Foundation Maths Seventh Edition PDF Textbook

Anthony Croft and Robert Davison Foundation Maths SEVENTH EDITION Foundation Maths  At Pearson, we have a simple mission: to help people make more of their lives through learning. We combine innovative learning technology with trusted content and educational expertise...

Anthony Croft and Robert Davison Foundation Maths SEVENTH EDITION Foundation Maths  At Pearson, we have a simple mission: to help people make more of their lives through learning. We combine innovative learning technology with trusted content and educational expertise to provide engaging and effective learning experiences that serve people wherever and whenever they are learning. From classroom to boardroom, our curriculum materials, digital learning tools and testing programmes help to educate millions of people worldwide – more than any other private enterprise. Every day our work helps learning flourish, and wherever learning flourishes, so do people. To learn more, please visit us at www.pearson.com/uk Foundation Maths Seventh edition Anthony Croft Loughborough University Robert Davison Harlow, England London New York Boston San Francisco Toronto Sydney Dubai Singapore Hong Kong Tokyo Seoul Taipei New Delhi Cape Town São Paulo Mexico City Madrid Amsterdam Munich Paris Milan PEARSON EDUCATION LIMITED KAO Two KAO Park Harlow CM17 9SR United Kingdom Tel: +44 (0)1279 623623 Web: www.pearson.com/uk First published 1995 (print) Third edition published 2003 (print) Fourth edition 2006 (print) Fifth edition 2010 (print) Sixth edition published 2016 (print and electronic) Seventh edition published 2020 (print and electronic) © Pearson Education Limited 1995, 2003, 2006, 2010 (print) © Pearson Education Limited 2016, 2020 (print and electronic) The rights of Anthony Croft and Robert Davison to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. 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ISBN: 978-1-292-28968-7 (print) 978-1-292-28973-1 (PDF) 978-1-292-28969-4 (ePub) British Library Cataloguing-in-Publication Data A catalogue record for the print edition is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for the print edition is available from the Library of Congress 10 9 8 7 6 5 4 3 2 1 24 23 22 21 20 Front cover image: © Shutterstock Premier / GrAI Print edition typeset in 10/12.5 pt Times LT pro by SPi Global Print edition printed and bound in Slovakia by Neografia NOTE THAT ANY PAGE CROSS REFERENCES REFER TO THE PRINT EDITION  Brief contents Preface xv Publisher’s acknowledgements xvii List of videos xix Mathematical symbols xxi Using mathematical and statistical computer software and apps in Foundation Maths xxiii 1 Arithmetic of whole numbers 1 2 Fractions 15 3 Decimal numbers 26 4 Percentage and ratio 34 5 Algebra 45 6 Indices 55 7 Simplifying algebraic expressions 71 8 Factorisation 80 9 Algebraic fractions 90 10 Transposing formulae 112 11 Solving equations 118 12 Sequences and series 130 13 Sets 144 14 Number bases 157 15 Elementary logic 168 16 Functions 179 17 Graphs of functions 191 18 The straight line 214 19 The exponential function 227 20 The logarithm function 236 21 Measurement 254 22 Introduction to trigonometry 275 23 The trigonometrical functions and their graphs 283 24 Trigonometrical identities and equations 301 25 Solution of triangles 314 26 Vectors 331 27 Matrices 348 28 Complex numbers 364 29 Tables and charts 383 vi Brief contents 30 Statistics 404 31 Probability 419 32 Correlation 428 33 Regression 444 34 Gradients of curves 451 35 Techniques of differentiation 468 36 Integration and areas under curves 479 37 Techniques of integration 496 38 Functions of more than one variable and partial differentiation 512 Solutions 529 Index 607 Supporting resources Visit go.pearson.com/uk/he/resources to find valuable online resources  Contents Preface xv Publisher’s acknowledgements xvii List of videos xix Mathematical symbols xxi Using mathematical and statistical computer software and apps in Foundation Maths xxiii 1 Arithmetic of whole numbers 1 1.1 Addition, subtraction, multiplication and division 1 1.2 The BODMAS rule 5 1.3 Prime numbers and factorisation 7 1.4 Highest common factor and lowest common multiple 10 Test and assignment exercises 1 14 2 Fractions 15 2.1 Introduction 15 2.2 Expressing a fraction in equivalent forms 16 2.3 Addition and subtraction of fractions 19 2.4 Multiplication of fractions 22 2.5 Division by a fraction 24 Test and assignment exercises 2 25 3 Decimal numbers 26 3.1 Decimal numbers 26 3.2 Significant figures and decimal places 30 Test and assignment exercises 3 33 4 Percentage and ratio 34 4.1 Percentage 34 4.2 Ratio 40 Test and assignment exercises 4 43 5 Algebra 45 5.1 What is algebra? 45 5.2 Powers or indices 47 5.3 Substitution and formulae 51 Test and assignment exercises 5 54 6 Indices 55 6.1 The laws of indices 55 6.2 Negative powers 59 6.3 Square roots, cube roots and fractional powers 62 viii Contents 6.4 Multiplication and division by powers of 10 66 6.5 Scientific notation 67 Challenge Exercise 6 69 Test and assignment exercises 6 69 7 Simplifying algebraic expressions 71 7.1 Addition and subtraction of like terms 71 7.2 Multiplying algebraic expressions and removing brackets 72 7.3 Removing brackets from a1b + c2, a1b - c2, 1a + b21c + d2 and 1a + b2 1c - d2 74 Challenge Exercise 7 78 Test and assignment exercises 7 78 8 Factorisation 80 8.1 Factors and common factors 80 8.2 Factorising quadratic expressions 82 8.3 Difference of two squares 88 Challenge Exercise 8 89 Test and assignment exercises 8 89 9 Algebraic fractions 90 9.1 Introduction 90 9.2 Cancelling common factors 91 9.3 Multiplication and division of algebraic fractions 96 9.4 Addition and subtraction of algebraic fractions 100 9.5 Partial fractions 103 Challenge Exercise 9 110 Test and assignment exercises 9 110 10 Transposing formulae 112 10.1 Rearranging a formula 112 Challenge Exercise 10 116 Test and assignment exercises 10 117 11 Solving equations 118 11.1 Solving linear equations 118 11.2 Solving simultaneous equations 121 11.3 Solving quadratic equations 122 Challenge Exercises 11 128 Test and assignment exercises 11 128 12 Sequences and series 130 12.1 Sequences 130 12.2 Arithmetic progressions 132 12.3 Geometric progressions 134 12.4 Infinite sequences 136 12.5 Series and sigma notation 137 12.6 Arithmetic series 140 12.7 Geometric series 141 12.8 Infinite geometric series 142 Challenge Exercises 12 143 Test and assignment exercises 12 143  Contents ix 13 Sets 144 13.1 Terminology 144 13.2 Sets defined mathematically 147 13.3 Venn diagrams 150 13.4 Number sets 153 Challenge Exercise 13 154 Test and assignment exercises 13 156 14 Number bases 157 14.1 The decimal system 157 14.2 The binary system 158 14.3 Octal system 161 14.4 Hexadecimal system 164 Challenge Exercise 14 167 Test and assignment exercises 14 167 15 Elementary logic 168 15.1 Logic and propositions 168 15.2 Symbolic logic 169 15.3 Truth tables 172 Test and assignment exercises 15 178 16 Functions 179 16.1 Definition of a function 179 16.2 Notation used for functions 180 16.3 Composite functions 185 16.4 The inverse of a function 187 Challenge Exercise 16 189 Test and assignment exercises 16 189 17 Graphs of functions 191 17.1 The x–y plane 191 17.2 Inequalities and intervals 193 17.3 Plotting the graph of a function 196 17.4 The domain and range of a function 200 17.5 Solving equations using graphs 204 17.6 Solving simultaneous equations graphically 207 Challenge Exercises 17 212 Test and assignment exercises 17 212 18 The straight line 214 18.1 Straight line graphs 214 18.2 Finding the equation of a straight line from its graph 218 18.3 Gradients of tangents to curves 222 Challenge Exercise 18 226 Test and assignment exercises 18 226 19 The exponential function 227 19.1 Exponential expressions 227 19.2 The exponential function and its graph 230 19.3 Solving equations involving exponential terms using a graphical method 233 x Contents Challenge Exercises 19 234 Test and assignment exercises 19 235 20 The logarithm function 236 20.1 Introducing logarithms 236 20.2 Calculating logarithms to any base 239 20.3 Laws of logarithms 240 20.4 Solving equations with logarithms 247 20.5 Properties and graph of the logarithm function 251 Challenge Exercises 20 252 Test and assignment exercises 20 253 21 Measurement 254 21.1 Introduction to measurement 254 21.2 Units of length 256 21.3 Area and volume 257 21.4 Measuring angles in degrees and radians 260 21.5 Areas of common shapes and volumes of common solids 265 21.6 Units of mass 269 21.7 Units of time 270 21.8 Dimensional analysis 270 Challenge Exercise 21 272 Test and assignment exercises 21 272 22 Introduction to trigonometry 275 22.1 The trigonometrical ratios 275 22.2 Finding an angle given one of its trigonometrical ratios 279 Challenge Exercise 22 281 Test and assignment exercises 22 282 23 The trigonometrical functions and their graphs 283 23.1 Extended definition of the trigonometrical ratios 283 23.2 Trigonometrical functions and their graphs 290 Challenge Exercise 23 299 Test and assignment exercises 23 299 24 Trigonometrical identities and equations 301 24.1 Trigonometrical identities 301 24.2 Solutions of trigonometrical equations 307 Challenge Exercises 24 311 Test and assignment exercises 24 312 25 Solution of triangles 314 25.1 Types of triangle 314 25.2 Pythagoras’ theorem 317 25.3 Solution of right-angled triangles 319 25.4 The sine rule 322 25.5 The cosine rule 327 Challenge Exercises 25 329 Test and assignment exercises 25 330  Contents xi 26 Vectors 331 26.1 Introduction to vectors and scalars 331 26.2 Multiplying a vector by a scalar 332 26.3 Adding and subtracting vectors 333 26.4 Representing vectors using Cartesian components 336 26.5 The scalar product 342 Challenge Exercise 26 345 Test and assignment exercises 26 345 27 Matrices 348 27.1 What is a matrix? 348 27.2 Addition, subtraction and multiplication of matrices 350 27.3 The inverse of a 2 * 2 matrix 356 27.4 Application of matrices to solving simultaneous equations 360 Challenge Exercises 27 362 Test and assignment exercises 27 362 28 Complex numbers 364 28.1 Introduction to complex numbers 364 28.2 Real and imaginary parts of a complex number 365 28.3 Addition, subtraction, multiplication and division of complex numbers 367 28.4 Representing complex numbers graphically – the Argand diagram 372 28.5 Modulus, argument and the polar form of a complex number 373 28.6 The exponential form of a complex number 378 Challenge Exercises 28 382 Test and assignment exercises 28 382 29 Tables and charts 383 29.1 Introduction to data 383 29.2 Frequency tables and distributions 385 29.3 Bar charts, pie charts, pictograms and histograms 391 Test and assignment exercises 29 401 30 Statistics 404 30.1 Introduction 404 30.2 Averages: the mean, median and mode 404 30.3 The variance and standard deviation 411 Challenge Exercises 30 416 Test and assignment exercises 30 417 31 Probability 419 31.1 Introduction 419 31.2 Calculating theoretical probabilities 420 31.3 Calculating experimental probabilities 423 31.4 Independent events 424 Challenge Exercise 31 426 Test and assignment exercises 31 426 32 Correlation 428 32.1 Introduction 428 32.2 Scatter diagrams 429 32.3 Correlation coefficient 433 xii Contents 32.4 Spearman's coefficient of rank correlation 439 Challenge Exercise 32 442 Test and assignment exercises 32 442 33 Regression 444 33.1 Introduction 444 33.2 The regression equation 445 Test and assignment exercises 33 450 34 Gradients of curves 451 34.1 The gradient function 451 34.2 Gradient function of y = x n 452 34.3 Some rules for finding gradient functions 457 34.4 Higher derivatives 459 34.5 Finding maximum and minimum points of a curve 460 Challenge Exercise 34 467 Test and assignment exercises 34 467 35 Techniques of differentiation 468 35.1 Introduction 468 35.2 The product rule 469 35.3 The quotient rule 471 35.4 The chain rule 474 Challenge Exercise 35 477 Test and assignment exercises 35 478 36 Integration and areas under curves 479 36.1 Introduction 479 36.2 Indefinite integration: the reverse of differentiation 480 36.3 Some rules for finding other indefinite integrals 483 36.4 Definite integrals 487 36.5 Areas under curves 490 Challenge Exercise 36 494 Test and assignment exercises 36 494 37 Techniques of integration 496 37.1 Products of functions 496 37.2 Integrating products of functions 497 37.3 Definite integrals 501 37.4 Integration by substitution 503 37.5 Integration by partial fractions 507 Challenge Exercise 37 509 Test and assignment exercises 37 510 38 Functions of more than one variable and partial differentiation 512 38.1 Functions of two independent variables 512 38.2 Representing a function of two independent variables graphically 515 38.3 Partial differentiation 518 38.4 Partial derivatives requiring the product or quotient rules 523  Contents xiii 38.5 Higher-order derivatives 524 38.6 Functions of several variables 526 Challenge Exercise 38 527 Test and assignment exercises 38 528 Solutions 529 Index 607 Preface Today, a huge variety of disciplines require their students to have knowledge of certain mathematical tools in order to appreciate the quantitative aspects of their subjects. At the same time, higher education institutions have widened access so that there is much greater variety in the pre-university mathematical experiences of the student body. Some students are returning to education after many years in the workplace or at home bringing up families. Foundation Maths has been written for those students in higher education who have not specialised in mathematics at A or AS level. It is intended for non-specialists who need some but not a great deal of mathematics as they embark upon their courses of higher education. It is likely to be especially use- ful to those students embarking upon a Foundation Degree with mathematical content. It takes students from around the lower levels of GCSE to a standard which will enable them to participate fully in a degree or diploma course. It is ideally suited for those studying marketing, business studies, management, sci- ence, engineering, social science, geography, computer science combined stud- ies and design. It will be useful for those who lack confidence and need careful, steady guidance in mathematical methods. Even for those whose mathematical expertise is already established, the book will be a helpful revision and refer- ence guide. The style of the book also makes it suitable for those who wish to engage in self-study or distance learning. We have tried throughout to adopt an informal, user-friendly approach and have described mathematical processes in everyday language. Mathemati- cal ideas are usually developed by example rather than by formal proof. This reflects our experience that students