Conquering Mathematics: From Arithmetic to Calculus PDF
Preview Conquering Mathematics: From Arithmetic to Calculus
Conquering Mathematics From Arithmetic to Calculus Conquering Mathematics From Arithmetic to Calculus Lloyd Motz and Jefferson Hane Weaver Springer Science+Business Media, LLC LIbrary of Congress CatalogIng-In-PublIcatIon Data Motz, Lloyd, 1910- ConquerIng lathelatlcs fro. arlthletlc to calculus I Lloyd Motz and ~efferson Hane Weaver. p. CI. Includes bIbliographIcal references and Index. t. MatheaatIcs. I. Weaver, ~efferson Hane. II. TItle. OA37.2.M68 1991 510--dc20 91-9329 CIP 10 9 8 7 6 ISBN 978-0-306-43768-7 ISBN 978-1-4899-2774-3(eBook) DOI 10.1007/978-1-4899-2774-3 (~I1991 Lloyd Motz and Jefferson Hane Weaver OriginallypublishedbyPlenumUSin1991. Allrights reserved No part ofthis book may be reproduced, stored ina retrieval system,or transmitted inany form or byany means, electronic, mechanical, photocopying, microfilming, recording,or otherwise, without written permission from the Publisher To Minne and Shelley Preface We have designed and written this book. not as a text nor for the professionalmathematician.but for the generalreader who is naturally attracted to mathematics as a great intellec tual challenge.and for the specialreaderwhose work requires him to have a deeper understanding ofmathematics than he acquired in school. Readers in the first group are drawn to mental recreational activities such as chess. bridge. and various types of puzzles. but they generally do not respond enthusiastically to mathematics because of their unhappy learning experiences with it during their school days. The readersinthe secondgrouptumtomathematicsasanecessity. but with painful resignation and considerable apprehension regarding their abilities to master the branch ofmathematics they need in their work. In either case. the fear of and revulsion to mathematics felt by these readers usually stem from their earlier frustrating encounters with it. vii viii PREFACE This book will show these readers that these fears, frustrations, and general antipathy are unwarranted, for, as stated, it is not a textbook full of long, boring proofs and hundreds of problems, rather it is an intellectual adventure, to be read with pleasure. Itwaswritten to be easilyaccessible and with concern for the mental tranquilityofthe readerwho willexperience considerable fulfillment when he/sheseesthe simplicity of basic mathematics. The emphasis throughout this book is on the clear explanation of mathematical con cepts. Wherever proofs are needed for clarity, they are developed using the kind of reasoning one applies to the solutionofproblemsingeneral. Wehaveincludedsuchproofs not only to clarify the mathematics, but also to demonstrate thatmathematicalreasoningisnot aspecial kind ofreasoning that requires a special kind of mind but can be produced by any mature mind. Since all mathematics isessentially arithmetic invarious guises, we have emphasized the importance of a thorough understanding of arithmetic as the basis for understanding all otherbranches ofmathematics.Accepting this asatruism, those who understand arithmetic, which we all do to some extent, can read this book with much more confidence, gratification,and agreatersenseofaccomplishmentthanthey would otherwise.With this in mind, wehave devoted the first two chapters of our book to the basic laws of arithmetic, since these laws apply to all branches of mathematics. The reader, mastering these laws, willbe well prepared to under stand and enjoy the chapters that follow. Arithmeticcan be fully understood only ifthe properties of numbers are known; therefore we discuss these in some detail, pointingoutthatnumbershavetwoaspects:theordinal (ordering) aspect, which leads to geometry, and the cardinal (quantitative) aspect, which leads to the traditional arith- PREFACE ix metic. In the first two chapters we discuss the various categories [integers, fractions, irrational numbers (algebraic and transcendental), prime numbers, and complex numbers] into which our number system has been divided. These various subgroups ofthe number system are all governed by the same arithmetic laws, but they differ from each other in certain basic properties we describe and analyze; these differences make arithmetic extremely interesting. We con clude these two chapters with a discussion of the different bases (e.g., base 10) on which a number system can be constructed and with a discussion of logarithms. In Chapter 3 we show that algebra is essentially the arithmetic ofletters or symbols, pointing out that it enlarges and generalizes arithmetic, releasing it