TITLE: Is there an intuitive way of visualising complex roots?
QUESTION [36 upvotes]: Consider the function $f(x)$ such that $f(x) = x^2-4x+13$. By considering the discriminant, it can immediately be seen that the function has no real roots, since $b^2-4ac = (-4)^2-4(13) = -36$ and $-36 < 0 $, and hence the graph looks as follows



With no real roots, and hence the $x$-axis is never intercepted.
However, with knowledge of complex numbers, it can be seen that the roots are $$x=\frac{4\pm\sqrt{-36}}{2}$$ $$x_1=2+3i$$ $$x_2=2-3i$$ $$\implies (x-2-3i)(x-2+3i)=0$$
and the question on the table is, are there any means of visualising this fact? I know the closest we generally come to visualising complex numbers is argand diagrams, but I've thus far failed to recognise whether or not this can be used to visualise solutions such as these.
An example of a visualisation such as this enquires looking at real solutions, so consider when $$x^2-4x+13=\frac{1}{2}x+10$$
Here, the graph of $0=2x^2-9x+6$ indicates the solutions to the system of equations, but is there any way this example can be applied when the roots are complex?



Any help is appreciated. Thank you.

REPLY [5 votes]: If we define a polynomial $f(z) = az^2 +bz +c$ (where $z \in \mathbb C$),
then $$f(z) = a(z - p)^2 - \frac{\Delta}{4a},$$
where $p = -\frac{b}{2a}$ and $\Delta = b^2 - 4ac.$

If $a,$ $b,$ and $c$ are all real, then $p$ and $\Delta$ are real.
It follows that $f(x)$ is real if and only if $(z-p)^2$ is real.
This can happen in either of two ways:
$z-p$ is real, or $z - p$ is purely imaginary.
In the case where $z-p$ is real, we can set $z = x$ where $x\in\mathbb R,$
and you can plot $y = f(x)$ over $x\in\mathbb R$ as the question
has done for $a=1,$ $b=-4,$ and $c=13.$
We get a parabola with a vertex at the coordinates
$\left(p, - \frac{\Delta}{4a}\right).$
In the case where $z - p$ is purely imaginary,
we can set $z = p + ix$ where $i$ is the imaginary unit and
$x\in\mathbb R.$
Then 
\begin{align}
f(z) &= a(p + ix)^2 + b(p + ix) + c \\
&= ap^2 + i2apx + i^2 ax^2 + bp + ibx + c \\
&= \frac{b^2}{4a} - ibx - ax^2 - \frac{b^2}{2a} + ibx + c \\
&= - ax^2 - \frac{b^2}{4a} + c \\
&= - ax^2 - \frac{\Delta}{4a} \\
\end{align}
If we set $g(x) = - ax^2 - \frac{\Delta}{4a},$
we can plot $y = g(x)$ in a Cartesian plane;
it is a parabola with a vertex at 
$\left(0, - \frac{\Delta}{4a}\right),$
oriented in an opposite direction to the parabola we got
for real values of $z.$
So imagine that we want to plot $y = f(z)$ over the complex plane,
but only for the values of $f(z)$ that are purely real.
We can visualize this in just three dimensions:
two dimensions to plot the complex input value, $z,$
and one dimension to plot the real output value $y.$
If we write $z = u + iv$ we can describe this three-dimensional space
by the coordinates $(u,v,y).$
The resulting plot consists of just two parabolas:
the parabola for $y = f(x)$ where $x$ is real,
presented as the plot $y = f(u)$ in the $u,y$ plane;
and the parabola for $y = g(x),$ translated $p$ units in the
$u$ direction and rotated $90$ degrees around the line $u = p,$ $v=0$
(a line through $(p,0,0)$ parallel to the $y$ axis)
so that it lies in the plane $u = p.$
That is, the parabolas have the same vertex, 
$\left(p, 0, - \frac{\Delta}{4a}\right),$
and the same axis, the line $u=p,$ $v=0,$
but they are in perpendicular planes.
In another answer to this question,
this same plot is "flattened" into two dimensions by rotating
the "imaginary" parabola by $90$ degrees around its axis.
Now, depending on whether $\frac{\Delta}{4a}$ is positive or negative,
we have the common vertex of the parabolas either below or above the
$u,v$ plane, and we have either two zeros on the upper parabola or
two zeros on the lower parabola.
That translates to two zeros on the line $v=0,y=0$
(two real solutions of $f(z) = 0$)
or two zeros on the line $u=p,y=0$
(two conjugate complex solutions of $f(z) = 0$).
Which of these pairs corresponds to the upper parabola and which to the lower parabola depends on the sign of $a.$
Of course if $\frac{\Delta}{4a} = 0$ then we get the double root $z = p.$