Problems and Proofs in Numbers and Algebra

1. Front Matter

   Pages i-x

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2. The Integers

   1. Front Matter

      Pages 1-1

      PDF

   2. Number Concepts, Prime Numbers, and the Division Algorithm

      • Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

      Pages 3-39

   3. Greatest Common Divisors, Diophantine Equations, and Combinatorics

      • Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

      Pages 41-78

   4. Equivalence Classes with Applications to Clock Arithmetic and Fractions

      • Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

      Pages 79-107

3. The Algebra of Polynomials and Linear Systems

   1. Front Matter

      Pages 109-109

      PDF

   2. Polynomials and the Division Algorithm

      • Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

      Pages 111-142

   3. Factoring Polynomials, Their Roots, and Some Applications

      • Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

      Pages 143-164

   4. Matrices and Systems of Linear Equations

      • Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

      Pages 165-216

4. Back Matter

   Pages 217-223

   PDF

Focusing on an approach of solving rigorous problems and learning how to prove, this volume is concentrated on two specific content themes, elementary number theory and algebraic polynomials. The benefit to readers who are moving from calculus to more abstract mathematics is to acquire the ability to understand proofs through use of the book and the multitude of proofs and problems that will be covered throughout. This book is meant to be a transitional precursor to more complex topics in analysis, advanced number theory, and abstract algebra. To achieve the goal of conceptual understanding, a large number of problems and examples will be interspersed through every chapter. The problems are always presented in a multi-step and often very challenging, requiring the reader to think about proofs, counter-examples, and conjectures. Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high-achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge. In the past, PNA has been taught in a "problem solving in middle school” course (twice), to a quite advanced high school students course (three semesters), and three times as a secondary resource for a course for future high school teachers. PNA is suitable for secondary math teachers who look for material to encourage and motivate more high achieving students.

• Algebra
• Combinatorics
• Euclidean Algorithms
• Number Theory
• Proofs

“Aimed at introducing postcalculus students to higher mathematics by way of solving rigorous problems and learning how to prove. … the content is less focused on basic mathematical concepts seen in upper-division college mathematics coursework and more so on topics that teachers might present in their classrooms, and on interesting applications … . Teachers of mathematics at the secondary level would be well served by taking a course based on this text. Summing Up: Recommended. Upper-division undergraduates through faculty.” (D. S. Larson, Choice, Vol. 53 (1), September, 2015)

• School of Mathematics, Georgia Institute of Technology, Atlanta, USA

  Richard S. Millman

• Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, USA

  Peter J. Shiue

• Department of Mathematics, Computer Science, and Statistics, Bloomsburg University, Bloomsburg, USA

  Eric Brendan Kahn

Richard S. Millman,Ph.D., Director, Center for Education Integrating Science, Mathematics, and Computing (CEISMC) Georgia Institute of Technology Atlanta, GA 30332-­0282

Peter J. Shiue Department of Mathematical Sciences University of Nevada, Las Vegas 4505 Maryland Pkwy Las Vegas, NV 89154-4020

Eric Brendan Kahn Department of Mathematics, Computer Science, and Statistics Bloomsburg University 400 East Second Street Bloomsburg, PA 17815

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