Maple and Mathematica, A Problem Solving Approach for Mathematics

In the history of mathematics there are many situations in which calculationswere performed incorrectly for important practical applications.Let us look at some examples, the history of computing the number πbegan in Egypt and Babylon about 2000 years BC, since then manymathematicians have calculated π (e.g., Archimedes, Ptolemy, Vi`ete,etc.). The first formula for computing decimal digits of π was discoveredby J. Machin (in 1706), who was the first to correctly compute100 digits of π. Then many people used his method, e.g., W. Shankscalculated π with 707 digits (within 15 years), although due to mistakesonly the first 527 were correct. For the next examples, we can mentionthe history of computing the fine-structure constant α (that was firstdiscovered by A. Sommerfeld), and the mathematical tables, exact solutions,and formulas, published in many mathematical textbooks, werenot verified rigorously [25]. These errors could have a large effect onresults obtained by engineers.