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Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory

Edward B. Burger and Robert Tubbs
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Álvaro Lozano Robledo
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Absolutely, read this one! Making Transcendence Transparent is one of those books that stand out from the crowd because the authors have put a lot of good work into it, and plenty of imagination and creativity. It is witty, funny at times, highly entertaining, very readable and interesting to both the casual and advanced reader.

As the title and subtitle explain, the book is an introduction to transcendental number theory. As anyone who has already studied the subject knows, the theorems in this area of mathematics are rather involved and usually quite obscure at first glance. In Making Transcendence Transparent the authors try (and succeed) to penetrate this "darkness" by building the intuition of the reader and providing clear expositions of the idea of the proof before presenting the actual proof.

The book is partitioned into 9 chapters and an appendix, covering the following topics (among others): the basic theory, mostly definitions, of rational, irrational, transcendental and algebraic numbers; Liouville's theorem and Liouville's numbers, Roth's theorem; polynomial vanishing and the transcendence of e; the Lindemann-Weierstrass theorem; Siegel's lemma; the Gelfond-Schneider theorem; Mahler's classification of transcendental numbers; the Weierstrass P -function and periods and transcendence in function fields.

As mentioned in the first paragraph, the book is beautifully written and very creatively put together (for example, each chapter is named after a well-known real number... except the last chapter, which is named after a well-known transcendental function). The text helps us understand the concepts by building a very strong intuition and also motivates the concepts from a historic point of view. Furthermore, the topics selected are interesting and provide a broad view of the subject. My only objection: not enough problems. However there are suggested problems along the chapters which the authors call challenges, some of them for a good reason. Conclusion: read this one!

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

A prequel to transcendence * Incredible numbers incredibly close to modest rational numbers * The powerful power series for e * Conjugation and symmetry as a means towards transcendence * The analytic adventures of exp(z) * Debunking conspiracy theories for independent functions * Class distinctions among complex numbers * Extending our reach through periodic functions * Transcending numbers and discovering a more formal e * Selected highlights from complex analysis