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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Equation complexe
Date: Tue, 25 Feb 2025 16:34:58 -0800
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On 2/25/2025 4:11 PM, Richard Hachel wrote:
> 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.

The fourth root of -81+0i wrt power of 4 is *:

roots[0] = (2.12132,2.12132)
roots[1] = (-2.12132,2.12132)
roots[2] = (-2.12132,-2.121321)
*roots[3] = (2.12132,-2.121321)

I don't know what you x=3i even means right now. Any of these roots 
raised to the 4'th power equals -81+0i.


> 
> 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.