Zhao Youqin's algorithm was an algorithm devised by

Yuan dynasty The Yuan dynasty (), officially the Great Yuan (; xng, , , literally "Great Yuan State"), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after its division. It was established by Kublai, the fift ...

Chinese astronomer and mathematician Zhao Youqin (, ? – 1330) to calculate the value of in his book ''Ge Xiang Xin Shu'' ().

Algorithm

Zhao Youqin started with an inscribed square in a circle with

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

r. If

\ell

denotes the length of a side of the square, draw a

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...

line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram: :

d=\sqrt

e=r-d=r-\sqrt.

Extend the perpendicular line d to dissect the circle into an octagon;

\ell_2

denotes the length of one side of octagon. :

\ell_2=\sqrt

\ell_2=\frac\sqrt

Let

l_3

denotes the length of a side of hexadecagon :

\ell_3=\frac\sqrt

similarly :

\ell_=\frac\sqrt

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 :

\pi =3.141592. \,

He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of , that is 3, 3.14, and , the last is the most exact.Yoshio Mikami, p136

References

{{DEFAULTSORT:Zhao Youqin's pi algorithm Chinese mathematics Pi algorithms

Algorithm

See also

References