Re: Equation complexe

Op 25/02/2025 om 21:07 schreef Richard Hachel:

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:
>

x^4=-81
>
What is x?

>
x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2, (-1-𝑖)⋅3/√2}

 Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans trop expliquer pourquoi.[..]

You have to understand that if we consider this equivalent case of
g(x) = x^4 + 9/4 and we visualize the complex roots as follows:
https://www.desmos.com/calculator/4mhohrcxlg
We're superimposing a real slice of the graph in the Cartesian plane on top of Complex plane where we visualize the complex roots of the polynomial.
In order to understand why these are the roots of the polynomial g(x)
in the complex numbers, we have to expand our view to 3D:
https://www.desmos.com/3d/ykhhcpb3xz
Where we can visualize that polynomial as a function that maps complex numbers to complex numbers as a combination of two surfaces in 3D
(one red surface for the real component of the output of the function and an orange surface for the imaginary component of the output of the function).
We can press the icons in front of the corresponding items in the list
on the left to temporarily hide those surfaces to identify the locations
in the complex plane where both components are equal to zero to see that those are exactly the complex roots of the polynomial g(x) in the complex number system.