2851:
724:
1392:
33:
1210:
175:
1371:
1003:
903:
1083:
1681:
668:
75:
569:
1215:
258:
758:
of a limit governs the number of terms of the expression needed to achieve a given number of digits of accuracy. In Viète's formula, the numbers of terms and digits are proportional to each other: the product of the first
1071:
1554:
921:
391:
282:, but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses and as a motivating example for the concept of
712:
gets arbitrarily large, these finite products have values that approach the value of Viète's formula arbitrarily closely. Viète did his work long before the concepts of limits and rigorous proofs of
813:
1220:
579:
1567:
574:
487:
1475:-gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a
1758:
1457:
180:
1205:{\displaystyle \pi =\lim _{k\to \infty }2^{k}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}}}} _{k{\text{ square roots}}},}
2203:
710:
690:
170:{\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 }
1487:
1014:
1492:
1366:{\displaystyle {\begin{aligned}\pi &=\lim _{k\to \infty }2^{k}{\sqrt {2-a_{k}}},\\a_{1}&=0,\\a_{k}&={\sqrt {2+a_{k-1}}}.\end{aligned}}}
347:
266:, who published it in 1593. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a
403:. As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation,
1403:, inscribed in a circle. The ratios between areas or perimeters of consecutive polygons in the sequence give the terms of Viète's formula.
313:, that has Viète's formula as a special case. Many similar formulas involving nested roots or infinite products are now known.
2021:
Maths for the
Mystified: An Exploration of the History of Mathematics and Its Relationship to Modern-day Science and Computing
2549:
2285:
2233:
2170:
1888:
1856:
1815:
336:
to (in principle) arbitrary accuracy had long been known. Viète's own method can be interpreted as a variation of an idea of
2344:
1701:
goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution
2029:
1383:
are now known, similar to Viète's in their use of either nested radicals or infinite products of trigonometric functions.
1971:
Cullerne, J. P.; Goekjian, M. C. Dunn (December 2011). "Teaching wave propagation and the emergence of Viète's formula".
998:{\displaystyle {\frac {2}{\pi }}=\cos {\frac {\pi }{4}}\cdot \cos {\frac {\pi }{8}}\cdot \cos {\frac {\pi }{16}}\cdots }
2802:
2588:
2500:
2479:
2098:
1765:
1479:(the diameter of the circle, counted twice) and a square, the ratio of perimeters of a square and an octagon, etc.
2054:
1471:
to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a
898:{\displaystyle {\frac {\sin x}{x}}=\cos {\frac {x}{2}}\cdot \cos {\frac {x}{4}}\cdot \cos {\frac {x}{8}}\cdots }
2718:
2318:
431:
2321:. Vol. 12. New York: John Wiley & Sons for the Mathematical Association of America. pp. 1–12.
1883:. Translated by Wilson, Stephen S. Providence, Rhode Island: American Mathematical Society. pp. 44–46.
472:, equal to the integral of products of the same functions, provides a motivating example for the concept of
2223:
1676:{\displaystyle \sin x=2^{n}\sin {\frac {x}{2^{n}}}\left(\prod _{i=1}^{n}\cos {\frac {x}{2^{i}}}\right).}
2887:
2586:
Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant".
716:
were developed in mathematics; the first proof that this limit exists was not given until the work of
663:{\displaystyle {\begin{aligned}a_{1}&={\sqrt {2}}\\a_{n}&={\sqrt {2+a_{n-1}}}.\end{aligned}}}
464:
in the limiting behavior of these speeds. Additionally, a derivation of this formula as a product of
1727:
2757:
1430:
473:
344:
of a circle by the perimeter of a many-sided polygon, used by
Archimedes to find the approximation
283:
434:
2210:
564:{\displaystyle \lim _{n\rightarrow \infty }\prod _{i=1}^{n}{\frac {a_{i}}{2}}={\frac {2}{\pi }},}
2631:
Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for
1007:
Then, expressing each term of the product on the right as a function of earlier terms using the
2892:
1557:
1483:
63:
2219:
2019:
395:
By publishing his method as a mathematical formula, Viète formulated the first instance of an
330:
325:
to two French kings, and amateur mathematician. He published this formula in 1593 in his work
2637:
2433:
Translated into
English by Thomas W. Polaski. See final formula. The same formula is also in
2423:[On various methods for expressing the quadrature of a circle with verging numbers].
