Re: Equation complexe

Le 26/02/2025 à 11:30, Barry Schwarz a écrit :

On Wed, 26 Feb 25 00:11:36 +0000, Richard Hachel <r.hachel@tiscali.fr>
wrote:

The answer is simple and obvious. x=3i.

 And incorrect.  In standard mathematics, (3i)^4 does not equal -81

That's what I'm saying.
But for me, (3i)^4=-81 (if always i^x=-1 ).
I'm just trying to understand things clearly at the base.
Standard mathematics poses i²=-1; which is justified, and it helps to solve quadratic equations without roots.
We then realize that the imaginary roots found for f(x) are the real roots of another curve g(x) which, on a Cartesian plane is the mirror curve of f(x) centered on the vertex of the other curve.
Thus, and conversely, the complex roots of one are always the real roots of the other.
So far, it's very useful for this kind of small mathematical maneuver, and very precise.
Now saying that i^x is -1 if x=2 is very good.
But I would like this very restrictive definition to be abandoned.
The goal is to define i^x whatever x, and the way it is done does not suit me.
It would be interesting to propose something like i^x=-1 whatever x, and in this sense make i invariant whatever the power we give it.
This would be a very important, general, and much more practical definition.
R.H.