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NNTP-Posting-Date: Fri, 28 Feb 2025 17:55:53 +0000
Subject: Re: New equation
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 28 Feb 2025 09:55:52 -0800
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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 !
>
>>> A BS-philosophical version of Hachel.
>>> Let's park them together.
>>
>> 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.