learn better from examples than from abstract development. Where appropriate, the examples contain a great deal of detail so that the student is not left wondering how one stage of a calculation leads to the next. In Foundation Maths, objectives are clearly stated at the begin- ning of each chapter, and key points and formulae are highlighted throughout the book. Self-assessment questions are provided at the end of most sections. These test understanding of important features in the section and answers are given at the back of the book. These are followed by exercises; it is essential that these are attempted as the only way to develop competence and understanding is through practice. Solutions to these exercises are given at the back of the book and should be consulted only after the exercises have been attempted. We have included in many of the chapters a number of challenge exercises. These exercises are intentionally demanding and require a considerable depth xvi Preface of understanding. Solutions to these exercises can be found at go.pearson.com/uk/he/resources. A further set of test and assignment exercises is given at the end of each chapter. These are provided so that the tutor can set regular assignments or tests throughout the course. Solutions to these are not provided. Feedback from students who have used earlier editions of this book indicates that they have found the style and pace of the book helpful in their study of mathematics at university. In order to keep the size of the book reasonable we have endeavoured to include topics which we think are most important, cause the most problems for students, and have the widest applicability. We have started the book with mate- rials on arithmetic including whole numbers, fractions and decimals. This is fol- lowed by several chapters which gradually introduce important and commonly used topics in algebra. There follows chapters on sets, number bases and logic, collectively known as discrete mathematics. The remaining chapters introduce functions, trigonometry, vectors, matrices, complex numbers, statistics, proba- bility and calculus. These will be found useful in the courses previously listed. The best strategy for those using the book would be to read through each section, carefully studying all of the worked examples and solutions. Many of these solutions develop important results needed later in the book. It is then a good idea to cover up the solution and try to work the example again inde- pendently. It is only by doing the calculation that the necessary techniques will be mastered. At the end of each section the self-assessment questions should be attempted. If these cannot be answered then the previous few pages should be worked through again in order to find the answers in the text, before checking with answers given at the back of the book. Finally, the exercises should be attempted and, again, answers should be checked regularly with those given at the back of the book. Foundation Maths is enhanced by video clips (see go.pearson.com/uk/he/ resources) in which we, the authors, work through some algebraic examples and exercises taken from the book, pointing out techniques and key points. The icon next to an exercise signifies that there is a corresponding video clip. VIDEO New to this 7th edition is the inclusion of many examples which illustrate how readily-available software can be used to tackle the mathematical problems you will meet in Foundation Maths. These examples are marked with symbol. Although many mathematical software packages and apps are available, the ones used here for the purposes of illustration are Excel and GeoGebra. Further details of this important aspect are given on p. xxiii. In conclusion, remember that learning mathematics takes time and effort. Carrying out a large number of exercises allows the student to experience a greater variety of problems, thus building up expertise and confidence. Armed with these the student will be able to tackle more unfamiliar and demanding problems that arise in other aspects of their course. We hope that you find Foundation Maths useful and wish you the very best of luck. Anthony Croft, Robert Davison 2020 Publisher’s acknowledgements Text credit(s): xxiv Advisory Committee on Mathematics Education: Mathematical Needs: Mathematics in the workplace and in Higher Education, June 2011. Advisory Committee on Mathematics Education. Photo credit(s): xxvi 9 13 87 88 109 295 297 298 341 344 353 354 358 377 464 476 502 GeoGebra: Screenshot of GeoGebra © GeoGebra 2019 xxv 199 206 211 264 293 294 438 441 430 448 Microsoft Corporation: Screenshot of Microsoft Excel © Microsoft 2019.  List of videos The following table lists the videos which accompany selected exercises and examples in the book. You can view the videos at go.pearson.com/uk/he/resources Name Reference Substitution of a value into a quadratic expression Exercise 5.3 Q13 Simplification of expressions requiring use of the first law of indices Exercise 6.1 Q8 Simplification of expressions requiring use of the second and third laws of indices Exercise 6.1 Q10 Simplification of expressions with negative powers Exercise 6.2 Q4 Removing the brackets from expressions 1 Example 7.18 Removing the brackets from expressions 2 Example 7.24 Factorising a quadratic expression 1 Example 8.6 Factorising a quadratic expression 2 Example 8.12 Simplifying an algebraic fraction 1 Example 9.4 Simplifying an algebraic fraction 2 Example 9.8 Simplifying the product of two algebraic fractions Example 9.19 Simplifying products and quotients of algebraic fractions Exercise 9.3 Q4 Adding algebraic fractions 1 Example 9.26 Adding algebraic fractions 2 Example 9.27 An example of partial fractions Example 9.30 Another example of partial fractions Example 9.31 Transposition of a formula Example 10.7 Solving simultaneous equations by elimination Example 11.6 Solving a quadratic equation by factorisation Example 11.10 Solving a quadratic equation using a formula Example 11.16 Mathematical symbols + plus - minus { plus or minus * multiply by # multiply by , divide by = is equal to K is identically equal to ≈ is approximately equal to ≠ is not equal to 7 is greater than ⩾ is greater than or equal to 6 is less than ⩽ is less than or equal to ∈ is a member of set e universal set t intersection d union ∅ empty set A complement of set A ⊆ subset ℝ all real numbers ℝ+ all numbers greater than 0 - ℝ all numbers less than 0 ℤ all integers ℕ all positive integers ℂ all complex numbers ℚ rational numbers xxii Mathematical symbols q irrational numbers 6 therefore ∞ infinity e the base of natural logarithms (2.718 …) ln natural logarithm log logarithm of base 10 a sum of terms 1 integral dy dx derivative of y with respect to x p ‘pi’ ≈ 3.14159 ¬ negation (not) ¿ conjunction (and) ¡ disjunction (or) S implication Using mathematical and statistical computer software and apps in Foundation Maths Foundation Maths has been written for students taking further and higher educa- tion courses who have not specialised in mathematics on post-16 qualifications and who need to use mathematics or statistics in their courses. Our intentions are to provide a thorough, carefully-paced foundation in the mathematical meth- ods needed for success, to develop understanding and to build confidence. So what has computer software to do with Foundation Maths? Computer software and apps available for