2152:
1876:
1807:
408:
271:
456:
Beyond its mathematical and historical significance, Viète's formula can be used to explain
2831:
2741:
2698:
2658:
2617:
2572:
2529:
2326:
2137:
2077:
1925:
1898:
1825:
2360:
2188:
399:
known in mathematics, and the first example of an explicit formula for the exact value of
8:
2685:
787:
755:
457:
267:
263:
24:
2439:
2377:(2007). "A simple geometric method of estimating the error in using Vieta's product for
2856:
2819:
2782:
2774:
2662:
2605:
2517:
2453:
2420:
2398:
2265:
2206:
2125:
2107:
1988:
1950:
1942:
1468:
1008:
713:
695:
675:
446:
302:
275:
2850:
2786:
2666:
2475:
2402:
2345:"Ueber die Convergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung"
2281:
2229:
2166:
2025:
1992:
1954:
1884:
1852:
1811:
469:
253:{\displaystyle {\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}.}
1984:
2811:
2766:
2727:
2683:(2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers".
2646:
2597:
2558:
2509:
2390:
2356:
2273:
2258:
2254:
2158:
2117:
2063:
1980:
1938:
1934:
412:
396:
322:
55:
2713:
2096:
Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums".
1080:
that still involves nested square roots of two, but uses only one multiplication:
750:
terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.
445:
digits and 16 decimal digits in 1424. Not long after Viète published his formula,
2865:
2827:
2737:
2694:
2680:
2654:
2613:
2568:
2525:
2474:(1st ed.). Oxford, United Kingdom: Oxford University Press. pp. 57–58.
2442:[Various observations about angles proceeding in geometric progression].
2374:
2340:
2322:
2277:
2133:
2073:
1894:
1821:
1798:
1721:
1412:
1396:
717:
2714:"Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products"
723:
2435:
2416:
807:
775:
310:
59:
2650:
2563:
2544:
2394:
2162:
2068:
2049:
2881:
2545:"Some closed-form evaluations of infinite products involving nested radicals"
2440:"Variae observationes circa angulos in progressione geometrica progredientes"
1794:
1391:
803:
423:
341:
2732:
2349:
Historisch-litterarische
Abteilung der Zeitschrift für Mathematik und Physik
2871:
2313:(1959). "Chapter 1: From Vieta to the notion of statistical independence".
1400:
1380:
306:
289:
The formula can be derived as a telescoping product of either the areas or
2347:[On the convergence of a special product expansion due to Vieta].
2383:
International
Journal of Mathematical Education in Science and Technology
2215:
1464:
1076:
It is also possible to derive from Viète's formula a related formula for
442:
47:
32:
2609:
1946:
2823:
2778:
2521:
2228:. Princeton, New Jersey: Princeton University Press. pp. 221–234.
2129:
1851:. Princeton, New Jersey: Princeton University Press. pp. 50, 140.
802:
Viète's formula may be obtained as a special case of a formula for the
337:
2472:
Euler's pioneering equation: the most beautiful theorem in mathematics
2213:(c. 1340 – 1425), but were not known in Europe until much later. See:
484:
Viète's formula may be rewritten and understood as a limit expression
2601:
2112:
290:
2815:
2770:
2513:
2121:
2315:
Statistical
Independence in Probability, Analysis and Number Theory
2310:
1844:
465:
449:
used a method closely related to Viète's to calculate 35 digits of
404:
2457:
2154:
An Atlas of
Functions: with Equator, the Atlas Function Calculator
460:
in an infinite chain of springs and masses, and the appearance of
1460:
1066:{\displaystyle \cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}}
294:
2421:"De variis modis circuli quadraturam numeris proxime exprimendi"
1549:{\displaystyle \sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}},}
1424:
774:
digits. This convergence rate compares very favorably with the
298:
453:, which were published only after van Ceulen's death in 1610.
2800:
Rummler, Hansklaus (1993). "Squaring the circle with holes".
1806:(2nd ed.). Boulder, Colorado: The Golem Press. pp.
1476:
386:{\displaystyle {\frac {223}{71}}<\pi <{\frac {22}{7}}.}
1463:, the second term is the ratio of areas of an octagon and a
2220:"7.3.1 Mādhava on the circumference and arcs of the circle"
1408:
426:. However, this was not the most accurate approximation to
2151:
Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010).
692:, the expression in the limit is a finite product, and as
23:. For formulas for symmetric functions of the roots, see
2272:. Berlin & Heidelberg: Springer. pp. 531–561.
2048:
Moreno, Samuel G.; García-Caballero, Esther M. (2013).
782:. Although Viète himself used his formula to calculate
415:
calls its appearance "the dawn of modern mathematics".
407:
highlights Viète's formula as marking the beginning of
67:
2867:
Variorum de rebus mathematicis responsorum, liber VIII
2047:
458:
the different speeds of waves of different frequencies
327:
Variorum de rebus mathematicis responsorum, liber VIII
39:
Variorum de rebus mathematicis responsorum, liber VIII
2259:"The life of Pi: From Archimedes to ENIAC and beyond"
2191:
1999:
1730:
1570:
1495:
1433:
1218:
1086:
1017:
924:
816:
698:
678:
577:
490:
350:
183:
78:
2846:
2150:
2874:. The formula is on the second half of p. 30.
