https://orcid.org/0000-0003-2023-8529
Mathematics is known for its rigor and beauty. While not everyone is able to appreciate the beauty that mathematics possesses, the rigor of mathematics is easily discernible. The book, Number systems: A path into rigorous mathematics, grew out of a module that the author taught at Loughborough University. Although no serious or conscious attempt to provide a coherent account of the history of number systems has been made, the book contains several snippets on the subject’s historical development, scattered throughout the book. Akin to a typical undergraduate text, the book contains new presentations of the existing knowledge on the subject but hardly any new research results.
The book is divided into 11 chapters and an appendix on how to read proofs. The chaptering is quite uneven. While some chapters are surprisingly brief, such as Chapter 11 with 2 pages, others are extremely long, such as Chapter 9 with 53 pages. However, equity is exercised throughout the book with regards to the role and the necessity of maintaining rigor in mathematics.
The first chapter is an introduction that explains the purpose of the book. It begins with a brief historical overview of how mathematical ideas were developed for practical applications in ancient times. The axiomatic method is discussed before the discussion of number systems in mathematics and mathematics writing, which discusses notations and terminology. The chapter concludes with an engaging section on logic and methods of proof, in which the author briefly mentions various methods of proof, such as proof by contradiction, proof by exhaustion, and the concept and the role of counterexamples.
Sets and relations are discussed in Chapter 2. It establishes the tone for several other chapters of the book. This chapter is about sets, relations, and binary operations. Sets are central to almost every branch of modern mathematics. The chapter has been written with this goal in mind, as well as the goal of exhibiting the rigor of mathematics.
Natural numbers, also known as counting numbers, are discussed in Chapter 3. The fact that 1 is the first number is obvious but still needs to be classified as an axiom demonstrates the extent of rigor that is evident in mathematics. This chapter begins with a description of Peano’s axioms, which were first published in 1889. Other fundamental concepts, such as addition, multiplication, and natural number exponentiation, are explained formally and rigorously. Order and boundedness have also been discussed in depth. The famous Hilbert’s hotel example beautifully explains the intriguing concept of infinite sets, which are sets that are equivalent to some of their proper subsets. The chapter concludes with a look at subtraction as the inverse of addition.
Chapter 4 discusses integers, including their definition, arithmetic, algebraic structure, order, cardinality, and boundedness. Notably, the section on the algebraic structure of integers goes beyond the scope of an elementary textbook on number systems by addressing integers from a group-theoretic perspective. Even beyond groups, the author introduces integers as a popular example of rings.
Chapter 5 is a brief but well-thought-out introduction to the theory of numbers. It starts with the famous division algorithm and discusses integer division, base representations, primes, and prime factorizations and congruences among others. The chapter ends with a surprisingly-long account of modular arithmetic.
Chapter 6 is devoted to rational numbers, which are the natural and legitimate successors to integers. The chapter discusses the denumerability of rationals after defining rational numbers and explaining their addition and multiplication. An interesting fact proved in the chapter is that rational numbers are countable, which means they are only as many as integers. The chapter concludes with a lovely but succinct presentation of sequences and series.
Chapter 7 is about real numbers. It begins with the intriguing question of the completeness of the set of rational numbers and proceeds to state the well-known Axiom of Completeness of Real Numbers. Dedekind cuts are also discussed and demonstrated. The uncountability of reals refers to the fact that there are more reals than rationals. The famous diagonal argument is used in the chapter to demonstrate this fact. Finally, the chapter discusses algebraic and transcendental numbers.
Quadratic extensions are discussed in Chapters 8 and 9. The subject matter covered in these two chapters is relatively complex, necessitating greater maturity and concentration on the part of the readers.
Chapter 10 attempts to convince readers that the world of numbers is vaster and more immense than they realize. It is divided into two sections: one on constructible numbers and one on hypercomplex numbers, which includes accounts of Hilbert’s quaternions and John Graves’ octonions.
Chapter 11 is a two-page guide to which readers should refer for a more comprehensive and advanced understanding of number systems. It implies that, in addition to abstract algebra, analysis is an excellent field for producing more elegant and unexpected results.
In all, the book is a great attempt to introduce number systems to an undergraduate audience with the main focus on the rigor of mathematics. While, on the one hand, it provides detailed but accessible explanations of theorems and their proof, on the other hand, it is an attempt to provide the level of explanation needed for a first-year mathematics course on the subject that most others fail to do.
Government Degree College Sopore, Baramulla,Jammu and Kashmir, India