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

Page 1  Page 2  Page3 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  
Proceed on this 2016 page 4: 

09 Feb 2016, seven properties of the number alpha in 3D  
22 Feb 2016, the cone theorem 
15 April 2016, debunking the Euler evaluation of zeta minus one 
06 May 2016, on the length of the product of two 3D numbers 
06 Sept 2016, systems of differential equations  





From 09 Feb 2016: Seven properties of the number alpha in 3D.

A few days back on the new website 3Dcomplexnumbers.net I said my next post would be about the elementary properties of the number alpha. Yet as I wrote it out it would become a bit too long for a post over there, it is about 5 A4 size pages. And since I did not post anything on this website, now it is February I thought: let's hang it in here.  

So all in all there is nothing new in here, but now all those elementary properties of the most important non-invertible number alpha are grouped together. Furthermore I gave a few ways of introducing this number.
Here are the five pages: 












From 22 Feb 2016: The Cone Theorem. 

In this update we take a look at all subspaces of the 3D complex and circular numbers that are shifted by a fixed angle if you multiply a 3D number X by an imaginary unit like j or j^2.
It turns out these are all cones.
The last 3 years every time we took a look at cones it was always those who include the 3 axis of 3D complex number space, of course if in that cone you multiply by j you always turn 90 degrees.
Now we are going to identify all other similar cones and the angle they get shifted under multiplication by j. 

This update is 12 pages long. 
























Link to the Polyray applet:  http://wims.math.leidenuniv.nl/wims/wims.cgi (likely you must search a little bit...)

Till updates. 


From 15 April 2016: Debunking the Euler evaluation of zeta at minus one. 

Last Monday this week I was at a public lecture given by a professional math professor named Frits Beukers. I went there because I had some vague hopes that I could observe some serious analytical continuation of the zeta function.
It did not happen, so the Euler valuation of 1 + 2 + 3 + 4 + 5 + etc etc = -1/12 was spoken about.

Now in the science of physics they use it all the time: If they encounter 1 + 2 + 3 + 4 + 5 + etc they simply replace it by -1/12 and after that they think the problems are solved. 

By doing so the science of physics is turned into the pseudo science of physics; also every time you see one of those overpaid professional professors seriously talking about this -1/12 you are observing tax payer money burned for no reason at all. 


In this update, only three pictures long, we are going to debunk this nonsense from Euler and other people. I do not blame Euler in any way, his record is clear and as far as I am concerned he is still standing at position number on of math people in the all time rankings... 

So I do not blame Euler that at present day we have all these idiots around using three or four brain cells and explaining that 1 + 2 + 3 + 4 + etc = -1/12. 


Update from 20 April: I emailed Daughter with a short version of the proof below and she told me she absolutely did not understand one thing from it. That made me scratch my head because in principle the proof below is not difficult at all.
So I selected a video from the university of Nottingham, I use exactly the kind of math they do in order to get the Euler result of -1/12. Check it out: 

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12 

Here are the three pictures: 






I could post many links to all kinds of files that say 1 + 2 + 3 + 4 + 5 + etc = -1/12, but look for yourselves if you want to see the brains of those idiots in action...  

Update from 21 Apr 2016: In the above three pictures I showed you a simple proof using very basic math and we first 'proved that' 1 + 1 + 1 + 1 + etc etc =0. And as such all constant arrays should be zero.

But you can talk that stuff to any value, look we first take the sum of all natural numbers (and once more first pretend Euler was right so we valuate it):

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + etc etc = -1/12.

Now all of a sudden from number 5 we raise everything with 1, so that gives:

1 + 2 + 3 + 4 + 6 + 7 + 8 + 9 + etc etc = ???

As you see on inspection: these are all natural numbers except for the number 5.
The new problem is: What is the value of ??? and what is the valuation of the added row 1 + 1 + 1 + 1 + etc?

If we stick to the valuation of the constant array or row as 1 + 1 + 1 + etc = 0, in that case ??? = -5 1/12.
If we stick to saying 'We had a row of -1/12 and we added nothing, we should have that ??? = -1/12'.

Tiny problem: The last detail would imply 1 + 1 + 1 + 1 + etc = 5 because that is what has gone missing...

Huge problem: We can repeat this line of reasoning for every natural number.

Conclusion: Likely there is something wrong with the assumption that it should be -1/12 in the first place... 

Till updates. 

Update from 29 April 2016: For some strange reason there were suddenly much more Youtube video's out there on the crazy Euler evaluation of the zeta function at minus one. I will not post any link or so, likely if you  read this you do a Youtube seach for yourself on for example 'The sum of all integers'. 

Two more pictures and that is definitely the very end of this update: 




The best video out there also debunking the Euler idea that 1 + 2 + 3 + etc = -1/12 was posted two years back and here is the link and stuff:  What is a function? Why 1+2+3+4+5+.... not equals -1/12 = Zeta(-1) 

At last I want to remark that Euler is one of my heroes when it comes to math, so for myself speaking it was great fun to debunk a tiny bit of his insights. Till updates. 


Added on 14 Jan 2018:

Finally the Mathologer guy is doing the right thing and is also debunking this nonsense from the Nottingham professors. He has a video of about 42 minutes where in detail he exposes the dumb lines of reasoning as done by the Nottingham professors. At the end of the video he also enters the realm of analytic continuation in order to evaluate the zeta function at zeta = -1.
Indeed the analytic continuation gives minus 1/12 but once more:

This does not mean that 1 + 2 + 3 + 4 + 5 + ..... = -1/12.

Here is the video:

Numberphile v. Math: the truth about 1+2+3+...=-1/12

Also do not forget to visit the new website: 3Dcomplexnumbers.net
that, like the name says, is mostly about three dimensional complex numbers.

Till updates. 


From 06 May 2016 : On the length of the product of two 3D numbers.

The honorable Gauss had a motto stating 'Few but ripe' and with that he meant only a few publications but they had to be ripe. Likely this was caused by the fact that Gauss, just like me, has experienced very negative things when dealing with professional math people. And Gauss, just like me, thought you are better of if you avoid all contacts with these weirdo's for the rest of your life. 

Now I disagree with the Gauss motto, I like to write as much as I see fit but one thing will be very clear:

Never in my life I will write a peer reviewed article, let it be in math or physics. I am really not waiting for the approval of some overpaid idiot commenting on my work. 

After all, the math professors have their stupid 2-4-8 theorem on higher dimensional numbers while the physics professors worship the electron that is supposedly to be a magnetic dipole. Idiots like that might be funny to look at just like a flock of monkeys on a rock is nice to look at, but just like I do not ask a chimpanzee if this kind of work might be worth publishing I do not ask any professional professor to do the same. 


Ok, this update is about the length of AX when A and X are both circular 3D numbers. 

In the complex plane we have this preservation of length meaning that for two complex numbers from the complex plane we always have |az| = |a| |z| or to put it in words: The length of the product of two complex numbers is the product of the length's of both complex numbers.

In three dimensions it is very different: We can have shrinkage, equality and extra growth. 

The equality stuff covers all thing related to 3D complex number use in quantum mechanics, so that is in the pocket. So all previous updates stay valid, let there be no worry. 


This update is eight pictures long, sized 550 x 1100 pixels each. 

Have fun reading it. 
















For myself speaking I consider it a great honor to construct a completely new way of crafting a coordinate system.

So beside the rectangular coordinates like the (x, y, z) notation, there are also 3D spherical coordinate systems and cylindrical coordinate systems. Now we also have this cone sweeping system and as usual my dear reader:

Zero reaction from professional professors, let it be math or physics.


A day later I realized I completely forget to give a link to the matrix log applet, here is a nice set of over 40 matrix applets: 
Linear algebra http://calculator.vhex.net/function-index/linear-algebra 

If you did choose the wrong a in your [a, r, t] coordinates you get a matrix that makes no sense, so the method of finding the [a, r, t] coordinates is foolproof I guess...;) 

Let's leave it with that. Till updates.  


From 06 Sept 2016: Systems of differential equations. 

This update is about the time evolutions on complex and circular number systems, we start with a simple example from the real line: 

Find a function x(t) such that the speed equals x(t)^2 and we start with x(0) = 1/2 so the speed in x = 1/2 equals 1/4. 

In Part 1 this simple example is expanded to the complex plane and the three dimensional complex number system.
It is important to emphasize that we can only solve this in 3D space by literally using the fact that it is a number system. 

Old methods related to overpaid math professors who think that it is impossible to craft a 3D complex number system fail although I do not offer a mathematical proof for this. Just look at history: They have no methods to solve for systems of differential equations like this. 

Anyway I am not aware of they having this capability. 

In Part 1 it is more or less like: If you can solve the differential equation on the real line, you can try to expand it to higher dimensions.

In Part 2 we focus on trying to find the Taylor coefficients in case you fail at solving the equation on the real line. 

The great thing about Part 2 is you understand that derivatives related to time can also be made from spatial derivatives. 
That is more or less why I investigated this kind of stuff in the first place; from present knowledge in the spatial domain all future is known given some initial condition...

Of course in practice like physics such systems do not exist, at best it will be some vague 'state space' so this really is a pure math update because choice of the origin in the used coordinate system will change the trajectories. 

Enough of this vague bla bla bla, this update is ten pictures long each size 550 by 1100 pixels. 
Have fun reading it!   





















This is the end of this update, I hope you learned a bit from it. 
After all retrieving future time info from present spatial derivatives is a special art...

Let's leave it with that, till updates.