Calculating Pi - Estimating as an Introduction to Technical...

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Estimating π as an Introduction to Technical Computation Mark French Department of Mechanical Engineering Technology Purdue University Abstract Engineering technology students often being their college experience with minimal backgrounds in technical computing. A simple exercise in computing an approximate value of π can introduce basic concepts and commonly used software while calculating a familiar number. Several different methods of approximation are presented starting with an intuitive geometric approach and progressing to more efficient methods based on series expansions. Background π is perhaps the most universally used mathematical constant, appearing in expressions from a huge range of technical fields. Every engineering technology student knows that π is approximately equal to 3.14 or 22/7. Given a dozen decimal places, it is possible to express the circumference of a circle 1 million km in diameter with an accuracy of 1mm. While there is no engineering reason to need more than about a dozen decimal places, the numerical value of π is currently known to more than a trillion decimal places. There are several reasons for this incredible level of precision. One is that number theorists are looking for patterns in the series of digits, though π has been proven to be irrational and, thus, cannot terminate in a repeating series of digits, no matter how long. Another, more
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practical use is to verify the accuracy of new computers by showing that they can correctly calculate π to a large number of digits [1]. Engineering technology students are generally just told that π is the ratio of the circumference to the diameter of a circle and that it has a certain approximate value. Common scientific calculators have π programmed to around a dozen significant figures and some students even take it upon themselves to memorize them. However, is it very uncommon for students to be shown where the numerical value of π comes from and even more rare for them to do the actual calculation. This situation offers an opportunity to both introduce technology students to some basic ideas of technical calculations and to remove some of the mystery about this universal constant. Fortunately, several means of calculating π are well within the mathematical abilities of technology students who are not yet familiar with the basics of calculus. A basic knowledge of calculus allows more efficient methods to be used. Many different methods have been proposed over the last few thousand years [2,3]; what follows is a description of several representative approaches in roughly increasing order of sophistication. Method of Polygons Among the first known approximations to π are 25/8 from the Babylonians and 256/81 from the Egyptians [4]. Both appear to date from around 1900 BC. The first known rigorous estimate was by Archimedes (287-212 BC). He showed that it can be
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approximated by inscribing and circumscribing regular polygons on a circle as shown in Figure 1.
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This note was uploaded on 02/21/2012 for the course MET 581 taught by Professor Staff during the Fall '08 term at Purdue University-West Lafayette.

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Calculating Pi - Estimating as an Introduction to Technical...

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