Le jeudi 24 février 2022 à 06:01:39 UTC+1, Ross A. Finlayson a écrit :
> On Sunday, February 20, 2022 at 6:42:01 AM UTC-8, Michael Moroney wrote:
> Cross product only exists in 3 and 7 dimensions.
>
> There are no knots in four dimensions.
>
> You'd be better off studying Cartan.
Thanks for your reply.

I know that there are theorems that show that there cannot be complex number with 3, 5, … dimensions. But a theorem has its domain of application. For example, triangle that does not satisfy Pythagorean theorem is forcefully not a triangle.

However, if you reject such triangle just with Pythagorean theorem, you will never get the geometry on the surface of a sphere. The domain of application of Pythagorean theorem is a plane, but it is no longer true on sphere or curved space.

So, if something does not obey some theorems, it is either wrong or a big discovery because it is outside the field of the theorems, like non-Euclidean geometry.

My complex number systems are constructed outside the field of the cited theorems and they work well without violating any theorem.

For example, my 3D complex numbers are in the form a+bi+cj, they can be added and multiplied which give result a’+b’i+c’j. They have conjugate and are in one to one correspondence with 3D Space.

PK

On Wednesday, February 23, 2022 at 8:27:40 PM UTC-5, tita...@gmail.com wrote:
> Le mercredi 23 février 2022 à 23:27:04 UTC+1, timba...@gmail.com a écrit :
> > On Sunday, February 20, 2022 at 12:05:26 PM UTC-5, tita...@gmail.com wrote:
> > > Le dimanche 20 février 2022 à 15:42:01 UTC+1, Michael Moroney a écrit :
> > > > On 2/20/2022 9:16 AM, Timothy Golden wrote:
> > > > If I understand correctly, there is a mathematical proof that a 3
> > > > dimensional field of mathematics with a set of 3 equivalent orthogonal
> > > > components which you call (h,i,j) is not mathematically possible? The
> > > > closest possible are the quaternions which have an unequal 4th
> > > > component, the real component. Is this correct?
> > > > I do as well.
> > >
> > > Thank you.
> > >
> > > The proof that "3 dimensional field of mathematics with a set of 3 equivalent orthogonal components which you call (h,i,j) is not mathematically possible" has a premise that the products of imaginary units are well defined, such as ij, jk, ki etc. In this case, 3 dimensional field cannot be put into complex space.
> > >
> > > But in my system, (h, i, j) is a space where ij is not defined and in higher space, ij, jk, ik etc do not exist. Multiplication of complex number with n dimensions are done in trigo or exponential forms. 3D complex numbers multiply together using the law I have defined, which state that the arguments of each number add together in the argument of the resulting number.. For example, if the argument of A is 2+3i and that of B is 5i+3j, then, the argument of the product is 2+3i+5i+3j=2+8i+3j. And the product equals e^(2+8i+3j), The modulus of A and B being 1.
> > >
> > > In this product, ij does not exist.
> > >
> > > PK
> > Is there no way to compute
> > ( a h + b i + c j )( d h + e i + f j ) ?
> > If there is a way to go through your trig and get this result; not necessarily in a compact format, but hopefully in a meaningful format, then I think this should be done.
> >
> > Of course my choice of values is primitive here; these could be z11 through z23 and be more systematic.
> The formula of the product ( a h + b i + c j )( d h + e i + f j ) would be too complex. So, I have used this instead
> v1=(a1+b1 i) c1+d1 j
> v2=(a2+b2 i) c2+d2 j
>
> I have derived the product below.
> v1 v2=((a1 a2-b1 b2 )+(b1 a2+a1 b2 )i)(c1 c2-d1 d2 )+(d1 c2+c1 d2 )j
>
> I have put these equations as eq. 1 and 2 in
> https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
> which is a draft. I will put it cleaner tomorrow.
> PK
Well, you've really gotten my ear with your Signet non défini.

Above don't you mean:
v1=(a1+b1 i) (c1+d1 j)
v2=(a2+b2 i) (c2+d2 j)

but now you are getting quite some sensible vector symmetry going.
Possibly you have to call this a new algebra?
Not really sure yet how you get half sensible statements like yours.
Waiting to find a kink in your armor, yet hoping there is none.
Almost as if you inserted a kink in your own armor as a gotcha.

On 2/24/2022 12:38 PM, PengKuan Em wrote:

> My complex number systems are constructed outside the field of the cited theorems and they work well without violating any theorem.
>
> For example, my 3D complex numbers are in the form a+bi+cj, they can be added and multiplied which give result a’+b’i+c’j. They have conjugate and are in one to one correspondence with 3D Space.
>
If we have two numbers, a+bi and c+dj, how do I multiply them together?
What happens with a "ij" term in the result?

On Thursday, February 24, 2022 at 12:53:18 PM UTC-5, Michael Moroney wrote:
> On 2/24/2022 12:38 PM, PengKuan Em wrote:
>
> > My complex number systems are constructed outside the field of the cited theorems and they work well without violating any theorem.
> >
> > For example, my 3D complex numbers are in the form a+bi+cj, they can be added and multiplied which give result a’+b’i+c’j. They have conjugate and are in one to one correspondence with 3D Space.
> >
> If we have two numbers, a+bi and c+dj, how do I multiply them together?
> What happens with a "ij" term in the result?

Incindentally (or not) standard vector systems that physics is working on fail to have any arithmetic product defined while we all abide them as respectable. Here to say you have a half-assed product defined will get you plenty farther. If some compelling physical correspondence exists, and I would have to give you Peng Kuan the credit for such an interpretation as I have no existing interpretation of your product; then your work could become relevant to physics rather than just being another odd-ball product.

I have to assure you that polysign in P4 are the 3D complex numbers, and they obey the algebraic principles just as C and R do. These are P3 and P2 respectively, and their little sibling P1 gets no shrift; even getting the boot from me early on. That has changed for some time and the singleton does carry persuasive force in the n-ary department of being.

Le jeudi 24 février 2022 à 18:53:18 UTC+1, Michael Moroney a écrit :
> On 2/24/2022 12:38 PM, PengKuan Em wrote:
>
> If we have two numbers, a+bi and c+dj, how do I multiply them together?
> What happens with a "ij" term in the result?

The product of these 2 numbers is the following:
(a+bi)(c+dj) =ac+bc i+d√(a^2+b^2 ) j

The derivation of this formula is in
https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
from eq. 18 to 23 and the final formula with my variable names is
(A1+B1 i)(A2+D2 j) =A1 A2+B1 A2 i+D2 √(A1^2+B1^2 ) j
and there is not ij.

Why there is not ij? This is because the multiplication is not defined as for 2D complex number which is classical algebraic operation. 3D multiplication is defined as a rotation in space, which is like for 2D complex numbers.. When two 2D complex numbers multiply, their argument add together, which corresponds to a rotation in the plane (1, i). 3D multiplication is a rotation in the 3D space (1,i,j).

a+bi is in the plane (1,i), c+dj is in the plane (1,j). When they multiply, the argument of a+bi add with that of c+dj. Let the argument of a+bi be alfa*i, that of c+dj be beta*j. The argument of the product equals alfa*i + beta*j. Then, the product equals
| a+bi | | c+dj | e^ alfa*i e^ beta*j
This is a point in the 3D space (1,i,j).

In the original paper, the expression for 3D complex number in Cartesian form is the equation 16, in trigonometric form is the equation 32, the multiplication formula is the equation 52. A 3D complex number is represented geometrically in fig 3, 3D multiplication-rotation is represented geometrically in fig 5 and 6. These equations and figs are in the original paper which is here:
https://www.academia.edu/71708344/Extending_complex_number_to_spaces_with_3_4_or_any_number_of_dimensions

PK

Le jeudi 24 février 2022 à 18:48:16 UTC+1, timba...@gmail.com a écrit :
> On Wednesday, February 23, 2022 at 8:27:40 PM UTC-5, tita...@gmail.com wrote:
> Well, you've really gotten my ear with your Signet non défini.
>
> Above don't you mean:
> v1=(a1+b1 i) (c1+d1 j)
> v2=(a2+b2 i) (c2+d2 j)
>
> but now you are getting quite some sensible vector symmetry going.
> Possibly you have to call this a new algebra?
> Not really sure yet how you get half sensible statements like yours.
> Waiting to find a kink in your armor, yet hoping there is none.
> Almost as if you inserted a kink in your own armor as a gotcha.

These numbers are in trigonometric form
v1=(a1+b1 i) (c1+d1 j)
v2=(a2+b2 i) (c2+d2 j)
which means: a1= cos theta, b1=sin theta, c1=cos psi, d1=sin psi
with the moduli of v1 and v2 = 1

These numbers are also expressed in Cartesian form as below:
v1=(a1+b1 i) c1+d1 j
v2=(a2+b2 i) c2+d2 j

Here, the imaginary units i and j are indicators of the cartesian components for i and j dimensions, like for A+Bi+Cj

But when multiplying, we have to do with the former expression:
v1v2=[ (a1+b1 i) (c1+d1 j) ] [ (a2+b2 i) (c2+d2 j) ]
= [ (a1+b1 i) (a2+b2 i) (c1+d1 j) (c2+d2 j)
This way, the parts of the dimension i are together and that of the dimension j are together. Then, we add the arguments of the dimension i together and the arguments of the dimension j together. There is no cross product ij.

Because this is not like classical algebraic multiplication, we can call it a new algebra.

"kink in your own armor"
In fact, I have just reproduced the geometrical rotation with algebra. So, I think if kink there is, it would be in geometry. But Euclid was right.

PK

Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> On Thursday, February 24, 2022 at 12:53:18 PM UTC-5, Michael Moroney wrote:
> Incindentally (or not) standard vector systems that physics is working on fail to have any arithmetic product defined while we all abide them as respectable. Here to say you have a half-assed product defined will get you plenty farther. If some compelling physical correspondence exists, and I would have to give you Peng Kuan the credit for such an interpretation as I have no existing interpretation of your product; then your work could become relevant to physics rather than just being another odd-ball product.
>
> I have to assure you that polysign in P4 are the 3D complex numbers, and they obey the algebraic principles just as C and R do. These are P3 and P2 respectively, and their little sibling P1 gets no shrift; even getting the boot from me early on. That has changed for some time and the singleton does carry persuasive force in the n-ary department of being.

In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.

My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.

