Essential Mathematics for Economic Analysis (4th Edition) PDF

Essential Mathematics for Economic Analysis Essential Mathematics for All the mathematical tools an economist needs are provided in this worldwide...

Essential Mathematics for Economic Analysis Essential Mathematics for All the mathematical tools an economist needs are provided in this worldwide bestseller. Economic Analysis Now fully updated, with new problems added for each chapter. New! Learning online with MyMathLab Global ‘Allows students to work at their own pace, get immediate feedback, and overcome problems by using the step-wise advice. This is an excellent tool for all students.’ FO U RT H E D I T I O N Jana Vyrastekova, University of Nijmegen, the Netherlands Go to www.mymathlab.com/global – your gateway to all the online resources for this book. r
MyMathLab Global provides you with the opportunity for unlimited practice, guided solutions with tips and hints to help you solve challenging questions, an interactive eBook, as well as a personalised study plan to help focus your revision efforts on the topics where you need most support. r
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and full instructions allowing you to register for access to MyMathLab Global. If you have purchased this text on its own, you can still purchase access online FOURTH at www.mymathlab.com/global. See the Guided Tour at the front of this EDITION text for more details. with StrØm Sydsæter & Hammond Knut Sydsæter is an Emeritus Professor of Mathematics in the Economics Department at the University of Oslo, where he has been teaching mathematics for economists since 1965. Peter Hammond is currently a Professor of Economics at the University of Warwick, where he moved in 2007 after becoming an Emeritus Professor at Stanford University. He has taught mathematics for economists at both universities. Arne Strøm has extensive experience in teaching mathematics for Knut Sydsæter & Peter Hammond economists in the Department of Economics at the University of Oslo. with Arne StrØm www.pearson-books.com E S S E N T I A L M AT HE M AT I C S F OR E C O NOMI C A N A LY S I S E S S E N T I A L M AT HE M AT I C S F OR E C O NOMI C A N A LY S I S F O UR T H E D I T I O N Knut Sydsæter and Peter Hammond with Arne Strøm Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearson.com/uk First published by Prentice-Hall, Inc. 1995 Second edition published 2006 Third edition published 2008 Fourth edition published by Pearson Education Limited 2012 © Prentice-Hall, Inc. 1995 © Knut Sydsæter and Peter Hammond 2002, 2006, 2008, 2012 The rights of Knut Sydsæter and Peter Hammond to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6−10 Kirby Street, London EC1N 8TS. Pearson Education is not responsible for the content of third-party internet sites. ISBN 978-0-273-76068-9 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 Typeset in 10/13 pt Times Roman by Matematisk Sats and Arne Strøm, Norway Printed and bound by Ashford Colour Press Ltd, Gosport, UK To the memory of my parents Elsie (1916–2007) and Fred (1916–2008), my first teachers of Mathematics, basic Economics, and many more important things. — Peter CONTENTS Preface xi 3.3 Double Sums 59 3.4 A Few Aspects of Logic 61 1 Introductory Topics I: Algebra 1 3.5 Mathematical Proofs 67 3.6 Essentials of Set Theory 69 1.1 The Real Numbers 1 1.2 Integer Powers 4 3.7 Mathematical Induction 75 1.3 Rules of Algebra 10 Review Problems for Chapter 3 77 1.4 Fractions 14 1.5 Fractional Powers 19 4 Functions of One Variable 79 1.6 Inequalities 24 4.1 Introduction 79 1.7 Intervals and Absolute Values 29 4.2 Basic Definitions 80 Review Problems for Chapter 1 32 4.3 Graphs of Functions 86 4.4 Linear Functions 89 2 Introductory Topics II: 4.5 Linear Models 95 Equations 35 4.6 Quadratic Functions 99 2.1 How to Solve Simple Equations 35 4.7 Polynomials 105 2.2 Equations with Parameters 38 4.8 Power Functions 112 2.3 Quadratic Equations 41 4.9 Exponential Functions 114 2.4 Linear Equations in Two Unknowns 46 4.10 Logarithmic Functions 119 2.5 Nonlinear Equations 48 Review Problems for Chapter 4 124 Review Problems for Chapter 2 49 5 Properties of Functions 127 3 Introductory Topics III: 5.1 Shifting Graphs 127 Miscellaneous 51 5.2 New Functions from Old 132 3.1 Summation Notation 51 5.3 Inverse Functions 136 3.2 Rules for Sums. Newton’s Binomial 5.4 Graphs of Equations 143 Formula 55 5.5 Distance in the Plane. Circles 146 viii CCOONN T ET N ENT ST S 5.6 General Functions 150 9 Integration 293 Review Problems for Chapter 5 153 9.1 Indefinite Integrals 293 9.2 Area and Definite Integrals 299 6 Differentiation 155 9.3 Properties of Definite Integrals 305 6.1 Slopes of Curves 155 9.4 Economic Applications 309 6.2 Tangents and Derivatives 157 9.5 Integration by Parts 315 6.3 Increasing and Decreasing Functions 163 9.6 Integration by Substitution 319 6.4 Rates of Change 165 9.7 Infinite Intervals of Integration 324 6.5 A Dash of Limits 169 9.8 A Glimpse at Differential Equations 330 6.6 Simple Rules for Differentiation 174 9.9 Separable and Linear Differential 6.7 Sums, Products, and Quotients 178 Equations 336 6.8 Chain Rule 184 Review Problems for Chapter 9 341 6.9 Higher-Order Derivatives 188 6.10 Exponential Functions 194 10 Topics in Financial 6.11 Logarithmic Functions 197 Economics 345 Review Problems for Chapter 6 203 10.1 Interest Periods and Effective Rates 345 10.2 Continuous Compounding 349 7 Derivatives in Use 205 10.3 Present Value 351 10.4 Geometric Series 353 7.1 Implicit Differentiation 205 10.5 Total Present Value 359 7.2 Economic Examples 210 10.6 Mortgage Repayments 364 7.3 Differentiating the Inverse 214 10.7 Internal Rate of Return 369 7.4 Linear Approximations 217 10.8 A Glimpse at Difference Equations 371 7.5 Polynomial Approximations 221 Review Problems for Chapter 10 374 7.6 Taylor’s Formula 225 7.7 Why Economists Use Elasticities 228 11 Functions of Many 7.8 Continuity 233 Variables 377 7.9 More on Limits 237 11.1 Functions of Two Variables 377 7.10 Intermediate Value Theorem. 11.2 Partial Derivatives with Two Variables 381 Newton’s Method 245 11.3 Geometric Representation 387 7.11 Infinite Sequences 249 11.4 Surfaces and Distance 393 7.12 L’Hôpital’s Rule 251 11.5 Functions of More Variables 396 Review Problems for Chapter 7 256 11.6 Partial Derivatives with More Variables 400 11.7 Economic Applications 404 8 Single-Variable 11.8 Partial Elasticities 406 Optimization 259 Review Problems for Chapter 11 408 8.1 Introduction 259 8.2 Simple Tests for Extreme Points 262 12 Tools for Comparative 8.3 Economic Examples 266 Statics 411 8.4 The Extreme Value Theorem 270 12.1 A Simple Chain Rule 411 8.5 Further Economic Examples 276 12.2 Chain Rules for Many Variables 416 8.6 Local Extreme Points 281 12.3 Implicit Differentiation along a 8.7 Inflection Points 287 Level Curve 420 Review Problems for Chapter 8 291 12.4 More General Cases 424  CO C ON NT E N T S ix 12.5 Elasticity of Substitution 428 15.5 The Transpose 562 12.6 Homogeneous Functions of 15.6 Gaussian Elimination 565 Two Variables 431 15.7 Vectors 570 12.7 Homogeneous and Homothetic 15.8 Geometric