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Making Transcendence Transparent: An intuitive approach to classical transcendental number theory [Paperback]

Edward B. Burger , Robert Tubbs

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Book Description

21 May 2011 1441919481 978-1441919489 Softcover reprint of hardcover 1st ed. 2004

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.


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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.)

From the Back Cover

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.


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Customer Reviews

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Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com: 3.8 out of 5 stars  4 reviews
8 of 9 people found the following review helpful
5.0 out of 5 stars Useful Introduction if supplemented 25 Sep 2012
By M.J. Headlee - Published on Amazon.com
Format:Paperback|Verified Purchase
I used this when writing my Master's Thesis on Transcendental Number Theory. The book is very thorough; it starts with proofs of the irrationality of root 2, then Liouville's result about transcendence (and his example). It then moves into proving the irrationality of both e and pi, using the classical results of Lambert, and then it uses the historical extensions to prove the Hermite-Lindemann-Weirstrass results that pi and e are transcendental. It goes on to discuss the works of Siegal, Mahler, and Baker, all important contributors to transcendental number theory.

Transcendental Number Theory is an area that has long been one almost impossible to approach for an undergraduate. There was a paucity of literature, and most of it devoted to (i) graduate students, or (ii) specialists in the field. Lang can elegantly prove the transcendence of e using complex analysis in about a page. Most undergraduates don't have the background or sophistication to read such proofs, nor to digest the monographs.
This book fills that niche. It is targeted to undergraduates. To that end they make you work for the results, and there are several gaps in the proofs that are left as "challenges". It is great for developing mathematical maturity, but some of them can be quite hard and frustrating. Additional references on topics like symmetric equations, continued fractions, or complex analysis can help you through some of those stumbling blocks. There are no exercises.

If you're an accomplished graduate student this book isn't targeted to you. It feels hand-holdie. Other books would be Mahler's lectures on Transcendental Number Theory or Gelfond's book on Transcendental and Algebraic Numbers that are more succinct. If you want an encyclopedic reference those books are better. Making Transcendence Transparent is ill suited as such a reference because of its tone, wordiness, and those gaps.

Finally, as the other reviewer noted, the bibliography is very small. There is much more literature out there than this, and the book gives few clues for future reading.

Despite that, this book fills the targeted niche precisely. At the very least, a curious undergraduate can get a healthy respect for the magnitude of work required to prove some of the classical results, and such an exposition is not normally attempted (or severely succinct) in most other books that are out there. While it's not a research monograph, it's not trying to be.
9 of 10 people found the following review helpful
4.0 out of 5 stars Review of Making Transcendence Transparent 31 Aug 2008
By Scott Tillinghast - Published on Amazon.com
Format:Hardcover|Verified Purchase
This is a good introduction to the study of transcendental numbers, and it is pretty new (2004).
The authors provide motivation for complex proofs by working up from simpler proofs for special cases. For example, they prove various properties of the exponential function, and these culminate in proof of the full Lindemann theorem. Likewise a series of special cases leads up to proof of the Gelfond-Schneider theorem.
There is a nice description of Mahler's classification of transcendental numbers.
Chapter 7 concerns elliptic functions and does a good job of introducing the concepts surrounding elliptic curves.
One weakness in this book is that it does not have a good bibliography. It could better fill its niche in the market if the authors had made one. Moreover, there are many proofs I would like to see laid out better, and for this reason I have decided to give just 4 stars.
0 of 2 people found the following review helpful
1.0 out of 5 stars making the opaque impenetrable 13 April 2014
By reader - Published on Amazon.com
Format:Hardcover|Verified Purchase
I can only envy the wisdom of those who completely ignored
this dross. First thing to be aware of is that transcendental
number theory is a sub sub specialty in pure mathematics and is
notorious for its difficult analytic estimates using obscure auxiliary
functions created ad hoc for each and every separate application.

So beware, anyone who is brass enough to slap on a bogus title like the
one above is someone who will lie to your face.
What they have actually accomplished instead is this: they have simply
copied the skeleton of proofs from previous books on the same subject
( though you might not be aware this from the stingy bibliography --
a miserly four items !!!! ) and cut out most of the technical lemmas.
These are pegged as LTR (left to reader) so be aware that what's
on offer here is really just half a book ( the outline ) and you gentle reader
are expected to supply the other half ( the proofs ). Just don't expect
to be paid for your efforts or your time.
1 of 4 people found the following review helpful
5.0 out of 5 stars I Liked this Book 19 Mar 2014
By Indikos - Published on Amazon.com
Format:Hardcover
This is a good book on transcendental numbers.
Covers a lot of stuff especially the Gelfand Schneider theorem.
I found it fairly easy to read.
A motivated junior should be able to read this book.
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