What is a real number (also rational, decimal, integer, natural, cardinal, ordinal...)? 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?
 A: Real numbers
Real numbers are any numbers you can locate (even approximately) on an infinite number line.  This is a theoretical number line with infinite "resolution" that extends infinitely in both positive and negative directions.

One neat property about real numbers is that they are orderable -- that is, given any two real numbers, you can tell which one is "higher" and which one is "lower" than the other.
Real numbers are closed under multiplication, addition, and subtraction.  That is, if you perform any of these operations on two real numbers, their result will always be real as well.  They are almost closed under division, except for the whole divide-by-zero issue.
Not real numbers:


*

*infinity

*the square root of -1

*1/0
Decimals
There isn't a rigorous definition of "decimals", because depending on where you use it, you'll get different definitions.
In the elementary sense, it means any number that has a "decimal part"; or a part after the radix (decimal point, etc.).
In a more advance sense, it means any number written in Base 10.
Natural numbers
Natural numbers are often also called "counting numbers", because they are the numbers you count with.  (0,) 1, 2, 3, 4, etc.
There is some disagreement in the mathematics community over whether or not 0 is a natural number.
Cardinals
In linguistics, this means the natural "numbers" themselves (1, 2, 3, etc.)  But you probably don't want to know about linguistics.
In Set Theory, two sets have the same cardinality if each element could be paired up with an element of the other set.
{1,2,3} and {4,5,6} share the same cardinality because you can pair up 1&4, 2&5, 3&6.
Ordinals
In linguistics, this means 1st, 2nd, 3rd, etc.  But you probably don't want to know about linguistics.
In Set Theory, an ordinal is a well-ordered set.
A: Going closer to the foundations of mathematics than most ever need or are comfortable with, we can also write down the classic model of $\mathbb{N_0}$ in ZF. This is of course going to be very bird's eye view and rather devoid of details.
"There exists a set, $I$, such that $\emptyset \in I$ and $x \in I \Rightarrow x \cup \{x\} \in I$." This is the axiom of infinity. We've the chosen the letter "I" to represent it since it is infinite and we can do induction over $I$ (thus sets satisfying this axiom are also called "inductive sets").
Then we construct the ordinals starting with $0 = \emptyset$, $1 = 0 \cup \{0\} = \{\emptyset, \{\emptyset\}\}, 2 = 1 \cup \{1\} \ldots$ Let us not discuss limit ordinals or why "the ordinals" is not a set - it is all rather complicated. Instead let me point out that from the existense of $I$ it's possible to show the existense of the smallest infinite ordinal, $\omega$, the first ordinal with infitely many preceding ordinals. Then we define $\mathbb{N}_0$ to be $\omega$ along with this "simple" definition of sum: $x+0 = x$, $x + 1 = x \cup \{x\}$ and $x + (y \cup \{y\}) = x+y \cup \{x+y\}$. 
The next couple of steps are easy if you're familiar with equivalence classes.
$\mathbb{Z} = \mathbb{N_0}\times\mathbb{N_0}/\sim$, where $(n,m) \sim (n',m')$ iff $(n+m') = (m+n')$. E.g. the integer -2 is the equivalence class consisting of $\{(0,2),(1,3),(2,4),\ldots\}$. Then you show that there is a natural inclusion $\mathbb{N_0} \hookrightarrow \mathbb{Z}$ so it makes sense to write things like $0 \in \mathbb{Z}$ even though we technically defined 0 to be $\emptyset$ and $\emptyset \notin \mathbb{Z}$.
