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Date: Thu, 06 Feb 25 22:15:30 +0000
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From: Richard Hachel <r.hachel@tiscali.fr>
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Le 06/02/2025 à 21:52, sobriquet a écrit :
> 
> Ok, but that's a bit like people saying that 3 + 5 = 7 and then claiming 
> that usually mathematicians say that 3 + 5 = 8, but they have different 
> concepts that are more correct.
> 
> Unless you're able to demonstrate that your alternative concepts are 
> superior than the conventional way of defining these concepts, it
> seems a bit silly.
> The concept of a complex number didn't fall from the sky.. it was 
> developed over many centuries by multiple generations of mathematicians. 
> So it seems unlikely that someone can come along and claim their way to 
> conceive of a complex number is superior or more correct.

You may be right.

But I also know that mathematicians and physicists can also be wrong.
Descartes' works are full of errors, Berkeley and Newton did not agree on 
the calculation of an infinitely small increment (I think Berkeley was 
right), Lorentz wandered for years on relativistic transformations before 
Poincaré gave it to him.

If you observe carefully, you will realize that basically, the notion of 
complex numbers is very quickly presented, then very quickly skipped. You 
are given an i²=-1, in order to square an awkward discriminant, and, if 
the principle is correct (we multiply by 1, then we pose that i²=-1 by 
convention and this allows us to have a positive root).

But no one has ever explained where this being comes from, which, in its 
being, is an entity whose square is equal to -1. It is a convention.

Personally, I wish that we put this being under the microscope once and 
for all, he fuuses imaginary.

It would seem that i is a special entity, which, like 1, can be used in 
such a way that, whatever its exponent, it remains identical to itself in 
its abstract being. -i is always equal to -i.

We can do as with 1^x give all the possible and imaginable powers to x, 
always, 1^x=1.

It seems that it is the same thing with i=-1.

i²=-1.
i^(-1/2)=-1
i^4=-1
i°=-1

etc...etc...etc...

So we have to define i, because then, in the calculations, big sign errors 
can appear. The biggest one can be that, sometimes, depending on the 
concept i²=-1 or i²=1 if we simplify too quickly.

Let's take z1=16+9i and z2=14+3i and make a product.
We have: aa'+i(ab'+ba')+(ib')(ib)

If we make the product as is, we see that if i=-1 then (-9)(-3)=+27,
but if we square it, we have i²=-1 and i²bb'=-27

The error is then colossal:

Hachel finds Z=251+174i.

Mathematicians find Z=197+174i

We need to think about this and check what is correct and why it is 
correct.

R.H.