I just found out about the hyperreals. Here is a visualization of the hyperreals:
The problem I have with this image, is that it conveys a very clear intuition, but often times, those intuitions are very misleading, so I don't know exactly how to interpret this.
My first idea is that the hyperreals are simply the same as $R^3$ (assuming there are is no fourth "level", as in the image here), because when we add $a\omega +b$ to $c\omega + d$, we get $(a+c)\omega + (b+d)$, just as if we added two vectors. If we multiply a hyperreal by a real number, we multiply both components.
It seems like the only addition that the hyperreals have, is an order. specifically it is a lexicographic order.
Am I correct that the hyperreals are simply $R^3$ with a lexicographic order? What's so special about them?
Bonus question: What is the difference between the hyperreals and the surreal numbers?
For bonus question.
The hyperreals (a "baby" name for nonstandard models of $\mathbb R$) have additional information: the transfer principle.
The surreals have addional information: simplicity (a technical notion, some numbers are "simpler" than others).
They are both nonarchimedean ordered fields containing the reals ... and all nonarchimedean ordered fields containing the reals admit structure such as you illustrate in your pictures.
There are others, too. Levi-Civita numbers. And (plug) transseries