Zhao Youqin's algorithm is an algorithm devised by

Yuan dynasty The Yuan dynasty ( ; zh, c=元朝, p=Yuáncháo), officially the Great Yuan (; Mongolian language, Mongolian: , , literally 'Great Yuan State'), was a Mongol-led imperial dynasty of China and a successor state to the Mongol Empire after Div ...

Chinese astronomer and mathematician

Zhao Youqin Zhao Youqin (趙友欽 1271-?) was a Chinese mathematician, astronomer, alchemist, and Taoist monk. He is most well known for his book ''Ge xiang xin shu (革象新书),'' translated either as ''New Elucidation of the Heavenly Bodies'' or ''New Wr ...

(, ? – 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 or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

r. If

\ell

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

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

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 In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a ...

;

\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 In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon. Regular hexadecagon A ''regular polygon, regular hexadecagon'' is a hexadecagon in which all angles are equal and all sides are congruent. It ...

\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 mathematical discoveries Pi algorithms

Algorithm

See also

References