We already have found some parametrizations for the complex exponential (that ellipse, see previous posts below) we do not really need such an identity. But they are always fun to make such identities in a new number system under study like lately those elliptical and hyperbolic multiplications in the plane. Lets recap what we have done all these posts:
1) We looked at the matrix representations for 2D numbers ruled by i^2 = -1 + i. The determinant of such matrices was x^2 + xy + y^2. Therefore we wanted to know more about the ellipse that gives a determinant of 1.
2) We found a way to take the logarithm of the imaginary unit i by taking the integral of the inverse from 1 to i. That is the number tau for this kind of multiplication. As such we had a complex exponential only now it covered that ellipse.
3) After that we had to find parametrizations of the complex exponential, actually we found two of them via very different idea’s. It was left totally unproven that the two were the same although in for example the Desmos graphing package they covered the same ellipse. (But that is not a real math proof of course).
4) That is this post: Penning down the de Moivre identity or formula for this particular kind of 2D multiplication. So we can end the recap here.
The post is seven pictures long and I made them a bit larger this time: 550×1200 pixels. There’s also two additional figures so all in all very much pictures for so little math. I like the end result a lot, ok ok it is not very deep math but it looks damn cute in my opinion. So I hope you will have fun reading it and thinking about the math involved in this post on 2D multiplications.
In the next figure you can see that three of the equations give rise to the same graph in the Desmos package while of course again remarking this is not what a math proof should be… Please ignore the typo (6) in the equation for the blue graph while remarking that Desmos still spits out the correct answer.
That was it for this post. May be the next post is on the conjugate or may be a post on magnetic stuff. I don’t know yet but I do know I want to thank you for your attention.
Some time ago I stopped writing posts about magnetism because the number of such posts would exceede the number of posts on the 3D complex numbers. And that was of couse not the long term strategic goal of this website, so I stopped posting it here. But on the other website I kept on writing small sniplets and what I consider the best two sniplets is now reposted here in a new post on magnetism.
For readers who are new: For the last 9 years I have been trying to figure out if electrons are truly tiny magnets yes or no. About 9 years ago I started to doubt that electrons or electron spin is indeed a bipolar magnet. At the time I tried to explain the results from the Stern-Gerlach experiment and I arrived at the conclusion that very likely electrons were magnetic monopoles. My main argument has been all those years: If electrons are magnetic dipoles, because they are so small they must be neutral under application of (large) magnetic fields.
Since the SG experiment it is know that lone electrons are not magnetically neutral but all and everything observed was always explained by electrons as tiny magnets. Why at the time (1922 and later) they never observed that there are all kinds of problems with electron spin as tiny magnets, is unknown to me. For example the scientists at the time had correspondence between each other and some of those letters literally started with the Gauss law for magnetism and stating that a solution must be found inside the framework of the Gauss law for magnetism…
It never dawned on them that doing science is that you must prove the Gauss law for magnetism does apply for lone or unpaired electrons. But they never did that, no one doubted that magnetism was without magnetic charges and as such even a very small particle like the electron had to be a tiny magnet.
Since last year I often phrase my view on electron magnetism as follows:
The magnetic properties of the electron are just like it’s electric properties: Permanent and Monopole.
It is a bit strange that after 9 years I still have to try and find nice sounding slogans like the above as if I were some marketing bureau.
Anyway one of the big mysteries of the official version of electron spin is that in an electron pair the spins must be opposite. Nobody remarks this is totally crazy because if we allow for that we also give up the observation that opposite charges atract while same charges repel. I made an extra picture for this weird official version of electron spin:
Well take your time to think about it, this is the official version of the electron pair if electrons were tiny magnets. The physics professors never ever mention such details, no you often get a boatload of complicated math but they never ever talk about what anti alignment for tiny magnets actually means.
I also want to remark that journalists never ever ask such questions when they interview physics professors on magnetic related stuff. It’s fucking taxpayer money and we must believe this kind of crap? Well yes, according to Cornell university we must. The next picture is one I actually used on the other website:
VESPR theory. VESPR stands for Valence Electron Shell Pair Repulsion. This theory comes a bit more from the chemical sciences where they try to explain the shapes of the electron clouds of atoms and molecules.
The important detail is that electron pairs are neutral to magnetism and that as such electron pairs around an atomic nucleus repel each other.