Interpretation of Vectors 573 Functions 435 15.9 Lines and Planes 578 12.8 Linear Approximations 440 Review Problems for Chapter 15 582 12.9 Differentials 444 12.10 Systems of Equations 449 16 Determinants and 12.11 Differentiating Systems of Equations 452 Inverse Matrices 585 Review Problems for Chapter 12 458 16.1 Determinants of Order 2 585 16.2 Determinants of Order 3 589 13 Multivariable 16.3 Determinants of Order n 593 Optimization 461 16.4 Basic Rules for Determinants 596 13.1 Two Variables: Necessary Conditions 461 16.5 Expansion by Cofactors 601 13.2 Two Variables: Sufficient Conditions 466 16.6 The Inverse of a Matrix 604 13.3 Local Extreme Points 470 16.7 A General Formula for the Inverse 610 13.4 Linear Models with Quadratic 16.8 Cramer’s Rule 613 Objectives 475 16.9 The Leontief Model 616 13.5 The Extreme Value Theorem 482 Review Problems for Chapter 16 621 13.6 Three or More Variables 487 13.7 Comparative Statics and the 17 Linear Programming 623 Envelope Theorem 491 17.1 A Graphical Approach 623 Review Problems for Chapter 13 495 17.2 Introduction to Duality Theory 629 17.3 The Duality Theorem 633 14 Constrained Optimization 497 17.4 A General Economic Interpretation 636 14.1 The Lagrange Multiplier Method 497 17.5 Complementary Slackness 638 14.2 Interpreting the Lagrange Multiplier 504 Review Problems for Chapter 17 643 14.3 Several Solution Candidates 507 14.4 Why the Lagrange Method Works 509 Appendix: Geometry 645 14.5 Sufficient Conditions 513 14.6 Additional Variables and Constraints 516 The Greek Alphabet 647 14.7 Comparative Statics 522 Answers to the Problems 649 14.8 Nonlinear Programming: A Simple Case 526 Index 739 14.9 Multiple Inequality Constraints 532 14.10 Nonnegativity Constraints 537 Review Problems for Chapter 14 541 15 Matrix and Vector Algebra 545 15.1 Systems of Linear Equations 545 15.2 Matrices and Matrix Operations 548 15.3 Matrix Multiplication 551 15.4 Rules for Matrix Multiplication 556 PREFACE I came to the position that mathematical analysis is not one of many ways of doing economic theory: It is the only way. Economic theory is mathematical analysis. Everything else is just pictures and talk. —R. E. Lucas, Jr. (2001) Purpose The subject matter that modern economics students are expected to master makes signi- ficant mathematical demands. This is true even of the less technical “applied” literature that students will be expected to read for courses in fields such as public finance, industrial organization, and labour economics, amongst several others. Indeed, the most relevant lit- erature typically presumes familiarity with several important mathematical tools, especially calculus for functions of one and several variables, as well as a basic understanding of mul- tivariable optimization problems with or without constraints. Linear algebra is also used to some extent in economic theory, and a great deal more in econometrics. The purpose of Essential Mathematics for Economic Analysis, therefore, is to help eco- nomics students acquire enough mathematical skill to access the literature that is most relevant to their undergraduate study. This should include what some students will need to conduct successfully an undergraduate research project or honours thesis. As the title suggests, this is a book on mathematics, whose material is arranged to allow progressive learning of mathematical topics. That said, we do frequently emphasize eco- nomic applications. These not only help motivate particular mathematical topics; we also want to help prospective economists acquire mutually reinforcing intuition in both math- ematics and economics. Indeed, as the list of examples on the inside front cover suggests, a considerable number of economic concepts and ideas receive some attention. We emphasize, however, that this is not a book about economics or even about mathemat- ical economics. Students should learn economic theory systematically from other courses, which use other textbooks. We will have succeeded if they can concentrate on the economics in these courses, having already thoroughly mastered the relevant mathematical tools this book presents. xii P R E FA F AC CEE Special Features and Accompanying Material All sections of the book, except one, conclude with problems, often quite numerous. There are also many review problems at the end of each chapter. Answers to almost all problems are provided at the end of the book, sometimes with several steps of the solution laid out. There are two main sources of supplementary material. The first, for both students and their instructors, is via MyMathLab Global. Students who have arranged access to this web site for our book will be able to generate a practically unlimited number of additional prob- lems which test how well some of the key ideas presented in the text have been understood. More explanation of this system is offered after this preface. The same web page also has a “student resources” tab with access to a Student’s Manual with more extensive answers (or, in the case of a few of the most theoretical or difficult problems in the book, the only answers) to problems marked with the symbol ⊂ ⊃. SM The second source, for instructors who adopt the book for their course, is an Instructor’s Manual that may be downloaded from the publisher’s Instructor Resource Centre. In addition, for courses with special needs, there is a brief online appendix on trigono- metric functions and complex numbers. This is also available via MyMathLab Global. Prerequisites Experience suggests that it is quite difficult to start a book like this at a level that is really too elementary.1 These days, in many parts of the world, students who enter college or uni- versity and specialize in economics have an enormous range of mathematical backgrounds and aptitudes. These range from, at the low end, a rather shaky command of elementary algebra, up to real facility in the calculus of functions of one variable. Furthermore, for many economics students, it may be some years since their last formal mathematics course. Accordingly, as mathematics becomes increasingly essential for specialist studies in eco- nomics, we feel obliged to provide as much quite elementary material as is reasonably possible. Our aim here is to give those with weaker mathematical backgrounds the chance to get started, and even to acquire a little confidence with some easy problems they can really solve on their own. To help instructors judge how much of the elementary material students really know before starting a course, the Instructor’s Manual provides some diagnostic test material. Although each instructor will obviously want to adjust the starting point and pace of a course to match the students’abilities, it is perhaps even more important that each individual student appreciates his or her own strengths and weaknesses, and receives some help and guidance in overcoming any of the latter. This makes it quite likely that weaker students will benefit significantly from the opportunity to work through the early more elementary chapters, even if they may not be part of the course itself. As for our economic discussions, students should find it easier to understand them if they already have a certain very rudimentary background in economics. Nevertheless, the text has often been used to teach mathematics for economics to students who are studying elementary economics at the same time. Nor do we see any reason why this material cannot 1 In a recent test for 120 first-year students intending to take an elementary economics course, there were 35 different answers to the problem of expanding (a + 2b)2.  