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and natural resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual hexadecimaldigit of π without calculating all the preceding digits.[97]Individual octal or binary digits may be extracted from the hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits.
What is the product of y4 and middot y6?
It is 24y^2.
What is The coefficient of 8 and middot 2N?
The coefficient is 1.6
The specific heat of a certain type of cooking oil is 1.75 cal(g and middot and degC). How much heat energy is needed to raise the temperature of 2.40 kg of this oil from 23 and degC to 191 and degC?
The energy is 103,6 kcal.
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Well, I went looking for a way to make an interpunct because I needed to use one and I found something useful for some situations. This website, http://www.statemaster.com/encyclopedia/Interpunct, has some helpful HTML codes for making interpuncts. I found that they don't work when I tried to send one through e-mail, but a couple did work when I was using their codes for an online custom bracelet. I don't know if this is helpful or not, but it has helped me. Symbol Character Entity Numeric Entity Unicode Code Point Notes · &middot; &#183; U+00B7 interpunct, middle dot · &#903; U+0387 Greek ano teleia ⋅ &sdot; &#8901; U+22C5 dot operator (mathematics) • &bull; &#8226; U+2022 bullet, often used to mark list items ‧ &#8231; U+2027 hyphenation point ・ &#12539; U+30FB fullwidth katakana middle dot ･ &#65381; U+FF65 halfwidth katakana middle dot ּ &#1468; U+05BC Hebrew point dagesh or mapiq Characters in the Symbol column, above, may not render correctly, if at all, in all browsers. For most North American Microsoft operating systems using OEM Code page 437, the Alt Code is 250
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What is 2(d 9)?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and Natural Resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual
What is 2-(4-3-22)?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and Natural Resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual
What is (227-23)2?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and natural resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual hexadecimaldigit of π without calculating all the preceding digits.[97]Individual octal or binary digits may be extracted from the hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits.[99]An important application of digit extraction algorithms is to validate new claims of record π comp
What is 2(13)?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and Natural Resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual
What is 2(53) equal?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and Natural Resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual
What is (x,20) and (20,-13)slope;0?
well you see Philabusting is when you talk for a long time with no propose to delay an organization and now I would like to say every diget of pie that I feel like and talk about ramdom things The U.S. Senate is the upper legislative chamber in the federal government. It's also the more powerful body, with just 100 members. Each state is granted two senators who represent the entire state; senators serve six-year terms and are popularly elected by their constituents.Leading the SenateThe vice President of the United States presides over the Senate and casts the deciding vote in the event of a tie. The Senate leadership also includes president pro tempore who presides in the absence of the vice president, a majority leader who appoints members to lead and serve on various committees, and a minority leader. Both parties -majority and minority-also have a whip who helps marshal senators' votes along party lines.The Powers of the SenateThe Senate's power derives from more than just its relatively exclusive membership; it also is granted specific powers in the Constitution. In addition to the many powers granted jointly to both houses of Congress, the Constitution enumerates the role of the upper body specifically in Article I, Section 3.While the House of Representatives has the power to recommend impeachment of a sitting president, vice president or other civic official such as a judge for "high crimes and misdemeanors," as written in the Constitution, the Senate is the sole jury once impeachment goes to trial. With a two-thirds majority, the Senate may thus remove an official from office. Two presidents, Andrew Johnson and Bill Clinton, have been tried; both were acquitted.The President of the United States has the power to negotiate treaties and agreements with other nations, but the Senate must ratify them by a two-thirds vote in order to take effect. This isn't the only way the Senate balances the power of the president. All presidential appointees, includingCabinet members, judicial appointees and ambassadors must be confirmed by the Senate, which can call any nominees to testify before it.The Senate also investigates matters of national interest. There have been special investigations of matters ranging from the Vietnam War to organized crime to the Watergate break-in and subsequent cover-up.The More "Deliberate" ChamberThe Senate is commonly the more deliberative of the two chambers of Congress; theoretically, a debate on the floor may go on indefinitely, and some seem to. Senators may filibuster, or delay further action by the body, by debating it at length; the only way to end a filibuster is through a motion of cloture, which requires the vote of 60 senators.The Senate Committee SystemThe Senate, like the House of