Description

For a first course in proof for Mathematics or Computer Science majors.

Besides giving students the techniques for solving polynomial equations and congruences, An Introduction to Mathematical Thinking provides preparation for more advanced courses in Linear and Modern Algebra, as well as Calculus. This text introduces the mathematics and computer science student to proofs and mathematical thinking while teaching basic algebraic skills involving number systems, including the integers and complex numbers. Ample questions at the end of each chapter provide opportunities for learning and practice; the Exercises are routine applications of the material in the chapter, while the Problems require more ingenuity, ranging from easy to nearly impossible. This extensive pedagogy offers extreme flexibility to instructors for class and homework assignments.

Features

• Classroom-tested for 10 years.

  • Enables students to use a text that is tried and true.

• Cryptography–Provides an interesting application of congruences.

  • Provides students with a topic that will hold their interest, while gently introducing them to the advanced mathematical concept of congruences.

• Complex numbers–Introduces a subject that is often taught hastily in more advanced courses.

  • Provides students with an in-depth treatment of a topic that will be useful in their future courses.

• Over 600 exercises and 270 carefully graded problems–Including some very challenging problems that will keep the very good student interested in the material.

• Provides both students and instructors with excellent pedagogical material.

• Balanced approach–Provides both concrete and abstract ideas.

  • Enables students to learn to think abstractly and mathematically.

Table of Contents

(NOTE: Each chapter contains exercise and problem sets.)

 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: Trigonometry.


Appendix: Inequalities.


Further Reading.


Answers.