Re: New equation

On 02/28/2025 05:50 AM, Jim Burns wrote:

On 2/27/2025 11:51 PM, Ross Finlayson wrote:

On 02/27/2025 01:46 AM, efji wrote:

Le 27/02/2025 à 05:19, Ross Finlayson a écrit :

>

Division in complex numbers is opinionated,
not unique.

>

:)
Hachel has a brother !

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A BS-philosophical version of Hachel.
Let's park them together.

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Division in complex numbers most surely is
non-unique, [...]

>
In the complex field,
division is unique, except by 0,
and division by 0 isn't at all.
Because the complex field is a field.
>
A field has operations '+' '⋅' such that
both are associative and commutative,
have identities 0 1 and inverses -x x⁻¹,
except 0⁻¹, and '⋅' distributes over '+'
>
Consider inverse(s?) x⁻¹ x⁻¹′
x⁻¹⋅x⋅x⁻¹′ = x⁻¹⋅x⋅x⁻¹′
x⁻¹⋅1 = 1⋅x⁻¹′
x⁻¹ = x⁻¹′
>
The complex field is a field,
in part, because 𝑖² = -1
from which it must follow that,
for (a+b𝑖)⁻¹ = x+y𝑖 such that
  (a+b𝑖)⋅(x+y𝑖) = 1
⎛
⎜ (a+b𝑖)⋅(x+y𝑖) = (ax-by)+(bx+ay)𝑖 = 1
⎜
⎜ ax-by = 1
⎜ bx+ay = 0
⎜
⎜ a²x-aby = a
⎜ b²x+aby = 0
⎜ (a²+b²)x = a
⎜ x = a/(a²+b²)
⎜
⎜ abx-b²y = b
⎜ abx+a²y = 0
⎜ (a²+b²)y = -b
⎜ y = -b/(a²+b²)
⎜
⎜ x+y𝑖 = (a-b𝑖)/(a²+b²)
⎜
⎜ (a+b𝑖)⁻¹ = (a-b𝑖)/(a²+b²)
⎝ uniquely, for a²+b² ≠ 0
>
It's possible that
you are confusing division with
logarithm or square root or some such.
>

Furthermore, if you don't know usual derivations
of Fourier-style analysis and for example about
that the small-angle approximation is a linearisation
and is an approximation and is after a numerical method,
you do _not_ know.
>
Then about integral analysis and this sort of
"original analysis" and about the identity line
being the envelope of these very usual integral
equations, it certainly is so.

>
Technobabble.
And not in a good way.
>

[...]

>
>

https://www.youtube.com/watch?v=Uv_6g__03_E&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=33
How about the "yin-yang ad-infinitum" bit
that shows directly a failure of inductive inference,
courtesy a simplest fact of geometry, and graphically.
Remember that?
Not.ultimately.untrue, say.