pi

 When a circle's diameter is 1, its circumference is pi. List of numbers – Irrational numbers γ – ζ(3) – √2 – √3 – √5 – φ – α – e – **π** – δ ||  ||  || **Number system ** || **Evaluation of π ** || Binary || 11.00100100001111110110…[1] || Decimal || 3.14159265358979323846264338327950288… || Hexadecimal || 3.243F6A8885A308D31319…[2] || Rational approximations || 3, 22⁄7, 333⁄106, 355⁄113, 103993/33102, ...[3] (listed in order of increasing accuracy) <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Continued fraction || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, … ][4] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 10pt;">(This continued fraction is not periodic. Shown in linear notation) <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Trigonometry || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">π radians = 180 degrees || <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">π is an irrational number, which means that its value cannot be expressed exactly as a fraction //m/////n//, where //m// and //n// are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The Greek letter π, often spelled out //pi// in text, was adopted for the number from the Greek word for //perimeter// "περίμετρος", first by William Jones in 1707, and popularized by Leonhard Euler in 1737.[6] The constant is occasionally also referred to as the **circular constant**, **Archimedes' constant** (not to be confused with an Archimedes number), or **Ludolph's number** (from a German mathematician whose efforts to calculate more of its digits became famous). <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Lower-case π is used to symbolize the constant. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Circumference = π × diameter <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The name of the Greek letter π is //pi//, and this spelling is commonly used in typographical contexts when the Greek letter is not available, or its usage could be problematic. It is not capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced /ˈpa ɪ <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">/, "pie" in English, which is the conventional English pronunciation of the Greek letter. In Greek, the name of this letter is pronounced [pi]. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.[7] π is Unicode character U+03C0 ("Greek small letter pi").[8] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Area of the circle = π × area of the shaded square <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:[7] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The ratio //C/////d// is constant, regardless of a circle's size. For example, if a circle has twice the diameter //d// of another circle it will also have twice the circumference //C//, preserving the ratio //C/////d//. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius:[7][9] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar. This can be considered a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive //x// for which cos(//x//) = 0.[10] The formulas below illustrate other (equivalent) definitions. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Squaring the circle: This was a problem proposed by the ancient geometers. In 1882, it was proven that π is transcendental, and consequently this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Being an irrational number, π cannot be written as the ratio of two integers. The belief in irrationality of π is mentioned by Muhammad ibn Mūsā al-Khwārizmī[11] in 9th century. Maimonides also mentions with certainty the irrationality of π in 12th century [12]. This was proved in 1768 by Johann Heinrich Lambert.[13] In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known.[14][15] A somewhat earlier similar proof is by Mary Cartwright.[16] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Furthermore, π is also transcendental, as was proved by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root.[17] An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[18] This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The numerical value of π truncated to 50 decimal places is:[19] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 //<span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">See the links below and those at sequence A000796 in OEIS for more digits. //<span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">While the value of π has been computed to more than a trillion (1012) digits,[20] elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.[21][22] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Because π is an irrational number, its decimal expansion never ends and does not repeat. This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties.[23] Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple base-10 pattern in the digits has ever been found.[24] Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to Archimedes,[25] is to calculate the perimeter, //Pn ,// of a regular polygon with //n// sides circumscribed around a circle with diameter //d.// Then <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range: 3 + 10 <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">⁄ <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">71 <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> < π < 3 + 1 <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">⁄ <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">7 <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">.[26] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:[27] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that nearly 300 terms are needed to calculate π correctly to 2 decimal places.[28] However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">and then define <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">then computing π10,10 will take similar computation time to computing 150 terms of the original series in a brute-force manner, and, correct to 9 decimal places. This computation is an example of the van Wijngaarden transformation.[29] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7th in the designs of the Old Kingdom pyramids in Egypt. The Great Pyramid at Giza, built c.2550-2500 B.C, was precisely 1760 cubits around with a height of 280 cubits (1760/280=2xPi). Egyptologists such as Professors Flinders Petrie [30] and I.E.S Edwards[31] have shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects[32][33]. The same apotropaic proportions were used earlier at the Pyramid of Meidum c.2600 B.C. This application is archaeologically evidenced, whereas textual evidence does not survive from this early period. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The early history of π from textual sources roughly parallels the development of mathematics as a whole.[34] Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.[35] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to Ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.