Comment by direwolf20
11 days ago
You can define that, but (if you don't already know about complex numbers) it's not obvious that it does anything mathematically interesting. It's just a cache for sin and cos, not a new type of anything. I could say that when evaluating 4th degree polynomials it's useful to have x, x^2 and x^3 immediately at hand, but the combination of those three isn't a new type of number, just a cache.
>it does anything mathematically interesting
You are right - its not interesting. You already know that rotation can be done through multiplication (i.e rotation matrix), and you are just simplifying it further.
After all, the only application of imaginary numbers outside their definition is roots of a polynomial. And if you think of rotation+scaling as simple movement through the complex plane to get back to the real one, it makes perfect sense.
You can apply this principle generically as well. Say you have an operation on some ordered set S that produces elements in a smaller subset of S called S' It then follows that the inverse operation of elements of the complement of S' with respect to the original set S is undefined.
But you can create a system where you enhance the dimension of the original set with another set, giving the definition of that inverse operation for compliment of S'. And if that extra set also has ordering, then you are by definition doing something analogous to rotation+scaling.
It seems obvious now only because of significant mathematical discoveries of prominent mathematicians.
If one is taught what those discoveries revealed then of course they would seem obvious.
Arguing as you are, it would appear one can call all and every theorem in mathematics that connects to different fields as something obvious. They weren't, till someone proved the connection and that knowledge percolated down to how maths is taught, to text books.
That the integral of
over the entire real line is sqrt pi can be surprising or obvious depending on how you were taught. At face value it has nothing to do with circles, unless you are taught the connection or you are a mathematician of high calibre who can see it without being taught the background information.