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

This textbook provides a rigorous presentation of mathematics before calculus, focusing on algebra, geometry, and number theory. It is designed for students seeking a deep and engaging mathematical challenge and covers topics such as the natural numbers, plane geometry, polynomials, rational expressions, exponential, logarithmic, and trigonometric functions. Axioms and limits are used throughout, and the book is suitable for mathematics circles and competition preparation.

Format: Hardback
Length: 527 pages
Publication date: 24 September 2021
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 chapters on polynomials cover arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is presented, followed by a comprehensive study of rational expressions. Exponential, logarithmic, and trigonometric functions are covered, with inequalities used to compare them with polynomial and rational functions. Axioms and limits serve as the underlying framework throughout, offering powerful tools and insights into non-trivial connections between topics.

Elements of Mathematics is particularly well-suited for students seeking a rigorous and engaging mathematical challenge using elementary tools. It can be used as an enhancement to the early undergraduate curriculum for high achievers or as a capstone course for senior students. The book assumes no formal prerequisites beyond high school algebra, making it accessible to mathematics circles and competition preparation. Advanced readers will appreciate the interleaving of ideas and the illuminating historical details.

In conclusion, this textbook offers a comprehensive and accessible introduction to mathematics, providing a solid foundation for further study in the field. Its rigorous approach, engaging examples, and historical insights make it an invaluable resource for students, educators, and enthusiasts alike.

Weight: 1050g
Dimension: 162 x 242 x 30 (mm)
ISBN-13: 9783030750503
Edition number: 1st ed. 2021