Feynstein100 said:
Not that I'm saying I'm better than everyone else. It's just that I feel like I'm trying to describe the difference between colors to someone who's colorblind. It makes me wish I was colorblind too.
I'm sorry to disabuse you of the notion that you are seeing deeply into mathematics. You are seeing far less than those of us with mathematical training. You lack a basic understanding of abstract mathematical concepts. Your question about "real" and "imaginary" dimensions misses the point about how real and complex numbers relate to each other mathematically.
The real plane ##\mathbb R^2## and the complex plane ##\mathbb C## are
isomorphic when considered as
real vector spaces. I.e. vector spaces over ##\mathbb R##. In this case, we are simply considering a complex number (vector in the complex plane) as an
ordered pair of real numbers with no further structure or properties.
The complex plane, however, is usually considered not as a real vector space, but an
algebraic field. That is to say, we define multiplication of complex numbers, which (as
@Pikkugnome has pointed out) distinguishes ##\mathbb C## from ##\mathbb R^2##.
Note that a
set in mathematics can be given
algebraic and/or
analytic structure. This means that the same underlying set can be the basis of different mathematical objects. Algebraic generally refers to
addition and/or
multiplication and analytic refers to properties of some fundamental notion of
distance or
length. Ultimately, leading to the notion of
analytic topology underpinned by the notion of a
collection of open sets.
It's perfectly acceptable, therefore, to see both ##\mathbb R^2## and ##\mathbb C## as built on the same underlying set of ordered pairs of real numbers. The difference is that ##\mathbb C## is endowed with a rule for multiplication that produces a closed
field.
If you introduce the same multiplication rule for ##\mathbb R^2##, then you do not create a new, distinct mathematical object, but simply ##\mathbb C## in a different notion. This is called an
isomorphism. And, to claim that this somehow avoids introducing complex numbers misses the point. You have created an object with the same properties as the complex numbers and have the complex numbers in all but name.
In fact, an interesting and sometimes useful isomorphism of the complex numbers is to a subset of real ##2 \times 2## matrices:
$$a + ib \leftrightarrow
\begin{bmatrix}
a& b\\
-b&a
\end{bmatrix}$$Finally, if we define distance between two points in ##\mathbb R^2## by the usual Euclidean metric:$$|(x_1, y_1) - (x_2, y_2)| \equiv \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$And we define the distance between two compelx numbers as$$|z_1 - z_2| = \sqrt{(z_1 - z_2)(z_1 - z_2)^*}$$Then, purely as simple topological spaces, the two are isomorphic.
However, when we look at complex-valued functions of a complex variable, we move into
complex analysis, which has no analogue in real analysis or multi-dimensional real vector spaces.