The logarithm of all 2D circular numbers (the split complex numbers).

Uncategorized

Yesterday I was editing the six pictures for this update and all of a sudden I realized I had made a dumb dumb mistake: The pictures count down from number 7 to number 2…

I had processed them in the wrong order; I had made seven background pictures but I filled in the math text in the wrong order.

All in all I decided to leave it this way; it might be a stupid mistake but it is not a critical mistake like making a critical math error or having wrong ideas about what is actually happening on the math level. It is just an editing error and also funny. So I leave it this way.

In this post we look at the so called split complex numbers, they are the cousin of the numbers from the complex plane. The only difference is that where in the complex plane the square of the imaginary unit equals minus one, for the split complex numbers this equals plus one.

Although this is a minor change, split complex numbers are not a field because it contains non-invertible numbers outside the number 0. All I do in this post is finding the eigenvalues and eigenvectors of all split complex numbers and via taking the log of the eigenvalues we calculate what the log of an arbitrary split complex number is.

In the speak of this website the split complex numbers are just the 2D circular numbers. Remeber in all dimensions numbers are complex or circular depending if the first imaginary unit equals -1 or +1. You can find many more ways of crafting a multiplication but the best math results are always found in the complex and circular version of the numbers in that particular dimension…













Ok, in this post I left all things out that talks about the 4D hybrid space that is a mixture of the 2D circular and complex numbers. But as you see on inspection of the above six pictures, the eigenvalues might be always real but they can be negative. As such always pay attention when you apply that function named the log…

That was it for this post, at this point in time I have no idea what the next post will be about. After all we had this long rout of over 20 posts on the 4D complex numbers and I left a whole lot of other stuff out in that period. Stuff like 3D Gauss integers or a general definition for integration that works in all dimensions. Till updates my dear reader.

And life? Life will go on.

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Originally I planned on showing you some numerical results from the circular 4D numbers while explaining there is also a number alpha in 4D. For me that would be a nice holiday away from all that 4D complex number stuff from the last months…

But the numerical applet did not work, it is still dead in the water:

http://calculator.vhex.net/calculator/linear-algebra/matrix-exponential-using-the-pade-approximation

Ok ok, I could have done those numerical showings also in rigid analysis but I guessed that calculating a 4D tau for circular numbers via analysis was too much. And I settled for a much more easy to understand thing:

The logarithmic function for every 2D circular number. In the field of professional math professors the 2D circular numbers are known as the split complex numbers.

So that is what the next post will be about: Finding log(z) for all invertible split complex numbers.

I only wrote one previous post on the 2D circular aka split complex numbers and that dates back to Nov 24 of the year 2016:

The second hybrid: a 4D mix of the complex and the circular plane.

Ha ha, now I can laugh about it but back in the time it was some hefty pain. Anyway to make a long story short: In that old post from 2016 I calculated the log for just one split complex number namely the first imaginary unit j.

Let me show you my favorite part of that old post from 2016:

So the next update will only contain 2×2 size matrices while I skip the detail that the log lives mainly in the hybrid number system from the old post.

Till updates.

Part 20: On the structure of non-invertible 4D complex numbers.

4D complex numbers

In general it is rather hard to find non-invertible 4D complex numbers because the determinant is non-negative everywhere. Just try it yourself, write down just one 4D complex number that is non-zero and not invertible.

That is not an easy task, after some time you will find some but do you have all?

But if you understand the concept of the eigenvalues that every 4D complex number Z has, it is easy to understand that if a 4D complex number is non-invertible at least one of the eigenvalues must be zero.

In previous posts we already unearthed the four eigenvalue functions that return the four eigenvalues each 4D complex number has.

In this post we will try to find where these eigenvalue functions are zero.

Since eigenvalue functions come in pairs whenever possible in the case of 4D complex numbers we only have two pairs of those eigenvalue functions.

Let’s stop the talking and just post the twelve pictures that make up part number 20 into the basics of the four dimensional complex numbers.

 

 

 

 

 

 

 

 

 

 

 

 

Ok, that was it for part 20 in the series that covers the basics of 4D complex numbers.

