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Problems and Proofs in Numbers and Algebra

  • Textbook
  • © 2015

Overview

Authors:
  1. Richard S. Millman

    1. School of Mathematics, Georgia Institute of Technology, Atlanta, USA

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  2. Peter J. Shiue

    1. Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Eric Brendan Kahn

    1. Department of Mathematics, Computer Science, and Statistics, Bloomsburg University, Bloomsburg, USA

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    You can also search for this author in PubMed   Google Scholar

  • Provides a foundation for solving proofs and problems as a transition to more abstract algebra and mathematics

  • Readers learn problem solving abilities concerning proofs through numbers and algebra

  • Problems and theorems are focused on many diverse areas of number theory and algebra

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-x
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  2. The Integers

    1. Front Matter

      Pages 1-1
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    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
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    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
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Keywords

About this book

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.

Reviews

“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)

Authors and Affiliations

  • 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

About the authors

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

Bibliographic Information

  • Book Title: Problems and Proofs in Numbers and Algebra

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

  • DOI: https://doi.org/10.1007/978-3-319-14427-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2015

  • Hardcover ISBN: 978-3-319-14426-9Published: 09 March 2015

  • Softcover ISBN: 978-3-319-35723-2Published: 06 October 2016

  • eBook ISBN: 978-3-319-14427-6Published: 09 February 2015

  • Edition Number: 1

  • Number of Pages: X, 223

  • Number of Illustrations: 9 b/w illustrations

  • Topics: General Algebraic Systems, Number Theory, Mathematical Logic and Foundations

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