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 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