# From Coulomb’s force to magnetic force and experiments that show magnetic force parallel to current

Abstract: The Lorentz force law is fundamental for electromagnetism. However, it is known long ago that the Lorentz forces between two current elements do not respect the Newton’s third law. This seemingly harmless flaw had never been corrected. In physical sciences a discrepancy often hides in it new understanding or unexpected breakthrough. For solving this problem, we give a purely theoretical derivation of magnetic force which respects the Newton’s third law in the case of current elements and is identical to the Lorentz force in the case of coils. This new law reveals how electric force is transformed into magnetic force by velocity and is supported by experimental evidences that we will explain and compute with the new law.

## 1.Introduction

The Figure 1 shows a case where dFa is perpendicular to dFb , so, dFa + dFb ¹ 0. This problem was known for longtime. People justify that the Lorentz forces that two closed loop currents act on each other do satisfy the Newton’s third law. Nevertheless, breaking the Newton’s third law does not fit scientific standard, even for the Lorentz forces law which is fundamental.

We will try to solve this problem with a new magnetic force law that we have derived with pure theory. The new law is derived from the Coulomb’s law which defines the Coulomb’s force for fixed charges. For moving electrons, the Coulomb’s force undergoes relativistic effects and varies with velocity.

## 2.     Consequences

### ·       The relation mu0 eps0 c2 = 1

Historically, the values of mu0, eps0 and the speed of light c were measured experimentally. It was James Clerk Maxwell who noticed that mu0eps0c2 = 1 . So, it was an empirical law.  In our derivation this relation emerged naturally from both relativistic dynamic effect and changing distance effect. So, we have theoretically proven this relation and in consequence, the relation mu0eps0c2 = 1 is now a theoretical law.

### ·       Biot–Savart law

The equation (58) is identical to the Biot–Savart law (59) but is derived with pure theory. So, the Biot–Savart law becomes a theoretical law too.

### ·       Lorentz force law

(61) is the Lorentz force that one dIb exerts on dIa . So, we have derived the Lorentz force law from the Coulomb’s law.

### ·       Magnetic force vs. Newton’s third law

The sum of the magnetic force (49) and its back force is zero. So, the magnetic force law (49) satisfies the Newton’s third law for current elements . Being an experimental law, the Lorentz force law does not describe a force that does not exist and thus, lacks this term. So, it cannot satisfy Newton’s third law. Thanks to the fully theoretical derivation, the magnetic force law (49) contains the missing last term and consequently, satisfies Newton’s third law.

## 3.     Experimental evidences

### ·       My experiments

The first experiment is «Continuous rotation of a circular coil experiment». The video of this experiment is: https://www.youtube.com/watch?v=9162Qw-wNow. In this video we see a round coil that rotates in its plane. Because the coil is round the driving force must be parallel to the wire, that is, the driving force is parallel to the current. This force cannot be Lorentz force which is perpendicular to the current. A detailed technical explanation is in the paper «Showing tangential magnetic force by experiment» .

I have also made a « Circular motor driven by tangential magnetic force » . The video of this experiment is: https://www.youtube.com/watch?v=JkGUaJqa6nU&list=UUuJXMstqPh8VY4UYqDgwcvQ. The technical details of this experiment is: « Detail of my circular motor using tangential force and the equivalence with homopolar motor » .

### ·       Experiment of wire fragmentation

In 1961, Jan Nasilowski in Poland has carried out an experiment which consisted of passing a huge current in a thin wire. The wire exploded into small pieces. The interesting thing is that the wires were not melted but teared apart by mechanical force.

## 4.     Conclusion

Because the new law gives the same prediction as the Lorentz force law for closed loop currents, it works for electromagnetism as the Lorentz force law. However, the component of magnetic force parallel to the current is new and shown to be rather significant. So, it could be used as the driving force for new devices.

Since the Biot–Savart law, the Lorentz force law and the relation mu0 eps0 c2 = 1are derived with pure theory, the deep mechanism that transforms electric force into magnetic force is revealed to be the two relativistic effects, electromagnetism is much better understood.

