Mathematics Read online

  MATHEMATICS

  A complete introduction

  Trevor Johnson and

  Hugh Neill

  Contents

  Welcome to Mathematics:

  A complete introduction!

  Introduction

  1 Number

  1.1 Introduction – place value

  1.2 Arithmetic – the four operations

  1.3 Order of operations

  1.4 Problems which use arithmetic

  1.5 Special numbers

  1.6 Multiples, factors and primes

  2 Angles

  2.1 Introduction

  2.2 Angles

  2.3 Measuring and drawing angles

  2.4 Using angle facts

  2.5 Parallel lines

  2.6 Bearings

  3 Fractions

  3.1 Introduction

  3.2 What is a fraction?

  3.3 Which fraction is bigger?

  3.4 Simplifying fractions

  3.5 Improper fractions

  3.6 Adding and subtracting fractions

  3.7 Multiplication of fractions

  3.8 Fractions of quantities

  3.9 Division of fractions

  3.10 A number as a fraction of another number

  4 Two-dimensional shapes

  4.1 Introduction

  4.2 Triangles

  4.3 Constructing triangles

  4.4 Quadrilaterals

  4.5 Polygons

  4.6 Interior and exterior angles

  4.7 Symmetries of regular polygons

  4.8 Congruent shapes

  4.9 Tessellations

  5 Decimals

  5.1 Introduction

  5.2 Place value

  5.3 Converting decimals to fractions

  5.4 Converting fractions to decimals 1

  5.5 Addition and subtraction

  5.6 Multiplication of decimals

  5.7 Division of decimals

  5.8 Converting fractions to decimals 2

  6 Statistics 1

  6.1 Introduction

  6.2 Collection of data

  6.3 Pictograms

  6.4 Bar charts

  6.5 Pie charts

  6.6 Line graphs

  6.7 Scatter graphs

  6.8 Discrete and continuous data

  6.9 Grouping data

  7 Directed numbers

  7.1 Introduction

  7.2 Ordering directed numbers

  7.3 Addition and subtraction

  7.4 Multiplication and division

  7.5 Using a calculator

  8 Graphs 1

  8.1 Coordinates

  8.2 Straight-line graphs

  8.3 Lines parallel to the axes

  9 Measurement

  9.1 The metric system

  9.2 Imperial units

  9.3 Converting between metric and imperial units

  9.4 Choosing suitable units

  10 Perimeter and area

  10.1 Perimeter

  10.2 Area

  10.3 Area of a rectangle

  10.4 Area of a parallelogram

  10.5 Area of a triangle

  10.6 Area of a trapezium

  11 Algebraic expressions

  11.1 Introduction – what is algebra?

  11.2 Writing expressions

  11.3 Simplifying expressions

  11.4 Evaluating expressions

  11.5 Squaring

  11.6 Brackets

  11.7 Factorizing expressions

  11.8 Indices

  11.9 Laws of indices

  11.10 Simplifying expressions with indices

  12 Approximation

  12.1 Introduction

  12.2 Rounding whole numbers

  12.3 Rounding with decimals

  12.4 Significant figures

  12.5 Estimates

  12.6 Rounding in practical problems

  12.7 Accuracy of measurements

  13 Equations 1

  13.1 Introduction

  13.2 Finding missing numbers

  13.3 Solving linear equations

  13.4 Equations with brackets

  13.5 Solving problems using equations

  13.6 Solving inequalities

  14 Percentages

  14.1 Introduction

  14.2 Percentages, decimals, fractions

  14.3 Percentages of quantities

  14.4 Increasing and decreasing quantities

  14.5 One quantity as a percentage of another quantity

  14.6 Percentage increases

  14.7 Using multipliers

  15 Formulae

  15.1 What is a formula?

  15.2 Evaluating terms other than the subject

  15.3 Changing the subject of a formula

  16 Circles

  16.1 Introduction

  16.2 Circumference of a circle

  16.3 Area of a circle

  16.4 Two properties of circles

  17 Probability

  17.1 Introduction

  17.2 Relative frequency

  17.3 Probability of a single event

  17.4 Two events

  17.5 Tree diagrams

  17.6 Expected frequency

  18 Three-dimensional shapes

  18.1 Introduction

  18.2 Nets and surface area

  18.3 Volume of a cuboid

  18.4 Volume of a prism

  18.5 Weight of a prism

  19 Ratio and proportion

  19.1 What is a ratio?

  19.2 Scales

  19.3 Using ratio

  19.4 Direct proportion

  19.5 Inverse proportion

  20 Pythagoras’ theorem and trigonometry

  20.1 Pythagoras’ theorem

  20.2 Using Pythagoras’ theorem

  20.3 Proof of Pythagoras’ theorem

  20.4 Pythagoras’ theorem problems

  20.5 Trigonometry

  20.6 The tangent ratio

  20.7 Values of the tangent

  20.8 Using tangents

  20.9 Sine and cosine

  21 Indices and standard form

  21.1 Indices

  21.2 Laws of indices

  21.3 Prime factors

  21.4 Highest common factor

  21.5 Lowest common multiple

  21.6 Standard form – large numbers

  21.7 Standard form – small numbers

  21.8 Standard form calculations

  22 Statistics 2

  22.1 Averages

  22.2 The mode

  22.3 The median

  22.4 The mean

  22.5 The range

  22.6 Frequency tables

  22.7 Grouped frequency tables

  23 Graphs 2

  23.1 Equations of straight lines

  23.2 Drawing straight-line graphs

  23.3 The gradient of a straight line

  23.4 Curved graphs

  24 Equations 2

  24.1 Simultaneous equations

  24.2 Algebraic methods

  24.3 Quadratic expressions

  24.4 Factorizing quadratic expressions

  24.5 Quadratic equations

  24.6 Solution by factorizing

  24.7 Cubic equations

  Answers

  Taking it further

  Welcome to Mathematics: A complete introduction!

