Help Your Kids with Math
DKReduce the stress of studying algebra, geometry, and statistics and help your child with their math homework, following Carol Vorderman's unique visual math book.
Help Your Kids with Math shows parents how to work with their kids to solve math problems stepbystep. Using pictures, diagrams, and easytofollow instructions and examples to cover all the important areas  covering everything from basic numeracy to more challenging subjects like statistics, trigonometry, and algebra  you'll learn to approach even the most complex mathproblems with confidence. This visual math guide has been updated and includes the latest changes to school curriculum and with additional content on roman numerals, time, fractions, and times tables. It also includes a glossary of key math terms and symbols.
Help Your Kids with Math is the perfect guide for every frustrated parent and desperate child, who wants to understand math and put it into practice.
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R NE evised co Mo W and pi re E es th D upd so an IT a ld 4 IO ted w 00 N o rld ,00 w 0 id e HELP YOUR KIDS WITH A UNIQUE STEPBYSTEP VISUAL GUIDE m ths a HELP YOUR KIDS WITH m th a HELP YOUR KIDS WITH A UNIQUE STEPBYSTEP VISUAL GUIDE LONDON, NEW YORK, MELBOURNE, MUNICH, AND DELHI Project Art Editor Mark Lloyd Designers Nicola Erdpresser, Riccie Janus, Maxine Pedliham, Silke Spingies, Rebecca Tennant Design Assistants Thomas Howey, Fiona Macdonald Production Editor Luca Frassinetti Production Erika Pepe Jacket Designer Duncan Turner Project Editor Nathan Joyce Editors Nicola Deschamps, Martha Evatt, Lizzie Munsey, Martyn Page, Laura Palosuo, Peter Preston, Miezan van Zyl US Editor Jill Hamilton US Consultant Alison Tribley Indexer Jane Parker Managing Editor Sarah Larter Managing Art Editor Michelle Baxter Publishing Manager Liz Wheeler Art Director Phil Ormerod Reference Publisher Jonathan Metcalf First American Edition, 2010 This Edition, 2014 Published in the United States by DK Publishing 345 Hudson Street New York, New York 10014 10 11 12 13 14 10 9 8 7 6 5 4 3 2 1 001–263995 – Jul/2014 Copyright © 2010, 2014 Dorling Kindersley Limited All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording, or otherwise) without prior written permission of the copyright owner and the above publisher of this book. Published in Great Britain by Dorling Kindersley Limited. A catalog record for this book is available from the Library of Congress. ISBN 9781465421661 DK books are available at special discounts when purchased in bulk for sales promotions, premiums, fundraising, or educational use. For details contact: DK Publishing Special Markets, 345 Hudson Street, New York, New York 10014 or SpecialSales@dk.com. Printed and bound by South China Printing Company, China Discover more at www.dk.com CARO; L VORDERMAN M.A.(Cantab), MBE is one of Britain’s bestloved TV personalities and is renowned for her excellent math skills. She has hosted numerous shows, from light entertainment with Carol Vorderman’s Better Homes and The Pride of Britain Awards, to scientific programs such as Tomorrow’s World, on the BBC, ITV, and Channel 4. Whether cohosting Channel 4’s Countdown for 26 years, becoming the secondbestselling female nonfiction author of the 2000s in the UK, or advising Parliament on the future of math education in the UK, Carol has a passion for and devotion to explaining math in an exciting and easily understandable way. BARRY LEWIS (Consultant Editor, Numbers, Geometry, Trigonometry, Algebra) studied math in college and graduated with honors. He spent many years in publishing, as an author and as an editor, where he developed a passion for mathematical books that presented this often difficult subject in accessible, appealing, and visual ways. He is the author of Diversions in Modern Mathematics, which subsequently appeared in Spanish as Matemáticas modernas. Aspectos recreativos. He was invited by the British government to run the major initiative Maths Year 2000, a celebration of mathematical achievement with the aim of making the subject more popular and less feared. In 2001 Barry became the president of the UK’s Mathematical Association, and was elected as a fellow of the Institute of Mathematics and its Applications, for his achievements in popularizing mathematics. He is currently the Chair of Council of the Mathematical Association and regularly publishes articles and books dealing with both research topics and ways of engaging people in this critical subject. ANDREW JEFFREY (Author, Probability) is a math consultant, well known for his passion and enthusiasm for the teaching and learning of math. A teacher for over 20 years, Andrew now spends his time training, coaching, and supporting teachers and delivering lectures for various organizations throughout Europe. He has written many books on the subject of math and is better known to many schools as the “Mathemagician.” MARCUS WEEKS (Author, Statistics) is the author of many books and has contributed to several reference books, including DK’s Science: The Definitive Visual Guide and Children’s Illustrated Encyclopedia. SEAN MCARDLE (Consultant) was head of math in two primary schools and has a Master of Philosophy degree in Educational Assessment. He has written or cowritten more than 100 mathematical textbooks for children and assessment books for teachers. Contents F O R E W O R D b y C a ro l Vo rd e r m a n 8 I N T R O D U C T I O N b y B a r r y Le w i s 10 1 NUMBERS 2 GEOMETRY Introducing numbers 14 What is geometry? 80 Addition 16 Tools in geometry 82 Subtraction 17 Angles 84 Multiplication 18 Straight lines 86 Division 22 Symmetry 88 Prime numbers 26 Coordinates 90 Units of measurement 28 Vectors 94 Telling the time 30 Translations 98 Roman numerals 33 Rotations 100 Positive and negative numbers 34 Reflections 102 Powers and roots 36 Enlargements 104 Surds 40 Scale drawings 106 Standard form 42 Bearings 108 Decimals 44 Constructions 110 Binary numbers 46 Loci 114 Fractions 48 Triangles 116 Ratio and proportion 56 Constructing triangles 118 Percentages 60 Congruent triangles 120 Area of a triangle 122 Converting fractions, decimals, and percentages 64 Similar triangles 125 Mental math 66 Pythagorean Theorem 128 Rounding off 70 Quadrilaterals 130 Using a calculator 72 Polygons 134 Personal finance 74 Circles 138 Business finance 76 Circumference and diameter 140 Area of a circle 142 The quadratic formula 192 Angles in a circle 144 Quadratic graphs 194 Chords and cyclic quadrilaterals 146 Inequalities 198 Tangents 148 Arcs 150 Sectors 151 Solids 152 What is statistics? 202 Volumes 154 Collecting and organizing data 204 Surface area of solids 156 Bar graphs 206 Pie charts 210 Line graphs 212 Averages 214 3 TRIGONOMETRY 5 STATISTICS What is trigonometry? 160 Moving averages 218 Using formulas in trigonometry 161 Measuring spread 220 Finding missing sides 162 Histograms 224 Finding missing angles 164 Scatter diagrams 226 4 6 PROBABILITY ALGEBRA What is algebra? 168 What is probability? 230 Sequences 170 Expectation and reality 232 Working with expressions 172 Combined probabilities 234 Expanding and factorizing expressions 174 Dependent events 236 Quadratic expressions 176 Tree diagrams 238 Formulas 177 Solving equations 180 Reference section 240 Linear graphs 182 Glossary 252 Simultaneous equations 186 Index 258 Factorizing quadratic equations 190 Acknowledgments 264 Foreword Hello Welcome to the wonderful world of math. Research has shown just how important it is for parents to be able to help children with their education. Being able to work through homework together and enjoy a subject, particularly math, is a vital part of a child’s progress. However, math homework can be the cause of upset in many households. The introduction of new methods of arithmetic hasn’t helped, as many parents are now simply unable to assist. We wanted this book to guide parents through some of the methods in early arithmetic and then for them to go on to enjoy some deeper mathematics. As a parent, I know just how important it is to be aware of it when your child is struggling and equally, when they are shining. By having a greater understanding of math, we can appreciate this even more. Over nearly 30 years, and for nearly every single day, I have had the privilege of hearing people’s very personal views about math and arithmetic. Many weren’t taught math particularly well or in an interesting way. If you were one of those people, then we hope that this book can go some way to changing your situation and that math, once understood, can begin to excite you as much as it does me. CAROL VORDERMAN π =3.1415926535897932384626433832 7950288419716939937510582097494 4592307816406286208998628034853 4211706798214808651328230664709 3844609550582231725359408128481 11745028410270193852110555964462 2948954930381964428810975665933 4461284756482337867831652712019 0914564856692346034861045432664 8213393607260249141273724587006 6063155881748815209209628292540 91715364367892590360011330530548 8204665213841469519451160943305 72703657595919530921861173819326 11793105118548074462379962749567 3518857527248912279381830119491 Introduction This book concentrates on the math tackled in schools between the ages of 9 and 16. But it does so in a gripping, engaging, and visual way. Its purpose is to teach math by stealth. It presents mathematical ideas, techniques, and procedures so that they are immediately absorbed and understood. Every spread in the book is written and presented so that the reader will exclaim, ”Ah ha—now I understand!” Students can use it on their own; equally, it helps parents understand and remember the subject and thus help their children. If parents too gain something in the process, then so much the better. At the start of the new millennium I had the privilege of being the director of the United Kingdom’s Maths Year 2000, a celebration of math and an international effort to highlight and boost awareness of the subject. It was supported by the British government and Carol Vorderman was also involved. Carol championed math across the British media, and is well known for her astonishingly agile ways of manipulating and working with numbers—almost as if they were her personal friends. My working, domestic, and sleeping hours are devoted to math—finding out how various subtle patterns based on counting items in sophisticated structures work and how they hang together. What united us was a shared passion for math and the contribution it makes to all our lives—economic, cultural, and practical. How is it that in a world ever more dominated by numbers, math—the subtle art that teases out the patterns, the harmonies, and the textures that make up the relationships between the numbers—is in danger? I sometimes think that we are drowning in numbers. As employees, our contribution is measured by targets, statistics, workforce percentages, and adherence to budget. As consumers, we are counted and aggregated according to every act of consumption. And in a nice subtlety, most of the products that we do consume come complete with their own personal statistics—the energy in a can of beans and its “low” salt content; the story in a newspaper and its swath of statistics controlling and interpreting the world, developing each truth, simplifying each problem. Each minute of every hour, each hour of every day, we record and publish ever more readings from our collective lifesupport machine. That is how we seek to understand the world, but the problem is, the more figures we get, the more truth seems to slip through our fingers. The danger is, despite all the numbers and our increasingly numerate world, math gets left behind. I’m sure that many think the ability to do the numbers is enough. Not so. Neither as individuals, nor collectively. Numbers are pinpricks in the fabric of math, blazing within. Without them we would be condemned to total darkness. With them we gain glimpses of the sparkling treasures otherwise hidden. This book sets out to address and solve this problem. Everyone can do math. BARRY LEWIS Former President, The Mathematical Association; Director Maths Year 2000. 