About Hachel's alternate "complex" numbers

[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:

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.

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.

It is also completely absurd to claim, as he did before shifting to
another incoherent idea (i^4 = i^2 = -1 ), that one of these three
structures would be the "correct" one while the others are "wrong".

To make a long story short, Hachel had an idea that, for once, was not
absurd. Strangely, he abandoned it in favor of another one that is
completely incoherent and contradictory.

[guido wugi]

Post by Python
I did a bit of research to see if the structure that Hachel originally
(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.
https://en.wikipedia.org/wiki/Split-complex_number
http://youtu.be/r5mccK8mNw8
He demonstrates there that there are only three associative R-algebras
- 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 :)

Post by Python
e^(i*theta) = cos(theta) + i*sin(theta)
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?

[Richard Hachel]

Post by guido wugi
I think he has problems with Lorentz features as well :)

Absolutely not.

Post by guido wugi
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?***@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?***@jntp/Data.Media:2>

R.H.

[Chris M. Thomasson]

Post by Richard Hachel

Post by guido wugi
I think he has problems with Lorentz features as well :)

Absolutely not.

Post by guido wugi
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.
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.

Saying i^2=-1 does define i? Also,

i = (0, 1)

or:

i = 0+1i

It's a unit going up the positive y axis.

where i^2 is:

(-1, 0) or -1+0i

That is a definition?

(0+1i)^2 = (-1+0i)

Post by Richard Hachel
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.
R.H.

[Richard Hachel]

Post by Chris M. Thomasson

What is unit i?

It's a unit going up the positive y axis.

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

Mon pétage de plomb n'est pas loin.

Continuez les mecs, continuez...

Si je fais sauter Notre-Dame à la nano-thermite, faudra pas venir
chialer si on vous en tient pour responsable, et si vous vous retrouvez
dans le box des co-inculpés, pour incitation au terrorisme, sur les
forums usenet.



R.H.

[Python]

Post by Richard Hachel

Post by Chris M. Thomasson

What is unit i?

It's a unit going up the positive y axis.

Mon pétage de plomb n'est pas loin.

Il a commencé il y a bien longtemps.

Post by Richard Hachel
Continuez les mecs, continuez...
Si je fais sauter Notre-Dame à la nano-thermite, faudra pas venir chialer si
on vous en tient pour responsable, et si vous vous retrouvez dans le box des
co-inculpés, pour incitation au terrorisme, sur les forums usenet.

Nous ne sommes pas responsables de ton total manque de logique et de
réflexion.

Chris travaille avec des nombres complexes dans le domaine du traitement
d'images planes. Pour ses applications en imagerie numérique la
définition de i en termes strictement géométriques (où la
multiplication par z = r*exp(i*theta) correspond à une combinaison d'une
rotation d'angle theta et d’homothétie de coefficient r) lui suffit
entièrement, et permet de faire tous les calculs pertinent sur un
ordinateur.

Un algébriste préférera la définition en terme de classes
d'équivalence de polynôme : R[X]/(X^2 + 1), ou encore en terme de
matrices 2x2 (idem en physique).

Il se trouve que *toutes* ces définitions sont strictement équivalentes,
c'est quelque chose qui se *démontre*.

Tu es comme un cul-de-jatte qui prétend pouvoir gagner le 100m, même pas
contre Usain Bolt, contre n'importe quel amateur de Saint Izan. Tes
chances sont faibles. D'ailleurs la course est finie, le stade est fermé
et tu continues à ramper comme un con sur la piste.

[Chris M. Thomasson]

Post by Richard Hachel
Mon pétage de plomb n'est pas loin.
Continuez les mecs, continuez...
Si je fais sauter Notre-Dame à la nano-thermite, faudra pas venir
chialer si on vous en tient pour responsable, et si vous vous retrouvez
dans le box des co-inculpés, pour incitation au terrorisme, sur les
forums usenet.

WTF! ;^o

[Python]

Post by Richard Hachel

Post by guido wugi
I think he has problems with Lorentz features as well :)

Absolutely not.

Post by guido wugi
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).

Post by Richard Hachel
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.

[Chris M. Thomasson]

Post by Python
I did a bit of research to see if the structure that Hachel originally
(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.
https://en.wikipedia.org/wiki/Split-complex_number

[...]

I wonder what Hachel thinks about the triplex numbers... His mind might
explode.