Re: New equation

On 2/28/2025 6:08 PM, Ross Finlayson wrote:

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

The complex field does not fail at being a field.

Oh, I read a definition of complex numbers
and point out that division is non-unique.

(c+d𝑖)/(a+b𝑖) = (c+d𝑖)⋅(a+b𝑖)⁻¹
x+y𝑖 such that
(a+b𝑖)⋅(x+y𝑖) = 1
is unique.
(x+y𝑖) = (a+b𝑖)⁻¹ = (a-b𝑖)/(a²+b²)
⎛ (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