23841:
3020:
1543:
4671:
7215:
15274:
14257:
3703:
6237:
766:
14904:
13463:
8937:
4239:
14920:
9125:
12163:
8561:
11088:
3692:
7155:
12330:
7831:
7854:
2618:
10727:
13945:
14472:
6253:
71:
7974:
16484:
13443:
11333:
4666:
2125:
13676:
9738:
10690:
2024:
17166:
4481:
10254:
10429:
2613:{\displaystyle {\begin{aligned}\pi &=3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}\end{aligned}}}
1766:
5249:
7576:
5683:
4466:
13940:{\displaystyle {\begin{aligned}\prod _{p}^{\infty }\left(1-{\frac {1}{p^{2}}}\right)&=\left(\prod _{p}^{\infty }{\frac {1}{1-p^{-2}}}\right)^{-1}\\&={\frac {1}{1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }}\\&={\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}\approx 61\%.\end{aligned}}}
7106:. 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
8485:
computations: After a new record is claimed, the decimal result is converted to hexadecimal, and then a digit extraction algorithm is used to calculate several randomly selected hexadecimal digits near the end; if they match, this provides a measure of confidence that the entire computation is
16610:
includes "3.14159". Pi Day in 2015 was particularly significant because the date and time 3/14/15 9:26:53 reflected many more digits of pi. In parts of the world where dates are commonly noted in day/month/year format, 22 July represents "Pi
Approximation Day", as 22/7 = 3.142857.
16969:
8465:
7777:
18827:
7392:
7233:
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. The fast iterative algorithms were anticipated in 1914, when Indian mathematician
5833:
5522:
5972:
9115:
6176:
13419:
16375:
4661:{\displaystyle {\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdots }
10205:
6681:
19476:
cf Hardy and Wright 1938 and 2000:177 footnote § 11.13–14. The proofs that e and π are transcendental can be found on pp. 170–176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pp. 417–419 for full
14153:
5375:
to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.
7121:
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
6802:
16665:, including 3.2. The bill is notorious as an attempt to establish a value of mathematical constant by legislative fiat. The bill was passed by the Indiana House of Representatives, but rejected by the Senate, and thus it did not become a law.
601:
to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test
12660:
7423:
18858:
5548:
4366:
12979:
6944:
4820:
1285:
13610:
9643:
15236:
14717:
12858:
8303:
10914:
10685:{\displaystyle \left(\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }\xi ^{2}|{\hat {f}}(\xi )|^{2}\,d\xi \right)\geq \left({\frac {1}{4\pi }}\int _{-\infty }^{\infty }|f(x)|^{2}\,dx\right)^{2}.}
16049:
7614:
5062:
10386:
7249:
5335:
5033:
15252:
on the
Hilbert space of square-integrable real-valued functions on the real line. The Hilbert transform, like the Fourier transform, can be characterized purely in terms of its transformation properties on the Hilbert space
11785:
5348:
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.
1735:". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the
14873:
2019:{\displaystyle \pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}}
17161:{\displaystyle {\overline {{\tfrac {16}{5}}-{\tfrac {4}{239}}}}-{\tfrac {1}{3}}{\overline {{\tfrac {16}{5^{3}}}-{\tfrac {4}{239^{3}}}}}+{\tfrac {1}{5}}{\overline {{\tfrac {16}{5^{5}}}-{\tfrac {4}{239^{5}}}}}-,\,\&c.=}
5410:
13250:
7033:
11921:
6489:", leading to speculation that Machin may have employed the Greek letter before Jones. Jones' notation was not immediately adopted by other mathematicians, with the fraction notation still being used as late as 1767.
7059:
computer. The record, always relying on an arctan series, was broken repeatedly (3089 digits in 1955, 7,480 digits in 1957; 10,000 digits in 1958; 100,000 digits in 1961) until 1 million digits were reached in 1973.
15082:
7134:
produced 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. Iterative methods were used by
Japanese mathematician
11642:
11036:
12143:
8849:
3919:
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. The
Chinese mathematician
3737:
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
16851:
22732:
the almost autistic satisfaction of the obsessive-compulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits
13148:
11336:
Complex analytic functions can be visualized as a collection of streamlines and equipotentials, systems of curves intersecting at right angles. Here illustrated is the complex logarithm of the Gamma function.
22584:
For instance, Pickover calls π "the most famous mathematical constant of all time", and
Peterson writes, "Of all known mathematical constants, however, pi continues to attract the most attention", citing the
13681:
1085:
20345:
15002:
9009:
6061:
2090:. These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to
22882:
5721:
3661:
11450:
8236:
13294:
3993:
16283:
3276:
15944:
4032:
10070:
3522:
11212:
3156:
1650:
14220:
7588:
calculations, including the first to surpass 1 billion (10) digits in 1989 by the
Chudnovsky brothers, 10 trillion (10) digits in 2011 by Alexander Yee and Shigeru Kondo, and 100 trillion digits by
13069:
12536:
11311:
6575:
17264:
15857:
10426:. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,
4119:
5862:
9925:
2130:
14023:
7958:
8283:
that are not reused after they are calculated. This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.
18630:
15738:
14608:
14565:
9254:
16156:
15389:
9309:
3954:
12280:
3683:, dating to an oral tradition from the first or second millennium BC, approximations are given which have been variously interpreted as approximately 3.08831, 3.08833, 3.004, 3, or 3.125.
6686:
12477:
10950:
12548:
8106:
15530:
15447:
15261:: up to a normalization factor, it is the unique bounded linear operator that commutes with positive dilations and anti-commutes with all reflections of the real line. The constant
9347:
23537:
14412:
12862:
11065:
as the eigenvalue associated with the
Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function. Equivalently,
4951:
12038:
6806:
6539:"). Because Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly, and the practice was universally adopted thereafter in the
6381:
15699:
4727:
837:
9473:
6420:
1178:
868:
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
20037:
Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus / ediderunt Adolf Krazer et
Ferdinand Rudio
15489:
13520:
12426:
9821:
8888:
18036:
15139:
14763:
14616:
12758:
12009:
10035:
938:
474:
16249:
10805:
15968:
4853:
3371:
15607:
3325:
13673:. For distinct primes, these divisibility events are mutually independent; so the probability that two numbers are relatively prime is given by a product over all primes:
9508:
7966:
is to draw a circle inscribed in a square, and randomly place dots in the square. The ratio of dots inside the circle to the total number of dots will approximately equal
6475:
6340:
3879:
1721:
10285:
4200:
4163:
1697:
893:
866:
11472:
of the curve, so that it can be deformed to a circle and then integrated explicitly in polar coordinates. More generally, it is true that if a rectifiable closed curve
4967:
1007:
15333:
at the cusp of the large "valley" on the right side of the
Mandelbrot set behaves similarly: the number of iterations until divergence multiplied by the square root of
9999:
9727:
11701:
9958:
3200:
22410:
18973:
16418:: The first word has three letters, the second word has one, the third has four, the fourth has one, the fifth has five, and so on. Such memorization aids are called
12058:
11952:
10776:
as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution. The
3419:
20337:
15556:
14779:
8255:
are very slow compared to other methods, and do not provide any information on the exact number of digits that are obtained. Thus they are never used to approximate
15654:
11972:
22874:
22812:
8544:
from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics,
4905:
4879:
19508:
the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters ... J.A. Segner ... in 1767, he represented
13157:
12362:
6953:
11853:
8908:
1739:. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.
21042:
See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, § 5.3.3, pp. 111–112.
19759:
5998:, whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of
15011:
5244:{\displaystyle \arctan x={\frac {x}{1+x^{2}}}+{\frac {2}{3}}{\frac {x^{3}}{(1+x^{2})^{2}}}+{\frac {2\cdot 4}{3\cdot 5}}{\frac {x^{5}}{(1+x^{2})^{3}}}+\cdots }
11546:
10969:
2627:
16426:, is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." When a poem is used, it is sometimes referred to as a
12078:
8776:
7571:{\displaystyle {\frac {1}{\pi }}={\frac {\sqrt {10005}}{4270934400}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!\,k!^{3}(-640320)^{3k}}}.}
3566:
in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period.
1132:
function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a
16782:
14236:. This functional determinant can be computed via a product expansion, and is equivalent to the Wallis product formula. The calculation can be recast in
5678:{\displaystyle \pi =3+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-{\frac {4}{8\times 9\times 10}}+\cdots }
4461:{\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }
23428:. In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.).
8290:
and Stanley Rabinowitz produced a simple spigot algorithm in 1995. Its speed is comparable to arctan algorithms, but not as fast as iterative algorithms.
23549:
22969:
17254:
5395:, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate
22382:
19090:
8481:
digits can be extracted from one or two hexadecimal digits. An important application of digit extraction algorithms is to validate new claims of record
5260:
1016:
14937:
7126:
the number of correct digits at each step. For example, the Brent–Salamin algorithm doubles the number of digits in each iteration. In 1984, brothers
20623:
1477:; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that
20010:
18657:
7174:, a handful of digits provide sufficient precision. According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most
21877:
18443:
11383:
8149:
7199:
7076:
4048:
applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value remained the most accurate approximation of
16526:. The digits are large wooden characters attached to the dome-like ceiling. The digits were based on an 1873 calculation by English mathematician
8460:{\displaystyle \pi =\sum _{k=0}^{\infty }{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right).}
7186:, Arndt concludes that a few hundred digits would suffice for any scientific application. Despite this, people have worked strenuously to compute
23039:
16407:, certified by Guinness World Records, is 70,000 digits, recited in India by Rajveer Meena in 9 hours and 27 minutes on 21 March 2015. In 2006,
7143:
between 1995 and 2002. This rapid convergence comes at a price: the iterative algorithms require significantly more memory than infinite series.
3221:
18234:
From Alexandria, through Baghdad: Surveys and studies in the ancient Greek and medieval Islamic mathematical sciences in honor of J. L. Berggren
15898:
3434:
17183:(among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr.
11159:
7772:{\displaystyle \pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right),}
3099:
1504:, and no evidence of a pattern was found. Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the
21379:
7387:{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{k!^{4}\left(396^{4k}\right)}}.}
7190:
to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with
4709:
in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."
22241:
21875:
Sondow, J. (1994). "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series".
19900:
12990:
11240:
19833:
15818:
7404:, setting a record of 17 million digits in 1985. Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (
6242:
The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician
20284:
23619:
23534:
19412:
Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted in
18029:
16411:, a retired Japanese engineer, claimed to have recited 100,000 decimal places, but the claim was not verified by Guinness World Records.
9869:
22645:
19875:
16940:
16936:
16930:
14160:
13089:
7914:
5828:{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}+\cdots }
5517:{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+{\frac {4}{13}}-\cdots }
4297:
20868:
19927:. Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche (in French). Vol. 19 (published 1886). p. 139.
19113:"Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae"
18143:
17475:"Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt"
16646:
has not made its way into mainstream mathematics, but since 2010 this has led to people celebrating Two Pi Day or Tau Day on June 28.
23061:
18736:
18510:
17437:
From a logical point of view, this is unsatisfactory at the present stage because we have not yet discussed the concept of arc length
6220:
and Euler. Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".
4300:
sometime in the 14th or 15th century. Around 1500 AD, a written description of an infinite series that could be used to compute
3002:
2985:
2952:
2898:
2829:
2782:
2772:
2623:
21256:
Del Pino, M.; Dolbeault, J. (2002). "Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions".
16103:
17503:
13965:
3612:
21975:
Tate, John T. (1950). "Fourier analysis in number fields, and Hecke's zeta-functions". In Cassels, J. W. S.; Fröhlich, A. (eds.).
21073:
20895:
17821:
18438:
9426:
3959:
3605:, dated around 1650 BC but copied from a document dated to 1850 BC, has a formula for the area of a circle that treats
8477:
without calculating all the preceding digits. Individual binary digits may be extracted from individual hexadecimal digits, and
8050:
621:, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as
13659:(for example, every 7th integer is divisible by 7.) Hence the probability that two numbers are both divisible by this prime is
12378:(normally defined only for non-negative integers) to all complex numbers, except the negative real integers, with the identity
8294:
4258:
3998:
694:
22400:
20045:
17477:[Representation of an analytical function of a complex variable, whose absolute value lies between two given limits].
1512:'s digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a
23468:
23437:
23414:
23377:
23345:
23320:
23293:
23258:
23237:
22921:
22788:
22700:
22639:
22617:
22569:
22322:
22281:
22199:
22174:
22117:
22078:
22035:
21984:
21804:
21621:
21596:
21546:
21018:
20470:
20255:
19607:
19501:
19382:
18705:
18564:
18249:
18137:
18083:
18023:
17815:
17786:
17551:
16603:
15866:
10390:
Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve
5967:{\displaystyle \pi ={3}+{\frac {4}{2\times 3\times 4}}-{\frac {4}{4\times 5\times 6}}+{\frac {4}{6\times 7\times 8}}-\cdots }
4324:, from around 1530 AD. Several infinite series are described, including series for sine (which Nilakantha attributes to
1728:
1589:
1559:
505:
597:
have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of
23898:
21495:
20930:
19095:
19008:
18596:
17389:
12482:
5257:
popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including
12014:
10048:. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.
5534:
as desired. It converges quite slowly, though – after 500,000 terms, it produces only five correct decimal digits of
4911:(that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.
4083:
22804:
13467:
9110:{\displaystyle \sin \theta =\sin \left(\theta +2\pi k\right){\text{ and }}\cos \theta =\cos \left(\theta +2\pi k\right).}
6522:
6171:{\displaystyle {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }
809:
22727:
22192:
Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups
20525:
18180:
14776:, which means that it transforms in a specific way. Certain identities hold for all automorphic forms. An example is
9784:
9663:
appears as an optimal constant in Wirtinger's inequality, and it follows that it is the smallest wavenumber, using the
8755:
as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular
20487:
11079:, the "whole business" of establishing the fundamental theorems of Fourier analysis reduces to the Gaussian integral.
8967:
plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2
1168:. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which
678:
The symbol used by mathematicians to represent the ratio of a circle's circumference to its diameter is the lowercase
23493:
23022:
22457:
22347:
22224:
21832:
21653:
21571:
20699:
17907:
16213:
15704:
15626:
14570:
14527:
13414:{\displaystyle \operatorname {Vol} ((n+1)\Delta _{n})={\frac {(n+1)^{n}}{n!}}\sim {\frac {e^{n+1}}{\sqrt {2\pi n}}}.}
11134:
1537:
714:
657:
makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to
593:, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and
137:
16730:
16370:{\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}\approx 1.2566370614\ldots \times 10^{-6}{\text{ N/A}}^{2}.}
15358:
6027:
3927:
3541:
3334:
1500:, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to
280:
16607:
12219:
10419:
7158:
As mathematicians discovered new algorithms, and computers became available, the number of known decimal digits of
769:
The circumference of a circle is slightly more than three times as long as its diameter. The exact ratio is called
371:
10200:{\displaystyle \pi \leq {\frac {\left(\int _{G}|\nabla u|^{2}\right)^{1/2}}{\left(\int _{G}|u|^{2}\right)^{1/2}}}}
1448:) is not precisely known; estimates have established that the irrationality measure is larger than the measure of
23612:
20807:
20442:
18046:
12435:
12187:
10921:
7092:
3065:
10258:
9156:
in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However,
8541:
116:
17300:
16738:
16567:
16438:
memorizers typically do not rely on poems, but instead use methods such as remembering number patterns and the
15392:
14435:
of this character. The other characters have derivatives whose magnitudes are positive integral multiples of 2
13423:
8769:
that describe circumference, area, or volume of shapes generated by circles typically have values that involve
7099:
6676:{\displaystyle \textstyle a_{0}=1,\quad b_{0}={\frac {1}{\sqrt {2}}},\quad t_{0}={\frac {1}{4}},\quad p_{0}=1.}
2119:
23546:
23146:
22938:
16530:, which included an error beginning at the 528th digit. The error was detected in 1946 and corrected in 1949.
11233:, which coincides with its radius, is also 1.) The Euler characteristic of a sphere can be computed from its
5545:(published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:
4908:
1108:
typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of
342:
22538:
22379:
16926:
15494:
14892:
12667:
10707:
6436:
6390:
6243:
4713:
4289:
1501:
579:
234:
17:
23746:
10711:
6193:
6019:
5056:
of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):
1423:
132:
23893:
21235:
L. Esposito; C. Nitsch; C. Trombetti (2011). "Best constants in Poincaré inequalities for convex domains".
17255:"Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud"
16548:
15005:
14432:
14356:
14148:{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).}
11362:
10781:
9201:
7115:
7088:
6567:
6044:
in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between
163:
23840:
19740:
19463:
Hardy and Wright 1938 and 2000: 177 footnote § 11.13–14 references Lindemann's proof as appearing at
16575:", an out-of-control computer is contained by being instructed to "Compute to the last digit the value of
13478:
12316:
assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the
9259:
4921:
153:
23331:
22483:"A slice of pi: An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist"
20007:
19165:
Chien-Lih, Hwang (2005). "89.67 An elementary derivation of Euler's series for the arctangent function".
18166:
17298:
Bailey, David H.; Plouffe, Simon M.; Borwein, Peter B.; Borwein, Jonathan M. (1997). "The quest for PI".
16619:
15782:
15665:
15301:
was discovered by David Boll in 1991. He examined the behaviour of the Mandelbrot set near the "neck" at
15118:
13151:
11321:
10959:
7242:, remarkable for their elegance, mathematical depth and rapid convergence. One of his formulae, based on
6359:
2111:
18584:
18433:
10694:
The physical consequence, about the uncertainty in simultaneous position and momentum observations of a
9432:
4685:
to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".
4045:
3908:
566:
made a five-digit approximation, both using geometrical techniques. The first computational formula for
189:
23605:
22875:"Pi Day: Why some mathematicians refuse to celebrate 14 March and won't observe the dessert-filled day"
22530:
15466:
12381:
9350:
8856:
8297:
7604:
6398:
2848:
23047:
21058:
19132:
Chien-Lih, Hwang (2004). "88.38 Some Observations on the Method of Arctangents for the Calculation of
16515:
16414:
One common technique is to memorize a story or poem in which the word lengths represent the digits of
14729:
11985:
11846:, representing the potential of a point source at the origin, whose associated field has unit outward
10004:
7182:
with a precision of one atom. Accounting for additional digits needed to compensate for computational
527:, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the
450:
184:
23888:
23815:
22066:
18311:, pp. 15–16, 175, 184–186, 205. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.
16277:
14460:
14305:
of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function
11500:
7593:
7202:); and within pure mathematics itself, providing data for evaluating the randomness of the digits of
7103:
7079:
that could multiply large numbers very rapidly. Such algorithms are particularly important in modern
6797:{\displaystyle \textstyle a_{n+1}={\frac {a_{n}+b_{n}}{2}},\quad \quad b_{n+1}={\sqrt {a_{n}b_{n}}},}
5526:
As individual terms of this infinite series are added to the sum, the total gradually gets closer to
4694:
4293:
4272:
techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite
3279:
959:. For example, one may directly compute the arc length of the top half of the unit circle, given in
910:
730:
447:, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as
21891:
21196:
17314:
6435:
alone to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician
21270:
20847:
19430:
16650:
16599:
15777:
appears routinely in equations describing fundamental principles of the universe, often because of
15561:
15397:
15249:
12655:{\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }{\frac {e^{z/n}}{1+z/n}}}
11839:
11499:
The general form of Cauchy's integral formula establishes the relationship between the values of a
11130:
9766:
7055:
that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the
4957:
to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.
4831:
4227:
3304:
19316:
Borwein, J.M.; Borwein, P.B.; Dilcher, K. (1989). "Pi, Euler Numbers, and Asymptotic Expansions".
18004:
17474:
9321:
6319:
5404:
4125:
3860:
1702:
1112:
that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by
578:
to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician
209:
22747:
20691:
19897:
18546:
17236:
17205:
15957:
13981:
13950:
12195:
11792:
11316:
The constant appears in many other integral formulae in topology, in particular, those involving
9137:
8736:
7039:
The development of computers in the mid-20th century again revolutionized the hunt for digits of
6451:
6205:
6181:
4325:
3733:
is sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of
3721:
was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician
1673:
1505:
204:
21725:
Benjamin Nill; Andreas Paffenholz (2014). "On the equality case in Erhart's volume conjecture".
19226:
9737:
7869:, which evaluate the results of multiple random trials, can be used to create approximations of
7083:
computations because most of the computer's time is devoted to multiplication. They include the
4226:
was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to
4172:
4135:
871:
844:
23676:
21922:
T. Friedmann; C.R. Hagen (2015). "Quantum mechanical derivation of the Wallis formula for pi".
21886:
21265:
21191:
20842:
20277:
17309:
16399:
16265:
16092:
15130:
14241:
14229:
13615:
12686:
11835:
11091:
10234:
9971:
9685:
9141:
8956:
8931:
8773:. For example, an integral that specifies half the area of a circle of radius one is given by:
8240:
This Monte Carlo method is independent of any relation to circles, and is a consequence of the
4273:
2910:
1470:
1415:
966:
904:
532:
481:
224:
23701:
23425:
22905:
20450:
20068:
19984:
19922:
19827:
19597:
19112:
18695:
18013:
16722:
12974:{\displaystyle S_{n-1}(r)={\frac {n\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n-1}.}
11103:
10052:
9940:
5994:
After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of
3532:
3169:
2639:
477:
158:
23857:
23639:
23227:
22778:
22629:
22441:
19856:
19722:
18506:
18172:
17259:
16507:
Perhaps because of the simplicity of its definition and its ubiquitous presence in formulae,
16083:
14611:
13985:
13497:
12682:
12145:
which has the 2-dimensional volume (i.e., the area) of the unit 2-sphere in the denominator.
12043:
11926:
11788:
11050:
10397:. The above is the most canonical definition, however, giving the unique unitary operator on
9316:
9196:
8490:
8241:
8122:
7787:
7396:
This series converges much more rapidly than most arctan series, including Machin's formula.
6939:{\displaystyle \textstyle t_{n+1}=t_{n}-p_{n}(a_{n}-a_{n+1})^{2},\quad \quad p_{n+1}=2p_{n}.}
6217:
6213:
6197:
6053:
4219:
4214:
was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist
3668:
3391:
1571:
1551:
1521:
1445:
1137:
1105:
960:
650:
489:
412:
19714:
19491:
17357:
16543:
it is suggested that the creator of the universe buried a message deep within the digits of
15535:
13964:
is connected in a deep way to the distribution of prime numbers. This is a special case of
9761:
serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned
6196:
exploited a continued-fraction representation of the tangent function. French mathematician
4815:{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots }
4348:
23862:
23852:
23337:
22609:
22603:
22599:
21994:
21941:
21687:. Athena series; selected topics in mathematics (1st ed.). Holt, Rinehart and Winston.
21402:
21310:
21301:
Payne, L.E.; Weinberger, H.F. (1960). "An optimal Poincaré inequality for convex domains".
21028:
20834:
20819:
20728:
19774:
19273:
18519:
18259:
18127:
17975:
17887:
17663:
16700:
15741:
15632:
15106:
14912:
14514:
14496:
12061:
11975:
11957:
11683:
11317:
11222:
10238:
8752:
8740:
8498:
7417:
7178:
calculations, because that is the accuracy necessary to calculate the circumference of the
7118:. As modified by Salamin and Brent, it is also referred to as the Brent–Salamin algorithm.
7107:
6007:
5361:
4061:
3726:
3211:
3081:
3038:
2103:
1724:
1432:
1280:{\displaystyle \{\dots ,-2\pi i,0,2\pi i,4\pi i,\dots \}=\{2\pi ki\mid k\in \mathbb {Z} \}}
571:
528:
364:
21083:
20757:
Rabinowitz, Stanley; Wagon, Stan (March 1995). "A spigot algorithm for the digits of Pi".
13605:{\displaystyle \zeta (2)={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots }
13073:
The gamma function can be used to create a simple approximation to the factorial function
8583:
appears in formulae for areas and volumes of geometrical shapes based on circles, such as
7214:
7194:
often make headlines around the world. They also have practical benefits, such as testing
6481:)", calculated for a circle with radius one. However, Jones writes that his equations for
8:
23696:
23686:
23669:
20121:
Reitwiesner, George (1950). "An ENIAC Determination of pi and e to 2000 Decimal Places".
19426:
16658:
16450:
16423:
16254:
15786:
15231:{\displaystyle Hf(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {f(x)\,dx}{x-t}}.}
14932:
14712:{\displaystyle \theta (z,\tau )=\sum _{n=-\infty }^{\infty }e^{2\pi inz+i\pi n^{2}\tau }}
14233:
12984:
12853:{\displaystyle V_{n}(r)={\frac {\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n},}
12542:
11843:
11465:
11345:
11230:
11099:
10773:
9664:
9182:
8919:
7866:
7413:
7235:
7218:
7179:
7084:
7068:
6003:
5053:
5042:
4884:
4858:
4825:
4344:
4333:
4320:
4315:
4215:
4166:
3846:
in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.
3839:
3563:
3328:
1736:
1732:
1663:
cannot be expressed using any finite combination of rational numbers and square roots or
1579:
1575:
1555:
1317:
1310:
1148:
952:
555:
501:
324:
259:
249:
219:
39:
35:
21945:
21314:
20998:
20838:
20321:
Schudel, Matt (25 March 2009). "John W. Wrench, Jr.: Mathematician Had a Taste for Pi".
20070:
Cursus Mathematicus: Elementorum Analyseos Infinitorum Elementorum Analyseos Infinitorvm
19778:
19277:
19124:
19100:
17979:
17667:
17493:
12344:
10909:{\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}
6303:) to form circle constants. (Before then, mathematicians sometimes used letters such as
4284:
and others who used geometrical techniques. Although infinite series were exploited for
3331:, celebrated in mathematics due to it containing five important mathematical constants:
951:
of the circle, a quantity which can be formally defined independently of geometry using
23478:
23109:
23014:
22961:
22855:
22756:
22507:
22482:
22463:
22267:
21957:
21931:
21904:
21793:
in 1881. For a more rigorous proof than the intuitive and informal one given here, see
21760:
21734:
21501:
21334:
21283:
21236:
21217:
21032:
20965:
20899:
20860:
20774:
20716:
20579:
20173:
20138:
19958:
19792:
19650:
19343:
19285:
19231:
19182:
19153:
19000:
18924:
18916:
18850:
18232:: from Archimedes to ENIAC and beyond". In Sidoli, Nathan; Van Brummelen, Glen (eds.).
17991:
17924:
17804:
17687:
17335:
17198:
s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
17191:
16682:
16044:{\displaystyle {\frac {1}{\tau }}=2{\frac {\pi ^{2}-9}{9\pi }}m_{\text{e}}\alpha ^{6},}
15460:
14724:
14328:
13642:. This probability is based on the observation that the probability that any number is
12294:
10792:
10695:
10273:
9851:
9160:
also appears in many natural situations having apparently nothing to do with geometry.
8893:
8139:
7874:
7836:
6506:, the ratio of periphery to radius, in this and some later writing. Euler first used
4223:
4207:
4203:
3680:
3215:
3046:
1760:
900:
563:
523:
For thousands of years, mathematicians have attempted to extend their understanding of
214:
21900:
21279:
20637:
20618:
20614:
18585:"Following in the footsteps of geometry: The mathematical world of Christiaan Huygens"
17481:(in German). Vol. 1. Berlin: Mayer & Müller (published 1894). pp. 51–66.
16586:
falls on 14 March (written 3/14 in the US style), and is popular among students.
14256:
7063:
Two additional developments around 1980 once again accelerated the ability to compute
23771:
23736:
23714:
23628:
23512:
23489:
23464:
23433:
23410:
23373:
23341:
23316:
23309:
23289:
23254:
23233:
23018:
22965:
22917:
22859:
22784:
22696:
22635:
22613:
22565:
22512:
22467:
22453:
22343:
22318:
22295:
22287:
22277:
22220:
22195:
22170:
22074:
22031:
21980:
21961:
21908:
21828:
21820:
21800:
21764:
21752:
21649:
21617:
21592:
21567:
21542:
21338:
21326:
21209:
21036:
21014:
20466:
20455:
Pi: The Next Generation, A Sourcebook on the Recent History of Pi and Its Computation
20251:
19603:
19497:
19378:
19186:
19157:
18928:
18854:
18701:
18560:
18245:
18176:
18133:
18079:
18019:
17995:
17811:
17782:
17691:
17679:
17547:
17540:
17327:
16770:, which implies a specific kind of statistical randomness on its digits in all bases.
16202:
15862:
15770:
15748:
can be characterized as the period of this map. This is notable in that the constant
15134:
15122:
14318:
14288:
14237:
11349:
10963:
10777:
10731:
10269:
9638:{\displaystyle \pi ^{2}\int _{0}^{1}|f(x)|^{2}\,dx\leq \int _{0}^{1}|f'(x)|^{2}\,dx,}
9502:
9354:
8748:
7163:
6189:
5388:
3698:
can be estimated by computing the perimeters of circumscribed and inscribed polygons.
3559:
3288:
1419:
1359:
544:
485:
444:
22835:
21790:
21393:
21374:
21221:
19065:
17806:
Science and Its Times: Understanding the Social Significance of Scientific Discovery
17675:
17339:
12681:
reduces to the Wallis product formula. The gamma function is also connected to the
10381:{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }\,dx.}
9353:
to take on only certain specific values. It must be positive, since the operator is
4964:
used the Gregory–Leibniz series to produce an algorithm that converged much faster:
4318:. The series are presented without proof, but proofs are presented in a later work,
4076:
computed 3.1418 using a polygonal method, independent of Archimedes. Italian author
1100:
Integration is no longer commonly used in a first analytical definition because, as
23808:
23803:
23781:
23766:
23731:
23649:
23277:
23101:
23006:
22953:
22847:
22586:
22502:
22494:
22445:
22132:
21949:
21896:
21744:
21388:
21318:
21287:
21275:
21201:
21006:
20864:
20852:
20766:
20708:
20632:
20571:
20458:
20446:
20165:
20130:
20035:
19950:
19782:
19642:
19576:
19565:
19555:
19545:
19333:
19325:
19281:
19236:
19174:
19145:
18996:
18992:
18908:
18842:
18653:
18649:
18552:
18542:
18412:
18237:
18225:
17983:
17916:
17671:
17621:
17600:
17470:
17319:
17250:
16674:
16572:
15953:
15812:
15102:
14915:(1718–1799), is a geometrical construction of the graph of the Cauchy distribution.
14880:
14773:
14518:
14272:
13635:
11979:
11827:
11823:
11341:
10715:
10253:
10056:
9863:
9742:
9418:
9168:
8545:
8272:
8267:
Two algorithms were discovered in 1995 that opened up new avenues of research into
7589:
7405:
7243:
7127:
7052:
6344:, to express the ratio of periphery and diameter in the 1647 and later editions of
6312:
6236:
5330:{\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},}
5028:{\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}
4352:
4257:
terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.
4132:
digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with
3207:
3030:
2099:
2028:
Truncating the continued fraction at any point yields a rational approximation for
1459:
1294:
A variation on the same idea, making use of sophisticated mathematical concepts of
1094:
700:
638:
594:
393:
106:
22140:
20856:
19704:
19037:. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.
18895:
Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for
13960:
The solution to the Basel problem implies that the geometrically derived quantity
4670:
229:
23553:
23541:
23248:
23212:
22957:
22851:
22722:
22688:
22405:
22386:
22271:
22263:
21990:
21398:
21115:
21024:
20922:
20891:
20724:
20156:
Nicholson, J. C.; Jeenel, J. (1955). "Some comments on a NORC Computation of π".
20014:
19904:
19435:
Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin
18754:
18255:
18241:
17883:
16654:
16555:
have also been incorporated into the lyrics of the song "Pi" from the 2005 album
16539:
16511:
has been represented in popular culture more than other mathematical constructs.
16459:
16422:. An early example of a mnemonic for pi, originally devised by English scientist
16408:
15892:
15869:, which shows that the uncertainty in the measurement of a particle's position (Δ
15125:
are associated with the asymptotics of the Poisson kernel. The Hilbert transform
15114:
15087:
14924:
14919:
14908:
14476:
14452:
14448:
13462:
12191:
12155:
11819:
11780:{\displaystyle \oint _{\gamma }g(z)\,dz=2\pi i\sum \operatorname {Res} (g,a_{k})}
11668:
11150:
10958:
equal to one, as is required for a probability distribution. This follows from a
10242:
10241:
problem in one dimension, the Poincaré inequality is the variational form of the
9745:
was the solution to an isoperimetric problem, according to a legend recounted by
8760:
8592:
8501:(a modification of the BBP algorithm) to compute the quadrillionth (10th) bit of
8026:
7183:
7136:
7048:
6261:
popularized the use of the Greek letter π in works he published in 1736 and 1748.
4678:
4269:
4165:
sides, which stood as the world record for about 180 years. French mathematician
3702:
3581:
dated 1900–1600 BC has a geometrical statement that, by implication, treats
1752:
1583:
1525:
1513:
1493:
1441:
1407:
1363:
1336:
513:
496:
involving only finite sums, products, powers, and integers. The transcendence of
357:
334:
303:
286:
264:
20244:, pp. 111 (5 times), pp. 113–114 (4 times). For details of algorithms, see
17385:
16434:
have been composed in several languages in addition to English. Record-setting
15273:
14903:
14268:
13447:
11460:
is not a circle, and hence does not have any obvious connection to the constant
7098:
The iterative algorithms were independently published in 1975–1976 by physicist
7075:, which were much faster than the infinite series; and second, the invention of
4238:
574:, was discovered a millennium later. The earliest known use of the Greek letter
23664:
23654:
22774:
22625:
21641:
21475:
21352:
21054:
20687:
19941:
Cajori, Florian (1913). "History of the Exponential and Logarithmic Concepts".
19108:
19086:
18556:
18417:
18400:
18018:. New Mathematical Library. Vol. 13. New York: Random House. p. 125.
17987:
17428:
16557:
16527:
16439:
16273:
16269:
16169:
16161:
15298:
15278:
15110:
14884:
14868:{\displaystyle \theta (z+\tau ,\tau )=e^{-\pi i\tau -2\pi iz}\theta (z,\tau ),}
14504:
14492:
14284:
14264:
14016:
The zeta function also satisfies Riemann's functional equation, which involves
13623:
12371:
12309:
12305:
12300:
12171:
12071:
are present because of a normalization by the n-dimensional volume of the unit
11831:
11493:
11234:
11107:
11057:, in probability and statistics. This theorem is ultimately connected with the
9753:
could enclose on all other sides within a single given oxhide, cut into strips.
9672:
9124:
8624:
7221:, working in isolation in India, produced many innovative series for computing
6284:
6258:
6040:
were aimed at increasing the accuracy of approximations. When Euler solved the
5368:
5353:
5254:
4705:. Newton himself used an arcsine series to compute a 15-digit approximation of
4475:
4329:
4310:
3382:
3163:
3053:
1340:
1328:
1159:
630:
244:
23040:"Life of pi in no danger – Experts cold-shoulder campaign to replace with tau"
22498:
22136:
21748:
21010:
21003:
Bodies of Constant Width: An Introduction to Convex Geometry with Applications
20462:
20338:"The Big Question: How close have we come to knowing the precise value of pi?"
20248:
Pi and the AGM: a Study in Analytic Number Theory and Computational Complexity
20041:
19992:
19928:
19871:
19760:"Ad Reverendum Virum D. Henricum Aldrich S.T.T. Decanum Aedis Christi Oxoniae"
19178:
19149:
18846:
16902:
Gupta, R. C. (1992). "On the remainder term in the Madhava–Leibniz's series".
12162:
7047:
and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an
1347:
is then defined as half the magnitude of the derivative of this homomorphism.
23877:
23719:
23681:
23281:
22717:
22023:
21756:
21330:
21213:
21111:
20815:
20811:
20514:
19045:
18786:
17683:
17517:
17331:
16862:
16767:
16638:
or the ratio of a circle's circumference to its radius, is more natural than
14522:
14001:
13631:
13619:
13482:
13431:
12428:. When the gamma function is evaluated at half-integers, the result contains
11142:
11095:
9680:
9133:
8510:
7600:
7409:
7195:
7131:
6540:
6276:
6041:
5045:, that were used to set several successive records for calculating digits of
4915:
4717:
3850:
3676:
3602:
3428:
3085:
3042:
1482:
1474:
1113:
789:
642:
603:
424:
329:
111:
23171:
22780:
Title An Engineer's Alphabet: Gleanings from the Softer Side of a Profession
22299:
19354:
15952:
is approximately equal to 3 plays a role in the relatively long lifetime of
11087:
23786:
23756:
23304:
23127:
22516:
21519:
19787:
19338:
19099:(in Latin). Academiae Imperialis Scientiarium Petropolitanae. p. 318.
19041:
18969:
18580:
17582:
16685:
16635:
15456:
14510:
14488:
14424:
14338:
13245:{\displaystyle \pi =\lim _{n\to \infty }{\frac {e^{2n}n!^{2}}{2n^{2n+1}}}.}
12329:
11353:
11058:
9774:
7175:
7154:
7028:{\displaystyle \textstyle \pi \approx {\frac {(a_{n}+b_{n})^{2}}{4t_{n}}}.}
6351:
6049:
4690:
4674:
4356:
3729:. This polygonal algorithm dominated for over 1,000 years, and as a result
3664:
1332:
1306:
1175:
is equal to one is then an (imaginary) arithmetic progression of the form:
1133:
614:
30:
This article is about the mathematical constant. For the Greek letter, see
22987:
22449:
16713:
has been added to several programming languages as a predefined constant.
14347:
of unit modulus complex numbers. It is a theorem that every character of
11916:{\displaystyle \Phi (\mathbf {x} )={\frac {1}{2\pi }}\log |\mathbf {x} |.}
8513:
application on one thousand computers over a 23-day period to compute 256
5530:, and – with a sufficient number of terms – can get as close to
4210:
reached 20 digits, a record he later increased to 35 digits (as a result,
23883:
23400:
22094:
21456:
21145:
19240:
18527:
His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 <
18009:
17185:
16595:
16390:
16385:
15265:
is the unique normalizing factor that makes this transformation unitary.
