This is a horrible simple post, after all for the complex numbers ruled by i^2 = -1 + 3i all you have to do is look where the determinant of the matrix representaitons equals zero. Well yes that is what we do, at present day there is that cute package or applet (in the past that was a so called computer program, why is all that kind of stuff an “applet” nowadays?) called DESMOS. With DESMOS the two lines that make up the set of non-invertibles is easy to graph. These two lines are interesting because they are the asymptotes of all those hyperboles in this space, anyway those hyperboles that can be written as det(z) = constant. Furthermore the two lines where det(z) = 0 separate the parts of this complex plane where det(z) > 0 from the parts where det(z) < 0.
I took the opportunity to introduce a more comprehensive notation to denote such spaces of 2D complex and split complex numbers. I wrote it much more as a set like in set theory and it includes the ‘rule’ for the imaginary component.
For readers who are new to this website and don’t have a clue what ‘hyperbolic’ or ‘elliptical’ 2D complex numbers are, it is all basically rather simple: These numbers are complex because i^2 = -1 + something, in the case of this post it is i^2 = -1 + 3i. These 2D complex numbers have matrix representations and the determinant of these matrices are constant along certain hyperboles. The case det(z) = 1 is very interesting for any of such a complex plane because that is a multiplicative group. Just like the unit circle in the standard or ordinary complex plane is a multiplicative group.
All in all this post has five pictures of size 550×1200 pixels and an additional two figures from graphs from the DESMOS applet.
For me it was funny to write some ‘high school math’ with just a few parabole kind of stuff in it. During the writing of this post I came across the idea of making a matrix representation of all that conjugating stuff, as such I found a beautiful but still extremely simple way to find the direction of the number tau in a particular space. The great thing is that this time it works in all dimensions so not only the 2D complex numbers but likely much much more. Lets try to upload my post to the internet and may I thank you for your attention.
Yes yes I know we already calculated the number tau for this space equipped with a hyperbolic multiplication. (That was a few posts back using matrix diagonalization.) But I had a few reasons to write this anyway, one reason was just curiosity. I wanted to know how those integrals looked and since we had calculated the number tau anyway we did not need to solve these integrals with pencil and paper. I also wanted you to show how you can write the product of such a complex number z against it’s conjugate. On the standard complex plane this defines a circle and on our hyperbolic space it is of course a hyperbole. At last I wanted to pen down the formula for finding the inverse on this particular hyperbolic complex number space. It looks just like the way this is done on the ordinary complex plane with the exception that if you calculate it the conjugate is a bit different. For me it is funny that we have exactly the same looking formula for the calculation of inverses. All in all it shows that the fixation the professional math professors have on all that “The norm of the product is the product of the norms” kind of stuff is only true because on the standard complex plane the determinant equals the square of the norm of a complex number z. In our present case of hyperbolic complex numbers we devide the conjugate by the determinant and those determinant define hyperboles and not circles. So nothing of that “The norm of the product is the product of the norms” kind of stuff. The deeper underlying mechanism is just always the determinant of the matrix representation.
It has to be remarked however that the study of normed spaces is important in itself and also in practice: If you can find a good norm for some difficult problem like the successive aproximations in say differential equations and you can prove using that norm the stuff converges, that is BINGO of course. Yet a norm is only a tool and not all there is inside that strange space known as human math.
The post itself is 6 images long, in it I have two (pairs of) integrals going from 1 to i. The integrals are of course the inverse of a complex number because the derivative of the logarithm is the inverse and we want to know the log of i because by definition that is the number tau.
I included a so called Figure 1 that show the evaluation of these integrals by the Wolfram package for definite integrals, it’s a handy online tool in case you don’t want to evaluate your integrals with pencil and paper. The last image is from the other website where once more I want to insult the math professors just a tiny tiny bit by using the standard formula of finding the inverso on the standard complex plane. So all in all this post is 8 pictures long.
That was it more or less for this post, I hope you are a bit more confident by now that you can actually integrate spaces like this more or less just like you do in the standard complex plane. The last image is from the other website, it contains a female robot. Why there are female robots is unknow to me, after all with other tools you never have females like have you ever heard of a female screw driver or a female mobile phone?
Ok, that was it for this post. Thanks for your attention and just like the female robot look up into the light and start thinking about the wisdom behind “The norm of the product is the product of the norms”.
This is now year nine or may be the tenth year that I started doubting that electrons were tiny bipolar magnets because it makes much more sense that they are magnetic monopoles. Over the years I have found out that logic just does not work and given the fact that physics people get a salery from tax payer money, that is weird behavior. But physics professors behave just like math professors who after 33 years of doing just nothing will keep on doing that and never ever talk about the three dimensional complex numbers. What explains that kind of behavior, after all it’s all tax payer money so they should be a bit more humble don’t you think? The way I see it is that university people like math and physics professors are some elite. And I don’t mean an elite in the sense they are the very best at their science, no it’s just a collection of overpaid snobs. You must not think I am emotional or so by using the word snobs, no it’s a cold hearted classification of their behavior. It is now 102 years since the original Stern Gerlach experiment and there is boatloads and boatloads of theory of how electrons should behave in case such an experiment is repeated (that is a squence of those magnetic fields) and it is easy to understand the very first experimental physics human that would do such a sequential SG experiment would likely be rewarded a Nobel prize. And in the physics community the Noble prize is what they all dream of. So in a century of time without doubt on many occasions such an attempt must have been undertaken. But there is no trace of any such experiment in the literature, the only experiment that was done was by Frisch and Segrè where they tried to flip the electron spin and that all failed big time. But when building their experimental setup Frisch and Segre got advice from Albert Einstein and likely because of that they got their (non) results published and as such we can find it back in the present day literature.
Now why should a succesful sequential Stern Gerlach experiment lead to an almost 100% probability of getting a Nobel prize? That is easy to explain: It would validate in a deep manner that quantum states like spin states are probabilistic in nature and as such would be a fundamental thing in say all the present day attempts there are in building quantum computers.
Another way of understanding there are just no successful sequential Stern Gerlach experiments done in the last 100 years is simple to do: Go to Youtube and search for it, all you find is animations that explain how it “should work”. But none of those videos give a hint of an experiment actually done…
Is it true there are no Nobel prizes rewarded in the last 100 years related to a repeated or sequential SG experiment? Well in this year 2024 the Nobel prize committee has a website and guess what? They have a search applet for their very website. If you search for “Stern Gerlach” you get something like 12 results and if you serach for “Stern Gerlach experiment” you only get 6 results. None of those results says anything about experimental validation of all that spinor crap or anything that shows you can actually flip the magnetic spin of an electron. I made a picture for the other website as you can see below:
If you want you can go to the website of the Nobel prize committee and look for yourself if you can find such a prize rewarded. Here is the link: https://www.nobelprize.org/. It’s all a big bunch of crap: Electrons are not tiny magnets, they carry magnetic charge just like they carry electric charge.
I am very well aware that logic does not work, but say to yourself about the crap of the electron pair they have over there in the physics community: The Pauli exclusion principle says that those electrons must have opposite spins so what does that mean if it is true that electrons are tiny magnets? Well if they have anti-parallel or opposite spins, doesn’t it look like this:
But again logic does not work so I do not expect that in this year 2024 the physics people will stop talking their usual bullshit. No way, after all as a social community they are just another bunch of overpaid snobs…
After having said that, after about only one century of time there is only recently an English translation made of the publication of the original Stern Gerlach experiment. The translation is done by Martin Bauer and here is a link to the pdf as you can find it on the preprint archive
