An Introduction to Mathematical Thinking:
Algebra and Number Systems

William J. Gilbert and Scott A. Vanstone

This book is designed for a first course in abstract mathematics for university students who wish to major in mathematics or computer science. At the same time as teaching algebra that will be useful in later studies, the book provides an introduction to mathematical thinking and to the art of writing proofs.

The algebra in the book is centered around the number systems, from the integers to the complex numbers, and the solution of polynomial equations in these systems. Even though the mathematics in the book is classical, we include a very up-to-date application to cryptography that appeals to students as being very relevant.

Contents

Preface
1. Logic and Proofs
   • The Language of Mathematics
   • Logic
   • Sets
   • Quantifiers
   • Proofs
   • Counterexamples
   • Exercises and Problems
2. Integers and Diophantine Equations
   The Division Algorithm
   • The Euclidean Algorithm
   • Linear Diophantine Equations
   • Integers in Different Bases
   • Prime Numbers
   • Exercises and Problems
3. Congruences
   • Congruence
   • Tests for Divisibility
   • Equivalence Relations
   • Modular Arithmetic
   • Linear Congruences
   • The Chinese Remainder Theorem
   • Euler-Fermat Theorem
   • Exercises and Problems
4. Induction and the Binomial Theorem
   • Mathematical Induction
   • Recursion
   • The Binomial Theorem
   • Exercises and Problems
5. Rational and Real Numbers
   • Rational Numbers
   • Real Numbers
   • Rational Exponents
   • Decimal Expansions
   • Exercises and Problems
6. Functions and Bijections
   • Functions
   • The Graph of a Function
   • Composition of Functions
   • Inverse Functions
   • The Inversion Theorem
   • Cardinality
   • Inverse Trigonometric Functions
   • Exponential and Logarithmic Functions
   • Permutations
   • Exercises and Problems
7. An Introduction to Cryptography
   Cryptography
   • Private-Key Cryptography
   • Public-Key Cryptography
   • The RSA Scheme
   • Exercises and Problems
8. Complex Numbers
   • Quadratic Equations
   • Complex Numbers
   • The Complex Plane
   • Properties of Complex Numbers
   • Polar Representation
   • De Moivre's Theorem
   • Roots of Complex Numbers
   • The Fundamental Theorem of Algebra
   • Exercises and Problems
9. Polynomial Equations
   Polynomials and Factoring
   • Complex Roots of a Polynomial
   • Rational Roots of a Polynomial
   • Approximating Real Roots
   • Polynomial Inequalities
   • Cubic Equations
   • Multiple Roots
   • Partial Fractions
   • Equations over a Finite Field
   • Exercises and Problems
   Appendix
   • Trigonometry
   • Inequalities
   Further Reading
   Answers to Odd-Numbered Exercises and Problems
   List of Symbols
   Index

Solutions

Complete solutions to all 911 exercises and problems are available from George Lobell, Acquisitions Editor at Pearson Prentice Hall.

There are ample questions at the end of each chapter. They are divided into two types; the Exercises are routine applications of the material in the chapter, while the Problems usually require more ingenuity and range from easy to nearly impossible.

Brief answers to all the odd-numbered questions are given at the back of the book.

Corrections

Errors that I know about are listed here.

Earlier Versions

William J. Gilbert and Scott A. Vanstone, Classical Algebra

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