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A Readable Introduction to Real Mathematics

Daniel Rosenthal, David Rosenthal, and Peter Rosenthal
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Allen Stenger
, on

This is an unfortunately-titled look at some classical areas of discrete mathematics. It includes a lot of elementary number theory, some transfinite cardinals, theory of equations (including some rudimentary theory of real and complex numbers, from an algebraic and not analytic viewpoint), and geometric constructions with straightedge and compass.

The intended purpose and audience of the book are unclear. This is a proofs book, and nearly everything is proved and most of the exercises are to prove something. It is based on 15 years of lectures at the University of Toronto, apparently in a bridge or transition course taken by math majors in the second year. The book’s preface promises a broad but very different scope; on p. ix it states

Since the only prerequisite for understanding this book is high school algebra, it is suitable as a textbook for a wide variety of courses. In particular, it is our view that it would be appropriate for courses for general arts and sciences students who want to get an appreciation of mathematics, for courses for prospective teachers, and for an introductory course for mathematics majors. 

It further states that “high school students who like mathematics might be directed to this book.”

The title is unfortunate on several counts. It suggests to the students that what they have been studying up to this point was not real mathematics, even though they have probably been studying geometry, trigonometry, solution of equations, and calculus. The term “real mathematics” is never defined, but it appears to mean “proofs”. Proofs are indeed a central feature of mathematics, but there are lots of valuable proof-free parts too, including heuristics, examples and counterexamples, survey papers, and math handbooks. No mathematician would turn up his nose at these as not being real mathematics. Although the book is indeed very readable, the term “Readable” in the title is puffery; no author would title a book “An Unreadable Introduction to ...”

The material on real and complex numbers is relatively weak. Contrary to what you might expect, the book does not construct the real numbers (although Dedekind cuts are in an extended exercise). It comes at them from a theory-of-equations perspective and deals with why we need something more than the rationals, but it doesn’t develop much of the reals; in particular, there’s no discussion of completeness. The book seems to assume that you already understand infinite decimal expansions.

The ordering of topics is a little peculiar, too. The Rational Roots theorem is proved very early, even before we have talked about the existence of irrational numbers, but instead of applying it, the book essentially uses specializations of the proof to show that numbers such as \(\sqrt{2}\) are irrational. Some of the proofs in this section are not careful. For example, the argument of a complex number is defined geometrically as an angle measured against the positive real axis, but in discussing multiplication of complex numbers using magnitudes and arguments, the book forgets to reduce the sum mod \(2 \pi\). For another example, on p. 138, when merging two field towers to make a new tower, it doesn’t note that some of the roots being adjoined may already be in the field at that point. Some of the terminology is non-standard also; for example, “algebraic number” only refers to real roots, and the old term “surd”, which usually means any irrational number, is used here to mean a constructible number, irrational or not.

The book is not very reliable about computers, although (perhaps happily) it makes little mention of them. On p. 23 it discusses the number \(3+2^{3,000,005}\) with the intent to illustrate the power of modular arithmetic. The book says,

This is a very big number. No computer that presently exists, or is even conceivable, would have sufficient capacity to display all the digits in that number.

It is indeed a very big number, at 903,092 digits, but Mathematica running on a desktop computer can calculate and display it in under a second. On p. 3 the book states,

Using refinements of this idea [trial division] and powerful computers, many very large numbers have been shown to be prime. For example, 100,000,559 is prime, as is 22,801,763,489.

Computers are very important in primality testing, but in real life they only use trial division to look for small factors and use much faster algorithms to test whether a given number is prime. Also, the examples given are not good. The first number, 100,000,559, is not prime, being \(53 \cdot 223 \cdot 8461\). The second number really is prime, although the way to go here would be the Brillhart–Lehmer–Selfridge test, which is based on a converse of Fermat’s little theorem. This second number is small enough that it could be proved by hand with this method and does not require computers.

There are a few fanciful statements. For example, on p. 5 we have,

If you are able to solve to Twin Primes Problem or determine the truth or falsity of Goldbach’s Conjecture, you will immediately become famous throughout the world and your name will remain famous as long as civilization endures.

On p. 106, after a good explanation of what it means that the Continuum Hypothesis is independent of the Zermelo–Fraenkel axioms, the book says,

However, many mathematicians believe that Zermelo–Fraenkel Set Theory captures all the reasonable properties of the real numbers of therefore conclude that no such proof is possible. We invite you to prove that those mathematicians are wrong by proving (or disproving) the Continuum Hypothesis.

I think the big weakness of using this book for a bridge course is that it never talks about proof per se; there is no discussion of what a proof is, no consideration of proof strategies, and only a couple of examples of fallacious proofs. It would have to be supplemented a good bit for a bridge course; as it stands, the students would have to learn proofs by osmosis, which defeats the purpose of a bridge course. Two better books for this purpose are Beck & Geoghegan’s The Art of Proof and Rotman’s Journey into Mathematics. Both cover much real mathematics but are much stronger on proof techniques, and both have a great deal of coverage of continuous math.

I think the big weakness of the book as a math-appreciation text, apart from a possible over-emphasis on proof, is that most of the material is very old (at least 200 years) and does not give a good idea of what mathematics is today. It also omits the whole area of continuous math. I think the best choice for this kind of course, although very expensive, is Burger & Starbird’s The Heart of Mathematics. It really does give the reader a good understanding of what mathematicians do today, and is full of real mathematics. A good cheaper alternative, although not very up-to-date, is Courant & Robbins & Stewart’s What is Mathematics?

Bottom line: a well-written book with much interesting material, but with a number of flaws and an uncertain audience.


​1. Introduction to the Natural Numbers
2. Mathematical Induction
3. Modular Arithmetic
4. The Fundamental Theorem of Arithmetic
5. Fermat's Theorem and Wilson's Theorem
6. Sending and Receiving Coded Messages
7. The Euclidean Algorithm and Applications
8. Rational Numbers and Irrational Numbers
9. The Complex Numbers
10. Sizes of Infinite Sets
11. Fundamentals of Euclidean Plane Geometry
12. Constructability