PP R E FFA ACE xiii be mastered by students interested in economics before they have begun studying the subject in a formal university course. Topics Covered After the introductory material in Chapters 1 to 3, a fairly leisurely treatment of single- variable differential calculus is contained in Chapters 4 to 8. This is followed by integration in Chapter 9, and by the application to interest rates and present values in Chapter 10. This may be as far as some elementary courses will go. Students who already have a thorough grounding in single variable calculus, however, may only need to go fairly quickly over some special topics in these chapters such as elasticity and conditions for global optimization that are often not thoroughly covered in standard calculus courses. We have already suggested the importance for budding economists of multivariable calculus (Chapters 11 and 12), of optimization theory with and without constraints (Chapters 13 and 14), and of the algebra of matrices and determinants (Chapters 15 and 16). These six chapters in some sense represent the heart of the book, on which students with a thorough grounding in single variable calculus can probably afford to concentrate. In addition, several instructors who have used previous editions report that they like to teach the elementary theory of linear programming, which is therefore covered in Chapter 17. The ordering of the chapters is fairly logical, with each chapter building on material in previous chapters. The main exception concerns Chapters 15 and 16 on linear algebra, as well as Chapter 17 on linear programming, most of which could be fitted in almost anywhere after Chapter 3. Indeed, some instructors may reasonably prefer to cover some concepts of linear algebra before moving on to multivariable calculus, or to cover linear programming before multivariable optimization with inequality constraints. Satisfying Diverse Requirements The less ambitious student can concentrate on learning the key concepts and techniques of each chapter. Often, these appear boxed and/or in colour, in order to emphasize their importance. Problems are essential to the learning process, and the easier ones should definitely be attempted. These basics should provide enough mathematical background for the student to be able to understand much of the economic theory that is embodied in applied work at the advanced undergraduate level. Students who are more ambitious, or who are led on by more demanding teachers, can try the more difficult problems. They can also study the material in smaller print. The latter is intended to encourage students to ask why a result is true, or why a problem should be tackled in a particular way. If more readers gain at least a little additional mathematical insight from working through these parts of our book, so much the better. The most able students, especially those intending to undertake postgraduate study in economics or some related subject, will benefit from a fuller explanation of some topics than we have been able to provide here. On a few occasions, therefore, we take the liberty of referring to our more advanced companion volume, Further Mathematics for Economic Analysis (usually abbreviated to FMEA). This is written jointly with our respective col- leagues Atle Seierstad and Arne Strøm in Oslo and, in a new forthcoming edition, with Andrés Carvajal at Warwick. In particular, FMEA offers a proper treatment of topics like xiv PPRREEFA F ACCEE second-order conditions for optimization, and the concavity or convexity of functions of more than two variables—topics that we think go rather beyond what is really “essential” for all economics students. Changes in the Fourth Edition We have been gratified by the number of students and their instructors from many parts of the world who appear to have found the first three editions useful.2 We have accordingly been encouraged to revise the text thoroughly once again. There are numerous minor changes and improvements, including the following in particular: (1) The main new feature is MyMathLab Global, explained on the page after this preface, as well as on the back cover. (2) New problems have been added for each chapter. (3) Some of the figures have been improved. Acknowledgements Over the years we have received help from so many colleagues, lecturers at other institutions, and students, that it is impractical to mention them all. Still, for some time now Arne Strøm, also at the Department of Economics of the Uni- versity of Oslo, has been an indispensable member of our production team. His mastery of the intricacies of the TEX typesetting system and his exceptional ability to spot errors and inaccuracies have been of enormous help. As long overdue recognition of his contribution, we have added his name on the front cover of this edition. Apart from our very helpful editors, with Kate Brewin at Pearson Education in charge, we should particularly like to thank Arve Michaelsen at Matematisk Sats in Norway for major assistance with the macros used to typeset the book, and for the figures. Very special thanks also go to professor Fred Böker at the University of Göttingen, who is not only responsible for translating previous editions into German, but has also shown exceptional diligence in paying close attention to the mathematical details of what he was translating. We appreciate the resulting large number of valuable suggestions for improvements and corrections that he continues to provide. To these and all the many unnamed persons and institutions who have helped us make this text possible, including some whose anonymous comments on earlier editions were forwarded to us by the publisher, we would like to express our deep appreciation and gratitude. We hope that all those who have assisted us may find the resulting product of benefit to their students. This, we can surely agree, is all that really matters in the end. Knut Sydsæter and Peter Hammond Oslo and Warwick, March 2012 2 Different English versions of this book have been translated into Albanian, German, Hungarian, Italian, Portuguese, Spanish, and Turkish.  