Examples of usage of transcendental numbers. 1] Transcendental numbers transcend human experience

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1 Examples of usage of transcendental numbers 1] Transcendental numbers transcend human experience Transcendental Numbers are in a word, profound. For they excel, surpass and transcend human experience. They are synonyms of peerless, incomparable, unequaled, matchless, unrivaled, unparalleled, unique, consummate, paramount, superior, surpassing, supreme, preeminent, sublime, excelling, superb, magnificent, marvelous, and... well, transcendental. Mathematically, they are, by definition, not capable of being produced by the algebraic operations of addition, multiplication, and involution, or any of the inverse operations. [1] Philosophically, they are existing apart from, not subject to the limitations of, the material universe, not to mention: 1 transcendent 2a (in Kantian philosophy) presupposed in and necessary to experience; a priori. b (in Schelling s philosophy) explaining matter and objective things as products of the subjective mind. c (esp. in Emerson s philosophy) regarding the divine as the guiding principle in man. 3 a visionary; abstract b vague; obscure. That should about cover it 2] Transcendental numbers are versatile I am in love with the mysterious transcendental numbers. Did you know that there are "more" transcendental numbers than the more familiar algebraic ones? Even so, only a few classes of transcendental numbers are known to humans, and it's very difficult to prove that a particular number is transcendental. In 1844, math genius Joseph Liouville ( ) was the first to prove the existence of transcendental numbers. (More precisely, he was the first to prove that a specific number was transcendental.) Hermite proved that the number e was transcendental in Lindeman proved that pi was transcendental in For more information, see my book Wonders of Numbers from which this is excerpted. Although they are not often recognized as such even by mathematicians, there are a lot of commonly-used numbers that are also transcendental, which can easily be shown by the Gelfond-Schneider theorem mentioned in the tenth item on the list. If a is algebraic, and c is algebraic, and b = logarithm (base a) of c is not rational, then b must be transcendental or else the theorem would imply that c must be transcendental--a contradiction. Then, with a=10 and c=2, the log of two, base ten, is transcendental, and so is any base ten logarithm of any rational number other than rational powers of ten. The same holds for any other rational logarithm base--so there are a lot of transcendental numbers that are in common use. I'd also like to point out that any number can be used to produce a transcendental by using Liouville's algorithm (see item number five). If the number is terminating, convert it to non-terminating by subtracting one from the last digit, and adding an infinite string of 9's to the end. Then just put each of its digits where 1

2 Liouville puts a one, even if the digit is zero. The result will be a transcendental number 3] Numbers known to be transcendental: e a if a is algebraic and nonzero (by the Lindemann Weierstrass theorem), and in particular, e itself. π (by the Lindemann Weierstrass theorem). e π, Gelfond's constant, as well as e -π/2 =i i (by the Gelfond Schneider theorem). a b where a is algebraic but not 0 or 1, and b is irrational algebraic (by the Gelfond Schneider theorem), in particular: o, the Gelfond Schneider constant (Hilbert number). sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann Weierstrass theorem). ln(a) if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann Weierstrass theorem). Γ(1/3), [9] Γ(1/4), [10] and Γ(1/6). [10] , the Champernowne constant. [11] Ω, Chaitin's constant (since it is a non-computable number). Prouhet Thue Morse constant where β > 1 and is the floor function. Two commentaries about transcendental numbers A] Making transcendental numbers (transcendence) transparent Quote from Mathematician Edward Burger (see biography below): 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. 2

3 Biography 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 tru Commentary By M.J. Headlee on September 24, 2012 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 3

4 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 B] Defining Transcendental Functions Question: Date: 02/05/2001 at 21:10:33 From: Ruth Subject: Defining transcendental functions I am looking for a precise definition of a transcendental function and of an elementary function. I BELIEVE that the set of elementary functions is just the set of algebraic functions and the set of transcendental functions combined, but I cannot find it specifically defined anywhere in my calculus book or at any of the Web sites I have visited. (Most sites with mathematics definitions do not include elementary functions.) My book (James Stewart's Calculus, 4th edition) defines transcendental functions as everything except algebraic functions (functions containing only addition, subtraction, division, multiplication, powers, and roots), then gives a few examples (trig and inverse trig functions, log and exponential functions, and hyperbolic and inverse hyperbolic functions), and then says, "it also includes a vast number of functions that have never been named...we will study transcendental functions that are defined as sums of infinite series." Are all infinite series transcendental functions? Later, when defining elementary functions, it lists examples and then says that the integral of an elementary function is usually not an elementary function. Can you define a non-elementary function by some other means than as the integral of another function? Can the unnamed transcendental functions that the book mentions be expressed symbolically, or would that be considered naming them? How am I to distinguish between an unnamed transcendental function and a nonelementary function? I am would LOVE a clear answer on this. Thank you, Ruth Costa Response: 4

5 Date: 02/08/2001 at 23:30:46 From: Doctor Fenton Subject: Re: Defining transcendental functions Hi Ruth, Thanks for writing to Dr. Math. You've asked some very good questions. The definition of elementary functions seems to be a matter of consensus within the mathematical community, and it basically consists of the "familiar" functions, and functions that can be generated from the standard ones by addition, subtraction, multiplication, division, and composition. Like most definitions, this is somewhat arbitrary, but I think there is pretty general agreement on which functions are "elementary." The definition of an algebraic function is also fairly precise, as you described (I might add that the powers and roots must be rational ones). A problem with "transcendental" is that it is a catch-all definition, defined by exclusion rather than by direct characterization. Anything that is not an algebraic function is, by definition, transcendental. In that regard, it's somewhat like irrational numbers, which are defined by exclusion: they aren't rational. That makes it difficult to make general statements about them. For example, we can say that the sum and product of rational numbers is rational, but sqrt(2) and (2-sqrt(2)) show that the sum of irrationals can be rational, and sqrt(2)*sqrt(2) shows that the product of irrationals can be rational. There is only a countably infinite number of algebraic functions, but it can be shown that there are uncountably many continuous functions on an interval, for example. Since there are only countably many finite strings of mathematical symbols, it isn't possible to describe "most" functions. These are some of the "unnamed transcendental functions" Professor Stewart is trying to describe. He can only give you a hint of the complexity of the set of transcendental functions. There are also "named" transcendental functions: Bessel functions and Hankel functions, to name a couple. These arise in certain specialized areas, and are well-known to experts in those fields, but the typical math student may never encounter them, so they are not put in the "elementary" list. So basically, there is just no way to describe a "typical" transcendental function. Many of them are indeed generated by infinite series, but on the other hand, some infinite series describe algebraic and elementary functions, too. Elementary and algebraic functions are relatively small, precisely defined collections of 5

6 functions, which can be characterized by common properties. Transcendental functions share only the property of being nonalgebraic. I'm sorry I can't be of more help, but there just isn't any way to characterize transcendental or non-elementary functions in general, other than that single exclusionary property. If you have further questions, please write us again. - Doctor Fenton, The Math Forum 6