Re: i²=-1

Le 16/02/2025 à 11:29, Alan Mackenzie a écrit :

MOI, j'explique. Le monde des imaginaires est tel que i^x=-1 quelque soit x.

 OK, let's assume that for the purposes of argument.
  i^2 = -1, and i^3 = -1.

 Yes.
 And also, i^(1/2), i^5, i^0, i^-5, i^(-5/3), etc...etc...etc...
 Always, i^x=-1.
 In mirror effect, we have the same thing in the reals with 1. The unit 1 remains invariably 1 whatever the exponent we attribute to it. It is invariable. 1^x=1.

Then

 Then what?

  (i^3 / i^2) = i,

 Yes.

and   (i^3 / i^2) = (-1 / -1) = 1.

 No.
 i^3/i^2=i
 i=-1 

So i = 1.

 Absolutely not.

But this contradicts i^2 = -1.

 No.

So your "system" is self contradictory.

 Absolutely not.
 You are making a conceptual error. You are multiplying imaginaries with the laws of real numbers.
In real numbers 1*1=1.
In imaginary numbers i*i=i
It is quite counter-intuitive, I admit, but it is the truth of things, and sometimes you have to know how to question your preconceptions.
Let's take the example of the very young driver, who must learn to use the rearview mirror. He is initially surprised that the rear of the vehicle turns to the right when he turns the steering wheel to the right. We have the same thing with the first outing in a boat. The preconception is that you have to force the right oar to turn right, but it is the opposite. The rower must force the left oar.
However, handling complex numbers is very simple if we get rid of a ridiculous preconception that we use an imaginary mirror numbering while using real mathematics.
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