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?