The Higher Arithmetic: An Introduction to the Theory of Numbers

Harold Davenport (1907–1969) was a British mathematician who worked in many areas of number theory. He wrote this book in 1952 as an introduction to number theory for a general audience. It has expanded over the years, and recent editions have been edited and added to by his son, James H. Davenport, a computer scientist at the University of Bath. The prerequisites have been kept very low: just high-school algebra and no calculus (but you do need a strong grasp of the algebra).

This is not a theorem-proof book. It’s more like a good expository talk, where you hear about the problems, see some examples, and then look at some statements of facts (results) along with arguments or explanations why they are true. There are in fact a lot of theorems and proofs here, but they are not packaged as such.

The chapter on continued fractions is especially good. It creeps up on the subject gradually, starting with fractions and the Euclidean algorithm, and lots of examples, but developing the algorithms in more generality than this. Then it makes the leap to irrational numbers (infinite continued fractions), concentrating on the case of quadratic irrationals (this case is especially nice because the fractions are periodic). It also goes into a good bit of detail about the approximation properties of the convergents. I believe this is the only part of the book that uses limits or infinite processes, although the way it is developed you don’t really have to understand those to follow the narrative.

There’s a lengthly and mostly up-to-date chapter on computers and number theory, that concentrates on several factorization methods and has a very clear section on cryptography with a good introduction (not just jumping directly to the Diffie-Hellman and RSA algorithms).

This eighth edition, published in 2008, is very polished. It has a support website with four pages of notes on new developments since publication, although it hasn’t been updated since 2013.

The market for this book is a little uncertain today. Originally it was aimed at the intelligent general reader who had a good knowledge of high-school mathematics, but I think there aren’t very many general readers today with a strong enough knowledge of algebra to follow the narrative. Most introductory number theory courses today are upper-division undergraduate courses and they are proofs courses, so this would not work well as a text. I do think it would work well as enrichment material; bright high-school math students would certainly be able to follow the book and many would be fascinated by it. It could also work for pre-service teachers.

There are a large number of short introductory number theory books at about the same level, most of which are more traditional textbooks than this one. Good examples of these include Vinogradov’s Elements of Number Theory, Dudley’s two books Elementary Number Theory and A Guide to Elementary Number Theory, LeVeque’s Fundamentals of Number Theory, and Andrews’s Number Theory. Some other books at the same level and with the same prerequisites as the present book, that also take an informal and exploratory approach, include Burn’s A Pathway into Number Theory (very IBL-oriented), Ore’s Number Theory and its History (lots of examples, almost no prerequisites), and Ogilvy & Anderson’s Excursions in Number Theory (a 1966 book that was also intended for a general audience but is probably too advanced today).

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.