1 Numbers 14 NUMBERS 2 Introducing numbers COUNTING AND NUMBERS FORM THE FOUNDATION OF MATHEMATICS. Numbers are symbols that developed as a way to record amounts or quantities, but over centuries mathematicians have discovered ways to use and interpret numbers in order to work out new information. each bead represents one unit What are numbers? units of 10, so two beads represent 20 Numbers are basically a set of standard symbols that represent quantities—the familiar 0 to 9. In addition to these whole numbers (also called integers) there are also fractions (see pp.48–55) and decimals (see pp.44–45). Numbers can also be negative, or less than zero (see pp.34–35). whole number 1 –2 decimal fraction negative number 1 3 units of 100, so one bead represents 100 0.4 △ Types of numbers Here 1 is a positive whole number and 2 is a negative number. The symbol 1⁄3 represents a fraction, which is one part of a whole that has been divided into three parts. A decimal is another way to express a fraction. LOOKING CLOSER Zero The use of the symbol for zero is considered an important advance in the way numbers are written. Before the symbol for zero was adopted, a blank space was used in calculations. This could lead to ambiguity and made numbers easier to confuse. For example, it was diﬃcult to distinguish between 400, 40, and 4, since they were all represented by only the number 4. The symbol zero developed from a dot ﬁrst used by Indian mathematicians to act a placeholder. 07:08 zero is important for 24hour timekeeping ◁ Abacus The abacus is a traditional calculating and counting device with beads that represent numbers. The number shown here is 120. ◁ Easy to read The zero acts as a placeholder for the “tens,” which makes it easy to distinguish the single minutes. ▽ First number One is not a prime number. It is called the “multiplicative identity,” because any number multiplied by 1 gives that number as the answer. 1 6 △ Perfect number This is the smallest perfect number, which is a number that is the sum of its positive divisors (except itself ). So, 1 + 2 + 3 = 6. ▽ Even prime number The number 2 is the only evennumbered prime number—a number that is only divisible by itself and 1 (see pp.26–27). 2 7 △ Not the sum of squares The number 7 is the lowest number that cannot be represented as the sum of the squares of three whole numbers (integers). INTRODUCING NUMBERS REAL WORLD Number symbols Many civilizations developed their own symbols for numbers, some of which are shown below, together with our modern Hindu–Arabic number system. One of the main advantages of our modern number system is that arithmetical operations, such as multiplication and division, are much easier to do than with the more complicated older number systems. Modern Hindu–Arabic 1 2 3 4 5 6 7 8 9 10 I II III IV V VI VII VIII IX X Mayan Ancient Chinese Ancient Roman Ancient Egyptian Babylonian ▽ Triangular number This is the smallest triangular number, which is a positive whole number that is the sum of consecutive whole numbers. So, 1 + 2 = 3. 3 8 △ Fibonacci number The number 8 is a cube number (23 = 8) and it is the only positive Fibonacci number (see p.171), other than 1, that is a cube. ▽ Composite number The number 4 is the smallest composite number —a number that is the product of other numbers. The factors of 4 are two 2s. 4 9 △ Highest decimal The number 9 is the highest singledigit whole number and the highest singledigit number in the decimal system. ▽ Prime number This is the only prime number to end with a 5. A 5sided polygon is the only shape for which the number of sides and diagonals are equal. 5 10 △ Base number The Western number system is based on the number 10. It is speculated that this is because humans used their fingers and toes for counting. 15 16 NUMBERS + Addition SEE ALSO NUMBERS ARE ADDED TOGETHER TO FIND THEIR TOTAL. THIS RESULT IS CALLED THE SUM. An easy way to work out the sum of two numbers is a number line. It is a group of numbers arranged in a straight line that makes it possible to count up or down. In this number line, 3 is added to 1. start at 1 0 move three steps along +1 +1 +1 Adding up 1 2 3 sign for addition ▷ What it means The result of adding 3 to the start number of 1 is 4. This means that the sum of 1 and 3 is 4. 17 Subtraction + 1+ total 4 = = NUMBER TO ADD 5 4 TOTAL, RESULT, OR SUM Adding large numbers Numbers that have two or more digits are added in vertical columns. First, add the ones, then the tens, the hundreds, and so on. The sum of each column is written beneath it. If the sum has two digits, the ﬁrst is carried to the next column. hundreds tens ones 928 + 191 space at foot of column for sum First, the numbers are written with their ones, tens, and hundreds directly above each other. carry 1 working from right, ﬁrst add ones 9 + 1 + the carried 1 = 11 add tens 1 1 928 + 191 9 928 + 191 19 Next, add the ones 1 and 8 and write their sum of 9 in the space underneath the ones column. The sum of the tens has two digits, so write the second underneath and carry the first to the next column. the ﬁrst 1 of 11 goes in the thousands column, while the second goes in the hundreds column ◁ Use a number line To add 3 to 1, start at 1 and move along the line three times—first to 2, then to 3, then to 4, which is the answer. equals sign leads to answer 3 FIRST NUMBER Positive and negative numbers 34–35 928 + 191 1,119 the answer is 1,119 Then add the hundreds and the carried digit. This sum has two digits, so the first goes in the thousands column. 17 ADDITION AND SUBTRACTION – Subtraction SEE ALSO 16 Addition Positive and negative numbers 34–35 A NUMBER IS SUBTRACTED FROM ANOTHER NUMBER TO FIND WHAT IS LEFT. THIS IS KNOWN AS THE DIFFERENCE. –1 –1 –1 Taking away A number line can also be used to show how to subtract numbers. From the ﬁrst number, move back along the line the number of places shown by the second number. Here 3 is taken from 4. 0 1 2 3 ◁ Use a number line To subtract 3 from 4, start at 4 and move three places along the number line, first to 3, then 2, and then to 1. start at 4, then move three places to left 4 5 equals sign leads to answer sign for subtraction – – 4 ▷ What it means The result of subtracting 3 from 4 is 1, so the difference between 3 and 4 is 1. 3 FIRST NUMBER = = 1 NUMBER TO SUBTRACT RESULT OR DIFFERENCE Subtracting large numbers Subtracting numbers of two or more digits is done in vertical columns. First subtract the ones, then the tens, the hundreds, and so on. Sometimes a digit is borrowed from the next column along. hundreds tens ones 928 – 191 number to be subtracted from number to subtract First, the numbers are written with their ones, tens, and hundreds directly above each other. subtract ones 928 – 191 7 Next, subtract the unit 1 from 8, and write their difference of 7 in the space underneath them. ﬁrst, borrow 1 from hundreds 8 1 928 – 191 37 then, carry 1 to tens In the tens, 9 cannot be subtracted from 2, so 1 is borrowed from the hundreds, turning 9 into 8 and 2 into 12. subtract 1 from 8 8 1 928 – 191 737 In the hundreds column, 1 is subtracted from the new, now lower number of 8. the answer is 737 18 NUMBERS × Multiplication SEE ALSO 16–17 Addition and Subtraction MULTIPLICATION INVOLVES ADDING A NUMBER TO ITSELF A NUMBER OF TIMES. THE RESULT OF MULTIPLYING NUMBERS IS CALLED THE PRODUCT. Division 22–25 Decimals 44–45 What is multiplication? The second number in a multiplication sum is the number to be added to itself and the ﬁrst is the number of times to add it. Here the number of rows of people is added together a number of times determined by the number of people in each row. This multiplication sum gives the total number of people in the group. 9 rows of people 13 people in each row 9 8 7 6 4 3 1 there are 13 people in each row 2 9 × 13 there are 9 rows of people 7 5 multiplication sign 1 2 3 4 5 8 9 6 △ How many people? The number of rows (9) is multiplied by the number of people in each row (13). The total number of people is 117. this sum means 13 added to itself 9 times 9 × 13 = 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 = 117 product of 9 and 13 is 117 10 11 1 19 M U LT I P L I C AT I O N Works both ways It does not matter which order numbers appear in a multiplication sum because the answer will be the same either way. Two methods of the same multiplication are shown here. 4 × 3 = 3 + 3 + 3 + 3 = 12 3 added to itself four times is 12 3 2 1 2 = 4 3 + + + 1 3 × 4 = 4 + 4 + 4 = 12 4 added to itself three times is 12 = 4 3 2 1 13 2 1 2 + + 3 1 Multiplying by 10, 100, 1,000 Patterns of multiplication Multiplying whole numbers by 10, 100, 1,000, and so on involves adding one zero (0), two zeroes (00), three zeroes (000), and so on to the right of the number being multiplied. There are quick ways to multiply two numbers, and these patterns of multiplication are easy to remember. The table shows patterns involved in multiplying numbers by 2, 5, 6, 9, 12, and 20. PAT T E R N S O F M U LT I P L I C AT I O N add 0 to end of number 34 × 10 = 340 add 00 to end of number To multiply How to do it 2 add the number to itself 2 × 11 = 11 + 11 = 22 5 the last digit of the number follows the pattern 5, 0, 5, 0 5, 10, 15, 20 6 multiplying 6 by any even number gives an 6 × 12 = 72 answer that ends in the same last digit as 6 × 8 = 48 the even number 9 multiply the number by 10, then subtract the number 9 × 7 = 10 × 7 – 7 = 63 12 multiply the original number first by 10, then multiply the original number by 2, and then add the two answers 12 × 10 = 120 12 × 2 = 24 120 + 24 = 144 20 multiply the number by 10 then multiply the answer by 2 14 × 20 = 14 × 10 = 140 140 × 2 = 280 72 × 100 = 7,200 add 000 to end of number 18 × 1,000 = 18,000 Example to multiply 20 NUMBERS MULTIPLES When a number is multiplied by any whole number the result (product) is called a multiple. For example, the first six multiples of the number 2 are 2, 4, 6, 8, 10, and 12. This is because 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, 2 × 4 = 8, 2 × 5 = 10, and 2 × 6 = 12. MULTIPLES OF 3 3×1= 3 3×2= 6 3×3= 9 3 × 4 = 12 3 × 5 = 15 first five multiples of 3 MULTIPLES OF 8 MULTIPLES OF 12 8×1= 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40 12 × 1 = 12 12 × 2 = 24 12 × 3 = 36 12 × 4 = 48 12 × 5 = 60 Common multiples Two or more numbers can have multiples in common. Drawing a grid, such as the one on the right, can help ﬁnd the common multiples of diﬀerent numbers. The smallest of these common numbers is called the lowest common multiple. 24 Lowest common multiple The lowest common multiple of 3 and 8 is 24 because it is the smallest number that both multiply into. 1 2 first five multiples of 8 3 4 5 6 first five multiples of 12 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 multiples of 3 51 52 53 54 55 56 57 58 59 60 multiples of 8 multiples of 3 and 8 ▷ Finding common multiples Multiples of 3 and multiples of 8 are highlighted on this grid. Some multiples are common to both numbers. 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 21 M U LT I P L I C AT I O N Short multiplication Multiplying a large number by a singledigit number is called short multiplication. The larger number is placed above the smaller one in columns arranged according to their value. 4 carried to tens column 4 196 × 7 2 6 carried to hundreds column 6 written in ones column 2 written in ones column 64 7 written in tens column To multiply 196 and 7, first multiply the ones 7 and 6. The product is 42, the 4 of which is carried. 64 1 written in hundreds column 196 × 7 72 196 × 7 1,372 3 written in hundreds column; 1 written in thousands column Next, multiply 7 and 9, the product of which is 63. The carried 4 is added to 63 to get 67. 1,372 is ﬁnal answer Finally, multiply 7 and 1. Add its product (7) to the carried 6 to get 13, giving a final product of 1,372. Long multiplication Multiplying two numbers that both contain at least two digits is called long multiplication. The numbers are placed one above the other, in columns arranged according to their value (ones, tens, hundreds, and so on). 428 multiplied by 1 428 × 111 428 First, multiply 428 by 1 in the ones column. Work digit by digit from right to left so 8 × 1, 2 × 1, and then 4 × 1. 428 × 111 428 4,280 428 multiplied by 10 add 0 when multiplying by 10 Multiply 428 digit by digit by 1 in the tens column. Remember to add 0 when multiplying by a number in the tens place. 428 multiplied by 100 428 × 111 428 4,280 42,800 add 00 when multiplying by 100 Multiply 428 digit by digit by 1 in the hundreds column. Add 00 when multiplying by a digit in the hundreds place. 