Algebra and Geometry Hung-Hsi Wu 10.1090/mbk/132 Algebra and Geometry Algebra and Geometry Hung-Hsi Wu 2010 Mathematics Subject Classification. Primary 97-01, 97-00, 97D99, 97-02, 00-01, 00-02. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-132 Library of Congress Cataloging-in-Publication Data Names: Wu, Hongxi, 1940- author. Title: Algebra and geometry / Hung-Hsi Wu. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Includes bibli- ographical references and index. Identifiers: LCCN 2020008693 | ISBN 9781470456764 (paperback) | ISBN 9781470460051 (ebook) Subjects: LCSH: Algebra. | Geometry. | AMS: Mathematics education – Instructional exposition (textbooks, tutorial papers, etc.). | Mathematics education – General reference works (hand- books, dictionaries, bibliographies, etc.). | Mathematics education – Education and instruction in mathematics – None of the above, but in this section. | Mathematics education – Research exposition (monographs, survey articles). | General – Instructional exposition (textbooks, tutorial papers, etc.). | General – Research exposition (monographs, survey articles). Classification: LCC QA154.3 .W84 2020 | DDC 512–dc23 LC record available at https://lccn.loc.gov/2020008693 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. 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Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 To NiuNiu with the hope that she will find it useful when she grows up Contents Contents of the Companion Volumes and Structure of the Chapters ix Preface xi To the Instructor xix To the Pre-Service Teacher xxxiii Prerequisites xxxvii Some Conventions xxxix Chapter1. LinearFunctions 1 1.1. Definition of a function and its graph 1 1.2. Why functions? 16 1.3. Linear functions of one variable 22 1.4. Linear inequalities and their graphs 30 1.5. Linear programming 42 1.6. Optimization: The general case 54 1.7. Appendix: Mathematical induction 57 Chapter 2. Quadratic Functions and Equations 63 2.1. Quadratic functions 63 2.2. A theorem on the graphs of quadratic functions 87 2.3. Graphs of equations of degree 2 96 2.4. The concept of an asymptote 116 Chapter 3. Polynomial and Rational Functions 121 3.1. Some basic facts about polynomials 121 3.2. Descartes’ rule of signs 128 3.3. Rational functions 131 Chapter 4. Exponential and Logarithmic Functions 135 4.1. An interpolation problem 136 4.2. Rational exponents 142 4.3. Exponential functions 153 4.4. Logarithms 162 Chapter 5. Polynomial Forms and Complex Numbers 175 5.1. Polynomial forms 175 5.2. Complex numbers 189 vii viii CONTENTS 5.3. Fundamental theorem of algebra 196 5.4. Binomial theorem 203 Chapter 6. Basic Theorems of Plane Geometry 213 6.1. Review 214 6.2. SSS and first consequences 220 6.3. Pedagogical comments 229 6.4. Proof of FTS 230 6.5. The angle sum of a triangle 238 6.6. Characterization of isometries 244 6.7. Some basic properties of a triangle 254 6.8. Basic properties of the circle 269 6.9. Power of a point with respect to a circle 297 6.10. Two interesting theorems about the circle 300 Chapter 7. Ruler and Compass Constructions 305 7.1. The basic constructions 306 7.2. The regular pentagon 318 7.3. A short history of the construction problems 325 Chapter 8. Axiomatic Systems 331 8.1. The concept of an axiomatic system 334 8.2. The role of axioms in school geometry 336 8.3. Hilbert’s axioms 339 8.4. Hyperbolic geometry 345 Appendix: Facts from [Wu2020a] 351 Glossary of Symbols 361 Bibliography 363 Index 367 Contents of the Companion Volumes and Structure of the Chapters Rational Numbers to Linear Equations [Wu2020a] Chapter 1: Fractions Chapter 2: Rational Numbers Chapter 3: The Euclidean Algorithm Chapter 4: Basic Isometries and Congruence Chapter 5: Dilation and Similarity Chapter 6: Symbolic Notation and Linear Equations Pre-Calculus, Calculus, and Beyond [Wu2020c] Chapter 1: Trigonometry Chapter 2: The Concept of Limit Chapter 3: The Decimal Expansion of a Number Chapter 4: Length and Area Chapter 5: 3-Dimensional Geometry and Volume Chapter 6: Derivatives and Integrals Chapter 7: Exponents and Logarithms, Revisited ix x CONTENTS OF THE COMPANION VOLUMES AND STRUCTURE OF THE CHAPTERS Structure of the chapters in this volume and its two companion volumes (RLE= Rational Numbers to Linear Equations A&G = Algebra and Geometry PCC = Pre-Calculus, Calculus, and Beyond) RLE-Chapter 1 ¨ ¨¨ ¨¨ ¨¨ ¨ RLE-Chapter 2 ¨ ``` ¨ ``` RLE-Chapter 4 ``` h ``` hhhh ` hh RLE-Chapter 5 RLE-Chapter 3 RLE-Chapter 6 A&G-Chapter 1 A&G-Chapter 2 A&G-Chapter 6 A&G-Chaptera 3 c a ¢ c aa ¢ c aa ¢ c a c A&G-Chapter 4 A&G-Chapter 5 ¢ c !! ¢ ! c !! ¢ c !! ¢ c ! A&G-Chapter 8 ¢ A&G-Chapter 7 PCC-Chapter 1 ¢ ¢ ! !! ¢ !! ¢ PCC-Chapter 2 ! !! ¢ !! ¢ !! ¢ !! ¢ PCC-Chapter 3 ¢ ¢ ¢ PCC-Chapter 4 ¢ ¢ ¢ ¢ PCC-Chapter 6 PCC-Chapter 5 PCC-Chapter 7 Preface Explain it! The most important thing is, that you are able to explain it! You will have exams, there you have to explain it. Eventually, you pass them, you get your diploma and you think, that’sit!