1. INTERNATIONAL BACCALAUREATEMATHEMATICSHIGHER LEVEL(CORE)i3RD EDITION3rd imprintSeries editor: Fabio CirritoContributing authors:Nigel BuckleIain Dunbar

2. MATHEMATICS Higher Level (Core)Copyright Nigel Buckle, Iain Dunbar, Key-Strokes Pty Ltd, Mifasa Pty Ltd.First published in 1997 by IBID Press2nd Edition published in 1999 by IBID Press,3rd Edition published in 2004 by IBID Press, 2nd imprint published in 2005Reprinted 2007Published by IBID Press, Victoria.Library Catalogue:Cirrito Fabio Editor., Buckle & Dunbar1. Mathematics, 2. International Baccalaureate. Series Title: InternationalBaccalaureate in DetailiiISBN: 1 876659 11 4 (10 digit)978 1 876659 11 0 (13 digit)All rights reserved except under the conditions described in the Copyright Act 1968 ofAustralia and subsequent amendments. No part of this publication may be reproduced,stored in a retrieval system, or transmitted in any form or by any means, without theprior permission of the publishers.While every care has been taken to trace and acknowledge copyright, the publisherstender their apologies for any accidental infringement where copyright has proveduntraceable. They would be pleased to come to a suitable arrangement with the rightfulowner in each case.This book has been developed independently of the International BaccalaureateOrganisation (IBO). The text is in no way connected with, or endorsed by, the IBO.This publication is independently produced for use by teachers and students. Althoughreferences have been reproduced with permission of the VCAA the publication is in noway connected or endorsed by the VCAA.We also wish to thank the Incorporated Association of Registered Teachers of Victoria forgranting us permission to reproduced questions from their October Examination Papers.Cover design by Adcore Creative.Published by IBID Press, at www.ibid.com.auFor further information contact fabio@ibid.com.auPrinted by SHANNON Books, Australia. 3. PREFACE TO 3RD EDITIONIt will be immediately obvious that the 3rd edition of the Mathematics Higher Level (Core) texthas been completely revised and updated. Sections of the previous two editions are still present,but much has happened to improve the text both in content and accuracy.In response to the many requests and suggestions from teachers worldwide the text wasextensively revised. There are more examples for students to refer to when learning the subjectmatter for the first time or during their revision period. There is an abundance of well-gradedexercises for students to hone their skills. Of course, it is not expected that any one student workthrough every question in this text - such a task would be quite a feat. It is hoped then thatteachers will guide the students as to which questions to attempt. The questions serve to developroutine skills, reinforce concepts introduced in the topic and develop skills in making appropriateuse of the graphics calculator.The text has been written in a conversational style so that students will find that they are notsimply making reference to an encyclopedia filled with mathematical facts, but rather find thatthey are in some way participating in or listening in on a discussion of the subject matter.Throughout the text the subject matter is presented using graphical, numerical, algebraic andverbal means whenever appropriate. Classical approaches have been judiciously combined withmodern approaches reflecting new technology - in particular the use of the graphics calculator.The book has been specifically written to meet the demands of the Higher Level (Core) section ofthe course and has been pitched at a level that is appropriate for students studying this subject.The book presents an extensive coverage of the syllabus and in some areas goes beyond what isrequired of the student. Again, this is for the teacher to decide how best to use these sections.Sets of revision exercises are included in the text. Many of the questions in these sets have beenaimed at a level that is on par with what a student could expect on an examination paper.However, some of the questions do go beyond the level that students may expect to find on anexamination paper. Success in examinations will depend on an individuals preparation and theywill find that making use of a selection of questions from a number of sources will be verybeneficial.I hope that most of the suggestions and recommendations that were brought forward have beenaddressed in this edition. However, there is always room for improvement. As always, I welcomeand encourage teachers and students to contact me with feedback, not only on their likes anddislikes but suggestions on how the book can be improved as well as where errors and misprintsoccur. There will be updates on the IBID Press website in relation to errors that have beenlocated in the book so we suggest that you visit the IBID website at www.ibid.com.au. If youbelieve you have located an error or misprint please email me at fabio@ibid.com.au.Fabio Cirrito, July 2004iii 4. MATHEMATICS Higher Level (Core)PREFACE TO 2ND EDITIONWe are grateful to all those teachers who have made comments and corrections on thefirst edition. We hope that these contributions have improved this second edition. Thisedition is now in line with the course whose first examinations will start in 2000. Asalways, we welcome all comments from teachers and with due time, will make use ofthem to further improve this book. Suggestions and comments can be directed to FabioCirrito via email: fabio@ibid.com.auFabio Cirrito, 1999PREFACE TO 1ST EDITIONThis text has been produced independently as a resource to support the teaching of theMathematics Higher Level Course of the International Baccalaureate. The examples andquestions do not necessarily reflect the views of the official senior examining teamappointed by the International Baccalaureate Organisation.The notation used is, as far as possible, that specified in the appropriate syllabusguidelines of the IB.The units of physical measurements are in S.I.The language and spelling are U.K. English.Currency quantities are specified in dollars, though these could be read as any currencythat is decimalised, such as Swiss francs, Lire etc.The graphic calculators covered directly in the text are the Texas TI/82 and 83.Supplementary material is available from the publisher for students using some othermakes and models of calculators. As it is important that students learn to interpretgraphic calculator output, the text and answers present a mixture of graphic calculatorscreens and conventional diagrams when discussing graphs.The text has been presented in the order in which the topics appear in the syllabus. Thisdoes not mean that the topics have to be treated in this order, though it is generally thecase that the more fundamental topics appear at the front of the book. Students arereminded that it is the IB Syllabus that specifies the contents of the course and not thistext. One of the keys to success in this course is to be thoroughly familiar with the coursecontents and the styles of questions that have been used in past examinations.Fabio Cirrito, August 1997.iv 5. CONTENTS1 THEORY OF KNOWLEDGE 11.1 Pure and Applied Mathematics 11.2 Axioms 21.3 Proof 41.3.1 Rules of Inference 51.3.2 Proof by Exhaustion 61.3.3 Direct Proof 71.3.4 Proof by Contradiction 81.4 Paradox 101.4.1 What is a Paradox? 101.4.2 Russells Paradox? 111.5 Mathematics and Other Disciplines 121.6 The nded say15 Exte Es2 ALGEBRA OF LINEAR AND QUADRATIC EXPRESSIONS 172.1 The Real Number Line 172.1.1 The Real Number Line 172.1.2 Set Builder Notation 172.1.3 Interval Notation 172.1.4 Number Systems 192.1.5 Irrational Numbers 202.1.6 The Absolute Value 222.2 Linear Algebra 242.2.1 Review of Linear Equations 242.2.2 Linear Inequations 292.3 Linear Functions 332.3.1 Graph of the Linear Function 332.3.2 Simultaneous Linear Equations in Two Unknowns 372.3.3 Simultaneous Linear Equations in Three Unknowns 422.4 Quadratics 452.4.1 Quadratic Equation 452.4.2 Quadratic Function 512.4.3 Quadratic Inequalities 582.4.4 Simultaneous Equations Involving Linear-Quadratic Equations 623 POLYNOMIALS 673.1 Algebra of Polynomials 673.1.1 Definition 673.1.2 Addition and Multiplication of Polynomials 673.1.3 Division of Polynomials 683.2 Synthetic Division 723.3 The Remainder Theorem 753.4 The Factor Theorem 773.5 Equations and Inequations 833.5.1 Polynomial Equations 833.5.2 Polynomial Inequations 863.6 Sketching Polynomials 883.6.1 Graphical Significance of Roots 883.6.2 Cubic Functions 89v 6. MATHEMATICS Higher Level (Core)4 THE BINOMIAL THEOREM 954.1 The Binomial Theorem 954.1.1 The Binomial Theorem 954.1.2 The General term 1004.2 Proof 1045 FUNCTIONS AND RELATIONS 1055.1 Relations 1055.1.1 Relations 1055.1.2 The Cartesian Plane 1075.1.3 Implied Domain 1085.1.4 Types of Relations 1095.1.5 Sketching with the Graphics Calculator 1105.2 Functions 1155.2.1 Definitions 1155.3 Some Standard Functions 1225.3.1 Hybrid Functions and Continuity 1225.3.2 The Absolute Value Function 1265.3.3 The Exponential Function 1315.3.4 The Logarithmic Function 1385.3.5 Equations of the Form y = xn, n = 1, 21445.4 Algebra of Functions 1485.4.1 Basic Operations and Composite Functions 1485.4.2 Identity and Inverse Functions 1576 TRANSFORMATIONS OF GRAPHS 1676.1 Translations 1676.1.1 Horizontal Translation 1676.1.2 Vertical Translation 1706.2 Dilations 1776.2.1 Dilation from the x-axis 1776.2.2 Dilation from the y-axis 1796.3 Reflections 1836.4 Reciprocal of a Function 1887 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 1977.1 Exponents 1977.1.1 Basic Rules of Indices 1977.1.2 Indicial Equations 2017.1.3 Equations of the form b f (x) = bg(x)2037.1.4 What if the Base is Not The Same? 2057.1.5 A Special Base (e) 2077.2 Exponential Modelling 2097.3 Logarithms 2177.3.1 What are Logarithms 2177.3.2 Can we find the Logarithm of a Negative Number? 2197.4 The Algebra of Logarithms 2217.5 Logarithmic Modelling 230REVISION SET A PAPER 1 AND PAPER 2 STYLE QUESTIONS 233vi 7. 