Representatives, sends bills to committees before bringing them before the full chamber; it also has committees which perform specific non-legislative functions as well. The Senate's committees include:agriculture, nutrition and forestry;appropriations;armed services;banking, housing and urban affairs;budget;commerce, science and transportation;energy and natural resources;environment and public works;finance;foreign relations;health, education, labor and pensions;homeland security and governmental affairs;judiciary;rules and administration;small business and entrepreneurship;and veterans' affairs.There are also special committees on aging, ethics, intelligence and Indian affairs; and joint committees with the House of Representatives.Government 101 Indexelkjflskdjfsdflakjsd;kghjalskdjfhlksjdhflakjshdflkasjdhflaksjdhflaksjdhflaksjdhflaksjdhflkasjdhflkasjhdflkajhsdflkhjThe number π (/aɪ/) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number - a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered. For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedesand Liu Huiused geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians includingMadhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.In the 20th century, mathematicians and computer scientists discovered new approaches that - when combined with increasing computational power - extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputersand high-precision multiplication algorithms.Because its definition relates to the circle, π is found in many formulae in trigonometryand geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such ascosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.Contents[hide] 1 Fundamentals1.1 Definition1.2 Name1.3 Properties1.4 Continued fractions1.5 Approximate value2 History2.1 Antiquity2.2 Polygon approximation era2.3 Infinite series2.4 Irrationality and transcendence2.5 Computer era and iterative algorithms2.6 Motivations for computing π2.7 Rapidly convergent series2.8 Spigot algorithms3 Use3.1 Geometry and trigonometry3.2 Complex numbers and analysis3.3 Number theory and Riemann zeta function3.4 Physics3.5 Probability and statistics3.6 Engineering and geology4 Outside the sciences4.1 Memorizing digits4.2 In popular culture5 See also6 Notes7 References8 Further reading9 External linksFundamentalsDefinitionThe circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called π.π is commonly defined as the ratio of a circle's circumferenceC to its diameterd:[1]The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle it will also have twice the circumference, preserving the ratio C/d. This definition of π is not universal, because it is only valid in flat (Euclidean) geometry and is not valid in curved (non-Euclidean) geometries.[1]For this reason, some mathematicians prefer definitions of π based on calculus or trigonometrythat do not rely on the circle. One such definition is: π is twice the smallest positive x for which cos(x) equals 0.[1][2]NameLeonhard Euler popularized the use of the Greek letter π in a work he published in 1748.The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the Greek letterπ. That letter (and therefore the number π itself) can be denoted by the Latin word pi.[3]In English, π ispronounced as "pie" ( /aɪ/, /ˈpaɪ/).[4]The lower-case letter π (or π in sans-seriffont) is not to be confused with the capital letter Π, which denotes a product of a sequence.The first mathematician to use the Greek letter π to represent the ratio of a circle's circumference to its diameter was William Jones, who used it in his work Synopsis Palmariorum Matheseos; or, a New Introduction to the Mathematics, of 1706.[5]Jones' first use of the Greek letter was in the phrase "1/2 Periphery (π)" in the discussion of a circle with radius one. He may have chosen π because it was the first letter in the Greek spelling of the wordperiphery.[6]Jones writes that his equations for π are from the "ready pen of the truly ingenious Mr. John Machin", leading to speculation that Machin may have employed the Greek letter before Jones.[7]The Greek letter had been used earlier for geometric concepts. For example, in 1631 it was used by William Oughtred to represent the half-circumference of a circle.[7]After Jones introduced the Greek letter in 1706, it was not adopted by other mathematicians until Euler used it in 1736. Before then, mathematicians sometimes used letters such as c or p instead.[7]Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly.[7]In 1748, Euler used π in his widely read work Introductio in analysin infinitorum (he wrote: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1") and the practice was universally adopted thereafter in the Western world.[7]Propertiesπ is an irrational number, meaning that it cannot be written as the ratio of two integers, such as 22/7 or other fractions that are commonly used to approximate π.[8]Since π is irrational, it has an infinite number of digits in itsdecimal representation, and it does not end with an infinitely repeating pattern of digits. There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique. The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln(2), but smaller than the measure of Liouville numbers.[9]Because π is a transcendental number,squaring the circle is not possible in a finite number of steps using the classical tools ofcompass and straightedge.π is a transcendental number, which means that it is not the solution of any non-constant polynomialwith rationalcoefficients, such as [10][11]The transcendence of π has two important consequences: First, πcannot be expressed using any combination of rational numbers and square roots or n-th roots such as or Second, since no transcendental number can be constructedwith compass and straightedge, it is not possible to "square the circle". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[12]Squaring a circle was one of the important geometry problems of the classical antiquity.[13]Amateur mathematicians in modern times have sometimes attempted to square the circle, and sometimes claim success, despite the fact that it is impossible.[14]The digits of π have no apparent pattern and pass tests for statistical randomness including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often.[15]The hypothesis that π is normal has not been proven or disproven.[15]Since the advent of computers, a large number of digits of π have been available on which to perform statistical analysis. Yasumasa Kanada has performed detailed statistical analyses on the decimal digits of π, and found them consistent with normality; for example, the frequency of the ten digits 0 to 9 were subjected to statistical significance tests, and no evidence of a pattern was found.[16]Despite the fact that π's digits pass statistical tests for randomness, π contains some sequences of digits that may appear non-random to non-mathematicians, such as the Feynman point, which is a sequence of six consecutive 9s that begins at the 762nd decimal place of the decimal representation of π.[17]Continued fractionsThe constant π is represented in thismosaicoutside the mathematics building at the Technische Universität Berlin.Like all Irrational Numbers, π cannot be represented as a simple fraction. But every irrational number, including π, can be represented by an infinite series of nested fractions, called a continued fraction:Truncating the continued fraction at any point generates a fraction that provides an approximation for π; two such fractions (22/7 and 355/113) have been used historically to approximate the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[18]Although the simple continued fraction for π (shown above) does not exhibit a pattern,[19]mathematicians have discovered several generalized continued fractions that do, such as:[20]Approximate valueSome approximations of π include:Fractions: Approximate fractions include (in order of increasing accuracy) 227, 333106, 355113, 5216316604, and 10399333102.[18]Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....[21]Binary: 11.001001000011111101101010100010001000010110100011 ....Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....[22]Sexagesimal: A base 60 approximation is 3:8:29:44:1HistorySee also: Chronology of computation of πAntiquityThe Great Pyramidat Giza, constructed c. 2589-2566 BC, was built with a perimeter of about 1760 cubits and a height of about 280 cubits; the ratio 1760/280 ≈ 6.2857 is approximately equal to 2π ≈ 6.2832. Based on this ratio, some Egyptologistsconcluded that the pyramid builders had knowledge of πand deliberately designed the pyramid to incorporate the proportions of a circle.[23]Others maintain that the suggested relationship to π is merely a coincidence, because there is no evidence that the pyramid builders had any knowledge of π, and because the dimensions of the pyramid are based on other factors.[24]The earliest written approximations of π are found in Egypt and Babylon, both within 1 percent of the true value. In Babylon, a clay tablet dated 1900-1600 BC has a geometrical statement that, by implication, treats π as 25/8 = 3.1250.[25]In Egypt, the Rhind Papyrus, dated around 1650 BC, but copied from a document dated to 1850 BC has a formula for the area of a circle that treats π as (16/9)2 ≈ 3.1605.[25]In India around 600 BC, the Shulba Sutras (Sanskrittexts that are rich in mathematical contents) treat π as (9785/5568)2 ≈ 3.088.[26]In 150 BC, or perhaps earlier, Indian sources treat π as ≈ 3.1622.[27]Two verses in the Hebrew Bible (written between the 8th and 3rd centuries BC) describe a ceremonial pool in the Temple of Solomon with a diameter of tencubits and a circumference of thirty cubits; the verses imply π is about three if the pool is circular.[28][29]Rabbi Nehemiah explained the discrepancy as being due to the thickness of the vessel. His early work of geometry, Mishnat ha-Middot, was written around 150 AD and takes the value of π to be three and one seventh.[30]Polygon approximation eraπ can be estimated by computing the perimeters of circumscribed and inscribed polygons.The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematicianArchimedes.[31]This polygonal algorithm dominated for over 1,000 years, and as a result πis sometimes referred to as "Archimedes' constant".[32]Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223/71 < π < 22/7 (3.1408 < π < 3.1429).[33]Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[34]Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga.[35]Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.[36]Archimedesdeveloped the polygonal approach to approximating π.In ancient China, values for π included 3.1547 (around 1 AD), (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[37]Around 265 AD, the Wei Kingdommathematician Liu Huicreated apolygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416.[38][39]Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4.[38]The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π ≈ 355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of πavailable for the next 800 years.[40]The Indian astronomer Aryabhataused a value of 3.1416 in his Āryabhaṭīya(499 AD).[41]Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[42]Italian author Danteapparently employed the value ≈ 3.14142.