use on tablets and smartphones which can be used to perform all of the mathematical and statistical calculations in Foundation Maths are now readily-available, either freely or at low-cost. You will probably come across a variety of such tools in your courses. They are able to go beyond arithmetical operations found on a calculator and can perform cal- culations using algebra. Two important topics that you will meet when studying calculus, namely di!erentiation and integration, give rise to problems which can be solved using software. Di!erentiation can be used to find the maximum and minimum values of a function, for example maximising profit or minimising cost in business analysis. Integration can be used, for example, in the solution of equations that describe the movement of a fluid or the vibration of an aerofoil. Software is used to apply these calculus techniques to such problems. Moreover, the software can produce visual representations of solutions which can supple- ment the information given in an algebraic answer and thereby provide more insight. For example, they can draw accurate graphs and enable the user to focus upon points of interest. Statistical apps can tackle the analysis of data sets and produce results in a wide variety of visually informative charts. With all this software, why do I need to learn Foundation Maths? At first sight it might appear that with access to these tools there is no longer a need to learn basic mathematical methods. However, and on the contrary, to be able to exploit their full capability and power, a firm understanding of the underlying mathematics is essential. In part, this is because to use such software requires the user to distinguish and understand mathematical and statistical ter- minology. For example, when using computer algebra software it is essential xxiv Using mathematical and statistical computer software and apps in Foundation Maths to understand the meaning of words such as simplify, factorise, solve, expand, …. Such words have precise mathematical meanings which inform important choices to be made by the user. Visualisation software is able to access user data in several ways (e.g. from a formula, from a set of data, from an external file) and display it using a variety of di!erent graphs (for example, polar, car- tesian and logarithmic graphs). So, it is important to understand what these words mean. Statistical software will allow you to interrogate large sets of data and to look for patterns in that data. You may be interested in whether two or more variables are associated, that is whether there is correlation, and if so, how strong is that association. Knowledge of the di!erent ways in which this strength is measured, that is through correlation coefficients, is important if you are to make sensible choices and correct inferences when using the software. Whilst exceptionally powerful, software is not infallible! It is important that you can look critically at the output and make a judgement as to whether it is likely to be correct or not. Even when the output is correct, this output may be in a form that you do not recognise. Acquiring the mathematical fluency to compare and contrast di!erent forms of output is a skill that you will develop by working through Foundation Maths. To quote from the ACME1 Mathematical Needs report: It is sometimes argued that the advent of computers has reduced the need for people to be able to do mathematics. Nothing could be further from the truth. Off-the-shelf and pur- pose-designed computer software packages are creating ever more data sets, statistics and graphs. Working with mathematical models, which people need to be able to understand, interpret, interrogate and use advantageously, is becom- ing commonplace. The use of quantitative data is now omnipresent and informs workplace practice. So how might I want to use software as I work through this book? Firstly, you are able to verify the solutions you have already obtained ‘by hand’, and this gives you confidence that your methods are appropriate and your solu- tions are correct. Secondly, you can explore the e!ect of changing some of the values or ‘parameters’ in a mathematical expression. For example, what will happen to the graphs of y = x2 + 2x + c or y = 1x - c2 2 when c is varied from negative to zero and then to positive values. Investigations like these are straightforward when you have access to software, and the results can be illu- minating and aid understanding. Thirdly, using software means that you can attempt more complicated, and often more realistic problems that would be too lengthy or time-consuming to tackle by hand. Finally, you are able to produce mathematical and statistical output, for example graphs or charts, in a way that looks attractive and professional. 1 ACME is the The Royal Society Advisory Committee on Mathematics Education, a distinguished body advising on mathematics education policy.  Using mathematical and statistical computer software and apps in Foundation Maths xxv One of the purposes of this book is to help you to understand the mathemati- cal foundations necessary to take advantage of this technology. We do not intend to teach you how to use the software – there are plenty of textbooks, user guides and on-line resources to help with this. However, we want to raise your aware- ness of what tools are available so that you become confident to explore these for yourself, or within your courses. Throughout the book we make reference to several pieces of software or apps outlined below. We do this solely for the pur- poses of illustration and are not making recommendations; there are numerous di!erent tools available and we would encourage you to explore the field for yourself and to take advice from within your own institution. The software and apps that we will refer to throughout Foundation Maths We illustrate how Microsoft Excel2 can be used for performing routine statistical calculations such as finding the mean and standard deviation of a set of data, for finding correlation coefficients and lines of best fit. Statistical charts and graphs such as the one shown in Figure i can be produced relatively easily from large sets of data. Figure i Using Microsoft Excel for producing a statistical chart We illustrate how GeoGebra3 can be used to perform calculations arising in algebra and calculus, and how it can be used to explore important mathemat- ical objects such as vectors and matrices. Figure ii shows a screenshot depict- ing several algebraic operations that you will learn about in Foundation Maths: commands for simplification, factorisation, expansion of brackets, and solving equations. Further examples are given as you work through the book. 2 Microsoft Excel is a spreadsheet developed by Microsoft Corporation 3 GeoGebra is an interactive geometry, algebra, statistics and calculus application (www.geogebra.org) xxvi Using mathematical and statistical computer software and apps in Foundation Maths Figure ii A selection of GeoGebra commands for algebraic manipulation As we have noted earlier, there are many other commonly used software packages and apps. If you are studying, or intend to study, engineering, physics or mathematics you are likely to come across Matlab, Mathematica or Maple; these are extremely powerful technical computing systems that include addi- tional toolboxes for tasks such as signal processing. If you are studying psychol- ogy or the social sciences it is likely that you will use statistical software such as SPSS, minitab or R in the analysis of large data sets. It will help your learning if you enquire about what packages are available for your use in the institution where you are studying and to explore how these can be put to use in the solu- tion of exercises in Foundation Maths. Examples illustrating use of software in Foundation Maths The ability to use modern software in the solution of mathematical and statisti- cal problems is an invaluable skill to develop. The first step in this development is to become aware of packages that are available, to appreciate how power- ful they are, and how you can make use of them, particularly once you have acquired the necessary fundamental mathematical knowledge. Throughout this edition of Foundation Maths we provide numerous illustra- tive examples of the wide-spread application and power of mathematical and statistical software. We would encourage you to try these and similar examples for yourself and to explore further.  