See you around my dear reader.

A teaser question: Can you prove this inequality?

4D complex numbers

Recently I am working on part 20 to the basics of the 4D complex numbers. Ok ok if you need 20 parts to explain ‘the basics’ how basic is it you can ask yourself.

You can argue long and short on this: are fresh Cauchy integral formula’s really ‘basic stuff’? I don’t know how a democratic vote among professional math professors would fall down.

Anyway, an important property of the determinant of 4D complex numbers is the fact that the determinant is always non-negavite. At least it is zero and at those points in space we have found a non-invertible number.

In part 20 on the basics to 4D complex numbers we will look when the eigenvalues of 4D complex numbers vanish; at those points the stuff is non-invertible & that is what we will be hunting on part 20.

In the next picture you see a difficult to understand inequality & the teaser question is:
Can you prove this inequality via math methods that do not use 4D complex number theory at all?

If so, you should definitely pop up a second pint of perfect beer on a late Friday evening.

Ok, that was it. Till updates in part 20 where we try to find all non-invertible 4D complex numbers in a not too difficult way.

The inverse theorem of Pythagoras (part 2).

Pythagoras stuff

Somewhere last year I just looked some nice video from the Mathologer about the theorem of Pythagoras. And since I myself have found a proof for the general theorem of Pythagoras in higher dimensions, I was puzzled about what the so called ‘inverse theorem of Pythagoras’ actually was.

Could I do that too in my general proof? And the answer was yes, but when I wrote that old proof of the general theorem of Pythagoras it was just a technical blip not worthwhile mentioning because it was a simple consequence of how those normal vectors work.

Anyway to make a long story short, a few days back I likely had nothing better to do and for some reason I did an internet search for ‘the inverse theorem of Pythagoras’. All I wanted to do is read a bit more about that from other people.

To my surprise my own writing popped up as search result number 3, that was weird because I wanted to read stuff written by other people… Here is a screenshot of the answers as given by the Google search machine:

Ok ok, not bad at search result number 3.

Now why bring this up? Well originally I forgot to post to the video that started my thinking in the first place. It is from the Mathologer and here at 16.00 minutes into his video is where my mind started to drift off:

The video from the Mathologer is here (title Visualizing Pythagoras: ultimate proofs and crazy contortions):

It is a very good video, my compliments.

After so much advertisements for the Mathologer, just a tiny advertisement for what I wrote on the subject of the inverse theorem of Pythagoras on March 20 in the year 2018:

What is the inverse Pythagoras theorem?

Ok, that was it. Till updates.

Part 19: Four integrals defining the 4D complex number tau.

4D complex numbers, Exponential circle, Exponential curves

It is a bit late but a happy new year anyway! In this post we will do a classic from the complex plane: calculation of the log of the first imaginary unit.

On the complex plane this is log i and on the complex 4D space this is log l .

Because this number is so important I have given it a separate name a long long time ago: These are the numbers tau in the diverse dimensions. In the complex plane it has no special name and it simply is i times pi/2.

On the real line it is pretty standard to define the log functions as the integral of the inverse 1/x. After all the derivative of log x on the real line is 1/x and as such you simply define the log to be the integral of the derivative…

On the complex plane you can do the same but depending of how your path goes around zero you can get different answers. Also in the complex plane (and other higher dimensional number systems) the log is ‘multi valued’. That is a reflection of the fact we can find exponential periodic functions also known as the exponential circles and curves.

The integrals in this part number 19 on the basics of 4D complex numbers are very hard to crack. I know of no way to find primitives and to crack them that way. May be that is possible, may be it is not, I just do not know. But because I developed the method of matrix diagonals for finding expressions for the value of those difficult looking integrals, more or less in an implicit manner we give the right valuations to those four integrals.

With the word ‘implicit’ I simply mean we skip the whole thing of caculating the number tau via matrix diagonalization. We only calculate what those integrals actually are in terms of a half circle with coordinates cos t and sin t.