P R E FA C E xv MyMathLab Global With your purchase of a new copy of this textbook, you may have received a Student Access Kit to MyMathLab Global. Follow the instructions on the card to register successfully and start making the most of the online resources. If you don’t have an access card, you can still access the resources by purchasing access online. Visit www. mymathlab.com/Global for details The Power of Practice MyMathLab Global provides a variety of tools to enable students to assess and progress their own learning, including questions and tests for each chapter of the book. A personalised study plan identifies areas to concentrate on to improve grades. MyMathLab Global gives you unrivalled resources: Sample tests for each chapter to see how much you have learned and where you still need practice A personalised study plan which constantly adapts to your strengths and weaknesses, taking you to exercises you can practise over and over again with different variables every time eText to click on to read the relevant parts of your textbook again See the guided tour on pp. xvi – xviii for more details. To activate your registration, go to www.mymathlab.com/Global  Guided t our of MyMathLab Global MyMathLab Global is an online assessment and revision tool that puts you in control of your learning through a suite of study and practice tools tied to the Pearson eText. SCREEN SHOT TO COME Why should I use MyMathLab Global? Since 2001, MyMathLab Global – along with MyMathLab, MyStatLab and MathXL – has helped over 9 million students succeed in more than 1,900 colleges and universities. MyMathLab Global engages students in active learning – it’s modular, self-paced, accessible anywhere with web access, and adaptable to each student’s learning style – and lecturers can easily customise MyMathLab Global to meet their students’ needs better.  GUIDED TOUR xvii How do I use MyMathLab Global? The Course home page is where you can view announcements from your lecturer and see an overview of your personal progress. View the Calendar to see the dates for online homework, quizzes and tests that your lecturer has set for you. Your lecturer may have chosen MyMathLab Global to provide online homework, quizzes and tests. Check here to access the homework that has been set for you. Keep track of your results in your own gradebook. Work through the questions in your personalised Study Plan at your own pace. Because the Study Plan is tailored to each student, you will be able to study more efficiently by only reviewing areas where you still need practice. The Study Plan also saves your results, helping you see at a glance exactly which topics you need to review. xviii GUIDED TOUR Additional instruction is provided in the form of detailed, step-by-step solutions to worked exercises. The figures in many of the exercises in MyMathLab Global are generated algorithmically, containing different values each time they are used. This means that you can practise individual problems as often as you like. There is also a link to the Pearson eText so you can easily review and master the content. Lecturer training and support Our dedicated team of technology specialists offer personalised training and support for MyMathLab Global, ensuring that you can maximise the benefits of MyMathLab Global. To find details of your local sales representatives, go to www.pearsoned.co.uk/replocator For a visual walkthrough of how to make the most of MyMathLab Global, visit www.MyMathLab.com/Global P UBLI S HER ’ S AC K N OW L E D G E ME N T S We are grateful to the following for permission to reproduce copyright material: Text Epigraph from the Preface from R.E. Lucas, Jr. (2001); Epigraph Chapter 9 by I.N. Stewart; Epigraph Chapter 15 by J.H. Drèze; Epigraph Chapter 16 by The Estate of Max Rosenlicht. In some instances we have been unable to trace the owners of copyright material, and we would appreciate any information that would enable us to do so.  1 INTRODUCTORY TOPICS I: ALGEBRA Is it right I ask; is it even prudence; to bore thyself and bore the students? —Mephistopheles to Faust (From Goethe’s Faust.) T his introductory chapter basically deals with elementary algebra, but we also briefly consider a few other topics that you might find that you need to review. Indeed, tests reveal that even students with a good background in mathematics often benefit from a brief review of what they learned in the past. These students should browse through the material and do some of the less simple problems. Students with a weaker background in mathematics, or who have been away from mathematics for a long time, should read the text carefully and then do most of the problems. Finally, those students who have considerable difficulties with this chapter should turn to a more elementary book on algebra. 1.1 The Real Numbers We start by reviewing some important facts and concepts concerning numbers. The basic numbers are 1, 2, 3, 4,... (natural numbers) also called positive integers. Of these 2, 4, 6, 8,... are the even numbers, whereas 1, 3, 5, 7,... are the odd numbers. Though familiar, such numbers are in reality rather abstract and advanced concepts. Civilization crossed a significant threshold when it grasped the idea that a flock of four sheep and a collection of four stones have something in common, namely “fourness”. This idea came to be represented by symbols such as the primitive :: (still used on dominoes or playing cards), the Roman numeral IV, and eventually the modern 4. This key notion is grasped and then continually refined as young children develop their mathematical skills. The positive integers, together with 0 and the negative integers −1, −2, −3, −4,... , make up the integers, which are 0, ±1, ±2, ±3, ±4,... (integers) Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 1  2 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA They can be represented on a number line like the one shown in Fig. 1 (where the arrow gives the direction in which the numbers increase). !5 !4 !3 !2 !1 0 1 2 3 4 5 Figure 1 The number line The rational numbers are those like 3/5 that can be written in the form a/b, where a and b are both integers. An integer n is also a rational number, because n = n/1. Other examples of rational numbers are 1 11 125 10 0 126 , , , − , 0= , −19, −1.26 = − 2 70 7 11 1 100 The rational numbers can also be represented on the number line. Imagine that we first mark 1/2 and all the multiples of 1/2. Then we mark 1/3 and all the multiples of 1/3, and so forth. You can be excused for thinking that “finally” there will be no more places left for putting more points on the line. But in fact this is quite wrong. The ancient Greeks already understood that “holes” would remain in the number line even after all the rational √ numbers had been √ marked off. For instance, there are no integers p and q such that 2 = p/q. Hence, 2 is not a rational number. (Euclid proved this fact in around the year 300 BC.) The rational numbers are therefore insufficient for measuring all possible lengths, let alone areas and volumes. This deficiency can be remedied by extending the concept of numbers to allow for the so-called irrational numbers. This extension can be carried out rather naturally by using decimal notation for numbers, as explained below. The way most people write numbers today is called the decimal system, or the base 10 system. It is a positional system with 10 as the base number. Every natural number can be written using only the symbols, 0, 1, 2,... , 9, which are called digits. You may recall that a digit is either a finger or a thumb, and that most humans have 10 digits. The positional system defines each combination of digits as a sum of powers of 10. For example, 1984 = 1 · 103 + 9 · 102 + 8 · 101 + 4 · 100 Each natural number can be uniquely expressed in this manner. With the use of the signs + and −, all integers, positive or negative, can be written in the same way. Decimal points also enable us to express rational numbers other than natural numbers. For example, 3.1415 = 3 + 1/101 + 4/102 + 1/103 + 5/104 Rational numbers that can be written exactly using only a finite number of decimal places are called finite decimal fractions. Each finite decimal fraction is a rational number, but not every rational number can be written as a finite decimal fraction. We also need to allow for infinite decimal fractions such as 100/3 = 33.333... where the three dots indicate that the digit 3 is repeated indefinitely. Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 2  SECTION 1.1 / THE REAL NUMBERS 3 If the decimal fraction is a rational number, then it will always be recurring or periodic— that is, after a certain place in the decimal expansion, it either stops or continues to repeat a finite sequence of digits. For example, 11/70 = 0.1 !571428"# $ 571428 ! "# $ 5... with the sequence of six digits 571428 repeated infinitely often. The definition of a real number follows from the previous discussion. We define a real number as an arbitrary infinite decimal fraction. Hence, a real number is of the form x = ±m.α1 α2 α3... , where m is a nonnegative integer, and αn (n = 1, 2...) is an infinite series of digits, each in the range 0 to 9. We have already identified the periodic decimal fractions with the rational numbers. In addition, there are infinitely many new numbers given by√ the nonperiodic √ √decimal fractions. These are called irrational numbers. Examples include 2, − 5, π, 2 2 , and 0.12112111211112.... We mentioned earlier that each rational number can be represented by a point on the number line. But not all points on the number line represent rational numbers. It is as if the irrational numbers “close up” the remaining holes on the number line after all the rational numbers have been positioned. Hence, an unbroken and endless straight line with an origin and a positive unit of length is a satisfactory model for the real numbers. We frequently state that there is a one-to-one correspondence between the real numbers and the points on a number line. Often, too, one speaks of the “real line” rather than the “number line”. The set of rational numbers as well as the set of irrational numbers are said to be “dense” on the number line. This means that between any two different real numbers, irrespective of how close they are to each other, we can always find both a rational and an irrational number—in fact, we can always find infinitely many of each. When applied to the real numbers, the four basic arithmetic operations always result in a real number. The only exception is that we cannot divide by 0.1 p is not defined for any real number p 0 This is very important and should not be confused with 0/a = 0, for all a ̸ = 0. Notice especially that 0/0 is not defined as any real number. For example, if a car requires 60 litres of fuel to go 600 kilometres, then its fuel consumption is 60/600 = 10 litres per 100 kilometres. However, if told that a car uses 0 litres of fuel to go 0 kilometres, we know nothing about its fuel consumption; 0/0 is completely undefined. 1 “Black holes are where God divided by zero.” (Steven Wright) Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 3  4 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA PROBLEMS FOR SECTION 1.1 1. Which of the following statements are true? (a) 1984 is a natural number. (b) −5 is to the right of −3 on the number line. (c) −13 is a natural number. (d) There is no natural number that is not rational. (e) 3.1415 is not rational. (f) The sum of two irrational numbers is irrational. (g) −3/4 is rational. (h) All rational numbers are real. 2. Explain why the infinite decimal expansion 1.01001000100001000001... is not a rational number. 1.2 Integer Powers You should recall that we often write 34 instead of the product 3 · 3 · 3 · 3, that 21 · 21 · 21 · 21 · 21 % &5 can be written as 21 , and that (−10)3 = (−10)(−10)(−10) = −1000. If a is any number and n is any natural number, then a n is defined by a n = !a · a "# ·... · a$ n factors The expression a n is called the nth power of a; here a is the base, and n is the exponent. We have, for example, a 2 = a · a, x 4 = x · x · x · x, and ' (5 p p p p p p = · · · · q q q q q q where a = p/q, and n = 5. By convention, a 1 = a, a “product” with only one factor. We usually drop the multiplication sign if this is unlikely to create misunderstanding. For example, we write abc instead of a · b · c, but it is safest to keep the multiplication sign in 1.053 = 1.05 · 1.05 · 1.05. We define further a0 = 1 for a ̸ = 0 Thus, 50 = 1, (−16.2)0 = 1, and (x · y)0 = 1 (if x · y ̸ = 0). But if a = 0, we do not assign a numerical value to a 0 ; the expression 00 is undefined. We also need to define powers with negative exponents. What do we mean by 3−2 ? It turns out that the sensible definition is to set 3−2 equal to 1/32 = 1/9. In general, 1 a −n = an whenever n is a natural number and a ̸ = 0. In particular, a −1 = 1/a. In this way we have defined a x for all integers x. Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 4  SECTION 1.2 / INTEGER POWERS 5 Calculators usually have a power key, denoted by y x or a x , which can be used to compute powers. Make sure you know how to use it by computing 23 (which is 8), 32 (which is 9), and 25−3 (which is 0.000064). Properties of Powers There are some rules for powers that you really must not only know by heart, but understand why they are true. The two most important are: (i) a r · a s = a r+s (ii) (a r )s = a rs Note carefully what these rules say. According to rule (i), powers with the same base are multiplied by adding the exponents. For example, a 3 · a 5 = !a · "# a · a$ · !a · a · "# · a · a · a · a$ = a 3+5 = a 8 a · a · a$ = !a · a · a · a"# 3 factors 5 factors 3 + 5 = 8 factors Here is an example of rule (ii): (a 2 )4 = a · a · a!"#$ !"#$ · a · a!"#$ · a · a!"