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Zhao Youqin's π algorithm
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<h2 id="algorithm">Algorithm</h2> <p>Zhao Youqin started with an inscribed square in a circle with <a href="/facts/Radius/9Gr9deYD">radius</a> r.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>If ℓ {\displaystyle \ell } denotes the length of a side of the square, draw a <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram: </p> d = r 2 − ( ℓ 2 ) 2 {\displaystyle d={\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}} e = r − d = r − r 2 − ( ℓ 2 ) 2 . {\displaystyle e=r-d=r-{\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}.} <p>Extend the perpendicular line d to dissect the circle into an <a href="/facts/Octagon/H5PuP3wP">octagon</a>; ℓ 2 {\displaystyle \ell _{2}} denotes the length of one side of octagon. </p> ℓ 2 = ( ℓ 2 ) 2 + e 2 {\displaystyle \ell _{2}={\sqrt {\left({\frac {\ell }{2}}\right)^{2}+e^{2}}}} ℓ 2 = 1 2 ℓ 2 + 4 ( r − 1 2 4 r 2 − ℓ 2 ) 2 {\displaystyle \ell _{2}={\frac {1}{2}}{\sqrt {\ell ^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell ^{2}}}\right)^{2}}}} <p>Let l 3 {\displaystyle l_{3}} denotes the length of a side of <a href="/facts/Hexadecagon/3jTUD0zD">hexadecagon</a> </p> ℓ 3 = 1 2 ℓ 2 2 + 4 ( r − 1 2 4 r 2 − ℓ 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}}}} <p>similarly </p> ℓ n + 1 = 1 2 ℓ n 2 + 4 ( r − 1 2 4 r 2 − ℓ 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}}}} <p>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 </p> π = 3.141592. {\displaystyle \pi =3.141592.\,} <p>He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of π, that is 3, 3.14, 22/7 and 355/113, the last is the most exact.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Liu_Hui%27s_%CF%80_algorithm/XWNMYrGv">Liu Hui's π algorithm</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Yoshio Mikami, Development of Mathematics in China and Japan, Chapter 20, The Studies about the Value of π etc., pp 135–138 <a href="/wiki/Yoshio_Mikami" target="_blank">/wiki/Yoshio_Mikami</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Yoshio Mikami, p136 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>