Exploring Mathematics With Computer

Preface

A computer is a powerful instrument for performing numerical experiments and simulations on an unprecedented scale. Many young mathematicians now use it for research from the very beginning. The computergenerated data lead to clues, then to solutions and can even suggest elegant proofs. Good examples are provided by the short final Section 65, which solves a difficult problem and by the very tough Problem 17, due to Euler, in Exercises to Sections 17-18; the computer makes it accessible.

This is a mathematics-, not a programming book. It is intended for students, mathematics teachers and mathematicians who are just starting to explore mathematics on their own computers. In studying it, and especially in working through its exercises, readers will get to know many new, elementary topics and learn as much from the extensive exercises as from the examples.

We use Turbo Pascal. We need only a fragment of this dialect, easily picked up by readers as they work their way through the examples and exercises. There is little difference between dialects of Pascal, so you will need only slight modifications for a different Pascal implementation. Most of our programs are complete. Each serves as a reading exercise to help you learn Pascal; for this reason the programs are short and, for the most part, comprehensible without comment. Yet, many are not trivial!

The book contains 65 topics, loosely structured into seven chapters; these, and even the individual topics, are independent of each other. If you know the rudiments of Pascal you can tackle them in any order; if not, you are advised to cover them in sequence since new features of Pascal are introduced when needed.

The book originated in an inservice course for high school mathematics teachers, and a seminar Mathematical Discoveries with a PC. The idea was to give teachers interesting and challenging material for use in their classes; teachers had complained that they had to learn a language by programming trivial examples. Boredom had prevented effective learning!

Arthur Engel, 1993

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