com The Math Handbook Everyday Math Made Simple Richard Elwes www.com New York • London © 2011 by Richard Elwes All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by reviewers, who may quote brief www.com passages in a review. Scanning, uploading, and electronic distribution of this book or the facilitation of the same without the permission of the publisher is prohibited. The picture credits constitute an extension to this copyright notice.
Please purchase only authorized electronic editions, and do not participate in or encourage www.com electronic piracy of copyrighted materials. Your support of the author’s rights is appreciated. Any member of educational institutions wishing to photocopy part or all of the work for classroom use or anthology should send inquiries to Permissions c/o Quercus Publishing Inc., 31 West 57th Street, 6th Floor, New York, NY 10019, or to permissions@quercus.com ISBN 978-1-62365-294-4 Distributed in the United States and Canada by Random House Publisher Services c/o Random House, 1745 Broadway New York, NY 10019 Designed and illustrated by Patrick Nugent www.com PICTURE CREDITS iStock: 4, 11, 66, 86, 140, 188, 205 Shutterstock: 18, 24, 35, 44, 73, 80, 95, 102, 110, 124, 174, 182, 197, 212 Thinkstock: 50, 57, 117, 148, 156, 166 Patrick Nugent: 132 www.com Contents Introduction The language of mathematics Addition Subtraction Multiplication Division Primes, factors and multiples www.com Negative numbers and the number line Decimals Fractions Arithmetic with fractions Powers The power of 10 Roots and logs www.com Percentages and proportions Algebra Equations Angles Triangles Circles Area and volume Polygons and solids www.com Pythagoras’ theorem Trigonometry Coordinates Graphs Statistics Probability Charts Answers to quizzes www.com Index www.com Introduction “I was never any good at mathematics.” I must have heard this sentence from a thousand different people. I cannot dispute that it may be true: people do have different strengths and weaknesses, different interests and priorities, different opportunities and obstacles.
But, all the same, an understanding of mathematics is not something www.com anyone is born with, not even Pythagoras himself. Like all other skills, from portraiture to computer programming, from knitting to playing cricket, mathematics can only be developed through practice, that is to say through actually doing it. Nor, in this age, is mathematics something anyone can afford to ignore. Few people stop to worry whether they are good at talking or www.com good at shopping.
Abilities may indeed vary, but generally talking and shopping are unavoidable parts of life. And so it is with mathematics. Rather than trying to hide from it, how about meeting it head on and becoming good at it? Sounds intimidating? Don’t panic! The good news is that just a handful of central ideas and techniques can carry you a very long way. So, I am pleased to www.com present this book: a no-nonsense guide to the essentials of the subject, especially written for anyone who “was never any good at mathematics.” Maybe not, but it’s not too late! Before we get underway, here’s a final word on philosophy.
Mathematical education is split between two rival camps. Traditionalists brandish rusty compasses and dusty books of log www.com tables, while modernists drop fashionable buzzwords like “chunking” and talk about the “number line.” This book has no loyalty to either group. I have simply taken the concepts I consider most important, and illustrated them as clearly and straightforwardly as I can. Many of the ideas are as ancient as the pyramids, though some have a more recent heritage.com modern presentation can bring a fresh clarity to a tired subject; in other cases, the old tried and tested methods are the best.
Richard Elwes www.com The language of mathematics • Writing mathematics • Understanding what the various mathematical symbols mean, and how to use them • Using BEDMAS to help with calculations www.com Let’s begin with one of the www.com commonest questions in any mathematics class: “Can’t I just use a calculator?” The answer is … of course you can! This book is not selling a puritanical brand of mathematics, where everything must be done laboriously by hand, and all help is turned down. You are welcome to use a calculator for arithmetic, just as you can use a word-processor for writing text. But handwriting is an essential skill, even in today’s hi-tech www. You can use a dictionary or a spell-checker too.
All the same, isn’t it a good idea to have a reasonable grasp of basic spelling? There may be times when you don’t have a calculator or a computer to hand. You don’t want to be completely lost without it! Nor do you want to have to consult it every time a few numbers need to be added together. After all, you don’t get out your dictionary every www.com time you want to write a simple phrase. So, no, I don’t want you to throw away your calculator.
But I would like to change the way you think about it. See it as a labor saving device, something to speed up calculations, a provider of handy shortcuts. The way I don’t want you to see it is as a mysterious black box which performs near-magical feats that www.com you alone could never hope to do. Some of the quizzes will show this icon , which asks you to have a go without a calculator.