2197:
1752:
1675:
1548:
1451:
1365:
1204:
1065:
997:
897:
790:version of his formula has been used to calculate
727:Comparison of the convergence of Viète's formula (
704:
684:
662:
563:
385:
252:
169:
2498:Servi, L. D. (2003). "Nested square roots of 2".
479:
2879:
2425:Commentarii Academiae Scientiarum Petropolitanae
1970:
1234:
1094:
492:
321:François Viète (1540–1603) was a French lawyer,
2755:Allen, Edward J. (1985). "Continued radicals".
2185:Very similar infinite trigonometric series for
1923:to thousands of digits from Vieta's formula".
309:leads to a generalized formula, discovered by
1486:and Euler's formula. Repeatedly applying the
733:) and several historical infinite series for
1407:Viète obtained his formula by comparing the
1914:
1912:
1910:
1908:
1875:Eymard, Pierre; Lafon, Jean Pierre (2004).
1874:
1459:, is the ratio of areas of a square and an
763:terms in the limit gives an expression for
37:
2460:. See the formula in numbered paragraph 3.
2249:
2247:
2245:
2091:
2089:
2087:
1966:
1964:
1789:
1787:
1785:
1783:
1781:
2731:
2711:
2705:
2562:
2111:
2067:
1918:
2493:
2491:
2452:Translated into English by Jordan Bell,
2095:
2017:
2005:
1905:
1870:
1868:
1793:
1482:Another derivation is possible based on
1390:
722:
31:
2799:
2793:
2253:
2242:
2214:
2179:
2084:
1961:
1778:
778:, a later infinite product formula for
36:Viète's formula, as printed in Viète's
2880:
2673:
2624:
2542:
2536:
2469:
2305:
2303:
2011:
1839:
1837:
1835:
270:expression and marks the beginning of
2754:
2748:
2679:
2630:
2585:
2579:
2550:Rocky Mountain Journal of Mathematics
2497:
2488:
2434:
2415:
2409:
2373:
2367:
2339:
2333:
2043:
2041:
1865:
301:. Alternatively, repeated use of the
1843:
1212:which can be rewritten compactly as
794:to hundreds of thousands of digits.
418:Using his formula, Viète calculated
278:and can be used for calculations of
19:This article is about a formula for
2309:
2300:
2144:
2024:. Leicester: Matador. p. 165.
1832:
797:
13:
2157:. New York: Springer. p. 15.
2038:
1244:
1104:
806:that has often been attributed to
786:only with nine-digit accuracy, an
767:that is accurate to approximately
746:is the approximation after taking
502:
213:
16:Infinite product converging to 2/π
14:
2904:
2842:
2803:The American Mathematical Monthly
2589:The American Mathematical Monthly
2501:The American Mathematical Monthly
2099:The American Mathematical Monthly
1427:. The first term in the product,
2849:
2270:From Alexandria, Through Baghdad
1766:List of trigonometric identities
1560:that, for all positive integers
1379:and other constants such as the
2463:
2055:Journal of Approximation Theory
1399:with numbers of sides equal to
316:
2719:Pacific Journal of Mathematics
2712:Stolarsky, Kenneth B. (1980).
1939:10.1080/0025570X.2008.11953549
1877:"2.1 Viète's infinite product"
1753:{\displaystyle x=2^{n}\alpha }
1241:
1101:
810:, more than a century later:
499:
480:Interpretation and convergence
177:It can also be represented as
1:
2319:Carus Mathematical Monographs
2018:De Smith, Michael J. (2006).
1771:
1452:{\displaystyle {\sqrt {2}}/2}
1386:
66:of the mathematical constant
2278:10.1007/978-3-642-36736-6_24
329:. At this time, methods for
7:
1715:
262:The formula is named after
10:
2909:
430:known at the time, as the
18:
2651:10.1007/s11139-005-4852-z
2564:10.1216/RMJ-2012-42-2-751
2470:Wilson, Robin J. (2018).
2395:10.1080/00207390601002799
2163:10.1007/978-0-387-48807-3
2069:10.1016/j.jat.2013.06.006
1985:10.1088/0031-9120/47/1/87
1919:Kreminski, Rick (2008). "
1467:, etc. Thus, the product
2758:The Mathematical Gazette
2050:"On Viète-like formulas"
1724:, same identity taking
1484:trigonometric identities
474:statistical independence
284:statistical independence
2733:10.2140/pjm.1980.89.209
2211:Madhava of Sangamagrama
1073:gives Viète's formula.
918:in this formula yields
441:to an accuracy of nine
422:to an accuracy of nine
62:representing twice the
2543:Nyblom, M. A. (2012).
