Complete Mathematics: Teach Yourself

For anyone who needs help grasping basic math concepts, this step-by-step guide offers clear, detailed explanations and numerous worked examples for quick learning. Teach Yourself Mathematics covers algebra and geometry; fractions and decimals; Pythagoras' theorem and trigonometry; ratio and proportion; and much more. This new edition addresses recent developments such as the greater prominence of probability and statistics in education and general usage. Throughout, exercises with answers are provided to test and reinforce understanding of mathematical processes and solutions to common problems. Cover......Page 1 Book title......Page 4 Contents......Page 6 Personal introduction......Page 12 Only got a minute?......Page 13 Only got five minutes?......Page 15 Only got ten minutes?......Page 17 1.1 Introduction – place value......Page 20 1.2 Arithmetic – the four operations......Page 22 1.3 Order of operations......Page 23 1.4 Problems which use arithmetic......Page 26 1.5 Special numbers......Page 27 1.6 Multiples, factors and primes......Page 29 2.1 Introduction......Page 34 2.2 Angles......Page 35 2.3 Measuring and drawing angles......Page 37 2.4 Using angle facts......Page 39 2.5 Parallel lines......Page 41 2.6 Bearings......Page 44 3.2 What is a fraction?......Page 48 3.4 Simplifying fractions......Page 50 3.5 Improper fractions......Page 52 3.6 Adding and subtracting fractions......Page 53 3.7 Multiplication of fractions......Page 55 3.8 Fractions of quantities......Page 56 3.9 Division of fractions......Page 57 3.10 A number as a fraction of another number......Page 59 4.2 Triangles......Page 63 4.3 Constructing triangles......Page 67 4.4 Quadrilaterals......Page 69 4.5 Polygons......Page 71 4.6 Interior and exterior angles......Page 72 4.8 Congruent shapes......Page 74 4.9 Tessellations......Page 76 5.1 Introduction......Page 82 5.2 Place value......Page 83 5.3 Converting decimals to fractions......Page 85 5.4 Converting fractions to decimals 1......Page 86 5.5 Addition and subtraction......Page 87 5.6 Multiplication of decimals......Page 88 5.7 Division of decimals......Page 91 5.8 Converting fractions to decimals 2......Page 93 6.1 Introduction......Page 99 6.2 Collection of data......Page 100 6.3 Pictograms......Page 101 6.4 Bar charts......Page 102 6.5 Pie charts......Page 106 6.6 Line graphs......Page 108 6.7 Scatter graphs......Page 112 6.8 Discrete and continuous data......Page 115 6.9 Grouping data......Page 116 7.1 Introduction......Page 123 7.2 Ordering directed numbers......Page 124 7.3 Addition and subtraction......Page 125 7.4 Multiplication and division......Page 127 7.5 Using a calculator......Page 130 8.1 Coordinates......Page 132 8.2 Straight-line graphs......Page 135 8.3 Lines parallel to the axes......Page 140 9.1 The metric system......Page 144 9.2 Imperial units......Page 146 9.3 Converting between metric and imperial units......Page 148 9.4 Choosing suitable units......Page 149 10.1 Perimeter......Page 153 10.2 Area......Page 155 10.3 Area of a rectangle......Page 157 10.5 Area of a triangle......Page 160 10.6 Area of a trapezium......Page 163 11.2 Writing expressions......Page 167 11.3 Simplifying expressions......Page 170 11.4 Evaluating expressions......Page 172 11.6 Brackets......Page 174 11.7 Factorizing expressions......Page 177 11.8 Indices......Page 178 11.9 Laws of indices......Page 179 11.10 Simplifying expressions with indices......Page 182 12.2 Rounding whole numbers......Page 188 12.3 Rounding with decimals......Page 190 12.4 Significant figures......Page 193 12.5 Estimates......Page 195 12.6 Rounding in practical problems......Page 197 12.7 Accuracy of measurements......Page 198 13.1 Introduction......Page 201 13.3 Solving linear equations......Page 202 13.4 Equations with brackets......Page 207 13.5 Solving problems using equations......Page 208 13.6 Solving inequalities......Page 210 14.2 Percentages, decimals, fractions......Page 214 14.3 Percentages of quantities......Page 218 14.4 Increasing and