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herrmann d number shape symmetry introduction number theory 2012

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Textbook in PDF format 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. The Triangle Game. The Beginnings of Number Theory. Setting the Table: Numbers, Sets and Functions. Rules of Arithmetic. One's Digit Arithmetic. Axioms in Number Theory. Consequences of the Rules of Arithmetic. Inequalities and Order. Divisibility and Primes. 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. Congruences and Groups. Congruences and Arithmetic of Residue Classes. Groups and Other Structures. Divisibility Tests. Rational Numbers and Real Numbers. Infinity. Rational Numbers. Irrational Numbers. Introduction to Geometry and Symmetry. Polygons and Their Construction. Polygons and Their Angles. 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. Euler's Formula. Symmetries of Regular Polyhedra. Reections and Rotations. Variations on a Theme: Other Polyhedra. Graph Theory. The Königsberg Bridge Problem. Colorability and Planarity. Graphs and Their Complements. Trees. Tessellations. 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

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herrmann d number shape symmetry introduction number theory 2012