“Archimedes' Recurrence Formula”的概念、定义、习题解题方法及翻译-数学辞典

单词

Archimedes' Recurrence Formula

释义

Archimedes' Recurrence Formula

Let and be the Perimeters of the Circumscribed andInscribed -gon and and the Perimeters of theCircumscribed and Inscribed -gon. Then

			(1)
			(2)

The first follows from the fact that side lengths of the Polygons on a Circle of Radius

are

			(3)
			(4)

			(5)
			(6)

But


			(7)

Using the identity

(8)

then gives

(9)

The second follows from

(10)

Using the identity

(11)

gives


	(12)

Successive application gives the Archimedes Algorithm, which can be used to provide successive approximationsto Pi (

See also Archimedes Algorithm, Pi

References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 186, 1965.

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