14720:
14500:
13427:
12313:
12198:
12183:
10765:
10248:
9746:
8936:
8915:
8569:
8470:
8276:
8022:
7397:
7044:
6486:
6269:
4961:
4701:, which led to the development of many infinite series for approximating
4471:
4242:
Comparison of the convergence of several historical infinite series for
4129:
3921:
3578:
3298:
3061:
3034:
2997:
2964:
1759:, can be represented by an infinite series of nested fractions, called a
1558:
is not possible in a finite number of steps using the classical tools of
1313:
679:
575:
436:
254:
239:
194:
168:
51:
31:
23515:
22760:
20720:
20518:
19747:(in Latin). Harvard University. Cambridge University press. p. 381.
18920:
15077:{\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}\,dx=\pi .}
13618:
for this infinite series was a famous problem in mathematics called the
12284:
The total curvature of a closed curve is always an integer multiple of 2
8729:
5687:
The following table compares the convergence rates of these two series:
4336:
or the Gregory–Leibniz series. Madhava used infinite series to estimate
3291:. This formula establishes a correspondence between imaginary powers of
2094:
than any other fraction with the same or a smaller denominator. Because
1516:
that begins at the 762nd decimal place of the decimal representation of
23113:
23010:
21680:
21322:
21205:
20790:
20788:
20778:
20583:
20177:
20142:
19991:(in Latin). Vol. 1. Academiae scientiarum Petropoli. p. 113.
19962:
19654:
19347:
19052:. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.
19004:
18912:
18794:
Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica)
17928:
17323:
16961:, which may very much facilitate the Practice; as for instance, in the
16534:
16496:
16173:
14245:
13451:
12210:
11637:{\displaystyle \oint _{\gamma }{f(z) \over z-z_{0}}\,dz=2\pi if(z_{0})}
11114:
11039:
11031:{\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\,du={\sqrt {\pi }}}
10769:
10706:
in the formulae of Fourier analysis is ultimately a consequence of the
10208:
9421:
of vibration of the string. One way to show this is by estimating the
9375:
9164:
8911:
8756:
8287:
7800:
7111:
5041:
with this formula. Other mathematicians created variants, now known as
4721:
4281:
3722:
3706:
2118:(shown above) also does not exhibit any other obvious pattern, several
944:
646:
547:
517:
23589:
23583:
22534:
21953:
19796:
16590:
and its digital representation are often used by self-described "math
12138:{\displaystyle \Phi (\mathbf {x} )=-{\frac {1}{4\pi |\mathbf {x} |}},}
11313:
reproducing the formula for the surface area of a sphere of radius 1.
8844:{\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}.}
8521:
at the two-quadrillionth (2×10th) bit, which also happens to be zero.
1542:
661:
have been published, and record-setting calculations of the digits of
641:. It also appears in areas having little to do with geometry, such as
23597:
23520:
22590:
22562:
Not A Wake: A dream embodying (pi)'s digits fully for 10,000 decimals
22073:(2nd ed.). Oxford University: Clarendon Press (published 1986).
20712:
16846:{\displaystyle \pi =\int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}.}
16562:
14423:, up to complex conjugation, that is a group isomorphism. Using the
13643:
12375:
12338:
12317:
12206:
11850:
through any smooth and oriented closed surface enclosing the source:
9396:
satisfies the boundary conditions and the differential equation with
8524:
In 2022, Plouffe found a base-10 algorithm for calculating digits of
6515:
4066:
4056:
3717:
The first recorded algorithm for rigorously calculating the value of
1496:
has performed detailed statistical analyses on the decimal digits of
948:
738:
634:
622:
199:
23105:
20785:
20770:
20575:
20169:
20134:
19954:
19646:
19583:
to represent the periphery (that is, the circumference) of a circle.
19329:
17920:
16677:, individuals and organizations frequently pay homage to the number
13284:
denote the simplex having all of its sides scaled up by a factor of
9823:
and equality is clearly achieved for the circle, since in that case
7973:
5344:
Machin-like formulae remained the best-known method for calculating
3577:
and Egypt, both within one percent of the true value. In Babylon, a
3019:
23218:
23131:
22432:
Danesi, Marcel (January 2021). "Chapter 4: Pi in Popular Culture".
22249:
21977:
Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965)
21936:
21066:
20970:
19194:
18541:
Brezinski, C. (2009). "Some pioneers of extrapolation methods". In
16419:
16067:
15876:
15796:
14471:
13143:{\textstyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}
12748:
12202:
12072:
11469:
11225:, which is an integer. An example is the surface area of a sphere
11138:
11110:
10038:
9961:
9862:
in many physical phenomena as well, for example those of classical
9765:, it can be characterized via its role as the best constant in the
9129:
8766:
8721:
6290:
4698:
4305:
4206:
arrived at 15 decimal places in 1593. In 1596, Dutch mathematician
3830:
3093:
1664:
1295:
1010:
956:
798:
618:
586:
493:
428:
23185:
22291:
21739:
21564:
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
21417:
An Introduction to Probability Theory and Its Applications, Vol. 1
21241:
19569:
19559:
19549:
17359:
Theorematum in libris Archimedis de sphaera et cylindro declarario
16518:(a science museum in Paris) there is a circular room known as the
13968:, which asserts the equality of similar such infinite products of
12324:
8986:
Common trigonometric functions have periods that are multiples of
8751:(formed by the intersection of three circles with the sides of an
8713:
Some of the formulae above are special cases of the volume of the
8469:
This formula, unlike others before it, can produce any individual
5356:, who employed a Machin-like formula to calculate 200 decimals of
3915:
of 3.1416. Liu later invented a faster method of calculating
2658:: Approximate fractions include (in order of increasing accuracy)
653:
can be defined without any reference to geometry. The ubiquity of
70:
22805:"Happy Pi Day! Watch these stunning videos of kids reciting 3.14"
22634:. MAA spectrum. Mathematical Association of America. p. 17.
22315:
Classical Theory of Structures Based on the Differential Equation
21437:
18991:(10): 657–664 Published by: Mathematical Association of America.
18000:
16453:, where the word lengths are required to represent the digits of
15294:
15008:. The total probability is equal to one, owing to the integral:
13264:
12075:. For example, in three dimensions, the Newtonian potential is:
8584:
7859:
Random dots are placed on a square and a circle inscribed inside.
6527:(he wrote: "for the sake of brevity we will write this number as
6252:
4328:), cosine, and arctangent which are now sometimes referred to as
3904:
3900:
3842:. Mathematicians using polygonal algorithms reached 39 digits of
3825:
3574:
2622:
The middle of these is due to the mid-17th century mathematician
1299:
1080:{\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}
765:
626:
440:
23372:. Translated by Wilson, Stephen. American Mathematical Society.
23062:"Forget Pi Day. We should be celebrating Tau Day | Science News"
22099:
Singular Integrals and Differentiability Properties of Functions
21234:
21182:
Talenti, Giorgio (1976). "Best constant in Sobolev inequality".
20667:
20665:
20208:
17710:
16066:
is present in some structural engineering formulae, such as the
14997:{\displaystyle g(x)={\frac {1}{\pi }}\cdot {\frac {1}{x^{2}+1}}}
7830:
5391:
faster than others. Given the choice of two infinite series for
4036:
3691:
500:
implies that it is impossible to solve the ancient challenge of
22594:
22194:. Cambridge Tracts in Mathematics. Cambridge University Press.
22001:
21724:
20820:"On the Rapid Computation of Various Polylogarithmic Constants"
19989:
Mechanica sive motus scientia analytice exposita. (cum tabulis)
18201:
16583:
16182:
15305:. When the number of iterations until divergence for the point
13989:
12716:
12312:
about the origin, or equivalently the degree of the map to the
11842:. Perhaps the simplest example of this is the two-dimensional
11069:
is the unique constant making the Gaussian normal distribution
11053:
explains the central role of normal distributions, and thus of
10726:
9422:
8960:
8944:
8714:
8588:
8506:
7051:(arctan) infinite series, a team led by George Reitwiesner and
6300:
6192:, meaning it is not equal to the quotient of any two integers.
3562:
were accurate to two decimal places; this was improved upon in
3069:
1121:
785:
725:
is distinguished from its capitalized and enlarged counterpart
420:
308:
20488:"How Google's Emma Haruka Iwao Helped Set a New Record for Pi"
20104:
20102:
19685:
19683:
19681:
19679:
19082:. Mathematical Association of America. 2014. pp. 109–118.
18935:
18675:
18475:
18314:
17362:(in Latin). Excudebat L. Lichfield, Veneunt apud T. Robinson.
13442:
13254:
As a geometrical application of Stirling's approximation, let
11332:
10051:
Wirtinger's inequality also generalizes to higher-dimensional
9846:
Ultimately, as a consequence of the isoperimetric inequality,
8560:
3838:
of 3.1416, which he may have obtained from Archimedes or from
3656:{\textstyle {\bigl (}{\frac {16}{9}}{\bigr )}^{2}\approx 3.16}
1492:
have been available on which to perform statistical analysis.
20740:
20738:
20662:
20225:
20223:
20184:
18719:
18717:
18465:
18463:
18461:
18382:
18380:
18132:. Wilfrid Laurier University Press. pp. 67–77, 165–166.
18109:
18107:
18092:
16661:
and contained text that implied various incorrect values for
16198:
16070:
formula derived by Euler, which gives the maximum axial load
13426:
is that this is the (optimal) upper bound on the volume of a
11445:{\displaystyle \oint _{\gamma }{\frac {dz}{z-z_{0}}}=2\pi i.}
8596:
8494:
8478:
8231:{\displaystyle \pi =\lim _{n\to \infty }{\frac {2n}{E^{2}}}.}
8025:, generated by a sequence of (fair) coin tosses: independent
7056:
6495:
started using the single-letter form beginning with his 1727
6492:
4077:
3899:(3rd century, approximately 3.1556). Around 265 AD, the
2852:
1303:
781:
416:
19933:
Car, soit π la circonference d'un cercle, dout le rayon est
19297:
18290:
17698:
16883:
16649:
In 1897, an amateur mathematician attempted to persuade the
12693:
9195:. The modes of vibration of the string are solutions of the
8922:), and the integral computes the area below the semicircle.
8531:
3988:{\textstyle \pi \approx {\frac {355}{113}}=3.14159292035...}
3911:
and used it with a 3,072-sided polygon to obtain a value of
23092:
Hallerberg, Arthur (May 1977). "Indiana's squared circle".
20196:
20099:
20087:
19803:
19676:
18740:
18697:
The Crest of the Peacock: Non-European Roots of Mathematics
17901:
Lange, L.J. (May 1999). "An Elegant Continued Fraction for
17654:
Salikhov, V. (2008). "On the Irrationality Measure of pi".
17637:
17635:
17522:
Einführung in die Differentialrechnung und Integralrechnung
17411:
17409:
17407:
17297:
16866:
16591:
16488:
15885:) cannot both be arbitrarily small at the same time (where
14898:
14772:
is the unique constant making the Jacobi theta function an
14521:
characterized by their transformation properties under the
14157:
Furthermore, the derivative of the zeta function satisfies
13454:
of analytic number theory are also localized in each prime
11847:
10785:
9750:
8940:
7400:
was the first to use it for advances in the calculation of
6052:
that later contributed to the development and study of the
6010:, the latter producing 14 correct decimal digits per term.
3271:{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi ,}
3006:
2989:
2956:
2902:
2833:
2786:
2776:
1488:
Since the advent of computers, a large number of digits of
1129:
706:
399:
23560:
21614:
Einstein's Field Equations and Their Physical Implications
21375:"On the role of the Heisenberg group in harmonic analysis"
20735:
20360:
20220:
19864:
Commentarii Academiae Scientiarum Imperialis Petropolitana
19623:
Archibald, R.C. (1921). "Historical Notes on the Relation
19445:
18876:
18714:
18487:
18458:
18377:
18365:
18353:
18341:
18278:
18266:
18189:
18104:
17966:
Kennedy, E.S. (1978). "Abu-r-Raihan al-Biruni, 973–1048".
17947:
17935:
16598:
among mathematically and technologically minded groups. A
16483:
15939:{\displaystyle \Delta x\,\Delta p\geq {\frac {h}{4\pi }}.}
13666:, and the probability that at least one of them is not is
13437:
11075:
equal to its own Fourier transform. Indeed, according to
9163:
In many applications, it plays a distinguished role as an
8599:. Below are some of the more common formulae that involve
4027:{\textstyle \pi \approx {\frac {22}{7}}=3.142857142857...}
1414:
is irrational, it has an infinite number of digits in its
22631:
Mathematical Treks: From Surreal Numbers to Magic Circles
20923:"Pi record smashed as team finds two-quadrillionth digit"
20562:
Ramaley, J.F. (October 1969). "Buffon's Noodle Problem".
19375:
The Penguin Dictionary of Curious and Interesting Numbers
19264:
Borwein, J.M.; Borwein, P.B. (1988). "Ramanujan and Pi".
17852:
17449:
17447:
17445:
16689:
16500:
16393:
is the practice of memorizing large numbers of digits of
13450:, which are arithmetic localizations of the circle. The
9866:. In two dimensions, the critical Sobolev inequality is
9417:
such value of the wavenumber, and is associated with the
8514:
8047:
with equal probabilities. The associated random walk is
7853:
6535:
is equal to half the circumference of a circle of radius
6448:. The Greek letter appears on p. 243 in the phrase "
6294:
3675:, this theory is not widely accepted by scholars. In the
3517:{\displaystyle e^{2\pi ik/n}\qquad (k=0,1,2,\dots ,n-1).}
2856:
1410:(ratio of whole numbers) can be its exact value. Because
1089:
An integral such as this was proposed as a definition of
903:; although the notion of a circle can be extended to any
23226:
Andrews, George E.; Askey, Richard; Roy, Ranjan (1999).
22914:
Numericon: A Journey through the Hidden Lives of Numbers
22242:"29.7 Probability: The Heisenberg Uncertainty Principle"
22215:
Halliday, David; Resnick, Robert; Walker, Jearl (1997).
21126:
20996:
18828:"Fast formulas for slowly convergent alternating series"
17734:
17722:
17632:
17404:
17279:
13634:
result that the probability of two random numbers being
13510:
is used in many areas of mathematics. When evaluated at
11207:{\displaystyle \int _{\Sigma }K\,dA=2\pi \chi (\Sigma )}
11102:
three and Euler characteristic −4, as a quotient of the
9675:
of the derivative operator on the space of functions on
6559:
3151:{\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi ),}
1645:{\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0}
21646:
Elliptic Partial Differential Equations of Second Order
20519:"Identities inspired by Ramanujan's Notebooks (part 2)"
19911:
is taken for the ratio of the radius to the periphery "
15532:
be the evaluation functional, which associates to each
12364:
is the product of all of the positive integers through
3023:
The association between imaginary powers of the number
3014:
2572:
2553:
2539:
2520:
2506:
2487:
2473:
2461:
2421:
2402:
2388:
2369:
2355:
2336:
2322:
2310:
2270:
2251:
2237:
2218:
2204:
2185:
2171:
2152:
1960:
1948:
1933:
1921:
1906:
1894:
1879:
1867:
1852:
1840:
1825:
1813:
1798:
1786:
907:, these new circles will no longer satisfy the formula
23275:
22745:
Rubillo, James M. (January 1989). "Disintegrate 'em".
22018:
22016:
21921:
20024:
denote the ratio of the diameter to the circumference"
19813:
19413:
18531:< 3.14159 26535 89793 23846 26433 83279 50288 4199.
17442:
17228:
17115:
17093:
17078:
17050:
17028:
17013:
16992:
16977:
16471:
contains 10,000 words, each representing one digit of
15355:
be the set of all twice differentiable real functions
15244:
is the unique (positive) normalizing factor such that
14317:
can be written as an infinite linear superposition of
13092:
13064:{\displaystyle 2\pi r={\frac {S_{n+1}(r)}{V_{n}(r)}}.}
12531:{\textstyle \Gamma (5/2)={\frac {3{\sqrt {\pi }}}{4}}}
12485:
11667:. Cauchy's integral formula is a special case of the
11306:{\displaystyle A(S)=\int _{S}1\,dA=2\pi \cdot 2=4\pi }
10926:
10249:
Fourier transform and Heisenberg uncertainty principle
7977:
Five random walks with 200 steps. The sample mean of
7584:
per term and has been used for several record-setting
6957:
6810:
6690:
6579:
6456:
6454:
6401:
6362:
6283:
in Latin) of a circle and was combined in ratios with
5313:
5289:
5265:
5263:
4924:
4834:
4175:
4138:
4086:
4001:
3962:
3615:
2575:
2556:
2542:
2523:
2509:
2490:
2476:
2464:
2424:
2405:
2391:
2372:
2358:
2339:
2325:
2313:
2273:
2254:
2240:
2221:
2207:
2188:
2174:
2155:
1963:
1951:
1944:
1936:
1924:
1917:
1909:
1897:
1890:
1882:
1870:
1863:
1855:
1843:
1836:
1828:
1816:
1809:
1801:
1789:
1782:
1592:
969:
913:
874:
847:
606:
as well as stress testing consumer computer hardware.
455:
22214:
22190:
Ovsienko, V.; Tabachnikov, S. (2004). "Section 1.3".
21480:
A Comprehensive Introduction to Differential Geometry
20040:(in Latin). Lipsae: B.G. Teubneri. pp. 133–134.
19315:
19035:
Series and Products in the Development of Mathematics
18431:
16972:
16785:
16286:
16216:
16106:
15971:
15901:
15852:{\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}.}
15821:
15707:
15668:
15635:
15564:
15538:
15497:
15469:
15400:
15361:
15142:
15014:
14940:
14782:
14732:
14619:
14573:
14530:
14359:
14163:
14026:
13679:
13523:
13297:
13160:
12993:
12865:
12761:
12551:
12438:
12384:
12347:
12222:
12081:
12046:
12017:
11988:
11960:
11929:
11856:
11704:
11549:
11468:, which implies that the integral is invariant under
11386:
11243:
11162:
10972:
10924:
10808:
10432:
10288:
10268:
also appears as a critical spectral parameter in the
10233:
of mean zero. Just as Wirtinger's inequality is the
10073:
10007:
9974:
9943:
9872:
9787:
9688:
9511:
9435:
9324:
9262:
9204:
9012:
8971: radians. The angle measure of 180° is equal to
8896:
8859:
8779:
8306:
8152:
8053:
7917:
7889:
times on a surface on which parallel lines are drawn
7617:
7426:
7252:
6956:
6809:
6689:
6578:
6322:
6064:
5865:
5724:
5551:
5413:
5065:
4970:
4887:
4861:
4730:
4716:, and independently, Leibniz in 1673, discovered the
4484:
4369:
4114:{\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142}
3930:
3863:
3437:
3394:
3337:
3307:
3224:
3172:
3102:
2128:
1769:
1705:
1676:
1181:
1019:
812:
715:
589:
soon led to the calculation of hundreds of digits of
453:
23529:
Demonstration by Lambert (1761) of irrationality of
23329:
22219:(5th ed.). John Wiley & Sons. p. 381.
21712:
21667:
21589:
The pleasures of pi, e and other interesting numbers
21431:
21098:
20984:
20806:
18061:
14466:
13980:
quantity: the reciprocal of the volume of a certain
7817:
are certain rational numbers that Plouffe computed.
5399:
to any given accuracy. A simple infinite series for
4276:. Infinite series allowed mathematicians to compute
3301:
centred at the origin of the complex plane. Setting
703:
396:
23405:: A Biography of the World's Most Mysterious Number
22189:
22013:
20896:"A new formula to compute the n binary digit of pi"
20278:"Some Background on Kanada's Recent Pi Calculation"
19971:
be the circumference (!) of a circle of unit radius
19294:, pp. 15–17, 70–72, 104, 156, 192–197, 201–202
18331:
18329:
17616:
17614:
16861:The polynomial shown is the first few terms of the
15101:The Cauchy distribution plays an important role in
14567:(or its various subgroups), a lattice in the group
13949:This probability can be used in conjunction with a
12699:The gamma function is used to calculate the volume
10747:. The coloured region between the function and the
9920:{\displaystyle 2\pi \|f\|_{2}\leq \|\nabla f\|_{1}}
7198:, testing numerical analysis algorithms (including
6485:are from the "ready pen of the truly ingenious Mr.
4218:reached 34 digits in 1621, and Austrian astronomer
1469:have no apparent pattern and have passed tests for
1120:is twice the smallest positive number at which the
23399:
23397:
23308:
23272:English translation by Catriona and David Lischka.
23086:
22695:. Johns Hopkins University Press. pp. 56–57.
22658:
22276:(2005 ed.). Mineola, NY: Dover Publications.
20692:"Unbounded spigot algorithms for the digits of pi"
20656:
20601:
20457:. Springer International Publishing. p. 469.
20329:
19400:
18003:used a three-sexagesimal-digit approximation, and
17803:
17752:
17539:
17435:. Vol. 1 (2nd ed.). Wiley. p. 102.
17160:
16845:
16369:
16243:
16150:
16043:
15938:
15851:
15732:
15693:
15648:
15601:
15550:
15524:
15483:
15441:
15383:
15230:
15076:
14996:
14879:transforms as a representation under the discrete
14867:
14757:
14711:
14602:
14559:
14406:
14215:{\displaystyle \exp(-\zeta '(0))={\sqrt {2\pi }}.}
14214:
14147:
13939:
13604:
13413:
13244:
13142:
13063:
12973:
12852:
12654:
12530:
12471:
12420:
12356:
12274:
12137:
12052:
12032:
12003:
11966:
11946:
11915:
11779:
11636:
11444:
11305:
11206:
11117:. The hyperbolic area of a fundamental domain is
11030:
10944:
10908:
10699:
10684:
10380:
10276:, that takes a complex-valued integrable function
10199:
10029:
9993:
9952:
9919:
9815:
9721:
9637:
9467:
9341:
9303:
9248:
9109:
8902:
8882:
8843:
8459:
8230:
8100:
7953:{\displaystyle \pi \approx {\frac {2n\ell }{xt}}.}
7952:
7771:
7570:
7386:
7027:
6938:
6796:
6675:
6469:
6414:
6375:
6334:
6204:is also irrational. In 1882, German mathematician
6170:
6013:
5966:
5827:
5677:
5516:
5360:in his head at the behest of German mathematician
5329:
5243:
5027:
4945:
4899:
4873:
4847:
4814:
4660:
4460:
4194:
4157:
4113:
4026:
3987:
3948:
3873:
3709:developed the polygonal approach to approximating
3655:
3516:
3413:
3365:
3319:
3270:
3194:
3150:
2612:
2018:
1715:
1691:
1644:
1430:; they generally require calculus and rely on the
1279:
1128:is also the smallest positive number at which the
1097:, who defined it directly as an integral in 1841.
1079:
1001:
932:
887:
860:
831:
468:
23591:approximation of π with rectangles and trapezoids
21859:
21635:
21633:
21255:
20624:Transactions of the American Mathematical Society
20123:Mathematical Tables and Other Aids to Computation
19713:] (in Latin). London: Thomas Harper. p.
17874:Mollin, R. A. (1999). "Continued fraction gems".
17842:
17840:
15459:, with two parameters corresponding to a pair of
14291:. Periodic functions are functions on the group
13626:solved it in 1735 when he showed it was equal to
12170:, and index/turning number 3, though it only has
9858:dimensions, which thus characterizes the role of
9850:appears in the optimal constant for the critical
8959:rely on angles, and mathematicians generally use
8540:is closely related to the circle, it is found in
8505:, which turned out to be 0. In September 2010, a
7901:of those times it comes to rest crossing a line (
4650:
4620:
4610:
4580:
4570:
4540:
4530:
4500:
3802:may have led to a widespread popular belief that
1723:). Second, since no transcendental number can be
23875:
23398:Posamentier, Alfred S.; Lehmann, Ingmar (2004).
22903:
22560:Keith, Michael; Diana Keith (17 February 2010).
22559:
22317:. Cambridge University Press. pp. 116–118.
21878:Proceedings of the American Mathematical Society
21639:
19812:, p. 165: A facsimile of Jones' text is in
18432:O'Connor, John J.; Robertson, Edmund F. (1999).
18326:
17611:
16779:The specific integral that Weierstrass used was
16280:. Before 20 May 2019, it was defined as exactly
15764:
15744:. This function is periodic, and the quantity
13638:(that is, having no shared factors) is equal to
13168:
8160:
7238:published dozens of innovative new formulae for
4332:. The series for arctangent is sometimes called
4288:most notably by European mathematicians such as
554:with arbitrary accuracy. In the 5th century AD,
23225:
22071:Introduction to the Theory of Fourier Integrals
21493:
21300:
20756:
20155:
19485:
19483:
18098:
16889:
16696:. The versions are 3, 3.1, 3.14, and so forth.
15733:{\displaystyle \mathbb {R} \to \mathbb {P} (V)}
14923:The Cauchy distribution governs the passage of
14603:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}
14560:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
13984:. In the case of the Basel problem, it is the
12325:The gamma function and Stirling's approximation
11237:and is found to be equal to two. Thus we have
10223:of diameter 1, and square-integrable functions
6554:
4169:in 1579 achieved nine digits with a polygon of
2633:
1302:, is the following theorem: there is a unique (
22053:Brownian motion and classical potential theory
21979:. Thompson, Washington, DC. pp. 305–347.
21860:Platonov, Vladimir; Rapinchuk, Andrei (1994).
21819:
21630:
21494:Kobayashi, Shoshichi; Nomizu, Katsumi (1996).
21110:
20271:
20269:
20267:
20245:
20079:notet peripheriam circuli, cuius diameter eſt
19921:Euler, Leonhard (1747). Henry, Charles (ed.).
19832:(in Latin). Halae Magdeburgicae. p. 282.
19591:
19589:
19396:
19394:
19263:
19117:Nova Acta Academiae Scientiarum Petropolitinae
18015:Episodes from the Early History of Mathematics
17837:
16547:. This part of the story was omitted from the
16151:{\displaystyle F={\frac {\pi ^{2}EI}{L^{2}}}.}
15956:. The inverse lifetime to lowest order in the
15384:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
14487:is connected in a deep way with the theory of
11809:
8743:, every curve of constant width has perimeter
7162:increased dramatically. The vertical scale is
3949:{\displaystyle 3.1415926<\pi <3.1415927}
1528:, although no connection to Feynman is known.
1350:
609:Because its definition relates to the circle,
23613:
23302:
22904:Freiberger, Marianne; Thomas, Rachel (2015).
22262:
22239:
21427:
21425:
21380:Bulletin of the American Mathematical Society
21150:Nature Series: Popular Lectures and Addresses
20441:
19534:Schepler, H.C. (1950) "The Chronology of Pi"
19522:, as did Oughtred more than a century earlier
18505:
18335:
18296:
18220:
18218:
18216:
18125:
17801:
17533:
17531:
14723:. This is sometimes written in terms of the
12275:{\displaystyle \int _{a}^{b}k(s)\,ds=2\pi N.}
11042:in the figure is equal to the square root of
10282:on the real line to the function defined as:
7907: > 0), then one may approximate
7877:is one such technique: If a needle of length
6543:, though the definition still varied between
6519:, and continued in his widely read 1748 work
4222:arrived at 38 digits in 1630 using 10 sides.
3636:
3618:
2114:. Although the simple continued fraction for
1520:. This is also called the "Feynman point" in
749:
721:). In mathematical use, the lowercase letter
535:, required fairly accurate approximations of
492:, meaning that it cannot be a solution of an
431:, approximately equal to 3.14159. The number
365:
23203:
22167:Chaos and fractals: new frontiers of science
21258:Journal de Mathématiques Pures et Appliquées
21052:
19857:"Tentamen explicationis phaenomenorum aeris"
19493:A History of Mathematical Notations: Vol. II
19480:
19259:
19257:
18787:"On the Leibnizian quadrature of the circle"
16945:There are various other ways of finding the
14459:. This is a version of the one-dimensional
11327:
10018:
10008:
9982:
9975:
9908:
9898:
9886:
9879:
8990:; for example, sine and cosine have period 2
8643:The area of an ellipse with semi-major axis
8555:
8251:These Monte Carlo methods for approximating
7592:in 2022. For similar formulae, see also the
7209:
3824:. Around 150 AD, Greek-Roman scientist
3686:
1274:
1245:
1239:
1182:
23510:
23365:Eymard, Pierre; Lafon, Jean Pierre (2004).
23363:
23330:Bronshteĭn, Ilia; Semendiaev, K.A. (1971).
23246:
23081:
22687:
22675:
22663:
22392:
22367:
22342:. Cambridge University Press. p. 233.
22240:Urone, Paul Peter; Hinrichs, Roger (2022).
21847:
21777:
21303:Archive for Rational Mechanics and Analysis
20884:
20794:
20744:
20682:
20680:
20671:
20651:
20596:
20549:
20426:
20414:
20402:
20390:
20378:
20366:
20308:
20264:
20241:
20229:
20214:
20202:
20190:
20120:
20108:
20093:
19809:
19689:
19586:
19451:
19391:
19360:
19303:
19291:
19212:
19200:
18956:
18941:
18882:
18813:
18772:
18723:
18700:. Princeton University Press. p. 264.
18681:
18609:
18493:
18481:
18469:
18386:
18371:
18359:
18347:
18320:
18308:
18284:
18272:
18207:
18195:
18113:
18078:. Princeton University Press. p. 160.
17953:
17941:
17858:
17846:
17764:
17740:
17728:
17716:
17704:
17641:
17469:
17415:
17285:
17249:
16965:, the Diameter is to Circumference as 1 to
12472:{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}
12333:Plot of the gamma function on the real axis
10945:{\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}
9171:can be modelled as the graph of a function
8300:, was discovered in 1995 by Simon Plouffe:
6520:
6446:; or, a New Introduction to the Mathematics
6441:
6431:The earliest known use of the Greek letter
6345:
6223:
6002:. Series that converge even faster include
4044:("approximate ratio"), respectively, using
2098:is transcendental, it is by definition not
1362:, meaning that it cannot be written as the
1287:and there is a unique positive real number
943:Here, the circumference of a circle is the
23620:
23606:
23091:
22783:. Cambridge University Press. p. 47.
22480:
22401:"How can anyone remember 100,000 numbers?"
22065:
21827:. Dover Publications Inc. pp. 29–35.
21455:
21422:
20952:Plouffe, Simon (2022). "A formula for the
20246:Borwein, Jonathan; Borwein, Peter (1987).
19891:Sumatur pro ratione radii ad peripheriem,
19235:. Vol. 102, no. 5. p. 342.
19218:
18756:Variorum de rebus mathematicis responsorum
18236:. Heidelberg: Springer. pp. 531–561.
18213:
18047:Section 8.5: Polar form of complex numbers
17528:
17388:. Dictionary.reference.com. 2 March 1993.
16642:and simplifies many formulae. This use of
16522:. On its wall are inscribed 707 digits of
11348:of a function over a positively oriented (
9931:a smooth function with compact support in
9315:is an eigenvalue of the second derivative
8720:and the surface area of its boundary, the
8607:The circumference of a circle with radius
8101:{\displaystyle W_{n}=\sum _{k=1}^{n}X_{k}}
7170:For most numerical calculations involving
7146:
6036:Not all mathematical advances relating to
372:
358:
69:
23354:
22833:
22553:
22506:
22337:
22115:
22050:
22007:
21935:
21890:
21738:
21443:
21392:
21269:
21240:
21195:
21132:
21120:Methods of mathematical physics, volume 1
20969:
20846:
20636:
19786:
19757:
19622:
19425:
19377:(revised ed.). Penguin. p. 35.
19337:
19254:
19164:
19131:
18768:
18766:
18631:"The Discovery of the Series Formula for
18540:
18416:
18405:Missouri Journal of Mathematical Sciences
17869:
17867:
17351:
17349:
17313:
17145:
16467:in this manner, and the full-length book
15908:
15717:
15709:
15518:
15477:
15377:
15369:
15204:
15058:
14719:which is a kind of modular form called a
14593:
14550:
14443:is the unique number such that the group
12730:-dimensional space, and the surface area
12250:
11991:
11727:
11596:
11272:
11176:
11038:which says that the area under the basic
11011:
10846:
10661:
10578:
10495:
10368:
9625:
9566:
9461:
8818:
8696:The surface area of a sphere with radius
8532:Role and characterizations in mathematics
7962:Another Monte Carlo method for computing
7526:
4268:was revolutionized by the development of
3881:(100 AD, approximately 3.1623), and
3000:(base 60) digits are 3;8,29,44,0,47 (see
1418:, and does not settle into an infinitely
1270:
23477:
23210:
22773:
22624:
22598:
22535:"Cadaeic Cadenza Notes & Commentary"
22363:
22361:
22359:
21799:. Oxford University Press. Theorem 332.
21797:An Introduction to the Theory of Numbers
21536:
21463:. Princeton University Press. p. 6.
21368:
21366:
21359:. Princeton University Press. p. 5.
21144:
21079:
20677:
20613:
20504:PSLQ means Partial Sum of Least Squares.
19720:
19702:
19659:It is noticeable that these letters are
19533:
19078:
19063:
18952:
18950:
18624:
18622:
18620:
18618:
18434:"Ghiyath al-Din Jamshid Mas'ud al-Kashi"
18042:
17776:
17653:
17620:
17599:
17502:] (in German). Hirzel. p. 195.
17355:
16482:
15656:is a one-dimensional linear subspace of
15463:for the differential equation. For any
15272:
15268:
14918:
14902:
14899:Cauchy distribution and potential theory
14470:
14255:
13466:Solution of the Basel problem using the
13461:
13441:
12328:
12161:
11692:and is continuous in a neighbourhood of
11331:
11086:
10725:
10403:that is also an algebra homomorphism of
10252:
9736:
9123:
8935:
8559:
7972:
7611:, conforming to the following template:
7213:
7200:high-precision multiplication algorithms
7153:
4669:
4237:
3701:
3690:
3018:
1541:
764:
23488:. Paris: Bibliothèque Pour la Science.
23455:
23423:
23247:Arndt, Jörg; Haenel, Christoph (2006).
22975:from the original on 28 September 2013.
22834:Rosenthal, Jeffrey S. (February 2015).
22744:
22164:
22022:
21518:
21482:. Vol. 3. Publish or Perish Press.
21351:
21181:
21094:
21092:
20951:
20686:
20561:
20513:
20320:
19050:The Mathematical Papers of Isaac Newton
18894:
18888:
18784:
18439:MacTutor History of Mathematics archive
18398:
18224:
18164:
17965:
17581:
17506:from the original on 14 September 2016.
17491:
17465:
17453:
17427:
16921:
16919:
16917:
16688:let the version numbers of his program
16551:adaptation of the novel. The digits of
16403:. The record for memorizing digits of
15525:{\displaystyle e_{t}:V\to \mathbb {R} }
15129:is the integral transform given by the
15090:of the Cauchy distribution is equal to
13438:Number theory and Riemann zeta function
11464:, a standard proof of this result uses
9140:. The associated eigenvalues form the
8568:times the shaded area. The area of the
8245:
6216:, confirming a conjecture made by both
3569:The earliest written approximations of
3210:can be related to the behaviour of the
1659:has two important consequences: First,
1147:can be defined using properties of the
1101:
14:
23876:
23627:
22985:
22936:
22730:from the original on 15 October 2006.
22648:from the original on 29 November 2016.
22431:
22111:
22109:
22030:. Boston: Birkhauser. pp. 1–117.
21874:
21611:
21474:
21184:Annali di Matematica Pura ed Applicata
21166:
20920:
20485:
20437:
20435:
20335:
20314:
20275:
20066:
19940:
19825:
19745:The mathematical works of Isaac Barrow
19738:
19489:
19040:
18968:
18825:
18775:, p. 188. Newton quoted by Arndt.
18763:
18693:
18171:. Princeton University Press. p.
18032:from the original on 29 November 2016.
17873:
17864:
17516:
17346:
16668:
16499:, making pie a frequent subject of pi
16445:A few authors have used the digits of
16076:that a long, slender column of length
15759:
15344:
11655:is analytic in the region enclosed by
10245:eigenvalue problem, in any dimension.
10063:-dimensional membrane. Specifically,
7820:
6497:Essay Explaining the Properties of Air
5379:
3924:, around 480 AD, calculated that
3671:was built with proportions related to
1742:
539:for practical computations. Around 250
486:enters a permanently repeating pattern
23601:
23585:approximation von π by lattice points
23511:
23126:
22398:
22356:
22093:
22051:Port, Sidney; Stone, Charles (1978).
21794:
21697:
21679:
21561:
21363:
20800:
20033:
19982:
19924:Lettres inédites d'Euler à d'Alembert
19920:
19854:
19595:
19372:
19224:
19107:
19085:
18947:
18901:Archive for History of Exact Sciences
18809:
18807:
18752:
18615:
18579:
18146:from the original on 29 November 2016
18057:
18055:
18008:
17900:
17824:from the original on 13 December 2019
17566:
17537:
17373: :: semidiameter. semiperipheria
17239:from the original on 6 December 2016.
17203:
16925:
16901:
16604:Massachusetts Institute of Technology
16478:
15785:. A simple formula from the field of
15781:'s relationship to the circle and to
15756:, appears naturally in this context.
14447:, equipped with its Haar measure, is
14407:{\displaystyle e_{n}(x)=e^{2\pi inx}}
13966:Weil's conjecture on Tamagawa numbers
12541:The gamma function is defined by its
10721:
9749:: those lands bordering the sea that
9667:of the eigenvalue. As a consequence,
9249:{\displaystyle f''(x)+\lambda f(x)=0}
9136:of the second derivative, and form a
8552:in some of their important formulae.
8021:using probability is to start with a
7607:to generate several new formulae for
7139:to set several records for computing
6560:Computer era and iterative algorithms
4946:{\textstyle z={\frac {1}{\sqrt {3}}}}
22715:
22541:from the original on 18 January 2009
21974:
21591:. World Scientific Pub. p. 21.
21497:Foundations of Differential Geometry
21461:Fourier analysis on Euclidean spaces
21372:
21148:(1894). "Isoperimetrical problems".
