Re: New way of dealing with complex numbers (roots of zero)

On 03/07/2025 05:23 AM, Ross Finlayson wrote:

On 03/07/2025 04:04 AM, Richard Hachel wrote:

Le 07/03/2025 à 12:50, Alan Mackenzie a écrit :

Richard Hachel <r.hachel@tiscali.fr> wrote:

Does my new way of dealing with complex numbers bring an advantage or
is it stupid?

>
It's stupid.  Your "complex numbers" are not complex numbers.  They're
something else altogether.  The term "complex number" has a meaning in
mathematics, science, engineering, etc., and you are being deliberately
ignorant of that meaning.
>

The goal is to find something more coherent (it already is), but above
all more useful.

>
You have failed at that goal.
>

I recall the idea: the imaginary number i is a unit such that it is
invariant whatever its power. For all x, we have i^x=-1.

>
That is just ignorance.  It's not even clear what you mean by the above.
Your ^ operator clearly has nothing to do with multiplicative powers.
>

This was already the case with the real unit n=1. For all x, 1^x=1.

>
And for all x apart from 0, x^0 = 1.  That includes i^0 = 1.
>

In this sense, not only is i²=-1 true, i^(1/2)=-1, but also i^4=-1,
i^5689=-1, i^(-3/2)=-1.

>
That isn't sense, it's nonsense.
>

Second, the real or complex roots of quadratic equations are:

>
They are well understood by virtually everybody but you, and have been
for many centuries.
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[ .... ]
>

Exemple : Roots of f(x)=x²+4x+5 ---> x'=i, x"=-5i

>
Wrong.
>

Finally, the complex roots of a function are the real roots of the
function in point symmetry $(0,y), and vice versa.

>
That's meaningless gibberish.
>

R.H.

>
I see you didn't understand anything.
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But it doesn't matter.
>
For those who are more open-minded and less stupid than Alan:
>
I'm not saying that mathematicians treat things like that. I'm saying
that I treat things like that.
Everyone does what they want.
>
Nothing prevents mathematicians from proposing their ideas, nothing
prevents me from proposing mine (validated in logic by AI).
Mathematicians pose i²=-1 and sqrt(i)=-1.
>
NOTHING prevents me from proposing a different law, encompassing these
two truths to bring them to i^x=-1 whatever x.
>
I affirm this as a new and universal law.
>
I am told that I am wrong, and that i^4=1.
>
I answer that they are wrong, and that they misunderstood me.
>
I tell them that if i^x=-1 (new global theory) then i^4=-1 and so on.
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They start telling me again: "No, no, i^4=1".
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This kind of knee-jerk response is stupid.
>
R.H.
>
>

>
Any "singularity theory" is yet a branch in a "multiplicity theory".
>
It's so that defining i^2 = -1 and root -1 = i symmetrically,
doesn't necessarily have that roots are unique, though powers
are generally considered unique, yet in terms of products and
inverses, there are inner and outer products and their inverses,
reflecting all manner of reflections, that it's already so
after expanding arithmetic from non-negative to positive and
negative or real to deMoivre/Euler/Gauss Argand/Wessel complex,
that introducing non-principal branches, and then picking
principal branches among what results those, i.e. that are
closed, does not eliminate those that aren't.
>
It's as closed-minded to ignore those as it is to ignore
the definition of an opening, a perestroika it's called
sometimes in algebra, or an opening, of what's closed.
>
So, of course mathematics already has names for these
sorts contemplations and deliberations, like openings
or perestroikas or catastrophes, sometimes, about
the inner and outer products, and complementary duals,
and these kinds of things.
>
>
Some for example even define arithmetic itself as,
instead of Peano's usual style, only defining increment
and division, and then that in the middle of those is
what results the modularity of integers and rationals,
with, "long subtraction", say, instead of, "long division".
Of course "fractions" is as old as "Egyptian mathematics".
>
Then, about roots of unity, yet really about "roots of zero",
here there's a sort of original analysis about an identity
dimension as I sort of ponder in my podcasts on the differential
analysis, among a sort of consideration that differential
analysis and integral analysis, while agreeing that the
anti-derivative is a thing, explores the non-uniqueness
and the multiplicity theory of the usual singularity theory.
>
>

It's kind of like Quine says in "Word & Object",
in showing how to dispense with a concept,
one needn't necessarily dispense with the concept.
Then he also sort of organizedly attributes all
these usual sorts notions of narrowing and compromise
in logic to particulars in the bibliography.
Anyways deMoivre/Euler/Gauss complex numbers
have quite a bit the broken bits at the ends,
split ends, say.