Re: Hello!

cOn 1/19/2025 2:41 AM, Richard Hachel wrote:

Le 19/01/2025 à 02:53, sobriquet a écrit :

Op 18/01/2025 om 11:34 schreef Richard Hachel:

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?

Let z=a+ib
It is a number that has
a real component and an imaginary component.
>
I wonder if the terms
"certain component" and "possible component" would not be as appropriate.

The terms 'real' and 'imaginary' are hand.me.downs
from the fabled.past of mathematics.
They might not always be a perfect fit.
If I decided to update our vocabulary,
for the sake of the children, perhaps,
I would select terms suggesting two directions.
For example, 'east' and 'north'.

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

It is a unit north.
Multiplication by i is a left turn.
Multiplication by i twice reverses direction.

Let's assume that i is a number, or rather a unit,
which is both its number and its opposite.

Let's not assume that a unit north is
both a unit east and a unit west.

Thus,
if we set z=9i
we see that z is both,
as in this story of Schrödinger's cat,
z=9 and z=-9
>
I remind you that we are in the imaginary.
So why not.

Because z = z  but  9 ≠ -9
⎛ Schrödinger's cat asks questions other than
⎜ how far north.south.east.west.
⎜
⎜ Experiments and Bell's theorem seem to say that,
⎜ before we look,
⎜ the cat doesn't have
⎜ either a past in which it died
⎜ or a past in which it survived.
⎜
⎜ But then we look, and _when we look_
⎜ we see a cat with one of those pasts.
⎜
⎜⎛
⎜⎜ The Moving Finger writes; and, having writ,
⎜⎜ Moves on: nor all thy Piety nor Wit
⎜⎜  Shall lure it back to cancel half a Line,
⎜⎜ Nor all thy Tears wash out a Word of it.
⎜⎝
⎜ -- Omar Khayyam, trans. Edward FitzGerald
⎜
⎜ What if, for just a moment,
⎜ we can hold the door to the cat shut, and,
⎜ for just that moment, the finger pauses,
⎜ and history isn't written -- yet.
⎜
⎜ But always our hand slips, the door opens,
⎜ the finger writes and moves on.
⎜ Holding that door shut
⎝ is very challenging, technically.

If we define complex multiplication
in the way you suggest
instead of the conventional way,
that would mean that
the operation of conjugation would no longer be
a homomorphism with respect to
the field of complex numbers under multiplication.

Conventionally,
when you stand on your head (conjugation),
all this makes (conjugated) sense.

So conj(z1*z2) would not be equal to
conj(z1)*conj(z2).
>
https://www.desmos.com/calculator/kqzgbliix1

Because as long as we do not know what i is worth,
which can be BOTH equal to 1 or -1
in this imaginary mathematics, we must pose i²=-1.

For ⅈ=1 or ⅈ=-1,  ⅈ²=1 or ⅈ²=1

You tell me:
yes, but it does not work with the conjugate.

I vaguely recall, from a course in complex analysis
I took approximately When Dinosaurs Ruled The Earth,
that, by requiring the complex field axioms
to be satisfied, one can generate ⅈ²=-1 algebraically.