y=f(x)=(x²)²+2x²+3

Post by Richard Hachel
Bonjour les amis !
I asked for the roots of the following equation on the French forums, I
only got one answer that didn't satisfy me, and the rest is just
contempt and insults.
So I'm trying my luck here.
y=f(x)=(x²)²+2x²+3
Il y a pour moi, deux racines très simples pour cette équation, dont
aucun n'est réelle.
Can the Anglo-Saxons find these two roots?
R.H.

Actually there are four complex roots.

https://www.wolframalpha.com/input?i=x%5E4%2B2x%5E2%2B3

https://www.desmos.com/calculator/in0q2rqzc9

Richard Hachel

2025-02-06 20:30:24 UTC

Actually there are four complex roots.
https://www.wolframalpha.com/input?i=x%5E4%2B2x%5E2%2B3

Yes, these are indeed the roots found in traditional development.

Mathematicians find four complex roots.

Personally, in this specific case, I only find two, because I think there
are only two.

But I use different concepts, and a different method.

For me, the roots are x'=-i and x"=i in this particular case, and I place
them on the y=0 axis, obviously, and on a simple Cartesian coordinate
system.

DON'T SHOUT!

I remind you that I use a different approach that I think is more correct
and in line with the very nature of i, and its precise definition, which
is not only i²=-1.

R.H.

sobriquet

2025-02-06 20:52:37 UTC

Post by Richard Hachel
Bonjour les amis !
I asked for the roots of the following equation on the French forums,
I only got one answer that didn't satisfy me, and the rest is just
contempt and insults.
So I'm trying my luck here.
y=f(x)=(x²)²+2x²+3
Il y a pour moi, deux racines très simples pour cette équation, dont
aucun n'est réelle.
Can the Anglo-Saxons find these two roots?
R.H.

Actually there are four complex roots.
https://www.wolframalpha.com/input?i=x%5E4%2B2x%5E2%2B3

Yes, these are indeed the roots found in traditional development.
Mathematicians find four complex roots.
Personally, in this specific case, I only find two, because I think
there are only two.
But I use different concepts, and a different method.
For me, the roots are x'=-i and x"=i in this particular case, and I
place them on the y=0 axis, obviously, and on a simple Cartesian
coordinate system.
DON'T SHOUT!
I remind you that I use a different approach that I think is more
correct and in line with the very nature of i, and its precise
definition, which is not only i²=-1.
R.H.

Ok, but that's a bit like people saying that 3 + 5 = 7 and then claiming
that usually mathematicians say that 3 + 5 = 8, but they have different
concepts that are more correct.

Unless you're able to demonstrate that your alternative concepts are
superior than the conventional way of defining these concepts, it
seems a bit silly.
The concept of a complex number didn't fall from the sky.. it was
developed over many centuries by multiple generations of mathematicians.
So it seems unlikely that someone can come along and claim their way to
conceive of a complex number is superior or more correct.

Chris M. Thomasson

2025-02-06 21:35:10 UTC

Post by Richard Hachel
Bonjour les amis !
I asked for the roots of the following equation on the French
forums, I only got one answer that didn't satisfy me, and the rest
is just contempt and insults.
So I'm trying my luck here.
y=f(x)=(x²)²+2x²+3
Il y a pour moi, deux racines très simples pour cette équation, dont
aucun n'est réelle.
Can the Anglo-Saxons find these two roots?
R.H.

Actually there are four complex roots.
https://www.wolframalpha.com/input?i=x%5E4%2B2x%5E2%2B3

Yes, these are indeed the roots found in traditional development.
Mathematicians find four complex roots.
Personally, in this specific case, I only find two, because I think
there are only two.
But I use different concepts, and a different method.
For me, the roots are x'=-i and x"=i in this particular case, and I
place them on the y=0 axis, obviously, and on a simple Cartesian
coordinate system.
DON'T SHOUT!
I remind you that I use a different approach that I think is more
correct and in line with the very nature of i, and its precise
definition, which is not only i²=-1.
R.H.

