linguistic:paraphrasing, linguistic:translation
Converting repeating decimals to fractions

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating decimals into fractions.

A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0. 789 = 0.789789789... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For example:

If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros:

If a non-repeating set of decimals precede the pattern (such as 0.1523 987), one may write the number as the sum of the non-repeating and repeating parts, respectively:

Then, convert both parts to fractions, and add them using the methods described above:

Alternatively, algebra can be used, such as below: