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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.

A recurring decimal is a fraction p/q that results in a sequence of digits that after some point starts repeating over and over again, e.g.
29/46 = 0,6 3043478260869565217391 3043478260869565217391 3043478260869565217391 ..

This can be written with a line over the final repeating part:

   ______________________
0,63043478260869565217391

hobyrne reminds me that it can also be written with a dot over the first and final digit of the repetition:

   .                    .
0,63043478260869565217391

The length of the repetition("recurrence") is always smaller than the value of the denominator q. This is because the rests in the division process are always between 0 and q, (excluding 0 and q).

If q is a factor of a power of ten, i.e. it has no factors other than the factors of 10, 2 and 5, then there is no recurrence, the fraction ends with all zeroes(which are not written).

There is another rule to the length("period") of the repetition(recurrence) in a recurring decimal in the fraction 1/x: The repetition is just as long as the smallest number made up of only the digit 9 ("999..") that is divisible by x.

For example, take 999999 (6 times). It is divisible by its factors 3*3*3*7*11*13*37= 999999. The factors 7 and 13 both appear first in a 9.. number here, and they have a repetition of length 6:

1/7= 0,142857142857..
1/13= 0,076923076923..

This is because
1/999999 = 0,000001000001...

Multiplying this by the other factors 3*3*3*7*11*37 (=76923) gives:
(3*3*3*7*11*37)/(3*3*3*7*11*13*37) = 0,076923076923..
or
1/13 = 0,076923076923..

Here is a small table of the factors which appear the first time in a 9.. number, sometimes it is one new factor, sometimes more:

      9 3,3       (period 1)
     99 11        (period 2)
    999 3,37      (period 3)
   9999 101       (period 4)
  99999 41,271    (period 5)
 999999 7,13      (period 6)
9999999 239,4649  (period 7) 

This works in other bases but 10, like hexadecimal too.

See: Can recurring digits be used to break RSA encryption?

All rational numbers fall into two categories, terminating and repeating. Terminating decimals are those said to have a definite last digit, such as -10, 2.5 or 1.793. Repeating decimals have a digit or series of digits that repeats ad infinitum, and include 0.673333... and 2.142857142857....

Although repeating decimals can be expressed in the ellipsis notation that I just used, there are easier ways. One such way is to put a vinculum bar over the repeating digits. Thus, 0.67333... and 2.142857... become

    _       ______
0.673 and 2.142857 

respectively. Another involves placing a dot over the first and last digits in the sequence. Using this format, the same two examples become:

    .       .    .
0.673 and 2.142857
In spoken English, the decimal would be called "two point one four two eight five seven repeating." Or, of course, you could just write out the fraction.

To find the fraction, follow these steps:

  1. Subtract the decimal's integral part. (Thus, 2.142857 repeating becomes 0.142857 repeating.) We will call this number n.
  2. Multiply n by a power of ten until the difference between n and it is a terminating decimal. This number can be expressed as a multiple of n. (In the example above, we would have 142,857.142857 repeating = 1,000,000n.)
  3. Subtract the two values to get a number equal to a multiple of n. (We now have 142,857 = 999,999n.)
  4. Now divide to find the value of n. (In the example, n = 142,857/999,999 = 1/7.)
  5. Finally, add the integer you subtracted at the beginning. The decimal can be expressed as a fraction or mixed number. (We now have 2+1/7 or 15/7 as our final answer.)

What makes a decimal repeating? A decimal is repeating when the denominator of its simplest fractional form cannot be decomposed into 2's and 5's; it is terminating, in other words, if its denominator is a factor of a power of ten. Thus, 387/1000 is terminating, but 2801/3600 is not. Of course (as N-Wing reminds me), in other bases than decimal, an n-mal will terminate with a denominator which is a factor of a power of n. Thus, a bimal will only terminate with a denominator which is a power of 2

But be warned! All terminating decimals can be expressed as a repeating decimal: even benign 2.5 can become 2.5000000... or 2.4999999.... While this may seem surprising, it is quite true and even useful—Cantor himself used it to show that the real numbers are uncountable.

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