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Pickover's sequence gives the starting positions in the decimal expansion of (ignoring the leading 3) in which the first digits of occur (counting the leading 2). So, since , the first digit "2" of occurs at position 6. Continuing, the sequence is given by 6, 28, 241, 11706, 28024, 33789, 1526800, 73154827, ... (OEIS A090898).Conversely, consider the sequence formed by the expansion of (ignoring the leading 2) in which the first digits of occur (counting the leading 3). So, since , the first digit "3" of occurs at position 17. Continuing, the sequence is given by 17, 189, 856, 17947, 53238, 1436935, 5000482, ... (OEIS A115234).

may be computed using a number of iterative algorithms. The best known such algorithms are the Archimedes algorithm, which was derived by Pfaff in 1800, and the Brent-Salamin formula. Borwein et al. (1989) discuss th-order iterative algorithms.The Brent-Salamin formula is a quadraticallyconverging algorithm.Another quadratically converging algorithm (Borweinand Borwein 1987, pp. 46-48) is obtained by defining(1)(2)and(3)(4)Then(5)with . decreases monotonically to with(6)for .A cubically converging algorithm which converges to the nearest multiple of to is the simple iteration(7)(Beeler et al. 1972). For example, applying to 23 gives the sequence 23, 22.1537796, 21.99186453, 21.99114858, ..., which converges to .A quartically converging algorithm is obtained by letting(8)(9)then defining(10)(11)Then(12)and converges to quartically with(13)(Borwein and Borwein 1987, pp. 170-171; Bailey 1988, Borwein..

Let and be the perimeters of the circumscribed and inscribed -gon and and the perimeters of the circumscribed and inscribed -gon. Then(1)(2)The first follows from the fact that side lengths of the polygons on a circle of radius are(3)(4)so(5)(6)But(7)(8)Using the identity(9)then gives(10)The second follows from(11)Using the identity(12)gives(13)(14)(15)(16)Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi ().

Successive application of Archimedes' recurrence formula gives the Archimedes algorithm, which can be used to provide successive approximations to (pi). The algorithm is also called the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation of by circumscribing and inscribing -gons on a circle. From Archimedes' recurrence formula, the circumferences and of the circumscribed and inscribed polygons are(1)(2)where(3)For a hexagon, and(4)(5)where . The first iteration of Archimedes' recurrence formula then gives(6)(7)(8)Additional iterations do not have simple closed forms, but the numerical approximations for , 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are(9)(10)(11)(12)(13)By taking (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result(14)..

Convergents of the pi continued fractions are the simplest approximants to . The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.Two approximations follow from the near-identity function evaluated at and , giving(1)(2)which are good to 2 and 3 digits, respectively.Kochanski's approximation is the rootof(3)given by(4)which is good to 4 digits.Another curious fact is the almost integer(5)which can also be written as(6)(7)Here, is Gelfond's constant. Applying cosine a few more times gives(8)Another approximation involving is given by(9)which is good to 2 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).An apparently interesting near-identity is given by(10)which becomes less surprising when it is noted that 555555 is a repdigit,so the above is..

The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The first few convergents are 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.The very large term 292 means that the convergent(1)is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). A nice expression for the third convergent of is given by(2)(Stoschek).The Engel expansion of is 1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ... (OEIS A006784).The following table summarizes some record computations of the continued fraction of pi.termsdatereference1977W. Gosper..

A numberwhere is an integer or rational number, is the inverse tangent, and is the inverse cotangent. Gregory numbers arise in the determination of Machin-like formulas. Every Gregory number can be expressed uniquely as a linear combination of s where the s are Størmer numbers.

A short mnemonic for remembering the first seven decimal digits of is "How I wish I could calculate pi" (C. Heckman, pers. comm., Feb. 3, 2005). Eight digits are given by "May I have a large container of coffee?" giving 3.1415926 (Gardner 1959; 1966, p. 92; Eves 1990, p. 122, Davis 1993, p. 9). "But I must a while endeavour to reckon right" gives nine correct digits (3.14159265). "May I have a white telephone, or pastel color" (M. Amling, pers. comm., Jul. 31, 2004) also gives nine correct digits.A more substantial mnemonic giving 15 digits (3.14159265358979) is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics," originally due to Sir James Jeans (Gardner 1966, p. 92; Castellanos 1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner 1997, p. 112). A slight extension..

There is a series of BBP-type formulas for in powers of ,(1)(2)(3)(4)(5)(6),(7)(8)(9)(10)some of which are noted by Bailey et al. (1997), and ,(11)(12)Another identity is(13)where is the polylogarithm. (13) is equivalent to(14)(Bailey et al. 1997).

There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..

The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging into the Leibniz series,(1)(Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular(2)where is the Riemann zeta function (Vardi 1991).Taking the partial series gives the analytic result(3)Rather amazingly, expanding about infinity gives the series(4)(Borwein and Bailey 2003, p. 50), where is an Euler number. This means that truncating the Gregory series at half a large power of 10 can give a decimal expansion for whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking gives where the sequence of differences is precisely twice the Euler (secant) numbers. In fact, just this pattern of digits was observed by J. R. North in 1988 before the closed form of the truncated series was known..

has decimal expansion given by(1)(OEIS A000796). The following table summarizes some record computations of the digits of .1999Kanada, Ushio and KurodaDec. 2002Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)Aug. 2012A. J. Yee (Yee)Aug. 2012S. Kondo and A. J. Yee (Yee)Dec. 2013A. J. Yee and S. Kondo (Yee)The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship..

The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating pidiscovered by Simon Plouffe in 1995,Amazingly, this formula is a digit-extraction algorithm for in base 16.Following the discovery of this and related formulas, similar formulas in other bases were investigated. This class of formulas are now known as BBP-type formulas.

The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter ,(1)(2) has decimal expansion given by(3)(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for .A brief history of notation for pi is given by Castellanos (1988ab). is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch calculator. The symbol was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing -gons on a circle using the Archimedes algorithm. Using (a 96-gon),..

The constant that Gelfond's theorem established to be transcendental seems to lack a generally accepted name. As a result, in this work, it will be dubbed Gelfond's constant. Both the Gelfond-Schneider constant and Gelfond's constant were singled out in the 7th of Hilbert's problems as examples of numbers whose transcendence was an open problem (Wells 1986, p. 45).Gelfond's constant has the numerical value(1)(OEIS A039661) and simplecontinued fraction(2)(OEIS A058287).Its digits can be computed efficiently using the iteration(3)with , and then plugging in to(4)(Borwein and Bailey 2003, p. 137).

The approximation for pi given by(1)(2)(3)In the above figure, let , and construct the circle centered at of radius 1. This intersects at point . Now construct the circle about with radius 1. The circles and intersect in , and the line intersects the perpendicular to through in the point . Now construct the point to be a distance 3 along . The line segment is then of length(4)This construction was given by the Polish Jesuit priest Kochansky (Steinhaus 1999).

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