428 × 111 428 + 4,280 42,800 = 47,508 Add together the products of the three multiplications. The answer is 47,508. LOOKING CLOSER Box method of multiplication ▷ The final step Add together the nine multiplications to find the final answer. 4 2 8 W R I T T E N I N 100 S , 10 S , A N D O N E S 111 W R I T T E N I N 100 S , 10S, AND ONES The long multiplication of 428 and 111 can be broken down further into simple multiplications with the help of a table or box. Each number is reduced to its hundreds, tens, and ones, and multiplied by the other. 400 20 8 100 400 × 100 = 40,000 20 × 100 = 2,000 8 × 100 = 800 10 400 × 10 = 4,000 20 × 10 = 200 8 × 10 = 80 1 400 × 1 = 400 20 × 1 = 20 8×1 =8 40,000 2,000 800 4,000 200 80 400 20 + 8 = 47,508 this is the ﬁnal answer 22 NUMBERS Division SEE ALSO 18–21 Multiplication Ratio and 16–17 Addition and subtraction DIVISION INVOLVES FINDING OUT HOW MANY TIMES ONE NUMBER GOES INTO ANOTHER NUMBER. 56–59 proportion There are two ways to think about division. The first is sharing a number out equally (10 coins to 2 people is 5 each). The other is dividing a number into equal groups (10 coins into piles containing 2 coins each is 5 piles). How division works ÷ Dividing one number by another ﬁnds out how many times the second number (the divisor) ﬁts into the ﬁrst (the dividend). For example, dividing 10 by 2 ﬁnds out how many times 2 ﬁts into 10. The result of the division is known as the quotient. / ◁ Division symbols There are three main symbols for division that all mean the same thing. For example, “6 divided by 3” can be expressed as 6 ÷ 3, 6/3, or –. 36 4 3 2 8 7 6 10 D is d EN hat are I D r t sh er I V be or b D num ided r num e iv e Th g d o t h in an b e by ÷ 5 1 ▽ Division as sharing Sharing equally is one type of division. Dividing 4 candies equally between 2 people means that each person gets the same number of candies: 2 each. = ÷ 4 CANDIES ÷ 2 PEOPLE = 2 CANDIES PER PERSON How division is linked to multiplication Division is the direct opposite or “inverse” of multiplication, and the two are always connected. If you know the answer to a particular division, you can form a multiplication from it and vice versa. 10÷ 2=5 5 × 2=10 ◁ Back to the beginning If 10 (the dividend) is divided by 2 (the divisor), the answer (the quotient) is 5. Multiplying the quotient (5) by the divisor of the original division problem (2) results in the original dividend (10). R s O at i e I S th ivid I V er d D u m b e d to e n d d n e us ivi Th i n g e d be th 3 LOOKING CLOSER DIVISION 23 Another approach to division Division can also be viewed as finding out how many groups of the second number (divisor) are contained in the first number (dividend). The operation remains the same in both cases. 10NDIES CA 10 ▽ Introducing remainders In this example, 10 candies are being divided among 3 girls. However, 3 does not divide exactly into 10—it fits 3 times with 1 left over. The amount left over from a division sum is called the remainder. DIV 9 This example shows 30 soccer balls, which are to be divided into groups of 3: group of three 3IRLS N ISIO G There are exactly 10 groups of 3 soccer balls, with no remainder, so 30 ÷ 3 = 10. DIVISION TIPS 3 3S 3 = IE ND CA EACH 3 1 AINING EM S R DIE N CA EN T r de ain m re 3 T I ult of n U O es io Q e r ivis Th e d th 1 A number is divisible by If... Examples 1 ER ver ot N D f t o ann her A I t le r c ot M un mbe o an R E e amo e nu ly int n t Th n o x a c he e w ide v di 2 the last digit is an even number 12, 134, 5,000 3 the sum of all digits when added together is divisible by 3 18 1+8 = 9 4 the number formed by the last two digits is divisible by 4 732 32 ÷ 4 = 8 5 the last digit is 5 or 0 25, 90, 835 6 the last digit is even and the sum of its digits when added together is divisible by 3 3,426 3+4+2+6 = 15 7 no simple divisibility test 8 the number formed by the last three digits is divisible by 8 7,536 536 ÷ 8 = 67 9 the sum of all of its digits is divisible by 9 6,831 6+8+3+1 = 18 10 the number ends in 0 30, 150, 4,270 24 NUMBERS LOOKING CLOSER Short division Converting remainders Short division is used to divide one number (the dividend) by another whole number (the divisor) that is less than 10. result is 132 start on the left with the ﬁrst 3 (divisor) 1 13 3 396 3 396 132 3 396 dividing line 396 is the dividend Divide the first 3 into 3. It fits once exactly, so put a 1 above the dividing line, directly above the 3 of the dividend. Move to the next column and divide 3 into 9. It fits three times exactly, so put a 3 directly above the 9 of the dividend. Divide 3 into 6, the last digit of the dividend. It goes twice exactly, so put a 2 directly above the 6 of the dividend. Carrying numbers 5 2,765 2,765 is the dividend Start with number 5. It does not divide into 2 because it is larger than 2. Instead, 5 will need to be divided into the first two digits of the dividend. 55 5 2,765 carry remainder 1 to next digit of dividend 2 2 1 carry remainder 2 to next digit of dividend 5 5 2,765 divide 5 into ﬁrst 2 digits of dividend divisor 1 22. 4 90.0 1 2 2 Divide 5 into 27. The result is 5 with a remainder of 2. Put 5 directly above the 7 and carry the remainder. the result is 553 553 5 2,765 2 2.5 4 90.0 1 Divide 4 into 20. It goes 5 times exactly, so put a 5 directly above the zero of the dividend and after the decimal point. 2 LOOKING CLOSER Making division simpler To make a division easier, sometimes the divisor can be split into factors. This means that a number of simpler divisions can be done. divisor is 6, which is 2 × 3. Splitting 6 into 2 and 3 simpliﬁes the sum 816÷6 816÷2 = 408 result is 136 408÷3 = 136 divide by ﬁrst factor of divisor 2 2 1 Divide 5 into 15. It fits three times exactly, so put 3 above the dividing line, directly above the final 5 of the dividend. 22 r 2 4 90 Carry the remainder (2) from above the dividing line to below the line and put it in front of the new zero. 2 divide by second factor of divisor This method of splitting the divisor into factors can also be used for more diﬃcult divisions. 405÷15 Divide 5 into 26. The result is 5 with a remainder of 1. Put 5 directly above the 6 and carry the remainder 1 to the next digit of the dividend. remainder Remove the remainder, 2 in this case, leaving 22. Add a decimal point above and below the dividing line. Next, add a zero to the dividend after the decimal point. 22. 4 90.0 1 When the result of a division gives a whole number and a remainder, the remainder can be carried over to the next digit of the dividend. start on the left When one number will not divide exactly into another, the answer has a remainder. Remainders can be converted into decimals, as shown below. 405÷5 = 81 splitting 15 into 5 and 3, which multiply to make 15, simpliﬁes the problem result is 27 81÷3 = 27 divide by ﬁrst factor of divisor divide result by second factor of divisor 25 DIVISION Long division Long division is usually used when the divisor is at least two digits long and the dividend is at least 3 digits long. Unlike short division, all the workings out are written out in full below the dividing line. Multiplication is used for ﬁnding remainders. A long division sum is presented in the example on the right. The answer (or quotient) goes in the space above the dividing line. the dividing line is used in place of ÷ or / sign DIVISOR 52 754 The calculations go in the space below the dividing line. number is used to divide dividend result is 1 1 52 754 multiply 1 (the number of times 52 goes into 75) by 52 to get 52 divide divisor into ﬁrst two digits of dividend subtract 52 from 75 Begin by dividing the divisor into the first two digits of the dividend. 52 fits into 75 once, so put a 1 above the dividing line, aligning it with the last digit of the number being divided. 14 52 754 –52 234 –208 26 multiply 4 (the number of times 52 goes into 234) by 52 to get 208 1 52 754 –52 23 amount left over from ﬁrst division add a decimal point then a zero amount left over from second division Work out the second remainder. The divisor, 52, does not divide into 234 exactly. To find the remainder, multiply 4 by 52 to make 208. Subtract 208 from 234, leaving 26. number that is divided by another number put result of second division above last digit being divided into divide divisor into 234 Work out the first remainder. The divisor 52 does not divide into 75 exactly. To work out the amount left over (the remainder), subtract 52 from 75. The result is 23. 14 52 754.0 –52 234 –208 260 14 52 754 –52 234 DIVIDEND bring down last digit of dividend and join it to remainder Now, bring down the last digit of the dividend and place it next to the remainder to form 234. Next, divide 234 by 52. It goes four times, so put a 4 next to the 1. add decimal point above other one 14.5 52 754.0 –52 234 –208 260 put result of last sum after decimal point bring down zero and join it to remainder There are no more whole numbers to bring down, so add a decimal point after the dividend and a zero after it. Bring down the zero and join it to the remainder 26 to form 260. Put a decimal point after the 14. Next, divide 260 by 52, which goes five times exactly. Put a 5 above the dividing line, aligned with the new zero in the dividend. 26 NUMBERS Prime numbers 11 SEE ALSO Multiplication 18–21 22–25 Division ANY WHOLE NUMBER LARGER THAN 1 THAT CANNOT BE DIVIDED BY ANY OTHER NUMBER EXCEPT FOR ITSELF AND 1. Introducing prime numbers 1 is not a prime number or a composite number Over 2,000 years ago, the Ancient Greek mathematician Euclid noted that some numbers are only divisible by 1 or the number itself. These numbers are known as prime numbers. A number that is not a prime is called a composite—it can be arrived at, or composed, by multiplying together smaller prime numbers, which are known as its prime factors. PICK A NUMBER FROM 1 TO 100 Is the number 2, 3, 5, or 7? NO YES Is it divisible by 2? NO YES THE NUMBER IS NOT PRIME Is it divisible by 3? NO YES YES Is it divisible by 7? NO YES 1 11 21 31 41 51 61 71 81 91 3 7 3 Is it divisible by 5? NO 2 is the only even prime number. No other even number is prime because they are all divisible by 2 THE NUMBER IS PRIME 3 △ Is a number prime? This flowchart can be used to determine whether a number between 1 and 100 is prime by checking if it is divisible by any of the primes 2, 3, 5, and 7. ▷ First 100 numbers This table shows the prime numbers among the first 100 whole numbers. 7 2 12 22 32 42 52 62 72 82 92 2 3 2 2 2 3 7 2 2 2 3 2 2 3 13 23 33 43 53 63 73 83 93 3 3 7 3 4 14 24 34 44 54 64 74 84 94 2 2 7 2 3 2 2 2 3 2 2 2 3 2 7 5 15 25 35 45 55 65 75 85 95 3 5 5 5 7 3 5 5 5 3 5 5 5 27 PRIME NUMBERS KEY Prime number A blue box indicates that the number is prime. It has no factors other than 1 and itself. 17 42 2 3 Composite number A yellow box denotes a composite number, which means that it is divisible by more than 1 and itself. 