Some questions to Nicolas: it seems from your proposed definition  that you do not suggest that , because if you did, you would not need to introduce Q. Is this so? Then I find it natural that also  be different from zero. (I think in terms of a free abelian group generated by the real numbers and divided by some subgroup of elements that should be equal to 1, if that helps.) But I am puzzled by your axioms. What about the following calculation: ? I am aware that I use a bit more than the axioms in your list, but it seems from your motivation that you accept implicitly rules like associativity and inverses of each element. I would be interested in knowing exactly at which step of my calculation I violate your intuition. Best,