
The real numbers represent almost the final step in the various expansions of the number system from the integers to the rational numbers to the algebraic numbers to the real numbers and the complex numbers. The real numbers form a field, they have an ordering, and they also have the property of â€˜completenessâ€™. This means that if there is any set X of real numbers such that they have an upper bound (a number that is bigger than everything in X), then they have a least upper bound (an upper bound which is smaller than any other). For example, suppose X is the set of numbers whose square is greater than 2. This has an upper bound (3 is bigger than anything in the set) and so has a least upper bound, which will be the square root of 2. The real numbers are (apart from just renaming the numbers) the only mathematical structure which is a complete ordered field.
The real numbers represent a considerable increase in size from the algebraic numbers. The algebraic numbers are countable (can be counted, or written in a list), whereas the real numbers cannot. This fact was first discovered by Georg Cantor (1845  1918), and was controversial because mathematicians felt that saying a set was infinite was all that could be said, there were no degrees of infinite (see set theory).
Why can the real numbers not be listed? Suppose that there is a way that they can be listed. Then there is a way of listing the numbers between 0 and 1 (as infinite decimals). But then there is a number which is not on the list: the one which in the nth decimal place has 1 if the nth decimal place of the nth number on the list is 2, and 2 if it is not. It is different from every number on the list, in the nth decimal place, and so no such list can be complete. So there is no way of listing the real numbers. SMcL 
