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.

Maths

Axioms.

Tell me more about these "axioms," as you call them.

>>1
1) Z, Q, R, and C are all defined in terms of the natural numbers, but the prime ring of any characteristic zero ring "is" Z also.

And there isn't a ring structure on the set of graphs. (that I know of)  Even if there was, it might not be compatible with the ring structure on any of those other rings. (i.e. you couldn't embed the set of graphs as a subset of C to identify a graph as a number)

I've been working with a hypothetical King of Math in my head

Please report to the nearest free clinic for your prescription for antipsychotics.  Mention this post and recieve your second month free!

>>5
I don't know much algebra so I'll just tell you what I got from that: a number is an element of any ring which contains Z as a subset (and whose rules for addition and multiplication are identical to that of Z for that subset?). I have yet to parse "the prime ring of any characteristic zero ring 'is' Z also." Wikipedia is little help.

Also, you seem to be implying that Z, Q, R, and C are not merely sets of numbers but instead those sets and the operations (+, x). Is this the correct way of looking at that?

Technically Z is just a set and (Z,+,x) is a ring, but it's customary to just write Z when it's implied which operations are meant.

A ring doesn't necessarily contain Z, Z/nZ (n =/= 0) is an example of this.

Making definitions in math is completely decentralized. There are conventions and traditions but you can actually make up new names and notations for every concept you use. For instance there are many notations for the derivative of a function f : f', d/dx(f), Df etc.

I think the closest you'll get to a "grand unified theory" of things that behave like numbers is commutative algebra, the study of commutative rings with unit and their modules. I'm sure wikipedia can help you out here.

>>6
Sorry, I was in a hurry and didn't explain things very well I guess.

a number is an element of any ring which contains Z as a subset.
Well, not quite...  I would say that something could be interpreted as being a "number" (in the sense I think you want) if it is a member of a ring which can be embedded (injective homomorphism) into the complex numbers.

the prime ring of any characteristic zero ring 'is' Z also.

The prime ring is the subring generated by 1 (the mult. identity of the ring).  So if the ring is characteristic 0, the prime ring is the set {...,-2,-1,0,1,2,...}, which is just Z.  That doesn't mean the whole ring can be embedded into C, though.  For instance the ring of polynomials with complex coefficients can't (I'm reasonably sure) be embedded into C.

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?").

>>9
Well whether or not you label something a "number" isn't really all that important.  One ring is just as good as the next, whether or not you can embed it in C.  AFAIK there isn't really a strict definition anywhere of what is or isn't a "number".

In "Number Theory", a "number" doesn't necessarily have to be a complex number, either.  Given a prime number p, you can put a different ("p-adic") topology on the rationals besides the usual one based on that prime, and construct the completion of Q with respect to that topology.  The field you get is called the "p-adic numbers" and can't be viewed as a subfield of C.