This is just for practice, rather than being a point of principle! Signs and symbols Mathematics has its own physical toolbox, full of calculators, compasses and protractors. We shall meet these in later chapters. Mathematics also comes with an www.com impressive lexicon of terms, from “radii” to “logarithms,” which we shall also get to know and love in the pages ahead. Perhaps the first barrier to mathematics, though, comes before these: it is the library of signs and symbols that are used.
Most obviously, there are the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is interesting that once we get to the number ten there is not a new www. Instead, the symbols for 0 and 1 are recycled and combined to produce the name “10.” Instead of having one symbol alone, we now have two symbols arranged in two columns. Which column the symbol is in carries just as much information as the symbol itself: the “1” in “13” does not only mean “one,” it means “one ten.” This method of representing numbers in columns is at the heart of the decimal system: the modern way of www.com representing numbers.
It is so familiar that we might not realize what an ingenious and efficient system it is. Any number whatsoever can be written using only the ten symbols 0–9. It is easy to read too: you don’t have to stop and wonder how much “41” is. This way of writing numbers has major consequences for the things that we do with them.
The best methods for addition, subtraction, www.com multiplication and division are based around understanding how the columns affect each other. We will explore these in depth in the coming chapters. There are many other symbols in mathematics besides numbers themselves. To start with, there are the four representing basic arithmetical procedures: +, −, ×, ÷.
In fact there are other symbols which mean the same things.com many situations, scientists prefer a dot, or even nothing at all, to indicate multiplication. So, in algebra, both ab and a · b, mean the same as a × b, as we shall see later. Similarly, division is just as commonly expressed by as by a ÷ b. This use of letters is perhaps the greatest barrier to mathematics.
How can you multiply and divide letters? (And why would you want www.com to?) These are fair questions, which we shall save until later. Writing mathematics Here is another common question: www.com “What is the point of writing out mathematics in a longwinded fashion? Surely all that matters is the final answer?” The answer is … no! Of course, the right answer is important. I might even agree that it is usually the most important thing. But it is certainly not the only important thing.
Why not? Because you will have a much better chance of reliably arriving at the right www.com answer if you are in command of the reasoning that leads you there. And the best way of ensuring that is to write out the intermediate steps, as clearly and accurately as possible. Writing out mathematics has two purposes. Firstly it is to guide and illuminate your own thought- processes.
You can only write things out clearly if you are thinking about them clearly, and it www.com is this clarity of thought that is the ultimate aim. The second purpose is the same as for almost any other form of writing: it is a form of communication with another human being. I suggest that you work under the assumption that someone will be along shortly to read your mathematics (whether or not this is actually true). Will they be able to tell what you are doing? Or is it a jumble of symbols, comprehensible only to you? www.com Mathematics is an extension of the English language (or any other language, but we’ll stick to English!), with some new symbols and words.
But all the usual laws of English remain valid. In particular, when you write out mathematics, the aim should be prose that another person can read and understand. So try not to end up with symbols scattered randomly around the page. That’s fine for rough working, while you are www.com trying to figure out what it is you want to write down.
But after you’ve figured it out, try to write everything clearly, in a way that communicates what you have understood to the reader, and helps them understand it too. The importance of equality The most important symbol in mathematics is “=.” Why? Because the number-one goal of mathematics is to discover the www.com value of unknown quantities, or to establish that two superficially different objects are actually one and the same. So an equation is really a sentence, an assertion. An example is “146 + 255 = 401,” which states that the value on the left-hand side of the “=” sign is the same as the value on the right.
It is amazing how often the “=” sign gets misused! If asked to calculate 13 + 12 + 8, many www.com people will write “13 + 12 = 25 + 8 = 33.” This may come from the use of calculators where the button can be interpreted to mean “work out the answer.” It may be clear what the line of thought is, but taken at face value it is nonsense: 13 + 12 is not equal to 25 + 8! A correct way to write this would be “13 + 12 + 8 = 25 + 8 = 33.” Now, every pair of quantities that are asserted to be equal reallywww.com are equal − a great improvement! The “=” sign has some lesser- known cousins, which make less powerful assertions: “<” and “>.” For example, the statement “A < B” says that the quantity A is less than B. An example might be 3 + 9 < 13. Flipping this around gives “B > A,” which says that B is greater than A, for example, 13 > 3 + 9. The statements “A < B” and “B > A” look different, but have exactly www.com the same meanings (in the same way that “A = B” and “B = A” mean essentially the same thing).
Other symbols in the same family are “≥” and “≤,” which stand for “is greater than or equal to” and “is less than or equal to” (otherwise known as “is at least” and “is at most”).