the a to z of mathematics, a basic guide - sidebotham

The A to Z of Mathematics The A to Z of Mathematics A Basic Guide Thomas H. Sidebotham St. Bede’s College Christchurch, New Zealand A JOHN WILEY & SONS, INC., PUBLICATION This book is printed on acid-free paper. ∞  Copyright C  2002 by John Wiley & Sons, Inc., New York. All rights reserved. Published simultaneously in Canada. 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, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per- copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750- 8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM. For ordering and customer service, call 1-800-CALL-WILEY. Library of Congress Cataloging-in-Publication Data is available The A to Z of Mathematics : A Basic Guide—Thomas H. Sidebotham ISBN 0-471-15045-2 Printed in the United States of America. 10987654321 To my wife, Patricia Who persuaded me to get started and supported me until it was finished Contents Preface ix Acknowledgments xi A 1 B 48 C 63 D 145 E 167 F 197 G 214 H 234 I 245 K 269 L 272 M 281 N 296 O 306 P 314 vii viii CONTENTS Q 358 R 371 S 399 T 433 U 463 V 464 W 471 X 472 Y 473 Z 474 Preface Throughout the world many people suffer from the same problem: math anxiety. No other area of skill seems to polarize people so readily into two contrasting groups, those who can do math and those who cannot. Of the two groups, the second one is by far the larger. To succeed in mathematics you need to understand the basics, and only then can you learn with confidence. Many people fall at this first hurdle and then struggle later. My aim in writing this book is to guide you through the basics so that you can develop an understanding of mathematical processes. As you study this book you will become aware of how mathematics relates to everyday life and situations with which you are familiar. Study this book in depth, simply browse, or search for the meaningof a word, andlearn yourmath again.Why shouldyou go to this trouble? Whatever your age, mathematics is one of the basic requirements of life. This study of mathematics will make a difference. This book is written in an appropriate language for explaining basic mathematics to the general reader, and uses examples drawn from everyday life. There are many worked examples with detailed steps of working. Each step of working is accompa- nied by an explanation. It is this process of showing HOW and explaining WHY that gives this book its unique style. Those mathematical abbreviations that often frustrate readers are written in full and the text is “user-friendly.” For quick reference the for- mat of the book is alphabetical, and it covers topics in basic mathematics. They are linked together with cross-references so that a theme can be followed through. This book is a great deal more than a dictionary. Under each entry there is a straightfor- ward explanation of the term, followed in many cases by carefully worked examples, showing the relevance of mathematics in the world around us. At the end of some entries the reader is directed to other references in the book if some prior knowledge is needed. The mathematics is reliable and up to date, and encompasses a wide range of topics so that everyone will find something of interest. ix x PREFACE The material in the book falls into three categories. 1. There are processes that explain specific skills; a typical example is the entry Algebra. 2. There are straightforward definitions of words with applications in the world around us, as in the entry Proportion. 3. There is a variety of enrichment material that has good entertainment value, like Hexomino. I believe there is something of interest for everyone. If you are curious about mathematics and it intrigues you, now may be the time to take the initiative and discover that you indeed have skills in this area of knowledge. Some people need the maturity of a few more years before they achieve success. If you are making a career change and need to revise your mathematical knowledge, then this book is for you. The book will appeal to everyone, even students, who may be interested in, or need to catch up on, basic mathematics. If you are a parent who desires to help your son or daughter and lack the expertise, then this book is for you also. The style and presentation of the book are chosen specifically to suit the lay reader. It is a useful resource for home schooling situations. I hope it is a rich source of ideas for mathematics teachers and also those who are in teacher training, whatever subject in which they are specializing. You will need a scientific calculator to follow through the steps of working in some of the examples. In statistical topics the reader is referred to the calculator handbook for its use, because brands of calculators can vary so much. The whole book is cross-referenced. If readers are not familiar with the explanations given in a specific entry, they are advised to first read the references at the end of the entry to prepare the groundwork. This book contains an abundance of diagrams, equations, tables, graphs, and worked examples. An emphasis is placed on SI units throughout. If you are keen to acquire the basic skills of mathematics in this book, I offer the following advice. Do not read it like you may read a novel in which you can skim and still enjoy the book and have a good grasp of the story and plot. To grasp mathematics you must examine the detail of every word and symbol. Have a pen, a calculator, and paper at hand to try out the processes and verify them for yourself. It is my hope that this book will be on a bookshelf in every home, and will be used by family members as a reference and guide. I am sure you will find it useful, interesting, and entertaining. Tom Sidebotham Acknowledgments I take this opportunity to thank the following persons, without whose help and guidance this book would not have been possible: Steve Quigley, for encouraging, enthusing, and redirecting me, and for sharing the vision; Heather Haselkorn, for all her efforts on my behalf to keep the project