For more detail of this study please read the complete paper here:

 Kuan Peng, 2017, Video https://www.youtube.com/watch?v=9162Qw-wNow

 Kuan Peng, 2014, Video https://www.youtube.com/watch?v=JkGUaJqa6nU&list=UUuJXMstqPh8VY4UYqDgwcvQ

 Kuan Peng, 2014, « Detail of my circular motor using tangential force and the equivalence with homopolar motor » , https://www.academia.edu/7879755/Detail_of_my_circular_motor_using_tangential_force_and_the_equivalence_with_homopolar_motor

# Is Hilbert’s Grand Hotel a paradox?

Hilbert’s Grand Hotel shows that a fully occupied hotel with infinitely many rooms can accommodate additional guests. But our analyze finds that this is not true. Let us illustrate Hilbert’s Grand Hotel in Figure 1 where each square is a room and is occupied. Suppose that the rooms of the hotel are numbered 1, 2, 3 … . We call the guest of the room 1 guest 1, the guest of the room 2 guest 2, the guest of the room n guest n and so on. The new guest is called guest G.

Let us analyze this paradox with a first case where the occupant of each room accepts to shift to the next room. When the guest G arrives and asks for a room, according to David Hilbert, the hotelkeeper will move the guest 1 to room 2 and accommodate the guest G in room 1, then move the guest 2 to room 3 to accommodate the guest 1, and so on. The general way is to move the guest n-1 to the room n to accommodate the guest n-2. This way the guest G is accommodated while all the old guests still has a room.

Let us show this procedure of room shifting with Figure 2. The room shifting is done step by step. The guest G takes the room 1, the guest 1 takes the room 2 and so on. At the step n-1, the guest n-1 is before the door of the room n. At each step from 1 to n-1, the guests 1 to n-1 are successively out of room. This is true for all n, however big n is. So, at any step one guest is out, which is shown in Figure 2.

Let us consider the case where no guest accepts to leave his room, see Figure 2. The guest G will knock successively every room. As no guest lets him in, he will knock the next room forever. In consequence, he will be out of room while going to infinitely far. This is illustrated by the letter G before the rows of rooms. So, the guest G is always out.

In the first case, it was the guests 1, 2, 3… that are out of room at each step. In the second case it is always the guest G who is out of room. So, in both cases one guest is out of room at every step, that is, there is a guest before the room n no matter how big the number n is. This means that one additional guest in Hilbert’s Grand Hotel is not accommodated even he goes to infinitely far. In other words, Hilbert’s Grand Hotel cannot accommodate additional guest in its infinitely many rooms.

For more detail of this study please read the complete paper here:

# Trajectory of ‘Oumuamua and wandering Sun, alien asteroids and comets detected by SOHO

The apparent non‑gravitational acceleration the extra-solar-system ‘Oumuamua exhibits is puzzling. We find that when the position and velocity of the Sun is correctly set in computing the predicted orbit, ‘Oumuamua’s trajectory can be explained with gravity and we have reproduced the unexpected gap by computation. We also propose to search for new extra-solar-system high speed asteroids with SOHO to check our method with their trajectories.

## 1.     ‘Oumuamua’s acceleration

‘Oumuamua, formally designated 1I/2017 U1, is an interstellar object passing through the Solar System which was first detected by Robert Weryk using the Pan-STARRS telescope on 19 October 2017 . It seems to exhibit non‑gravitational acceleration, making it go further than expected .

In the article « Our Solar System’s First Known Interstellar Object Gets Unexpected Speed Boost » June 27, 2018 , it was reported that “’Oumuamua had been boosted by 25,000 miles (40,000 kilometers) compared to where it would have been if only gravitational forces were affecting its motion”, see Figure 1 and Figure 2 which are screenshots of the animation in this article and show the predicted orbit and the unexpected gap of 40 000 kilometers between the last observation of ‘Oumuamua and the predicted spot.

Although “the Canada-France-Hawaii Telescope (CFHT) and, in the following days, the ESO Very Large Telescope (VLT) and the Gemini South Telescope, both8-meter-class facilities, found no sign of coma despite optimal seeing conditions”, the authors still “find outgassing to be the most physically plausible explanation” .

However, because of the lack of coma we think that the boost of ‘Oumuamua can still be attributed to some overlooked effects. For example, what if the Sun is moving in the solar system? What if, due to the motion, the Sun is not exactly at the focus of the predicted hyperbolic orbit? In these cases, the real trajectory of ‘Oumuamua will not fit the predicted one.

We know that the Sun is not exactly at the barycenter of the solar system and moves relative to it, as shown by Figure 3  in which the Sun is the central point and the barycenter wanders around it. But in reality the solar system is an isolated system the center of which is its barycenter. The frame of the barycenter is inertial, in this frame the barycenter is fixed and the Sun wanders around it. So, the Sun is always at a distance from the barycenter and moves at nonzero velocity.

Pursuing this direction we propose this hypothesis: the unexpected gap could be the consequence of erroneous position and velocity of the Sun with which the predicted orbit was computed.

For checking this hypothesis we will compute the trajectory of ‘Oumuamua by adjusting the position and velocity of the Sun such as to reproduce approximately the gap of 40 000 km.

## 2.     Static and shifting orbits

The basic parameters of the predicted orbit for ‘Oumuamua are published by JPL / NASA in the page ‘Oumuamua (A/2017 U1) . The eccentricity e, the semi-major axis a, the orbital elements  and the standard gravitational parameter of the Sun GM  are given in Table 1. The semi-latus rectum l and specific relative angular momentum h are computed in (2) and (3). The predicted orbit is a hyperbolic orbit which is expressed by equation (1) .

Just for the purpose of checking our hypothesis, we put the focus of the predicted orbit at the barycenter of the solar system. This orbit is static and we call it the static orbit. The orbit of ‘Oumuamua is a hyperbola the focus of which is the moving Sun. So this orbit shifts in the frame of the barycenter and is called shifting orbit.

## 3.     Search for high speed asteroids near the Sun

Beside of computing the trajectory of ‘Oumuamua, a better way to check our hypothesis is by experiment, that is, by observing new high speed asteroids and compare their trajectory with prediction. However, recorded asteroids and comets coming into the solar system from the outside are scarce. But I think that in reality such objects are not so rare, only that far from the Sun they are too faint to be detected. When they are near the Sun they become very bright and can be detected by Sun gazing satellites such as SOHO (Solar and Heliospheric Observatory).

Many such alien asteroids and comets are already recorded by SOHO and are dormant in great number in the archives of SOHO. Thanks to NASA Goddard’s YouTube Movie “Decades of Sun from ESA & NASA’s SOHO” , which is a video made with all the photos taken by LASCO/C3 between 1998/01/06 and 2020/10/23,  I have found several high speed asteroids and comets in it.

For example, the asteroid that was recorded from 2004/02/26 to 2004/02/28. I have computed its visual velocity which is the velocity of the dot on the image and got 160 km/s, see Figure 7. The comet recorded from 2015/02/18 to 2015/02/21 is measured at 182 km/s, see Figure 8. For comparison, the speed of ‘Oumuamua at perihelion is 87.71 km/s . I used the diameter of the view field of LASCO/C3 which equals 30 radii of the Sun  and the time printed on the images to compute its visual velocity. As the actual trajectories make an angle with the plane of the image, their real velocity are forcefully bigger.  Only objects coming from the outside of the solar system can be as speedy.

I have made a clip of the asteroid and put it here: Super-fast alien asteroid (160 km/s), taken by SOHO in 2004, https://youtu.be/GTGuEKndNIc

The alien asteroids and comets in the archives of SOHO are interesting. We can count their number, mapping their direction and measure their record breaking speed and size. On the other hand, we can monitor in real time the appearance of new alien asteroids and comets, work out their orbits for observing them later.

In searching asteroids in the Movie “Decades of Sun from ESA & NASA’s SOHO” , the background stars and the streaks left by space particles are very dizzying which makes the researcher miss interesting asteroids and comets. So, I suggest that the background stars and streaks be removed for this research.

For those who are interested in seeing the asteroids and comets that I have found I have put in the appendices the links that point to the frames of the Movie “Decades of Sun from ESA & NASA’s SOHO”  where they appear. I have seen more comets because they are brighter and easy to see. I have also put the few visual velocities that I have computed, which show that some comets are very fast and could have come from the outside of the solar system.

## 4.     Discussion

The analysis above shows that the unexpected gap can be well explained by the gravity of the Sun provided that the position and velocity of the Sun be correctly set in computing the predicted orbit. What matters is that the position and velocity of ‘Oumuamua be given with respect to the Sun’s actual position and velocity. If ‘Oumuamua were located with respect to the barycenter of the solar system while the Sun is not there, the predicted orbit would be wrong.

We have shown that because the combined trajectory is not a hyperbola and has moving focus the non‑gravitational acceleration may be unnecessary to explain that “the observed orbital arc cannot be fit in its entirety by a trajectory governed solely by gravitational forces due to the Sun, the eight planets, the Moon, Pluto, the 16 biggest bodies in the asteroid main belt, and relativistic effects” . This is corroborated by the lack of coma.

We have also discovered that even the smallest error on the position and velocity of the Sun is enough to create an unexpected gap. Indeed, the distance used to reproduce the unexpected gap is only 12% of the radius of the Sun. So, the position and velocity of the Sun and the data of ‘Oumuamua must be precisely set.

On the other hand, our hypothesis must be checked again and again before being validated. Indeed, other value was given to this gap, for example, in the article « THIS INTERSTELLAR ASTEROID IS ACCELERATING » , 100 000 km has been given to the gap rather than 40 000 km. Obviously, there are several different predicted trajectories for ‘Oumuamua with different results. The correct theoretical trajectory for ‘Oumuamua must give Gap = 0 and our method can help for finding it.

We have proposed to check our hypothesis using the SOHO satellite in real time to find new alien asteroids and comets, work out their orbits for observing them later. It will be interesting to search for alien asteroids and comets in the archives of SOHO, which I have done partly.

For more detail of this study please read the complete paper here:

# Determination of the relative roll, pitch and yaw between arbitrary objects using 3D complex number

The roll, pitch and yaw of an object relative to another is complex to compute. We use 3D complex number to compute them which makes the computation easier and more intuitive.

Roll, pitch and yaw are angles of orientation of an object in space and the conversion of these angles among different reference frames is not easy, as this example illustrates:

https://math.stackexchange.com/questions/1884215/how-to-calculate-relative-pitch-roll-and-yaw-given-absolutes

We will use 3D complex number to compute the roll, pitch and yaw of a telephon relative to a car. We label the car as object a and the telephon as object b. Their reference frames are labeled as frame A and B and these frames are orientated with respect to the ground whose frame is labeled as frame G. The base vectors of the frame A are ax, ay and az, that of the frame Bare bx, by and bz and that of the frame G are gx, gy and gz. The vectors ax, ay and az and bx, by and bz are expressedwith gx, gy and gz in equations (1) and (2) with Maand Mb being the matrices of transformation.

The roll, pitch and yaw of the object b relative to the object a define the orientation of the frame B in the frame A. This orientation is defined by the angles of rotation q, j and y, see Figure 1. For computing these angles we express the vectors bx, by and bz with the vectors ax, ay and az in (3), (4) and (5). In (10), (11) and (12) the coefficients in the parentheses are dot products between ax, ay and az and bx, by and bz. The equations (10) , (11) and (12) are written in matrix form in (13) from which we extract the matrix of transformation Mba shown in (14), see (3), (4) and (5).

From the 5 underscored coefficients in (14) we derive the angles q, j and y using (15), (16) and (17), knowing that  is always positive because j is between -p/2 and p/2, see Figure 1. Roll, pitch and yaw of an object are the rotation angles of the object around the x, y and z axis respectively, see Figure 1 and the page https://en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft).

So, the conversion formula between roll, pitch and yaw and the angles q, j and y is equation (18).

The main advancement given by our method using 3D complex number is to have related the roll, pitch and yaw of an object to the 3D complex number that represent the x axis of the object. This mathematical discovery makes the conversion of roll, pitch and yaw between arbitrary frames easier and more intuitive.

The angles of rotation q, j and y are computed with the 5 underscored coefficients of the matrix Mba in (14) saving thus the computation time for the 4 other coefficients which are not needed. This will make the computation faster and the motion of moving images on screen smoother. So, our method using 3D complex number would be beneficial to applications such as video games or street view of digital map etc.

For more detail of this study please read the complete paper here:

«Determination of the relative roll, pitch and yaw between arbitrary objects using 3D complex number»

# Real numbers and points on the number line with regard to Cantor’s diagonal argument

Cantor’s diagonal argument claims that ℝ is uncountable. When we see real numbers as points on the number line, we can put a name on each point and put the names into a list without contravening Cantor’s diagonal argument because we cannot create a diagonal from a list of names.

However, we do not need such a impossible list, but just to split R into two parts, S2 and S10. the members of S2 are real numbers expressed in binary, those of S10 in decimal. We create a list of real numbers by picking one member from S2 and one member from S10 alternately and forever. This list is a composite list whose members are in binary and decimal alternately. The diagonal of this list is a sequence of binary and decimal digits alternately and out-of-the-list-number cannot be constructed from it.

In fact, composite list can be created in splitting R into many subsets in numeral systems of different bases from which no out-of-the-list-number can be created and there is no real number excluded from the composite list. Because the composite list is constructed from the whole R and no real number is found outside, the composite list contains R.

If there is one list that contains R we can already conclude that R is countable. But the permutation of the subsets of R can create a huge number of different composite lists which all contain R. So, we conclude with confidence that R is countable. Then Cantor’s diagonal argument fails.

Cantor’s diagonal argument expresses real numbers only in one numeral system, which restricts the used list. If a binary list is shown not to contain R, this can be caused either by “list” or by “binary”. Because Cantor has focused only on “list” overlooking “binary”, this is the flaw that breaks Cantor’s diagonal argument which then does not prove ℝ uncountable.

For more detail of this study please read the complete paper here:

# Examination of Cantor’s proofs for uncountability and axiom for counting infinite sets

An analysis of Cantor’s theory of uncountable sets: The logic of his proofs has some weaknesses. Cantor assumes for both his proofs that all real numbers (set R) are in a list (list L). Considering L as a set this assumption assumes R belongs to L. This makes the claim “a real number is constructed but is not in the list L” questionable. We propose a solution to this problem, an axiom for counting infinite sets and a solution to continuum hypothesis.

# Examination of Cantor’s proofs for uncountability and axiom for counting infinite sets

I do a detailed analysis of Cantor’s theory of uncountable sets. The logic of his proofs has some weaknesses. I propose an axiom and a solution to continuum hypothesis.

The main idea is:
Assumption of Cantor’s proofs: All real numbers (set R) are in a list (list L).

This assumption means R=L, considering L as a set. This makes the claim “a real number is created but is not in the list L” wrong. Indeed, if a number is outside L, it is outside R too. So, the statement “the created real number is not in the list L” means it is not in R and is not a real number, which is equivalent to claim that a real number is not real number. This is absurd but Cantor’s diagonal argument and nested intervals proof both claim that a real number is not in the list L and thus, is not a real number, which make them wrong.

On the other hand, Cantor’s both proofs search for contradiction. Can “this real number is not a real number” be the contradiction? No. The contradiction of the proofs is in the third step: sout “is not in the list”. By failing to create sout the second step collapses before the third step declares the contradiction.

# For Newtonian mechanics

The Schwarzschild radius is the radius of the event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics, which is explained below. Consider a big mass M which creates the gravitational acceleration a for a small mass m at the distance r from M, see figure 1 and equation (1) for gravitational acceleration a=GM/r2. For computing v the radial velocity of m in the gravitational field of M we integrate equation (1).

We compute for the case where m freefalls from infinitely far starting with zero velocity, see (7). With these conditions the radial velocity of m at the distance r2 from M is computed in (8), v2 = 2GM/r2. Reversing (8), r2 = 2GM/v2. The Schwarzschild radius of the event horizon of M is rs such that the Schwarzschild factor equals infinity, see (10).

When v2 equals the speed of light c, we apply v2 = c into (9) and we obtain (11) where r2 = rs. So, r2 equals rs and the Schwarzschild radius is computed with Newtonian mechanics.

# For relativity

Although the Schwarzschild radius rs is a relativistic quantity, in the above it is derived completely with Newtonian mechanics, which is somewhat weird. What will be its value if we apply relativistic principle?

In the following derivation we will use the formula for relativistic transformation of acceleration which is derived in the paper « Relativistic kinematics » linked here: https://www.academia.edu/44582027/Relativistic_kinematics

The formula is the equation (18) of the paper.

Here, this formula is given by (12) in which the gravitational acceleration of m is a and the acceleration in space is ar and the radial velocity is computed by integration. With the same conditions as (7), the constant of integration k equals 0, see (16). Then, using k = 0 in (15), v is expressed with r in (17).

In the case where the small mass m approaches M, the distance r approaches 0, the radial velocity of m approaches the speed of light c, see (18). So, v the radial velocity of m does not become bigger than the speed of light c for r > 0. The Schwarzschild radius rs is the radius such that v = c. So, rs = r = 0, see (19).

# Conclusion

When the relativistic principles are applied correctly, the Schwarzschild radius rs equals zero. The gravitational force on m approaches infinity near M. But the speed of m never reaches the speed of light c, which is true for any force however strong it is and for time of acceleration however long it is. The Schwarzschild radius rs must obey relativistic principle and is shown to be zero.

That the Schwarzschild radius rs equals zero means that the geometrical size of a black hole should be zero and thus, a black hole should not have an interior. I have reached this conclusion and explained it in the paper « Gravitational time dilation and black hole » in which I have also shown that point masses could not coalesce to form a black hole whose volume is zero and that observations support this conclusion. This paper is linked here:

The equations and figure are in the paper below

For more detailed information, I invite you to read the two cited papers:

# Computing orientation with complex multiplication but without trigonometric function

Today’s methods for computing orientation are quaternion and rotation matrix. However, their efficiencies are tarnished by the complexity of the rotation matrix and the counterintuitivity of quaternion. A better method is presented here. It uses complex multiplication for rotating vectors in 3D space and can compute orientation without angle and trigonometric functions, which is simple, intuitive and fast.

1. Basic orientation 1
2. Rotation using complex multiplication 1
a. Complex multiplication 1
b. Mixed multiplication 2
3. Reference frames 2
a. Ground frame and proper frame 2
b. Direction frame 3
4. Base vectors of the direction frame 3
a. Rotation around the z axis 3
b. Rotation around the y axis 3
c. The 3 axes of the direction frame 4
d. Direction frame and 3d complex number 4
5. Roll, Pitch and Yaw 4
a. Roll 4
b. Pitch and Yaw 4
6. Determination of the angles 5
a. Direction angles 5
b. Roll angle 5
c. Direction frame angles without trigonometric functions 5
d. Roll angle without trigonometric functions 6
7. Computation for oriented points 6
a. Position of one point 6
b. Computation without trigonometric functions 6
c. Procedure of computation without trigonometric function 7
8. Discussion 7
See the article with figures and equations here
https://pengkuanonmaths.blogspot.com/2022/05/computing-orientation-with-complex.html
9. Basic orientation
A rigid body can rotate around 3 orthogonal axes in space, see Figure 1. The state of these 3 rotations is the orientation of this body. The commonly used orientation systems are Euler angles and Tait–Bryan angles. Two methods are usually used to compute orientation: quaternion  and rotation matrix . But they have their drawbacks. For quaternion, the computation for rotating a vector p needs to multiply p on the left by the rotation vector q and on the right by its conjugate q-1, which implies two 4D multiplications with p in between, see equation (1). The weird thing is that the rotation vector q is not computed with the rotation angle  but its half /2, see (2), . The half angle and the “sandwich multiplication” make the quaternion method counterintuitive.

For rotation matrix, the 3 angles of orientation are mingled in the 9 elements of the matrix where one gets easily lost. For example, in «Step by step rotation in normal and high dimensional space and meaning of quaternion», the transformation matrix is equation (3) which is confusing with the messed trigonometric functions.

Can we find a better method? In fact, I have constructed a 3D complex number system in «Extending complex number to spaces with 3, 4 or any number of dimensions»  which computes easily the rotation of a vector in 3D space. The combination of this system with the step by step rotation described in «Step by step rotation in normal and high dimensional space and meaning of quaternion» gives birth to a new method. This method can use directly the coordinates of points to rotate an object without using trigonometric function. We will explain first this method that uses complex multiplication and trigonometric function.

1. Rotation using complex multiplication
a. Mixed multiplication
We notice that when a 3D vector rotates around an axis that is perpendicular to it, the vector rotates in a plane and we call it the rotation plane. For using 2D complex multiplication in 3D space, we consider the rotation plane as a complex plane and use the 2D complex multiplication to rotate a vector in this plane.

Let (e1, e2) be a plane in 3D space, e1 and e2 the base vectors of the plane, so they 3D vectors. We make this plane equivalent to the complex plane using equation (7), that is, e1 corresponds to the real axis and e2 to the imaginary axis. u is a vector in the plane and is expressed in (8), with a and b being its components. Because the plane (e1, e2) is equivalent to the complex plane, u has an equivalent complex number u, which is expressed in (9). Let v be an other complex number which is expressed in (10).

We multiply u with v and the complex product uv is given in (11). The real and imaginary parts of uv are written in (12). Because the complex plane is equivalent to the plane (e1, e2), we replace the 1 and i that are in (11) with e1 and e2 and obtain uv in (13) which is a vector in the plane (e1, e2).

So, we have created a new type of multiplication: the vector u multiplied by the complex number v. The vector u is a 3D vector because e1 and e2 are 3D vectors. The product of this multiplication is uv which is a 3D vector too. We call this multiplication “mixed multiplication” and the product “mixed product”. We state the Definition 1.

Definition 1: Mixed multiplication and mixed product
The plane (e1, e2) is equivalent to the complex plane. u is a vector in this plane and equals ae1+be2. u is a complex number and equals a+bi. v is an other complex number. u is multiplied by v and the result of this multiplication is denoted as uv and equals real(uv)e1+ imag (uv)e2, with real(uv) and imag (uv) being the real and imaginary parts of the complex product uv. uv is a vector in the plane (e1, e2). This multiplication is called the mixed multiplication and uv the mixed product.

Let us use the newly defined mixed multiplication to rotate the vector u given in (14). The complex number u given in (4) is the complex equivalent of u and the complex number v is the ei given in (5). The complex product uv is computed in (6). Then the mixed product uv is given in (15) whose components equal the real and imaginary parts in (6). So, uv is well the vector u rotated by the angle . We see that mixed multiplication is a very easy way to compute the rotation of a 3D vector in the rotation plane.

See the article with figures and equations here
https://pengkuanonmaths.blogspot.com/2022/05/computing-orientation-with-complex.html

# Extending complex number to spaces with 3, 4 or any number of dimensions

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