30 May 2008

Shoup: A computational introduction to number theory and algebra

A Computational Introduction to Number Theory and Algebra, online book, by Victor Shoup. (There's also a print edition, but the online PDF-book will remain freely available.) It is what it sounds like; "computational" doesn't mean "non-mathematical" but rather means that a lot of the applications are chosen with regard to their usability in computer science, specifically cryptography.

From reddit, where various commenters have pointed out things like "the author says that book is elementary but it really isn't." Of course, this is fairly common. The author actually says
The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed.
The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course on algorithms and data structures.

As usual, "elementary" means something like "a smart, well-trained undergrad could read and understand it"; it's a term of art like anything else. But in some provinces of the Internet there seems to be an idea that everything should be immediately understandable to all readers at first glance. (I use Reddit for the links it gives me to interesting news, not for the comments.)


Blake Stacey said...

Feynman had a wonderful line which I stuck in my e-mail signature list: "Now, elementary does not mean easy to understand." The way he meant it, an "elementary" proof might have had a lot of steps and thus be difficult to follow, but each step would not require sophisticated background knowledge.

So, yeah: a derivation is "elementary" if a smart, patient undergrad could follow it, and it is exceedingly elementary if a high-school student who has just survived geometry class could do so.

Aaron said...

@ blake---

Three cheers for Feynman! I'm becoming more and more convinced that the confusion of "elementary" with "simple" is one of the reasons that math-based classes for non-mathematicians often suck so much. By using elementary methods to protect students from math, teachers end up obfuscating important ideas and making everything seem turbid and arbitrary.


A good example is the AP macroeconomics course that I took one summer during high school. None of the math in that course was more difficult than solving a two-variable linear system, but a lot of people still had trouble with it---not because they couldn't have understood it, but because the teacher treated math like magic instead of like algebra. This may not be an isolated incident: a friend of mine who's taking a college economics course this summer has had at least one lecture in which the professor made simple concepts unintelligible by using seriously inappropriate notation. It's possible that the notation is standard---goodness knows bad notation is rampant everywhere---but I suspect it may have been motivated at least partly by the desire to avoid explicitly introducing new mathematical concepts.

Another example is the folk linear algebra that gets taught in a lot of high school math courses and college physics courses. You might call it dehydrated linear algebra: it's the desiccated bag of index-summing tricks that remain after the subject has been sucked dry of all intuition and understanding. It was especially harmful for me because it convinced me for a long time that I didn't need to take linear algebra, since I already knew it... even though I'd never even heard of a linear map! When I finally took a linear algebra class for mathematicians, it was revelatory, and the mechanical lessons on matrix and tensor arithmetic I'd been fed previously suddenly made a lot more sense!


Unknown said...

Mathematics of Linear Algebra