$\mathbb{Q} = \mathbb{Z}\times\mathbb{Z}\setminus\{0\}/\sim$, where $(a,b) \sim (c,d)$ iff $ad=bc$. Again we have natural inclusions $\mathbb{N}_0 \hookrightarrow \mathbb{Q}$ and $\mathbb{Z} \hookrightarrow \mathbb{Q}$ and each equivalence class is one rational number, i.e. a fraction. E.g. $\tfrac{2}{3}$ is the set $\{(2,3),(4,6),(6,9),\ldots\}$.
When we want the real numbers it gets hard. Typically in a first course in analysis you would give the students the reals in the form of an extra axiom:
There exists a field, $\mathbb{R}$, together with an order ("inequality sign") $\leq$ such that:
1) $\mathbb{R}$ is an ordered field, and
2) Every subset of $\mathbb{R}$ that has an upper bound has a least upper bound.
Why don't you start from scratch? Well... first of all there's the issue of time, but also that of mathematical maturity. Any construction of $\mathbb{R}$ is going to require quite a bit of machinery. Some algebra, some topology and being able to manipulate things like "sequences of equivalence classes of sequences of rational numbers" without getting lost. Even formulating the notion of uniqueness of the real numbers can be a bit tricky. We want not just that any two fields satisfying 1) and 2) to be isomorphic (both as fields and w.r.t. order), but also that the isomorphism should be unique. 
So what different constructions are there? Quite a lot actually. My 3 personal favourites are:
A) Dedekind cuts. Based on the "obvious" observation that a real number is uniquely determined by which rational numbers it is smaller than and which it is larger than. Pros: Easy to prove that this construction has the least upper bound property. Difficulties: It has to be a field, so we need to define multiplication and division. How do you divide subsets of rational numbers? Especially when the divisor contains $0$?
B) Equivalence classes of rational cauchy sequences. Based on the observation that the only way a rational cauchy sequence can not converge is if the limit is somehow "not there". So we just set the limit to be the sequence itself! This is of course nonsense, but it can be made precise. Pros: How easy it is to show it's a field: The set of rational cauchy sequences under coordinate-wise addition and multiplication is a ring and the rational sequences that converge to $0$ are a maximal ideal! Difficulties: Defining the order (note that $(\tfrac{1}{n})$ belongs to the same equivalence class as $0$) and then showing that the order is in fact total. Keeping track of sequences of equivalence classes of rational cauchy sequences when you're trying to prove that your field is complete can be a bit tricky.
C) "Almost-homomorphisms" of $\mathbb{Z}$. Based on the observation (at least since we got pixel-based displays) that a real number $\alpha$ is in some sense the same as choosing a rule for how to draw a line with slope $\alpha$ on an infinite ($\mathbb{Z}\times\mathbb{Z}$) computer display. Pros and cons? Not sure actually. It could be a fun little project to work out the details I guess.
A: The natural numbers are defined by the Peano axioms, as in the answer of Isaac. 
You can also view the natural numbers as the cardinalities of finite sets, which implies that zero is a natural number.
Now the other number domains arise because mathematicians want to give values for certain operations which otherwise are only defined partially. Alternatively, they want to give solutions to equations of certain forms, which cannot always be solved in the smaller domain.