If you use the idea that electrons are magnetic monopoles this all is very logical: Coulomb forces pull electrons in and the electrons form pairs because they have opposite monopole magnetic charges.
If you use the idea that electrons are tiny magnets this all is very crazy: Coulomb forces pull electrons in and they only form pairs? Why not form other configurations that are possible with tiny magnets? Why only electron pairs my dear physics professors?
My dear reader you have a brain for yourself so look in the picture below as why this particular atomic nucleus has two electron pairs that repel each other. And don’t mind the female robot or ponder the question as why there are female robots at all… Just think a bit around the nonsense that comes along with electrons being tiny bipolar magnets. Here is the picture as used on the other website:
It’s time to publish this post, thanks for your attention and see you in a next post.
Basically this post is just two parametrizations of an ellipse, so all in all it should be a total cakewalk… So I don’t know why it took me so long to write it, ok ok there are more hobbies as math competing for my time. But all in all for the level of difficulty it took more time as estimated before. In the last post we looked at the number tau that is the logarithm for the imaginary unit i and as such I felt obliged to at least base one of the parametrizations on that. So that will be the first parametrization shown in this post. The second one is a projection of the 3D complex exponential on the xy-plane. So I just left the z-coordinate out and see what kind of ellipse you get when you project the 3D exponential circle on the 2D plane. Acually I did it with the 3D circular multiplication but that makes no difference only the cosines are now more easy to work with. Anyway the surprise was that I got the same ellipse back, so there is clearly a more deeper lying connection between these two spaces (the 3D circular numbers and these 2D complex multiplication defined by i^2 = -1 + i). A part of the story as why there is a connection between these spaces is of course found into looking at their eigenvalues. And they are the same although 3D complex numbers have of course 3 eigenvalues while the 2D numbers have two eigen values. A lot of people have never done the calculation but the complex plane has all kinds of complex numbers z that each have eigenvalues too… Anyway I felt that out of this post otherwise it would just become too long to read because all in all it’s now already 10 images. Seven images with math made with LaTex and three additional figures with sceenshots from the DESMOS graphical package. By the way it has nothing to do with this post but lately I did see a video where a guy claimed he calculated a lot of the Riemann zeta function zero’s with DESMOS. I was like WTF but it is indeed possible, you can only make a finite approximation and the guy used the first 200 terms of the Riemann zeta thing. At this point in time I have no idea what the next post will be about, may be it’s time for a new magnetism post or whatever what. We’ll wait and see, there will always pop something up because otherwise this would not be post number 254 or so. Well here is the stuff, I hope you like it or enjoy it.
That was it for this post, of course one of the reasons to write is that I could now file it under the two categories “3D complex numbers” and “2D multiplications” because we now have some connection going on here. And I also need some more posts related to 3D complex numbers because some time ago I found out that the total number of posts on magnetism would exceed those of the 3D complex numbers.
And we can’t have that of course, the goal of starting this website was to promote 3D complex numbers via offering all kinds of insights of how to look at them. The math professors had a big failure on that because about 150 years since Hamilton they shout that they can’t find the 3D complex numbers. Ok ok, they also want it as a field where any non-zero number is invertible and that shows they just don’t know what they are talking about. The 3D complex numbers are interesting simply because they have all those non-invertible numbers in them.
It is time to split my dear reader so we can both go our own way so I want to thank you for your attention.
This way of doing the complex multiplication keeps on drawing my attention because of the funny property that i^3 = -1. As such it has interesting parallels to the 3D complex number. For example the eigenvalues of this defining imaginary unit is the third root out of -1 (and it’s conjugate). That is in line with the results from the 3D numbers although over there 3D numbers have 3 eigenvalues and not 2.
In this post I want to show you a way to find the logarithm of this imaginary unit i via integrating the inverse from 1 to i. Just like on the real line if you integrate 1/x from 1 to say some positive a, you get log a. It is important to remark there are more methods to find such logarithms. For example you can diagonalize the multiplication and take the log of the eigenvalues and as such you can find the log of the imaginary unit.
Anyway back in the time I did craft my first complex exponential for the 3D complex numbers this way (using the integral of the inverse) so for me it is a bit of a walk down memory lane. You always get integrals that are hard to crack but if you use the WolframAlpha website it’s easy to find. Remarkably enough the two values for the integrals we will find below are also found in 3D and even the 6D complex numbers. So for me that was something new.
