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A Readable Introduction to Real Mathematics

  • Daniel Rosenthal
  • David Rosenthal
  • Peter Rosenthal

Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 1-7
  3. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 9-22
  4. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 23-29
  5. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 31-35
  6. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 37-42
  7. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 43-48
  8. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 49-61
  9. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 63-72
  10. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 73-88
  11. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 89-113
  12. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 115-132
  13. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 133-164
  14. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 165-192
  15. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 193-213
  16. Back Matter
    Pages 215-218

About this book

Introduction

Designed for an undergraduate course or for independent study, this text presents sophisticated mathematical ideas in an elementary and friendly fashion. The fundamental purpose of this book is to teach mathematical thinking while conveying the beauty and elegance of mathematics. The book contains a large number of exercises of varying difficulty, some of which are designed to help reinforce basic concepts and others of which will challenge virtually all readers. The sole prerequisite for reading this text is high school algebra. Topics covered include: * mathematical induction * modular arithmetic * the Fundamental Theorem of Arithmetic * Fermat's Little Theorem * RSA encryption * the Euclidean algorithm * rational and irrational numbers * complex numbers * cardinality * Euclidean plane geometry * constructibility (including a proof that an angle of 60 degrees cannot be trisected with a straightedge and compass)* infinite series * higher dimensional spaces.

This textbook is suitable for a wide variety of courses and for a broad range of students of mathematics and other subjects. Mathematically inclined senior high school students will also be able to read this book.

From the reviews of the first edition:

“It is carefully written in a precise but readable and engaging style… I thoroughly enjoyed reading this recent addition to the Springer Undergraduate Texts in Mathematics series and commend this clear, well-organised, unfussy text to its target audiences.” (Nick Lord, The Mathematical Gazette, Vol. 100 (547), 2016)

 “The book is an introduction to real mathematics and is very readable. … The book is indeed a joy to read, and would be an excellent text for an ‘appreciation of mathematics’ course, among other possibilities.” (G.A. Heuer, Mathematical Reviews, February, 2015)

“Many a benighted book misguidedly addresses the need [to teach mathematical thinking] by framing reasoning, or narrowly, proof, not as pervasive modality but somehow as itself an autonomous mathematical subject. Fortunately, the present book gets it right.... [presenting] well-chosen, basic, conceptual mathematics, suitably accessible after a K-12 education, in a detailed, self-conscious way that emphasizes methodology alongside content and crucially leads to an ultimate clear payoff. … Summing Up:  Recommended. Lower-division undergraduates and two-year technical program students; general readers.” (D.V. Feldman, Choice, Vol. 52 (6), February, 2015)

Keywords

Fermat's Theorem RSA method cardinality complex numbers mathematical induction natural numbers trisection of angles

Authors and affiliations

  • Daniel Rosenthal
    • 1
  • David Rosenthal
    • 2
  • Peter Rosenthal
    • 3
  1. 1.TorontoCanada
  2. 2.Department of Mathematics and Computer ScienceSt. John’s UniversityQueensUSA
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-00632-7
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-00631-0
  • Online ISBN 978-3-030-00632-7
  • Series Print ISSN 0172-6056
  • Series Online ISSN 2197-5604
  • Buy this book on publisher's site
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