[7] The Indian text //Shatapatha Brahmana// gives π as 339/108 ≈ 3.139. The Hebrew Bible appears to suggest, in the Book of Kings, that π = 3, which is notably worse than other estimates available at the time of writing (600 BC). The interpretation of the passage is disputed,[36][37] as some believe the ratio of 3:1 is of an interior circumference to an exterior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls (See: Biblical value of π). <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">**Archimedes' Pi aproximation** <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Archimedes (287–212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters:[37] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">By using the equivalent of 96-sided polygons, he proved that 3 + 10/71 < π < 3 + 1/7.[37] The average of these values is about 3.14185. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In the following centuries further development took place in India and China. Around AD 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to a 3072-gon and obtained an approximate value for π of 3.1416, as follows: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π < 3.1415927 using Liu Hui's algorithm applied to a 12288-gon. This value was the most accurate approximation of π available for the next 900 years. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Until the second millennium, π was known to fewer than 10 decimal digits. The next major advance in π studies came with the development of infinite series and subsequently with the discovery of calculus, which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, Madhava of Sangamagrama found the first known such series: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">This is now known as the Madhava–Leibniz series[38][39] or Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Madhava was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī, who determined 16 decimals of π. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometric method to compute 35 decimals of π. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.[40] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">found by François Viète in 1593. Another famous result is Wallis' product, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">by John Wallis in 1655. Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."[41] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In 1706 John Machin was the first to compute 100 decimals of π, using the formula <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">with <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. 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 Gauss. The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.) <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien-Marie Legendre also proved in 1794 π2 to be irrational. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre and Euler speculated that π might be transcendental, which was finally proved in 1882 by Ferdinand von Lindemann. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">William Jones' book //A New Introduction to Mathematics// from 1706 is said to be the first use of the Greek letter π for this constant, but the notation became particularly popular after Leonhard Euler adopted it in 1737.[42] He wrote: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 − 4/239) − 1/3(16/53 − 4/2393) + ... = 3.14159... = π[7]}} <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The advent of digital computers in the 20th century led to an increased rate of new π calculation records. John von Neumann et al. used ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours. [43][44] Additional thousands of decimal places were obtained in the following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance and mathematical depth.[45] One of his formulas is the series, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">and the related one found by the Chudnovsky brothers in 1987, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">which deliver 14 digits per term.[45] The Chudnovskys used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that //multiply// the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step.[46] The algorithm consists of setting <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">and iterating <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">until //an// and //bn// are close enough. Then the estimate for π is given by <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein.[47] The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.[48] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Simon Plouffe.[49] The formula, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones.[49] Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0.[50] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">If a formula of the form <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">were found where //b// and //c// are positive integers and //p// and //q// are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of π at any position in base //bc// without computing all the preceding digits in that base, in a time just depending on the size of the integer //k// and on the fixed degree of the polynomials. Plouffe also describes such formulas as the interesting ones for computing numbers of class **SC***, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer //k//, and requiring modest computing resources. The previous formula (found by Plouffe for π with //b//=2 and //c//=4, but also found for log(9/10) and for a few other irrational constants), implies that π is a **SC*** number. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In 2006, Simon Plouffe, using the integer relation algorithm PSLQ, found a series of beautiful formulas.[51] Let //q// = eπ (Gelfond's constant), then <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">and others of form, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">where //k// is an odd number, and //a//, //b//, //c// are rational numbers. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In the previous formula, if //k// is of the form 4//m// + 3, then the formula has the particularly simple form, <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">for some rational number //p// where the denominator is a highly factorable number, though no rigorous proof has yet been given. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:[52] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">or <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">However, there are generalized continued fractions for π with a perfectly regular structure, such as: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">[[image:file:///C:/DOCUME%7E1/PERSONAL/LOCALS%7E1/Temp/msohtmlclip1/01/clip_image043.gif width="757" height="204" caption="\pi=\cfrac{4}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}}= 3+\cfrac{1^2}{6+\cfrac{3^2}{6+\cfrac{5^2}{6+\cfrac{7^2}{6+\cfrac{9^2}{6+\ddots}}}}}= \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ddots}}}}}"]] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Recent decades have seen a surge in the record for number of digits memorized. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Even long before computers have calculated π, memorizing a //record// number of digits became an obsession for some people. In 2006, Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places.