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Vieta’s product for pi from the Archimedean algorithm

Published online by Cambridge University Press:  25 August 2015

Thomas J. Osler  and
Sky Waterpeace
Thomas J. Osler
Affiliation:
Mathematics Department, Rowan University, Glassboro NJ 08028 USA e-mail: osler@rowan.edu
Sky Waterpeace
Affiliation:
Mathematics Department, Rowan University, Glassboro NJ 08028 USA e-mail: osler@rowan.edu
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Extract

In this paper we show how to derive Vieta’s famous product of nested radicals for π [1] from the Archimedean iterative algorithm for π, [2, 3]. Only simple algebraic manipulations are needed.

The Archimedean iterative algorithm for calculating π uses a method involving the two equations:

and

In using this algorithm, we start with a circle of diameter 1. Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to this circle. The larger one has perimeter a0, the smaller has perimeter b0. Since the perimeter of the circle is π we have b0 < π < a0. Now consider regular polygons with twice the number of sides that circumscribe and inscribe the circle and call their perimeters a1 and b1 respectively. These can be calculated from (1) and (2). Continuing in this way we generate the perimeters of inscribed and circumscribed regular polygons, and in each case the number of sides is twice the sides of the previous polygon. Clearly the sequence {an} approaches π from above and {bn} approaches π from below.

Type
Research Article
Information
The Mathematical Gazette , Volume 98 , Issue 543 , November 2014 , pp. 429 - 431
Copyright
Copyright © The Mathematical Association 2014

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References

1. Heath, T.L., The works of Archimedes, Cambridge University Press (2004).Google Scholar
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