A primer on the three dimensional complex numbers.
Author: Reinko Venema. 

Page 1  Page 2  Page 3 Page 4


Introduction: This file is a collection of the pictures from Feb 2012 until now that are related to the 3D complex numbers.
3D complex numbers are in many ways just like the ordinary complex numbers from the complex plane; only in the 3D case you simply 'create' an 'imaginary number' j who's third power equals minus one.

Recall that in the ordinary complex plane the 'imaginary unit' i has a square of minus one.

Just like ordinary complex numbers (written often as z = x + iy) it looks rather elementary to use rectangular coordinates in real numbers (x, y, z).

Therefore the number one is written as 1 = (1, 0, 0)
the number j is j = (0, 1, 0) and the square of j is in the z-axis direction (0, 0, 1).


So only pictures in this stuff below... 

Remark: Everywhere where in these pictures it says 'see below' you must read that as 'see above' in some other picture. (Because this file is chronological while the homepage of this website is anti-chronological.) 

Updates in this html file not mentioned on the homepage
(these updates are a bit more technical): 

12 May 2013; the number i is not in 3D to be found   
21 May 2013, the wedge product 
30 May 2013, the log for j squared 
13 June 2013, tau calculus 
13 June 2013, making chopped meat of the 1, 2, 4, 8 theorem 
31 July 2013, a 6 by 6 matrix for log j 
08 Sept 2013, 10 properties for the exponential tau 
23 Sept 2013, without comment just 3 circles 
06 Oct 2013, exponential circles  
18 Oct 2013, determinant graphs in 3D + a glimpse of the general theory 
01 Nov 2013, eigenvalues and eigenvectors 
08 Nov 2013, the wrath of Diophantus  
28 Nov 2013, relations with linear control theory 
08 Dec 2013, correction and addendum Fourier transforms 
24 Dec 2013, general 2D stuff  & the song of omega 

Proceed on this 2014 page 2

05 Jan 2014, the song of omega reloaded  
18 Jan 2014, Cauchy integrals 
30 Jan 2014, 5D complex numbers and the golden ratio  
03 March 2014, a simple model for the proton and the neutron   
29 April 2014, do electrons have spin?  
28 May 2014, on the quantumization of magnetism 
02 Aug 2014,
the outer product in 3D 
15 Sept 2014, superposition of quantum states using 3D numbers 
26 Nov 2014, an oversight of 10 exponential circles & curves 
16 Dec 2014, what did Hamilton do wrong in 3D? 

Proceed on this 2015 page 3: 

13 Feb 2015, 3 phase electricity and 3D circular numbers 
23 Feb 2015, punishment for funny Nottingham professors 
05 March 2015, the gamma function in 3D 
26 March 2015, zeta on the critical strip (3D version only) 
06 April 2015, magnetic monopoles (Reinko versus Paul Dirac) 
18 April 2015, the curves of grace in 3D 
08 May 2015, the Aurora Borealis and magnetic monopoles 
14 June 2015, Strong Interaction and 3D circular numbers  
14 July 2015, the missing equations 
04 Sept 2015, the geometric series in 3D  
21 Nov 2015, integral calculus done with matrix diagonalization 
06 Dec 2015, inclusion and extension of complex spaces  
Go to page 4 (page 4 is the last one because I opened a new website on all things related to higher dimensional complex numbers and a few bits of other stuff like magnetism, here is the link:   http://3dcomplexnumbers.net/)


From 13 Feb 2015: 3 phase electricity and 3D circular numbers. 

This update contains no new mathematics but the goal is to show that you can apply the higher dimensional complex and circular numbers to almost everything just like it is always done with complex numbers from the complex plane.  

Industrial electricity or 3 phase electricity is just screaming bloody murder for 3D numbers, just look at the way the voltages behave and you understand why. 

In this update I propose a new design for the placement of the coils, as far as I can see this will have no serious practical application anywhere, but once more: I want to show you that there is no reason to skip 3D numbers in practical applications... 

Enough of the bla bla introduction, here is the six page long update:   
















Once more: this update only serves as an example that you can use 3D numbers just as easily as 2D complex numbers or 1D real numbers... Till updates. 



From 23 Feb 2015: Punishment for funny Nottingham professors.

Almost 3 years of writing on higher dimensional complex & circular numbers & of course the Nottingham professors still do not have any clue whatsoever... Anyway, at the Nottingham university they have those Numberphile video series and one of them is about math humor. 

