Post by sobriquetPost by Richard HachelWhat if Descartes and Gauss were completely wrong?
No, not about everything, obviously, but about some important details?
What if there were two blunders hidden, correcting each other,
according to the theory of compensated errors?
First blunder: after having understood that the real roots were
revealed by x=[-b(+/-)sqrt(b²-4ac)]/2a, which is true and which is
easily demonstrated, generalizing the same discriminant too quickly,
without paying attention to the signs (complexes being complex to
handle) and setting i²=-1 (which is true) then
x=[-b(+/-)i.sqrt(b²-4ac)]/2a instead of x=[-b(+/-)i.sqrt(b²+4ac)]/2a.
The complex root is no longer the same. There would therefore be a
first error due to a misunderstood sign.
The error is then compensated by another sign error, during the proof
by check via the reverse path. Thus, for me, the correct roots of
f(x)=x²-2x+8 are x'=4i, and x"=2i which can easily be placed on the
usual x'Ox axis of Cartesian coordinate systems, roots found elsewhere
by using x=[-b(+/-)i.sqrt(b²+4ac)]/2a without being trapped by a sign
error (we are no longer in real roots, but in complex roots, where
x=-i on the x'Ox axis and vice versa).
R.H.
If you think you have a superior theory of complex numbers, you're
better off making an engaging video on the subject and then you can
http://youtu.be/5PcpBw5Hbwo
https://www.oclc.org/
The textual and abstract nature of mathematical learning
...
Well when you look at de Moivre theorem then Euler identity
then into Gauss and the Gaussian, then there's Argand and
Wessel, and they're different already, as with regards to
things like not really needing Hilbert space according to
Zariski and Kodaira and Lescop, though working in it,
why it's not so much that Gauss was wrong is that simply
he expands the range of definition on finding a non-principal,
then picks some principal branch among all the sticks of those,
the so many other non-principal branches.
So, saying things like "complex division is defined only one
way" is as closed-minded as "i doesn't exist".
Then again many people still have problems considering
the fact that the regular singular points of the hypergeometric
are 0, 1, and infinity, and 0, 1, and infinity form an expression
together, and the only operation is division.
Of course many people are familiar with the theorem that
the complex numbers are a complete ordered field and so
are the real numbers and that's unique up to isomorphism,
yet for example I wrote field operations equipping [-1, 1]
with field operations, another, different, complete ordered
field.
Things like singular integrals and all sorts of results
in function theory for example the convolutive: don't
actually need complex numbers at all, and, can be re-built
in any number of other ways.