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Which numbers are real? (English)
Washington, DC: The Mathematical Association of America (MAA) (ISBN 978-0-88385-777-9/hbk; 978-1-61444-107-6/ebook). x, 219~p. (2012).
The series “Classroom Resource Materials” of the Mathematical Association of America is intended to provide supplementary classroom material for both students and teachers, including historical information on special topics, exploratory examples for basic theories, textbooks with unusual approaches for presenting mathematical ideas, and other projects. The book under review, another volume within this series, introduces the reader to a variety of number systems in mathematics. The focus is on alternative real number systems, that is, on number systems that generalize and extend the real numbers, and that still share some fundamental algebraic and geometric properties with the latter. As the author points out, the main goal of the book is to present some interesting, partially even exotic, mathematics to undergraduate students, thereby demonstrating the immense range of mathematics, where even a well-established concept such as the real numbers has several alternatives. As for the contents, the book consists of three parts. Part I is titled “The reals” and comprises the first two chapters. Chapter 1 gives the axioms for the real number systems, i.e., the field axioms, the order axioms, and the completeness axiom. The necessary background material from set theory and algebraic field theory is developed along the way, and the relationship between order completeness, Cauchy completeness, and the Archimedean property is carefully analyzed. Chapter 2 describes both Cantor’s construction and Dedekind’s construction of the reals (from the rational numbers), turns then to the uniqueness of the real number system as a complete ordered field, and concludes the discussion with a few comments on the historical foundation of the real numbers. Part II is devoted to number systems (containing the reals) that are still complete (skew-)fields, but not linearly ordered. According to the elementary level of the book, only systems with the geometry of a finite-dimensional ${\bbfR}$-vector space are considered. Chapter 3 discusses the only field with this property, namely the field of complex numbers, while Chapter 4 describes the remaining other candidate; the skew field of quaternions. As for applications, Chapter 3 briefly touches upon complex calculus, whereas Chapter 4 illustrates the use of quaternions in the theory of geometric transformations, in the special theory of relativity, and in Hamilton’s calculus of the nabla operator. Part III turns to alternative real lines. In the remaining three chapters, the author depicts number systems that contain the real numbers and have the same algebraic and geometric properties as the reals. These number systems represent some eccentric and relatively unexplored parts of mathematics, and they embody different ideas of number, different philosophies of mathematics, and also conflicting visions about the logical framework for dealing with these objects. In this context, Chapter 5 explains the theory of constructive real numbers. This viewpoint was formulated by E. Bishop (1929‒1983), one of the great expositors of constructive mathematics in the second half of the 20th century, and appeared as a variant of Cantor’s construction of the reals via Cauchy sequences of rational numbers. Basically, the constructive reals have the algebra of a field and the geometry of linear order, but they do not satisfy the trichotomy law of the classical reals. At the end of this chapter, a brief outlook to the corresponding constructive calculus is given, with special reference to the standard book [{\it E. Bishop} and {\it D. Bridges}, Constructive analysis. Grundlehren der Mathematischen Wissenschaften, 279. Berlin etc.: Springer-Verlag. XII, 477 p. (1985; Zbl 0656.03042)]. Chapter 6 introduces the system of hyperreal numbers, a central concept in non-standard analysis. Invented by A. Robinson in the early 1960s, the hyperreal numbers form an ordered field containing ${\bbfR}$ as well as infinitely large and infinitely small numbers. However, the field of hyperreal numbers is not order complete. This chapter is based on the author’s and {\it D. Klingenberg’s} earlier textbook [Infinitesimal Calculus, Dover Publications (2003)], and is certainly the most detailed discourse in the present volume. Finally, Chapter 7 turns to another, also relatively recent invention: the surreal numbers. These objects were discovered in the early 1970s by John H. Conway, basically as part of his analysis of combinatorial games in [{\it H. J. Conway}, On Numbers and Games, 2nd ed., A. K. Peters/CRC (2000; Zbl 0972.11002)], and later described by {\it D. E. Knuth} in his novel “Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness” [Reading, Mass. etc.: Addison-Wesley Publishing Company. 119 p. (1974; Zbl 0334.00005)]. After a brief introduction to combinatorial games, the surreal numbers are constructed as a special class of such games, without using the rational numbers or even the integers. This leads to a proof technique for the surreals, which amounts to playing games, and it tuns out that the class of surreal numbers includes the real numbers as well as certain infinite and infinitesimal numbers. As for the intricate proof of the fact that the surreal numbers have the properties of a linearly ordered field, the reader is referred to H. J. Conway’s original text cited above. The book ends with an application to analyzing games with surreal numbers. A particular feature of the book under review is that it encourages readers to participate in the development of the material themselves. In fact, the proofs of many results are either contained in the large number of problems or depend on results proved in problems. The prerequisites are standard undergraduate courses: discrete mathematics, multivariable calculus, and basic linear algebra, while all the further background material is introduced in the text. All together, this book lucidly presents several non-standard topics that are of general interest to mathematics students. Also, the text is perfectly suitable for a course on the foundations of number systems in analysis, or for student projects in advanced calculus classes. Both students and teachers will find this versatile text very educating, entertaining, inspiring, and generally useful at the same time.
Reviewer: Werner Kleinert (Berlin)
Classification: F55 A15