XPost: fr.sci.maths

x^4=-81

What is x?

R.H.

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On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2, (-1-𝑖)⋅3/√2}

https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
⎛ ...
⎜ every non-constant single-variable polynomial
⎜ with complex coefficients has at least one complex root.
⎜
⎜ The theorem is also stated as follows:
⎜ every non-zero, single-variable, degree n polynomial
⎜ with complex coefficients has,
⎝ counted with multiplicity, exactly n complex roots.

https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Proofs Real-analytic proofs
Complex-analytic proofs
Topological proofs
Algebraic proofs
By induction
From Galois theory
Geometric proofs

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Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2, (-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans trop expliquer pourquoi.

Personally, I propose only one root, but it is not in conformity with
what is said about "complex numbers".
I remind you that I do not admit the definition i²=-1, which, in itself,
is not false, but so narrow that I do not understand its semantic
interest.
For me, the definition must be extended to all powers of x such that
i^x=-1.
In short, it is necessary to go through a constancy of i^x=-1, like 1^x=1.
This defined, we easily have x^4=-81 if x=3i.
The other four roots being incorrect (in the proposed system).

R.H.

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Le 25/02/2025 à 21:07, Richard Hachel a écrit :

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2,
(-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans trop expliquer pourquoi.

Personally, I propose only one root, but it is not in conformity with what is
said about "complex numbers".

So it is not about complex as it is not only said but proven from the definition of complex numbers.

I remind you that I do not admit the definition i²=-1,

It is not the definition, it is a consequence of a definition that I've
shown you numerous times. You are a liar.

which, in itself, is not false, but so narrow that I do not understand its semantic interest
For me, the definition must be extended to all powers of x such that i^x=-1.

Which leads to immediate inconsistencies. This property is contradictory.

In short, it is necessary to go through a constancy of i^x=-1, like 1^x=1. This defined, we easily have x^4=-81 if x=3i.
The other four roots being incorrect (in the proposed system).

This is utter bullshit. As are all your your fantasies. Build on top of
your ignorance, hypocrisies, stupidity and arrogance.

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Le 25/02/2025 à 22:47, Richard Hachel a écrit :

Le 25/02/2025 à 21:38, Python a écrit :

Le 25/02/2025 à 21:07, Richard Hachel a écrit :

Le 25/02/2025 à 18:21, Jim Burns a écrit :

For me, the definition must be extended to all powers of x such that i^x=-1.

This property is contradictory.

Absolutely not.

Tu n'as toujours pas compris (comme tu n'as toujours pas compris toutes les équations que j'ai données
pour bonifier et rendre plus compréhensible la relativité restreinte).

This is not inserting a contradictory property, contradictory properties, in all my special relativity, I have knocked them out.
It is the complete opposite of my personality to propose contradictory properties.

It is actually a perfectly accurate definition of your personality. In
addition to fatuity, ignorance and dishonesty.

On the contrary, it is a fantastic basic principle that is affirmed.
We extend the property of the imaginary number i to all powers of i. The definition of i is then clearly named, and given.
The imaginary number i is a very useful imaginary number to perform many operations and whose definition is the invariance of its nature whatever the power
that we give it.
i^x=-1 whatever x. We see that i is the antithesis of 1.
It then becomes obvious that i^(1/2)=-1 and that i²=-1 as many say. But this is
not enough if they believe that i²*i²=1.
Here, there is a bias, we use a real operation on an imaginary structure. We must set i²*i²=i^4=-1.
This is NOT contradictory. It is an intrinsic part of my (Nazi) ideology.

R.H.

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Le 25/02/2025 à 21:38, Python a écrit :

Le 25/02/2025 à 21:07, Richard Hachel a écrit :

Le 25/02/2025 à 18:21, Jim Burns a écrit :

For me, the definition must be extended to all powers of x such that i^x=-1.

This property is contradictory.

Absolutely not.

Tu n'as toujours pas compris (comme tu n'as toujours pas compris toutes
les équations que j'ai données
pour bonifier et rendre plus compréhensible la relativité restreinte).

This is not inserting a contradictory property, contradictory properties,
in all my special relativity, I have knocked them out.
It is the complete opposite of my personality to propose contradictory properties.
On the contrary, it is a fantastic basic principle that is affirmed.
We extend the property of the imaginary number i to all powers of i. The definition of i is then clearly named, and given.
The imaginary number i is a very useful imaginary number to perform many operations and whose definition is the invariance of its nature whatever
the power that we give it.
i^x=-1 whatever x. We see that i is the antithesis of 1.
It then becomes obvious that i^(1/2)=-1 and that i²=-1 as many say. But
this is not enough if they believe that i²*i²=1.
Here, there is a bias, we use a real operation on an imaginary structure.
We must set i²*i²=i^4=-1.
This is NOT contradictory. It is an intrinsic part of my (Nazi) ideology.

R.H.

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Le 25/02/2025 à 22:49, "Chris M. Thomasson" a écrit :

On 2/25/2025 6:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

Try to forgive the floating point precision aspects, but, the roots are
the r's, raising them to the 4'th power gives the p's:
_____________________
r0 = (2.12132,2.12132)
r1 = (-2.12132,2.12132)
r2 = (-2.12132,-2.12132)
r3 = (2.12132,-2.12132)

p0 = (-81,-7.08124e-06)
p1 = (-81,-1.93183e-06)
p2 = (-81,-7.53158e-05)
p3 = (-81,4.57051e-05)
_____________________

To gain a root, here is my code:
_____________________
ct_complex
root_calc(
ct_complex const& z,
int p,
int n
) {
float radius = std::pow(std::abs(z), 1.0 / p);
float angle_base = std::arg(z) / p;
float angle_step = (CT_PI * 2.0) / p;
float angle = angle = angle_step * n;

ct_complex c = {
std::cos(angle_base + angle) * radius,
std::sin(angle_base + angle) * radius
};

return c;
}
_____________________

Also, this is not using floating point for roots, just signed integers.

This is quite complicated, where I propose to use the nature of the
imaginary number i in a somewhat particular way, and according to the new
idea that i is not only defined by i²=-1 or i=sqrt(-1), but rather with
the generalized idea that for all x, i^x=-1.
A bit like if this imaginary was the antithesis of 1 where for all x, then 1^x=1.

With this technique, we immediately have x=3i.

R.H.

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Op 25/02/2025 om 21:07 schreef Richard Hachel:

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2, (-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans
trop expliquer pourquoi.
Personally, I propose only one root, but it is not in conformity with
what is said about "complex numbers".
I remind you that I do not admit the definition i²=-1, which, in
itself, is not false, but so narrow that I do not understand its
semantic interest.
For me, the definition must be extended to all powers of x such that
i^x=-1.

Contradictory, thus impossible.
If i^2=-1 then (i^2)^2 = (-1)^2 = 1 =  i^4

--
guido wugi

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Le 25/02/2025 à 23:05, guido wugi a écrit :

Op 25/02/2025 om 21:07 schreef Richard Hachel:

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2,
(-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans
trop expliquer pourquoi.
Personally, I propose only one root, but it is not in conformity with
what is said about "complex numbers".
I remind you that I do not admit the definition i²=-1, which, in
itself, is not false, but so narrow that I do not understand its
semantic interest.
For me, the definition must be extended to all powers of x such that
i^x=-1.

Contradictory, thus impossible.
If i^2=-1 then (i^2)^2 = (-1)^2 = 1 =  i^4

Absolutely not...

Pfffff...

You are using real powers on imaginary numbers.

You don't seem to understand what I'm saying.

I am stating the precise, formal, and definitive definition that in this imaginary system, the fundamental law is i^x=-1.

A fundamental law is not negotiable.

Now, you start to negotiate, like Python, and to tell me while crying:
i^4=1.

This is not serious behavior.

R.H.

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Le 25/02/2025 à 23:08, "Chris M. Thomasson" a écrit :

On 2/25/2025 6:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

what about x^7 = -81 ?

Using my own code:

r0 = (1.68791,0.812857)
r1 = (0.41688,1.82647)
r2 = (-1.16807,1.46472)
r3 = (-1.87344,-1.63782e-07)
r4 = (-1.16807,-1.46472)
r5 = (0.416881,-1.82647)
r6 = (1.68792,-0.812856)

p0 = (-81,-2.63931e-05)
p1 = (-81,-1.93183e-06)
p2 = (-81,2.25295e-05)
p3 = (-81,-4.69908e-05)
p4 = (-81,-0.000177025)
p5 = (-81,-7.53158e-05)
p6 = (-81,-0.000263285)

x=-1.873444

R.H.

