Problems and proofs in numbers and algebra / Richard Millman.

Includes bibliographical references.

I. The Integers
1. Number Concepts, Prime Numbers, and the Division Algorithm
2. Greatest Common Divisors, Diophantine Equations, and Combinatorics
3. Equivalence Classes with Applications to Clock Arithmetics and Fractions
II. The Algebra of Polynomials and Linear Systems
4. Polynomials and the Division Algorithm
5. Factoring Polynomials, Their Roots, and Some Applications 6. Matrices and Systems of Linear Equations.

Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets in-depth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications in-depth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles. The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to prove or solve complex problems. This method of instruction is augmented by examining applications of number theory in systems such as RSA cryptography, Universal Product Code (UPC), and International Standard Book Number (ISBN). The numerous problems and examples included in each section reward curiosity and insightfulness over more simplistic approaches. Each problem set begins with a few easy problems, progressing to problems or proofs with multi-step solutions. Exercises in the text stay close to the examples of the section, allowing students the immediate opportunity to practice developing techniques.