decreasing quantities......Page 220 14.5 One quantity as a percentage of another quantity......Page 221 14.6 Percentage increases......Page 222 14.7 Using multipliers......Page 224 15.1 What is a formula?......Page 228 15.2 Evaluating terms other than the subject......Page 230 15.3 Changing the subject of a formula......Page 232 16.1 Introduction......Page 235 16.2 Circumference of a circle......Page 237 16.3 Area of a circle......Page 240 16.4 Two properties of circles......Page 243 17.1 Introduction......Page 248 17.2 Relative frequency......Page 249 17.3 Probability of a single event......Page 250 17.4 Two events......Page 253 17.5 Tree diagrams......Page 258 17.6 Expected frequency......Page 262 18.1 Introduction......Page 265 18.2 Nets and surface area......Page 268 18.3 Volume of a cuboid......Page 270 18.4 Volume of a prism......Page 272 18.5 Weight of a prism......Page 275 19.1 What is a ratio?......Page 279 19.2 Scales......Page 281 19.3 Using ratio......Page 284 19.4 Direct proportion......Page 286 19.5 Inverse proportion......Page 288 20.1 Pythagoras’ theorem......Page 292 20.2 Using Pythagoras’ theorem......Page 293 20.3 Proof of Pythagoras’ theorem......Page 297 20.4 Pythagoras’ theorem problems......Page 298 20.5 Trigonometry......Page 299 20.6 The tangent ratio......Page 300 20.7 Values of the tangent......Page 301 20.8 Using tangents......Page 302 20.9 Sine and cosine......Page 304 21.1 Indices......Page 311 21.2 Laws of indices......Page 312 21.3 Prime factors......Page 313 21.4 Highest common factor......Page 314 21.5 Lowest common multiple......Page 316 21.6 Standard form – large numbers......Page 317 21.7 Standard form – small numbers......Page 320 21.8 Standard form calculations......Page 321 22.2 The mode......Page 325 22.3 The median......Page 326 22.4 The mean......Page 328 22.5 The range......Page 331 22.6 Frequency tables......Page 332 22.7 Grouped frequency tables......Page 334 23.1 Equations of straight lines......Page 341 23.2 Drawing straight-line graphs......Page 344 23.3 The gradient of a straight line......Page 347 23.4 Curved graphs......Page 350 24.1 Simultaneous equations......Page 356 24.2 Algebraic methods......Page 358 24.3 Quadratic expressions......Page 362 24.4 Factorizing quadratic expressions......Page 364 24.5 Quadratic equations......Page 367 24.6 Solution by factorizing......Page 368 24.7 Cubic equations......Page 371 Answers......Page 377 Taking it further......Page 407 C......Page 412 F......Page 413 M......Page 414 P......Page 415 S......Page 416 Z......Page 417 Cover 1 Book title 4 Contents 6 Personal introduction 12 Only got a minute? 13 Only got five minutes? 15 Only got ten minutes? 17 1 Number 20 1.1 Introduction – place value 20 1.2 Arithmetic – the four operations 22 1.3 Order of operations 23 1.4 Problems which use arithmetic 26 1.5 Special numbers 27 1.6 Multiples, factors and primes 29 2 Angles 34 2.1 Introduction 34 2.2 Angles 35 2.3 Measuring and drawing angles 37 2.4 Using angle facts 39 2.5 Parallel lines 41 2.6 Bearings 44 3 Fractions 48 3.1 Introduction 48 3.2 What is a fraction? 48 3.3 Which fraction is bigger? 50 3.4 Simplifying fractions 50 3.5 Improper fractions 52 3.6 Adding and subtracting fractions 53 3.7 Multiplication of fractions 55 3.8 Fractions of quantities 56 3.9 Division of fractions 57 3.10 A number as a fraction of another number 59 4 Two-dimensional shapes 63 4.1 Introduction 63 4.2 Triangles 63 4.3 Constructing triangles 67 4.4 Quadrilaterals 69 4.5 Polygons 71 4.6 Interior and exterior angles 72 4.7 Symmetries of regular polygons 74 4.8 Congruent shapes 74 4.9 Tessellations 76 5 Decimals 82 5.1 Introduction 82 5.2 Place value 83 5.3 Converting decimals to fractions 85 5.4 Converting fractions to decimals 1 86 5.5 Addition and subtraction 87 5.6 Multiplication of decimals 88 5.7 Division of decimals 91 5.8 Converting fractions to decimals 2 93 6 Statistics 1 99 6.1 Introduction 99 6.2 Collection of data 100 6.3 Pictograms 101 6.4 Bar charts 102 6.5 Pie charts 106 6.6 Line graphs 108 6.7 Scatter