Elements of Mathematics: A Problem-Centered Approach to History and Foundations

This textbook presents a rigorous introduction to mathematics before calculus, covering algebra, geometry, and number theory. It builds from set theory foundations and includes thought-provoking examples and challenging problems. The book covers the natural numbers, plane geometry, polynomials, rational expressions, exponential, logarithmic, and trigonometric functions, and emphasizes axioms and limits. It is ideal for students seeking a deep mathematical challenge and is suitable for various levels, including high achievers and competition preparation.

Format: Paperback / softback
Length: 527 pages
Publication date: 24 September 2022
Publisher: Springer Nature Switzerland AG

This comprehensive textbook provides a rigorous introduction to mathematics before the advent of calculus. It delves into fundamental concepts in algebra, geometry, and number theory, building upon the foundations of set theory in an elementary and inquiry-driven manner. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theoretical development, while frequent historical asides unveil the historical origins of the ideas.

The book begins with a thorough exploration of the natural numbers, employing Peano's axioms to establish the natural, integral, rational, and real number systems. Plane geometry is introduced through Birkhoff's axioms of metric geometry, and subsequent chapters delve into arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is presented, followed by a comprehensive study of rational expressions. The treatment concludes with an in-depth exploration of exponential, logarithmic, and trigonometric functions, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits serve as the underlying framework throughout, offering not only powerful tools but also insights into non-trivial connections between topics.

Elements of Mathematics is particularly well-suited for students seeking a rigorous and engaging mathematical challenge utilizing elementary tools. Whether used as an enhancement to the early undergraduate curriculum for high achievers or as a capstone course for senior students, instructors will find ample material to engage inquisitive mathematics majors. No formal prerequisites beyond high school algebra are assumed, making the book accessible to mathematics circles and competition preparation. Advanced readers will appreciate the interleaving of ideas and the illuminating historical details.

In summary, this textbook offers a comprehensive and accessible introduction to mathematics, providing a solid foundation for students embarking on further mathematical studies. Its rigorous approach, engaging examples, and historical insights make it an invaluable resource for both educators and students alike.

Weight: 968g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030750534
Edition number: 1st ed. 2021