For myself speaking I loved the way the inverse of a complex number based on i^2 = -1 + i looks. You have to divide by the determinant once more proving that norms do not have very much to do with it. (In standard lessons on complex numbers it is always told that the norm of the product is the product of the norms, but that’s only so for the complex plane and the quaternions. So if you keep on trying such idea’s you won’t come very far…)
This post is five pictures long, lets go:
Ok, that was it more or less for this post. Since we are now getting more and more posts on two dimensional complex (and split complex) numbers may be I will open a new category for those posts. On the other hand you must not open a new category every time you things that are a bit different from what you usually do…
Last year the Nobel prize in physics went to a bunch of people that did experiments that gave rise to a so called violation of the Bell inequalities. As a consequence we are told we are living in a so called ‘non local’ universe and if you measure the quantum state of a particle here, this particle can be entangled with a particle in another galaxy and instantly that entangled particle will change or jump into another quantum state. So the idea is that entangled particles can influence each other at a speed that is infinite so information travel is faster as the speed of light. Well that is very interesting but when it comes to electrons and their spin state I just don’t buy that kind of crap. Why the Nobel prize committee thinks this is science worthy of their famous prize is unknown to me.
Now what is a ‘violation of the Bell inequalities’? Informally said there is too much correlation observed that cannot be explained by so called ‘hidden variables’ that are unknown. To focus the mind let me give you a simple example:
Two electrons in an electron pair get separated, one electron stays here on earth while the physics professors transport the other electron to another galaxy. It is claimed that if you measure the spin state of the electron here on earth into a particular direction, in that case the spin state of the electron that was transported by the physics professors to another galaxy instantly jumps into the other spin state. And the Nobel prize committee handed out a Nobel prize for that.
Well that is all very interesting but I think electrons carry a monopole magnetic charge and as such it is impossible to flip the spin. Take for example any chemical stuff based on binding via electron pairs, say your own body. Now if we put a magnetic field through your body tissue, does any electron flip it’s spin state? Do half of your electron pairs turn from a binding pair into a non-binding pair? No that never happens, electron spin state is not a fragile thing, it is permanent and cannot be altered.
And that is where all those experiments where they try to violate the Bell inequalities go wrong: They all assume that the photons they produce are coming from an electron that is in a superposition of spin up and spin down. But that is never the case if it is true that electron monopole charge is a permanent feature.
I have a video for you and the preprint pdf from 24 Aug 2015 that was published by the TU Delft group that did this loophole free Bell test. The video is easy and it even has music, what more do you want? The pdf is hard to read and it takes some time to grasp what is going on. I made four pictures from screenshots with a bit comment from me in it. After that the video and the pdf paper.
I do not know if they used the same electrons over and over again. Likely not because in quantum mechanics there is also the idea that if you measure some quantum property of a particle, after that in stays in that quantum state.
If electrons have a permanent magnetic charge, this must have a profound effect on the photons they produce. To be precise, the magnetic phase will be shifted 180 degrees if you compare the two different kind of photons. Without good solid proof I always assume that is why we have left and right circular photons. But information on that important detail is hard to find for years and years.
It has been a while since I updated this website with a post on magnetism. But some months back all of a sudden I would have more posts on magnetism compared to the main category of this website: The 3D complex numbers. So that is why in the last couple of months I did post nothing about magnetism over here but only on the other website.
At the end let me link you to the pdf from the preprint archive
At last I want to remark that even if you think it is very likely there is something wrong with the official version of electron spin (the tiny bipolar magnet model). In that case you must not think that for example the Nobel prize committee will come out saying they were wrong on electron spin in say the next 3 to 5 years. That’s not going to happen and that in itself is an interesting social phenomenon. Well thanks for your attention and see you in the next post.
If you change the way the multiplication in the complex plane works, instead of a unit circle as the complex exponential you get ellipses and hyperbola. In this post I give a few examples, where usually the complex plane is ruled by i^2 = -1 we replace that by i^2 = -1 + i and i^2 = -1 + 3i. In the complex plane the unit circle is often defined as the solution to the complex variable z multiplied against it’s conjugate and then solve where this product is one. There is nothing wrong with that, only it leads to what is often told in class or college and that is: The norm of a product of two complex numbers is the product of the norms. And ok ok, on the complex plane this is true but in all other spaces the I equipped with a multiplication it was never true. It is the determinant that does all the work because after all on the complex plane the determinant of a matrix representation of the complex variable z is x^2 + y^2. (Here as usual z = x + iy for real valued variables x and y.)
