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