Using mathematical and statistical computer software and apps in Foundation Maths xxvii Purpose Page To prime factorise an integer and perform related prime number calculations 9 To find the highest common factor and lowest common multiple of a set of numbers 13 To factorise a quadratic expression 87 To express an algebraic fraction in partial fractions 109 To perform conversions such as radians to degrees 263 To draw graphs of functions 198 To produce graphs of trigonometrical functions 293/4 To find graphical solutions of an equation 205 To find graphical solutions of simultaneous equations 210 To explore the effect of changing k in y = sin kx, y = cos kx, y = tan kx 295 To explore the effect of changing a in y = sin1x + a2, y = cos1x + a2, y = tan1x + a2 296 To explore the effect of changing A in y = A sin kx, y = A cos kx, y = A tan kx 297 To explore composite transformations of trigonometric graphs e.g. y = A cos1kx + a2 as A, k 298 and a are varied To visualise vectors 340 To calculate scalar and vector products of vectors 344 To multiply matrices 353 To find the inverse of a matrix 357 To perform calculations with complex numbers such as finding their moduli and arguments; 376 finding the complex roots of equations To find and evaluate derivatives 476 To find turning points and other special points on the graph of a function 464 To find indefinite and definite integrals 502 To draw a scatter diagram and line of best fit 430 To calculate a product-moment correlation coefficient 438 To calculate the Spearman coefficient of rank correlation 440 To calculate a line of best fit 448  Arithmetic of whole numbers 1 Objectives: This chapter: explains the rules for adding, subtracting, multiplying and dividing positive and negative numbers explains what is meant by an integer explains what is meant by a prime number explains what is meant by a factor explains how to prime factorise an integer explains the terms ‘highest common factor’ and ‘lowest common multiple’ 1.1 Addition, subtraction, multiplication and division Arithmetic is the study of numbers and their manipulation. A clear and firm understanding of the rules of arithmetic is essential for tackling everyday calculations. Arithmetic also serves as a springboard for tackling more abstract mathematics such as algebra and calculus. The calculations in this chapter will involve mainly whole numbers, or integers as they are often called. The positive integers are the numbers 1, 2, 3, 4, 5... and the negative integers are the numbers c -5, -4, -3, -2, -1 The dots (...) indicate that this sequence of numbers continues indefinitely. The number 0 is also an integer but is neither positive nor negative. To find the sum of two or more numbers, the numbers are added together. To find the difference of two numbers, the second is subtracted from the first. The product of two numbers is found by multiplying 2 Arithmetic of whole numbers the numbers together. Finally, the quotient of two numbers is found by dividing the first number by the second. WORKED EXAMPLE 1.1 (a) Find the sum of 3, 6 and 4. (b) Find the difference of 6 and 4. (c) Find the product of 7 and 2. (d) Find the quotient of 20 and 4. Solution (a) The sum of 3, 6 and 4 is 3 + 6 + 4 = 13 (b) The difference of 6 and 4 is 6 - 4 = 2 (c) The product of 7 and 2 is 7 * 2 = 14 (d) The quotient of 20 and 4 is 20 4 , that is 5. When writing products we sometimes replace the sign * by ‘ # ’ or even omit it completely. For example, 3 * 6 * 9 could be written as 3 # 6 # 9 or (3)(6)(9). On occasions it is necessary to perform calculations involving negative num- bers. To understand how these are added and subtracted consider Figure 1.1, which shows a number line. Figure 1.1 The number line Any number can be represented by a point on the line. Positive numbers are on the right-hand side of the line and negative numbers are on the left. From any given point on the line, we can add a positive number by moving that number of places to the right. For example, to find the sum 5 + 3, start at the point 5 and move 3 places to the right, to arrive at 8. This is shown in Figure 1.2. Figure 1.2 To add a positive number, move that number of places to the right  1.1 Addition, subtraction, multiplication and division 3 To subtract a positive number, we move that number of places to the left. For example, to find the difference 5 - 7, start at the point 5 and move 7 places to the left to arrive at -2. Thus 5 - 7 = -2. This is shown in Figure 1.3. The result of finding -3 - 4 is also shown to be -7. Figure 1.3 To subtract a positive number, move that number of places to the left To add a negative number we move to the left. The result of finding 2 + (-3) is shown in Figure 1.4. Starting at 2, we move 3 places to the left, to arrive at -1. Figure 1.4 Adding a negative number involves moving to the left We see that 2 + (-3) = -1. Note that this is the same as the result of finding 2 - 3, so that adding a negative number is equivalent to subtracting a positive number. For example 9+(-4) = 9-4 = 5, 3+(-7) = 3-7 = -4, -6+(-10) = -6-10 = -16 To subtract a negative number we move to the right. The result of finding 5 - (-3) is shown in Figure 1.5 Figure 1.5 Subtracting a negative number involves moving to the right We see that 5 - ( -3) = 8. This is the same as the result of finding 5+3, so subtracting a negative number is equivalent to adding a positive number. For example 6-(-2) = 6+2 = 8, -5-(-3) = -5 + 3 = -2, -1-(-1) = -1+1 = 0 Key point Adding a negative number is equivalent to subtracting a positive number. Subtracting a negative number is equivalent to adding a positive number. WORKED EXAMPLE 1.2 Evaluate (a) 8 + ( -4), (b) -15 + (-3), (c) -15 - ( -4). Solution (a) 8 + ( -4) is equivalent to 8 - 4, that is 4. 