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© 2014

A Readable Introduction to Real Mathematics

Benefits

  • Presents sophisticated ideas in algebra and geometry in an elementary fashion

  • Includes exercises of varying difficulty to help motivate and teach the reader

  • Develops mathematical thinking that will be useful for future mathematics teachers and mathematics majors

  • Solutions to selected exercises are freely available in PDF

Textbook

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

Table of contents

  1. Front Matter
    Pages i-xii
  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 31-34
  6. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 35-40
  7. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 41-46
  8. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 47-60
  9. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 61-70
  10. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 71-84
  11. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 85-108
  12. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 109-126
  13. Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 127-157
  14. Back Matter
    Pages 159-161

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 engage the reader and to teach a real understanding of mathematical thinking while conveying the beauty and elegance of mathematics. The text focuses on teaching the understanding of mathematical proofs. The material covered has applications both to mathematics and to other subjects. The book contains a large number of exercises of varying difficulty, designed to help reinforce basic concepts and to motivate and challenge the reader. The sole prerequisite for understanding the text is basic high school algebra; some trigonometry is needed for Chapters 9 and 12. 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 * constructability (including a proof that an angle of 60 degrees cannot be trisected with a straightedge and compass) This textbook is suitable for a wide variety of courses and for a broad range of students in the fields of education, liberal arts, physical sciences and mathematics. Students at the senior high school level who like mathematics will also be able to further their understanding of mathematical thinking by reading this book.

Keywords

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

Authors and affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Dept. of Mathematics & Computer ScienceSt. John's UniversityJamaicaUSA
  3. 3.Department of MathematicsUniversity of TorontoTorontoCanada

About the authors

Daniel Rosenthal is a mathematics student at the University of Toronto. David Rosenthal is Associate Professor of Mathematics at St. John's University in New York City. Peter Rosenthal is Professor Emeritus of Mathematics at the University of Toronto.

Bibliographic information

  • Book Title A Readable Introduction to Real Mathematics
  • Authors Daniel Rosenthal
    David Rosenthal
    Peter Rosenthal
  • Series Title Undergraduate Texts in Mathematics
  • Series Abbreviated Title Undergraduate Texts Mathematics
  • DOI https://doi.org/10.1007/978-3-319-05654-8
  • Copyright Information Springer International Publishing Switzerland 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-05653-1
  • Softcover ISBN 978-3-319-37950-0
  • eBook ISBN 978-3-319-05654-8
  • Series ISSN 0172-6056
  • Series E-ISSN 2197-5604
  • Edition Number 1
  • Number of Pages XII, 161
  • Number of Illustrations 50 b/w illustrations, 0 illustrations in colour
  • Topics Mathematics, general
    Number Theory
    Geometry
  • Buy this book on publisher's site
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Reviews

“It is carefully written in a precise but readable and engaging style and is tightly organised into eight short ‘core’ chapters and four longer standalone ‘extension’ chapters. … 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)

“Daniel Rosenthal and Peter Rosenthal (both, Univ. of Toronto) and David Rosenthal (St. John's Univ.) present 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)