At the very beginning of my Analysis book, the axioms for the real number system are given, beginning with the addition axioms. The sum of x and y is commutative, and associative. There exists a zero identity element, but it isn't claimed to be unique. And there exists a negative element, so the sum of x and it's negative is the zero identity element. Any operation which can satisfy these properties can be called addition, and the objects it operates on can potentially be called real numbers, right? The notion of integers and counting hasn't entered into anything yet. Likewise multiplication has axioms of commutativity, associativity, an identity element different from the additive identity element, but not claimed to be unique. There's distributivity over addition. Then there's reciprocity, for all elements except the addition identity. I hate exceptions! Is the exclusion absolutely necessary, or is it possible to have some kind of number system where the additive identity element can have a multiplicative reciprocal? On page 4 in Shilov's book, it says that the distributive property connects the operation of multiplication with addition, but isn't the distinctiveness of the two operator's identity elements and the reciprocal exclusion of zero a more forceful connection? Perhaps it's the same thing.