8 SEQUENCES AND SERIES 2418.1 Arithmetic Sequences and Series 2418.1.1 Arithmetic Sequences 2418.1.2 Arithmetic Series 2458.1.3 Sigma Notation 2498.2 Geometric Sequences and Series 2528.2.1 Geometric Sequences 2528.2.2 Geometric Series 2578.2.3 Combined A.Ps and G.Ps 2618.2.4 Convergent Series 2638.3 Compound Interest and Superannuation 2678.3.1 Compound Interest 2678.3.2 Superannuation 2689 MENSURATION 2739.1 Trigonometric Ratios 2739.1.1 Review of Trigonometric Functions for Right-angled Triangles 2739.1.2 Exact Values 2739.2 Applications 2789.2.1 Angle of Elevation and Depression 2789.2.2 Bearings 2809.3 Right Angles in 3Dimensions 2839.4 Area of a Triangle 2879.5 Non Right-Angled Triangles 2909.5.1 The Sine Rule 2909.5.2 The Ambiguous Case 2949.5.3 Applications of The Sine Rule 2989.5.4 The Cosine Rule 3009.5.5 Applications of The Cosine Rule 3029.6 More Applications in 3D 3079.7 Arc, Sectors and Segments 3099.7.1 Radian Measure of an Angle 3099.7.2 Arc Length 3109.7.3 Area of Sector 31110 CIRCULAR TRIGONOMETRIC FUNCTIONS 31510.1 Trigonometric Ratios 31510.1.1 The Unit Circle 31510.1.2 Angle of Any Magnitude 31610.2 Trigonometric Identities 32810.2.1 The Fundamental Identity 32810.2.2 Compound Angle Identities 33310.3 Trigonometric Functions 34410.3.1 The Sine, Cosine and Tangent Functions 34410.3.2 Transformations of Trigonometric Functions 34710.4 Inverse Trigonometric Functions 35710.4.1 The Inverse Sine Function 35710.4.2 The Inverse Cosine Function 35910.4.3 The Inverse Tangent Function 36110.5 Trigonometric Equations 36610.5.1 Solution to sinx = a, cosx = a and tanx = a36610.6 Applications 377vii 8. MATHEMATICS Higher Level (Core)11 COMPLEX NUMBERS 38311.1 Complex Numbers 38311.1.1 Introduction 38311.1.2 Notations and 38311.1.3 Using the TI83 with Complex Numbers 38411.1.4 The Algebra of Complex Numbers 38511.2 Geometrical Representation of Complex Numbers 39211.2.1 The Argand Diagram 39211.2.2 Geometrical Properties of Complex Numbers 39311.3 Polar Form of Complex Numbers 40211.4 Polynomials over the Complex Field 41311.4.1 Quadratic Equations 41311.4.2 Polynomial Equations (of order3) 41411.4.3 Solutions to Equations of the Form 41911.5 Miscellaneous Proofs 42511.5.1 The Triangle Inequality 42511.5.2 De Moivres Theorem 42511.5.3 Fundamental Theorem of Algebra 42611.5.4 Conjugate Root Theorem 42612 MATHEMATICAL INDUCTION 42712.1 Mathematical Induction 42712.1.1 The Principles of Mathematical Induction and Proofs 42712.2 Further Examples Part 1 43112.3 Further Examples Part 2 43312.4 Forming Conjectures 43612.4.1 The nth Term of a Sequence 43612.4.2 The Sum to n Terms of a Sequence 440REVISION SET B PAPER 1 AND PAPER 2 STYLE QUESTIONS 44513 STATISTICS 45513.1 Describing Data 45513.1.1 Data Collection 45513.1.2 Types of Data13.1.3 Discrete and Continuous Data 45613.2 Frequency Diagrams 45713.2.1 Producing a Frequency Diagram 45713.2.2 Using a Graphics Calculator 45813.3 Statistical Measures 1 46013.3.1 Measure of Central Tendency 46013.3.2 Mode 46013.3.3 Mean 46013.3.4 Median 46213.4 Statistical Measures 2 46313.4.1 Measures of Spread 46313.4.2 Variance and Standard Deviation 46413.4.3 Using a Graphics Calculator 46613.5 Statistical Measures 3 46813.5.1 Quartiles 46813.5.2 Box Plot 470viiii2 = 1zn = x + iy 9. 14 COUNTING PRINCIPLES 47714.1 Multiplication Principle 47714.1.1 Definition 47714.1.2 Multiplication Principle 47714.1.3 Permutations 47914.2 Combinations 48415 PROBABILITY 48915.1 Probability48915.1.1 Probability as a LongTerm Relative Frequency 48915.1.2 Theoretical Probability 48915.1.3 Laws of Probability 49015.1.4 Definition of Probability 49015.1.5 Problem Solving Strategies in Probability 49215.2 ProbabilityVenn Diagrams 49415.3 Conditional robability P 49815.3.1 Informal Definition of Conditional Probability 49815.3.2 Formal Definition of Conditional Probability 49815.3.3 Independence 50015.4 Bayes Theorem 50515.4.1 Law of Total Probability 50515.4.2 Bayes Theorem for Two Elements 50615.5 Using Permutations and Combinations in Probability 51016 DISCRETE RANDOM VARIABLES 51316.1 Discrete Random Variables 51316.1.1 Concept of a Random Variable 51316.1.2 Discrete Random Variable 51316.1.3 Probability Distributions 51416.1.4 Properties of The Probability Function 51516.1.5 Constructing Probability Functions 51716.2 Mean and Variance 52116.2.1 Central Tendency and Expectation 52116.2.2 So, what exactly does E(X)measure? 52116.2.3 Properties of The Expectation (Function) 52316.2.4 Variance 52516.2.5 Properties of The Variance 52616.3 The Binomial Distribution 53016.3.1 The Binomial Experiment 53016.3.2 Bernoulli Trials 53016.3.3 Properties of The Binomial Experiment 53016.3.4 Binomial Distribution 53016.3.5 Expectation, Mode and Variance for the Binomial Distribution 53316.4 Hypergeometric Distribution 53916.4.1 Hypergeometric Distribution Function 53916.4.2 Mean and Variance of The Hypergeometric Distribution 54116.5 Poisson Distribution 54516.5.1 Poisson Distribution Function 54516.5.2 Poisson Recurrence Formula 54816.5.3 Mean and Variance of The Poisson Distribution 54916.5.4 Poisson s imit nomial a a L of 551 ix 10. MATHEMATICS Higher Level (Core)17 THE NORMAL DISTRIBUTION 55517.1 The Normal Distribution 55517.1.1 Why The Normal Distribution? 55517.1.2 The Standard Normal Curve 55517.1.3 Using The Standard Normal Table 55617.2 Formalising The Definition of The Normal Distribution 55917.2.1 The Normal Distribution 55917.2.2 Properties of This Curve 56017.2.3 Finding Probabilities Using The Normal Distribution 56017.2.4 The Standard Normal Distribution 56017.2.5 Finding Probabilities 56117.2.6 Standardising any Normal Distribution 56217.2.7 Inverse Problems 56617.2.8 Finding Quantiles 568REVISION SET C PAPER 1 AND PAPER 2 STYLE QUESTIONS 57518 RATES OF CHANGE 58518.1 Quantitative Measure 58518.1.1 Functional Dependence 58518.1.2 Quantitative Aspects of Change 58618.1.3 Average Rate of Change 58618.1.4 Determining the Average Rate of Change 58718.1.5 Velocity as a Measure of Rate of Change of Displacement 58818.2 Qualitative Measure 59218.2.1 Qualitative Aspects of Change 59218.2.2 Describing the Behaviour of a Graph 59218.2.3 Producing a Graph from a Physical Situation 59218.3 Instantaneous Rate of Change 59518.3.1 Informal Idea of Limits 59518.4 Differentiation Process 60118.4.1 The Derivative and The Gradient Function 60118.4.2 Notation and Language 60219 DIFFERENTIAL CALCULUS 60519.1 Differentiation 60519.1.1 Review 60519.1.2 Power Rule for Differentiation 60819.1.3 Derivative of a Sum or Difference 61019.2 Graphical Interpretation of The Derivative 61319.2.1 The Value of The Derivative at a Particular Point on a Curve 61319.2.2 Gradient Function from a Graph 61819.2.3 Differentiating with Variables other than x and y 62119.3 Derivative of Transcendental Functions 62219.3.1 Derivative of Circular Trigonometric Functions 62219.3.2 Derivative of The Exponential Functions 62319.3.3 Derivative of The Natural Log Functions 62419.3.4 Derivative of a Product of Functions 62419.3.5 Derivative of a Quotient of Functions 62619.3.6 The Chain Rule 62719.3.7 Derivative of Reciprocal Circular Functions 63419.4 Derivative of Inverse Trigonometric Functions 639x 11. 19.4.1 Derivative of 63919.4.2 Derivative of 64019.4.3 Derivative of 64119.4.4 Generalisations for The Derivative of The Inverse Circular Functions 64119.5 Derivative of and 64419.5.1 Differentiating 64419.5.2 Differentiating 64619.5.3 Generalisation of Exponential and Logarithmic Derivatives 64719.6 Second Derivative 65019.7 Implicit Differentiation 65219.7.1 Implicit Relations 65219.7.2 Second Derivative 65819.8 Proofs 66019.8.1 Derivative of where n is a Negative Integer 66019.8.2 Derivative of where n is a Fraction 66019.8.3 Product Rule and Quotient Rule 66119.8.4 Derivative of Some Trigonometric Functions 66119.8.5 Exponential and where n s al i Re 66420 DIFFERENTIAL CALCULUS AND CURVE SKETCHING 66520.1 Tangents and Normals 66520.1.1 Equation of Tangent 66520.1.2 Equation ormal of 667 20.2 Curve ketching S 67120.2.1 Increasing nd ecreasing unctions a D 671 F20.2.2 Stationary oints P 67320.2.3 Global Maxima and Minima 68420.2.4 Vertical Tangents and Cusps 68820.2.5 Summary 69020.3 The Second Derivative and its Application 69520.3.1 Definition 69520.4 Rational Functions 70220.4.1 Sketching the Graph of 70220.4.2 Other Rational Functions 70420.4.3 Oblique and Curved Asymptotes 70621 APPLICATIONS OF DIFFERENTIAL CALCULUS 71121.1 Rates Of Change 71121.1.1 Definitions 71121.1.2 Rates of Change and Their Sign 71221.2 Applied Rates of Change 713Application to Economics 714Application to Kinematics 71521.3 Kinematics 71820.3.1 Motion Along a Straight Line 71821.4 Related Rates 72721.5 Applied Maxima and Minima Problems 74121.5.1 MaximaMinima Problems 741xiSin1(x)Cos1(x)Tan1(x)ax logaxy ax =y= logaxy xn =y= xny = xnx ax + b-c---x----+-----d--, cx + d0 12. MATHEMATICS Higher Level (Core)21.5.2 End point Maxima/Minima Problems 75221.5.3 Optimisation for Integer Valued Variables 754REVISION SET D PAPER 1 AND PAPER 2 STYLE QUESTIONS 76322 INTEGRATION AND ITS APPLICATIONS 77322.1 Integration 77322.1.1 Antidifferentiation The finite egralInde Int 77322.1.2 Language nd otation a N 77322.1.3 Determining The finite egral Inde Int 77422.1.4 Properties The finite egral of Inde Int 77622.2 Solving or c f 77822.3 Standard ntegrals I 78122.4 The efinite ntegral D I 79022.4.1 Why The Definite Integral? 79022.4.2 Language and Notation 79022.4.3 Properties The efinite egral of D Int 79322.5 Applications ntegration of 798 I22.5.1 Introduction o he Area neath e t t Be a C7u9r8v22.5.2 In earch tter Approximation S of Be 799 a22.5.3 Towards an Exact Value 80022.5.4 The Definite Integral and Areas 80122.5.5 Further Observation about Areas 80222.5.6 The Signed Areas 80322.5.7 Steps for Finding Areas 80422.5.8 Area Between Two Curves 80522.6 Applications to Kinematics 81222.7 Applications robability to 816 22.7.1 Continuous Random Variables 81622.7.2 Mode, Mean, Median and Variance 82022.8 Volumes (Solid of Revolution) 82623 FURTHER INTEGRATION 83323.1 Integration by Substitution 83323.1.1 Indefinite Integrals 83323.1.2 Substitution Rule 84123.2 Integration by Parts 84823.2.1 The Basics 84823.2.2 Repeated Integration by Parts 85023.3 Applications 85224 DIFFERENTIAL EQUATIONS 85324.1 Differential Equations 85324.1.1 What are Differential Equations? 85324.1.2 Verifying a Solution 85424.2 Solving Differential Equations 85724.3 Applications 86625 MATRICES 87525.1 Introduction to Matrices 87525.1.1 Definitions 87525.1.2 Matrix Multiplication 881xii 13. 25.2 Inverses and Determinants 88825.2.1 Inverse and Determinant of 2 by 2 Matrices 88825.2.2 Inverse and Determinant of 3 by 3 Matrices 89125.3 Simultaneous Equations 90026 VECTORS 90926.1 Introduction to Vectors 90926.1.1 Scalar and Vector Quantities 90926.2 Representing Vectors 91026.2.1 Directed Line Segment 91026.2.2 Magnitude of a Vector 91026.2.3 Equal Vectors 91026.2.4 Negative Vectors 91126.2.5 Zero Vector 91126.2.6 Orientation and Vectors 91126.3 Algebra and Geometry of Vectors 91526.3.1 Addition of Vectors 91526.3.2 Subtraction of Vectors 91626.3.3 Multiplication of Vectors by Scalars 91626.3.4 Angle Between Vectors 91726.3.5 Applications to Geometry 91726.4 Cartesian Representation of Vectors in 2-D and 3-D 92326.4.1 Representation in 2-D 92326.4.2 Unit Vector and Base Vector Notation 92326.4.3 Representation in 3-D 92426.4.4 Vector Operations 92526.5 Further Properties of Vectors in 2-D and 3-D 92926.5.1 Magnitude of a Vector 92926.5.2 Unit Vectors 93026.6 Scalar Product of 2 Vectors 93226.6.1 Definition of the Scalar Product 93226.6.2 Properties of the Scalar Product 93426.6.3 Special Cases of the Scalar Product 93526.6.4 Direction Cosine 93826.6.5 Using a Graphics Calculator 94326.7 Vector Equation of a Line 94426.7.1 Vector Equation of a Line in 2-D 94426.7.2 Lines in Three Dimensions 95227 3-D GEOMETRY 96127.1 Vector Product 96127.1.1 Vector Product 96127.1.2 Vector Form of the Vector Product 96427.1.3 Applications of the Vector Product 96827.2 Planes in 3-Dimensions 97227.2.1 Vector Equation of a Plane 97227.2.2 Cartesian Equation of a Plane 97327.2.3 Normal Vector Form of a Plane 97527.2.4 The Normal Form 97927.3 Intersecting Lines and Planes 98227.3.1 Intersection of Two Lines 98227.3.2 Intersection of a Line and a Plane 983xiii 14. MATHEMATICS Higher Level (Core)27.3.3 Intersection of Two Planes 98527.3.4 Intersection of Three Planes 986REVISION SET E PAPER 1 AND PAPER 2 STYLE QUESTIONS 991ANSWERS 1001xiv 15. NOTATIONThe list below represents the signs and symbols which are recommended by the InternationalOrganization for Standardization as well as other symbols that are used in the text.the set of positive integers and zero, {0, 1, 2, 3,...