Recurring decimal

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Recurring decimals are a way of representing as decimals certain fractions which are not of the form p/(2ⁿ5^m) in lowest terms. 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 the part of the sequence that extends infinitely, dots should be placed above the numerals to be repeated. Where this is impossible, the extension may be represented by an ellipsis (...) although this may introduce uncertainty as to exactly which digits should be repeated:

1/9 = 0.111111111111...
1/7 = 0.142857142857...
1/3 = 0.333333333333...
2/3 = 0.666666666666...
7/12 = 0.58333333333...

Calculating the fraction

Given a repeating decimal, it is possible to calculate the fraction which produced it. For example:

x = 0.333333...

10x = 3.33333...

9x = 3 so that x = 1/3

y = 0.18181818...

100x = 18.181818...

99x = 18 so that x = 2/11

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 10^k-1 is divisible by d.

For example, the fraction 2/7 has d=7, and the smallest k that makes 10^k-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 sometimes contested by the mathematically naive:

      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, the RHS does not terminate (it is recurring).

For a more formal proof, consider the formula:

It follows that

On the other hand we can evaluate this limit easily as 1, also, by dividing top and bottom by 10ⁿ.

Generalising this, any number with a finite decimal expression (a decimal fraction) also has an expression as a recurring decimal.

For example 3/4 = 0.75 = 0.74999999...

See also: Decimal