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?

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.
