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The real number system is a mathematical field. To start with, the Real number system is a Group. This means that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element which is also in the set. The Group satisfies four axioms: closure, associativity, identity and invertibility. In addition, it is a Ring. A ring is an Abelian group (that is, addition is commutative) and it has a second binary operation (multiplication) that is defined on its elements. This second operation is distributive over the first. And finally, a Field is a Ring over which division - by non-zero numbers - is defined. There are several mathematical terms above which have been left undefined to keep the answer to a manageable size. All these algebraic structures are more than a term's worth of studying. You can find out more about them using Wikipedia but be sure to select the hit that has "mathematical" in it!

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Q: What are the rules of the real number system?
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