Re: y=f(x)=(x²)²+2x²+3

Richard Hachel <r.hachel@tiscali.fr> wrote:

[ .... ]

I think a new approach to complex numbers may be possible, and it starts
by redefining what the imaginary i is.

No, such a "new approach" to complex numbers is not possible.  You can
define what you like, but you are not free to call that "complex numbers"
unless it conforms to actual complex number theory.  You seem to be
ignorant of all mathematics, so you are not in a position to say what is
possible, and what not.

It is defined in a dramatically stupid way.

That is a spectacularly stupid thing to write.

And we say, stuttering: "it is the... the number... uh... that... which....
that if you square it, it becomes -1."

This is not very reasonable.

By contrast, it is you who is not very reasonable.

But this dramatic and narrow definition turns downright horrific when we
say: "Let's square the square".

Then everything becomes dreadful. We say (i²)(i²)=1 because (-1)(-1)=1.

That is true, and not dreadful.

And we attribute to an imaginary structure the same property as to a real
structure.

Each has many properties, many (but not all) of them shared.  Among those
which are shared is the ability to do normal arithmetic on them.  What is
not shared, for example, is the < relationship (not present in complex
numbers), and closure for polynomial equations (doesn't hold for real
numbers)

But, hold on tight, friends, this is false.

(i²)²=-1, and not 1.

Where do you get that garbage from?  i^4 = 1.

And there, the whole structure that we thought we had defined by a simple
i²=-1, which was true, collapses for everything else.

Garbage.  Nothing "collapses".  The theory of complex numbers is, as far
as mathematicians can determine, consistent.  It is vast and fascinating
in its own right.  It is also useful to scientists and engineers.

I suggest you make more humble efforts to learn and understand it.

R.H.

--
Alan Mackenzie (Nuremberg, Germany).