is there a way to graph imaginary values on a cartesian coordinate system?

for that matter, is there any other way to graph them?

is there a way to graph imaginary values on a cartesian coordinate system?

for that matter, is there any other way to graph them?

ok, imaginary plane. y axis is replaced with the imaginary axis and the x axis is replaced with the real axis. how do you use it?

x axis has the non imaginary parts while the y axis has the imaginary parts.

 

So if you have a number 25 + 32i you would graph a point at (25, 32i).

 

Does that make sense?

so, in the imaginary plane, (x,y) is x+y?

 

how do you plot things like parabolas with imaginary roots(or is it zeros? i can't remember)?

You're extending your problem, so we'll extend the solution. Instead of 2 axes, you really need 2 axes PER variable. Parabolas require 2 variables, so your gunna need 2x2=4 axes to graph it properly this way. Since it's nearly imposible to think in 4 dimentions, I suggest you restrict one of the variables to the real numbers and graph the parabola as a curved plane in 3D space. For y<real> = x<complex>^2 you get a "saddle" centered at the origin (I'm pretty sure, but didn't really do it).

 

This may sound really complex, and that's the reason it's not done very often, it's just too difficult to visualize 4 dimentions.

so, imaginary planes are just used for points? no lines, conic sections, trigonometry, ...?

You may draw these things in the complex plane, but you are simply describing a set of complex numbers, unlike the real plane where you are describing a relation between pairs of real numbers.

 

You can have equations in complex variables such as w=cos(z) and so on but to draw them requires two complex planes, ie 4 dimensional real space.

 

Functions and "graphs" of complex variables are very important (Fermat's Last Theorem is proved by considering certain fucntions of complex vaiables called modular forms).

 

Loci of points in the complex plane are also of interest. Look up Julia Sets and Benoit Mandelbrot