### Synopses & Reviews

While the study of transcendental numbers is a fundamental pursuit within number theory, the general mathematics community is familiar only with its most elementary results. The aim of Making Transcendence Transparent is to introduce readers to the major "classical" results and themes of transcendental number theory and to provide an intuitive framework in which the basic principles and tools of transcendence can be understood. The text includes not just the myriad of technical details requisite for transcendence proofs, but also intuitive overviews of the central ideas of those arguments so that readers can appreciate and enjoy a panoramic view of transcendence. In addition, the text offers a number of excursions into the basic algebraic notions necessary for the journey. Thus the book is designed to appeal not only to interested mathematicians, but also to both graduate students and advanced undergraduates. Edward Burger is Professor of Mathematics and Chair at Williams College. His research interests are in Diophantine analysis, and he is the author of over forty papers, books, and videos. The Mathematical Association of America has honored Burger on a number of occasions including, most recently, in awarding him the prestigious 2004 Chauvenet Prize. Robert Tubbs is a Professor at the University of Colorado in Boulder. He has written numerous papers in transcendental number theory. Tubbs has held visiting positions at the Institute for Advanced Study, MSRI, and at Paris VI. He has recently completed a book on the cultural history of mathematical truth.

#### Review

From the reviews: "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. ... The text helps us understand the concepts by building a very strong intuition and also motivates the concepts from a historical point of view. ... Conclusion: read this one!" (Álvaro Lozano-Robledo, MathDL, January, 2006) "One of the goals of the authors is to provide the reader with an intuitive framework in which the major classical results of transcendental number theory can be appreciated. ... This book is an introduction to the subject which is supposed to enable the reader to pursue later his study with more modern results. ... An appendix provides basic facts from complex analysis which are required for the proofs. This book is aimed at beginners who like to have examples and detailed proofs." (Zentralblatt MATH, August, 2005) "The book under review covers much wonderful material, heads in several directions, and could be used in many ways." (MAA reviews, D'Angelo, John P.)

#### Review

From the reviews:

"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. ... The text helps us understand the concepts by building a very strong intuition and also motivates the concepts from a historical point of view. ... Conclusion: read this one!" (Álvaro Lozano-Robledo, MathDL, January, 2006)

"One of the goals of the authors is to provide the reader with an intuitive framework in which the major classical results of transcendental number theory can be appreciated. ... This book is an introduction to the subject which is supposed to enable the reader to pursue later his study with more modern results. ... An appendix provides basic facts from complex analysis which are required for the proofs. This book is aimed at beginners who like to have examples and detailed proofs." (Zentralblatt MATH, August, 2005)

"The book under review covers much wonderful material, heads in several directions, and could be used in many ways." (MAA reviews, D'Angelo, John P.)

#### Synopsis

While the theory of transcendental numbers is a fundamental and important branch of number theory, most mathematicians know only its most elementary results. The aim of "Making Transcendence Transparent" is to provide the reader with an understanding of the basic principles and tools of transcendence theory and an intuitive framework within which the major results can be appreciated and their proofs can be understood. The book includes big picture overviews of the over-arching ideas, and general points of attack in particular arguments, so the reader will enjoy and appreciate the panoramic view of transcendence. It is designed to appeal to interested mathematicians, graduate students, and advanced undergraduates.

#### Synopsis

The Journey Ahead At the heart of transcendental number theory lies an intriguing paradox: While essen tially all numbers are transcendental, establishing the transcendence of a particular number is a monumental task. Thus transcendental numbers are an enigmatic species of number: We know they are all around us and yet it requires enormous effort to catch one. More often than not, they slip through our fingers and dissappear back into the dense jungle of numbers. Here we will venture to tame a few of these incredible creatures. In the pages ahead we offer an approach to transcendence that not only includes the intricate analysis but also the beautiful ideas behind the technical details. The phrase "classical transcendental number theory" in the title of this book refers to the most widely known results that were obtained in the nineteenth and early twentieth centuries. The reason for this focus is threefold. Firstly, this body of work requires only the mathematical techniques and tools familiar to advanced undergraduate mathematics students, and thus this area can be appreciated by a wide range of readers. Secondly, the ideas behind modem transcendence results are almost always an elaboration of the classical arguments we will explore here. And finally, and perhaps more importantly, this early work yields the transcendence of such admired and well-known numbers as e, rr, and even 2v'2."

#### Synopsis

This is the first book that makes the difficult and important subject of transcendental number theory accessible to undergraduate mathematics students. Edward Burger is one of the authors of The Heart of Mathematics, winner of a 2001 Robert W. Hamilton Book Award. He will also be awarded the 2004 Chauvenet Prize, one of the most prestigious MAA prizes for outstanding exposition.

#### Synopsis

Edward Burger is one of the authors of "The Heart of Mathematics," Winner of a 2001 Robert W. Hamilton Book Award. He will also be awarded the 2004 Chauvenet Prize, one of the most prestigious MAA prizes acknowledging an outstanding expository article. To read more about Ed Burger go to http://www.williams.edu/Mathematics/eburger/.

### Table of Contents

* 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