Preface 
heu-ris-tic [adjective]
1. serving to indicate or point out; stimulating interest as a means of furthering investigation.
2. encouraging a person to
learn, discover, understand, or solve problems on his or her own, as by
experimenting, evaluating possible answers or solutions, or by trial and
error: a heuristic teaching method.
[Source: Dictionary.com]
Introduction
This book is an outgrowth of
classes given at the University of California,Santa Barbara, mainly for
students who had little mathematical background. Many of the students
indicated they never understood what mathematics was all about (beyond
what they learned in algebra and geometry). Was there any more
mathematics to be discovered or created? How could one actually discover
or create new mathematics?
In order to give these students
some sort of answers to such questions, we designed a course in which
the students could actually participate in the discovery of mathematics.
The class was not presented in the usual lecture fashion. And it did
not deal with topics that the students had seen before. Ordinary
algebra, geometry, and arithmetic played minor roles in most of the
problems we addressed. Whatever algebra and geometry that did appear was
relatively easy and straightforward.
Our objective was to give the
students an appreciation of mathematics, rather than to provide tools
they would need in some field that required mathematics. In that sense,
the course was like a course in music appreciation or art appreciation.
Such courses don’t attempt to train students to become pianists,
composers, or artists. Instead, they attempt to give the students a
sense of the subject.
Why do so many intelligent
people have so little sense of the field of mathematics? A partial
explanation involves the difficulty in communicating mathematics to the
general public. Without special training in astronomy, medicine, or
other scientific areas, a person can still get a sense of what goes on
in those areas just by reading newspapers. But this is much more
difficult in mathematics. This may be so because much of modern
mathematics involves very technical language that is difficult to
express in ordinary English. Even professional mathematicians often have
difficulty communicating their work to other professional
mathematicians who work in different areas.
This isn’t surprising when one
realizes how many areas and sub areas there are in mathematics.
Mathematical Reviews (MR) is a journal that provides short reviews of
mathematical papers that appear in over 2000 journals from around the
world. The subject classification used by MR has over 50 subject areas,
each of which has several subareas. Each of these subareas has many
sub-sub areas. A research mathematician might be an expert in several of
the sub-sub areas, be conversant in several areas, and know very little
about the other areas.
Objectives
Our objective is to impart some of the flavor of mathematics. We do this
in several ways. First, by actively participating in the discovery
process, a reader will get a sense of how mathematicians discover new
mathematics.
A problem arises. Discovery often begins with some experimentation to
help give a sense of what is involved in the problem. After a while one
might have enough understanding of the problem to be able to make a
plausible conjecture, which one then tries to prove. The attempt to
prove the conjecture can have several different outcomes. Sometimes the
proof works. Other times it doesn’t work, but in trying to prove it one
learns much more about the problem and identifies some stumbling blocks.
Sometimes these stumbling blocks seem insurmountable and one tries to
prove they actually are insurmountable—the conjecture is false. That may
create its own stumbling blocks. All the time one learns more and more
about the problem. Finally one either proves the conjecture or disproves
it. (Or simply gives up!).
We shall see all of this unfolding in the several chapters in the book.
Our discovery process will be similar to that of a research
mathematician’s, though our problems will be much less technical.
The first part of each chapter deals with a problem we wish to consider.
We then go into the discovery mode and eventually obtain some answers.
After this we turn to other aspects of mathematics related to the
material of the chapter. What is the history of the problem? Who solved
it? What are some related problems ? How can other areas of mathematics
be brought to bear on the problem? Do computers have any role in solving
the problems raised? What about conjectures that seemed to be true, but
were eventually proven false? Or remain unsolved?
We have tried to find some balance between discovery and instruction.
This is not always possible: it is impossible to resist the many
occasions when some idea leads naturally to another wonderful idea. The
reader will not discover the connection, even with prodding, so we drop
our heuristic approach and explain the new ideas. This is probably in
the nature of things. When we look back on everything we have learned,
certainly it is all a combination of stuff we figured out for ourselves
and other stuff that we learned from others. It is the combination of
the two that makes learning rewarding and productive. It is likely the
stress on just the instruction part that explains the many people in
this world who claim to dislike or fear mathematics.
Prerequisites
The main prerequisite for getting much from this book is curiosity and a
willingness to attempt the problems we present. These problems usually
set things up for the next stage in the discovery process. This is
different from most text books, where the problems at the end of a
section are intended to firm the readers’ knowledge of the material just
presented.
Almost all problems have answers supplied at the end of the chapter. The
word ANSWER following a problem indicates that an answer is supplied.
For readers using a PDF file on a computer or laptop screen, that word
is hyperlinked to the answer. Readers working on a paperback version
will have to scan the end of the chapter to find the appropriate answer.
When the book is read in a self-study manner, rather than in a classroom
setting with an instructor to set the pace, there may be a temptation
to move ahead quickly, to get to the end of the process to learn the
result. (Did the butler commit the crime?). We urge that one resist the
temptation. The students who got the most out of the class were the ones
who participated actively in the discovery process. This included
working the problems as they arose. They said that understanding this
process was of more value to them than learning the answer.
In order to understand the material in most of the chapters, one needs a
bit of algebra (just enough to be able to manipulate some simple
algebraic expressions, though such manipulations play only a very minor
role), a bit of geometry, and a little arithmetic.
One topic that is not usually covered in a first course in algebra is
mathematical induction. This tool appears in several places. Readers not
familiar with mathematical induction can reasonably work through a
chapter that has an induction argument until that argument is needed. At
that point, one can consult the Appendix where induction is
discussed and induction proofs are given that are relevant to various
problems we discuss. Induction does not take part in the discovery
process—it is used only to verify that certain statements are true.
Rigor versus intuition
Professional mathematicians must be rigorous in their work. This
involves giving careful definitions, even of apparently familiar
objects. This often involves a great deal of “technical machinery.” A
mathematician needs to know such things as exactly what a “curve” is,
what it means to “go around a curve so that the inside is to the left,”
how to math- ematically describe the number of “holes” in a pretzel and
the meaning of area.
It should be understood, however, that this is not the situation when a
mathematician first starts thinking of a problem and working out a
solution. Things are rather vague and intuitive in the early stages. The
polish and rigor appear in full force only in the final drafts.
Since this book is not intended for mathematicians, who would require
formal definitions and proofs, we can relax these requirements
con- siderably. Everything we say in an informal way can be said in a
mathematically rigorous way, but that is not our purpose. Our purpose is
to provide some of the flavor of mathematics and introduce the reader
to topics that some students were surprised to find involved
mathematics. Thus we can take for granted that readers intuitively
understand concepts such as curves, inside, left, holes, and area. We
will occasionally describe a concept with which the reader may not be
familiar, but our overall style is primarily a leisurely, informal one.
Acknowledgements
The authors are grateful to many who have contributed in one way or
another to the creation of this book. In particular, the many students
who participated in the classes contributed greatly. They pointed out
difficulties, found interesting solutions to problems that were
different from our solutions, provided reasonable approaches to problems
that sometimes worked and sometimes didn’t, but were often creative
whether or not they worked.
We also want to acknowledge Professor Steve Agronsky who used
preliminary versions of this book in his classes at Cal. Poly.
University in San Luis Obispo, California. He made a number of useful
suggestions based on his teachings.
Most of the figures were prepared by us using Mathematica.TM A number of the figures were found on the internet and, naturally
enough, it has proved to be difficult to give proper attribution. Any
person who is the original author of such figures is invited to write to
us with instructions as to whom to give proper credit. We are thankful
for those who have released such figures to the public domain and will
be equally thankful to those who wish credit.