4 Arithmetic of whole numbers (b) Because adding a negative number is equivalent to subtracting a positive number we find -15 + ( -3) is equivalent to -15 - 3, that is -18. (c) -15 - ( -4) is equivalent to -15 + 4, that is -11. When we need to multiply or divide negative numbers, care must be taken with the sign of the answer; that is, whether the result is positive or negative. The following rules apply for determining the sign of the answer when multiplying or dividing positive and negative numbers. Key point (positive) * (positive) = positive and positive (positive) * (negative) = negative = positive positive (negative) * (positive) = negative (negative) * (negative) = positive positive = negative negative negative = negative positive negative = positive negative WORKED EXAMPLE 1.3 Evaluate 12 -8 -6 (a) 3 * (-2) (b) ( -1) * 7 (c) (-2) * ( -4) (d) (e) (f) (-4) 4 -2 Solution (a) We have a positive number, 3, multiplied by a negative number, -2, and so the result will be negative: 3 * (-2) = -6 (b) (-1) * 7 = -7 (c) Here we have two negative numbers being multiplied and so the result will be positive: (-2) * (-4) = 8 (d) A positive number, 12, divided by a negative number, -4, gives a negative result: 12 = -3 -4  1.2 The BODMAS rule 5 (e) A negative number, -8, divided by a positive number, 4, gives a negative result: -8 = -2 4 (f) A negative number, -6, divided by a negative number, -2, gives a positive result: -6 = 3 -2 Self-assessment questions 1.1 1. Explain what is meant by an integer, a positive integer and a negative integer. 2. Explain the terms sum, difference, product and quotient. 3. State the sign of the result obtained after performing the following calculations: (a) ( -5) * ( -3) (b) (-4) * 2 (c) -72 (d) -- 84. Exercise 1.1 1. Without using a calculator, evaluate each of 3. Without using a calculator, evaluate the following: (a) -153 (b) 21 7 (c) -721 (d) --21 7 (e) 21 -7 (a) 6 + ( -3) (b) 6 - ( -3) - 12 - 12 12 (f) 2 (g) -2 (h) -2 (c) 16 + ( -5) (d) 16 - ( -5) (e) 27 - ( -3) (f) 27 - ( -29) 4. Find the sum and product of (a) 3 and 6, (g) -16 + 3 (h) -16 + (-3) (b) 10 and 7, (c) 2, 3 and 6. (i) -16 - 3 (j) -16 - (-3) 5. Find the difference and quotient of (a) 18 (k) -23 + 52 (l) -23 + (-52) and 9, (b) 20 and 5, (c) 100 and 20. (m) -23 - 52 (n) -23 - (-52) 2. Without using a calculator, evaluate (a) 3 * (-8) (b) (-4) * 8 (c) 15 * (-2) (d) ( -2) * (-8) (e) 14 * (-3) 1.2 The BODMAS rule When evaluating numerical expressions we need to know the order in which addition, subtraction, multiplication and division are carried out. As a simple example, consider evaluating 2 + 3 * 4. If the addition is carried out first we get 2 + 3 * 4 = 5 * 4 = 20. If the multiplication is carried out first 6 Arithmetic of whole numbers we get 2 + 3 * 4 = 2 + 12 = 14. Clearly the order of carrying out numer- ical operations is important. The BODMAS rule tells us the order in which we must carry out the operations of addition, subtraction, multiplication and division. Key point BODMAS stands for Brackets ( ) First priority Of * Second priority Division , Second priority Multiplication * Second priority Addition + Third priority Subtraction - Third priority This is the order of carrying out arithmetical operations, with bracketed expressions having highest priority and subtraction and addition having the lowest priority. Note that ‘Of’, ‘Division’ and ‘Multiplication’ have equal priority, as do ‘Addition’ and ‘Subtraction’. ‘Of’ is used to show multiplication when dealing with fractions: for example, find 12 of 6 means 1 2 * 6. If an expression contains only multiplication and division, we evaluate by working from left to right. Similarly, if an expression contains only addition and subtraction, we also evaluate by working from left to right. WORKED EXAMPLES 1.4 Evaluate (a) 2 + 3 * 4 (b) (2 + 3) * 4 Solution (a) Using the BODMAS rule we see that multiplication is carried out first. So 2 + 3 * 4 = 2 + 12 = 14 (b) Using the BODMAS rule we see that the bracketed expression takes prior- ity over all else. Hence (2 + 3) * 4 = 5 * 4 = 20 1.5 Evaluate (a) 4 - 2 , 2 (b) 1 - 3 + 2 * 2  1.3 Prime numbers and factorisation 7 Solution (a) Division is carried out before subtraction, and so 2 4 - 2 , 2 = 4 - = 3 2 (b) Multiplication is carried out before subtraction or addition: 1 - 3 + 2 * 2 = 1 - 3 + 4 = 2 1.6 Evaluate (a) (12 , 4) * 3 (b) 12 , (4 * 3) Solution Recall that bracketed expressions are evaluated first. 12 (a) (12 , 4) * 3 = ¢ ≤ * 3 = 3 * 3 = 9 4 (b) 12 , (4 * 3) = 12 , 12 = 1 Example 1.6 shows the importance of the position of brackets in an expression. Self-assessment questions 1.2 1. State the BODMAS rule used to evaluate expressions. 2. The position of brackets in an expression is unimportant. True or false? Exercise 1.2 1. Evaluate the following expressions: 2. Place brackets in the following expressions (a) 6 - 2 * 2 (b) (6 - 2) * 2 to make them correct: (c) 6 , 2 - 2 (d) (6 , 2) - 2 (a) 6 * 12 - 3 + 1 = 55 (e) 6 - 2 + 3 * 2 (f) 6 - (2 + 3) * 2 (b) 6 * 12 - 3 + 1 = 68 16 -24 (c) 6 * 12 - 3 + 1 = 60 (g) (6 - 2) + 3 * 2 (h) (i) -2 -3 (d) 5 * 4 - 3 + 2 = 7 (j) ( -6) * ( -2) (k) (-2)(-3)(-4) (e) 5 * 4 - 3 + 2 = 15 (f) 5 * 4 - 3 + 2 = -5 1.3 Prime numbers and factorisation A prime number is a positive integer, larger than 1, which cannot be expressed as the product of two smaller positive integers. To put it another way, a prime number is one that can be divided exactly only by 1 and itself. 8 Arithmetic of whole numbers For example, 6 = 2 * 3, so 6 can be expressed as a product of smaller numbers and hence 6 is not a prime number. However, 7 is prime. Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23. Note that 2 is the only even prime. Factorise means ‘write as a product’. By writing 12 as 3 * 4 we have facto- rised 12. We say 3 is a factor of 12 and 4 is also a factor of 12. The way in which a number is factorised is not unique: for example, 12 may be expressed as 3 * 4 or 2 * 6. Note that 2 and 6 are also factors of 12. When a number is written as a product of prime numbers we say the number has been prime factorised. To prime factorise a number, consider the technique used in the following examples. WORKED EXAMPLES 1.7 Prime factorise the following numbers: (a) 12 (b) 42 (c) 40 (d) 70 Solution (a) We begin with 2 and see whether this is a factor of 12. Clearly it is, so we write 12 = 2 * 6 Now we consider 6. Again 2 is a factor so we write 12 = 2 * 2 * 3 All the factors are now prime, that is the prime factorisation of 12 is 2 * 2 * 3. (b) We begin with 2 and see whether this is a factor of 42. Clearly it is and so we can write 42 = 2 * 21 Now we consider 21. Now 2 is not a factor of 21, so we examine the next prime, 3. Clearly 3 is a factor of 21 and so we can write 42 = 2 * 3 * 7 All the factors are now prime, and so the prime factorisation of 42 is 2 * 3 * 7. (c) Clearly 2 is a factor of 40, 40 = 2 * 20 Clearly 2 is a factor of 20, 40 = 2 * 2 * 10  1.3 Prime numbers and factorisation 9 Again 2 is a factor of 10, 40 = 2 * 2 * 2 * 5 All the factors are now prime. The prime factorisation of 40 is 2 * 2 * 2 * 5. (d) Clearly 2 is a factor of 70, 70 = 2 * 35 We consider 35: 2 is not a factor, 3 is not a factor, but 5 is: 70 = 2 * 5 * 7 All the factors are prime. The prime factorisation of 70 is 2 * 5 * 7. 1.8 Prime factorise 2299. Solution We note that 2 is not a factor and so we try 3. Again 3 is not a factor and so we try 5. This process continues until we find the first prime factor. It is 11: 2299 = 11 * 209 We now consider 209. The first prime factor is 11: 2299 = 11 * 11 * 19 All the factors are prime. The prime factorisation of 2299 is 11 * 11 * 19. Here we illustrate how software can be used to perform the prime factorisation of 2299 using GeoGebra Classic v6 with the CAS (Computer Algebra System) interface. Numerous other packages are available that perform the same calcu- lation. Figure 1.6 illustrates the output following the command Factor( ) which performs the prime factorisation. Compare this output to the solution of Worked Example 1.8, particularly the use of the power 2 in 112. Powers, or indi- ces as they are also called, are explained in Chapter 5. Figure 1.6 Syntax used to perform prime number calculations. In addition, Figure 1.6 shows the commands IsPrime( ) and NextPrime( ) which will test whether a given number is prime and find the first prime number greater than a given number. You should consult the on-line help provided with your software to explore other prime number commands. 