Re: Equation complexe

On 2/25/2025 5:11 PM, Richard Hachel wrote:

Le 26/02/2025 à 01:34, "Chris M. Thomasson" a écrit :

On 2/25/2025 4:11 PM, Richard Hachel wrote:

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.

 We are not talking about the same thing, nor are we using the same mathematics.
 If I ask what are the complex roots of f(x)=x²+4x+5,
you will tell me that we must use [-b$sqrt(b²-4ac)]/2a using i.
 And you will give me x'=-2+i and x"=-2-i.
 Coordinates on x'Ox that I will immediately place in A(-3,0) and B(-1,0) and which are the imaginary roots on y=0, the equation having no real roots.
 Well, I do the same to find the fourth root of -81.
 x^4=-81.
 For me, i ^x=-1 whatever 1.
 x^4=-81 ---> x^4=-(i^4)(-81)=81(i^4)
x=3i

List out all of the four roots of -81+0i using your system such that when they are raised to the 4'th power equal -81. I know how to find them using good ol' complex numbers.

 Conversely, x^4=(3i)^4 = 81(i^4) with i^4=-1 by definition of i for me.
 But that has nothing to do with the Argand frame, which is something completely different.
 R.H.