21089:
20531:from the original on 14 January 2012
20048:from the original on 16 October 2017
19836:from the original on 15 October 2017
19814:Berggren, Borwein & Borwein 1997
19602:. Courier Corporation. p. 312.
19414:Berggren, Borwein & Borwein 1997
19206:
19096:Institutiones Calculi Differentialis
18597:Digital Library for Dutch Literature
18302:
18073:
17968:Journal for the History of Astronomy
17802:Schlager, Neil; Lauer, Josh (2001).
17267:from the original on 19 October 2019
16914:
16749:
16397:, and world-records are kept by the
16379:
14475:Theta functions transform under the
13972:quantities, localized at each prime
13481:area of a fundamental domain of the
12209:along a curve taken with respect to
12033:{\displaystyle \Delta \Phi =\delta }
11076:
10055:that provide best constants for the
9304:{\displaystyle f''(t)=-\lambda f(x)}
8735:Apart from circles, there are other
8275:because, like water dripping from a
8262:
6376:{\textstyle {\frac {\pi }{\delta }}}
6311:instead.) The first recorded use is
5337:with which he computed 20 digits of
4918:used the Gregory–Leibniz series for
4689:In the 1660s, the English scientist
4296:, the approach also appeared in the
3857:included 3.1547 (around 1 AD),
3214:of a complex variable, described by
3015:Complex numbers and Euler's identity
1335:), onto the multiplicative group of
550:created an algorithm to approximate
22872:
22413:from the original on 18 August 2013
22312:
22106:
21864:. Academic Press. pp. 262–265.
21586:
21541:. Universities Press. p. 166.
20921:Palmer, Jason (16 September 2010).
20890:
20555:
20432:
20067:Segner, Johann Andreas von (1761).
19032:
18694:Joseph, George Gheverghese (1991).
18635:by Leibniz, Gregory and Nilakantha"
18628:
18603:
18551:. World Scientific. pp. 1–22.
18401:"al-Risāla al-muhītīyya: A Summary"
17542:Principles of Mathematical Analysis
17524:(in German). Noordoff. p. 193.
16268:, which describe the properties of
15805:, swinging with a small amplitude (
15694:{\displaystyle t\mapsto \ker e_{t}}
14883:. General modular forms and other
14353:is one of the complex exponentials
10067:is the greatest constant such that
8665:The volume of a sphere with radius
8259:when speed or accuracy is desired.
8121:is drawn from a shifted and scaled
6523:Introductio in analysin infinitorum
3060:, can be expressed using a pair of
832:{\displaystyle \pi ={\frac {C}{d}}}
24:
23448:
23152:from the original on 13 April 2016
23028:from the original on 22 June 2012.
22885:from the original on 24 April 2019
22815:from the original on 15 March 2015
22399:Otake, Tomoko (17 December 2006).
21862:Algebraic Groups and Number Theory
20933:from the original on 17 March 2011
20874:from the original on 22 July 2012.
20290:from the original on 15 April 2012
19286:10.1038/scientificamerican0288-112
18804:
18446:from the original on 12 April 2011
18052:
18007:expanded this to nine digits; see
17781:. St. Martin's Press. p. 37.
17146:
16805:
16800:
16634:, as the number of radians in one
16463:contains the first 3835 digits of
15909:
15902:
15867:Heisenberg's uncertainty principle
15813:earth's gravitational acceleration
15184:
15179:
15028:
15023:
14660:
14655:
14579:
14576:
14536:
14533:
14427:on the circle group, the constant
14275:makes heavy use of this machinery.
14103:
13927:
13755:
13694:
13323:
13271:-dimensional Euclidean space, and
13178:
12920:
12805:
12606:
12552:
12486:
12439:
12385:
12148:
12082:
12021:
12018:
11961:
11857:
11198:
11168:
10986:
10981:
10952:makes the area under the graph of
10710:, asserting the uniqueness of the
10627:
10622:
10525:
10520:
10451:
10446:
10326:
10321:
10103:
10011:
9944:
9901:
9468:{\displaystyle f:\to \mathbb {C} }
8548:, and number theory, also include
8329:
8170:
7647:
7468:
7301:
6568:Gauss–Legendre iterative algorithm
6415:{\textstyle {\frac {\pi }{\rho }}}
4824:This series, sometimes called the
4359:, which is more typically used in
4253:is the approximation after taking
4233:
4052:available for the next 800 years.
3601: = 3.125. In Egypt, the
2796:: The first 50 decimal digits are
1485:has not been proven or disproven.
25:
23910:
23504:
22529:
22340:An Introduction to Fluid Dynamics
21901:10.1090/s0002-9939-1994-1172954-7
20997:Martini, Horst; Montejano, Luis;
20700:The American Mathematical Monthly
20638:10.1090/s0002-9947-1960-0114110-9
20564:The American Mathematical Monthly
20348:from the original on 2 April 2012
20073:(in Latin). Renger. p. 374.
19943:The American Mathematical Monthly
19881:from the original on 1 April 2016
19635:The American Mathematical Monthly
17908:The American Mathematical Monthly
17392:from the original on 28 July 2014
17206:"William Jones: The First Use of
15484:{\displaystyle t\in \mathbb {R} }
14499:involves in an essential way the
14467:Modular forms and theta functions
14267:(shown), which are elements of a
14251:
12421:{\displaystyle \Gamma (n)=(n-1)!}
12067:In higher dimensions, factors of
11135:differential geometry of surfaces
9816:{\displaystyle 4\pi A\leq P^{2},}
9679:vanishing at both endpoints (the
8925:
8883:{\displaystyle {\sqrt {1-x^{2}}}}
4909:it converges impractically slowly
4280:with much greater precision than
3956:and suggested the approximations
3909:polygon-based iterative algorithm
3554:The best-known approximations to
1574:, which means that it is not the
1566:In addition to being irrational,
1402:are commonly used to approximate
23839:
23178:
23164:
23132:"The Future of TeX and Metafont"
23120:
23075:
23054:
23032:
22979:
22930:
22897:
22866:
22827:
22797:
22767:
22738:
22709:
22681:
22669:
22652:
22578:
22523:
22474:
22425:
22373:
22331:
22306:
22256:
22233:
22208:
22183:
22158:
22087:
22059:
22044:
21968:
21915:
21868:
21853:
21841:
21813:
21783:
21771:
21718:
21713:Bronshteĭn & Semendiaev 1971
21706:
21691:
21673:
21668:Bronshteĭn & Semendiaev 1971
21661:
21605:
21580:
21555:
21530:
21512:
21487:
21468:
21449:
21432:Bronshteĭn & Semendiaev 1971
21409:
21357:Harmonic analysis in phase space
21345:
21294:
21249:
21228:
21175:
21160:
21138:
21104:
21099:Bronshteĭn & Semendiaev 1971
21046:
20990:
20985:Bronshteĭn & Semendiaev 1971
20978:
20945:
20914:
20878:
20750:
20645:
20607:
20590:
20543:
20507:
20498:
20479:
20420:
20408:
20396:
20384:
20372:
20336:Connor, Steve (8 January 2010).
20302:
20276:Bailey, David H. (16 May 2003).
20235:
20149:
20114:
20060:
20027:
20008:English translation by Ian Bruce
20004:rationem diametri ad peripheriam
19976:
19914:
19898:English translation by Ian Bruce
19848:
19826:Segner, Joannes Andreas (1756).
19819:
19751:
19732:
19695:
19616:
19527:
19470:
18062:Bronshteĭn & Semendiaev 1971
17216:. McGraw–Hill. pp. 346–347.
16707:for use in programs. Similarly,
16608:Rensselaer Polytechnic Institute
14758:{\displaystyle q=e^{\pi i\tau }}
14479:of periods of an elliptic curve.
12166:This curve has total curvature 6
12154:This section is an excerpt from
12120:
12089:
12004:{\displaystyle \mathbb {R} ^{2}}
11901:
11864:
10784:of the normal distribution with
10420:Heisenberg uncertainty principle
10259:geodesic in the Heisenberg group
10030:{\displaystyle \|\nabla f\|_{1}}
8279:, they produce single digits of
7852:
7829:
7112:arithmetic–geometric mean method
6251:
6235:
3431:" and are given by the formula:
1531:
1508:. Thus, because the sequence of
1458:but smaller than the measure of
933:{\textstyle \pi ={\frac {C}{d}}}
699:
665:often result in news headlines.
469:{\displaystyle {\tfrac {22}{7}}}
435:appears in many formulae across
392:
23566:2 billion searchable digits of
23355:Dym, H.; McKean, H. P. (1972).
23232:. Cambridge: University Press.
22481:Raz, A.; Packard, M.G. (2009).
21924:Journal of Mathematical Physics
21394:10.1090/S0273-0979-1980-14825-9
19564:Part 3. May/Jun. (5): 279-283.
19554:Part 2. Mar/Apr. (4): 216-228.
19544:Part 1. Jan/Feb. (3): 165–170.
19457:
19419:
19406:
19366:
19309:
19225:Hayes, Brian (September 2014).
19057:
19026:
18974:"On Arccotangent Relations for
18962:
18819:
18778:
18746:
18729:
18687:
18573:
18548:The Birth of Numerical Analysis
18534:
18499:
18425:
18392:
18158:
18119:
18067:
17959:
17894:
17795:
17770:
17758:
17746:
17676:10.1070/RM2008v063n03ABEH004543
17647:
17626:Fonctions d'une variable réelle
17593:
17575:
17560:
17510:
17485:
17459:
17421:
17378:
16855:
16773:
16244:{\displaystyle F=6\pi \eta Rv.}
14417:There is a unique character on
14020:as well as the gamma function:
12188:differential geometry of curves
9762:
9732:
9357:, so it is convenient to write
8890:represents the height over the
8853:In that integral, the function
7580:It produces about 14 digits of
7093:Fourier transform-based methods
6899:
6898:
6746:
6745:
6655:
6628:
6599:
6014:Irrationality and transcendence
5367:In 1853, British mathematician
4914:In 1699, English mathematician
4474:published what is now known as
4347:published what is now known as
3465:
2770:. (List is selected terms from
2120:generalized continued fractions
1436:technique. The degree to which
1327:of real numbers under addition
668:
23087:Posamentier & Lehmann 2004
23046:. 30 June 2011. Archived from
22999:The Mathematical Intelligencer
22937:Abbott, Stephen (April 2012).
22716:Gill, Andy (4 November 2005).
22659:Posamentier & Lehmann 2004
22169:. Springer. pp. 801–803.
21700:Partial Differential Equations
21500:. Vol. 2 (New ed.).
20956:th decimal digit or binary of
20657:Posamentier & Lehmann 2004
20602:Posamentier & Lehmann 2004
20486:Cassel, David (11 June 2022).
19496:. Cosimo, Inc. pp. 8–13.
19401:Posamentier & Lehmann 2004
18997:10.1080/00029890.1938.11990873
18654:10.1080/0025570X.1990.11977541
18129:The Shape of the Great Pyramid
17753:Posamentier & Lehmann 2004
17301:The Mathematical Intelligencer
17291:
17243:
17221:
16932:Synopsis Palmariorum Matheseos
16895:
16756:
16739:List of mathematical constants
16657:, which described a method to
15727:
15721:
15713:
15672:
15596:
15590:
15581:
15575:
15514:
15430:
15424:
15415:
15409:
15393:ordinary differential equation
15373:
15201:
15195:
15155:
15149:
14950:
14944:
14859:
14847:
14804:
14786:
14635:
14623:
14597:
14589:
14554:
14546:
14376:
14370:
14193:
14190:
14184:
14170:
14139:
14127:
14118:
14106:
14036:
14030:
13957:using a Monte Carlo approach.
13895:
13889:
13630:. Euler's result leads to the
13533:
13527:
13354:
13341:
13332:
13319:
13307:
13304:
13175:
13052:
13046:
13031:
13025:
12888:
12882:
12778:
12772:
12561:
12555:
12503:
12489:
12456:
12442:
12412:
12400:
12394:
12388:
12247:
12241:
12125:
12115:
12093:
12085:
11906:
11896:
11868:
11860:
11774:
11755:
11724:
11718:
11631:
11618:
11572:
11566:
11253:
11247:
11201:
11195:
10898:
10882:
10868:
10855:
10818:
10812:
10651:
10646:
10640:
10633:
10568:
10563:
10557:
10551:
10541:
10485:
10480:
10474:
10467:
10340:
10334:
10307:
10301:
10295:
10166:
10157:
10111:
10099:
9716:
9704:
9615:
9610:
9604:
9592:
9556:
9551:
9545:
9538:
9457:
9454:
9442:
9328:
9298:
9292:
9277:
9271:
9237:
9231:
9219:
9213:
9119:
8947:functions repeat with period 2
8564:The area of the circle equals
8293:Another spigot algorithm, the
8213:
8208:
8193:
8189:
8167:
7550:
7540:
7520:
7511:
7506:
7491:
7485:
7476:
7339:
7324:
7318:
7309:
7077:fast multiplication algorithms
7067:. First, the discovery of new
6994:
6967:
6886:
6853:
6443:Synopsis Palmariorum Matheseos
5223:
5203:
5149:
5129:
4848:{\textstyle {\frac {\pi }{4}}}
4080:apparently employed the value
4059:used a value of 3.1416 in his
3784:). Archimedes' upper bound of
3508:
3466:
3366:{\displaystyle e^{i\pi }+1=0.}
3327:in Euler's formula results in
3142:
3115:
2842:Digits in other number systems
2032:; the first four of these are
1755:. But every number, including
1514:sequence of six consecutive 9s
1502:statistical significance tests
905:curve (non-Euclidean) geometry
13:
1:
22438:) in Nature, Art, and Culture
22101:. Princeton University Press.
22055:. Academic Press. p. 29.
21280:10.1016/s0021-7824(02)01266-7
21171:. Cambridge University Press.
20857:10.1090/S0025-5718-97-00856-9
20759:American Mathematical Monthly
19743:. In Whewell, William (ed.).
19318:American Mathematical Monthly
18985:American Mathematical Monthly
18399:Azarian, Mohammad K. (2010).
18126:Herz-Fischler, Roger (2000).
18099:Andrews, Askey & Roy 1999
17628:(in French). Springer. §II.3.
16890:Andrews, Askey & Roy 1999
16744:
16731:Chronology of computation of
16614:Some have proposed replacing
16060:is the mass of the electron.
15789:gives the approximate period
15765:Describing physical phenomena
15602:{\displaystyle e_{t}(f)=f(t)}
15442:{\displaystyle f''(x)+f(x)=0}
14439:. As a result, the constant
14431:is half the magnitude of the
14246:spectrum of the hydrogen atom
13446:Each prime has an associated
12983:Further, it follows from the
11485:, then the above integral is
9645:with equality precisely when
9167:. For example, an idealized
5352:In 1844, a record was set by
5037:Machin reached 100 digits of
4202:sides. Flemish mathematician
4070:
3542:Chronology of computation of
3320:{\displaystyle \varphi =\pi }
3202:. The frequent appearance of
3096:from the positive real line:
1538:Lindemann–Weierstrass theorem
1422:of digits. There are several
760:
613:is found in many formulae in
23357:Fourier series and integrals
22958:10.4169/mathhorizons.19.4.34
22852:10.4169/mathhorizons.22.3.22
22608:. Wiley & Sons. p.
22165:Peitgen, Heinz-Otto (2004).
21434:, pp. 106–107, 744, 748
20451:"15.2 Computational records"
19431:"Über die Ludolph'sche Zahl"
18507:Grienbergerus, Christophorus
18242:10.1007/978-3-642-36736-6_24
17656:Russian Mathematical Surveys
17546:. McGraw-Hill. p. 183.
17214:A Source Book in Mathematics
17204:Smith, David Eugene (1929).
17134:
17069:
17004:
16935:. London: J. Wale. pp.
16876:
16602:variously attributed to the
16491:are circular, and "pie" and
16100:can carry without buckling:
15783:spherical coordinate systems
15119:Conjugate harmonic functions
15006:probability density function
14891:, once again because of the
11954:is necessary to ensure that
11661:and extends continuously to
11229:of curvature 1 (so that its
10782:probability density function
9665:variational characterization
9342:{\displaystyle f\mapsto f''}
8509:employee used the company's
7110:, in what is now termed the
6555:Modern quest for more digits
6470:{\textstyle {\tfrac {1}{2}}}
6335:{\displaystyle \delta .\pi }
6268:In the earliest usages, the
4478:, also an infinite product:
4259:(click for detail)
3874:{\displaystyle {\sqrt {10}}}
3549:
3421:, and these are called the "
2634:Approximate value and digits
1716:{\displaystyle {\sqrt {10}}}
748:is discussed in the section
7:
23899:Real transcendental numbers
23594:(interactive illustrations)
23392:(in French). Hermann. 1999.
23333:A Guide Book to Mathematics
21825:Excursions in Number Theory
21789:This theorem was proved by
21526:. McGraw-Hill. p. 115.
21419:, Wiley, 1968, pp. 174–190.
21059:"Section 5.5, Exercise 316"
17876:Nieuw Archief voor Wiskunde
17500:The Elements of Mathematics
17495:Die Elemente der Mathematik
16716:
16449:to establish a new form of
15861:One of the key formulae of
15325:approaches zero. The point
15281:can be used to approximate
15105:because it is the simplest
13424:Ehrhart's volume conjecture
11810:Vector calculus and physics
11082:
9152:Many of the appearances of
8489:Between 1998 and 2000, the
5054:accelerated the convergence
4195:{\textstyle 3\times 2^{17}}
4158:{\textstyle 3\times 2^{28}}
2112:periodic continued fraction
1751:cannot be represented as a
1692:{\displaystyle {\sqrt{31}}}
1351:Irrationality and normality
888:{\textstyle {\frac {C}{d}}}
861:{\textstyle {\frac {C}{d}}}
780:is commonly defined as the
685:, sometimes spelled out as
10:
23915:
23424:Remmert, Reinhold (2012).
22380:"Most Pi Places Memorized"
22118:"Pi in the Mandelbrot set"
21823:; Anderson, J. T. (1988).
21169:Isoperimetric inequalities
21122:. Wiley. pp. 286–290.
20827:Mathematics of Computation
19767:Philosophical Transactions
19721:Oughtred, William (1694).
19703:Oughtred, William (1648).
19663:used separately, that is,
18753:Vieta, Franciscus (1593).
18557:10.1142/9789812836267_0001
18518:(in Latin). Archived from
17988:10.1177/002182867800900106
17589:. McGraw-Hill. p. 46.
17356:Oughtred, William (1652).
16383:
15740:from the real line to the
15455:is a two-dimensional real
14455:of integral multiples of 2
14283:also appears naturally in
12677:and squared, the equation
12153:
10712:Schrödinger representation
10037:refer respectively to the
9132:of a vibrating string are
8929:
8298:digit extraction algorithm
7605:integer relation algorithm
6385:to represent the constant
6017:
4340:to 11 digits around 1400.
3539:
3530:
3526:
3088:, and the other (angle or
2628:§ Brouncker's formula
1535:
1136:, or as the solution of a
1002:{\textstyle x^{2}+y^{2}=1}
29:
27:Number, approximately 3.14
23848:
23837:
23635:
23204:General and cited sources
22939:"My Conversion to Tauism"
22499:10.1080/13554790902776896
22389:, Guinness World Records.
22338:Batchelor, G. K. (1967).
22137:10.1142/S0218348X01000828
22116:Klebanoff, Aaron (2001).
21749:10.1515/advgeom-2014-0001
21011:10.1007/978-3-030-03868-7
20463:10.1007/978-3-319-32377-0
19758:Gregorius, David (1695).
19671:used for 'Semiperipheria'
19179:10.1017/S0025557200178404
19150:10.1017/S0025557200175060
18847:10.1017/S0025557200002928
18336:Boyer & Merzbach 1991
18297:Boyer & Merzbach 1991
17777:Beckmann, Peter (1989) .
17571:. McGraw-Hill. p. 2.
17569:Real and complex analysis
17492:Baltzer, Richard (1870).
16278:electromagnetic radiation
16253:In electromagnetics, the
16172:, which approximates the
14893:Stone–von Neumann theorem
14461:Poisson summation formula
14263:appears in characters of
14228:can be obtained from the
12668:Euler–Mascheroni constant
11501:complex analytic function
11363:Cauchy's integral formula
11328:Cauchy's integral formula
11059:spectral characterization
10708:Stone–von Neumann theorem
10422:also contains the number
9994:{\displaystyle \|f\|_{2}}
9781:satisfies the inequality
9722:{\displaystyle H_{0}^{1}}
8963:as units of measurement.
8556:Geometry and trigonometry
8017:Another way to calculate
7210:Rapidly convergent series
6442:
5851:
5384:Some infinite series for
4695:Gottfried Wilhelm Leibniz
4693:and German mathematician
4294:Gottfried Wilhelm Leibniz
3687:Polygon approximation era
1747:As an irrational number,
1731:, it is not possible to "
901:flat (Euclidean) geometry
744:The choice of the symbol
164:Madhava's correction term
23540:31 December 2014 at the
23463:. Walker & Company.
23311:A History of Mathematics
23172:"PEP 628 – Add math.tau"
22916:. Quercus. p. 159.
22385:14 February 2016 at the
22028:Tata Lectures on Theta I
21698:Evans, Lawrence (1997).
21537:Joglekar, S. D. (2005).
20797:, pp. 117, 126–128.
20034:Euler, Leonhard (1922).
19983:Euler, Leonhard (1736).
19855:Euler, Leonhard (1727).
19596:Smith, David E. (1958).
19490:Cajori, Florian (2007).
19363:, Formula 16.10, p. 223.
19203:, pp. 192–196, 205.
18785:Horvath, Miklos (1983).
18545:; Cools, Ronald (eds.).
18418:10.35834/mjms/1312233136
18076:E: The Story of a Number
17229:"π trillion digits of π"
17189:; and by means thereof,
15701:defines a function from
15317:, the result approaches
15250:linear complex structure
14433:Radon–Nikodym derivative
14327:. That is, continuous
13152:Stirling's approximation
12689:, in which the constant
11840:Einstein field equations
11787:where the sum is of the
11340:One of the key tools in
9953:{\displaystyle \nabla f}
9767:isoperimetric inequality
9743:ancient city of Carthage
9349:, and is constrained by
9144:of integer multiples of
8918:is a consequence of the
8737:curves of constant width
7116:Gauss–Legendre algorithm
7089:Toom–Cook multiplication
4228:Richardson extrapolation
3667:have theorized that the
3195:{\displaystyle i^{2}=-1}
3092:) the counter-clockwise
1729:compass and straightedge
1560:compass and straightedge
899:implicitly makes use of
508:. The decimal digits of
506:compass and straightedge
343:Other topics related to
23456:Blatner, David (1999).
23384:English translation of
23082:Arndt & Haenel 2006
22986:Palais, Robert (2001).
22748:The Mathematics Teacher
22693:Math Goes to the Movies
22676:Arndt & Haenel 2006
22664:Arndt & Haenel 2006
22368:Arndt & Haenel 2006
22217:Fundamentals of Physics
21848:Arndt & Haenel 2006
21778:Arndt & Haenel 2006
21616:. Springer. p. 7.
21612:Ehlers, Jürgen (2000).
21459:; Weiss, Guido (1971).
20885:Arndt & Haenel 2006
20795:Arndt & Haenel 2006
20745:Arndt & Haenel 2006
20672:Arndt & Haenel 2006
20652:Arndt & Haenel 2006
20597:Arndt & Haenel 2006
20550:Arndt & Haenel 2006
20427:Eymard & Lafon 2004
20415:Arndt & Haenel 2006
20403:Arndt & Haenel 2006
20391:Arndt & Haenel 2006
20379:Arndt & Haenel 2006
20367:Arndt & Haenel 2006
20309:Arndt & Haenel 2006
20242:Arndt & Haenel 2006
20230:Arndt & Haenel 2006
20215:Arndt & Haenel 2006
20203:Arndt & Haenel 2006
20191:Arndt & Haenel 2006
20109:Arndt & Haenel 2006
20094:Arndt & Haenel 2006
19985:"Ch. 3 Prop. 34 Cor. 1"
19939:English translation in
19810:Arndt & Haenel 2006
19690:Arndt & Haenel 2006
19452:Arndt & Haenel 2006
19361:Arndt & Haenel 2006
19304:Arndt & Haenel 2006
19292:Arndt & Haenel 2006
19227:"Pencil, Paper, and Pi"
19213:Arndt & Haenel 2006
19201:Arndt & Haenel 2006
19080:How Euler Did Even More
19046:Whiteside, Derek Thomas
18957:Arndt & Haenel 2006
18942:Arndt & Haenel 2006
18883:Arndt & Haenel 2006
18814:Eymard & Lafon 2004
18773:Arndt & Haenel 2006
18724:Arndt & Haenel 2006
18682:Arndt & Haenel 2006
18610:Arndt & Haenel 2006
18512:Elementa Trigonometrica
18494:Arndt & Haenel 2006
18482:Arndt & Haenel 2006
18470:Arndt & Haenel 2006
18387:Arndt & Haenel 2006
18372:Arndt & Haenel 2006
18360:Arndt & Haenel 2006
18348:Arndt & Haenel 2006
18321:Arndt & Haenel 2006
18309:Arndt & Haenel 2006
18285:Arndt & Haenel 2006
18273:Arndt & Haenel 2006
18208:Arndt & Haenel 2006
18196:Arndt & Haenel 2006
18114:Arndt & Haenel 2006
17954:Arndt & Haenel 2006
17942:Arndt & Haenel 2006
17859:Arndt & Haenel 2006
17847:Eymard & Lafon 2004
17765:Eymard & Lafon 2004
17741:Arndt & Haenel 2006
17729:Arndt & Haenel 2006
17717:Arndt & Haenel 2006
17705:Arndt & Haenel 2006
17642:Arndt & Haenel 2006
17416:Arndt & Haenel 2006
17286:Arndt & Haenel 2006
16766:is conjectured to be a
16516:Palais de la Découverte
16430:. Poems for memorizing
16191:, moving with velocity
15958:fine-structure constant
13982:locally symmetric space
13951:random number generator
12694:plays an important role
12685:and identities for the
12374:extends the concept of
12053:{\displaystyle \delta }
11947:{\displaystyle 1/2\pi }
11686:the region enclosed by
11365:states that if a point
11322:Chern–Weil homomorphism
8957:trigonometric functions
7599:In 2006, mathematician
6275:was used to denote the
6224:Adoption of the symbol
6206:Ferdinand von Lindemann
6182:Johann Heinrich Lambert
5541:An infinite series for
4326:Madhava of Sangamagrama
4124:The Persian astronomer
3414:{\displaystyle z^{n}=1}
3076:) is used to represent
3066:polar coordinate system
2963:The first 20 digits in
2909:The first 36 digits in
1506:infinite monkey theorem
1440:can be approximated by
751:Adoption of the symbol
673:
562:to seven digits, while
516:, but no proof of this
23211:Abramson, Jay (2014).
22691:; Ross, Marty (2012).
21648:, New York: Springer,
21508:Characteristic classes
21167:Chavel, Isaac (2001).
20619:"Projection Constants"
20158:Math. Tabl. Aids. Comp
19788:10.1098/rstl.1695.0114
19739:Barrow, Isaac (1860).
19724:Key of the Mathematics
19719:(English translation:
19711:The key to mathematics
19599:History of Mathematics
17567:Rudin, Walter (1986).
17538:Rudin, Walter (1976).
17162:
16847:
16582:In the United States,
16504:
16400:Guinness World Records
16371:
16245:
16152:
16093:area moment of inertia
16045:
15940:
15853:
15734:
15695:
15650:
15603:
15552:
15551:{\displaystyle f\in V}
15526:
15485:
15443:
15385:
15286:
15232:
15131:Cauchy principal value
15094:, which also involves
15078:
14998:
14928:
14916:
14869:
14759:
14713:
14664:
14604:
14561:
14480:
14408:
14276:
14230:functional determinant
14224:A consequence is that
14216:
14149:
13941:
13759:
13698:
13606:
13493:
13459:
13415:
13246:
13144:
13065:
12975:
12854:
12753:−1)-dimensional sphere
12687:functional determinant
12656:
12610:
12532:
12473:
12422:
12358:
12334:
12276:
12179:
12139:
12054:
12034:
12005:
11968:
11948:
11917:
11781:
11638:
11530:at any interior point
11446:
11337:
11318:characteristic classes
11307:
11208:
11141:. Specifically, if a
11122:
11032:
10946:
10910:
10761:
10686:
10382:
10261:
10201:
10031:
9995:
9954:
9921:
9817:
9754:
9723:
9639:
9469:
9427:Wirtinger's inequality
9351:Sturm–Liouville theory
9343:
9305:
9250:
9149:
9142:arithmetic progression
9111:
8952:
8932:Units of angle measure
8904:
8884:
8845:
8726:−1)-dimensional sphere
8577:
8461:
8333:
8232:
8102:
8087:
8014:
7954:
7793:(Gelfond's constant),
7773:
7651:
7572:
7472:
7388:
7305:
7226:
7167:
7147:Motives for computing
7036:
7029:
6940:
6798:
6677:
6521:
6471:
6416:
6377:
6346:
6336:
6172:
5968:
5829:
5679:
5518:
5405:Gregory–Leibniz series
5331:
5245:
5029:
4947:
4901:
4875:
4849:
4826:Gregory–Leibniz series
4816:
4686:
4662:
4462:
4261:
4196:
4159:
4115:
4055:The Indian astronomer
4034:, which he termed the
4028:
3989:
3950:
3875:
3714:
3699:
3657:
3518:
3415:
3367:
3321:
3272:
3196:
3152:
3049:
2614:
2020:
1717:
1693:
1646:
1586:coefficients, such as
1563:
1473:, including tests for
1471:statistical randomness
1416:decimal representation
1281:
1081:
1003:
934:
889:
862:
833:
774:
556:Chinese mathematicians
482:decimal representation
478:used to approximate it
470:
59:mathematical constant
34:. For other uses, see
23315:(2 ed.). Wiley.
22600:Pickover, Clifford A.
22450:10.1163/9789004433397
22008:Dym & McKean 1972
21795:Hardy, G. H. (2008).
21562:Schey, H. M. (1996).
21444:Dym & McKean 1972
21133:Dym & McKean 1972
20889:Bellards formula in:
19467:. 20 (1882), 213–225.
19373:Wells, David (1997).
19064:Sandifer, Ed (2009).
19033:Roy, Ranjan (2021) .
18826:Cooker, M.J. (2011).
18228:(2014). "The life of
18165:Plofker, Kim (2009).
17719:, pp. 22, 28–30.
17260:Google Cloud Platform
17210:for the Circle Ratio"
17163:
16848:
16701:programming languages
16486:
16372:
16246:
16153:
16084:modulus of elasticity
16046:
15941:
15854:
15735:
15696:
15651:
15649:{\displaystyle e_{t}}
15604:
15553:
15527:
15486:
15444:
15386:
15276:
15269:In the Mandelbrot set
15233:
15079:
14999:
14922:
14906:
14870:
14760:
14714:
14641:
14612:Jacobi theta function
14610:. An example is the
14605:
14562:
14515:holomorphic functions
14474:
14409:
14259:
14217:
14150:
13986:hyperbolic 3-manifold
13942:
13745:
13684:
13607:
13517:it can be written as
13498:Riemann zeta function
13465:
13445:
13416:
13247:
13145:
13066:
12976:
12855:
12747:of its boundary, the
12683:Riemann zeta function
12657:
12590:
12533:
12474:
12423:
12359:
12332:
12277:
12165:
12140:
12055:
12035:
12006:
11969:
11967:{\displaystyle \Phi }
11949:
11918:
11782:
11639:
11447:
11335:
11308:
11209:
11090:
11051:central limit theorem
11033:
10947:
10911:
10798:, naturally contains
10729:
10702:. The appearance of
10687:
10383:
10256:
10202:
10053:Poincaré inequalities
10032:
9996:
9955:
9922:
9818:
9740:
9724:
9640:
9470:
9344:
9306:
9251:
9197:differential equation
9177:on the unit interval
9127:
9112:
8939:
8905:
8885:
8846:
8747:times its width. The
8563:
8491:distributed computing
8462:
8313:
8242:central limit theorem
8233:
8146:can be calculated by
8138:defines a (discrete)
8123:binomial distribution
8103:
8067:
7976:
7955:
7911:based on the counts:
7847:are dropped randomly.
7774:
7631:
7594:Ramanujan–Sato series
7573:
7452:
7420:developed in 1987 is
7389:
7285:
7217:
7157:
7030:
6946:Then an estimate for
6941:
6799:
6678:
6564:
6472:
6417:
6378:
6337:
6198:Adrien-Marie Legendre
6173:
6054:Riemann zeta function
5969:
5830:
5680:
5519:
5332:
5246:
5030:
4948:
4902:
4876:
4850:
4817:
4673:
4663:
4463:
4241:
4220:Christoph Grienberger
4197:
4160:
4116:
4040:(''close ratio") and
4029:
3990:
3951:
3876:
3705:
3694:
3669:Great Pyramid of Giza
3658:
3560:before the Common Era
3519:
3416:
3368:
3322:
3273:
3197:
3153:
3080:'s distance from the
3022:
2615:
2021:
1718:
1694:
1655:The transcendence of
1647:
1572:transcendental number
1552:transcendental number
1545:
1522:mathematical folklore
1446:irrationality measure
1364:ratio of two integers
1282:
1143:In a similar spirit,
1138:differential equation
1106:differential calculus
1082:
1004:
961:Cartesian coordinates
935:
895:. This definition of
890:
863:
834:
768:
731:product of a sequence
651:mathematical analysis
564:Indian mathematicians
490:transcendental number
471:
413:mathematical constant
117:Use in other formulae
23552:2 April 2015 at the
23483:Le fascinant nombre
23409:. Prometheus Books.
23338:Verlag Harri Deutsch
23094:Mathematics Magazine
22273:Quantum Field Theory
21727:Advances in Geometry
21587:Yeo, Adrian (2006).
21539:Mathematical Physics
21373:Howe, Roger (1980).
20902:on 12 September 2007
20447:Borwein, Jonathan M.
20217:, pp. 132, 140.
20013:10 June 2016 at the
19903:10 June 2016 at the
19536:Mathematics Magazine
19241:10.1511/2014.110.342
19167:Mathematical Gazette
19138:Mathematical Gazette
19123:: 133–149, 167–168.
18835:Mathematical Gazette
18642:Mathematics Magazine
18629:Roy, Ranjan (1990).
18595:: 83–93 – via
18226:Borwein, Jonathan M.
18210:, pp. 175, 205.
18168:Mathematics in India
17607:. Springer. §VIII.2.
16970:
16783:
16681:. For instance, the
16284:
16214:
16104:
15969:
15899:
15819:
15742:real projective line
15705:
15666:
15633:
15562:
15536:
15495:
15467:
15398:
15359:
15140:
15012:
14938:
14780:
14730:
14617:
14571:
14528:
14497:Chudnovsky algorithm
14495:. For example, the
14357:
14242:variational approach
14161:
14024:
13677:
13521:
13430:containing only one
13295:
13158:
13090:
12991:
12863:
12759:
12549:
12483:
12436:
12382:
12345:
12220:
12079:
12062:Dirac delta function
12044:
12015:
11986:
11976:fundamental solution
11958:
11927:
11854:
11702:
11684:meromorphic function
11547:
11513:on the Jordan curve
11384:
11241:
11223:Euler characteristic
11160:
11131:Gauss–Bonnet formula
10970:
10922:
10806:
10430:
10286:
10239:Dirichlet eigenvalue
10071:
10005:
9972:
9941:
9870:
9785:
9773:enclosed by a plane
9686:
9509:
9433:
9322:
9260:
9202:
9138:harmonic progression
9010:
8894:
8857:
8777:
8753:equilateral triangle
8649:and semi-minor axis
8304:
8150:
8051:
7915:
7895:units apart, and if
7615:
7424:
7250:
7108:Carl Friedrich Gauss
7069:iterative algorithms
6954:
6807:
6687:
6576:
6452:
6399:
6360:
6320:
6293:or semidiameter) or
6200:proved in 1794 that
6184:in 1768 proved that
6062:
5863:
5722:
5693:Infinite series for
5549:
5411:
5362:Carl Friedrich Gauss
5261:
5063:
5043:Machin-like formulae
4968:
4922:
4885:
4859:
4855:when evaluated with
4832:
4728:
4482:
4367:
4173:
4136:
4084:
3999:
3960:
3928:
3861:
3727:method of exhaustion
3613:
3435:
3392:
3335:
3305:
3222:
3212:exponential function
3170:
3100:
2126:
2104:quadratic irrational
1767:
1703:
1674:
1590:
1578:of any non-constant
1433:reductio ad absurdum
1366:. Fractions such as
1291:with this property.
1179:
1116:, is the following:
1017:
967:
911:
872:
845:
810:
514:randomly distributed
480:. Consequently, its
451:
52:a series of articles
23894:Mathematical series
23479:Delahaye, Jean-Paul
23288:. Springer-Verlag.
23276:Berggren, Lennart;
23253:. Springer-Verlag.
22313:Low, Peter (1971).
21946:2015JMP....56k2101F
21702:. AMS. p. 615.
21315:1960ArRMA...5..286P
21084:Section 5.1: Angles
20987:, pp. 200, 209
20839:1997MaCom..66..903B
20405:, pp. 104, 206
20323:The Washington Post
19829:Cursus Mathematicus
19816:, pp. 108–109.
19779:1695RSPT...19..637G
19278:1988SciAm.258b.112B
19266:Scientific American
18944:, pp. 192–193.
18684:, pp. 185–186.
18589:De Zeventiende Eeuw
18525:on 1 February 2014.
18484:, pp. 182–183.
18323:, pp. 176–177.
17980:1978JHA.....9...65K
17668:2008RuMaS..63..570S
17479:Mathematische Werke
16809:
16669:In computer culture
16651:Indiana legislature
16451:constrained writing
16266:Maxwell's equations
16255:vacuum permeability
15787:classical mechanics
15760:Outside mathematics
15345:Projective geometry
15188:
15107:Furstenberg measure
15032:
14933:Cauchy distribution
14927:through a membrane.