#$ · a · a · a · a$ = a 2 · 4 = a 8 · a = !a · a · a · a"# 2 factors 2 factors 2 factors 2 factors 2 · 4 = 8 factors Division of two powers with the same base goes like this: ar 1 ar ÷ as = s = a r s = a r · a −s = a r−s a a Thus we divide two powers with the same base by subtracting the exponent in the denom- inator from that in the numerator. For example, a 3 ÷ a 5 = a 3−5 = a −2. Finally, note that (ab)r = !ab · ab"# ·... · a$ · !b · b ·"#... · b$ = a r br ·... · ab$ = !a · a "# r factors r factors r factors and r factors ) a *r # $! " a a a a · a ·... · a ar = · ·... · = = r = a r b−r b !b b "# b$ !b · b ·"#... · b$ b r factors r factors These rules can be extended to cases where there are several factors. For instance, (abcde)r = a r br cr d r er We saw that (ab)r = a r br. What about (a + b)r ? One of the most common errors committed in elementary algebra is to equate this to a r + br. For example, (2 + 3)3 = 53 = 125, but 23 + 33 = 8 + 27 = 35. Thus, (a + b)r ̸ = a r + br (in general) Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 5  6 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA t p t q−1 EXAMPLE 1 Simplify2 (a) x p x 2p (b) t s ÷ t s−1 (c) a 2 b3 a −1 b5 (d). t r t s−1 Solution: (a) x p x 2p = x p+2p = x 3p (b) t s ÷ t s−1 = t s−(s−1) = t s−s+1 = t 1 = t (c) a 2 b3 a −1 b5 = a 2 a −1 b3 b5 = a 2−1 b3+5 = a 1 b8 = ab8 t p · t q−1 t p+q−1 (d) = = t p+q−1−(r+s−1) = t p+q−1−r−s+1 = t p+q−r−s t r · t s−1 t r+s−1 EXAMPLE 2 If x −2 y 3 = 5, compute x −4 y 6 , x 6 y −9 , and x 2 y −3 + 2x −10 y 15. Solution: In computing x −4 y 6 , how can we make use of the assumption that x −2 y 3 = 5? A moment’s reflection might lead you to see that (x −2 y 3 )2 = x −4 y 6 , and hence x −4 y 6 = 52 = 25. Similarly, x 6 y −9 = (x −2 y 3 )−3 = 5−3 = 1/125 x 2 y −3 + 2x −10 y 15 = (x −2 y 3 )−1 + 2(x −2 y 3 )5 = 5−1 + 2 · 55 = 6250.2 NOTE 1 An important motivation for introducing the definitions a 0 = 1 and a −n = 1/a n is that we want the rules for powers to be valid for negative and zero exponents as well as for positive ones. For example, we want a r · a s = a r+s to be valid when r = 5 and s = 0. This requires that a 5 · a 0 = a 5+0 = a 5 , so we must choose a 0 = 1. If a n · a m = a n+m is to be valid when m = −n, we must have a n · a −n = a n+(−n) = a 0 = 1. Because a n · (1/a n ) = 1, we must define a −n to be 1/a n. NOTE 2 It is easy to make mistakes when dealing with powers. The following examples highlight some common sources of confusion. (a) There is an important difference between (−10)2 = (−10)(−10) = 100, and −102 = −(10 · 10) = −100. The square of minus 10 is not equal to minus the square of 10. (b) Note that (2x)−1 = 1/(2x). Here the product 2x is raised to the power of −1. On the other hand, in the expression 2x −1 only x is raised to the power −1, so 2x −1 = 2 · (1/x) = 2/x. (c) The volume of a ball with radius r is 43 πr 3. What will the volume be if the % radius & is doubled? The new volume is 43 π(2r)3 = 43 π(2r)(2r)(2r) = 43 π8r 3 = 8 43 πr 3 , so the volume is 8 times the initial one. (If we made the mistake of “simplifying” (2r)3 to 2r 3 , the result would imply only a doubling of the volume; this should be contrary to common sense.) Compound Interest Powers are used in practically every branch of applied mathematics, including economics. To illustrate their use, recall how they are needed to calculate compound interest. 2 Here and throughout the book we strongly suggest that when you attempt to solve a problem, you cover the solution and then gradually reveal the proposed answer to see if you are right. Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 6  SECTION 1.2 / INTEGER POWERS 7 Suppose you deposit $1000 in a bank account paying 8% interest at the end of each year.3 After one year you will have earned $1000 · 0.08 = $80 in interest, so the amount in your bank account will be $1080. This can be rewritten as ' ( 1000 · 8 8 1000 + = 1000 1 + = 1000 · 1.08 100 100 Suppose this new amount of $1000 · 1.08 is left in the bank for another year at an interest rate of 8%. After a second year, the extra interest will be $1000 · 1.08 · 0.08. So the total amount will have grown to 1000 · 1.08 + (1000 · 1.08) · 0.08 = 1000 · 1.08(1 + 0.08) = 1000 · (1.08)2 Each year the amount will increase by the factor 1.08, and we see that at the end of t years it will have grown to $1000 · (1.08)t. If the original amount is $K and the interest rate is p% per year, by the end of the first year, the amount will be K + K · p/100 = K(1 + p/100) dollars. The growth factor per year is thus 1 + p/100. In general, after t (whole) years, the original investment of $K will have grown to an amount ) p *t K 1+ 100 when the interest rate is p% per year (and interest is added to the capital every year—that is, there is compound interest). This example illustrates a general principle: A quantity K which increases by p% per year will have increased after t years to ) p *t K 1+ 100 p Here 1 + is called the growth factor for a growth of p%. 100 If you see an expression like (1.08)t you should immediately be able to recognize it as the amount to which $1 has grown after t years when the interest rate is 8% per year. How should you interpret (1.08)0 ? You deposit $1 at 8% per year, and leave the amount for 0 years. Then you still have only $1, because there has been no time to accumulate any interest, so that (1.08)0 must equal 1. NOTE 3 1000·(1.08)5 is the amount you will have in your account after 5 years if you invest $1000 at 8% interest per year. Using a calculator, you find that you will have approximately $1469.33. A rather common mistake is to put 1000 · (1.08)5 = (1000 · 1.08)5 = (1080)5. This is 1012 (or a trillion) times the right answer. 3 Remember that 1% means one in a hundred, or 0.01. So 23%, for example, is 23 · 0.01 = 0.23. 23 To calculate 23% of 4000, we write 4000 · 100 = 920 or 4000 · 0.23 = 920. Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 7  8 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA EXAMPLE 3 A new car has been bought for $15 000 and is assumed to decrease in value (depreciate) by 15% per year over a six-year period. What is its value after 6 years? Solution: After one year its value is down to ' ( 15 000 · 15 15 15 000 − = 15 000 1 − = 15 000 · 0.85 = 12 750 100 100 After two years its value is 15 000 · (0.85)2 = 10 837.50, and so on. After six years we realize that its value must be 15 000 · (0.85)6 ≈ 5 657. This example illustrates a general principle: A quantity K which decreases by p% per year, will after t years have decreased to ) p *t K 1− 100 p Here 1 − is called the growth factor for a decline of p%. 100 Do We Really Need Negative Exponents? How much money should you have deposited in a bank 5 years ago in order to have $1000 today, given that the interest rate has been 8% per year over this period? If we call this amount x, the requirement is that x · (1.08)5 must equal $1000, or that x · (1.08)5 = 1000. Dividing by 1.085 on both sides yields 1000 x= = 1000 · (1.08)−5 (1.08)5 (which is approximately $681). Thus, $(1.08)−5 is what you should have deposited 5 years ago in order to have $1 today, given the constant interest rate of 8%. In general, $P (1 + p/100)−t is what you should have deposited t years ago in order to have $P today, if the interest rate has been p% every year. PROBLEMS FOR SECTION 1.2 1. Compute: (a) 103 (b) (−0.3)2 (c) 4−2 (d) (0.1)−1 2. Write as powers of 2: (a) 4 (b) 1 (c) 64 (d) 1/16 3. Write as powers: % &% &% & (a) 15 · 15 · 15 (b) − 13 − 13 − 13 (c) 1 10 (d) 0.0000001 (e) t t t t t t (f) (a − b)(a − b)(a − b) (g) a a b b b b (h) (−a)(−a)(−a) In Problems 4–6 expand and simplify. 4. (a) 25 · 25 (b) 38 · 3−2 · 3−3 (c) (2x)3 (d) (−3xy 2 )3 Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 8  SECTION 1.2 / INTEGER POWERS 9 p 24 p3 a 4 b−3 34 (32 )6 p γ (pq)σ 5. (a) (b) (c) (d) p4 p (a 2 b−3 )2 (−3)15 37 p 2γ +σ q σ −2 ' (3 4 4 2 · 62 6. (a) 20 · 21 · 22 · 23 (b) (c) 3 33 · 23 (d) x 5 x 4 (e) y 5 y 4 y 3 (f) (2xy)3 102 · 10−4 · 103 (k 2 )3 k 4 (x + 1)3 (x + 1)−2 (g) (h) (i) 100 · 10−2 · 105 (k 3 )2 (x + 1)2 (x + 1)−3 7. The surface area of a sphere with radius r is 4πr 2. (a) By what factor will the surface area increase if the radius is tripled? (b) If the radius increases by 16%, by how many % will the surface area increase? 