Category Archives: 2D multiplications

A look at the set of non invertible hyperbolic 2D complex numbers.

2D multiplications, Last Fermat theorem

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.

Integrals for the number tau for the 2D multiplication defined by i^2 = -1 + 3i.

2D multiplications

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”.

Parametrization of the hyperbole where the multiplication is defined by i^2 = -1 + 3i.

2D multiplications

With a new number tau (see previous post) there is always a log of an suitable imaginary unit found, but that does not mean you have a parametrization instantly. And with ‘suitable’ I mean the determinant of the imaginary unit must be one because I always want to make some complex exponential, in this case a complex exponential hyperbole.
The post is relatively short, I always try to write short posts and I always fail fail and fail in that. But now it’s only four pictures long so this comes close to a tiktok version of my ususal math postings. By the way it has nothing to do with this post, but do you like tiktok? I don’t like it very much, it is more for people that have a bit different mindset compared to the way my old brain works…
In this post I first neatly write down the parametrization using the sinh and cosh to express the two coordinate functions. And after that I more or less express it all as much as possible into the two eigenvalues of our beloved imaginary unit i that rules this plane via: i^2 = -1 + 3i.
May be you have never thought about imaginary units as having eigen values themselves. In that case I invite you to calculate the eigenvalues of the ordinary numbers z from the complex plane. You know that plane that is ruled by an imaginary unit i via the rule: i^2 = -1. You will find a very interesting answer and of course after that you wonder WTF are the eigen vectors?

But let us not digress and enjoy the beauty of a complex exponential that is a hyperbole in this case. Here we go:

Figure1: Don’t mind the ‘female robot’ because all female robots are fake.

I am sorry for those bad looking accolades, it is some small fault in the Latex math package or some other old computer feature. You see it in much more texts written in Latex; the { and the } are just not presented properly.
That was it for this post, as always thanks for your attention.

The number tau for the hyperbolic multiplication i^2 = -1 + 3i.

2D multiplications, Matrix representation

Some posts ago I showed you how you can calculate the number tau (always the logarithm of a suitable imaginary unit) using integrals for an elliptic multiplication. To be precise you can integrate the inverse of numbers along a path and that gives you the log. Just like on the real line if you start integrating in 1 and integrate 1/x you will get log(x). If you have read that post you know or remember those integrals look rather scary. And the method of using integrals is in it’s simplest on the 2D plane, in 3D real space those integrals are a lot harder to crack. And if the dimension is beyond 3 it gets worse and worse.
That is why many years ago I developed a method that would always work in all dimensions and that is using matrix diagonalization. If you want the log of an imaginary unit, you can diagonalize it’s matrix representation. And ok ok that too becomes a bit more cumbersome when the dimensions rise. I once calculated the number tau for seven dimensional circular numbers or if you want for 7D split complex numbers. As you might have observed for yourself: For a human such calculations are a pain in the ass because just the tiniest of mistakes lead to the wrong answer. It is just like multiplying two large numbers by hand with paper and pencil, one digit wrong and the whole thing is wrong.
Now we are going to calculate a log in a 2D space so wouldn’t it be handy if at least beforehand we know in what direction this log will go? After all a 2D real space is also known as a plane and in a plane we have vectors and stuff.

So for the very first tme after 12 years of not using it, I decided to include a very simple idea of a guy named Sophus Lie. When back in the year 2012 I decided to pick up my idea’s around higher complex numbers again of course I looked up if I could use anything from the past. And without doubt the math related to Sophus Lie was the most promising one because all other stuff was contaminated by those evil algebra people that at best use the square of an imaginary unit.
But I decided not to do it because yes indeed those Lie groups were smooth so it was related to differentiation but it also had weird stuff like the Lie bracket that I had no use for. Beside that in Lie groups and Lie algebra’s there are no Cauchy-Riemann equations. As such I just could not use it and I decided to go my own way.
Yet in this post I use a simple idea of Sophus Lie: If you differentiate the group at 1, that vector will point into the direction of the logarithm of the imaginary unit. It’s not a very deep math result but it is very helpful. Compare it to a screwdriver, a screwdriver is not a complicated machinery but it can be very useful in case you need to screw some screws…

Anyway for the mulitiplication in the complex plane ruled by
i^2 = -1 + 3i I used the method of matrix diagonalization to get the log of the imaginary unit i. So all in all it is very simple but I needed 8 pictures to pen it all down and also one extra picture know as Figure 1.

Figure1.

That was it for this post, we now have a number tau that is the logarithm of the imaginary unit i that rules the multiplication on this complex plane. The next post is about finding the parametrization for the hyperbole that has a determinant of 1 using this number tau.
As always thanks for your attention and see you in the next post.

Comparison of the conjugate on five different spaces.

