An excursion through elementary mathematics. Volume I. Real numbers and functions
This book develops several basic concepts in elementary mathematics and which are needed in the preparation of mathematical Olympiads around the world. The work expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This first volume mainly focuses on real numbers, elementary algebra, and real functions. The book starts with a non-axiomatic discussion of the most elementary properties of real numbers, followed by a detailed study of basic algebraic identities, equations and systems of equations, elementary sequences, mathematical induction, and the binomial theorem. The author is also concerned with an exhaustive elementary study of functions, which culminates with a first look at implicitly defined functions. The volume also includes basic properties of several fundamental concepts such as continuity, differentiability, and integrability. The abstract notions and properties are fully illustrated with examples, counter-examples and applications. Throughout the book, the author includes examples taken from mathematical competitions. The reviewer points out the connections with several important applications, such as the proof and several applications of the Jensen convexity inequality, the Lambert theorem on the irrationality of \(\pi\), the Stirling formula on the asymptotic behavior of \(n!\), examples of continuous and nowhere differentiable functions, the generating function method, and the Weierstrass approximation theorem. Throughout the book, the author puts great emphasis on explaining the issues discussed. Instructors of teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. The presentation is pedagogical and coherent throughout with emphasis on concepts and underlying ideas, however without forsaking mathematical rigor. The author has succeeded in crafting a comprehensive book to give a well-rounded picture of the role of real numbers and functions in elementary mathematics.
scientific article; zbMATH DE number 6696129