An Introduction to Mathematical Thinking:
Algebra and Number Systems

William J. Gilbert and Scott A. Vanstone

An Introduction to Mathematical Thinking: Algebra and Number Systems,
Pearson Prentice Hall, Upper Saddle River, New Jersey: 2005.

[312 pages. ISBN 0131848682]

Math Thinking cover

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.


  1. Logic and Proofs
  2. Integers and Diophantine Equations
  3. Congruences
  4. Induction and the Binomial Theorem
  5. Rational and Real Numbers
  6. Functions and Bijections
  7. An Introduction to Cryptography
  8. Complex Numbers
  9. Polynomial Equations Appendix Further Reading
    Answers to Odd-Numbered Exercises and Problems
    List of Symbols


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.


Errors that I know about are listed here.

Earlier Versions

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

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