Number Systems: A Path into Rigorous Mathematics

Description

Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs. The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding. The author’s motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems. Features Approachable for students who have not yet studied mathematics beyond school Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof Draws attention to connections with other areas of mathematics Plenty of exercises for students, both straightforward problems and more in-depth investigations Introduces many concepts that are required in more advanced topics in mathematics. Cover Half Title Title Page Copyright Page Contents Preface and Acknowledgments 1. Introduction: The Purpose of This Book 1.1. A Very Brief Historical Context 1.2. The Axiomatic Method 1.3. The Place of Number Systems within Mathematics 1.4. Mathematical Writing, Notation, and Terminology 1.5. Logic and Methods of Proof 2. Sets and Relations 2.1. Sets 2.1.1. Quantifiers 2.1.2. Subsets and Equality 2.1.3. Union, Intersection, and Complement 2.1.4. Ordered Sets 2.2. Relations between Sets 2.2.1. Relations in General 2.2.2. Functions 2.3. Relations on a Set 2.3.1. Equivalence Relations 2.3.2. Order Relations 2.3.3. Transitivity and Proofs 2.4. Binary Operations and Algebraic Structures 3. Natural Numbers, N 3.1. Peano's Axioms 3.2. Addition of Natural Numbers 3.3. Multiplication of Natural Numbers 3.4. Exponentiation (Powers) of Natural Numbers 3.5. Order in the Natural Numbers 3.6. Bounded Sets in N 3.7. Cardinality, Finite and Infinite Sets 3.7.1. Some Useful Notations 3.7.2. Finite Sets, Their Subsets and Injections 3.7.3. Finiteness and Boundedness of Sets 3.7.4. Infinite Sets 3.8. Subtraction: The Inverse of Addition 4. Integers, Z 4.1. Definition of the Integers 4.2. Arithmetic on Z 4.3. Algebraic Structure of Z 4.3.1. An Abelian Group 4.3.2. A Commutative Ring 4.4. Order in Z 4.4.1. How to Solve Inequalities 4.5. Finite, Infinite, and Bounded Sets in Z 5. Foundations of Number Theory 5.1. Integer Division 5.2. Expressing Integers in Any Base 5.3. Prime Numbers and Prime Factorisation 5.3.1. Prime Numbers and Prime Factorisation in N 5.3.2. Primes in Z and Other Number Systems 5.4. Congruence 5.5. Modular Arithmetic 5.6. Zd as an Algebraic Structure 6. Rational Numbers, Q 6.1. Definition of the Rationals 6.2. Addition and Multiplication on Q 6.3. Countability of Q 6.4. Exponentiation and Its Inverse(s) on Q 6.4.1. Integer Powers 6.4.2. Roots and Fractional Powers 6.4.3. Logarithms 6.5. Order in Q 6.6. Bounded Sets in Q 6.7. Expressing Rational Numbers in Any Base 6.7.1. Terminating Base-b Representations 6.7.2. Repeating Base-b Representations 6.7.3. Fractions from Repeating Base-b Representations 6.8. Sequences and Series 7. Real Numbers, R 7.1. The Requirements for Our Next Number System 7.2. Dedekind Cuts 7.3. Order and Bounded Sets in R 7.4. Addition in R 7.5. Multiplication in R 7.6. Exponentiation in R 7.7. Expressing Real Numbers in Any Base 7.8. Cardinality of R 7.9. Algebraic and Transcendental Numbers 8. Quadratic Extensions I: General Concepts and Extensions of Z and Q 8.1. General Concepts of Quadratic Extensions 8.2. Introduction to Quadratic Rings: Extensions of Z 8.3. Units in Z[p k] 8.4. Primes in Z[p k] 8.4.1. Basic Theorems about Primes 8.4.2. Associates Classes and Conjugates of Primes 8.4.3. How to Search for Primes 8.5. Prime Factorisation in Z[p k] 8.6. Quadratic Fields: Extensions of Q 8.6.1. Algebraic Numbers in Quadratic Fields 8.6.2. Quadratic Integers 8.7. Norm-Euclidean Rings and Unique Prime Factorisation 9. Quadratic Extensions II: Complex Numbers, C 9.1. Complex Numbers as a Quadratic Extension 9.2. Exponentiation by Real Powers in C: A First Approach 9.3. Geometry of C; the Principal Value of the Argument, and the Number ˇ 9.3.1. The Unit Circle and the Principal Value of the Argument of a Complex Number 9.3.2. The Number ˇ 9.4. Use of the Argument to De ne Real Powers in C 9.4.1. The PVA of a Product 9.4.2. The Multiple-Valued Argument and the Definition of Real Powers 9.4.3. Evaluating Rational Powers of Complex Numbers 9.5. Exponentiation by Complex Powers; the Number e 9.5.1. The Number e and Its Powers 9.5.2. General Exponentiation and Logarithms in C 9.5.3. Trigonometric Functions 9.6. The Fundamental Theorem of Algebra 9.6.1. Factorisation of Polynomials 9.7. Cardinality of C 10. Yet More Number Systems 10.1. Constructible Numbers 10.2. Hypercomplex Numbers 11. Where Do We Go from Here? 11.1. Number Theory and Abstract Algebra 11.2. Analysis A. How to Read Proofs: The "Self-Explanation" Strategy Bibliography Index