This post is 8 pictures long in the usual size of 550 by 775 pixels (I had to enlarge the latest picture a little bit). I hope it is not loaded with typo’s any more and you have a more or less clean mathematical experience:

End of this post.

At year’s end: What is the most nice magnetic result of this year?

Magnetism

There is more than one candidate; the possible explanation of those solar loops via the rotating plasma under it is indeed very very nice.

But it is very hard to find experimental evidence for that, how to check that on the sun where there is one of those solar loops, the solar plasma underneath it is rotating?

So my choice for this year most beautiful insight is how those magnetic domains in materials like iron actually work. If my thinking on electrons as magnetic monopoles is correct, you could view those magnetic domains as surplusses of either north-pole electrons or south-pole charged electrons.

This should be much more easy to verify experimentally. After all there is still plenty of magnetic tape around and the best way of checking if magnetic domains are suplusses of one of the two magnetic charges is much more handy if you have a flat surface like magnetic tape.

Furthermore at present day it should be possible to measure very small magnetic charges. After all in most computers there still are spinning hard disks that use magnetism as the basic information storage.

Anyway to make a long story short: If in a flat material like magnetic tape you can go around a magnetic domain with a tiny compass, all of the time the tiny compass should point towards that magnetic domain with the same side of the compass needle.

So it should always point with the north-pole of the compass needle or the south-pole of the compass needle. Remark that according to standard physics theory going around a magnetic domain should always give different readings with a tiny compass needle…

Also in that line of thinking, the domain walls shoud have surplusses of electron pairs that all are spinning around to compensate or neutralize the magnetic forces they feel. An important clue to that lies in the fact you cannot really move or transport domain walls in a wire as the engeneers of IBM tried with making their nano wire racetrack memory.

This year we heard more or less nothing from IBM with progress into the concept of nano wire racetrack memory. Yeah yeah, the price of not understanding electron spin is huge, if we could have fast computer memory that uses very little energy that would be great…

This year I also gave up on my fantasies of trying to make an official publication in some physics yournal. I don’t think such a publication will ever pass the peer review that those scientific yournals use. Those peer review people just want the electron is a magnetic dipole and that’s it. So I did not try that this year nor will I try next year.

Not that I am aginst peer review. Suppose there would be zero peer review and think for example medical scientific publications. They would be filled with all kind of weird benefits that homeopathy therapy has, or the healing of your chakra’s with mineral chrystals… Of course we cannot have that. As such there must always be the so called peer review…

Ok, the word count counter says 500+ words written so I have to stop writing.

Here is the link to what I self more or less consider as the best improving insight on my behalf on the behavior of all things magnetic. It is from 7 July this year:

07 July 2018: Reason 64: Bloch and Neel walls explained.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff03.htm#07July2018

Ok, that was it. Happy new year.

Just a short video on the Fourier stuff.

Exponential circle, Uncategorized

This is the shortest post ever written on this website.

I found one of those video’s where the Fourier series is explained as the summation of a bunch of circles. Likely when you visit a website like this one, you already know how to craft a Fourier series of some real valued function on a finite domain.

You can enjoy a perfect visualization of that in the video below:

Only one small screen shot from the video:

Oh oh, the word count counter says 80+ words. Let me stop typing silly words because that would destroy my goal of the ‘shortest post ever’. Till updates.

Part 18: Calculating the 4D number tau it’s inverse in a very simple way.

4D complex numbers, Matrix representation

It is the shortest day of the year today and weirdly enough I like this kind of wether better compared to the extreme heat of last summer. Normally I dislike those long dark days but after so much heat for so long I just don’t mind the darkness and the tiny amounts of cold.

In the previous post we found a general way of finding all inverses possible in the space of the 4D complex numbers. Furthermore in the post with the new Cauchy integral representation we had to make heavy use of 1/8tau and as such it is finally time to look at what the inverse of tau actually is.

I found a very simple way of calculating the inverse of the number tau. It boils down to solving a system of two linear equations in two variables. As far as I know reality, most math professionals can actually do this. Ok ok, for the calculation to be that simple you first must assume that the inverse ‘looks like’ the number tau in the sense it has no real component and it is just like tau a linear combination of those so called ‘imitators of i‘.

