Addition operator

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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.
 

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of course the identity elements are unique and it's simple to prove that they are. just assume that there is another one with the same properties and you'll see that they have to be one and the same.
 

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