Finally I think I have a chance of not just appreciating, but actually understanding, the most beautiful equation.
https://www.youtube.com/watch?v=f8CXG7dS-D0
BTW, The Welch Labs YouTube series on imaginary numbers is still the best I've come across.
This complex number book is based on the innovative work of deriving the imaginary unit and Euler formula from first principles. Complex numbers formed from imaginary numbers have widespread applications, but have remained puzzling to many learners. The work opens up the opportunity to explore, extend and share with the general public the newly gained comprehensive understanding of complex numbers for the thorough demystificatio, popularization and empowerment to broader educational levels.
just in case anyone is interested in how to do complex linear algebra, including complex tensor products, without using complex numbers - using real linear operators \(J\) with \(J\circ J=-I\) instead of complex scalars - here are some links.
The TLDR version on a poster:
http://mwaa.math.indianapolis.iu.edu/Slides/coffman.pdf
Using a basis, matrices, and summing over indices - sections 4&5 of this paper:
https://users.pfw.edu/CoffmanA/pdf/basis1.pdf
Without a basis (but still a lot of notation) - Chapter 5 starting on page 195, with tensor products starting with Example 5.74 on page 208:
https://users.pfw.edu/CoffmanA/pdf/book.pdf
#LinearAlgebra #ComplexNumbers #NotASubToot
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new representations are shown to be unique, and to have useful properties. They also admit a geometric cutting sequence interpretation.
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Decagon (fractal version)
\(z_{n+1}=fold(z_n)^2+c\)
where fold is a generalized absolute value function. A complex number has two components: a real and an imaginary part.
If we take the absolute value of one of these parts, we can interpret this as a fold in the complex plane. For example, |re(z)| causes a fold of the complex plane around the imaginary axis, which means that the left half ends up on the right half. If we do this for the imaginary component |im(z)|, we fold the complex plane around the real axis which means that the bottom half ends up on the top half.
These two operations are quite similar, because the imaginary fold is just like the real fold of the plane, except that it was previously rotated 90 degrees (z * i). But what if we rotate the plane by an arbitrary number of degrees?
An arbitrary rotation of the complex plane can be expressed as rot(z, radians) = z * (cos(radians) + sin(radians) * i), where radians encodes the rotation.
The image here is produced, by rotating the plane exactly five times, and folding the imaginary part each time.
I found this algorithm in the Fractal Formus under the name “Correction for the Infinite Burning Ship Fractal Algorithm”.
It can be seen as a generalization of the burning ship obtained by folding the complex plane twice with a rotation of 90 degrees, i.e. folding both the real and the imaginary part.
#fractalfriday #fractal #burningship #mandelbrot #complexplane #complexnumbers #mathart #math #escapetimefractals
Some relationships just don’t add up.
But in math? It’s all about real and imaginary compatibility.
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Is Nature complex or hypercomplex?
#quantumphysics #quantummechanics #complexnumbers #research #science
🤓
https://phys.org/news/2025-03-quantum-mechanics-hypercomplex-complex.html
Today, physicists are still asking themselves whether quantum mechanics needs hypercomplex numbers. FAU researchers Ece Ipek Saruhan, Prof. Dr. Joachim von Zanthier and Dr. Marc Oliver Pleinert have been investigating this question in their research.
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