Number, shape, and symmetry : an introduction to number theory, geometry, and group theory

Preface that students keep a pencil and paper handy as they read the text so that the Practice Problems can be solved as soon as they are encountered. The second category of problems presented in the text is that of Challenge Problems. These are intended to be more difficult than the Practice Problems, and their solutions may require considerable ingenuity. Finally, at the end of each chapter, we present a large set of exercises. Some of these exercises test the students' understanding of the material in the text. Others are designed to extend the textual material and project students into more sophisticated aspects of number theory and geometry. This book first emerged from a two-quarter course for nonscience majors at the University of Chicago. That course covered number theory in the first quarter and geometry in the second. We expect that, by expanding on the textual material through projects contained in the exercises, this text could constitute a two-semester course. At Chicago, we have also met the need to present this material in a single quarter. The selected topics for a coherent and interesting one-quarter (or one-semester) course include Chapters 0-4, Chapter 6, Chapters 9-14. Chapter 5, Variations on a Theme, and Chapter 15, Tessellations, contain material that can be assigned as projects for student investigation and presentation in class. A one-quarter class would also omit Chapters 8 and 16, which deal with ideas of the infinite. We would like to thank those who provided help for getting this text in print. First, we had support from the College of the University of Chicago. This support allowed us to hire course assistants, without whose help the completion of the text would have taken considerably more time. Second, we owe a tremendous debt of gratitude to Ashley Reiter Ahlin, Emily Peters, and Alex Zorn, who not only prepared the manuscript, but also provided valuable suggestions and feedback about the organization and content of the book. We also thank Barbara Csima, John Zekos, David Schmitz, Ryan Reich, and Kevin Tucker for their help in writing and collecting exercises, and for typing part of the text. Those who used the preliminary notes in the Math 110 course at Chicago, including Chris Degni, Chris French, Kaj Gartz, Chris Hallstrom, Moon Duchin, and "This beautifully produced book shows how number theory and geometry are essential components to understanding mathematics, with emphasis on teaching and learning such topics. The presentation is excellent and the approach to logic and proofs exemplary. ... The book accomplishes the rare feat of presenting some real mathematics in a clear and accessible manner, thereby showing some of the most fundamental ideas of mathematics. It is an engaging text offering the opportunity to a beginner to learn and savor the many ideas involved, and it is also a good resource for readers interested in exploring such ideas. ... It is suitable for school teachers and their more able students, particularly those who want enrichment activities for school mathematical societies. It is also an excellent text for liberal arts students at university, and perhaps even for students in science and engineering. Thus, students already familiar with topics such as calculus and differential equations will find the book an enjoyable read to complement what they are used to."-Mathematical Gazette"Well-rounded approaches to logic and proofs have been achieved in Number, Shape, & Symmetry. ... The proofs in this book guide the student from simple ideas ... to more advanced ventures ... It is good to see the arithmetic developed in detail from the fundamental axioms so that students have a clear understanding of each consequence. It is also good that the authors do not take for granted how to solve equations ... The text has a nice, natural build-up in difficulty of problems. ... Diane L. Herrmann and Paul J. Sally, Jr., have dedicated a great deal of time to writing the text. ... Each section is written to be manageable for students to learn, with just the correct amount of content. When I was reading the text, I thought it was my own personal professor who was not only teaching and presenting "This textbook shows how number theory and geometry are the essential components in the teaching and learning of mathematics for students in primary grades. The book synthesizes basic ideas that lead to an appreciation of the deeper mathematical ideas that grow from these foundations. The authors reflect their extensive experience teaching undergraduate nonscience majors, students in the Young Scholars Program, and public school K-8 teachers in the Seminars for Endorsement of Science and Mathematics Educators (SESAME)."-- Provided by publisher Through a careful treatment of number theory and geometry, Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofs. Classroom-tested, the book draws on the authors' successful work with undergraduate students at the University of Chicago, seventh