IJAM: Volume 37, No. 6 (2024)

DOI: 10.12732/ijam.v37i6.2

A FORMULA FOR \pi WITH A NESTED RADICAL

Dubravko Sabolic

University of Zagreb, FER

Unska 3

10000 Zagreb, CROATIA

Abstract. A formula for $\pi$ is derived, using successive approximation of the area of a wedge of arbitrary central angle cut out from a unit-radius circle. The resulting formula takes the form of

$2^n$, multiplied by a nested radical of the order n. In general case, this radical splits into two separate nested radicals of the same shape and order, which are symmetric in a way that one of them stems from an arbitrary seed value, $s \in (0,1)$, while the other is started from the complementary seed, $1-s$. A reduced formula is also derived, with only one nested radical. The main computational characteristics of both types of the algorithm were briefly discussed, and compared to that of several other widely known $\pi$-formulas.

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How to cite this paper?
DOI: 10.12732/ijam.v37i6.2
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 6

References

[1] E.W. Weisstein, Pi Formulas, From: MathWorld - A Wolfram Web Resource,
URL: http://mathworld.wolfram.com/PiFormulas.html.

[2] Approximations of \pi, From: Wikipedia,
URL: https://en.wikipedia.org/wiki/Approximations_of_pi.

[3] D.H. Bailey, A collection of mathematical formulas involving \pi,
URL: https://www.davidhbailey.com/dhbpapers/pi-formulas.pdf.

[4] K.A. Arras, An introduction to error propagation, Swiss Federal Institute of Technology Lausanne, URL: https://infoscience.epfl.ch/record/97374/files/TR-98-01R3.pdf.

[5] D. Sabolic and R. Malaric, Another formula for \pi with an infinitely nested radical (In Croatian language), Poucak, 80 (2019), 6-26, URL: https://hrcak.srce.hr/file/347642.

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