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Understanding a mathematical concept doesn't mean being able to perform computations using it. It means having an intuitive understanding for what it can repres
by lmkg 14y ago
Understanding a mathematical concept doesn't mean being able to perform computations using it. It means having an intuitive understanding for what it can represent and how to use it, and how to interpret concepts that use it.
Defining complex numbers as a bunch of arbitrary arithmetic operations on tuples lends nearly zero understanding, no matter how good you get at performing that arithmetic. Understanding complex numbers as rotations is much more useful in being able to understand, say, electromagnetism. In fact, I'll go so far as to say it makes the physics easier even if it makes the math more cumbersome.
Teaching mathematics as simple rules for manipulation of symbols is basically like telling only the punchline to a joke. Not even saying the punchline early, but just saying it on its own and providing no context. The fact that math can be (not is) just a bunch of mechanical rules is amazing when those rules can be mapped to complex and abstract phenomenon, and used to provide deeper understanding of them. The mechanical rules, in themselves, are boring. Their boringness is actually what makes them a good punchline, when contrasted with the intricacy of the systems they describe, but without that set-up, they're just boring.
- aaronbrethorst 14y agoThank you for explaining in three simple paragraphs why I hated every calculus professor I ever had.
- jasomill 14y agoI'd highly recommend Tristan Needham's Visual Complex Analysis to anyone interested in the subject, for exactly this reason . . . and, subsequently, Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables as proof that, nevertheless, "algebraic" need not imply "boring" or "computational". As a silly example, I'll never forget Cartan's definition of "2π" as the unique positive real number such that the kernel of the homomorphism "t -> e^it" from the additive group of reals to the multiplicative group of unit-length complexes is the set of integer multiples of 2π.
- pm90 14y agoMaybe It's because my mind is different, but I actually found his explanation very nice. I understand that it looks like just "symbol manipulation" but that is just one way to look at it. He shows how to define complex numbers using the bare minimum assumptions. As he mentioned, the reason why complex numbers are useful is precisely because of their properties under addition, subtraction and multiplication. Like I said, it just might be because our minds are different. But this seems like a good foundation, i.e you first show that this is all there is to it. Now, you can go ahead and explain the nice physical/geometric interpretations. Incidentally, this is the way that most books introduce Algebra (well, mathematical algebra, with groups, rings etc, not the algebra from high school)