MASTERING MATHEMATICS MACMILLAN MASTER SERIES Basic Management Biology Chemistry Commerce Computer Programming Computers Data Processing Economics Electronics English Language French German Italian Marketing Mathematics Modern World History Office Practice Physics Principles of Accounts Sociology Spanish Statistics Study Skills OTHER BOOKS BY THE SAME AUTHORS INCLUDE Mathematics I New Syllabus Mathematics for O-level, I and 2 Multiple Choice Tests for O-level Mathematics Mathematics for O-level, Part 1 (perry and Naish) Numerical Examples for Advanced Physics (perry and Wales) Enjoying Science (Perry and Stafford) Statistics Alive (Moss and Perry) Numbers Working (Moss and Perry) MASTERING MATHEMATICS o. & J. PERRY M Text © O. Perry and J. Perry 1982. Artwork © The Macmi11an Press Ltd., 1982. Softcover reprint of the hardcover 1st edition 1982 All rights reserved. No part of this pUblication may be reproduced or transmitted, in any form or by any means, without permission. First published 1982 by THE MACMILLAN PRESS LTD London and Basingstoke Companies and representatives throughout the world Typeset by Reproduction Drawings Ltd, Sutton, Surrey ISBN 978-0-333-31043-4 ISBN 978-1-349-16709-8 (eBook) DOI 10.1007/978-1-349-16709-8 ISBN 978-0-333-31070-0 export The paperback edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. This book is also available under the title Basic Mathematics published by Macmillan Education. CONTENTS Notation xii Preface xiii 1 Integers 1.1 Numbers 1.2 Positive integers 1 1.3 Place value 2 1.4 Addition and subtraction of natural numbers 3 1.5 Multiplication and division of natural numbers 4 1.6 Mixed operations 6 1.7 Multiples and factors 7 1.8 Negative integers 9 1.9 Order relations and the number line 9 1.10 Arithmetic operations with directed numbers 10 2 Common fractions 2.1 Rational numbers 14 2.2 Common fractions 14 2.3 Order relations for common fractions 16 2.4 Improper fractions and mixed numbers 17 2.5 Operations defined on common fractions 18 3 Decimal fractions 3.1 Relating decimals to common fractions 24 3.2 Decimal places and significant figures 26 3.3 Numbers in standard index form 27 3.4 Decimal currency and metric units 28 3.5 Addition and subtraction of decimal fractions 28 3.6 Multiplication of decimal fractions 30 3.7 Division of decimal fractions 31 CONTENTS 4 Roots, indices, four-figure 4.1 Irrational numbers 33 tables,caJculators 4.2 Roots and surds 33 4.3 Indices 34 4.4 limits of accuracy and maximum error 36 4.5 Aids to computation 37 4.6 Obtaining values from four- figure tables 37 4.7 Squares 37 4.8 Reciprocals 39 4.9 Square roots 40 4.10 Logarithms and antilogarithms 42 4.11 Multiplication and division using logarithms 43 4.12 Calculation of powers and roots using logarithms 45 4.13 The use of electronic calculators 47 5 Percentage, ratio and 5.1 Percentage 48 proportion 5.2 Ratio and proportion 50 5.3 Changing a quantity in a given ratio 51 5.4 Change of units 52 5.5 Profit and loss 54 5.6 Simple interest 55 5.7 Dividing a quantity in a given ratio 57 6 Other number bases 6.1 Counting in bases other than ten 59 6.2 Changing bases 60 6.3 Prime numbers and factors 63 6.4 Arithmetic operations in other number bases 64 Progress test 1 67 7 Algebraic expressions 7.1 Algebraic notation 68 7.2 Indices in algebra 70 7.3 Multiplication and division of algebraic terms 72 7.4 Substitution 73 7.5 Addition and subtraction of algebraic terms 74 7.6 Removing brackets using the distributive law 76 vii 7.7 Algebraic fractions 76 7.8 Factorising algebraic expressions 78 8 Algebraic equations and 8.1 The use of equality and inequalities inequality symbols 84 8.2 The solution of linear equations and inequalities 84 8.3 Forming an equation from given information 90 8.4 The solution of simultaneous linear equations 91 8.5 Problems leading to simultaneous equations 93 8.6 The solution of quadratic equations and inequalities 95 8.7 Problems involving quadratic equations 99 8.8 The solution of simultaneous linear and quadratic equations 101 8.9 Transposition of formulae 102 9 Sets 9.1 Set language and notation 106 9.2 Subsets 107 9.3 Venn diagrams 108 9.4 Methods of combining sets, union and intersection 109 9.5 The solution on problems involving sets 113 9.6 Binary operations 116 9.7 Operation tables 118 10 Matrices 10.1 The size of a matrix 121 10.2 Operations defined on matrices 122 10.3 Some special matrices 128 10.4 Two by two matrices 129 10.5 The use of a matrix to present information 134 Progress test 2 136 11 Introduction to 11.1 Points, lines and planes 137 geometry 11.2 Angles 137 11.3 Equal angles formed by intersecting lines 138 11.4 Polygons 141 11.5 Triangles 143 CONTENTS 11.6 Similar and congruent triangles 146 11.7 Quadrilaterals 150 11.8 Symmetrical properties of polygons 153 12 Geometrical constructions 12.1 To construct the bisector of a given angle 156 12.2 To construct the perpendicular bisector of a given straight line 157 12.3 To coI1struct a line perpen- dicular to a given line from a given point not on the line 158 12.4 To construct an angle equal to a given angle 159 12.5 To construct angles of 60° and 30° 159 12.6 To construct an angle of 45° 160 12.7 To construct triangles of given dimensions 161 12.8 To construct quadrilaterals of given dimensions 165 13 Perimeter, area and 13.1 Perimeters of plane figures 168 volume 13.2 Areas of plane figures 169 13.3 The ratio of the areas of similar figures 174 13.4 Volumes of geometrical solids 177 13.5 Surface areas of geometrical solids 181 13.6 Ratio of areas and volumes of similar solids 183 14 Mappings and 14.1 Mappings 187 functions, variation 14.2 Mapping diagrams 190 14.3 Combining functions 192 14.4 Inverse functions 193 14.5 Factor theorem 196 14.6 Variation 198 Progress test 3 202 15 Cartesian graphs of 15.1 Cartesian coordinates 204 functions 15.2 Graphs of linear functions 206 15.3 Determining the gradient of a graph 206 ix 15.4 Determining the gradient and intercept of a line from the equation 208 15.5 Determining the equation of a straight line 209 15.6 The graphical solution of simultaneous linear equations and inequalities 211 15.7 The graphs of quadratic functions 214 15.8 The graphs of cubic and reciprocal functions 217 16 Applications of graphs 16.1 linear programming 222 16.2 Obtaining a linear law from experimental data 225 16.3 Estimation of the area under a curve 228 16.4 Graphical kinematics 231 17 Differential calculus 17.1 The differential coefficient 238 17.2 Differentiation of powers by a formula 238 17.3 A derived function as a gradient 240 17.4 Maximum and minimum points on a curve 242 17.5 Rates of change 243 17.6 Differentiation applied to kinematics 245 18 Integral calculus 18.1 Integration as the inverse of differentiation 248 18.2 Definite integration 249 18.3 The area under a curve 251 18.4 The volume of a solid of revolution 254 18.5 Integral calculus applied to kinematics 256 19 Statistics 19.1 Statistical data 259 19.2 Diagrammatic representation of data 260 19.3 Histograms 264 19.4 Measures of location 267 19.5 Graphs of cumulative frequency 270 19.6 Measures of dispersion 271