[内容简介]

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.

[目录]

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.