2199:
1849:Trigonometric Delights
1754:
1677:
1641:
1558:mathematical induction
1550:
1453:
1404:
1367:
1206:
1067:
999:
899:
751:
706:
686:
664:
565:
527:
387:
254:
217:
171:
43:
38:
2638:The Ramanujan Journal
2264:. In Sidoli, Nathan;
2200:
1755:
1678:
1621:
1551:
1454:
1423:sides inscribed in a
1394:
1368:
1207:
1068:
1000:
900:
726:
707:
687:
665:
566:
507:
432:Persian mathematician
409:mathematical analysis
388:
340:of approximating the
272:mathematical analysis
255:
197:
172:
35:
2255:Borwein, Jonathan M.
2225:Mathematics in India
2205:appeared earlier in
2198:{\displaystyle \pi }
2189:
1926:Mathematics Magazine
1728:
1568:
1556:leads to a proof by
1493:
1488:double-angle formula
1431:
1216:
1084:
1015:
922:
814:
696:
676:
575:
488:
348:
181:
76:
2686:Fibonacci Quarterly
2266:Van Brummelen, Glen
756:rate of convergence
672:For each choice of
2857:Mathematics portal
2444:Opuscula Analytica
2207:Indian mathematics
2195:
1760:on Viète's formula
1750:
1673:
1546:
1449:
1405:
1375:Many formulae for
1363:
1361:
1248:
1202:
1198:
1194: square roots
1186:
1108:
1063:
1009:half-angle formula
995:
895:
752:
702:
682:
660:
658:
561:
506:
447:Ludolph van Ceulen
383:
303:half-angle formula
276:linear convergence
250:
167:
44:
2888:Infinite products
2287:978-3-642-36735-9
2235:978-0-691-12067-6
2209:, in the work of
2172:978-0-387-48807-3
1973:Physics Education
1890:978-0-8218-3246-2
1858:978-1-4008-4282-7
1817:978-0-88029-418-8
1663:
1614:
1541:
1525:
1439:
1354:
1277:
1233:
1195:
1182:
1180:
1178:
1176:
1174:
1172:
1121:
1119:
1093:
1061:
1060:
1032:
990:
971:
952:
933:
890:
871:
852:
833:
705:{\displaystyle n}
685:{\displaystyle n}
651:
604:
556:
543:
491:
470:Rademacher system
378:
359:
245:
192:
162:
158:
156:
154:
127:
123:
121:
102:
98:
87:
54:is the following
2900:
2859:
2854:
2853:
2836:
2835:
2797:
2791:
2790:
2765:(450): 261–263.
2752:
2746:
2745:
2735:
2709:
2703:
2702:
2681:Osler, Thomas J.
2677:
2671:
2670:
2634:
2628:
2622:
2621:
2602:10.2307/27641976
2583:
2577:
2576:
2566:
2540:
2534:
2533:
2495:
2486:
2485:
2467:
2461:
2451:
2432:
2413:
2407:
2406:
2380:
2375:Osler, Thomas J.
2371:
2365:
2364:
2337:
2331:
2330:
2307:
2298:
2297:
2295:
2294:
2263:
2251:
2240:
2239:
2204:
2202:
2201:
2196:
2183:
2177:
2176:
2148:
2142:
2141:
2115:
2093:
2082:
2081:
2071:
2045:
2036:
2035:
2031:978-1905237-81-4
2015:
2009:
2003:
1997:
1996:
1968:
1959:
1958:
1922:
1916:
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1902:
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1841:
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1803:
1791:
1759:
1757:
1756:
1751:
1746:
1745:
1711:
1709:
1700:
1697:in the limit as
1696:
1692:
1682:
1680:
1679:
1674:
1669:
1665:
1664:
1662:
1661:
1649:
1640:
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1615:
1613:
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1600:
1592:
1591:
1563:
1555:
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1552:
1547:
1542:
1534:
1526:
1518:
1474:
1458:
1456:
1455:
1450:
1445:
1440:
1435:
1422:
1418:
1413:regular polygons
1397:regular polygons
1378:
1372:
1370:
1369:
1364:
1362:
1355:
1353:
1352:
1331:
1322:
1321:
1295:
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1278:
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1260:
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1247:
1211:
1209:
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1187:
