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.
For a couple of years I have a few pdf files in my possession written by other people about the subject of higher dimensional complex and circular numbers. In the post we will take a look at the work of Shlomo Jacobi, the pdf is not written by him because Shlomo passed away before it was finished. It is about the 3D complex numbers so it is about the main subject of this website.
Link used: http://search.arxiv.org:8081/paper.jsp?r=1509.01459&qid=1603841443251ler_nCnN_1477984027&qs=Shlomo+Jacobi&in=math
Weirdly enough if you search for ‘3D hypercomplex number’ the above pdf does not pop up at all at the preprint archive. But via his name (Shlomo Jacobi) I could find it back. Over the years I have found three other people who have written about complex numbers beyond the 2D complex plane. I consider the work of Mr. Jacobi to be the best so I start with that one. So now we are with four; four people who have looked at stuff like 3D complex numbers. One thing is directly curious: None of them is a math professional, not even a high school teacher or something like that. I think that when you are a professional math professor and you start investigating higher dimensional complex numbers; you colleagues will laugh about it because ‘they do not exist’. And in that manner it are the universities themselves that ensure they are stupid and they stay stupid. There are some theorems out there that say a 3D complex field is not possible. That is easy to check, but the math professionals make the mistake that they think 3D complex numbers are not possible. But no, the 2-4-8 theorem of say Hurwitz say only a field is not possible or it says the extension of 2D to 3D is not possible. That’s all true but it never says 3D complex numbers are not possible…
Because Shlomo Jacobi passed away an unknown part of the pdf is written by someone else. So for me it is impossible to estimate what was found by Shlomo but is left out of the pdf. For example Shlomo did find the Cauchy-Riemann equations for the 3D complex numbers but it is only in an epilogue at the end of the pdf.
The content of the pdf can be used for a basic introduction into the 3D complex numbers. It’s content is more or less the ‘algebra approach’ to 3D complex numbers while I directly and instantly went into the ‘analysis approach’ bcause I do not like algebra that much. The pdf contains all the basic stuff: definition of a 3D complex number, the inverse, the matrix representation and stuff he names ‘invariant spaces’. Invariant spaces are the two sets of 3D complex numbers that make up all the non-invertible numbers. Mr. Jacobi understands the concept of divisors of zero (a typical algebra thing that I do like) and he correctly indentifies them in his system of ‘novel hypercomplex numbers’. There is a rudimentary approach towards analysis found in the pdf; Mr. Jacobi defines three power series named sin1, sin2 and sin3 . I remember I looked into stuff like that myself and somewhere on this website it must be filed under ‘curves of grace’.
A detail that is a bit strange is the next: Mr. Jacobi found the exponential circle too. He litarally names it ‘exponential circle’ just like I do. And circles always have a center, they have a midpoint and guess how he names that center? It is the number alpha…
Because Mr. Jacobi found the exponential circle I applaud him long and hard and because he named it’s center the number alpha, at the end I included a more or less new Euler identity based on a very simple property of the important number alpha: If you square alpha it does not change. Just like the square of 1 is 1 and the square of 0 is 0. Actually ‘new’ identity is about five years old, but in the science of math that is a fresh result.
The content of this post is seven pictures long, please read the pdf first and I hope that the mathematical parts of your brain have fun digesting it all. Most pictures are of the standard size of 550×775 pixels.
Ok ok, may be you need to turn this into exponential circles first in order to craft the proof that a human brain could understand. And I am rolling from laughter from one side of the room to the other side; how likely is it that professional math professors will find just one exponential circle let alone higher dimensional curves?
I have to laugh hard; that is a very unlikely thing.
End of this post, see you around & see if I can get the above stuff online.
This post took me a long time to write, not that it was so very difficult or so but lately I am learning that graphics program named GIMP. And that absorbs a lot of time and because I am only sitting behind my computer a few hours a day, doing GIMP goes at the expense of writing math…
I always make my pictures with an old graphics program named Picture Publisher 10. It is so old that on most windows 7 and windows 10 it does not run but it has all kinds of features that even the modern expensive graphics programs simply still don’t have. Silently I was hoping that I could use GIMP for my math texts and yes that could be done but in that case I have to use old background pictures forever. Or I have to craft a ‘new style’ for making the background in the math pictures that can last at least one decade.
