Re: What is i ? was: Hello!

On 1/31/2025 6:31 AM, guido wugi wrote:

Op 30/01/2025 om 22:55 schreef Jim Burns:

There are different values possible for 𝒆₂∘𝒆₂ = [-μ₁ -μ₂],
but, as long as μ₁ > μ₂²/4,
⟨ℝ²,⨢,∘⟩ is a field extending ⟨ℝ,+,⋅⟩.
>
For two plane.multiplications ∘′ and ∘″
we can map 𝒊′ ⟷ 𝒊″

𝒊′ = ±[μ′₂/2 1]/(μ′₁-μ′₂²/4)¹ᐟ²
𝒊″ = ±[μ″₂/2 1]/(μ″₁-μ″₂²/4)¹ᐟ²

and then
(a𝒆₁⨢b𝒊′)∘′(c𝒆₁⨢d𝒊′) = (ac-bd)𝒆₁⨢(ad+bc)𝒊′
(a𝒆₁⨢b𝒊″)∘″(c𝒆₁⨢d𝒊″) = (ac-bd)𝒆₁⨢(ad+bc)𝒊″
>
And the two ℝ.extending plane.multiplications
are isomorphic.
Therefore, there is
only one extension of ℝ to ℝ², up to isomorphism,
and, for that extension, 𝒊² = -1
>
⎛ What I got wrong initially was that 𝒊 ≠ 𝒆₂,
⎜ at least, not necessarily equal.
⎜ That it's not doesn't matter, though.
⎜ All the different '∘' with their different 𝒊
⎝ map to each other very neatly.

>
Not sure I "got" it all.

My explanations evolve.
To a higher, purer state, I hope,
but to a different state, at least.
Richard Hachel's question "What is i?"
is a good one.
Others will ask it, others have asked it.
I think I might have, a million or so years ago.
When I spot another excuse to try,
I'll likely try explaining again.
It'll likely be different. Again.
Maybe even better.
Thank you for your attention.

I once did, I guess, a similar thinkthing about not necessarily fields, but
multidimensional numbers alright,
as n-vectors and
as "autovariant" nxn matrix families:
https://www.wugi.be/hypereal.htm

That deserves more than a glance.
I'm a great fan of vector spaces.
So many theorems, so broadly applicable.
Fourier transforms are rotations in function space!
You:
/ I called these the "Autovariance conditions",
| assuring that the matrix family embraces
| any product of its members.
| Without these conditions,
| a random product would “leave”
| the n-dim matrix family into
\ the n x n matrix space!
Perhaps you are talking about
an n.dimensional subspace closed under
the usual matrix multiplication?
Yes,
it reminds me of how I describe complex numbers.
For me, "mediating" 2x2 matrices are inserted
to define the not.the.usual product.
[a b]∘[c d] :=
[ [a b]ᵀ𝑴₁[c d] [a b]ᵀ𝑴₂[c d] ]
[ 𝑴₁ 𝑴₂ ] has eight degrees of freedom, enough,
we find, to impose field conditions on '∘'
I wonder what we can get if
something like that is done with nxn matrices.