Practicality and Pedagogy

Quoting from R.W. Hamming's (he of the Hamming code fame) Numerical Methods for Scientists and Engineers:

"Numbers are the basis of numerical methods, Thus logically, they belong at the beginning of a course on numerical methods. On the other hand, psychologically they occur rather late in development.

The situation in numerical methods is very like that in the calculus course where the real number system is basic to the limit process. The calculus course, therefore, often begins with a detailed, fairly rigorous discussion of the real number system. Unfortunately, at this point in his education the student has little reason to care about the topic, and it always turns out to be the most difficult part of the entire course. Furthermore, this topic generally repels the student, and this attitude carries over to the rest of the course.

The history of mathematics further shows that the real number system was very late in developing. The discoverers and developers of the calculus ignored the niceties of the number system for many years. The biological principal "ontogeny recapitulates phylogeny" means that "the development of the individual tends to repeat the development of the species" This is very relevant to teaching; the history of a subject gives important clues as to the ordering and relative difficulties of the material being taught.

History likewise shows that for a long time the number system used in computing was essentially ignored and taken for granted. Putting the topic first, therefore, requires justification, because we are asking the beginner to learn material whose importance he is not psychologically prepared to accept. The justification is the same as that for the calculus course. If we are to make rapid progress and not to have to repeat some material several times, then it is necessary to start with a firm foundation. The author's own experience was that only after many years of computing did he come to understand how the number system used by machines affected what was obtained and how at times it led him astray.

Thus we are asking the beginner to accept on faith that the material in this chapter is basic and to put aside his natural psychological prejudices in favor of the logical approach. Of course he wants to get on to solving real problems and not to fuss with aparently trivial, irrelevant details of the number system used by machines, which he thinks he understands anyway. In compensation we will try to make the material a bit more dramatic than usual in order to sustain his interest through this desert of logical presentation. Probably he should plan to review this chapter later several times until he becomes thoroughly familiar with many of the various peculiar features of the number system, since they are the only numbers that can occur in the computation; there are no other numbers, and the mathematician's "real" number system is purely fictitious."