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?
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:
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:
With ‘outer product’ I mean the 3D cross product.
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.
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.
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.
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.
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… Link: Split-complex number.
This is the last part on this oversight of counter examples to the last theorem of Pierre de Fermat and it contains only the two dimensional split complex numbers. When I wrote the previous post I realized that I had completely forgetten about the 2D split numbers. And indeed the math results as found in this post are not very deep, it’s importance lies in the fact that the counter examples now are unbounded. All counter examples based on modular arithmetic are always bounded, periodic to be precise, so professional math professors could use that as a reason to declare that all a bunch of nonsense because the real integers are unbounded. And my other counter examples that are unbounded are only on 3D complex & circular number spaces and the 4D complex numbers so that will be neglected and talked into insignificance because ‘That is not serious math’ or whatever kind of nonsense those shitholes come up with.
All in all despite the lack of mathematical depth I am very satisfied with this very short update. The 2D split numbers have a history of say 170 years so all those smart math assholes can think a bit about why they never formulated such simple counter examples to the last theorem of Fermat… May be the simplicity of the math results posted is a good thing in the long run: compare it to just the natural numbers or the counting numbers. That is a set of numbers that is very simple too, but they contain prime numbers and all of a sudden you can ask thousands and thousands of complicated and difficult questions about natural numbers. So I am not ashamed at all by the lack of math depth in this post, I only point to the fact that over the course of 170 years all those professional math professors never found counter examples on that space.
This post is just 3 pictures long although I had to enlarge the lastest one a little bit. The first two pictures are 550×825 pixels and the last one is 550×975 pixels. Here we go:
That was it for this post, one of the details as why this post is significant is the use of those projector numbers. You will find that nowhere on the entire internet just like the use of 3D complex numbers is totally zero. Let’s leave it with that, likely the next post is about magnetism and guess what? The physics professors still think there is no need at all to give experimental proof to their idea of the electron having two magnetic poles. So it are not only the math professors that are the overpaid idiots in this little world of monkeys that think they are the masters of the planet.
I finished the proof that was originally meant to be an appendix to the previous post. And I have two more or less small things I want to share with you so lets get started with the first thing:
Thing 1: Tibees comes up with a very cute program of graphing 3D surfaces. It’s name is surfer, the software is very simple to use and it has the giant benefit of making graphs from implicit equations like f(x, y, z) = 0. For example if you want the unit sphere in 3D space you must do x^2 + y^2 + z^2 – 1 = 0. Now for this website I always used an internet applet that uses ray tracing and by doing so over the years such graphs always look the same. But this surfer program has cute output too and it has the benefit you don’t need to be online. Here is how such a graph looks, it is the determinant in the space of 3D complex numbers, to be precise it shows the numbers with a determinant of 1:
By the way, the surface of this graph is a multiplicative group on it’s own in 3D space. I never do much group stuff but if you want it, here you have it. And for no reason at all I used GIMP to make one of those cubes from the above graph. It serves no reason beside looking cute:
The Tibees female had a video out last week where she discusses a lot of such surfaces in three dimensional space using that surfer software. And she is a pleasant thing to look at, it is not you are looking at all those extravert males drowning in self-importance only lamentating shallow thoughts. The problem posed in the video is an iteresting one, I don’t have a clue how to solve it. Title of the video: The Shape No One Thought was Possible. It is a funny title because if you start thinking about all the things that math professors thought were not possible you can wonder if there is enough paper in the entire universe to write that all out.. Link to the Surfer program in case you want to download & install it: https://www.imaginary.org/program/surfer.
So far for thing 1.
Thing 2: The last weeks it is more and more dawning on me that all those centuries those idiots (the math professors) did not find counter examples to the last theorem of Fermat. Nor was there any improvement on the little theorem of Fermat. Only Euler did some stuff on the little theorem with his totient function, but for the rest it is not much… Well since Jan of this year I found many counter example to the last theorem of Fermat and in my view I made a serious improvement on the little theorem of Fermat. So is the improvement serious or not? Here is a picture that shows the change:
So it’s modulo ap instead of modulo p.