}the set of integers, {0, 1, 2, 3...}the set of positive integers, {1, 2, 3,...}the set of rational numbersthe set of positive rational numbers,the set of real numbersthe set of positive real numbersthe set of complex numbers, 16. x|x = ab---, b0, a, b xvz a complex numberz* the complex conjugate of z|z| the modulus of zarg z the argument of zRe z the real part of zIm z the imaginary part of zthe set with elementsn(A) the number of elements in the finite set Athe set of all x such that...is an element ofis not an element of the empty (null) setU the universal set union intersectionis a proper subset ofis a proper subset ofthe complement of set Athe Cartesian product of sets AB,a|b a divides ba to the power or the nth root of a{x|x , x0}+ {x|x , x0}C {a + bi|a, b }{x1, x2} x1, x2{x| }A'A B (A B = {(a, b)|a A, b B})a1 n / a n , 1n--- 17. MATHEMATICS Higher Level (Core)a1 2 / a , 12a to the power or the square root of a0the modulus or absolute value of xidentityis approximately equal tox x x 0,xx 0, 18. Sn u1 + u2 + u3 ++ unS u1 + u2 + u3 +uii = 1xvi--- is greater than is greater than or equal to is less than is less than or equal to[a, b] axb]a, b[ axbunis not greater thanis not less thanthe closed intervalthe open intervalthe nth term of a sequence or seriesd the common difference of an arithmetic sequencer the common ratio of an geometric sequencethe sum of the first n terms of a sequencethe sum to infinity of a sequence!# $ n!f is a function under which each element of set A has an image in set Bf is a function under which x is mapped to ythe image of x under the function fthe inverse function of the function fthe composite function of f and gthe limit of as x tends to athe derivative of y with respect to xthe derivative of with respect to xnu1 + u2 ++ unuin i = 1u1 u2 unnr-r--!--(---n---------r---)--!-f :A % Bf :x % yf (x)f 1(x)f oglim x % a f (x) f (x)dy-d---x--f'(x) f (x) 19. d2ydx -------2-f''(x) f (x)dnydxn --------f (n)(x) f (x)ydxydx athe second derivative of y with respect to xthe second derivative of with respect to xthe nth derivative of y with respect to xthe nth derivative of with respect to xthe indefinite integral of y with respect to xthe indefinite integral of y with respect to x between the limits x = a and x = bthe exponential function of xlogarithm to the base a of xthe natural logarithm of x,xviib exlogaxlnx logexsin,cos,tan the circular functionsthe inverse circular functionscsc,sec,cot the reciprocal circular functionsthe point A in the plane with Cartesian coordinates x and y[AB] the line segment with endpoints A and BAB the length of [AB](AB) the line containing points A and Bthe angle at Athe angle between the lines [CA] and [AB]the triangle whose vertices are A, B and Cv the vector vthe vector represented in magnitude and direction by the directed line segmentfrom A to Ba the position vectori,j,k unit vectors in the directions of the Cartesian coordinate axes|a| the magnitude of athe magnitude ofthe scalar product of v and wthe vector product of v and warcsinarccosarctan ' ' A(x, y)ACA B(ABCABOAAB ABv w ))vw 20. MATHEMATICS Higher Level (Core)the inverse of the non-singular matrix Athe transpose of the matrix AdetA the determinant of the square matrix AI the identity matrixP(A) probability of event AP(A) probability of the event not AP(A|B) probability of the event A given Bobservationsfrequencies with which the observations occurprobability distribution function P(X = x) of the discrete random variable Xprobability density function of the continuous random variable Xcumulative distribution function of the continuous random variable Xthe expected value of the random variable Xthe variance of the random variable Xk= -----------------n------------------ n f ik= ----------------n------------------ n f ixviii population meankpopulation variance, wherepopulation standard deviationsample meanksample variance, wherestandard deviation of the sampleunbiased estimate of the population variance orB(n,p) binomial distribution with parameters n and pPo(m) Poisson distribution with mean mnormal distribution with mean and variancek------------n---------1-------------X~B(n,p) the random variable X has a binomial distribution with parameters n and pX~Po(m) the random variable X has a Poisson distribution with mean mX~ the random variable X has a normal distribution with mean and variancecumulative distribution function of the standardised normal variable: N(0,1)A1ATx1, x2,f 1, f 2, x1, x2,Pxf (x)F(x)E(x)Var(x)*2 *2f i(xi )2i = 1i = 1=*xs2n *2f i(xi x)2i = 1i = 1=sns2n 1 s2n 1n= -n---------1--s2nf i(xi x)2i = 1N(, *2) *2N(, *2) *2, 21. -2-2calc -2calcxixv number of degrees of freedomthe chi-squared distributionthe chi-squared test statistic, whereAB the difference of the sets A and Bthe symmetric difference of the sets A and Ba complete graph with n verticesa complete bipartite graph with n vertices and another set of m verticesthe set of equivalence classes {0,1,2,...,p1} of integers modulo pgcd(a,b) the greatest common divisor of the integers a and blcm(a,b) the least common multiple of the integers a and bthe adjacency matrix of graph Gthe cost adjacency matrix of graph G( f o f e)2f e= ------------------------(AB = AB' = {x|x A and xB})A(B (A(B = (AB)(BA)).n.n, mpAGCG 22. MATHEMATICS Higher Level (Core)xx 23. Theory of Knowledge CHAPTER 11.1 PURE AND APPLIED MATHEMATICS1Mathematics has clearly played a significantpart in the development of many past andpresent civilisations.There is good evidence that mathematical,and probably astronomical techniques, wereused to build the many stone circles ofEurope which are thought to be at least threethousand years old (Thom). It is likely thatthe Egyptian pyramids and constructions onAztec and Mayan sites in South Americawere also built by mathematicallysophisticated architects. Similarly, culturesin China, India and throughout the MiddleEast developed mathematics a very long time ago. It is also the case that there have been verysuccessful cultures that have found little use for mathematics. Ancient Rome, handicapped, as itwas, by a non-place value number system did not develop a mathematical tradition at anythinglike the same level as that of Ancient Greece. Also, the Australian Aborigines who have one of themost long lasting and successful cultures in human history did not find much need formathematical methods. The same is true of the many aboriginal cultures of Africa, Asia and theAmericas. This may well be because these aboriginal cultures did not value ownership in the waythat western culture does and had no need to count their possessions. Instead, to aboriginalcultures, a responsible and sustainable relationship with the environment is more important thanacquisition and exploitation. Maybe we should learn from this before it is too late!Mathematics has developed two distinct branches. Pure mathematics, which is studied for its ownsake, and applied mathematics which is studied for its usefulness. This is not to say that the twobranches have not cross-fertilised each other, for there have been many examples in which theyhave.The pure mathematician Pierre de Fermat (1601-1665) guessed that the equationhas whole numbered solutions for n = 2 only. To the pure mathematician, this type of problem isinteresting for its own sake. To study it is to look for an essential truth, the majestic clockworkof the universe. Pure mathematicians see beauty and elegance in a neat proof. To puremathematicians, their subject is an art.Applied mathematics seeks to develop mathematical objects such as equations and computeralgorithms that can be used to predict what will happen if we follow a particular course of action.This is a very valuable capability. We no longer build bridges without making careful calculationsas to whether or not they will stand. Airline pilots are able to experience serious failures incommercial jets without either risking lives or the airlines valuable aeroplanes or, indeed,without even leaving the ground.CHAPTER 1xn + yn = zn 24. MATHEMATICS Higher Level (Core)1.2 AXIOMSMathematics is based on axioms. These are facts that are assumed to be true. An axiom is astatement that is accepted without proof. Early sets of axioms contained statements that appearedto be obviously true. Euclid postulated a number of these obvious axioms.2Example:Things equal to the same thing are equal to each other;That is, if y = a and x = a then y = x.Euclid was mainly interested in geometry and we still call plane geometry Euclidean. InEuclidean space, the shortest distance between two points is a straight line. We will see later thatit is possible to develop a useful, consistent mathematics that does not accept this axiom.Most axiom systems have been based on the notion of a set, meaning a collection of objects. Anexample of a set axiom is the axiom of specification. In crude terms, this says that if we have aset of objects and are looking at placing some condition or specification on this set, then the setthus specified must exist. We consider some examples of this axiom.Example:Assume that the set of citizens of China is defined. If we impose the condition that the membersof this set must be female, then this new set (of Chinese females) is defined.As a more mathematical example, if we assume that the set of whole numbers exists, then the setof even numbers (multiples of 2) must also exist.A second example of a set axiom is the axiom of powers:Example:For each set, there exists a collection of sets that contains amongst its elements all the subsets ofthe original set. If we look at the set of cats in Bogota, then there must be a set that contains all thefemale cats in Bogota, another that contains all the cats with green eyes in Bogota, another thatcontains all the Bogota cats with black tails, etc. A good, but theoretical, account of axiomatic settheory can be found in Halmos, 1960.Mathematics has, in some sense, been a search for the smallest possible set of consistent axioms.In the section on paradox, we will look further at the notion of axioms and the search for a set ofassumptions that does not lead to contradictions. There is a very strong sense in whichmathematics is an unusual pursuit in this respect. Pure mathematics is concerned with absolutetruth only in the sense of creating a self-consistent structure of thinking.As an example of some axioms that may not seem to be sensible, consider a geometry in whichthe shortest path between two points is the arc of a circle and all parallel lines meet. Theseaxioms do not seem to make sense in normal geometry. The first mathematicians to investigatenon-Euclidean geometry were the Russian, Nicolai Lobachevsky (1793-1856) and the Hungarian,Janos Bolyai (1802-1860). Independently, they developed self consistent geometries that did notinclude the so called parallel postulate which states that for every line AB and point C outside ABthere is only one line through C that does not meet AB. 25. Theory of Knowledge CHAPTER 1A BSince both lines extend to infinity in both directions, this seems to be obvious. Non-Euclideangeometries do not include this postulate and assume either that there are no lines through C thatdo not meet AB or that there is more than one such line. It was the great achievement ofLobachevsky and Bolyai that they proved that these assumptions lead to geometries that are selfconsistent and thus acceptable as true to pure mathematicians. In case you are thinking that thissort of activity is completely useless, one of the two non-Euclidean geometries discussed abovehas actually proved to be useful; the geometry of shapes drawn on a sphere. This is useful becauseit is the geometry used by the navigators of aeroplanes and ships.The first point about this geometry is that it is impossible to travel in straight lines. On the surfaceof a sphere, the shortest distance between two points is an arc of a circle centred at the centre ofthe sphere (a great circle). The shortest path from Rome to Djakarta is circular. If you want to seethis path on a geographers globe, take a length of sewing cotton and stretch it tightly between thetwo cities. The cotton will follow the approximate great circle route between the two cities.If we now think of the arcs of great circles as our straight lines, what kind of geometry will weget? You can see some of these results without going into any complex calculations. For example,what would a triangle look like?The first point is that the angles of this triangle add up to more than 180. There are many otherodd features of this geometry. However, fortunately for the international airline trade, thegeometry is self consistent and allows us to navigate safely around the surface of the globe. Thusnon-Euclidean geometry is an acceptable pure mathematical structure.While you are thinking about unusual geometries, what are the main features of the geometry ofshapes drawn on the saddle surface?3CRomeDjakarta 26. MATHEMATICS Higher Level (Core)One final point on the subject of non-Euclidean geometries; it seems to be the case that our threedimensional universe is also curved. This was one of the great insights of Albert Einstein (1879-1955). We do not yet know if our universe is bent back on itself rather like a sphere or whetheranother model is appropriate. A short account of non-Euclidean Geometries can be found inCameron (pp31-40).By contrast, applied mathematics is judged more by its ability to predict the future, than by itsself-consistency. Applied mathematics is also based on axioms, but these are judged more on theirability to lead to calculations that can predict eclipses, cyclones, whether or not a suspensionbridge will be able to support traffic loads, etc. In some cases such mathematical models can bevery complex and may not give very accurate predictions. Applied mathematics is about getting aprediction, evaluating it (seeing how well it predicts the future) and then improving the model.In summary, both branches of mathematics are based on axioms. These may or may not bedesigned to be realistic. What matters to the pure mathematician is that an axiom set should notlead to contradictions. The applied mathematician is looking for an axiom set and a mathematicalstructure built on these axioms that can be used to model the phenomena that we observe innature. As we have seen, useful axiom sets need not start out being sensible.The system of deduction that we use to build theother truths of mathematics is known as proof.ABCRS QT PProof has a very special meaning in mathematics. We use the word generally to mean proofbeyond reasonable doubt in situations such as law courts when we accept some doubt in averdict. For mathematicians, proof is an argument that has no doubt at all. When a new proof ispublished, it is scrutinised and criticised by other mathematicians and is accepted when it isestablished that every step in the argument is legitimate. Only when this has happened does aproof become accepted.Technically, every step in a proof rests on the axioms of the mathematics that is being used. As wehave seen, there is more than one set of axioms that could be chosen. The statements that weprove from the axioms are known as theorems. Once we have a theorem, it becomes a statementthat we accept as true and which can be used in the proof of other theorems. In this way we buildup a structure that constitutes a mathematics. The axioms are the foundations and the theoremsare the superstructure. In the previous section we made use of the idea of consistency. This meansthat it must not be possible to use our axiom set to prove two theorems that are contradictory.There are a variety of methods of proof available. This section will look at three of these in detail.We will mention others.4A' B' C'V P, Q, R are collinear1.3 PROOF 27. Theory of Knowledge CHAPTER 11.3.1 RULES OF INFERENCEAll proofs depend on rules of inference. Fundamental to these rules is the idea of implication.As an example, we can say that 2x = 4 (which is known as a proposition) implies that x = 2(provided that x is a normal real number and that we are talking about normal arithmetic).In mathematical shorthand we would write this statement as .This implication works both ways because x = 2 implies that 2x = 4 also.This is written as or the fact that the implication is both ways can be written asx = 22x = 4ab bc ac5.2x = 4x = 2x = 22x = 4The symbol is read as If and only if or simply as Iff, i.e., If with two fs.Not every implication works both ways in this manner:If x = 2 then we can conclude that x2 = 4.However, we cannot conclude the reverse:i.e., x2 = 4implies that x = 2 is false because x might be 2.Sothat x = 2x2 = 4is all that can be said in this case.There are four main rules of inference:ab1. The rule of detachment: from a is true and is true we can infer that b is true. a andb are propositions.Example: If the following propositions are true:It is raining.If it is raining, I will take an umbrella.We can infer that I will take an umbrella.2. The rule of syllogism: from is true and is true, we can conclude thatis true. a, bc are propositions.Example: If we accept as true that:abbcif x is an odd number then x is not divisible by 4 ( )and,if x is not divisible by 4 then x is not divisible by 16 ( )We can infer that the proposition;if x is an odd number then x is not divisible by 16 ( ) is true.ac 28. MATHEMATICS Higher Level (Core)3. The rule of equivalence: at any stage in an argument we can replace any statement by an6equivalent statement.Example: If x is a whole number, the statement x is even could be replaced by thestatement x is divisible by 2.4. The rule of substitution: If we have a true statement about all the elements of a set, thenthat statement is true about any individual member of the set.Example: If we accept that all lions have sharp teeth then Benji, who is a lion, musthave sharp teeth.Now that we have our rules of inference, we can look at some of the most commonly usedmethods of proof1.3.2 PROOF BY EXHAUSTIONThis method can be, as its name implies, exhausting! It depends on testing every possible case ofa theorem.Example:Consider the theorem: Every year must contain at least one Friday the thirteenth.There are a limited number of possibilities as the first day of every year must be a Monday or aTuesday or a Wednesday.... or a Sunday (7 possibilities). Taking the fact that the year might ormight not be a leap year (with 366 days) means that there are going to be fourteen possibilities.Once we have established all the possibilities, we would look at the calendar associated with eachand establish whether or not it has a Friday the thirteenth. If, for example, we are looking at anon-leap year in which January 1st is a Saturday, there will be a Friday the thirteenth in May.Take a look at all the possibilities (an electronic organiser helps!). Is the theorem true? 29. Theory of Knowledge CHAPTER 11.3.3 DIRECT PROOFThe following diagrams represent a proof of the theorem of Pythagoras described in The Ascentof Man (Bronowski pp 158-161). The theorem states that the area of a square drawn on thehypotenuse of a right angled triangle is equal to the sum of the areas of the squares drawn on thetwo shorter sides. The method is direct in the sense that it makes no assumptions at the start. Canyou follow the steps of this proof and draw the appropriate conclusion?7 30. MATHEMATICS Higher Level (Core)1.3.4 PROOF BY CONTRADICTIONThis method works by assuming that the proposition is false and then proving that thisassumption leads to a contradiction.Example:The number greatly interested classical Greek mathematicians who were unable to find anumber that, when it was squared, gave exactly 2.Modern students are often fooled into thinking that their calculatorsgive an exact square root for 2 as when 2 is entered and the squareroot button is pressed, a result (depending on the model ofcalculator) of 1.414213562 is produced. When this is squared,exactly 2 results. This is not because we have an exact square root. Itresults from the way in which the calculator is designed to calculatewith more figures than it actually displays. The first answer is storedto more figures than are shown, the result is rounded and then displayed. The same is true of thesecond result which only rounds to 2. Try squaring 1.414213562, the answer is not 2.The theorem we shall prove is that there is no fraction that when squared gives 2. This alsoimplies that there is no terminating or recurring decimal that, when squared, gives exactly 2, butthis further theorem requires more argument.The method begins by assuming that there is a fraction (pq are integers) which has beencancelled to its lowest terms, such that . From the assumption, the argument proceeds:As with most mathematical proofs, we have used simple axioms and theorems of arithmetic.The most complex theorem used is that if is even, then p is even. Can you prove this?The main proof continues with the deduction that if p is even there must be another integer, r, thatis half p.We now have our contradiction as we assumed that was in its lowest terms so pq cannotboth be even. This proves the result, because we have a contradiction.This theorem is a very strong statement of impossibility.There are very few other areas of knowledge in which we can make similar statements. We mightbe virtually certain that we will never travel faster than the speed of light but it would be a bravephysicist who would state with certainty that it is impossible. Other methods of proof includeproof by induction which is mainly used to prove theorems involving sequences of statements.Whilst on the subject of proof, it is worth noting that it is much easier to disprove a statement than82pq---p-q-- = 2p-q-- = 2 p2 -q---2- = 2p2 = 2q2p2 is evenp is evenp2p = 2rp2 = 4r22q2 = 4r2q2 = 2r2q2 is evenq is evenpq--- 31. Theory of Knowledge