14875:which implies that
14329:group homomorphisms
14240:, specifically the
14234:harmonic oscillator
12985:functional equation
12543:Weierstrass product
12237:
11844:Newtonian potential
11836:Maxwell's equations
11454:Although the curve
11346:contour integration
11231:radius of curvature
10990:
10960:change of variables
10774:normal distribution
10772:frequently use the
10631:
10529:
10455:
10330:
9703:
9590:
9536:
8994:, so for any angle
8920:Pythagorean theorem
8797:
8763:of constant width.
7867:Monte Carlo methods
7821:Monte Carlo methods
7414:Chudnovsky brothers
7236:Srinivasa Ramanujan
7219:Srinivasa Ramanujan
7180:observable universe
7085:Karatsuba algorithm
6347:Clavis Mathematicae
6008:Chudnovsky's series
5380:Rate of convergence
4900:{\displaystyle z=1}
4874:{\displaystyle z=1}
4316:Nilakantha Somayaji
4264:The calculation of
4216:Willebrord Snellius
4046:Liu Hui's algorithm
3840:Apollonius of Perga
3834:, gave a value for
3725:, implementing the
3564:Chinese mathematics
3287:is the base of the
2574:
2555:
2541:
2522:
2508:
2489:
2475:
2463:
2423:
2404:
2390:
2371:
2357:
2338:
2324:
2312:
2272:
2253:
2239:
2220:
2206:
2187:
2173:
2154:
2102:and so cannot be a
1962:
1950:
1935:
1923:
1908:
1896:
1881:
1869:
1854:
1842:
1827:
1815:
1800:
1788:
1743:Continued fractions
1737:classical antiquity
1580:polynomial equation
1556:squaring the circle
1149:complex exponential
1124:function equals 0.
1043:
733:, analogous to how
695:pronounced as "pie"
595:computer scientists
545:Greek mathematician
502:squaring the circle
325:Squaring the circle
260:Chudnovsky brothers
250:Srinivasa Ramanujan
36:Pi (disambiguation)
23629:Irrational numbers
23513:Weisstein, Eric W.
23426:"Ch. 5 What is π?"
23130:(3 October 1990).
23084:, pp. 211–212
23011:10.1007/BF03026846
22718:"Review of Aerial"
22597:as examples. See:
22564:. Vinculum Press.
22246:College Physics 2e
22146:on 27 October 2011
21685:The Gamma Function
21670:, pp. 191–192
21502:Wiley Interscience
21323:10.1007/BF00252910
21206:10.1007/BF02418013
21101:, pp. 210–211
20659:, pp. 105–108
20417:, pp. 110–111
20381:, pp. 103–104
19706:Clavis Mathematicæ
19701:See, for example,
19416:, pp. 129–140
19232:American Scientist
19215:, pp. 194–196
18913:10.1007/BF00384331
18612:, pp. 185–191
18074:Maor, Eli (2009).
17605:Topologie generale
17324:10.1007/BF03024340
17158:
17131:
17109:
17087:
17066:
17044:
17022:
17001:
16986:
16843:
16792:
16723:Approximations of
16683:computer scientist
16505:
16479:In popular culture
16367:
16241:
16185:objects of radius
16181:exerted on small,
16148:
16041:
15936:
15849:
15730:
15691:
15646:
15621:. Then, for each
15615:at the real point
15599:
15548:
15522:
15481:
15461:initial conditions
15439:
15381:
15287:
15228:
15171:
15113:associated with a
15074:
15015:
14994:
14929:
14925:Brownian particles
14917:
14865:
14755:
14709:
14600:
14557:
14481:
14404:
14319:unitary characters
14289:periodic functions
14277:
14212:
14145:
13937:
13935:
13602:
13494:
13460:
13411:
13242:
13182:
13150:which is known as
13140:
13061:
12971:
12850:
12652:
12528:
12469:
12418:
12357:{\displaystyle n!}
12354:
12335:
12295:index of the curve
12272:
12223:
12180:
12135:
12050:
12030:
12001:
11964:
11944:
11913:
11777:
11634:
11442:
11338:
11303:
11204:
11133:which relates the
11123:
11121:, by Gauss–Bonnet.
11028:
10973:
10942:
10940:
10906:
10793:standard deviation
10762:
10722:Gaussian integrals
10696:quantum mechanical
10682:
10614:
10512:
10438:
10378:
10313:
10274:integral transform
10262:
10257:An animation of a
10197:
10027:
9991:
9950:
9917:
9852:Sobolev inequality
9813:
9755:
9719:
9689:
9635:
9576:
9522:
9465:
9425:, which satisfies
9339:
9301:
9246:
9150:
9107:
8953:
8900:
8880:
8841:
8780:
8767:Definite integrals
8578:
8457:
8271:. They are called
8228:
8174:
8140:stochastic process
8108:so that, for each
8098:
8015:
7950:
7769:
7568:
7418:Chudnovsky formula
7384:
7227:
7168:
7025:
7024:
6936:
6935:
6794:
6793:
6673:
6672:
6467:
6465:
6412:
6373:
6332:
6168:
5964:
5825:
5675:
5514:
5327:
5322:
5298:
5274:
5241:
5025:
4943:
4897:
4871:
4845:
4812:
4687:
4658:
4458:
4262:
4224:Christiaan Huygens
4208:Ludolph van Ceulen
4204:Adriaan van Roomen
4192:
4155:
4111:
4024:
3985:
3946:
3871:
3715:
3700:
3681:Indian mathematics
3653:
3533:Approximations of
3514:
3411:
3363:
3317:
3297:and points on the
3268:
3192:
3148:
3050:
2855:2) digits (called
2640:approximations of
2610:
2608:
2602:
2597:
2592:
2587:
2569:
2536:
2503:
2470:
2451:
2446:
2441:
2436:
2418:
2385:
2352:
2319:
2300:
2295:
2290:
2285:
2267:
2234:
2201:
2168:
2016:
2015:
2011:
2009:
2005:
2003:
1999:
1997:
1993:
1991:
1987:
1985:
1981:
1979:
1975:
1957:
1930:
1903:
1876:
1849:
1822:
1795:
1761:continued fraction
1713:
1689:
1642:
1564:
1277:
1077:
1026:
999:
930:
885:
858:
829:
775:
729:, which denotes a
466:
464:
407:; spelled out as "
215:Ludolph van Ceulen
23871:
23870:
23772:Supersilver ratio
23737:Supergolden ratio
23697:Twelfth root of 2
23470:978-0-8027-7562-7
23439:978-1-4612-1005-4
23416:978-1-59102-200-8
23387:Autour du nombre
23379:978-0-8218-3246-2
23359:. Academic Press.
23347:978-3-87144-095-3
23322:978-0-471-54397-8
23295:978-0-387-20571-7
23286:Pi: a Source Book
23278:Borwein, Jonathan
23260:978-3-540-66572-4
23239:978-0-521-78988-2
23229:Special Functions
22923:978-1-62365-411-5
22873:Griffin, Andrew.
22811:. 14 March 2015.
22790:978-1-139-50530-7
22702:978-1-421-40484-4
22641:978-0-88385-537-9
22619:978-0-471-11857-2
22571:978-0-9630097-1-5
22440:. Brill. p.
22324:978-0-521-08089-7
22283:978-0-486-44568-7
22201:978-0-521-83186-4
22176:978-0-387-20229-7
22080:978-0-8284-0324-5
22037:978-3-7643-3109-2
21986:978-0-9502734-2-6
21954:10.1063/1.4930800
21806:978-0-19-921986-5
21623:978-3-540-67073-5
21598:978-981-270-078-0
21548:978-81-7371-422-1
21146:Thompson, William
21020:978-3-030-03866-3
20999:Oliveros, Déborah
20812:Borwein, Peter B.
20674:, pp. 77–84.
20472:978-3-319-32375-6
20257:978-0-471-31515-5
20193:, pp. 15–17.
19609:978-0-486-20430-7
19503:978-1-60206-714-1
19384:978-0-14-026149-3
19306:, pp. 69–72.
18759:. Vol. VIII.
18707:978-0-691-13526-7
18566:978-981-283-625-0
18543:Bultheel, Adhemar
18251:978-3-642-36735-9
18139:978-0-88920-324-2
18085:978-0-691-14134-3
18025:978-0-88385-613-0
17817:978-0-7876-3933-4
17788:978-0-88029-418-8
17707:, pp. 22–23.
17622:Bourbaki, Nicolas
17601:Bourbaki, Nicolas
17553:978-0-07-054235-8
17471:Weierstrass, Karl
17253:(14 March 2019).
17251:Haruka Iwao, Emma
17137:
17130:
17108:
17086:
17072:
17065:
17043:
17021:
17007:
17000:
16985:
16865:expansion of the
16838:
16750:Explanatory notes
16659:square the circle
16380:Memorizing digits
16356:
16325:
16203:dynamic viscosity
16143:
16025:
16016:
15980:
15931:
15863:quantum mechanics
15844:
15843:
15771:physical constant
15391:that satisfy the
15313:is multiplied by
15289:An occurrence of
15223:
15169:
15135:singular integral
15123:Hilbert transform
15117:in a half-plane.
15056:
14992:
14964:
14238:quantum mechanics
14207:
14123:
14102:
14094:
14070:
13919:
13899:
13868:
13859:
13839:
13786:
13725:
13594:
13574:
13554:
13406:
13405:
13372:
13237:
13167:
13154:. Equivalently,
13128:
13113:
13056:
12950:
12936:
12835:
12821:
12720:-dimensional ball
12650:
12588:
12526:
12520:
12467:
12130:
11887:
11826:, for example in
11818:is ubiquitous in
11594:
11519:and the value of
11476:does not contain
11425:
11026:
10964:Gaussian integral
10939:
10938:
10844:
10841:
10778:Gaussian function
10732:Gaussian function
10612:
10554:
10298:
10270:Fourier transform
10195:
9651:is a multiple of
9503:square integrable
9429:: for a function
9413:is, in fact, the
9355:negative definite
9059:
8903:{\displaystyle x}
8878:
8836:
8816:
8749:Reuleaux triangle
8741:Barbier's theorem
8718:-dimensional ball
8499:Bellard's formula
8447:
8423:
8399:
8375:
8349:
8273:spigot algorithms
8263:Spigot algorithms
8223:
8159:
7945:
7759:
7728:
7697:
7667:
7563:
7450:
7446:
7435:
7379:
7283:
7277:
7261:
7244:modular equations
7043:. Mathematicians
7019:
6788:
6740:
6650:
6623:
6622:
6551:as late as 1761.
6513:in his 1736 work
6499:, though he used
6464:
6439:in his 1706 work
6410:
6371:
6160:
6140:
6120:
6100:
6080:
6032:is transcendental
5992:
5991:
5956:
5929:
5902:
5817:
5804:
5791:
5778:
5765:
5752:
5739:
5667:
5640:
5613:
5586:
5506:
5493:
5480:
5467:
5454:
5441:
5428:
5321:
5297:
5273:
5233:
5188:
5159:
5114:
5101:
5020:
5001:
4979:
4941:
4940:
4843:
4804:
4784:
4764:
4646:
4633:
4606:
4593:
4566:
4553:
4526:
4513:
4493:
4453:
4449:
4447:
4445:
4418:
4414:
4412:
4393:
4389:
4378:
4103:
4099:
4022:3.142857142857...
4016:
3977:
3869:
3631:
3289:natural logarithm
2624:William Brouncker
2604:
2599:
2594:
2589:
2573:
2554:
2540:
2521:
2507:
2488:
2474:
2462:
2453:
2448:
2443:
2438:
2422:
2403:
2389:
2370:
2356:
2337:
2323:
2311:
2302:
2297:
2292:
2287:
2271:
2252:
2238:
2219:
2205:
2186:
2172:
2153:
2013:
2007:
2001:
1995:
1989:
1983:
1977:
1961:
1949:
1934:
1922:
1907:
1895:
1880:
1868:
1853:
1841:
1826:
1814:
1799:
1787:
1733:square the circle
1711:
1687:
1628:
1608:
1460:Liouville numbers
1420:repeating pattern
1360:irrational number
1072:
1071:
928:
883:
856:
827:
585:The invention of
463:
445:irrational number
382:
381:
16:(Redirected from
23906:
23889:Complex analysis
23843:
23831:
23821:
23809:Square root of 7
23804:Square root of 6
23799:
23782:Square root of 5
23777:
23767:Square root of 3
23762:
23752:
23742:
23732:Square root of 2
23725:
23710:
23692:
23660:
23645:
23622:
23615:
23608:
23599:
23598:
23580:
23579:
23573:
23569:
23563:
23532:
23526:
23525:
23499:
23486:
23474:
23461:
23443:
23420:
23408:
23404:
23394:
23393:
23390:
23383:
23370:
23360:
23351:
23326:
23314:
23305:Merzbach, Uta C.
23303:Boyer, Carl B.;
23299:
23271:
23269:
23267:
23243:
23222:
23197:
23196:
23194:
23192:
23182:
23176:
23175:
23168:
23162:
23161:
23159:
23157:
23151:
23136:
23124:
23118:
23117:
23089:, pp. 36–37
23079:
23073:
23072:
23070:
23068:
23058:
23052:
23051:
23050:on 13 July 2013.
23036:
23030:
23029:
23027:
22996:
22991:
22983:
22977:
22976:
22974:
22943:
22934:
22928:
22927:
22909:
22901:
22895:
22894:
22892:
22890:
22870:
22864:
22863:
22831:
22825:
22824:
22822:
22820:
22801:
22795:
22794:
22771:
22765:
22764:
22742:
22736:
22735:
22713:
22707:
22706:
22689:Polster, Burkard
22685:
22679:
22673:
22667:
22656:
22650:
22649:
22623:
22605:Keys to Infinity
22582:
22576:
22575:
22557:
22551:
22550:
22548:
22546:
22527:
22521:
22520:
22510:
22478:
22472:
22471:
22437:
22429:
22423:
22422:
22420:
22418:
22396:
22390:
22377:
22371:
22370:, pp. 44–45
22365:
22354:
22353:
22335:
22329:
22328:
22310:
22304:
22303:
22260:
22254:
22253:
22237:
22231:
22230:
22212:
22206:
22205:
22187:
22181:
22180:
22162:
22156:
22155:
22153:
22151:
22145:
22139:. Archived from
22122:
22113:
22104:
22102:
22091:
22085:
22084:
22063:
22057:
22056:
22048:
22042:
22041:
22020:
22011:
22005:
21999:
21998:
21972:
21966:
21965:
21939:
21919:
21913:
21912:
21894:
21872:
21866:
21865:
21857:
21851:
21845:
21839:
21838:
21817:
21811:
21810:
21787:
21781:
21780:, pp. 41–43
21775:
21769:
21768:
21742:
21722:
21716:
21710:
21704:
21703:
21695:
21689:
21688:
21677:
21671:
21665:
21659:
21658:
21637:
21628:
21627:
21609:
21603:
21602:
21584:
21578:
21577:
21559:
21553:
21552:
21534:
21528:
21527:
21524:Complex analysis
21516:
21510:
21505:
21491:
21485:
21483:
21472:
21466:
21464:
21453:
21447:
21441:
21435:
21429:
21420:
21413:
21407:
21406:
21396:
21370:
21361:
21360:
21349:
21343:
21342:
21298:
21292:
21291:
21273:
21253:
21247:
21246:
21244:
21232:
21226:
21225:
21199:
21179:
21173:
21172:
21164:
21158:
21157:
21142:
21136:
21130:
21124:
21123:
21116:Courant, Richard
21108:
21102:
21096:
21087:
21077:
21071:
21070:
21050:
21044:
21040:
20994:
20988:
20982:
20976:
20975:
20973:
20963:
20959:
20955:
20949:
20943:
20942:
20940:
20938:
20918:
20912:
20911:
20909:
20907:
20898:. Archived from
20892:Bellard, Fabrice
20882:
20876:
20875:
20873:
20850:
20833:(218): 903–913.
20824:
20808:Bailey, David H.
20804:
20798:
20792:
20783:
20782:
20754:
20748:
20742:
20733:
20732:
20713:10.2307/27641917
20696:
20684:
20675:
20669:
20660:
20649:
20643:
20642:
20640:
20611:
20605:
20599:, pp. 39–40
20594:
20588:
20587:
20559:
20553:
20547:
20541:
20540:
20538:
20536:
20530:
20523:
20511:
20505:
20502:
20496:
20495:
20483:
20477:
20476:
20443:Bailey, David H.
20439:
20430:
20424:
20418:
20412:
20406:
20400:
20394:
20388:
20382:
20376:
20370:
20364:
20358:
20357:
20355:
20353:
20333:
20327:
20326:
20318:
20312:
20311:, pp. 17–19
20306:
20300:
20299:
20297:
20295:
20289:
20282:
20273:
20262:
20261:
20239:
20233:
20227:
20218:
20212:
20206:
20200:
20194:
20188:
20182:
20181:
20153:
20147:
20146:
20118:
20112:
20106:
20097:
20091:
20085:
20084:
20082:
20078:
20064:
20058:
20057:
20055:
20053:
20031:
20025:
20023:
20006:
20003:
19980:
19974:
19973:
19970:
19938:
19936:
19918:
19912:
19910:
19896:
19894:
19888:
19886:
19880:
19861:
19852:
19846:
19845:
19843:
19841:
19823:
19817:
19807:
19801:
19800:
19790:
19773:(231): 637–652.
19764:
19755:
19749:
19748:
19736:
19730:
19728:
19718:
19699:
19693:
19687:
19674:
19673:
19666:
19632:
19620:
19614:
19613:
19593:
19584:
19582:
19579:used the letter
19577:William Oughtred
19573:
19531:
19525:
19524:
19521:
19511:
19487:
19478:
19474:
19468:
19461:
19455:
19449:
19443:
19442:
19423:
19417:
19410:
19404:
19398:
19389:
19388:
19370:
19364:
19358:
19352:
19351:
19341:
19313:
19307:
19301:
19295:
19289:
19261:
19252:
19251:
19249:
19247:
19222:
19216:
19210:
19204:
19198:
19192:
19190:
19173:(516): 469–470.
19161:
19144:(512): 270–278.
19135:
19128:
19104:
19083:
19076:
19073:How Euler Did It
19070:
19061:
19055:
19053:
19038:
19030:
19024:
19023:
19021:
19019:
19013:
19007:. Archived from
18982:
18977:
18966:
18960:
18959:, pp. 72–74
18954:
18945:
18939:
18933:
18932:
18898:
18892:
18886:
18880:
18874:
18873:
18871:
18869:
18863:
18857:. Archived from
18841:(533): 218–226.
18832:
18823:
18817:
18816:, pp. 53–54
18811:
18802:
18801:
18791:
18782:
18776:
18770:
18761:
18760:
18750:
18744:
18743:
18733:
18727:
18721:
18712:
18711:
18691:
18685:
18679:
18673:
18672:
18670:
18668:
18663:on 14 March 2023
18662:
18656:. Archived from
18639:
18634:
18626:
18613:
18607:
18601:
18600:
18581:Yoder, Joella G.
18577:
18571:
18570:
18538:
18532:
18530:
18526:
18524:
18517:
18503:
18497:
18491:
18485:
18479:
18473:
18467:
18456:
18455:
18453:
18451:
18429:
18423:
18422:
18420:
18396:
18390:
18384:
18375:
18369:
18363:
18357:
18351:
18345:
18339:
18333:
18324:
18318:
18312:
18306:
18300:
18294:
18288:
18282:
18276:
18270:
18264:
18263:
18231:
18222:
18211:
18205:
18199:
18193:
18187:
18186:
18162:
18156:
18155:
18153:
18151:
18123:
18117:
18111:
18102:
18096:
18090:
18089:
18071:
18065:
18059:
18050:
18040:
18034:
18033:
18005:Jamshīd al-Kāshī
17999:
17963:
17957:
17951:
17945:
17939:
17933:
17932:
17904:
17898:
17892:
17891:
17871:
17862:
17856:
17850:
17844:
17835:
17833:
17831:
17829:
17809:
17799:
17793:
17792:
17774:
17768:
17762:
17756:
17750:
17744:
17738:
17732:
17726:
17720:
17714:
17708:
17702:
17696:
17695:
17651:
17645:
17639:
17630:
17629:
17618:
17609:
17608:
17597:
17591:
17590:
17587:Complex analysis
17579:
17573:
17572:
17564:
17558:
17557:
17545:
17535:
17526:
17525:
17514:
17508:
17507:
17489:
17483:
17482:
17468:, p. 148.
17463:
17457:
17451:
17440:
17439:
17425:
17419:
17413:
17402:
17401:
17399:
17397:
17382:
17376:
17375:
17372:
17353:
17344:
17343:
17317:
17295:
17289:
17283:
17277:
17276:
17274:
17272:
17247:
17241:
17240:
17225:
17219:
17217:
17209:
17200:
17197:
17178:
17167:
17165:
17164:
17159:
17138:
17133:
17132:
17129:
17128:
17116:
17110:
17107:
17106:
17094:
17090:
17088:
17079:
17073:
17068:
17067:
17064:
17063:
17051:
17045:
17042:
17041:
17029:
17025:
17023:
17014:
17008:
17003:
17002:
16993:
16987:
16978:
16974:
16923:
16912:
16911:
16899:
16893:
16887:
16870:
16859:
16853:
16852:
16850:
16849:
16844:
16839:
16837:
16836:
16835:
16819:
16811:
16808:
16803:
16777:
16771:
16765:
16760:
16734:
16726:
16711:
16706:
16695:
16680:
16675:internet culture
16673:In contemporary
16664:
16645:
16641:
16633:
16628:
16617:
16589:
16578:
16573:Wolf in the Fold
16554:
16546:
16525:
16510:
16494:
16474:
16466:
16456:
16448:
16437:
16433:
16417:
16406:
16396:
16376:
16374:
16373:
16368:
16363:
16362:
16357:
16354:
16351:
16350:
16326:
16323:
16321:
16320:
16296:
16295:
16250:
16248:
16247:
16242:
16209:
16196:
16190:
16180:
16174:frictional force
16167:
16157:
16155:
16154:
16149:
16144:
16142:
16141:
16132:
16125:
16124:
16114:
16099:
16090:
16081:
16075:
16065:
16059:
16050:
16048:
16047:
16042:
16037:
16036:
16027:
16026:
16023:
16017:
16015:
16007:
16000:
15999:
15989:
15981:
15973:
15964:
15954:orthopositronium
15951:
15945:
15943:
15942:
15937:
15932:
15930:
15919:
15890:
15884:
15874:
15858:
15856:
15855:
15850:
15845:
15836:
15835:
15810:
15804:
15794:
15780:
15776:
15755:
15751:
15747:
15739:
15737:
15736:
15731:
15720:
15712:
15700:
15698:
15697:
15692:
15690:
15689:
15661:
15655:
15653:
15652:
15647:
15645:
15644:
15620:
15614:
15609:of the function
15608:
15606:
15605:
15600:
15574:
15573:
15557:
15555:
15554:
15549:
15531:
15529:
15528:
15523:
15521:
15507:
15506:
15490:
15488:
15487:
15482:
15480:
15454:
15448:
15446:
15445:
15440:
15408:
15390:
15388:
15387:
15382:
15380:
15372:
15354:
15340:
15336:
15332:
15324:
15320:
15316:
15312:
15304:
15292:
15284:
15264:
15260:
15243:
15237:
15235:
15234:
15229:
15224:
15222:
15211:
15190:
15187:
15182:
15170:
15162:
15121:and so also the
15109:, the classical
15103:potential theory
15097:
15093:
15083:
15081:
15080:
15075:
15057:
15055:
15048:
15047:
15034:
15031:
15026:
15003:
15001:
15000:
14995:
14993:
14991:
14984:
14983:
14970:
14965:
14957:
14890:
14881:Heisenberg group
14878:
14874:
14872:
14871:
14866:
14843:
14842:
14774:automorphic form
14771:
14764:
14762:
14761:
14756:
14754:
14753:
14718:
14716:
14715:
14710:
14708:
14707:
14703:
14702:
14663:
14658:
14609:
14607:
14606:
14601:
14596:
14588:
14587:
14582:
14566:
14564:
14563:
14558:
14553:
14545:
14544:
14539:
14519:upper half plane
14486:
14458:
14442:
14438:
14430:
14422:
14413:
14411:
14410:
14405:
14403:
14402:
14369:
14368:
14352:
14346:
14336:
14326:
14316:
14310:
14304:
14282:
14262:
14227:
14221:
14219:
14218:
14213:
14208:
14200:
14183:
14154:
14152:
14151:
14146:
14121:
14100:
14099:
14095:
14090:
14082:
14068:
14067:
14066:
14051:
14050:
14019:
14012:
13963:
13956:
13946:
13944:
13943:
13938:
13936:
13920:
13918:
13917:
13905:
13900:
13898:
13881:
13873:
13869:
13867:
13860:
13858:
13857:
13845:
13840:
13838:
13837:
13825:
13813:
13805:
13801:
13800:
13792:
13788:
13787:
13785:
13784:
13783:
13761:
13758:
13753:
13731:
13727:
13726:
13724:
13723:
13711:
13697:
13692:
13672:
13668:1 − 1/
13665:
13658:
13651:
13641:
13636:relatively prime
13629:
13611:
13609:
13608:
13603:
13595:
13593:
13592:
13580:
13575:
13573:
13572:
13560:
13555:
13553:
13552:
13540:
13516:
13509:
13491:
13489:
13476:
13420:
13418:
13417:
13412:
13407:
13395:
13394:
13393:
13378:
13373:
13371:
13363:
13362:
13361:
13339:
13331:
13330:
13290:
13283:
13277: + 1)Δ
13265:standard simplex
13262:
13251:
13249:
13248:
13243:
13238:
13236:
13235:
13234:
13212:
13211:
13210:
13198:
13197:
13184:
13181:
13149:
13147:
13146:
13141:
13139:
13138:
13133:
13129:
13121:
13114:
13103:
13085:
13079:
13070:
13068:
13067:
13062:
13057:
13055:
13045:
13044:
13034:
13024:
13023:
13007:
12980:
12978:
12977:
12972:
12967:
12966:
12951:
12949:
12948:
12944:
12937:
12929:
12918:
12917:
12916:
12912:
12895:
12881:
12880:
12859:
12857:
12856:
12851:
12846:
12845:
12836:
12834:
12833:
12829:
12822:
12814:
12803:
12802:
12798:
12785:
12771:
12770:
12746:
12714:
12692:
12680:
12676:
12670:. Evaluated at
12665:
12661:
12659:
12658:
12653:
12651:
12649:
12645:
12630:
12629:
12625:
12612:
12609:
12604:
12589:
12584:
12583:
12568:
12537:
12535:
12534:
12529:
12527:
12522:
12521:
12516:
12510:
12499:
12478:
12476:
12475:
12470:
12468:
12463:
12452:
12431:
12427:
12425:
12424:
12419:
12369:
12363:
12361:
12360:
12355:
12287:
12281:
12279:
12278:
12273:
12236:
12231:
12177:
12169:
12144:
12142:
12141:
12136:
12131:
12129:
12128:
12123:
12118:
12103:
12092:
12070:
12059:
12057:
12056:
12051:
12039:
12037:
12036:
12031:
12010:
12008:
12007:
12002:
12000:
11999:
11994:
11980:Poisson equation
11973:
11971:
11970:
11965:
11953:
11951:
11950:
11945:
11937:
11922:
11920:
11919:
11914:
11909:
11904:
11899:
11888:
11886:
11875:
11867:
11824:potential theory
11817:
11805:
11786:
11784:
11783:
11778:
11773:
11772:
11714:
11713:
11697:
11691:
11681:
11666:
11660:
11654:
11643:
11641:
11640:
11635:
11630:
11629:
11595:
11593:
11592:
11591:
11575:
11561:
11559:
11558:
11542:
11538:
11529:
11518:
11512:
11491:
11484:
11475:
11466:Morera's theorem
11463:
11459:
11451:
11449:
11448:
11443:
11426:
11424:
11423:
11422:
11406:
11398:
11396:
11395:
11379:
11373:
11360:
11342:complex analysis
11312:
11310:
11309:
11304:
11268:
11267:
11220:
11213:
11211:
11210:
11205:
11172:
11171:
11148:
11128:
11120:
11104:hyperbolic plane
11074:
11068:
11064:
11056:
11045:
11037:
11035:
11034:
11029:
11027:
11022:
11010:
11009:
11008:
11007:
10989:
10984:
10957:
10951:
10949:
10948:
10943:
10941:
10931:
10927:
10915:
10913:
10912:
10907:
10902:
10901:
10897:
10896:
10881:
10876:
10875:
10845:
10843:
10842:
10834:
10825:
10801:
10797:
10790:
10759:
10758:
10757:
10746:
10716:Heisenberg group
10705:
10691:
10689:
10688:
10683:
10678:
10677:
10672:
10668:
10660:
10659:
10654:
10636:
10630:
10625:
10613:
10611:
10600:
10589:
10585:
10577:
10576:
10571:
10556:
10555:
10547:
10544:
10539:
10538:
10528:
10523:
10506:
10502:
10494:
10493:
10488:
10470:
10465:
10464:
10454:
10449:
10425:
10414:
10408:
10402:
10393:
10387:
10385:
10384:
10379:
10367:
10366:
10329:
10324:
10300:
10299:
10291:
10281:
10267:
10232:
10222:
10216:
10206:
10204:
10203:
10198:
10196:
10194:
10193:
10189:
10180:
10176:
10175:
10174:
10169:
10160:
10155:
10154:
10139:
10138:
10134:
10125:
10121:
10120:
10119:
10114:
10102:
10097:
10096:
10081:
10066:
10057:Dirichlet energy
10045:
10041:
10036:
10034:
10033:
10028:
10026:
10025:
10000:
9998:
9997:
9992:
9990:
9989:
9959:
9957:
9956:
9951:
9936:
9926:
9924:
9923:
9918:
9916:
9915:
9894:
9893:
9864:potential theory
9861:
9849:
9842:
9832:
9822:
9820:
9819:
9814:
9809:
9808:
9780:
9772:
9760:
9728:
9726:
9725:
9720:
9702:
9697:
9678:
9671:is the smallest
9670:
9662:
9658:
9650:
9644:
9642:
9641:
9636:
9624:
9623:
9618:
9603:
9595:
9589:
9584:
9565:
9564:
9559:
9541:
9535:
9530:
9521:
9520:
9500:
9499:
9491:
9485:
9474:
9472:
9471:
9466:
9464:
9419:fundamental mode
9412:
9405:
9395:
9373:
9366:
9348:
9346:
9345:
9340:
9338:
9314:
9310:
9308:
9307:
9302:
9270:
9255:
9253:
9252:
9247:
9212:
9194:
9180:
9176:
9169:vibrating string
9159:
9155:
9147:
9116:
9114:
9113:
9108:
9103:
9099:
9060:
9057:
9055:
9051:
9005:
9000:and any integer
8999:
8993:
8989:
8982:
8980:
8974:
8970:
8966:
8950:
8909:
8907:
8906:
8901:
8889:
8887:
8886:
8881:
8879:
8877:
8876:
8861:
8850:
8848:
8847:
8842:
8837:
8829:
8817:
8815:
8814:
8799:
8796:
8791:
8772:
8761:algebraic curves
8746:
8708:
8701:
8692:
8687:
8685:
8684:
8681:
8678:
8670:
8661:
8654:
8648:
8639:
8632:
8625:area of a circle
8619:
8612:
8602:
8582:
8575:
8567:
8551:
8546:Fourier analysis
8539:
8527:
8520:
8504:
8484:
8476:
8466:
8464:
8463:
8458:
8453:
8449:
8448:
8446:
8429:
8424:
8422:
8405:
8400:
8398:
8381:
8376:
8374:
8357:
8350:
8348:
8347:
8335:
8332:
8327:
8282:
8270:
8258:
8254:
8237:
8235:
8234:
8229:
8224:
8222:
8221:
8220:
8211:
8206:
8205:
8196:
8184:
8176:
8173:
8145:
8137:
8128:
8120:
8111:
8107:
8105:
8104:
8099:
8097:
8096:
8086:
8081:
8063:
8062:
8046:
8036:
8027:random variables
8020:
8012:
8008:
8004:
7996:
7989:
7987:
7969:
7965:
7959:
7957:
7956:
7951:
7946:
7944:
7936:
7925:
7910:
7906:
7900:
7894:
7888:
7882:
7872:
7856:
7833:
7816:
7798:
7792:
7784:
7778:
7776:
7775:
7770:
7765:
7761:
7760:
7758:
7751:
7750:
7734:
7729:
7727:
7720:
7719:
7703:
7698:
7696:
7689:
7688:
7675:
7668:
7666:
7665:
7653:
7650:
7645:
7627:
7626:
7610:
7590:Emma Haruka Iwao
7587:
7583:
7577:
7575:
7574:
7569:
7564:
7562:
7561:
7560:
7539:
7538:
7509:
7474:
7471:
7466:
7451:
7442:
7441:
7436:
7428:
7403:
7393:
7391:
7390:
7385:
7380:
7378:
7377:
7373:
7372:
7356:
7355:
7342:
7307:
7304:
7299:
7284:
7279:
7278:
7273:
7267:
7262:
7254:
7241:
7232:
7224:
7205:
7193:
7189:
7184:round-off errors
7173:
7161:
7150:
7142:
7114:(AGM method) or
7082:
7074:
7066:
7053:John von Neumann
7042:
7034:
7032:
7031:
7026:
7020:
7018:
7017:
7016:
7003:
7002:
7001:
6992:
6991:
6979:
6978:
6965:
6949:
6945:
6943:
6942:
6937:
6931:
6930:
6915:
6914:
6894:
6893:
6884:
6883:
6865:
6864:
6852:
6851:
6839:
6838:
6826:
6825:
6803:
6801:
6800:
6795:
6789:
6787:
6786:
6777:
6776:
6767:
6762:
6761:
6741:
6736:
6735:
6734:
6722:
6721:
6711:
6706:
6705:
6682:
6680:
6679:
6674:
6665:
6664:
6651:
6643:
6638:
6637:
6624:
6618:
6614:
6609:
6608:
6589:
6588:
6550:
6546:
6538:
6534:
6530:
6526:
6512:
6510:
6505:
6484:
6480:
6476:
6474:
6473:
6468:
6466:
6457:
6445:
6444:
6434:
6427:
6423:
6421:
6419:
6418:
6413:
6411:
6403:
6388:
6384:
6382:
6380:
6379:
6374:
6372:
6364:
6349:
6343:
6341:
6339:
6338:
6333:
6310:
6306:
6297:
6287:
6273:
6255:
6239:
6227:
6211:
6203:
6187:
6180:Swiss scientist
6177:
6175:
6174:
6169:
6161:
6159:
6158:
6146:
6141:
6139:
6138:
6126:
6121:
6119:
6118:
6106:
6101:
6099:
6098:
6086:
6081:
6076:
6075:
6066:
6047:
6039:
6031:
6023:
6001:
5997:
5973:
5971:
5970:
5965:
5957:
5955:
5935:
5930:
5928:
5908:
5903:
5901:
5881:
5876:
5854:
5834:
5832:
5831:
5826:
5818:
5810:
5805:
5797:
5792:
5784:
5779:
5771:
5766:
5758:
5753:
5745:
5740:
5732:
5696:
5690:
5689:
5684:
5682:
5681:
5676:
5668:
5666:
5646:
5641:
5639:
5619:
5614:
5612:
5592:
5587:
5585:
5565:
5544:
5537:
5533:
5529:
5523:
5521:
5520:
5515:
5507:
5499:
5494:
5486:
5481:
5473:
5468:
5460:
5455:
5447:
5442:
5434:
5429:
5421:
5402:
5398:
5394:
5387:
5374:
5359:
5347:
5340:
5336:
5334:
5333:
5328:
5323:
5314:
5299:
5290:
5275:
5266:
5250:
5248:
5247:
5242:
5234:
5232:
5231:
5230:
5221:
5220:
5201:
5200:
5191:
5189:
5187:
5176:
5165:
5160:
5158:
5157:
5156:
5147:
5146:
5127:
5126:
5117:
5115:
5107:
5102:
5100:
5099:
5098:
5079:
5048:
5040:
5034:
5032:
5031:
5026:
5021:
5013:
5002:
4994:
4980:
4972:
4956:
4952:
4950:
4949:
4944:
4942:
4936:
4932:
4906:
4904:
4903:
4898:
4880:
4878:
4877:
4872:
4854:
4852:
4851:
4846:
4844:
4836:
4821:
4819:
4818:
4813:
4805:
4800:
4799:
4790:
4785:
4780:
4779:
4770:
4765:
4760:
4759:
4750:
4708:
4704:
4684:
4667:
4665:
4664:
4659:
4654:
4653:
4647:
4639:
4634:
4626:
4624:
4623:
4614:
4613:
4607:
4599:
4594:
4586:
4584:
4583:
4574:
4573:
4567:
4559:
4554:
4546:
4544:
4543:
4534:
4533:
4527:
4519:
4514:
4506:
4504:
4503:
4494:
4486:
4467:
4465:
4464:
4459:
4454:
4448:
4446:
4441:
4433:
4425:
4424:
4419:
4413:
4408:
4400:
4399:
4394:
4385:
4384:
4379:
4371:
4362:
4355:(rather than an
4353:infinite product
4339:
4334:Gregory's series
4304:was laid out in
4303:
4287:
4279:
4267:
4245:
4213:
4201:
4199:
4198:
4193:
4191:
4190:
4164:
4162:
4161:
4156:
4154:
4153:
4126:Jamshīd al-Kāshī
4120:
4118:
4117:
4112:
4104:
4095:
4094:
4075:
4072:
4051:
4033:
4031:
4030:
4025:
4017:
4009:
3994:
3992:
3991:
3986:
3983:3.14159292035...
