Numbers

So, um, is there some grand unified theory of numbers? What makes an integer, a fraction, a real, and a complex number all the same thing? What makes, say, a graph, not one of these things?

Also could you tell me how these definitions get decided, because I've been working with a hypothetical King of Math in my head and it just doesn't feel right. He's kind of a jerk.

Ok cool, thanks for answering my question. The king seems pretty happy right now, so I'm glad.

To be honest, though, I was hoping for something a little more...axiomatic. Having all numbers partially dependent on the complex numbers is kind of like basing all numbers on the rationals. That got overturned because the limitations of the rationals made certain lengths immeasurable. The Banach-Tarski paradox established the existence of immeasurable volumes. Why can't the same thing happen all over again? (I know very little about the Banach-Tarski paradox so I apologize for the ignorance that line of thought undoubtably reveals, its just the similarity of the two situations struck me and was in fact what prompted me to ask the question "What is a number?").