Re: New equation

On 02/25/2025 07:50 PM, Chris M. Thomasson wrote:

On 2/25/2025 6:26 PM, sobriquet wrote:

Op 26/02/2025 om 02:47 schreef Richard Hachel:

Le 26/02/2025 à 01:47, sobriquet a écrit :

Op 26/02/2025 om 00:58 schreef Richard Hachel:

Le 26/02/2025 à 00:03, "Chris M. Thomasson" a écrit :

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https://www.desmos.com/calculator/c3gntlt7kq
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the function g1(x) is the reflection of the function g0(x) in your
approach to yield the magenta colored 'roots', while the
conventional approach with complex numbers yields the complex roots
colored in yellow.
How about the function g2(x), which has (approximated) roots colored
in green.

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I don't understand how you mix Cartesian and Argand coordinate
systems. They are not the same thing.
The curve f(x)=x²+4x+5 must be represented on a Cartesian coordinate
system, no relation to a complex Argand coordinate system.

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Not necessarily. You can consider that equation in many number systems.
You can consider it in the integers, in the rational numbers, in the
real numbers, in complex numbers and more (like quaternions).
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Similarly, the curve g(x)=-x²-4x-3 which is the mirror curve must be
represented on a Cartesian coordinate system.

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No, it is just a polynomial equation, but that doesn't dictate which
number system we have to use. We can consider the same polynomial
equation in many number systems.
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We breathe, we blow.
I will then represent the real roots of the curve g(x), since f(x)
does not have any. So I find x'=-3 and x"=-1. These are the two small
majenta points that you represented. The coordinates are A(-3,0) and
B(-1,0).
There is nothing here that is difficult to understand.
Now, I must also place my two small yellow points which are the
complex roots of f(x). Now, I told you that the complex roots of a
function are the real roots of the mirror function, and conversely,
the real roots of a function are the complete roots of its mirror
function.
You must therefore place your small yellow points ON your small
magenta points.
The way you place them seems to represent an Argand reference frame
whose interest here is nil.

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The Argand plane is the plane of complex numbers.
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https://en.wikipedia.org/wiki/Complex_plane
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The complex roots of f(x) are therefore x'=3i and x"=i.
They are the same as A and B.
But as for g(x), we write them in real form, that is to say A(-3,0)
and B(-1,0); for f(x), we must write them in such a way that we know
that they are complex roots and the simplest notation is A(3i,0) and
B(i,0).
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You can, of course, write in the form A(-2+i,0) and B(-2-i,0) but it
is the same thing (+i=-1 ; -i=+1).
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Your yellow and majenta points must however be confused, they are the
same roots in fact, but one seen from f(x) and complex; the other
seen from g(x) and real.
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Placing the majenta or yellow points elsewhere than on x'Ox" has
absolutely no interest here, except to show that we are mixing a
concrete Cartesian frame with an Argand frame which is an abstract
frame where i is isolated from the complex, then placed vertically.
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R.H.

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We're talking about some sort of extension of the real numbers here
and the complex numbers are such an extension that ensures that
polynomials
always have roots in that number system.

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It's strange. I am not sure he would be able to implement my Multi Julia
using his, "system"...
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The complex number system has a wide range of applications in science
(like in quantum physics) and your alternative system is not a system
at all, because it fails to yield anything meaningful and consistent.
This is evident when you try out your approach by tweaking the
functions a bit and (for instance) considering rational functions (so
allowing for both negative and positive exponents) or fractional
exponents. In that case the conventional approach will hold up,
because it's a *system* that hangs together in a meaningful and
consistent way.
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Division in complex numbers is opinionated, not unique.
So, the natural products and alll their combinations
don't necessarily arrive at "staying in the system".
Furthermore, in things like Fourier-style analysis,
which often enough employ numerical methods a.k.a.
approximations here the small-angle approximation
in their derivations, _always have a non-zero error_.
Then, something like the "identity dimension", sees
instead of going _out_ in the numbers, where complex
numbers and their iterative products may neatly model
reflections and rotations, instead go _in_ the numbers,
making for the envelope of the linear fractional equation,
Clairaut's and d'Alembert's equations, and otherwise
with regards to _integral_ analysis vis-a-vis the
_differential_ analysis.
These each have things the other can't implement,
yet somehow they're part of one thing.
It's called completions since mathematics is replete.