Re: About Hachel's alternate "complex" numbers

Op 1/03/2025 om 15:15 schreef Python:

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I did a bit of research to see if the structure that Hachel originally proposed, with these multiplication rules:
(a, b) * (a', b') = (aa' + bb', ab' + a'b)
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had already been studied. Since it is clearly a ring (but not a field, as it has divisors of zero), it seemed likely to me.
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And indeed, it has! This is called the set of split-complex numbers:
https://en.wikipedia.org/wiki/Split-complex_number
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I came across it while watching a video by Michael Penn:
https://www.youtube.com/watch?v=r5mccK8mNw8
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He demonstrates there that there are only three associative R-algebras over R^2:
- Dual numbers R(epsilon) with epsilon^2 = 0 (i.e. R[X]/(X^2))  - Complex numbers R(i) with i^2 = -1 \) (i.e. R[X]/(X^2 + 1)  - Split-complex numbers R(j) with j^2 = 1 (i.e.R[X]/(X^2 - 1))
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Among these three, only the complex numbers form a field. All three also have a 2x2 matrix representation.
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What should please Hachel is that split-complex numbers naturally express Lorentz transformations, since their isometries are hyperbolic rotations.

I think he has problems with Lorentz features as well :)

There is even an analogue to Euler’s identity:
e^(i*theta) = cos(theta) + i*sin(theta)
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which is:
e^(j*theta) = cosh(theta) + j*sinh(theta)
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However, note that while R(j) corresponds to Hachel’s *first* proposed structure, it has *nothing to do* with his *second* proposal of introducing an element such that (i^2 = i^4 = -1 ). As was pointed out to him (both here and on fr.sci.maths), this immediately leads to contradictions.

At least there are idempotent numbers e and e* so that e=ee (=eee...) and e* ditto. A bit of a compensation for his absurd i-rules?
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