I was just thinking about how strange it is that multiplicative inversion bifurcates the rational numbers into such asymmetric segments.
Additive inversion bifurcates the number line cleanly into a left half and a right half, but multiplicative inversion bifurcates it into a finite length segment ($-1$ to $1$) and an infinite length segment (everything else). I understand that it is a paradox of infinity that all multiplicative inverses of elements in an infinite length interval can fit into a finite length interval, but would it have been possible for civilization to have chosen a numeric paradigm where multiplicative inversion was more symmetric, i.e., via some combination of...
- A different total ordering of numbers
- A different number base
- As I understand, some (all?) modular arithmetic systems have the property that their multiplicative inverses are spread across the entire finite range of numbers; perhaps we could start to arrive at an infinite numeric paradigm similar to our own by thinking of the $n$ in a mod-$n$ arithmetic system as a variable and taking its limit as it approaches infinity?
...or is this asymmetry a logically necessary fact of mathematics?
Let $A=(-1,0)\cup (1,\infty)$. Then $\{\,\frac1x\mid x\in A\,\}=\{-x\mid x\in A\,\}$ and all we left out are $\{-1,0,1\}$.