The Art of Proof
Basic Training for Deeper Mathematics
Authors: Beck, Matthias, Geoghegan, Ross
Free Preview Presents fundamental mathematics, integers and real numbers, in a way that asks for student participation, while teaching how mathematics is done
 Provides students with methods and ideas they can use in future courses
 Primarily for: undergraduates who have studied calculus or linear algebra; mathematics teachers and teachersintraining; scientists and social scientists who want to strengthen their command of mathematical methods
 Extra topics in appendices give instructor flexibility
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 About this Textbook

The Art of Proof is designed for a onesemester or twoquarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail, while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book  Discrete and Continuous  be given equal attention. The book ends with short essays on further topics suitable for seminarstyle presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.
 About the authors

Matthias Beck received his initial training in mathematics in Würzburg, Germany, received his Ph.D. in mathematics from Temple University, and is now associate professor of mathematics at San Francisco State University. He is the recipient of the 2013 MAA Haimo Award for Distinguished College or University Teaching of Mathematics. He is the author of a previously published Springer book, Computing the Continuous Discretely (with Sinai Robins).
Ross Geoghegan received his initial training in mathematics in Dublin, Ireland, received his Ph.D. in mathematics from Cornell University, and is now professor of mathematics at the State University of New York at Binghamton. He is the author of a previously published Springer book, Topological Methods in Group Theory.
 Reviews

From the reviews:
"The Art of Proof is a surprising union of rigor with taste and wit. The authors take a hardcore axiomatic approach, but the writing is never dry. Instead, topics are carefully chosen and meticulously developed with grace and humor, careful attention to detail, and just the right number of skillbuilding exercises and thoughtprovoking problems.
"The text is spare—well under two hundred pages—but contains a thorough axiomatic development of the integers and the reals, along with nonstandard optional topics such as Cayley graphs and generating functions. Instead of the standard scattershot "symbolic logicset theoryfunctionsproof by contradictionzzzz..." books, this text keeps its focus on just a few fundamental ideas, of which induction is the most important. This helps my students to feel that they are participants in a grand undertaking—the construction of a number system—rather than passive victims of one proof technique after another." —Paul Zeitz (Mathematics Professor at the University of San Francisco)“This qualitative transition presents a most acute pedagogical challenge. … This book does feature definite mathematical content, contrasting with works that aim at decoupling purely logical apparatus from strictly mathematical concerns. … The authors write with the authority of research mathematicians and clearly mean to open that avenue to students. Summing Up: Recommended. Upperdivision undergraduates through professionals.” (D. V. Feldman, Choice, Vol. 48 (8), April, 2011)
“This book offers an approach wellbalanced between rigor and clarifying simplification. Dilbert and Foxtrot cartoons with philosophical quotes presage the introduction of axioms and preliminary propositions. This graceful and witty blend succeeds well in a textbook for a postcalculus course transitioning a student to higher mathematics. The Art of Proof can also well serve independent readers looking for a solitary path to a vista on higher mathematics.” (Tom Schulte, The Mathematical Association of America, November, 2010)
“This is an undergraduate text to extend, in a deeper and formal way, the usual initial knowledge of mathematics. The book deals with classical topics like integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, uncountable sets … . The publication may be useful for people using the book to teach a course on the above mentioned topics. … The aim behind this textbook is teaching how to read and write mathematics as well as understanding key methods and concepts.” (Claudi Alsina, Zentralblatt MATH, Vol. 1198, 2010)
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Bibliographic Information
 Bibliographic Information

 Book Title
 The Art of Proof
 Book Subtitle
 Basic Training for Deeper Mathematics
 Authors

 Matthias Beck
 Ross Geoghegan
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2010
 Publisher
 SpringerVerlag New York
 Copyright Holder
 Matthias Beck and Ross Geoghegan
 eBook ISBN
 9781441970237
 DOI
 10.1007/9781441970237
 Hardcover ISBN
 9781441970220
 Softcover ISBN
 9781493940868
 Series ISSN
 01726056
 Edition Number
 1
 Number of Pages
 XXI, 182
 Topics