Is there any operation that really exists beyond just a Rotation and a Reflection?

Question

I'm looking for fundamental operations between numbers. Kind of really really essential operations which can be done on $\mathbb N$ and $\mathbb Z$, following Kroenecker motto "Naturals where invented by God, all the rest by humans". But anyway if the argument is insightful you're allowed to extend the field until $\mathbb C$ or whatever exotic field you like.

This thinking was heuristic and so it's not really developed, is just to give you an insight of what I'm interested into. I was thinking about sum (and subtraction); multiplication (and division); power (and extracting radicals). At the end we can morally define all these three operation by starting by addition. So I was thinking "is there, in essence anything else beyond additioning number?"

It's the kind of question that jumps into your mind in the early morning before the first coffee.

I then thought that in fact adding integer numbers together (namely $x+a$) may be thought as a permutation of the elements of $\mathbb Z$ where the permutation is a just and infinite rotation of $a$ elements, which could appear as a shifting right or left depending if you're summing or subtracting.

This made me think that if rotation are represented by sums, then reflections could not be derived as rotation on $\mathbb Z$, so I concluded that maybe beyond "shifting or rotate" $\mathbb Z$ with addition, there also should be a natural operation very basic operation which is "reflecting" $\mathbb Z$ which clearly is a $-1$ multiplication.

Now I'm asking is there anything else really basic I'm missing? Really rotation and reflection are all of what exists? Do you have any insightful picture about it?

Answer 1

Geometrically speaking, treating $R$ as a 1 dimensional geometric space (a line), multiplication furnishes scaling and reflection, and addition furnishes translation.

You can say the same about any subring, but he problem is that such rings describe geometries that aren't so nice.

If you work with just $\Bbb Z$, then you can't scale some points to other points. You can't stretch $3$ to $2$ with multiplication. And, of course, multiplication does not create a permutation of the elements. You need at least a field to do these things, so $\Bbb Q$ is the simplest subring that allows you to scale each nonzero thing to each other nonzero thing.

The only thing about $\Bbb Q$ or any other proper subfield of the reals is that they will contain 'holes' where things are missing. This becomes clearer when you look upward to planes, because there you can see that if the lines aren't as "complete" as the real numbers, some pairs of lines that you thought should intersect don't intersect because their intersection point is absent.

As for 'other operations', you could enforce meaning upon any operation you pick, it just wouldn't be particularly natural. The main one you might not have thought of is inversion $x\mapsto 1/x$ which also requires at least the rational numbers. This is better viewed as a geometric operation on the projective line over the field, so that you can interchange the point at infinity with $0$ using the inversion map, and the thing is really a permutation of the elements.

I think you might find this solution and this solution informative as well.