Through a careful treatment of number theory and geometry, this text helps readers understand serious mathematical ideas and proofs. It shows how number theory and geometry are essential components to understanding mathematics. Classroom-tested, the book draws on the authors’ successful work with undergraduate students at the University of Chica

The Triangle Game. The Beginnings of Number Theory. Axioms in Number Theory. Divisibility and Primes. The Division and Euclidean Algorithms. Variations on a Theme. Congruences and Groups. Applications of Congruences. Rational Numbers and Real Numbers. Introduction to Geometry and Symmetry. Polygons and Their Construction. Symmetry Groups. Permutations. Polyhedra. Graph Theory. Tessellations. Connections. Appendix. Glossary. Bibliography. Index.

"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 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