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The Bright Child's Book of Numbers William C. Schulz

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The Bright Child's Book of Numbers William C. Schulz The Bright Child’s Book of Numbers William C. Schulz March 18, 2020 Transgalactic Publishing Company Flagstaff, Vienna, Cosmopolis ii c 2018 by William C. Schulz Every creator painfully experiences the chasm between his inner vision and its ultimate expression. Isaac Bashevis Singer iii To my daughters Alexia and Danae and my grandchildren Jack and Klara: all a source of interest, pride and joy. iv Contents Introduction ix 1 NUMBERSWITHOUTGEOMETRY 1 1.1 Introduction.............................. 2 1.2 Exponents............................... 3 1.3.ALittleAlgebra............................ 5 1.4 Modular Rings and Fields . 18 1.5 AlgebraicNumbers.......................... 22 1.6 MathematicalInduction . 29 1.7 HistoricalNotes............................ 35 1.8 ProblemsforChapter1 ....................... 38 2NUMBERSWITHGEOMETRY 43 2.1 ALittleHistory............................ 44 2.2 The Babylonian and the Decimal Systems of Representing Numbers 45 2.3 GeometricRepresentationoftheDecimals . 48 2.4 ConstructionoftheRealNumbers . 52 2.5 Appendix.DedekindCuts . 58 2.6 ComplexNumbers .......................... 59 2.7 Appendix: Radian measure and Euler’s formula . 71 2.8 ProblemsforChapter2 ....................... 79 3 QUATERNIONS 87 3.1 Introduction.............................. 88 3.2 Algebraic Definition and Properties of Quaternions . 91 3.3 RelationsofQuaternionstoGeometry . 96 3.3.1 PureVectorAlgebra . .. .. .. .. .. .. .. .. 97 3.3.2 Connections Between the Various Products . 100 3.4 VectorIdentitiesViaQuaternions. 105 3.5 ProblemsforChapter3 . 108 4 INFINITE NUMBERS 109 4.1 Introduction.............................. 110 4.2 TheAlgebraofSets ......................... 112 v vi CONTENTS 4.3 EquivalenceforSets ......................... 117 4.4 DefinitionoftheNaturalNumbers . 118 4.5 CardinalArithmetic ......................... 125 4.6 AppendixonSetPhilosophy. 132 4.6.1 Too liberal a policy . 132 4.6.2 TheAxiomaticApproach . 133 4.7 SomeHistory ............................. 136 4.8 Early Calculus an Infinitesimals . 137 4.9 TheFateofInfinitesimals . 139 4.10FormalLogic ............................. 140 4.11 SyntacticsandSemantics . 142 4.12 ModelsandConsistency . 143 4.13Introduction.............................. 147 4.14 ProblemsforChapter4 . 149 5 MATRICES 155 5.1 Introduction.............................. 156 5.2 BasicPropertiesandArithmetic . 157 5.3 Determinants ............................. 159 5.3.1 Cramer’sRule ........................ 163 5.3.2 Inverse Matrices and Applications . 164 5.4 MatrixRepresentationofNumbers . 167 5.4.1 Representationofthe complexnumbers . 169 5.4.2 RealRepresentationofthe Quaternions . 171 5.4.3 Complex Representation of the Quaternions . 172 5.5 ProblemsforChapter5 . 174 6 SOME NUMBER THEORY 181 6.1 Introduction.............................. 182 6.2 Dotpatterns ............................. 182 6.3 Euclidean Algorithm and Associated Things . 184 6.3.1 Basic Algorithm . 185 6.3.2 Matrix Form of Euclidean Algorithm . 187 6.3.3 Two-Variable Linear Diophantine Equations . 189 6.3.4 ContinuedFractions . 192 6.4 PrimeNumbers............................ 199 6.4.1 Introduction ......................... 199 6.4.2 Units .............................200 6.4.3 The Definition of Prime . 201 6.5 TheFundamentalTheoremofArithmetic . 204 6.5.1 Euclidean Functions . 204 6.5.2 UnitsAgain.......................... 205 6.5.3 TheFundamentalTheorem . 206 6.6 NumberTheoryandGeometry . 209 6.7 The Distribution of Primes . 211 6.7.1 ThePrimeNumberTheorem . 211 CONTENTS vii 6.7.2 Primes in Arithmetic Progressions . 213 6.8 Indices.................................216 6.9 QuadraticReciprocity . 218 6.10 TheChineseRemainderTheorem. 227 viii CONTENTS Introduction WHO IS THIS BOOK FOR? This is the little book that I wanted to read when I was a child. It is written for the reasonably bright child who is also curious. Such a child sits in school and listens to the teacher present the rules of numbers, without much effort to explain where the numbers come from and why the rules work. This is part of the reason for this book. I explain what kinds of numbers there are, how they came to be, what stupid ideas were prominent in history about them and how some of the ideas continue to plague teaching and learning. I write in what I hope is a colloquial style that a reasonably bright child can cope with and enjoy. This is a math book and it takes some thinking and some effort, so in addition to intelligence and curiosity the child must be willing to put forth a little effort, but I hope I am not going to strain anyone. If you are a girl, people may discourage you from reading this book. These are people who think women should spend their lives washing dishes, canning fruit and changing diapers. There is nothing wrong with these activities but this book will give you a glimpse of ways to enrich your life with mathematics or some other science, and whatever