3978:
3970:
3955:
3953:
3952:
3947:
3918:
3914:
3898:
3897:
3895:
3894:
3891:
3888:
3880:
3878:
3877:
3872:
3870:
3865:
3856:
3845:
3837:
3823:
3822:
3820:
3819:
3816:
3813:
3805:
3801:
3800:
3798:
3797:
3794:
3791:
3783:
3781:
3775:
3774:
3772:
3771:
3768:
3765:
3758:
3754:
3752:
3751:
3748:
3745:
3736:
3732:
3720:
3712:
3697:
3674:
3663:. Although some
3662:
3660:
3659:
3654:
3646:
3645:
3640:
3639:
3632:
3624:
3622:
3621:
3608:
3600:
3598:
3597:
3594:
3591:
3584:
3572:
3557:
3545:
3536:
3523:
3521:
3520:
3515:
3464:
3463:
3459:
3426:
3420:
3418:
3417:
3412:
3404:
3403:
3387:
3380:
3372:
3370:
3369:
3364:
3350:
3349:
3329:Euler's identity
3326:
3324:
3323:
3318:
3296:
3285:
3277:
3275:
3274:
3269:
3237:
3236:
3208:complex analysis
3205:
3201:
3199:
3198:
3193:
3182:
3181:
3161:
3157:
3155:
3154:
3149:
3091:
3079:
3075:
3059:
3028:
3009:
2992:
2982:
2981:
2978:
2975:
2972:
2959:
2949:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2905:
2895:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2836:
2826:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2789:
2779:
2769:
2767:
2766:
2763:
2760:
2753:
2751:
2750:
2747:
2744:
2737:
2735:
2734:
2731:
2728:
2721:
2719:
2718:
2715:
2712:
2705:
2703:
2702:
2699:
2696:
2689:
2687:
2686:
2683:
2680:
2673:
2671:
2670:
2667:
2664:
2619:
2617:
2616:
2611:
2609:
2605:
2603:
2601:
2600:
2598:
2596:
2595:
2593:
2591:
2590:
2588:
2586:
2570:
2568:
2567:
2566:
2551:
2537:
2535:
2534:
2533:
2518:
2504:
2502:
2501:
2500:
2485:
2471:
2469:
2459:
2454:
2452:
2450:
2449:
2447:
2445:
2444:
2442:
2440:
2439:
2437:
2435:
2419:
2417:
2416:
2415:
2400:
2386:
2384:
2383:
2382:
2367:
2353:
2351:
2350:
2349:
2334:
2320:
2318:
2308:
2303:
2301:
2299:
2298:
2296:
2294:
2293:
2291:
2289:
2288:
2286:
2284:
2268:
2266:
2265:
2264:
2249:
2235:
2233:
2232:
2231:
2216:
2202:
2200:
2199:
2198:
2183:
2169:
2167:
2166:
2165:
2150:
2117:
2109:
2097:
2093:
2089:
2088:
2086:
2085:
2082:
2079:
2071:
2070:
2068:
2067:
2064:
2061:
2053:
2052:
2050:
2049:
2046:
2043:
2035:
2031:
2025:
2023:
2022:
2017:
2014:
2012:
2010:
2008:
2006:
2004:
2002:
2000:
1998:
1996:
1994:
1992:
1990:
1988:
1986:
1984:
1982:
1980:
1978:
1976:
1974:
1958:
1956:
1946:
1931:
1929:
1919:
1904:
1902:
1892:
1877:
1875:
1865:
1850:
1848:
1838:
1823:
1821:
1811:
1796:
1794:
1784:
1758:
1750:
1722:
1720:
1719:
1714:
1712:
1707:
1698:
1696:
1695:
1690:
1688:
1686:
1678:
1662:
1658:
1651:
1649:
1648:
1643:
1629:
1624:
1623:
1614:
1609:
1604:
1603:
1594:
1569:
1549:
1519:
1511:
1499:
1491:
1480:
1468:
1457:
1453:
1442:rational numbers
1439:
1427:
1413:
1405:
1401:
1400:
1398:
1397:
1394:
1391:
1383:
1382:
1380:
1379:
1376:
1373:
1357:
1346:
1343:one. The number
1290:
1286:
1284:
1283:
1278:
1273:
1174:
1167:
1157:
1146:
1127:
1119:
1111:
1095:Karl Weierstrass
1092:
1086:
1084:
1083:
1078:
1073:
1070:
1069:
1054:
1053:
1045:
1042:
1037:
1008:
1006:
1005:
1000:
992:
991:
979:
978:
963:by the equation
939:
937:
936:
931:
929:
921:
898:
894:
892:
891:
886:
884:
876:
867:
865:
864:
859:
857:
849:
838:
836:
835:
830:
828:
820:
805:
796:
779:
772:
754:
747:
736:
728:
724:
718:
713:
712:
709:
708:
705:
692:
683:
664:
660:
656:
649:, and in modern
639:electromagnetism
612:
600:
592:
569:
561:
553:
542:
538:
526:
520:has been found.
511:
499:
484:never ends, nor
475:
473:
472:
467:
465:
456:
434:
406:
405:
402:
401:
398:
388:
374:
367:
360:
346:
338:
210:Jamshīd al-Kāshī
107:Area of a circle
93:
92:
89:
86:
83:
73:
63:
47:
46:
21:
23914:
23913:
23909:
23908:
23907:
23905:
23904:
23903:
23874:
23873:
23872:
23867:
23844:
23835:
23829:
23819:
23798:
23790:
23775:
23760:
23750:
23740:
23723:
23705:
23690:
23658:
23643:
23631:
23626:
23577:
23575:
23571:
23567:
23561:
23554:Wayback Machine
23542:Wayback Machine
23530:
23507:
23502:
23496:
23484:
23471:
23459:
23451:
23449:Further reading
23446:
23440:
23417:
23402:
23388:
23385:
23380:
23368:
23364:
23348:
23323:
23296:
23265:
23263:
23261:
23240:
23206:
23201:
23200:
23190:
23188:
23184:
23183:
23179:
23170:
23169:
23165:
23155:
23153:
23149:
23134:
23125:
23121:
23106:10.2307/2689499
23090:
23085:
23080:
23076:
23066:
23064:
23060:
23059:
23055:
23044:Telegraph India
23038:
23037:
23033:
23025:
22994:
22989:
22984:
22980:
22972:
22941:
22935:
22931:
22924:
22907:
22906:"Tau – the new
22902:
22898:
22888:
22886:
22879:The Independent
22871:
22867:
22832:
22828:
22818:
22816:
22803:
22802:
22798:
22791:
22775:Petroski, Henry
22772:
22768:
22743:
22739:
22723:The Independent
22714:
22710:
22703:
22686:
22682:
22674:
22670:
22662:
22657:
22653:
22642:
22626:Peterson, Ivars
22620:
22583:
22579:
22572:
22558:
22554:
22544:
22542:
22528:
22524:
22479:
22475:
22460:
22435:
22430:
22426:
22416:
22414:
22406:The Japan Times
22397:
22393:
22387:Wayback Machine
22378:
22374:
22366:
22357:
22350:
22336:
22332:
22325:
22311:
22307:
22284:
22261:
22257:
22238:
22234:
22227:
22213:
22209:
22202:
22188:
22184:
22177:
22163:
22159:
22149:
22147:
22143:
22120:
22114:
22107:
22092:
22088:
22081:
22064:
22060:
22049:
22045:
22038:
22021:
22014:
22006:
22002:
21987:
21973:
21969:
21920:
21916:
21892:10.1.1.352.5774
21873:
21869:
21858:
21854:
21846:
21842:
21835:
21818:
21814:
21807:
21788:
21784:
21776:
21772:
21723:
21719:
21711:
21707:
21696:
21692:
21678:
21674:
21666:
21662:
21656:
21642:Trudinger, Neil
21638:
21631:
21624:
21610:
21606:
21599:
21585:
21581:
21574:
21566:. W.W. Norton.
21560:
21556:
21549:
21535:
21531:
21517:
21513:
21492:
21488:
21476:Spivak, Michael
21473:
21469:
21465:; Theorem 1.13.
21454:
21450:
21442:
21438:
21430:
21423:
21414:
21410:
21371:
21364:
21353:Folland, Gerald
21350:
21346:
21299:
21295:
21254:
21250:
21233:
21229:
21197:10.1.1.615.4193
21180:
21176:
21165:
21161:
21143:
21139:
21131:
21127:
21109:
21105:
21097:
21090:
21078:
21074:
21065:. Vol. 1.
21055:Strang, Gilbert
21053:Herman, Edwin;
21051:
21047:
21021:
20995:
20991:
20983:
20979:
20961:
20957:
20953:
20950:
20946:
20936:
20934:
20919:
20915:
20905:
20903:
20888:
20883:
20879:
20871:
20822:
20805:
20801:
20793:
20786:
20771:10.2307/2975006
20755:
20751:
20743:
20736:
20694:
20688:Gibbons, Jeremy
20685:
20678:
20670:
20663:
20655:
20650:
20646:
20612:
20608:
20600:
20595:
20591:
20576:10.2307/2317945
20560:
20556:
20548:
20544:
20534:
20532:
20528:
20521:
20512:
20508:
20503:
20499:
20484:
20480:
20473:
20440:
20433:
20425:
20421:
20413:
20409:
20401:
20397:
20389:
20385:
20377:
20373:
20365:
20361:
20351:
20349:
20342:The Independent
20334:
20330:
20319:
20315:
20307:
20303:
20293:
20291:
20287:
20280:
20274:
20265:
20258:
20240:
20236:
20228:
20221:
20213:
20209:
20201:
20197:
20189:
20185:
20170:10.2307/2002052
20164:(52): 162–164.
20154:
20150:
20135:10.2307/2002695
20119:
20115:
20107:
20100:
20092:
20088:
20080:
20076:
20065:
20061:
20051:
20049:
20032:
20028:
20018:
20015:Wayback Machine
19998:
19981:
19977:
19968:
19955:10.2307/2973441
19934:
19919:
19915:
19908:
19905:Wayback Machine
19892:
19884:
19882:
19878:
19859:
19853:
19849:
19839:
19837:
19824:
19820:
19808:
19804:
19762:
19756:
19752:
19737:
19733:
19727:. J. Salusbury.
19700:
19696:
19688:
19677:
19664:
19647:10.2307/2972388
19624:
19621:
19617:
19610:
19594:
19587:
19580:
19574:
19570:10.2307/3029000
19563:
19560:10.2307/3029832
19553:
19550:10.2307/3029284
19543:
19532:
19528:
19513:
19509:
19504:
19488:
19481:
19475:
19471:
19462:
19458:
19450:
19446:
19424:
19420:
19411:
19407:
19399:
19392:
19385:
19371:
19367:
19359:
19355:
19339:1959.13/1043679
19330:10.2307/2324715
19314:
19310:
19302:
19298:
19290:
19262:
19255:
19245:
19243:
19223:
19219:
19211:
19207:
19199:
19195:
19133:
19109:Euler, Leonhard
19087:Euler, Leonhard
19068:
19062:
19058:
19031:
19027:
19017:
19015:
19014:on 7 March 2023
19011:
18980:
18975:
18967:
18963:
18955:
18948:
18940:
18936:
18896:
18893:
18889:
18881:
18877:
18867:
18865:
18861:
18830:
18824:
18820:
18812:
18805:
18789:
18783:
18779:
18771:
18764:
18751:
18747:
18735:
18734:
18730:
18722:
18715:
18708:
18692:
18688:
18680:
18676:
18666:
18664:
18660:
18637:
18632:
18627:
18616:
18608:
18604:
18578:
18574:
18567:
18539:
18535:
18528:
18522:
18515:
18504:
18500:
18492:
18488:
18480:
18476:
18468:
18459:
18449:
18447:
18430:
18426:
18397:
18393:
18385:
18378:
18370:
18366:
18358:
18354:
18346:
18342:
18334:
18327:
18319:
18315:
18307:
18303:
18295:
18291:
18283:
18279:
18271:
18267:
18252:
18229:
18223:
18214:
18206:
18202:
18194:
18190:
18183:
18163:
18159:
18149:
18147:
18140:
18124:
18120:
18112:
18105:
18097:
18093:
18086:
18072:
18068:
18060:
18053:
18041:
18037:
18026:
17964:
17960:
17952:
17948:
17940:
17936:
17921:10.2307/2589152
17902:
17899:
17895:
17872:
17865:
17857:
17853:
17845:
17838:
17827:
17825:
17818:
17800:
17796:
17789:
17775:
17771:
17763:
17759:
17751:
17747:
17739:
17735:
17727:
17723:
17715:
17711:
17703:
17699:
17652:
17648:
17640:
17633:
17619:
17612:
17598:
17594:
17580:
17576:
17565:
17561:
17554:
17536:
17529:
17515:
17511:
17490:
17486:
17464:
17460:
17452:
17443:
17426:
17422:
17414:
17405:
17395:
17393:
17384:
17383:
17379:
17364:
17354:
17347:
17315:10.1.1.138.7085
17296:
17292:
17284:
17280:
17270:
17268:
17248:
17244:
17227:
17226:
17222:
17207:
17195:
17169:
17168:
17124:
17120:
17114:
17102:
17098:
17092:
17091:
17089:
17077:
17059:
17055:
17049:
17037:
17033:
17027:
17026:
17024:
17012:
16991:
16976:
16975:
16973:
16971:
16968:
16967:
16966:
16943:. p. 263:
16924:
16915:
16900:
16896:
16888:
16884:
16879:
16874:
16873:
16860:
16856:
16831:
16827:
16820:
16812:
16810:
16804:
16796:
16784:
16781:
16780:
16778:
16774:
16763:
16762:In particular,
16761:
16757:
16752:
16747:
16732:
16724:
16719:
16709:
16704:
16693:
16678:
16671:
16662:
16655:Indiana Pi Bill
16643:
16639:
16631:
16630:, arguing that
16620:
16615:
16587:
16576:
16552:
16544:
16523:
16508:
16492:
16487:A pi pie. Many
16481:
16472:
16464:
16460:Cadaeic Cadenza
16454:
16446:
16435:
16431:
16415:
16409:Akira Haraguchi
16404:
16394:
16388:
16382:
16358:
16353:
16352:
16343:
16339:
16322:
16313:
16309:
16291:
16287:
16285:
16282:
16281:
16263:
16215:
16212:
16211:
16205:
16192:
16186:
16176:
16165:
16137:
16133:
16120:
16116:
16115:
16113:
16105:
16102:
16101:
16095:
16086:
16077:
16071:
16063:
16058:
16052:
16032:
16028:
16022:
16018:
16008:
15995:
15991:
15990:
15988:
15972:
15970:
15967:
15966:
15960:
15949:
15923:
15918:
15900:
15897:
15896:
15893:Planck constant
15886:
15880:
15870:
15834:
15820:
15817:
15816:
15806:
15800:
15790:
15778:
15774:
15769:Although not a
15767:
15762:
15753:
15752:, rather than 2
15749:
15745:
15716:
15708:
15706:
15703:
15702:
15685:
15681:
15667:
15664:
15663:
15657:
15640:
15636:
15634:
15631:
15630:
15616:
15610:
15569:
15565:
15563:
15560:
15559:
15537:
15534:
15533:
15517:
15502:
15498:
15496:
15493:
15492:
15476:
15468:
15465:
15464:
15450:
15401:
15399:
15396:
15395:
15376:
15368:
15360:
15357:
15356:
15350:
15347:
15338:
15334:
15326:
15322:
15318:
15314:
15306:
15302:
15290:
15282:
15271:
15262:
15254:
15241:
15212:
15191:
15189:
15183:
15175:
15161:
15141:
15138:
15137:
15115:Brownian motion
15095:
15091:
15088:Shannon entropy
15043:
15039:
15038:
15033:
15027:
15019:
15013:
15010:
15009:
14979:
14975:
14974:
14969:
14956:
14939:
14936:
14935:
14909:Witch of Agnesi
14901:
14888:
14885:theta functions
14876:
14814:
14810:
14781:
14778:
14777:
14769:
14743:
14739:
14731:
14728:
14727:
14698:
14694:
14669:
14665:
14659:
14645:
14618:
14615:
14614:
14592:
14583:
14575:
14574:
14572:
14569:
14568:
14549:
14540:
14532:
14531:
14529:
14526:
14525:
14493:theta functions
14484:
14469:
14456:
14449:Pontrjagin dual
14440:
14436:
14428:
14418:
14386:
14382:
14364:
14360:
14358:
14355:
14354:
14348:
14341:
14332:
14322:
14312:
14306:
14292:
14280:
14260:
14254:
14225:
14199:
14176:
14162:
14159:
14158:
14083:
14081:
14077:
14056:
14052:
14046:
14042:
14025:
14022:
14021:
14017:
14005:
13993:
13988:
13961:
13954:
13953:to approximate
13934:
13933:
13913:
13909:
13904:
13885:
13880:
13871:
13870:
13853:
13849:
13844:
13833:
13829:
13824:
13817:
13812:
13803:
13802:
13793:
13776:
13772:
13765:
13760:
13754:
13749:
13744:
13740:
13739:
13732:
13719:
13715:
13710:
13703:
13699:
13693:
13688:
13680:
13678:
13675:
13674:
13667:
13660:
13653:
13647:
13639:
13627:
13616:simple solution
13588:
13584:
13579:
13568:
13564:
13559:
13548:
13544:
13539:
13522:
13519:
13518:
13511:
13500:
13487:
13486:
13471:
13470:: the value of
13468:Weil conjecture
13440:
13383:
13379:
13377:
13364:
13357:
13353:
13340:
13338:
13326:
13322:
13296:
13293:
13292:
13285:
13282:
13272:
13261:
13255:
13221:
13217:
13213:
13206:
13202:
13190:
13186:
13185:
13183:
13171:
13159:
13156:
13155:
13134:
13120:
13116:
13115:
13102:
13091:
13088:
13087:
13081:
13074:
13040:
13036:
13035:
13013:
13009:
13008:
13006:
12992:
12989:
12988:
12956:
12952:
12928:
12927:
12923:
12919:
12908:
12904:
12900:
12896:
12894:
12870:
12866:
12864:
12861:
12860:
12841:
12837:
12813:
12812:
12808:
12804:
12794:
12790:
12786:
12784:
12766:
12762:
12760:
12757:
12756:
12740:
12731:
12708:
12700:
12690:
12678:
12671:
12663:
12641:
12631:
12621:
12617:
12613:
12611:
12605:
12594:
12573:
12569:
12567:
12550:
12547:
12546:
12515:
12511:
12509:
12495:
12484:
12481:
12480:
12462:
12448:
12437:
12434:
12433:
12432:. For example,
12429:
12383:
12380:
12379:
12365:
12346:
12343:
12342:
12327:
12322:
12321:
12285:
12232:
12227:
12221:
12218:
12217:
12192:total curvature
12175:
12167:
12159:
12156:Total curvature
12151:
12149:Total curvature
12124:
12119:
12114:
12107:
12102:
12088:
12080:
12077:
12076:
12068:
12045:
12042:
12041:
12016:
12013:
12012:
11995:
11990:
11989:
11987:
11984:
11983:
11959:
11956:
11955:
11933:
11928:
11925:
11924:
11905:
11900:
11895:
11879:
11874:
11863:
11855:
11852:
11851:
11838:, and even the
11820:vector calculus
11815:
11812:
11796:
11768:
11764:
11709:
11705:
11703:
11700:
11699:
11693:
11687:
11672:
11669:residue theorem
11662:
11656:
11645:
11625:
11621:
11587:
11583:
11576:
11562:
11560:
11554:
11550:
11548:
11545:
11544:
11540:
11537:
11531:
11520:
11514:
11503:
11486:
11483:
11477:
11473:
11461:
11455:
11418:
11414:
11407:
11399:
11397:
11391:
11387:
11385:
11382:
11381:
11375:
11374:is interior to
11372:
11366:
11356:
11330:
11263:
11259:
11242:
11239:
11238:
11235:homology groups
11215:
11167:
11163:
11161:
11158:
11157:
11151:Gauss curvature
11146:
11129:appears in the
11126:
11118:
11098:, a surface of
11085:
11070:
11066:
11062:
11054:
11043:
11021:
11003:
10999:
10995:
10991:
10985:
10977:
10971:
10968:
10967:
10953:
10925:
10923:
10920:
10919:
10892:
10888:
10877:
10871:
10867:
10851:
10847:
10833:
10829:
10824:
10807:
10804:
10803:
10799:
10795:
10788:
10780:, which is the
10755:
10753:
10752:
10751:-axis has area
10734:
10730:A graph of the
10724:
10703:
10700:discussed below
10673:
10655:
10650:
10649:
10632:
10626:
10618:
10604:
10599:
10598:
10594:
10593:
10572:
10567:
10566:
10546:
10545:
10540:
10534:
10530:
10524:
10516:
10511:
10507:
10489:
10484:
10483:
10466:
10460:
10456:
10450:
10442:
10437:
10433:
10431:
10428:
10427:
10423:
10410:
10404:
10398:
10391:
10347:
10343:
10325:
10317:
10290:
10289:
10287:
10284:
10283:
10277:
10272:. This is the
10265:
10251:
10228:
10218:
10212:
10185:
10181:
10170:
10165:
10164:
10156:
10150:
10146:
10145:
10141:
10140:
10130:
10126:
10115:
10110:
10109:
10098:
10092:
10088:
10087:
10083:
10082:
10080:
10072:
10069:
10068:
10064:
10043:
10039:
10021:
10017:
10006:
10003:
10002:
9985:
9981:
9973:
9970:
9969:
9942:
9939:
9938:
9932:
9911:
9907:
9889:
9885:
9871:
9868:
9867:
9859:
9847:
9834:
9824:
9804:
9800:
9786:
9783:
9782:
9778:
9770:
9758:
9735:
9698:
9693:
9687:
9684:
9683:
9676:
9668:
9660:
9652:
9646:
9619:
9614:
9613:
9596:
9591:
9585:
9580:
9560:
9555:
9554:
9537:
9531:
9526:
9516:
9512:
9510:
9507:
9506:
9497:
9493:
9487:
9476:
9460:
9434:
9431:
9430:
9410:
9397:
9379:
9368:
9358:
9331:
9323:
9320:
9319:
9312:
9263:
9261:
9258:
9257:
9205:
9203:
9200:
9199:
9185:
9178:
9172:
9157:
9153:
9145:
9122:
9083:
9079:
9058: and
9056:
9035:
9031:
9011:
9008:
9007:
9001:
8995:
8991:
8987:
8978:
8976:
8972:
8968:
8964:
8948:
8934:
8928:
8895:
8892:
8891:
8872:
8868:
8860:
8858:
8855:
8854:
8828:
8810:
8806:
8798:
8792:
8784:
8778:
8775:
8774:
8770:
8744:
8703:
8697:
8682:
8679:
8676:
8675:
8673:
8672:
8666:
8656:
8650:
8644:
8634:
8628:
8614:
8608:
8600:
8580:
8573:
8565:
8558:
8549:
8537:
8534:
8525:
8518:
8502:
8482:
8474:
8433:
8428:
8409:
8404:
8385:
8380:
8361:
8356:
8355:
8351:
8343:
8339:
8334:
8328:
8317:
8305:
8302:
8301:
8286:Mathematicians
8280:
8268:
8265:
8256:
8252:
8216:
8212:
8207:
8201:
8197:
8192:
8185:
8177:
8175:
8163:
8151:
8148:
8147:
8143:
8135:
8130:
8126:
8118:
8113:
8109:
8092:
8088:
8082:
8071:
8058:
8054:
8052:
8049:
8048:
8043:
8038:
8034:
8029:
8018:
8010:
8006:
7998:
7991:
7986:
7980:
7978:
7967:
7963:
7937:
7926:
7924:
7916:
7913:
7912:
7908:
7902:
7896:
7890:
7884:
7878:
7875:Buffon's needle
7870:
7864:
7863:
7862:
7861:
7860:
7857:
7849:
7848:
7837:Buffon's needle
7834:
7823:
7804:
7794:
7786:
7780:
7743:
7739:
7738:
7733:
7712:
7708:
7707:
7702:
7684:
7680:
7679:
7674:
7673:
7669:
7661:
7657:
7652:
7646:
7635:
7622:
7618:
7616:
7613:
7612:
7608:
7585:
7581:
7553:
7549:
7534:
7530:
7510:
7475:
7473:
7467:
7456:
7440:
7427:
7425:
7422:
7421:
7401:
7365:
7361:
7357:
7351:
7347:
7343:
7308:
7306:
7300:
7289:
7272:
7268:
7266:
7253:
7251:
7248:
7247:
7239:
7230:
7222:
7212:
7203:
7191:
7187:
7171:
7159:
7152:
7148:
7140:
7137:Yasumasa Kanada
7080:
7072:
7064:
7049:inverse tangent
7040:
7037:
7012:
7008:
7004:
6997:
6993:
6987:
6983:
6974:
6970:
6966:
6964:
6955:
6952:
6951:
6947:
6926:
6922:
6904:
6900:
6889:
6885:
6873:
6869:
6860:
6856:
6847:
6843:
6834:
6830:
6815:
6811:
6808:
6805:
6804:
6782:
6778:
6772:
6768:
6766:
6751:
6747:
6730:
6726:
6717:
6713:
6712:
6710:
6695:
6691:
6688:
6685:
6684:
6660:
6656:
6642:
6633:
6629:
6613:
6604:
6600:
6584:
6580:
6577:
6574:
6573:
6571:
6562:
6557:
6548:
6544:
6536:
6532:
6528:
6508:
6507:
6500:
6482:
6478:
6455:
6453:
6450:
6449:
6432:
6425:
6402:
6400:
6397:
6396:
6394:
6386:
6363:
6361:
6358:
6357:
6355:
6321:
6318:
6317:
6315:
6308:
6304:
6295:
6285:
6271:
6266:
6265:
6264:
6263:
6262:
6256:
6248:
6247:
6240:
6229:
6225:
6209:
6201:
6194:Lambert's proof
6185:
6154:
6150:
6145:
6134:
6130:
6125:
6114:
6110:
6105:
6094:
6090:
6085:
6071:
6067:
6065:
6063:
6060:
6059:
6045:
6037:
6034:
6029:
6021:
6016:
6004:Machin's series
5999:
5995:
5939:
5934:
5912:
5907:
5885:
5880:
5872:
5864:
5861:
5860:
5852:
5809:
5796:
5783:
5770:
5757:
5744:
5731:
5723:
5720:
5719:
5694:
5650:
5645:
5623:
5618:
5596:
5591:
5569:
5564:
5550:
5547:
5546:
5542:
5535:
5531:
5527:
5498:
5485:
5472:
5459:
5446:
5433:
5420:
5412:
5409:
5408:
5400:
5396:
5392:
5385:
5382:
5372:
5357:
5345:
5338:
5312:
5288:
5264:
5262:
5259:
5258:
5226:
5222:
5216:
5212:
5202:
5196:
5192:
5190:
5177:
5166:
5164:
5152:
5148:
5142:
5138:
5128:
5122:
5118:
5116:
5106:
5094:
5090:
5083:
5078:
5064:
5061:
5060:
5046:
5038:
5012:
4993:
4971:
4969:
4966:
4965:
4954:
4931:
4923:
4920:
4919:
4886:
4883:
4882:
4860:
4857:
4856:
4835:
4833:
4830:
4829:
4795:
4791:
4789:
4775:
4771:
4769:
4755:
4751:
4749:
4729:
4726:
4725:
4706:
4702:
4682:
4679:infinite series
4649:
4648:
4638:
4625:
4619:
4618:
4609:
4608:
4598:
4585:
4579:
4578:
4569:
4568:
4558:
4545:
4539:
4538:
4529:
4528:
4518:
4505:
4499:
4498:
4485:
4483:
4480:
4479:
4440:
4432:
4423:
4407:
4398:
4383:
4370:
4368:
4365:
4364:
4363:calculations):
4360:
4349:Viète's formula
4337:
4301:
4285:
4277:
4270:infinite series
4265:
4251:
4243:
4236:
4234:Infinite series
4211:
4186:
4182:
4174:
4171:
4170:
4149:
4145:
4137:
4134:
4133:
4093:
4085:
4082:
4081:
4073:
4065:(499 AD).
4049:
4008:
4000:
3997:
3996:
3969:
3961:
3958:
3957:
3929:
3926:
3925:
3916:
3912:
3892:
3889:
3886:
3885:
3883:
3882:
3864:
3862:
3859:
3858:
3854:
3843:
3835:
3817:
3814:
3811:
3810:
3808:
3807:
3803:
3795:
3792:
3789:
3788:
3786:
3785:
3779:
3777:
3769:
3766:
3763:
3762:
3760:
3756:
3749:
3746:
3743:
3742:
3740:
3739:
3734:
3730:
3718:
3710:
3695:
3689:
3672:
3665:pyramidologists
3641:
3635:
3634:
3633:
3623:
3617:
3616:
3614:
3611:
3610:
3606:
3595:
3592:
3589:
3588:
3586:
3582:
3570:
3555:
3552:
3547:
3543:
3538:
3534:
3529:
3455:
3442:
3438:
3436:
3433:
3432:
3422:
3399:
3395:
3393:
3390:
3389:
3385:
3383:complex numbers
3376:
3342:
3338:
3336:
3333:
3332:
3306:
3303:
3302:
3292:
3281:
3229:
3225:
3223:
3220:
3219:
3216:Euler's formula
3203:
3177:
3173:
3171:
3168:
3167:
3159:
3101:
3098:
3097:
3089:
3077:
3073:
3057:
3047:Euler's formula
3037:centred at the
3024:
3017:
3001:
2996:The first five
2984:
2979:
2976:
2973:
2970:
2968:
2967:(base 16) are
2951:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2914:
2897:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2860:
2828:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2797:
2781:
2771:
2764:
2761:
2758:
2757:
2755:
2748:
2745:
2742:
2741:
2739:
2732:
2729:
2726:
2725:
2723:
2716:
2713:
2710:
2709:
2707:
2700:
2697:
2694:
2693:
2691:
2684:
2681:
2678:
2677:
2675:
2668:
2665:
2662:
2661:
2659:
2636:
2607:
2606:
2576:
2571:
2562:
2558:
2557:
2552:
2550:
2543:
2538:
2529:
2525:
2524:
2519:
2517:
2510:
2505:
2496:
2492:
2491:
2486:
2484:
2477:
2472:
2465:
2460:
2458:
2425:
2420:
2411:
2407:
2406:
2401:
2399:
2392:
2387:
2378:
2374:
2373:
2368:
2366:
2359:
2354:
2345:
2341:
2340:
2335:
2333:
2326:
2321:
2314:
2309:
2307:
2274:
2269:
2260:
2256:
2255:
2250:
2248:
2241:
2236:
2227:
2223:
2222:
2217:
2215:
2208:
2203:
2194:
2190:
2189:
2184:
2182:
2175:
2170:
2161:
2157:
2156:
2151:
2149:
2136:
2129:
2127:
2124:
2123:
2115:
2107:
2095:
2091:
2083:
2080:
2077:
2076:
2074:
2073:
2065:
2062:
2059:
2058:
2056:
2055:
2047:
2044:
2041:
2040:
2038:
2037:
2033:
2029:
1964:
1959:
1952:
1947:
1945:
1937:
1932:
1925:
1920:
1918:
1910:
1905:
1898:
1893:
1891:
1883:
1878:
1871:
1866:
1864:
1856:
1851:
1844:
1839:
1837:
1829:
1824:
1817:
1812:
1810:
1802:
1797:
1790:
1785:
1783:
1768:
1765:
1764:
1756:
1753:common fraction
1748:
1745:
1706:
1704:
1701:
1700:
1682:
1677:
1675:
1672:
1671:
1660:
1656:
1619:
1615:
1613:
1599:
1595:
1593:
1591:
1588:
1587:
1567:
1547:
1540:
1534:
1526:Richard Feynman
1517:
1509:
1497:
1494:Yasumasa Kanada
1489:
1478:
1466:
1455:
1449:
1437:
1425:
1411:
1408:common fraction
1403:
1395:
1392:
1389:
1388:
1386:
1385:
1377:
1374:
1371:
1370:
1368:
1367:
1355:
1353:
1344:
1337:complex numbers
1288:
1269:
1180:
1177:
1176:
1169:
1163:
1152:
1144:
1125:
1117:
1109:
1090:
1065:
1061:
1046:
1044:
1038:
1030:
1018:
1015:
1014:
987:
983:
974:
970:
968:
965:
964:
920:
912:
909:
908:
896:
875:
873:
870:
869:
848:
846:
843:
842:
819:
811:
808:
807:
801:
792:
777:
770:
763:
752:
745:
734:
726:
722:
716:
702:
698:
690:
681:
676:
671:
662:
658:
654:
610:
598:
590:
572:infinite series
567:
559:
551:
540:
536:
524:
509:
497:
454:
452:
449:
448:
432:
395:
391:
386:
378:
344:
336:
304:Indiana pi bill
287:A History of Pi
265:Yasumasa Kanada
90:
87:
84:
81:
79:
61:
43:
28:
23:
22:
15:
12:
11:
5:
23912:
23902:
23901:
23896:
23891:
23886:
23869:
23868:
23866:
23865:
23860:
23858:Transcendental
23855:
23849:
23846:
23845:
23838:
23836:
23834:
23833:
23823:
23812:
23811:
23806:
23801:
23794:
23784:
23779:
23769:
23764:
23754:
23744:
23734:
23728:
23727:
23717:
23715:Cube root of 2
23712:
23699:
23694:
23684:
23679:
23677:Logarithm of 2
23673:
23672:
23667:
23662:
23652:
23647:
23636:
23633:
23632:
23625:
23624:
23617:
23610:
23602:
23596:
23595:
23581:
23558:
23527:
23506:
23505:External links
23503:
23501:
23500:
23494:
23475:
23469:
23452:
23450:
23447:
23445:
23444:
23438:
23421:
23415:
23395:
23378:
23361:
23352:
23346:
23327:
23321:
23300:
23294:
23282:Borwein, Peter
23273:
23259:
23244:
23238:
23223:
23207:
23205:
23202:
23199:
23198:
23177:
23163:
23119:
23100:(3): 136–140.
23074:
23053:
23031:
22978:
22929:
22922:
22896:
22865:
22826:
22796:
22789:
22766:
22737:
22708:
22701:
22680:
22668:
22651:
22640:
22618:
22577:
22570:
22552:
22522:
22493:(5): 361–372.
22473:
22458:
22424:
22391:
22372:
22355:
22348:
22330:
22323:
22305:
22282:
22255:
22232:
22225:
22207:
22200:
22182:
22175:
22157:
22131:(4): 393–402.
22105:
22086:
22079:
22067:Titchmarsh, E.
22058:
22043:
22036:
22024:Mumford, David
22012:
22000:
21985:
21967:
21930:(11): 112101.
21914:
21885:(2): 421–424.
21867:
21852:
21840:
21833:
21812:
21805:
21791:Ernesto Cesàro
21782:
21770:
21733:(4): 579–586.
21717:
21705:
21690:
21672:
21660:
21654:
21629:
21622:
21604:
21597:
21579:
21572:
21554:
21547:
21529:
21511:
21506:; Chapter XII
21504:. p. 293.
21486:
21467:
21448:
21446:, Section 2.7.
21436:
21421:
21408:
21387:(2): 821–844.
21362:
21344:
21309:(1): 286–292.
21293:
21271:10.1.1.57.7077
21264:(9): 847–875.
21248:
21227:
21190:(1): 353–372.
21174:
21159:
21137:
21125:
21112:Hilbert, David
21103:
21088:
21072:
21069:. p. 594.
21045:
21019:
21005:. Birkhäuser.
20989:
20977:
20960:and powers of
20944:
20913:
20877:
20848:10.1.1.55.3762
20818:(April 1997).
20816:Plouffe, Simon
20799:
20784:
20765:(3): 195–203.
20749:
20734:
20707:(4): 318–328.
20676:
20661:
20644:
20631:(3): 451–465.
20606:
20589:
20570:(8): 916–918.
20554:
20542:
20517:(April 2006).
20515:Plouffe, Simon
20506:
20497:
20478:
20471:
20431:
20419:
20407:
20395:
20383:
20371:
20359:
20328:
20313:
20301:
20263:
20256:
20234:
20219:
20207:
20205:, p. 131.
20195:
20183:
20148:
20113:
20111:, p. 197.
20098:
20096:, p. 205.
20086:
20059:
20026:
19975:
19913:
19847:
19818:
19802:
19750:
19741:"Lecture XXIV"
19731:
19694:
19692:, p. 166.
19675:
19641:(3): 116–121.
19615:
19608:
19585:
19526:
19502:
19479:
19469:
19456:
19454:, p. 196.
19444:
19418:
19405:
19390:
19383:
19365:
19353:
19324:(8): 681–687.
19308:
19296:
19272:(2): 112–117.
19253:
19217:
19205:
19193:
19066:"Estimating π"
19056:
19025:
18961:
18946:
18934:
18887:
18885:, p. 189.
18875:
18818:
18803:
18777:
18762:
18745:
18728:
18726:, p. 187.
18713:
18706:
18686:
18674:
18648:(5): 291–306.
18614:
18602:
18572:
18565:
18533:
18498:
18496:, p. 183.
18486:
18474:
18472:, p. 182.
18457:
18424:
18391:
18389:, p. 180.
18376:
18374:, p. 179.
18364:
18362:, p. 178.
18352:
18350:, p. 177.
18340:
18325:
18313:
18301:
18299:, p. 168.
18289:
18287:, p. 176.
18277:
18275:, p. 171.
18265:
18250:
18212:
18200:
18198:, p. 170.
18188:
18182:978-0691120676
18181:
18157:
18138:
18118:
18116:, p. 167.
18103:
18091:
18084:
18066:
18051:
18035:
18024:
17958:
17956:, p. 242.
17946:
17944:, p. 240.
17934:
17915:(5): 456–458.
17893:
17882:(3): 383–405.
17863:
17851:
17836:
17816:
17810:. Gale Group.
17794:
17787:
17769:
17757:
17745:
17733:
17721:
17709:
17697:
17662:(3): 570–572.
17646:
17631:
17610:
17592:
17574:
17559:
17552:
17527:
17518:Landau, Edmund
17509:
17484:
17458:
17456:, p. 129.