This is a short post, only five pictures long. I started the 4D complex number stuff somewhere in April of this year so it is only 8 month down the timeline that we look at the 4D complex numbers. It is interesting to compare the behavior of the average math professor to back in the time to Hamilton who found the 4D quaternions.

Hamilton became sir Hamilton rather soon (although I do not know why he became a noble man) and what do I get? Only silence year in year out. You see the difference between present day and past centuries is the highly inflated ego of the present day university professors. Being humble is not something they are good at…

After having said that, here are the five pictures:

All in all I have begun linking the 4D complex numbers more and more in the last 8 months. On details the 4D complex numbers are very different compared to 3D and say 5D complex numbers but there are always reasons for that. For example the number tau has an inverse in the space of 4D complex numbers but this is not the case in 3 or 5D complex numbers space.

Well, have a nice Christmas & likely see you in the next year 2019.

Part 17: The inverse of a 4D complex number old school style (via minor matrices).

4D complex numbers, Matrix representation

Ha, a couple of weeks back I met an old colleague and it was nice to see him. We made a bit of small talk and more or less all of a sudden he said: ‘But you still can always do this’. And he meant getting a PhD in math. 

I was a bit surprised he did bring this up, for me that was a station passed long ago. But he made me thinking a bit, why am I not interested in getting a math degree? 

And when I thought it out I also had to laugh: Those people cannot go beyond the complex plane for let’s say 250 years. And the only people I know of that have studied complex numbers beyond the complex plane are all non-math people. Furthermore inside math there is that cultural thing that more or less says that if you try to find complex numbers beyond the complex plane, you must have a ‘mental thing’ because have you never heard of the 2-4-8 theorem? 

Beside this, if I tried it in the years 1990 and 1991 with very simple: Here this is how the 3D Cauchy Riemann equations look… And you look them in the eyes, but there is nothing happening behind those eyes or in the brain of that particular math professor. Why the hell should I return and under the perfect guidance of such a person get a PhD? 

I am not a masochist. If complex numbers beyond the complex plane are ignored, why try to change this? After all this is a free world and most societies run best when people can do what they are good at. Apparently math like I make simply falls off the radar screen, I do not have much problems with that. 

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After having said that, this update Part 17 in the basics to the 4D complex numbers is as boring as possible. Just finding the inverse of a matrix just like in linear algebra with the method of minor matrices.

Believe me it is boring as hell. And after all that boring stuff only one small glimmer of light via crafting a very simple factorization of the determinant inside the 4D complex numbers. So that is very different from the previous factorization where we multiplied the four eigenvalue functions. From the math point it is a shallow result because it is so easy to find but when before your very own eyes you see the determinant arising from those calculations, it is just beautiful. And may be we should be striving a tiny bit more upon mathematical beauty…

This post is nine pictures long in the usual size of 550×775 pixels.

 

As an antidote against so much polynomials like det(Z), with 2 dimensions like a flatscreen television, you can do a lot of fun too. The antidote is a video from the standupmaths guy, it is very funny and has the title ‘Infinite DVD unboxing video: Festival of the Spoken Nerd’. Here is the vid:

End of this update, see you around.

Ok ok, a few days later I decided to write a small appendix to this post and in order too keep it simple let’s calculate the determinant of the 3D circular numbers. I have to admit this is shallow math but despite being shallow it gives a crazy way to calculate the determinant of a 3D circular number…

So a small appendix, here it is:

And now you are really at the end of this post.

In the next post let’s calculate the inverse of the 4D complex tau number. After all a few months back I gave you the new Cauchy integral representation and I only showed that the determinant of tau was nonzero.

But the fact that the Cauchy integral representation is so easy to craft on the 4D complex numbers arises from the fact the inverse of tau exists in the first place. In 3D the number tau is not invertible, and Cauchy integral representations are much more harder to find.

Ok, drink a green tea or pop up a fresh pint, till updates.