Synopses & Reviews
The Magic of Numbers was written with two goals in mind: first, to introduce the reader to some of the beauty of numbersthe patterns in their behavior that have fascinated mathematicians for millennia, and some surprising applications of those patterns; second, and equally important, to teach the reader something of the mathematical mode of thought: the feeling of exploration, excitement, and discovery that are part of how mathematics is developed.
The book, written originally for the course Quantitative Reasoning 28 that the authors developed and taught at Harvard, draws the reader into the content through an engaging and informal writing style. Example-driven, it reduces to a minimum the abstract notation and formal argument that often creates a barrier between mathematicians and students, focusing more instead on the experimental aspect of the subject. Above all, the authors communicate to the reader a sense of the joy and fascination of learning mathematics.
Additional exercises, problems, and sample exams are available at: www.prenhall.com/gross Principal topics include:
- Counting and basic combinatorics, with applications to probability and games
- The arithmetic of natural numbers: the Euclidean Algorithm and the unique factorization theorem
- Modular arithmetic, including Fermat's Theorem, Euler's Theorem, and how to take powers and roots
- Codes: how the special properties of ordinary and modular arithmetic in combination allow us to construct the public-key codes that help make data transmission secure.
About the Author
Benedict Gross is the Leverett Professor of Mathematics and Dean of Harvard College.
Joe Harris is the Higgins Professor of Mathematics and Chair of the Mathematics Department at Harvard.
Table of Contents
I. COUNTING. 1. Simple Counting. 2. The Multiplication Principle.
3. The Subtraction Principle.
4. Collections.
5. Probability.
6. The Binomial Theorem.
7. Advanced Counting.
II. ARITHMETIC. 8. Divisibility.
9. Combinations.
10. Primes.
11. Factorization.
12. Consequences.
13. Relatively Prime.
III. MODULAR ARITMETIC. 14. What is a Number?
15. Modular Arithmetic.
16. Congruences.
17. Division.
18. Powers.
19. Roots.
20. Euler's Theorem.
IV. CODES AND PRIMES. 21. Codes.
22. Public-Key Cryptography.
23. Finding Primes.
24. Generators, Roots, and Passwords.