Understanding of Mathematics is Prerequisite

In the last fifteen years we have frequently encountered the opinion that, thanks to the new technologies, we need to learn mathematics only to the very basic level. According to this extreme opinion sufficient understanding of mathematics is such as to enable one to realize three main steps: to formulate an engineering problem in mathematical language, to perform suitable commands into a computer (input), and to read the needed information from the obtained output. Some schools follow such recommendations, without any critical appraisal, perhaps motivated by the reduced teaching loads. When doing just basic mathematics we do not need deep understanding of math-ematics subject. However, as tasks become more complicated, deeper understanding is needed. First of all, we must prepare a suitable input, if we wish the computer to work properly. After the computer’s work is finished, we have to simplify or to control the obtained results. This last work is extremely important when dealing with demanding task. The reason is that computer can give a wrong result, even without any warning. It seems that the most important finding is the fact that we must control the computer itself. However, to control the computer we need a fairly good understanding of mathematics. We shall illustrate this assertion with some instructive examples, which show the vital interplay between mathematics and computers. The first three examples are relatively simple, but the last one is quite complicated. Example 1. Using Mathematica [2] and sequences f(n), ϕ(n), f ∗(n), g(n), and 8