  Mathematics: A complete introduction aims to give you a broad mathematical experience and a firm foundation for further study. The book will also be a useful source of reference for homework or revision for students who are studying a mathematics course.

  This book is primarily aimed at students who do not have teacher support. Consequently, the explanations are detailed and there are numerous worked examples. Throughout this new edition are author insight boxes which provide additional support, including memory aids and
tips on how to avoid common pitfalls. Also new to this book are end of chapter summaries of all the key points and formulae.

  Access to a calculator has been assumed throughout the book. Generally, a basic calculator is adequate but, for Chapters 20 and 21, a scientific calculator, that is, a calculator which includes the trigonometric functions of sine, cosine and tangent, is essential. Very little else, other than some knowledge of basic arithmetic, has been assumed.

  To derive maximum benefit from Mathematics: A complete introduction, work through and don’t just read through the book. This applies not only to the exercises but also to the examples. Doing this will help your mathematical confidence grow.

  The authors would like to thank the staff of Hodder Education for the valuable advice they have given.

  Trevor Johnson, Hugh Neill

  Introduction

  Mathematics is probably the most versatile subject there is. It has applications in a wide range of other subjects from science and geography through to art. In science, without mathematics there would be no formulae or equations describing the laws of physics which have enabled the technology that we use each day to develop. Mathematics enables psychologists to find out more about human behaviour, looking for correlations between sets of statistics. Doctors and pharmaceutical companies use mathematics to interpret the data resulting from a drug trial. An understanding of mathematics is essential in architecture and engineering. Mathematics is also used to model situations such as the stock market or the flow of traffic on motorways.

  Mathematics is also intrinsically beautiful. Many famous artists used mathematics as a basis for their work – Kandinsky made extensive use of geometry in his paintings and Escher is famed for his complicated tessellations and optical illusions. Mathematics also occurs naturally. For example, the patterns on the wings of a butterfly or moth are symmetrical, and the pattern of seeds on a sunflower head, the scales of a pinecone and the spiral of a snail’s shell all follow the same sequence of numbers – called the Fibonacci sequence.

  Maths really is ubiquitous and having an understanding of the basics of mathematics is invaluable to many aspects of modern life.

  The word ‘mathematics’ originally comes from the Greek for knowledge and science. To most people maths means arithmetic – addition, subtraction, multiplication and division – which are the building blocks for maths. Without these we couldn’t interpret large sets of data, solve an equation or work out the area of a shape.

  We live in a technological age and mathematics underpins all of this technology. It is hard to think of an aspect of modern life that is not reliant in some way on technology and therefore mathematics. Mathematics is also intrinsic to many careers: civil engineers rely on mathematics to work out whether the bridge they are building will actually support a given load and surveyors use trigonometry to calculate the distance between two points. Insurance companies use complex probability tables for their life assurance policies, the self-employed use percentages to file their tax returns, while pharmacists use formulae to work out the correct dosage for a particular drug and sportspeople are constantly using forms of intuitive mathematics.

  Many people lack confidence in mathematics and feel that they are not very good in this subject. It seems acceptable to say, ‘Oh, I’m hopeless at maths’, in a way that would not be accepted about reading or writing. However, without realising it, every day we are using maths. Working out which box of cereal is best value for money involves proportion. Placing a bet on a horse requires an understanding of probability. Calculating the best mobile phone tariff involves using simple formulae. Working out how much tax you should pay for your tax return needs a good grasp of percentages. When reading the newspaper we are bombarded with statistics – results of surveys, charts and graphs – which we need to be able to interpret so that we can make informed choices about our daily lives.

  Maths is also fun, a fact which is often forgotten! Most newspapers run a daily maths type puzzle: Sudoku, Futoshiki and Kakuro are three popular puzzles which involve numbers, logical thinking and problem-solving skills. Researchers are discovering more and more health benefits from solving the ubiquitous brain training maths style puzzles. Many of these puzzles are not only fun but are also positively good for you. Recent research suggests that keeping the mind stimulated with activities such as Sudoku and crossword puzzles may help prevent Alzheimer’s disease and can improve your cognitive abilities.

