TITLE: What is a real number (also rational, decimal, integer, natural, cardinal, ordinal...)?
QUESTION [19 upvotes]: In mathematics, there seem to be a lot of different types of numbers. What exactly are:

Real numbers
Integers
Rational numbers
Decimals
Complex numbers
Natural numbers
Cardinals
Ordinals

And as workmad3 points out, some more advanced types of numbers (I'd never heard of)

Hyper-reals
Quaternions
Imaginary numbers

Are there any other types of classifications of a number I missed?

REPLY [3 votes]: You tossed out a ton of names associated with number systems, and I thought you might like to know this:

With the exception of the natural numbers, ordinals and cardinals, all of the things in your post are examples of rings. Rings are a type of algebraic object that gather up most of the important properties of number systems.

By "important properties" I'm thinking of addition, subtraction, multiplication and distributivity. It turns out we can do quite well without division and commutativity :)
Rather than thinking of number systems as a menagerie of names like the one in your post, this might help you get a handle on most of them all at once. (Actually after the next paragraph we'll be able to pick the natural numbers back up.)
What are the technicalities keeping the three exceptions I mentioned from being rings? For one thing, the natural numbers, cardinals and ordinals don't really have any appropriate notion of subtraction which is part of the definition of rings. Secondly, all rings are supposed to be made up of a set of points, and while the natural numbers are a set of points, the cardinals and ordinals are not.
The natural numbers actually form a semiring, which is basically a ring that might not have subtraction. In this sense they are just a number system that is not quite as nice as a ring.
The cardinals and ordinals have addition and multiplication operations too which might qualify them to be semirings if they weren't so darn big (being a set is part of the definition of a semiring, too). But if you were willing to accept a semiring-that-is-a-proper-class then you would have a type of object which encompasses all the examples you gave.