you learn some of it will get passed on to your children. Domestic chores are necessary but the good life has room for other things, and if this book encourages you to seek them out one of my goals will have been accomplished. This book in its initial stages talks about things that are well known to most kids by the 6th grade; whole numbers, signed whole numbers (integers), frac- tions and decimals (real numbers). I also talk about complex numbers, or to use the common slur, imaginary numbers. These are important for many pur- poses nowadays in science and technology, but there is another reason. From experience I know that past the age of 14 it is difficult psychologically to in- troduce new numbers. The experience can be frightening and troublesome for some. A child who accepted fractions and decimals without the least qualm, on encountering complex numbers at age 16, may be initially tripped up, and some students may be able to operate with them while simultaneously maintaining a disbelief in their existence. A few years using them will blunt this effect but we can much more easily ward it off by simply teaching them earlier along with ix x INTRODUCTION their representation in the plane. I learned them at about 12 and they seemed to me a natural extension of the real numbers. No mental trauma at all. This book is very systematic. We first introduce the various numbers in an algebraic context, with virtually no geometry. Later when the algebra is comfortable we then explain numbers in a geometric context, which is much more difficult than it at first appears. The later sections talk about various ideas of great interest and little difficulty that are not now a part of the ordinary curriculum. Many of these ideas have some uses and show there is much fun in mathematics beyond the stuff in the standard curriculum. The hope is that they may encourage students to learn more about these ideas when they become available and of course I hope to suggest to some of the children to later consider careers in mathematics and science or related areas. Some of this book is challenging but the book need not be read in order; I have been careful about this. If the reader gets to a point where she is no longer interested, skip to another part. It will be perfectly comprehensible at the beginning. Not everyone needs to know everything in the book and it’s better to skip to another part than to flounder in things no longer interesting. We are all different and things that greatly interest some people will have zero or less) interest for others. Make the book your own by reading the parts of it you like. You can always come back later when you may want to take another look. There are many fine books which present mathematics in a cultural context, as a part of general education. Some of these books present a good deal of philosophy along with the mathematics. This is not one of those books. This book is about mathematics itself. It may be elementary mathematics, (though some would claim I go beyond elementary things,) but it is still mathematics. I do from time to time present some of the history of the mathematics being discussed, but this is not an end in itself; it is for amusement and it is not systematic. For a systematic cultural and philosophical perspective on numbers I suggest the book WHAT IS NUMBER by Robert Tubbs. Let me say a few words about what this book is not. It is not a book that will help you do your taxes or realign your sewer pipe. It will not get you to Calculus any faster; the book is virtually Calculus free. However, I can promise you that if you read this book you will be in a lot better position than many other students when you get to college math because you will have learned some mathematical ways of thinking and will have seen some things already that will be new to your fellow students. You will occasionally recognize old mathematical friends and you will not be nearly as prone to misunderstand what the teacher is telling you. And if I am at all successful in the project, your math classes will seem like friendly places rather than hostile territory. Of course not everything in a math course can be fun, but maybe you will have learned that if you persevere through the less amusing stuff you will then be in a position where math again becomes fun and you’ll enjoy sense of achievement that comes with being able to do difficult things that have frightened off many an otherwise brave-hearted adventurer. Also, knowledge, like magic tricks or a never-fail ambrosia salad, is handy at parties and eventually generates a larger xi paycheck. One last word. Although this book does not cover any statistics and proba- bility, never miss a chance to learn them. They are handy in every walk of life. And a note of caution: take them in the math/stat department; don’t be fooled by classes with names like Economic Statistics, which will give you all of the fog and little of the clarity of the subject, and thus much less of the utility. A NOTE ON PROBLEMS This book contains a fair number of problems.
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