An Excursion through Elementary Mathematics, Volume I: Real Numbers and Functions

✍ Scribed by Antonio Caminha Muniz Neto (auth.)

• Publisher: Springer International Publishing
• Year: 2017
• Tongue: English
• Leaves: 657
• Series: Problem Books in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This first volume covers Real Numbers, Functions, Real Analysis, Systems of Equations, Limits and Derivatives, and much more.

As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level.

The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.

✦ Table of Contents

Front Matter....Pages i-xiii
The Set of Real Numbers....Pages 1-18
Algebraic Identities, Equations and Systems....Pages 19-59
Elementary Sequences....Pages 61-86
Induction and the Binomial Formula....Pages 87-110
Elementary Inequalities....Pages 111-141
The Concept of Function....Pages 143-200
More on Real Numbers....Pages 201-244
Continuous Functions....Pages 245-273
Limits and Derivatives....Pages 275-345
Riemann’s Integral....Pages 347-445
Series of Functions....Pages 447-510
Back Matter....Pages 511-652

✦ Subjects

Real Functions;General Algebraic Systems;Linear and Multilinear Algebras, Matrix Theory