Problems and Proofs in Numbers and Algebra

ISBN-13: 9783319357232 / Angielski / Miękka / 2016 / 223 str.

Richard S. Millman; Peter J. Shiue; Eric Brendan Kahn

Opis

In the past, PNA has been taught in a "problem solving in middle school" course (twice), to a quite advanced high school students course (three semesters), and three times as a secondary resource for a course for future high school teachers.

Recenzje

"Aimed at introducing postcalculus students to higher mathematics by way of solving rigorous problems and learning how to prove. ... the content is less focused on basic mathematical concepts seen in upper-division college mathematics coursework and more so on topics that teachers might present in their classrooms, and on interesting applications ... . Teachers of mathematics at the secondary level would be well served by taking a course based on this text. Summing Up: Recommended. Upper-division undergraduates through faculty." (D. S. Larson, Choice, Vol. 53 (1), September, 2015)

Spis treści

​ ​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

O autorze

Richard S. Millman,Ph.D., Director, Center for Education Integrating Science, Mathematics, and Computing (CEISMC) Georgia Institute of Technology Atlanta, GA 30332-­0282

Peter J. Shiue Department of Mathematical Sciences University of Nevada, Las Vegas 4505 Maryland Pkwy Las Vegas, NV 89154-4020

Eric Brendan Kahn Department of Mathematics, Computer Science, and Statistics Bloomsburg University 400 East Second Street Bloomsburg, PA 17815

Blurb

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.  Beyond the undergraduate mathematics student audience, the text can also offer a rigorous treatment of mathematics content (numbers and algebra) for high achieving high school students. Furthermore, prospective teachers will add to the breadth of the audience as math education majors, will understand more thoroughly methods of proof, and will add to the depth of their mathematical knowledge.