Thread: complex roots of quadratic

Hi there,

I've been looking at complex numbers, really fascinating stuff!

While I understand the general idea behind complex numbers, I'm struggling to see how the complex roots of the quadratic work.

See image below of the plot of the quadratic, I've drawn a red point where the complex number lies, but what about the complex number ?

Am I looking at this the wrong way? What exactly do the complex roots mean in the solution of the above quadratic?



Looking forward to a response!

The complex roots are zeros of the polynomial: if you substitute them into the polynomial it would be equal to zero when evaluated. However, it's not so easy to visual complex roots, because you can't just plot them like you would two real roots. The complex numbers lie in the complex plane, but your parabola is graphed in the x-y plane. So you can't plot your roots in this case, at least not along with your parabola.

And for , it would lie four directly below , because in the complex plane the "x" axis is the real part and the "y" axis is the imaginary part of the complex number.

Thanks for the reply Chemboy!

Cool, yeah I understand that lies directly below , I obviously did it wrong. I assumed that I could take the to be the same as the imaginary axis...

How would I be able to visualize the complex roots? Any way?

Well, you can still certainly graph the complex roots like that and treat the plane you have like the x-y plane and the complex plane, but the graph of your parabola is in the x-y while the roots are in the complex plane so it won't really do you any good. There won't be a visual relationship between your quadratic and its roots when you do it that way. So if you want to picture where the roots lie in the complex plane you might as well graph them separately from the graph.

I don't know of a way to really visualize the complex roots in terms of where they lie geometrically in relation to the quadratic. Just considering that you're talking two different planes here, even though graphing in one is similar to graphing in the other, I'm not sure there is a way at all.

Sure that does make sense. Pity though

I've seen a lot of complex graphs done with color. To fully graph a complex function with 1 free variable, you need 4 dimensions x(real), x(imag), y(real) and y(imag). (y is a dependent variable, BTW.) So, to visualize this, you'll need some way of putting that on the screen.

Since you'll end up with some kind of 2D object in that 4D space, that can be done with projection. Another way would be to use color for the two y dimensions. So map some square's x to x(real) and its y to x(imag) then map y(real) to say blue and y(imag) to red. You'll get a square colored with reds, blues and purples that will represent your function over some complex interval. (BTW, if you look up some complex functions on Wikipedia or Mathworld, you'll sometimes see graphs like this, but maybe with different colors.)

Thanks for the reply MagiMaster. That certainly sounds interesting! Don't think I'm quite prepared to look into that just yet - I mean 4D space?

4-dimensional space doesn't really work any differently than 3-space, it's just difficult to do things in it because we can't visualize it, since we're stuck in 3-space (that's always been an annoyance of mine). And since complex numbers are 2-dimensional to begin with, you will never end up with any less than 4 dimensions when you have a function , since you need to represent your original 2-dimensional input complex number as well as your 2-dimensional output complex number, just as when we graph a simple function we plot an x-coordinate and a y-coordinate, and graph the points according to the input and the output.

Something that may interest you... The complex plane is of complex dimension 1, and of real dimension 2. So if you have the graph of a complex function in four dimensions of real space, as it must be, then it is in two complex dimensions. Makes sense, right?

Hey Chemboy,

Thanks for the reply! I totally get what you are saying and message certainly helped me gain a bit more understanding

Originally Posted by Chemboy

The complex plane is of complex dimension 1, and of real dimension 2.

That is indeed very interesting to think about