Congratulations to Dr David Angell upon the release of his new book, "Irrationality and Transcendence in Number Theory".
Irrationality and Transcendence in Number Theory tells the story of irrational numbers from their discovery in the days of Pythagoras to the ideas behind the work of Baker and Mahler on transcendence in the 20th century.
It focuses on themes of irrationality, algebraic and transcendental numbers, continued fractions, approximation of real numbers by rationals, and relations between automata and transcendence.
This book serves as a guide and introduction to number theory for advanced undergraduates and early postgraduates, featuring developments from ancient to modern times.
It includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material. Techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computation are utilised. The book is accessible to readers without any specific specialist background.
We spoke to Dr Angell about his new book.
David, what prompted you to write on this topic?
"It's a field that has fascinated me more or less forever. I wrote an Honours thesis on one topic in this area, did my Ph.D. in another. I gave a fourth-year course on Irrationality and Transcendence a few times 10 or 20 years ago and wrote up lecture notes for the students. One student said that they were the best lecture notes she had ever seen - and she went on to win a University Medal, so I think we can trust her opinion! I was very happy myself with the notes and thought I would like to see them in print. It took a while, but eventually the opportunity arose when in early 2020 Taylor and Francis/CRC Press expressed an interest.
"Well, we all know what happened in 2020... but at least writing the book gave me something to do during repeated lockdowns. I say "writing the book" because, although it was based on my earlier work, it didn't take me long to discover the immense difference between a successful set of lecture notes and a publishable book. I put a lot of careful thought into expressing things as clearly as possible and ended up doing a huge amount of rewriting. I kept on thinking of exciting new topics to add too - this didn't exactly speed up the process! There were exercises and hints/solutions to be added - I had a large collection of these, which had been used for assessment when the course was given, but I still had to select the best, refine them and add them to the manuscript. I also wanted to make sure everything was referenced properly, including providing original sources for readers who might like to chase up the history of various topics."
What is unique about Irrationality and Transcendence in Number Theory?
"Although there are already many books in this area, I believe I have still managed to do some worthwhile new things. One of the chapters links number theory with automata theory - this was the topic of my Ph.D. thesis and a lot of work has been done in the field, but to the best of my knowledge it has never appeared in any book aimed at undergraduates. The basic idea is that if you can produce the digits of an infinite decimal by some automatic procedure - a simple computer programme, if you like - then it makes sense to say that the digits must form a "pattern" of some sort. Under these circumstances, we expect the decimal to be either a rational or a transcendental number - the question, of course, is whether we can prove it.
"I hope that the format of the book will also be found to be, in an innovative way, very friendly to readers. One of the most fascinating aspects of this subject is that it uses techniques from widely varying areas of mathematics - complex integration, automata theory, linear algebra, basic number theory among others. That's a lot of background. To make it all accessible to readers, the necessary facts are given in an appendix to each chapter.
"For readers who have studied, let's say, complex analysis already, the appendix should just serve as a checklist, or a reminder, of what they know. But I am firmly convinced that for readers who do not have this background knowledge, it is not only possible but appropriate for them to read the results, take them on trust without proof, and continue with the main arguments of my book. They can pick up more details later when they study complex analysis more formally.
"By doing things this way, I found in lectures that not only fourth year students (the "official" audience) but also talented third and even sometimes second year students were able to follow my arguments. In fact, I don't see any reason why a talented HSC student should not be able to get a great deal from the book, as long as they are prepared to accept the material given in the appendices. (And to work hard of course!) With all respect to people who are concerned with applications of mathematics to "real life", as far as I am concerned, the main thing about studying irrationality and transcendence is that it is fun!
"So, I have tried to assemble a collection of exercises and extra topics to emphasise this. There are relatively few routine exercises; I have concentrated on those which extend the material of the text. Readers can ponder the connections of the subject with music, the calendar, exposing dodgy investment companies, and analysing Sherlock Holmes stories. (That's why he's on the cover!) There are also connections with geometric topics - for example, can you cut a triangular pyramid into pieces and reassemble them to form a cube? - which on the face of it have nothing to do with irrationality."