New Zealand Maths Olympiad Committee online » Links

A very useful list …

Michael — Wed, 15 Sep 2010 19:40:05 +0000

Having the right mental habits is one of the keys to being a successful problem solver. One of the best lists I’ve seen was recently posted at a math education blog. The list is supposedly for sixth grade (i.e. year six) students, but is much more universal than that. You should go there and read the whole thing, but even the excerpted headings are worth knowing:

Pattern Sniff
Experiment, Guess and Conjecture
Organize and Simplify
Describe
Tinker and Invent
Visualize
Strategize, Reason and Prove
Connect
Listen and Collaborate
Contextualize, Reflect and Persevere

Michael

Chrystal’s Algebra

Michael — Wed, 09 Dec 2009 20:31:18 +0000

A classic (pre 1900) textbook on algebra by Chrystal has recently been scanned and made available electronically. Volume 1 is not likely to be of much interest for training purposes but chapters 23 and 24 (on combinatorics), 32 through 34 (continued fractions), and 35 (number theory), of volume 2 are. However, be warned, the scanned PDF is about 30M, so don’t try this on a slow or expensive connection!

Michael

Polynomials, pure mathematics, Princeton Companion

Heather — Thu, 02 Apr 2009 00:23:49 +0000

Round about sixth form one learns that every polynomial can be factorized, as a product of linear factors. Why? Well, here’s a polynomial, see. It’s probably a cubic with integer coefficients — after all, most nontrivial polynomials that one encounters are. You play with it until you discover a root, likely by looking at integer factors of the highest and lowest coefficients. Then you polynomial-divide through by the linear factor which that root gives you, and get a quadratic, whose roots there’s a formula for finding. Tada!

Of course, there’s a problem with this algorithm: it depends on figuring out how to break down your polynomial into only linear and quadratic factors.

Perhaps you know (at least in theory) the formulae for roots of general cubics and quartics. Then you can carry out the above algorithm even if initially you can only see how to break down your polynomial into linear, quadratic, cubic and quartic factors. That certainly covers most of the polynomials you’ll ever encounter at school.

But then problems arise. The Fundamental Theorem of Algebra is the aforementioned factorization theorem. More precisely, it states that every polynomial with integer, rational, real, and, importantly, even complex coefficients has a unique decomposition into linear complex-coefficiented factors. Its proof is one of the coolest things you’ll learn in university complex analysis. On the other hand, Galois theory, one of the coolest things you’ll learn in university abstract algebra, shows that there’s no general formula for the roots of greater-than-fourth degree polynomials. The result is the kind of paradox that logicians love to hate: polynomials with roots that you know exist, but which you have no algorithm for constructing.

The two links in the paragraph above are to sample articles of the Princeton Companion to Mathematics, a recently-published “encyclopedia of modern pure mathematics” edited by mathematician, Fields medallist, British 1981 IMO perfect scorer, and blogger Timothy Gowers. More sample articles available here (log in with username Guest and password PCM, and click the link to Resources). Two others which I especially like are: a quick tour of fundamental definitions and structures (Section Four is a great introduction to the notions of limit and continuity, which, as an Olympiad hopeful, you may well have struggled through lately); and a discussion of how professional mathematicians go about problem-solving.

Olympiad mathematics is full of beautiful ideas, which show up all the better for being restricted to within a tightly-contained syllabus. Modern pure mathematics, as the Companion does an exhilarating job of showing, takes the beautiful ideas and lets them loose.

Number theory texts

Heather — Tue, 24 Feb 2009 03:26:44 +0000

(At least!) a couple of good, comprehensive introductions to elementary number theory are available online. These notes by Jim Hefferon and W. Edwin Clark are nicely written and gently-paced. These ones by Naoki Sato are a bit more Olympiad-focused.

Proof style guide

Heather — Sat, 24 Jan 2009 13:29:32 +0000

Clear mathematical writing often leads to clearer thinking. Of course, in Olympiad problems, it also reduces the risk of losing marks for small errors!

If you want to work on your exposition, I heartily recommend this style guide for writing up solutions to mathematical problems, by John M. Lee. It’s aimed mainly at people writing university mathematics assignments, with lots of time to spare, so some of the details in the last section are more pedantic than Olympiad contestants will need to bother with. But the general instructions in the first half are excellent.

One sometimes-helpful trick (Lee mentions it at the start of the second page) is to try to break down your solution to a question into several sub-arguments. In long complicated questions, clarity, and keeping track of small details, seem to arise much more easily out of a nice sequence of Propositions and Lemmas — each stated clearly, then proved before going on to the next — than out of a single essay-style answer.

Linkage: Abstract Math

Michael — Thu, 22 Jan 2009 22:09:55 +0000

The site Abstract Math by Charles Wells presents some useful insights on writing and doing proofs. The author’s opinions in some areas are arguable (so you should read and think about them, rather than follow the advice slavishly — the only piece of advice which you should in general follow slavishly!) For instance, I wouldn’t recommend the “Languages of Math” section which is far too detailed at present. However, there are little gems to be found everywhere, and from within that section comes the link to Timothy Gowers article, “The language and grammar of mathematics.”. By the way, the “Tricki” which he (Gowers) promises should be live soon will certainly be a great resource.

Returning to Abstract Math, I particularly recommend the section entitled Doing Math. Just in case you’re too lazy to click a link and explore I will briefly quote one particularly important concept from the “Useful Behaviors” section:

METHOD: To prove that a statement involving a concept is true,start by rewriting the statement using the definition of the concept.

So very simple, and yet so true. But, don’t wait, go and explore Abstract Math now!

Michael