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INTRODUCTION

One of the authors once had a friend, whom we will call Jack, who said that the first day in his high school algebra course the teacher said, "Now, let x be any number," and Jack said, "Wait a minute!"---and that's right where he was at the end of the year. He never quite caught up with the teacher. All he got out of high school algebra was a vague dislike of mathematical symbols and a distrust of abstract reasoning.

There are a number of people in the world who have had experiences like Jack. It's too bad. Not only are these people denied the practical use of algebra (because they never learned it), but also the whole wonderful world of mathematics is a mystery to those who have never learned to be comfortable with mathematical symbols. And modern mathematics is a wonderful world, replete with beautiful creations of the disciplined human imagination.

This book is addressed to people like Jack in the hope that we can convince these people that there is really no sinister motive behind such a phrase as "Let x be any number." The practical value of a grasp of elementary algebra should need no comment in this age of Sputniks and nuclear reactors (to say nothing of time payments, complicated retirement plans, and income taxes). More important, however, is the matter of showing how, by a disciplined use of symbols, some rather elegant pieces of reasoning can be accomplished, and decisions reached in questions that appear at first sight to be com- pletely beyond our grasp. We have included several such examples in this book.

In particular we attempt to show how a consideration of the elementary processes of algebra leads to a deeper understanding of the concept of numbers, proceeding from the so-called "natural numbers"---the positive integers---to the entire set of numbers used in modern mathematics.

To establish the requirement for numbers other than the positive whole numbers, we repeat some classic proofs which are milestones of mathematical thought, both in their form and in the importance of the concepts they establish. If we succeed in creating a taste for such morsels, then the book has served its purpose.

Note to the Reader