Re: New equation

On 02/28/2025 10:46 AM, Jim Burns wrote:

On 2/28/2025 12:55 PM, Ross Finlayson wrote:

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

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

>

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.

>

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)⁻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.

>

https://www.youtube.com/watch?
v=Uv_6g__03_E&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=33

>
What do you say in that video which, in your opinion,
contributes to the discussion?
>

How about the "yin-yang ad-infinitum" bit
that shows directly a failure of inductive inference,
courtesy a simplest fact of geometry, and graphically.

>
The complex field does not fail at being a field.
>
Reliable inductive inferences do not fail at being correct.
>
A reliable inductive inference
concludes, from a subset being inductive,
that that inductive subset is the whole superset.
It is a _reliable_ inductive inference
only where there is only one inductive subset:
the whole superset.
>
A failure of inductive inference would be
x and c such that  x ∈ {c} ∧ x ≠ c
>
Are you claiming you have such x and c?
>

Not.ultimately.untrue, say.

>
In a finite sequence of claims,
either there is no false claim
or there is a first false claim.
>
In a finite sequence of claims in which
each claim is true.or.not.first.false,
there is no first.false claim.
>
In a finite sequence of claims in which
each claim is true.or.not.first.false,
there is no false claim.
>
>

Oh, I read a definition of complex numbers
and point out that division is non-unique.
There are plenty of Zeno's arguments
using induction that never get anywhere.