Handbook of Measurement Error Models

Abstract

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.