*

*integers arise from subtraction (solutions of a + x = b)

*rational numbers arise from division (solutions of a * x = b)

*real numbers arise from taking limits or upper bounds

*complex numbers arise from taking roots (solutions of polynomial equations)

*cardinal numbers arise as the sizes of sets. They can be constructed as isomorphism classes of sets, e.g. the number one would then be the class of all sets containing one element. They are useful for talking about the different sizes of infinite sets, which can get extremely large. Some set theorists compete over who can define the largest cardinal.

*ordinal numbers arise as the sizes of well-orderings. They can be constructed as isomorphism classes of well-orderings, e.g. the number two would then be the class of all well-orderings containing two elements (with the first smaller than the second in the ordering). Interestingly, the ordinals are themselves well-ordered in a single well-ordering, so they come in a well-defined sequence, and an ordinal number can be viewed as describing its position in this sequence.
A: You might as well start at the 'lowest':


*

*Integers are all the whole numbers,

*Natural numbers (N) are the set of positive integers, {1, 2, ...} (0 optional),

*Rational numbers (Q) are any number that can be represented a/b with a and b being Integers (|b| < 0.

*Real numbers are all the Rational numbers and all the others.

*Cardinality is the number of elements in a set.

*An Ordinal is a well ordered set.

A: Ordinals aren't simply sets well-ordered by the membership relation, but they're also transitive i.e. such a relation is transitive on them ( meaning that  $y$ is transitive iff $ x \in y \Rightarrow x \subset y$ )
A: 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.
A: The natural numbers can be defined by Peano's Axioms (sometimes called the Peano Postulates):


*

*Zero is a number.

*If n is a number, the successor of n is a number.

*zero is not the successor of a number.

*Two numbers of which the successors are equal are themselves equal.

*(induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.


(This definition includes 0 in the natural numbers; altering rules 1, 3, and 5 to refer to one instead of zero excludes 0 from the natural numbers.  Whether or not 0 is a natural number varies in various texts.)
The whole numbers are the natural numbers with the additive identity element called 0.
The integers are the whole numbers and their additive inverses.
The rational numbers are numbers that can be expressed as a ratio of an integer to a non-zero integer.
The real numbers are the set of numbers that are limits of Cauchy sequences of rational numbers.
The irrational numbers are the real numbers that are not rational numbers.
The complex numbers are the numbers that can be expressed as a + b * i where a and b are real numbers and i behaves like a real number under addition/multiplication/distribution/etc., with the added rule that i2 = -1.
The imaginary numbers are sometimes defined to be the "pure imaginary" numbers--complex numbers for which the "real part" a = 0, sometimes with the added restriction that b is not zero--and are sometimes defined to be the non-real complex numbers.
The algebraic numbers are numbers that are solutions to polynomial equations with integer coefficients.
The transcendental numbers are complex numbers (sometimes limited to real numbers) that are not algebraic.
A: I think you were being a little too hard on Isaac.  The truth is that the real numbers are a sophisticated mathematical construction and that any explanation of what they "are" which pretends otherwise is a convenient fiction.  Mathematicians need these kind of sophisticated constructions because they are what is required for rigorous proofs.  Before people explicitly constructed the real numbers and used them to define and prove things about other concepts, it was never totally clear what was true or what was false, and everybody was very confused.
For example, Cantor proved that the number of points in the plane is the same as the number of  points on a line.  Many people thought that this was impossible before he did it; they had an intuition that you couldn't possibly "fit" the plane into the line.  More generally, people were pretty sure you couldn't fit $\mathbb{R}^n$ into $\mathbb{R}^m$ if $n$ was greater than $m$.  It wasn't until quite a bit later that mathematicians formalized and proved a rigorous mathematical statement which justified this intuition called invariance of domain, which says you can't do this in a continuous way.  One of the many mathematical constructions you need to even state this theorem is the construction of the real numbers.  (Another is a formal definition of what "continuous" means, but one thing at a time.)
So, what are the real numbers?  They are a formal way to fill "holes" in the rational numbers, which is necessary for all sorts of things.  The most basic thing they are necessary for is doing geometry.  You probably know that the square root of 2 is irrational.  What this means is that it is impossible to think about the diagonal of a square as being the same kind of object as the sides of a square while only using rational numbers.  But you can rotate a diagonal, and it looks just like the side of a square, only a bit longer.  So you'd like a number system in which you can sensibly talk about any number you can construct geometrically.  You'd also like to be able to talk about rotation!  You can't do that with just rational numbers, either.
So how do you fill in enough holes to do geometry?  Dedekind came up with a very clever way to do this.  It starts by observing that a rational number $q$ is completely determined by the set of rational numbers greater than it and the set of rational numbers less than it.  For example, 1/2 is completely determined by the fact that it's always between 1/2 + 1/n and 1/2 - 1/n.  (For the initiated, this is a special case of the Yoneda lemma.)  But there are "numbers," such as the square root of 2, which aren't rational, and yet have the property that we can always tell what rational numbers are greater than it and what rational numbers are less than it.  For the square root of 2, these are precisely the fractions p/q such that 2q^2 < p^2 and such that 2q^2 > p^2, respectively.  Dedekind's brilliant idea was the following:

Define a real number to be a partition of the rational numbers!

In Dedekind's construction, the square root of 2 quite literally is the set of rational numbers that are greater than it, and the set of rational numbers that are less than it. You can define all the usual arithmetic operations on these "numbers," called Dedekind cuts, and prove all the wonderful theorems you'll find in a standard book on real analysis.  In particular, the property that guarantees that all the holes are filled is called completeness.   

