Overview
- Develops fundamental concepts in algebra, geometry, and number theory from the foundations of set theory
- Engages readers through challenging examples and problems inspired by mathematical contests
- Illuminates the historical context of key mathematical developments
Part of the book series: Undergraduate Texts in Mathematics (UTM)
Part of the book sub series: Readings in Mathematics (READINMATH)
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Table of contents (12 chapters)
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Front Matter
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Back Matter
Keywords
- Axiomatic mathematics before calculus
- Axiomatic precalculus
- Rigorous precalculus textbook
- Mathematics competition textbook
- Math circles textbook
- Mathematics senior capstone textbook
- History of mathematics textbook
- Peano's axioms
- Dedekind cut
- Real number system via Cauchy sequences
- Stolz-Cesaro theorems
- Birkhoff metric geometry
- Arithmetic operations on polynomials
- Bernoulli numbers
- Basel problem
About this book
This textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed.
Beginning with a thorough treatment of the natural numbers via Peano’s axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoff’s axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics.
Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.
Reviews
“Transparency of explanation and gradually built material are outstanding features of the textbook. In addition, solutions to some problems are designed using more than one approach, making it adaptable to various students' backgrounds. … The book makes itself accessible to a vast population of students. The book can enhance the undergraduate curriculum or serve as a reflective resource for graduate mathematics students.” (Andrzej Sokolowski, MAA Reviews, March 20, 2022)
“A historical concern is present throughout,with pieces of information on the history of concepts and theorems.” (Victor V. Pambuccian, zbMATH 1479.00002, 2022)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Elements of Mathematics
Book Subtitle: A Problem-Centered Approach to History and Foundations
Authors: Gabor Toth
Series Title: Undergraduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-030-75051-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-75050-3Published: 24 September 2021
Softcover ISBN: 978-3-030-75053-4Published: 25 September 2022
eBook ISBN: 978-3-030-75051-0Published: 23 September 2021
Series ISSN: 0172-6056
Series E-ISSN: 2197-5604
Edition Number: 1
Number of Pages: XV, 527
Number of Illustrations: 30 b/w illustrations, 3 illustrations in colour
Topics: Real Functions, Number Theory, Field Theory and Polynomials, Geometry, History of Mathematical Sciences