8. Which of the following equalities are true and which are false? Justify your answers. (Note: a and b are positive, m and n are integers.) (a) a 0 = 0 (b) (a + b)−n = 1/(a + b)n (c) a m · a m = a 2m (d) a m · bm = (ab)2m (e) (a + b)m = a m + bm (f) a n · bm = (ab)n+m 9. Complete the following: (a) xy = 3 implies x3y3 =... (b) ab = −2 implies (ab)4 =... (c) a 2 = 4 implies (a 8 )0 =... (d) n integer implies (−1)2n =... 10. Compute the following: (a) 13% of 150 (b) 6% of 2400 (c) 5.5% of 200 11. A box containing 5 balls costs $8.50. If the balls are bought individually, they cost $2.00 each. How much cheaper is it, in percentage terms, to buy the box as opposed to buying 5 individual balls? 12. Give economic interpretations to each of the following expressions and then use a calculator to find the approximate values: (a) 50 · (1.11)8 (b) 10 000 · (1.12)20 (c) 5000 · (1.07)−10 13. (a) $12 000 is deposited in an account earning 4% interest per year. What is the amount after 15 years? (b) If the interest rate is 6% each year, how much money should you have deposited in a bank 5 years ago to have $50 000 today? 14. A quantity increases by 25% each year for 3 years. How much is the combined percentage growth p over the three year period? 15. (a) A firm’s profit increased from 1990 to 1991 by 20%, but it decreased by 17% from 1991 to 1992. Which of the years 1990 and 1992 had the higher profit? (b) What percentage decrease in profits from 1991 to 1992 would imply that profits were equal in 1990 and 1992? Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 9  10 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA 1.3 Rules of Algebra You are certainly already familiar with the most common rules of algebra. We have already used some in this chapter. Nevertheless, it may be useful to recall those that are most important. If a, b, and c are arbitrary numbers, then: (a) a+b =b+a (g) 1·a =a (b) (a + b) + c = a + (b + c) (h) aa −1 = 1 for a ̸ = 0 (c) a+0=a (i) (−a)b = a(−b) = −ab (d) a + (−a) = 0 (j) (−a)(−b) = ab (e) ab = ba (k) a(b + c) = ab + ac (f) (ab)c = a(bc) (l) (a + b)c = ac + bc These rules are used in the following examples: 5 + x2 = x2 + 5 (a + 2b) + 3b = a + (2b + 3b) = a + 5b x 13 = 13 x (xy)y −1 = x(yy −1 ) = x (−3)5 = 3(−5) = −(3 · 5) = −15 (−6)(−20) = 120 3x(y + 2z) = 3xy + 6xz (t 2 + 2t)4t 3 = t 2 4t 3 + 2t4t 3 = 4t 5 + 8t 4 The algebraic rules can be combined in several ways to give: a(b − c) = a[b + (−c)] = ab + a(−c) = ab − ac x(a + b − c + d) = xa + xb − xc + xd (a + b)(c + d) = ac + ad + bc + bd Figure 1 provides a geometric argument for the last of these algebraic rules for the case in which the numbers a, b, c, and d are all positive. The area (a + b)(c + d) of the large rectangle is the sum of the areas of the four small rectangles. c"d a ac ad a"b b bc bd c d Figure 1 Recall the following three “quadratic identities”, which are so important that you should definitely memorize them. (a + b)2 = a 2 + 2ab + b2 (a − b)2 = a 2 − 2ab + b2 (a + b)(a − b) = a 2 − b2 Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 10  SECTION 1.3 / RULES OF ALGEBRA 11 The last of these is called the difference-of-squares formula. The proofs are very easy. For example, (a + b)2 means (a + b)(a + b), which equals aa + ab + ba + bb = a 2 + 2ab + b2. EXAMPLE 1 Expand: (a) (3x + 2y)2 (b) (1 − 2z)2 (c) (4p + 5q)(4p − 5q). Solution: (a) (3x + 2y)2 = (3x)2 + 2(3x)(2y) + (2y)2 = 9x 2 + 12xy + 4y 2 (b) (1 − 2z)2 = 1 − 2 · 1 · 2 · z + (2z)2 = 1 − 4z + 4z2 (c) (4p + 5q)(4p − 5q) = (4p)2 − (5q)2 = 16p2 − 25q 2 We often encounter parentheses with a minus sign in front. Because (−1)x = −x, −(a + b − c + d) = −a − b + c − d In words: When removing a pair of parentheses with a minus in front, change the signs of all the terms within the parentheses—do not leave any out. We saw how to multiply two factors, (a +b) and (c +d). How do we compute such products when there are several factors? Here is an example: + , % & (a + b)(c + d)(e + f ) = (a + b)(c + d) (e + f ) = ac + ad + bc + bd (e + f ) = (ac + ad + bc + bd)e + (ac + ad + bc + bd)f = ace + ade + bce + bde + acf + adf + bcf + bdf + , Alternatively, write (a + b)(c + d)(e + f ) = (a + b) (c + d)(e + f ) , then expand and show that you get the same answer. EXAMPLE 2 Expand (r + 1)3. Solution: + , (r + 1)3 = (r + 1)(r + 1) (r + 1) = (r 2 + 2r + 1)(r + 1) = r 3 + 3r 2 + 3r + 1 Illustration: A ball with radius r metres has a volume of 43 πr 3 cubic metres. By how much does the volume expand if the radius increases by 1 metre? The solution is 4 3 π(r + 1)3 − 43 πr 3 = 43 π(r 3 + 3r 2 + 3r + 1) − 43 πr 3 = 43 π(3r 2 + 3r + 1) Algebraic Expressions Expressions involving letters such as 3xy − 5x 2 y 3 + 2xy + 6y 3 x 2 − 3x + 5yx + 8 are called algebraic expressions. We call 3xy, −5x 2 y 3 , 2xy, 6y 3 x 2 , −3x, 5yx, and 8 the terms in the expression that is formed by adding all the terms together. The numbers 3, −5, 2, 6, −3, and 5 are the numerical coefficients of the first six terms. Two terms where only the Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 11  12 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA numerical coefficients are different, such as −5x 2 y 3 and 6y 3 x 2 , are called terms of the same type. In order to simplify expressions, we collect terms of the same type. Then within each term, we put numerical coefficients first and place the letters in alphabetical order. Thus, 3xy − 5x 2 y 3 + 2xy + 6y 3 x 2 − 3x + 5yx + 8 = x 2 y 3 + 10xy − 3x + 8 EXAMPLE 3 Expand and simplify: (2pq − 3p 2 )(p + 2q) − (q 2 − 2pq)(2p − q). Solution: (2pq − 3p 2 )(p + 2q) − (q 2 − 2pq)(2p − q) = 2pqp + 2pq2q − 3p 3 − 6p 2 q − (q 2 2p − q 3 − 4pqp + 2pq 2 ) = 2p2 q + 4pq 2 − 3p 3 − 6p 2 q − 2pq 2 + q 3 + 4p 2 q − 2pq 2 = −3p3 + q 3 Factoring When we write 49 = 7 · 7 and 672 = 2 · 2 · 2 · 2 · 2 · 3 · 7, we have factored these numbers. Algebraic expressions can often be factored in a similar way. For example, 6x 2 y = 2·3·x·x·y and 5x 2 y 3 − 15xy 2 = 5 · x · y · y(xy − 3). EXAMPLE 4 Factor each of the following: (a) 5x 2 + 15x (b) − 18b2 + 9ab (c) K(1 + r) + K(1 + r)r (d) δL−3 + (1 − δ)L−2 Solution: (a) 5x 2 + 15x = 5x(x + 3) (b) −18b2 + 9ab = 9ab − 18b2 = 3 · 3b(a − 2b) (c) K(1 + r) + K(1 + r)r = K(1 + r)(1 + r) = K(1 + r)2 (d) δL−3 + (1 − δ)L−2 = L−3 [δ + (1 − δ)L] The “quadratic identities” can often be used in reverse for factoring. They sometimes enable us to factor expressions that otherwise appear to have no factors. EXAMPLE 5 Factor each of the following: (a) 16a 2 − 1 (b) x 2 y 2 − 25z2 (c) 4u2 + 8u + 4 (d) x 2 − x + 1 4 Solution: (a) 16a 2 − 1 = (4a + 1)(4a − 1) (b) x 2 y 2 − 25z2 = (xy + 5z)(xy − 5z) (c) 4u2 + 8u + 4 = 4(u2 + 2u + 1) = 4(u + 1)2 1 (d) x 2 − x + 4 = (x − 21 )2 Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 12  SECTION 1.3 / RULES OF ALGEBRA 13 NOTE 1 To factor an expression means to express it as a product of simpler factors. Note that 9x 2 − 25y 2 = 3 · 3 · x · x − 5 · 5 · y · y does not factor 9x 2 − 25y 2. A correct factoring is 9x 2 − 25y 2 = (3x − 5y)(3x + 5y). Sometimes one has to show a measure of inventiveness to find a factoring: 4x 2 − y 2 + 6x 2 + 3xy = (4x 2 − y 2 ) + 3x(2x + y) = (2x + y)(2x − y) + 3x(2x + y) = (2x + y)(2x − y + 3x) = (2x + y)(5x − y) Although it might be difficult, or impossible, to find a factoring, it is very easy to verify that an algebraic expression has been factored correctly by simply multiplying the factors. For example, we check that x 2 − (a + b)x + ab = (x − a)(x − b) by expanding (x − a)(x − b). Most algebraic expressions cannot be factored. For example, there is no way to write x 2 + 10x + 50 as a product of simpler factors.4 PROBLEMS FOR SECTION 1.3 In Problems 1–5, expand and simplify. 