17441:
17420:
17403:
17377:
17345:
17290:
17278:
17242:
17220:
17170:3.14159, &
17157:
17154:
17151:
17148:
17144:
17141:
17136:
17127:
17123:
17119:
17113:
17105:
17101:
17097:
17085:
17082:
17076:
17071:
17062:
17058:
17054:
17048:
17040:
17036:
17032:
17020:
17017:
17011:
17006:
16999:
16996:
16990:
16984:
16981:
16953:of particular
16927:Jones, William
16913:
16904:Ganita Bharati
16894:
16881:
16880:
16878:
16875:
16872:
16871:
16854:
16842:
16834:
16830:
16826:
16823:
16818:
16815:
16807:
16802:
16799:
16795:
16791:
16788:
16772:
16754:
16753:
16751:
16748:
16746:
16743:
16742:
16741:
16736:
16728:
16718:
16715:
16670:
16667:
16565:. In the 1967
16537:'s 1985 novel
16528:William Shanks
16480:
16477:
16440:method of loci
16384:Main article:
16381:
16378:
16366:
16361:
16349:
16346:
16342:
16338:
16335:
16332:
16329:
16319:
16316:
16312:
16308:
16305:
16302:
16299:
16294:
16290:
16261:
16240:
16237:
16234:
16231:
16228:
16225:
16222:
16219:
16162:fluid dynamics
16147:
16140:
16136:
16131:
16128:
16123:
16119:
16112:
16109:
16056:
16040:
16035:
16031:
16021:
16014:
16011:
16006:
16003:
15998:
15994:
15987:
15984:
15979:
15976:
15948:The fact that
15935:
15929:
15926:
15922:
15917:
15914:
15911:
15907:
15904:
15848:
15842:
15839:
15833:
15830:
15827:
15824:
15766:
15763:
15761:
15758:
15729:
15726:
15723:
15719:
15715:
15711:
15688:
15684:
15680:
15677:
15674:
15671:
15643:
15639:
15598:
15595:
15592:
15589:
15586:
15583:
15580:
15577:
15572:
15568:
15547:
15544:
15541:
15520:
15516:
15513:
15510:
15505:
15501:
15479:
15475:
15472:
15438:
15435:
15432:
15429:
15426:
15423:
15420:
15417:
15414:
15411:
15407:
15404:
15379:
15375:
15371:
15367:
15364:
15346:
15343:
15299:Mandelbrot set
15279:Mandelbrot set
15270:
15267:
15227:
15221:
15218:
15215:
15210:
15207:
15203:
15200:
15197:
15194:
15186:
15181:
15178:
15174:
15168:
15165:
15160:
15157:
15154:
15151:
15148:
15145:
15111:Poisson kernel
15073:
15070:
15067:
15064:
15061:
15054:
15051:
15046:
15042:
15037:
15030:
15025:
15022:
15018:
14990:
14987:
14982:
14978:
14973:
14968:
14963:
14960:
14955:
14952:
14949:
14946:
14943:
14900:
14897:
14864:
14861:
14858:
14855:
14852:
14849:
14846:
14841:
14838:
14835:
14832:
14829:
14826:
14823:
14820:
14817:
14813:
14809:
14806:
14803:
14800:
14797:
14794:
14791:
14788:
14785:
14752:
14749:
14746:
14742:
14738:
14735:
14706:
14701:
14697:
14693:
14690:
14687:
14684:
14681:
14678:
14675:
14672:
14668:
14662:
14657:
14654:
14651:
14648:
14644:
14640:
14637:
14634:
14631:
14628:
14625:
14622:
14599:
14595:
14591:
14586:
14581:
14578:
14556:
14552:
14548:
14543:
14538:
14535:
14505:elliptic curve
14468:
14465:
14401:
14398:
14395:
14392:
14389:
14385:
14381:
14378:
14375:
14372:
14367:
14363:
14285:Fourier series
14265:p-adic numbers
14253:
14252:Fourier series
14250:
14211:
14206:
14203:
14198:
14195:
14192:
14189:
14186:
14182:
14179:
14175:
14172:
14169:
14166:
14144:
14141:
14138:
14135:
14132:
14129:
14126:
14120:
14117:
14114:
14111:
14108:
14105:
14098:
14093:
14089:
14086:
14080:
14076:
14073:
14065:
14062:
14059:
14055:
14049:
14045:
14041:
14038:
14035:
14032:
14029:
14003:
13991:
13932:
13929:
13926:
13923:
13916:
13912:
13908:
13903:
13897:
13894:
13891:
13888:
13884:
13879:
13876:
13874:
13872:
13866:
13863:
13856:
13852:
13848:
13843:
13836:
13832:
13828:
13823:
13820:
13816:
13811:
13808:
13806:
13804:
13799:
13796:
13791:
13782:
13779:
13775:
13771:
13768:
13764:
13757:
13752:
13748:
13743:
13738:
13735:
13733:
13730:
13722:
13718:
13714:
13709:
13706:
13702:
13696:
13691:
13687:
13683:
13682:
13624:Leonhard Euler
13601:
13598:
13591:
13587:
13583:
13578:
13571:
13567:
13563:
13558:
13551:
13547:
13543:
13538:
13535:
13532:
13529:
13526:
13439:
13436:
13410:
13404:
13401:
13398:
13392:
13389:
13386:
13382:
13376:
13370:
13367:
13360:
13356:
13352:
13349:
13346:
13343:
13337:
13334:
13329:
13325:
13321:
13318:
13315:
13312:
13309:
13306:
13303:
13300:
13289: + 1
13278:
13257:
13241:
13233:
13230:
13227:
13224:
13220:
13216:
13209:
13205:
13201:
13196:
13193:
13189:
13180:
13177:
13174:
13170:
13166:
13163:
13137:
13132:
13127:
13124:
13119:
13112:
13109:
13106:
13101:
13098:
13095:
13060:
13054:
13051:
13048:
13043:
13039:
13033:
13030:
13027:
13022:
13019:
13016:
13012:
13005:
13002:
12999:
12996:
12970:
12965:
12962:
12959:
12955:
12947:
12943:
12940:
12935:
12932:
12926:
12922:
12915:
12911:
12907:
12903:
12899:
12893:
12890:
12887:
12884:
12879:
12876:
12873:
12869:
12849:
12844:
12840:
12832:
12828:
12825:
12820:
12817:
12811:
12807:
12801:
12797:
12793:
12789:
12783:
12780:
12777:
12774:
12769:
12765:
12735:
12704:
12648:
12644:
12640:
12637:
12634:
12628:
12624:
12620:
12616:
12608:
12603:
12600:
12597:
12593:
12587:
12582:
12579:
12576:
12572:
12566:
12563:
12560:
12557:
12554:
12525:
12519:
12514:
12508:
12505:
12502:
12498:
12494:
12491:
12488:
12466:
12461:
12458:
12455:
12451:
12447:
12444:
12441:
12417:
12414:
12411:
12408:
12405:
12402:
12399:
12396:
12393:
12390:
12387:
12372:gamma function
12353:
12350:
12326:
12323:
12310:tangent vector
12306:winding number
12301:turning number
12292:is called the
12283:
12282:
12271:
12268:
12265:
12262:
12259:
12256:
12253:
12249:
12246:
12243:
12240:
12235:
12230:
12226:
12172:winding number
12160:
12152:
12150:
12147:
12134:
12127:
12122:
12117:
12113:
12110:
12106:
12101:
12098:
12095:
12091:
12087:
12084:
12049:
12029:
12026:
12023:
12020:
11998:
11993:
11963:
11943:
11940:
11936:
11932:
11923:The factor of
11912:
11908:
11903:
11898:
11894:
11891:
11885:
11882:
11878:
11873:
11870:
11866:
11862:
11859:
11811:
11808:
11776:
11771:
11767:
11763:
11760:
11757:
11754:
11751:
11748:
11745:
11742:
11739:
11736:
11733:
11730:
11726:
11723:
11720:
11717:
11712:
11708:
11633:
11628:
11624:
11620:
11617:
11614:
11611:
11608:
11605:
11602:
11599:
11590:
11586:
11582:
11579:
11574:
11571:
11568:
11565:
11557:
11553:
11535:
11496:of the curve.
11494:winding number
11481:
11441:
11438:
11435:
11432:
11429:
11421:
11417:
11413:
11410:
11405:
11402:
11394:
11390:
11370:
11329:
11326:
11302:
11299:
11296:
11293:
11290:
11287:
11284:
11281:
11278:
11275:
11271:
11266:
11262:
11258:
11255:
11252:
11249:
11246:
11203:
11200:
11197:
11194:
11191:
11188:
11185:
11182:
11179:
11175:
11170:
11166:
11108:symmetry group
11092:Uniformization
11084:
11081:
11025:
11020:
11017:
11014:
11006:
11002:
10998:
10994:
10988:
10983:
10980:
10976:
10937:
10934:
10930:
10918:The factor of
10905:
10900:
10895:
10891:
10887:
10884:
10880:
10874:
10870:
10866:
10863:
10860:
10857:
10854:
10850:
10840:
10837:
10832:
10828:
10823:
10820:
10817:
10814:
10811:
10764:The fields of
10723:
10720:
10681:
10676:
10671:
10667:
10664:
10658:
10653:
10648:
10645:
10642:
10639:
10635:
10629:
10624:
10621:
10617:
10610:
10607:
10603:
10597:
10592:
10588:
10584:
10581:
10575:
10570:
10565:
10562:
10559:
10553:
10550:
10543:
10537:
10533:
10527:
10522:
10519:
10515:
10510:
10505:
10501:
10498:
10492:
10487:
10482:
10479:
10476:
10473:
10469:
10463:
10459:
10453:
10448:
10445:
10441:
10436:
10377:
10374:
10371:
10365:
10362:
10359:
10356:
10353:
10350:
10346:
10342:
10339:
10336:
10333:
10328:
10323:
10320:
10316:
10312:
10309:
10306:
10303:
10297:
10294:
10250:
10247:
10192:
10188:
10184:
10179:
10173:
10168:
10163:
10159:
10153:
10149:
10144:
10137:
10133:
10129:
10124:
10118:
10113:
10108:
10105:
10101:
10095:
10091:
10086:
10079:
10076:
10024:
10020:
10016:
10013:
10010:
9988:
9984:
9980:
9977:
9949:
9946:
9914:
9910:
9906:
9903:
9900:
9897:
9892:
9888:
9884:
9881:
9878:
9875:
9812:
9807:
9803:
9799:
9796:
9793:
9790:
9734:
9731:
9718:
9715:
9712:
9709:
9706:
9701:
9696:
9692:
9677:[0, 1]
9673:singular value
9634:
9631:
9628:
9622:
9617:
9612:
9609:
9606:
9602:
9599:
9594:
9588:
9583:
9579:
9575:
9572:
9569:
9563:
9558:
9553:
9550:
9547:
9544:
9540:
9534:
9529:
9525:
9519:
9515:
9463:
9459:
9456:
9453:
9450:
9447:
9444:
9441:
9438:
9374:is called the
9337:
9334:
9330:
9327:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9279:
9276:
9273:
9269:
9266:
9245:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9221:
9218:
9215:
9211:
9208:
9179:[0, 1]
9134:eigenfunctions
9121:
9118:
9106:
9102:
9098:
9095:
9092:
9089:
9086:
9082:
9078:
9075:
9072:
9069:
9066:
9063:
9054:
9050:
9047:
9044:
9041:
9038:
9034:
9030:
9027:
9024:
9021:
9018:
9015:
8930:Main article:
8927:
8926:Units of angle
8924:
8899:
8875:
8871:
8867:
8864:
8840:
8835:
8832:
8827:
8824:
8821:
8813:
8809:
8805:
8802:
8795:
8790:
8787:
8783:
8711:
8710:
8694:
8663:
8641:
8621:
8557:
8554:
8533:
8530:
8456:
8452:
8445:
8442:
8439:
8436:
8432:
8427:
8421:
8418:
8415:
8412:
8408:
8403:
8397:
8394:
8391:
8388:
8384:
8379:
8373:
8370:
8367:
8364:
8360:
8354:
8346:
8342:
8338:
8331:
8326:
8323:
8320:
8316:
8312:
8309:
8264:
8261:
8227:
8219:
8215:
8210:
8204:
8200:
8195:
8191:
8188:
8183:
8180:
8172:
8169:
8166:
8162:
8158:
8155:
8133:
8116:
8095:
8091:
8085:
8080:
8077:
8074:
8070:
8066:
8061:
8057:
8041:
8032:
7984:
7949:
7943:
7940:
7935:
7932:
7929:
7923:
7920:
7858:
7851:
7850:
7835:
7828:
7827:
7826:
7825:
7824:
7822:
7819:
7768:
7764:
7757:
7754:
7749:
7746:
7742:
7737:
7732:
7726:
7723:
7718:
7715:
7711:
7706:
7701:
7695:
7692:
7687:
7683:
7678:
7672:
7664:
7660:
7656:
7649:
7644:
7641:
7638:
7634:
7630:
7625:
7621:
7603:used the PSLQ
7567:
7559:
7556:
7552:
7548:
7545:
7542:
7537:
7533:
7529:
7525:
7522:
7519:
7516:
7513:
7508:
7505:
7502:
7499:
7496:
7493:
7490:
7487:
7484:
7481:
7478:
7470:
7465:
7462:
7459:
7455:
7449:
7445:
7439:
7434:
7431:
7383:
7376:
7371:
7368:
7364:
7360:
7354:
7350:
7346:
7341:
7338:
7335:
7332:
7329:
7326:
7323:
7320:
7317:
7314:
7311:
7303:
7298:
7295:
7292:
7288:
7282:
7276:
7271:
7265:
7260:
7257:
7211:
7208:
7196:supercomputers
7151:
7145:
7102:and scientist
7100:Eugene Salamin
7071:for computing
7023:
7015:
7011:
7007:
7000:
6996:
6990:
6986:
6982:
6977:
6973:
6969:
6963:
6960:
6934:
6929:
6925:
6921:
6918:
6913:
6910:
6907:
6903:
6897:
6892:
6888:
6882:
6879:
6876:
6872:
6868:
6863:
6859:
6855:
6850:
6846:
6842:
6837:
6833:
6829:
6824:
6821:
6818:
6814:
6792:
6785:
6781:
6775:
6771:
6765:
6760:
6757:
6754:
6750:
6744:
6739:
6733:
6729:
6725:
6720:
6716:
6709:
6704:
6701:
6698:
6694:
6671:
6668:
6663:
6659:
6654:
6649:
6646:
6641:
6636:
6632:
6627:
6621:
6617:
6612:
6607:
6603:
6598:
6595:
6592:
6587:
6583:
6563:
6561:
6558:
6556:
6553:
6463:
6460:
6426:6.28...
6409:
6406:
6370:
6367:
6354:likewise used
6331:
6328:
6325:
6281:semiperipheria
6259:Leonhard Euler
6257:
6250:
6249:
6241:
6234:
6233:
6232:
6231:
6230:
6228:
6222:
6214:transcendental
6167:
6164:
6157:
6153:
6149:
6144:
6137:
6133:
6129:
6124:
6117:
6113:
6109:
6104:
6097:
6093:
6089:
6084:
6079:
6074:
6070:
6015:
6012:
5990:
5989:
5986:
5983:
5980:
5977:
5974:
5963:
5960:
5954:
5951:
5948:
5945:
5942:
5938:
5933:
5927:
5924:
5921:
5918:
5915:
5911:
5906:
5900:
5897:
5894:
5891:
5888:
5884:
5879:
5875:
5871:
5868:
5857:
5856:
5850:
5847:
5844:
5841:
5838:
5835:
5824:
5821:
5816:
5813:
5808:
5803:
5800:
5795:
5790:
5787:
5782:
5777:
5774:
5769:
5764:
5761:
5756:
5751:
5748:
5743:
5738:
5735:
5730:
5727:
5716:
5715:
5714:Converges to:
5712:
5711:After 5th term
5709:
5708:After 4th term
5706:
5705:After 3rd term
5703:
5702:After 2nd term
5700:
5699:After 1st term
5697:
5674:
5671:
5665:
5662:
5659:
5656:
5653:
5649:
5644:
5638:
5635:
5632:
5629:
5626:
5622:
5617:
5611:
5608:
5605:
5602:
5599:
5595:
5590:
5584:
5581:
5578:
5575:
5572:
5568:
5563:
5560:
5557:
5554:
5513:
5510:
5505:
5502:
5497:
5492:
5489:
5484:
5479:
5476:
5471:
5466:
5463:
5458:
5453:
5450:
5445:
5440:
5437:
5432:
5427:
5424:
5419:
5416:
5381:
5378:
5369:William Shanks
5354:Zacharias Dase
5326:
5320:
5317:
5311:
5308:
5305:
5302:
5296:
5293:
5287:
5284:
5281:
5278:
5272:
5269:
5255:Leonhard Euler
5252:
5251:
5240:
5237:
5229:
5225:
5219:
5215:
5211:
5208:
5205:
5199:
5195:
5186:
5183:
5180:
5175:
5172:
5169:
5163:
5155:
5151:
5145:
5141:
5137:
5134:
5131:
5125:
5121:
5113:
5110:
5105:
5097:
5093:
5089:
5086:
5082:
5077:
5074:
5071:
5068:
5024:
5019:
5016:
5011:
5008:
5005:
5000:
4997:
4992:
4989:
4986:
4983:
4978:
4975:
4939:
4935:
4930:
4927:
4896:
4893:
4890:
4870:
4867:
4864:
4842:
4839:
4811:
4808:
4803:
4798:
4794:
4788:
4783:
4778:
4774:
4768:
4763:
4758:
4754:
4748:
4745:
4742:
4739:
4736:
4733:
4720:expansion for
4657:
4652:
4645:
4642:
4637:
4632:
4629:
4622:
4617:
4612:
4605:
4602:
4597:
4592:
4589:
4582:
4577:
4572:
4565:
4562:
4557:
4552:
4549:
4542:
4537:
4532:
4525:
4522:
4517:
4512:
4509:
4502:
4497:
4492:
4489:
4476:Wallis product
4457:
4452:
4444:
4439:
4436:
4431:
4428:
4422:
4417:
4411:
4406:
4403:
4397:
4392:
4388:
4382:
4377:
4374:
4345:François Viète
4330:Madhava series
4311:Tantrasamgraha
4249:
4235:
4232:
4189:
4185:
4181:
4178:
4167:François Viète
4152:
4148:
4144:
4141:
4128:produced nine
4110:
4107:
4102:
4098:
4092:
4089:
4023:
4020:
4015:
4012:
4007:
4004:
3984:
3981:
3976:
3973:
3968:
3965:
3945:
3942:
3939:
3936:
3933:
3903:mathematician
3868:
3688:
3685:
3652:
3649:
3644:
3638:
3630:
3627:
3620:
3551:
3548:
3531:Main article:
3528:
3525:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3462:
3458:
3454:
3451:
3448:
3445:
3441:
3429:roots of unity
3410:
3407:
3402:
3398:
3362:
3359:
3356:
3353:
3348:
3345:
3341:
3316:
3313:
3310:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3235:
3232:
3228:
3191:
3188:
3185:
3180:
3176:
3164:imaginary unit
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3068:, one number (
3054:complex number
3016:
3013:
3012:
3011:
2994:
2961:
2907:
2839:
2838:
2791:
2653:
2635:
2632:
2585:
2582:
2579:
2565:
2561:
2549:
2546:
2532:
2528:
2516:
2513:
2499:
2495:
2483:
2480:
2468:
2457:
2434:
2431:
2428:
2414:
2410:
2398:
2395:
2381:
2377:
2365:
2362:
2348:
2344:
2332:
2329:
2317:
2306:
2283:
2280:
2277:
2263:
2259:
2247:
2244:
2230:
2226:
2214:
2211:
2197:
2193:
2181:
2178:
2164:
2160:
2148:
2145:
2142:
2139:
2137:
2135:
2132:
2131:
2110:cannot have a
1973:
1970:
1967:
1955:
1943:
1940:
1928:
1916:
1913:
1901:
1889:
1886:
1874:
1862:
1859:
1847:
1835:
1832:
1820:
1808:
1805:
1793:
1781:
1778:
1775:
1772:
1744:
1741:
1710:
1685:
1681:
1641:
1638:
1635:
1632:
1627:
1622:
1618:
1612:
1607:
1602:
1598:
1533:
1530:
1465:The digits of
1352:
1349:
1341:absolute value
1331:integers (the
1276:
1272:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1076:
1068:
1064:
1060:
1057:
1052:
1049:
1041:
1036:
1033:
1029:
1025:
1022:
998:
995:
990:
986:
982:
977:
973:
955:—a concept in
927:
924:
919:
916:
882:
879:
855:
852:
826:
823:
818:
815:
762:
759:
675:
672:
670:
667:
631:thermodynamics
604:supercomputers
462:
459:
380:
379:
377:
376:
369:
362:
354:
351:
350:
349:
348:
340:
332:
327:
319:
318:
317:Related topics
314:
313:
312:
311:
306:
298:
297:
293:
292:
291:
290:
283:
275:
274:
270:
269:
268:
267:
262:
257:
252:
247:
245:William Shanks
242:
237:
232:
227:
222:
220:François Viète
217:
212:
207:
202:
197:
192:
187:
179:
178:
174:
173:
172:
171:
166:
161:
159:Approximations
156:
154:Less than 22/7
148:
147:
143:
142:
141:
140:
135:
127:
126:
122:
121:
120:
119:
114:
109:
101:
100:
96:
95:
75:
74:
66:
65:
56:
55:
26:
9:
6:
4:
3:
2:
23911:
23900:
23897:
23895:
23892:
23890:
23887:
23885:
23882:
23881:
23879:
23864:
23863:Trigonometric
23861:
23859:
23856:
23854:
23853:Schizophrenic
23851:
23850:
23847:
23842:
23827:
23824:
23817:
23814:
23813:
23810:
23807:
23805:
23802:
23797:
23793:
23788:
23785:
23783:
23780:
23773:
23770:
23768:
23765:
23758:
23755:
23748:
23747:Erdős–Borwein
23745:
23738:
23735:
23733:
23730:
23729:
23721:
23720:Plastic ratio
23718:
23716:
23713:
23708:
23703:
23700:
23698:
23695:
23688:
23685:
23683:
23680:
23678:
23675:
23674:
23671:
23668:
23666:
23663:
23656:
23653:
23651:
23648:
23641:
23638:
23637:
23634:
23630:
23623:
23618:
23616:
23611:
23609:
23604:
23603:
23600:
23593:
23592:
23587:
23586:
23582:
23565:
23564:Search Engine
23559:
23556:
23555:
23551:
23548:
23544:and analysed
23543:
23539:
23536:
23528:
23523:
23522:
23517:
23514:
23509:
23508:
23497:
23495:2-902918-25-9
23491:
23487:
23480:
23476:
23472:
23466:
23462:
23454:
23453:
23441:
23435:
23431:
23427:
23422:
23418:
23412:
23407:
23406:
23396:
23391:
23381:
23375:
23371:
23362:
23358:
23353:
23349:
23343:
23339:
23335:
23334:
23328:
23324:
23318:
23313:
23312:
23306:
23301:
23297:
23291:
23287:
23283:
23279:
23274:
23262:
23256:
23252:
23251:
23245:
23241:
23235:
23231:
23230:
23224:
23220:
23216:
23215:
23209:
23208:
23187:
23181:
23173:
23167:
23148:
23144:
23140:
23133:
23129:
23128:Knuth, Donald
23123:
23115:
23111:
23107:
23103:
23099:
23095:
23088:
23083:
23078:
23063:
23057:
23049:
23045:
23041:
23035:
23024:
23020:
23016:
23012:
23008:
23004:
23000:
22993:
22982:
22971:
22967:
22963:
22959:
22955:
22951:
22947:
22946:Math Horizons
22940:
22933:
22925:
22919:
22915:
22911:
22900:
22884:
22880:
22876:
22869:
22861:
22857:
22853:
22849:
22845:
22841:
22840:Math Horizons
22837:
22830:
22814:
22810:
22806:
22800:
22792:
22786:
22782:
22781:
22776:
22770:
22762:
22758:
22754:
22750:
22749:
22741:
22734:
22729:
22725:
22724:
22719:
22712:
22704:
22698:
22694:
22690:
22684:
22677:
22672:
22665:
22661:, p. 118
22660:
22655:
22647:
22643:
22637:
22633:
22632:
22627:
22621:
22615:
22611:
22607:
22606:
22601:
22596:
22592:
22588:
22581:
22573:
22567:
22563:
22556:
22540:
22536:
22532:
22526:
22518:
22514:
22509:
22504:
22500:
22496:
22492:
22488:
22484:
22477:
22469:
22465:
22461:
22459:9789004433373
22455:
22451:
22447:
22443:
22439:
22428:
22412:
22408:
22407:
22402:
22395:
22388:
22384:
22381:
22376:
22369:
22364:
22362:
22360:
22351:
22349:0-521-66396-2
22345:
22341:
22334:
22326:
22320:
22316:
22309:
22301:
22297:
22293:
22289:
22285:
22279:
22275:
22274:
22269:
22265:
22259:
22251:
22247:
22243:
22236:
22228:
22226:0-471-14854-7
22222:
22218:
22211:
22203:
22197:
22193:
22186:
22178:
22172:
22168:
22161:
22142:
22138:
22134:
22130:
22126:
22119:
22112:
22110:
22103:; Chapter II.
22100:
22096:
22090:
22082:
22076:
22072:
22068:
22062:
22054:
22047:
22039:
22033:
22029:
22025:
22019:
22017:
22009:
22004:
21996:
21992:
21988:
21982:
21978:
21971:
21963:
21959:
21955:
21951:
21947:
21943:
21938:
21933:
21929:
21925:
21918:
21910:
21906:
21902:
21898:
21893:
21888:
21884:
21880:
21879:
21871:
21863:
21856:
21849:
21844:
21836:
21834:0-486-25778-9
21830:
21826:
21822:
21821:Ogilvy, C. S.
21816:
21808:
21802:
21798:
21792:
21786:
21779:
21774:
21766:
21762:
21758:
21754:
21750:
21746:
21741:
21736:
21732:
21728:
21721:
21715:, p. 190
21714:
21709:
21701:
21694:
21686:
21682:
21676:
21669:
21664:
21657:
21655:3-540-41160-7
21651:
21647:
21643:
21640:Gilbarg, D.;
21636:
21634:
21625:
21619:
21615:
21608:
21600:
21594:
21590:
21583:
21575:
21573:0-393-96997-5
21569:
21565:
21558:
21550:
21544:
21540:
21533:
21525:
21521:
21520:Ahlfors, Lars
21515:
21509:
21503:
21499:
21498:
21490:
21481:
21477:
21471:
21462:
21458:
21452:
21445:
21440:
21433:
21428:
21426:
21418:
21412:
21404:
21400:
21395:
21390:
21386:
21382:
21381:
21376:
21369:
21367:
21358:
21354:
21348:
21340:
21336:
21332:
21328:
21324:
21320:
21316:
21312:
21308:
21304:
21297:
21289:
21285:
21281:
21277:
21272:
21267:
21263:
21259:
21252:
21243:
21238:
21231:
21223:
21219:
21215:
21211:
21207:
21203:
21198:
21193:
21189:
21185:
21178:
21170:
21163:
21155:
21151:
21147:
21141:
21135:, p. 47.
21134:
21129:
21121:
21117:
21113:
21107:
21100:
21095:
21093:
21085:
21081:
21080:Abramson 2014
21076:
21068:
21064:
21060:
21056:
21049:
21043:
21038:
21034:
21030:
21026:
21022:
21016:
21012:
21008:
21004:
21000:
20993:
20986:
20981:
20972:
20967:
20948:
20932:
20928:
20924:
20917:
20901:
20897:
20893:
20886:
20881:
20870:
20866:
20862:
20858:
20854:
20849:
20844:
20840:
20836:
20832:
20828:
20821:
20817:
20813:
20809:
20803:
20796:
20791:
20789:
20780:
20776:
20772:
20768:
20764:
20760:
20753:
20747:, p. 77.
20746:
20741:
20739:
20730:
20726:
20722:
20718:
20714:
20710:
20706:
20702:
20701:
20693:
20689:
20683:
20681:
20673:
20668:
20666:
20658:
20654:, pp. 43
20653:
20648:
20639:
20634:
20630:
20626:
20625:
20620:
20616:
20610:
20604:, p. 105
20603:
20598:
20593:
20585:
20581:
20577:
20573:
20569:
20565:
20558:
20551:
20546:
20527:
20520:
20516:
20510:
20501:
20493:
20492:The New Stack
20489:
20482:
20474:
20468:
20464:
20460:
20456:
20452:
20448:
20444:
20438:
20436:
20429:, p. 254
20428:
20423:
20416:
20411:
20404:
20399:
20393:, p. 104
20392:
20387:
20380:
20375:
20369:, p. 18.
20368:
20363:
20347:
20343:
20339:
20332:
20325:. p. B5.
20324:
20317:
20310:
20305:
20286:
20279:
20272:
20270:
20268:
20259:
20253:
20249:
20243:
20238:
20232:, p. 87.
20231:
20226:
20224:
20216:
20211:
20204:
20199:
20192:
20187:
20179:
20175:
20171:
20167:
20163:
20159:
20152:
20144:
20140:
20136:
20132:
20129:(29): 11–15.
20128:
20124:
20117:
20110:
20105:
20103:
20095:
20090:
20083:
20072:
20071:
20063:
20047:
20043:
20039:
20038:
20030:
20022:
20017: : "Let
20016:
20012:
20009:
20005:
20002:
19994:
19990:
19986:
19979:
19972:
19964:
19960:
19956:
19952:
19948:
19944:
19937:
19930:
19926:
19925:
19917:
19906:
19902:
19899:
19895:
19877:
19873:
19869:
19865:
19858:
19851:
19835:
19831:
19830:
19822:
19815:
19811:
19806:
19798:
19794:
19789:
19784:
19780:
19776:
19772:
19768:
19761:
19754:
19746:
19742:
19735:
19726:
19725:
19716:
19712:
19708:
19707:
19698:
19691:
19686:
19684:
19682:
19680:
19672:
19670:
19662:
19656:
19652:
19648:
19644:
19640:
19636:
19631:
19627:
19619:
19611:
19605:
19601:
19600:
19592:
19590:
19578:
19571:
19567:
19561:
19557:
19551:
19547:
19541:
19537:
19530:
19523:
19520:
19516:
19505:
19499:
19495:
19494:
19486:
19484:
19473:
19466:
19460:
19453:
19448:
19440:
19436:
19432:
19428:
19427:Lindemann, F.
19422:
19415:
19409:
19403:, p. 284
19402:
19397:
19395:
19386:
19380:
19376:
19369:
19362:
19357:
19349:
19345:
19340:
19335:
19331:
19327:
19323:
19319:
19312:
19305:
19300:
19293:
19287:
19283:
19279:
19275:
19271:
19267:
19260:
19258:
19242:
19238:
19234:
19233:
19228:
19221:
19214:
19209:
19202:
19197:
19191:
19188:
19184:
19180:
19176:
19172:
19168:
19162:
19159:
19155:
19151:
19147:
19143:
19139:
19129:
19126:
19122:
19118:
19114:
19110:
19105:
19102:
19098:
19097:
19092:
19088:
19081:
19077:Reprinted in
19074:
19067:
19060:
19054:
19051:
19047:
19043:
19042:Newton, Isaac
19036:
19029:
19010:
19006:
19002:
18998:
18994:
18990:
18986:
18979:
18971:
18970:Lehmer, D. H.
18965:
18958:
18953:
18951:
18943:
18938:
18930:
18926:
18922:
18918:
18914:
18910:
18906:
18902:
18891:
18884:
18879:
18864:on 4 May 2019
18860:
18856:
18852:
18848:
18844:
18840:
18836:
18829:
18822:
18815:
18810:
18808:
18799:
18795:
18788:
18781:
18774:
18769:
18767:
18758:
18757:
18749:
18742:
18738:
18732:
18725:
18720:
18718:
18709:
18703:
18699:
18698:
18690:
18683:
18678:
18659:
18655:
18651:
18647:
18643:
18636:
18625:
18623:
18621:
18619:
18611:
18606:
18598:
18594:
18590:
18586:
18582:
18576:
18568:
18562:
18558:
18554:
18550:
18549:
18544:
18537:
18521:
18514:
18513:
18508:
18502:
18495:
18490:
18483:
18478:
18471:
18466:
18464:
18462:
18445:
18441:
18440:
18435:
18428:
18419:
18414:
18410:
18406:
18402:
18395:
18388:
18383:
18381:
18373:
18368:
18361:
18356:
18349:
18344:
18338:, p. 202
18337:
18332:
18330:
18322:
18317:
18310:
18305:
18298:
18293:
18286:
18281:
18274:
18269:
18261:
18257:
18253:
18247:
18243:
18239:
18235:
18227:
18221:
18219:
18217:
18209:
18204:
18197:
18192:
18184:
18178:
18174:
18170:
18169:
18161:
18145:
18141:
18135:
18131:
18130:
18122:
18115:
18110:
18108:
18101:, p. 14.
18100:
18095:
18087:
18081:
18077:
18070:
18064:, p. 592
18063:
18058:
18056:
18048:
18044:
18043:Abramson 2014
18039:
18031:
18027:
18021:
18017:
18016:
18011:
18006:
18002:
17997:
17993:
17989:
17985:
17981:
17977:
17973:
17969:
17962:
17955:
17950:
17943:
17938:
17930:
17926:
17922:
17918:
17914:
17910:
17909:
17897:
17889:
17885:
17881:
17877:
17870:
17868:
17861:, p. 33.
17860:
17855:
17848:
17843:
17841:
17823:
17819:
17813:
17808:
17807:
17798:
17790:
17784:
17780:
17779:History of Pi
17773:
17767:, p. 129
17766:
17761:
17754:
17749:
17742:
17737:
17730:
17725:
17718:
17713:
17706:
17701:
17693:
17689:
17685:
17681:
17677:
17673:
17669:
17665:
17661:
17657:
17650:
17643:
17638:
17636:
17627:
17623:
17617:
17615:
17606:
17602:
17596:
17588:
17584:
17583:Ahlfors, Lars
17578:
17570:
17563:
17555:
17549:
17544:
17543:
17534:
17532:
17523:
17519:
17513:
17505:
17501:
17497:
17496:
17488:
17480:
17476:
17472:
17467:
17462:
17455:
17450:
17448:
17446:
17438:
17434:
17430:
17424:
17417:
17412:
17410:
17408:
17391:
17387:
17381:
17374:
17371:
17367:
17361:
17360:
17352:
17350:
17341:
17337:
17333:
17329:
17325:
17321:
17316:
17311:
17307:
17303:
17302:
17294:
17288:, p. 17.
17287:
17282:
17266:
17262:
17261:
17256:
17252:
17246:
17238:
17234:
17230:
17224:
17218:
17215:
17211:
17202:Reprinted in
17199:
17194:
17193:
17188:
17187:
17182:
17177:
17173:
17155:
17152:
17149:
17142:
17139:
17125:
17121:
17117:
17111:
17103:
17099:
17095:
17083:
17080:
17074:
17060:
17056:
17052:
17046:
17038:
17034:
17030:
17018:
17015:
17009:
16997:
16994:
16988:
16982:
16979:
16964:
16960:
16956:
16952:
16948:
16942:
16938:
16934:
16933:
16928:
16922:
16920:
16918:
16910:(1–4): 68–71.
16909:
16905:
16898:
16892:, p. 59.