7 6 16 26 36 46 56 66 76 86 96 3 2 2 2 3 2 2 7 2 3 2 2 2 3 7 17 27 37 47 57 67 77 87 97 3 3 7 3 8 18 28 38 48 58 68 78 88 98 2 2 3 2 7 2 2 3 2 2 2 3 2 2 7 9 19 29 39 49 59 69 79 89 99 3 30 3 7 3 3 10 20 30 40 50 60 70 80 90 100 2 5 2 5 2 3 5 2 5 2 5 2 3 2 5 5 2 2 7 5 3 2 5 5 Every number is either a prime or the result of multiplying together prime numbers. Prime factorization is the process of breaking down a composite number into the prime numbers that it is made up of. These are known as its prime factors. remaining factor prime factor smaller numbers show whether the number is divisible by 2, 3, 5, or 7, or a combination of them 2 Prime factors = 5 × 6 To find the prime factors of 30, find the largest prime number that divides into 30, which is 5. The remaining factor is 6 (5 x 6 = 30), which needs to be broken down into prime numbers. largest prime factor 6 = 3 × 2 Next, take the remaining factor and find the largest prime number that divides into it, and any smaller prime numbers. In this case, the prime numbers that divide into 6 are 3 and 2. list prime factors in descending order 30 = 5 × 3 × 2 It is now possible to see that 30 is the product of multiplying together the prime numbers 5, 3, and 2. Therefore, the prime factors of 30 are 5, 3, and, 2. REAL WORLD Encryption Many transactions in banks and stores rely on the Internet and other communications systems. To protect the information, it is coded using a number fldjhg83asldkfdslkfjour523ijwli that is the product of just two huge eorit84wodfpflciry38s0x8b6lkj qpeoith73kdicuvyebdkciurmol primes. The security relies on the fact wpeodikrucnyr83iowp7uhjwm that no “eavesdropper” can factorize the kdieolekdoripasswordqe8ki number because its factors are so large. mdkdoritut6483kednffkeoskeo ▷ Data protection To provide constant security, mathematicians relentlessly hunt for ever bigger primes. kdieujr83iowplwqpwo98irkldil ieow98mqloapkijuhrnmeuidy6 woqp90jqiuke4lmicunejwkiuyj length △ Weight and mass Weight is how heavy something is in relation to the force of gravity acting upon it. Mass is the amount of matter that makes up the object. Both are measured in the same units, such as grams and kilograms, or ounces and pounds. this is the height of the building width area is made up of two of the same units, because width is also a length area = length × width ◁ Area Area is measured in squared units. The area of a rectangle is the product of its length and width; if they were both measured in meters (m) its area would be m × m, which is written as m2. A compound unit is made up of more than one of the basic units, including using the same unit repeatedly. Examples include area, volume, speed, and density. Compound measures △ Time Time is measured in milliseconds, seconds, minutes, hours, days, weeks, months, and years. Different countries and cultures may have calendars that start a new year at a different time. these three units are heavier these two units are lighter A unit is any agreed or standardized measurement of size. This allows quantities to be accurately measured. There are three basic units: time, weight (including mass), and length. Basic units h leng th this is the length of the building width len gth A B distance between cities A and B plane ﬂies set distance between two cities Distance is the amount of space between two points. It expresses length, but is also used to describe a journey, which is not always the most direct route between two points . Distance LOOKING CLOSER Reference 177–179 242–245 154–155 Formulas SEE ALSO Volumes volume is a compound of three of the same units, because width and height are technically lengths volume = length × width × height ◁ Volume Volume is measured in cubed units. The volume of a cuboid is the product of its height, width, and length; if they were all measured in meters (m), its area would be m × m × m, or m³. △ Length Length is how long something is. It is measured in centimeters, meters, and kilometers in the metric system, or in inches, feet, yards, and miles in the imperial system (see pp.242–245). widt this is the width of the building UNITS OF MEASUREMENT ARE STANDARD SIZES USED TO MEASURE TIME, MASS, AND LENGTH. Units of measurement height height S T D D T= S time = distance ÷ speed D=S×T this line acts as a multiplication sign S T D S T D distance = speed × time mass volume D= M D V M D V M D V V= volume = mass ÷ density M=D×V M D mass = density × volume this line acts as a multiplication sign M D V this line acts as a division sign density = mass ÷ volume M V ▷ Density formula triangle The relationships between density, mass, and volume can be shown in a triangle. The position of each unit of measurement in the triangle shows how to calculate that unit using the other two measurements. Density = Density measures how much matter is packed into a given volume of a substance. It involves two units—mass and volume. The formula for measuring density is mass ÷ volume. If this is measured in grams and centimeters, the unit for density will be g/cm³. Density D this line acts as a division sign speed = distance ÷ time D S= T S T distance time ▷ Speed formula triangle The relationships between speed, distance, and time can be shown in a triangle. The position of each unit in the triangle indicates how to use the other two measurements to calculate that unit. Speed = Speed measures the distance (length) traveled in a given time. This means that the formula for measuring speed is length ÷ time. If this is measured in kilometers and hours, the unit for speed will be km/h. Speed 20 60 = 1 3 hour 20 km D T time is 1/3 hour = 60 km/h M D density is 0.0113 kg/cm³ = 44.25 cm g 0.5 k △ Using the formula Substitute the values for mass and density into the formula for volume. Divide the mass (0.5 kg) by the density (0.0113 kg/cm³) to find the volume, in this case 44.25 cm³. V= mass is 0.5 kg ▷ Finding volume Lead has a density of 0.0113 kg/cm³. With this measurement, the volume of a lead weight that has a mass of 0.5 kg can be found. density of lead is constant, regardless of mass Then, substitute the values for distance and time into the formula for speed. Divide the distance (20 km) by the time (1/3 hour) to find the speed, in this case 60 km/h. S= distance is 20 km First, convert the minutes into hours. To convert minutes into hours, divide them by 60, then cancel the fraction—divide the top and bottom numbers by 20. This gives an answer of 1/3 hour. 20 minutes = divide 20 by 60 to ﬁnd its value in hours ▷ Finding speed A van travels 20 km in 20 minutes. From this information its speed in km/h can be found. 30 NUMBERS Telling the time SEE ALSO Introducing numbers 14–15 28–29 Units of measurement TIME IS MEASURED IN THE SAME WAY AROUND THE WORLD. THE MAIN UNITS ARE SECONDS, MINUTES, AND HOURS. Telling the time is an important skill and one that is used in many ways: What time is breakfast? How long until my birthday? Which is the quickest route? Measuring time Units of time measure how long events take and the gaps between the events. Sometimes it is important to measure time exactly, in a science experiment for example. At other times, accuracy of measurement is not so important, such as when we go to a friend’s house to play. For thousands of years time was measured simply by observing the movement of the sun, moon, or stars, but now our watches and clocks are extremely accurate. 1 2 second 11 12 21 22 31 32 41 42 51 52 3 13 23 33 43 53 4 14 24 34 44 54 5 15 25 35 45 55 6 16 26 36 46 56 7 17 27 37 47 57 8 18 28 38 48 58 9 19 29 39 49 59 10 20 30 40 50 60 Bigger units of time This is a list of the most commonly used bigger units of time. Other units include the Olympiad, which is a period of 4 years and starts on January 1st of a year in which the summer Olympics take place. 7 days is 1 week ◁ Units of time The units we use around the world are based on 1 second as measured by International Atomic Time. There are 86,400 seconds in one day. Fortnight is short for 14 nights and is the same as 2 weeks Between 28 and 31 days is 1 month 365 days is 1 year (366 in a leap year) There are 60 seconds in each minute. 1 minute 10 years is a decade There are 60 minutes in each hour. 100 years is a century 1 minute 2 3 4 5 6 7 8 9 10 1000 years is a millennium 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 1 hour There are 24 hours in each day. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 hour 1 day 31 TELLING THE TIME Reading the time The short hand indicates what hour it is. This hour hand shows 11. The time can be told by looking carefully at where the hands point on a clock or watch. The hour hand is shorter and moves around slowly. The minute hand is longer than the hour hand and points at minutes “past” the hour or “to” the next one. Most clock faces show the minutes in groups of ﬁve and the inbetween minutes are shown by a short line or mark. The second hand is usually long and thin, and sweeps quickly around the face every minute, marking 60 seconds. 12 2 10 Between each number is 5 minutes. 9 4 8 The second hand shows 40 seconds on this clock face. 7 When the minute hand points to 12, the time is “on the hour” as shown by the hour hand. 5 5 past 12 1 11 10 past 10 to If the minute hand is pointing to 9, the time is “quarter to” the hour. When the minute hand points to 3, the time is “quarter past” the hour. 2 10 to t the hou 9 as the hour p 3 quarter past ◁ Quarters and halves A clock can show the time as a “quarter past” or a “quarter to.” The quarter refers to a quarter of an hour, which is 15 minutes. Although we say “quarter” and “half,” we do not normally say “threequarters” in the same way. We might say something took “threequarters of an hour,” though, meaning 45 minutes. r 4 8 20 past 20 to 7 If the minute hand points to 6, the time is “half past” the hour. 5 6 25 to 25 past half past 1 9 4 5 9 4 7 5 9 4 7 5 2 10 3 8 1 11 2 10 3 8 12 1 11 2 10 3 8 1 11 2 10 7 12 12 12 11 The long hand shows the minutes. On this clock face 20 minutes have passed. △ A clock face A clock face is a visual way to show the time easily and clearly. There are many types of clock and watch faces. o’clock 5 to The number of small marks are the number of minutes or seconds. 3 6 quarter to The hands move in this direction. This is called the clockwise direction. 1 11 9 3 4 8 7 5 6 6 6 6 10 o’clock Quarter past 1 Half past 3 Quarter to 7 32 NUMBERS Analogue time Most clocks and watches only go up to 12 hours, but there are 24 hours in one day. To show the diﬀerence between morning and night, we use AM or PM. The middle of the day (12 o’clock) is called midday or noon. Before noon we say the time is AM 12 noon After noon we say PM 6 AM 6 PM △ AM or PM The initials AM and PM stand for the Latin words ante meridiem (meaning “before noon”) and post meridiem (meaning “after noon”). The first 12 hours of the day are called AM and the second 12 hours of the day are called PM. Digital time Traditional clock faces show time in an analoge format but digital formats are also common, especially on electrical devices such as computers, televisions, and mobile phones. Some digital displays show time in the 24hour system; others use the analoge system and also show AM or PM. △ Hours and minutes On a digital clock, the hours are shown first followed by a colon and the minutes. Some displays may also show seconds. △ 24hour digital display If the hours or minutes are single digit numbers, a zero (called a leading zero) is placed to the left of the digit. AM PM △ Midnight When it is midnight, the clock resets to 00:00. Midnight is an abbreviated form of “middle of the night.” △ 12hour digital display This type of display will have AM and PM with the relevant part of the day highlighted. 24hour clock The 24hour system was devised to stop confusion between morning and afternoon times, and runs continuously from midnight to midnight. It is often used in computers, by the military, and on timetables. To convert from the 12hour system to the 24hour system, you add 12 to the hour for times after noon. For example, 11 PM becomes 23:00 (11 + 12) and 8:45 PM becomes 20:45 (8:45 + 12). 12hour clock 24hour clock 12:00 midnight 00:00 1:00 AM 01:00 2:00 AM 02:00 3:00 AM 03:00 4:00 AM 04:00 5:00 AM 05:00 6:00 AM 06:00 7:00 AM 07:00 8:00 AM 08:00 9:00 AM 09:00 10:00 AM 10:00 11:00 AM 11:00 12:00 noon 12:00 1:00 PM 13:00 2:00 PM 14:00 3:00 PM 15:00 4:00 PM 16:00 5:00 PM 17:00 6:00 PM 18:00 7:00 PM 19:00 8:00 PM 20:00 9:00 PM 21:00 10:00 PM 22:00 11:00 PM 23:00 ROMAN NUMERALS XVll Roman numerals 33 SEE ALSO 14–15 Introducing numbers DEVELOPED BY THE ANCIENT ROMANS, THIS SYSTEM USES LETTERS FROM THE LATIN ALPHABET TO REPRESENT NUMBERS. Number Roman numeral 1 I 2 II 3 III 4 IV 5 V 6 VI Forming numbers 7 VII Some key principles were observed by the ancient Romans to “create” numbers from the seven letters. 8 VIII 9 IX 10 X 11 XI 12 XII 13 XIII 14 XIV 15 XV 16 XVI 17 XVII 18 XVIII 19 XIX 20 XX Understanding Roman numerals The Roman numeral system does not use zero. To make a number, seven letters are combined. These are the letters and their values: I V X L C D M 1 5 10 50 100 500 1000 First principle When a smaller number appears after a larger number, the smaller number is added to the larger number to find the total value. XI = X + I = 11 XVll = X + V + l + l = 17 Second principle When a smaller number appears before a larger number, the smaller number is subtracted from the larger number to find the total value. lX = X – I = 9 CM = M – C = 900 Third principle Each letter can be repeated up to three times. XX = X + X = 20 XXX = X + X + X = 30 30 XXX Using Roman numerals 40 XL Although Roman numerals are not widely used today, they still appear on some clock faces, with the names of monarchs and popes, and for important dates. 