In the same series Mathematics Physics Chemistry French English Typing Intermediate Algebra & Analytic Geometry Electronic Computers INTERMEDIATE ALGEBRA & ANALYTIC GEOMETRY Made Simple William R. Gondin, Ph.D. and Bernard Sohmer, Ph.D. Advisory editor Patrick Murphy, B.Sc., A.F.I.M.A. Made Simple Books W. H. ALLEN London in association with DONALD MOORE PRESS LTD Asia TUDOR DISTRIBUTORS Australia © 1959 by Doubleday & Company, Inc. and 1967 by W. H. Allen & Company Made and printed in Great Britain by Richard Clay (The Chaucer Press), Ltd., Bungay, Suffolk for the publishers W. H. Allen & Company Essex Street, London W.C.2 This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser Foreword It is estimated that the pace of modern scientific progress will double our present-day knowledge within the next few years; the world is in the middle of a 'knowledge explosion'. Always accepted as the 'Queen of the Sciences', Mathematics is now, more than ever before, the indispensable key to the mastery of man's own environment, either through the work of the engineer and scientist or through the logical thinking of the custodians of society. An ever-increasing number of people who never studied the subject beyond the elementary stage now find they need some command of its more advanced techniques in the jobs by which they earn their living. If only to keep abreast of the times they must have more than an elementary knowledge of Mathematics, and this book is designed to help meet such needs. It must be realized that Mathematics is an activity, a way of thinking. The subject has a reputation for being abstruse, but this is usually due to an unsatisfactory introduction to the jargon of its abbreviated peculiarities, as though the symbols themselves were the subject rather than the thoughts behind them. Mathematics requires the confidence of good groundwork upon which to develop its general ideas, so patience is advised at each stage of self-instruction. The beauty of the subject lies in its abbreviated presentation of exact detail. One associates elegance with the simplicity of formal proofs, but like all tastes it is acquired only through effort, and it is as well to bear in mind that a few crisp lines of 'proof probably summarize the trial-and-error research of generations of mathematicians. Small wonder you might have to read them through once or twice more! Although this book is designed principally for self-study, it is invaluable as a companion reader for students taking one of the academic courses, such as the G.C.E.—either at O or A level. Whatever the context of study, the reader is helped by the book's technique of introducing each new topic by a consideration of its underlying ideas in terms of the problems from which these ideas arise. Only after these have been carefully discussed is the pace later quickened by the formal methods of presentation. Even then, more space than usual is devoted to such questions as: does the method always work? If not, why does it fail? There are helpful summaries at the end of chapters, and a very detailed set of answers which form an integral part of a text that is both readable and instructive. PATRICK MURPHY OA v CHAPTER I PRELIMINARY REVIEW The Purpose of These Books Once the student of mathematics has a reasonably good command of elementary algebra and geometry, he stands at the threshold of a whole new world of mathematical discovery and experience. Perhaps the greater subtleties of higher mathematics are still beyond him. But here at hand, under the general heading of advanced mathematics, are some of the most important analytic tools of modern science. With relatively little more time and effort, the student may now reach out and grasp those tools. The purpose of this book and of its companion volume, Advanced Algebra and Calculus Made Simple, is to help you, the reader, make the most of that opportunity. But first of all, a few words of general explanation.... Before you start a new enterprise, it is always a good idea to take a glance ahead, a glance around, and then a last glance back. That is the purpose of this present chapter. What is Advanced Mathematics! Although the term is sometimes used differently, advanced mathematics is most often understood to be the content of first courses in such subjects as algebra, analytic geometry, vector analysis, differential calculus, and integral calculus. Of these, calculus is by far the most powerful as a basic analytic tool of modern science. For that reason you will find the most interesting practical problems in the second of these two volumes where both differential calculus and integral calculus are introduced. For the same reason, the other mathe- matical subjects in these two books are presented in such a way as best to lead up to their later applications in calculus. Before you can concentrate on the essentially new ideas of calculus, how- ever, you first need to understand the use of analytic methods in geometry. And before you can handle either calculus or analytic geometry effectively you need more algebraic technique than the reader who has studied only ele- mentary mathematics is likely to command. This present volume therefore continues with a section on those relatively advanced topics in algebra which are applied directly in analytic geometry. And the next volume begins with a second section on algebra dealing with further topics required in calculus. Advanced Approaches to Mathematics In what ways do the above-mentioned subjects differ from more elementary ones? 3 4 Intermediate Algebra and Analytic Geometry Made Simple Most obvious is the fact that their 'advanced techniques' are ways for solving more difficult kinds of problems, or