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Readings in Mathematics
cover

Elements of Mathematics

A Problem-Centered Approach to History and Foundations

Authors: Toth, Gabor

  • 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
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eBook $59.99
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  • The eBook version of this title will be available soon
  • Due: October 8, 2021
  • ISBN 978-3-030-75051-0
  • Digitally watermarked, DRM-free
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Hardcover $79.99
price for USA in USD
  • Customers within the U.S. and Canada please contact Customer Service at +1-800-777-4643, Latin America please contact us at +1-212-460-1500 (24 hours a day, 7 days a week). Pre-ordered printed titles are excluded from promotions.
  • Due: September 10, 2021
  • ISBN 978-3-030-75050-3
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
About this Textbook

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.

About the authors

Gabor Toth is Distinguished Professor of Mathematics at Rutgers University, Camden. His research interests include convex geometry and differential geometry, in particular, harmonic maps and minimal immersions. Beyond mathematics, he teaches Ancient Egyptian Grammar and the history of precolonial Africa. He regularly trains gifted high school students for mathematical contests in Princeton. His previous books include Measures of Symmetry for Convex Sets and Stability, Glimpses of Algebra and Geometry, and Finite Möbius Groups, Spherical Minimal Immersions, and Moduli.

Buy this book

eBook $59.99
price for USA in USD
  • The eBook version of this title will be available soon
  • Due: October 8, 2021
  • ISBN 978-3-030-75051-0
  • Digitally watermarked, DRM-free
  • Included format:
  • ebooks can be used on all reading devices
Hardcover $79.99
price for USA in USD
  • Customers within the U.S. and Canada please contact Customer Service at +1-800-777-4643, Latin America please contact us at +1-212-460-1500 (24 hours a day, 7 days a week). Pre-ordered printed titles are excluded from promotions.
  • Due: September 10, 2021
  • ISBN 978-3-030-75050-3
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
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Bibliographic Information

Bibliographic Information
Book Title
Elements of Mathematics
Book Subtitle
A Problem-Centered Approach to History and Foundations
Authors
Series Title
Readings in Mathematics
Copyright
2021
Publisher
Springer International Publishing
Copyright Holder
The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
eBook ISBN
978-3-030-75051-0
DOI
10.1007/978-3-030-75051-0
Hardcover ISBN
978-3-030-75050-3
Edition Number
1
Number of Pages
XV, 531
Number of Illustrations
30 b/w illustrations, 3 illustrations in colour
Topics