A recurring decimal is a representation for rational numbers which have a denominator with a prime factor other than 2 or 5. (The decimal notation for rationals that has a denominator with only 2 and/or 5 as prime factors is trivial.) The digits to the left of the decimal point represents the integer part, as with regular decimal notation, then on the right of the decimal point there is a string of 0 or more digits, then the recurring string. The recurring string takes two forms: if it is a single digit, that digit is written with a dot over it. If it is more than one digit, dots are put over the first and last digits. Some examples:
.
1/6 = 0.16
. .
9/7 = 1.285714
..
3147/990 = 3.178
The set of numbers which can be represented by recurring decimals is the same as the set of rational numbers. Every rational number has a recurring decimal representation, and every recurring decimal notation is a rational number.
A very interesting thing to note is that if a rational number can be written with a denominator of N, then there are fewer than N digits between the decimal point and the recurring string (assuming the recurring string is marked at it's earliest occurrance), and there are fewer than N digits in the recurring string - in any base, not just in decimal.
Another important thing about this notation is that nothing follows the recurring string. The recurrance goes on forever, there is nothing that can be put 'after the last one' because there is no 'last one'. The notation does not permit that. Therefore, if someone writes something like:
.
0 = 0.01
then that person is misusing the notation.
Gorgonzola reminds me that there is another common notation for recurring decimals: a vertical bar over the digit sequence that recurs. The examples above in this form are:
_
1/6 = 0.16
______
9/7 = 1.285714
__
3147/990 = 3.178
_
0 = 0.01
Again, the last one is incorrect use of the notation, and meaningless.