from the restraints imposed on it by the specific quantitative features that num bers introduce. Algebra thus carries usfrom a numericallogic to a pure logic of entities that mayor may not represent real, concrete, physical objects or numbers. Indeed, algebras have been created that have their own rules ofaddition and multi plication. Thus, in the algebra ofvectors, the law ofaddition isquite different from thatofarithmetic,and innoncommuta tive algebras the product ab of two factors a and b does not havetoequal ba (multiplicationisnoncommutative;theorder ofthe factors inaproductisimportant).Clearly,inanoncom mutative algebra the factors of a product are not ordinary numbers but represent some kind of operation which must be performed in a definite order. In another algebra, called Boolean algebra, the elements may be abstract objects, such aslogicalpropositionsandsetsofabstractentitiesorelectrical networks. Boolean algebra is thus the algebra of pure logic. We do not discuss these esoteric algebras in our text but mention them here to indicate the vast wealth of algebra. x PREFACE Although any algebra can be developed as a logical system of abstract concepts and propositions without refer ence to any kind of concrete representation, it can be under stood more easily ifit can be described by such a representa tion. We do this in Chapter 4 where we develop the basic algebraic concepts graphically, which leads one quite naturally to the function concept. The graph (the plot ofthe dependence of one quantity on another) shows the relation ship between the two entities being plotted and thus defines a function. The function concept leads to the polynomial and to the algebraic equation, which we illustrate by the contour of an imaginary road, laid off in one direction only (east west), but rising and falling with respect to sea level. This contourreveals allthe properties ofafunctional relationship, no matter how complex it may be. It also gives us a simple understanding of the solutions of the algebraic equation associated with a polynomial ifwepicturethe road as cutting sea-level points to the east or west of our position. The distance from this position (the origin of our graph) of the sea-level cutting point is a solution ofalgebraicequation; the number of such solutions that may exist depends on the complexity of the functional relationship. We discuss these from the simple to the complex. The limitation ofthe number system to points on a line must be removed ifwe are to go from arithmeticto geometry. We do this in Chapter 5 where we introduce the concept of the dimensionality of a manifold. Limiting ourselves to the points on a plane-a two-dimensional manifold of points we develop the geometry of triangles, which requires the introduction of the angle concept. We complete this chapter by showing how most theorems in geometry, such as the theorem of Pythagoras, can be deduced with elementary reasoning and a few geometric properties oftriangles. PREFACE xi Continuing with geometry, we devote Chapter 6 to the geometry ofthe circle and to trigonometry. These two facets of mathematics go together because all the trigonometric properties ofangles can be deduced from the unit circle, that is,a circle whose radius isone unit oflength (one centimeter, one inch, one foot, etc.). This method of introducing trigonometry removes all mystery associated with it and per mits the reader to grasp its basic elements quite easily. Since the circleisaveryspecial caseofaclassofgeometricfigures called conic sections-that play a very important role in physical phenomena from astronomy to atomic physics, as well as in mathematics itself, wedescribe the mathematics of conic sections in Chapter 7which we call analytic geometry. This phase of mathematics, which was developed by Descartes, deals with the application of algebra to geometry; the most interesting results of this work are the algebraic equations of conic sections-the ellipse, the parabola, and the hyperbola. We derive these equations not by long alge braic and geometric arguments but from simple symmetry arguments, and then show, in a very elementary way, that the orbits of the planets are ellipses. In the penultimate chapter, Chapter 8, we show how differential calculus arose simultaneously from Newton's study of the instantaneous speed of a body and Leibnitz's analysis of the rate of change of a function with respect to the entity on which it depends. We also discuss how the calculus made possible much of what we now know as modem science and technology. Lloyd Motz Jefferson Hane Weaver
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