But let’s not nag at what GIMP cannot do, if you install just one large addon you have about 500 filters extra and my old program PP10 comes from an era when the word ‘addon’ was not a word used ever. Before we jump to the math, let me show you a nice picture you can make with the tiling filter inside GIMP. It is about my total bicycle distance since I bought this bicycle computer, it says 77 thousand km so the Tour the France racers can suck a tip on that:
Ok, let us look at the math of this post. This is part 22 in the introduction to the 4D complex numbers. The 4D complex numbers have three imaginary units, l, l^2 and l^3. And the stuff that makes it ‘complex’ is the fact that l^4 = -1, you can compare that to the complex plane where the square of the imaginary unit equals -1.
On the complex plane, if you know what the logarithm of i is, you can use that to find the exponential circle also known as the complex exponential. This is what the number tau always is in all kinds of spaces: It is always the logarithm of the first imaginary unit that has a determinant of +1. In this post we will calculate the eigenvalues of this important number tau. That will be done with two methods. In the first method we simply use the eigenvalue functions, plug in the number tau and voila: out come the four eigenvalues. In the second method we first calculate the four eigenvalues of the imaginary unit l and ‘simply’ take the logarithm of those four eigenvalues.
It is not much of a secret that my style of work is rather sloppy, I never order my work in theorems, lemma’s or corrolaries. It is not only that such an approach if too much a straight jacket for me, it also frees me from a lot of planning. I simply take some subject, like in this case the eigenvalues of the number tau and start working on explaining that. While writing that out there always comes more stuff around that I could include yes or no. In this post what came around was that only after writing down the four eigenvalues I realized that you can use them to prove that the exponential curve (the 4D complex exponential) has a determinant of 1 for all points on that curve. That was an important result or an important idea so I included it because that makes proving that the determinant is 1 much more easy.
Now a few posts back with that video from that German physics guy Alexander Unzicker I said that he (and of course all other physics professionals) could always use the 4D complex exponential curve for the ‘phase shifts’ that those physics people always do. But for doing such 4D ‘phase shifts’ or unitary transformations in general, you need of course some kind of proof that determinant values are always +1. Well Alexander, likely you will never read this post but below you can find that very proof.
The previous post was from the end of August and now I think about it: Have I done so little math during the last four weeks? Yes there were no results simply left out, it was only penning down these eigenvalues of tau and the idea you can use these eigenvalues for proving the 4D exponential curve always has a determinant of 1. It is amazing that GIMP can hinder the creation of fresh math… 😉
The math pictures are seven in number, all in the usual size of 550×775 pixels. I hope you like it and see you in the next post.
So these are the four eigenvalues of the number tau and based on that the four eigenvalues of the 4D complex exponential for a values of time.
That’s it for this post. See you in a future post.
Added on 27 Sept 2020: This proceeds the two pictures made with GIMP that started this post. I just made the whole stuff on a cube (actually it is a beam because the starting picture is not a square). It is amazing how good such filters in GIMP are:
Ah, finally it is finished. This work grew longer than expected but with a bit of hindsight that is also logical: for example I spell out in detail once more how to find the eigenvalue functions for a arbitrary number X. After all that is an important detail so it is worth repeating. But I skipped the proces of diagonalization because we do not need it in this post. Yet if you teach math and the time has come to do the complex number stuff, you could show the students how to diagonalize the complex multiplication for numbers from the complex plane. Most of the time students only diagonalize just one matrix with some numbers in it and that’s all, they never diagonalize an entire family of matrices. So that is why that would be useful, on the other hand the eigenvalues for a number z from the complex plane is z itself and it’s conjugate… And say for yourself: diagonalizing a number z so that later you must multiply the eigenvalues (also z) is very useless, as a matter of fact it is hard to find anything that is more useless… And once you have explained that diag stuff is usefull and utterly un-usefull at the same time, you can point to the live of the average math professor: also utterly useless…
Of course in higher dimensions the proces of diagonalization is very handy because it gives you for example a way of calculating the logarithm of higher dimensional numbers. And that way can be used in any dimension while all other methods for finding a logarithm get more and more difficult (as far as I know).
In this post I also worked out in detail what the eigenvalues of non-invertible numbers are; the non zero numbers with a determinant of zero have at least one eigenvalue being zero. I calcualted the eigenvalues for the numbers tau and alpha for both the complex and circular 3D multiplication.
This is post number 150 so all in all on average I write just about 30 posts a year. That is a cost of about 2€ per post… 😉 Luckily this hobby of 3D complex numbers is a rather cheap hobby while at the same time it keeps the mind sharp. A disadvantage is that if it takes me just 5 or 10 minutes to do some calculations with pencil and paper, it often takes 5 to 10 hours before it is turned into a post that is more or less readable for other people… And that is something I value highly; so often you come across sloppy explanations that are not carfully thought through. I don’t like that.
