[Coda]

Originally Posted by Potironette

Ohh, so f(x) means, the function where real number x is put into it, whereas f(x + yi) means the function where a complex number is put into it, in which the real number is on the x axis and the imaginary numbers are represented by the y axis?

Yup, bingo.

So..basically for things like y=x² you can't plot imaginary numbers because when every x is an imaginary, x squared would be a real and thus not on a graph where both axis are imaginary..? But then y = x would be graphable and maybe y = x³ would be too..? But then there is a way to plot it--I think I'm misunderstanding something :/

You can of course find specific examples that you would be able to graph. (And yes, I think x³ works, because i³ = -i. The graph ends up being upside down relative to the real-valued version.)

Geometrically, this means you're taking that four-dimensional graph and taking a cross-section of it.

Err, what's the difference between a curve and a surface? In fact, what is a curve and what is a surface?

Whee, this starts getting into some crazy abstract geometry, and I've gotta be careful in defining this or else we're going to dive into another rabbit hole. XD

Okay, so... A point is a zero-dimensional object. No matter how many dimensions of space you have, a point in it is infinitely small.

A curve is a stretched-out point. It's infinitely thin, but it has a defined length. Imagine an infinitely thin piece of wire. It can bend around in however many dimensions you have available in the model -- in a one-dimensional system (for example, the real number line) all curves are straight line segments, but in a two-dimensional system (a graph), curves can bend around as long as they stay flat (because if they didn't stay flat then they wouldn't fit in the two-dimensional space). In a three-dimensional system (space), a curve can bend around in all three dimensions, et cetera. We say that a curve is a one-dimensional object embedded in an n-dimensional space.

A surface is a stretched-out curve. It has a defined area, but it's infinitely shallow, so it has no volume. Imagine a rubber balloon -- you can stretch it and bend it into any shape you want, but the rubber itself is still two-dimensional. It gets hard to visualize taking this into four dimensions, but you CAN bend it in four dimensions, and that's what counts here.

Anything higher gets called a hypersurface or an n-surface as long as it's embedded in a space with at least one dimension higher than itself. If it's not, then it's called a hypersolid / n-solid (or just a solid, if it's in three dimensions), because it's completely space-filling. (Yes, you could call a two-dimensional object in a two-dimensional space a 2-solid.)

I'm... pretty good at higher-dimensional geometry. >.> I have a reasonably easy time visualizing 2-surfaces embedded in 4-space. Klein bottles are fun. So are tesseracts. (Though tesseracts are 4-solids, not 2-surfaces.)