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1177:
1175:
1173:
1168:
1154:
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1138:
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1107:
1079:
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1004:
1002:
1001:
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991:
983:
972:
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953:
945:
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926:
917:
915:
904:
902:
901:
896:
891:
883:
872:
864:
853:
845:
834:
829:
818:
798:Related formulas
793:
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773:
766:
762:
749:
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711:
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703:
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435:Jamshīd al-Kāshī
429:
421:
413:Jonathan Borwein
402:
397:infinite product
392:
390:
389:
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371:
360:
352:
334:
323:privy councillor
297:converging to a
281:
259:
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56:infinite product
41:
25:Vieta's formulas
22:
2908:
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2903:
2902:
2901:
2899:
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2897:
2878:
2877:
2855:
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2845:
2840:
2839:
2816:10.2307/2324662
2798:
2794:
2771:10.2307/3617569
2753:
2749:
2710:
2706:
2678:
2674:
2632:
2629:
2625:
2584:
2580:
2541:
2537:
2514:10.2307/3647881
2496:
2489:
2482:
2468:
2464:
2436:Euler, Leonhard
2417:Euler, Leonhard
2414:
2410:
2378:
2372:
2368:
2338:
2334:
2308:
2301:
2292:
2290:
2288:
2261:
2252:
2243:
2236:
2190:
2187:
2186:
2184:
2180:
2173:
2149:
2145:
2122:10.2307/2974641
2094:
2085:
2046:
2039:
2032:
2016:
2012:
2004:
2000:
1969:
1962:
1920:
1917:
1906:
1891:
1873:
1866:
1859:
1842:
1833:
1818:
1801:
1792:
1779:
1774:
1741:
1737:
1729:
1726:
1725:
1718:
1707:
1702:
1698:
1694:
1686:
1657:
1653:
1648:
1636:
1625:
1620:
1616:
1608:
1604:
1599:
1587:
1583:
1569:
1566:
1565:
1561:
1533:
1517:
1494:
1491:
1490:
1472:
1441:
1434:
1432:
1429:
1428:
1420:
1416:
1389:
1376:
1360:
1359:
1342:
1338:
1330:
1323:
1317:
1313:
1310:
1309:
1296:
1290:
1286:
1283:
1282:
1271:
1267:
1259:
1253:
1249:
1237:
1226:
1219:
1217:
1214:
1213:
1192:
1188:
1167:
1153:
1145:
1137:
1129:
1120:
1113:
1109:
1097:
1085:
1082:
1081:
1077:
1040:
1037:
1024:
1016:
1013:
1012:
982:
963:
944:
925:
923:
920:
919:
913:
908:
882:
863:
844:
819:
817:
815:
812:
811:
800:
791:
783:
779:
768:
764:
760:
747:
743:
738:
734:
728:
718:Ferdinand Rudio
697:
694:
693:
677:
674:
673:
657:
656:
639:
635:
627:
620:
614:
610:
607:
606:
599:
592:
586:
582:
578:
576:
573:
572:
548:
534:
530:
528:
522:
511:
495:
489:
486:
485:
482:
461:
450:
438:
437:had calculated
427:
419:
400:
370:
351:
349:
346:
345:
332:
319:
279:
233:
229:
224:
212:
201:
184:
182:
179:
178:
149:
141:
132:
116:
107:
92:
79:
77:
74:
73:
68:
60:nested radicals
52:Viète's formula
28:
20:
17:
12:
11:
5:
2906:
2896:
2895:
2890:
2876:
2875:
2861:
2860:
2844:
2843:External links
2841:
2838:
2837:
2810:(9): 858–860.
2792:
2747:
2726:(1): 209–227.
2704:
2693:(3): 202–204.
2672:
2645:(3): 305–324.
2623:
2596:(6): 510–520.
2578:
2557:(2): 751–758.
2535:
2508:(4): 326–330.
2487:
2480:
2462:
2408:
2389:(1): 136–142.
2366:
2332:
2299:
2286:
2241:
2234:
2194:
2178:
2171:
2143:
2106:(8): 716–724.
2083:
2037:
2030:
2010:
1998:
1960:
1933:(3): 201–207.