As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”

On 2/17/2022 12:42 PM, PengKuan Em wrote:
> Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional complex systems are extensions of the classical complex number system.
> ***
>
> In about 500 years after the birth of complex number, there were several attempts to extend complex number to more than 2 dimensions, for example we have theories such as quaternions, tessarines, coquaternions, biquaternions, and octonions. But none has reached the success of the classical complex number in 2 dimensions. Among these theories the most famous is quaternion which has found use in computational geometry. But quaternion is a 4 dimensional complex number but is used in 3 dimensional vector space, which is somewhat awkward.
>
> In this article we will show that multidimensional complex number with 3, 4 or more dimensions exist and will explain how to construct them. Like classical complex number system, a multidimensional complex number system possesses algebraic operations in its complex space that have geometrical meaning in the corresponding vector space.
>
> In the following exposition, spaces with 3, 4 or n dimensions will be referred to as 3D, 4D and nD spaces and the corresponding complex numbers as 3D, 4D and nD complex numbers. Since a complex number corresponds to a vector, a complex number will be referred to as a vector when convenient. We will begin with constructing 3D complex number system. Then we will generalize to spaces with 4 and more dimensions.
>
> The 3D complex number system is constructed from a 2D complex number system which is the classical complex number system. So, let us see how classical complex number works.
>
> Classical complex number
>
> Classical complex space is a plane with two orthogonal axes, see Figure 1:
> The axis of real numbers which is labeled as h.
> The axis of imaginary numbers which is labeled as i.
>
> This plane is labeled as (h, i). On this plane a complex number is both a point and a vector, for example the vector u in Figure 1. u makes the angle  with the axis h and and its length is |u|. As complex number, u’s argument is  and its modulus is |u|. In polar coordinate system the complex number u is expressed in equation (1), where i is the imaginary unit, see (3). Equation (1) is referred to as the trigonometric form of u.
>
> We develop (1) into (2) in which we introduce (4) and obtain (5) where the numbers ‘a’ and b are the Cartesian coordinates of u. So, equation (5) is referred to as the Cartesian form of u.
>
> Equation (6) is the Euler’s formula for  and i, and is introduced into (1) which becomes (7). Equation (7) expresses u in the form of an exponential function and is referred to as the exponential form of u. So, a classical complex number can be expressed in Cartesian, trigonometric or exponential form and has a geometrical meaning which is the vector u in Figure 1.
>
> 3D complex number
>
> 3D space and vector
> A 3D complex number is also a vector, which we will construct from the 2D plane (h, i). For doing so, we add the axis j perpendicularly to the plane (h, i) and obtain the 3D space whose axes are labeled as h, i and j, see Figure 2. We refer to this space as (h, i, j).
>
> We attach the unit vectors eh, ei and ej to the axes h, i and j respectively. The 3D space based on these vectors is referred to as (eh, ei, ej). We have then two 3D spaces: the complex space (h, i, j) and the vector space (eh, ei, ej). We will create a vector labeled as v in (eh, ei, ej) which corresponds to a 3D complex number in (h, i, j) labeled also as v.
>
> With the help of Figure 2 we create the vector v in the desired form by starting with a vector u which is expressed in (8) with |u| being its modulus and  the angle it makes with the axis h. So, u is in the horizontal plane (eh, ei). Dividing u by |u| gives the unit vector eu, see (9). The unit vectors eu and ej are the basis vectors of the vertical plane (eu, ej), see Figure 2. The vector v is created by rotating the vector u in this plane toward the axis j. The angle of rotation is , so v is expressed with the angle  on the basis vectors eu and ej in (10).
>
> As the length of u stays the same during the rotation, the modulus of u and v are equal, see (11). Introducing the expression of eu (9) into (10) gives (12) which is developed into (13) using (11). The vector v is expressed with its modulus |v| and the angles  and  on the basis vectors eh, ei and ej, see (13).
>
> Notice that the angle  is between the vector v and the horizontal plane (eh, ei), see Figure 2, which is different from the usual spherical coordinate system where the angle  is between the vector v and the axis j. So, when u is horizontal the angle  equals zero rather then /2.
>
> See the article with figures and equations here
> https://www.academia.edu/71708344/Extending_complex_number_to_spaces_with_3_4_or_any_number_of_dimensions
> https://pengkuanonmaths.blogspot.com/2022/02/extending-complex-number-to-spaces-with.html
>

Post it over on https://fractalforums.org. There are a lot of people who
will try to create fractals with it. I do not have the time to give it a
really deep dive right now. Fwiw, here is some of my work:

https://fractalforums.org/index.php?action=gallery;sa=view;id=6714

On Wednesday, February 23, 2022 at 11:33:11 PM UTC-5, Jan wrote:
> On Thursday, February 17, 2022 at 12:45:10 PM UTC-8, tita...@gmail.com wrote:
> > Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional complex systems are extensions of the classical complex number system.
> > ***
> >
> > In about 500 years after the birth of complex number, there were several attempts to extend complex number to more than 2 dimensions, for example we have theories such as quaternions, tessarines, coquaternions, biquaternions, and octonions. But none has reached the success of the classical complex number in 2 dimensions. Among these theories the most famous is quaternion which has found use in computational geometry. But quaternion is a 4 dimensional complex number but is used in 3 dimensional vector space, which is somewhat awkward.
> >
> > In this article we will show that multidimensional complex number with 3, 4 or more dimensions exist and will explain how to construct them. Like classical complex number system, a multidimensional complex number system possesses algebraic operations in its complex space that have geometrical meaning in the corresponding vector space.
> >
> > In the following exposition, spaces with 3, 4 or n dimensions will be referred to as 3D, 4D and nD spaces and the corresponding complex numbers as 3D, 4D and nD complex numbers. Since a complex number corresponds to a vector, a complex number will be referred to as a vector when convenient. We will begin with constructing 3D complex number system. Then we will generalize to spaces with 4 and more dimensions.
> >
> > The 3D complex number system is constructed from a 2D complex number system which is the classical complex number system. So, let us see how classical complex number works.
> >
> > Classical complex number
> >
> > Classical complex space is a plane with two orthogonal axes, see Figure 1:
> > The axis of real numbers which is labeled as h.
> > The axis of imaginary numbers which is labeled as i.
> >
> > This plane is labeled as (h, i). On this plane a complex number is both a point and a vector, for example the vector u in Figure 1. u makes the angle  with the axis h and and its length is |u|. As complex number, u’s argument is  and its modulus is |u|. In polar coordinate system the complex number u is expressed in equation (1), where i is the imaginary unit, see (3). Equation (1) is referred to as the trigonometric form of u.
> >
> > We develop (1) into (2) in which we introduce (4) and obtain (5) where the numbers ‘a’ and b are the Cartesian coordinates of u. So, equation (5) is referred to as the Cartesian form of u.
> >
> > Equation (6) is the Euler’s formula for  and i, and is introduced into (1) which becomes (7). Equation (7) expresses u in the form of an exponential function and is referred to as the exponential form of u. So, a classical complex number can be expressed in Cartesian, trigonometric or exponential form and has a geometrical meaning which is the vector u in Figure 1.
> >
> > 3D complex number
> >
> > 3D space and vector
> > A 3D complex number is also a vector, which we will construct from the 2D plane (h, i). For doing so, we add the axis j perpendicularly to the plane (h, i) and obtain the 3D space whose axes are labeled as h, i and j, see Figure 2. We refer to this space as (h, i, j).
> >
> > We attach the unit vectors eh, ei and ej to the axes h, i and j respectively. The 3D space based on these vectors is referred to as (eh, ei, ej). We have then two 3D spaces: the complex space (h, i, j) and the vector space (eh, ei, ej). We will create a vector labeled as v in (eh, ei, ej) which corresponds to a 3D complex number in (h, i, j) labeled also as v.
> >
> > With the help of Figure 2 we create the vector v in the desired form by starting with a vector u which is expressed in (8) with |u| being its modulus and  the angle it makes with the axis h. So, u is in the horizontal plane (eh, ei). Dividing u by |u| gives the unit vector eu, see (9). The unit vectors eu and ej are the basis vectors of the vertical plane (eu, ej), see Figure 2. The vector v is created by rotating the vector u in this plane toward the axis j. The angle of rotation is , so v is expressed with the angle  on the basis vectors eu and ej in (10).
> >
> > As the length of u stays the same during the rotation, the modulus of u and v are equal, see (11). Introducing the expression of eu (9) into (10) gives (12) which is developed into (13) using (11). The vector v is expressed with its modulus |v| and the angles  and  on the basis vectors eh, ei and ej, see (13).
> >
> > Notice that the angle  is between the vector v and the horizontal plane (eh, ei), see Figure 2, which is different from the usual spherical coordinate system where the angle  is between the vector v and the axis j. So, when u is horizontal the angle  equals zero rather then /2.
> >
> > See the article with figures and equations here
> > https://www.academia.edu/71708344/Extending_complex_number_to_spaces_with_3_4_or_any_number_of_dimensions
> > https://pengkuanonmaths.blogspot.com/2022/02/extending-complex-number-to-spaces-with.html
> It was proved many years ago that the only dimensions that allow algebraic structures (or even
> merely homotopy-equivalent structures) can only be in dimensions 1, 2, 4, and 8 (reals, complexes,
> quaternions, and octonions, respectively).
>
> So something is wrong somewhere with your argument.
>
> --
> Jan

Jan, this is definitely not the case. It may be some twisted argument that all those algebraic structures will fit into these containers you mention; thus making those containers fundamental... but this sort of logic is flawed. Native construction deserve to be considered on their own terms. Obviously the 3D system already is proven as P4 of polysign. That is now decades old. Polysign develop n-ary algebra by generalizing sign. Any kid could do it, but two-signed morons rule this day and their placebo effect has gone global. At least it seems this way from where I sit, and yet I am willing to entertain a curtain of some sort. We are motion sensing creatures so when they place moving objects in front of your eyes; flash things rapidly at you; the timings of these events seem to be speeding up generation by generation. I suppose humanity will be at a breaking point. Ultimately we are engaged in a progression and I do look forward to what comes next.

If facts ever matter then the situation globally right now all comes back onto us; as in the US; as in the USA as in the United States of America; as in complicity in a democracy never felt so good, eh?

On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > On Thursday, February 24, 2022 at 12:53:18 PM UTC-5, Michael Moroney wrote:
> > Incindentally (or not) standard vector systems that physics is working on fail to have any arithmetic product defined while we all abide them as respectable. Here to say you have a half-assed product defined will get you plenty farther. If some compelling physical correspondence exists, and I would have to give you Peng Kuan the credit for such an interpretation as I have no existing interpretation of your product; then your work could become relevant to physics rather than just being another odd-ball product.
> >
> > I have to assure you that polysign in P4 are the 3D complex numbers, and they obey the algebraic principles just as C and R do. These are P3 and P2 respectively, and their little sibling P1 gets no shrift; even getting the boot from me early on. That has changed for some time and the singleton does carry persuasive force in the n-ary department of being.
> In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method.. Then, I tumbled on multidimensional complex number.
>
> My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
>
> As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”

That was from you own work....
https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
Hah! Must have been a typo that your interface printed up to me.

