Can you please fill me in on more useful practical applications of
complex numbers than just the Mandelbrot visualization?
Johann 'Myrkraverk' Oskarsson <johann@myrkraverk.invalid> wrote or quoted:
Can you please fill me in on more useful practical applications of
complex numbers than just the Mandelbrot visualization?
Physics is full of them, especially in quantum theory.
The following example does not actually require quantum theory:
A /glass plate/ can shift the phase of a light beam travelling
through it. A plate with a certain thickness d will change the
amplitude from a to -a.
Change of amplitude from a to -a
Before
| .....---...
| ...'' ''.. ..
| ..' ''.. ..'
| ..'' `'. ..'
|.' `'. ..' |-------------------------------`----------------------------.''--------
| `'.. ..'
| `.. ..'
| ''.. ...'
| ''-.......''
After
|
| .....'--...
| ..'' '''..
| ..'' ''.
| ..' `'.
| ..' '.. |-.---------------------------:'-------------------------------`--------
| `'. ..' `'.
| `'.. ..' `'..
| ''.. ...'' ''
| ''--......''
|
Introducing /two/ such plates, we get -(-a), which is the original
wave again.
Each plate multiplies the amplitude by -1, and their combination has
the effect of /two/ multiplications by -1, i.e., of (-1)*(-1)=1.
So, a plate of thickness d multiplies by -1.
If we combine /two/ plates of thickness d/2 we still get a multipli-
cation by -1. So what does /one/ plate of thickness d/2 multiply
the amplitude with?
Dear sci.math,
Can you please fill me in on more useful practical applications of
complex numbers than just the Mandelbrot visualization?
I was just reading about them (again) in the /Linear Algebra Done
Right/ book, but it seems the first chapter doesn't have an example
of a practical applications.-a And I'm out of touch with higher
mathematics.
Thank you.
Johann 'Myrkraverk' Oskarsson <johann@myrkraverk.invalid> wrote or quoted:
This is some beautiful ASCII art rendering of waves. Do you have a tool
that does this, or do you keep this pre-rendered in a text file for just
such occasions?
I've written a Python script that renders pixels to an array
and then tries to match rectangles with such pixels to ASCII
characters; it is using a specific raster font. The raw results
of this approach did not look very good, and I found out that
I can improve the result by restricting the set of characters to
just a few selected characters like ".". Also, I take the slope
of the curve into account. For example, the downward moving
accent "`" is only used where the curve does move downward with
approximately this angle (as can be seen in the sine plots).
But this Python script is not yet ready for publication. I also
edited two characters of the plots manually in my previous post.
If we combine /two/ plates of thickness d/2 we still get a multipli-I have a feeling the answer should be /i/, but I'm not sure. Feel free
cation by -1. So what does /one/ plate of thickness d/2 multiply
the amplitude with?
to recommend books, websites, or PDF files where I can brush up on light
physics.
This example was taken from a book about quantum physics that is
as easy and readable as a book about this topic can possibly be:
"Quantum Processes, Systems, and Information" (2010) -
Benjamin Schumacher and Michael D. Westmoreland.
(Schumacher is known for his coinage of the word "qubit".)
The authors write in section 2.1:
|Glass plates can be made in a continuous range of thicknesses,
|producing a continuous range of phase shifts. For this to be
|possible, the beam phases a must be complex quantities, with
|both real and imaginary parts. A plate with thickness d/2 may
|multiply the amplitude by a factor of i = sqrt reA1. This does not
|change the magnitude of the complex phase a, since |a| = |ia|.
|Two such plates (or a single plate of thickness d) multiply
|the phase by i^2 = reA1, as required.
Dear sci.math,
Can you please fill me in on more useful practical applications of
complex numbers than just the Mandelbrot visualization?
I was just reading about them (again) in the /Linear Algebra Done
Right/ book, but it seems the first chapter doesn't have an example
of a practical applications. And I'm out of touch with higher
mathematics.
Thank you.
On 08/01/2026 05:30 AM, Johann 'Myrkraverk' Oskarsson wrote:
Dear sci.math,
Can you please fill me in on more useful practical applications of
complex numbers than just the Mandelbrot visualization?
I was just reading about them (again) in the /Linear Algebra Done
Right/ book, but it seems the first chapter doesn't have an example
of a practical applications. And I'm out of touch with higher
mathematics.
Thank you.
Complex numbers with deMoivre from trigonometry then Euler's theorem
which is after telescoping limits then "the Gaussian" make for that
the deMoivre-Euler-Gauss is used a lot in the things like contour
integrals and Gauss-Bonnet-Ostogradsky, and about the roots of unity,
has that what they do is agree on the diagram of the complex plane, C,
on R^2, the two-dimensional real plane with a Cartesian basis. So,
other ways to do this include starting with the Cartanian, after
Elie Cartan makes for reflections and rotations, where this is then
often directly, yet not necessarily, connected to the complex numbers
then later the hypercomplex numbers and Clifford algebras and geometric algebras, point being the reflect the geometric about reflections
and rotations, the Cartanian. Then, the accounts of Argand and Wessell
(sp.) planes, if complex numbers are around some Bombelli, usually are
said to agree, yet have different motivations and perspectives.
Complex numbers have one aspect that's non-Gaussian, it's that
division, the operation, is not unique, and admits two quotients,
here called "left-complex" and right-complex", that, the Gaussian
branch of division for complex analysis is usual, yet not unique.
So, complex analysis is useful, yet not necessary, per se. Then,
it's ubiquitous, yet, also open, or a multiplicity, about division,
so all the derivations have implicit openings and are only closed
by convention.
This is then sometimes summed up as about "complex-analyticity",
that measure and the analytical basis exists, since "real-analyticity",
is about measure and its relation to the geometric, as the "uniqueness
up to isomorphism of the complete ordered field", including both
R and C as complete ordered fields, when that, as mentioned above,
division is not unique. Then also there are counterexamples of
complete ordered fields with field operations in [-1, 1], which
makes another account of analytical basis.
So, accounts of complex-analyticity, are open at least two ways,
then that's sometimes called "almost-analytic", which is "not quite",
then usually called good enough and ignored, which doesn't satisfy
everyone's requirements called rigor.
Then, often used in functional analysis, for example about Hilbert
space, for convenience, since after Euler's identity then in the
differential that members of C are in terms of e, about the differential
and e and addition-formulae and so on and Euler's identity involving e
and pi relating to trigonometry and involving e, pi, and -1
involving subtraction-formulae, is pretty much what it's about.
Johann 'Myrkraverk' Oskarsson wrote:
Dear sci.math,
Can you please fill me in on more useful practical applications of
complex numbers than just the Mandelbrot visualization?
I was just reading about them (again) in the /Linear Algebra Done
Right/ book, but it seems the first chapter doesn't have an example
of a practical applications.-a And I'm out of touch with higher
mathematics.
Thank you.
https://study.com/cimages/multimages/16/complex-number-representation1439546104989929617.png
Complex numbers turn hard AC (alternating current) circuit math into
simple algebra by tracking both signal size and timing shifts (phase) at
the same time. Engineers use the symbol \(j\) instead of \(i\) for the imaginary unit so it does not get mixed up with current (\(i\)).
Key Concepts
The \(j\)-operator: Represents a 90-degree phase shift. Multiplying by
\(j\) turns a value along the vertical axis, matching how inductors and capacitors offset waves.
Impedance (\(Z\)): Combines resistance (\(R\), real part) and reactance (\(X\), imaginary part) into a single complex value: \(Z = R + jX\).
Forms of Expression:
-a- Rectangular Form: \(A + jB\) (best for adding and subtracting).
-a- Polar/Exponential Form: \(Magnitude \angle \theta\) or \(Ae^{j\theta }\) (best for multiplying and dividing, like using Ohm's Law with AC).
Main Uses
AC Circuit Analysis: Changes messy differential calculus equations into
easy linear algebra using complex impedances.
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