CHAPTER 1to prove it. When we succeed in disproving a statement, we have succeeded in proving itsnegation or reverse. To disprove a statement, all we need is a single example of a case in whichthe theorem does not hold. Such a case is known as a counter-example.Example:The theorem all prime numbers are odd is false. This can be established by noting that 2 is aneven prime and, therefore, is the only counter-example we need to give. By this method we haveproved the theorem that not every prime number is odd.This is another example of the way in which pure mathematicians think in a slightly different wayfrom other disciplines. Zoo-keepers (and indeed the rest of us) might be happy with the statementthat all giraffes have long necks and would not be very impressed with a pure mathematicianwho said that the statement was false because there was one giraffe (with a birth defect) who hasa very short neck. This goes back to the slightly different standards of proof that are required inmathematics.Counter-examples and proofs in mathematics may be difficult to find.Consider the theorem that every odd positive integer is the sum of a prime number and twice thesquare of an integer. Examples of this theorem that do work are:5 = 3 + 2 12, 15 = 13 + 2 12, 35 = 17 + 2 329.The theorem remains true for a very large number of cases and we do not arrive at a counter-exampleuntil 5777.Another similar theorem is known as the Goldbach Conjecture. Christian Goldbach (1690-1764) stated that every even number larger than 2 can be written as the sum of two primes. Forexample, 4 = 2 + 2, 10 = 3 + 7, 48 = 19 + 29etc. No-one has every found a counter-exampleto this simple conjecture and yet no accepted proof has ever been produced, despite the fact thatthe conjecture is not exactly recent!Finally, whilst considering proof, it would be a mistake to think that mathematics is a completeset of truths that has nothing which needs to be added. We have already seen that there areunproved theorems that we suspect to be true. It is also the case that new branches of mathematicsare emerging with a fair degree of regularity. During this course you will study linearprogramming which was developed in the 1940s to help solve the problems associated with thedistribution of limited resources. Recently, both pure and applied mathematics have beenenriched by the development of Chaos Theory. This has produced items of beauty such as theMandelbrot set and insights into the workings of nature. It seems, for example, that the results ofChaos Theory indicate that accurate long term weather forecasts will never be possible(Mandelbrot). 32. MATHEMATICS Higher Level (Core)1.4 PARADOX1.4.1 WHAT IS A PARADOX?Pure mathematics is a quest for a structure that does not contain internal contradictions. Asatisfactory mathematics will contain no nonsense.Consider the following proof:Try substituting x = 1 to check this line.Factorising using the difference of two squares.Dividing both sides by x 1.Let x = 1Then x2 1 = x 1(x + 1)(x 1) = x 1x + 1 = 12 = 1 Substituting x = 1.There is obviously something wrong here as this is the sort of inconsistency that we havediscussed earlier in this chapter, but what is wrong? To discover this, we must check each line ofthe argument for errors or faulty reasoning.Line 1 must be acceptable as we are entitled to assign a numerical value to a pronumeral.Line 2 is true because the left hand and right hand sides are the same if we substitute the10given value of the pronumeral.Line 3 is a simple factorisation of the left hand side.Line 4 is obtained from line 3 by dividing both sides of the equation by x 1 and shouldbe acceptable as we have done the same thing to both sides of the equation.Line 5 is obtained from line 4 by substituting x = 1 and so should give the correct answer.Obviously we have an unacceptable conclusion from a seemingly watertight argument. Theremust be something there that needs to be removed as an acceptable operation in mathematics.The unacceptable operation is dividing both sides by x 1 and then using a value of 1 for x. Whatwe have effectively done is divide by a quantity that is zero. It is this operation that has allowed usto prove that 2 = 1, an unacceptable result. When a paradox of this sort arises, we need to look atthe steps of the proof to see if there is a faulty step. If there is, then the faulty step must beremoved. In this case, we must add this rule to the allowed operations of mathematics:Never divide by a quantity that is, or will become, zero.This rule, often ignored by students, has important implications for Algebra and Calculus.Some paradoxes are arguments that seem to be sound but contain a hidden error and thus do notcontain serious implications for the structure of mathematical logic. An amusing compilation ofsimple paradoxes can be found in Gardner (1982). An example is the elevator paradox.Why does it always seem that when we are waiting for an elevator near the bottom of a tallbuilding and wanting to go up, the first elevator to arrive is always going down? Also, when wewant to go back down, why is the first elevator to arrive always going up? Is this a realphenomenon or is it just a subjective result of our impatience for the elevator to arrive? Or is itanother example of Murphys Law; whatever can go wrong will go wrong? 33. Theory of Knowledge CHAPTER 1This is quite a complex question, but a simple explanation might run as follows:If we are waiting near the bottom of a tall building, there are asmall number of floors below us from which elevators that aregoing up might come and then pass our floor.By contrast, there are more floors above us from which elevatorsmight come and then pass our floor going down.On the basis of this and assuming that the elevators are randomlydistributed amongst the floors, it is more likely that the nextelevator to pass will come from above and will, therefore, begoing down.By contrast, if we are waiting near the top of a tall building, thereare a small number of floors above us from which elevators thatare going down might come and then pass our floor.Also, there are more floors below us from which elevators mightcome and then pass our floor going up.It is more likely that the next elevator to pass will come frombelow and will, therefore, be going up.A fuller analysis of this paradox can be found in Gardner (pp96-97).The elevator paradox does not contain serious implication for the structure of mathematics likeour first example. We will conclude this section with a look at a modern paradox that did cause are-evaluation of one of the basic ideas of mathematics, the set.1.4.2 RUSSELLS PARADOXBertrand Russell (1872-1970) looked in detail at the basic set axioms of mathematics. We doregard the existence of sets as axiomatic in all mathematical structures. Does this mean that wecan make a set that contains everything? There would seem to be no difficulty with this as wejust move around the universe and sweep everything that we meet into our set, numbers, words,whales, motorcycles etc. and the result is the set that contains everything.Russell posed the following question which we will relate in the context of library catalogues.Every library has a catalogue. There are various forms thatthis catalogue might take; a book, a set of cards, a computerdisc etc. Whatever form the catalogue in your local librarytakes, there is a sense in which this catalogue is a book (orpublication) owned by the library and, as such, shouldappear as an entry in the catalogue:Of course, many librarians will decide that it is silly to include the catalogue as an entry in thecatalogue because people who are already looking at the catalogue know where to find it in the11CATALOGUE NEWEL LIBRARYCastle, The.F Kafka 231.72CatalogueAt receptionCatcher in the RyeJD Salinger 123.64Catherine the GreatA BiographyJ Nelson 217.42CatullusThe complete worksEdited by F Wills312.42 34. MATHEMATICS Higher Level (Core)library! It follows that library catalogues can be divided into two distinct groups:Catalogues that do contain an entry describing themselves.Catalogues that do not contain an entry describing themselves.Next, let us make a catalogue of all the catalogues of type two, those that do not containthemselves.This gives us a problem. Should we include an entry describing our new catalogue? If we do, thenour catalogue ceases to be a catalogue of all those catalogues that do not contain themselves. Ifwe do not, then our catalogue is no longer a complete catalogue of all those catalogues that do notcontain themselves.The conclusion is that making such a catalogue is impossible. This does not mean that the librarycatalogues themselves cannot exist. We have, however, defined an impossible catalogue.In set terms, Russells paradox says that sets are of two types:Type 1 Sets that do contain themselves.Type 2 Sets that do not contain themselves.The set of all sets of type 2 cannot be properly defined without reaching a contradiction.The most commonly accepted result of Russells paradox is the conclusion that we have to bevery careful when we talk about sets of everything. The most usual way out is to work within acarefully defined universal set, chosen to be appropriate to the mathematics that we areundertaking. If we are doing normal arithmetic, the universal set is the set of real numbers.1.5 MATHEMATICS AND OTHER DISCIPLINESWhen writing Theory of Knowledge essays, students are required to develop their arguments in across disciplinary way. For more details on this, you are strongly advised to read the taskspecifications and the assessment criteria that accompany the essay title. You are reminded that itis these statements that define what is expected of a good essay, not the contents of this Chapterwhich have been provided as a background resource. A good essay will only result if you developyour own ideas and examples in a clear and connected manner. Part of this process may includecomparing the mathematical method described earlier with the methods that are appropriate toother systems of knowledge.As we have seen, mathematics rests on sets of axioms. This is true of many other disciplines.There is a sense in which many ethical systems also have their axioms such as Thou shalt notkill.The Ancient Greeks believed that beauty and harmony are based, almost axiomatically, onmathematical proportions. The golden mean is found by dividing a line in the following ratio:A B CThe ratio of the length AB to the length BC is the same as the ratio of the length BC to the whole12 35. Theory of Knowledge CHAPTER 1length AC. The actual ratio is 1: or about 1:1.618. The Greek idea was that if this lineis converted into a rectangle, then the shape produced would be in perfect proportion:Likewise, the correct place to put the centre ofinterest in a picture is placed at the golden meanposition between the sides and also at the goldenmean between top and bottom. Take a look at theway in which television pictures are composed tosee if we still use this idea:In a similar way, the Ancient Greeks believed that ratio determined harmony in music. If twosimilar strings whose lengths bear a simple ratio such