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.
Rezensionen / Stimmen
"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)
Produkt-Info
Previously published in hardcover
Reihe
Auflage
Softcover reprint of the original 1st ed. 2014
Sprache
Verlagsort
Verlagsgruppe
Springer International Publishing
Zielgruppe
Illustrationen
50
50 s/w Abbildungen
50 Illustrations, black and white; XII, 161 p. 50 illus.
Maße
Höhe: 23.5 cm
Breite: 15.5 cm
Gewicht
ISBN-13
978-3-319-37950-0 (9783319379500)
DOI
10.1007/978-3-319-05654-8
Schweitzer Klassifikation
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.
1. Introduction to the Natural Numbers.- 2. Mathematical Induction.- 3. Modular Arithmetic.- 4. The Fundamental Theorem of Arithmetic.- 5. Fermat's Theorem and Wilson's Theorem.- 6. Sending and Receiving Coded Messages.- 7. The Euclidean Algorithm and Applications.- 8. Rational Numbers and Irrational Numbers.- 9. The Complex Numbers.- 10. Sizes of Infinite Sets.- 11. Fundamentals of Euclidean Plane Geometry.- 12. Constructability.