graphs 112 6.8 Discrete and continuous data 115 6.9 Grouping data 116 7 Directed numbers 123 7.1 Introduction 123 7.2 Ordering directed numbers 124 7.3 Addition and subtraction 125 7.4 Multiplication and division 127 7.5 Using a calculator 130 8 Graphs 1 132 8.1 Coordinates 132 8.2 Straight-line graphs 135 8.3 Lines parallel to the axes 140 9 Measurement 144 9.1 The metric system 144 9.2 Imperial units 146 9.3 Converting between metric and imperial units 148 9.4 Choosing suitable units 149 10 Perimeter and area 153 10.1 Perimeter 153 10.2 Area 155 10.3 Area of a rectangle 157 10.4 Area of a parallelogram 160 10.5 Area of a triangle 160 10.6 Area of a trapezium 163 11 Algebraic expressions 167 11.1 Introduction – what is algebra? 167 11.2 Writing expressions 167 11.3 Simplifying expressions 170 11.4 Evaluating expressions 172 11.5 Squaring 174 11.6 Brackets 174 11.7 Factorizing expressions 177 11.8 Indices 178 11.9 Laws of indices 179 11.10 Simplifying expressions with indices 182 12 Approximation 188 12.1 Introduction 188 12.2 Rounding whole numbers 188 12.3 Rounding with decimals 190 12.4 Significant figures 193 12.5 Estimates 195 12.6 Rounding in practical problems 197 12.7 Accuracy of measurements 198 13 Equations 1 201 13.1 Introduction 201 13.2 Finding missing numbers 202 13.3 Solving linear equations 202 13.4 Equations with brackets 207 13.5 Solving problems using equations 208 13.6 Solving inequalities 210 14 Percentages 214 14.1 Introduction 214 14.2 Percentages, decimals, fractions 214 14.3 Percentages of quantities 218 14.4 Increasing and decreasing quantities 220 14.5 One quantity as a percentage of another quantity 221 14.6 Percentage increases 222 14.7 Using multipliers 224 15 Formulae 228 15.1 What is a formula? 228 15.2 Evaluating terms other than the subject 230 15.3 Changing the subject of a formula 232 16 Circles 235 16.1 Introduction 235 16.2 Circumference of a circle 237 16.3 Area of a circle 240 16.4 Two properties of circles 243 17 Probability 248 17.1 Introduction 248 17.2 Relative frequency 249 17.3 Probability of a single event 250 17.4 Two events 253 17.5 Tree diagrams 258 17.6 Expected frequency 262 18 Three-dimensional shapes 265 18.1 Introduction 265 18.2 Nets and surface area 268 18.3 Volume of a cuboid 270 18.4 Volume of a prism 272 18.5 Weight of a prism 275 19 Ratio and proportion 279 19.1 What is a ratio? 279 19.2 Scales 281 19.3 Using ratio 284 19.4 Direct proportion 286 19.5 Inverse proportion 288 20 Pythagoras’ theorem and trigonometry 292 20.1 Pythagoras’ theorem 292 20.2 Using Pythagoras’ theorem 293 20.3 Proof of Pythagoras’ theorem 297 20.4 Pythagoras’ theorem problems 298 20.5 Trigonometry 299 20.6 The tangent ratio 300 20.7 Values of the tangent 301 20.8 Using tangents 302 20.9 Sine and cosine 304 21 Indices and standard form 311 21.1 Indices 311 21.2 Laws of indices 312 21.3 Prime factors 313 21.4 Highest common factor 314 21.5 Lowest common multiple 316 21.6 Standard form – large numbers 317 21.7 Standard form – small numbers 320 21.8 Standard form calculations 321 22 Statistics 2 325 22.1 Averages 325 22.2 The mode 325 22.3 The median 326 22.4 The mean 328 22.5 The range 331 22.6 Frequency tables 332 22.7 Grouped frequency tables 334 23 Graphs 2 341 23.1 Equations of straight lines 341 23.2 Drawing straight-line graphs 344 23.3 The gradient of a straight line 347 23.4 Curved graphs 350 24 Equations 2 356 24.1 Simultaneous equations 356 24.2 Algebraic methods 358 24.3 Quadratic expressions 362 24.4 Factorizing quadratic expressions 364 24.5 Quadratic equations 367 24.6 Solution by factorizing 368 24.7 Cubic equations 371 Answers 377 Taking it further 407 Index 412 A 412 B 412 C 412 D 413 E 413 F 413 G 414 H 414 I 414 K 414 L 414 M 414 N 415 O 415 P 415 Q 416 R 416 S 416 T 417 U 417 V 417 W 417 X 417 Y 417 Z 417 A step-by-step introduction to the subject of mathematics, packed full of worked examples and useful exercises. The authors guide you through the essentials quickly and easily, giving you the knowledge you need to gain confidence