Making Transcendence Transparent

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.