Saturday, July 28, 2007

Philosophers like numbers

A while back I wrote about how I was working through a few yellow UTM books to learn a bit of math that I didn't know. I got sidetracked and didn't write the follow ups that I wanted to. I'm being driven by procrastination, so it seemed like a great time to write. The next book I want to mention is John Stillwell's Elements of Number Theory. I liked his book on geometry, which I'll talk about soon, so I thought I'd try his number theory book. I feel like I know a little bit about natural numbers after all the time spent playing around in Peano arithmetic. This turned out not to be terribly true. The book covers a fair amount. It starts with the natural numbers, integers and diophantine equations. The following chapters are on factorization, the Euclidean algorithm, congruence arithmetic, the application of factorization to cryptology, gaussian integers (complex numbers a+bi with both a,b as integers), the Pell equation, quadratic integers, and the four square theorem. There are a few more chapters on more algebraic topics like rings and ideals, which seem like they would be very neat. I didn't finish this book because I got frustrated with it. There are a decent number of topics in the book which the author relates to the others very well. There is a good bit of narrative connecting the different parts of the book together. This nice feature is unfortunately combined with few proofs and a dearth of good exercises after the first chapter. It seemed like a book that would be fun to go through once you already knew some number theory, but as a book to learn number theory, it falls short. The proofs and exercises are sparser than one would expect. Lame. On another note, Stillwell seems to like quaternions, which are an extension of the complex numbers. From what I gather, quaternions are a relatively obscure domain of math that aren't used much. Both this book and Stillwell's geometry book have at least one section on them. I will hazard a guess that his other books mention them as well. This is just a little curiosity I wanted to mention. It is kind of cute.

In summary, I'd recommend avoiding this book. If anyone has a recommendation for a good number theory book, I'm all ears.


joyrexus said...

"If anyone has a recommendation for a good number theory book, I'm all ears."

Yes. André Weil's magisterial "Number Theory". A truly beautiful book put out by Birkhäuser. (Note: do NOT buy his book "Basic Number Theory", which is rather deceptively titled.)

Anecdotally, A.W. was brother of Simone and a close friend of that great polycephalic mathematician, Nicolas Bourbaki.

Robbie said...

I think quaternions are used in the calculations for the guidance systems for space-shuttles. Peter Simons has some stuff on the history of these things which is pretty interesting.

I found studying number theory really hard: the proofs, once you have them, don't seem to fit into natural patterns. That makes it really fun, in some ways, but also kind of puzzling. I think my tutor said that it made a bit more systematic strength when you get up into elliptical curves and suchlike.

One thing I found really useful was Dickson's History of Number theory (a mammoth book: libraries may have it.) You can look up some topic, and get some of the backdrop, plus sketch proofs.

I used the OUP books when doing the course: Rose's "A course in number theory" and Hardy's "An introduction to the theory of numbers". Long time since I looked at them, but I remember both being pretty good (the Hardy once is pretty tough at times).

Shawn said...

Thank you both for the recommendations.

Quaternions came up in a couple more places in my reading recently. Apparently they are getting popular in some areas of physics. I forget which exactly. I think most sources have said that vectors usurped the role of quaternions in most applications, which is why they have kind of hidden in relative obscurity.

I was also receommended Stark's An Introduction to Number Theory.

Adam B said...

Niven, Zuckerman, and Montgomery's _Introduction to the Theory of Numbers_ is a pretty standard introductory text. I remember doing lots of interesting exercises when I took number theory, and I think that a lot of them were from that text. Check it out!