
06186008
j
2013d.00809
Dion, Peter
Ho, Anthony
A new iterative method to calculate $\pi$.
Aust. Sr. Math. J. 26, No. 1, 4149 (2012).
2012
Australian Association of Mathematics Teachers (AAMT), Adelaide, SA
EN
N55
N45
U75
computers
teaching methods
geometric concepts
programming
computation
undergraduate students
college students
numbers
approximation
$\pi$
equations
NewtonRaphson method
http://www.aamt.edu.au/Webshop/Entirecatalogue/AustralianSeniorMathematicsJournal
Summary: For at least 2000 years people have been trying to calculate the value of $\pi$, the ratio of the circumference to the diameter of a circle. People know that $\pi$ is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real accuracy did not come until the use of infinite series techniques, in which one can, by calculating more and more terms, obtain smaller and smaller corrections all leading to a precise value. In this paper, the authors use an iterative approach to calculate $\pi$, in which a computer is also used. This method initially appears to be very impressive, providing more than sixteen decimal places with only three iterations. This article addresses first year undergraduate university students who will encounter in their mathematics courses the subjects of differentiation, Taylor series, exponentials and natural logarithms, and the idea of array storage in basic computer programming. In terms of teaching concepts the authors describe the NewtonRaphson method for solving equations in detail as a very useful and lesser known application of differentiation, and then apply it in an unexpected manner to solve a seemingly unrelated problem, the determination of the value of $\pi$, thus introducing a new method of approaching one of the most famous endeavours in mathematics. In pursuing this they delve into practical problems in using power series to calculate functions and also into in certain aspects of numerical analysis. (ERIC)