TITLE: Can numbers exist outside of the number space?
QUESTION [10 upvotes]: I was unsure about how to actually write the question. 
I know that in the real number axis we have the naturals, integers, rationals, irrationals and reals; and in the imaginary number axis we have the imaginary numbers, together forming the complex number plane. I also know that adding dimensions to that plane we can have numbers of dimension $2^n$ (quaternions, octonions, etc). 
What I want to know is: Are more numbers outside of that? Has a type of numbers ever been created from scratch just to satisfy or demonstrate a given property, even if they are not used outside of that? Are there numbers that are not numbers by the previous definition?
Any reference or links to lists of number types other than those I have mentioned would be greatly appreciated.

REPLY [3 votes]: Are [there] more numbers outside of [real division algebras]?

Nowadays, mathematicians don't really speak of "numbers", but of algebraic structures, non-empty sets equipped with one or more binary operations satisfying axioms. The binary operations are typically viewed as generalizing "addition" and "multiplication"; the axioms include the associative axiom, the existence of a neutral element and possibly of inverses, the commutative axiom, a distributive axiom for multiplication with respect to addition, and so forth.
Depending on the axioms satisfied, the resulting structure (non-empty set plus one or more operations) may be called a group, a ring, a field, or any of a plethora of similar terms.
If a collection of axioms is fixed, one can attempt to classify (up to isomorphism) the structures satisfying the axioms. Only relatively rarely is there a structure characterized completely by abstract properties. One famous example is the ordered field of real numbers, which is the unique complete, ordered field.
There are more types of algebraic structure, and more examples of such structures, than you can shake a stick at.