[53] This, however, has yet to be verified by Guinness World Records. The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China.[54] It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.[55] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">On June, 17th, 2009 Andriy Slyusarchuk, a Ukrainian neurosurgeon, medical doctor and professor claimed to have memorized 30 million digits of pi, which were printed in 20 volumes of text.[56] He has been officially congratulated by the President of Ukraine Viktor Yuschenko. A possibility of financing a dedicated research center for development of Mr. Slyusarchuk's methodology had been discussed.[57][58] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">There are many ways to memorize π, including the use of "piems", which are poems that represent π in a way such that the length of each word (in letters) represents a digit. Here is an example of a piem, originally devised by Sir James Jeans: //How I need// (or: //want//) //a drink, alcoholic in nature// (or: //of course//)//, after the heavy lectures// (or: //chapters//) //involving quantum mechanics.//[59][60] Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on. The //Cadaeic Cadenza// contains the first 3834 digits of π in this manner.[61] Piems are related to the entire field of humorous yet serious study that involves the use of mnemonic techniques to remember the digits of π, known as piphilology. In other languages there are similar methods of memorization. However, this method proves inefficient for large memorizations of π. Other methods include remembering patterns in the numbers and the method of loci.[62][63] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions.[17] Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π.[64] The more terms included in a calculation, the closer to π the result will get. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Consequently, numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more precision. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. The approximation 355⁄113 (3.1415929…) is the best one that may be expressed with a three-digit or four-digit numerator and denominator; the next good approximatioion 103993/33102 (3.14159265301...) requires much bigger numbers, due to the large number number 292 in the continued fraction expansion.[3] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The earliest numerical approximation of π is almost certainly the value 3.[37] In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The most pressing open question about π is whether it is a normal number—whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in //every// integer base, not just base 10.[65] Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π.[66] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory.[67] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">It is also unknown whether π and //e// are algebraically independent, although Yuri Nesterenko proved the algebraic independence of {π, //e//π, Γ(1/4)} in 1996.[68] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.[69] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">For any circle with radius //r// and diameter //d// = 2//r//, the circumference is π//d// and the area is π//r//2. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as ellipses, spheres, cones, and tori.[70] Accordingly, π appears in definite integrals that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the unit disk is given by:[71] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">and <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">gives half the circumference of the unit circle.[70] More complicated shapes can be integrated as solids of revolution.[72] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">From the unit-circle definition of the trigonometric functions also follows that the sine and cosine have period 2π. That is, for all //x// and integers //n//, sin(//x//) = sin(//x// + 2π//n//) and cos(//x//) = cos(//x// + 2π//n//). Because sin(0) = 0, sin(2π//n//) = 0 for all integers //n//. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In modern mathematics, π is often //defined// using trigonometric functions, for example as the smallest positive //x// for which sin //x// = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the inverse trigonometric functions, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as power series is the easiest way to derive infinite series for π. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Euler's formula depicted on the complex plane. Increasing the angle //φ// to //π// radians (180°) yields Euler's identity. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">A complex number //z// can be expressed in polar coordinates as follows: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">where //i// is the imaginary unit satisfying //i//2 = −1 and //e// ≈ 2.71828 is Euler's number. This formula implies that imaginary powers of //e// describe rotations on the unit circle in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation //φ// = π results in the remarkable Euler's identity <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Euler's identity is famous for linking several basic mathematical constants and operators. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">There are //n// different //n//-th roots of unity <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">The Gaussian integral <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">A consequence is that the gamma function of a half-integer is a rational multiple of √π. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Although not a physical constant, π appears routinely in equations describing fundamental principles of the Universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems. Using units such as Planck units can sometimes eliminate π from formulae. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">In probability and statistics, there are many distributions whose formulas contain π, including: <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Note that since for any probability density function //f//(//x//), the above formulas can be used to produce other integral formulas for π.[80] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Buffon's needle problem is sometimes quoted as a empirical approximation of π in "popular mathematics" works. Consider dropping a needle of length //L// repeatedly on a surface containing parallel lines drawn //S// units apart (with //S// > //L//). If the needle is dropped //n// times and //x// of those times it comes to rest crossing a line (//x// > 0), then one may approximate π using the Monte Carlo method:[81][82][83][84] <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of π //by experiment//. Reliably getting just three digits (including the initial "3") right requires millions of throws,[81] and the number of throws grows exponentially with the number of digits desired. Furthermore, any error in the measurement of the lengths //L// and //S// will transfer directly to an error in the approximated π. For example, a difference of a single atom in the length of a 10-centimeter needle would show up around the 9th digit of the result. In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits. <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;"> <span style="color: #f79646; font-family: "Times New Roman","serif"; font-size: 12pt;">Retrieved from "http://en.wikipedia.org/wiki/Pi"
