Comment by tsimionescu
9 hours ago
> Not quite. If it did, the -i-i would be i^2, not -1.
It is i², though, but that is equal to -1. Just like 22 = 2² = 4. I also maintain that the historical view is that this something extra comes from the properties of polynomials and their roots, not from geometric orthogonality.
> Exponentiation where you have i exponent however, is not defined solely in the complex field.
We can take another tack for defining the complex exponential function, if you'd prefer. One of the definitions of the exponential function is that e^x is the only function that respects f'(x) = f(x) (well, up to constant multiplication).
So, we need to look for a function f(z) such that f'(z) = f(z). There are various ways to do this (for example, using the Taylor series expansion and noting that all of the f derived n times factors are equal to f(z), which yields the power series definition). You don't need to appeal to limits of e^x to get there this way.
> Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
Understood, you are using a different sense of "algebraic" than what I was - I was thinking more of the abstract algebraic definition of C.
Still, the sense you are using seems to be the concept of algebraic numbers, which is more formalized - the algebraic numbers are all those that represent the root of a polynomial with integer or rational coefficients. Interestingly, while pi and e are not algebraic numbers, i or i+7 are still algebraic.
However, I'm not sure what the point of bringing this up is. Exponentiation is simply not defined over the algebraic numbers, especially not e^x where x is algebraic - so if we restrict ourselves to algebraic numbers, e^i is not defined, true, but neither is e^1. And while 2^2 is defined, of course, I'm not even sure you can define 2^sqrt(2), so I wouldn't be surprised if 2^i doesn't make sense either.
Either way, the algebraic numbers are not "a way of thinking about numbers", they are a restricted subset of what is generally meant by "number", and many famous and useful results from many branches of math do not work over this subset (for example, you can't even use the same set of algebraic numbers to refer to the length of a circle and the length of a square).
I still think you misunderstanding what im saying. i don't have a problem with the math working out, its that i have a problem with the math being able to be applied in the first place.
Let me try a different way.
Suppose you start with just the i number line with the rules that exist. You have zero, integer i's, rational is, and even irrational i's. All seems good. Then you start to define operations. i*i is undefined (because it goes to -1, and the reals are outside of your domain that you are working with currently). And this means you can't effectively do any sort of futher work in defining exponentiation, or limits, because you can have purely complex polynomials with just undefined terms.
So like you said, complex is R+, not something like RxR. The definition of complex numbers is intrinsic to real numbers - its an enhancement on real numbers. And by extension, all the math works out when you do taylor series with e^i and such.
But this pretty much means its a rigid definition, i.e you are defining something in a certain construction to supplement reals.
And as for geometric claim, my argument with that is that just like when you have x, and then you add <x,y> in some form and way, you are defining geometry. So in defining a+bi, you are defining geometry.*