50 L 60 LX 70 LXX 80 LXXX 90 XC 100 C 500 D 1000 M Time Names Dates Henry VIII MMXIV Henry the eighth 2014 34 NUMBERS +– Positive and negative numbers A POSITIVE NUMBER IS A NUMBER THAT IS MORE THAN ZERO, WHILE A NEGATIVE NUMBER IS LESS THAN ZERO. SEE ALSO Introducing numbers 14–15 16–17 Addition and subtraction A positive number is shown by a plus sign (+), or has no sign in front of it. If a number is negative, it has a minus sign (–) in front of it. Why use positives and negatives? Positive numbers are used when an amount is counted up from zero, and negative numbers when it is counted down from zero. For example, if a bank account has money in it, it is a positive amount of money, but if the account is overdrawn, the amount of money in the account is negative. negative numbers –5 –4 –3 –2 number line continues forever Adding and subtracting positives and negatives Use a number line to add and subtract positive and negative numbers. Find the ﬁrst number on the line and then move the number of steps shown by the second number. Move right for addition and left for subtraction. start at 6 6–1=5 start at –5 –1 0 1 +1 –6 –5 2 –1 –4 move four places to left –3 – 4 = –7 0 –4 –7 5 1 2 –2 –5 –2 3 7 –1 0 4 1 2 –– 1 –– 2 5 6 7 move four places left from 3 to 7 –1 –4 6 move three places right from 5 to 2 +3 –3 4 move two places right from 5 to 7 –3 –6 3 +2 double negative is same as adding together, so move 2 places to the right 5 –– 2 = 7 start at –3 –1 move three places to right –5 + 3 = –2 start at 5 move one place left from 6 to 5 move one place to left –3 –2 –1 0 1 LOOKING CLOSER Double negatives If a negative or minus number is subtracted from a positive number, it creates a double negative. The ﬁrst negative is cancelled out by the second negative, so the result is always a positive, for example 5 minus –2 is the same as adding 2 to 5. ––=+ △ Like signs equal a positive If any two like signs appear together, the result is always positive. The result is negative with two unlike signs together. 35 P O S I T I V E A N D N E G AT I V E N U M B E R S °C REAL WORLD 50 40 30 20 10 + 0 – 10 20 30 Thermometer Negative numbers are necessary to record temperatures, as during the winter they can fall well below 32°F (0°C), which is freezing point. The lowest temperature ever recorded is –128.6°F (–89.2°C), in Antarctica. ▽ Number line A number line is a good way to get to grips with positive and negative numbers. Draw the positive numbers to the right of 0, and the negative numbers to the left of 0. Adding color makes them easier to tell apart. 0 means nothing; it separates positive numbers from negative numbers –1 0 1 2 Multiplying and dividing To multiply or divide any two numbers, ﬁrst ignore whether they are positive or negative, then work out if the answer is positive or negative using the diagram on the right. 2×4=8 –1 × 6 = –6 –4 ÷ 2 = –2 –2 × 4 = –8 –2 × –4 = 8 –10 ÷ –2 = 5 number line continues forever positive numbers 8 is positive because +×+=+ –6 is negative because –×+=– –2 is negative because –÷+=– –8 is negative because –×+=– 8 is positive because –×–=+ 5 is positive because –÷–=+ 3 +×+ –×– +÷+ –÷– +× – –×+ +÷ – –÷+ 4 5 like signs give a positive answer =+ unlike signs give a negative answer =– △ Positive or negative answer The sign in the answer depends on whether the signs of the values are alike or not. °F 120 10 0 80 60 40 20 + 0 – 20 36 NUMBERS Powers and roots SEE ALSO Multiplication 18–21 22–25 Division 42–43 Standard form A POWER IS THE NUMBER OF TIMES A NUMBER IS MULTIPLIED BY ITSELF. THE ROOT OF A NUMBER IS A NUMBER THAT, MULTIPLIED BY ITSELF, EQUALS THE ORIGINAL NUMBER. 5 Introducing powers A power is the number of times a number is multiplied by itself. This is indicated as a smaller number positioned to the right above the number. Multiplying a number by itself once is described as “squaring” the number; multiplying a number by itself twice is described as “cubing” the number. this is the power, which shows how many times to multiply the number (5 means 5 × 5 × 5 × 5) this is the number that the power relates to ▷ Squared number This image shows how many units make up 5. There are 5 rows, each with 5 units—so 5 × 5 = 25. 5 4 3 2 5 △ The cube of a number Multiplying a number by itself twice gives its cube. The power for a cube number is 3, for example 5³, which means there are 3 number 5’s being multiplied: 5 × 5 × 5. 5 rows with 5 units in each row 4 this is the power; 5³ is called “5 cubed” 3 5×5×5=5 = 125 2 △The square of a number Multiplying a number by itself gives the square of the number. The power for a square number is 2, for example 52 means that 2 number 5’s are being multiplied. 1 this is the power; 5² is called “5 squared” 4 72–73 1 5×5=5 = 25 Using a calculator 3 1 2 3 4 5 1 4 2 3 3 5 5 horizontal rows 2 4 ▷ Cubed number This image shows how many units make up 5³. There are 5 horizontal rows and 5 vertical rows, each with 5 units in each one, so 5 × 5 × 5 = 125. 5 1 5 vertical rows 5 blocks of units POWERS AND ROOTS Square roots and cube roots this is the square root symbol A square root is a number that, multiplied by itself once, equals a given number. For example, one square root of 4 is 2, because 2 × 2 = 4. Another square root is –2, as (–2) × (–2) = 4; the square roots of numbers can be either positive or negative. A cube root is a number that, multiplied by itself twice, equals a given number. For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27. this is the cube root symbol 25 125 this is the number for which the cube root is being found this is the number for which the square root is being found this is the square root of 25 square root symbol 37 25 is 52 25 = 5 because 5 × 5 = 25 △ The square root of a number The square root of a number is the number which, when squared (multiplied by itself ), equals the number under the square root sign. cube root symbol 125 = 5 125 is 53 this is the cube root of 125 because 5 × 5 × 5 = 125 △ The cube root of a number The cube root of a number is the number that, when cubed (multiplied by itself twice), equals the number under the cube root sign. COMMON SQUARE ROOTS Square root Answer Why? 1 1 Because 1 × 1 = 1 4 2 Because 2 × 2 = 4 9 3 Because 3 × 3 = 9 16 4 Because 4 × 4 = 16 25 5 Because 5 × 5 = 25 36 6 Because 6 × 6 = 36 49 7 Because 7 × 7 = 49 64 8 Because 8 × 8 = 64 81 9 Because 9 × 9 = 81 100 10 Because 10 × 10 = 100 121 11 Because 11 × 11 = 121 144 12 Because 12 × 12 = 144 169 13 Because 13 × 13 = 169 LOOKING CLOSER Using a calculator Calculators can be used to ﬁnd powers and square roots. Most calculators have buttons to square and cube numbers, buttons to ﬁnd square roots and cube roots, and an exponent button, which allows them to raise numbers to any power. Xy △ Exponent This button allows any number to be raised to any power. △ Square root This button allows the square root of any number to be found. 3 = 3 Xy = 243 25 = =5 25 5 ◁ Using exponents First enter the number to be raised to a power, then press the exponent button, then enter the power required. ◁ Using square roots On most calculators, find the square root of a number by pressing the square root button first and then entering the number. 38 NUMBERS add the powers Multiplying powers of the same number the ﬁrst power To multiply powers that have the same base number, simply add the powers. The power of the answer is the sum of the powers that are being multiplied. the second power + 6² × 6³ = 6⁵ the power of the answer is: 2 + 3 = 5 because ▷ Writing it out Writing out what each of these powers represents shows why powers are added together to multiply them. (6×6)×(6×6×6) 6² is 6 × 6 6³ is 6 × 6 × 6 = 6×6×6×6×6 6 × 6 × 6 × 6 × 6 is 6 subtract the second power from the ﬁrst Dividing powers of the same number the ﬁrst power To divide powers of the same base number, subtract the second power from the ﬁrst. The power of the answer is the diﬀerence between the ﬁrst and second powers. − the second power 4⁴ ÷ 4 ² = 4 ² the power of the answer is: 4 – 2 = 2 because 4 is 4 × 4 × 4 × 4 ▷ Writing it out Writing out the division of the powers as a fraction and then canceling the fraction shows why powers to be divided can simply be subtracted. 4×4×4×4 4×4 4 is 4 × 4 4×4×4×4 4×4 = 4×4 cancel the fraction to its simplest terms 4 × 4 is 4 LOOKING CLOSER Zero power Any number raised to the power 0 is equal to 1. Dividing two equal powers of the same base number gives a power of 0, and therefore the answer 1. These rules only apply when dealing with powers of the same base number. the ﬁrst power the second power the power of the answer is: 3 – 3 = 0 8³ ÷ 8³ = 8⁰ = 1 any number to the power 0 = 1 because ▷ Writing it out Writing out the division of two equal powers makes it clear why any number to the power 0 is always equal to 1. 8 is 8 × 8 × 8 8×8×8 8×8×8 1 = 512 512 = any number divided by itself = 1 POWERS AND ROOTS 39 Finding a square root by estimation It is possible to ﬁnd a square root through estimation, by choosing a number to multiply by itself, working out the answer, and then altering the number depending on whether the answer needs to be higher or lower. 32 = ? 25 = 5 1,000 = ? 36 = 6, 1,600 = 40 900 = 30, and so the answer must be somewhere between 5 and 6. Start with the midpoint between the two, 5.5: and so the answer must be between 40 and 30. 1,000 is closer to 900 than 1,600, so start with a number closer to 30, such as 32: 5.5 × 5.5 = 30.25 5.75 × 5.75 = 33.0625 5.65 × 5.65 = 31.9225 5.66 × 5.66 = 32.0356 32 × 32 = 1,024 Too high Too low 31 × 31 = 961 31.5 × 31.5 = 992.25 Too low 31.6 × 31.6 = 998.56 Too low Too high 31.65 × 31.65 = 1,001.72 this would round up to 31.62 × 31.62 = 999.8244 1,000 as the nearest the square root of 32 is approximately 5.66 Too low Too high Too low this would round down to 32 the square root of 1,000 is approximately 31.62 whole number Finding a cube root by estimation Cube roots of numbers can also be estimated without a calculator. Use round numbers to start with, then use these answers to get closer to the ﬁnal answer. 32 = ? 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so 800 = ? 9 × 9 × 9 = 729 10 × 10 × 10 = 1,000, the answer is somewhere between 3 and 4. Start with the midpoint between the two, 3.5: and so the answer is somewhere between 9 and 10. 800 is closer to 729 than 1000, so start with a number closer to 9, such as 9.1: 3.5 × 3.5 × 3.5 = 42.875 Too high Too high 3.3 × 3.3 × 3.3 = 35.937 3.1 × 3.1 × 3.1 = 29.791 Too low Too high 3.2 × 3.2 × 3.2 = 32.768 would be 32.2 3.18 × 3.18 × 3.18 = 32.157432 this rounded to the Too low 9.1 × 9.1 × 9.1 = 753.571 9.3 × 9.3 × 9.3 = 804.357 Too high 9.27 × 9.27 × 9.27 = 796.5979 Too low 9.28 × 9.28 × 9.28 = 799.1787 Very close 9.284 × 9.284 × 9.284 = 800.2126 the cube root of 32 is approximately 3.18 tenths place, which would round to 32 the cube root of 800 is approximately 9.284 this would round down to 800 40 NUMBERS Surds SEE ALSO roots 36–39 Powers and 48–55 Fractions A SURD IS A SQUARE ROOT THAT CANNOT BE WRITTEN AS A WHOLE NUMBER. IT HAS AN INFINITE NUMBER OF DIGITS AFTER THE DECIMAL POINT. Introducing surds Some square roots are whole numbers and are easy to write. But some are irrational numbers—numbers that go on forever after the decimal point. These numbers cannot be written out in full, so the most accurate way to express them is as square roots. rational number irrational number 4=2 5 = 2.2360679774... △ Surd The square root of 5 is an irrational number—it goes on forever. It cannot accurately be written out in full, so it is most simply expressed as the surd √5. △Not a surd The square root of 4 is not a surd. It is the number 2, a whole, or rational, number. Simplifying surds Some surds can be made simpler by taking out factors that can be written as whole numbers. A few simple rules can help with this. ▷ Square roots A square root is the number that, when multiplied by itself, equals the number inside the root. a × a = a 3× 3 = 3 multiply the surd by itself to get the number inside the square root ▷ Multiplying roots Multiplying two numbers together and taking the square root of the result equals the same answer as taking the square roots of the two numbers and mutiplying them together. look for factors that are square numbers ab = a × b √16 = 4, so this can be written as 4 × √3 48 = 16 × 3 = 16 × 3 = 4 × 3 48 can be written as 16 × 3 the square root of 16 is a whole number the square root of 3 is an irrational number, so it stays in surd form SURDS ▷ Dividing roots Dividing one number by another and taking the square root of the result is the same as dividing the square root of the first number by the square root of the second. ▷ Simplifying further When dividing square roots, look out for ways to simplify the top as well as the bottom of the fraction. a = a b b √7 is irrational (2.6457...), so leave as a surd 7= 7= 7 16 16 4 16 is 4 squared √9 = 3 (3 x 3 = 9) 8 = 8 = 8 9 9 3 2× 2 3 = ﬁnal, simpliﬁed form 8 = 4× 2 = 2× 2 8 is 4 × 2 4 is 2 squared Surds in fractions When a surd appears in a fraction, it is helpful to make sure it appears in the numerator (top of the fraction) not the denominator (bottom of the fraction). This is called rationalizing, and is done by multiplying the whole fraction by the surd on the bottom. ▷ Rationalizing The fraction stays the same if the top and bottom are multiplied by the same number. 