In that it was stated that you cannot take the cross product of a scalar and a vector and that was viewed upon as a joke. Later I realized once more how dumb and retarded you become when rejecting that large ocean of math known as the higher dimensional numbers. 

But Nottingham is not the only university, the sleazeballs from my own university here in Groningen do a perfct job hanging out the silent coward. Well, apparently that is the choice they have made...  


In this update: 

The good old cross product in 3D plus a 4D version of it.
And for the rest we are looking at pairs of imaginary numbers that imitate the number i from the complex plane. All calculations are performed in 3D and 5D using the complex version of vector multiplication.

Let's get started:   




















In the second appendix we look a bit deeper into why this root mapping stuff between different dimensions actually works. Here is appendix number two: 



 End of this update, till updates.  



From 05 March 2015: The gamma function in 3D (using the complex multiplication in 3D, the circular multiplication goes similar). 

All functions that are analytic on the real line can be extended to three dimensional complex space; in the previous update I did that for the Riemann zeta function, in this update we take a look at the gamma function.  

Here we go, it is seven pages long: 














I did not change one little detail at that Gauss representation thing, normally I never include stuff from other math folks unchanged but for Gauss I make an exception because Gauss was just like me a math amateur. 
And for the obvious reasons Gauss, just like me, stayed away from those incompetents known as the 'professional math professors'.  

Here is the link to the Youtubber: 

Gamma Function - Part 2 - Gauss Representation 

Also including much info on the gamma function in one & two dimensions: 

Gamma function 


End of this update.  

Till updates. 



From 26 March 2015: Zeta on the critical strip (3D version only).

This update is about numerical math to find the zero's of the Riemann zeta function in 3D.
This update is about 3D complex numbers, the 3D circular numbers will have similar properties.
This update used the alternating Riemann zeta function, this is also known as the Dirichlet eta function. 

For myself speaking, I think the zero's are also on a critical line in 3D complex space.
But you never know; for example it is a fundamental result the number i from the 2D complex space does not exist inside 3D complex & 3D circular space... 

Because all universities completely neglect me, I have no access to reliable computers. Therefore I have no clue if my insights are indeed correct, but if they are correct you should find multiples of the number alpha on the places a Riemann zeta zero is expected.

Despite the coward and retarded behavior of the 'official universities', for me it was fun to write a bit into the direction of numerical math. But I refuse to write computer code to find the zero's of the Riemann zeta function; how can a reasonable person trust the Microsoft C sharp language? 
You just cannot be sure if your results are what they are supposed to be... 


Enough of the introductionary bla bla, this update is nine pages long. Here we go: 


















A small appendix telling you the obvious:  


End of this update. 



From 06 April 2015: When thinking about the Stern Gerlach experiment where ionized silver atoms were send through an inhomogeneous magnetic field, I suddenly thought; this can only happen if the electrons are magnetic monopoles.

Because if they were magnetic dipoles, because of their very little weight they would always align themselves towards the stronger part of the inhomogeneous magnetic field. So the outcome of the amazing Stern Gerlach experiment can perfectly be explained if electrons were in fact magnetic monopoles. 

Once you are on that track, you realize soon that if electrons were magnetic dipoles if would be very strange to observe only pairs of electrons; with ordinary bar magnets you can make circles of magnets with any kind of number. For example 3 bar magnets in an equilateral triangle. 

If you would have a triple electrons, on one side you can put a hydrogen nucleus (a proton) and at the other side a helium nucleus to form HHe or the well known helium hydrogen molecule...   

Of course this molecule does not exist because as a matter of fact Helium is one of the inert gasses because the outer orbital is completely filled with electron pairs.  

Nowhere in the last year or this year I have found one little piece of evidence that electrons are magnetic dipoles; ok ok if the are in orbit they create the standard dipole magnetic field that comes with electrical currents that circle around. But on there own electrons are magnetic monopoles. 

Anyway, that is what I think of it.  


Now what pushed me over the edge in penning these two pages down is an article I found via the website of CERN; it is available at the preprint archive and has the title: 

The Search for Magnetic Monopoles.
James L. Pinfold, Physics Department, University of Alberta.  

In the article I found on what equation Dirac concluded that if a magnetic monopole does exist, in that case all electrical forces would be quantasized. That is: all electrical charges come in multiples on one smallest charge. The charge of an electron is often denoted as e, we know now that quarks have charges of e/3 and 2e/2. So for the time being this smallest charge possible is e/3 and all other charges in our macroscopic would are multiples of that... 


