Introduction to Real Mathematics. A Handbook for Mathematics Teachers Based on the Course "100 Lessons in Mathematics"

The publication presents an expanded and revised outline of the lecture course "100 Lessons in Mathematics" by Alexey Savvateev, which was delivered at the Filippov School (Moscow) from 2014 to 2018. The book outlines the concept of numbers at different... levels of rigor and complexity. Starting with simple geometric representations that describe basic arithmetic operations, and ending with complex algebraic concepts, the authors introduce the reader to the fundamentals of number theory, group theory, linear algebra, and complex analysis. The book focuses on the following topics: movements and similarities of lines and planes, linear equations in integers, arithmetic of remainders, polynomial rings, permutation groups, complex numbers, models of real numbers, and limit theory. A number of well-known mathematical facts are discussed in the book: The Fundamental Theorem of Arithmetic, Shalay's Theorem, Fermat's Theorem for n = 4, the unsolvable problem of duplicating the cube, Euler's formula, and Cantor's Theorem. Special attention is given to the group-theoretical approach to describing mathematical concepts, with a detailed examination of the structure of the permutation group and related problems. Additionally, the axiom of completeness (the principle of continuity) of real numbers is studied in detail, as well as the construction of the real and complex exponentials. The book is equipped with a large number of auxiliary sketches and illustrations (over 100), as well as problems of varying degrees of difficulty for independent solution (over 800). The content covers topics such as: geometry, linear algebra, movements, group theory, complex numbers, mathematical analysis, polynomials, and the ring of Gaussian numbers.