Therefore in this post we will solve for det(z) = 1 for the two modified multiplications we will look at. I did choose the two multiplications so that in both cases det(i) = 1. That has the property that if we multiply and z against i, the determinant stays the same; det(iz) = det(z).
I simply name complex z with integer x and y also integers, a more precise name would be Gaussian integers to distinguish them from the integers we use on the real line. Anyway I do not think it is confusing, it is rather logical to expect a point in the plane with integers coordinates to be an integer point or an integer 2D complex number z.
Beside the ellipses and hyperbola defined by det(z) = 1, or course there are many more as for example defined by det(z) = 3. Suppose we have some integer point or z on say det(z) = 3, if we multiply that z by i you stay on that curve. Furthermore such a point iz will always be an integer point to because after all the multiplication of integers is always an integer itself. That is more or less the main result of this post; by multiplication with the modified imaginary unit i you hop through all other integer points of such an ellipse or hyperbole. (By the way I use the word hyperbola to be the plural of hyperbole but I do not know if that is the ‘official’ plural for a hyperbole.)
What I found curious at first is the fact that expressions like z = -3 + 8i can have an integer inverse. But it has it’s own unavoidable logic: The 2×2 matrix representation contains only (real) integers and if the determinant is one, the inverse matrix will have no fractions whatsoever. The same goes for any square matrix with integer entries, if the determinant is one the inverse will also be a matrix with only integer entries.
This post is six pictures long, each size 550×1100 and three additional screen shots where I used the desmos graphics package for drawing ellipses and a hyperbole. At last I want to remark that I estimate these results as shown here are not new, the math community is investigating so called Diophantine equations (those are equations where you look for integer solutions) and as such a lot of people have likely found that there are simple linear relations between those integer solutions. Likely the only thing new here is that I modify the way the complex number i behaves as a square, as far as I know math folks never do that. So let me try to upload the pictures and I hope you have fun reading it.
Ok, that was it for this post. I hope you liked it and learned a bit of math from it. I do not have a good category for 2D numbers so I only file this under ‘matrix representations’ because those determinants do not fall from the sky. And file it under ‘uncategorized’. Thanks for your attention and see you in a new post.
Addendum added 09 Dec 2023: I made a picture for the other website but since I made it already why not hang it in here too? See picture 05 above where we looked at when you get an elliptical multiplication and when the hyperbole version. In the picture below you see a rather weird complex exponential: a straight line. And the powers of i just hop over all those integer values on that line. The multiplication here is defined by i^2 = -1 + 2i. All positive powers hop to the left and upwards, the inverses go the other way. For example the inverse of i equals 2 – i.
Ok, that was it for this post. Thanks for your attention.
Even if you change the multiplication in the plane away from the complex or split complex multiplication it is always easy to find the famous CR-equations. And if you have those in the pocket you can differentiate functions defined on the space you made ‘just like’ on the real line.
After all the CR-equations only need that the numbers commute, you also need to make sure that all basis vectors have an inverse but that are the only restrictions. As far as I know in the math world of the universities the CR-equations are only used in the complex plane, may be some stuff with multiple complex variables and that’s all there is in that part of the math universe.
Originally I wrote the post in just one go and one day later when I read it I was rewarded with a lot of stupid typo’s. Stuff like mindlessly typing x and y where it should have been a and b, also I added the inverse of the imaginary unit because after all I said it was important in the previous post where we looked at CR-equations in the case of a more general n-dimensional space.
An interesting feature of the two dimensional plane is that the two basis vectors 1 and i always commute no matter what you come up with for i^2. I hope the reader is familiar with the fact that on the complex numbers you have the square of i being minus one and plus one for the split complex numbers. In this post we will look at a 2D multiplication that is ruled by i^2 = 1 + i, so you can view this as a minor modification of the split complex numbers.
