Re: Equation complexe

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.