as 1:2 or 2:3 are plucked together theresulting sound will be pleasant (harmonious). If the ratio of string lengths is awkward, such as17:19, then the notes will be discordant. The same principle of simple ratios is used in tuningmusical instruments (in most cultures) today.The most common connection between mathematics and other disciplines is the use ofmathematics as a tool. Examples are: the use of statistics by insurance actuaries, probability byquality control officers and of almost all branches of mathematics by engineers. Every timemathematics is used in this way, there is an assumption that the calculations will be done usingtechniques that produce consistent and correct answers. It is here that pure mathematicaltechniques, applied mathematical modelling and other disciplines interface.In some of these examples, we apply very precise criteria to our calculations and are prepared toaccept only very low levels of error. Navigation satellite systems work by measuring the positionof a point on or above the Earth relative to the positions of satellites orbiting the Earth.Navigator1312---(1 + 5)A B CCentre of interestSatelliteSatellite 36. MATHEMATICS Higher Level (Core)This system will only work if the positions of the satellites are known with very great precision.By contrast, when calculations are made to forecast the weather, whilst they are done with asmuch precision as necessary, because the data is incomplete and the atmospheric models used areapproximate, the results of the calculations are, at best, only an indication of what might happen.Fortunately, most of us expect this and are much more tolerant of errors in weather forecastingthan we would be if airlines regularly failed to find their destinations!There are, therefore a large number of ways in which mathematics complements otherdisciplines. In fact, because computers are essentially mathematical devices and we areincreasingly dependent on them, it could be argued that mathematics and its methods underpinthe modern world.That is not to say that mathematics is everywhere. Many very successful people have managedto avoid the subject altogether. Great art, music and poetry has been produced by people forwhom mathematical ideas held little interest.In using mathematical ideas in essays, remember that you should produce original examples, lookat them in a mathematical context and then compare the ways in which the example might appearto a mathematician with the way in which the same example might appear to a thinker fromanother discipline.As a very simple example, what should we think of gambling?To the mathematician (Pascal was one of the first to look at this activity from the mathematicalperspective), a gambling game is a probability event. The outcome of a single spin of a roulettewheel is unknown. If we place a single bet, we can only know the chances of winning, notwhether or not we will win. Also, in the long run, we can expect to lose one thirtyseventh of anymoney that we bet every time we play. To the mathematician, (or at least to this mathematician)this rather removes the interest from the game!Other people look at gambling from a different standpoint. To the politician, a casino is a sourceof revenue and possibly a focus of some social problems. To a social scientist, the major concernmight be problem gamblers and the effect that gambling has on the fabric of society. A theologianmight look at the ethical issues as being paramount. Is it ethical to take money for a service suchas is provided by a casino? Many of these people might use mathematics in their investigations,but they are all bringing a slightly different view to the discussion.As we can see, there are many sides to this question as there are many sides to most questions.Mathematics can often illuminate these, but will seldom provide all the answers. When youchoose an essay title, you do not have to use mathematical ideas or a mathematical method todevelop your analysis. However, we hope that if you do choose to do this, you will find the briefsketch of the mathematical method described in this Chapter helpful.We will finish with one observation.Mathematics and mathematicians are sometimes viewed as dry and unimaginative. This may betrue in some cases, but definitely not all.We conclude with some remarks by the mathematician Charles Dodgson (1832-1898), otherwiseknown as Lewis Carroll:14 37. Theory of Knowledge CHAPTER 1The time has come, the Walrus said,To talk of many things:Of shoes and ships and sealing wax,Of cabbages and kings,Of why the sea is boiling hotAnd whether pigs have wings.Through the Looking GlassReferences:Megalithic Sites in Britain. Thom, A. (1967). U.K. Oxford University Press.Heritage Mathematics. Cameron, M. (1984).U.K. E.J. Arnold.The Ascent of Man, Bronowski, J. (1973).U.K. BBC.The Fractal Geometry of Nature, Mandelbrot, B (1977), U.S. W.H. FreemanCo. 1977.Gotcha!, Gardner, M. (1977). U.S.A. W.H. FreemanCo.1.6 THE EXTENDED ESSAYWe would like to encourage students to consider Mathematics as a choice of subject for theirextended essay.Whilst there is a requirement that these have solid academic content, it is not necessary to recordan original discovery to produce an excellent essay! That said, many of the great originaldiscoveries of mathematics are the work of comparatively young individuals with a modest levelof experience.An excellent example is Evariste Galois who struggled to enter university and whose life was cutshort by a duel in 1832 at age 21. Galois left a set of memoirs, many of which were written onthe night before the duel, that are regarded as some of the most original ideas ever contributed tomathematics.It is fashionable today to regard this sort of original thought as the preserve of experts. We assertthat it is not and encourage all our students to believe that they are capable of original ideas andhope that, if they do have a new idea, they have the courage to explore it.Students might choose to look at some of the many simply stated but as yet unproved conjecturesof mathematics:1. There are an infinite number of prime numbers. Pairs of primes such as 57, 1113that are separated by one even number are called twin primes.How many twin primes are there?2. The Goldbach conjecture: Every even natural number greater than 2 is equal to thesum of two prime numbers remains unproved.3. If an infinite number of canon-balls are stacked in an infinite pyramid, what is thebiggest proportion of the space they can fill?4. What are Mersenne primes and can you find a new one? The 25th26th Mersenneprimes were found by high school students.15... and a lot more! 38. MATHEMATICS Higher Level (Core)As a short case study, we will outline the work of a student who undertook a mathematical essay.The topic she chose was The Mathematics of Knots.To begin with, the student displayed an understanding of two of the major ways of thinkingthat are characteristic of mathematicians: EXISTENCE and CLASSIFICATION.EXISTENCE means developing tests for when a knot does or does not exist. It is neither possiblenor appropriate to explore all these ideas in an introduction such as this, but the essentials areillustrated below.Take a look at these two photographs of knots:These look similar, but if the ends are pulled apart, the results are quite different:The left hand arrangement was a tangle whereas the right was a knot.The student investigated and skillfully explained the tests that can be applied to such ropearrangements to determine if a knot EXISTS.The second feature of this essay was a look at the CLASSIFICATION of knots. These twophotographs show knots that have two different applications:Sheet bend BowlineThese knots have two different uses. The sheet bend is used to join two lengths of rope and isdesigned not to slip. The bowline (pronounced bo-lin) is the knot you tie around your waist ifyou are drowning and a rescuer throws you a rescue rope. It will not slip of or tighten around youand, irrespective of mathematics, is well worth knowing!These knots have different uses and belong to different CLASSES of knot. There are knotssimilar to the bowline that are intended to slip (such as the noose) that belong to yet anotherclass of knots. What are the classes of knots and what are their mathematical characteristics?16 39. Algebra of Linear and Quadratic Expressions CHAPTER 22.1 THE REAL NUMBER LINE2.1.1 THE REAL NUMBER LINEA visual method to represent the set of real numbers, , is to use a straight line. Thisgeometrical representation is known as the real number line. It is usually drawn as a horizontalstraight line extending out indefinitely in two directions with a point of reference known as theorigin, usually denoted by the letter O. Corresponding to every real number x there is a point P,on the line, representing this value. If x0, the point P lies to the right of O. If x0 the pointP lies to the left of O. If x = 0, the point P is at the origin, O.2.1.2 SET BUILDER NOTATIONThe set of points on the real number line can also be written in an algebraic form:This means that any real number set can be expressed algebraically. For example, the set ofpositive real numbers = ,negative real numbers =Note that .Similarly we can construct any subset of the real number line. The great thing about using setnotation is that we can quickly identify if a point on the number line is included or excluded in theset. How do we represent these inclusions and exclusions on the real number line?If the number is included in the set, you black-out a circle at thatpoint on the number line this is called a closed circle.For example, the set has the following representation:If the number is excluded from the set, you place a circle at thatpoint on the number line this is called an open circle.For example, the set has the following representation:2.1.3 INTERVAL NOTATIONAnother notation that is used to describe subsets of the real numbers is interval notation. Thisform of notation makes use of only square brackets or square brackets and round brackets toindicate if a number is included or excluded. For the examples above we have:= [3, ) or [3, [and for = (3, ) or ]3, [Notice that is never included!17CHAPTER 2O P. . . 6 5 4 3 2 1 0 1 2 3 4 5 6 . . . x= {x : x}+= {x : x0}= {x : x0}={0}+0 30 3xx{x : x3}{x : x3}{x : x3}{x : x3} 40. MATHEMATICS Higher Level (Core)It should be noted that can also be expressed as . Hence the reasonfor having in the interval notation representation.A summary of the different possible intervals is given in the table below:Real Number Line Set Notation Interval Notation Example18orororororororor ororor ororor ororor oror{x : x3} {x : 3 x}a ba bxx{x : a x b} x [a, b] 3 8 x{x : 3 x 8}x [3, 8]a ba bxx{x : a xb} x [a, b) x [a, b[ 3 8 x{x : 3 x8}x [3, 8)x [3, 8[a ba bxx{x : ax b} x (a, b] x ]a, b] 3 8 x{x : 3x 8}x (3, 8]x ]3, 8]a ba bxx{x : axb} x (a, b) x ]a, b[ 3 8 x{x : 3x8}x (3, 8)x ]3, 8[aa bxx{x : xa}{x : a x}x [a, ) x [a, [ 3 x{x : x3}x [3, )x [3, [aa bxx{x : xa}{x : ax}x (a, ) x ]a, [ 3 x{x : x3}x (3, )x ]3, [aaxx{x : x a}{x : x a}x (, a] x ], a] 8 x{x : x 8}x (, 8]x ], 8]aaxx{x : xa}{x : xa}x (, a) x ], a[ 8 x{x : x8}x (, 8)x ], 8[ 41. Algebra of Linear and Quadratic Expressions CHAPTER 2We also make the following point in relation to set notation. Rather than using the colon : inexpressions such as {x : x3} we can also use the separator |. That is, {x x3}. Eithernotation can be used.2.1.4 NUMBER SYSTEMSThe set of real numbers can be broken down into two subsets, namely, the set of rationalnumbers and the set of irrational numbers. The set of rational numbers can itself be brokendown into two sets, the set of integers and the set of fractions. The set of integers can then bebroken down into the set of positive integers, the set of negative integers and the set that includesthe number zero. Each of these sets can be represented by a different symbol.Real NumbersIrrational Numbers Rational NumbersFractions IntegersPositive Zero NegativeIntegers IntegersIn this book we will use the following notation and definitions:Set of positive integers and zero = = {0, 1, 2, 3, . . .