Re: New equation

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