Re: Equation complexe

Am 26.02.2025 um 13:11 schrieb sobriquet:

Op 26/02/2025 om 05:12 schreef sobriquet:

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?

@RH:
If x ∈ IR, then x^2 ∈ IR and x^2 >= 0, and hence x^4 = (x^2)^2 >= 0.
So x can't be in IR.
Now let x = ir with r ∈ IR (and i, the imaginary unit, ∈ C).
Then x^2 = i^2 * r^2 = -1 * r^2 ∈ IR. Hence x^4 = (x^2)^2 >= 0.
Hence neither x ∈ IR nor x = ir with r ∈ IR will work.
Actually, any x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2, (-1-𝑖)⋅3/√2} will work.
Hope this helps.