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

Diane L. Herrmann, Paul J. Sally, Jr.

October 18, 2012 by A K Peters/CRC Press
Textbook - 444 Pages - 319 B/W Illustrations
ISBN 9781466554641 - CAT# K15328

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  • 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

Watch Diane L. Herrmann discuss the book.


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.