
Where the complex points are (2016) - ColinWright
https://blogs.adelaide.edu.au/maths-learning/2016/08/05/where-the-complex-points-are/
======
ohazi
I find this less intuitive than simply looking at the real component,
imaginary component, magnitude, and phase plots of simple, and then
progressively more complicated functions of interest.

For example, starting with y = x^2+1:

[https://www.wolframalpha.com/input/?i=plot+z%5E2+%2B+1](https://www.wolframalpha.com/input/?i=plot+z%5E2+%2B+1)

You can see that the real component has a hyperbolic parabaloid shape. The
imaginary component does too, just rotated by 45 degrees. This shape clearly
has zeroes, since it's continuous and has infinite range. You can try to look
for spots where both components are zero, or you can plot the magnitude:

[https://www.wolframalpha.com/input/?i=plot+Magnitude%28z%5E2...](https://www.wolframalpha.com/input/?i=plot+Magnitude%28z%5E2+%2B+1%29+from+-2i+to+2i)

You can then clearly see that the zeros are "out" and below the global minimum
of the real plot.

You can only really look at a slice at a time, but I find this easier than the
crumpled-paper-at-every-point approach, because it's a lot harder to see
patterns emerge in nearby bits of crumpled paper.

------
cmehdy
It's always appreciable to try to comprehend stuff through new ways and tools,
but I fail to see the aim of the post. Isn't it a matter of properly defining
the domain of application and associating a way to visualize it?

Representing the result of a function from R to R requires two orthogonal axes
because the element pre-transformation is 1-dimensional and the result is also
1-dimensional.

For C to R, this would require 2+1=3 orthogonal axes, so it can be visualized
with a 3D representation. Likewise for R to C.

From C to C that would be 4-dimensional and becomes already trickier without
some effort to conceptualize it, and certainly becomes less intuitive without
resorting to alternate ways to conceptualize dimensions.

It quite probable that beyond that, one would really encounter decreasing
returns on trying to visualize the situation because the cost of abstraction
would increase in order to rely on multisensorial approaches to compensate for
our inability to visually perceive much beyond 3D.

It's entirely possible that the whole post flew way over my head and I
absolutely did not get it though, in which case I am truly just a confused
commenter.

~~~
perl4ever
One of the things that is painful because it's out of reach for me
intellectually and yet tantalizingly straightforward sounding is extending the
concept of an n-dimensional space to infinite dimensions.

The idea of a mathematical function of a function doesn't sound like a big
deal; it sounds vaguely similar to mundane abstractions in programming. This
sort of thing is beyond my ability to cope with and yet it sounds like
counting 1, 2, 3, not like abracadabra...

[https://en.wikipedia.org/wiki/Functional_analysis](https://en.wikipedia.org/wiki/Functional_analysis)
[https://en.wikipedia.org/wiki/Hilbert_space](https://en.wikipedia.org/wiki/Hilbert_space)

~~~
JadeNB
One of the very important things to know is that, while an infinite-
dimensional, separable Hilbert space is basically "n-dimensional space, but
more so", there are more general kinds of infinite-dimensional spaces (I'm
thinking of Banach spaces, but you can certainly get still more general than
that) that are much more general than that. The prototypical examples of these
are the L^p spaces, where p stands for ∞, or a real number p ≥ 1—but p _isn
't_ the dimension, as in ℝ^n (they're all infinite dimensional, at least for a
reasonable underlying metric space). Rather, it's a parameter that controls,
in some sense, how close the geometry is to being governed by the Pythagorean
theorem, so that only p = 2 gives (a Hilbert space, and hence) the 'usual'
geometry.

I think your point with functional programming is spot on. Just as one learns,
to pick that bête noire, monads not by reading yet another clever re-packaging
that somehow only manages to make them sounds _more_ difficult, but rather by
finding a problem for which they're relevant and getting a feel for them by
using them, so too does one learn about infinite-dimensional space not by
treating it as some sort of philosophical profundity, but by finding a problem
for which it's the right setting, and realising that it's just a mathematical
tool like any other.

------
btilly
I think about it quite differently.

A function in the complex plane is a 4-dimensional thing. A graph of that
function on the real plane is a 2-D slice of that 4-D thing.

That said, the one visualization of complex numbers that I wish more
understood was a complex number in polar coordinates. In polar coordinates,
addition is complicated. But multiplication is simple. Every complex number is
a magnitude and an angle. You multiply the magnitudes and add the angles.

What this means is that -1 is (1, 180 degrees). Literally a turn halfway
around the circle. And now what are its square roots? Well i is (1, 90
degrees) and -i is (1, -90 degrees). Now stand up and actually do those turns.

The result is that i is a turning motion that takes you off the real line. But
that visualization helps build intuition about why in the complex plane there
should be a close connection between exponential functions and sin/cos.
(Specifically e^(ix) = cos(x) + i sin(x) - in other words it is a turn by x
radians.)

~~~
gliese1337
I'm pretty sure that is in fact _the same_ way of thinking about it. A 2D
space with extra 2D spaces attached at every point _is_ a 4D space.

------
elteto
Not a mathematician so correct away:

I think the initial confusion stemmed from not pinning down the domain of x
and y.

When he says “I don’t see the complex points” in the xy graph that is because
they aren’t there: the xy graph is in R^2 and x^2 + 1 = y has no solution in
the reals for y = 0.

But if you imagine, as he did, an extra dimension attached at each point then
that is now a complex space, is it not?

~~~
potiuper
The domains of x and y both appear defined as the real numbers. But, the reals
do not have a root for every non-constant polynomial defined in terms of x or
y as the real numbers are an not algebraically closed field. The proposed
concept attaches two extra (complex) dimensions to each two dimensional real
point. This is close to visualizing R^4 by attaching an R^2 to each point of
R^2, so in some sense the tangent bundle of R^2.

------
galaxyLogic
Does it matter what is the orientation of the "iPlane"?

