Monthly Archives: August 2026

Two times the number tau for the 2D elliptic complex numbers.

2D multiplications, Exponential circle, Exponential curves, Integration, Uncategorized

Throughout this entire website numbers tau are always the logarithm of a suitable imaginary unit. Of course this depends on the kind of space you are in, the higher the dimension the more difficult they become to calculate. Here is the dimension as low as possible with only 2 dimensions, in this post and all other posts on the elliptics they are always 2D and in this post I only look at the imaginary unit i and the multiplication they define are as usual:
i^2 = -1 + i, this gives the elliptic complex numbers. And if we do
i^2 = -1 – i that gives the split elliptic numbers.

Try it yourself: The third power of these imaginary units are -1 in the first case and the third power is +1 for the split elliptic numbers. That is why I name those the split complex numbers, just like there are ordinary complex and split complex numbers are defined by i^2 = +1.

The basic idea behind this post is that the derivative of the log equals the inverse and the other way around: integration of the inverse must yield the logarithm. Of course outside the real line complex exponentials can have periods and sometimes people use that to say the log is a multi valued function. That is also why people often leave a cut out of the complex plane, for example the complex plane without the negative real numbers. Now the log is not multi valued. Well I skip all such considerations and beside that in this post we only calculate a number and we do not study the log in itself.

So integration of the inverse in a 2D number system, that’s basically what this post is. It is more or less the royal road paved with silver and gold that leads to defining complex exponentials while as we will see in the next post that if we want to parametrize these complex exponentials we don’t need knowledge about the actual number tau at all.

This post is seven images in itself and one addendum so 8 images in total.

That was it for this post, thanks for your attention and I hope you liked it a bit. Or may be you didn’t vomit at all and that would also be a great thing, that would still be progress although may be you didn’t like it like I hoped for… Enough of this silly talk, let me upload the whole bunch and see you in a future post.

Two versions of the 2D elliptic complex numbers.

2D multiplications, Matrix representation

The main reason for me writing this post is once more showing you how to take a conjugate when it is not as simple as “A reflection into the x-axis”. I am now in my sixties when it comes to my age and to be honest in my entire life I have never seen just one person doing it right. All they do is flip it into the x-axis and when their calculations derail they just throw away the whole thing and never think twice about what they are doing.

A better way of formulating a conjugate is the next: Replace all imaginary units by their inverse. And yes, in case of the standard 2D complex plane that is a reflection in the x-axis. But that does not mean that in every space where a conjugate is a meaningful thing it will be a reflection. In the case of this post it surely is not a reflection and likely that is the reason that these kind of elliptic complex numbers are completely unknown to the professional math profession. In that regard it is the same as say the 3D complex numbers; also completely unknown because they are to stupid to adjust the math when that’s needed. In some highly autistic version of reality the conjugate must be a flip in the real axis and these people are totally blind for what they do wrong.

This post is five images long and an extra addendum with the two graphs of the ellipses that form the complex exponential on these two relatively simple spaces. May be the next post is about finding those famous numbers tau, on this entire website a number tau is always the logarithm of a suitable imaginary unit. Because a 2D number space is very simple we have only one imaginary unit of course written as i and tau is the log of i.

The direction of the tangents in the point z = 1 is the direction of the number tau. That is a simple yet effective idea from Sophus Lie who a long long time ago invented those things we now know as Lie algebra’s and Lie groups. So on the one hand if you already know what tau is, all you do is differentiate the complex exponential. And if you haven’t found the precise value for tau you can always use implicit differentiation on the equations that define the complex exponential, in this case the equations for the two ellipses. You can use that for checking your calculations for the number tau because those calculations have a strong tendency to run if the rails but if you get the wrong direction you now know there must be a calculation error somewhere…

Ok that was it for this post and this is the 300-th post on this website. As you might have noticed I don’t write more or less two posts a months, only about one. With 300 posts on higher dimensional number systems I have done enough and there is no use in repeating things over and over again. That was it for this post, as always thanks for your attention.