10 Arithmetic of whole numbers Self-assessment questions 1.3 1. Explain what is meant by a prime number. 2. List the first 10 prime numbers. 3. Explain why all even numbers other than 2 cannot be prime. Exercise 1.3 1. State which of the following numbers are prime numbers: 1 (a) 13 (b) 1000 (c) 2 (d) 29 (e) 2 2. Prime factorise the following numbers: (a) 26 (b) 100 (c) 27 (d) 71 (e) 64 (f) 87 (g) 437 (h) 899 3. Prime factorise the two numbers 30 and 42. List any prime factors which are common to both numbers. 1.4 Highest common factor and lowest common multiple Highest common factor Suppose we prime factorise 12. This gives 12 = 2 * 2 * 3. From this prime factorisation we can deduce all the factors of 12: 2 is a factor of 12 3 is a factor of 12 2 * 2 = 4 is a factor of 12 2 * 3 = 6 is a factor of 12 Hence 12 has factors 2, 3, 4 and 6, in addition to the obvious factors of 1 and 12. Similarly we could prime factorise 18 to obtain 18 = 2 * 3 * 3. From this we can list the factors of 18: 2 is a factor of 18 3 is a factor of 18 2 * 3 = 6 is a factor of 18 3 * 3 = 9 is a factor of 18 The factors of 18 are 1, 2, 3, 6, 9 and 18. Some factors are common to both 12 and 18. These are 2, 3 and 6. These are common factors of 12 and 18. The highest common factor of 12 and 18 is 6.  1.4 Highest common factor and lowest common multiple 11 The highest common factor of 12 and 18 can be obtained directly from their prime factorisation. We simply note all the primes common to both factorisations: 12 = 2 * 2 * 3 18 = 2 * 3 * 3 Common to both is 2 * 3. Thus the highest common factor is 2 * 3 = 6. Thus 6 is the highest number that divides exactly into both 12 and 18. Key point Given two or more numbers the highest common factor (h.c.f.) is the largest (highest) number that is a factor of all the given numbers. The highest common factor is also referred to as the greatest common divisor (g.c.d). WORKED EXAMPLES 1.9 Find the h.c.f. of 12 and 27. Solution We prime factorise 12 and 27: 12 = 2 * 2 * 3 27 = 3 * 3 * 3 Common to both is 3. Thus 3 is the h.c.f. of 12 and 27. This means that 3 is the highest number that divides both 12 and 27. 1.10 Find the h.c.f. of 28 and 210. Solution The numbers are prime factorised: 28 = 2 * 2 * 7 210 = 2 * 3 * 5 * 7 The factors that are common are identified: a 2 is common to both and a 7 is common to both. Hence both numbers are divisible by 2 * 7 = 14. Since this number contains all the common factors it is the highest common factor. 1.11 Find the h.c.f. of 90 and 108. Solution The numbers are prime factorised: 90 = 2 * 3 * 3 * 5 108 = 2 * 2 * 3 * 3 * 3 The common factors are 2, 3 and 3 and so the h.c.f. is 2 * 3 * 3, that is 18. This is the highest number that divides both 90 and 108. 12 Arithmetic of whole numbers 1.12 Find the h.c.f. of 12, 18 and 20. Solution Prime factorisation yields 12 = 2 * 2 * 3 18 = 2 * 3 * 3 20 = 2 * 2 * 5 There is only one factor common to all three numbers: it is 2. Hence 2 is the h.c.f. of 12, 18 and 20. Lowest common multiple Suppose we are given two or more numbers and wish to find numbers into which all the given numbers will divide. For example, given 4 and 6 we see that they both divide exactly into 12, 24, 36, 48, 60 and so on. The smallest number into which they both divide is 12. We say 12 is the lowest common multiple of 4 and 6. Key point The lowest common multiple (l.c.m.) of a set of numbers is the smallest (lowest) number into which all the given numbers will divide exactly. WORKED EXAMPLE 1.13 Find the l.c.m. of 6 and 10. Solution We seek the smallest number into which both 6 and 10 will divide exactly. There are many numbers into which 6 and 10 will divide, for example 60, 120, 600, but we are seeking the smallest such number. By inspection, the smallest such number is 30. Thus the l.c.m. of 6 and 10 is 30. A more systematic method of finding the l.c.m. involves the use of prime factorisation. WORKED EXAMPLES 1.14 Find the l.c.m. of 15 and 20. Solution As a first step, the numbers are prime factorised: 15 = 3 * 5 20 = 2 * 2 * 5 Since 15 must divide into the l.c.m., then the l.c.m. must contain the factors of 15, that is 3 * 5. Similarly, as 20 must divide into the l.c.m., then the l.c.m. must also contain the factors of 20, that is 2 * 2 * 5. The l.c.m. is the smallest  1.4 Highest common factor and lowest common multiple 13 number that contains both of these sets of factors. Note that the l.c.m. will contain only 2s, 3s and 5s as its prime factors. We now need to determine how many of these particular factors are needed. To determine the l.c.m. we ask ‘How many factors of 2 are required?’, ‘How many factors of 3 are required?’, ‘How many factors of 5 are required?’ The highest number of 2s occurs in the factorisation of 20. Hence the l.c.m. requires two factors of 2. Consider the number of 3s required. The high- est number of 3s occurs in the factorisation of 15. Hence the l.c.m. requires one factor of 3. Consider the number of 5s required. The highest number of 5s is 1 and so the l.c.m. requires one factor of 5. Hence the l.c.m. is 2 * 2 * 3 * 5 = 60. Hence 60 is the smallest number into which both 15 and 20 will divide exactly. 1.15 Find the l.c.m. of 20, 24 and 25. Solution The numbers are prime factorised: 20 = 2 * 2 * 5 24 = 2 * 2 * 2 * 3 25 = 5 * 5 By considering the prime factorisations of 20, 24 and 25 we see that the only primes involved are 2, 3 and 5. Hence the l.c.m. will contain only 2s, 3s and 5s. Consider the number of 2s required. The highest number of 2s required is three from factorising 24. The highest number of 3s required is one, again from factorising 24. The highest number of 5s required is two, found from factorising 25. Hence the l.c.m. is given by l.c.m. = 2 * 2 * 2 * 3 * 5 * 5 = 600 Hence 600 is the smallest number into which 20, 24 and 25 will all divide exactly. GeoGebra can be used to find the highest common factor, HCF( ), and lowest common multiple, LCM( ), of a set or list of numbers, entered using braces, for example as a={12,18,20}. Use of this software for verification of Worked Examples 1.12 and 1.15 is shown in Figure 1.7. Figure 1.7 Using software to find the h.c.f and l.c.m. 14 Arithmetic of whole numbers Self-assessment questions 1.4 1. Explain what is meant by the h.c.f. of a set of numbers. 2. Explain what is meant by the l.c.m. of a set of numbers. Exercise 1.4 1. Calculate the h.c.f. of the following sets of numbers: (a) 12, 15, 21 (b) 16, 24, 40 (c) 28, 70, 120, 160 (d) 35, 38, 42 (e) 96, 120, 144 2. Calculate the l.c.m. of the following sets of numbers: (a) 5, 6, 8 (b) 20, 30 (c) 7, 9, 12 (d) 100, 150, 235 (e) 96, 120, 144 Test and assignment exercises 1 1. Evaluate (a) 6 , 2 + 1 (b) 6 , (2 + 1) (c) 12 + 4 , 4 (d) (12 + 4) , 4 (e) 3 * 2 + 1 (f) 3 * (2 + 1) (g) 6 - 2 + 4 , 2 (h) (6 - 2 + 4) , 2 (i) 6 - (2 + 4 , 2) (j) 6 - (2 + 4) , 2 (k) 2 * 4 - 1 (l) 2 * (4 - 1) (m) 2 * 6 , (3 - 1) (n) 2 * (6 , 3) - 1 (o) 2 * (6 , 3 - 1) 2. Prime factorise (a) 56, (b) 39, (c) 74. 