1 1. (a) −3 + (−4) − (−8) (b) (−3)(2 − 4) (c) (−3)(−12)(− ) 2 (d) −3[4 − (−2)] (e) −3(−x − 4) (f) (5x − 3y)9 ' ( 3 2 (g) 2x (h) 0 · (1 − x) (i) −7x 2x 14x 2. (a) 5a 2 − 3b − (−a 2 − b) − 3(a 2 + b) (b) −x(2x − y) + y(1 − x) + 3(x + y) 2 2 (c) 12t − 3t + 16 − 2(6t − 2t + 8) (d) r 3 − 3r 2 s + s 3 − (−s 3 − r 3 + 3r 2 s) 3. (a) −3(n2 − 2n + 3) (b) x 2 (1 + x 3 ) (c) (4n − 3)(n − 2) (d) 6a 2 b(5ab − 3ab2 ) (e) (a 2 b − ab2 )(a + b) (f) (x − y)(x − 2y)(x − 3y) 4. (a) (ax + b)(cx + d) (b) (2 − t 2 )(2 + t 2 ) (c) (u − v)2 (u + v)2 ⊂⊃ 5. SM (a) (2t − 1)(t 2 − 2t + 1) (b) (a + 1)2 + (a − 1)2 − 2(a + 1)(a − 1) (c) (x + y + z)2 (d) (x + y + z)2 − (x − y − z)2 4 If we introduce complex numbers, however, then x 2 + 10x + 50 can be factored. Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 13  14 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA 6. Expand each of the following: ' (2 1 (a) (x + 2y)2 (b) −x (c) (3u − 5v)2 (d) (2z − 5w)(2z + 5w) x (a + 1)2 − (a − 1)2 7. (a) 2012 − 1992 = (b) If u2 − 4u + 4 = 1, then u = (c) = (b + 1)2 − (b − 1)2 8. Compute 10002 /(2522 − 2482 ) without using a calculator. 9. Verify the following cubic identities, which are occasionally useful: (a) (a + b)3 = a 3 + 3a 2 b + 3ab2 + b3 (b) (a − b)3 = a 3 − 3a 2 b + 3ab2 − b3 (c) a 3 − b3 = (a − b)(a 2 + ab + b2 ) (d) a 3 + b3 = (a + b)(a 2 − ab + b2 ) In Problems 10 to 15, factor the given expressions. 10. (a) 21x 2 y 3 (b) 3x − 9y + 27z (c) a 3 − a 2 b (d) 8x 2 y 2 − 16xy 11. (a) 28a 2 b3 (b) 4x + 8y − 24z (c) 2x 2 − 6xy (d) 4a 2 b3 + 6a 3 b2 (e) 7x 2 − 49xy (f) 5xy 2 − 45x 3 y 2 (g) 16 − b2 (h) 3x 2 − 12 12. (a) x 2 − 4x + 4 (b) 4t 2 s − 8ts 2 (c) 16a 2 + 16ab + 4b2 (d) 5x 3 − 10xy 2 ⊂⊃ 13. SM (a) a 2 + 4ab + 4b2 (b) K 2 L − L2 K (c) K −4 − LK −5 (d) 9z2 − 16w 2 (e) − 15 x 2 + 2xy − 5y 2 (f) a 4 − b4 14. (a) 5x + 5y + ax + ay (b) u2 − v 2 + 3v + 3u (c) P 3 + Q3 + Q2 P + P 2 Q 15. (a) K 3 − K 2 L (b) KL3 + KL (c) L2 − K 2 (d) K 2 − 2KL + L2 (e) K 3 L − 4K 2 L2 + 4KL3 (f) K −3 − K −6 1.4 Fractions Recall that a ← numerator a÷b = b ← denominator For example, 5 ÷ 8 = 58. For typographical reasons we often write 5/8 instead of 58. Of course, 5 ÷ 8 = 0.625. In this case, we have written the fraction as a decimal number. The fraction 5/8 is called a proper fraction because 5 is less than 8. The fraction 19/8 is an improper fraction because the numerator is larger than (or equal to) the denominator. An improper fraction can be written as a mixed number: 19 3 3 =2+ =2 8 8 8 Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 14  SECTION 1.4 / FRACTIONS 15 Here 2 38 means 2 plus 3/8. On the other hand, 2 · 38 = 2·3 3 8 = 4 (by the rules reviewed in what follows). Note, however, that 2 8 means 2 · 8 ; the notation 2x x x 8 or 2x/8 is obviously preferable in this case. Indeed, 198 or 19/8 is probably better than 2 3 8 because it also helps avoid ambiguity. The most important properties of fractions are listed below, with simple numerical ex- amples. It is absolutely essential for you to master these rules, so you should carefully check that you know each of them. Rule: Example: a · \c a 21 7 · 3\ 7 (1) = (b ̸ = 0 and c ̸ = 0) = = b · \c b 15 5 · 3\ 5 −a (−a) · (−1) a −5 5 (2) = = = −b (−b) · (−1) b −6 6 a a (−1)a −a 13 13 (−1)13 −13 (3) − = (−1) = = − = (−1) = = b b b b 15 15 15 15 a b a+b 5 13 18 (4) + = + = =6 c c c 3 3 3 a c a·d +b·c 3 1 3·6+5·1 23 (5) + = + = = b d b·d 5 6 5·6 30 b a·c+b 3 5·5+3 28 (6) a+ = 5+ = = c c 5 5 5 b a·b 3 21 (7) a· = 7· = c c 5 5 a c a·c 4 5 4·5 4\ · 5 5 (8) · = · = = = b d b·d 7 8 7·8 7 · 2 · 4\ 14 a c a d a·d 3 6 3 14 3\ · 2\ · 7 7 (9) ÷ = · = ÷ = · = = b d b c b·c 8 14 8 6 2\ · 2 · 2 · 2 · 3\ 8 Rule (1) is very important. It is the rule used to reduce fractions by factoring the numerator and the denominator, then cancelling common factors (that is, dividing both the numerator and denominator by the same nonzero quantity). 5x 2 yz3 x 2 + xy 4 − 4a + a 2 EXAMPLE 1 Simplify: (a) (b) (c) 25xy 2 z x2 − y2 a2 − 4 Solution: 5x 2 yz3 5\ · x\ · x · y\ · \z · z · z xz2 x 2 + xy x(x + y) x (a) = = (b) = = 25xy 2 z 5\ · 5 · x\ · y\ · y · \z 5y 2 x −y 2 (x − y)(x + y) x−y 4 − 4a + a 2 (a − 2)(a − 2) a−2 (c) = = a2 − 4 (a − 2)(a + 2) a+2 Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 15  16 CHAPTER 1 / INTRODUCTORY TOPICS I: ALGEBRA When we use rule (1) in reverse, we are expanding the fraction. For example, 5/8 = 5 · 125/8 · 125 = 625/1000 = 0.625. When we simplify fractions, only common factors can be removed. A frequently occur- ring error is illustrated in the following example. 2x\ + 3y 2 + 3y\ 2+3 Wrong! → = = =5 x\y y\ 1 In fact, the numerator and the denominator in the fraction (2x + 3y)/xy do not have any common factors. But a correct simplification is this: (2x + 3y)/xy = 2/y + 3/x. Another error is shown in the next example. x x x 1 1 Wrong! → = 2+ = + x 2 + 2x x 2x x 2 A correct way of simplifying the fraction is to cancel the common factor x, which yields the fraction 1/(x + 2). Rules (4)–(6) are those used to add fractions. Note that (5) follows from (1) and (4): a c a·d c·b a·d +b·c + = + = b d b·d d ·b b·d and we see easily that, for example, a c e adf cbf ebd adf − cbf + ebd − + = − + = (∗) b d f bdf bdf bdf bdf If the numbers b, d, and f have common factors, the computation carried out in (∗) involves unnecessarily large numbers. We can simplify the process by first finding the least common denominator (LCD) of the fractions. To do so, factor each denominator completely; the LCD is the product of all the distinct factors that appear in any denominator, each raised to the highest power to which it gets raised in any denominator. The use of the LCD is demonstrated in the following example. EXAMPLE 2 Simplify the following: 1 1 1 2+a 1−b 2b x−y x 3xy (a) − + (b) + − 2 2 (c) − + 2 2 3 6 a2b ab2 a b x+y x−y x − y2 Solution: 1 1 1 1·3 1·2 1 3−2+1 2 1 (a) The LCD is 6 and so − + = − + = = = 2 3 6 2·3 2·3 2·3 6 6 3 (b) The LCD is a 2 b2 and so 2+a 1−b 2b (2 + a)b (1 − b)a 2b 2 + 2 − 2 2 = 2 2 + 2 2 − 2 2 a b ab a b a b a b a b 2b + ab + a − ba − 2b a 1 = = 2 2 = 2 a 2 b2 a b ab Essential Math. for Econ. Analysis, 4th edn EME4_C01.TEX, 16 May 2012, 14:24 Page 16  SECTION 1.4 / FRACTIONS 17 (c) The LCD is (x + y)(x − y) and so x−y x 3xy (x − y)(x − y) (x + y)x 3xy − + 2 = − + x+y x−y x − y2 (x − y)(x + y) (x + y)(x − y) (x − y)(x + y) x 2 − 2xy + y 2 − x 2 − xy + 3xy y2