Re: Today's curve.

Le 28/03/2025 à 17:10, Jim Burns a écrit :

On 3/28/2025 8:33 AM, Richard Hachel wrote:
 

Let the function f(x)=x⁴-2x²+8 be given.

 Define '⋅'
⟨0,1⟩⋅⟨0,1⟩ = ⟨-1,0⟩
⟨1,0⟩⋅⟨0,1⟩ = ⟨0,1⟩
⟨0,1⟩⋅⟨1,0⟩ = ⟨0,1⟩
⟨1,0⟩⋅⟨1,0⟩ = ⟨1,0⟩
 f(⟨x,y⟩) = ⟨x,y⟩⁴-2⟨x,y⟩²+8⟨1,0⟩
 It follows that
f(⟨x,y⟩) = ⟨0,0⟩  ⇔
⟨x,y⟩ ∈ {⟨a,b⟩,⟨-a,b⟩,⟨a,-b⟩,⟨-a,-b⟩}
 f(⟨x,y⟩) = ⟨x-a,y-b⟩⋅⟨x+a,y-b⟩⋅⟨x-a,y+b⟩⋅⟨x+a,y+b⟩
 a = 2³ᐟ⁴cos(½tan⁻¹(√7))
b = 2³ᐟ⁴sin(½tan⁻¹(√7))

This is indeed what mathematicians say, but it's neither true nor beautiful.
In good mathematical logic, there are only two roots, both of the complex root type.
Let x'=2i and x"=-2i.
The way mathematicians deal with all this is as ridiculous as it is shameful.
Almost criminal.
R.H.