1904:
1889:
1864:
1857:
1831:
1816:
1795:Beckmann, Petr
1776:
1775:
1773:
1770:
1769:
1768:
1762:
1761:
1749:
1744:
1740:
1736:
1733:
1717:
1714:
1672:
1668:
1660:
1656:
1652:
1647:
1644:
1639:
1634:
1631:
1628:
1624:
1619:
1611:
1607:
1603:
1598:
1595:
1590:
1586:
1582:
1579:
1576:
1573:
1545:
1540:
1537:
1532:
1529:
1524:
1521:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1448:
1444:
1438:
1395:A sequence of
1388:
1385:
1358:
1351:
1348:
1345:
1341:
1337:
1334:
1329:
1326:
1324:
1320:
1316:
1312:
1311:
1308:
1305:
1302:
1299:
1297:
1293:
1289:
1285:
1284:
1281:
1274:
1270:
1266:
1263:
1256:
1252:
1246:
1243:
1240:
1236:
1232:
1229:
1227:
1225:
1222:
1221:
1201:
1191:
1185:
1171:
1166:
1163:
1160:
1157:
1152:
1149:
1144:
1141:
1136:
1133:
1128:
1125:
1116:
1112:
1106:
1103:
1100:
1096:
1092:
1089:
1059:
1055:
1052:
1049:
1046:
1043:
1036:
1031:
1028:
1023:
1020:
994:
989:
986:
981:
978:
975:
970:
967:
962:
959:
956:
951:
948:
943:
940:
937:
932:
929:
894:
889:
886:
881:
878:
875:
870:
867:
862:
859:
856:
851:
848:
843:
840:
837:
832:
828:
825:
822:
808:Leonhard Euler
799:
796:
776:Wallis product
741:
701:
681:
655:
648:
645:
642:
638:
634:
631:
626:
623:
621:
617:
613:
609:
608:
603:
598:
595:
593:
589:
585:
581:
580:
560:
555:
552:
547:
542:
537:
533:
525:
520:
517:
514:
510:
504:
501:
498:
494:
481:
478:
468:involving the
424:decimal digits
382:
377:
374:
369:
366:
363:
358:
355:
331:approximating
318:
315:
311:Leonhard Euler
264:François Viète
249:
242:
239:
236:
232:
228:
223:
220:
215:
210:
207:
204:
200:
196:
191:
188:
166:
161:
153:
148:
145:
140:
137:
131:
126:
120:
115:
112:
106:
101:
97:
91:
86:
83:
15:
9:
6:
4:
3:
2:
2905:
2894:
2893:Pi algorithms
2891:
2889:
2886:
2885:
2883:
2873:
2869:
2868:
2863:
2862:
2858:
2852:
2847:
2833:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2804:
2796:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2759:
2751:
2743:
2739:
2734:
2729:
2725:
2721:
2720:
2715:
2708:
2700:
2696:
2692:
2688:
2687:
2682:
2676:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2640:
2639:
2627:
2619:
2615:
2611:
2607:
2603:
2599:
2595:
2591:
2590:
2582:
2574:
2570:
2565:
2560:
2556:
2552:
2551:
2546:
2539:
2531:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2502:
2494:
2492:
2483:
2481:9780198794929
2477:
2473:
2466:
2459:
2455:
2449:
2445:
2441:
2437:
2430:
2426:
2422:
2418:
2412:
2404:
2400:
2396:
2392:
2388:
2384:
2376:
2370:
2362:
2358:
2354:
2351:(in German).
2350:
2346:
2342:
2336:
2328:
2324:
2320:
2316:
2312:
2306:
2304:
2289:
2283:
2279:
2275:
2271:
2267:
2260:
2256:
2250:
2248:
2246:
2237:
2231:
2227:
2226:
2221:
2217:
2212:
2208:
2192:
2182:
2174:
2168:
2164:
2160:
2156:
2155:
2147:
2139:
2135:
2131:
2127:
2123:
2119:
2114:
2109:
2105:
2101:
2100:
2092:
2090:
2088:
2079:
2075:
2070:
2065:
2061:
2057:
2056:
2051:
2044:
2042:
2033:
2027:
2023:
2022:
2014:
2008:, p. 67.
2007:
2006:Beckmann 1971
2002:
1994:
1990:
1986:
1982:
1978:
1974:
1967:
1965:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1927:
1915:
1913:
1911:
1909:
1900:
1896:
1892:
1886:
1882:
1881:The Number pi
1878:
1871:
1869:
1860:
1854:
1850:
1846:
1840:
1838:
1836:
1827:
1823:
1819:
1813:
1809:
1805:
1804:
1800:A History of
1796:
1790:
1788:
1786:
1784:
1782:
1777:
1767:
1764:
1763:
1747:
1742:
1738:
1734:
1731:
1723:
1720:
1719:
1713:
1705:
1690:
1683:
1670:
1666:
1658:
1654:
1650:
1645:
1642:
1637:
1632:
1629:
1626:
1622:
1617:
1609:
1605:
1601:
1596:
1593:
1588:
1584:
1580:
1577:
1574:
1571:
1559:
1543:
1538:
1535:
1530:
1527:
1522:
1519:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1489:
1485:
1480:
1478:
1470:
1466:
1462:
1446:
1442:
1436:
1426:
1414:
1410:
1402:
1401:powers of two
1398:
1393:
1384:
1382:
1373:
1356:
1349:
1346:
1343:
1339:
1335:
1332:
1327:
1325:
1318:
1314:
1306:
1303:
1300:
1298:
1291:
1287:
1279:
1272:
1268:
1264:
1261:
1254:
1250:
1238:
1230:
1228:
1223:
1199:
1189:
1183:
1169:
1164:
1161:
1158:
1155:
1150:
1147:
1142:
1139:
1134:
1131:
1126:
1123:
1114:
1110:
1098:
1090:
1087:
1074:
1057:
1053:
1050:
1047:
1044:
1041:
1034:
1029:
1026:
1021:
1018:
1010:
1005:
992:
987:
984:
979:
976:
973:
968:
965:
960:
957:
954:
949:
946:
941:
938:
935:
930:
927:
911:
907:Substituting
905:
892:
887:
884:
879:
876:
873:
868:
865:
860:
857:
854:
849:
846:
841:
838:
835:
830:
826:
823:
820:
809:
805:
804:sinc function
795:
789:
777:
772:
757:
744:
731:
725:
721:
719:
715:
699:
679:
670:
653:
646:
643:
640:
636:
632:
629:
624:
622:
615:
611:
601:
596:
594:
587:
583:
558:
553:
550:
545:
540:
535:
531:
523:
518:
515:
512:
508:
496:
477:
475:
471:
467:
459:
454:
448:
444:
436:
433:
425:
416:
414:
410:
406:
398:
393:
380:
375:
372:
367:
364:
361:
356:
353:
343:
342:circumference
339:
335:
328:
324:
314:
312:
308:
304:
300:
296:
292:
287:
285:
277:
273:
269:
265:
260:
247:
240:
237:
234:
230:
226:
221:
218:
208:
205:
202:
198:
194:
189:
186:
164:
159:
151:
146:
143:
138:
135:
129:
124:
118:
113:
110:
104:
99:
95:
89:
84:
81:
71:
65:
61:
57:
53:
49:
40:
34:
30:
26:
2872:Google Books
2866:
2807:
2801:
2795:
2762:
2756:
2750:
2723:
2717:
2707:
2690:
2684:
2675:
2642:
2636:
2626:
2593:
2587:
2581:
2554:
2548:
2538:
2505:
2499:
2471:
2465:
2447:
2446:(in Latin).
2443:
2428:
2427:(in Latin).
2424:
2411:
2386:
2382:
2369:
2352:
2348:
2335:
2314:
2291:. Retrieved
2269:
2224:
2216:Plofker, Kim
2181:
2153:
2146:
2113:math/0411380
2103:
2097:
2059:
2053:
2020:
2013:
2001:
1979:(1): 87–91.
1976:
1972:
1930:
1924:
1880:
1848:
1799:
1722:Morrie's law
1703:
1688:
1684:
1481:
1406:
1381:golden ratio
1374:
1075:
1006:
909:
906:
801:
770:
753:
739:
729:
671:
483:
455:
417:
394:
326:
320:
317:Significance
307:trigonometry
288:
261:
51:
45:
29:
2355:: 139–140.
1465:hexadecagon
788:accelerated
714:convergence
443:sexagesimal
48:mathematics
2882:Categories
2870:(1593) on
2450:: 345–352.
2431:: 222–236.
2361:23.0263.02
2293:2024-08-20
2062:: 90–112.
1772:References
1469:telescopes
1387:Derivation
338:Archimedes
293:of nested
291:perimeters
64:reciprocal
2787:250441699
2667:123023282
2458:1009.1439
2403:120145020
2341:Rudio, F.
2311:Kac, Mark
2193:π
1993:122368450
1955:125362227
1845:Maor, Eli
1748:α
1685:The term
1646:
1623:∏
1597:
1575:
1531:
1515:
1500:
1347:−
1265:−
1245:∞
1242:→
1224:π
1184:⏟
1162:⋯
1127:−
1105:∞
1102:→
1088:π
1051:
1022:
993:⋯
985:π
980:
974:⋅
966:π
961:
955:⋅
947:π
942:
931:π
893:⋯
880:
874:⋅
861:
855:⋅
842:
824:
720:in 1891.
644:−
554:π
509:∏
503:∞
500:→
466:integrals
365:π
274:. It has
227:π
222:
214:∞
199:∏
190:π
165:⋯
130:⋅
105:⋅
85:π
2864:Viète's
2610:27641976
2438:(1783).
2419:(1738).
2343:(1891).
2268:(eds.).
2257:(2014).
2218:(2009).
1947:27643107
1847:(2011).
1797:(1971).
1716:See also
1693:goes to
405:Eli Maor
295:polygons
2832:1247533
2824:2324662
2779:3617569
2742:0596932
2699:2437033
2659:2193382
2618:2231136
2573:2915517
2530:1984573
2522:3647881
2327:0110114
2138:1357488
2130:2974641
2078:3090772
1899:2036595
1826:0449960
1461:octagon
2830:
2822:
2785:
2777:
2740:
2697:
2665:
2657:
2616:
2608:
2571:
2528:
2520:
2478:
2401:
2359:
2325:
2284:
2232:
2169:
2136:
2128:
2076:
2028:
1991:
1953:
1945:
1897:
1887:
1855:
1824:
1814:
1687:2 sin(
1425:circle
571:where
299:circle
42:(1593)
2820:JSTOR
2783:S2CID
2775:JSTOR
2663:S2CID
2606:JSTOR
2518:JSTOR
2454:arXiv
2399:S2CID
2262:(PDF)
2126:JSTOR
2108:arXiv
1989:S2CID
1951:S2CID
1943:JSTOR
1808:94–95
1477:digon
1415:with
1409:areas
305:from
268:limit
2476:ISBN
2282:ISBN
2230:ISBN
2167:ISBN
2026:ISBN
1885:ISBN
1853:ISBN
1812:ISBN
1419:and
754:The
411:and
368:<
362:<
2812:doi
2808:100
2767:doi
2728:doi
2647:doi
2635:".
