My text book Irrationality and Transcendence in Number Theory has been published by Taylor and Francis/CRC Press and is now available. It 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. Readers are led through the developments in number theory from ancient to modern times. The book includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material.

Irrationality and Transcendence in Number Theory is now available from the publishers, or from the UNSW bookshop, or through any of the usual online retailers, for example, Book Depository and Amazon.

I am delighted to thank Helena Brusic of The Imagination Agency for her wonderful cover image!

I have been teaching in the School of Mathematics and Statistics at the University of New South Wales since 1989. In recent years I have attempted to provide printed notes for all of my higher year lecture courses, and have made them accessible on the web where possible. This page serves as the top level of an archive which I hope will ultimately make all of these notes available to users of the Internet.

As everyone knows, stealing documents by means of the Internet is perfectly simple. **Please don't do it!!!** These notes are copyright, which means that you are very welcome to view them online, and even to download them for your own personal use, but **not** to distribute them to other people without my permission. If you contact me by email I will be very glad to give this permission unless I see some good reason why I should not.

Please be aware that my lecture notes are intended to be used as a basis, not as a substitute, for lectures. There may be various spots at which the reader feels some extra explanation is needed: such additional material would have been given verbally in lectures. Nevertheless, enquiries and comments about the lecture notes are welcome, and may be emailed to me here.

For course enquiries, consult the School of Mathematics and Statistics website. Follow the link for current students or for future students.

Read about The Bourbaki Ensemble, possibly the only orchestra in the world to be named after a polycephalic French mathematician! |
Also Orchestra 143, a classical chamber orchestra whose name involves a fundamental error in combinatorics. |

All of my lecture notes are available in PostScript and may be read using Ghostview or Gview or GV. If you are a student at UNSW and are reading this through your account at the School of Mathematics and Statistics then one of these should be available and the lecture notes should open automatically. If you cannot read the notes you want, try downloading the appropriate software from here. Some of my lecture notes are also available in PDF, but I recommend using the PostScript version.

- Some examples of basic proofs which could be useful for students in MATH1081, MATH3421 and other subjects.
- MATH1231 algebra: supplementary notes, examples and exercises.
- MATH2410 Automata and Algorithms is a course which addresses at an elementary level various topics of importance in mathematical computer science…
- MATH2501 Linear Algebra is a second year course which develops and extends the work on vector spaces, linear transformations and eigenvalues begun in MATH1231…
- MATH3421 Logic and Computability gives an introduction to a subject which may be seen as arising from mathematicians' attempts early in the twentieth century to understand
*precisely*what it is we do when we write a proof or perform a calculation… - 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…
- MATH5505 Advanced Combinatorics was given as an Honours level course in pure mathematics in 1997. The lecturer was Norman Wildberger. David Angell attended the course, wrote up course notes and added some extra material…

Besides teaching university courses I have given talks on various topics at the Mathematical Association of NSW annual Talented Students' Day, and have published articles in Parabola, a mathematics journal for secondary students and teachers published by the Australian Mathematics Trust in association with the School of Mathematics. Here is a list of some talks and articles - more links will be added soon.

Arithmetic and music in twelve easy steps. Why are there twelve semitones in an octave and not, say, eleven or thirteen? This article shows that there are good mathematical reasons behind this basic musical fact.

22/7 and all that.

Currency, crows and unexpected examinations. Some perplexing questions about mathematics and ornithology…

Waiter! A table for infinitely many please.

Beginning algebraic number theory. The recent proof of Fermat's Last Theorem is a mathematical mountain which few can scale. However, a ramble through the foothills surrounding it will provide many fascinating glimpses of the subject.

Cubics.

Ordering complex numbers... not. At school you have probably been taught that it makes no sense to say that one complex number is less than another. However, there are various plausible ways in which we might attempt to do just that. Is it really true that none of them works? And if not, why not?

Last modified 17 January 2022

David Angell, david.angell@unsw.edu.au, [61] (2) 9385 7061

School of Mathematics and Statistics, University of New South Wales

UNSW Sydney NSW 2052, Australia