Herrmann / Sally, Jr. Number, Shape, & Symmetry

An Introduction to Number Theory, Geometry, and Group Theory
1. Auflage 2013
ISBN: 978-1-4665-5465-8
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 444 Seiten

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 to tenth grade mathematically talented students in the University of Chicago’s Young Scholars Program, and elementary public school teachers in the Seminars for Endorsement in Science and Mathematics Education (SESAME).

The first half of the book focuses on number theory, beginning with the rules of arithmetic (axioms for the integers). The authors then present all the basic ideas and applications of divisibility, primes, and modular arithmetic. They also introduce the abstract notion of a group and include numerous examples. The final topics on number theory consist of rational numbers, real numbers, and ideas about infinity.

Moving on to geometry, the text covers polygons and polyhedra, including the construction of regular polygons and regular polyhedra. It studies tessellation by looking at patterns in the plane, especially those made by regular polygons or sets of regular polygons. The text also determines the symmetry groups of these figures and patterns, demonstrating how groups arise in both geometry and number theory.
The book is suitable for pre-service or in-service training for elementary school teachers, general education mathematics or math for liberal arts undergraduate-level courses, and enrichment activities for high school students or math clubs.

Zielgruppe

Students and instructors in math education; students in gifted math programs; general readers interested in mathematics and education.

Autoren/Hrsg.

Herrmann, Diane L.

Sally, Jr., Paul J.

Fachgebiete

• Sozialwissenschaften Pädagogik Lehrerausbildung, Unterricht & Didaktik Allgemeine Didaktik Naturwissenschaften, Mathematik (Unterricht & Didaktik)
• Mathematik | Informatik Mathematik Algebra Zahlentheorie
• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

The Triangle Game

The Beginnings of Number Theory

Setting the Table: Numbers, Sets and Functions

Rules of Arithmetic

A New System

One's Digit Arithmetic

Axioms in Number Theory

Consequences of the Rules of Arithmetic

Inequalities and Order

Divisibility and Primes

Divisibility

Greatest Common Divisor

Primes

The Division and Euclidean Algorithms

The Division Algorithm

The Euclidean Algorithm and the Greatest Common Divisor
The Fundamental Theorem of Arithmetic

Variations on a Theme

Applications of Divisibility

More Algorithms

Congruences and Groups

Congruences and Arithmetic of Residue Classes
Groups and Other Structures

Applications of Congruences

Divisibility Tests

Days of the Week

Check Digits

Rational Numbers and Real Numbers

Fractions to Decimals
Decimals to Fractions
Infinity

Rational Numbers

Irrational Numbers

How Many Real Numbers?

Introduction to Geometry and Symmetry
Polygons and Their Construction

Polygons and Their Angles
Constructions

Symmetry Groups

Symmetric Motions of the Triangle

Symmetric Motions of the Square

Symmetries of Regular n-gons

Permutations

Symmetric Motions as Permutations

Counting Permutations and Symmetric Groups

Even More Economy of Notation

Polyhedra

Regular Polyhedra

Euler’s Formula

Symmetries of Regular Polyhedra

Reections and Rotations

Variations on a Theme: Other Polyhedra

Graph Theory

Introduction

The Königsberg Bridge Problem

Colorability and Planarity

Graphs and Their Complements

Trees

Tessellations

Tessellating with a Single Shape

Tessellations with Multiple Shapes

Variations on a Theme: Polyominoes

Frieze Patterns
Infinite Patterns in Two and Three Dimensions

Connections

The Golden Ratio and Fibonacci Numbers

Constructible Numbers and Polygons

Appendix: Euclidean Geometry Review

Glossary

Bibliography
Index
Practice Problem Solutions and Hints as well as Exercises appear at the end of each chapter.

Über Autor(innen)

Diane L. Herrmann is a senior lecturer and associate director of undergraduate studies in mathematics at the University of Chicago. Dr. Herrmann is a member of the American Mathematical Society, Mathematical Association of America, Association for Women in Mathematics, Physical Sciences Collegiate Division Governing Committee, and Society for Values in Higher Education. She is also involved with the University of Chicago’s Young Scholars Program, Summer Research Opportunity Program (SROP), and Seminars for Elementary Specialists and Mathematics Educators (SESAME).

Paul J. Sally, Jr. is a professor and director of undergraduate studies in mathematics at the University of Chicago, where he has directed the Young Scholars Program for mathematically talented 7-12 grade students. Dr. Sally also founded SESAME, a staff development program for elementary public school teachers in Chicago. He is a member of the U.S. Steering Committee for the Third International Mathematics and Science Study (TIMSS) and has served as Chairman of the Board of Trustees for the American Mathematical Society.