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 24pt;">Pi **<span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt; text-decoration: none;"> <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 24pt;">
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;">Pi **<span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 12pt;"> or **π** is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159 in the usual decimal notation (see the table for its representation in some other bases). π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π.[5]
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Fundamentals **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Fundamentals **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">The letter π **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Definition **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Irrationality and transcendence **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Numerical value **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Calculating π **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">History **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Geometrical period **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Classical period **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Computation in the computer age **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Pi and continued fraction **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Memorizing digits **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Advanced properties **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Numerical approximations **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Open questions **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">Use in mathematics and science **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Geometry and trigonometry **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Complex numbers and calculus **
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Physics **
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">The cosmological constant:[73]
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (Δ//x//) and momentum (Δ//p//) can not both be arbitrarily small at the same time:[74]
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Einstein's field equation of general relativity:[75]
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Coulomb's law for the electric force, describing the force between two electric charges (//q1// and //q2//) separated by distance //r//:[76]
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Magnetic permeability of free space:[77]
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Kepler's third law constant, relating the orbital period (//P//) and the semimajor axis (//a//) to the masses (//M// and //m//) of two co-orbiting bodies:
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 13.5pt;">Probability and statistics **
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">the probability density function for the normal distribution with mean μ and standard deviation σ, due to the Gaussian integral:[78]
 * <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">the probability density function for the (standard) Cauchy distribution:[79]
 * <span style="color: #404040; font-family: "Times New Roman","serif"; font-size: 18pt;">References **
 * 1) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Alexander D. Poularikas (1999). //The handbook of formulas and tables for signal processing//. CRC Press. p. 9.8. ISBN 9780849385797. http://books.google.com/books?id=aaStYSe6WVcC&pg=PT165&dq=11.001001+different-number-bases&ei=FBXiStjlIZKalASsgoWjDA#v=onepage&q=11.001001%20different-number-bases&f=false.
 * 2) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> "Sample digits for hexa decimal digits of pi". Dec. 6, 2002. http://www.super-computing.org/pi-hexa_current.html.
 * 3) <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **//a//** **//b//** Gourdon, Xavier; Pascal Sebah. "Collection of approximations for π". Numbers, constants and computation. http://numbers.computation.free.fr/Constants/Pi/piApprox.html. Retrieved 2007-11-08.
 * 4) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> A001203: Continued fraction for Pi, On-Line Encyclopedia of Integer Sequences
 * 5) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Howard Whitley Eves (1969). //An Introduction to the History of Mathematics//. Holt, Rinehart & Winston. http://books.google.com/books?id=LIsuAAAAIAAJ&q=%22important+numbers+in+mathematics%22&dq=%22important+numbers+in+mathematics%22&pgis=1.
 * 6) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Comanor, Milton; Ralph P. Boas (1976). "Pi". in William D. Halsey. //Collier's Encyclopedia//. **19**. New York: Macmillan Educational Corporation. pp. 21-22.
 * 7) <span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **//a//** **//b//** **//c//** **//d//** **//e//** "About Pi". //Ask Dr. Math FAQ//. http://mathforum.org/dr.math/faq/faq.pi.html. Retrieved 2007-10-29.
 * 8) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> "Characters Ordered by Unicode". W3C. http://www.w3.org/TR/MathML2/bycodes.html. Retrieved 2007-10-25.
 * 9) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Richmond, Bettina (1999-01-12). "Area of a Circle". Western Kentucky University. http://www.wku.edu/~tom.richmond/Pir2.html. Retrieved 2007-11-04.
 * 10) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Rudin, Walter (1976) [1953]. //Principles of Mathematical Analysis// (3e ed.). McGraw-Hill. pp. 183. ISBN 0-07-054235-X.
 * 11) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Glimpses in the history of a great number: Pi in Arabic mathematicsby Mustafa Mawaldi
 * 12) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Commentary to Mishna, beginning of Eruvin
 * 13) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Lambert, Johann Heinrich (1761), "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques", //Histoire de l'Académie,// (Berlin) **XVII**: 265–322, 1768
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 * 85) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> See, //e.g//, Lennart Berggren, Jonathan M. Borwein, and Peter B. Borwein (eds.), //Pi: A Source Book//. Springer, 1999 (2nd ed.). ISBN 978-0-387-98946-4.
 * 86) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> See Alfred S. Posamentier and Ingmar Lehmann, //Pi: A Biography of the World's Most Mysterious Number//. Prometheus Books, 2004. ISBN 978-1-59102-200-8.
 * 87) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> //E.g.//, MSNBC, Man recites pi from memory to 83,431 places July 3, 2005; Matt Schudel, Obituaries: "John W. Wrench, Jr.: Mathematician Had a Taste for Pi" //The Washington Post//, March 25, 2009, p. B5.
 * 88) **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">^ **<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"> Pi Day activities.
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