Liu Hui's π algorithm: Difference between revisions

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{{pi box}}

[[Image:Cutcircle2.svg|thumb|right|Liu Hui's method of calculating the area of a circle]]

'''Liu Hui's {{pi}} algorithm''' was invented by [[Liu Hui]] (fl. 3rd century), a mathematician of the [[Cao Wei|Wei Kingdom]]. Before his time, the ratio of the circumference of a circle to its diameter was often taken experimentally as three in China, while [[Zhang Heng]] (78&ndash;139) rendered it as 3.1724 (from the proportion of the celestial circle to the diameter of the earth, {{math|92/29}}) or as <math>\pi \approx \sqrt{10} \approx 3.162</math>. Liu Hui was not satisfied with this value. He commented that it was too large and overshot the mark. Another mathematician [[Wan Fan]] (219&ndash;257) provided {{math|1=π ≈ 142/45 ≈ 3.156}}.<ref>Schepler, Herman C. (1950), “The Chronology of Pi”, Mathematics Magazine 23 (3): 165–170, ISSN {{issn|0025-570X}}.</ref> All these empirical {{pi}} values were accurate to two digits (i.e. one decimal place). Liu Hui was the first Chinese mathematician to provide a rigorous algorithm for calculation of {{pi}} to any accuracy. Liu Hui's own calculation with a 96-gon provided an accuracy of five digits: {{math|π ≈ 3.1416}}.

Liu Hui remarked in his commentary to the ''[[The Nine Chapters on the Mathematical Art]]'',<ref>Needham, Volume 3, 66.</ref> that the ratio of the circumference of an inscribed hexagon to the diameter of the circle was three, hence {{pi}} must be greater than three. He went on to provide a detailed step-by-step description of an iterative algorithm to calculate {{pi}} to any required accuracy based on bisecting polygons; he calculated {{pi}} to between 3.141024 and 3.142708 with a 96-gon; he suggested that 3.14 was a good enough approximation, and expressed {{pi}} as 157/50; he admitted that this number was a bit small. Later he invented an ingenious [[#Quick method|quick method]] to improve on it, and obtained {{math|π ≈ 3.1416}} with only a 96-gon, with an accuracy comparable to that from a 1536-gon. His most important contribution in this area was his simple iterative {{pi}} algorithm.

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==Significance of Liu Hui's algorithm==

Liu Hui's {{pi}} algorithm was one of his most important contributions to ancient Chinese mathematics. It was based on calculation of {{math|N}}-gon area, in contrast to the Archimedean algorithm based on polygon circumference. Archimedes used a circumscribed 96-gon to obtain an upper limit <math>\pi <22/7=3.142857</math>, and an inscribed 96-gon to obtain the lower limit <math>223/71=3.140845</math>. Liu Hui was able to obtain both his upper limit 3.142704 and lower limit 3.141024 with only an inscribed 96-gon. Furthermore, both the Liu Hui limits were tighter than Archimedes's: 3.140845 < 3.141024 < {{pi}} < 3.142704 < 3.142857. With his method Zu Chongzhi obtained the eight-digit result: 3.1415926 < {{pi}} < 3.1415927, which held the world record for the most accurate value of {{pi}} for 1200 years, even by 1600 in Europe, mathematician [[Adriaan Anthonisz]] and his son obtained {{pi}} value of 3.1415929, accurate only to 7 digits.<ref>Robert Temple, The Genius of China, a refined value of pi, p144-145, ISBN {{isbn|1-85375-292-4}}</ref>

== See also ==

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== Further reading ==

*Needham, Joseph (1986). ''Science and Civilization in China'': Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.

* Wu Wenjun ed, ''History of Chinese Mathematics'' Vol III (in Chinese) ISBN {{isbn|7-303-04557-0}}

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