I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
Few here can even give each other that much credit.

So if I told you that there can be a four-signed number system would it be any wonder that:
- 1 + 1 * 1 # 1 = 0 ,
especially given that the two-signed real numbers obey:
- 1 + 1 = 0
and the three-signed numbers obey:
- 1 + 1 * 1 = 0 ?

The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.

The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.

Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba...@gmail.com a écrit :
> On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> >
> > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> >
> > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> That was from you own work....
> https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> Hah! Must have been a typo that your interface printed up to me.
>
> I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> Few here can even give each other that much credit.
>
> So if I told you that there can be a four-signed number system would it be any wonder that:
> - 1 + 1 * 1 # 1 = 0 ,
> especially given that the two-signed real numbers obey:
> - 1 + 1 = 0
> and the three-signed numbers obey:
> - 1 + 1 * 1 = 0 ?
>
> The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
>
> The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.

I see. “Signet non défini” is a failed cross reference in my document.docx

I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.

PK

Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a écrit :
> On 2/17/2022 12:42 PM, PengKuan Em wrote:
> Post it over on https://fractalforums.org. There are a lot of people who
> will try to create fractals with it. I do not have the time to give it a
> really deep dive right now. Fwiw, here is some of my work:
>
> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714

Thanks.

On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail.com wrote:
> Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba...@gmail.com a écrit :
> > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > >
> > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number..
> > >
> > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > That was from you own work....
> > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > Hah! Must have been a typo that your interface printed up to me.
> >
> > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > Few here can even give each other that much credit.
> >
> > So if I told you that there can be a four-signed number system would it be any wonder that:
> > - 1 + 1 * 1 # 1 = 0 ,
> > especially given that the two-signed real numbers obey:
> > - 1 + 1 = 0
> > and the three-signed numbers obey:
> > - 1 + 1 * 1 = 0 ?
> >
> > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> >
> > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> I see. “Signet non défini” is a failed cross reference in my document.docx
>
> I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
>
> PK

No. P2 are the real numbers. They are two-signed numbers. Instances:
-1.23 , + 0.01 , - 2.3 + 0.01= -2.29
P3 are the complex numbers, but they are in a new format:
-1.23 + 2.3 * 0.1= -1.13 + 2.2
but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
- 1 + 1 * 1 = 0
no different than back in P2:
- 1 + 1 = 0 .
The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
- 1 + 1 * 1 # 1 = 0 .
These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
- - = +
- - - = *
- - - - = #
- - - - - = -
so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
- - = +
- - - = -
so the modulo two nature of the real value sign is exactly consistent with P2.

The real numbers are two-signed numbers. They are balanced by these signs. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.

Le samedi 26 février 2022 à 13:47:13 UTC+1, timba...@gmail.com a écrit :
> On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail.com wrote:
> > Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba...@gmail.com a écrit :
> > > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail..com a écrit :
> > > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > > >
> > > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> > > >
> > > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > > That was from you own work....
> > > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > > Hah! Must have been a typo that your interface printed up to me.
> > >
> > > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > > Few here can even give each other that much credit.
> > >
> > > So if I told you that there can be a four-signed number system would it be any wonder that:
> > > - 1 + 1 * 1 # 1 = 0 ,
> > > especially given that the two-signed real numbers obey:
> > > - 1 + 1 = 0
> > > and the three-signed numbers obey:
> > > - 1 + 1 * 1 = 0 ?
> > >
> > > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> > >
> > > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> > I see. “Signet non défini” is a failed cross reference in my document.docx
> >
> > I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
> >
> > PK
> No. P2 are the real numbers. They are two-signed numbers. Instances:
> -1.23 , + 0.01 , - 2.3 + 0.01= -2.29
> P3 are the complex numbers, but they are in a new format:
> -1.23 + 2.3 * 0.1= -1.13 + 2.2
> but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
> - 1 + 1 * 1 = 0
> no different than back in P2:
> - 1 + 1 = 0 .
> The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
> - 1 + 1 * 1 # 1 = 0 .
> These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
> - - = +
> - - - = *
> - - - - = #
> - - - - - = -
> so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
> - - = +
> - - - = -
> so the modulo two nature of the real value sign is exactly consistent with P2.
>
> The real numbers are two-signed numbers. They are balanced by these signs.. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.
Thanks.

On Monday, February 28, 2022 at 6:13:52 PM UTC-5, tita...@gmail.com wrote:
> Le samedi 26 février 2022 à 13:47:13 UTC+1, timba...@gmail.com a écrit :
> > On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail.com wrote:
> > > Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba...@gmail.com a écrit :
> > > > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > > > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > > > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > > > >
> > > > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> > > > >
> > > > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > > > That was from you own work....
> > > > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > > > Hah! Must have been a typo that your interface printed up to me.
> > > >
> > > > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > > > Few here can even give each other that much credit.
> > > >
> > > > So if I told you that there can be a four-signed number system would it be any wonder that:
> > > > - 1 + 1 * 1 # 1 = 0 ,
> > > > especially given that the two-signed real numbers obey:
> > > > - 1 + 1 = 0
> > > > and the three-signed numbers obey:
> > > > - 1 + 1 * 1 = 0 ?
> > > >
> > > > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> > > >
> > > > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> > > I see. “Signet non défini” is a failed cross reference in my document.docx
> > >
> > > I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
> > >
> > > PK
> > No. P2 are the real numbers. They are two-signed numbers. Instances:
> > -1.23 , + 0.01 , - 2.3 + 0.01= -2.29
> > P3 are the complex numbers, but they are in a new format:
> > -1.23 + 2.3 * 0.1= -1.13 + 2.2
> > but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
> > - 1 + 1 * 1 = 0
> > no different than back in P2:
> > - 1 + 1 = 0 .
> > The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
> > - 1 + 1 * 1 # 1 = 0 .
> > These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
> > - - = +
> > - - - = *
> > - - - - = #
> > - - - - - = -
> > so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
> > - - = +
> > - - - = -
> > so the modulo two nature of the real value sign is exactly consistent with P2.
> >
> > The real numbers are two-signed numbers. They are balanced by these signs. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.
> Thanks.

Presuming you accept these details there is quite a lot more that unfolds. P1 you see is consistent with time as a unidirectional feature. The oddity is that P1 is zero dimensional as well. Time actually does obey this seeming conundrum, and in hindsight it is real valued time that is to blame for so much of the stupidity. As we observe the three dimensions of space via experimental practice we do in fact observe time as zero dimensional: we observe no freedom to move an object in time. It is this method which allows us to claim the three dimensions of space. Even this language though of dimension is built on the back of the real value as if it was fundamental. In polysign the real value is P2 and it is no more fundamental than P3 or P4. These are n-ary siblings.

As you may know there is a quest for emergent spacetime under way within at least one branch of physics. Polysign numbers are yielding an emergent spacetime without even doing any physics. How? So far you have seen how they are general dimensional. So where is the breakpoint? Have a look at the behavior:
| z1 z2 | = | z1 | | z2 |
In polysign it only holds through P3. In effect we have a natural breakpoint :
P1 P2 P3 | P4 P5 P6 ...
and now whether you look at the bottom three as spacetime, or throw space in as P4 even; there are two emergent spacetime candidates there. The one which included P4 is a 6D system:
P1 : 0D
P1 P2 : 1D
P1 P2 P3 : 3D
P1 P2 P3 P4 : 6D
though again the usage of 'dimension' is faulty from the context of polysign. Well, I shouldn't quite push it that far: P2 are the first graphically renderable system with distinct features. Still, the inclusion of unidirectional time as natural within polysign is quite a high merit; something untouchable by two-signed morons (TSMs).

Polysign has led me on a journey that goes into abstract algebra and back onto operator theory and out the other side to the construction of the real value whereupon I have found fault with the rational value; the irrational value merely being a teaser to get beyond the dubious nature of the rational value. It sort of ends in the quagmire of the engineer versus the mathematician upon the value
2.01
for instance, whereby the engineer sees this value as gray and the mathematician sees a perfect value. The continuum and its analysis is quite relevant here. Epsilon/delta shows the way for the mathematician by chasing digits but for them the notion that close enough is good enough was forgotten. Their immunity provides quite a gap. Sadly the consequences of their ignorance travel through the subject and much of it comes out rotten even as the beast carries onward in accumulation. Oh, what an accumulation. So we must carry on and try not to become carrion.

As capitalist backed Nazis are prepared to burn communists alive in the streets fears awaken. As to who is at the reigns: did you get your grant? Will they burn socialists too? These are questions slowly coming to the minds of mathematicians in a free world that is exposed as a series of lies. Freedom is something that we take as individuals; it is not something granted from on up high. That humans are great followers is well exposed in music; in sports; in politics; in the media. That humans are programmable beings... this is what we have to deal with now.