16891:
16886:
16882:
16868:
16864:
16863:Taylor series
16858:
16840:
16832:
16828:
16824:
16821:
16816:
16813:
16797:
16793:
16789:
16786:
16776:
16769:
16768:normal number
16759:
16755:
16740:
16737:
16735:
16729:
16727:
16721:
16720:
16714:
16712:
16702:
16697:
16691:
16687:
16684:
16676:
16666:
16660:
16656:
16652:
16647:
16637:
16629:
16627:
16623:
16612:
16609:
16605:
16601:
16600:college cheer
16597:
16593:
16585:
16580:
16574:
16570:
16569:
16564:
16560:
16559:
16550:
16542:
16541:
16536:
16531:
16529:
16521:
16517:
16512:
16502:
16498:
16490:
16485:
16476:
16470:
16462:
16461:
16452:
16443:
16441:
16429:
16425:
16421:
16412:
16410:
16402:
16401:
16392:
16387:
16377:
16364:
16359:
16347:
16344:
16340:
16336:
16333:
16330:
16327:
16317:
16314:
16310:
16306:
16303:
16300:
16297:
16292:
16288:
16279:
16275:
16271:
16267:
16260:
16256:
16251:
16238:
16235:
16232:
16229:
16226:
16223:
16220:
16217:
16208:
16204:
16200:
16195:
16189:
16184:
16179:
16175:
16171:
16163:
16160:The field of
16158:
16145:
16138:
16134:
16129:
16126:
16121:
16117:
16110:
16107:
16098:
16094:
16089:
16085:
16080:
16074:
16069:
16061:
16055:
16038:
16033:
16029:
16019:
16012:
16009:
16004:
16001:
15996:
15992:
15985:
15982:
15977:
15974:
15963:
15959:
15955:
15946:
15933:
15927:
15924:
15920:
15915:
15912:
15905:
15894:
15889:
15883:
15878:
15873:
15868:
15864:
15859:
15846:
15840:
15837:
15831:
15828:
15825:
15822:
15814:
15809:
15803:
15798:
15793:
15788:
15784:
15772:
15757:
15743:
15724:
15686:
15682:
15678:
15675:
15669:
15660:
15641:
15637:
15628:
15624:
15619:
15613:
15593:
15587:
15584:
15578:
15570:
15566:
15545:
15542:
15539:
15511:
15508:
15503:
15499:
15473:
15470:
15462:
15458:
15453:
15436:
15433:
15427:
15421:
15418:
15412:
15405:
15402:
15394:
15365:
15362:
15353:
15342:
15330:
15310:
15300:
15296:
15280:
15275:
15266:
15258:
15251:
15247:
15240:The constant
15238:
15225:
15219:
15216:
15213:
15208:
15205:
15198:
15192:
15176:
15172:
15166:
15163:
15158:
15152:
15146:
15143:
15136:
15132:
15128:
15124:
15120:
15116:
15112:
15108:
15104:
15099:
15089:
15084:
15071:
15068:
15065:
15062:
15059:
15052:
15049:
15044:
15040:
15035:
15020:
15016:
15007:
14988:
14985:
14980:
14976:
14971:
14966:
14961:
14958:
14953:
14947:
14941:
14934:
14926:
14921:
14914:
14910:
14905:
14896:
14894:
14887:also involve
14886:
14882:
14862:
14856:
14853:
14850:
14844:
14839:
14836:
14833:
14830:
14827:
14824:
14821:
14818:
14815:
14811:
14807:
14801:
14798:
14795:
14792:
14789:
14783:
14775:
14768:The constant
14766:
14750:
14747:
14744:
14740:
14736:
14733:
14726:
14722:
14704:
14699:
14695:
14691:
14688:
14685:
14682:
14679:
14676:
14673:
14670:
14666:
14652:
14649:
14646:
14642:
14638:
14632:
14629:
14626:
14620:
14613:
14584:
14541:
14524:
14523:modular group
14520:
14516:
14512:
14511:Modular forms
14508:
14506:
14502:
14498:
14494:
14490:
14489:modular forms
14483:The constant
14478:
14473:
14464:
14462:
14454:
14450:
14446:
14434:
14426:
14421:
14415:
14399:
14396:
14393:
14390:
14387:
14383:
14379:
14373:
14365:
14361:
14351:
14344:
14340:
14335:
14330:
14325:
14320:
14315:
14309:
14303:
14299:
14295:
14290:
14286:
14279:The constant
14274:
14273:Tate's thesis
14270:
14266:
14258:
14249:
14247:
14243:
14239:
14235:
14231:
14222:
14209:
14204:
14201:
14196:
14187:
14180:
14177:
14173:
14167:
14164:
14155:
14142:
14136:
14133:
14130:
14124:
14115:
14112:
14109:
14096:
14091:
14087:
14084:
14078:
14074:
14071:
14063:
14060:
14057:
14053:
14047:
14043:
14039:
14033:
14027:
14014:
14011:
14009:
13999:
13997:
13987:
13983:
13979:
13975:
13971:
13967:
13958:
13952:
13947:
13930:
13924:
13921:
13914:
13910:
13906:
13901:
13892:
13886:
13882:
13877:
13875:
13864:
13861:
13854:
13850:
13846:
13841:
13834:
13830:
13826:
13821:
13818:
13814:
13809:
13807:
13797:
13794:
13789:
13780:
13777:
13773:
13769:
13766:
13762:
13750:
13746:
13741:
13736:
13734:
13728:
13720:
13716:
13712:
13707:
13704:
13700:
13689:
13685:
13671:
13664:
13657:
13650:
13645:
13637:
13633:
13632:number theory
13625:
13621:
13620:Basel problem
13617:
13612:
13599:
13596:
13589:
13585:
13581:
13576:
13569:
13565:
13561:
13556:
13549:
13545:
13541:
13536:
13530:
13524:
13514:
13507:
13503:
13499:
13484:
13483:modular group
13480:
13474:
13469:
13464:
13457:
13453:
13449:
13444:
13435:
13433:
13432:lattice point
13429:
13425:
13421:
13408:
13402:
13399:
13396:
13390:
13387:
13384:
13380:
13374:
13368:
13365:
13358:
13350:
13347:
13344:
13335:
13327:
13316:
13313:
13310:
13301:
13298:
13288:
13281:
13276:
13270:
13266:
13260:
13252:
13239:
13231:
13228:
13225:
13222:
13218:
13214:
13207:
13203:
13199:
13194:
13191:
13187:
13172:
13164:
13161:
13153:
13135:
13130:
13125:
13122:
13117:
13110:
13107:
13104:
13099:
13096:
13093:
13084:
13077:
13071:
13058:
13049:
13041:
13037:
13028:
13020:
13017:
13014:
13010:
13003:
13000:
12997:
12994:
12986:
12981:
12968:
12963:
12960:
12957:
12953:
12945:
12941:
12938:
12933:
12930:
12924:
12913:
12909:
12905:
12901:
12897:
12891:
12885:
12877:
12874:
12871:
12867:
12847:
12842:
12838:
12830:
12826:
12823:
12818:
12815:
12809:
12799:
12795:
12791:
12787:
12781:
12775:
12767:
12763:
12754:
12752:
12744:
12738:
12734:
12729:
12726:in Euclidean
12725:
12721:
12719:
12712:
12707:
12703:
12697:
12695:
12688:
12684:
12674:
12669:
12646:
12642:
12638:
12635:
12632:
12626:
12622:
12618:
12614:
12601:
12598:
12595:
12591:
12585:
12580:
12577:
12574:
12570:
12564:
12558:
12545:development:
12544:
12539:
12523:
12517:
12512:
12506:
12500:
12496:
12492:
12464:
12459:
12453:
12449:
12445:
12415:
12409:
12406:
12403:
12397:
12391:
12377:
12373:
12368:
12351:
12348:
12340:
12331:
12320:for surfaces.
12319:
12315:
12311:
12307:
12303:
12302:
12297:
12296:
12291:
12269:
12266:
12263:
12260:
12257:
12254:
12251:
12244:
12238:
12233:
12228:
12224:
12216:
12215:
12214:
12212:
12208:
12204:
12200:
12197:
12193:
12189:
12186:study of the
12185:
12173:
12164:
12157:
12146:
12132:
12111:
12108:
12104:
12099:
12096:
12074:
12065:
12063:
12047:
12027:
12024:
11996:
11981:
11977:
11941:
11938:
11934:
11930:
11910:
11892:
11889:
11883:
11880:
11876:
11871:
11849:
11845:
11841:
11837:
11833:
11829:
11828:Coulomb's law
11825:
11821:
11814:The constant
11807:
11803:
11799:
11794:
11790:
11769:
11765:
11761:
11758:
11752:
11749:
11746:
11743:
11740:
11737:
11734:
11731:
11728:
11721:
11715:
11710:
11706:
11696:
11690:
11685:
11679:
11675:
11670:
11665:
11659:
11652:
11648:
11626:
11622:
11615:
11612:
11609:
11606:
11603:
11600:
11597:
11588:
11584:
11580:
11577:
11569:
11563:
11555:
11551:
11534:
11527:
11523:
11517:
11510:
11506:
11502:
11497:
11495:
11490:
11480:
11471:
11467:
11458:
11452:
11439:
11436:
11433:
11430:
11427:
11419:
11415:
11411:
11408:
11403:
11400:
11392:
11388:
11378:
11369:
11364:
11361:. A form of
11359:
11355:
11351:
11347:
11343:
11334:
11325:
11323:
11319:
11314:
11300:
11297:
11294:
11291:
11288:
11285:
11282:
11279:
11276:
11273:
11269:
11264:
11260:
11256:
11250:
11244:
11236:
11232:
11228:
11224:
11218:
11192:
11189:
11186:
11183:
11180:
11177:
11173:
11164:
11155:
11152:
11144:
11140:
11136:
11132:
11125:The constant
11116:
11112:
11109:
11105:
11101:
11097:
11096:Klein quartic
11093:
11089:
11080:
11078:
11073:
11060:
11052:
11047:
11041:
11023:
11018:
11015:
11012:
11004:
11000:
10996:
10992:
10978:
10974:
10965:
10961:
10956:
10935:
10932:
10928:
10916:
10903:
10893:
10889:
10885:
10878:
10872:
10864:
10861:
10858:
10852:
10848:
10838:
10835:
10830:
10826:
10821:
10815:
10809:
10794:
10787:
10783:
10779:
10775:
10771:
10767:
10750:
10745:
10741:
10737:
10733:
10728:
10719:
10717:
10713:
10709:
10701:
10697:
10692:
10679:
10674:
10669:
10665:
10662:
10656:
10643:
10637:
10619:
10615:
10608:
10605:
10601:
10595:
10590:
10586:
10582:
10579:
10573:
10560:
10548:
10535:
10531:
10517:
10513:
10508:
10503:
10499:
10496:
10490:
10477:
10471:
10461:
10457:
10443:
10439:
10434:
10421:
10416:
10413:
10407:
10401:
10396:
10388:
10375:
10372:
10369:
10363:
10360:
10357:
10354:
10351:
10348:
10344:
10337:
10331:
10318:
10314:
10310:
10304:
10292:
10280:
10275:
10271:
10264:The constant
10260:
10255:
10246:
10244:
10240:
10236:
10231:
10226:
10221:
10215:
10210:
10190:
10186:
10182:
10177:
10171:
10161:
10151:
10147:
10142:
10135:
10131:
10127:
10122:
10116:
10106:
10093:
10089:
10084:
10077:
10074:
10062:
10058:
10054:
10049:
10047:
10022:
10014:
9986:
9978:
9967:
9963:
9947:
9935:
9930:
9912:
9904:
9895:
9890:
9882:
9876:
9873:
9865:
9857:
9853:
9844:
9841:
9837:
9831:
9827:
9810:
9805:
9801:
9797:
9794:
9791:
9788:
9777:of perimeter
9776:
9768:
9764:
9752:
9748:
9744:
9739:
9730:
9713:
9710:
9707:
9699:
9694:
9690:
9682:
9681:Sobolev space
9674:
9666:
9656:
9649:
9632:
9629:
9626:
9620:
9607:
9600:
9597:
9586:
9581:
9577:
9573:
9570:
9567:
9561:
9548:
9542:
9532:
9527:
9523:
9517:
9513:
9504:
9496:
9490:
9483:
9479:
9451:
9448:
9445:
9439:
9436:
9428:
9424:
9420:
9416:
9407:
9404:
9400:
9393:
9390:
9386:
9382:
9377:
9371:
9365:
9361:
9356:
9352:
9335:
9332:
9325:
9318:
9295:
9289:
9286:
9283:
9280:
9274:
9267:
9264:
9243:
9240:
9234:
9228:
9225:
9222:
9216:
9209:
9206:
9198:
9192:
9188:
9184:
9175:
9170:
9166:
9161:
9143:
9139:
9135:
9131:
9126:
9117:
9104:
9100:
9096:
9093:
9090:
9087:
9084:
9080:
9076:
9073:
9070:
9067:
9064:
9061:
9052:
9048:
9045:
9042:
9039:
9036:
9032:
9028:
9025:
9022:
9019:
9016:
9013:
9004:
8998:
8984:
8975:radians, and
8962:
8958:
8946:
8942:
8938:
8933:
8923:
8921:
8917:
8913:
8897:
8873:
8869:
8865:
8862:
8851:
8838:
8833:
8830:
8825:
8822:
8819:
8811:
8807:
8803:
8800:
8793:
8788:
8785:
8781:
8768:
8764:
8762:
8758:
8754:
8750:
8742:
8738:
8733:
8731:
8727:
8725:
8719:
8717:
8707:
8700:
8695:
8691:
8669:
8664:
8660:
8653:
8647:
8642:
8638:
8631:
8626:
8622:
8618:
8611:
8606:
8605:
8604:
8598:
8594:
8590:
8586:
8571:
8562:
8553:
8547:
8543:
8542:many formulae
8529:
8522:
8516:
8512:
8508:
8500:
8496:
8492:
8487:
8480:
8472:
8467:
8454:
8450:
8443:
8440:
8437:
8434:
8430:
8425:
8419:
8416:
8413:
8410:
8406:
8401:
8395:
8392:
8389:
8386:
8382:
8377:
8371:
8368:
8365:
8362:
8358:
8352:
8344:
8340:
8336:
8324:
8321:
8318:
8314:
8310:
8307:
8299:
8296:
8291:
8289:
8284:
8278:
8274:
8260:
8249:
8247:
8243:
8238:
8225:
8217:
8202:
8198:
8186:
8181:
8178:
8164:
8156:
8153:
8141:
8136:
8124:
8119:
8093:
8089:
8083:
8078:
8075:
8072:
8068:
8064:
8059:
8055:
8044:
8035:
8028:
8024:
8002:
7994:
7983:
7975:
7971:
7960:
7947:
7941:
7938:
7933:
7930:
7927:
7921:
7918:
7905:
7899:
7893:
7887:
7881:
7876:
7868:
7855:
7846:
7842:
7838:
7832:
7818:
7815:
7811:
7807:
7802:
7797:
7791:
7790:
7783:
7766:
7762:
7755:
7752:
7747:
7744:
7740:
7735:
7730:
7724:
7721:
7716:
7713:
7709:
7704:
7699:
7693:
7690:
7685:
7681:
7676:
7670:
7662:
7658:
7654:
7642:
7639:
7636:
7632:
7628:
7623:
7619:
7606:
7602:
7601:Simon Plouffe
7597:
7595:
7591:
7578:
7565:
7557:
7554:
7546:
7543:
7535:
7531:
7527:
7523:
7517:
7514:
7503:
7500:
7497:
7494:
7488:
7482:
7479:
7463:
7460:
7457:
7453:
7447:
7443:
7437:
7432:
7429:
7419:
7415:
7411:
7407:
7399:
7394:
7381:
7374:
7369:
7366:
7362:
7358:
7352:
7348:
7344:
7336:
7333:
7330:
7327:
7321:
7315:
7312:
7296:
7293:
7290:
7286:
7280:
7274:
7269:
7263:
7258:
7255:
7245:
7237:
7220:
7216:
7207:
7201:
7197:
7185:
7181:
7177:
7165:
7156:
7144:
7138:
7133:
7132:Peter Borwein
7129:
7125:
7119:
7117:
7113:
7109:
7105:
7104:Richard Brent
7101:
7096:
7094:
7090:
7086:
7078:
7070:
7061:
7058:
7054:
7050:
7046:
7035:
7021:
7013:
7009:
7005:
6998:
6988:
6984:
6980:
6975:
6971:
6961:
6958:
6932:
6927:
6923:
6919:
6916:
6911:
6908:
6905:
6901:
6895:
6890:
6880:
6877:
6874:
6870:
6866:
6861:
6857:
6848:
6844:
6840:
6835:
6831:
6827:
6822:
6819:
6816:
6812:
6790:
6783:
6779:
6773:
6769:
6763:
6758:
6755:
6752:
6748:
6742:
6737:
6731:
6727:
6723:
6718:
6714:
6707:
6702:
6699:
6696:
6692:
6669:
6666:
6661:
6657:
6652:
6647:
6644:
6639:
6634:
6630:
6625:
6619:
6615:
6610:
6605:
6601:
6596:
6593:
6590:
6585:
6581:
6569:
6552:
6542:
6541:Western world
6525:
6524:
6518:
6517:
6503:
6498:
6494:
6490:
6488:
6461:
6458:
6447:
6438:
6437:William Jones
6429:
6424:to represent
6407:
6404:
6393:instead used
6392:
6368:
6365:
6353:
6348:
6329:
6326:
6323:
6314:
6302:
6298:
6292:
6288:
6282:
6278:
6277:semiperimeter
6274:
6270:Greek letter
6260:
6254:
6245:
6244:William Jones
6238:
6221:
6219:
6215:
6207:
6199:
6195:
6191:
6183:
6178:
6165:
6162:
6155:
6151:
6147:
6142:
6135:
6131:
6127:
6122:
6115:
6111:
6107:
6102:
6095:
6091:
6087:
6082:
6077:
6072:
6068:
6057:
6055:
6051:
6050:prime numbers
6043:
6042:Basel problem
6033:
6025:
6024:is irrational
6011:
6009:
6005:
5987:
5984:
5981:
5978:
5975:
5961:
5958:
5952:
5949:
5946:
5943:
5940:
5936:
5931:
5925:
5922:
5919:
5916:
5913:
5909:
5904:
5898:
5895:
5892:
5889:
5886:
5882:
5877:
5873:
5869:
5866:
5859:
5858:
5855:= 3.1415 ...
5848:
5845:
5842:
5839:
5836:
5822:
5819:
5814:
5811:
5806:
5801:
5798:
5793:
5788:
5785:
5780:
5775:
5772:
5767:
5762:
5759:
5754:
5749:
5746:
5741:
5736:
5733:
5728:
5725:
5718:
5717:
5713:
5710:
5707:
5704:
5701:
5698:
5692:
5691:
5688:
5685:
5672:
5669:
5663:
5660:
5657:
5654:
5651:
5647:
5642:
5636:
5633:
5630:
5627:
5624:
5620:
5615:
5609:
5606:
5603:
5600:
5597:
5593:
5588:
5582:
5579:
5576:
5573:
5570:
5566:
5561:
5558:
5555:
5552:
5539:
5524:
5511:
5508:
5503:
5500:
5495:
5490:
5487:
5482:
5477:
5474:
5469:
5464:
5461:
5456:
5451:
5448:
5443:
5438:
5435:
5430:
5425:
5422:
5417:
5414:
5406:
5390:
5377:
5370:
5365:
5363:
5355:
5350:
5342:
5341:in one hour.
5324:
5318:
5315:
5309:
5306:
5303:
5300:
5294:
5291:
5285:
5282:
5279:
5276:
5270:
5267:
5256:
5238:
5235:
5227:
5217:
5213:
5209:
5206:
5197:
5193:
5184:
5181:
5178:
5173:
5170:
5167:
5161:
5153:
5143:
5139:
5135:
5132:
5123:
5119:
5111:
5108:
5103:
5095:
5091:
5087:
5084:
5080:
5075:
5072:
5069:
5066:
5059:
5058:
5057:
5055:
5052:Isaac Newton
5050:
5044:
5035:
5022:
5017:
5014:
5009:
5006:
5003:
4998:
4995:
4990:
4987:
4984:
4981:
4976:
4973:
4963:
4958:
4937:
4933:
4928:
4925:
4917:
4916:Abraham Sharp
4912:
4910:
4894:
4891:
4888:
4868:
4865:
4862:
4840:
4837:
4827:
4822:
4809:
4806:
4801:
4796:
4792:
4786:
4781:
4776:
4772:
4766:
4761:
4756:
4752:
4746:
4743:
4740:
4737:
4734:
4731:
4723:
4719:
4718:Taylor series
4715:
4714:James Gregory
4710:
4700:
4696:
4692:
4680:
4676:
4672:
4668:
4655:
4643:
4640:
4635:
4630:
4627:
4615:
4603:
4600:
4595:
4590:
4587:
4575:
4563:
4560:
4555:
4550:
4547:
4535:
4523:
4520:
4515:
4510:
4507:
4495:
4490:
4487:
4477:
4473:
4468:
4455:
4450:
4442:
4437:
4434:
4429:
4426:
4420:
4415:
4409:
4404:
4401:
4395:
4390:
4386:
4380:
4375:
4372:
4358:
4354:
4350:
4346:
4341:
4335:
4331:
4327:
4323:
4322:
4317:
4313:
4312:
4307:
4299:
4298:Kerala school
4295:
4291:
4290:James Gregory
4283:
4275:
4271:
4260:
4256:
4252:
4240:
4231:
4229:
4225:
4221:
4217:
4209:
4205:
4187:
4183:
4179:
4176:
4168:
4150:
4146:
4142:
4139:
4131:
4127:
4122:
4108:
4105:
4100:
4096:
4090:
4087:
4079:
4068:
4064:
4063:
4058:
4053:
4047:
4043:
4039:
4038:
4021:
4018:
4013:
4010:
4005:
4002:
3982:
3979:
3974:
3971:
3966:
3963:
3943:
3940:
3937:
3934:
3931:
3923:
3910:
3906:
3902:
3866:
3853:, values for
3852:
3851:ancient China
3847:
3841:
3833:
3832:
3827:
3728:
3724:
3708:
3704:
3693:
3684:
3682:
3678:
3677:Shulba Sutras
3670:
3666:
3650:
3647:
3642:
3628:
3625:
3604:
3603:Rhind Papyrus
3580:
3576:
3573:are found in
3567:
3565:
3561:
3546:
3537:
3524:
3511:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3460:
3456:
3452:
3449:
3446:
3443:
3439:
3430:
3425:
3408:
3405:
3400:
3396:
3384:
3379:
3373:
3360:
3357:
3354:
3351:
3346:
3343:
3339:
3330:
3314:
3311:
3308:
3300:
3295:
3290:
3286:
3284:
3280:the constant
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3233:
3230:
3226:
3217:
3213:
3209:
3189:
3186:
3183:
3178:
3174:
3165:
3145:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3112:
3109:
3106:
3103:
3095:
3087:
3086:complex plane
3083:
3071:
3067:
3063:
3055:
3048:
3044:
3043:complex plane
3040:
3036:
3032:
3027:
3021:
3008:
3004:
2999:
2995:
2991:
2987:
2966:
2962:
2958:
2954:
2913:(base 3) are
2912:
2908:
2904:
2900:
2858:
2854:
2850:
2847:The first 48
2846:
2845:
2844:
2843:
2835:
2831:
2795:
2792:
2788:
2784:
2778:
2774:
2657:
2654:
2651:
2648:
2647:
2646:
2644:
2643:
2631:
2629:
2625:
2620:
2583:
2580:
2577:
2563:
2559:
2547:
2544:
2530:
2526:
2514:
2511:
2497:
2493:
2481:
2478:
2466:
2455:
2432:
2429:
2426:
2412:
2408:
2396:
2393:
2379:
2375:
2363:
2360:
2346:
2342:
2330:
2327:
2315:
2304:
2281:
2278:
2275:
2261:
2257:
2245:
2242:
2228:
2224:
2212:
2209:
2195:
2191:
2179:
2176:
2162:
2158:
2146:
2143:
2140:
2138:
2133:
2122:do, such as:
2121:
2113:
2106:. Therefore,
2105:
2101:
2026:
1971:
1968:
1965:
1953:
1941:
1938:
1926:
1914:
1911:
1899:
1887:
1884:
1872:
1860:
1857:
1845:
1833:
1830:
1818:
1806:
1803:
1791:
1779:
1776:
1773:
1770:
1762:
1754:
1740:
1738:
1734:
1730:
1726:
1708:
1683:
1679:
1669:
1667:
1653:
1639:
1636:
1633:
1630:
1625:
1620:
1616:
1610:
1605:
1600:
1596:
1585:
1581:
1577:
1573:
1561:
1557:
1553:
1544:
1539:
1532:Transcendence
1529:
1527:
1523:
1515:
1507:
1503:
1495:
1486:
1484:
1476:
1472:
1463:
1461:
1452:
1447:
1443:
1435:
1434:
1429:
1428:is irrational
1421:
1417:
1409:
1365:
1361:
1348:
1342:
1338:
1334:
1330:
1326:
1322:
1319:
1315:
1312:
1308:
1305:
1301:
1297:
1292:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1242:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1173:
1166:
1161:
1156:
1150:
1141:
1139:
1135:
1131:
1123:
1115:
1114:Edmund Landau
1107:
1103:
1098:
1096:
1087:
1074:
1066:
1062:
1058:
1055:
1050:
1047:
1039:
1034:
1031:
1027:
1023:
1020:
1012:
996:
993:
988:
984:
980:
975:
971:
962:
958:
954:
950:
946:
941:
925:
922:
917:
914:
906:
902:
880:
877:
853:
850:
839:
824:
821:
816:
813:
804:
800:
795:
791:
790:circumference
787:
783:
767:
758:
756:
755:
742:
740:
732:
720:
719:
711:
696:
688:
684:
680:Greek letter
666:
652:
648:
644:
643:number theory
640:
636:
632:
628:
624:
620:
616:
607:
605:
596:
588:
583:
581:
580:William Jones
577:
573:
565:
558:approximated
557:
549:
546:
534:
530:
521:
519:
515:
512:appear to be
507:
503:
495:
491:
487:
483:
479:
476:are commonly
460:
457:
446:
442:
438:
430:
426:
425:circumference
422:
418:
414:
410:
404:
389:
375:
370:
368:
363:
361:
356:
355:
353:
352:
347:
341:
339:
335:Six nines in
333:
331:
330:Basel problem
328:
326:
323:
322:
321:
320:
316:
315:
310:
307:
305:
302:
301:
300:
299:
295:
294:
289:
288:
284:
282:
279:
278:
277:
276:
272:
271:
266:
263:
261:
258:
256:
253:
251:
248:
246:
243:
241:
238:
236:
235:William Jones
233:
231:
228:
226:
225:Seki Takakazu
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
182:
181:
180:
176:
175:
170:
167:
165:
162:
160:
157:
155:
152:
151:
150:
149:
145:
144:
139:
138:Transcendence
136:
134:
133:Irrationality
131:
130:
129:
128:
124:
123:
118:
115:
113:
112:Circumference
110:
108:
105:
104:
103:
102:
98:
97:
94:
77:
76:
72:
68:
67:
64:
58:
57:
53:
49:
48:
45:
41:
37:
33:
19:
18:History of pi
23825:
23795:
23791:
23787:Silver ratio
23757:Golden ratio
23706:
23590:
23584:
23545:
23519:
23482:
23457:
23432:. Springer.
23429:
23401:
23386:
23366:
23356:
23332:
23310:
23285:
23264:. Retrieved
23250:Pi Unleashed
23249:
23228:
23213:
23189:. Retrieved
23180:
23166:
23154:. Retrieved
23142:
23138:
23122:
23097:
23093:
23077:
23065:. Retrieved
23056:
23048:the original
23043:
23034:
23002:
22998:
22981:
22949:
22945:
22932:
22913:
22899:
22887:. Retrieved
22878:
22868:
22843:
22839:
22836:"Pi Instant"
22829:
22817:. Retrieved
22809:USAToday.com
22808:
22799:
22779:
22769:
22752:
22746:
22740:
22731:
22721:
22711:
22692:
22683:
22678:, p. 14
22671:
22666:, p. 50
22654:
22630:
22604:
22580:
22561:
22555:
22543:. Retrieved
22525:
22490:
22486:
22476:
22433:
22427:
22415:. Retrieved
22404:
22394:
22375:
22339:
22333:
22314:
22308:
22272:
22268:Zuber, J.-B.
22264:Itzykson, C.
22258:
22245:
22235:
22216:
22210:
22191:
22185:
22166:
22160:
22148:. Retrieved
22141:the original
22128:
22124:
22098:
22095:Stein, Elias
22089:
22070:
22061:
22052:
22046:
22027:
22010:, Chapter 4.
22003:
21976:
21970:
21927:
21923:
21917:
21882:
21876:
21870:
21861:
21855:
21850:, p. 43
21843:
21824:
21815:
21796:
21785:
21773:
21730:
21726:
21720:
21708:
21699:
21693:
21684:
21675:
21663:
21645:
21613:
21607:
21588:
21582:
21563:
21557:
21538:
21532:
21523:
21514:
21507:
21496:
21489:
21484:; Chapter 6.
21479:
21470:
21460:
21457:Stein, Elias
21451:
21439:
21416:
21411:
21384:
21378:
21356:
21347:
21306:
21302:
21296:
21261:
21257:
21251:
21230:
21187:
21183:
21177:
21168:
21162:
21153:
21149:
21140:
21128:
21119:
21106:
21075:
21062:
21048:
21041:
21002:
20992:
20980:
20947:
20935:. Retrieved
20926:
20916:
20904:. Retrieved
20900:the original
20887:, p. 20
20880:
20830:
20826:
20802:
20762:
20758:
20752:
20704:
20698:
20647:
20628:
20622:
20615:Grünbaum, B.
20609:
20592:
20567:
20563:
20557:
20552:, p. 39
20545:
20533:. Retrieved
20509:
20500:
20491:
20481:
20454:
20422:
20410:
20398:
20386:
20374:
20362:
20350:. Retrieved
20341:
20331:
20322:
20316:
20304:
20292:. Retrieved
20247:
20237:
20210:
20198:
20186:
20161:
20157:
20151:
20126:
20122:
20116:
20089:
20074:
20069:
20062:
20050:. Retrieved
20036:
20029:
20020:
20000:
19996:
19988:
19978:
19966:
19949:(3): 75–84.
19946:
19942:
19932:
19923:
19916:
19890:
19883:. Retrieved
19867:
19866:(in Latin).
19863:
19850:
19838:. Retrieved
19828:
19821:
19805:
19770:
19769:(in Latin).
19766:
19753:
19744:
19734:
19723:
19710:
19705:
19697:
19668:
19660:
19658:
19638:
19634:
19629:
19625:
19618:
19598:
19575:See p. 220:
19539:
19535:
19529:
19518:
19514:
19507:
19492:
19472:
19464:
19459:
19447:
19438:
19434:
19421:
19408:
19374:
19368:
19356:
19321:
19317:
19311:
19299:
19269:
19265:
19244:. Retrieved
19230:
19220:
19208:
19196:
19170:
19166:
19163:
19141:
19137:
19130:
19120:
19116:
19106:
19094:
19084:
19079:
19072:
19059:
19049:
19039:
19034:
19028:
19016:. Retrieved
19009:the original
18988:
18984:
18964:
18937:
18904:
18900:
18890:
18878:
18866:. Retrieved
18859:the original
18838:
18834:
18821:
18797:
18793:
18780:
18755:
18748:
18731:
18696:
18689:
18677:
18665:. Retrieved
18658:the original
18645:
18641:
18605:
18592:
18588:
18575:
18547:
18536:
18520:the original
18511:
18501:
18489:
18477:
18448:. Retrieved
18437:
18427:
18411:(2): 64–85.
18408:
18404:
18394:
18367:
18355:
18343:
18316:
18304:
18292:
18280:
18268:
18233:
18203:
18191:
18167:
18160:
18148:. Retrieved
18128:
18121:
18094:
18075:
18069:
18038:
18014:
18010:Aaboe, Asger
17971:
17967:
17961:
17949:
17937:
17912:
17906:
17896:
17879:
17875:
17854:
17849:, p. 78
17826:. Retrieved
17805:
17797:
17778:
17772:
17760:
17755:, p. 25
17748:
17743:, p. 6.
17736:
17731:, p. 3.
17724:
17712:
17700:
17659:
17655:
17649:
17644:, p. 5.
17625:
17604:
17595:
17586:
17577:
17568:
17562:
17541:
17521:
17512:
17499:
17494:
17487:
17478:
17466:Remmert 2012
17461:
17454:Remmert 2012
17436:
17432:
17429:Apostol, Tom
17423:
17418:, p. 8.
17394:. Retrieved
17380:
17369:
17365:
17363:
17358:
17308:(1): 50–56.
17305:
17299:
17293:
17281:
17269:. Retrieved
17258:
17245:
17232:
17223:
17213:
17201:
17190:
17184:
17180:
17175:
17171:
16962:
16958:
16954:
16950:
16946:
16944:
16931:
16907:
16903:
16897:
16885:
16857:
16775:
16758:
16708:
16698:
16686:Donald Knuth
16672:
16653:to pass the
16648:
16625:
16621:
16613:
16596:inside jokes
16581:
16566:
16556:
16538:
16532:
16519:
16513:
16506:
16468:
16458:
16444:
16427:
16413:
16398:
16389:
16331:1.2566370614
16258:
16252:
16206:
16193:
16187:
16177:
16159:
16096:
16087:
16078:
16072:
16062:
16053:
15961:
15947:
15887:
15881:
15871:
15860:
15807:
15801:
15795:of a simple
15791:
15768:
15658:
15622:
15617:
15611:
15457:vector space
15451:
15351:
15348:
15328:
15308:
15288:
15256:
15245:
15239:
15126:
15100:
15085:
14930:
14913:Maria Agnesi
14911:, named for
14767:
14509:
14482:
14444:
14425:Haar measure
14419:
14416:
14349:
14342:
14339:circle group
14333:
14323:
14313:
14307:
14301:
14297:
14293:
14278:
14269:Prüfer group
14223:
14156:
14015:
14007:
13995:
13977:
13973:
13969:
13959:
13948:
13669:
13662:
13655:
13648:
13613:
13512:
13505:
13501:
13495:
13472:
13455:
13448:Prüfer group
13422:
13286:
13279:
13274:
13268:
13258:
13253:
13082:
13075:
13072:
12982:
12750:
12742:
12736:
12732:
12727:
12723:
12717:
12710:
12705:
12701:
12698:
12672:
12540:
12366:
12336:
12308:of the unit
12304:– it is the
12299:
12293:
12289:
12184:mathematical
12181:
12066:
11813:
11801:
11797:
11694:
11688:
11677:
11673:
11663:
11657:
11650:
11646:
11532:
11525:
11521:
11515:
11508:
11504:
11498:
11488:
11478:
11456:
11453:
11376:
11367:
11357:
11354:Jordan curve
11339:
11315:
11226:
11216:
11153:
11124:
11071:
11048:
10954:
10917:
10763:
10748:
10743:
10739:
10735:
10693:
10417:
10411:
10405:
10399:
10394:
10389:
10278:
10263:
10237:form of the
10229:
10224:
10219:
10213:
10060:
10050:
9965:
9933:
9928:
9855:
9845:
9839:
9835:
9829:
9825:
9775:Jordan curve
9756:
9733:Inequalities
9654:
9647:
9494:
9488:
9481:
9477:
9414:
9408:
9402:
9398:
9391:
9388:
9384:
9380:
9369:
9363:
9359:
9190:
9186:
9173:
9162:
9151:
9002:
8996:
8985:
8981:/180 radians
8954:
8852:
8765:
8734:
8723:
8715:
8712:
8705:
8698:
8689:
8667:
8658:
8651:
8645:
8636:
8629:
8627:with radius
8616:
8609:
8579:
8535:
8523:
8488:
8468:
8292:
8285:
8266:
8250:
8244:, discussed
8239:
8131:
8114:
8039:
8030:
8016:
8000:
7992:
7981:
7961:
7903:
7897:
7891:
7885:
7879:
7865:
7844:
7840:
7813:
7809:
7805:
7795:
7788:
7781:
7598:
7579:
7395:
7228:
7176:cosmological
7169:
7123:
7120:
7097:
7062:
7038:
6950:is given by
6565:
6514:
6501:
6496:
6491:
6440:
6430:
6280:
6267:
6208:proved that
6179:
6058:
6035:
5993:
5686:
5540:
5525:
5383:
5366:
5351:
5343:
5253:
5051:
5036:
4959:
4913:
4823:
4711:
4691:Isaac Newton
4688:
4675:Isaac Newton
4469:
4357:infinite sum
4342:
4319:
4309:
4263:
4254:
4247:
4123:
4060:
4054:
4041:
4035:
3848:
3829:
3806:is equal to
3778:3.1408 <
3716:
3568:
3553:
3423:
3377:
3374:
3293:
3282:
3062:real numbers
3051:
3025:
2841:
2840:
2793:
2655:
2649:
2641:
2637:
2621:
2027:
1746:
1665:
1654:
1565:
1487:
1464:
1450:
1444:(called the
1431:
1424:proofs that
1354:
1333:circle group
1324:
1320:
1307:automorphism
1293:
1171:
1164:
1154:
1142:
1134:power series
1102:Remmert 2012
1099:
1088:
942:
840:
802:
793:
776:
750:
743:
689:In English,
686:
677:
669:Fundamentals
615:trigonometry
608:
584:
522:
415:that is the
408:
385:
383:
285:
230:Takebe Kenko
169:Memorization
78:
60:
44:
23458:The Joy of
23367:The Number
23214:Precalculus
23186:"Crate tau"
23156:17 February
22589:π perfume,
22531:Keith, Mike
21681:Artin, Emil
21415:Feller, W.
19018:21 February
18907:(1): 1–14.
18868:23 February
18667:21 February
17828:19 December
17186:John Machin
16955:Curve Lines
16424:James Jeans
16391:Piphilology
16386:Piphilology
16276:fields and
16264:appears in
16170:Stokes' law
15297:called the
14721:Jacobi form
14501:j-invariant
13978:geometrical
13646:by a prime
13452:L-functions
13428:convex body
13263:denote the
12314:unit circle
12199:plane curve
11832:Gauss's law
11350:rectifiable
11077:Howe (1980)
10766:probability
10698:system, is
10235:variational
9769:: the area
9757:The number
9747:Lord Kelvin
9505:, we have:
9120:Eigenvalues
8916:square root
8910:-axis of a
8570:unit circle
8471:hexadecimal
8023:random walk
7883:is dropped
7398:Bill Gosper
7164:logarithmic
7045:John Wrench
6572:Initialize
6487:John Machin
6477:Periphery (
6028:Proof that
6020:Proof that
5988:3.1396 ...
5371:calculated
4962:John Machin
4953:to compute
4697:discovered
4681:to compute
4472:John Wallis
4130:sexagesimal
4074: 1220
4062:Āryabhaṭīya
3922:Zu Chongzhi
3901:Wei Kingdom
3782:< 3.1429
3579:clay tablet
3388:satisfying
3299:unit circle
3166:satisfying
3035:unit circle
2998:sexagesimal
2965:hexadecimal
1725:constructed
1314:isomorphism
947:around the
570:, based on
533:Babylonians
443:. It is an
437:mathematics
384:The number
255:John Wrench
240:John Machin
195:Zu Chongzhi
32:Pi (letter)
23878:Categories
23687:Lemniscate
23191:6 December
23145:(1): 145.
23005:(3): 7–8.
22992:Is Wrong!"
22889:2 February
22417:27 October
22292:2005053026
21937:1510.07813
21156:: 571–592.
20971:2201.12601
20906:27 October
20344:. London.
20052:15 October
19893:I : π
19885:15 October
19840:15 October
19510:3.14159...
19477:citations.
19441:: 679–682.
19246:22 January
17192:Van Ceulen
16745:References
16535:Carl Sagan
16497:homophones
16469:Not a Wake
15799:of length
15558:the value
15303:(−0.75, 0)
15248:defines a
13970:arithmetic
13614:Finding a
13479:hyperbolic
13080:for large
12722:of radius
12679:Γ(1/2) = π
12211:arc length
11671:, that if
11492:times the
11115:Fano plane
11040:bell curve
10770:statistics
9751:Queen Dido
9409:The value
9376:wavenumber
9183:fixed ends
9165:eigenvalue
8912:semicircle
8288:Stan Wagon
8037:such that
8005:is within
7839:. Needles
7801:odd number
7448:4270934400
7412:) and the
6313:Oughtred's
6190:irrational
6018:See also:
5985:3.1452 ...
5982:3.1333 ...
5979:3.1666 ...
5849:3.3396 ...
5846:2.8952 ...
5843:3.4666 ...
5840:2.6666 ...