Originally I prepared 10 pictures to write the post on but I had so much text that I started expanding those pictures and in the end I made an 11-th picture to get it all on. So I just expanded those pictures to make the text fit more or less precise of most of them have weird sizes. May be it is better to just stick to the size of 550×775 pixels and just make more of them if needed and not this chaotic expansion on the fly. Ok, here we go:
I expect that when you made it this far, you already know what the Cauchy/Schwarz inequality is. But in case you never heard of it, please try to understand that beautiful but very simple inequality. Here is a wiki: Cauchy-Schwarz Inequality. Link used: https://brilliant.org/wiki/cauchy-schwarz-inequality/
Ok, this is more or less what I had to say on the subject of crafting a norm from eigenvalues. Don´t forget in the complex plane the square of the norm is also the product of the eigenvalues of a complex number z. So for centuries the math professors are already doing this although I do not think they are aware that they use a product of eigenvalues. For them likely it is just some stuff that is ´Just like Pythagoras´.
Ok ok I was a bit lazy but it is finished now so let’s finally post this scalar replacement theorem. Never in this post I formulate or proof this scalar replacement theorem, but basically this theorem says that if you replace the real numbers (scalars) in the way you describe say 2D split complex numbers by numbers from the complex plane, the result is a space who’s numbers also commute and it even has viable Cauchy Riemann equations. In this post I will write z = x + yj for the 2D circular numbers (also known as the split complex numbers) and write z = x + yi for numbers from the complex plane. If you combine such spaces it must have imaginary units that are different in notation, so j is the imaginary unit that does j^2 = 1 while the good ol i from the complex plane is known for it’s important property that i^2 = -1.
If we replace the x and y in z = x + yj by complex numbers we get a new 4D space where both j and i place there role. All in all those 4D numbers will be written as Z = a + bi + cj + dij. Of course the a, b, c and d are real numbers and as such this new space is 4D.
A long time ago I once used this to calculate the logatithm of j, it worked perfectly and that is why I more or less gave idea’s like that the name of ‘scalar replacement’. Later I found that way of using diagonalization of the matrix representations in order to calculate the logarithm, that is a far more general useable way of calculating logarithms but anyway the original calculation for log j was so cute, I could not abondon it and say to that calculation: From now on you are a poor orphan and no one will help you survive from day to day… How could I abandon such a calculation, better loose the UK a 100 times on a row than abandoning such nice calculations… 😉
But let’s go back to being a serious and responsible adult; the post is relatively long with 10 pictures. As usual I had to leave a lot out and I hope it is more or less easily readable. After all a lot of math out there looks like it is written by people who eat a plate of coal for breakfast. And if you eat coal for breakfast, likely this has an influence on the math you will produce on a particular day… Ok, here we go:
Ok, the goal of this post is of course to make you think a little bit about this 4D space and compare it to the quaternions and stuff. But last year on 2 March I posted the diagonalization method for finding the logarithm of an arbitrary split complex number. Below is a link.
Let me end this post with a funny mathematical joke about how to NOT WRITE MATH. Using a fucking lot of indices is not a way to make your work readable, here is a picture of what I view as some kind of mathematical joke.
In case you desire a serious headache, go read that file.
It is about time for a new post on 3D numbers, circular and complex. In this post I write the sphere-cone equation in a matrix notation so see the previous post on conjugates if you feel confused. The sphere-cone equation gives us two equations, as the name suggests these are a sphere and a cone and on the intersection we find the famous exponential circle.
Beside the sphere-cone equation I also demand that the determinant equals 1, now we have three equations and every intersection of those 3 equations has as it’s solution the exponential circle. Can it become more crazy? Yes because it is possible to factorize the third degree determinant into a linear and a quadratic factor. Those factors must also be 1 and now we have five equations! And since you can pick 10 pairs out of five, we now have 10 ways of solving for the intersection where the exponential circle lives…
It is strange that after all these years it is still easy to find 10 video’s where so called ‘professional math professors’ sing their praise upon the exponential circle in the complex plane. They really go beserk over the fact that e to the power it gives the cosine and sine thing. And after all those years still silent, yeah yeah those hero’s really deserve the title of honorable shithole… It is honorable because they often have relatively large salaries and they are shitholes because of their brave behavior when it comes to 3D complex numbers. Bah, I am getting a bad taste in my mouth when I think about the behavior of professional math professors. Let me stop writing about that low form of life.
