Foreword
The physicist Richard Feynman (1918-1988) gave three public lectures at McGill University in 1976. His work on physics has been followed by many scientists and students. In the lectures, partly tongue-in-cheek, he suggested that physics was based on two easily described operations, namely the addition and multiplication of arrows in the plane. His description of arrow addition and multiplication for a general, non-mathematical audience was a model for the informal, very visual, most adequate, presentation of mathematical ideas. But he gave it under the guise of describing physics. And he avoided panic among the mathematically shy by not saying that the arrows, with their addition and multiplication, represent what pure and applied mathematicians (since Gauss) regard as the complex numbers. No mastery of the algebraic way of writing and thinking was required to understand his live description of addition and multiplication. When I attended Feynman‘s lectures, I thought his description of arrows in the plane could be an excellent way to introduce complex numbers. The chapters on complex numbers elaborate on Feynman’s live presentation, although their on-paper presentation employs the algebraic way of writing and reasoning. With Feynman's energetic presentation as a model, I looked for and found in 1983, a preview and simple tour of calculus (slope-related calculations) which likewise required a minimal knowledge of algebra. Just the definition of a slope to a straight line needs to be understood to follow it. The why slopes chapters extend this tour and provide a geometric motivation for calculus, easy to describe and to repeat without a great dependence on algebra and without requiring a mastery of the rules of differentiation, that is slope calculation, for nonlinear functions. This book is one of three volumes on understanding and explaining reasoning skills and mathematics. The objective of this volume is to complement other texts in algebra, trigonometry and calculus. Students may be able to read the first part of this book during their high school days and keep the rest of this work for consultation during their college studies. The first why slopes chapters gradually illustrate the algebraic or symbolic way of writing and thinking. The later is employed more deeply in some later chapters and at full strength in proper calculus courses. The aim of the first chapters is to provide a simple image-based preview or review of calculus. In it, dependence on symbols or algebra is kept to a minimum. The images may help readers to see and physically grasp the simplest slope-related ideas in calculus. The remaining chapters cover more topics – see the table of contents. Appendices present the most advanced topics. Theorems in first courses on calculus are often stated without proof. The appendices state the theorems and give or indicate the proofs. This should provide a context for the decimal-free approach favored in advance calculus or modern mathematical analysis. This is a book which a student could begin reading in high school and continuing reading through further college math courses. Material elementary to advanced is covered.
Alan Selby
Montreal March 1996
Copyright © 1995, 1996 by A. M. Selby
Canadian Cataloguing in Publication Data
Selby, Alan M,
Understanding and Explaining reason and math Contents: v. 1. Elements of Reason - v. 2. Three Skills for algebra - v.3. Why Slopes and more math. ISBN 0-9697564-4-5 (set) - ISBN 0-9697564-1-0 (v. 1) - ISBN 0-9697564-2-9 (v. 2) - ISBN 0-9697564-3-7 (v. 3) - 1. Mathematics–Philosophy. 2. Reason. 3. Algebra. 4. Calculus. I. Title. II. Title: Elements of reason. III.Three Skills for algebra. IV. Title: Why Slopes and more math. QA8.4.S44 1995 510’.1 C95-900945-0 Reprinting may lead to new ISBN numbers |
Units in Calculations: Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area. |
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