Elements of Mathematics

This textbook offers a thorough exploration of mathematics prior to calculus, making it a valuable resource for students and instructors alike.

This textbook provides a comprehensive exploration of mathematics prior to the introduction of calculus. It carefully develops fundamental concepts in algebra, geometry, and number theory, utilizing the foundations of set theory along an elementary and inquiry-driven approach. The text is enriched with thought-provoking examples and challenging problems inspired by mathematical contests, which serve to motivate the theoretical concepts presented. Additionally, frequent historical insights illuminate the evolution of these mathematical ideas, offering readers a deeper appreciation of their significance.

Beginning with a detailed examination of natural numbers through Peano’s axioms, the initial chapters establish the natural, integral, rational, and real number systems. Plane geometry is introduced using Birkhoff’s axioms of metric geometry, while subsequent chapters on polynomials cover essential topics such as arithmetical operations, roots, and the factoring of multivariate expressions. An elementary classification of conics is also provided, followed by an extensive study of rational expressions. The text culminates with discussions on exponential, logarithmic, and trigonometric functions, all framed by inequalities that compare these functions with polynomial and rational counterparts. Axioms and limits are consistently employed throughout, offering powerful tools and insights into the intricate connections between various mathematical topics.

Elements of Mathematics is particularly suited for students seeking a rigorous and engaging mathematical challenge grounded in elementary principles. Whether aimed at enhancing the undergraduate curriculum for high-achieving students or serving as a reflective capstone for seniors, instructors will find a wealth of material for curious mathematics majors. With no formal prerequisites beyond high school algebra, this book is ideal for mathematics circles and competition preparation, appealing to both novice and advanced readers with its interweaving of ideas and illuminating historical context.