MATH5535 Irrationality and Transcendence is a course whose roots go back to about 500 B.C., when Pythagoras or one of his followers proved that, contrary to "common sense", some numbers cannot be expressed as a ratio of integers. While the Ancient Greeks succeeded in proving various surd expressions to be irrational, little further progress was made until the eighteenth century, when Euler and Lambert proved the irrationality of e, π and related numbers. We look first at more modern proofs of these results, deferring Lambert's work until later.
The question of transcendence is deeper, and harder, than that of irrationality. After giving a survey of the basic ideas regarding algebraic numbers, we shall prove the existence of transcendentals, firstly (following Cantor) without exhibiting any particular example! The simplest approach to showing that a specific number is transcendental is to study its approximations by rational numbers; continued fractions provide an important tool for doing so. Taking another look at e and π, we shall adapt Hermite's method to prove the transcendence of these numbers.
A recent and fascinating topic connects transcendence with deterministic finite automata, a kind of very elementary computing device. Ideas concerning such automata can be used to investigate the transcendence of numbers which display some sort of "pattern" in their decimal expansions or continued fractions.
One of the most exciting aspects of this subject is that it uses techniques from widely diverse areas of mathematics: number theory, calculus, set theory, complex analysis, linear algebra and the theory of computation will all be touched upon. Each chapter concludes with an appendix setting out the basic facts needed from these topics, so that the notes are accessible to readers without any specialist background in these areas.
The notes are divided into seven chapters (could be more later!). They were initially written, designed and typeset by David Angell in 2000, and were comprehensively revised in 2002. The current version includes a small number of further revisions from 2005 and 2007. Older versions are not available. Some portraits of mathematicians are included in the notes; these were obtained from the History of Mathematics Archive maintained by the University of St.Andrews, Scotland. A visit to this marvellous site is warmly recommended!
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