Preface

This book is a guide to discovering mathematics.

Every mathematic s textboo k i s filled with result s an d technique s whic h

once were unknown. Th e results were discovered by mathematicians who exper-

imented, conjectured, discussed their work with others, and then experimented

some more. Man y promising ideas turned out to be dead-ends, and lots of hard

work resulted i n little output. Ofte n th e first progress was the understandin g

of some special cases. Continue d work led to greater understanding, and some-

times a complex picture began to be seen as simple and familiar. B y the time

the work reaches a textbook, it bears no resemblance to its early form, and the

details of its birth and adolescence have been lost. Th e precise and methodical

exposition of a typical textbook is often the first contac t one has with the topic,

and this leads many people to mistakenly think that mathematics is a dry, rigid,

and unchanging subject.

We believe tha t th e mos t excitin g par t o f mathematic s i s th e proces s of

invention an d discovery. Th e aim of this boo k is to introduc e tha t proces s to

you, th e reader. B y mean s of a wide variety of tasks, thi s book will lea d you

to discover some real mathematics. Ther e are no formulas to memorize. Ther e

are no procedures to follow. B y looking at examples, searching for patterns in

those examples, an d then searching for the reasons behind those patterns, you

will develop your own mathematical ideas . Th e book is only a guide; its job is

to start you in the right direction , an d to bring you back if you stray too far.

The discovery is left to you.

This book is suitable for a one semester course at the beginning undergrad-

uate level. Ther e are no prerequisites. An y college student interested in discov-

ering the beauty of mathematics can enjoy a course taught from this book. A n

interested high school student will find this book to be a pleasant introductio n

to some modern areas of mathematics.

While preparin g thi s boo k w e were fortunat e t o hav e acces s t o excellen t

notes taken by Hui-Chun Lee. We thank Klaus Peters and Gretchen Wright for

helpful comments on an early version of this book.

David W. Farmer

Theodore B. Stanford

September, 1995

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