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Irrationality and Transcendence in Number Theory

David Angell
Chapman and Hall/CRC
Publication Date: 
Number of Pages: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on
This excellent book not only helps fill a substantial gap in the undergraduate mathematics literature, but it does so in a way that most students will, I think, find interesting, inviting and accessible. The gap I refer to concerns transcendental numbers. Most of us, I think, would have no difficulty in immediately providing an interested upper-level mathematics major with a list of sources for, say, a proof of the Law of Quadratic Reciprocity, but if a student were to ask for an accessible source for a proof of the transcendence of \( \pi \) , it might be somewhat harder to come up with one. A few books on Galois theory, such as Ian Stewart's Galois Theory, and Hadlock's Field Theory and Its Classical Problems, contain such proofs, though the titles of these books do not reveal that. The only book that I know of, off the top of my head, that is devoted entirely to the subject of transcendental numbers and is intended to be accessible to undergraduates is  Making Transcendence Transparent by Burger and Tubbs, and while that is an excellent book, it is also a moderately demanding one. 
The book now under review is in some (but not, as we will see, all) ways somewhat less ambitious than is the book by Burger and Tubbs. Some results proved in the latter text (e.g., the Gelfond-Schneider theorem on transcendence) are stated without proof here. However, this book does cover a lot of interesting results that most undergraduates never get to see, and would, I think, make an excellent text for a senior-level seminar or capstone course.
Basic prerequisites for the text are calculus and some prior exposure to elementary number theory, preferably including continued fractions. Readers who want to follow the details of the proofs of some basic facts about algebraic numbers rather than just take the results on faith should know some basic abstract and linear algebra. Complex analysis is sometimes used, but only in a way that parallels real analysis. Basic ideas of set theory are used on occasion as well. However, to make the book as accessible as possible, every chapter has one or more Appendices discussing background material.
The book begins with an introductory chapter that discusses some easy irrationality results (irrationality of square roots and other quadratic expressions, and relating rationality to decimal expansions) and concludes with a proof of the irrationality of the number \( e \), proved via its infinite series expansion. 
It was Lambert, in the 18th century, who first proved the irrationality of \( \pi \) . Lambert also proved the irrationality of \( e^{r} \) for any nonzero rational number \( r \). A century later, Hermite revisited, and gave different proofs of, these results; these more modern proofs (modified by Niven) are presented in chapter 2. Hermite’s ideas can also be exploited to prove the irrationality of some trigonometric values; these proofs round out the chapter. 
Chapter 3 introduces transcendental numbers. Algebraic numbers, algebraic integers, and transcendental numbers are defined and it is proved that the set of algebraic numbers is a subfield of the complex numbers and that the set of algebraic integers is an integral domain. Cantor’s set-theoretic argument is then given to prove the existence of transcendental numbers (without actually exhibiting one). The first actual example of a transcendental number that is given is Liouville’s number, which appears as part of a discussion of approximation of real numbers by rational ones.  The chapter concludes with a sketch of the proof of the transcendence of the number \( \zeta(3)=\sum 1/n^{3} \), with a number of the more difficult technical details omitted. 
Chapter 4 introduces continued fractions as a tool for finding approximations by rational numbers, and chapter 5 establishes the transcendence of \( e \) and \( \pi \) and also proves the more general result that \( e^{a} \) is transcendental if \( a \) is a nonzero algebraic number. 
Chapter 6, perhaps the most unusual of the book, discusses connections with automata theory. This is material that I have not seen elsewhere. Finally, in chapter 7, the author returns to the ideas of chapter 4 and uses generalized continued fractions to give Lambert’s proofs of the irrationality of \( e \) and \( \pi \), as well as some trigonometric values.
This material is, of course, very nontrivial, but Angell goes to great lengths to make it accessible. He writes slowly and clearly and spends a lot of time motivating results.  As previously noted, he also includes background Appendices in each chapter.
There are other useful pedagogical features. Each chapter ends with an extensive collection of exercises, most of them non-routine; a 20-page section at the end of the book offers hints to these. The book also contains a five-page bibliography (one that, surprisingly, omits the Burger/Tubbs book mentioned earlier) that directs a reader to useful sources. 
The subject matter of this book is interesting and beautiful and deserves to be made accessible to well-prepared senior undergraduates. Angell has done an excellent job in helping to do so.

Mark Hunacek ([email protected]) is a Teaching Professor Emeritus at Iowa State University.