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Date: Fri, 07 Feb 25 21:12:18 +0000
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From: Richard Hachel <r.hachel@tiscali.fr>
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Le 07/02/2025 à 14:27, Alan Mackenzie a écrit :
> Richard Hachel <r.hachel@tiscali.fr> wrote:

>> (i²)²=-1, and not 1.
> 
> Where do you get that garbage from?  i^4 = 1.
> 
>> And there, the whole structure that we thought we had defined by a simple 
>> i²=-1, which was true, collapses for everything else.
> 
> Garbage.  Nothing "collapses".  The theory of complex numbers is, as far
> as mathematicians can determine, consistent.  It is vast and fascinating
> in its own right.  It is also useful to scientists and engineers.
> 
> I suggest you make more humble efforts to learn and understand it.
> 
>> R.H. 

No, it is not consistent.

When I speak in real terms, I set (-x)(-x)=x².

If x=-5, then x²=25.

So far, it is perfectly consistent.

But if I use another form of mathematics, and I set 1=-i², in order to 
get rid of both the (-) sign and the square root, I have to go to the end 
of the structure used. I can no longer do everything I want and anyhow. I 
have to use this structure in a consistent way (it is, and it is 
magnificent if we know how to use it well).

In an imaginary universe, it is a bit like practicing in a mirror, the 
real becomes imaginary, and the imaginary becomes real.

But once in the imaginary, you have to stay WITH the laws of imaginary 
operations, and the mistake you make is to say: if (-n)(-n)=n² in 
reality, then it is obvious that (-i)(-i)=i² in the imaginary, then that 
(i²)²=1 and so on.

This is not how to proceed, it is incoherent.

It is like hammering a nail with a screwdriver. Some can do it, but a 
simple little hammer blow is more useful.

THIS is what mathematicians do without paying attention to their blunder.

But it is not a mathematically correct notion, and we play with numbers 
without knowing what we are doing.

R.H.