Elements of Mathematics

Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

This textbook provides a comprehensive and systematic exposition of mathematics before the development of calculus. An elementary, inquiry-driven path is followed to arrive at fundamental algebra, geometry, and number theory concepts. These fundamental notions are derived from the foundations of set theory. The theory is motivated by thought-provoking examples and tough problems, both of which are inspired by mathematical competitions. Additionally, many historical asides describe the tale of how the ideas were initially formed.

In the first few chapters, after introducing Peano’s axioms and providing a comprehensive discussion of the natural numbers, the emphasis shifts to establishing the natural, integral, rational, and real number systems. Axioms of Birkhoff’s metric geometry are used to introduce plane geometry, and chapters on polynomials cover arithmetical operations, roots, and factoring multivariate expressions. The discussion begins with a fundamental categorization of conics and then moves on to an in-depth analysis of rational formulations. Inequalities that compare them to polynomial and rational functions drive the addition of exponential, logarithmic, and trigonometric functions, which round out the picture and complete it. The treatment is underpinned throughout by axioms and limitations, which provide powerful tools and insights into relationships between issues that are not trivial.

Students who are looking for a rigorous and thought-provoking mathematical challenge that makes use of fundamental concepts will find Elements of Mathematics to be an excellent choice. Enquiring mathematics majors will discover a wealth of material available for their instructors to use in various contexts, including strengthening the early undergraduate curriculum for high achievers and developing thoughtful senior capstones. Because there are no formal requirements assumed beyond high school algebra, this book is perfect for mathematics circles as well as training for mathematical competitions. Readers who are more experienced in their mathematics studies will appreciate how ideas and information that illuminates the past are woven together throughout the text.