Recurring decimal
Recurring decimals are a way of representing as decimals certain fractions which are not of the form p/(2n5m). These decimal representations include an infinitely repeated pattern at the end of the fraction (this pattern may be as short as a single digit).
To indicate that the pattern extends infinitely, it is represented by an ellipsis (...):
- 1/9 = .111111111111...
- 1/7 = .142857142857...
- 1/3 = .333333333333...
- 2/3 = .666666666666...
Calculating the fraction
Given a repeating decimal, it is possible to calculate the fraction which produced it. For example:
- x = .333333...
- 10x = 3.33333...
- 9x = 3 so that x = 1/3
From this kind of argument, we can see that the period of the repeating decimal of a fraction n/d will be (at most) the smallest number k such that 10k-1 is divisible by d.
For example, the fraction 2/7 has d=7, and the smallest k that makes 10k-1 divisible by 7 is k=6, because 999999 = 7×142857. The period of the fraction 2/7 is therefore 6.
The case of .99999...
The method of calculating fractions from repeated decimals, especially the case of 1 = .99999..., is often contested by amateur mathematicians.
x = .99999...
10x = 9.9999...
10x - x = 9.9999... - .99999...
9x = 9
x = 1
Some argue that in the second step of the equation given above, 10x = 9.9999...0. This is not the case, and involves a fallacy: in fact the RHS is meaaningless.
For a more formal proof, consider the formula:
If n = 1, 
If n = 2, 
It follows that
On the other hand we can evaluate this limit easily as 1, also, by dividing top and bottom by 10n.
See also: Decimal
Referenced By
List of mathematical topics (P-R) | List of number theory topics
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