[42]The Persian astronomer Jamshīd al-Kāshī produced 16 digits in 1424 using a polygon with 3×228 sides,[43][44]which stood as the world record for about 180 years.[45]French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[45]Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[45]In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century).[46]Dutch scientist Willebrord Snellius reached 34 digits in 1621,[47]and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630,[48]which remains the most accurate approximation manually achieved using polygonal algorithms.[47]Infinite seriesThe calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence.[49]Infinite series allowed mathematicians to compute π with much greater precision than Archimedesand others who used geometrical techniques.[49]Although infinite series were exploited for π most notably by European mathematicians such as James Gregoryand Gottfried Wilhelm Leibniz, the approach was first discovered in Indiasometime between 1400 and 1500 AD.[50]The first written description of an infinite series that could be used to compute π was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji in his Tantrasamgraha, around 1500 AD.[51]The series are presented without proof, but proofs are presented in a later Indian work, Yuktibhāṣā, from around 1530 AD. Nilakantha attributes the series to an earlier Indian mathematician, Madhava of Sangamagrama, who lived c. 1350 - c. 1425.[51]Several infinite series are described, including series for sine, tangent, and cosine, which are now referred to as the Madhava series or Gregory-Leibniz series.[51]Madhava used infinite series to estimate π to 11 digits around 1400, but that record was beaten around 1430 by the Persian mathematician Jamshīd al-Kāshī, using a polygonal algorithm.[52]Isaac Newtonused infinite series to compute π to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".[53]The first infinite sequence discovered in Europe was an infinite product (rather than an infinite sum, which are more typically used in π calculations) found by French mathematician François Viète in 1593:[54]The second infinite sequence found in Europe, by John Wallis in 1655, was also an infinite product.[54]The discovery of calculus, by English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz in the 1660s, led to the development of many infinite series for approximating π. Newton himself used an arcsin series to compute a 15 digit approximation of π in 1665 or 1666, later writing "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[53]In Europe, Madhava's formula was rediscovered by Scottish mathematician James Gregoryin 1671, and by Leibniz in 1674:[55][56]This formula, the Gregory-Leibniz series, equals when evaluated with z = 1.[56]In 1699, English mathematician Abraham Sharp used the Gregory-Leibniz series to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.[57]The Gregory-Leibniz series is simple, butconvergesvery slowly (that is, approaches the answer gradually), so it is not used in modern π calculations.[58]In 1706 John Machinused the Gregory-Leibniz series to produce an algorithm that converged much faster:[59]Machin reached 100 digits of π with this formula.[60]Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for π digits.[60]Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson - the best approximation achieved without the aid of a calculating device.[61]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 at the behest of German mathematician Carl Friedrich Gauss.[62]British mathematician William Shanks famously took 15 years to calculate π to 707 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect.[62]Rate of convergenceSome infinite series for π convergefaster than others. Given the choice of two infinite series for π, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate π to any given accuracy.[63]A simple infinite series for π is the Gregory-Leibniz series:[64]As individual terms of this infinite series are added to the sum, the total gradually gets closer to π, and - with a sufficient number of terms - can get as close to π as desired. It converges quite slowly, though - after 500,000 terms, it produces only five correct decimal digits of π.[65]An infinite series for π (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory-Leibniz series is:[66]The following table compares the convergence rates of these two series:Infinite series for π After 1st term After 2nd term After 3rd term After 4th term After 5th term Converges to: 4.0000 2.6666... 3.4666... 2.8952... 3.3396... π = 3.1415... 3.0000 3.1666... 3.1333... 3.1452... 3.1396... After five terms, the sum of the Gregory-Leibniz series is within 0.2 of the correct value of π, whereas the sum of Nilakantha's series is within 0.002 of the correct value of π. Nilakantha's series converges faster and is more useful for computing digits of π. Series that converge even faster include Machin's seriesand Chudnovsky's series, the latter producing 14 correct decimal digits per term.[63]Irrationality and transcendenceNot all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problemin 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:[67]Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational, meaning it is not equal to the quotient of any two whole numbers.[8]Lambert's proof exploited a continued-fraction representation of the tangent function.[68]French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.