Ok, but that's a bit like people saying that 3 + 5 = 7 and then claiming
that usually mathematicians say that 3 + 5 = 8, but they have different
concepts that are more correct.

Basically. :^)

Post by sobriquet
Unless you're able to demonstrate that your alternative concepts are
superior than the conventional way of defining these concepts, it
seems a bit silly.

Ditto.

Post by sobriquet
The concept of a complex number didn't fall from the sky.. it was
developed over many centuries by multiple generations of mathematicians.
So it seems unlikely that someone can come along and claim their way to
conceive of a complex number is superior or more correct.

For some reason, "some" people seem to want to say complex numbers are
not "real" because of the word "imaginary" used to define the y axis?

FromTheRafters

2025-02-06 21:44:59 UTC

Post by Chris M. Thomasson

Actually there are four complex roots.
https://www.wolframalpha.com/input?i=x%5E4%2B2x%5E2%2B3

Yes, these are indeed the roots found in traditional development.
Mathematicians find four complex roots.
Personally, in this specific case, I only find two, because I think there
are only two.
But I use different concepts, and a different method.
For me, the roots are x'=-i and x"=i in this particular case, and I place
them on the y=0 axis, obviously, and on a simple Cartesian coordinate
system.
DON'T SHOUT!
I remind you that I use a different approach that I think is more correct
and in line with the very nature of i, and its precise definition, which
is not only i²=-1.
R.H.

Ok, but that's a bit like people saying that 3 + 5 = 7 and then claiming
that usually mathematicians say that 3 + 5 = 8, but they have different
concepts that are more correct.

Basically. :^)

Post by sobriquet
Unless you're able to demonstrate that your alternative concepts are
superior than the conventional way of defining these concepts, it
seems a bit silly.

Ditto.

Post by sobriquet
The concept of a complex number didn't fall from the sky.. it was developed
over many centuries by multiple generations of mathematicians. So it seems
unlikely that someone can come along and claim their way to conceive of a
complex number is superior or more correct.

For some reason, "some" people seem to want to say complex numbers are not
"real" because of the word "imaginary" used to define the y axis?

Some people thought (some still do) that the negative numbers aren't
real either. Yet, there's no harm in pretending they are as long as
correct answers are the result.

Richard Hachel

2025-02-07 13:46:13 UTC

Post by Chris M. Thomasson
For some reason, "some" people seem to want to say complex numbers are
not "real" because of the word "imaginary" used to define the y axis?

Complex roots must be placed on the x'Ox axis since that is the definition
of roots. It is particularly stupid to want to place them anywhere other
than on that axis.
We will say: yes, but we can't, there are no real roots.

As for the equation f(x)=x²+4x+5 or for the equation g(x)=(x²)²+2x²+3

We simply need to clearly define what a complex root is, and no one does
it (it is still Dr. Hachel who must explain it).

Complex roots are the roots of the mirror curve.

Thus we have as roots for f(x)=-2+i and -2-i which correspond perfectly to
-3 and -1 of the mirror curve, and that is where we must place the two
roots on the Cartesian coordinate system.

For the curve g(x) the roots are -i and +i. Which corresponds to the roots
of the mirror curve at the top which is g'(x)=-(x²)²-2x²+3 and whose
roots are 1 and -1.

I don't understand all this crazy stuff that mathematicians say, and that
artificial intelligence stupidly relays.

All this comes from a misunderstanding of what i is and what an imaginary
number is. For example, I am told that (i²)=-1 and therefore that
(i²)²=1. This is completely false. Here, the mathematician is
multiplying imaginaries with the laws of real numbers.
This is particularly stupid. We must set (i²)²=-1 which will avoid huge
mathematical blunders.

This seems surprising, but it is obvious when we have the keys to the
concepts to use correctly.

R.H.

FromTheRafters

2025-02-07 14:40:50 UTC

Post by Richard Hachel
All this comes from a misunderstanding of what i is and what an imaginary
number is.

That much is clear, since those are the same thing.

Richard Hachel

2025-02-06 22:15:30 UTC

Post by sobriquet
Ok, but that's a bit like people saying that 3 + 5 = 7 and then claiming
that usually mathematicians say that 3 + 5 = 8, but they have different
concepts that are more correct.
Unless you're able to demonstrate that your alternative concepts are
superior than the conventional way of defining these concepts, it
seems a bit silly.
The concept of a complex number didn't fall from the sky.. it was
developed over many centuries by multiple generations of mathematicians.
So it seems unlikely that someone can come along and claim their way to
conceive of a complex number is superior or more correct.

You may be right.

But I also know that mathematicians and physicists can also be wrong.
Descartes' works are full of errors, Berkeley and Newton did not agree on
the calculation of an infinitely small increment (I think Berkeley was
right), Lorentz wandered for years on relativistic transformations before
Poincaré gave it to him.

If you observe carefully, you will realize that basically, the notion of
complex numbers is very quickly presented, then very quickly skipped. You
are given an i²=-1, in order to square an awkward discriminant, and, if
the principle is correct (we multiply by 1, then we pose that i²=-1 by
convention and this allows us to have a positive root).

But no one has ever explained where this being comes from, which, in its
being, is an entity whose square is equal to -1. It is a convention.

Personally, I wish that we put this being under the microscope once and
for all, he fuuses imaginary.

It would seem that i is a special entity, which, like 1, can be used in
such a way that, whatever its exponent, it remains identical to itself in
its abstract being. -i is always equal to -i.

We can do as with 1^x give all the possible and imaginable powers to x,
always, 1^x=1.

It seems that it is the same thing with i=-1.

i²=-1.
i^(-1/2)=-1
i^4=-1
i°=-1

etc...etc...etc...

So we have to define i, because then, in the calculations, big sign errors
can appear. The biggest one can be that, sometimes, depending on the
concept i²=-1 or i²=1 if we simplify too quickly.

Let's take z1=16+9i and z2=14+3i and make a product.
We have: aa'+i(ab'+ba')+(ib')(ib)

If we make the product as is, we see that if i=-1 then (-9)(-3)=+27,
but if we square it, we have i²=-1 and i²bb'=-27

The error is then colossal:

Hachel finds Z=251+174i.

Mathematicians find Z=197+174i

We need to think about this and check what is correct and why it is
correct.

R.H.

Python

2025-02-06 22:44:39 UTC

Le 06/02/2025 à 23:15, Richard Hachel a écrit :
..

But no one has ever explained where this being comes from, which, in its being,
is an entity whose square is equal to -1. It is a convention.

Why do you keep lying Richard?

It is not a "convention" it is what it is: in the set R[X]/(X^2+1) i is
the equivalence class of X.

sobriquet

2025-02-07 00:24:25 UTC

Post by sobriquet
Ok, but that's a bit like people saying that 3 + 5 = 7 and then
claiming that usually mathematicians say that 3 + 5 = 8, but they have
different concepts that are more correct.
Unless you're able to demonstrate that your alternative concepts are
superior than the conventional way of defining these concepts, it
seems a bit silly.
The concept of a complex number didn't fall from the sky.. it was
developed over many centuries by multiple generations of
mathematicians. So it seems unlikely that someone can come along and
claim their way to conceive of a complex number is superior or more
correct.

You may be right.
But I also know that mathematicians and physicists can also be wrong.
Descartes' works are full of errors, Berkeley and Newton did not agree
on the calculation of an infinitely small increment (I think Berkeley
was right), Lorentz wandered for years on relativistic transformations
before Poincaré gave it to him.
If you observe carefully, you will realize that basically, the notion of
complex numbers is very quickly presented, then very quickly skipped.
You are given an i²=-1, in order to square an awkward discriminant, and,
if the principle is correct (we multiply by 1, then we pose that i²=-1
by convention and this allows us to have a positive root).
But no one has ever explained where this being comes from, which, in its
being, is an entity whose square is equal to -1. It is a convention.
Personally, I wish that we put this being under the microscope once and
for all, he fuuses imaginary.
It would seem that i is a special entity, which, like 1, can be used in
such a way that, whatever its exponent, it remains identical to itself
in its abstract being. -i is always equal to -i.
We can do as with 1^x give all the possible and imaginable powers to x,
always, 1^x=1.
It seems that it is the same thing with i=-1.
i²=-1.
i^(-1/2)=-1
i^4=-1
i°=-1
etc...etc...etc...
So we have to define i, because then, in the calculations, big sign
errors can appear. The biggest one can be that, sometimes, depending on
the concept i²=-1 or i²=1 if we simplify too quickly.
Let's take z1=16+9i and z2=14+3i and make a product.
We have: aa'+i(ab'+ba')+(ib')(ib)
If we make the product as is, we see that if i=-1 then (-9)(-3)=+27,
but if we square it, we have i²=-1 and i²bb'=-27
Hachel finds Z=251+174i.
Mathematicians find Z=197+174i
We need to think about this and check what is correct and why it is
correct.
R.H.

But what would be a compelling reason to accept one way to define
multiplication of complex numbers over another way?

With the standard way of defining complex numbers, if we pick any point
on the unit circle and we multiply it with any other point on the unit
circle, the result will end up on the unit circle. With your method,
this would fail (in the sense that the product would not end up on the
unit circle if the factors of the product are on the unit circle).
This seems like a compelling reason to me to prefer the standard way to
define multiplication of complex numbers over your alternative way.

You might say it's irrelevant whether complex numbers on the unit
circle multiply in a way that ends up on the unit circle, but other nice
properties also fail, like the way we can multiply complex numbers in
polar form by multiplying their modulus and adding their argument.

Richard Hachel

2025-02-07 11:14:26 UTC

Post by sobriquet
Ok, but that's a bit like people saying that 3 + 5 = 7 and then
claiming that usually mathematicians say that 3 + 5 = 8, but they have
different concepts that are more correct.
Unless you're able to demonstrate that your alternative concepts are
superior than the conventional way of defining these concepts, it
seems a bit silly.
The concept of a complex number didn't fall from the sky.. it was
developed over many centuries by multiple generations of
mathematicians. So it seems unlikely that someone can come along and
claim their way to conceive of a complex number is superior or more
correct.

You may be right.
But I also know that mathematicians and physicists can also be wrong.
Descartes' works are full of errors, Berkeley and Newton did not agree
on the calculation of an infinitely small increment (I think Berkeley
was right), Lorentz wandered for years on relativistic transformations
before Poincaré gave it to him.
If you observe carefully, you will realize that basically, the notion of
complex numbers is very quickly presented, then very quickly skipped.
You are given an i²=-1, in order to square an awkward discriminant, and,
if the principle is correct (we multiply by 1, then we pose that i²=-1
by convention and this allows us to have a positive root).
But no one has ever explained where this being comes from, which, in its
being, is an entity whose square is equal to -1. It is a convention.
Personally, I wish that we put this being under the microscope once and
for all, he fuuses imaginary.
It would seem that i is a special entity, which, like 1, can be used in
such a way that, whatever its exponent, it remains identical to itself
in its abstract being. -i is always equal to -i.
We can do as with 1^x give all the possible and imaginable powers to x,
always, 1^x=1.
It seems that it is the same thing with i=-1.
i²=-1.
i^(-1/2)=-1
i^4=-1
i°=-1
etc...etc...etc...
So we have to define i, because then, in the calculations, big sign
errors can appear. The biggest one can be that, sometimes, depending on
the concept i²=-1 or i²=1 if we simplify too quickly.
Let's take z1=16+9i and z2=14+3i and make a product.
We have: aa'+i(ab'+ba')+(ib')(ib)
If we make the product as is, we see that if i=-1 then (-9)(-3)=+27,
but if we square it, we have i²=-1 and i²bb'=-27
Hachel finds Z=251+174i.
Mathematicians find Z=197+174i
We need to think about this and check what is correct and why it is
correct.
R.H.

