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

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
>

Have you tried computing the Mandelbulb with these?

Le jeudi 17 février 2022 à 22:11:27 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
> >
> Have you tried computing the Mandelbulb with these?

Thank you for replying me.

I'm sorry that I'm not a specialist of that. I have just written my theory but not more. But I will be pleased that this can help doing other things.

PK

On Thursday, February 17, 2022 at 1:21:10 PM UTC-8, tita...@gmail.com wrote:
> Le jeudi 17 février 2022 à 22:11:27 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
> > >
> > Have you tried computing the Mandelbulb with these?
> Thank you for replying me.
>
> I'm sorry that I'm not a specialist of that. I have just written my theory but not more. But I will be pleased that this can help doing other things..
>
> PK

If our dimensional universe dies does the infinite dimensional multiverse?
So why did we exist? or it? Why would science look forward to nonexistence?

Mitchell Raemsch

On 2/17/2022 1:20 PM, PengKuan Em wrote:
> Le jeudi 17 février 2022 à 22:11:27 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
>>>
>> Have you tried computing the Mandelbulb with these?
>
> Thank you for replying me.
>
> I'm sorry that I'm not a specialist of that. I have just written my theory but not more. But I will be pleased that this can help doing other things.

No problem. However, you just might like this:

https://www.soler7.com/Fractals/Matrices%20to%20Triplex.pdf

The triplex is a way to extend 2d complex numbers into 3d. It is used to
create a Mandelbulb. Here is some actual code:

https://www.shadertoy.com/view/ltfSWn

read it.

On Thursday, February 17, 2022 at 3:45:10 PM UTC-5, 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

Why is your product buried?
Can you simply jot it down here?
Is it associative? commutative?

On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> On Thursday, February 17, 2022 at 3:45:10 PM UTC-5, 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
> Why is your product buried?
> Can you simply jot it down here?
> Is it associative? commutative?

found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).

Le jeudi 17 février 2022 à 22:40:43 UTC+1, mitchr...@gmail.com a écrit :
> On Thursday, February 17, 2022 at 1:21:10 PM UTC-8, tita...@gmail.com wrote:
> > Le jeudi 17 février 2022 à 22:11:27 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
> > > >
> > > Have you tried computing the Mandelbulb with these?
> > Thank you for replying me.
> >
> > I'm sorry that I'm not a specialist of that. I have just written my theory but not more. But I will be pleased that this can help doing other things.
> >
> > PK
> If our dimensional universe dies does the infinite dimensional multiverse?
> So why did we exist? or it? Why would science look forward to nonexistence?
>
> Mitchell Raemsch
Because when sciences look, nonexistence becomes existent.

PK

Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
> On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).

Product is associative and commutative in trigonometric and exponential forms.

PK

On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:
> Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
> > On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> > found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).
> Product is associative and commutative in trigonometric and exponential forms.
>
> PK

The product can be presented as
z1 z2
and the format of its encryption is not relevant. If the product obeys
z1( z2 + z3) = z1 z2 + z1 z3
and your numbers can be broken apart as sums then I'm afraid your system may be quite broken.
Can I ask you what is
( 1i + 5j )( 1i + 4h ) ?
I don't mean to be discouraging and I am open to your answer. Hopefully this will be easy too.

Are you aware of polysign numbers? They are general dimensional and carry the reals as P2 and the complex numbers as P3. P4 are 3D in ordinary terms, but polysign does away with orthogonality and has no need of the Cartesian product in its construction. Simply balance in the signs will do:
P2 : - 1 + 1 = 0 .
P3 : - 1 + 1 * 1 = 0 .
where '*' is a third sign. Clearly the - and + take different roles in P3 compared to P2. Always a firm expression ought to declare what signature it is working in. For instance
+ 2.45
without a specification of P2, or P3, or P4 for instance is ambiguous. It turns out that
-1
or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.

