Re: Hello!

Le 19/01/2025 à 17:04, Moebius a écrit :

Am 19.01.2025 um 16:50 schrieb Moebius:

Am 19.01.2025 um 16:39 schrieb Moebius:

Am 19.01.2025 um 16:30 schrieb Richard Hachel:

Le 19/01/2025 à 15:28, Moebius a écrit :

Am 19.01.2025 um 12:37 schrieb Richard Hachel:

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Complex numbers can be defined as (ordered) pairs of real numbers.
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Then
         i := (0, 1) .
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Hence i =/= 1 and i =/= -1.

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No.

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Yes.

 Hint: Then (in this context) i * i = (0, 1) * (0, 1) = (-1, 0), the latter is "identified" with -1, hence we may write:
                   i^2 = -1 .
 That's exactly what we want.

 Hint: No "extremely fine mathematical error" at all.
 On the other hand, this one is an "extremely severe mathematical error":

Richard "Hachel" Lengrand suffers of a lot of mental diseases. One of them is pathological hubris. When he fails to understand something (and he failed to understand complex numbers when in high schools, then he studied... medicine) instead of thinking and studying the subject he systematically decides that everyone is wrong and he pretend to reinvent the subjet.
He's doing this for Special Relativity for decades and ended up with an atrocious bunch of nonsense and contradiction.
He cannot realize that there cannot be an "error" in a definition, such as the ones for complex numbers.
Either it is consistent, compatible with general algebraic structures and useful or not.
Complex numbers are consistent, they form a Field extending R, allowing to factorize all polynomials better than in R[X]. They allow to utterly simplify a lot of geometric problems and deduce trigonometric identity.
They are not the only interesting structures on R^2 btw, dual numbers defined by a+b*epsilon where epsilon^2 = 0 (more rigorously this is the ring R[X]/X^2, while epsilon is the equivalence class of the polynomial X, quite a similar approach to the algebraic construction of C as R[X]/(X^1+1))
Dual numbers provide a very elegant introduction of calculus without having to deal with limites (at least for polynomials).
Hachel/Lengrand's "ideas" may have had some sense, but they are 100% unrelated to complex numbers. Moreover the idea of something having two distinct values -1 and 1 is, of course, totally absurd and contradictory. Given his deranged mental states Richard does not bother.