Number, Shape, & Symmetry

Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory

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Features

  • Provides the foundation for teaching and learning mathematics, showing how number theory and geometry are essential components to understanding mathematics
  • Synthesizes basic ideas that lead to an appreciation of the deeper mathematical ideas that grow from these foundations
  • Reflects the authors’ extensive experience teaching undergraduates, students in the Young Scholars Program, and public school teachers in SESAME
  • Includes relatively simple practice problems, more challenging problems, and sets of exercises

Forthcoming solutions manual available upon qualifying course adoption

Watch Diane L. Herrmann discuss the book.

Summary

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.

Table of Contents

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.

Author Bio(s)

Editorial Reviews

"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 material, but was guiding me through each step of the lesson through clear examples, as if presented in a face-to-face class. … On the college level, this is a great book to use as either a primary or supplementary book for a number theory class."
—Peter Olszewski, MAA Reviews, August 2013

"All budding mathematicians should have the opportunity to savour this marvelously engaging book. The authors bring to the text an extensive background working with students and have mastered the fine art of both motivating and delighting them with mathematics. Their experience is evident on every page: creative practice problems draw the reader into the discussion, while frequent examples and detailed diagrams keep each section lively and appealing. Herrmann and Sally have carefully charted a course that takes the reader through number theory, introductory group theory, and geometry, with an emphasis on symmetries in the latter two subjects. The result is a labour of love that should inspire young minds for years to come."
—Sam Vandervelde, author of Bridge to Higher Mathematics and coordinator of the Mandelbrot Competition

"Number, Shape, & Symmetry accomplishes the rare feat of presenting real and deep mathematics in a clear and accessible manner. This book distills the beauty of some of the most fundamental ideas of mathematics and is a terrific resource for anyone interested in exploring these subjects."
—Bridget Tenner, Associate Professor of Mathematics, DePaul University

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