4881:. But for
4722:arctangent
4321:Yuktibhāṣā
4282:Archimedes
3907:created a
3776:(that is,
3723:Archimedes
3707:Archimedes
3540:See also:
3381:different
3375:There are
1570:is also a
1536:See also:
1311:continuous
1104:explains,
945:arc length
841:The ratio
761:Definition
647:statistics
548:Archimedes
518:conjecture
488:. It is a
296:In culture
281:Chronology
185:Archimedes
125:Properties
23650:Liouville
23640:Chaitin's
23521:MathWorld
23019:120965049
22966:126179022
22952:(4): 34.
22860:218542599
22846:(3): 22.
22755:(1): 10.
22591:Pi (film)
22487:Neurocase
22468:224869535
21962:119315853
21909:122276856
21887:CiteSeerX
21765:119125713
21757:1615-7168
21740:1205.1270
21339:121881343
21331:0003-9527
21266:CiteSeerX
21242:1110.2960
21214:1618-1891
21192:CiteSeerX
21037:127264210
20843:CiteSeerX
20250:. Wiley.
20075:Si autem
20019:1 :
19999:1 :
19465:Math. Ann
19187:123395287
19158:123532808
19111:(1798) .
19091:"§2.2.30"
18929:121087222
18855:123392772
18450:11 August
17996:126383231
17834:, p. 185.
17692:250798202
17684:0036-0279
17332:0343-6993
17310:CiteSeerX
17147:&
17140:−
17135:¯
17112:−
17070:¯
17047:−
17010:−
17005:¯
16989:−
16877:Citations
16869:function.
16806:∞
16801:∞
16798:−
16794:∫
16787:π
16692:approach
16571:episode "
16568:Star Trek
16563:Kate Bush
16420:mnemonics
16355: N/A
16345:−
16337:×
16334:…
16328:≈
16324: H/m
16315:−
16307:×
16304:π
16289:μ
16257:constant
16230:η
16227:π
16183:spherical
16164:contains
16118:π
16030:α
16013:π
16002:−
15993:π
15978:τ
15928:π
15916:≥
15910:Δ
15903:Δ
15832:π
15826:≈
15714:→
15679:
15673:↦
15662:. Hence
15543:∈
15515:→
15474:∈
15374:→
15337:tends to
15217:−
15185:∞
15180:∞
15177:−
15173:∫
15167:π
15069:π
15029:∞
15024:∞
15021:−
15017:∫
14967:⋅
14962:π
14857:τ
14845:θ
14834:π
14828:−
14825:τ
14819:π
14816:−
14802:τ
14796:τ
14784:θ
14751:τ
14745:π
14705:τ
14692:π
14674:π
14661:∞
14656:∞
14653:−
14643:∑
14633:τ
14621:θ
14391:π
14205:π
14178:ζ
14174:−
14168:
14134:−
14125:ζ
14113:−
14104:Γ
14085:π
14075:
14061:−
14054:π
14028:ζ
13928:%
13922:≈
13911:π
13887:ζ
13865:⋯
13795:−
13778:−
13770:−
13756:∞
13747:∏
13708:−
13695:∞
13686:∏
13644:divisible
13600:⋯
13525:ζ
13400:π
13375:∼
13324:Δ
13302:
13179:∞
13176:→
13162:π
13108:π
13100:∼
12998:π
12961:−
12921:Γ
12902:π
12875:−
12806:Γ
12788:π
12607:∞
12592:∏
12578:γ
12575:−
12553:Γ
12518:π
12487:Γ
12465:π
12440:Γ
12407:−
12386:Γ
12376:factorial
12341:function
12339:factorial
12318:Gauss map
12264:π
12225:∫
12207:curvature
12112:π
12100:−
12083:Φ
12048:δ
12028:δ
12022:Φ
12019:Δ
11962:Φ
11942:π
11893:
11884:π
11858:Φ
11753:
11747:∑
11741:π
11711:γ
11707:∮
11644:provided
11610:π
11581:−
11556:γ
11552:∮
11434:π
11412:−
11393:γ
11389:∮
11301:π
11289:⋅
11286:π
11261:∫
11199:Σ
11193:χ
11190:π
11169:Σ
11165:∫
11137:to their
11024:π
10997:−
10987:∞
10982:∞
10979:−
10975:∫
10936:π
10890:σ
10865:μ
10862:−
10853:−
10839:π
10831:σ
10628:∞
10623:∞
10620:−
10616:∫
10609:π
10591:≥
10583:ξ
10561:ξ
10552:^
10532:ξ
10526:∞
10521:∞
10518:−
10514:∫
10452:∞
10447:∞
10444:−
10440:∫
10395:somewhere
10364:ξ
10355:π
10349:−
10327:∞
10322:∞
10319:−
10315:∫
10305:ξ
10296:^
10148:∫
10104:∇
10090:∫
10078:≤
10075:π
10019:‖
10012:∇
10009:‖
9983:‖
9976:‖
9945:∇
9909:‖
9902:∇
9899:‖
9896:≤
9887:‖
9880:‖
9877:π
9798:≤
9792:π
9578:∫
9574:≤
9524:∫
9514:π
9458:→
9329:↦
9287:λ
9284:−
9226:λ
9130:overtones
9094:π
9085:θ
9077:
9068:θ
9065:
9046:π
9037:θ
9029:
9020:θ
9017:
8866:−
8831:π
8804:−
8786:−
8782:∫
8759:and even
8486:correct.
8473:digit of
8426:−
8402:−
8378:−
8330:∞
8315:∑
8308:π
8171:∞
8168:→
8154:π
8069:∑
7997:, and so
7934:ℓ
7922:≈
7919:π
7753:−
7722:−
7691:−
7648:∞
7633:∑
7620:π
7544:−
7501:545140134
7469:∞
7454:∑
7433:π
7302:∞
7287:∑
7259:π
6962:≈
6959:π
6867:−
6841:−
6516:Mechanica
6511:= 3.14...
6504:= 6.28...
6408:ρ
6405:π
6369:δ
6366:π
6330:π
6324:δ
6166:⋯
6069:π
5962:⋯
5959:−
5950:×
5944:×
5923:×
5917:×
5905:−
5896:×
5890:×
5867:π
5823:⋯
5794:−
5768:−
5742:−
5726:π
5673:⋯
5661:×
5655:×
5643:−
5634:×
5628:×
5607:×
5601:×
5589:−
5580:×
5574:×
5553:π
5512:⋯
5509:−
5483:−
5457:−
5431:−
5415:π
5310:
5286:
5268:π
5239:⋯
5182:⋅
5171:⋅
5070:
5010:
5004:−
4991:
4974:π
4960:In 1706,
4838:π
4828:, equals
4810:⋯
4787:−
4747:−
4735:
4712:In 1671,
4656:⋯
4636:⋅
4616:⋅
4596:⋅
4576:⋅
4556:⋅
4536:⋅
4516:⋅
4488:π
4470:In 1655,
4456:⋯
4421:⋅
4396:⋅
4376:π
4343:In 1593,
4308:verse in
4180:×
4143:×
4106:≈
4067:Fibonacci
4057:Aryabhata
4006:≈
4003:π
3967:≈
3964:π
3944:3.1415927
3938:π
3932:3.1415926
3828:, in his
3648:≈
3550:Antiquity
3503:−
3494:…
3447:π
3347:π
3315:π
3309:φ
3263:φ
3260:
3248:φ
3245:
3234:φ
3187:−
3140:φ
3137:
3125:φ
3122:
3113:⋅
3064:. In the
3045:given by
2759:245850922
2656:Fractions
2645:include:
2584:⋱
2433:⋱
2282:⋱
2134:π
2100:algebraic
1972:⋱
1771:π
1670:(such as
1668:-th roots
1611:−
1475:normality
1406:, but no
1316:from the
1267:∈
1261:∣
1252:π
1237:…
1228:π
1216:π
1198:π
1192:−
1186:…
1162:variable
1059:−
1032:−
1028:∫
1021:π
1009:, as the
949:perimeter
915:π
814:π
739:summation
635:mechanics
623:cosmology
582:in 1706.
529:Egyptians
200:Aryabhata
23550:Archived
23538:Archived
23481:(1997).
23307:(1991).
23284:(1997).
23219:OpenStax
23147:Archived
23023:Archived
22970:Archived
22883:Archived
22819:14 March
22813:Archived
22777:(2011).
22761:27966082
22728:Archived
22646:Archived
22628:(2002).
22602:(1995).
22587:Givenchy
22539:Archived
22517:19585350
22411:Archived
22383:Archived
22300:61200849
22270:(1980).
22250:OpenStax
22150:14 April
22125:Fractals
22097:(1970).
22069:(1948).
22026:(1983).
21683:(1964).
21644:(1983),
21522:(1966).
21478:(1999).
21355:(1989).
21222:16923822
21118:(1966).
21067:OpenStax
21063:Calculus
21057:(2016).
21001:(2019).
20937:26 March
20931:Archived
20927:BBC News
20869:Archived
20721:27641917
20690:(2006).
20617:(1960).
20535:10 April
20526:Archived
20449:(2016).
20352:14 April
20346:Archived
20294:12 April
20285:Archived
20046:Archived
20011:Archived
19997:Denotet
19967:Letting
19901:Archived
19876:Archived
19834:Archived
19429:(1882).
19089:(1755).
19044:(1971).
18972:(1938).
18921:41133896
18800:: 75–83.
18583:(1996).
18509:(1630).
18444:Archived
18144:Archived
18030:Archived
18012:(1964).
17822:Archived
17624:(1979).
17603:(1981).
17585:(1966).
17520:(1934).
17504:Archived
17473:(1841).
17433:Calculus
17431:(1967).
17390:Archived
17340:14318695
17271:12 April
17265:Archived
17237:Archived
16929:(1706).
16717:See also
16703:include
16274:magnetic
16270:electric
16068:buckling
15877:momentum
15797:pendulum
15449:. Then
15406:″
15327:(0.25 +
15307:(−0.75,
14181:′
13976:, and a
13485:, times
13291:. Then
12288:, where
12203:integral
12196:immersed
12174:2 about
12073:n sphere
11789:residues
11470:homotopy
11320:via the
11145:surface
11139:topology
11111:PSL(2,7)
11083:Topology
10211:subsets
10207:for all
9962:gradient
9601:′
9387:) = sin(
9378:. Then
9367:, where
9336:″
9317:operator
9311:. Thus
9268:″
9210:″
8728:, given
8585:ellipses
8536:Because
8493:project
8142:. Then
8129:varies,
8045:∈ {−1,1}
8009:of
7495:13591409
7406:Jonathan
7124:multiply
6683:Iterate
6389:, while
6291:diameter
6218:Legendre
6048:and the
5389:converge
4699:calculus
4306:Sanskrit
4274:sequence
3831:Almagest
3094:rotation
2824:37510...
2765:78256779
2650:Integers
1584:rational
1576:solution
1546:Because
1524:, after
1296:topology
1011:integral
957:calculus
799:diameter
737:denotes
627:fractals
619:geometry
587:calculus
543:BC, the
494:equation
429:diameter
411:") is a
91:26433...
50:Part of
23816:Euler's
23702:Apéry's
23576:√
23430:Numbers
23139:TeX Mag
23114:2689499
22545:29 July
22508:4323087
21995:0217026
21942:Bibcode
21403:0578375
21311:Bibcode
21288:8409465
21029:3930585
20865:6109631
20835:Bibcode
20779:2975006
20729:2211758
20584:2317945
20178:2002052
20143:2002695
19963:2973441
19870:: 351.
19775:Bibcode
19655:2972388
19348:2324715
19274:Bibcode
19048:(ed.).
19005:2302434
18741:A060294
18739::
18260:3203895
18001:Ptolemy
17976:Bibcode
17929:2589152
17888:1743850
17664:Bibcode
17396:18 June
17233:pi2e.ch
17179:. This
16947:Lengths
16606:or the
16540:Contact
16520:pi room
16514:In the
15891:is the
15811:is the
15295:fractal
15293:in the
15133:of the
14517:in the
14477:lattice
14453:lattice
14451:to the
14337:to the
14244:to the
14232:of the
13477:is the
12715:of the
12666:is the
12201:is the
12060:is the
11978:of the
11974:is the
11791:at the
11698:, then
11380:, then
11221:is the
11156:, then
11143:compact
11113:of the
11106:by the
11094:of the
10962:in the
10754:√
10714:of the
10243:Neumann
9960:is the
9659:. Here
9484:(1) = 0
9193:(1) = 0
9181:, with
8961:radians
8686:
8674:
8589:spheres
7229:Modern
6549:6.28...
6545:3.14...
6531:; thus
6391:Gregory
6387:3.14...
6246:in 1706
5403:is the
4109:3.14142
3905:Liu Hui
3896:
3884:
3826:Ptolemy
3821:
3809:
3799:
3787:
3773:
3761:
3753:
3741:
3599:
3587:
3575:Babylon
3558:dating
3527:History
3162:is the
3084:of the
3041:in the
3033:on the
3007:A060707
3005::
2990:A062964
2988::
2980:1319...
2957:A004602
2955::
2911:ternary
2903:A004601
2901::
2893:0011...
2861:11.0010
2834:A000796
2832::
2798:3.14159
2787:A063673
2785::
2777:A063674
2775::
2768:
2756:
2752:
2740:
2736:
2724:
2720:
2708:
2704:
2692:
2688:
2676:
2672:
2660:
2087:
2075:
2069:
2057:
2051:
2039:
1399:
1387:
1381:
1369:
1300:algebra
1160:complex
1158:, of a
797:to its
504:with a
441:physics
427:to its
273:History
205:Madhava
190:Liu Hui
80:3.14159
23682:Dottie
23557:(PDF).
23547:BibNum
23535:online
23492:
23467:
23436:
23413:
23376:
23344:
23319:
23292:
23266:5 June
23257:
23236:
23112:
23017:
22964:
22920:
22858:
22787:
22759:
22699:
22638:
22616:
22595:Pi Day
22593:, and
22568:
22515:
22505:
22466:
22456:
22346:
22321:
22298:
22290:
22280:
22223:
22198:
22173:
22077:
22034:
21993:
21983:
21960:
21907:
21889:
21831:
21803:
21763:
21755:
21652:
21620:
21595:
21570:
21545:
21401:
21337:
21329:
21286:
21268:
21220:
21212:
21194:
21035:
21027:
21017:
20863:
20845:
20777:
20727:
20719:
20582:
20469:
20254:
20176:
20141:
19961:
19797:102382
19795:
19653:
19606:
19500:
19381:
19346:
19185:
19156:
19003:
18927:
18919:
18853:
18704:
18563:
18258:
18248:
18179:
18150:5 June
18136:
18082:
18022:
17994:
17974:: 65.
17927:
17886:
17814:
17785:
17690:
17682:
17550:
17338:
17330:
17312:
17181:Series
16963:Circle
16959:Planes
16594:" for
16584:Pi Day
16558:Aerial
16457:. The
16091:, and
16051:where
15875:) and
15627:kernel
15625:, the
15491:, let
15092:ln(4π)
14503:of an
14122:
14101:
14069:
12662:where
12370:. The
12194:of an
12190:, the
12040:where
11214:where
10209:convex
10059:of an
9968:, and
9653:sin(π
9480:(0) =
9423:energy
9372:> 0
9189:(0) =
8945:cosine
8757:smooth
8595:, and
8511:Hadoop
8507:Yahoo!
8277:spigot
8125:. As
8003:≈ 3.19
7999:2(200)
7995:= 56/5
7988:|
7979:|
7803:, and
7799:is an
7779:where
7547:640320
7416:. The
7091:, and
6352:Barrow
6301:radius
5976:3.0000
5837:4.0000
5307:arctan
5283:arctan
5067:arctan
5007:arctan
4988:arctan
4732:arctan
3278:where
3158:where
3082:origin
3070:radius
3056:, say
3039:origin
3031:points
2969:3.243F
2947:220...
2915:10.010
2859:) are
2849:binary
2794:Digits
2754:, and
2743:104348
2727:103993
2626:, see
2072:, and
1483:normal
1358:is an
1329:modulo
1122:cosine
953:limits
786:circle
637:, and
541:
421:circle
309:Pi Day
177:People
54:on the
23670:Cahen
23665:Omega
23655:Prime
23150:(PDF)
23135:(PDF)
23110:JSTOR
23067:2 May
23026:(PDF)
23015:S2CID
22995:(PDF)
22973:(PDF)
22962:S2CID
22942:(PDF)
22856:S2CID
22757:JSTOR
22733:long)
22464:S2CID
22144:(PDF)
22121:(PDF)
21958:S2CID
21932:arXiv
21905:S2CID
21761:S2CID
21735:arXiv
21335:S2CID
21284:S2CID
21237:arXiv
21218:S2CID
21033:S2CID
20966:arXiv
20872:(PDF)
20861:S2CID
20823:(PDF)
20775:JSTOR
20717:JSTOR
20695:(PDF)
20580:JSTOR
20529:(PDF)
20522:(PDF)
20288:(PDF)
20281:(PDF)
20174:JSTOR
20139:JSTOR
19959:JSTOR
19879:(PDF)
19860:(PDF)
19793:JSTOR
19763:(PDF)
19709:[
19661:never
19651:JSTOR
19344:JSTOR
19183:S2CID
19154:S2CID
19125:E 705
19101:E 212
19069:(PDF)
19012:(PDF)
19001:JSTOR
18981:(PDF)
18925:S2CID
18917:JSTOR
18862:(PDF)
18851:S2CID
18831:(PDF)
18790:(PDF)
18661:(PDF)
18638:(PDF)
18523:(PDF)
18516:(PDF)
17992:S2CID
17925:JSTOR
17688:S2CID
17498:[
17336:S2CID
17196:'
16951:Areas
16949:, or
16699:Many
16592:geeks
16201:with
16199:fluid
16197:in a
15004:is a
14331:from
12987:that
12675:= 1/2
11793:poles
11682:is a
11100:genus
10046:-norm
9763:above
9501:both
9475:with
9415:least
9256:, or
8977:1° =
8914:(the
8739:. By
8730:below
8593:cones
8497:used
8495:PiHex
8479:octal
8246:below
7444:10005
7410:Peter
7334:26390
7246:, is
7057:ENIAC
6493:Euler
6299:(for
6289:(for
4677:used
4351:, an
4078:Dante
4042:Yuelü
3759:<
3755:<
2983:(see
2950:(see
2896:(see
2827:(see
2821:69399
2818:41971
2815:50288
2812:83279
2809:26433
2806:23846
2803:89793
2800:26535
2749:33215
2733:33102
2717:16604
2711:52163
2638:Some
1727:with
1582:with
1550:is a
1318:group
1304:up to
784:of a
782:ratio
419:of a
417:ratio
146:Value
88:23846
85:89793
82:26535
23588:and
23574:and
23516:"Pi"
23490:ISBN
23465:ISBN
23434:ISBN
23411:ISBN
23374:ISBN
23342:ISBN
23317:ISBN
23290:ISBN
23268:2013
23255:ISBN
23234:ISBN
23193:2022
23158:2017
23069:2023
22918:ISBN
22891:2019
22821:2015
22785:ISBN
22697:ISBN
22636:ISBN
22614:ISBN
22566:ISBN
22547:2009
22513:PMID
22454:ISBN
22434:Pi (
22419:2007
22344:ISBN
22319:ISBN
22296:OCLC
22288:LCCN
22278:ISBN
22221:ISBN
22196:ISBN
22171:ISBN
22152:2012
22075:ISBN
22032:ISBN
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13531:2
13528:(
13513:s
13508:)
13506:s
13504:(
13502:ζ
13492:.
13488:π
13473:ζ
13458:.
13456:p
13409:.
13403:n
13397:2
13391:1
13388:+
13385:n
13381:e
13369:!
13366:n
13359:n
13355:)
13351:1
13348:+
13345:n
13342:(
13336:=
13333:)
13328:n
13320:)
13317:1
13314:+
13311:n
13308:(
13305:(
13287:n
13280:n
13275:n
13273:(
13269:n
13259:n
13256:Δ
13240:.
13232:1
13229:+
13226:n
13223:2
13219:n
13215:2
13208:2
13204:!
13200:n
13195:n
13192:2
13188:e
13173:n
13165:=
13136:n
13131:)
13126:e
13123:n
13118:(
13111:n
13105:2
13097:!
13094:n
13083:n
13078:!
13076:n
13059:.
13053:)
13050:r
13047:(
13042:n
13038:V
13032:)
13029:r
13026:(
13021:1
13018:+
13015:n
13011:S
13004:=
13001:r
12995:2
12969:.
12964:1
12958:n
12954:r
12946:)
12942:1
12939:+
12934:2
12931:n
12925:(
12914:2
12910:/
12906:n
12898:n
12892:=
12889:)
12886:r
12883:(
12878:1
12872:n
12868:S
12848:,
12843:n
12839:r
12831:)
12827:1
12824:+
12819:2
12816:n
12810:(
12800:2
12796:/
12792:n
12782:=
12779:)
12776:r
12773:(
12768:n
12764:V
12751:n
12749:(
12745:)
12743:r
12741:(
12737:n
12733:S
12728:n
12724:r
12718:n
12713:)
12711:r
12709:(
12706:n
12702:V
12691:π
12673:z
12664:γ
12647:n
12643:/
12639:z
12636:+
12633:1
12627:n
12623:/
12619:z
12615:e
12602:1
12599:=
12596:n
12586:z
12581:z
12571:e
12565:=
12562:)
12559:z
12556:(
12524:4
12513:3
12507:=
12504:)
12501:2
12497:/
12493:5
12490:(
12460:=
12457:)
12454:2
12450:/
12446:1
12443:(
12430:π
12416:!
12413:)
12410:1
12404:n
12401:(
12398:=
12395:)
12392:n
12389:(
12367:n
12352:!
12349:n
12290:N
12286:π
12270:.
12267:N
12261:2
12258:=
12255:s
12252:d
12248:)
12245:s
12242:(
12239:k
12234:b
12229:a
12178:.
12176:p
12168:π
12158:.
12133:,
12126:|
12121:x
12116:|
12109:4
12105:1
12097:=
12094:)
12090:x
12086:(
12069:π
12025:=
11997:2
11992:R
11939:2
11935:/
11931:1
11911:.
11907:|
11902:x
11897:|
11881:2
11877:1
11872:=
11869:)
11865:x
11861:(
11816:π
11804:)
11802:z
11800:(
11798:g
11775:)
11770:k
11766:a
11762:,
11759:g
11756:(
11744:i
11738:2
11735:=
11732:z
11729:d
11725:)
11722:z
11719:(
11716:g
11695:γ
11689:γ
11680:)
11678:z
11676:(
11674:g
11664:γ
11658:γ
11653:)
11651:z
11649:(
11647:f
11632:)
11627:0
11623:z
11619:(
11616:f
11613:i
11607:2
11604:=
11601:z
11598:d
11589:0
11585:z
11578:z
11573:)
11570:z
11567:(
11564:f
11541:γ
11536:0
11533:z
11528:)
11526:z
11524:(
11522:f
11516:γ
11511:)
11509:z
11507:(
11505:f
11489:i
11482:0
11479:z
11474:γ
11462:π
11457:γ
11440:.
11437:i
11431:2
11428:=
11420:0
11416:z
11409:z
11404:z
11401:d
11377:γ
11371:0
11368:z
11358:γ
11298:4
11295:=
11292:2
11283:2
11280:=
11277:A
11274:d
11270:1
11265:S
11257:=
11254:)
11251:S
11248:(
11245:A
11227:S
11217:χ
11202:)
11196:(
11187:2
11184:=
11181:A
11178:d
11174:K
11154:K
11147:Σ
11127:π
11072:e
11067:π
11063:π
11055:π
11044:π
11019:=
11016:u
11013:d
11005:2
11001:u
10993:e
10955:f
10933:2
10929:1
10904:.
10899:)
10894:2
10886:2
10883:(
10879:/
10873:2
10869:)
10859:x
10856:(
10849:e
10836:2
10827:1
10822:=
10819:)
10816:x
10813:(
10810:f
10800:π
10796:σ
10789:μ
10760:.
10756:π
10749:x
10744:e
10740:x
10738:(
10736:ƒ
10704:π
10680:.
10675:2
10670:)
10666:x
10663:d
10657:2
10652:|
10647:)
10644:x
10641:(
10638:f
10634:|
10606:4
10602:1
10596:(
10587:)
10580:d
10574:2
10569:|
10564:)
10558:(
10549:f
10542:|
10536:2
10509:(
10504:)
10500:x
10497:d
10491:2
10486:|
10481:)
10478:x
10475:(
10472:f
10468:|
10462:2
10458:x
10435:(
10424:π
10412:L
10406:L
10400:L
10392:π
10376:.
10373:x
10370:d
10361:x
10358:i
10352:2
10345:e
10341:)
10338:x
10335:(
10332:f
10311:=
10308:)
10302:(
10293:f
10279:f
10266:π
10230:G
10225:u
10220:R
10214:G
10191:2
10187:/
10183:1
10178:)
10172:2
10167:|
10162:u
10158:|
10152:G
10143:(
10136:2
10132:/
10128:1
10123:)
10117:2
10112:|
10107:u
10100:|
10094:G
10085:(
10065:π
10061:n
10044:L
10040:L
10023:1
10015:f
9987:2
9979:f
9966:f
9948:f
9934:R
9929:f
9913:1
9905:f
9891:2
9883:f
9874:2
9860:π
9856:n
9848:π
9840:r
9836:P
9830:r
9826:A
9811:,
9806:2
9802:P
9795:A
9789:4
9779:P
9771:A
9759:π
9717:]
9714:1
9711:,
9708:0
9705:[
9700:1
9695:0
9691:H
9669:π
9661:π
9657:)
9655:x
9648:f
9633:,
9630:x
9627:d
9621:2
9616:|
9611:)
9608:x
9605:(
9598:f
9593:|
9587:1
9582:0
9571:x
9568:d
9562:2
9557:|
9552:)
9549:x
9546:(
9543:f
9539:|
9533:1
9528:0
9518:2
9498:′
9495:f
9489:f
9482:f
9478:f
9462:C
9455:]
9452:1
9449:,
9446:0
9443:[
9440::
9437:f
9411:π
9403:π
9399:ν
9394:)
9392:x
9389:π
9385:x
9383:(
9381:f
9370:ν
9364:ν
9360:λ
9333:f
9326:f
9313:λ
9299:)
9296:x
9293:(
9290:f
9281:=
9278:)
9275:t
9272:(
9265:f
9244:0
9241:=
9238:)
9235:x
9232:(
9229:f
9223:+
9220:)
9217:x
9214:(
9207:f
9191:f
9187:f
9174:f
9158:π
9154:π
9148:.
9146:π
9105:.
9101:)
9097:k
9091:2
9088:+
9081:(
9071:=
9053:)
9049:k
9043:2
9040:+
9033:(
9023:=
9003:k
8997:θ
8992:π
8988:π
8979:π
8973:π
8969:π
8965:π
8951:.
8949:π
8898:x
8874:2
8870:x
8863:1
8839:.
8834:2
8826:=
8823:x
8820:d
8812:2
8808:x
8801:1
8794:1
8789:1
8771:π
8745:π
8724:n
8722:(
8716:n
8709:.
8706:r
8699:r
8693:.
8690:r
8688:π
8683:3
8680:/
8677:4
8668:r
8662:.
8657:π
8652:b
8646:a
8640:.
8637:r
8635:π
8630:r
8620:.
8617:r
8610:r
8601:π
8581:π
8576:.
8574:π
8566:π
8550:π
8538:π
8526:π
8519:π
8503:π
8483:π
8475:π
8455:.
8451:)
8444:6
8441:+
8438:k
8435:8
8431:1
8420:5
8417:+
8414:k
8411:8
8407:1
8396:4
8393:+
8390:k
8387:8
8383:2
8372:1
8369:+
8366:k
8363:8
8359:4
8353:(
8345:k
8337:1
8325:0
8322:=
8319:k
8311:=
8281:π
8269:π
8257:π
8253:π
8226:.
8218:2
8214:]
8209:|
8203:n
8199:W
8194:|
8190:[
8187:E
8182:n
8179:2
8165:n
8157:=
8144:π
8134:n
8132:W
8127:n
8117:n
8115:W
8110:n
8094:k
8090:X
8084:n
8079:1
8076:=
8073:k
8065:=
8060:n
8056:W
8042:k
8040:X
8033:k
8031:X
8019:π
8013:.
8011:π
8001:μ
7993:μ
7982:W
7964:π
7948:.
7942:t
7939:x
7931:n
7928:2
7909:π
7904:x
7898:x
7892:t
7886:n
7880:ℓ
7871:π
7845:b
7841:a
7814:c
7810:b
7806:a
7796:k
7789:e
7782:q
7767:,
7763:)
7756:1
7748:n
7745:4
7741:q
7736:c
7731:+
7725:1
7717:n
7714:2
7710:q
7705:b
7700:+
7694:1
7686:n
7682:q
7677:a
7671:(
7663:k
7659:n
7655:1
7643:1
7640:=
7637:n
7629:=
7624:k
7609:π
7586:π
7582:π
7566:.
7558:k
7555:3
7551:)
7541:(
7536:3
7532:!
7528:k
7524:!
7521:)
7518:k
7515:3
7512:(
7507:)
7504:k
7498:+
7492:(
7489:!
7486:)
7483:k
7480:6
7477:(
7464:0
7461:=
7458:k
7438:=
7430:1
7402:π
7382:.
7375:)
7370:k
7367:4
7359:(
7353:4
7349:!
7345:k
7340:)
7337:k
7331:+
7325:(
7322:!
7319:)
7316:k
7313:4
7310:(
7297:0
7294:=
7291:k
7275:2
7270:2
7264:=
7256:1
7240:π
7231:π
7225:.
7223:π
7204:π
7192:π
7188:π
7172:π
7166:.
7160:π
7149:π
7141:π
7081:π
7073:π
7065:π
7041:π
7022:.
7014:n
7010:t
7006:4
6999:2
6995:)
6989:n
6985:b
6981:+
6976:n
6972:a
6968:(
6948:π
6933:.
6928:n
6924:p
6920:2
6917:=
6912:1
6909:+
6906:n
6902:p
6896:,
6891:2
6887:)
6881:1
6878:+
6875:n
6871:a
6862:n
6858:a
6854:(
6849:n
6845:p
6836:n
6832:t
6828:=
6823:1
6820:+
6817:n
6813:t
6791:,
6784:n
6780:b
6774:n
6770:a
6764:=
6759:1
6756:+
6753:n
6749:b
6743:,
6738:2
6732:n
6728:b
6724:+
6719:n
6715:a
6708:=
6703:1
6700:+
6697:n
6693:a
6667:=
6662:0
6658:p
6653:,
6648:4
6645:1
6640:=
6635:0
6631:t
6626:,
6620:2
6616:1
6611:=
6606:0
6602:b
6597:,
6594:1
6591:=
6586:0
6582:a
6570::
6537:1
6533:π
6529:π
6509:π
6502:π
6483:π
6479:π
6462:2
6459:1
6433:π
6422:"
6395:"
6383:"
6356:"
6342:"
6327:.
6316:"
6309:p
6305:c
6296:ρ
6286:δ
6279:(
6272:π
6226:π
6210:π
6202:π
6186:π
6163:+
6156:2
6152:4
6148:1
6143:+
6136:2
6132:3
6128:1
6123:+
6116:2
6112:2
6108:1
6103:+
6096:2
6092:1
6088:1
6083:=
6078:6
6073:2
6046:π
6038:π
6030:π
6022:π
6000:π
5996:π
5953:8
5947:7
5941:6
5937:4
5932:+
5926:6
5920:5
5914:4
5910:4
5899:4
5893:3
5887:2
5883:4
5878:+
5874:3
5870:=
5853:π
5820:+
5812:4
5807:+
5799:4
5789:9
5786:4
5781:+
5776:7
5773:4
5763:5
5760:4
5755:+
5750:3
5747:4
5737:1
5734:4
5729:=
5695:π
5670:+
5658:9
5652:8
5648:4
5637:8
5631:7
5625:6
5621:4
5616:+
5610:6
5604:5
5598:4
5594:4
5583:4
5577:3
5571:2
5567:4
5562:+
5559:3
5556:=
5543:π
5536:π
5532:π
5528:π
5501:4
5496:+
5488:4
5478:9
5475:4
5470:+
5465:7
5462:4
5452:5
5449:4
5444:+
5439:3
5436:4
5426:1
5423:4
5418:=
5401:π
5397:π
5393:π
5386:π
5373:π
5358:π
5346:π
5339:π
5325:,
5316:3
5304:2
5301:+
5295:7
5292:1
5280:5
5277:=
5271:4
5236:+
5228:3
5224:)
5218:2
5214:x
5210:+
5207:1
5204:(
5198:5
5194:x
5185:5
5179:3
5174:4
5168:2
5162:+
5154:2
5150:)
5144:2
5140:x
5136:+
5133:1
5130:(
5124:3
5120:x
5112:3
5109:2
5104:+
5096:2
5092:x
5088:+
5085:1
5081:x
5076:=
5073:x
5047:π
5039:π
5023:.
5015:1
4999:5
4996:1
4985:4
4982:=
4977:4
4955:π
4938:3
4934:1
4929:=
4926:z
4895:1
4892:=
4889:z
4869:1
4866:=
4863:z
4841:4
4807:+
4802:7
4797:7
4793:z
4782:5
4777:5
4773:z
4767:+
4762:3
4757:3
4753:z
4744:z
4741:=
4738:z
4707:π
4703:π
4683:π
4651:)
4644:9
4641:8
4631:7
4628:8
4621:(
4611:)
4604:7
4601:6
4591:5
4588:6
4581:(
4571:)
4564:5
4561:4
4551:3
4548:4
4541:(
4531:)
4524:3
4521:2
4511:1
4508:2
4501:(
4496:=
4491:2
4451:2
4443:2
4438:+
4435:2
4430:+
4427:2
4416:2
4410:2
4405:+
4402:2
4391:2
4387:2
4381:=
4373:2
4361:π
4338:π
4302:π
4286:π
4278:π
4266:π
4255:n
4250:n
4248:S
4244:π
4212:π
4184:2
4177:3
4147:2
4140:3
4097:2
4091:+
4088:3
4050:π
4019:=
4014:7
3980:=
3917:π
3913:π
3890:/
3855:π
3844:π
3836:π
3818:7
3815:/
3804:π
3796:7
3793:/
3780:π
3770:7
3767:/
3757:π
3747:/
3735:π
3731:π
3719:π
3713:.
3711:π
3696:π
3673:π
3643:2
3637:)
3629:9
3619:(
3607:π
3596:8
3593:/
3583:π
3571:π
3556:π
3544:π
3535:π
3512:.
3509:)
3506:1
3500:n
3497:,
3491:,
3488:2
3485:,
3482:1
3479:,
3476:0
3473:=
3470:k
3467:(
3461:n
3457:/
3453:k
3450:i
3444:2
3440:e
3424:n
3409:1
3406:=
3401:n
3397:z
3386:z
3378:n
3358:=
3355:1
3352:+
3344:i
3340:e
3312:=
3294:e
3283:e
3266:,
3254:i
3251:+
3239:=
3231:i
3227:e
3204:π
3190:1
3184:=
3179:2
3175:i
3160:i
3146:,
3143:)
3131:i
3128:+
3116:(
3110:r
3107:=
3104:z
3090:φ
3078:z
3074:r
3058:z
3026:e
3010:)
2993:)
2960:)
2906:)
2851:(
2837:)
2762:/
2746:/
2730:/
2714:/
2698:/
2682:/
2669:7
2666:/
2581:+
2578:7
2564:2
2560:3
2548:+
2545:5
2531:2
2527:2
2515:+
2512:3
2498:2
2494:1
2482:+
2479:1
2467:4
2456:=
2430:+
2427:2
2413:2
2409:5
2397:+
2394:2
2380:2
2376:3
2364:+
2361:2
2347:2
2343:1
2331:+
2328:1
2316:4
2305:=
2279:+
2276:6
2262:2
2258:7
2246:+
2243:6
2229:2
2225:5
2213:+
2210:6
2196:2
2192:3
2180:+
2177:6
2163:2
2159:1
2147:+
2144:3
2141:=
2116:π
2108:π
2096:π
2092:π
2081:/
2063:/
2048:7
2045:/
2034:3
2030:π
1969:+
1966:1
1954:1
1942:+
1939:1
1927:1
1915:+
1912:1
1900:1
1888:+
1873:1
1861:+
1858:1
1846:1
1834:+
1819:1
1807:+
1804:7
1792:1
1780:+
1777:3
1774:=
1757:π
1749:π
1684:3
1666:n
1661:π
1657:π
1640:0
1637:=
1634:x
1631:+
1626:6
1621:3
1617:x
1601:5
1597:x
1568:π
1562:.
1548:π
1518:π
1510:π
1498:π
1490:π
1479:π
1467:π
1451:e
1438:π
1426:π
1412:π
1404:π
1393:/
1378:7
1375:/
1356:π
1345:π
1325:Z
1323:/
1321:R
1289:π
1275:}
1271:Z
1264:k
1258:i
1255:k
1249:2
1246:{
1243:=
1240:}
1234:,
1231:i
1225:4
1222:,
1219:i
1213:2
1210:,
1207:0
1204:,
1201:i
1195:2
1189:,
1183:{
1172:z
1165:z
1155:z
1145:π
1126:π
1118:π
1110:π
1091:π
1075:.
1067:2
1063:x
1056:1
1051:x
1048:d
1040:1
1035:1
1024:=
997:1
994:=
989:2
985:y
981:+
976:2
972:x
926:d
923:C
918:=
897:π
881:d
878:C
854:d
851:C
825:d
822:C
817:=
803:d
794:C
778:π
773:.
771:π
753:π
746:π
735:Σ
727:Π
723:π
710:/
704:p
701:/
697:(
691:π
682:π
663:π
659:π
655:π
611:π
599:π
591:π
576:π
568:π
560:π
552:π
537:π
525:π
510:π
498:π
461:7
433:π
403:/
397:p
394:/
390:(
387:π
373:e
366:t
359:v
345:π
337:π
62:π
42:.
20:)
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