Le mardi 1 mars 2022 à 16:24:47 UTC+1, timba...@gmail.com a écrit :
> On Monday, February 28, 2022 at 6:13:52 PM UTC-5, tita...@gmail.com wrote:
> > Le samedi 26 février 2022 à 13:47:13 UTC+1, timba...@gmail.com a écrit :
> > > On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail.com wrote:
> > > > Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba...@gmail.com a écrit :
> > > > > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail..com wrote:
> > > > > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > > > > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > > > > >
> > > > > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> > > > > >
> > > > > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > > > > That was from you own work....
> > > > > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > > > > Hah! Must have been a typo that your interface printed up to me.
> > > > >
> > > > > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > > > > Few here can even give each other that much credit.
> > > > >
> > > > > So if I told you that there can be a four-signed number system would it be any wonder that:
> > > > > - 1 + 1 * 1 # 1 = 0 ,
> > > > > especially given that the two-signed real numbers obey:
> > > > > - 1 + 1 = 0
> > > > > and the three-signed numbers obey:
> > > > > - 1 + 1 * 1 = 0 ?
> > > > >
> > > > > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> > > > >
> > > > > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> > > > I see. “Signet non défini” is a failed cross reference in my document.docx
> > > >
> > > > I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
> > > >
> > > > PK
> > > No. P2 are the real numbers. They are two-signed numbers. Instances:
> > > -1.23 , + 0.01 , - 2.3 + 0.01= -2.29
> > > P3 are the complex numbers, but they are in a new format:
> > > -1.23 + 2.3 * 0.1= -1.13 + 2.2
> > > but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
> > > - 1 + 1 * 1 = 0
> > > no different than back in P2:
> > > - 1 + 1 = 0 .
> > > The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
> > > - 1 + 1 * 1 # 1 = 0 .
> > > These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
> > > - - = +
> > > - - - = *
> > > - - - - = #
> > > - - - - - = -
> > > so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
> > > - - = +
> > > - - - = -
> > > so the modulo two nature of the real value sign is exactly consistent with P2.
> > >
> > > The real numbers are two-signed numbers. They are balanced by these signs. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.
> > Thanks.
> Presuming you accept these details there is quite a lot more that unfolds.. P1 you see is consistent with time as a unidirectional feature. The oddity is that P1 is zero dimensional as well. Time actually does obey this seeming conundrum, and in hindsight it is real valued time that is to blame for so much of the stupidity. As we observe the three dimensions of space via experimental practice we do in fact observe time as zero dimensional: we observe no freedom to move an object in time. It is this method which allows us to claim the three dimensions of space. Even this language though of dimension is built on the back of the real value as if it was fundamental. In polysign the real value is P2 and it is no more fundamental than P3 or P4. These are n-ary siblings.
>
> As you may know there is a quest for emergent spacetime under way within at least one branch of physics. Polysign numbers are yielding an emergent spacetime without even doing any physics. How? So far you have seen how they are general dimensional. So where is the breakpoint? Have a look at the behavior:
> | z1 z2 | = | z1 | | z2 |
> In polysign it only holds through P3. In effect we have a natural breakpoint :
> P1 P2 P3 | P4 P5 P6 ...
> and now whether you look at the bottom three as spacetime, or throw space in as P4 even; there are two emergent spacetime candidates there. The one which included P4 is a 6D system:
> P1 : 0D
> P1 P2 : 1D
> P1 P2 P3 : 3D
> P1 P2 P3 P4 : 6D
> though again the usage of 'dimension' is faulty from the context of polysign. Well, I shouldn't quite push it that far: P2 are the first graphically renderable system with distinct features. Still, the inclusion of unidirectional time as natural within polysign is quite a high merit; something untouchable by two-signed morons (TSMs).
>
> Polysign has led me on a journey that goes into abstract algebra and back onto operator theory and out the other side to the construction of the real value whereupon I have found fault with the rational value; the irrational value merely being a teaser to get beyond the dubious nature of the rational value. It sort of ends in the quagmire of the engineer versus the mathematician upon the value
> 2.01
> for instance, whereby the engineer sees this value as gray and the mathematician sees a perfect value. The continuum and its analysis is quite relevant here. Epsilon/delta shows the way for the mathematician by chasing digits but for them the notion that close enough is good enough was forgotten. Their immunity provides quite a gap. Sadly the consequences of their ignorance travel through the subject and much of it comes out rotten even as the beast carries onward in accumulation. Oh, what an accumulation. So we must carry on and try not to become carrion.
>
> As capitalist backed Nazis are prepared to burn communists alive in the streets fears awaken. As to who is at the reigns: did you get your grant? Will they burn socialists too? These are questions slowly coming to the minds of mathematicians in a free world that is exposed as a series of lies. Freedom is something that we take as individuals; it is not something granted from on up high. That humans are great followers is well exposed in music; in sports; in politics; in the media. That humans are programmable beings.... this is what we have to deal with now.
I see that you have a huge project which is mathematical as well as philosophical. But I think that the 0 dimensional nature of time is an interesting idea, which will be useful in relativity.

PK

On Thursday, March 3, 2022 at 1:59:22 PM UTC-5, tita...@gmail.com wrote:
> Le mardi 1 mars 2022 à 16:24:47 UTC+1, timba...@gmail.com a écrit :
> > On Monday, February 28, 2022 at 6:13:52 PM UTC-5, tita...@gmail.com wrote:
> > > Le samedi 26 février 2022 à 13:47:13 UTC+1, timba...@gmail.com a écrit :
> > > > On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail.com wrote:
> > > > > Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba...@gmail.com a écrit :
> > > > > > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > > > > > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > > > > > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > > > > > >
> > > > > > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> > > > > > >
> > > > > > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > > > > > That was from you own work....
> > > > > > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > > > > > Hah! Must have been a typo that your interface printed up to me..
> > > > > >
> > > > > > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > > > > > Few here can even give each other that much credit.
> > > > > >
> > > > > > So if I told you that there can be a four-signed number system would it be any wonder that:
> > > > > > - 1 + 1 * 1 # 1 = 0 ,
> > > > > > especially given that the two-signed real numbers obey:
> > > > > > - 1 + 1 = 0
> > > > > > and the three-signed numbers obey:
> > > > > > - 1 + 1 * 1 = 0 ?
> > > > > >
> > > > > > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> > > > > >
> > > > > > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> > > > > I see. “Signet non défini” is a failed cross reference in my document.docx
> > > > >
> > > > > I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
> > > > >
> > > > > PK
> > > > No. P2 are the real numbers. They are two-signed numbers. Instances:
> > > > -1.23 , + 0.01 , - 2.3 + 0.01= -2.29
> > > > P3 are the complex numbers, but they are in a new format:
> > > > -1.23 + 2.3 * 0.1= -1.13 + 2.2
> > > > but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
> > > > - 1 + 1 * 1 = 0
> > > > no different than back in P2:
> > > > - 1 + 1 = 0 .
> > > > The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
> > > > - 1 + 1 * 1 # 1 = 0 .
> > > > These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
> > > > - - = +
> > > > - - - = *
> > > > - - - - = #
> > > > - - - - - = -
> > > > so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
> > > > - - = +
> > > > - - - = -
> > > > so the modulo two nature of the real value sign is exactly consistent with P2.
> > > >
> > > > The real numbers are two-signed numbers. They are balanced by these signs. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.
> > > Thanks.
> > Presuming you accept these details there is quite a lot more that unfolds. P1 you see is consistent with time as a unidirectional feature. The oddity is that P1 is zero dimensional as well. Time actually does obey this seeming conundrum, and in hindsight it is real valued time that is to blame for so much of the stupidity. As we observe the three dimensions of space via experimental practice we do in fact observe time as zero dimensional: we observe no freedom to move an object in time. It is this method which allows us to claim the three dimensions of space. Even this language though of dimension is built on the back of the real value as if it was fundamental. In polysign the real value is P2 and it is no more fundamental than P3 or P4.. These are n-ary siblings.
> >
> > As you may know there is a quest for emergent spacetime under way within at least one branch of physics. Polysign numbers are yielding an emergent spacetime without even doing any physics. How? So far you have seen how they are general dimensional. So where is the breakpoint? Have a look at the behavior:
> > | z1 z2 | = | z1 | | z2 |
> > In polysign it only holds through P3. In effect we have a natural breakpoint :
> > P1 P2 P3 | P4 P5 P6 ...
> > and now whether you look at the bottom three as spacetime, or throw space in as P4 even; there are two emergent spacetime candidates there. The one which included P4 is a 6D system:
> > P1 : 0D
> > P1 P2 : 1D
> > P1 P2 P3 : 3D
> > P1 P2 P3 P4 : 6D
> > though again the usage of 'dimension' is faulty from the context of polysign. Well, I shouldn't quite push it that far: P2 are the first graphically renderable system with distinct features. Still, the inclusion of unidirectional time as natural within polysign is quite a high merit; something untouchable by two-signed morons (TSMs).
> >
> > Polysign has led me on a journey that goes into abstract algebra and back onto operator theory and out the other side to the construction of the real value whereupon I have found fault with the rational value; the irrational value merely being a teaser to get beyond the dubious nature of the rational value. It sort of ends in the quagmire of the engineer versus the mathematician upon the value
> > 2.01
> > for instance, whereby the engineer sees this value as gray and the mathematician sees a perfect value. The continuum and its analysis is quite relevant here. Epsilon/delta shows the way for the mathematician by chasing digits but for them the notion that close enough is good enough was forgotten.. Their immunity provides quite a gap. Sadly the consequences of their ignorance travel through the subject and much of it comes out rotten even as the beast carries onward in accumulation. Oh, what an accumulation. So we must carry on and try not to become carrion.
> >
> > As capitalist backed Nazis are prepared to burn communists alive in the streets fears awaken. As to who is at the reigns: did you get your grant? Will they burn socialists too? These are questions slowly coming to the minds of mathematicians in a free world that is exposed as a series of lies. Freedom is something that we take as individuals; it is not something granted from on up high. That humans are great followers is well exposed in music; in sports; in politics; in the media. That humans are programmable beings.... this is what we have to deal with now.
> I see that you have a huge project which is mathematical as well as philosophical. But I think that the 0 dimensional nature of time is an interesting idea, which will be useful in relativity.
>
> PK

Thank you. There is much to discuss. Electromagnetics has been claimed to come out cleanly in 6D systems right? The progression
0D 1D 3D 6D 10D 15D 21D
encompasses rather a lot of theories.

Also, the electromagnetic tensor is antisymmetric, and this means that when the redundancy is removed that you have a triangular form. So do these polysign tatrices (singular tatrix):
P1 : 0D : a10
P1 P2 : 1D : a10 a20 a21
P1 P2 P3 : 3D : a10 a20 a21 a30 a31 a32
P1 P2 P3 P4 : 6D :
a10
a20 a21
a30 a31 a32
a40 a41 a42 a43

