Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-28T20:59:03.202Z Has data issue: false hasContentIssue false
  • English
  • Français

An infinite product of nested radicals for log x from the Archimedean algorithm

Published online by Cambridge University Press:  14 June 2016

Thomas J. Osler ,
Walter Jacob  and
Ryo Nishimura
Thomas J. Osler
Affiliation:
Mathematics Department, Rowan University, Glassboro NJ 08028USA e-mail: osler@rowan.edu
Walter Jacob
Affiliation:
Mathematics Department, Temple University, Philadelphia, Pa. USA
Ryo Nishimura
Affiliation:
Department of Frontier Materials, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya Aichi, 466-8555, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

The so-called Archimedean iterative algorithm for calculating π uses a method involving the two equations

(1)

and

(2)

(Note that (1) is the harmonic mean.) Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to a circle of diameter one. The larger one has perimeter a0, the smaller has perimeter b0. (Archimedes used hexagons with and b0 = 3, but regular polygons of any number of sides can begin the iterations.) 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 (2) and (3). (See [1] for a derivation or many explanations on the web.) 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.

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 548 , July 2016 , pp. 274 - 278
Copyright
Copyright © Mathematical Association 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Osler, T. J., Five historic formulas for pi, Mathematics and Computer Education, 43 (2009) pp. 250–258.Google Scholar
2.Phillips, G. M., Archimedes the Numerical Analyst, Amer. Math. Monthly 88 (1981) pp. 165–169.Google Scholar
3.Berggren, L., Borwein, J., Borwein, P., Pi: A Source Book (3rd edn.) Springer (2004).CrossRefGoogle Scholar
4.Vieta, F., Variorum de Rebus Mathematicis Reponsorum Liber VII, (1593) in: Opera Mathematica, (reprinted) Georg Olms Verlag, Hildesheim, New York (1970) pp. 398–400, 436-446.Google Scholar
5.Heath, T. L., The works of Archimedes, Dover Publications, Mineola, NY (2002).Google Scholar
6.Miel, G., Of calculations past and present: The Archimedean algorithm, Amer. Math. Monthly 90 (1) (January 1983) pp. 17–35.Google Scholar
7.Stillwell, J., Mathematics and its history (3rd edn.) Springer, New York (2010).CrossRefGoogle Scholar
8.Nishimura, R., New inequalities and infinite product formulas for the trigonometric and the lemniscate functions, Math. Ineq. Appl. 18 (2) (2015) pp. 529–540.Google Scholar
9.Nishimura, R., New properties of the lemniscate function and its transformation, J. Math. Anal. Appl. 427 (1) (2015) pp. 460–468.CrossRefGoogle Scholar