  In today’s society being numerate is at least as important a skill as being literate. The key mathematics skills – such as basic numeracy and problem solving – are essential to many careers and highly valued by employers. Being proficient in mathematics has a positive impact on career development and employability, as well as self-esteem. Mathematics: A complete introduction takes you through worked examples and has carefully graded exercises designed to improve confidence and problem-solving skills.

  1

  Number

  In this chapter you will learn:

  • about place value

  • about the four operations of arithmetic

  • about the order in which arithmetic operations should be carried out

  • about some special numbers.

  1.1 Introduction – place value

  Over the centuries, many systems of writing numbers have been used. For example, the number which we write as 17 is written as XVII using Roman numerals.

  Our system of writing numbers is called the decimal system, because it is based on ten, the number of fingers and thumbs we have.

  The decimal system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

  The place of the digit in a number tells you the value of that digit.

  When you write a number, the value of the digit in each of the first four columns, starting from the right, is

  Thus, in the number 3652, the value of the 2 is two units and the value of the 6 is six hundreds.

  The decimal system uses zero to show that a column is empty. So 308 means three hundreds and eight units; that is, there are no tens. The number five thousand and twenty-three would be written 5023, using the zero to show that there are no hundreds.

  Extra columns may be added to deal with larger numbers, the value of each column being ten times greater than that of the column on its immediate right. It is usual, however, to split large numbers up into groups of three digits.

  You read 372 891 as three hundred and seventy-two thousand, eight hundred and ninety-one.

  In words, 428 763 236 is four hundred and twenty-eight million, seven hundred and sixty-three thousand, two hundred and thirty-six.

  Exercise 1.1

  1 What is the value of the 5 in each of the following numbers?

  (a) 357

  (b) 598

  (c) 5842

  (d) 6785

  2 Write these numbers in figures.

  (a) seventeen

  (b) seventy

  (c) ninety-seven

  (d) five hundred and forty-six

  (e) six hundred and three

  (f) eight hundred and ten

  (g) four hundred and fifty

  (h) ten thousand

  (i) eight thousand, nine hundred and thirty-four

  (j) six thousand, four hundred and eighty

  (k) three thousand and six

  3 Write these numbers completely in words.

  (a) 52

  (b) 871

  (c) 5624

  (d) 980

  (e) 7001

  (f) 35013

  (g) 241001

  (h) 1001312

  4 Write down the largest number and the smallest number you can make using

  (a) 5, 9 and 7

  (b) 7, 2, 1 and 9.

  5 Write these numbers completely in words.

  (a) 342785

  (b) 3783194

  (c) 17021209

  (d) 305213097

  6 Write these numbers in figures.

  (a) five hundred and sixteen thousand, two hundred and nineteen
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  (b) two hundred and six thousand, and twenty-four

  (c) twenty-one million, four hundred and thirty-seven thousand, eight hundred and sixty-nine

  (d) seven million, six hundred and four thousand, and thirteen

  1.2 Arithmetic – the four operations

  In this book, the authors assume that you have access to a calculator, and that you are able to carry out simple examples of the four operations of arithmetic – addition, subtraction, multiplication and division – either in your head or on paper. If you have difficulty with this, you may wish to consult Teach Yourself Basic Mathematics.

  Each of the operations has a special symbol, and a name for the result.

  Adding 4 and 3 is written 4 + 3, and the result, 7, is called the sum of 4 and 3.

  Subtracting a number, 4, from a larger number, 7, is written 7 − 4, and the result, 3, is their difference.

  Addition and subtraction are ‘reverse’ processes:

  4 + 3 = 7 and 7 − 4 = 3.

  Multiplying 8 by 4 is written 8 × 4, and the result, 32, is called the product of 8 by 4.

  Dividing 8 by 4 is written 8 ÷ 4, or , and the result, 2, is called their quotient.

  Multiplication and division are ‘reverse’ processes:

  4 × 2 = 8 and 8 ÷ 4 = 2.

  Sometimes divisions are not exact. For example, if you divide 7 by 4, the quotient is 1 and the remainder is 3, because 4 goes into 7 once and there is 3 left over.

  Exercise 1.2 gives you practice with the four operations. You should be able to do all of this exercise without a calculator.

  Exercise 1.2

  1 Work out the results of the following additions.

  (a) 23 + 6

  (b) 33 + 15

  (c) 45 + 9

  (d) 27 + 34

  2 Find the sum of

  (a) 65, 24 and 5

  (b) 27, 36 and 51.

  3 Work out the following differences.

  (a) 73 − 9

  (b) 49 − 35

  (c) 592 − 76

  (d) 128 − 43

  (e) 124 − 58

  (f) 171 − 93

  4 Find the difference between

  (a) 152 and 134

  (b) 317 and 452.

  5 Work out

  (a) 5 × 9

  (b) 43 × 7

  (c) 429 × 6

  (d) 508 × 7

  (e) 27 × 11

  (f) 12 × 28

  (g) 13 × 14

  (h) 21 × 21

  6 Find the product of 108 and 23.

  7 Work out

  (a) 375 ÷ 5

  (b) 846 ÷ 3