Re: What is i ? was: Hello!

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

On 1/18/2025 5:34 AM, Richard Hachel wrote:
>

I was recently thinking,
because of a poster named Python,
about what complex numbers were,
wondering if teaching them was so important and useful,
especially in kindergarten
where children are only learning to read.
What is a complex number?

>

What is i?
>
It is an imaginary unit, such that i*i=-1.

>
⟨ℝ,+,⋅⟩ is a field, which means:
'+','⋅': ℝ×ℝ → ℝ
are associative and commutative, and
have identities 0,1 and inverses -x,x⁻¹,
except there's no 0⁻¹,
and '⋅' distributes over '+'.
>
⟨ℝ²,⨢,∘⟩ is a field, which means:
'⨢','∘': ℝ²×ℝ² → ℝ²
are associative and commutative, and
have identities 𝟎,𝒆₁ = [1 0] and inverses -𝐱,𝐱⁻¹,
except there's no 𝟎⁻¹,
and '∘' distributes over '⨢'.
>
⟨ℝ²,⨢,∘⟩ is defined to be
an extension of ⟨ℝ,+,⋅⟩, which means:
'⨢' and '∘' agree with '+' and '⋅' on ℝ×{0}
and, on the whole ℝ², is a field.
>
'∘' is bilinear, and
𝒆₁ is left.unit and right.unit.
⎛ (c⋅𝐫)∘𝐬 = c⋅(𝐫∘𝐬)
⎜ (𝐫⨢𝐭)∘𝐬 = 𝐫∘𝐬 ⨢ 𝐭∘𝐬
⎜ 𝐫∘(c⋅𝐬) = c⋅(𝐫∘𝐬)
⎜ 𝐫∘(𝐬⨢𝐭) = 𝐫∘𝐬 ⨢ 𝐫∘𝐭
⎜ 𝒆₁∘𝐬 = 𝐬
⎝ 𝐫∘𝒆₁ = 𝐫
>
That's enough information to determine a lot of,
but not all of the definition of '∘'.
>
Let 𝒆₂ = [0 1]
and 𝒆₂∘𝒆₂ = [-μ₁ -μ₂].
(a𝒆₁⨢b𝒆₂)∘(c𝒆₁⨢d𝒆₂) = (ac-μ₁bd)𝒆₁⨢(ad+bc-μ₂bd)𝒆₂
>
μ₁ > μ₂²/4
 iff
a point 𝒊 exists such that 𝒊∘𝒊 = -𝒆₁
𝒊 = ±[μ₂/2 1]/(μ₁-μ₂²/4)¹ᐟ²
and
(a𝒆₁⨢b𝒊)∘(c𝒆₁⨢d𝒊) = (ac-bd)𝒆₁⨢(ad+bc)𝒊
and
the constants μ₁,μ₂ disappear from view,
disappear into 𝒊, in effect,
and
for each 𝐫 ≠ 𝟎, 𝐫⁻¹ exists, 𝐫∘𝐫⁻¹ = 𝒆₁
and
⟨ℝ²,⨢,∘⟩ is a field extending ⟨ℝ,+,⋅⟩.
>
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″
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. 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
--
guido wugi