for solving simple problems more efficiently. This is illustrated throughout these volumes, but first in Chapters IV and VII below. Less commonly realized is the fact that much of the power of advanced mathematical techniques derives from a difference of approach—a broader point of view concerning the nature of mathematical problems, with resulting deeper insights into connexions between seemingly unrelated mathematical subjects. This is best illustrated by the section on analytic geometry, but first in Chapter III below. An approach to mathematics may also be more advanced, however, only in that it is a more careful re-examination of ways of dealing with problems— including even very simple ones which the elementary student may feel he already fully understands. It may raise such questions, for instance, as : does a given method always work? If it does not work, under what conditions does it fail ? Why does it fail ? And what interpretation are we to place upon the conditions of our problem and the results of our mathematical operations in each case? This is first illustrated in Chapter II. Recommended Sequence of Study The usual subjects of advanced mathematics are arranged in these two volumes to form a natural study sequence. Each section, as a unit, leads on to the next. So the reader may begin at any point which his previous prepara- tion will permit and proceed systematically thereafter. However, the chapters in both of these volumes have been so arranged that they may also be studied in other sequences if the reader wishes. After finish- ing Chapter III in this book, for instance, you may proceed at once to the first two chapters on analytic geometry, and complete the rest of the algebra section later. Since alternative sequences like these may be preferred by some students, the reader is informed in the text wherever a reasonable possibility of choice occurs. But if you have any doubt at all as to which way to proceed at such points you are strongly advised to elect the sequence which temporarily skips you ahead in the text. The reason is this ... In these volumes the usual topics of advanced mathematics are organized in separate sections under the conventional headings of different 'subjects' only because some school curricula still parcel them out that way, and because some readers may therefore feel that they should concentrate on one subject in one section at a time. But in some respects, the concurrent study of two, or possibly even three, of the conventionally separated subjects is more practicable, and indeed easier, than the study of any one of them separately. For instance, the reader who has carefully covered Chapters II and III of the following algebra section should find the analytic geometry of points, distances, lines, and slopes, in Chapters IX and X, little more than clarifying exercises in the application of simple linear equations and elementary graphing technique with which he has just become familiar. And very similar relation- ships exist between several other parallel sets of chapters. Consequently, even though you may start with a more limited objective in Preliminary Review 5 the use of these books, by taking recommended alternative choices of chapters in later sections, you may be able to accomplish that objective better. Mean- while, for little, if any, additional cost in time and effort, you will also have acquired command of a more advanced subject matter as an extra-dividend, so to speak. Recommended Pace of Study The reader who begins his study with the first chapter in any major section of either of these two volumes may perhaps feel that the text's pace—the rate at which new material is introduced—is 'somewhat slow'. But the reader whose mathematical preparation encourages him to begin his study in the middle of a section may perhaps have the opposite feeling that its pace is 'a bit fast'. This is explained by another important feature of the deliberate plan of these books. As a new subject is introduced in each major section, careful consideration is given to basic underlying ideas which should be clearly understood before they are likely to be lost to view in a mass of technical details. But once the student has grasped these fundamentals, he is thereafter able to cover the remaining material much more quickly. A sound study practice, then, is this : if the pace of the text seems to you to be too slow at any point, make sure you are really getting its main ideas before you begin to skip ahead in the same section. But, if the pace of the text seems to you to be too fast at any point, look back to earlier chapters which contain explanations that will enable you to cover the later material more rapidly. Recommended Use of Practice Exercises The best test of your understanding of mathematics is your ability to apply it in solving problems. For that reason, practice exercises are given at regular intervals throughout the text, with answers at the end of the book. Moreover, the solutions of many of the problems in these exercises are applied again later, either in examples illustrating more advanced techniques or in further problems. It is a good idea, therefore, to do these exercises regularly. Check your re- sults with the answers at the end of the book. Make sure that you can do most of the problems before you go on further. And keep your work organized systematically in a notebook so that you can refer back to it later. Assumed Background of the Reader In this book and its companion volume, Advanced Algebra and Calculus Made Simple, the reader is assumed already to be familiar with the following: Elementary Topics (Chapter in Mathematics Made Simple) Signed numbers and algebraic