After this long introduction, here are the two pages of this update: 





An important detail: if electrons are in fact magnetic monopoles, one of the laws of Gauss about magnetism is no longer valid. 
That is why I end it with saying 'So the master, so the slaves'. 

I do not want to discredit Gauss in any way; he was a great guy because just like me he did not want any kind of career in professional mathematics because basically these are all retards. Although highly intelligent, all these math professors only serve their masters and are afraid to think outside the box. 

The average math professor is only worried about what the colleagues would think and because of that fundamental emotional hinder they perform far beyond their potential. 


May be my vision on professional math is a bit over the top, but with my own skin I have felt how they behave...     



The Search for Magnetic Monopoles 

Also funny is the 'Understanding permanent magnets' from this collection of links: 

Tech Notes 

Till updates.   



From 18 April 2015: The curves of grace in 3D.

Of course all readers will wonder what īcurves of graceī are; this is simple to understand: 

They are an arbitrary imaginary number in the exponent and we let run an exponential time evolution.
So in Latex code it is written as e^{j t} or e^{j^2 t}. 

Beside a scaling factor in the exponent, these curves of grace factorize the exponential circles in 3D.
We look only at the 3D version of complex and circular numbers; but higher dimensions go similar. 

In appendix 1 I propose a very simple and completely unreliable model for the stream of matter around a black hole. But that simple flow of matter makes clear to anyone how black holes probably work. 
So I would like to ask you not to view this simple model as a tool, but more as a toy to play with and subtract some ideas from it.  

Enough intro talk, here we go: 



































A few extra words on appendix 1 with the black hole evolutions; Once more realize this simple model is likely far from the truth, yet suppose matter is streaming in along a curve that meets the event horizon of the black hole. That matter will be sucked in. 

Yet it is important to realize that a black hole and it's event horizon are relatively small objects compared to all that space the in falling matter occupies, so a lot of matter 'will miss' the event horizon. The matter piles up fast and the pressure get tremendously high.  

The pressure gets so high that a rudimentary version of the Pauli exclusion principle sets in: 
Two electrons or protons cannot be in the same place at the same time.
So the matter related to the Pauli principle will disintegrate into for example photons (you can stack any amount of photons onto each other, photons have no mass and as such occupy no space). 

It is claimed that about 40% of the in falling matter is transformed into energy, I know no way of validating that but if true it is expected it will go more or less as I describe it here: The relatively small size of the event horizon is the main culprit in the building up of the pressure that causes the disintegration of the stuff with mass in it. 


Used applets: 

The polyray applet for the determinant layer on page 1 can be found on: 
http://wims.unice.fr/~wims/wims.cgi? (select the online calculators and function plotter link over there). 

The screen shots for the exponential matrix using the Pade approximation can be found at: 

Matrix exponential using the Pade approximation calculator 


The was it for this update, till updates.  



From 08 May 2015; The Aurora Borealis and magnetic monopoles      

The Aurora Borealis is the polar light on the Northern hemisphere. In this update that is not on higher dimensional complex numbers but wholly on electrons and protons as being magnetic monopoles. (As a matter of fact I think that all spin half particles are magnetic monopoles.) 

The longer I think about it, the more obvious is that at least electrons are magnetic monopoles.
The main reason is of course the so called Stern Gerlach experiment where it was observed that electrons move against the inhomogeneous magnetic field. But if a small mass particle like an electron would be a bipolar magnet, it would simply turn and never move away from the stronger side of the magnet.   

Furthermore, the entire Stern Gerlach experiment was done with ionized electrons; once more if that lonely electron that is unpaired is a magnetic dipole, how can only one electron steer the entire silver ion away from the stronger parts of the magnetic field. 


In this update we will do two so called thought experiments; we pretend to be an electron coming from the sun via the solar wind and we will be either a bipolar magnetic thing and a monopole magnetic thing.
We ponder questions like why a magnetic bipolar electron would have a preference for the earth magnetic North or South pole. (They have not of course, buy magnetic monopoles do have a strong preference.)

It is five pages long, at the end I give two simple stochastic walks because I view the earth magnetic field as some giant complicated stochastic walk. We try to shine a bit of light on why the earth magnetic field is always changing direction over the millennia. But it is a geological fact the earth magnetic field always changes in a rather chaotic fashion...  

Here are the five pages: 










In this update I avoided all calculations, yet on the internet there are dozens and dozens files around that show it could very well be that electrons are magnetic monopoles. 
Here are a few additional links: 

Electron spin resonance (Source BU Chem on Youtube).

