A Readable Introduction to Real Mathematics
Authors: Rosenthal, Daniel, Rosenthal, David, Rosenthal, Peter
Free Preview 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
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 About this Textbook

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, wellorganised, 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] wellchosen, basic, conceptual mathematics, suitably accessible after a K12 education, in a detailed, selfconscious way that emphasizes methodology alongside content and crucially leads to an ultimate clear payoff. … Summing Up: Recommended. Lowerdivision undergraduates and twoyear technical program students; general readers.” (D.V. Feldman, Choice, Vol. 52 (6), February, 2015)
 About the authors

Daniel Rosenthal obtained his mathematics degree from 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.  Reviews

“The book was quite an enjoyable read … . It would undoubtedly help students just entering the world of theoretical mathematics, though perhaps after more advanced preparatory material than just high school algebra and trigonometry.” (Meghan De Witt, MAA Reviews, October 6, 2019)
Reviews of the first edition:
“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, wellorganised, 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 wellchosen, basic, conceptual mathematics, suitably accessible after a K12 education, in a detailed, selfconscious way that emphasizes methodology alongside content and crucially leads to an ultimate clear payoff. … Summing Up: Recommended. Lowerdivision undergraduates and twoyear technical program students; general readers.” (D. V. Feldman, Choice, Vol. 52 (6), February, 2015)
 Table of contents (14 chapters)


Introduction to the Natural Numbers
Pages 17

Mathematical Induction
Pages 922

Modular Arithmetic
Pages 2329

The Fundamental Theorem of Arithmetic
Pages 3135

Fermat’s Little Theorem and Wilson’s Theorem
Pages 3742

Table of contents (14 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 A Readable Introduction to Real Mathematics
 Authors

 Daniel Rosenthal
 David Rosenthal
 Peter Rosenthal
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2018
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer Nature Switzerland AG
 eBook ISBN
 9783030006327
 DOI
 10.1007/9783030006327
 Hardcover ISBN
 9783030006310
 Series ISSN
 01726056
 Edition Number
 2
 Number of Pages
 XVIII, 218
 Number of Illustrations
 63 b/w illustrations
 Topics