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Liu Hui proved an inequality involving {{pi}} by considering the area of inscribed polygons with {{math|''N''}} and 2{{math|''N''}} sides.
In the diagram, the yellow area represents the area of an {{math|''N''}}-gon, denoted by <math>A_N</math>, and the yellow area plus the green area represents the area of a 2{{math|''N''}}-gon, denoted by <math>A_{2N}</math>. Therefore, the green area represents the difference between the areas of the 2{{math|''N''}}-gon and the ''N''-gon:
:<math>D_{2N}=A_{2N}-A_N.</math>
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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 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 Adriaen Anthoniszoom and his son obtained {{pi}} value of 3.1415929, accurate only to 7 digit, still 3 digits short of Zu's result<ref>Robert Temple, The Genius of China, a refined value of pi, p144-145, ISBN 1-85375-292-4</ref>
== See also ==▼
* [[Method of exhaustion]]▼
*[[Zhao Youqin's π algorithm]]▼
==Notes==
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<br>
Liu Hui's quick method was potentially able to deliver almost the same result of 12288-gon (3.141592516588) with only 96-gon.
▲== See also ==
▲* [[Method of exhaustion]]
▲*[[Zhao Youqin's π algorithm]]
== References ==
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