Category Archives: 3D complex numbers

What is the repeated conjugate ‘determinant style’? (Also: Repeated adjoint matrix.)

3D complex numbers, Matrix representation

This post is easy going for most people who have mastered the art of finding the inverse of a square matrix using the so called adjoint matrix. I was curious what happens to a 3D circular number if you take the conjugate ‘determinant style’ twice. In terms of standard linear algebra this is the same as taking the adjoint of the adjoint of a square matrix.

It is well known by now (I hope anyway) that 3D complex and circular numbers contain a set of numbers with a determinant of zero, you can’t find an inverse for them. To be precise, if you take some circular 3D number, say X, and you make some limit where you send X into a not invertible number, you know the inverse will blow up to infinity.
But the conjugate ‘determinant style’ does not blow up, on the contrary in the previous post we observed that taking this kind of conjugate gave an extra zero eigenvalue in this conjugate.

In terms of linear algebra: If a square matrix is not invertible, it’s adjoint is ‘even more’ non invertible because a lot of the eigenvalues of that matrix turn to zero.

And although the inverse blows up to infinity, the cute result found is that it blows up in a very specific direction. After all it is the fact that the determinant goes to zero that blows the whole thing up, the conjugate ‘determinant style’ is as continuous as can be around zero…

It’s a miracle but the math is not that hard this time.

Four pictures for now and I plan on a small one picture addendum later.
So lets go:

Isn’t it cute? This infinity has a direction namely the number alpha…;)
Small correction: It should be taking the conjugate TWICE…

All in all this is not a deep math post but it was fun to look at anyway. May be a small appendix will be added later, so may be till updates inside this post or otherwise in some new post.
Added Sunday 16 April: A small appendix where you can see what the adjoint taking process is doing with the eigenvalues of a 5×5 diagonal matrix. The appendix was just over one picture long so I had to spread it out over two pictures. You understand fast what the point is if you calculate a few of the determinants of those minor matrices. Remark here with a 5×5 matrix all such minors are 4×4 matrices so it is the standard setting and not like that advanced theorem of Pythagoras stuff.
Well it all speaks for itself:

Ok, that was it for this update. Thanks for the attention and see you in another post.

Factorization of the determinant inside the space of 3D circular numbers. Aka: The conjugate ‘determinant style’.

3D complex numbers, Matrix representation

A few weeks back I was thinking in writing finally some post about general theory for spaces with arbitrary dimension. It soon dawned on me that the first post should be about the impossibility of solving X^2 = -1 on spaces of odd dimension for both the complex and the circular method of mulitplication on those spaces. So post number one should be about the fact the famous number i does not exist in spaces with dimensions 1, 3, 5 etc.
And what about the second post? Well you can always factorize the determinant inside such spaces, that is a very interesting observation because the determinant is also the product of all eigenvalues. These eigenvalues live traditionally in the complex plane and as such a naive math professor could easily think that the determinant can only be factorized inside the complex plane. So that would be a reasonable post number two.
Since all these years I only did such a factorization once I decided to do it again and that is this post. The basic idea is very simple: If you want to find an expression for the inverse of a general 3D circular number, you need the determinant of that number. From that you can easily find a factorization of the determinant. It’s as simple as efficient.

But now I have repeated it in the space of 3D circular numbers I discovered that part of this factorization behaves very interesting when you restrict yourself to the subset of all 3D circular number that are not invertible. That is that taking the conjugate ‘determinant style’. The weird result is that taking this kind of a conjugate increases the number of eigenvalues that are zero. So this form of conjugation transports circular numbers with only one eigenvalue zero to the sub-space of numbers with two eigenvalues zero.

For years I have been avoiding writing general theory because I considered it better to take one space at a time and look at the details on just that one space. May be that still is the best way to go because now I have this new transporting detail for only what would be the second post of a general theory, it looks like it is very hard to prove such a thing in a general setting.

Luckily the math content of this post is not deep in the sense if you know how to find the inverse of a square matrix, you understand fast what is going on at the surface. But what happens at the level of non-invertibles is mind blowing: What the hell is going on there and is it possible to catch that into some form of general theory?

I tried to keep it short but all in all it grew to a nice patch of math that is 8 pictures long. Here is the stuff:

At the end of this post I want to remark that the quadratic behaviour of our conjugate ‘determinant style’ is caused by the fact it was done on a 3D space. If for example you are looking at 17 dimensional number, complex or circular, this method of taking a conjugate is a 16 degree beast in 17 variables. how to prove all non-invertible numbers get transported to more and more eigenvalues zero?