1 2 = 1× 2 2× 2 2 2 = the surd √2 is now on top of the fraction multiply top and bottom by the surd √2 ▷ Simplifying further Sometimes rationalizing a fraction gives us another surd that can be simplified further. 12 and 15 can both be divided by 3 to simplify further 12 = 12 × 15 = 12 × 15 = 4 × 15 5 15 × 15 15 15 multiply both top and bottom by √15 multiplying √15 by √15 gives 15 41 42 NUMBERS 3 4×10 Standard Form SEE ALSO Multiplication 18–21 22–25 36–39 Division Powers and roots STANDARD FORM IS A CONVENIENT WAY OF WRITING VERY LARGE AND VERY SMALL NUMBERS. this is the power of 10 Introducing standard form 4 × 10 Standard form makes very large or very small numbers easier to understand by showing them as a number multiplied by a power of 10. This is useful because the size of the power of 10 makes it possible to get an instant impression of how big the number really is. 3 ◁ Using standard form This is how 4,000 is written as standard form—it shows that the decimal place for the number represented, 4,000, is 3 places to the right of 4. How to write a number in standard form To write a number in standard form, work out how many places the decimal point must move to form a number between 1 and 10. If the number does not have a decimal point, add one after its ﬁnal digit. very large number ▷ Take a number Standard form is usually used for very large or very small numbers. ▷ Add the decimal point Identify the position of the decimal point if there is one. Add a decimal point at the end of the number, if it does not already have one. ▷ Move the decimal point Move along the number and count how many places the decimal point must move to form a number between 1 and 10. ▷ Write as standard form The number between 1 and 10 is multiplied by 10, and the small number, the “power” of 10, is found by counting how many places the decimal point moved to create the ﬁrst number. 1,230,000 add decimal point 3 2 1 0.0006 1 2 3 4 1,230,000. 0.0006 the decimal point moves 4 places to the right the decimal point moves 6 places to the left the power is 6 because the decimal point moved six places; the power is positive because the decimal point moved to the left 1.23 × 10 0.0006 decimal point is already here 1,230,000. 6 5 4 very small number 6 the ﬁrst number must always be between 1 and 10 the power is negative because the decimal point moved to the right 6 × 10 –4 the power is 4 because the decimal point moved four places S TA N D A R D F O R M 43 Standard form in action Sometimes it is diﬃcult to compare extremely large or small numbers because of the number of digits they contain. Standard form makes this easier. The mass of Earth is 5,974,200,000,000,000,000,000,000 kg 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5,974, 20 0,0 0 0,0 0 0,0 0 0,0 0 0,0 0 0,0 0 0.0 kg The decimal point moves 24 places to the left. The mass of the planet Mars is 23 22 21 20 19 18 17 16 15 14 13 12 11 10 6 5 4 9 8 7 3 2 1 6 41,910,0 0 0,0 0 0,0 0 0,0 0 0,0 0 0,0 0 0.0 kg The decimal point moves 23 places to the left. Written in standard form these numbers are much easier to compare. Earth’s mass in standard form is 5.9742 × 10 24 kg 23 kg ▷ Comparing planet mass It is immediately evident that the mass of the Earth is bigger than the mass of Mars because 10 is 10 times larger than 10²³. The mass of Mars in standard form is 6.4191 × 10 LOOKING CLOSER E X A M P L E S O F S TA N D A R D F O R M Standard form 7.36 × 10 kg Example Decimal form Weight of the Moon 73,600,000,000,000,000,000,000 kg Humans on Earth 6,800,000,000 Speed of light 300,000,000 m/sec 6.8 × 10 3 × 10 m/sec 384,000 km 3.8 × 10 km 365,000 tons 3.65 × 10 tons Distance around the Equator 40,075 km Height of Mount Everest 8,850 m 4 × 10 km 8.850 × 10 m Speed of a bullet 710 m/sec Speed of a snail 0.001 m/sec Width of a red blood cell 0.00067 cm Length of a virus 0.000 000 009 cm 6.7 × 10 cm 9 × 10 cm Weight of a dust particle 0.000 000 000 753 kg 7.53 × 10 kg Distance of the Moon from the Earth Weight of the Empire State building Standard form and calculators The exponent button on a calculator allows a number to be raised to any power. Calculators give very large answers in standard form. Xy △ Exponent button This calculator button allows any number to be raised to any power. Using the exponent button: 7.1 × 10 m/sec 1 × 10 m/sec 4 × 10 4 × 2 is entered by pressing 10 Xy 2 On some calculators, answers appear in standard form: 1234567 × 89101112 = 1.100012925 × 10 14 so the answer is approximately 110,001,292,500,000 44 NUMBERS Decimals SEE ALSO Multiplication 18–21 22–25 Using a Division NUMBERS WRITTEN IN DECIMAL FORM ARE CALLED DECIMAL NUMBERS OR, MORE SIMPLY, DECIMALS. calculator 72–73 Decimal numbers In a decimal number, the digits to the left of the decimal point are whole numbers. The digits to the right of the decimal point are not whole numbers. The ﬁrst digit to the right of the decimal point represents tenths, the second hundredths, and so on. These are called fractional parts. fractional part is 56 whole number part is 1,234 1,234 . 56 △ Whole and fractional parts The whole numbers represent – moving left from the decimal point – ones, tens, hundreds, and thousands. The fractional numbers – moving right from the decimal place – are tenths then hundredths. decimal point separates the whole numbers (on the left) from the fractional numbers (on the right) Multiplication To multiply decimals, ﬁrst remove the decimal point. Then perform a long multiplication of the two numbers, before adding the decimal point back in to the answer. Here 1.9 (a decimal) is multiplied by 7 (a whole number). 6 is carried to the tens column multiply 7 by 9 1.9 19 decimal point is removed First, remove any decimal points, so that both numbers can be treated as whole numbers. × 6 19 7 3 6 7 × 9 = 63, the ﬁrst digit, 6, is carried to the tens column Then multiply the two numbers, starting in the ones column. Carry ones to the tens if necessary. 19 × 7 133 1 × 7 + 6 = 13, which is written across two columns Next multiply the tens. The product is 7, which, added to the carried 6, makes 13. Write this across two columns. multiply 7 by 1 decimal point is put back in 133 13.3 Finally, count the decimal digits in the original numbers – there is 1. The answer will also have 1 decimal digit. 45 DECIMALS DIVISION Dividing one number by another often gives a decimal answer. Sometimes it is easier to turn decimals into whole numbers before dividing them. Short division with decimals Many numbers do not divide into each other exactly. If this is the case, a decimal point is added to the number being divided, and zeros are added after the point until the division is solved. Here 6 is divided by 8. add a decimal point on the answer line 8 goes into 60 7 times with 4 left over answer is 0.75 carry 4 carry 6 Both numbers are whole. As 8 will not divide into 6, put in a decimal point with a 0 after it and carry the 6. 0. 8 6.0 6 add a 0 after the decimal point add a decimal point after 6 Dividing 60 by 8 gives 7, with a remainder of 4. Write the 7 on the answer line, add another 0 after the decimal place, and carry the 4. 0.7 8 6.00 6 Dividing 40 by 8 gives 5 exactly, and the division ends (terminates). The answer to 6 ÷ 8 is 0.75. 4 add another 0 divide 60 by 8 0.75 8 6.0 0 6 divide 40 by 8 Dividing decimals 0.7 8 6.0 0 60 –56 4 Above, short division was used to ﬁnd the decimal answer for the sum 8 ÷ 6. Long division can be used to achieve the same result. 0 8 6.0 0 8 ﬁts into 6 0 times, so write 0 here multiply 8 times 0 to get 0 First, divide 8 into 6. It goes 0 times, so put a 0 above the 6. Multiply 8 × 0, and write the result (0) under the 6. 0.7 8 6.0 0 60 add decimal point bring down a 0 divide 60 by 8 Subtract 0 from 6 to get 6, and bring down a 0. Divide 8 into 60 and put the answer, 7, after a decimal point. 4 multiply 8 times 7 to get 56 ﬁrst remainder is 4 Work out the first remainder by multiplying 8 by 7 and subtracting this from 60. The answer is 4. 0.75 8 6.0 0 0 60 56 40 8 goes into 40 exactly 5 times bring down a 0 divide 40 by 8 Bring down a zero to join the 4 and divide the number by 8. It goes exactly 5 times, so put a 5 above the line. LOOKING CLOSER Decimals that do not end Sometimes the answer to a division can be a decimal number that repeats without ending. This is called a "repeating" decimal. For example, here 1 is divided by 3. Both the calculations and the answers in the division become identical after the second stage, and the answer repeats endlessly. 3 goes into 10 three times, with 1 left over add a decimal point to the answer line 0. 1 3 1.0 carry 1 3 does not divide into 1 3 does not divide into 1, so enter 0 on the answer line. Add a decimal point after 0, and carry 1. 0.3 1 1 3 1.00 3 goes into 10 three times, with 1 left over divide 10 by 3 10 divided by 3 gives 3, with a remainder of 1. Write the 3 on the answer line and carry the 1 to the next 0. 0.3 3 3 1.010 – 0.3 symbol for a repeating decimal Dividing 10 by 3 again gives exactly the same answer as the last step. This is repeated infinitely. This type of repeating decimal is written with a line over the repeating digit. 46 NUMBERS 10101 Binary numbers SEE ALSO Introducing numbers 14–15 33 Roman numerals NUMBERS ARE COMMONLY WRITTEN USING THE DECIMAL SYSTEM, BUT NUMBERS CAN BE WRITTEN IN ANY NUMBER BASE. What is a binary number? Decimal numbers The decimal system uses the digits 0 through to 9, while the binary system, also known as base 2, uses only two digits—0 and 1. Binary numbers should not be thought of in the same way as decimal numbers. For example, 10 is said as “ten” in the decimal system but must be said as “one zero” in the binary system. This is because the value of each “place” is diﬀerent in decimal and binary. 0 1 2 3 4 5 6 7 8 9 Binary numbers a single binary digit is called a “bit,” which is short for binary digit 0 1 Counting in the decimal system Decimal When using the decimal system for column sums, numbers are written from right to left (from lowest to highest). Each column is worth ten times more than the column to the right of it. The decimal number system is also known as base 10. 0 0 0 1 1 1 one 2 10 1 two 3 11 1 two + 1 one 4 100 1 four 5 101 1 four + 1 one 6 110 1 four + 1 two 7 111 1 four + 1 two + 1 one 8 1000 1 eight 9 1001 1 eight + 1 one 10 1010 1 eight + 1 two Counting in binary 11 1011 1 eight + 1 two + 1 one Each column in the binary system is worth two times more than the column to the right of it and, as in the decimal system, 0 represents zero value. A similar system of headings may be used with binary numbers but only two symbols are used (0 and 1). 12 1100 1 eight + 1 four 13 1101 1 eight + 1 four + 1 one 14 1110 1 eight + 1 four + 1 two 15 1111 1 eight + 1 four + 1 two + 1 one 16 10000 1 sixteen 17 10001 1 sixteen + 1 one 18 10010 1 sixteen + 1 two 19 10011 1 sixteen + 1 two + 1 one 20 10100 1 sixteen + 1 four 50 110010 100 1100100 ×10 ×10 ×10 Thousands Hundreds Tens Ones 1000 100 10 1 6 4 5 2 6 0 0 0 + 4 0 0 + 50 + 2 = 6452 ×2 ×2 ×2 ×2 ×2 Thirtytwos Sixteens Eights Fours Twos Ones 32 16 8 4 2 1 1 1 1 0 0 1 32 + 16 + 8 + 0 + 0 + 1 = 57 written in the decimal system Binary 1 thirtytwo + 1 sixteen + 1 two 1 sixtyfour + 1 thirtytwo + 1 four BINARY NUMBERS 47 Adding in binary Numbers written in binary form can be added together in a similar way to the decimal system, and column addition may be done like this: Fours Twos Ones 111 + 101 1+0+1=2 (this is 10 in binary) 1+1=2 (this is 10 in binary) (4 + 2 + 1 in decimal) (4 + 0 + 1 in decimal) 111 + 101 0 1 Align the numbers under their correct placevalue columns as in the decimal system. It may be helpful to write in the column headings when first learning this system. 111 + 101 1100 111 + 101 00 1 carry 1 Add the ones column. The answer is 2, which is 10 in binary. The twos are shown in the next column so carry a 1 to the next column and leave 0 in the ones column. the answer is 12 in the decimal system (8 + 4 + 0 + 0 = 12) 1 carry 1 Now the digits in the twos column are added together with the 1 carried over from the ones column. The total is 2 again (10 in binary) so a 1 needs to be carried and a 0 left in the twos column. carry 1 and place in the eights column Finally, add the 1s in the fours column, which gives us 3, (11 in binary). This is the end of the equation so the final 1 is placed in the eights column. Subtracting in binary Subtraction works in a similar way to the decimal system but “borrows” in diﬀerent units to the decimal system—borrowing by twos instead of tens. Eights Fours it is helpful to add zeros Twos Ones 1101 – 11 (8 + 4 + 1 in decimal) (2 + 1 in decimal) The numbers are written in their correct placevalue columns as in the decimal system. we put a 2 in the twos column because the borrowed number represents 2 lots of twos subtract the units 0 2 0 2 1101 – 0011 0 1101 – 0011 10 Add zeros so that there are the same number of digits in each column. Then begin by subtracting the ones column: 1 minus 1 is 0, so place a 0 in the answer space. Now move on to the twos column on the left. The lower 1 cannot be subtracted from the 0 above it so borrow from the fours column and replace it with a 0. Then put a 2 above the twos column. Subtract the lower 1 from the upper 2. This leaves 1 as the answer. the answer is 10 in the decimal system (8 + 0 + 2 + 0 = 10) 1101 – 0011 1010 Now subtract the digits in the fours column, which gives us 0. Finally, in the eights column we have nothing to subtract from the upper 1, so 1 is written in the answer space. 48 NUMBERS Fractions SEE ALSO Division 22–25 44–45 Decimals Ratio and A FRACTION REPRESENTS A PART OF A WHOLE NUMBER. THEY ARE WRITTEN AS ONE NUMBER OVER ANOTHER NUMBER. proportion 56–59 Percentages 60–61 Converting fractions, decimals, and percentages 64–65 Writing fractions The number on the top of a fraction shows how many equal parts of the whole are being dealt with, while the number on the bottom shows the total number of equal parts that the whole has been divided into. 1 2 Quarter One fourth, or 1⁄4 (a quarter), shows 1 part out of 4 equal parts in a whole. Numerator The number of equal parts examined. 4 Dividing line This is also written as ∕. Denominator Total number of equal parts in the whole. 8 Eighth 1 ⁄8 (one eighth) is 1 part out of 8 equal parts in a whole. 1 1 1 16 1 32 1 64 6 1 4 Sixteenth 1 ⁄16 (one sixteenth) is 1 part out of 16 equal parts in a whole. ▷ Equal parts of a whole The circle on the right shows how parts of a whole can be divided in different ways to form different fractions. One thirtysecond 1 ⁄32 (one thirtysecond) is 1 part out of 32 equal parts in a whole. One sixtyfourth 1 ⁄64 (one sixtyfourth) is 1 part out of 64 equal parts in a whole. 49 FRACTIONS Types of fractions A proper fraction—where the numerator is smaller than the denominator—is just one type of fraction. When the number of parts is greater than the whole, the result is a fraction that can be written in two ways— either as an improper fraction (also known as a topheavy fraction) or a mixed fraction. numerator has higher value than denominator numerator has lower value than denominator 1 4 35 4 ◁ Proper fraction In this fraction the number of parts examined, shown on top, is less than the whole. ◁ Improper fraction The larger numerator indicates that the parts come from more than one whole. whole number fraction 1 3 10 ◁ Mixed fraction A whole number is combined with a proper fraction. Depicting fractions Fractions can be illustrated in many ways, using any shape that can be divided into an equal number of parts. 1 2 1 3 1 3 = 1 3 1 3 1 3 1 5 1 6 1 6 Half 1 ⁄2 (one half ) is 1 part out of 2 equal parts in a whole. 1 3 ◁ Split equally The shapes show that there is more than one way of depicting a fraction. 1 7 1 7 1 7 1 7 1 5 1 5 1 5 1 6 1 6 1 5 = 1 6 1 6 1 7 1 7 1 7 = = 1 1 1 1 1 5 5 5 5 5 1 6 1 6 1 6 1 6 1 6 1 6 1 1 1 7 7 1 7 7 1 1 1 7 7 7 50 NUMBERS Turning topheavy fractions into mixed fractions A topheavy fraction can be turned into a mixed fraction by dividing the numerator by the denominator. 35 is 4 1 2 5 6 9 10 3 4 7 8 11 12 13 14 17 18 21 22 15 16 19 20 23 24 25 26 29 30 33 34 27 28 31 32 35 is 8 3 4 35 = 35 ÷ 4 = 8 r3 = 4 8 3 4 denominator 3 equal parts of 1 whole left over each group of 4 represents 1 whole whole number of 8 is produced with 3 left over numerator Draw groups of four numbers—each group represents a whole number. The fraction is eight whole numbers with 3 ⁄4 (three quarters) left over. Divide the numerator by the denominator, in this case, 35 by 4. The result is the mixed fraction 83⁄4 made up of the whole number 8 and 3 parts—or 3⁄4 (three quarters)—left over. Turning mixed fractions into topheavy fractions A mixed fraction can be changed into a topheavy fraction by multiplying the whole number by the denominator and adding the result to the numerator. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 whole number 10 1 is 3 add to numerator is 19 20 21 22 23 24 25 26 27 28 29 30 31 31 3 each group of 3 parts represents 1 whole 1 ⁄3 (one third) of a whole left over Draw the fraction as ten groups of three parts with one part left over. In this way it is possible to count 31 parts in the fraction. multiply whole number by denominator 10 1 10 × 3 + 1 31 = = 3 3 3 denominator Multiply the whole number by the denominator—in this case, 10 × 3 = 30. Then add the numerator. The result is the topheavy fraction 31⁄3, with a numerator (31) greater than the denominator (3). FRACTIONS 51 Equivalent fractions The same fraction can be written in diﬀerent ways. These are known as equivalent (meaning “equal”) fractions, even though they look diﬀerent. 3 highlighted sectors take up same space as 9 highlighted sectors in circle on left = numerator ÷3 9 12 3 4 = ÷3 denominator 2 highlighted rectangles take up same space as 4 highlighted rectangles on left = divide numerator and denominator by same number ÷2 4 6 2 3 = ÷2 cancellation results in equivalent fraction with smaller numerator and denominator △ Cancellation Cancellation is a method used to find an equivalent fraction that is simpler than the original. To cancel a fraction divide the numerator and denominator by the same number. 16 highlighted triangles take up same space as 4 highlighted triangles in square on left = numerator ×4 4 8 = ×4 denominator 16 32 2 highlighted triangles take up same space as highlighted triangle on left = ×2 1 3 multiply numerator and denominator by same number 2 6 = ×2 △ Reverse cancellation Multiplying the numerator and denominator by the same number is called reverse cancellation. This results in an equivalent fraction with a larger numerator and denominator. Table of equivalent fractions / = / / / / / / / / / / = / / / / / / / / / / = / / / / / / / / / / = / / / / / / / / / / = / / / / / / / / / / = / / / / / / / / / / = / / / / / / / / / / = / / / / / / / / / 52 NUMBERS Finding a common denominator When ﬁnding the relative sizes of two or more fractions, ﬁnding a common denominator makes it much easier. A common denominator is a number that can be divided exactly by the denominators of all of the fractions. Once this has been found, comparing fractions is just a matter of comparing their numerators. ▷ Comparing fractions To work out the relative sizes of fractions, it is necessary to convert them so that they all have the same denominator. To do so, first look at the denominators of all the fractions being compared. 2 3 denominator 5 8 multiples of 3 ▷ Make a list List the multiples – all the whole number products of each denominator – for all of the denominators. Pick a sensible stopping point for the list, such as 100. ▷ Find the lowest common denominator List only the multiples that are common to all three sets. These numbers are called common denominators. Identify the lowest one. ▷ Convert the fractions Find out how many times the original denominator goes into the common denominator. Multiply the numerator by the same number. It is now possible to compare the fractions. denominator 7 12 denominator multiples of 12 multiples of 8 3, 6, 9, 12, 8, 16, 24, 32, 12, 24, 36, 40, 48, 56, 48, 60, 72, 15, 18, 21, 84, 96… 24, 27, 30… 64, 72… lowest common denominator of 3, 8, and 12 common denominators 24, 48, 72, 96… largest fraction smallest fraction 2 ×8 16 5 ×3 15 = = 3 ×8 24 8 ×3 24 original denominator goes into common denominator 8 times, so multiply both numerator and denominator by 8 original denominator goes into common denominator 3 times, so multiply both numerator and denominator by 3 7 ×2 14 = 12 ×2 24 original denominator goes into common denominator 2 times, so multiply both numerator and denominator by 2 FRACTIONS ADDING AND SUBTRACTING FRACTIONS Just like whole numbers, it is possible to add and subtract fractions. How it is done depends on whether the denominators are the same or diﬀerent. Adding and subtracting fractions with the same denominator To add or subtract fractions that have the same denominator, simply add or subtract their numerators to get the answer. The denominators stay the same. + 1 4 = 2 4 + = = – 3 4 To add fractions, add together only the numerators. The denominator in the result remains unchanged. 7 8 4 8 – = 3 8 To subtract fractions, subtract the smaller numerator from the larger. The denominator in the result stays the same. Adding fractions with different denominators To add fractions that have diﬀerent denominators, it is necessary to change one or both of the fractions so they have the same denominator. This involves ﬁnding a common denominator (see opposite). multiply whole number by denominator then add numerator / can now be added to / as both have same denominator 6 is a common denominator of both 3 and 6 remainder becomes numerator of fraction ×2 4 1 5 + 3 6 4×3+1 3 13 5 + 3 6 13 26 5 = + 3 6 6 ×2 denominator stays same First, turn any mixed fractions that are being added into improper fractions. The two fractions have different denominators, so a common denominator is needed. 31 6 31 ÷ 6 = 5r1 = 5 16 denominator goes into common denominator 2 times, so multiply both numerator and denominator by 2 Convert the fractions into fractions with common denominators by multiplying. If necessary, divide the numerator by the denominator to turn the improper fraction back into a mixed fraction. Subtracting fractions with different denominators To subtract fractions with diﬀerent denominators, a common denominator must be found. multiply whole number by denominator 4 is a common denominator then add numerator of both 2 and 4 / can be subtracted from / because both have same denominator remainder becomes numerator of fraction ×2 6 12 – 34 6×2+1 2 13 3 − 4 2 ×2 denominator stays same First, turn any mixed fractions in the equation into improper fractions by multiplying. 13 26 3 = − 2 4 4 The two fractions have different denominators, so a common denominator is needed. 23 4 23 ÷ 4 = 5r1 = 5 34 denominator goes into common denominator 2 times, so multiply both numerator and denominator by 2 Convert the fractions into fractions with common denominators by multiplying. If necessary, divide the numerator by the denominator to turn the improper fraction back into a mixed fraction. 53 54 NUMBERS MULTIPLYING FRACTIONS Fractions can be multiplied by other fractions. To multiply fractions by mixed fractions or whole numbers, they first need to be converted into improper (topheavy) fractions. two equal parts multiplying / by 3 is same as adding / to / to / × 3= × 3= 1 2 + 1 2 + + 1 2 + 1 2 whole number an improper fraction with whole number as numerator and 1 as denominator = remainder becomes numerator of fraction × = 1 12 1 2 × 3 1 = 3 2 3÷2 = 1r1 = denominator stays the same × Convert the whole number to a fraction. Next, multiply both numerators together and then both denominators. Imagine multiplying a fraction by a whole number as adding the fraction to itself that many times. Alternatively, imagine multiplying a whole number by a fraction as taking that portion of the whole number, here ½ of 3. 