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Subject: Re: Equation complexe
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Date: Wed, 26 Feb 25 00:11:36 +0000
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From: Richard Hachel <r.hachel@tiscali.fr>
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Le 26/02/2025 à 00:48, "Chris M. Thomasson" a écrit :
> On 2/25/2025 2:20 PM, Richard Hachel wrote:
>> Le 25/02/2025 à 23:08, "Chris M. Thomasson" a écrit :
>>> On 2/25/2025 6:23 AM, Richard Hachel wrote:
>>>> x^4=-81
>>>>
>>>> What is x?
> [...]
>> x=-1.873444
> 
> To the 13'th power with higher precision:
> 
> roots[0] = (1.01898,0.251156)
> roots[1] = (0.7855438,0.6959311)
> roots[2] = (0.3721492,0.9812768)
> roots[3] = (-0.1265003,1.041824)
> roots[4] = (-0.5961701,0.8637015)
> roots[5] = (-0.9292645,0.4877156)
> roots[6] = (-1.049476,5.945845e-16)
> roots[7] = (-0.9292645,-0.4877156)
> roots[8] = (-0.5961701,-0.8637015)
> roots[9] = (-0.1265003,-1.041824)
> roots[10] = (0.3721492,-0.9812768)
> roots[11] = (0.7855438,-0.6959311)
> roots[12] = (1.01898,-0.251156)
> 
> raised[0] = (-1.873444,2.294307e-16)
> raised[1] = (-1.873444,4.016197e-15)
> raised[2] = (-1.873444,4.475059e-15)
> raised[3] = (-1.873444,1.606015e-15)
> raised[4] = (-1.873444,2.064877e-15)
> raised[5] = (-1.873444,9.179548e-15)
> raised[6] = (-1.873444,9.63841e-15)
> raised[7] = (-1.873444,4.132072e-15)
> raised[8] = (-1.873444,4.590934e-15)
> raised[9] = (-1.873444,1.170561e-14)
> raised[10] = (-1.873444,2.214818e-14)
> raised[11] = (-1.873444,1.262333e-14)
> raised[12] = (-1.873444,2.306591e-14)

I think that for the moment, we are making things terribly complicated.
If I ask you the cube root of 27?
Are you going to make a computer program?
Why make a computer program if I ask you the fourth root of -81?

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

All these misunderstandings come from the fact that no clear and 
universally usable definition of the imaginary number i has ever been 
given.

Against all expectations, in analytical mathematics, i is an imaginary 
unit such that, for all x, i^x=-1.

We see that saying that i²=-1 is completely legal.

Or that sqrt(i)=i^(1/2)=-1.

Certainly.

But we also see that (i²)² is not equal to 1, and that those who believe 
it are corrupting themselves.

R.H.