This post is 8 pictures long. May be, I have not decided yet, is the next post about parametrizations of the exponential circle. In these 8 pictures I work out the case for the circular multiplication, that is the case where the imaginary unit j behaves like j^3 = 1. At the end I only give the 3D complex version of the matrix form of the sphere-cone equation and the rest you are supposed to do yourself.
Ok, again do not confuse this with quadratic forms. A matrix equation as written above has a real and two imaginary components while quadratic forms are often just real valued.
Let´s try to upload this stuff. See you in the next post.
This is another lightweight easy going summer update. It is about matrix representations and how to find the conjugate of a 3D complex or circular number. I use the case of the complex plane of 2D conplex numbers to show that conjugation is not some silly reflection just always but rather simple will always be the upper row of a proper matrix representation. As a matter of fact it is so easy to understand that even the biggest idiots on this planet could understand it if they wanted. Of course math professors don’t want to understand 3D numbers so also this new school year nothing will happen on that front…
Did you know that math professors study the periodic system? Yes they do, anyway in my home country the Netherlands they do because every year they get a pay rise and that pay rise is called a periodic. And as such they study the periodic system deep and hard…
I classified this post only under the categories ‘3D complex numbers’ and ‘matrix representations’ and left all stuff related to exponential circles out. Yet the exponential circle stuff is interesting; after reading this post try to find out if the numbers alpha (the midpoint of the exponential circles) are symmetrix (yes). And the two numbers tau (the log of the first imaginary unit on the circular and complex 3D space) are anti-symmetrix (yes).
This post is just over 7 pictures long. As the background picture I used the one I crafted for the general theorem of Pythagoras. (Never read that one? Use the search funtion for this website please!) All pictures are of the usual size namely 550×775 pixels.
Ok, that was it for this update. Although it is so very simple (for years I did not want to write of just two simple planes that contain all the self-conjugate numbers) but why make it always so difficult? Come on it is summer time and in the summer almost all things are more important than math. For example goalkeeper cat is far more important compared to those stupid 3D numbers. So finally I repost a video about a cat and that makes me very similar to about 3 billion other people.
This is a lazy easy going summer post, it does not have much mathematical depth. Let’s say the depth of a bird bath. But with most posts I write you also need a lot of knowledge about what was in previous posts and for the average person coming along that is often too time consuming… So we keep it simple today; quadratic forms on 3D space.
If you have had one or two courses of linear algebra you likely have encountered quadratic forms. They are often denoted as Q(X) where the X is a column matrix and the quadratic form is defined as Q(X) = XT A X. Here XT is the transponent of X so that would be a matrix row. As you might guess, the X column matrix contains the variables while the constant square matrix A is the source of coefficients in the quadratic form Q(X). In most literature it is told the matrix A is symmetric, of course there is no reason at all for that; any square matrix will do. On the other hand it is easy to see or to show that if a square matrix is anti-symmetric the corresponding quadratic form will always be zero everywhere.
In this post we will take matrices that are always the matrix representation of 3D complex & circular numbers. Matrix representations are a complete category on this website so if you don’t know them you must look that up first. (Oh oh, here I go again: this was supposed to be easy but now the average reader must first try to understand matrix representations of higher dimensional multiplications…)
Compared to the previous update on the likely failure of all fusion reactors this post is far less dramatic. If in the future I am right and we will never have fusion power, that will be the difference between life and death of hundreds of millions of people in the long run… So in order to be a bit less depressing let’s lift the spirits by a lightweight new post on quadratic forms! Why not enjoy life as long as it lasts?
Ok, the actual post is seven pictures long, all in the usual size of 550×775 pixels.
I have to admit that for me the use of the number alpha was important because that is at the center of the exponential circles in the 3D complex and circular spaces. So I have a legitimate reason to post this also under the category ´exponential circle´. And from the non-bird bath deep math, that is the big math ocean that is very deep, I like to classify as much posts under that category ´exponential circles´.
Ok, let´s leave it with that and try to upload this post. Till updates my dear reader.
It is a bit vague what exactly a multiplication is, but I always use things that ‘rotate over the dimensions’. For example on the 3D complex space the imiginary unit is written as j and the powers of j simply rotate over the dimensions because:
j = (0, 1, 0) j^2 = (0, 0, 1) and j^3 = (-1, 0, 0). Etc, the period becomes 6 in this way because after the sixth power everything repeats.