[69]Computer era and iterative algorithmsJohn von Neumann was part of the team that first used a digital computer, ENIAC, to compute π.The development of computers in the mid-20th century again revolutionized the hunt for digits of π. American mathematicians John Wrenchand Levi Smith reached 1,120 digits in 1949 using a desk calculator.[70]Using an arctan infinite series, a team led by George Reitwiesner and John von Neumann achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIACcomputer.[71]The record, always relying on arctan series, was broken repeatedly (7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits was reached in 1973.[72]Two additional developments around 1980 once again accelerated the ability to compute π. First, the discovery of new iterative algorithms for computing π, which were much faster than the infinite series; and second, the invention of fast multiplication algorithms that could multiply large numbers very rapidly.[73]Such algorithms are particularly important in modern π computations, because most of the computer's time is devoted to multiplication.[74]They include the Karatsuba algorithm, Toom-Cook multiplication, and Fourier transform-based methods.[75]The Gauss-Legendre iterative algorithm:InitializeIterateThen an estimate for π is given byThe iterative algorithms were independently published in 1975-1976 by American physicist Eugene Salamin and Australian scientist Richard Brent.[76]These avoid reliance on infinite series. An iterative algorithm repeats a specific calculation, each iteration using the outputs from prior steps as its inputs, and produces a result in each step that converges to the desired value. The approach was actually invented over 160 years earlier by Carl Friedrich Gauss, in what is now termed the arithmetic-geometric mean method (AGM method) or Gauss-Legendre algorithm.[76]As modified by Salamin and Brent, it is also referred to as the Brent-Salamin algorithm.The iterative algorithms were widely used after 1980 because they are faster than infinite series algorithms: whereas infinite series typically increase the number of correct digits additively in successive terms, iterative algorithms generally multiplythe number of correct digits at each step. For example, the Brent-Salamin algorithm doubles the number of digits in each iteration. In 1984, the Canadian brothers John and Peter Borweinproduced an iterative algorithm that quadruples the number of digits in each step; and in 1987, one that increases the number of digits five times in each step.[77]Iterative methods were used by Japanese mathematicianYasumasa Kanada to set several records for computing π between 1995 and 2002.[78]This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.[78]Motivations for computing πAs mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of π increased dramatically.For most numerical calculations involving π, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmologicalcalculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom.[79]Despite this, people have worked strenuously to compute π to thousands and millions of digits.[80]This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world.[81][82]They also have practical benefits, such as testingsupercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.[83]Rapidly convergent seriesSrinivasa Ramanujan, working in isolation in India, produced many innovative series for computing π.Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[78]The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth, and rapid convergence.[84]One of his formulae, based on modular equations:This series converges much more rapidly than most arctan series, including Machin's formula.[85]Bill Gosperwas the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985.[86]Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers and the Chudnovsky brothers.[87]The Chudnovsky formula developed in 1987 isIt produces about 14 digits of π per term,[88]and has been used for several record-setting π calculations, including the first to surpass (109) digits in 1989 by the Chudnovsky brothers, 2.7 trillion (2.7×1012) digits by Fabrice Bellardin 2009, and 10 trillion (1013) digits in 2011 by Alexander Yee and Shigeru Kondo.[89][90]In 2006, Canadian mathematician Simon Plouffe used the PSLQ integer relation algorithm[91]to generate several new formulae for π, conforming to the following template:where is eπ (Gelfond's constant), is an odd number, and are certain rational numbers that Plouffe computed.[92]Spigot algorithmsTwo algorithms were discovered in 1995 that opened up new avenues of research into π. They are called spigot algorithms because, like water dripping from a spigot, they produce single digits of π that are not reused after they are calculated.[93][94]This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[93]American mathematicians Stan Wagonand Stanley Rabinowitz produced a simple spigot algorithm in 1995.[94][95][96]Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.[95]Another spigot algorithm, the digit extraction algorithm, was discovered in 1995 by Simon Plouffe:[97][98]This formula, unlike others before it, can produce any individual hexadecimaldigit of π without calculating all the preceding digits.[97]Individual octal or binary digits may be extracted from the hexadecimal digits. Variations of the algorithm have been discovered, but no digit extraction algorithm has yet been found that rapidly produces decimal digits.
What is philabusting filibuster?
A philabuster is when a politician uses a filibuster to delay or block legislative action by speaking for an extended period. The term is a play on words blending filibuster and Philadelphia. It may specifically refer to filibusters carried out in Philadelphia or by Philadelphia politicians.
