29 August 2009


There's a series of math talks on Octonions this semester in the math department. The first one was yesterday afternoon. It dawned on me there that I do still have an interest in math for math's sake, particularly when it has interesting applications. Octonions turn out to be useful in string theory, which I found in a general hunt yesterday, and the Wikipedia article mentions special relativity and quantum logic as well.

It's also the first group I've come across that is neither commutative nor associative. Commutative means the order doesn't matter: 3*5 = 5*3, 6+8=8+6, etc. Subtraction and division by themselves are noncommutative, but since subtraction can be seen as adding a negative, and division is multiplying by the inverse, you can get around that. Matrices are the first things most people see where order really does matter. AB need not equal BA for two matrices A and B. Associativity means, roughly, that where you "put the parentheses" doesn't matter. So:

3(5*4) = (3*4)5

In the first case, I get 3 times 20, which is 60. In the second I get 12 times 5, which is still 60. Multiplication of real numbers is associative. In fact, I can't recall ever encountering a group operation that was not associative until reading up on the octonions.

The other thing I noticed at the talk was that, despite not taking any upper level math for the past several years, everything still made sense to me. Also I was a bit amused that the current grad students, who have all taken algebra more recently than I, didn't immediately recall that you generally use divisibility to establish that if a prime divides a product, it must divide at least one of the multiplicands of the product (Yes, multiplicand is a word; I just checked). The other nice thing about math is that I know the majority of people in the department and I might get away with not moving my office again, particularly since we've had several people retire and the temporary dean of Arts and Sciences (the bastard who sent out the gods-bedamned 'terminal' notices) is against hiring any more lecturers.

Still, I'm going to keep working through my old physics books. I wonder if it's possible to do a crossover math-physics doctorate here...

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