expressions (VII) Algebraic formulae and equations (VIII) Roots, powers, and exponents (IX and X) Factoring (XI) Common logarithms (XII) Number series (XIII) Elementary geometry (XIV) Trigonometry for angles between 0° and 90° (XV) 6 Intermediate Algebra and Analytic Geometry Made Simple NOTE: The Mathematics Made Simple references in parentheses here are to chapters of the book by that title in the same publisher's Made Simple series. Further references to this source are made with the abbreviation, 'MMS\ Whenever you are in doubt about such elementary subjects, you should, of course, review them before attempting to go on in either of the two present volumes. In the long run, that will save both time and effort. And it will help to assure a sounder foundation for all further study. Note on Tables of Formulae There is a summary of main points at the end of most chapters in this book. Some of these summaries contain tables of formulae which you will find very useful, not only for your first study of the text but also for reference purposes later. The meaning and application of all tabulated formulae in later chapters is explained at length. However, this present chapter concludes with a table of formulae which are only briefly described because they summarize definitions, principles, and other elementary mathematical relationships with which you are assumed already to be familiar. Many are referred to at relevant points in the text below. But you are advised to scan the entire table now as a final review of your previous study of elementary mathematics. The first few formulae on this table may seem so obvious as hardly to be worth stating. For instance, the fifth—reading: kxy = yx9—means only that 'the ordinary quantities of elementary mathematics may be multiplied to- gether in any order'. As an example, 3(7) = 21 and 7(3) = 21 ; hence it makes no difference whether we multiply 3 and 7 together as 3 times 7 or as 7 times 3. But you will see later on in vector analysis (Advanced Algebra and Calculus Made Simple) that this is not true for all kinds of quantities, since the vector product of two vector quantities depends upon the order in which these quantities are multiplied. In some of these formulae the value, 0, is excluded for one of the quantities by a parenthetical inequality in the form, x Φ 0, to be read: 4JC not equal to 0'. Reasons for this special treatment of the value, 0, are explained later in the text. Note that the formulae in this table are written with the designator-letter R (Review). This is for convenience in distinguishing them from other series of formulae tabulated, and referred to, in the text. REVIEW TABLE OF ELEMENTARY FORMULAE R-Series Formulae Fundamental Operations RI: x + y = y + x Quantities may be added in any order. R2: x + (y + z) = (x + y) + z=+y + z ... or in any grouping. Preliminary Review 7 R3: x + (-y) = x - y Adding —y to x is the same as subtracting y from x. R4: * — (—y) = x + y Subtracting — y from x is the same as adding y to x. R5: xy = >>* Quantities may be multiplied in any order. R6: x(yz) = (xy)z ... or in any grouping. R7: x(—y) = —xy The product of a negative quantity and a positive quantity is negative. R8: (-x)(-y) = *v The product of two negative quantities is positive. Operations Involving Parentheses R9: a(x + y) = ax + ay All terms within a parenthesis are to be multiplied by a coefficient outside the parenthesis. RIO: -(* - ; , ) = -* -(-y) = — x + y A minus sign preceding a parenthesis applies to all terms within the parenthesis. Rll: x — [a{y — z)] = x — [ay — az] = JC — ay + az or * - [a(y - z)] = Λ: - «(y - z) = x — öy + az Sets of parentheses, or brackets, are to be removed 'from the inside out' or 'from the outside in'. Operations with Fractions $>(^o) R12: - = x y Dividing by y is the same as multiplying by 1/y, provided y Φ 0. R13: Ξ =^ = ^,(„*0) y ny y/n The numerator and denominator of a fraction may both be multiplied or divided by the same quantity, n, without changing the fraction's value, provided n Φ 0. R14: 4- = y- x/y x Dividing by a fraction is the same as multiplying by the reciprocal of that frac- tion. R15: î + y- = ±±1 a a a The sum of two fractions with a common denominator is the sum of the numera- tors over the common denominator. 8 Intermediate Algebra and Analytic Geometry Made Simple R16: Fractions with different denominators must be reduced to a common denomina- tor before being added. m)- K": l=-H*l-S The product of two fractions is the product of their numerators over the product of their denominators. Operations with Exponents R18: (xy)n = xnyn (XY = Ξ2 R19: \y) yn An exponent outside a parenthesis is understood to apply to each factor within the parenthesis. R20: xmxn = xm+n The product of two powers of the same base quantity is the same base quantity with an exponent equal to the sum of the original exponents. R21: ±- = m-n x xn The quotient of two such powers is the same base quantity with an exponent equal to the exponent of the numerator minus the exponent of the denominator. R22: x1 = x Quantities written without exponents are understood to have the exponent, 1. R23: x° = 1,(JC φ 0) Any quantity, except 0, with the exponent, 0, equals 1. R24: x~n = — n xn A base quantity with a negative exponent is equal to the same quantity in the denominator with the sign of its exponent changed. R25: (xm)n = xmn The nth power of the mth power of a quantity is the mnth power ofthat quantity. R26: xV" = ψχ R27: xm/n = (x^n)m = {xmfin = ^(x)m = tfxm The numerator of a fractional exponent indicates a power, and the denominator a root, of the base quantity. Special Factors R28 ax + ay = a(x + y) R29 x2 - y2 = (χ - y) (x + y) R30 x2 + 2xy + y2 = (JC + ^)2