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Maple and Mathematica: A Problem Solving Approach for Mathematics

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In the history of mathematics there are many situations in which calculations were performed incorrectly for important practical applications. Let us look at some examples, the history of computing the number π began in Egypt and Babylon about 2000 years BC, since then many mathematicians have calculated π (e.g., Archimedes, Ptolemy, Vi`ete, etc.). The first formula for computing decimal digits of π was discovered by J. Machin (in 1706), who was the first to correctly compute 100 digits of π. Then many people used his method, e.g., W. Shanks calculated π with 707 digits (within 15 years), although due to mistakes only the first 527 were correct. For the next examples, we can mention the history of computing the fine-structure constant α (that was first discovered by A. Sommerfeld), and the mathematical tables, exact solutions, and formulas, published in many mathematical textbooks, were not verified rigorously [25]. These errors could have a large effect on results obtained by engineers.

But sometimes, the solution of such problems required such technology that was not available at that time. In modern mathematics there exist computers that can perform various mathematical operations for which humans are incapable. Therefore the computers can be used to verify the results obtained by humans, to discovery new results, to improve the results that a human can obtain without any technology. With respect to our example of computing π, we can mention that recently (in 2002) Y. Kanada, Y. Ushiro, H. Kuroda, and M. Kudoh, calculated π over 1.241 trillion digits (explicitly, 1, 241, 100, 000, 000).
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Though we did not know it at the time, this book’s genesis began with the arrival of Cris Calude in New Zealand. Cris has always had an intense interest in algorithmic information theory. The event that led to much of the recent research presented here was the articulation by Cris of a seemingly innocuous question. This question...

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