id: 06059850
dt: b
an: 2013a.00535
au: Henle, Michael
ti: Which numbers are real?
so: 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).
py: 2012
pu: Washington, DC: The Mathematical Association of America (MAA)
la: EN
cc: F55 A15
ut: number systems; real numbers; complex numbers; quaternions; hyperreal
numbers; surreal numbers; constructive real numbers; game theory
ci: Zbl 0656.03042; Zbl 0334.00005; Zbl 0972.11002
li: http://www.maa.org/ebooks/crm/WNR.html
ab: 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.
rv: Werner Kleinert (Berlin)