1 12 Divide the numerator of the resulting fraction by the denominator. The answer is given as a mixed fraction. Multiplying two proper fractions Proper fractions can be multiplied by each other. It is useful to imagine that the multiplication sign means “of”—the problem below can be expressed as “what is ½ of ¾?” and “what is ¾ of ½?”. = × 1 2 × make a fraction half as big by doubling the total number of parts—the numerator is unchanged 3 4 = 3 8 imagine multiplication sign means “of” × 1 3 = 3 × 8 2 4 increasing value of denominator decreases value of fraction × one fraction splits another to increase number of parts in result Multiply the numerators and the denominators. The resulting fraction answers both questions: “what is ½ of ¾?” and “what is ¾ of ½?”. Visually, the result of multiplying two proper fractions is that the space taken by both together is reduced. Multiplying mixed fractions To multiply a proper fraction by a mixed fraction, it is necessary to ﬁrst convert the mixed fraction into an improper fraction. multiply whole number by denominator 3 2 5 3×5+2 × 5 6 5 add to numerator First, turn the mixed fraction into an improper fraction. remainder becomes numerator of fraction × 17 5 × 5 6 = 85 30 to show in its lowest form divide both numbers by 5 to get 5⁄6 85 ÷ 30 = 2r25 = × Next, multiply the numerators and denominators of both fractions to get a new fraction. 2 2530 denominator stays the same Divide the numerator of the new improper fraction by its denominator. The answer is shown as a mixed fraction. 55 FRACTIONS DIVIDING FRACTIONS Fractions can be divided by whole numbers. Turn the whole number into a fraction and find the reciprocal of this fraction by turning it upside down, then multiply it by the first fraction. each part is 1⁄8 (one eighth) dividing by 2 means splitting in half 1 ⁄4 (one quarter) whole number ﬁrst turned into improper fraction ÷2 = 1 4 ÷ 2 = 1÷2 4 1 1 8 denominator is doubled, so value is halved sw i tc h around ÷ sign becomes × sign 1 1 1 × = 4 2 8 sw i tc h a round To divide a fraction by a whole number, convert the whole number into a fraction, turn that fraction upside down, and multiply both the numerators and the denominators. Picture dividing a fraction by a whole number as splitting it into that many parts. In this example, ¼ is split in half, resulting in twice as many equal parts. Dividing two proper fractions Proper fractions can be divided by other proper fractions by using an inverse operation. Multiplication and division are inverse operations—they are the opposite of each other. imagine the multiplication sign means “of” is same as saying ÷ sw 3 multiplied by /, or / of 3, gives / × 1÷1 4 3 = same as 3⁄1 1 ÷ 1 4 3 is same as saying 1 × 4 3 ÷ sign becomes × sign = 3 4 Dividing one fraction by another is the same as turning the second fraction upside down and then multiplying the two. i ou tc h a r n d denominator is now numerator 1 3 3 × = 4 1 4 sw i tc h a round To divide two fractions use the inverse operation— turn the last fraction upside down, then multiply the numerators and the denominators. Dividing mixed fractions To divide mixed fractions, ﬁrst convert them into improper fractions, then turn the second fraction upside down and multiply it by the ﬁrst. whole number 1 ÷ 3 1 2 multiply whole number by denominator 1 4 1×3+1 3 denominator First, turn each of the mixed fractions into improper fractions by multiplying the whole number by the denominator and adding the numerator. sw 4÷9 3 4 2×4+1 4 add to numerator ÷ sign becomes × sign i tc h around denominator is now numerator 4 4 16 × = 3 9 27 sw i tc h a round Divide the two fractions by turning the second fraction upside down, then multiplying the numerators and the denominators. 56 NUMBERS : Ratio and proportion SEE ALSO RATIO COMPARES THE SIZE OF QUANTITIES. PROPORTION COMPARES THE RELATIONSHIP BETWEEN TWO SETS OF QUANTITIES. Multiplication 18–21 22–25 48–55 Division Fractions Ratios show how much bigger one thing is than another. Two things are in proportion when a change in one causes a related change in the other. Writing ratios Ratios are written as two or more numbers with a colon between each. For example, a fruit bowl in which the ratio of apples to pears is 2 : 1 means that there are 2 apples for every 1 pear in the bowl. ◁ Supporters This group represents fans of two football clubs, the “greens” and the “blues.” these are the fans of the “greens” ▷Forming a ratio To compare the numbers of people who support the two different clubs, write them as a ratio. This makes it clear that for every 4 green fans there are 3 blue fans. there are 4 green supporters 4 ▽ More ratios The same process applies to any set of data that needs to be compared. Here are more groups of fans, and the ratios they represent. 1: 2 1: △1:2 One fan of the greens and 2 fans of the blues can be compared as the ratio 1 : 2. This means that in this case there are twice as many fans of the blues as of the greens. 3 this is the symbol for the ratio between the fans : 3 2 : △1:3 One fan of the greens and 3 fans of the blues can be shown as the ratio 1 : 3, which means that, in this case, there are three times more blue fans than green fans. there are 3 blue supporters 5 △2:5 Two fans of the greens and 5 fans of the blues can be compared as the ratio 2 : 5. There are more than twice as many fans of the blues as of the greens. 57 R AT I O A N D P R O P O R T I O N Finding a ratio 20 minutes is / of an hour Large numbers can also be written as ratios. For example, to ﬁnd the ratio between 1 hour and 20 minutes, convert them into the same unit, then cancel these numbers by ﬁnding the highest number that divides into both. 1 hour is the same as 60 minutes, so convert minutes are the smaller unit ratios show information in the same way as fractions do this is the symbol for ratio 60 ÷ 20 = 3 12 20 ÷ 20 = 1 9 20 mins, 60 mins 20 : 60 1 : 3 Convert one of the quantities so that both have the same units. In this example use minutes. Write as a ratio by inserting a colon between the two quantities. 3 6 Cancel the units to their lowest terms. Here both sides divide exactly by 20 to give the ratio 1 : 3. Working with ratios Ratios can represent real values. In a scale, the small number of the ratio is the value on the scale model, and the larger is the real value it represents. scale = 1 : 50,000 ▷ Scaling down 1 : 50,000 cm is used as the scale on a map. Find out what a distance of 1.5 cm represents on this map. 1.5 distance on map scale shows what real distance is represented by each distance on the map cm scale on map actual distance represented by the map 1.5 cm × 50,000 = 75,000 cm = 750 m ▷ Scaling up The plan of a microchip has the scale 40 : 1. The length of the plan is 18 cm. The scale can be used to find the length of the actual microchip. length of plan divide by scale to ﬁnd actual size actual length of microchip 18 cm ÷ 40 = 0.45 cm the answer is converted into a more suitable unit—there are 100 cm in a meter Comparing ratios compare the numerators Converting ratios into fractions allows their size to be compared. To compare the ratios 4 : 5 and 1 : 2, write them as fractions with the same denominator. 1:2= and 4:5= 1 2 4 5 fraction that represents ratio 1 : 2 fraction that represents ratio 4 : 5 First write each ratio as a fraction, placing the smaller quantity in each above the larger quantity. 5 × 2 is 10, the common denominator 2 × 5 is 10, the common denominator ×2 ×5 1 2 = 105 ×5 4 5 = 108 ×2 Convert the fractions so that they both have the same denominator, by multiplying the first fraction by 5 and the second by 2. 5 10 is smaller than 8 10 so 1:2 is smaller than 4:5 Because the fractions now share a denominator, their sizes can be compared, making it clear which ratio is bigger. 58 NUMBERS PROPORTION Two quantities are in proportion when a change in one causes a change in the other by a related number. Two examples of this are direct and indirect (also called inverse) proportion. Direct proportion ▷ Direct proportion This table and graph show the directly proportional relationship between the number of gardeners and the number of trees planted. Two quantities are in direct proportion if the ratio between them is always the same. This means, for example, that if one quantity doubles then so does the other. each gardener can plant 2 trees in a day ×2 ×2 doubling number of gardeners doubles number of trees planted 2:4 ×112 ×112 5 4 3 6 3 line showing direct proportion is always straight 2 0 Indirect proportion ▷ Indirect proportion This table and graph show the indirectly proportional relationship between the vans used and the time taken to deliver the parcels. Two quantities are in indirect proportion if their product (the answer when they are multiplied by each other) is always the same. So if one quantity doubles, the other quantity halves. 1 2 3 NUMBER OF GARDENERS Vans Days 1 8 2 4 4 2 1:8 ×2 ÷2 1 van takes 8 days to deliver the parcels 8 6 2:4 ×2 ÷2 2 vans take 4 days to deliver the parcels TIME DAYS if the number of vans doubles then it takes half the time to deliver the parcels 2 2 1 3:6 ▷ Delivering parcels The number of vans used to deliver parcels determines how many days it takes to deliver the parcels. Twice as many vans means half as many days to deliver. 1 4 the ratio is always the same when reduced to its simplest terms, in this case 1 : 2 1 van takes 8 days to deliver some parcels Trees 2 gardeners can plant 4 trees in a day 6 TREES PLANTED IN A DAY ▷Planting trees The number of gardeners used to plant trees determines how many trees can be planted in a day: twice as many gardeners means twice as many trees can be planted. 1:2 Gardeners the line showing indirect proportion is always curved 4 2 vans take 4 days to deliver the parcels 2 4:2 the product of the number of vans and days is always the same: 8 0 1 2 3 4 NUMBER OF VANS 5 59 R AT I O A N D P R O P O R T I O N Dividing in a given ratio A quantity can be divided into two, three, or more parts, according to a given ratio. This example shows how to divide 20 people into the ratios 2 : 3 and 6 : 3 : 1. DIVIDING INTO A TWOPART RATIO DIVIDING INTO A THREEPART RATIO These are the ratios to divide the people into. 2:3 6:3:1 total number of parts in the ratio 2+3=5 number of parts in the ratio total number of people 20 ÷ 5 = 4 2 in the ratio 3 in the ratio 8 people represented by 2 in the ratio total number of parts in the ratio total number of people 20 ÷ 10 = 2 Divide the number of people by the parts of the ratio. 2×4=8 3 × 4 = 12 12 people represented by 3 in the ratio 6 + 3 + 1 = 10 Add the different parts of the ratio to find the total parts. 6 in the ratio 3 in the ratio Multiply each part of the ratio by this quantity to find the size of the groups the ratios represent. 1 in the ratio 6 × 2 = 12 3×2=6 1×2=2 12 people represented by 6 in the ratio 6 people represented by 3 in the ratio 2 people represented by 1 in the ratio Proportional quantities Proportion can be used to solve problems involving unknown quantities. For example, if 3 bags contain 18 apples, how many apples do 5 bags contain? total number of apples bags apples per bag 18 ÷ 3 = 6 There is a total of 18 apples in 3 bags. Each bag contains the same number of apples. To find out how many apples there are in 1 bag, divide the total number of apples by the number of bags. apples per bag number of bags 6 × 5 = 30 total To find the number of apples in 5 bags, multiply the number of apples in 1 bag by 5. 60 0 0 NUMBERS Percentages SEE ALSO Decimals 44–45 48–55 Fractions Ratio and A PERCENTAGE SHOWS AN AMOUNT AS A PART OF 100. Any number can be written as a part of 100 or a percentage. Percent means “per hundred,” and it is a useful way of comparing two or more quantities. The symbol “%” is used to indicate a percentage. proportion 56–59 Rounding oﬀ 70–71 F TEAEMALE CHE RS 10 o ut o or f 10 0 Parts of 100 10 % The simplest way to start looking at percentages is by dealing with a block of 100 units, as shown in the main image. These 100 units represent the total number of people in a school. This total can be divided into diﬀerent groups according to the proportion of the total 100 they represent. 100% 50% ▷ This group is equally divided between 50 blue and 50 purple ﬁgures. Each represents 50 out of 100 or 50% of the total. This is the same as half. 1% ▷ In this group there is only 1 blue ﬁgure out of 100, or 1%. 2 3 4 TS DEN U T or LE S MA 0 f 10 o t u 19 o 19 % 5 6 7 RS CHE A E or LE T MA 0 f 10 o t 5 ou 5% 8 9 10 1 1 2 ▷ This is simply another