Figured I might as well add something about the complex numbers.  The story here is beautiful, and if you're really interested you should check out Tristan Needham's Visual Complex Analysis.  Some people say that the point of the complex numbers is to let you solve polynomials, but this is really selling them short.  The complex numbers are an inherently geometric construction, and should be understood as such.  Their geometry and topology just happens to be responsible for the fact that you can solve polynomials with them, but it's also responsible for much more.
Here is a quick sketch.  Now that you've got the real numbers on your hands, you can rigorously talk about plane geometry.  In plane geometry, an important notion is that of similarity.  Informally, two figures are similar if they have the same shape.  More formally, two figures are similar if you can rotate, translate, and scale one figure so that it matches up with the other.  So similarity is all about a certain collection of transformations of the points in the plane.  It was Klein who first realized that the important features of different flavors of "geometry" are captured in what kind of transformations are allowed.  So to do geometry the modern way we should focus our attention on these transformations, which form a group.
To make this easier, let's ignore the translations for now.  We'll pick an origin for our plane, and we'll only allow rotations and scalings about this origin.  Rotations and scalings have the property that they are both linear transformations; this means that if you know what the transformation does to two points $u, v$, you also know what it does to the vector sum $u + v$.  In particular, a linear transformation is determined by what it does to the point $(1, 0)$ and to the point $(0, 1)$.  
However, rotations and scalings satisfy an extra property: they are, in fact, determined by what they do to the point $(1, 0)$.  This is because $(0, 1)$ can be obtained from $(1, 0)$ by a rotation by 90 degrees, and rotations and scalings commute with each other: if you rotate x degrees then y degrees, that's the same as rotating y degrees then x degrees, which is the same as rotating x+y degrees.  Similarly, if you rotate x degrees then scale by 2, that's the same as scaling by 2, then rotating x degrees.  So if you know what a rotation-and-scaling does to $(1, 0)$, you just rotate that vector by 90 degrees, and you know what it did to $(0, 1)$.
So to every rotation-and-scaling, we can assign two real numbers: the coordinates of the image of the point $(1, 0)$.  In general, a rotation by $\theta$ angles followed by a scaling by $r$ sends $(1, 0)$ to $(r \cos \theta, r \sin \theta)$.  A different transformation, say a rotation by $\phi$ angles followed by a scaling by $s$, sends $(1, 0)$ to $(s \cos \phi, s \sin \phi)$.  And their composition sends $(1, 0)$ to $(rs \cos (\theta + \phi), rs \sin (\theta + \phi))$.  In other words, composition of rotations-and-scalings defines a multiplication law on pairs of real numbers.  What is this law, exactly?  Well, by the angle addition formulas, it's
$$(a, b) * (c, d) = (ac - bd, ad + bc).$$
And this is precisely the rule for multiplication in the complex numbers, where $(a, b)$ corresponds to $a + bi$.  You get the rule for addition by observing that not only can you compose two rotations-and-scalings, you can also add their results. 
Together, the real numbers and the complex numbers provide a foundation for much of modern mathematics and physics.  For example, the complex numbers turn out (for reasons which are still not well understood) to be fundamental in the description of quantum mechanics.
A: *

*Natural numbers
The "counting" numbers.  (That is, all integers, that are one or greater).

*Whole numbers
The Natural numbers, and zero.

*Integers
The Whole numbers, and the negatives of the Natural numbers.

*Rational numbers
Any number that may be expressed by any integer A divided by any integer B, where B is not zero.

*Irrational numbers
Any number that cannot be expressed as a rational number, but is not imaginary.  All irrational numbers have an infinite decimal representation.

*Real numbers
All of the Rational and Irrational numbers.

*Imaginary numbers
All Real numbers, multiplied by the square root of negative one.  Imaginary numbers are signified by the letter i.

*Complex numbers
Numbers composed of the sum of a Real and an Imaginary number.  This includes all Real and all Imaginary numbers.