Le samedi 19 février 2022 à 00:16:58 UTC+1, timba...@gmail.com a écrit :
> On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:
> > Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
> > > On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> > > found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).
> > Product is associative and commutative in trigonometric and exponential forms.
> >
> > PK
> The product can be presented as
> z1 z2
> and the format of its encryption is not relevant. If the product obeys
> z1( z2 + z3) = z1 z2 + z1 z3
> and your numbers can be broken apart as sums then I'm afraid your system may be quite broken.
> Can I ask you what is
> ( 1i + 5j )( 1i + 4h ) ?
> I don't mean to be discouraging and I am open to your answer. Hopefully this will be easy too.
>
> Are you aware of polysign numbers? They are general dimensional and carry the reals as P2 and the complex numbers as P3. P4 are 3D in ordinary terms, but polysign does away with orthogonality and has no need of the Cartesian product in its construction. Simply balance in the signs will do:
> P2 : - 1 + 1 = 0 .
> P3 : - 1 + 1 * 1 = 0 .
> where '*' is a third sign. Clearly the - and + take different roles in P3 compared to P2. Always a firm expression ought to declare what signature it is working in. For instance
> + 2.45
> without a specification of P2, or P3, or P4 for instance is ambiguous. It turns out that
> -1
> or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.

Thank you for reading my article and giving me reply for discussion, which is a good occasion to show the 3D complex number with concrete numbers.

I have put the figure and equations in this page.
https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html

What is the product that you posted (1i + 5j )(1i + 4h ) ? First, we do not multiply in Cartesian form. We have to convert the complex numbers into trigonometric form.
(1i + 5j ) is in the plane (i,j), (1i + 4h ) in (h,i), see Figure 1

(1i + 4h ) is converted from eq. 1 to 3, (1i + 5j ) is converted from eq. 4 to 6. The used formulas for conversion are the eq. 16 of the original paper.
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

Then we multiply them. I have commuted the two 3D complex numbers in eq. (7) because multiplication is commutative.
Then I multiply (1i + 5j )(1i + 4h ) in trigonometric form in eq. (8). The used formulas for multiplication are the eq. 51, 52 of the original paper.
It is easier to use exponential form, see eq. (9)(10)
Then we develop eq. 8 into Cartesian form in eq. (11) which is simplified in to (12)

If we multiply directly your numbers, we would have
(1i + 5j )(1i + 4h ) = i (1i + 4h ) + 5j (1i + 4h )
= 1ii + 4ih + 5 ji + 4*5jh = -1 + 4ih + 5ji + 20jh
It is interesting to compare this with the eq. 12
If we commute your number, we would have
(1i + 4h ) (1i + 5j ) = (1i + 4h )i+ (1i + 4h ) 5j
=1ii + 4hi + 5ij + 20hj

“Are you aware of polysign numbers?”
Sorry, I do not know this. I will get a look.

PK

Le samedi 19 février 2022 à 00:16:58 UTC+1, timba...@gmail.com a écrit :
> On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:

> or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.

Sorry, I have made a mistake in my example for " Extending...".
https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
It was in eq. 11 where I forgot to multiply 1 with square root of 4^2+1.
The numerical result was wrong, but the theory is not affected.
I have already corrected it.

Apology to you and all people who have read this page of example.

PK

Le samedi 19 février 2022 à 00:16:58 UTC+1, timba...@gmail.com a écrit :
> On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:
> > Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
> > > On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> > > found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).
> > Product is associative and commutative in trigonometric and exponential forms.
> >
> > PK
> The product can be presented as
> z1 z2
> and the format of its encryption is not relevant. If the product obeys
> z1( z2 + z3) = z1 z2 + z1 z3
> and your numbers can be broken apart as sums then I'm afraid your system may be quite broken.
> Can I ask you what is
> ( 1i + 5j )( 1i + 4h ) ?
> I don't mean to be discouraging and I am open to your answer. Hopefully this will be easy too.
>
> Are you aware of polysign numbers? They are general dimensional and carry the reals as P2 and the complex numbers as P3. P4 are 3D in ordinary terms, but polysign does away with orthogonality and has no need of the Cartesian product in its construction. Simply balance in the signs will do:
> P2 : - 1 + 1 = 0 .
> P3 : - 1 + 1 * 1 = 0 .
> where '*' is a third sign. Clearly the - and + take different roles in P3 compared to P2. Always a firm expression ought to declare what signature it is working in. For instance
> + 2.45
> without a specification of P2, or P3, or P4 for instance is ambiguous. It turns out that
> -1
> or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.

