id: 06292992
dt: b
an: 2016f.00814
au: Rosenthal, Daniel; Rosenthal, David; Rosenthal, Peter
ti: A readable introduction to real mathematics.
so: Undergraduate Texts in Mathematics. Cham: Springer (ISBN
978-3-319-05653-1/hbk; 978-3-319-05654-8/ebook). xii, 161~p. (2014).
py: 2014
pu: Cham: Springer
la: EN
cc: F15 H15 I15 E65
ut:
ci: Zbl 0843.15002; Zbl 0886.15001; Zbl 1304.15001
li: doi:10.1007/978-3-319-05654-8
ab: Book titles like [{\it S. Axler}, Linear algebra done right. New York, NY:
Springer-Verlag (1995; Zbl 0843.15002), Linear algebra done right. 2nd
ed. New York, NY: Springer (1997; Zbl 0886.15001); Linear algebra done
right. 3rd ed. Cham: Springer (2015; Zbl 1304.15001)] are highly
suspicious and deterrent. The same criticism applies to this book
entitled “A readable introduction to real mathematics”: the
decision of whether or not a book is readable can be left only to the
reader, not to the authors; moreover, it is highly subjective. Another
questionable point is the word “real” in the title: does it mean
that the authors treat analysis on the real line, or the book refers to
the real number system, or simply that it contains “mathematics that
matters”? Even worse, the authors claim that their work “presents
sophisticated mathematical ideas in an elementary and friendly
fashion”. This is really weird: every person with a minimal
mathematical background knows that sophisticated results usually
require sophisticated methods and hard work and, vice versa, what can
be taught in an elementary way is usually not very deep. At this point
the reviewer could disregard the book and turn to his usual day work.
However, closer scrutiny reveals that the book is not as bad as these
superficialities suggest. It is intended as an introductory textbook
for fresh (wo)men at various university faculties who want to get in
touch for the first time with the art of theorems and proofs, and then
perhaps start appreciating mathematics. It covers the following topics:
natural numbers, induction, the fundamental theorem of arithmetic,
Fermat’s little theorem, cryptography, the Euclidean algorithm,
rational and irrational numbers, real and complex numbers, cardinality,
Euclidean geometry in the plane, construct ability of geometric
objects. This summary shows that the authors’ aims and scope seems to
be sketching some basic principles of a first-year mathematics course,
with a particular emphasis on elementary number theory and elementary
geometry. The “leitmotiv” which all chapters have in common is to
teach the young reader to understand mathematical thinking (or, rather,
mathematical reasoning). Although this may be taught by virtually any
good textbook in any field and on any level of mathematics, the
authors’ choice of topics and way to discuss them is quite
successful. reasonable, and satisfactory. The reviewer found the
exercises at the end of each chapter particularly useful, in particular
because the authors did not include solutions. The subdivision into
“basic”, “interesting”, and “challenging” problems is
somewhat artificial: sometimes the challenging problems are almost
trivial even for beginners. The authors’ statement “You [= the
reader] may be reading this book on your own or as a text for a course
you are enrolled in. Regardless of your reason for reading this book,
we hope that you will find it understandable and interesting” in the
Preface somewhat calms down the irritation about what the reviewer has
written at the beginning. In fact, this (and only this) is the right
way to leave the final judgement on the quality of a book to the
reader, and it is a pity that the authors did not restrict themselves
to such a modest attitude right from the beginning.
rv: Jürgen Appell (Würzburg)