Re: New equation

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Sujet : Re: New equation
De : dohduhdah (at) *nospam* yahoo.com (sobriquet)
Groupes : sci.math
Date : 26. Feb 2025, 03:26:14
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <vplu46$294n9$1@dont-email.me>
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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 :
 
https://www.desmos.com/calculator/c3gntlt7kq
>
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.
 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.
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).

Similarly, the curve g(x)=-x²-4x-3 which is the mirror curve must be represented on a Cartesian coordinate system.
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.

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.
The Argand plane is the plane of complex numbers.
https://en.wikipedia.org/wiki/Complex_plane

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).
 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).
 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.
 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.
 R.H.
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.
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.

Date Sujet#  Auteur
24 Feb 25 * New equation57Richard Hachel
24 Feb 25 `* Re: New equation56Barry Schwarz
24 Feb 25  +- Re: New equation1Richard Hachel
24 Feb 25  `* Re: New equation54Richard Hachel
24 Feb 25   +- Re: New equation1sobriquet
25 Feb 25   `* Re: New equation52Barry Schwarz
25 Feb 25    +* Re: New equation50Richard Hachel
25 Feb 25    i`* Re: New equation49Chris M. Thomasson
25 Feb 25    i `* Re: New equation48Richard Hachel
26 Feb 25    i  `* Re: New equation47Chris M. Thomasson
26 Feb 25    i   +* Re: New equation2Chris M. Thomasson
26 Feb 25    i   i`- Re: New equation1Chris M. Thomasson
26 Feb 25    i   `* Re: New equation44Richard Hachel
26 Feb 25    i    +* Re: New equation2Chris M. Thomasson
26 Feb 25    i    i`- Re: New equation1Richard Hachel
26 Feb 25    i    `* Re: New equation41sobriquet
26 Feb 25    i     `* Re: New equation40Richard Hachel
26 Feb 25    i      `* Re: New equation39sobriquet
26 Feb 25    i       `* Re: New equation38Chris M. Thomasson
27 Feb 25    i        `* Re: New equation37Ross Finlayson
27 Feb 25    i         +- Re: New equation1Chris M. Thomasson
27 Feb 25    i         `* Re: New equation35efji
28 Feb 25    i          `* Re: New equation34Ross Finlayson
28 Feb 25    i           +* Re: New equation3Chris M. Thomasson
28 Feb 25    i           i`* Re: New equation2Ross Finlayson
28 Feb 25    i           i `- Re: New equation1Chris M. Thomasson
28 Feb 25    i           +- Re: New equation1efji
28 Feb 25    i           `* Re: New equation29Jim Burns
28 Feb 25    i            `* Re: New equation28Ross Finlayson
28 Feb 25    i             `* Re: New equation27Jim Burns
1 Mar 25    i              `* Re: New equation26Ross Finlayson
1 Mar 25    i               +* Re: New equation24Jim Burns
1 Mar 25    i               i`* Re: New equation23Ross Finlayson
2 Mar 25    i               i `* Re: New equation22Jim Burns
2 Mar 25    i               i  +* Re: New equation4Richard Hachel
2 Mar 25    i               i  i`* Re: New equation3Python
2 Mar 25    i               i  i `* Re: New equation2Richard Hachel
2 Mar 25    i               i  i  `- Re: New equation1Python
2 Mar 25    i               i  `* Re: New equation17Ross Finlayson
2 Mar 25    i               i   `* Re: New equation16Jim Burns
2 Mar 25    i               i    `* Re: New equation15Richard Hachel
2 Mar 25    i               i     +* Re: New equation10Python
2 Mar 25    i               i     i+* Re: New equation2efji
2 Mar 25    i               i     ii`- Re: New equation1Chris M. Thomasson
2 Mar 25    i               i     i`* Re: New equation7Richard Hachel
2 Mar 25    i               i     i +* Re: New equation5efji
2 Mar 25    i               i     i i`* Re: New equation4Richard Hachel
2 Mar 25    i               i     i i `* Re: New equation3Python
2 Mar 25    i               i     i i  `* Re: New equation2Richard Hachel
2 Mar 25    i               i     i i   `- Re: New equation1Python
2 Mar 25    i               i     i `- Re: New equation1Python
2 Mar 25    i               i     `* Re: New equation4Jim Burns
2 Mar 25    i               i      `* Re: New equation3Chris M. Thomasson
2 Mar 25    i               i       `* Re: New equation2Jim Burns
2 Mar 25    i               i        `- Re: New equation1Chris M. Thomasson
1 Mar 25    i               `- Re: New equation1Jim Burns
26 Feb 25    `- Re: New equation1Chris M. Thomasson

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