—No,thewholelifeisanexam,.... Solearntoexplainit! A quote attributed to Rudolf M¨ossbauer (Nobel Prize in Physics, 1961) [Wiki-M¨ossbauer] This volume and its two companion volumes, [Wu2020a]and[Wu2020c], are written for high school teachers and mathematics education researchers. Formally, they assume only a knowledge of whole numbers. Informally, they also assume a level of mathematical maturity that welcomes reasoning, i.e., proofs. Teachers have to be at ease with such reasoning in their teaching before they can convince their students to do likewise, and mathematics educators must base their research on correct content knowledge that takes reasoning for granted. It is unfortunate that, despite a plethora of publications on high school mathematics, none has given a systematic, grade-level appropriate exposition of all the mathematical topics (i.e., not including probability and statistics) in the curriculum of grades 9–12 that also embraces mathematical integrity. The main purpose of these volumes is to make an initial attempt to fill this void.1 Because this is the second volume of a three-volume set, there are copious references throughout to the first volume, [Wu2020a]. In an effort to make this volume as self-contained as possible, we have collected the relevant assumptions, definitions, and theorems from [Wu2020a] in an appendix (page 351ff.) These three volumes, together with [Wu2011a], [Wu2016a], and [Wu2016b], give a complete exposition of the mathematics of K–12 that respects the normal progression of topics from grade to grade. Given that there is no lack of publications on school mathematics, from school textbooks to education research monographs on proofs, one may question why these six volumes (totaling some 2,500 pages) needed to be written. A simple answer is the presence of what we call Textbook School Mathematics (TSM) in the school curriculum. TSM is the irreparably flawed mathematics underlying the overwhelming majority of standard school mathemat- ics textbooks and professional development materials for teachers in roughly the five decades after 1970; it is characterized by a lack of clarity, persistent replacement of reasoning by rote memorization, and overall incoherence. This is the body of knowl- edge passed from teachers to students, so the cycle is repeated when some of these 1While these three volumes touch on probability only lightly in Section 5.4 of this volume, we should point out that there is a long section (Section 1.10) on finite probability in [Wu2016a]. xi xii PREFACE same students become teachers. The fact that TSM is unlearnable is a main cause of the current crisis in school mathematics education. It is hoped that these six volumes will help break this vicious cycle by offering, in sufficient detail, a usable replacement for TSM. They demonstrate, in a systematic and grade-appropriate manner, how the mathematics of the school curriculum can be developed coher- ently and purposefully by the use of precise definitions and reasoning. They open a curtain to reveal how school mathematics can be transformed into learnable math- ematics in the school classroom. In fact, parts of these six volumes2 have served as a blueprint for the writing of CCSSM, the Common Core State Standards for Mathematics ([CCSSM]). We refer to the Preface of [Wu2020a]aswellas pp. xx–xxv below for a more in-depth discussion of the issues involved. This volume covers high school algebra and high school geometry. In so doing, it has to confront two major instructional problems in the school mathematics cur- riculum. A current battle cry in school mathematics education is “Algebra for All” (compare [NMAP]), but algebra is also known as a notorious gatekeeper course that has kept too many students away from mathematics and other STEM subjects. In fact, the quadratic formula has become the poster child of arcane information in the curriculum—it is claimed—that only serves the purpose of thwarting the rising ambitions of many deserving students.3 A second problem is the high school ge- ometry course; its reputation for incomprehensibility and unlearnability is too well known for further comments here (see, e.g., Part II of the book review [Wu2004]).