}.The set of integers = = {0, 1, 2, 3, . . . }The set of positive integers = = {1, 2, 3, . . . } [Also known as Natural numbers]The set of rational numbers == ---, b0 and a and b are integers =++ {x x , x0}19Definition:+x x abThe set of positive rational numbers =The set of positive real numbers = =The empty set = = The set with no members.(a) Write each of the following using interval notation.(b) Represent the sets on the real number line.EXAMPLE 2.1i.ii.iii.{x 1x 4}{x x3}{x x6}{x : x8}{5} 42. MATHEMATICS Higher Level (Core)i. (a) = or using round bracket: (1, 4].ii. (a) = or [3, 6) using round bracket.iii. (a) = ], 8[{5} or (, 8){5} using round brackets.2.1.5 IRRATIONAL NUMBERSWe provided a definition of rational numbers earlier. The question then remains, what is anirrational number? The obvious answer is Whatever is not a rational number! So, what do thesenumbers look like? The best way to answer this is to say that it is a number that cannot bewritten in the form where a and b are integers. Examples of irrational numbers are20and so on.How do we know that is an irrational number?We can show this as follows a process known as reductio ad absurdum meaning to prove bycontradiction:Assume that is a rational number. Then by the definition of rational numbers there existintegers a and b (where a and b have no common factors) such thatUpon squaring we have:Then, must be even [because is even i.e., any number multiplied by 2 is even] and soa must also be even.This means we can express a as 2k. So, setting a = 2k we have:But, , so thatAnd so, must also be even, meaning that b is even.Then, since both a and b are even it follows that a and b have a common factor (i.e., 2).This is contrary to our original hypothesis. Therefore is not a rational number and musttherefore be an irrational number.One subset of irrational numbers is known as the set of surds. Surds can be expressed in the formwhere and . A commonly encountered surd is (i.e., the square root).Solution{x 1x 4} ]1, 4](b) 1 0 4 x{x x3}{x x6} [3, [],6[ = [3, 6[(b) 0 3 6 x{x : x8}{5}(b) 0 5 8 xab---, b02, 3, 222 ab= ---, b02 a2= -b---2-a2 = 2b2a2 2b2a2 = 4k2a2 = 2b2 2b2 = 4k2b2 = 2k2b22n a n + a + a 43. Algebra of Linear and Quadratic Expressions CHAPTER 2The laws of operations apply to surds in the same way that they apply to real numbers. Wesummarise some results involving surds:a b = abab------- ab= ---a2b = a b, a0a b c d = ac bd( a + b)( a b) = a bNotice that the last result shows that we obtain a rational number! The surds a bandare know as conjugate pairs. Whenever conjugate pairs are multiplied together they21produce a rational number.EXAMPLE 2.2(a) ====(b) ===-2----+---------2-- 1 + 2-2----+---------2-- 2 2(a) = [multiplying numerator and denominator by conjugate]==(b) == [cannot be simplified further]a + bExpand the following(a) ( 2 + 3)( 6 3) (b) ( 5 2 3)( 5 3)Solution( 2 + 3)( 6 3) 2 6 3 2 + 3 6 3 32 ( 2 3) 3 2 + 3 ( 3 2) 3 32 3 3 2 + 3 2 3 3 3( 5 2 3)( 5 3) 5 5 5 3 2 3 5 + 2 3 35 15 2 15 + 2 311 3 15Rationalise the denominator of (a) 1 (b)-2-------3---------1--EXAMPLE 2.3Solution1-2----+---------2-- 1 -2-------------2--2 2---4---------2----1 12 --- 2-2-------3---------1-- 2 3 + 11 + 2-2-------3---------1-- 1 + 2 -2-------3-----+----1-- 2 3 + 1 + 2 6 + 2= ---------------4---------3---------1-----------------2 3 + 1 + 2 6 + 2-----------------------1---1------------------------ 44. MATHEMATICS Higher Level (Core)2.1.6 THE ABSOLUTE VALUEThe absolute value or modulus of a number x, denoted by , is defined as follows:If and if .This means that the absolute value of any number will always be positive.E.g., : seeing as 40, we use the value of 4. Whereas, : bytaking the negative of a negative number we obtain a positive number.{x : x = 7}{x : x 3} {x : x1}(a) We are looking for value(s) of x such that when we take the absolute value of x it is 7.From the definition of the absolute value, we must have that x = 7 or x = 7.That is, . Therefore, the solution set is {7, 7}(b) i This time we want all values of x such that their absolute value is less than or equalto 3. For example, if x = 2.5 then which is less than 3. However, ifx = 4 then which is greater than 3. So we cannot have x = 4.Working along these lines we must have:Using interval notation we have = {x : 3 x 3} = [3, 3]ii. This time we want numbers for which their absolute value is greater than 1.For example,1 and . We then have:Using interval notation we have = ], 1[]1, [1. Show the following sets on the real number line.(a) (b) (c) {4}(d) ]2, 7]]4, 8[ (e) (, 4)[2, 5) (f)2. Write the following using interval notation.(a) (b)(c) (d)(e) (f)22xx0x = x x0x = x4 = 4 2 = (2) = 2(a) Find .(b) Use the number line to represent the setsi. ii.Express these sets using interval notation.EXAMPLE 2.4Solutionx = 7 x = 72.5 = 2.54 = 43 0 3 x{x : x 3}1.2 = 1.2 3.2 = 3.211 0 1 x{x : x1}EXERCISES 2.1{x 2 x 8} {x x7} {x 2x 6}{x : x6}{x 2 x 7} {x x9}{x 0x 5} {x : x 0}{x : x8}{x : x4} {x : x1}{x : x2} 45. Algebra of Linear and Quadratic Expressions CHAPTER 2---x 2{x : x4}{x : 2x12}233. Simplify the following.(a) (b) (c)4. Simplify the following.(a) (b)(c) (d)5. Rationalise the denominator in each of the following.(a) (b) (c)(d) (e) (f)+ --- x2 16. (a) If , find the value of i. ii. --- x2 1(b) If , find the value of i. ii.7. Find the value of x if(a) (b) (c)(d) (e) (f)8. Represent each of the following on the real number line.(a) (b)(c) (d)9. Write the following using interval notation.(a) (b) (c)10. Prove each of the following if .(a)(b)(c)(d)(e)--- ab11. Find the square root of the following.(a)(b)3 5 + 20 2 3 27 2 + 3 + 8 18( 5 + 1)( 5 1) (2 3 2)( 2 + 3)(3 2 6)( 3 + 3) (2 + 3 3)21-2----+---------3-- 3-----7---------2-- 3-----5---------2--2 5 + 1-------3---------2--- 2 + 3-----3-------------5-- 2 3-2-------5---------3--------2--x 5 3 + = x 1x+ -x---2-x 4 3 + = x 1x+ -x---2-{x x = 3} {x x = 10} {x x = 2}{x x + 1 = 3} {x x + 2 = 10} {x x 2 = 2}{x : x 5} {x x2}{x : 2 x5} {x : 2 x8}x x 1 0 { } x12a , bab = a bab= -----, b0a + b a + ba b a + ba b a b9 + 2 1812 2 32 46. MATHEMATICS Higher Level (Core)2.2 LINEAR ALGEBRA2.2.1 REVIEW OF LINEAR EQUATIONSA linear equation in the variable x (say) takes on the form where a, b and c are realconstants. The equation is linear because x is raised to the power of one. To solve such equationswe use the rules of transposition:Solving produces a solution that can be represented on the real number line.EXAMPLE 2.5(a)(b)(c) [dont forget to multiply the 3 and 2]Sometimes equations might not appear to be linear, but with some algebra, they form into linearequations. The following examples shows how this works.x 3-----2------ 1 = x -----2------ = 124(a)(b)ax + b = cax + b = cax = c b x c b= -----a------ax + b = cSolve the following linear equations(a) 4x + 5 = 21 (b) 9 2x = 7 (c) 3(5x 2) = 12Solution4x + 5 = 214x = 16 x = 49 2x = 72x = 2 x = 13(5x 2) = 1215x 6 = 1215x = 18x65= ---(a) Solve for x, .(b) Find .x x3--- 2 xEXAMPLE 2.6Solutionx 3-----2------ 1 x x 3=-----2------ = x + 1 x 3 = 2(x + 1) x 3 = 2x + 2x = 5x = 5 -----2------ 1 2xx3--- 2 x--6---- 3(2 x)= ---------6----------- = 12x 3(2 x) = 6 47. Algebra of Linear and Quadratic Expressions CHAPTER 2EXAMPLE 2.7(a) [taking b out as a common factor]25[dividing both sides by b](b)Linear Equations involving Absolute valuesRecall that if (where a0) then, . Using this result we can solve similarlinear equations.That is,orNotice that this time we have two solutions!(a) or(b)(c)2x 6 + 3x = 65x = 12 x 12= --5----Solve the following literal equations for x, where a and b are real constants.(a) bx b2 = ab (b) bx = a(b x)Solutionbx b2 = abbx = ab + b2bx = b(a + b) x = a + bbx = a(b x)bx = ab axbx + ax = ab(b + a)x = ab x ab= -b----+-----a--x = a x = a or aax + b = cax + b = c or ax + b = cax = c b or ax = c b x c b= -----a------ x (c + b)= -------a---------Solve the following.(a) (b) (c) 2x 6 = x 1 5 = 3 12 ---x = 2EXAMPLE 2.8Solution2x = 62x = 6 2x = 6 x = 3 or x = 3x 1 = 5 x 1 = 5 or x 1 = 5 x = 6 or x = 4---x 2 3 123 12---x 2 or 3 12= = ---x = 2 48. MATHEMATICS Higher Level (Core)---x 1 or 1212= ---x = 5 x = 2 or x = 10Examples such as those that follow require careful consideration of the restrictions placed on theabsolute values of the variables. We work through a number of such equations.Solve the following.EXAMPLE 2.9(a) x = 2x + 1 (b) x 1 = x (c) 3 x = x 1(a) By definition, if x0 and x if x0. Therefore we have two separate equationsx 1 ( ) if x 126to solve, one for x0 and one for x0.Case 1 ( x0):Now, our solution is that x = 1, however, we must also satisfy the condition that x0.As both statements do not agree with each other, we conclude that for x0 there is nosolution.Case 2 ( x0):This time our solution is that x = , and we must also satisfy the condition that x0.As both statements agree with each other, we conclude that for x0 there is a solution.Namely, .Therefore, has only one solution, .Notice that unlike Case 1, Case 2 has only one solution.(b) Using our definition for the absolute value, we have:1. if . i.e., if x1.2. if x 10. i.e., if x1.Notice we can combine these two expressions into one expression, namely:[this is known as a hybrid expression]x 1 ( ) if x 1Meaning that solving is the same as solving .Case 1 ( x1):.And so, there is no solution for the case when x1.Solutionx = xx = 2x + 1, x0x = 1, x0 x = 1, x0x = 2x + 1, x03x = 1, x0x13= ---, x013---x 13= ---x 2x 1 + = x 13= ---x 1 = x 1 x 10 x 1 = x 1x 1 = (x 1) x 1 = (x 1)x 1 =x 1 if x1x 1 = x x =x 1 if x1x 1 = x if x11 = 0 if x1 49. Algebra of Linear and Quadratic Expressions CHAPTER 2(x 1) = x if x1 x + 1 = x if x12x = 1 if x13 x ( ) if 3 x 027Case 2 ( x1):x12Then, as lies in the interval x1, it is a solution.So, .