Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory

ISBN-13: 9781461270713 / Angielski / Miękka / 2012 / 344 str.

Jiri Herman; Radan Kucera; Jaromir Simsa

Opis

A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.

Recenzje

"In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here." (Olaf Ninnemann, zbMATH 1055.00006, 2021)

From the reviews:

MATHEMATICAL INTELLIGENCER

"I have nothing but praise for this book, and I can't imagine a mathamtician who wouldn't want to own it."

MAA ONLINE

"This book contains over 300 examples and then over 700 exercises for the reader. The authors have even included a fourth chapter with hints and answers to the exercises. So, even if the solution isn't given in the back, at least the reader will have some hint about which direction to start looking. The book provides a nice introduction into many of the problems-solvers' 'tricks of the trade.' There is a lot presented here, so the reader (especially an undergraduate) may want to take the following approach: read just a few pages at a time, work out some of the exercises, then take some time to digest the material before going on."

 

Spis treści

1 Algebraic Identities and Equations.- 1 Formulas for Powers.- 2 Finite Sums.- 3 Polynomials.- 4 Symmetric Polynomials.- 5 Systems of Equations.- 6 Irrational Equations.- 7 Some Applications of Complex Numbers.- 2 Algebraic Inequalities.- 1 Definitions and Properties.- 2 Basic Methods.- 3 The Use of Algebraic Formulas.- 4 The Method of Squares.- 5 The Discriminant and Cauchy’s Inequality.- 6 The Induction Principle.- 7 Chebyshev’s Inequality.- 8 Inequalities Between Means.- 9 Appendix on Irrational Numbers.- 3 Number Theory.- 1 Basic Concepts.- 2 Prime Numbers.- 3 Congruences.- 4 Congruences in One Variable.- 5 Diophantine Equations.- 6 Solvability of Diopha,ntine Equations.- 7 Integer Part and Fractional Part.- 8 Base Representations.- 9 Dirichlet’s Principle.- 10 Polynomials.- 4 Hints and Answers.- 1 Hints and Answers to Chapter 1.- 2 Hints and Answers to Chapter 2.- 3 Hints and Answers to Chapter 3.