# Proof that Z^2 = -1 cannot be solved on real spaces with an odd dimension. (General theory part 1.)

Finally after all those years something of a more general approach to multiplication in higher dimensions? Yes but at the same time I remark you should not learn or study higher dimensional numbers that way. You better pick a particular space like 3D complex numbers and find a lot out about them and then move on to say 4D or 5D complex numbers and repeat that process.
Problem with a more general approach is that those spaces are just too different from each other so it is hard to find some stuff all of those spaces have. It is like making theory for the complex plane and the split complex numbers at the same time: It is not a good idea because they behave very differently.
The math in this post is utterly simple, basically I use only that the square of a real number, this time a determinant, cannot be negative. The most complicated thing I use of the rule that says the determinant of a square is the square of the determinant like in det(Z^2) = det(Z)^2.

This post is only 3.5 pictures long so I added some extra stuff like the number tau for the 4D complex numbers and my old proof from 2015 that on the space of 3D complex numbers you can’t solve X^2 = -1.

I hope it’s all a bit readable so here we go: Oops, this is the circular multiplication… Well replace j^3 = 1 by j^3 = -1 and do it yourself if you want to.

So all in all my goal was to use the impossibility of x^2 being negative on the real line to the more general setting of n-dimensional numbers. As such the math in this post is not very deep, it is as shallow as possible. Ok ok may be that 4D tau is some stuff that makes math professors see water burning because they only have the complex plane.
Let me end this post with thanking you to make it till the end, you have endured weird looking robots without getting mentally ill! Congratulations!
At the end a link to that old file from 2015:

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

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:

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’.

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.

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

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:

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 detailed sketch of the full theorem of Pythagoras (that matrix version). Part 2 of 2.

The reason I name these two posts a sketch of (a proof) of the full theorem of Pythagoras is that I want to leave out all the hardcore technical details. After all it should be a bit readable and this is not a hardcore technical report or so. Beside this, making those extra matrix columns until you have a square matrix is so simple to understand: It is just not possible this method does not return the volume of those parallelepiped.
I added four more pictures to this sketch, that should cover more or less what I skipped in the previous post. For example we start with a non-square matrix A, turn it into a square matrix M and this matrix M always has a non-negative determinant. But in many introductionary courses on linear algebra you are thought that if you swap two columns in a matrix, the determinant changes sign. So why does this not happen here? Well if you swap two columns in the parallelepiped A, the newly added columns to make it a square matrix change too. So the determinant always keeps on returning the positive value of the volume of any parallelepiped A. (I never mentioned that all columns of A must be lineary independent, but that is logical we only look at stuff with a non-zero volume.)

Just an hour ago I found another pdf on the Pythagoras stuff and the author Melvin Fitting has also found the extension of the cross product to higher dimensions. At the end of this post you can view his pdf.

Now what all these proof of the diverse versions of Pythagorean theorems have in common is that you have some object, say a simplex or a parallelepiped, these proof always need the technical details that come with such an object. But a few posts back when I wrote on it for the first time I remarked that those projections of a parallelogram in 3D space always make it shrink by a constant factor. See the post ‘Is this the most simple proof ever?’ for the details. And there it simply worked for all objects you have as long as they are flat. May be in a next post we will work that out for the matrix version of the Pythagoras stuff.

Ok, four more pictures as a supplement to the first part. I am to lazy to repair it but this is picture number 8 and not number 12:

That was it for this post. Thanks for your attention.

# A detailed sketch of the full theorem of Pythagoras (that matrix version).Part 1 of 2.

For me it was a strange experience to construct a square matrix and if you take the determinant of the thing the lower dimensional volume of some other thing comes out. In this post I will calculate the length of a 3D vector using the determinant of a 3×3 matrix.
Now why was this a strange experience? Well from day one in the lectures on linear algebra you are thought that the determinant of say a 3×3 matrix always returns the 3D volume of the parallelepiped that is spanned by the three columns of that matrix.
But it is all relative easy to understand: Suppose I have some vector A in three dimensional space. I put this vector in the first column of a 3×3 matrix. After I add two more columns that are both perpendicular to each other and to A. After that I normalize the two added columns to one. And if I now take the determinant you get the length of the first column.
That calculation is actually an example below in the pictures.

Well you can argue that this is a horrible complicated way to calculate the length of a vector but it works with all parallelepiped in all dimensions. You can always make a non-square matrix, say an nxd matrix with n rows and d columns square. Such a nxd matrix can always be viewed as some parallelepiped if it doesn’t have too many columns. So d must never exceed n because that is not a parallelepiped thing.

Orthogonal matrices. In linear algebra the name orthogonal matrix is a bit strange. It is more then the columns being orthogonal to each other; the columns must also be normalized. Ok ok there are reasons that inside linear algebra it is named orthogonal because if you name it an orthonormal matrix it now more clear that norms must be one but then it is more vague that the columns are perpendicular. So an orthogonalnormalized matrix would be a better name but in itself that is a very strange word and people might thing weird things about math people.
Anyway loose from the historical development of linear algebra, I would like to introduce the concept of perpendicular matrices where the columns are not normalized but perpendicular to each other. In that post we will always have some non-square matrix A and we add perpendicular columns until we have a square matrix.

Another thing I would like to remark is that I always prefer to give some good examples and try not to be too technical. So I give a detailed example of a five dimensional vector and how to make a 5×5 matrix from that who’s determinant is the length of our starting 5D vector.
I hope that is much more readable compared to some highly technical writing that is hard to read in the first place and the key idea’s are hard to find because it is all so hardcore.

This small series of posts on the Pythagoras stuff was motivated by a pdf from Charles Frohman, you can find downloads in previous posts, and he proves what he names the ‘full’ theorem of Pythagoras via calculating the determinant of A^tA (here A^t represents the transpose of A) in terms of a bunch of minors of A.
A disadvantage of my method is that it is not that transparant as why we end up with that bunch of minors of A. On the other hand the adding of perpendicular columns is just so cute from the mathematical point of view that it is good to compose this post about it.

The post is eight pictures long so after a few days of thinking you can start to understand why this expansion of columns is say ‘mathematical beautiful’ where of course I will not define what ‘math beauty’ is because beauty is not a mathmatical thing. Here we go:

Inside linear algebra you could also name this the theorem of the marching minors. But hey it is time to split and see you in the next post.

# Comparing the two sphere-cone equations.

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.

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:

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.

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:

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.

# Oversight of all counter examples to the last theorem of Pierre de Fermat, Part 3.

It is late at night, my computer clock says it is 1.01 on a Sunday night. But I am all alone so why not post this update? This post does not have much mathematical depth, it is all very easy to understand if you know what split complex numbers are.
In the language of this website, the split complex numbers are the 2D circular numbers, In the past I named a particular set of numbers complex or circular. I did choose for circular because the matrix representations of circular numbers are the so called circulant matrices. It is always better to give mathematical stuff some kind of functional name so people can make sense of what the stuff is about. For me no silly names like ‘3D Venema positive numbers’ or ‘3D Venema complex numbers’. In math the objects should have names that describe them, the name of a person should not be hanged on such an object. For example the Cayley-Hamilton theorem is a total stupid name, the names of the humans who wrote it out are not relevant at all. Further reading on circulant matrices: Circulant matrix.
I also have a wiki on split complex numbers for you, but like all common sources they have the conjugate completely wrong. Professional math professors always think that taking a conjugate is just replacing a + by a – but that is just too simplistic. That’s one of the many reasons they never found 3D complex numbers for themselves, if you do that conjugate thing in the silly way all your 3D complex math does not amount to much…