2598:doi
2594:113
2559:doi
2510:doi
2506:110
2391:doi
2381:".
2357:JFM
2274:doi
2159:doi
2118:doi
2104:102
2064:doi
2060:174
1981:doi
1935:doi
1691:/2)
1643:cos
1594:sin
1572:sin
1528:cos
1512:sin
1497:sin
1411:of
1235:lim
1095:lim
1048:cos
1019:cos
977:cos
958:cos
939:cos
877:cos
858:cos
839:cos
821:sin
769:0.6
493:lim
354:223
219:cos
58:of
46:In
2884::
2828:MR
2826:.
2818:.
2806:.
2781:.
2773:.
2763:69
2761:.
2738:MR
2736:.
2724:89
2722:.
2716:.
2695:MR
2691:45
2689:.
2661:.
2655:MR
2653:.
2643:10
2641:.
2614:MR
2612:.
2604:.
2592:.
2569:MR
2567:.
2555:42
2553:.
2547:.
2526:MR
2524:.
2516:.
2504:.
2490:^
2397:.
2387:38
2385:.
2353:36
2323:MR
2317:.
2302:^
2280:.
2244:^
2222:.
2165:.
2134:MR
2132:.
2124:.
2116:.
2102:.
2086:^
2074:MR
2072:.
2058:.
2052:.
2040:^
1987:.
1977:47
1975:.
1963:^
1949:.
1941:.
1931:81
1929:.
1907:^
1895:MR
1893:.
1879:.
1867:^
1834:^
1822:MR
1820:.
1810:.
1780:^
1712:.
1710:/2
1706:=
1564:,
1011::
988:16
916:/2
912:=
737:.
476:.
373:22
357:71
286:.
72::
50:,
2834:.
2814::
2789:.
2769::
2744:.
2730::
2701:.
2669:.
2649::
2633:π
2620:.
2600::
2575:.
2561::
2532:.
2512::
2484:.
2456::
2448:1
2429:9
2405:.
2393::
2379:π
2363:.
2329:.
2296:.
2276::
2238:.
2175:.
2161::
2140:.
2120::
2110::
2080:.
2066::
2034:.
1995:.
1983::
1957:.
1937::
1921:π
1901:.
1861:.
1828:.
1802:π
1743:n
1739:2
1735:=
1732:x
1708:π
1704:x
1699:n
1695:x
1689:x
1671:.
1667:)
1659:i
1655:2
1651:x
1638:n
1633:1
1630:=
1627:i
1618:(
1610:n
1606:2
1602:x
1589:n
1585:2
1581:=
1578:x
1562:n
1544:,
1539:2
1536:x
1523:2
1520:x
1509:2
1506:=
1503:x
1473:2
1447:2
1443:/
1437:2
1421:2
1417:2
1377:π
1357:.
1350:1
1344:k
1340:a
1336:+
1333:2
1328:=
1319:k
1315:a
1307:,
1304:0
1301:=
1292:1
1288:a
1280:,
1273:k
1269:a
1262:2
1255:k
1251:2
1239:k
1231:=
1200:,
1190:k
1170:2
1165:+
1159:+
1156:2
1151:+
1148:2
1143:+
1140:2
1135:+
1132:2
1124:2
1115:k
1111:2
1099:k
1091:=
1078:π
1058:2
1054:x
1045:+
1042:1
1035:=
1030:2
1027:x
969:8
950:4
936:=
928:2
914:π
910:x
888:8
885:x
869:4
866:x
850:2
847:x
836:=
831:x
827:x
792:π
784:π
780:π
771:n
765:π
761:n
748:n
742:n
740:S
735:π
730:×
700:n
680:n
654:.
647:1
641:n
637:a
633:+
630:2
625:=
616:n
612:a
602:2
597:=
588:1
584:a
559:,
551:2
546:=
541:2
536:i
532:a
524:n
519:1
516:=
513:i
497:n
462:π
451:π
439:π
428:π
420:π
401:π
381:.
376:7
333:π
280:π
248:.
241:1
238:+
235:n
231:2
209:1
206:=
203:n
195:=
187:2
160:2
152:2
147:+
144:2
139:+
136:2
125:2
119:2
114:+
111:2
100:2
96:2
90:=
82:2
69:π
27:.
21:π
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