At first I named the multiplication a ‘strange multiplication’ but one day later I realized that the age old golden ratio has the same property as my imaginary unit i. If you square the golden ratio, that is also the same as the golden ratio plus one. So I renamed it to the golden ratio multiplication. I know it is a little click baity because the golden ratio itself is not used but only the polynomial equation you need to calculate the golden ratio. Universities have their multi-million marketing budgets, still can’t find 3D complex numbers by the way and I have my free tiny click baity golden ratio multiplication. I think it is an allowed sin.
Did you know that if there is something wrong with a product, it always needs massive marketing budgets. Just look at Coca Cola, without the advertisements the stuff should gradually sell less and less because it is not a healthy product. You can say it is an unhealthy product so there is something wrong with it and as such it needs all that marketing stuff in order to survive.
This post is only four pictures long, I hope it is a bit more easy to digest because it is only two dimensions. So lets go to the math in the four pictures:
Ok, that was it for this post. Thanks for your attention.
There are many ways to introduce CR-equations for higher dimensional complex and circular numbers. For example you could remark that if you have a function, say f(X), defined on a higher dimensional number space, it’s Jacobian matrix should nicely follow the matrix representation of that particular higher dimensional number space. I didn’t do that, I tried to formulate in what I name CR-equations chain rule style. A long time ago and I did not remember what text it was but it was an old text from Riemann and it occured he wrote the equations also chain rule style. That was very refreshing to me and it showed also that I am still not 100% crazy…;) Even if you know nothing or almost nothing about say 3D complex numbers and you only have a bit of math knowledge about the complex plane, the way Riemann wrote it is very easy to understand. Say you have a function f(z) defined on the complex plane and as usual we write z = x + iy for the complex number, likely you know that the derivative f'(z) is found by a partial differentiation to the real variable x. But what happens if you take the partial differential to the variable y? That is how Rieman formulated it in that old text: you get f'(z) times i. And that is of course just a simple application of the chain rule that you know from the real line. And that is also the way I mostly wrote it because if you express it only in the diverse partial differentials, that is a lot of work in my Latex math typing environment and for you as a reader it is hard to read and understand what is going on. In the case of 3D complex or circular numbers you already have 9 partial differentials that fall apart into three groups of three differentials each. In this post I tried much more to hang on to how differentiation was orginally formulated, of course I don’t do it in the ways Newton and Leibniz did it with infitesimals and so on but in a good old limit. And in order to formulate it in limits I constantly need to divide by vectors from higher dimensional real spaces like 3D, 4D or now in the general case n-dimensional numbers. That should serve as an antidote to what a lot of math professors think: You cannot divide by a vector. Well may be they can’t but I can and I am very satisfied with it. Apperently for the math professors it is too difficult to define multiplications on higher dimensional spaces that do the trick. (Don’t try to do that with say Clifford algebra’s, they are indeed higher dimensional but as always professional math professors turn the stuff into crap and indeed on Clifford algebra’s you can’t divide most of the time.)
May be I should have given more examples or work them out a bit more but the text was already rather long. It is six pictures and picture size is 550×1100 so that is relatively long but I used a somehow larger font so it should read a bit faster.
Of course the most important feature of the CR-equations is that in case a function defined on a higher dimensional space obeys them, you can differentiate just like you do on the real line. Just like we say that on the complex plane the derivative of f(z) = z^2 is given by f'(z) = 2z. Basically all functions that are analytic on the real line can be expanded into arbitrary dimension, for example the sine and cosine funtions live in every dimension. Not that math professors have only an infitesimal amount of interest into stuff like that, but I like it. Here are the six pictures that compose this post, I hope it is comprihensible enough and more or less typo free:
Ok that was it, thanks for your attention and I hope that in some point in your future life you have some value to this kind of math.
On this entire website when I talked about a matrix representation it was always meant as a representation that mimics the multiplipaction on a particular space as say the 3D complex numbers. And making such matrices has the benefit you can apply all kinds of linear algebra like matrix diagonalization, or finding eigenvalues (the eigenvalue functions) and so on and so on. So the matrix representation was always the representation of a higher dimensional number. Big E is very different, this matrix describes the multiplication itself. As such it contains all possible products of two basis vectors and since this is supposed general theory I wrote it in the form of an nxn matrix. For people who like writing computer code, if you can implement this thing properly you can make all kinds of changes to the multiplication. As a matter of fact you can choose whatever you want the product of two basis vectors to be. So in that sense it is much more general as just the complex or the circular multiplication. I do not like writing computer code myself that much but I can perfectly understand people who like to write that. After all every now and then even I use programs like PARI and without people that like to write code such free programs are just not there. The math in this post is highly descriptive, it is the kind of math that I do not like most of the time but now I finally wrote this matrix down it was fun to do. If you are just interested in some fixed number space as say the 3D or 4D complex numbers, this concept of big E is not very useful. It is handy when you want to compare a lot of different multiplication in the same dimension and as such it could be a tool that comes in handy.
