TITLE: imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?
QUESTION [12 upvotes]: In another question here, about roots of equations being imaginary, the accepted answer 
said something interesting about "imaginary" (as a technical word in math) not meaning "not real".  I understand that imaginary numbers are "manipulable and usable to do all kinds of things".   And thus, "they exist" at least, in the realm of mathematics.
So my question is this. I was playing around in wxMaxima, and solving polynomials, to get their roots.  In maxima, $\sqrt{-1}$ is represented as %i.
Editor's note: all "%i"s have been replaced with $i$.
Here's an example of a polynomial, say $3x^6 + 4x^4$:
solve($3x^6 + 4x^4$,$x$);
Maxima's results:
[$x=\displaystyle -\frac{2i}{\sqrt{3}}$,$x=\displaystyle \frac{2i}{\sqrt{3}}$,$x=0$]
My exact question:

Is it common to have the same imaginary result twice with only the sign different (both q and -q are imaginary roots)?  (At first I thought it had reported the same root twice as I overlooked the negative sign.)
One can easily plot a real root and see how it intercepts the x axis.  How does one "visualize" or think of an imaginary root -- does it correspond to a local minimum that does not occur at the x axis? 
What uses do imaginary roots of polynomials have in math?

REPLY [5 votes]: You might want to take a look at this as well: Do complex numbers really exist?.
Additionally, you can feel proud of yourself for inquiring along the same lines (with the aid of a computer) that Cardano & other mathematicians did 500 years ago, which led to the acceptance and elaboration of %i. Your observation about $\pm i$ is perspicacious and important in the theory of differential equations (dynamical systems), as well as other places.


We think about $\sqrt{-1}$ as part of a larger entity called the complex plane (google is your friend). $\sqrt{-1}$ is 90 degrees around the complex plane, whereas $-1$ is 180 degrees around the complex plane -- a fully turned-around $+1$. The 180-degree turn is the number line you learned in school; going only 90 degrees enters the complex plane.
The unifying idea is that a pair of imaginary and "real" numbers $x + i \cdot y$ are equivalent to a radius and an angle. Simple enough if you're familiar with rectangular and polar coordinates. But the key observation by de Moivre which unlocked a deep and mysterious world is that this can all be done with the natural base $e$.

$\exp (i \cdot x) = \cos x + i \sin x$.

Therefore $x+\sqrt{-1} \cdot y = \sqrt{x^2 + y^2} \cdot \exp (\sqrt{-1} \cdot \arctan {y \over x})$, which we think of as 

$\mathrm{radius} \cdot e^{i \cdot \mathrm{angle}}$.

Here are some concrete examples of writing %i numbers in polar ($\exp$) form:

$3 + 4i = 5 \cdot e^{\pi \over 4}$
$-1 = e^{ \sqrt{-1} \cdot \pi }$
$\sqrt{-1} = e^{\sqrt{-1} \cdot {\pi \over 2}}$
$\sqrt{-25} = 5 \cdot e^{\sqrt{-1} \cdot {\pi \over 2}}$

As far as uses within mathematics, %i:

makes any polynomial equation soluble
matrices with %i model certain groups (google hermitian matrix)
complex eigenvalues tell you about the kind of solution to a (system of) differential equation(s)
helps model fluid flow
%i reveals more about the derivative (google amplitwist or Cauchy-Riemann equations)
%i reveals a fundamental to unity to the Moebius transformations (watch this)
you need %i for Fourier decomposition
the "angle" allows us to model the phase of a signal or quantum state as well as its magnitude
some previously non-integrable functions become integrable when you make use of the complex plane (google Laurent series or calculus of residues)
twistors and spinors in mathematical physics use %i
you can factor any polynomial all the way down using %i
the characteristic function of a probability distribution uses %i

If you've got the time, look at Tristan Needham's Visual Complex Analysis or Roger Penrose's The Road to Reality. Two of the best maths books that have been written, period. And they both show really cool things you can do with %i.
You're asking all the right questions to begin a magnificent journey of discovery.