Recurring decimal

Recurring decimal

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 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 = 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