2D multiplications, 3D complex numbers, Exponential circle, Exponential curves, Matrix representation

To be a bit precise: I think two spaces are different if they have a different form of multiplication defined on them. Now everybody knows the conjugate, you have some complex number z = x + iy and the conjugate is given by z = x – iy. As such it is very simple to understand; real numbers stay the same under conjugation and if a complex numbers has an imaginary component, that gets flipped in the real axis.

But a long long time ago when I tried to find the conjugate for 3D complex numbers, this simple flip does not work. You only get advanced gibberish so I took a good deep look at it. And I found that the matrix representation of some complex z = x + iy number has an upper row that you can view as the conjugate. So I tried the upper row of my matrices for the 3D complex and circular numbers and voila instead of gibberish for the very first time I found what I at present day name the “Sphere-cone equation”.

I never gave it much thought anymore because it looked like problem solved and this works forever. But a couple of months ago when I discovered those elliptic and hyperbolic versions of 2D numbers, my solution of taking the upper row does not work. It does not work in the sense it produces gibberish so once more I had to find out why I was so utterly stupid one more time. At first I wanted to explain it via exponential curves or as we have them for 2D and 3D complex numbers: a circle that is the complex exponential. And of course what you want in you have some parametrization of that circle, taking the conjugate makes stuff run back in time. Take for example e^it in the standard complex plane where the multiplication is ruled by i^2 = -1. Of course you want the conjugate of
e^it to be e^-it or time running backwards.

But after that it dawned on me there is a more simple explanation that at the same time covers the explanation with complex exponentials (or exponential circles as I name them in low dimensions n = 2, 3). And that more simple thing is that taking the conjugate of any imaginary unit always gives you the inverse of that imaginary unit.

And finding the inverse of imaginary units in low dimensions like 2D or 3D complex numbers is very simple. An important reason as why I look into those elliptic complex 2D numbers lately is the cute fact that if you use the multiplication rule i^2 = -1 + i, in that case the third power is minus one: i^3 = -1. And you do not have to be a genius to find out that the inverse of this imaginary unit i is given by -i^2 .
If you use the idea of the conjugate is the inverse of imaginary units on those elliptic and hyperbolic version of the complex plane, if you multiply z against it’s conjugate you always get the determinant of the matrix representation.
For me this is a small but significant win over the professional math professors who like a broken vinyl record keep on barking out: “The norm of the product is the product of the norms”. Well no no overpaid weirdo’s, it’s always determinants. And because the determinant on the oridinary complex plane is given as x^2 + y^2, that is why the math professors bark their product norm song out for so long.

Anyway because I found this easy way of explaining I was able to cram in five different spaces in just seven images. Now for me it is very easy to jump in my mind from one space to the other but if you are a victim of the evil math professors you only know about the complex plane and may be some quaternion stuff but for the rest you mind is empty. That could cause you having a bit of trouble of jumping between spaces yourself because say 3D circular numbers are not something on the forefront of your brain tissue, in that case only look at what you understand and build upon that.

All that’s left for me to do is to hang in the seven images that make up the math kernel of this post. I made them a tiny bit higher this time, the sizes are 550×1250. A graph of the hyperbolic version of the complex exponential can be found at the seventh image. Have fun reading it and let me hope that you, just like me, have learned a bit from this conjugate stuff.
The picture text already starts wrong: It’s five spaces, not four…

At last I want to remark that the 2D hyperbolic complex numbers are beautiful to see. But why should that be a complex exponential while the split complex numbers from the overpaid math professors does not have a complex exponential?
Well that is because the determinant of the imaginary unit must be +1 and not -1 like we have for those split complex numbers from the overpaid math professors. Lets leave it with that and may I thank you for your attention if you are still awake by now.

A de Moivre identiy for the i^2 = -1 + i multiplication.

2D multiplications

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.

Often when found for the first time, the math was formulated very differently.

Here is a link to a more general wiki upon Abraham de Moivre:
https://en.wikipedia.org/wiki/Abraham_de_Moivre.

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.

Once more: This is not a proof but you can use it for a bit of relief that you’re on the right road.

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.

Two parametrizations for the ‘unit’ ellipse in the i^2 = -1 + i kind of multiplication.

2D multiplications, 3D complex numbers, Exponential circle, Exponential curves

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.

Figure 1: This parametrization is based on the number tau.
Figure 2: The projection in red, stuff without 1/3 and 2/3 in blue.
Figure 3: The end should read (t – 1.5) but I was to lazy to repair 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.

An inverse and a number tau for the i^2 = -1 + i multiplication.

2D multiplications

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:

May be this parametrization is the next post.

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…

2D elliptical and hyperbolic multiplications.

2D multiplications, Matrix representation

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.

Funny detail: i^3 = -1.

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.

Who would have thought that a complex exponential can be a line?

Ok, that was it for this post. Thanks for your attention.

Cauchy-Riemann equations for a ‘golden ratio’ 2D multiplication.

2D multiplications, CR equations

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.