May be it is better to skip the whole idea of crafting a general theory once more and only look at the beautiful specifics of the individual spaces under consideration.

End of this post and thanks for your attention.

Comparison of the ‘Speed = the Square’ equation on 7 different spaces.

3D complex numbers, 4D complex numbers, Uncategorized

This post is very simlilar to a few back when we calculated the results on 4 different spaces. This time I hardly pen down any calculation but only give the results so we can compare them a little bit.
The way most professional math professors tell the story of complex numbers it goes a bit like this: We have the real number line, the complex plane and on top of that a genius named Hamilton found the quaternions. On top of that there are a bunch of so called Clifford algebra’s and oh we math professors are just so good. There is no comparison to us, we are the smartest professionals in the world!

Well that is very interesting because it is well known these so called ‘professionals’ could not find the 3D complex numbers for about 150 years. So how come they all say we have this and that (complex plane and quaternions) and that’s enough, we are just perfect! Why they keep on saying rubbish like that is the so called Dunning-Kruger effect. That’s something from psychology and it says that people who lack understanding of some complicated stuff also lack the insight that they are stupid to the bone when it comes to that particular complicated stuff. So the views of professional math professors is very interesting but can be neglected one 100 percent, it’s just Dunning-Kruger effect…

If you look at the seven results of the ‘Speed = the Square’ equations, the solutions form a strickt pattern that only depends of the number of dimensions and if it is the complex or the circular multiplication. So every time a math professor goes from the complex plane to the wonderful world of quaternions you now know you are listening to a weirdo.

I said I only give results but since I have never ever introduced the 4D circular numbers I just extrapolated the other six spaces to the solution that lives in that beautiful space. So the last example is a bit longer.

Anyway although the math depth of this post is not that very deep (solving a differential equation that wants the derivative to be the square of what you differentiate), it clearly demonstrates solutions of all 7 different spaces look strikingly similar.
But because of the Dunning-Kruger effect likely the math professors will keep on telling total crap when it comes to complex numbers. Why am I wasting my time on explaining math professor behaviour? Better go to the five pictures of our post. Here we go & bye bye math professors.

May be I should write some posts about general complex number theory on spaces of arbitrary dimension. On the other hand I found the 3D complex numbers back in the year 1990. So if after all those years I will once more try to write some general theory one thing will be clear: Math professors will keep on trying to convince you of the beauty of quaternions or that garbage from the Clifford algebra’s.

Why, as a society, do we keep on wating tax payer money on math professors? Ok, they do not everything wrong but all in all it is not a great science or so where the participants are capable of weeding the faults out and grow more of the good stuff.
Let me end this post and thank you for your attention.

Solving the ‘Speed = the Square’ equation on the space of 4D complex numbers.

3D complex numbers, 4D complex numbers

Unavoidable I had to write some post after the video on the quaternion from Hamilton. Now my 4D complex numbers commute so they are very different from the standard version of quaternions. Just like in the complex plane the multiplication is ruled by the imaginary unit i that has the defining property of i^2 = -1. On the space of four dimensional complex number I mostly write l for the first imaginary component, the defining property is of course that now the fourth power equals minus one: l^4 = -1.
In 2018 I wrote about 20 introdutionary posts about the 4D complex numbers. That is much more as you would need for the quaternions of Hamilton but on the quaternions you can’t do complex analysis and that explains almost all of the difference.
You can view the quaternions as three complex planes fused together by the common use of the real line. My 4D complex numbers can be viewed as a merge of two complex planes in the sense that there are two planes clearly ‘the same’ as a complex plane.
This post is once more one of the ‘Speed = the Square’ equations and just as on the other spaces we looked at we choose the initial condition such that it is the first imaginary unit l. As such our solution is easily found to be f(t) = l / (1 – lt) because if you differentiate that you get the square. So from the mathematical point of view this is all rather shallow math because all we have to do is find the four coordinate functions of our solution f(t). For that you need to calculate the inverse of 1 – lt and to be honest after so much years I think almost all math professors are just to fucking stupid to find the inverse of any non real 4D compex number Z let alone if you have something with a variable t in it like in 1/(1 – lt).

I did my best to write this as transparant as possible while also keeping it as short as possible. For an indepth look at how to find the inverse of a 4D complex number, look for Part 17 in the intro series to the 4D complex numbers. (Just use the search function for this website for that.)

This post is just three pictures long so lets hope that is inside your avarage attention span. And it’s math so without doubt a lot of people will digest this with a speed of one picture a week! No I am not being sarcastic or so, I just like as how I evolved to the math place I am now. Often that also goes very slow but it has to be remarked the math professors are much more slow slow slow because they could not find the 3D complex numbers in all of human history.
Let’s dive into the picture stuff:

One of the funny things of the math of this post is that on the one hand it is very simple: You only need high school math like the quotient rule for checking my claims are true and differentiation mimics the multiplication on the 4D complex numbers. On the other hand you have those math professors likely not capable of finding these easy coordinate functions for themselves.
But this post is not meant as an anti math professor rant but more upon the beauty of simple math you can do on say the space of 4D complex numbers.
See you in the next post.

Why could Hamilton not find the three dimensional complex numbers?

3D complex numbers, 4D complex numbers

This very short post was written because of a video from the video channel Kathy loves physics. It is one of those “Quaternions are fantastic” video’s. And Kathy just like a lot of other physics people think indeed that quaternions are fantastic. But you cannot differentiate or integrate on the quaternions so I guess this stronly limits it’s use in physics.
But quaternions are very handy in describing rotations in 3D space, I never studied the details but it was said that on the space shuttle it was used for nagvigation. And because of these rotation properties at present day they are used in the games industry.

Anyway in the video Kathy explains that Hamilton did try for a long time to find the three dimensional complex numbers. And he never succeeded in that. Of course I know this for decades right now but in the past I never looked into a tiny bit more detail in what Hamilton was actually doing.
And he was looking at complex numbers of the form X = x + yi + zj where the imaginary components both equal to minus one: i^2 = j^2 = -1.

If you check the easy calculations in this post for yourself, it is amazing how much it already looks like the stuff as found on the quaternions. As such it is all of a sudden much less a surprise that Hamilton found the quaternions. As a matter of fact it was only waiting until he would stumble across them. But at the time the concept of a four dimensional space was something that made you look like a crazy lunatic, there were even vector wars and lots of crazy emotional stuff.

At present day it is accepted that 3D complex numbers do not exist, in my experience the professional math community is still emotionally laden but now into the direction of total neglect. Stupid shallow thought like “If Hamilton could not find them, they likely don’t exist”.

Back in the 19-th century they were always looking for an extenstion of the complex plane to three dimensional space. Of course they failed in that attempt because it is a fact of math life that you cannot solve the equation X^2 = -1 on the space of 3D complex (and also circular) numbers.

The content of this post is just two pictures, after that two more pictures as I used them on the other website and after that you can finally dive into the Video from Kathy. If you are interested in physics and also the history of physics, Kathy her channel is a thing you should take a look at if you’ve never seen it. Here we go:

YES, that is what he should have done. Hamilton tried for about one decade to find the numbers that form the title of this very website, so may be he tried this kind of approach. I don’t know, but the 3D complex numbers are not some extension of the complex plane because 2 is not a divisor of 3. You know that prime number stuff is going on here.
But the math professors are not interested in that kind of stuff.

Here is how I used it on the other website:

As you see in the above picture I was already working on the previous post because if you differentiate the three functions that mimics the 3D circular multiplication. You can also mimic the multiplication on the complex plane, that is in the next picture:

At last you can view the famous video of Kathy! It’s only 30 minutes or so but if you see too many so called TIKTOK videos that is infinitely long: Wow 30 minutes long looking at just one video?

End of this post, likely the next post is about 4D complex numbers.

Solving the ‘Speed = The Square’ equation on four different spaces.

3D complex numbers, Matrix representation, Uncategorized

With ‘speed = square’ I simply mean that the speed is a vector made up of the square of where you are. The four spaces are:
1) The real line,
2) The complex plane (2D complex numbers),
3) The 3D circular numbers and
4) The 3D complex numbers.

I will write the solutions always as dependend on time, so on the real line a solution is written as x(t), on the complex plane as z(t) and on both 3D number spaces as X(t). And because it looks rather compact I also use the Newtonian dot notation for the derivative with respect to time. It has to be remarked that Newton often used this notation for natural objects with some kind of speed (didn’t he name it flux or so?).
Anyway this post has nothing to do with physics, here we just perform an interesting mathematical ecercise: We look at what happens when points always have a speed that is the square of their position.

On every space I give only one solution, that is a curve with a specific initital value, mostly the first imaginary component on that space. Of course on the real line the initial condition must be a real number because it lacks imaginary stuff.

If you go through the seven pictures of this post, ask in the back of your mind question as why is this all working? Well that is because the time domains we are using are made of real numbers and, that is important, the real line is also a part of the complex and circular number systems.
The other way you can argue that the geometric series stuff we use can also be extended from the real line to the three other spaces. To be precise: we don’t use the geometric series but the fractional function that represents it.

Ok, lets go to the seven pictures:

That Newton dot notation just looks so cute…
The words ‘Analytic continuation’ are not completely correct…

Remark: This post is not deep mathematics or so. We start every time with a function we know that if you differentiate it you will get the square. After that we look at it’s coordinate functions and shout in bewilderment: Wow that gives the square, it is a God given miracle!

No these are not God given miracles but I did an internet search on the next phrase of Latex code: \dot{z} = z^2. To my surprise nothing of interest popped up in the Google search results. So I wonder if this is just one more case of low hanging math fruits that are not plucked by math professors? Who knows?

End of this post, thanks for your attention.

A simple theorem on the zero’s of polynomials on the space of 3D complex numbers.

3D complex numbers

In this post we look in detail at a very simple yet important polynomial namely

p(X) = X (X – 1).

Why does it have four zero’s in the space of 3D complex numbers? Well if you solve for the zero’s of p so try to solve p(X) = 0, that is you are looking for all numbers such that X^2 = X.
These numbers are their own square, on the real line or on the complex plane there are only two numbers that are their own square namely 0 and 1.
On the space of 3D complex numbers we also have an exponential circle and the midpoint of that circle is the famous number alpha. It is a cakewalk to calculate that alpha is it’s own square just like (1 – alpha).

This post is four pictures long in the size 550×825 pixels so it is not such a long read this time. In case you are not familiar with this number alpha, use the search function on this website and search for the post “Seven properties of the number alpha”. Of course since it is math you will also need a few days time of thinking the stuff out, after all the human brain is not very good at mathematics…
Well have fun reading it.

The last crazy calculation shows that a polynomial in it’s factor representation is not unique. Those zero’s at zeta one and zeta two are clearly different from 0 and 1 but they give rise to the same polynomial.

At last I want to remark that unlike on the complex plane there is no clean cut way to tell how many zero’s a given polymial will have. On the complex plane it is standard knowledge that an n-degree polynomial always has n roots (although these roots can all have the same value). But on the complex 3D numbers it is more like the situation on the real line. On the real line the polynomial p(x) = x^2 + 1 has no solution just like it has no solution on the space of 3D complex numbers.
That was it for this post, thanks for your attention.

Comparing the two sphere-cone equations.

3D complex numbers, Exponential circle, Matrix representation

This channel is of course not meant for political statements but this fucking war is a fucking distraction from doing math. While writing this post in small pieces I was constantly dissatisfied with the level of math (too simple, done too often in the past etc). But when I was finished and read it all over, all in all it was not bad. It is a short oversight of how to find shere-cone equations and once more how to find a conjugate.
And once more: The math professors are doing it wrong when it comes to finding the conjugate for 32 years now & the clock keeps on ticking. On the one hand this is remarkable because if you do internet searches a lot of people understand that the Jacobian matrix should be the matrix representation for the derivative of a complex valued function in say three dimensional space. So that goes good, but when it comes to taking the conjugate for some strage reason they all keep on doing it wrong wrong and wrong again so they will never find serious math when it comes to number systems outside the complex plane or for that matter the quaternions.

The setup of this post is as next:
1) Explaining (once more) how to find the conjugate.
2) Calculating the two sphere-cone equations.
3) The solution of these S-C equations is the exponential circle that is,
4) parametrisized by three so called coordinate functions that we
5) substitute into both S-C equations in order to get
6) just one equation.

Basically this says that the complex and circular multiplication on our beloved three dimensional space are ‘very similar’. Just like that old problem of solving X^2 = -1 is impossible in these spaces while the cubic problem of X^3 = -1 has only trivial solutions like basis vectors. That too is ‘very similar’ behaviour.

Anyway this post is six pictures long.

That was it for this post on my beloved three dimensional complex numbers.

Addendum to the previous post: The new de Moivre identity for the 3D circular numbers + 2 videos.

3D complex numbers, Exponential circle, Matrix representation

I know I know I have published stuff like this before and over again. But that was also years ago and now I do it again it is still not boring to me. After all the professional math professors still are not capable of finding those beautiful exponential circles and curves simply because they all imitate each other. And they imitate each other with how to use and find a so called conjugate. And if you use the conjugate only as some form of ‘flipping a number into the real axis’ all your calculation will turn into garbage. Anyway by sheer coincidence I came across two videos of math folks doing it all wrong. One of the videos is even about the 3D circular numbers although that guy names them triplex numbers.

You can do a lot with exponential circles and curves. A very basic thing is making new de Moivre identities. From a historical point of view these are important because the original de Moivre identity predates the first exponential circle from Euler by about 50 years. In that sense new de Moivre identities are very seldom so you might expect some interest of the professional math community…

Come on, give me a break, professional math professors do a lot of stuff but paying attention to new de Moivre identities is not among what they do. But that is well known so lets move on to the four pictures of our update. After that I will show you the two video’s.

Let us proceed with the two video’s. Below you see a picture from the first video that is about 3D circular numbers and of course the conjugate is done wrong because math folks can only do that detail wrong:

Below you can see the video:

By all standards the above video is very good. Ok the conjugate is not correct and may be the logarithm is handled very sloppy because a good log is also a way to craft exponential circles. But hey: after 30 years I have learned not to complain that much…

The next video is from Michael Penn. He has lots of videos out and if you watch them you might think there is nothing wrong with that guy. And yes most of the time there is nothing wrong with him until he starts doing all kinds of algebra’s and of course doing the conjugate thing wrong. Michael is doing only two dimensional albebra’s in the next video but if you deviate from the complex plane very soon you must use the conjugate as it is supposed to be: The upper row of the matrix representation.

Here a screen shot with the content of the crimes commited:

Most of his other video’s are better, but his knowledge is just a reflection of what professional math professors think about conjugation. It is always just a flip in the real axis.

Here is his vid:

Ok, that was it for this appendix to the previous post.

Once more: The sphere-cone equation.

3D complex numbers, Exponential circle, Matrix representation

It is past midnight, this evening I brewed hopefully a lovely beer. It is late so let me keep the intro short. The last time I often lack stuff for new posts because most of the theory of 3D complex and circular numbers has been posted in this collection of 200+ posts. And you cannot keep it repeating over and over again, if all those years in the past the math professionals did just nothing, why would they change their behaviour in the future? Beside that I do not want have anything to do with them any more, it is and stays a collection of overpaid weirdo’s and there is nothing that can change that.
On the other hand one of the most famous expressions in math is and stays the exponential circle in the complex plane.
That stuff like e^it = cos t + isin t is what makes many hearts beat a tiny bit faster. So when someone comes along stating that he found an exponential circle in spaces like 3D complex numbers, you might expect some kind of attention. But no, once more the math professionals prove they are not very professional. Whatever happens over there I do not know. May be they think because they could not find this in about 350 years no one can so it must all be faulty. For me it was a big disappointment to get discriminated so much, on the other hand it validates that math professors just are not scientists. Ok they have their salary, their social standing, their list of publications and so on and so on. But putting lickstick on a pig does not make it a shining beauty, it stays a pig. So a math professor can have his or her prized title of professor, that does not make such a person a scientist of course. At best they show some form of imitating how a scientist should behave but again does such behaviour make these people scientists?
Anyway a couple of days back at the end of a long day I typed in a search phrase in a website with the cute name duckduckgo.com. Sometimes I check if websites like that track this very website and I just searched for “3D complex numbers”. The first picture that emerged was indeed from this website and it was from the year 2017. I looked at it and yes deep in my brain it said I had seen it before but what was it about? Well it was the product of two coordinate functions of the exponential circle in 3D. It is a very cute graph, you can compare it to say the product of the sine and cosine function in the complex plane.
So I want to avoid repeating all that has been written in the past of this website but why not one more post about the 3D exponential circles?

In the end I decided to show you how likely one of those deeply incompetent “professional” math professors would handle the concept of conjugation. Of course one hundred % of these idiots and imbeciles would do it as “This is just a flip in the real axis or in the x-axis” and totally spoil the shere-cone equation and only find weird garbage that indeed better cannot be published. After all our overpaid idiots still haven’t found the 3D complex numbers, I am still living on my tax payer unemployment benefit and life, well life will go on. But it is not only math, with physics there are similar problems and they all boil down to that often an idiot does not realize he or she is an idiot.

But let’s post the six pictures, may I will add an addendum in a few days, may be not. Here we go:

Isn’t that a cute graph or not?

Ok, may be in will write one more appendix about how these kind of coordinate functions of exponential circles give rise to also new de Moivre identities. That is of interest because the original de Moivre identity predates the Euler exponential circle by about 50 years.

Yet once more: Likely there is just nothing that will wake up the branch of overpaid weirdo’s known as the math professors…
So for today & late at night that was it.
Thanks for your attention.