Le vendredi 4 mars 2022 à 22:08:28 UTC+1, timba...@gmail.com a écrit :
> On Thursday, March 3, 2022 at 1:59:22 PM UTC-5, tita...@gmail.com wrote:
> > Le mardi 1 mars 2022 à 16:24:47 UTC+1, timba...@gmail.com a écrit :
> > > On Monday, February 28, 2022 at 6:13:52 PM UTC-5, tita...@gmail.com wrote:
> > > > Le samedi 26 février 2022 à 13:47:13 UTC+1, timba...@gmail.com a écrit :
> > > > > On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail.com wrote:
> > > > > > Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba....@gmail.com a écrit :
> > > > > > > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > > > > > > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba....@gmail.com a écrit :
> > > > > > > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > > > > > > >
> > > > > > > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> > > > > > > >
> > > > > > > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > > > > > > That was from you own work....
> > > > > > > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > > > > > > Hah! Must have been a typo that your interface printed up to me.
> > > > > > >
> > > > > > > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > > > > > > Few here can even give each other that much credit.
> > > > > > >
> > > > > > > So if I told you that there can be a four-signed number system would it be any wonder that:
> > > > > > > - 1 + 1 * 1 # 1 = 0 ,
> > > > > > > especially given that the two-signed real numbers obey:
> > > > > > > - 1 + 1 = 0
> > > > > > > and the three-signed numbers obey:
> > > > > > > - 1 + 1 * 1 = 0 ?
> > > > > > >
> > > > > > > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> > > > > > >
> > > > > > > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> > > > > > I see. “Signet non défini” is a failed cross reference in my document.docx
> > > > > >
> > > > > > I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
> > > > > >
> > > > > > PK
> > > > > No. P2 are the real numbers. They are two-signed numbers. Instances:
> > > > > -1.23 , + 0.01 , - 2.3 + 0.01= -2.29
> > > > > P3 are the complex numbers, but they are in a new format:
> > > > > -1.23 + 2.3 * 0.1= -1.13 + 2.2
> > > > > but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
> > > > > - 1 + 1 * 1 = 0
> > > > > no different than back in P2:
> > > > > - 1 + 1 = 0 .
> > > > > The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
> > > > > - 1 + 1 * 1 # 1 = 0 .
> > > > > These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
> > > > > - - = +
> > > > > - - - = *
> > > > > - - - - = #
> > > > > - - - - - = -
> > > > > so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
> > > > > - - = +
> > > > > - - - = -
> > > > > so the modulo two nature of the real value sign is exactly consistent with P2.
> > > > >
> > > > > The real numbers are two-signed numbers. They are balanced by these signs. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.
> > > > Thanks.
> > > Presuming you accept these details there is quite a lot more that unfolds. P1 you see is consistent with time as a unidirectional feature. The oddity is that P1 is zero dimensional as well. Time actually does obey this seeming conundrum, and in hindsight it is real valued time that is to blame for so much of the stupidity. As we observe the three dimensions of space via experimental practice we do in fact observe time as zero dimensional: we observe no freedom to move an object in time. It is this method which allows us to claim the three dimensions of space. Even this language though of dimension is built on the back of the real value as if it was fundamental. In polysign the real value is P2 and it is no more fundamental than P3 or P4. These are n-ary siblings.
> > >
> > > As you may know there is a quest for emergent spacetime under way within at least one branch of physics. Polysign numbers are yielding an emergent spacetime without even doing any physics. How? So far you have seen how they are general dimensional. So where is the breakpoint? Have a look at the behavior:
> > > | z1 z2 | = | z1 | | z2 |
> > > In polysign it only holds through P3. In effect we have a natural breakpoint :
> > > P1 P2 P3 | P4 P5 P6 ...
> > > and now whether you look at the bottom three as spacetime, or throw space in as P4 even; there are two emergent spacetime candidates there. The one which included P4 is a 6D system:
> > > P1 : 0D
> > > P1 P2 : 1D
> > > P1 P2 P3 : 3D
> > > P1 P2 P3 P4 : 6D
> > > though again the usage of 'dimension' is faulty from the context of polysign. Well, I shouldn't quite push it that far: P2 are the first graphically renderable system with distinct features. Still, the inclusion of unidirectional time as natural within polysign is quite a high merit; something untouchable by two-signed morons (TSMs).
> > >
> > > Polysign has led me on a journey that goes into abstract algebra and back onto operator theory and out the other side to the construction of the real value whereupon I have found fault with the rational value; the irrational value merely being a teaser to get beyond the dubious nature of the rational value. It sort of ends in the quagmire of the engineer versus the mathematician upon the value
> > > 2.01
> > > for instance, whereby the engineer sees this value as gray and the mathematician sees a perfect value. The continuum and its analysis is quite relevant here. Epsilon/delta shows the way for the mathematician by chasing digits but for them the notion that close enough is good enough was forgotten. Their immunity provides quite a gap. Sadly the consequences of their ignorance travel through the subject and much of it comes out rotten even as the beast carries onward in accumulation. Oh, what an accumulation. So we must carry on and try not to become carrion.
> > >
> > > As capitalist backed Nazis are prepared to burn communists alive in the streets fears awaken. As to who is at the reigns: did you get your grant? Will they burn socialists too? These are questions slowly coming to the minds of mathematicians in a free world that is exposed as a series of lies. Freedom is something that we take as individuals; it is not something granted from on up high. That humans are great followers is well exposed in music; in sports; in politics; in the media. That humans are programmable beings... this is what we have to deal with now.
> > I see that you have a huge project which is mathematical as well as philosophical. But I think that the 0 dimensional nature of time is an interesting idea, which will be useful in relativity.
> >
> > PK
> Thank you. There is much to discuss. Electromagnetics has been claimed to come out cleanly in 6D systems right? The progression
> 0D 1D 3D 6D 10D 15D 21D
> encompasses rather a lot of theories.
>
> Also, the electromagnetic tensor is antisymmetric, and this means that when the redundancy is removed that you have a triangular form. So do these polysign tatrices (singular tatrix):
> P1 : 0D : a10
> P1 P2 : 1D : a10 a20 a21
> P1 P2 P3 : 3D : a10 a20 a21 a30 a31 a32
> P1 P2 P3 P4 : 6D :
> a10
> a20 a21
> a30 a31 a32
> a40 a41 a42 a43
>
> where these asubemen (amn) are unsigned magnitudes.
> It turns out that in turning over the ground within fundamental mathematics, as for instance to what degree do I having generalized sign have to define very carefully what I mean by magnitude? Early on I left this problem alone, but now coming through operator theory and numerical construction of the reals I see that there are definite serious issues when a value such as 2.01 could mean one thing to an engineer and another thing entirely to the mathematician. You see for the engineer this value is gray, whereas for the mathematician it is already perfect. Yet we know where the perfection lays: it is in the chasing of the digits. That the concept of continuum that the mathematicians have built is not consistent with the continuum that engineers and physicists work in is readily proven. In effect this offense takes many interpretations and this is not at all where I started the analysis. Stepping back one more stage I found that the rational value is illegitimate and is simply laid followed up by the irrational value which you are supposed to take a big gulp over and then get on and call these things real.
> The diversionary tactic I have seen elsewhere as well, with one flawed concept covered up by a rapid changeup like the ideal and quotient concepts in abstract algebra.
>
> So the stage prior to the rational analysis is operator theory where clearly division is not a fundamental thing; it is by definition the inverse operation of the product and furthermore carries peculiar exceptions which obliterate its purity. That this is the operator which constructs the rational numbers: here was my lead. Strangely operator theory was put up just before they get to the ideal and quotient in abstract algebra. Closure is always stated. Rational values offend closure. Polynomials offend closure. These polynomials I am speaking of do not have defined X within their construction: instead they treat it as abstract and claim to build up for instance the complex numbers as ( X / XX + 1 ) or some such quotient. I didn't quite state that right and I don't care to go back and do the research. This is the means by which they wrap an unending polynomial by assigning XX to 1 so that XXX goes to X and so forth. This behavior is extremely close to the behavior of polysign numbers in that modulo behaviors are present. Funny thing is we've just gotten into embedded MU and they do practically the same thing as these X's. In P3 when you've got zzzz = z then you've got a MU candidate, MU being minus unity, as in the first sign which is capable of generating all the others as MU^n. This peculiarity of polysign as a handy notational feature may be quite a bit more in that a single mu vector arguably does code the others, and yet geometrically the rendering will always be a matter of balancing the components, and as to where you chose to put those rays... they are perfectly symmetrical with no chirality requirement. This is somewhat an open area and I am fairly sure you might have an opinion. In some ways you could argue that some sort of relativity theory ought to be built up here. Whatever, polysign are vector behaved, but the tatrix form carries the family in a structure that does not obey the principal of isotropy. This is to say that if polysign are emergent spacetime that they argue for a structured spacetime, and this then is the next step, and sadly a conflict with existing theory. If we allow that the tensor worked well in 3D space but that the introduction of bidirectional time into an arbitrary reference frame as X4 was a boo-boo. Yet if the 3D space argument is taken seriously then I suppose it is an argument for taking
> P1 P2 P3 | P4
> very seriously; that 3D physical space being P4, which then means there are additional dimensions just as Maxwell described for magnetism, and I believe P2 will be charge, P3 magnetism, and possibly P1 as mass (certainly naively this is the case).
>
> The unit shells of these spaces for some reason are appropriate. When we do it in P2 we get just the two point -1 and +1. You could almost argue that as charge we happened to get it right, with the electon as -1 being freakily active while +1 is quite inactive... from a product standpoint, which is fairly abstract from the ordinary multiplication, but here we are in this strange unitized form. In P3 we see a loop suggestive of magnetic flux. Yet three poles develop it. There is a neutral pole that goes unrecognized in modernity while the minus and plus poles are exact symmetrical opposites of each other just as the active north and south poles behave. Indeed we are caught with the quips of the south pole being at the north pole, or your compass needle pointing south to north and so forth. We mark the south side of the needle north. This is a logical conflict. Meanwhile what we call spin is known to be intrinsic and here under the polysign progression (tatrix) form this is the case if this interpretation holds up.

On Thursday, March 10, 2022 at 7:28:53 PM UTC-5, tita...@gmail.com wrote:
> Le vendredi 4 mars 2022 à 22:08:28 UTC+1, timba...@gmail.com a écrit :
> > On Thursday, March 3, 2022 at 1:59:22 PM UTC-5, tita...@gmail.com wrote:
> > > Le mardi 1 mars 2022 à 16:24:47 UTC+1, timba...@gmail.com a écrit :
> > > > On Monday, February 28, 2022 at 6:13:52 PM UTC-5, tita...@gmail.com wrote:
> > > > > Le samedi 26 février 2022 à 13:47:13 UTC+1, timba...@gmail.com a écrit :
> > > > > > On Friday, February 25, 2022 at 5:17:47 PM UTC-5, tita...@gmail..com wrote:
> > > > > > > Le vendredi 25 février 2022 à 16:04:55 UTC+1, timba....@gmail.com a écrit :
> > > > > > > > On Thursday, February 24, 2022 at 4:34:54 PM UTC-5, tita...@gmail.com wrote:
> > > > > > > > > Le jeudi 24 février 2022 à 19:46:34 UTC+1, timba...@gmail.com a écrit :
> > > > > > > > > In fact, I’m working on physics, more precisely in relativity and Electromagnetic. While computing the trajectory of Oumuamua, I wanted to use complex number for numerical root searching with Newton’s method. Then, I tumbled on multidimensional complex number.
> > > > > > > > >
> > > > > > > > > My 3D complex number is a vector whose angles are that of a reference of orientation, like Euler’s reference, and these angles are the arguments of the complex number. When we multiply one complex number with another complex number, this is equivalent to rotate the first vector with the angles of the second, that is, the arguments of the second complex number.
> > > > > > > > >
> > > > > > > > > As I’m not a mathematician, I do not know the meaning of “Signet non défini”, nor that of “polysign in P4”
> > > > > > > > That was from you own work....
> > > > > > > > https://www.proz.com/kudoz/french-to-english/other/250526-signet-non-d%C3%A9fini.html
> > > > > > > > Hah! Must have been a typo that your interface printed up to me.
> > > > > > > >
> > > > > > > > I get that you are coherent. Sparring here on usenet is a thing and you are holding up just fine.
> > > > > > > > Few here can even give each other that much credit.
> > > > > > > >
> > > > > > > > So if I told you that there can be a four-signed number system would it be any wonder that:
> > > > > > > > - 1 + 1 * 1 # 1 = 0 ,
> > > > > > > > especially given that the two-signed real numbers obey:
> > > > > > > > - 1 + 1 = 0
> > > > > > > > and the three-signed numbers obey:
> > > > > > > > - 1 + 1 * 1 = 0 ?
> > > > > > > >
> > > > > > > > The product is trivial. They are rotational in nature. They naturally generate simplex geometry without any need of the cross product; the Cartesian product, so to speak. They are n-ary, and goodness: P1 hasn't even gotten under the lens yet.
> > > > > > > >
> > > > > > > > The fact that you are into emag is great. I'm trying to work it into polysign. Rather hoping it will come out of polysign.
> > > > > > > I see. “Signet non défini” is a failed cross reference in my document.docx
> > > > > > >
> > > > > > > I see what polysign is about. Complex number can be seen as a P2, 3D complex P3 etc.
> > > > > > >
> > > > > > > PK
> > > > > > No. P2 are the real numbers. They are two-signed numbers. Instances:
> > > > > > -1.23 , + 0.01 , - 2.3 + 0.01= -2.29
> > > > > > P3 are the complex numbers, but they are in a new format:
> > > > > > -1.23 + 2.3 * 0.1= -1.13 + 2.2
> > > > > > but you see there is no further reduction in this P3 value. P3 are two dimensional just as P2 were one dimensional, yet there is no orthogonality in use here. To truly generalize sign exposes the balance that P3 have achieved:
> > > > > > - 1 + 1 * 1 = 0
> > > > > > no different than back in P2:
> > > > > > - 1 + 1 = 0 .
> > > > > > The P3 signs are in a plane and are equally spaced 120 degrees apart. They form a simplex, and of course P4 are the 3D version forming rays from the center of a tetrahedron out to its vertices. Do not draw the frame of the tet. Just the rays are needed. They are perfectly balanced:
> > > > > > - 1 + 1 * 1 # 1 = 0 .
> > > > > > These are the 3D version that polysign yields with no further rule system than was already in use in P2. It's just that in four-signed math the sign mechanics are modulo four:
> > > > > > - - = +
> > > > > > - - - = *
> > > > > > - - - - = #
> > > > > > - - - - - = -
> > > > > > so you see we reuse the signs but their meaning is eccentric to the signature of the system. This is as it should be. This notation is completely consistent with the real value and ordinary nomenclature around it:
> > > > > > - - = +
> > > > > > - - - = -
> > > > > > so the modulo two nature of the real value sign is exactly consistent with P2.
> > > > > >
> > > > > > The real numbers are two-signed numbers. They are balanced by these signs. This balance yields their geometry. This same is true of P3, P4, and strangely enough of P1 as well. Oddly, in Pn the angle will tend toward ninety degrees for high signature.
> > > > > Thanks.
> > > > Presuming you accept these details there is quite a lot more that unfolds. P1 you see is consistent with time as a unidirectional feature. The oddity is that P1 is zero dimensional as well. Time actually does obey this seeming conundrum, and in hindsight it is real valued time that is to blame for so much of the stupidity. As we observe the three dimensions of space via experimental practice we do in fact observe time as zero dimensional: we observe no freedom to move an object in time. It is this method which allows us to claim the three dimensions of space. Even this language though of dimension is built on the back of the real value as if it was fundamental. In polysign the real value is P2 and it is no more fundamental than P3 or P4. These are n-ary siblings.
> > > >
> > > > As you may know there is a quest for emergent spacetime under way within at least one branch of physics. Polysign numbers are yielding an emergent spacetime without even doing any physics. How? So far you have seen how they are general dimensional. So where is the breakpoint? Have a look at the behavior:
> > > > | z1 z2 | = | z1 | | z2 |
> > > > In polysign it only holds through P3. In effect we have a natural breakpoint :
> > > > P1 P2 P3 | P4 P5 P6 ...
> > > > and now whether you look at the bottom three as spacetime, or throw space in as P4 even; there are two emergent spacetime candidates there. The one which included P4 is a 6D system:
> > > > P1 : 0D
> > > > P1 P2 : 1D
> > > > P1 P2 P3 : 3D
> > > > P1 P2 P3 P4 : 6D
> > > > though again the usage of 'dimension' is faulty from the context of polysign. Well, I shouldn't quite push it that far: P2 are the first graphically renderable system with distinct features. Still, the inclusion of unidirectional time as natural within polysign is quite a high merit; something untouchable by two-signed morons (TSMs).
> > > >
> > > > Polysign has led me on a journey that goes into abstract algebra and back onto operator theory and out the other side to the construction of the real value whereupon I have found fault with the rational value; the irrational value merely being a teaser to get beyond the dubious nature of the rational value. It sort of ends in the quagmire of the engineer versus the mathematician upon the value
> > > > 2.01
> > > > for instance, whereby the engineer sees this value as gray and the mathematician sees a perfect value. The continuum and its analysis is quite relevant here. Epsilon/delta shows the way for the mathematician by chasing digits but for them the notion that close enough is good enough was forgotten. Their immunity provides quite a gap. Sadly the consequences of their ignorance travel through the subject and much of it comes out rotten even as the beast carries onward in accumulation. Oh, what an accumulation. So we must carry on and try not to become carrion.
> > > >
> > > > As capitalist backed Nazis are prepared to burn communists alive in the streets fears awaken. As to who is at the reigns: did you get your grant? Will they burn socialists too? These are questions slowly coming to the minds of mathematicians in a free world that is exposed as a series of lies. Freedom is something that we take as individuals; it is not something granted from on up high. That humans are great followers is well exposed in music; in sports; in politics; in the media. That humans are programmable beings... this is what we have to deal with now.
> > > I see that you have a huge project which is mathematical as well as philosophical. But I think that the 0 dimensional nature of time is an interesting idea, which will be useful in relativity.
> > >
> > > PK
> > Thank you. There is much to discuss. Electromagnetics has been claimed to come out cleanly in 6D systems right? The progression
> > 0D 1D 3D 6D 10D 15D 21D
> > encompasses rather a lot of theories.
> >
> > Also, the electromagnetic tensor is antisymmetric, and this means that when the redundancy is removed that you have a triangular form. So do these polysign tatrices (singular tatrix):
> > P1 : 0D : a10
> > P1 P2 : 1D : a10 a20 a21
> > P1 P2 P3 : 3D : a10 a20 a21 a30 a31 a32
> > P1 P2 P3 P4 : 6D :
> > a10
> > a20 a21
> > a30 a31 a32
> > a40 a41 a42 a43
> >
> > where these asubemen (amn) are unsigned magnitudes.
> > It turns out that in turning over the ground within fundamental mathematics, as for instance to what degree do I having generalized sign have to define very carefully what I mean by magnitude? Early on I left this problem alone, but now coming through operator theory and numerical construction of the reals I see that there are definite serious issues when a value such as 2.01 could mean one thing to an engineer and another thing entirely to the mathematician. You see for the engineer this value is gray, whereas for the mathematician it is already perfect. Yet we know where the perfection lays: it is in the chasing of the digits. That the concept of continuum that the mathematicians have built is not consistent with the continuum that engineers and physicists work in is readily proven. In effect this offense takes many interpretations and this is not at all where I started the analysis. Stepping back one more stage I found that the rational value is illegitimate and is simply laid followed up by the irrational value which you are supposed to take a big gulp over and then get on and call these things real.
> > The diversionary tactic I have seen elsewhere as well, with one flawed concept covered up by a rapid changeup like the ideal and quotient concepts in abstract algebra.
> >
> > So the stage prior to the rational analysis is operator theory where clearly division is not a fundamental thing; it is by definition the inverse operation of the product and furthermore carries peculiar exceptions which obliterate its purity. That this is the operator which constructs the rational numbers: here was my lead. Strangely operator theory was put up just before they get to the ideal and quotient in abstract algebra. Closure is always stated. Rational values offend closure. Polynomials offend closure. These polynomials I am speaking of do not have defined X within their construction: instead they treat it as abstract and claim to build up for instance the complex numbers as ( X / XX + 1 ) or some such quotient. I didn't quite state that right and I don't care to go back and do the research. This is the means by which they wrap an unending polynomial by assigning XX to 1 so that XXX goes to X and so forth. This behavior is extremely close to the behavior of polysign numbers in that modulo behaviors are present. Funny thing is we've just gotten into embedded MU and they do practically the same thing as these X's. In P3 when you've got zzzz = z then you've got a MU candidate, MU being minus unity, as in the first sign which is capable of generating all the others as MU^n. This peculiarity of polysign as a handy notational feature may be quite a bit more in that a single mu vector arguably does code the others, and yet geometrically the rendering will always be a matter of balancing the components, and as to where you chose to put those rays... they are perfectly symmetrical with no chirality requirement. This is somewhat an open area and I am fairly sure you might have an opinion. In some ways you could argue that some sort of relativity theory ought to be built up here. Whatever, polysign are vector behaved, but the tatrix form carries the family in a structure that does not obey the principal of isotropy. This is to say that if polysign are emergent spacetime that they argue for a structured spacetime, and this then is the next step, and sadly a conflict with existing theory. If we allow that the tensor worked well in 3D space but that the introduction of bidirectional time into an arbitrary reference frame as X4 was a boo-boo. Yet if the 3D space argument is taken seriously then I suppose it is an argument for taking
> > P1 P2 P3 | P4
> > very seriously; that 3D physical space being P4, which then means there are additional dimensions just as Maxwell described for magnetism, and I believe P2 will be charge, P3 magnetism, and possibly P1 as mass (certainly naively this is the case).
> >
> > The unit shells of these spaces for some reason are appropriate. When we do it in P2 we get just the two point -1 and +1. You could almost argue that as charge we happened to get it right, with the electon as -1 being freakily active while +1 is quite inactive... from a product standpoint, which is fairly abstract from the ordinary multiplication, but here we are in this strange unitized form. In P3 we see a loop suggestive of magnetic flux. Yet three poles develop it. There is a neutral pole that goes unrecognized in modernity while the minus and plus poles are exact symmetrical opposites of each other just as the active north and south poles behave. Indeed we are caught with the quips of the south pole being at the north pole, or your compass needle pointing south to north and so forth. We mark the south side of the needle north. This is a logical conflict. Meanwhile what we call spin is known to be intrinsic and here under the polysign progression (tatrix) form this is the case if this interpretation holds up.
> Sorry for replying you so later. In fact, I do not know how to reply. After your 2 lengthy comments, I see your strong desire of getting recognition for your theory of ploysign number, just like me for my theory of relativity. But I cannot understand your theory because, as said in my article, I’m not a professional mathematician and lack the knowledge necessary to be of any help to you.
>
> So, sorry again and I wish all the good to you and good luck.
>
> PK

Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a écrit :
> On 2/17/2022 12:42 PM, PengKuan Em wrote:
> > Multidimensional complex systems with 3, 4 or more dimensions are constructed. They possess algebraic operations which have geometrical meanings. Multidimensional complex numbers can be written in Cartesian, trigonometric and exponential form and can be converted from one form to another. Each complex numbers has a conjugate. Multidimensional complex systems are extensions of the classical complex number system.
> > ***
> >
> > In about 500 years after the birth of complex number, there were several attempts to extend complex number to more than 2 dimensions, for example we have theories such as quaternions, tessarines, coquaternions, biquaternions, and octonions. But none has reached the success of the classical complex number in 2 dimensions. Among these theories the most famous is quaternion which has found use in computational geometry. But quaternion is a 4 dimensional complex number but is used in 3 dimensional vector space, which is somewhat awkward.
> >
> > In this article we will show that multidimensional complex number with 3, 4 or more dimensions exist and will explain how to construct them. Like classical complex number system, a multidimensional complex number system possesses algebraic operations in its complex space that have geometrical meaning in the corresponding vector space.
> >
> > In the following exposition, spaces with 3, 4 or n dimensions will be referred to as 3D, 4D and nD spaces and the corresponding complex numbers as 3D, 4D and nD complex numbers. Since a complex number corresponds to a vector, a complex number will be referred to as a vector when convenient. We will begin with constructing 3D complex number system. Then we will generalize to spaces with 4 and more dimensions.
> >
> > The 3D complex number system is constructed from a 2D complex number system which is the classical complex number system. So, let us see how classical complex number works.
> >
> > Classical complex number
> >
> > Classical complex space is a plane with two orthogonal axes, see Figure 1:
> > The axis of real numbers which is labeled as h.
> > The axis of imaginary numbers which is labeled as i.
> >
> > This plane is labeled as (h, i). On this plane a complex number is both a point and a vector, for example the vector u in Figure 1. u makes the angle  with the axis h and and its length is |u|. As complex number, u’s argument is  and its modulus is |u|. In polar coordinate system the complex number u is expressed in equation (1), where i is the imaginary unit, see (3). Equation (1) is referred to as the trigonometric form of u.
> >
> > We develop (1) into (2) in which we introduce (4) and obtain (5) where the numbers ‘a’ and b are the Cartesian coordinates of u. So, equation (5) is referred to as the Cartesian form of u.
> >
> > Equation (6) is the Euler’s formula for  and i, and is introduced into (1) which becomes (7). Equation (7) expresses u in the form of an exponential function and is referred to as the exponential form of u. So, a classical complex number can be expressed in Cartesian, trigonometric or exponential form and has a geometrical meaning which is the vector u in Figure 1.
> >
> > 3D complex number
> >
> > 3D space and vector
> > A 3D complex number is also a vector, which we will construct from the 2D plane (h, i). For doing so, we add the axis j perpendicularly to the plane (h, i) and obtain the 3D space whose axes are labeled as h, i and j, see Figure 2. We refer to this space as (h, i, j).
> >
> > We attach the unit vectors eh, ei and ej to the axes h, i and j respectively. The 3D space based on these vectors is referred to as (eh, ei, ej). We have then two 3D spaces: the complex space (h, i, j) and the vector space (eh, ei, ej). We will create a vector labeled as v in (eh, ei, ej) which corresponds to a 3D complex number in (h, i, j) labeled also as v.
> >
> > With the help of Figure 2 we create the vector v in the desired form by starting with a vector u which is expressed in (8) with |u| being its modulus and  the angle it makes with the axis h. So, u is in the horizontal plane (eh, ei). Dividing u by |u| gives the unit vector eu, see (9). The unit vectors eu and ej are the basis vectors of the vertical plane (eu, ej), see Figure 2. The vector v is created by rotating the vector u in this plane toward the axis j. The angle of rotation is , so v is expressed with the angle  on the basis vectors eu and ej in (10).
> >
> > As the length of u stays the same during the rotation, the modulus of u and v are equal, see (11). Introducing the expression of eu (9) into (10) gives (12) which is developed into (13) using (11). The vector v is expressed with its modulus |v| and the angles  and  on the basis vectors eh, ei and ej, see (13).
> >
> > Notice that the angle  is between the vector v and the horizontal plane (eh, ei), see Figure 2, which is different from the usual spherical coordinate system where the angle  is between the vector v and the axis j. So, when u is horizontal the angle  equals zero rather then /2.
> >
> > See the article with figures and equations here
> > https://www.academia.edu/71708344/Extending_complex_number_to_spaces_with_3_4_or_any_number_of_dimensions
> > https://pengkuanonmaths.blogspot.com/2022/02/extending-complex-number-to-spaces-with.html
> >
> Post it over on https://fractalforums.org. There are a lot of people who
> will try to create fractals with it. I do not have the time to give it a
> really deep dive right now. Fwiw, here is some of my work:
>
> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714

Following your advice, I have put my post in https://fractalforums.org which is here
https://fractalforums.org/fractal-mathematics-and-new-theories/28/extending-complex-number-to-spaces-with-3-4-or-any-number-of-dimensions/4675/new#new

PK

On 3/11/2022 1:59 PM, PengKuan Em wrote:
> Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a écrit :
>> On 2/17/2022 12:42 PM, PengKuan Em wrote:
[...]
>> Post it over on https://fractalforums.org." rel="nofollow" target="_blank">https://fractalforums.org. There are a lot of people who
>> will try to create fractals with it. I do not have the time to give it a
>> really deep dive right now. Fwiw, here is some of my work:
>>
>> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
>
> Following your advice, I have put my post in https://fractalforums.org which is here
> https://fractalforums.org/fractal-mathematics-and-new-theories/28/extending-complex-number-to-spaces-with-3-4-or-any-number-of-dimensions/4675/new#new

I see it. You should get some really good help there.

Le dimanche 13 mars 2022 à 08:15:22 UTC+1, Chris M. Thomasson a écrit :
> On 3/11/2022 1:59 PM, PengKuan Em wrote:
> > Le vendredi 25 février 2022 à 03:24:19 UTC+1, Chris M. Thomasson a écrit :
> >> On 2/17/2022 12:42 PM, PengKuan Em wrote:
> [...]
> >> Post it over on https://fractalforums.org." rel="nofollow" target="_blank">https://fractalforums.org. There are a lot of people who
> >> will try to create fractals with it. I do not have the time to give it a
> >> really deep dive right now. Fwiw, here is some of my work:
> >>
> >> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
> >
> > Following your advice, I have put my post in https://fractalforums.org which is here
> > https://fractalforums.org/fractal-mathematics-and-new-theories/28/extending-complex-number-to-spaces-with-3-4-or-any-number-of-dimensions/4675/new#new
> I see it. You should get some really good help there.
Yes, I did get some interesting opinions. THank you again.

PK

Le vendredi 11 mars 2022 à 13:52:21 UTC+1, timba...@gmail.com a écrit :
“ Well PK you are too modest. Your construction is interesting. I do not mean to step on your toes but to discuss the topic at greater length with you. I did refer back to your online doc and see the form:
( a + b i ) c + d j
recurring so that this cannot be a typo. Why is the c term in there at all?”
This is indeed a very sharp question. I have reread my explanation and found that it was not well written. So, I have made some correction to make it clearer.

“Why c is there?”
In fact, simple form of 3D complex number is a+bi+cj. But as multiplication is very complex with a, b and c, I have express it in the form ( a + b i ) c + d j with
a=cos theta, b=sin theta, c=cos psi, d=sin psi
and the form of 3D complex number becomes
v= ( cos theta + sin theta i ) cos psi + sin psi j

This way, I can use the exponential form to multiply v1 and v2, that is, by adding the arguments, then transform the result with arguments back to Cartesian components. The correction that I have done is an explanation of this idea and is between equations 76 and 99.

Here is the link for download.
https://drive.google.com/file/d/159FE7mCrLcjGz7MXqCEvRHAKBH6sWiKX/view?usp=sharing

> I do believe that your pursuit is valid. I am like you really. Mathematics for mathematics sake is not my cup of tea. That emergent spacetime does yield from the polysign progression; and without doing a bit of physics; now that is a basis.
>
> Of course I'd like to win you over to polysign, and you'd like to win me over to your construction. As some level we've given them away here on usenet. One odd realization: you can't even give away a good idea.

Yes. I have written this article for nD complex number just not to through this idea away. I think this is a good idea which may help the sciences. And I have get some good reaction in the fractal forum
https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
that Chris M. Thomasson had advised me.

Like you have complained, I have not being able to make my work published as they are against mainstream theories. For example, I get rejected by journals for electromagnetic work because I give a magnetic force law that respect Newton’s third law while Lorentz force law does not. This defect is what makes physics failed to progress in the last 100 years. If you search in the wrong direction, you will never find the treasure.

The scientists are like the keeper of a temple. If you go in and show them that their God is not the good one, they will surely through you out even though their theory is wrong. What they search is to keep their science in the same track as the past. They judge a new idea by looking if it is in the old track. But the good way is always to search for new track, not things in the old track.

PK

On Monday, March 14, 2022 at 5:47:50 PM UTC-4, tita...@gmail.com wrote:
> Le vendredi 11 mars 2022 à 13:52:21 UTC+1, timba...@gmail.com a écrit :
> “ Well PK you are too modest. Your construction is interesting. I do not mean to step on your toes but to discuss the topic at greater length with you. I did refer back to your online doc and see the form:
> ( a + b i ) c + d j
> recurring so that this cannot be a typo. Why is the c term in there at all?”
> This is indeed a very sharp question. I have reread my explanation and found that it was not well written. So, I have made some correction to make it clearer.
>
> “Why c is there?”
> In fact, simple form of 3D complex number is a+bi+cj. But as multiplication is very complex with a, b and c, I have express it in the form ( a + b i ) c + d j with
> a=cos theta, b=sin theta, c=cos psi, d=sin psi
> and the form of 3D complex number becomes
> v= ( cos theta + sin theta i ) cos psi + sin psi j
>
> This way, I can use the exponential form to multiply v1 and v2, that is, by adding the arguments, then transform the result with arguments back to Cartesian components. The correction that I have done is an explanation of this idea and is between equations 76 and 99.
>
> Here is the link for download.
> https://drive.google.com/file/d/159FE7mCrLcjGz7MXqCEvRHAKBH6sWiKX/view?usp=sharing
> > I do believe that your pursuit is valid. I am like you really. Mathematics for mathematics sake is not my cup of tea. That emergent spacetime does yield from the polysign progression; and without doing a bit of physics; now that is a basis.
> >
> > Of course I'd like to win you over to polysign, and you'd like to win me over to your construction. As some level we've given them away here on usenet. One odd realization: you can't even give away a good idea.
> Yes. I have written this article for nD complex number just not to through this idea away. I think this is a good idea which may help the sciences. And I have get some good reaction in the fractal forum
> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
> that Chris M. Thomasson had advised me.
>
> Like you have complained, I have not being able to make my work published as they are against mainstream theories. For example, I get rejected by journals for electromagnetic work because I give a magnetic force law that respect Newton’s third law while Lorentz force law does not. This defect is what makes physics failed to progress in the last 100 years. If you search in the wrong direction, you will never find the treasure.
>
> The scientists are like the keeper of a temple. If you go in and show them that their God is not the good one, they will surely through you out even though their theory is wrong. What they search is to keep their science in the same track as the past. They judge a new idea by looking if it is in the old track. But the good way is always to search for new track, not things in the old track.
>
> PK

I understand that your math comes out of the angular form and I've sort of been pushing on the vector form. At some level the ultimate form is the single variable form z so that we can simply discuss z1z2 for instance as the product and z1+z2 as the sum. We don't generally expect any surprises in the sum but the product always seems to be of interest. For instance division as the reverse operator of the product and its mechanics can get difficult. This certainly is true of polysign. Only just recently did division get an algorithm, and it is not as if it is easy. The old familiar behavior:
| z1 z2 | = | z1 | | z2 |
does not hold in polysign and there are nonzero products yielding zero such as:
( - 1 + 1 )( + 1 # 1 ) = 0 [P4]
Supposedly Dedekind proved that associative ( z1(z2z3) = (z1z2)z3 ) systems will always portray images of R and C as RxC (Cartesian product), CxCxR, CxCxC, etc., these being higher dimensional forms. P4 does definitely portray this RxC three dimensional behavior. If your system is algebraic in the usual sense then it will as well, unless something is amiss. This is not at all the end of the system, and afaict Dedekind's work is only available in german. A kind review:
https://projecteuclid.org/download/pdf_1/euclid.bams/1183494211
and the best online English translation though it doesn't include the stuff we are after:
https://archive.org/details/cu31924001586282
There was a group on Yahoo that was dedicated to higher dimensional algebras but it has disappeared with yahoo groups disappearance. I want to say it was Hans Koeplinger or some such name who was the moderator and the title of the group was quite clever too, though he never was receptive to polysign numbers. It's escaped me now, but possibly somebody else will chime in. That era of activity was probably close to a decade ago.

I do agree with your stance on academia. The level of regurgitation that is expected of the masters makes me want to barf. Dedekind does stand out as a plain spoken mathematician who was not afraid to use words of his own liking, and this shows through in the English translation above. It turns out that the interpretation of the continuum as if resolved by the rational value is problematic according to my own analysis. As is common in algorithmic coding false assumptions tend to carry bugs along within the system in a way that causes the system to deteriorate in multiple ways. It comes through operator theory yet lands clearly in the lap of any plain spoken person who cares to question a value such as
2.01
and the mathematician's version of this value versus the engineer's version.. Clearly they do not use the same number to mean the same thing. This then reflects back onto Dedekind and the interpretation of the continuum. Chasing digits is epsilon/delta theory. The square root of two is a value and an operator. So is one half. That these things are confused with pure numbers is only a trick of the eye, to reuse Descartes.

On Monday, March 14, 2022 at 5:47:50 PM UTC-4, tita...@gmail.com wrote:
> Le vendredi 11 mars 2022 à 13:52:21 UTC+1, timba...@gmail.com a écrit :
> “ Well PK you are too modest. Your construction is interesting. I do not mean to step on your toes but to discuss the topic at greater length with you. I did refer back to your online doc and see the form:
> ( a + b i ) c + d j
> recurring so that this cannot be a typo. Why is the c term in there at all?”
> This is indeed a very sharp question. I have reread my explanation and found that it was not well written. So, I have made some correction to make it clearer.
>
> “Why c is there?”
> In fact, simple form of 3D complex number is a+bi+cj. But as multiplication is very complex with a, b and c, I have express it in the form ( a + b i ) c + d j with
> a=cos theta, b=sin theta, c=cos psi, d=sin psi
> and the form of 3D complex number becomes
> v= ( cos theta + sin theta i ) cos psi + sin psi j
>
> This way, I can use the exponential form to multiply v1 and v2, that is, by adding the arguments, then transform the result with arguments back to Cartesian components. The correction that I have done is an explanation of this idea and is between equations 76 and 99.
>
> Here is the link for download.
> https://drive.google.com/file/d/159FE7mCrLcjGz7MXqCEvRHAKBH6sWiKX/view?usp=sharing
> > I do believe that your pursuit is valid. I am like you really. Mathematics for mathematics sake is not my cup of tea. That emergent spacetime does yield from the polysign progression; and without doing a bit of physics; now that is a basis.
> >
> > Of course I'd like to win you over to polysign, and you'd like to win me over to your construction. As some level we've given them away here on usenet. One odd realization: you can't even give away a good idea.
> Yes. I have written this article for nD complex number just not to through this idea away. I think this is a good idea which may help the sciences. And I have get some good reaction in the fractal forum
> https://fractalforums.org/index.php?action=gallery;sa=view;id=6714
> that Chris M. Thomasson had advised me.
>
> Like you have complained, I have not being able to make my work published as they are against mainstream theories. For example, I get rejected by journals for electromagnetic work because I give a magnetic force law that respect Newton’s third law while Lorentz force law does not. This defect is what makes physics failed to progress in the last 100 years. If you search in the wrong direction, you will never find the treasure.
>
> The scientists are like the keeper of a temple. If you go in and show them that their God is not the good one, they will surely through you out even though their theory is wrong. What they search is to keep their science in the same track as the past. They judge a new idea by looking if it is in the old track. But the good way is always to search for new track, not things in the old track.
>
> PK

Another quick thought and it is just my opinion, but you are roaming back and forth between (h,i,j) in R and R1 + R2i + R3j, and this you may be consistent at, but may be annoying particularly to people of the quaternion variety, which is likely a large part of your readership future or present. I do see the amount of work you have into your document and I wouldn't rush it in, but rather think it over and try to optimize notation. Possibly you are correct to stay where you are. In polysign the ordered form does work though the values are not signed at all. So a value like (1.2,2.3,3.4) is unambiguously a value in P3 which is equivalent to -1.2+2.3*3.4, or a value *1.2-2.3+3.4, depending on where the identity product goes. These sorts of notational difficulties should be repeatedly addressed consistently so that confusion cannot arise. I don't need the ordered form very often and there is actually a need of a zero sign '@' in polysign so that the identity sign does not shift around, at which point the identity position is (1,0,0,...) as in:
(1,0,0,0,...) z = z
or:
( @ 1 ) z = z
In P2 this zero sign '@' is equal to '+' with two strokes to draw the sign and modulo two these are the same thing yet in P3 the '*' sign with three strokes to draw it is the same as the zero sign. If polysign were ever seriously accepted this zero sign would need to be introduced, yet it can coexist with the traditional real value's usage of '+', which is really merely coincidence. And of course our usage of '+' to mean summation has to be carefully dealt with in polysign; the '@' symbol is unambiguously the sum in Pn using what is left of traditional notation.

This idea that notation matters: to some they will cast it off, yet our usage of notation with a sum requiring a symbol yet the product requiring mere juxtaposition is a matter of convention only. At some level that we communicate here in a singular series of characters could be an impediment to such things as general dimensional notions as we are working on. The Cartesian product as well proves to be problematic. Functional analysis relies heavily on a Cartesian product to define sum and product within the modern math curriculum. Also these are defined as binary operators whereas the n-ary form would be more general. Strangely enough in the n-ary form we see the singleton show its face as well. Translating the usual binary operator into n-ary would suggest that a sum of three values is a three dimensional problem in say RxRxR for real values. I assure you that this is not the case, and that even the binary form is a fraud. Closure as a mathematical concept requires just one set to work in. For instance if I offer you a sum of a and b yielding c where is the need of a Cartesian product?
a+b=c ; a in R, b in R, c in R.
It's one set only.

Ooh, it seems somebody has been at this page:
https://en.wikipedia.org/wiki/Ring_(mathematics)
fixing it up to hide this problem.

Still, here we see the old problem:
https://en.wikipedia.org/wiki/Binary_operation

and the earlier page does reference the binary operator still. I know these concepts are subtle, but if there is room for reinterpretation at some fundamental level then possibly the upset will unshuffle the deck, so to speak.. The accumulation in mathematics is extremely annoying and I certainly prefer to distance myself from it as you do, though we can't help but bump our heads straight back into it. Possibly the upside is that it is a tolerant place, and academia needs room enough for so many PhDs now that this tolerance could be obscenely permissive. On the downside claims of perfection within mathematics are definitely false. Escapism is more the truth of the matter. Escape into your specialty mostly with no regard for physical correspondence. For those who want physics to be born this cannot be good. Still as individuals room for everybody is established.

I don't think this tension will go away until we have a TOE or some unification and even then the idea that such a theory will carry parallel interpretations is entirely plausible. So a necessary tension between mimicry and freedom somehow will lead us into a progression, but if an invalid assumption lays beneath our works than all are flawed. I do see the Cartesian product as a candidate for this analysis. Since when are you allowed two copies of the real value? since the Cartesian product, really. So then what are two independent systems doing tied together at zero and perfectly perpendicular to each other? Is this really a natural occurrence? And here polysign do form an entirely different means of achieving the general dimensional situation. Indeed, the word 'dimension' as tied to the real line has to be carefully noted as old language. We can still use it, but the nonorthogonal coordinate systems that polysign natively develops are based on a balance amongst signs.