MATHEMATICS: A Second Start, 2nd edition "Talking of education, people have now a-days" (said he) "got a strange opinion that every thing should be taught by lectures. Now, I cannot see that lectures can do so much good as reading the books from which the lectures are taken. I know nothing that can be best taught by lectures, except where experiments are to be shewn. You may teach chymestry by lectures — You might teach making of shoes by lectures!" James Boswell: Life ofSamuelJohnson, 1766 The direction in which education starts a man will determine his future life. Plato (427-347 BC): The Republic Mathematics possesses not only truth, but supreme beauty, cold and austere like that of sculpture, and capable of stern perfection, such as only great art can show. Bertrand Russell ( 1872-1970): The Principles of Mathematics MATHEMATICS: A Second Start, 2nd edition SHEILA PAGE Lecturer in Mathematics, University of Bradford JOHN BERRY Professor of Mathematics Education, University of Plymouth and HOWARD HAMPSON Department of Mathematics, Torquay College of Further Education WP WOOD H LAD PUBLISHING Oxford Cambridge Philadelphia New Delhi Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia. PA 19102-3406, USA Woodhead Publishing India Private Limited, G-2, Vardaan House. 7/28 Ansari Road. Daryaganj, New Delhi - 110002, India www.woodheadpublishingindia.com First published in 1986 by Ellis Horwood Limited Second edition published by Horwood Publishing Limited, 2002 Reprinted by Woodhead Publishing Limited, 2011 © S. Page, J. Berry and H. Hampson, 2002 The authors have asserted their moral rights This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials. Neither the authors nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-1-898563-04-4 Printed by Lightning Source. Contents Preface to Second Edition vii Introduction to the First Edition vii Chapter 0 Using a Scientific Calculator 1 Chapter 1 The Set of Real Numbers 9 Chapter 2 Number Skills 37 Chapter 3 Algebra: a Basic Toolkit 55 Chapter 4 Algebra: More Tools 75 Chapter 5 Products 90 Chapter 6 Factors 106 Chapter 7 Equations and Inequalities 121 Chapter 8 Manipulation of Formulae 153 Chapter 9 Quadratic Equations 170 Chapter 10 Simultaneous Equations 186 Chapter 11 Indices and Logarithms 196 Chapter 12 Functions and Graphs 211 Chapter 13 Linear Graphs and Their Use in Experimental Work 230 Chapter 14 Calculus - the Mathematics of Change 245 Chapter 15 Applications of Differentiation 266 vi Contents Chapter 16 Integration 286 Chapter 17 Introduction to Trigonometry 302 Chapter 18 Calculus of Trigonometry 333 Chapter 19 The Law of Natural Growth 345 Chapter 20 Some Applications of Logarithms 362 Chapter 21 A First Look at Statistics 377 Chapter 22 Probability 400 Chapter 23 Probability Distributions 415 Appendix 439 Answers 440 Index 466 Preface to Second Edition In revising this text we have attempted to retain the original aims of Mrs Page's book; that is to provide a second start at mathematics for those students who 'never could do maths'. The text provides the basic algebra, calculus and statistics for students in further and higher education who require mathematics to support their main study area. Howard Hampson John Berry Introduction to the First Edition This book is intended for the student 'who never could do maths', the student who for one reason or another missed his or her way at school, either by absence from some of the O-level course, or by having too many changes of masters or schools. Such a student has lost confidence in his own ability to tackle the subject. It is intended for the student whose knowledge extends only to O-level mathematics, probably obtained several years ago, but who would now like to be able to communicate with mathematicians, i.e. to know what is meant when one talks about 'an integral' or 'a differential equation'. It is based on a course of tutorials and discussions with students of this type, to whom I am indebted for their patient endeavours and encouragement. So let us begin at the beginning. The student who does not require the elementary work, nevertheless, is advised to see that the examples at the end of the chapter are worked through conscientiously, as each step depends on the previous steps being fully under­ stood. One must be completely competent in dealing with mathematical 'shorthand', i.e. the method of expressing an idea in as neat a way as possible, so that it can be handled easily and quickly. ο Using a Scientific Calculator INTRODUCTION Few people attempt to study and use mathematics without taking the drudgery out of the arithmetical operation by using a calculator. The speed and accuracy of modern calculators can benefit everyone from scientists and engineers, to business people and to those in the home. Children are encouraged to use simple calculators from an early age in school. Students in colleges and universities use advanced scientific calculators, often with graph drawing facilities, in most courses. You will need a calculator to study and use the mathematics in this book. A natural question to ask is 'which calculator should I buy?' There are many different calculators on the market. We have decided not to recommend a particular brand name or model because their features regularly change, but we do offer the following guidelines: 1. The calculator should be a scientific calculator containing trigonometric, logarithmic and exponential functions. Look for keys marked sin, cos, tan, ex, In, log. You may not understand their meaning yet; we will develop the theory in later chapters. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) 2. Look for simple statistics measures such as the mean and standard deviation. 3. Although a calculator with graph drawing facilities will help your understanding of functions and their properties, it is not essential to include this feature. Avoid a calculator with advanced programmable features. When you have purchased a new calculator you should study the booklet provided with it very carefully. Explore the features of the different keys. This chapter of the book is provided for two reasons: