in the complex plane...
it doesn't change i.
Its formula is defined as having no solution.

Mitchell Raemsch

On 7/20/2023 10:39 AM, mitchr...@gmail.com wrote:
> in the complex plane...
> it doesn't change i.
> Its formula is defined as having no solution.
>

i = the point 0+1i

We can plot it at the 2-ary point (0, 1). Notice the radius? Its one,
the unit.

Mitchr, you may think in branch the business into
https://en.wikipedia.org/wiki/Split-complex_number#Definition
https://en.wikipedia.org/wiki/Dual_number
not i, rather j or ε

Greetings ;^)

torsdag 20 juli 2023 kl. 19:39:34 UTC+2 skrev mitchr...@gmail.com:
> in the complex plane...
> it doesn't change i.
> Its formula is defined as having no solution.
>
> Mitchell Raemsch
i is defined as one of the solutions to x²+1=0.

On Friday, July 21, 2023 at 3:49:33 AM UTC-4, markus...@gmail.com wrote:
> torsdag 20 juli 2023 kl. 19:39:34 UTC+2 skrev mitchr...@gmail.com:
> > in the complex plane...
> > it doesn't change i.
> > Its formula is defined as having no solution.
> >
> > Mitchell Raemsch
> i is defined as one of the solutions to x²+1=0.

Allow me to explain, gentlemen.
As to what domain you polynomial expression is in: back in Descartes day he didn't even have the negative value, and so those roots which presented themselves in his polynomial studies were called 'false', though he was able to play sign games that would invert everything. Another class of roots, like the one above which he had no hope of solving he called 'imaginary'. Possibly I am attributing to Descartes what others of his time as well were working on, but this is how a reading of La Geometrie goes.

So there has been a four hundred year hiccup in mathematics. This came about due to the resistance of the introduction of the negative sign; that built the other half of the real number; that we today take as fundamental in so many constructions, which were refused by the pure mind which sought the pure number, without any additional glyphs attached. Upon establishing the two-signed system:
- a + a = 0, or even more simply -1+1=0
it should have been trivial to develop the three-signed system
- a + a * a = 0
but it either did not occur, or possibly it was buried. Here in this three-signed system we require the plane to admit the geometry, just as back in P2 we required the line. These two systems should not be confused with each other; they are not subsets as modern mathematics describes. They are each as fundamental as the other; merely in a series of such number systems:
P1 P2 P3 P4 ...
and so, as the original polynomial and its supposed space of operation: you can choose any of these. You can put a z in Pn and it will work algebraically. Yes, the dreaded zero divisors will occur, but so what: without them you won't have any claim on emergent spacetime. We actually needed that breakpoint:
P1 P2 P3 | P4 P5 P6 ...
and you've got the one-signed numbers; that magical unidirectional system that nobody else seems to get... why, exactly?
I'm pretty sure that humanity still has not gotten past the negative value. Philosophically the choice to reject it ought still to have some room, I would think. After all, by what right do you insert a binary discrete symbol ahead of a value that is meant to provide a continuous sense? Why, by golly, yes, that doesn't make much sense, does it? We could almost bottom out with a belief that the ray is more fundamental than the line here. Then too, what is all this new fancy vector stuff but rays? Built atop lines? Or built atop rays?

Returning to the front of the discussion it can be seen that in P3 the square root of minus one is plus one, but of course these are three-signed number, and so the meaning of this plus sign and this minus sign are different that the P2 version. There is no need to conjure up additional symbols ahead of your signs, and force this weird set growth that results out of a polynomial.

Is it a problem that there is no solution in P2? It is a fact, it seems, but that doesn't require that P2 not exist on its own. The polynomial is something that you plug these numbers into; not the other way around. I keep finding ways that modern mathematics has built a farce. May the farce be with you, and not with me. In Godzilla Wee Trust. Fake America, Fake Again.