On a wiki with a lot of proofs for the little theorem of Fermat they start with a so called ‘simplification’. The simplification says that you must pick the number a between 0 and p. So if you have an odd prime, say a = 113, does the little theorem only make sense for exponents above 113? And can’t we say anything about let’s say the square 113^2?
With the new version of the little theorem we don’t have such problems any longer. Here is a screen shot from the start of that wiki, the upper part shows you the improvement:
Here is a link to that wiki that is interesting anyway.
If you follow that link you can also scroll down to the bottom of the wiki where you can find the notes they used. It is an impressive list of names like Dirichlet, André Weil, Hardy & Wright and so on and so on. All I want to remark is that non of them found counter examples to the last theorem nor did they improve on the little theorem of Fermat. Now I don’t want to be negative on Dirichlet because without his kernel I could never have crafted my modified Dirichlet kernel that is more or less the biggest math result I ever found. But the rest of these people it is just another batch of overpaid non performers. It’s just an opinion so you don’t have to agree with it, but why do so many people get boatloads of money while they contribute not that much?
End of thing 2.
Now we are finally ready to post the main dish in this post: the proof that the subset of four dimensional rational numbers form a field. Math professors always think it is ‘very important’ if something is a field while in my life I was never impressed that much by it. And now I am thinking about it a few weeks more, the less impressed I get by this new field of four dimensional complex numbers. Inside the theory of higher dimensional complex numbers the concept of ‘imitators of i‘ is important: these are higher dimensional numbers that if you square them they have at least some of the properties of the number i from the complex plane. They rotate everything by 90 degrees or even better they actually square to minus one. Well one of the imitators of i in the space of 4D complex number is dependent of the square root of 2. As such it is not a 4D rational complex number. That detail alone severely downsizes my enthousiasm. But anyway, the next pictures are also a repeat of old important knowledge like the eigenvalue functions. Instead of always trying to get the eigenvalues from some 4×4 matrix, with the eigenvalue functions with two fingers in your nose you can pump out the eigenvalues you need fast. This post is six pictures long each size 550×825 pixels. Here we go:
Yes that is the end of this post that like always grew longer than expected. If you haven’t fallen asleep by now, thanks for your attention and don’t forget to hunt the math professors until they are all dead! Well may be that is not a good idea, but never forget they are too stupid to improve on the little Fermat theorem and of course we will hear nothing from that line of the profession…
This year starting in January I found more and more counter examples to the last theorem of Fermat. As a by product when we looked at the stuff on the 4D complex numbers, we found that if we restrict ourselves to the 4D rationales, they were always invertible. And as such they form a field, this is a surprising result because the official knowledge is that the only possible 4D number system are the quaternions from Hamilton. So how this relates to those stupid theorems of Hurwitz and Fröbenius about higher dimensional complex numbers is something I haven’t studied yet. But that Hurwitz thing is based on some quadratic form so likely he missed this new field of 4D complex rationales because the 4D complex numbers are ruled by a 4 dimensional thing namely the fourth power of the first imaginary unit equals minus one: l^4 = -1. Compare that to the complex plane that has all of it’s properties related to that defining equation i^2 = -1.
And because we now have a 4D field I thought like let’s repeat how you find the inverse of a 4D complex rational number. And also prove that we have a field as basic a proof can be. But while writing this post I had to abandon the second thing otherwise this post would grow too long. Of course in the past I have crafted a post for finding the non-invertible 4D complex numbers but in that post I never remarked that rational 4D complex numbers are always invertible. To be honest in the past it has never dawned on me that it was a field, for me this is not extremely important but for the professional math people it is.
When back in Jan of this year I found the first counter example to the last theorem of Fermat I was a bit hesitant to post it because it was so easy to find for me. But now four months further down the time line I only found two examples where other people use some form of my idea’s around those counter examples and both persons have no clue whatsoever that they are looking at a counter example to the last theorem of Fermat. But in a pdf from Gerhard Frey (that is the Frey from the Frey elliptic curve that plays an important role in the proof to the last theorem of Fermat by Andrew Wiles) it was stated as: (X + Y)^p = X^p + Y^p modulo p. That’s all those professionals have, it is of a devastating minimal content but at least it is something that you could classify as a rudimentary counter example to the last theorem of Fermat. It only works when your exponent if precisely that prime number p and it lacks the mathematical beauty that for example we have in expressions like: 12^n = 5^n + 7^n modulo 35.
Anyway this post contains nothing new but there is some value in repeating how to find inverses of higher dimensional complex numbers. All you need is a ton of linear algebra and for that let me finish this intro on a positive note: Without the professional math professors crafting linear algebra in the past, at present day for me it would be much harder to make progress in higher dimensional complex numbers. And it is amazing: Why is linear algebra relatively good while in higher dimensional number systems we only look at a rather weird collection of idea’s? This post is made up of seven pictures each of size 550×800 pixels.
Stupid typo: Z = 1 + l +… so the real part must be one.
Ok ok this post is not loaded with all kinds of deep math results. But if you have a properly functioning brain you will have plenty of paths to explore. And the professional math professors? Well those overpaid weirdo’s will keep on neglecting the good side of math and that is important too: That behavior validates they are overpaid weirdo’s…
For example the new and improved little theorem of Fermat: The overpaid weirdo’s will neglect it year in year out. That’s the way it is, here is once more a manifestation of the new and improved little theorem of Fermat:
Let’s leave it with that. Thanks for your attention.
A couple of days ago all of a sudden while riding my bicycle I calculated what the so called directional derivative is for 3D & 4D complex numbers. And it is a cute calculation but I decided not to write a post about it. After all rather likely I had done stuff like that many years ago.
Anyway a day later I came across a few Youtube video’s about the directional derivative and all those two guys came up with was an inner product of the gradient and a vector. Ok ok that is not wrong or so, but that is only the case for scalar valued functions on say 3D space. A scalar field as physics people would say it. The first video was from the Kahn academy and the guy from 3Blue1Brown has been working over there lately. It is amazing that just one guy can lift such a channel up in a significant manner. The second video was from some professional math professor who went on talking a full 2.5 hour about the directional derivative of just a scalar field. I could not stand it; how can you talk so long about something that is so easy to explain? Now I do not blame that math professor, may be he was working in the USA and had to teach first year math students. Now in the USA fresh students are horrible at math because in the USA the education before the universities is relatively retarded.
Furthermore I tried to remember when I should have done the directional derivative. I could not remember it and in order to get rid of my annoyance I decided to write a small post about it. Within two hours I was finished resulting in four pictures of the usual 550×775 pixel size. So when I work hard I can produce say 3 to 4 pictures in two hours of time. I did not know that because most of the time I do not work that fast or hard. After all this is supposed to be a hobby so most of my writing is done in a relaxed way without any hurry. I have to say that may be I should have taken a bit more time at the end where the so called Cauchy-Riemann equations come into play. I only gave the example for the identiy function and after that jumped to the case of a general function. May be for the majority of professional math professors that is way to fast, but hey just the simple 3D complex numbers are ‘way to fast’ for those turtles in the last two centuries…
Anyway, here is the short post of only 4 pictures:
Should I have made the explanation longer? After all so often during the last years I have explained that the usual derivative f'(X) is found by differentiating into the direction of the real numbers. At some point in time I have the right to stop explaining that 1 + 1 = 2.
Also I found a better video from the Kahn academy that starts with a formal definition of the directional derivative:
At last let me remark that this stuff easily works for vector valued functions because in the above limit you only have to subtract two vectors and that is always allowed in any vector space. And only if you hang in a suitable multiplication like the complex multiplication of 3D or 4D real space you can tweak it like in the form of picture number 4 above.
That was all I had for you today, this is post number 166 already so I am wondering if this website is may be becoming too big? If people find something, can they find what they are searching for or do they get lost in the woods? So see you in another post, take care of yourself & till the next post.
