The Magic of Numbers

by Gross, Benedict; Harris, Joe

• ISBN13: 9780131777217
• ISBN10: 0131777211
• Edition: 1st
• Format: Paperback
• Copyright: 2004-01-01
• Publisher: Pearson College Div

Summary

The Magic of Numbers was written with two goals in mind: first, to introduce the reader to some of the beauty of numbers-the 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: bull; bull;Counting and basic combinatorics, with applications to probability and games bull;The arithmetic of natural numbers: the Euclidean Algorithm and the unique factorization theorem bull;Modular arithmetic, including Fermat's Theorem, Euler's Theorem, and how to take powers and roots bull;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.

Table of Contents

Counting

Simple Counting

The Multiplication Principle

The Subtraction Principle

Collections

Probability

The Binomial Theorem

Advanced Counting

Arithmetic

Divisibility

Combinations

Primes

Factorization

Consequences

Relatively Prime

Modular Aritmetic

What is a Number?

Modular Arithmetic

Congruences

Division

Powers

Roots

Euler's Theorem

Codes and Primes

Codes

Public-Key Cryptography

Finding Primes

Generators, Roots, and Passwords

Table of Contents provided by Publisher. All Rights Reserved.

Excerpts

The primarily purpose of the preface to a textbook is to convey a sense of the goals of the book, and to a lesser extent its level, pace and language. Since this book is based on a course, Quantitative Reasoning 28, that we developed at Harvard University, it seems reasonable to start by describing the goals of that course. There are, it seems to us, two strains in our educational system, reflecting two disparate aims. One is preprofessional: among the courses you take in high school and in college are those that will, it's hoped, provide you with the basis of the discipline that will become your vocation in later life. The other strain, by contrast, seeks to enrich your life: to expose you to ideas and modes of thought that you might not come in contact with otherwise. This is one of the ideas underlying Harvard's Core Program, and QR28 was developed as a Core course. It's not a technical course, designed to prepare you for the next course; rather, it's simply a collection of topics that we find fascinating, and that make up a coherent whole. Our hope is that we will be able to communicate to you some idea of the mathematical view of the world, and of what attracts people to math in the first place. Probably the best way to describe the course is by analogy: you might think of it as a math appreciation course, to be taken in the same spirit as you would a music appreciation course. We're not trying to teach you how to write a symphony, or to play the violin; we simply want you to be able to hear the music. Or you might think of it like an introductory language course--say, Italian--that you take for fun and because it's such a beautiful language. This analogy is particularly apt in one respect. The heart of a language course is not the memorization of a lot of vocabulary and verb tenses--though there's inevitably a lot of that involved--but rather the experience of thinking and speaking in a different tongue. In the same way, in this text there are of necessity a fair number of techniques to learn and calculations to carry out, but that's just the means to an end: our goal, ultimately, is to give you the experience of thinking in math. What sort of prerequisites does this book have? Well, the technical answer to that is "virtually none": junior high school algebra will cover it handily. (To be concrete, if you can add fractions, and are reasonably comfortable with the use of letters to stand for numbers, you should be solid.) Probably more important, though, is a less quantifiable requirement: we would ask that the reader be prepared to approach the book in a spirit of adventure and exploration, and with the understanding that, while some work will be required, the experience will be worth it. Additional exercises, problems, and sample exams are available at www.prenhall.com/gross .