afloat and for maintaining the lines of communication; David Byatt, for scrutinizing the text and offering judicious alterna- tives (the accuracy is entirely the responsibility of the author); Dr. David Sidebotham, for his computer skills in enabling me to transplant the book in New York; Stephanie Lentz and her team at TechBooks for transforming my manuscript so wonderfully well; and lastly, the Angel at my elbow throughout the writing. xi A ABSOLUTE VALUE The absolute value of a number is the distance of the number from the origin 0, measured along the number line. In the accompanying figure the absolute value of 3 at point B is the distance of point B from 0, which is 3. The absolute value of −2at point A is the distance of point A from 0, which is 2. −3 −2 −1 0 1 2 3 AB 4 The symbol for the absolute value of a number is two vertical parallel lines placed around the number. At the point B, the absolute value of 3 is written as | point B −point 0 | =|3 − 0| =|3| = 3 If the subtraction had taken place in the opposite way, the answer would still be 3, but the working would be Absolute value of 3 = | point 0 −point B | = | 0 − 3 | = | −3 | = 3 as before Similarly for the point A: Absolute value of −2 = | point A − point 0 | = | −2 − 0 | = | −2 | = 2 1 The A to Z of Mathematics: A Basic Guide. Thomas H. Sidebotham Copyright ¶ 2002 John Wiley & Sons, Inc. ISBN: 0-471-15045-2 [...]... compasses and insert the point at O, and draw an arc AB Using the same radius, or longer if you wish, insert the point of the compass at A and draw an arc Then insert the compass point at B, and using the same radius, draw another arc to meet the arc from A at X The line OX is the angle bisector of angle AOB, and divides the pizza equally The point X is equidistant from the lines OA and OB References: Arc,... h−1 , to one decimal place References: Average Speed, Canceling, Equivalent Fractions, Rational Expression, Reciprocal ALTERNATE ANGLES Figure a shows two parallel lines indicated by arrows The line cutting across them is called a transversal A pair of angles such as a and b that are on alternate sides of the transversal and lie between the parallel lines are equal in size and are called alternate angles... shows two more pairs of corresponding angles that are equal in size: Angle e = angle f Angle g = angle h Example A ladder leans against a vertical wall, and the angle the ladder makes with the horizontal ground is 65◦ Find the angle the ladder makes with the top of the wall (figure g) Solution Write x = 65◦ Corresponding angles are equal The ladder makes an angle of 65◦ with the top of the wall x 65° (g)... find the angle, marked x, that the laser beam makes with the line of the seabed The seabed is parallel to the sea level 47 ° x (d) Solution Write x = 47◦ Alternate angles are equal The laser beam makes an angle of 47◦ with the seabed There are two more geometry theorems that relate to two parallel lines and their transversal They are as follows: 1 Corresponding angles are equal in size 2 Cointerior angles... projection of ON onto the plane can be considered to be the shadow cast by the nail ON when parallel rays of light shine at right angles to the plane N W O (a) The angle NOW is the angle between the line ON and the plane of the wood The method of calculating the angle between a line and a plane is explained in the following example Example The longest diagonal of a cuboid, which is a box, is the line... roughly the same size as 208 So a zero must be inserted to make sure the answer of 210 is approximately the same size as 208 Alternatively, the answer can be expressed in standard form as 2.1 × 102 , which is rounded to 2 sf 200 208 210 (h) References: Approximation, Decimal, Limits of Accuracy, Standard Form ACRE The acre is a unit of area for measuring the size of a piece of land, and is an imperial unit... between a line and a plane, the line is projected onto the plane, and then the angle between the projected line and the original line is calculated This angle is the angle between the line and the plane Suppose a straight nail ON is hammered into a piece of wood at an angle so that the nail is not upright (see figure a) The projection of the nail ON onto the plane of the wood is OW, as shown in figure a The. .. Combining the 7 with the 3 to make 10 = 15 Reference: Associative Law ADJACENT ANGLES Angles such as x and y in figure a that lie side by side are called adjacent angles x y a (a) b (b) 10 ALGEBRA Any number of adjacent angles that form a straight line add up to 180◦ When two adjacent angles form a straight line they add up to 180◦ In figure b, angles a and b are supplementary angles: a + b = 180◦ Example... Cointerior angles are supplementary, which means they add together to equal 180◦ With reference to figure e, a pair of angles that are on the same side of the transversal and occupy a similar position are equal in size and are called corresponding angles Angles a and b are corresponding angles, and so are angles c and d: Angle a = angle b Angle c = angle d c a g e d b h f (e) (f) ALTERNATE ANGLES 19 Figure... ruler-and-compass construction, which is described in the example below Example A piece of pizza has been left for Jacob and Nathan’s supper Doing a drawing of the pizza, show how to construct a line, using ruler and compasses only, that equally divides the piece of pizza into two X A O B Solution The drawing of the pizza is shown in the figure with the construction lines drawn dashed Open the compasses . available The A to Z of Mathematics : A Basic Guide Thomas H. Sidebotham ISBN 0-4 7 1-1 504 5-2 Printed in the United States of America. 10987654321 To my wife, Patricia Who persuaded me to get started and supported. can skim and still enjoy the book and have a good grasp of the story and plot. To grasp mathematics you must examine the detail of every word and symbol. Have a pen, a calculator, and paper at. The A to Z of Mathematics The A to Z of Mathematics A Basic Guide Thomas H. Sidebotham St. Bede’s College Christchurch, New Zealand A JOHN WILEY & SONS, INC., PUBLICATION This