Problems and Proofs in Numbers and Algebra

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

Table of contents

  1. Front Matter
    Pages i-x
  2. The Integers

    1. Front Matter
      Pages 1-1
    2. Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
      Pages 3-39
    3. Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
      Pages 41-78
    4. 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
    2. Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
      Pages 111-142
    3. Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
      Pages 143-164
    4. Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn
      Pages 165-216
  4. Back Matter
    Pages 217-223

About this book

Introduction

Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets in-depth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications in-depth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles.

The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to “prove” or “solve” complex problems. This method of instruction is augmented by examining applications of number theory in systems such as RSA cryptography, Universal Product Code (UPC), and International Standard Book Number (ISBN). The numerous problems and examples included in each section reward curiosity and insightfulness over more simplistic approaches. Each problem set begins with a few easy problems, progressing to problems or proofs with multi-step solutions. Exercises in the text stay close to the examples of the section, allowing students the immediate opportunity to practice developing techniques.  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.

Keywords

Algebra Combinatorics Euclidean Algorithms Number Theory Proofs

Authors and affiliations

  • Richard S. Millman
    • 1
  • Peter J. Shiue
    • 2
  • Eric Brendan Kahn
    • 3
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Mathematical SciencesUniversity of Nevada, Las VegasLas VegasUSA
  3. 3.Department of Mathematics, Computer Science, and StatisticsBloomsburg UniversityBloomsburgUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-14427-6
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-14426-9
  • Online ISBN 978-3-319-14427-6
  • About this book