Electron spin resonance example (Source BU Chem on Youtube). 

Hyperfine coupling (Source BU Chem on Youtube).

Zeeman splitting example 1 (Source BU Chem on Youtube).

It is needless to say that in my view the hyperfine spectrum structure is caused by electrons being magnetic monopoles and atomic nuclei are a sum of north and south magnetic monopoles. This also explains why isotopes have different levels of spin in there nucleus: the neutron is also the sum of three magnetic monopoles and as such a magnetic monopole... 


More on the dynamics driving the changes in the earth magnetic field: 

Geomagnetic reversal (Source WikipediA)

2012: Magnetic Pole Reversal Happens All The (Geologic) Time (Source NASA)
May be the link does not work because it looks like the NASA websites has dynamic links, in that case do a Google search on, for example, earth magnetic field reversal. And the NASA link will pop up...


Let me end this update with a picture for this update, it is from a video from the Aurora Borealis: 


Ok, that was it for this update. Till updates.  



From 14 June 2015: Strong Interaction and 3D circular numbers.

The strong interaction is what the protons and neutrons bind into nuclear particles and via the exchange of pions it holds the nucleus together. That is why relatively small nuclei are so strong compared to large and-or instable nuclei because those are hold together by the electro/magnetic forces. 

How this last thing actually happens is never good explained by the people of physics because according to the standard model of particle physics spin half particles are not magnetic monopoles...  

This update is only about the strong interaction inside the proton and the neutron and it is 15 pages long and has three appendices at the end.  

By sheer coincidence I did include the exponential curve for the 11-dimensional real space equipped with the circular multiplication. That was meant as comparisment to the folks from string theory that use vibrating strings to explain everything; tiny problem is that the scale where these vibrating strings operate are completely outside experimental validation.
Well it turns out yesterday when I was watching some small historical oversight of string theory that originally it was invented in order to explain the strong interaction...

The simple model as I craft it here is the 3D exponential circle that passes through the three basis vectors 
(1, 0, 0), (0, 1, 0) and (0, 0, 1).  The quarks are these basis vectors and the gluons that transport the color charge move round and round at the speed of light via the exponential circle.

So you can view this update as one the things you can do with exponential circles and curves. 

In the appendices I give also a way to construct the exponential circle from the complex plane using 2D modified Dirichlet kernels and in the last appendix there is experimental proof you can use the 11D exponential curve as probability amplitudes as they do in quantum physics. 

All in all this update is much more mathematical physics and not pure math; but that is ok because every now and then some applications must come along of course. 

Let's get started; 15 pictures on A4 size: 






























You can use the next text with a bit of cut and paste in order to check that the above math experiment is indeed correct and you can use the values of the coordinate functions as probability amplitudes because the squares always add up to 1.

((1/11)*sin(11*(x))/sin(x))^2 +
((1/11)*sin(11*(x - pi/11))/sin(x - pi/11))^2 +
((1/11)*sin(11*(x - 2*pi/11))/sin(x - 2*pi/11))^2 +
((1/11)*sin(11*(x - 3*pi/11))/sin(x - 3*pi/11))^2 +
((1/11)*sin(11*(x - 4*pi/11))/sin(x - 4*pi/11))^2 +
((1/11)*sin(11*(x - 5*pi/11))/sin(x - 5*pi/11))^2 +
((1/11)*sin(11*(x - 6*pi/11))/sin(x - 6*pi/11))^2 +
((1/11)*sin(11*(x - 7*pi/11))/sin(x - 7*pi/11))^2 +
((1/11)*sin(11*(x - 8*pi/11))/sin(x - 8*pi/11))^2 +
((1/11)*sin(11*(x - 9*pi/11))/sin(x - 9*pi/11))^2 +
((1/11)*sin(11*(x - 10*pi/11))/sin(x - 10*pi/11))^2 

You can cut and paste that in some internet application for drawing graphs, I have a strong preference for that WWW Interactive Multipurpose Server or WIMS. They are always session bounded so you must find the applet for yourself but that is not so hard. Here is the link: 


This is the end of this update, till updates. 


From 14 July 2015: The missing equations.

This is a long update of 21 pages; that is an expansion of precisely 10% of all combined updates from the last 3 years although it does not contain 10% more math. 

This update is more like a physics approach to math; because we have missing equations we simply cannot give mathematical proofs because we do not know what is missing. The equations that we study are those who's solution is an exponential curve, or an exponential circle, and we constantly use the modified Dirichlet kernels to construct the exponential curves/circles. 

And with each construction we show that from the equations that we know of, they fulfill those equations. 

Simple example in the complex plane: 
Take the unit circle around 0, the equation the we know is x^2 + y^2 = 1.
Then show that cos^2 t + sin^2 t = 1. 

Furthermore we only use internet apps to draw graphs of the parameterizations of our known equations, so in that sense the internet app is the experiment in the science of physics. 

Every graph of such an equation in this 21 page update shows a new trigonometric indentity.
Let's go: 














































In case you want to make some of those screen shots yourself, you must cut and paste the next stuff that is from the fifth cone equation into the 'Animated drawing' applet in the next link. (Search for graph plotters, 2D explicit variant) and cut & paste this: 

((1/11)*sin(11*(x))/sin(x))*((1/11)*sin(11*(x - 6*pi/11))/sin(x - 6*pi/11)) +
((1/11)*sin(11*(x - 6*pi/11))/sin(x - 6*pi/11))*((1/11)*sin(11*(x - pi/11))/sin(x - pi/11)) +
((1/11)*sin(11*(x - pi/11))/sin(x - pi/11))*((1/11)*sin(11*(x - 7*pi/11))/sin(x - 7*pi/11)) +
((1/11)*sin(11*(x - 7*pi/11))/sin(x - 7*pi/11))*((1/11)*sin(11*(x - 2*pi/11))/sin(x - 2*pi/11)) +
((1/11)*sin(11*(x - 2*pi/11))/sin(x - 2*pi/11))*((1/11)*sin(11*(x - 8*pi/11))/sin(x - 8*pi/11)) +
((1/11)*sin(11*(x - 8*pi/11))/sin(x - 8*pi/11))*((1/11)*sin(11*(x - 3*pi/11))/sin(x - 3*pi/11)) +
((1/11)*sin(11*(x - 3*pi/11))/sin(x - 3*pi/11))*((1/11)*sin(11*(x - 9*pi/11))/sin(x - 9*pi/11)) +
((1/11)*sin(11*(x - 9*pi/11))/sin(x - 9*pi/11))*((1/11)*sin(11*(x - 4*pi/11))/sin(x - 4*pi/11)) +
((1/11)*sin(11*(x - 4*pi/11))/sin(x - 4*pi/11))*((1/11)*sin(11*(x - 10*pi/11))/sin(x - 10*pi/11)) +
((1/11)*sin(11*(x - 10*pi/11))/sin(x - 10*pi/11))*((1/11)*sin(11*(x - 5*pi/11))/sin(x - 5*pi/11)) +
((1/11)*sin(11*(x - 5*pi/11))/sin(x - 5*pi/11))*((1/11)*sin(11*(x))/sin(x))

And the sum of the 11 coordinate functions has the next txt coding: 

((1/11)*sin(11*(x))/sin(x)) +
((1/11)*sin(11*(x - pi/11))/sin(x - pi/11)) +
((1/11)*sin(11*(x - 2*pi/11))/sin(x - 2*pi/11)) +
((1/11)*sin(11*(x - 3*pi/11))/sin(x - 3*pi/11)) +
((1/11)*sin(11*(x - 4*pi/11))/sin(x - 4*pi/11)) +
((1/11)*sin(11*(x - 5*pi/11))/sin(x - 5*pi/11)) +
((1/11)*sin(11*(x - 6*pi/11))/sin(x - 6*pi/11)) +
((1/11)*sin(11*(x - 7*pi/11))/sin(x - 7*pi/11)) +
((1/11)*sin(11*(x - 8*pi/11))/sin(x - 8*pi/11)) +
((1/11)*sin(11*(x - 9*pi/11))/sin(x - 9*pi/11)) +
((1/11)*sin(11*(x - 10*pi/11))/sin(x - 10*pi/11)) 

The applet ignores the line breaks (the 'return' or 'enter' button from your computer keyboard). 

The Animated drawing applet can be found here: http://wims.unice.fr/~wims/wims.cgi?   


May be I will write one or two appendices, if not till updates.


Ok ok, on 18 July this is the first appendix where I more or less in detail try to explain how I found the modified Dirichlet kernels. And you can believe me: this modified stuff is awesome... 






I guess till the next appendix...   


And indeed on 20 July 2015 we have four more pages on the abundance of equations in three dimensional space.
They will speak for themselves.  








May be I should not have posted the 'fun equation' because it focuses not on higher dimensional complex numbers.

Yet it was fun to write it down. Till the next appendix update that will contain only hard core mathematics because after all those fancy screen shots of funny equations it is about time to once more beat the shit out of the math departments on a global scale.   

So the next and latest appendix will beat some shit out & lets leave it with that. 


The promised link with the Polyray applet can be found at: 

The Polyray applet can be found here: http://wims.unice.fr/~wims/wims.cgi?     

Till updates.


On 05 August 2015 there is appendix number 3: Going hardcore nine pages long. 

This appendix contains mostly math so almost no screen shots, to be precise: two screen shots.
It is more or less a hardcore update so the majority of so called 'professional professors' will absolutely understand nothing from it and they will judge it as having zero value.

Yet it  contains one of the most important results of this year number 2015:
The discovery that inside the factors of matrix representations all kinds of geometric
figures live. Look at the last page nr 34, you see the picture of a cylinder. 

The equation that defines that cylinder is a factor of 3D matrix representations of circular numbers. 

In this appendix I am again using the method of eigenvalue functions, this is a powerful method that by far is not exhausted. In the future we will proceed with that.. 

Well here is the stuff: 


















Once more: The fact there is a cylinder hiding inside the equation for the determinant as one of the factors is a crazy result. But I said it so often the last four years: the higher dimensional complex numbers are a giant ocean where the two dimensional complex plane from those 'professional professors' is a small fishbowl.. 

Till updates.  



From 04 Sept 2015: The geometric series in 3D.

This is a small six page long update and the goal of this update for me was to write a text that could be understood in large part by students of mathematics in their second year. 

So if you are a student of the science of math, what must you know before you can grasp the butter & meat of this six page small update? Very simple: 

1) You must know the complex plane a bit. 
2) Knowledge of a bit of linear algebra like eigen values comes in handy too.
3) It is also handy if you already know what the geometric series is.
4) The meat part: It would be great if you know the spectral mapping theorem concerning eigen values... 

So here we go, only six pages (or jpg pictures) long: 












Further reading: Geometric series 

Ok, end of this update. Live well & work well my dear reader. 


From 21 Nov 2015: Integral calculus done with matrix diagonalization. 

About two and a half year back I gave the first proof for integrals like below having the analytical value they have, back in the time I named it the principle of massive modified madness. That sounds strange, I understand that, but if you understand the math behind the stuff you also understand humans are only dumb monkeys when it comes to complex numbers... 

Anyway, to make a long story short, in the next six pages you can find a more elementary proof that does not use īmassive modified madnessī at all but only subtle stuff from inside the toolbox of modern math. 

All you need is basic insight in what you can do with 3D complex numbers, so good luck with it.  












Ok, that was it. It is the year 2015 and there is about zero point zero reaction from the estimated one hundred thousand math professors there are our small planet... Letīs leave it with that, till updates my dear reader. 



From 06 Dec 2015: This update from 8 pages long is partly based on an idea from Dennis Morris who wrote a book with the title: Complex numbers the higher dimensional forms -2nd edition.

For me it is very funny that Dennis is not a professional math worker, he does physics in particular the quantum variant as far as I know. Anyway if you are interested you can buy this book from Amazon.
For myself speaking this book has the mathematical depth of a bird bath, that is why it is that good for professional math professors who are known to easily get suffocated in deep water like a bird bath... 


After having said that, in this update I do the following: 

1) We start with 6D complex numbers and observe that like expected the complex plane is included and the 3D numbers too.
2) A bit of matrix representation theory, I only use matrices when I need results from linear algebra but Dennis does matrices all the time. Well each author shout make a choice on this, but I prefer it to hang as close as possible to the complex plane where we write z = x + yi and only use matrix representations when needed.
3) Making clear that ij = ji so that
4) We can expand the complex plane together with the 3D complex numbers to 6D complex numbers.
5) Every now and then I recall the sledge hammer theorem and at the end we see a capture of Miley Cyrus licking a sledge hammer. Of course every female singer/performer should make her own choice in that: To lick a sledge hammer or not to lick a sledge hammer is the question... 


Enough of the bla bla bla, here are 8 fresh pages:
















Indeed this is the end of this update, donīt forget to bookmark my new website 3dcomplexnumbers.net (ok I am a bit lazy with placing updates over there but it is more or less long term policy that everything will go over there).
But the new website is a wordpress format website, so I still have not made the final decision on that because with all these pre cooked formats you also have the setbacks that come with that.
On the other hand I also try to avoid typing html too much so pre cooked stuff can be very good if it fits my needs. 

Till updates. 


From xx xxx 2015: