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## Über dieses Buch

Focusing on an approach of solving rigorous problems and learning how to prove, this volume is concentrated on two specific content themes, elementary number theory and algebraic polynomials. The benefit to readers who are moving from calculus to more abstract mathematics is to acquire the ability to understand proofs through use of the book and the multitude of proofs and problems that will be covered throughout. This book is meant to be a transitional precursor to more complex topics in analysis, advanced number theory, and abstract algebra. To achieve the goal of conceptual understanding, a large number of problems and examples will be interspersed through every chapter. The problems are always presented in a multi-step and often very challenging, requiring the reader to think about proofs, counter-examples, and conjectures. 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. 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. PNA is suitable for secondary math teachers who look for material to encourage and motivate more high achieving students.

## Inhaltsverzeichnis

### Chapter 1. Number Concepts, Prime Numbers, and the Division Algorithm

Abstract
Introduction The topic of this section is the divisibility of integers, the basic building blocks (called prime numbers) for integers, and how to apply this foundation to problem solving in combinatorics and word problems in which integers solutions are sought (Diophantine equations). Brief Description The operations of addition and multiplication are two ways to combine integers to get a third integer (this is called the closure property). We’ll now talk a bit about trying to do the operations in reverse (subtraction and division) and discuss the formal definitions which will give rise to the big question of this half of the text and the big theorem, Fundamental Theorem of Arithmetic, and its applications.
Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

### Chapter 2. Greatest Common Divisors, Diophantine Equations, and Combinatorics

Abstract
Introduction This chapter deals with the structure of how prime numbers are put together (Prime Factorization) and the tools of number theory which include greatest common divisors (GCD), least common multiples (LCM). These tools provide the beginning of important theorems such as the Fundamental Theorem of Arithmetic and the Euclidean Algorithm and its byproduct. Problems which have only solutions in integers (Diophantine equations) are discussed. Application to combinatorics is also mentioned.
Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

### Chapter 3. Equivalence Classes with Applications to Clock Arithmetic and Fractions

Abstract
Introduction We will first define a quite unifying concept: what it means for two objects to be equivalent (rather than equal), and apply it in the next section to modular (or clock) arithmetic and in the following one to establish a rigorous definition of fractions. The notion of equivalence is pervasive in mathematics and will be a part of many courses you will take in the future, including abstract algebra and geometry. The last section shows that there are fascinating applications from number theory.
Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

### Chapter 4. Polynomials and the Division Algorithm

Abstract
Introduction At this point, you are quite adept with algebra and, especially, polynomials.
Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

### Chapter 5. Factoring Polynomials, Their Roots, and Some Applications

Abstract
This chapter focuses on the relationship between polynomials with real coefficients, roots which may be complex numbers or rational roots, and $$\mathrm{GCD}$$ and $$\mathrm{LCM}$$ for polynomials.
Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

### Chapter 6. Matrices and Systems of Linear Equations

Abstract
This chapter discusses the solutions of a system of linear equations by viewing the solutions as a set and interpreting that set geometrically. Matrices and operations on matrices are briefly reviewed and their effect on the related solution sets is also described geometrically.
Richard S. Millman, Peter J. Shiue, Eric Brendan Kahn

### Backmatter

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