(c) This time we will reduce the amount of setting out:Case 1 (x 3): .3 x ( ) if 3 xAs x = 2 lies in the interval x 3, then it is a solution.Case 2 (x3): .For which there is no solution.Therefore,Solving Equations with the TI-83Equations such as those we have just looked at can also be solved using the solve( option on theTI83. We do this by calling up the Catalogue and then1. locating the solve( option2. enter the relevant equation[The equation must be entered in the form Equation = 0. So, to solve the equation2x + 6 = 15, we must rewrite it as 2x + 6 15 = 0 so that the equation that isentered into the TI83 is 2x + 6 15 = 0]3. indicate the variable we are solving for4. provide a reasonable guess (for the answer)To obtain the solve( option we use the following sequence:then use the arrow key to reach solve( and then press :We look at some of the problems we have aleady solved:= --- if x1x 12= ---x 1 x x12= = ---3 x x 1 3 x if 3 x0x 1 3 x if 3x= = = x 13 x = x 1, x 34 = 2x, x 3 x = 2, x 3(3 x) = x 1, x32 = 0, x33 x = x 1 x = 22nd 0 LN ENTER 50. MATHEMATICS Higher Level (Core)Solve the following linear equationsEXAMPLE 2.94x + 5 = 21 3(5x 2) = 12 x 3(a) (b) (c)(d)x = 2x + 1(a) (b) (c)--3---- 1228(d)-----2------ 1 = xAnother method is to use the Equation solver facility. The expression must still be entered inthe form Equation = 0. To call up the Equation solver screen1. press2. Enter the equation in the form Equation = 03. Move the cursor over the variable for which youwant to solve and then press .It is important that you become familiar with both modes of solving equations, althougheventually you will prefer one method over the other.1. Solve the following linear equations.(a) (b) (c)(d) (e) (f)2. Solve the following equations.--- = 2x 14(a) (b) (c)(d) (e) (f)SolutionNotice that in each case we have used a guess of 5.MATH 0ALPHA ENTEREXERCISES 2.2.12x 8 = 5x 3 12 = 2 13 ---x = 43 2x-------7-------- = 2 5x--- + 23+ --- = 15 x 1 ( ) 12 = 3 2 12---x ! 4= 2(2x + 1) = 12x 1 3 x = 3 5 1320 = 5 2x 3 ( ) 8 14---x ! x= ---x 51. Algebra of Linear and Quadratic Expressions CHAPTER 2------------ 8 + 1 u = u 1 4= --- x 2-----3------ + 1 1 x------------ 3 u3---------5----------- 2 (u + 1)= -y--------1-- 3(u + 1)-y--------1-- + 1 2x b = b 2 a(x b) = b + a ax = b(a x)--- b = 1a--- a xa-------b-------- 1 bx= -a---------x-- 1 ax-b---------x- b+ -------a-------- = 0 a2x 8 = 5x 3 12 = 2 13--- + 23--3---- 125 x 1 ( ) 12 = 3 2 12= 23---x ! 4a 2 x b = a 1b ---x = b 2ax b = 3b2x x 1 = 1 x 12= --- x x + 2 = 3x 1 ---x = 2x + 1 1 + 2x = 2 x 3x + 4 = 3 xx + a = 2x + a, a0 2x a = 2a x, a02x a + x a a 0, = 12x + 1 + x 1 = 3 x + 1 x 1 = 2293. Solve the following equations.2 u 6(a) (b) (c)(d) (e) (f)= -x----+-----1-- 15-x----+-----1-- + 2 14. Solve the following equations for x.(a) (b) (c)(d) (e) (f)(g) (h) (i)xa--- a b = xbb x-a----+-----x-- b + x5. Solve for x.(a) (b) (c)(d) -------3 7-------- 2x= 2 (e) 5x(f)(g) (h) (i)(j) (k) (l)6. Solve the following equations.--- = 2x 14(a) (b) (c)(d) (e) (f)1 137. Solve the following for x.= -----4--------- 1x--- + 1b ---x = 4+ --- = 1---x 1 +! 1(a) (b)(c) (d) , a08. Solve the following for x.(a) (b)= --------5---------= ---= -a---------x-=---x + a = x a9. Make use of your graphics calculator to solve a selection of equations from questions 1 to8.2.2.2 LINEAR INEQUATIONSInequalities are solved in the same way as equalities, with the exception that when both sides aremulitplied or divided by a negative number, the direction of the inequality sign reverses.EXAMPLE 2.10 Find (a) {x : x + 14} (b) {x 2x 51} 52. MATHEMATICS Higher Level (Core)-------7-------- 4x 3 -------2-------- E XAMPLE 2.1130(a) .Therefore, the solution set (s.s) is .(b)Therefore, the solution set (s.s) is .(a)Therefore, s.s. =(b) [multiply both sides by 14][notice the reversal of the inequality as we dividedby a negative number]32 ------ Therefore, s.s. is .When dealing with inequalities that involve absolute values, we need to keep in mind thefollowing:Solutionx + 14 x3{x : x3}2x 512x6 x3{x : x3}Find (a) {x : x + 23 2x} (b) x : 3 2xSolutionx + 23 2x3x + 233x1x 13---x : x 13--- 3 2x-------7-------- 4x 3-------2-------- 14 3 2x!-------7--------14 4x 3 !-------2-------- 2(3 2x) 7(4x 3)6 4x 28x 2132x 27x 27-3---2--x : x 271.2.xaaxaxa xa or xaEXAMPLE 2.12 Find (a) {x : x + 14} (b) {x 2x 5 1} 53. Algebra of Linear and Quadratic Expressions CHAPTER 2---x 3 x--- x------x : x 1331(a)[subtracting 1 from both inequalities]Therefore, s.s. is .(b)[adding 5 to both sides of inequality][dividing both sides by 2]---x 3 1 12(a) ororor [Note the reversal of inequality sign, i.e., by 2]Therefore, s.s. is .(b)orororTherefore, s.s. is .As with equalities, we need to consider the conditions of the absolute value.For x0 we have: .As both conditions are satisfied, i.e., x0 and , one s.s. is .For x0 we have: .However, this time the conditions do not agree with each other, i.e., we cannot have thesolution x1 with the restriction that x0.Therefore, the only s.s. is .Solutionx + 144x + 145x3{x : 5x3}2x 5 11 2x 5 14 2x 62 x 3Find (a) (b) x : 1 12{ 3x 2 15}EXAMPLE 2.13Solution1 12---x 3 1 12 ---x312---x 212---x4 x4 x8{x : x4}{x : x8}3x 2 15 3x 263x 26 3x 2 63x8 3x 4x 83--- x 43 ---x : x 43: x 83EXAMPLE 2.14 Find {x : x1 2x}Solutionx 1 2x 3x 1 x 13 ---x 13--- x1 2x x1x : x --- 13 54. MATHEMATICS Higher Level (Core)-----3------2x 1 x 1 x 3 +22x + 1x 3 x 4-----5------ 2 xx3(x + 4) x 4 -----2------ 1 3x5x 2--- 2 3x+ -------3--------2------------ 1 x+ -----4------ 1 x5 -----3------3 1 x +2a(x + 1)2a, a0 a x+ -a---2- ba0 x x 1--- + 34--- --- 9 3x 12--- 5 2 x42x 1 45 2x 21 x2x2+ ---34x12x45 3 6 4x + 110 12 4 x22 ---9 3x + ------ 3 3x--2---- 7 p 3---x + 1x 3 x2x 2x 1x + 1ax x a x ax 1 +xa32EXERCISES 2.2.21. Solve the following inequalities.(a) (b) (c)(d) (e) (f)2. Solve the following inequalities.2x + 1-------5-------- 2 x(a) (b) (c)3. Solve the following inequalities.+-----------------2------ + 1a, a0(a) (b)(c) , (d) , a0-a---2- 4x--- bxa--a---- b4. Solve the following inequalities.-a----+-----1-- x + 1+-a----+-----1-- ax(a) |4x + 2| 6 (b) |2x 1| 5 (c) |4x 2| 8(d) |4x + 2| 0 (e) |x 1| 8 (f) |3x + 3| 12(g) (h) (i)3 x25. Solve the following inequalities.(a) (b) (c)(d) 3 13(e) (f)(g) (h) (i)6. For what value(s) of p does have no solutions?7. Solve the following inequalities.12(a) (b) (c)8. Solve the following inequalities where 0a1. ---7 ---5(a) (b) (c) ---x 1x 9. Find (a) (b) .---x + ax : 4 x 2x { } x : 13 55. Algebra of Linear and Quadratic Expressions CHAPTER 22.3 LINEAR FUNCTIONS2.3.1 GRAPH OF THE LINEAR FUNCTIONThe study of functions and relations is dealt with in detail in Chapter 5, however, we give a basicdefinition of the term function at this point.A function is an algebraic expression that will generate only one valueof y for any one value of x.For example, if x = 5, then will only generate one value of y. Consider the functionthen, . i.e., only one yvalue has been generated.There are three possible outcomes when a linear function is graphed:Case 1: m0 Case 2: m0 Case 3: m = 0c c cHowever, sometimes the linear function is expressed in different forms. The reason is that it issometimes more convenient to express it in a form other than the standard form. For example, thefollowing linear functions are all the same:, , ,They have simply been rewritten into different formats.(a) The function represents a straight linewith a gradient of 2 and a yintercept of 1.i.e., the graph will cut the yaxis at the point (0, 1).The xintercept is obtained by solving :33.y = f (x)f (5)f (x) = x + 3 f (5) = 5 + 3 = 8A linear function has the formy = f (x) = mx + cIts graph is a straight line such that1. m is the slope of the line.2. c is the yintercept (i.e., where the line cuts the yaxis).y y yO x O x O xf (x) = 5x 2 y = 5x 2 y + 2 = 5x 5x + y + 2 = 0Sketch the graphs of the following linear functions.(a) (b) (c) f x ( ) 2x 1 + = y 3 12= ---x 4x 2y = 5EXAMPLE 2.15y10.5xSolutionf (x) = 2x + 1f (x) = 02x + 1 = 02x = 1 x = 0.5 56. MATHEMATICS Higher Level (Core)y 3 12= ---x(b) The function represents a straight line12with a gradient of and a yintercept of 3.i.e., the graph will cut the yaxis at the point (0, 3).The xintercept is obtained by solving :y = 04x 2y = 54x 2y 5 2y4x 5 y 2x 52= = = ---y = 0f (x) = kSolving for x is equivalent tofinding the xvalue where the graphs of34.---3 12---x 0 12= ---x = 3 x = 6(c) We first rewrite the equation :.So, we have a straight line with gradient 2 andyintercept at 2.5.The xintercept is obtained by solving .Using the original equation we haveyx3O 6yx-2.51.25O4x 2 0 ( ) 5 x54= = --- = 1.25Geometrical interpretation of solving equationsIt is interesting to note that although we have been solving equations like 2x + 1 = 7, we hadnot provided (apart from using the real number line) a geometrical representation corresponding tothe equation. We can now provide such an interpretation.Solving 2x + 1 = 7for x, is the same as consideringy = 2x + 1the function f (x) y= 2x + 1and then asking the question:7y = 7When will the straight line y = 2x + 1have a yvalue of 7.Or,When will the straight line y = 2x + 1meet the straight line y = 7.O ?xIn general we have the following geometrical interpretation for any equation.and y = k intersect.y = f (x)y = kyy = f (x)xa bNote that although with linear equations there will only be one solution, for non-linear functionsthere may very well be more than one solution.We complete this section by providing a summary of other properties of straight lines: 57. Algebra of Linear and Quadratic Expressions CHAPTER 2= -x---2--------x---1- m Rise= ( ) m x x1l1 m1l2 m2 m1 = m2l1 // l2 iff m1 = m2l1 m1l2 m2 m1 m2 = 1l1 $ l2 iff m1 m2 1 or m1 1= = ------Find the equation of the line that passes through the point (1,3) and isEXAMPLE 2.16parallel to the line with equation 2x y + 7 = 0 .35Properties of straight lines1. Gradient of a lineThe gradient, m, of the line through two pointsand is given by or(x1, y1)(x2, y2)Rise = y2 y1Run = x2 x1xy(x1, y1)(x2, y2) m y2 y1= -R----u---n--From this we can obtain the pointgradient form of a line.That is, if (x, y) is any point on a straight line having a gradient m, and is another fixedpoint on that line then the equation of that line is given by2. Parallel linesy y1The straight line with gradient is parallel to thestraight line with gradient if and only if .That is,Notice that if the two lines are parallel, they also make equal angleswith the xaxis.3. Perpendicular linesThe straight line with gradient is perpendicular to thestraight line with gradient if and only if .That is, .(x1, y1)l2xyl1m1 = m2# #y l1xl290m2The gradient of the line is found by rearranging to the form to get:. The gradient is 2 and so all the lines parallel to this will also have gradient 2. Theequation of the required line is . The value of the constant c can be found by using thefact that the line passes through the point (1,3).That is,Therefore the equation of the straight line is .Solution2x y + 7 = 0 y = mx + cy = 2x + 7y = 2x + cy = 2x + c3 = 2 1 + cc = 5y = 2x + 5Find the equation of the line which passes through the point (1,4) andEXAMPLE 2.17which is perpendicular to the line with equation 2x + 5y + 2 = 0 . 58. MATHEMATICS Higher Level (Core)The gradient form of is .So the gradient is . The gradient of all lines perpendicular to this line is found using the factthat the product of the gradients of perpendicular lines is 1: .Then, the equation of the line is . The constant c is found in the same way as theprevious example: Using the point (1, 4) we have .Therefore the equation of the straight line is .1. Sketch the graph of the following straight lines.(i) (ii) (iii)(iv) (v) (vi)(vii)