Fall 2004
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Calculating Pi
Paul Bamberg
| Location: Quincy House Bullitt Room |
| Meeting time: W., 3-6 |
| Catalog number: 4737 |
| This seminar will focus on the mathematical, computational, and historical aspects of calculating pi. Many early civilizations found an approximate solution to this problem, and the ancient Greeks posed the problem of “squaring the circle”—constructing, using nothing but a straightedge and compass, in a finite number of operations, a square whose area equals the area of a given circle. They failed to solve the problem, for the simple reason that the construction is impossible, though it took about two thousand years to prove this. In the meantime, many of the great mathematicians of history, including Archimedes, Newton, Gauss, and Euler, worked on the problem. Since pi cannot be calculated in a finite number of operations, the methods they employed—taking limits, summing infinite series, calculating definite integrals, evaluating infinite products—go to the heart of the calculus. The seminar will explore a wide variety of methods for computing pi and their implementation in C++ and Mathematica on a personal computer. Members of the seminar will use geometry and calculus to prove the correctness of these methods and assess their accuracy, and they then will use these methods to calculate pi to a large number of decimal places. Everyone in the seminar will need to have studied calculus. The participants can modify and enhance existing C++ programs to implement algorithms for computing pi or can learn to use Mathematica, a powerful interactive software tool for doing symbolic mathematics. There will be scope for a variety of interests and talents: people who enjoy reading Newton’s or Euler’s work in its original form, or who excel at proofs, or who are skilled at user-interface design or object-oriented programming, or who have a strong interest in the history of mathematics and the lives of great mathematicians.
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