A Readable Introduction to Real Mathematics

1. Front Matter

   Pages i-xii

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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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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.

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

“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)

• 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

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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