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Le 25/02/2025 à 23:15, Richard Hachel a écrit :

Le 25/02/2025 à 23:05, guido wugi a écrit :

Op 25/02/2025 om 21:07 schreef Richard Hachel:

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2,
(-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans
trop expliquer pourquoi.
Personally, I propose only one root, but it is not in conformity with
what is said about "complex numbers".
I remind you that I do not admit the definition i²=-1, which, in
itself, is not false, but so narrow that I do not understand its
semantic interest.
For me, the definition must be extended to all powers of x such that
i^x=-1.

Contradictory, thus impossible.
If i^2=-1 then (i^2)^2 = (-1)^2 = 1 =  i^4

Absolutely not...

Pfffff...

You are using real powers on imaginary numbers.

You don't seem to understand what I'm saying.

I am stating the precise, formal, and definitive definition that in this imaginary system, the fundamental law is i^x=-1.

A fundamental law is not negotiable.

Now, you start to negotiate, like Python, and to tell me while crying: i^4=1.

This is not serious behavior.

Yours is not.

You are contradicting this :

If a = b then a*a = b*b

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Le 25/02/2025 à 23:18, Python a écrit :

Le 25/02/2025 à 23:15, Richard Hachel a écrit :

Le 25/02/2025 à 23:05, guido wugi a écrit :

Op 25/02/2025 om 21:07 schreef Richard Hachel:

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2,
(-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans
trop expliquer pourquoi.
Personally, I propose only one root, but it is not in conformity with
what is said about "complex numbers".
I remind you that I do not admit the definition i²=-1, which, in
itself, is not false, but so narrow that I do not understand its
semantic interest.
For me, the definition must be extended to all powers of x such that
i^x=-1.

Contradictory, thus impossible.
If i^2=-1 then (i^2)^2 = (-1)^2 = 1 =  i^4

Absolutely not...

Pfffff...

You are using real powers on imaginary numbers.

You don't seem to understand what I'm saying.

I am stating the precise, formal, and definitive definition that in this
imaginary system, the fundamental law is i^x=-1.

A fundamental law is not negotiable.

Now, you start to negotiate, like Python, and to tell me while crying: i^4=1.

This is not serious behavior.

Yours is not.

You are contradicting this :

If a = b then a*a = b*b

What is impressive, pathologically, is that you do so, and couldn't spot
how silly it is, because of your self-indulgence.

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On Tue, 25 Feb 25 14:23:28 +0000, Richard Hachel <r.hachel@tiscali.fr>
wrote:

x^4=-81

What is x?

R.H.

One of the values of x is 1.5*sqrt(2)+1.5*sqrt(2)*i. You can find the
others yourself if you are willing to use standard math.

--
Remove del for email

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Le 26/02/2025 à 00:48, "Chris M. Thomasson" a écrit :

On 2/25/2025 2:20 PM, Richard Hachel wrote:

Le 25/02/2025 à 23:08, "Chris M. Thomasson" a écrit :

On 2/25/2025 6:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

[...]

x=-1.873444

To the 13'th power with higher precision:

roots[0] = (1.01898,0.251156)
roots[1] = (0.7855438,0.6959311)
roots[2] = (0.3721492,0.9812768)
roots[3] = (-0.1265003,1.041824)
roots[4] = (-0.5961701,0.8637015)
roots[5] = (-0.9292645,0.4877156)
roots[6] = (-1.049476,5.945845e-16)
roots[7] = (-0.9292645,-0.4877156)
roots[8] = (-0.5961701,-0.8637015)
roots[9] = (-0.1265003,-1.041824)
roots[10] = (0.3721492,-0.9812768)
roots[11] = (0.7855438,-0.6959311)
roots[12] = (1.01898,-0.251156)

raised[0] = (-1.873444,2.294307e-16)
raised[1] = (-1.873444,4.016197e-15)
raised[2] = (-1.873444,4.475059e-15)
raised[3] = (-1.873444,1.606015e-15)
raised[4] = (-1.873444,2.064877e-15)
raised[5] = (-1.873444,9.179548e-15)
raised[6] = (-1.873444,9.63841e-15)
raised[7] = (-1.873444,4.132072e-15)
raised[8] = (-1.873444,4.590934e-15)
raised[9] = (-1.873444,1.170561e-14)
raised[10] = (-1.873444,2.214818e-14)
raised[11] = (-1.873444,1.262333e-14)
raised[12] = (-1.873444,2.306591e-14)

I think that for the moment, we are making things terribly complicated.
If I ask you the cube root of 27?
Are you going to make a computer program?
Why make a computer program if I ask you the fourth root of -81?

The answer is simple and obvious. x=3i.

All these misunderstandings come from the fact that no clear and
universally usable definition of the imaginary number i has ever been
given.

Against all expectations, in analytical mathematics, i is an imaginary
unit such that, for all x, i^x=-1.

We see that saying that i²=-1 is completely legal.

Or that sqrt(i)=i^(1/2)=-1.

Certainly.

But we also see that (i²)² is not equal to 1, and that those who believe
it are corrupting themselves.

R.H.

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Le 26/02/2025 à 01:34, "Chris M. Thomasson" a écrit :

On 2/25/2025 4:11 PM, Richard Hachel wrote:

To the 13'th power with higher precision:

roots[0] = (1.01898,0.251156)
roots[1] = (0.7855438,0.6959311)
roots[2] = (0.3721492,0.9812768)
roots[3] = (-0.1265003,1.041824)
roots[4] = (-0.5961701,0.8637015)
roots[5] = (-0.9292645,0.4877156)
roots[6] = (-1.049476,5.945845e-16)
roots[7] = (-0.9292645,-0.4877156)
roots[8] = (-0.5961701,-0.8637015)
roots[9] = (-0.1265003,-1.041824)
roots[10] = (0.3721492,-0.9812768)
roots[11] = (0.7855438,-0.6959311)
roots[12] = (1.01898,-0.251156)

raised[0] = (-1.873444,2.294307e-16)
raised[1] = (-1.873444,4.016197e-15)
raised[2] = (-1.873444,4.475059e-15)
raised[3] = (-1.873444,1.606015e-15)
raised[4] = (-1.873444,2.064877e-15)
raised[5] = (-1.873444,9.179548e-15)
raised[6] = (-1.873444,9.63841e-15)
raised[7] = (-1.873444,4.132072e-15)
raised[8] = (-1.873444,4.590934e-15)
raised[9] = (-1.873444,1.170561e-14)
raised[10] = (-1.873444,2.214818e-14)
raised[11] = (-1.873444,1.262333e-14)
raised[12] = (-1.873444,2.306591e-14)

I think that for the moment, we are making things terribly complicated.
If I ask you the cube root of 27?
Are you going to make a computer program?
Why make a computer program if I ask you the fourth root of -81?

The answer is simple and obvious. x=3i.

The fourth root of -81+0i wrt power of 4 is *:

roots[0] = (2.12132,2.12132)
roots[1] = (-2.12132,2.12132)
roots[2] = (-2.12132,-2.121321)
*roots[3] = (2.12132,-2.121321)

I don't know what you x=3i even means right now. Any of these roots
raised to the 4'th power equals -81+0i.

We are not talking about the same thing, nor are we using the same
mathematics.

If I ask what are the complex roots of f(x)=x²+4x+5,
you will tell me that we must use [-b$sqrt(b²-4ac)]/2a using i.

And you will give me x'=-2+i and x"=-2-i.

Coordinates on x'Ox that I will immediately place in A(-3,0) and B(-1,0)
and which are the imaginary roots on y=0, the equation having no real
roots.

Well, I do the same to find the fourth root of -81.

x^4=-81.

For me, i ^x=-1 whatever 1.

x^4=-81 ---> x^4=-(i^4)(-81)=81(i^4)
x=3i

Conversely, x^4=(3i)^4 = 81(i^4) with i^4=-1 by definition of i for me.

But that has nothing to do with the Argand frame, which is something
completely different.

R.H.

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Op 25/02/2025 om 21:07 schreef Richard Hachel:

Le 25/02/2025 à 18:21, Jim Burns a écrit :

On 2/25/2025 9:23 AM, Richard Hachel wrote:

x^4=-81

What is x?

x ∈ {(+1+𝑖)⋅3/√2, (+1-𝑖)⋅3/√2, (-1+𝑖)⋅3/√2, (-1-𝑖)⋅3/√2}

Oui, c'est ce que dis aussi l'Intelligence artificielle, mais sans trop expliquer pourquoi.[..]

You have to understand that if we consider this equivalent case of
g(x) = x^4 + 9/4 and we visualize the complex roots as follows:

https://www.desmos.com/calculator/4mhohrcxlg

We're superimposing a real slice of the graph in the Cartesian plane on
top of Complex plane where we visualize the complex roots of the polynomial.

In order to understand why these are the roots of the polynomial g(x)
in the complex numbers, we have to expand our view to 3D:

https://www.desmos.com/3d/ykhhcpb3xz

Where we can visualize that polynomial as a function that maps complex
numbers to complex numbers as a combination of two surfaces in 3D
(one red surface for the real component of the output of the function
and an orange surface for the imaginary component of the output of the function).
We can press the icons in front of the corresponding items in the list
on the left to temporarily hide those surfaces to identify the locations
in the complex plane where both components are equal to zero to see that
those are exactly the complex roots of the polynomial g(x) in the
complex number system.

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Le 26/02/2025 à 01:37, Chris M. Thomasson a écrit :

Why do you seem to misunderstand complex numbers so much?

Don't lose your time answering this disturbed and provocative person. He
mainly needs to be admitted to a psychiatric hospital. The sooner, the
better.

--
F.J.

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Op 27/02/2025 om 17:20 schreef Richard Hachel:

Le 27/02/2025 à 08:16, Moebius a écrit :

((+1+𝑖)⋅3/√2)^4 = -81
((+1-𝑖)⋅3/√2)^4 = -81
((-1+𝑖)⋅3/√2)^4 = -81
((-1-𝑖)⋅3/√2)^4 = -81

((+1+𝑖)⋅3/√2)^4 = -81
((+1+𝑖)^4(3/√2)^4 = -81
(1+2i+i²)(1+2i+i²)(3/√2)^4=-81
i²=-1
(2i²)(81/4)=(i²)(81/2)=-81

Here you go wrong.. it should say:
(2i)²(81/4)=(i²)81=-81

i²=-1
-40.5=-81

C'est absurde.

Etc....
Non, non, non, non...

(3i²)^4=-81
i^4=-1 (Hachel copyrights)
3^4=81

R.H.

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On 2/27/2025 2:35 PM, Jim Burns wrote:

On 2/27/2025 10:42 AM, Richard Hachel wrote:

Le 27/02/2025 à 08:47, joes a écrit :

It seems that the imaginary unit i is
a special unit such that i^x=-1 whatever x.

Multiplication by 𝑖 is a ¹/₄.turn of the complex plane.

Multiplication twice by 𝑖 is a ¹/₂.turn, which is
multiplication by -1

The four solutions to x⁴ = -1 are
a ¹/₈.turn, a ³/₈.turn, a ⁵/₈.turn, and a ⁷/₈.turn
Repeated four times, they are
a ¹/₂.turn, a 1¹/₂.turn, a 2¹/₂.turn, and a 3¹/₈.turn,
which are multiplication by -1, -1, -1, and -1.

Multiplication by 3 is left as an exercise for the reader.

There is a very nice formula expressing all this.
𝑒ⁱᶿ = sin(θ) + 𝑖⋅cos(θ)

D'oh!
𝑒ⁱᶿ = cos(θ) + 𝑖⋅sin(θ)

x⁴ = -81
x ∈ { 3𝑒ⁱᐧᵖⁱᐟ⁴, 3𝑒³ⁱᐧᵖⁱᐟ⁴, 3𝑒⁵ⁱᐧᵖⁱᐟ⁴, 3𝑒⁷ⁱᐧᵖⁱᐟ⁴ }

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Le 27/02/2025 à 21:53, Jim Burns a écrit :

On 2/27/2025 2:35 PM, Jim Burns wrote:

There is a very nice formula expressing all this.
𝑒ⁱᶿ = sin(θ) + 𝑖⋅cos(θ)

D'oh!

𝑒ⁱᶿ = cos(θ) + 𝑖⋅sin(θ)

Yes, it's more better. :))

R.H.

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Le 28/02/2025 à 00:53, "Chris M. Thomasson" a écrit :

Claro que si.
Pero, i*i=i²=-1 ; (i²)²=-1
It seems that the imaginary unit i is a special unit such that i^x=-1
whatever x.

Mathematicians are right when they say that i=-1, that i^(-1/2)=-1, that
i²=-1.
But if we understand Dr. Hachel's idea, we see that these three true
statements are not enough.
Hachel imposes that i is an imaginary unit such that i^x=-1 whatever x.
This confuses the mathematician, who is used to working with real
numbers, and who sets a²*a²=a^4 with systematically a>0.
But here we are not working with real numbers, but with the imaginary i.
It is not the same thing: we must systematically set i^x=-1 for all x.

It kind of seems like you deny that the y axis even exists?

Where do you see that I deny the existence of the y-axis?
I do not deny the existence of the y-axis, on the contrary, I affirm that
in a Cartesian coordinate system, there are two axes, and that we can draw
a coordinate system Ox, Oy.
In this coordinate system, we can draw straight lines, curves, etc...
We can draw, for example f(x)=2x+1, or g(x)=sqrt(x)+2, or h(x)=x²+2x+1.
Now, we can look for roots, that is to say the place where these functions cross the y=0 axis. If there are no roots, we can sometimes look for
complex roots.
Simply, in Hachel, all real or complex roots must be on the x'Ox axis,
that is to say on the y=0 line.
Trying to place roots "elsewhere" is particularly absurd, since by
definition, the place where a function crosses x'Ox is on x'Ox.
We will then say, but if the function does not cross x'Ox, what happens,
like for example f(x)=x²+4x+5 which has no real roots. We must then
rotate the curve in such a way that roots will appear, which will be the
real roots of the mirror curve, and at the same time the complex roots of
the original curve.
Thus, the complex roots of a curve are the real roots of the mirror curve,
and vice versa.
We will find here that f(x) has two complex roots which are x'=3i, and
x"=i which are the same points as x'=-3 and x'=-1 but noted differently, depending on whether we indicate that they are the real roots of g(x) or
the complex roots of f(x).
The points A and B thus noted on the Cartesian coordinate system are
A(-3,0) and B(-1,0), or, in mirror, complex coordinates, A(3i,0) and
B(i,0).

Thus we note that the complex axis IS the x'Ox axis, but inverted, and
that it only concerns the abscissas. The y-axis remains in its place, and
is used to position the ordinate.

Then, we can resort, for other reasons, to an Argand-Gauss coordinate
system, which we use in a completely different way (orthogonalization of x
in a+ib), but these are two very different things, and two totally
different reference points that should not be confused and even less used
as confused on the same diagram.

R.H.

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Op 28/02/2025 om 01:50 schreef Richard Hachel:

Le 28/02/2025 à 00:53, "Chris M. Thomasson" a écrit :

Claro que si.
Pero, i*i=i²=-1 ; (i²)²=-1
It seems that the imaginary unit i is a special unit such that i^x=-1
whatever x.

Mathematicians are right when they say that i=-1, that i^(-1/2)=-1,
that i²=-1.
But if we understand Dr. Hachel's idea, we see that these three true
statements are not enough.
Hachel imposes that i is an imaginary unit such that i^x=-1 whatever x.
This confuses the mathematician, who is used to working with real
numbers, and who sets a²*a²=a^4 with systematically a>0.
But here we are not working with real numbers, but with the imaginary i. >>> It is not the same thing: we must systematically set i^x=-1 for all x.

It kind of seems like you deny that the y axis even exists?

Where do you see that I deny the existence of the y-axis?
I do not deny the existence of the y-axis, on the contrary, I affirm
that in a Cartesian coordinate system, there are two axes, and that we
can draw a coordinate system Ox, Oy.
In this coordinate system, we can draw straight lines, curves, etc...
We can draw, for example f(x)=2x+1, or g(x)=sqrt(x)+2, or h(x)=x²+2x+1.
Now, we can look for roots, that is to say the place where these
functions cross the y=0 axis. If there are no roots, we can sometimes
look for complex roots.
Simply, in Hachel, all real or complex roots must be on the x'Ox axis,
that is to say on the y=0 line.
Trying to place roots "elsewhere" is particularly absurd, since by definition, the place where a function crosses x'Ox is on x'Ox.
We will then say, but if the function does not cross x'Ox, what happens,
like for example f(x)=x²+4x+5 which has no real roots. We must then
rotate the curve in such a way that roots will appear, which will be the
real roots of the mirror curve, and at the same time the complex roots
of the original curve.
Thus, the complex roots of a curve are the real roots of the mirror
curve, and vice versa.
We will find here that f(x) has two complex roots which are x'=3i, and
x"=i which are the same points as x'=-3 and x'=-1 but noted differently, depending on whether we indicate that they are the real roots of g(x) or
the complex roots of f(x).
The points A and B thus noted on the Cartesian coordinate system are
A(-3,0) and B(-1,0), or, in mirror, complex coordinates, A(3i,0) and
B(i,0).

Thus we note that the complex axis IS the x'Ox axis, but inverted, and
that it only concerns the abscissas. The y-axis remains in its place,
and is used to position the ordinate.

Then, we can resort, for other reasons, to an Argand-Gauss coordinate
system, which we use in a completely different way (orthogonalization of
x in a+ib), but these are two very different things, and two totally different reference points that should not be confused and even less
used as confused on the same diagram.

R.H.

Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2.

https://youtu.be/GZegwJVC_Pc?t=67

You claim that -1 * -1 = -1
since you claim that i^4 = -1 = i^2 * i^2 = -1 * -1

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Le 28/02/2025 à 02:40, sobriquet a écrit :

Op 28/02/2025 om 01:50 schreef Richard Hachel:

Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2.

Ce n'est pas ce que j'ai dit.

You claim that -1 * -1 = -1

Ce n'est pas non plus ce que j'ai dit.

since you claim that i^4 = -1 = i^2 * i^2

Oui, ça, je l'ai dit.

= -1 * -1

Non.

R.H.

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Op 28/02/2025 om 02:49 schreef Richard Hachel:

Le 28/02/2025 à 02:40, sobriquet a écrit :

Op 28/02/2025 om 01:50 schreef Richard Hachel:

Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2.

Ce n'est pas ce que j'ai dit.

You claim that -1 * -1 = -1

Ce n'est pas non plus ce que j'ai dit.

since you claim that i^4 = -1 = i^2 * i^2

Oui, ça, je l'ai dit.

= -1 * -1

Non.

R.H.

Ok, so which of the following statements are false according to you?

i^4 = -1

i^2 = -1

i^2 = i * i

i^4 = i * i * i * i

i^4 = i^2 * i^2

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Le 28/02/2025 à 03:24, sobriquet a écrit :

Op 28/02/2025 om 02:49 schreef Richard Hachel:

Le 28/02/2025 à 02:40, sobriquet a écrit :

Op 28/02/2025 om 01:50 schreef Richard Hachel:

Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2.

Ce n'est pas ce que j'ai dit.

You claim that -1 * -1 = -1

Ce n'est pas non plus ce que j'ai dit.

since you claim that i^4 = -1 = i^2 * i^2

Oui, ça, je l'ai dit.

= -1 * -1

Non.

R.H.

Ok, so which of the following statements are false according to you?

i^4 = -1

C'est vrai.

i^2 = -1

C'est vrai.

i^2 = i * i

C'est vrai.

i^4 = i * i * i * i

C'est vrai.

i^4 = i^2 * i^2

C'est vrai.

R.H.

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Op 28/02/2025 om 14:05 schreef Richard Hachel:

Le 28/02/2025 à 03:24, sobriquet a écrit :

Op 28/02/2025 om 02:49 schreef Richard Hachel:

Le 28/02/2025 à 02:40, sobriquet a écrit :

Op 28/02/2025 om 01:50 schreef Richard Hachel:

Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2.

Ce n'est pas ce que j'ai dit.

You claim that -1 * -1 = -1

Ce n'est pas non plus ce que j'ai dit.

since you claim that i^4 = -1 = i^2 * i^2

Oui, ça, je l'ai dit.

= -1 * -1

Non.

R.H.

Ok, so which of the following statements are false according to you?

i^4 = -1

C'est vrai.

i^2 = -1

C'est vrai.

i^2 = i * i

C'est vrai.

i^4 = i * i * i * i

C'est vrai.

i^4 = i^2 * i^2

C'est vrai.

R.H.

So basically you're saying

-1 * -1 = -1

Because that is the way the equality relation works in math.

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Le 28/02/2025 à 14:16, sobriquet a écrit :

Op 28/02/2025 om 14:05 schreef Richard Hachel:

Le 28/02/2025 à 03:24, sobriquet a écrit :

Op 28/02/2025 om 02:49 schreef Richard Hachel:

Le 28/02/2025 à 02:40, sobriquet a écrit :

Op 28/02/2025 om 01:50 schreef Richard Hachel:

Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2.

Ce n'est pas ce que j'ai dit.

You claim that -1 * -1 = -1

Ce n'est pas non plus ce que j'ai dit.

since you claim that i^4 = -1 = i^2 * i^2

Oui, ça, je l'ai dit.

= -1 * -1

Non.

R.H.

Ok, so which of the following statements are false according to you?

i^4 = -1

C'est vrai.

i^2 = -1

C'est vrai.

i^2 = i * i

C'est vrai.

i^4 = i * i * i * i

C'est vrai.

i^4 = i^2 * i^2

C'est vrai.

R.H.

So basically you're saying

-1 * -1 = -1

No.

Because that is the way the equality relation works in math.

No.

Not in complex mathematics.

If you want to use complex numbers, you have to use the laws of complex numbers.
So you have to use i^x=-1 whatever x is.
Otherwise, it's like eating soup with a fork.

R.H.

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Le 28/02/2025 à 14:26, Richard Hachel a écrit :

If you want to use complex numbers

Please, stop using the term "complex" for your shitty delirium. Thanks.

--
F.J.

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Le 28/02/2025 à 14:29, efji a écrit :

Le 28/02/2025 à 14:26, Richard Hachel a écrit :

If you want to use complex numbers

Please, stop using the term "complex" for your shitty delirium. Thanks.

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Le 28/02/2025 à 14:29, efji a écrit :

Le 28/02/2025 à 14:26, Richard Hachel a écrit :

If you want to use complex numbers

Please, stop using the term "complex" for your shitty delirium. Thanks.

I refuse.
I call a complex number an imaginary number that involves the imaginary
unit i.
And I don't see why I should be forbidden from voting for the imaginary candidate I have chosen.

R.H.

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On 2/28/2025 8:57 AM, Richard Hachel wrote:

Le 28/02/2025 à 14:29, efji a écrit :

Le 28/02/2025 à 14:26, Richard Hachel a écrit :

If you want to use complex numbers

Please, stop using the term "complex"
for your shitty delirium. Thanks.

I refuse.
I call a complex number
an imaginary number that involves
the imaginary unit i.
And I don't see why
I should be forbidden from voting for
the imaginary candidate I have chosen.

https://en.wikipedia.org/wiki/Cooperative_principle
⎛
⎜ These describe specific rational principles observed
⎜ by people who follow the cooperative principle
⎜ in pursuit of effective communication.
⎜ Applying the Gricean maxims is therefore
⎜ a way to explain the link between utterances and
⎝ what is understood from them.

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Am 28.02.2025 um 14:16 schrieb sobriquet:

So basically you're saying

-1 * -1 = -1

From this we get (with division by -1 on both sides):

-1 = 1.

And from this (with addition of 1 on both sides):

0 = 2 .

Division by 2 gives:

0 = 1 .

That's a fine result!

Moreover it implies

1 * 1 = 0 * 0 = 0 = 2 .

Just as you said:

"Your approach to math closely resembles Terrence Howard who claims

1 * 1 = 2."

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Am 28.02.2025 um 14:29 schrieb efji:

Le 28/02/2025 à 14:26, Richard Hachel a écrit :

If you want to use complex numbers

Please, stop using the term "complex" for your shitty delirium. Thanks.

I suggest /delusional/ instead.

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Le 28/02/2025 à 15:26, Moebius a écrit :

Am 28.02.2025 um 14:29 schrieb efji:

Le 28/02/2025 à 14:26, Richard Hachel a écrit :

If you want to use complex numbers

Please, stop using the term "complex" for your shitty delirium. Thanks.

I suggest /delusional/ instead.

Shut up.

When you are up to it you will be able to speak.

And that is far from being your case.

You're a buffoon like the others.

A small shrill bell carried by the wind.

R.H

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