I have rethought about the example and have rewritten the page of explanation here.
https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
I have put much more detail to make the reading easier while having corrected another mistake.

I have proofread the page again and again, hoping that there may not be other error.

PK

On Sunday, February 20, 2022 at 6:52:27 AM UTC-5, tita...@gmail.com wrote:
> Le samedi 19 février 2022 à 00:16:58 UTC+1, timba...@gmail.com a écrit :
> > On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:
> > > Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
> > > > On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> > > > found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).
> > > Product is associative and commutative in trigonometric and exponential forms.
> > >
> > > PK
> > The product can be presented as
> > z1 z2
> > and the format of its encryption is not relevant. If the product obeys
> > z1( z2 + z3) = z1 z2 + z1 z3
> > and your numbers can be broken apart as sums then I'm afraid your system may be quite broken.
> > Can I ask you what is
> > ( 1i + 5j )( 1i + 4h ) ?
> > I don't mean to be discouraging and I am open to your answer. Hopefully this will be easy too.
> >
> > Are you aware of polysign numbers? They are general dimensional and carry the reals as P2 and the complex numbers as P3. P4 are 3D in ordinary terms, but polysign does away with orthogonality and has no need of the Cartesian product in its construction. Simply balance in the signs will do:
> > P2 : - 1 + 1 = 0 .
> > P3 : - 1 + 1 * 1 = 0 .
> > where '*' is a third sign. Clearly the - and + take different roles in P3 compared to P2. Always a firm expression ought to declare what signature it is working in. For instance
> > + 2.45
> > without a specification of P2, or P3, or P4 for instance is ambiguous. It turns out that
> > -1
> > or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.
> I have rethought about the example and have rewritten the page of explanation here.
> https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
> I have put much more detail to make the reading easier while having corrected another mistake.
>
> I have proofread the page again and again, hoping that there may not be other error.
>
> PK

OK, nice work PK. You are clearly working your system very seriously.
However as I see it you've got:
( 1i + 5j )( 1i + 4h ) = - 5 + 20 i + sqrt(17) j
and now the (h,i,j) system has turned out to have a real component? Isn't this then a 4D result?

I do take interest in your system though I don't understand it well yet.

On 2/20/2022 9:16 AM, Timothy Golden wrote:
> On Sunday, February 20, 2022 at 6:52:27 AM UTC-5, tita...@gmail.com wrote:
>> Le samedi 19 février 2022 à 00:16:58 UTC+1, timba...@gmail.com a écrit :
>>> On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:
>>>> Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
>>>>> On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
>>>>> found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).
>>>> Product is associative and commutative in trigonometric and exponential forms.
>>>>
>>>> PK
>>> The product can be presented as
>>> z1 z2
>>> and the format of its encryption is not relevant. If the product obeys
>>> z1( z2 + z3) = z1 z2 + z1 z3
>>> and your numbers can be broken apart as sums then I'm afraid your system may be quite broken.
>>> Can I ask you what is
>>> ( 1i + 5j )( 1i + 4h ) ?
>>> I don't mean to be discouraging and I am open to your answer. Hopefully this will be easy too.
>>>
>>> Are you aware of polysign numbers? They are general dimensional and carry the reals as P2 and the complex numbers as P3. P4 are 3D in ordinary terms, but polysign does away with orthogonality and has no need of the Cartesian product in its construction. Simply balance in the signs will do:
>>> P2 : - 1 + 1 = 0 .
>>> P3 : - 1 + 1 * 1 = 0 .
>>> where '*' is a third sign. Clearly the - and + take different roles in P3 compared to P2. Always a firm expression ought to declare what signature it is working in. For instance
>>> + 2.45
>>> without a specification of P2, or P3, or P4 for instance is ambiguous. It turns out that
>>> -1
>>> or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.
>> I have rethought about the example and have rewritten the page of explanation here.
>> https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
>> I have put much more detail to make the reading easier while having corrected another mistake.
>>
>> I have proofread the page again and again, hoping that there may not be other error.
>>
>> PK
>
> OK, nice work PK. You are clearly working your system very seriously.
> However as I see it you've got:
> ( 1i + 5j )( 1i + 4h ) = - 5 + 20 i + sqrt(17) j
> and now the (h,i,j) system has turned out to have a real component? Isn't this then a 4D result?

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 take interest in your system though I don't understand it well yet.
>
I do as well.

Le dimanche 20 février 2022 à 15:16:57 UTC+1, timba...@gmail.com a écrit :
> On Sunday, February 20, 2022 at 6:52:27 AM UTC-5, tita...@gmail.com wrote:
> OK, nice work PK. You are clearly working your system very seriously.
> However as I see it you've got:
> ( 1i + 5j )( 1i + 4h ) = - 5 + 20 i + sqrt(17) j
> and now the (h,i,j) system has turned out to have a real component? Isn't this then a 4D result?
>
> I do take interest in your system though I don't understand it well yet.

Thanks.

In fact, h is not an imaginary number, but the unit vector on the real line.. This is because it is easier to put the space as (h, i, j) rather than (1, i, j) and make each dimension more distinguishable in the equations.

So, (h, i, j) is a 3D space with h: real, i:1st imaginary, j: second imaginary.

By the way, I have put a cleaner version of the example in the same page.
https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html

And sadly, I have found an other numerical error. Sigh. I cannot avoid error by writing on the fly in discussion.

So, the result is ( 1i + 5j )( 1i + 4h ) = - 1 + 4 i + 5 sqrt(17) j

PK

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

On Sunday, February 20, 2022 at 11:54:37 AM UTC-5, tita...@gmail.com wrote:
> Le dimanche 20 février 2022 à 15:16:57 UTC+1, timba...@gmail.com a écrit :
> > On Sunday, February 20, 2022 at 6:52:27 AM UTC-5, tita...@gmail.com wrote:
> > OK, nice work PK. You are clearly working your system very seriously.
> > However as I see it you've got:
> > ( 1i + 5j )( 1i + 4h ) = - 5 + 20 i + sqrt(17) j
> > and now the (h,i,j) system has turned out to have a real component? Isn't this then a 4D result?
> >
> > I do take interest in your system though I don't understand it well yet..
> Thanks.
>
> In fact, h is not an imaginary number, but the unit vector on the real line. This is because it is easier to put the space as (h, i, j) rather than (1, i, j) and make each dimension more distinguishable in the equations.
>
> So, (h, i, j) is a 3D space with h: real, i:1st imaginary, j: second imaginary.
>
> By the way, I have put a cleaner version of the example in the same page.
> https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
>
> And sadly, I have found an other numerical error. Sigh. I cannot avoid error by writing on the fly in discussion.
>
> So, the result is ( 1i + 5j )( 1i + 4h ) = - 1 + 4 i + 5 sqrt(17) j
>
> PK

So,
( 1i + 5j )( 1i + 4h ) = - 1 h + 4 i + 5 sqrt(17) j

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.

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

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

On Sunday, February 20, 2022 at 6:42:01 AM UTC-8, Michael Moroney wrote:
> On 2/20/2022 9:16 AM, Timothy Golden wrote:
> > On Sunday, February 20, 2022 at 6:52:27 AM UTC-5, tita...@gmail.com wrote:
> >> Le samedi 19 février 2022 à 00:16:58 UTC+1, timba...@gmail.com a écrit :
> >>> On Friday, February 18, 2022 at 4:26:06 PM UTC-5, tita...@gmail.com wrote:
> >>>> Le vendredi 18 février 2022 à 21:28:34 UTC+1, timba...@gmail.com a écrit :
> >>>>> On Friday, February 18, 2022 at 8:39:50 AM UTC-5, Timothy Golden wrote:
> >>>>> found: Rule 3: The products of imaginary units ij and ji are not defined in the 3D complex space (h, i, j).
> >>>> Product is associative and commutative in trigonometric and exponential forms.
> >>>>
> >>>> PK
> >>> The product can be presented as
> >>> z1 z2
> >>> and the format of its encryption is not relevant. If the product obeys
> >>> z1( z2 + z3) = z1 z2 + z1 z3
> >>> and your numbers can be broken apart as sums then I'm afraid your system may be quite broken.
> >>> Can I ask you what is
> >>> ( 1i + 5j )( 1i + 4h ) ?
> >>> I don't mean to be discouraging and I am open to your answer. Hopefully this will be easy too.
> >>>
> >>> Are you aware of polysign numbers? They are general dimensional and carry the reals as P2 and the complex numbers as P3. P4 are 3D in ordinary terms, but polysign does away with orthogonality and has no need of the Cartesian product in its construction. Simply balance in the signs will do:
> >>> P2 : - 1 + 1 = 0 .
> >>> P3 : - 1 + 1 * 1 = 0 .
> >>> where '*' is a third sign. Clearly the - and + take different roles in P3 compared to P2. Always a firm expression ought to declare what signature it is working in. For instance
> >>> + 2.45
> >>> without a specification of P2, or P3, or P4 for instance is ambiguous.. It turns out that
> >>> -1
> >>> or minus unity, or MU, is quite relevant and can be spoken of specially as it is in the first sign. Its activity in product can yield all the other signs as well. In real analysis the tendency is to treat +1 as the more fundamental of the two.
> >> I have rethought about the example and have rewritten the page of explanation here.
> >> https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
> >> I have put much more detail to make the reading easier while having corrected another mistake.
> >>
> >> I have proofread the page again and again, hoping that there may not be other error.
> >>
> >> PK
> >
> > OK, nice work PK. You are clearly working your system very seriously.
> > However as I see it you've got:
> > ( 1i + 5j )( 1i + 4h ) = - 5 + 20 i + sqrt(17) j
> > and now the (h,i,j) system has turned out to have a real component? Isn't this then a 4D result?
> 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 take interest in your system though I don't understand it well yet..
> >
> I do as well.

Cross product only exists in 3 and 7 dimensions.

There are no knots in four dimensions.

You'd be better off studying Cartan.

torsdag 17 februari 2022 kl. 21:45:10 UTC+1 skrev tita...@gmail.com:
> 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

Look up quaternions/hamiltonians

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:
> 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.

I have written the cleaner derivation for the expression of v1v2 in
https://pengkuanonmaths.blogspot.com/2022/02/example-for-extending-complex-number-to.html
THe complex numbers are
v1=(A1+B1 i) C1+D1 j
v2=(A2+B2 i) C2+D2 j

The formula of v1v2 is
v1 v2=((A1 A2-B1 B2 )+(B1 A2+A1 B2 )i)(C1 C2-(D1 D2)/|v1| |v2| )+(D1 |v2| C2+|v1| C1 D2 )j
PK

Le jeudi 24 février 2022 à 05:33:11 UTC+1, Jan a écrit :
> On Thursday, February 17, 2022 at 12:45:10 PM UTC-8, tita...@gmail.com wrote:
> 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
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