3. Find the h.c.f. of (a) 8, 12, 14 (b) 18, 42, 66 (c) 20, 24, 30 (d) 16, 24, 32, 160 4. Find the l.c.m. of (a) 10, 15 (b) 11, 13 (c) 8, 14, 16 (d) 15, 24, 30  Fractions 2 Objectives: This chapter: explains what is meant by a fraction defines the terms ‘improper fraction’, ‘proper fraction’ and ‘mixed fraction’ explains how to write fractions in different but equivalent forms explains how to simplify fractions by cancelling common factors explains how to add, subtract, multiply and divide fractions 2.1 Introduction The arithmetic of fractions is very important groundwork which must be mas- tered before topics in algebra such as formulae and equations can be understood. The same techniques that are used to manipulate fractions are used in these more advanced topics. You should use this chapter to ensure that you are confi- dent at handling fractions before moving on to algebra. In all the examples and exercises it is important that you should carry out the calculations without the use of a calculator. Fractions are numbers such as 12, 34, 11 8 and so on. In general a fraction is a p number of the form q, where the letters p and q represent whole numbers or integers. The integer q can never be zero because it is not possible to divide by zero. p In any fraction q the number p is called the numerator and the number q is called the denominator. Key point numerator p fraction = = denominator q Suppose that p and q are both positive numbers. If p is less than q, the frac- tion is said to be a proper fraction. So 12 and 34 are proper fractions since the 16 Fractions numerator is less than the denominator. If p is greater than or equal to q, the 7 3 fraction is said to be improper. So 11 8 , 4 and 3 are all improper fractions. If either of p or q is negative, we simply ignore the negative sign when deter- mining whether the fraction is proper or improper. So - 35, -217 and -421 are proper fractions, but -33, -28 and - 11 2 are improper. Note that all proper fractions have a value less than 1. The denominator of a fraction can take the value 1, as in 31 and 71. In these cases the result is a whole number, 3 and 7. A fraction is inverted by interchanging its numerator and denominator. When 3 2 1 4 2 is inverted this results in 3. If 4 is inverted this results in 4 since 4 = 1. The reciprocal of a number is found by inverting it, so, for example, the reciprocal of 45 is 54. Self-assessment questions 2.1 1. Explain the terms (a) fraction, (b) improper fraction, (c) proper fraction. In each case give an exam- ple of your own. 2. Explain the terms (a) numerator, (b) denominator. Exercise 2.1 1. Classify each of the following as proper or improper: 9 (a) 17 (b) -179 (c) 88 (d) - 78 (e) 11077 2.2 Expressing a fraction in equivalent forms Given a fraction, we may be able to express it in a different form. For example, you will know that 12 is equivalent to 24. Note that multiplying both numerator and denominator by the same number leaves the value of the fraction unchanged. So, for example, 1 1 * 2 2 = = 2 2 * 2 4 We say that 12 and 24 are equivalent fractions. Although they might look differ- ent, they have the same value. 8 Similarly, given the fraction 12 we can divide both numerator and denomina- tor by 4 to obtain 8 8>4 2 = = 12 12>4 3 8 so 12 and 23 have the same value and are equivalent fractions.  2.2 Expressing a fraction in equivalent forms 17 Key point Multiplying or dividing both numerator and denominator of a fraction by the same number produces a fraction having the same value, called an equivalent fraction. A fraction is in its simplest form when there are no factors common to both 5 numerator and denominator. For example, 12 is in its simplest form, but 36 is not since 3 is a factor common to both numerator and denominator. Its simplest form is the equivalent fraction 12. To express a fraction in its simplest form we look for factors that are common to both the numerator and denominator. This is done by prime factorising both of these. Dividing both the numerator and denominator by any common factors removes them but leaves an equivalent fraction. This is equivalent to cancelling any common factors. For example, to simplify 46 we prime factorise to produce 4 2 * 2 = 6 2 * 3 Dividing both numerator and denominator by 2 leaves 23. This is equivalent to cancelling the common factor of 2. WORKED EXAMPLES 2.1 Express 24 36 in its simplest form. Solution We seek factors common to both numerator and denominator. To do this we prime factorise 24 and 36: Prime factorisation has 24 = 2 * 2 * 2 * 3 36 = 2 * 2 * 3 * 3 been described in §1.3. The factors 2 * 2 * 3 are common to both 24 and 36 and so these may be cancelled. Note that only common factors may be cancelled when simplifying a fraction. Hence Finding the highest common factor (h.c.f.) of 24 2 * 2 * 2 * 3 2 = = two numbers is detailed 36 2 * 2 * 3 * 3 3 in §1.4. In its simplest form 24 2 36 is 3. In effect we have divided 24 and 36 by 12, which is their h.c.f. 2.2 Express 49 21 in its simplest form. Solution Prime factorising 49 and 21 gives 49 = 7 * 7 21 = 3 * 7 Their h.c.f. is 7. Dividing 49 and 21 by 7 gives 49 7 = 21 3 Hence the simplest form of 49 7 21 is 3. 18 Fractions Before we can start to add and subtract fractions it is necessary to be able to convert fractions into a variety of equivalent forms. Work through the following examples. WORKED EXAMPLES 2.3 Express 34 as an equivalent fraction having a denominator of 20. Solution To achieve a denominator of 20, the existing denominator must be multiplied by 5. To produce an equivalent fraction both numerator and denominator must be multiplied by 5, so 3 3 * 5 15 = = 4 4 * 5 20 2.4 Express 7 as an equivalent fraction with a denominator of 3. Solution Note that 7 is the same as the fraction 71. To achieve a denominator of 3, the existing denominator must be multiplied by 3. To produce an equivalent fraction both numerator and denominator must be multiplied by 3, so 7 7 * 3 21 7 = = = 1 1 * 3 3 Self-assessment questions 2.2 1. All integers can be thought of as fractions. True or false? 2. Explain the use of h.c.f. in the simplification of fractions. 3. Give an example of three fractions that are equivalent. Exercise 2.2 1. Express the following fractions in their sim- 5. Express 2 as an equivalent fraction with a plest form: denominator of 4. (a) 18 (b) 12 (c) 15 45 (d) 80 25 (e) 15 27 90 20 15 16 60 6. Express 6 as an equivalent fraction with a (f) 200 (g) 20 2 (h) 18 (i) 24 (j) 30 65 denominator of 3. 100 6 13 (k) 12 (l) (m) (n) 12 (o) 21 13 45 9 16 42 7. Express each of the fractions 23, 54 and 56 as an (p) 39 (q) 11 33 (r) 14 30 (s) - 12 11 16 (t) - 33 equivalent fraction with a denominator of 12. - 14 (u) - 30 8. Express each of the fractions 49, 12 and 56 as an 3 2. Express as an equivalent fraction having a 4 equivalent fraction with a denominator of denominator of 28. 18. 3. Express 4 as an equivalent fraction with a 9. Express each of the following numbers as denominator of 5. an equivalent fraction with a denominator of 5 4. Express 12 as an equivalent fraction having a 12: denominator of 36. (a) 12 (b) 34 (c) 52 (d) 5 (e) 4 (f) 12  2.3 Addition and subtraction of fractions 19 2.3 Addition and subtraction of fractions To add and subtract fractions we first rewrite each fraction so that they all hav