The entries of this matrix big E are the products of two basis vectors so this is very different from your usual matrix that often only contain real numbers or in more advanced cases complex numbers from the complex plane. I think it could lead to some trouble if you try to write code where the matrix entries are vectors, an alternative would be to represent big E as n square nxn matrices but that makes it a bit less overseeable.
May be in linear algebra you have seen so called quadratic forms using a symmetric matrix. It is a way to represent all quadratic polymonials in n variables. Big E looks a lot like that only you now have vectors as entries.
I did choose the number 1 to be the very first basis vector, so that would give the real line in that particular space. Of course one of the interesting details is that all analytic functions that you know from the real line can easily be extended to all other spaces you want. For example the sine and cosine or the exponential function live in all kinds of spaces in all kinds of dimensions. As such it is much much broader as only a sine on the real line and the complex plane. This post is five pictures long each 550×1100 pixels. I made them a bit larger because I use a larger font compared to a lot old posts. There are hardly mathematical results in this post because it is so descriptive. Compare it to the defenition of what a group is without many examples, the definition is often boring to read and only comes alive when you see good examples of the math involved.
If you want to try yourself and do a bit of complex analysis on higher dimensional spaces, ensure your big E matrix is symmetric. In that case the multiplication commutes, that is AB = BA always. If you also ensure all basis vectors are invertible you can find the so called Cauchy-Riemann equations for your particular multiplication. Once you have your set of CR equations you can differentiate all you want and also define line integrals (line integral are actually along a curve but that does not matter). A simple counter example would the 4D quaternions, they do not commute and as such it is not possible to conduct any meaningful complex analysis on the space of quaternions. End of this post and thanks for your attention.
I had a bit of computer problems lately, my oldest computer did not allow me to enter the BIOS and as such I could not tell from what hard disk to boot and that was the end of the motherboard in that old computer. It was my last Windows XP computer and looking at the latest Windows 7, 10 or 11 stuff there is not that much progress. Windows OS systems become more and more a fashion thing because most of it is outdeveloped and hard to improve upon… But lets go to this small post about the long title above: total angular momentum.
To put it simple: In an atom or molecule with unpaired electrons in it physics people often think of the spin of the unpaired electrons as vectors that get projected against the vertical direction if a magnetic field is applied. Those projections are treated as scalars and as you might have guessed it is either +1/2 or -1/2. (Here I leave the constant of Planck out for simplicity.)
Ok my version of magnetism is that electrons are magnetic monopoles so they have either a north pole charge or a south pole magnetic charge. Adding these up works just like adding up the electrical charges in an atom or a molecule; because after the humble opinion of your servant electron magnetism looks just like electron electricity: permanent and monopole. So that is all to know, basically it is very simple once you have seen a few times what they do and mean with that “Total angular momentum in the x-direction” because that sounds so complicated. But all in all the calculations they do are just like adding up monopole magnetic charges so they were on the right track more or less a long time ago. This post is four pictures and for me that is a bit hard because I always made the backgrounds on the computer that broke down. And that old graphics program can’t run on Windows 7 or higher while modern graphics programs like GIMP just don’t have the features to make such a background in a few clicks. Of course a long time ago this doomsday day was foreseen and I saved a lot of backgrounds as png files. So I opened that folder and what? It’s fucking empty…:( Well lets go to the pictures:
The most hilaric picture is the fourth one where two electrons with opposite spins are in different energy levels in an atom or molecule. And if you want your electron spin as a vector, how the hell should you get opposite pointing vectors while the electrons are in different parts of the atom or molecule? That is very weird behavior and as such it always goes unexplained to the extend it is never mentioned as a problem. Just like all that other stuff that is wrong with electron magnetism as you view them as tiny magnets and not as magnetic monopoles.
Well thanks for your attention and see you in another post.