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  • Re: About Hachel's alternate "complex" numbers

    From Richard Hachel@21:1/5 to All on Mon Mar 3 17:53:00 2025
    Le 03/03/2025 à 18:31, guido wugi a écrit :
    Op 1/03/2025 om 15:15 schreef Python:

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

    Absolutely not.


    A bit of a compensation for his absurd i-rules?

    I have no problem with Mr. Poincaré's transformations given for the first
    time in their positive form well before Mr. Einstein's plagiarism.
    I don't even have a problem extending them from uniform media to rotating media.

    <http://nemoweb.net/jntp?aR0_yJtT5efYPVxrttKBVvw5pAY@jntp/Data.Media:1>

    My ideas on the nature of the imaginary i are to be classified in the
    clarity and beauty of mathematics. What do mathematicians say? That
    i²=-1? What a great deal! They do not define i, but its square.

    This amounts to saying that we must define the number 3 by saying that "3
    is the number that is the cubic root of 27". It's not that it's wrong,
    it's that it's ridiculous. As ridiculous and childish as saying that a
    swallow is a swallow and that a square is not round.

    I define i as the unit whose exponent x will always make equal to -1.
    So it is something completely different from what mathematicians say.
    As for (-i) depending on the even or odd exponent, its value changes sign continuously. (-i)^8=-1 ; (-i)^9=+1, etc...
    That is what I said.

    <http://nemoweb.net/jntp?aR0_yJtT5efYPVxrttKBVvw5pAY@jntp/Data.Media:2>

    R.H.

    --
    <https://www.nemoweb.net/?DataID=aR0_yJtT5efYPVxrttKBVvw5pAY@jntp>

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  • From guido wugi@21:1/5 to All on Mon Mar 3 18:31:04 2025
    Op 1/03/2025 om 15:15 schreef Python:

    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)

    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.

    And indeed, it has! This is called the set of split-complex numbers: https://en.wikipedia.org/wiki/Split-complex_number

    I came across it while watching a video by Michael Penn: https://www.youtube.com/watch?v=r5mccK8mNw8

    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))

    Among these three, only the complex numbers form a field. All three
    also have a 2x2 matrix representation.

    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)

    which is:
    e^(j*theta) = cosh(theta) + j*sinh(theta)

    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?

    --
    guido wugi

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  • From Python@21:1/5 to All on Mon Mar 3 21:20:01 2025
    Le 03/03/2025 à 18:53, Richard Hachel a écrit :
    Le 03/03/2025 à 18:31, guido wugi a écrit :
    Op 1/03/2025 om 15:15 schreef Python:

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

    Absolutely not.


    A bit of a compensation for his absurd i-rules?

    I have no problem with Mr. Poincaré's transformations given for the first time
    in their positive form well before Mr. Einstein's plagiarism.

    There is no plagiarism. Both Einstein and Poincaré would kick your ass together if they could encounter you.

    The irony is that your first ideas, when it comes to "complex" numbers (aa'+bb', ab'+a'b), is... the mathematical structure of Minkowki
    space-time. That you despise for some reason without any understanding of
    what it is (same with Einstein and Poincaré, btw, you don't understand a single line of their work).

    My ideas on the nature of the imaginary i are to be classified in the clarity and beauty of mathematics. What do mathematicians say? That i²=-1? What a great
    deal! They do not define i, but its square.

    Same lie again and again Richard? Mathematicians define C first, then
    define i as a specific member of this set and *then* they demonstrate that
    i^2 = -1.

    I, and others, have shown you what these definitions are. You are a damn hypocrite.

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