But what would be a compelling reason to accept one way to define
multiplication of complex numbers over another way?
With the standard way of defining complex numbers, if we pick any point
on the unit circle and we multiply it with any other point on the unit
circle, the result will end up on the unit circle. With your method,
this would fail (in the sense that the product would not end up on the
unit circle if the factors of the product are on the unit circle).
This seems like a compelling reason to me to prefer the standard way to
define multiplication of complex numbers over your alternative way.
You might say it's irrelevant whether complex numbers on the unit
circle multiply in a way that ends up on the unit circle, but other nice
properties also fail, like the way we can multiply complex numbers in
polar form by multiplying their modulus and adding their argument.

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

It is defined in a dramatically stupid way.

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.

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.

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

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

(i²)²=-1, and not 1.

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

R.H.

Alan Mackenzie

2025-02-07 13:27:05 UTC

Richard Hachel <***@tiscali.fr> wrote:

[ .... ]

Post by Richard Hachel
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.

Post by Richard Hachel
It is defined in a dramatically stupid way.

That is a spectacularly stupid thing to write.

Post by Richard Hachel
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.

Post by Richard Hachel
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.

Post by Richard Hachel
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)

Post by Richard Hachel
But, hold on tight, friends, this is false.
(i²)²=-1, and not 1.

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

Post by Richard Hachel
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.

Post by Richard Hachel
R.H.

--
Alan Mackenzie (Nuremberg, Germany).

Richard Hachel

2025-02-07 13:48:41 UTC

Post by Alan Mackenzie
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.

:))

R.H.

Python

2025-02-07 19:45:32 UTC

:))

Nothing is funny here, Richard. You're showing how a crank you are. It may
have been funny at first, it is not anymore.

Everyone noticed that you removes 90% of Alan's debunking of your claims.
In addition to be a crank you have absolutely no intellectual integrity.

Richard Hachel

2025-02-07 20:59:07 UTC

:))

Nothing is funny here, Richard. You're showing how a crank you are. It may have
been funny at first, it is not anymore.
Everyone noticed that you removes 90% of Alan's debunking of your claims. In
addition to be a crank you have absolutely no intellectual integrity.

Ah oui, tu as raison.

J'ai merdouillé, j'ai pas vu la suite du post, attends je vais voir.

R.H.

Moebius

2025-02-21 23:15:02 UTC

Post by Python
In addition to be a crank you have absolutely no intellectual integrity.

Talking about Mückenheim? :-o

Richard Hachel

2025-02-07 21:12:18 UTC

Post by Alan Mackenzie

Post by Richard Hachel
(i²)²=-1, and not 1.

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

Post by Richard Hachel
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.

Post by Richard Hachel
R.H.

No, it is not consistent.

When I speak in real terms, I set (-x)(-x)=x².

If x=-5, then x²=25.

So far, it is perfectly consistent.

But if I use another form of mathematics, and I set 1=-i², in order to
get rid of both the (-) sign and the square root, I have to go to the end
of the structure used. I can no longer do everything I want and anyhow. I
have to use this structure in a consistent way (it is, and it is
magnificent if we know how to use it well).

In an imaginary universe, it is a bit like practicing in a mirror, the
real becomes imaginary, and the imaginary becomes real.

But once in the imaginary, you have to stay WITH the laws of imaginary
operations, and the mistake you make is to say: if (-n)(-n)=n² in
reality, then it is obvious that (-i)(-i)=i² in the imaginary, then that
(i²)²=1 and so on.

This is not how to proceed, it is incoherent.

It is like hammering a nail with a screwdriver. Some can do it, but a
simple little hammer blow is more useful.

THIS is what mathematicians do without paying attention to their blunder.

But it is not a mathematically correct notion, and we play with numbers
without knowing what we are doing.

R.H.

Moebius

2025-02-21 23:24:45 UTC

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

A joke, right?!

Alan Mackenzie

2025-02-22 15:47:07 UTC

Post by Moebius

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

[ "It" being the theory of complex numbers. ]

Post by Moebius
A joke, right?!

Not at all.

--
Alan Mackenzie (Nuremberg, Germany).

sobriquet

2025-02-07 13:51:17 UTC