© 2013

An Invitation to Abstract Mathematics


  • Gives a broad view of the field of mathematics without the artificial division of subjects???

  • Provides students with a broad exposure to mathematics by including an unusually diverse array of topics

  • Discusses important milestones in the history of mathematics and features some of the most interesting recent accomplishments in the field

  • Goes beyond the typical elementary text by providing a more thorough and deeper treatment whenever feasible

  • The text can be studied at various levels and depths, with, hopefully, plenty of fun for students and instructors alike


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

Table of contents

  1. Front Matter
    Pages i-xiv
  2. What’s Mathematics?

    1. Front Matter
      Pages 1-1
    2. Béla Bajnok
      Pages 3-10
    3. Béla Bajnok
      Pages 11-22
    4. Béla Bajnok
      Pages 23-31
    5. Béla Bajnok
      Pages 33-44
    6. Béla Bajnok
      Pages 45-54
    7. Béla Bajnok
      Pages 55-68
  3. How to Solve It?

    1. Front Matter
      Pages 69-69
    2. Béla Bajnok
      Pages 71-81
    3. Béla Bajnok
      Pages 83-94
    4. Béla Bajnok
      Pages 95-108
    5. Béla Bajnok
      Pages 109-123
    6. Béla Bajnok
      Pages 135-147
    7. Béla Bajnok
      Pages 149-160
    8. Béla Bajnok
      Pages 161-173
    9. Béla Bajnok
      Pages 175-187
    10. Béla Bajnok
      Pages 189-195
  4. Advanced Math for Beginners

    1. Front Matter
      Pages 197-197

About this book


This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind.  The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse.  The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader.  With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible.  This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians.

Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research projects.


abstract mathematics bridge course cardinalities decision trees infinity logic mathematical proof mathematics milestones number systems prime number theorem pythagoream theorem set theory surreal numbers transition course

Authors and affiliations

  1. 1., Department of MathematicsGettysburg CollegeGettysburgUSA

About the authors

Béla Bajnok is a native of Hungary, and he remains greatly influenced by the excellent education he received there. He currently holds the endowed position of Alumni Chair Professor at Gettysburg College. He is also the recipient of the Mathematical Association of America's 2012 James P. Crawford EPADEL Teaching Award.

Bibliographic information

Industry Sectors
Finance, Business & Banking


From the reviews:

“Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … Bajnok’s book can most certainly be used as a text for a traditional transition course designed for mathematics majors. He wrote the book with a broader audience in mind, and personally uses it at Gettysburg College for anyone interested in deepening his or her mathematical knowledge … .” (Jill Dietz, MAA Reviews, May, 2014)

“This book can be added to the growing population of ‘transition’ books--texts aimed at bridging the leap from the traditional calculus sequence to upper-division coursework in mathematics. The topics typically associated with such a text are all present: basic logic, predicates and quantifiers, induction, relations, and functions. … any reader able to complete this work and its problems will be suitably prepared for further studies in mathematics. Summing Up: Recommended. Lower- and upper-division undergraduates.” (D. S. Larson, Choice, Vol. 51 (5), January, 2014)

“This textbook aims for the spot in many departments’ curriculum where students are introduced to advanced mathematics. … the author introduces the game in an applied setting that the reader will likely find both compelling and intriguing: a set of options for two corporations in competition. … This text definitely focuses on mathematics. The material covers a wide range of material, probably more than most instructors will cover in one semester … .” (John Perry, zbMATH, Vol. 1274, 2013)