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 state of [[Cao Wei|Cao Wei dynasty]]. 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 [[Wang 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|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 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.