Number Systems A Path into Rigorous Mathematics
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.
- 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.
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. 2.2. Relations between Sets. 2.3. Relations on a Set. 3. Natural Number, 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. 3.8. 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. 4.4. Order in Z. 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. 5.4. 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. 6.5. Order in Q. 6.6. Bounded Sets in Q. 6.7. Expressing Rational Numbers in any Base. 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[√k]. 8.4. Primes in Z[√k]. 8.5. Prime Factorisation in Z[√k. 8.6. Quadratic Fields: Extensions of Q. 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. Geometry of C; the Principal Value of the Argument, and the Number π. 9.4. Use of the Argument to Define Real Powers in C. 9.5. Exponentiation by Complex Powers; the Number e. 9.6. The Fundamental Theorem of Algebra. 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.
"This is not exactly a standard elementary number theory textbook, nor is it focused only on proof-writing skills. Rather, it’s a fascinating look at the properties of various number systems, beginning with the natural numbers and wending a fascinating path all the way through to a brief look at the octonions. Along the way, there’s an appreciation for rigor and considerable effort dedicated to aiding the reader in thinking mathematically. With that in mind, it seems like this book would be a great choice for a transition course. Abelian groups and Dedekind cuts both make appearances, providing a possible bridge into later courses in abstract algebra or real analysis
But its appeal is not limited to prospective mathematics majors. Number Systems has the potential to serve as an excellent introduction for college students–at any level; the book grew out of a course for first-year students–to the non-computational side of our subject and to encourage them to think deeply about mathematics in a way that we’d all like to encourage. At the same time, this is also a book that holds a lot of appeal for seasoned professionals who want to revisit some ideas in a more recreational setting. It’s always a pleasure to encounter familiar ideas in a novel setting, and this book does a fine job of providing that pleasure."
– Mark Bollman, MAA Reviews