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A Readable Introduction to Real Mathematics

  • Textbook
  • © 2014

Overview

Authors:
  1. Daniel Rosenthal

    1. Department of Mathematics, University of Toronto, Toronto, Canada

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  2. David Rosenthal

    1. Dept. of Mathematics & Computer Science, St. John's University, Jamaica, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Peter Rosenthal

    1. Department of Mathematics, University of Toronto, Toronto, Canada

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    You can also search for this author in PubMed   Google Scholar

  • 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
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Introduction to the Natural Numbers

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 1-7

  3. Mathematical Induction

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 9-22

  4. Modular Arithmetic

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 23-29

  5. The Fundamental Theorem of Arithmetic

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 31-34

  6. Fermat’s Theorem and Wilson’s Theorem

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 35-40

  7. Sending and Receiving Secret Messages

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 41-46

  8. The Euclidean Algorithm and Applications

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 47-60

  9. Rational Numbers and Irrational Numbers

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 61-70

  10. The Complex Numbers

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 71-84

  11. Sizes of Infinite Sets

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 85-108

  12. Fundamentals of Euclidean Plane Geometry

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 109-126

  13. Constructability

    • Daniel Rosenthal, David Rosenthal, Peter Rosenthal
    Pages 127-157

  14. Back Matter

    Pages 159-161
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Keywords

About this book

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.

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)

Authors and Affiliations

  • Department of Mathematics, University of Toronto, Toronto, Canada

    Daniel Rosenthal, Peter Rosenthal

  • Dept. of Mathematics & Computer Science, St. John's University, Jamaica, USA

    David Rosenthal

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.

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