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Approximations for the mathematical constant pi () in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes). In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was made only from the 15th century (Jamshīd al-Kāshī), and early modern mathematicians reached an accuracy of 35 digits by the 18th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega), surpassing the accuracy required for any conceivable application outside of pure mathematics. The record of manual approximation of is held by William Shanks, who calculated 527 digits correctly in the years preceding 1873. Since the mid 20th century, approximation of has been the task of electronic digital computers; the current record (as of May 2015) is at 13.3 trillion digits, calculated in October 2014.〔http://www.numberworld.org/y-cruncher/〕 ==Early history== The best known approximations to dating to before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics 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. Some Egyptologists〔Petrie, W.M.F. ''Wisdom of the Egyptians'' (1940)〕 have claimed that the ancient Egyptians used an approximation of as from as early as the Old Kingdom.〔 Based on the Great Pyramid of Giza, supposedly built so that the circle whose radius is equal to the height of the pyramid has a circumference equal to the perimeter of the base (it is 1760 cubits around and 280 cubits in height). Verner, Miroslav. ''The Pyramids: The Mystery, Culture, and Science of Egypt's Great Monuments.'' Grove Press. 2001 (1997). ISBN 0-8021-3935-3〕 This claim has met with skepticism.〔Rossi, ''Corinna Architecture and Mathematics in Ancient Egypt,'' Cambridge University Press. 2007. ISBN 978-0-521-69053-9.〕〔Legon, J. A. R. ''On Pyramid Dimensions and Proportions'' (1991) Discussions in Egyptology (20) 25-34 ()〕 Babylonian mathematics usually approximated to 3, sufficient for the architectural projects of the time (notably also reflected in the description of Solomon's Temple in the Hebrew Bible).〔See #Imputed biblical value. There has been concern over the apparent biblical statement of ≈ 3 from the early times of rabbinical Judaism, addressed by Rabbi Nehemiah in the 2nd century. Petr Beckmann, ''A History of Pi'', St. Martin's (1971).〕 The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of as , about 0.5 percent below the exact value.〔 David Gilman Romano, ''Athletics and Mathematics in Archaic Corinth: The Origins of the Greek Stadion'', American Philosophical Society, 1993, ( p. 78 ). "A group of mathematical clay tablets from the Old Babylonian Period, excavated at Susa in 1936, and published by E.M. Bruins in 1950, provide the information that the Babylonian approximation of was 3 1/8 or 3.125." E. M. Bruins, ''(Quelques textes mathématiques de la Mission de Suse )'', 1950. E. M. Bruins and M. Rutten, ''Textes mathématiques de Suse'', Mémoires de la Mission archéologique en Iran vol. XXXIV (1961). See also "in 1936, a tablet was excavated some 200 miles from Babylon. () The mentioned tablet, whose translation was partially published only in 1950, () states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60+36/(60)2 ( = 3/0.96 = 25/8 )". Jason Dyer , (On the Ancient Babylonian Value for Pi ), 3 December 2008.〕 At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the Second Intermediate Period, c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of as ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle by approximating the circle by an octagon.〔〔Katz, Victor J. (ed.), Imhausen, Annette et al. ''The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook,'' Princeton University Press. 2007. ISBN 978-0-691-11485-9〕 Astronomical calculations in the ''Shatapatha Brahmana'' (c. 4th century BCE) use a fractional approximation of .〔Chaitanya, Krishna. (A profile of Indian culture. ) Indian Book Company (1975). P. 133.〕 In the 3rd century BCE, Archimedes proved the sharp inequalities < < , by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively). In the 2nd century CE, Ptolemy, used the value , the first known approximation accurate to three decimal places (accuracy 2·10−5).〔http://uzweb.uz.ac.zw/science/maths/zimaths/pi.htm〕 The Chinese mathematician Liu Hui in 263 CE computed to between and by inscribing an 96-gon and 192-gon; the average of these two values is 3.141864 (accuracy 9·10−5). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result (accuracy 2·10−6), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi.〔. Reprinted in . See in particular pp. 333–334 (pp. 28–29 of the reprint).〕 Zu Chongzhi is known to have computed between 3.1415926 and 3.1415927, which was correct to seven decimal places. He gave two other approximations of : and . The latter fraction is the best possible rational approximation of using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's result surpasses the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium. In Gupta-era India (6th century), mathematician Aryabhata in his astronomical treatise Āryabhaṭīya calculated the value of to five significant figures ().〔Āryabhaṭīya (): :. :"Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, (4 + 100) × 8 + 62000 is the circumference of a circle with diameter 20000. This provides a value of , 〕 using it to calculate an approximation of the earth's circumference. Aryabhata stated that his result "approximately" (' "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of astronomy and mathematics) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational).〔 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「approximations of π」の詳細全文を読む スポンサード リンク
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