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 . 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).