In this post we will look at a more general formulation of what the third power of j is. The Cayley-Hamilton theorem says that you can write the third power of 3 by 3 matrices always as some linear combination of the lower powers.
That is what we do in this post; we take a look at j^3 = a + bj + cj^2. Here the a, b and c are real numbers. The allowed values that j^3 can take is what I call the ‘parameter space’. This parameter space is rather big, it is almost 3D real space but if you want the 3D Cauchy-Riemann equations to fly it has to be that a is always non zero. There is nothing mysterious about that demand of being non zero: if the constant a = 0, the imaginary unit is no longer invertible and that is the root cause of a whole lot of trouble and we want to avoid that.
It is well known that sir Hamilton tried to find the 3D complex numbers for about a full decade. Because he wanted this 3D complex number space as some extension of the complex plane, he failed in this detail and instead found the quaternions… But if the 3D numbers were some extension of the 2D complex plane, there should be at least one number X in 3D such that it squares to minus one. At the end I give a simple proof why the equation X^2 = -1 cannot be solved in 3D space for all allowed parameters. So although we have a 3D ocean of parameters and as such an infinite amount of different multiplications, none of them contains a number that squares to minus one…
I gave a small theorem covering the impossibility of solving X^2 = -1 a relative harsh name: Trashing the Hamilton approach for 3D complex numbers. This should not be viewed as some emotional statement about the Hamilton guy. It is just what it says: trashing that kind of approach…
This post is 7 pictures long, each of the usual size of 550×775 pixels.
Sorry for the test picture, but the seven jpg pictures refused to upload. And that is strange because they are just seven clean jpg’s. Now it is repaired although I do not understand this strange error.
That is strange, if you don’t know the Cayley-Hamilton theorem; it is the finding that every square matrix A, if you calculate the characteristic polynomial for the matrix A it is always zero. At first this is a very surprising result, but it is easy to prove. It’s importance lies in the fact that in this way you can always break down higher powers of the matrix A in lower powers. In the study of higher dimensional complex and circular numbers we do this all the time. If in 3D space I say that the third power of the imaginary component is minus one, j^3 = -1, we only write the third power as a multiple of the zero’th power…
In this post I will give two simple proofs of the Cayley-Hamilton theorem and although in my brain this is just a one line proof, if you write it down it always gets longer than anticipated.
At the end I show you an old video from the year 1986 from the London Mathematical Society where it is claimed that the CH theorem was neglected for 25 years. Now Hamilton is also famous for having sought the 3D complex numbers for about a full decade before he gave up. And I still do not understand why Hamilton tried this for so long but likely he wanted to include the imaginary unit i from the complex plane in it and that is impossible. Or may be he wanted a 3D complex number system that is also a field (in a field all elements or numbers that are non-zero have an inverse, in algebra wordings; there are no divisors of zero). A 3D field is also impossible and in this post I included a small proof for that.
Furthermore in this post at some point may be you read the words ‘total incompetents’ and ‘local university’. You must not view that as some emotional wording, on the contrary it is a cold clinical description of how math goes over there. So you must not think I am some kind of frustrated person, for me it is enough that I know how for example to craft a 3D complex number system. If they don’t want to do that, be my guest. After all this is a free country and we also have this concept of ‘academic freedom’ where the high shot math professors can do what they want.
And what is this ‘academic freedom’ anyway? If for example unpaired electrons are never magnetically neutral but electron pairs always are magnetically neutral, can the physical reality be that electrons are magnetic dipoles? Of course not, that is a crazy idea to begin with. But 97 years of academic freedom since the Stern-Gerlach experiment have never ever brought any meaningful understanding of the magnetic properties of the electron. If it acts as a magnetic charge and you say it is not a charge it is easy to understand how you can fool yourself for about one century of time.
This post is seven pictures long although the last picture is empty. The two proofs of the Cayley/Hamilton theorem is how I would prove such a thing but good theorems always have many proofs. All pictures are of the size 550×775 pixels.
Here is the old video from 1986 where it is claimed the Cayley-Hamilton theorem was neglected for about 25 years. Oh oh oh what a deep crime. But the human mind is not made to produce or understand math, so in my view 25 years is a short period of time if in the good old days math professors were equally smart as the present day math professors. The title of the video is The Rise and Fall of Matrices.
Ok let me leave it with that an not post a link to the top wiki on the Cayley-Hamilton theorem where all kinds of interesting proofs are given. Till updates my dear reader.