Re: Division of two complex numbers

Le 20/01/2025 à 20:33, Moebius a écrit :

Am 20.01.2025 um 19:27 schrieb Python:

Le 20/01/2025 à 19:23, Richard Hachel  a écrit :

Le 20/01/2025 à 19:10, Python a écrit :

Le 20/01/2025 à 18:58, Richard Hachel  a écrit :

Mathematicians give:
>
z1/z2=[(aa'+bb')/(a'²+b'²)]+i[(ba'-ab')/(a'²+b'²)]
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It was necessary to write:
z1/z2=[(aa'-bb')/(a'²-b'²)]+i[(ba'-ab')/(a'²-b'²)]

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I've explained how i is defined in a positive way in modern algebra. i^2 = -1 is not a definition. It is a *property* that can be deduced from a definition of i.

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 That is what I saw.
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 Is not a definition.
 It doesn't explain why.
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We have the same thing with Einstein and relativity.
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[snip unrelated nonsense about your idiotic views on Relativity]

 

It is clear that i²=-1, but we don't say WHY. It is clear however that if i is both 1 and -1 (which gives two possible solutions) we can consider its square as the product of itself by its opposite, and vice versa.

 I've posted a definition of i (which is NOT i^2 = -1) numerous times. A "positive" definition as you asked for.

 I've already told this idiot:
 Complex numbers can be defined as (ordered) pairs of real numbers.
 Then we may define (in this context):
           i := (0, 1) .
  From this we get: i^2 = -1.

The concept of context is far out of Hachel/Lengrand's ability. He's a complete idiot as you noticed, and a stubborn crank.
When it comes to definition of C your definition stands, anyway I do prefer the one in term of equivalence classes of R[X]. As we all know (except Richard which is absolutely incapable of learning anything and out of rational thinking) they are equivalent i.e. defines isomorphic sets and structures.