
Visualizing Complex Functions - vankessel
https://vankessel.io/blog/2019/01/06/visualizing-complex-functions
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Jedi72
If I had a dollar for every explanation of complex numbers that is basically
just "A complex number is a real number plus an imaginary component, where i =
sqrt(1)" I would almost have enough money to go back to uni and study math.
It's far enough through the curriculum that most people get through the class
by symbolic pattern matching and algorithmic question-answering rather than
actual understanding (I studied EE), and I am pretty certain that even most
(note, most) professors only really understand them as a quirky trick of our
mathematical system. Ask them to explain it in English, not just as a
mathematical definition, they all seem to come unstuck.

This old but very good lecture series really helped me - I can at least accept
that complex numbers are not some ficticious hack now - but I confess I still
dont have an intuitive grasp of complex numbers.

[https://youtu.be/BOx8LRyr8mU](https://youtu.be/BOx8LRyr8mU)

~~~
enriquto
a complex number is a point in the plane (in the same way as a real number is
a point in a straight line)

~~~
adrian_b
While it is true that 2 real numbers can determine both a complex number and a
point in plane and it is also true that 1 real number can determine both a
real number and a point in a straight line, these are different mathematical
things.

The correct view is that the real numbers, also known as scalars, are
quotients of 1-dimensional vectors, while the complex numbers are quotients of
2-dimensional vectors.

This means that given two 1-dimensional vectors, the second being non-null,
there is always a real number by which you can multiply the second vector to
obtain the first vector.

The same for two 2-dimensional vectors and a complex number. In this case the
magnitude of the complex number changes the magnitude of the vector, while the
phase rotates the vector.

When you understand this fact about complex numbers, than it becomes obvious
that the imaginary unit is not imaginary but just a rotation with a right
angle and it is trivial that its square, i.e. a rotation with 180 degrees is
equivalent with a multiplication by -1.

The point are a third, different kind of mathematical entities, distinct from
both real & complex numbers and from vectors.

In fact points (i.e. members of 1-dimensional, 2-dimensional and so on affine
spaces) are the primitive objects.

From points you can define the vectors as differences of points (i,e,
translations). Then from vectors you can define real numbers, complex numbers,
quaternions and higher-dimensional matrices as quotients of vectors.

(Such definitions define e.g. a 2-dimensional vector as a class of equivalence
of pairs of points in plane that are transformed into each other by a
translation, and a complex number as a class of equivalence of pairs of
2-dimensional vectors which are transformed into each other by a proportional
change in magnitude and a rotation with a fixed angle.)

Obviously, my explanation here is very simplified, but it is useful to be
aware that e.g. a point in plane, a 2-dimensional vector and a complex number
are 3 very different kinds of mathematical objects, even if all 3 are
determined by a pair of real numbers.

They are very different because the set of operations that can be applied to
each is different. For example only the complex numbers are members of a
field. You can add 2-dimensional vectors (i.e. compose 2 translations), but
you cannot add 2 points from a plane.

~~~
ttoinou
How do you define a 1D vector then ?

~~~
adrian_b
Exactly as I have written above, a 1D vector is the difference between 2
points on a straight line.

In most mathematical handbooks it is now traditional to define first the real
numbers deriving them from integer numbers, via rational numbers, without any
geometric interpretations, then to define the straight line and the mapping
between the real numbers and the straight line. Then usually the vectors are
derived from tuples of real numbers.

I believe that this methodology is very wrong because it occults the real
meaning and usefulness of vectors, real numbers, complex numbers and of many
other important mathematical entities. This sequence of the derivations of the
main mathematical objects leads to many confusions and mistakes and it also
does not correspond with the historical development of mathematics, where the
real numbers, that is the "measures", as they were initially called, were
indeed obtained since the earliest times as quotients of differences between
points, i.e. as quotients of vectors (e.g. the quotient between a measured
length and a standard length, e.g. a foot), and not as some limits of rational
sequences.

It is perfectly possible and much clearer in my opinion, to start from axioms
of the affine spaces (i.e. the spaces of points) and to derive the real
numbers (and everything else in geometry) from that.

The result that the real numbers correspond to limits of rational sequences is
very important in practice, but it is not the motivation for the introduction
of real numbers. Why would you believe a priori that the limits of rational
number sequences are of any interest for you?

The points are interesting, because they model your environment. Then the
vectors and then the real numbers, complex numbers, matrices etc. become
interesting to be able to model the geometric transformations of the points.

~~~
ttoinou
So you're defining rational number, not reals. For your explanation, why not,
I also like visual exploration like this

------
ttoinou
The images are missing a way to identify zeroes and poles. Here are my takes
on this subject with this shader "Complex Maps"
[https://www.shadertoy.com/view/Ms2Bz3](https://www.shadertoy.com/view/Ms2Bz3)

And this video from a few years ago "Obama deformed by holomorphic complex
functions (conformal map)"
[https://www.youtube.com/watch?v=CMMrEDIFPZY](https://www.youtube.com/watch?v=CMMrEDIFPZY)

------
newprint
www.visual.wegert.com is beautiful yearly calendar of visualizations of
Complex functions. There is also an entire book devoted to exploring C.A.
using visualization.

