Zhao Youqin's Ο algorithm
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Zhao Youqin's Ο algorithm is an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (θ΅΅ει¦, ? β 1330) to calculate the value of Ο in his book Ge Xiang Xin Shu (ι©θ±‘ζ°δΉ¦).
Contents
β’ Algorithm
β’ See also
β’ References
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Algorithm
If β {\displaystyle \ell } denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r β d. Then from the diagram:
d = r 2 β ( β 2 ) 2 {\displaystyle d={\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}}
e = r β d = r β r 2 β ( β 2 ) 2 . {\displaystyle e=r-d=r-{\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}.}
Extend the perpendicular line d to dissect the circle into an octagon; β 2 {\displaystyle \ell _{2}} denotes the length of one side of octagon.
β 2 = ( β 2 ) 2 + e 2 {\displaystyle \ell _{2}={\sqrt {\left({\frac {\ell }{2}}\right)^{2}+e^{2}}}}
β 2 = 1 2 β 2 + 4 ( r β 1 2 4 r 2 β β 2 ) 2 {\displaystyle \ell _{2}={\frac {1}{2}}{\sqrt {\ell ^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell ^{2}}}\right)^{2}}}}
Let l 3 {\displaystyle l_{3}} denotes the length of a side of hexadecagon
β 3 = 1 2 β 2 2 + 4 ( r β 1 2 4 r 2 β β 2 2 ) 2 {\displaystyle \ell _{3}={\frac {1}{2}}{\sqrt {\ell _{2}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{2}^{2}}}\right)^{2}}}}
similarly
β n + 1 = 1 2 β n 2 + 4 ( r β 1 2 4 r 2 β β n 2 ) 2 {\displaystyle \ell _{n+1}={\frac {1}{2}}{\sqrt {\ell _{n}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{n}^{2}}}\right)^{2}}}}
Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or
Ο = 3.141592. {\displaystyle \pi =3.141592.\,}
He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of Ο, that is 3, 3.14, β 22/7β and β 355/113β , the last is the most exact.cite-ref-2[2]
See also
References
cite-note-11. β Yoshio Mikami, Development of Mathematics in China and Japan, Chapter 20, The Studies about the Value of Ο etc., pp 135β138
cite-note-22. β Yoshio Mikami, p136