decimal point. Thus 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 = 0.789789789... 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: = 5/9 = 62/99 = 264/999 = 6291/9999 If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros: = 5/90 = 392/999000 = 12/9900 If a non-repeating set of decimals precede the pattern (such as ), one may write the number as the sum of the non-repeating and repeating parts, respectively: 0.1523 + Then, convert both parts to fractions, and add them using the methods described above: 1523 / 10000 + 987 / 9990000 = 1522464 / 9990000 Alternatively, algebra can be used, such as below: Let x = the repeating decimal: x = Multiply both sides by the power of 10 just great enough (in this case 104) to move the decimal point just before the repeating part of the decimal number: 10,000x = Multiply both sides by the power of 10 (in this case 103) that is the same as the number of places that repeat: 10,000,000x = Subtract the two equations from each other (if a = b and c = d, then a − c = b − d): 10,000,000x − 10,000x = − Continue the subtraction operation to clear the repeating decimal: 9,990,000x = 1,523,987 − 1,523 9,990,000x = 1,522,464 Divide both sides by 9,990,000 to represent x as a fraction x = Fractions in abstract mathematics In addition to being of great practical importance, fractions are also studied by mathematicians, who check that the rules for fractions given above are consistent and reliable. Mathematicians define a fraction as an ordered pair of integers and for which the operations addition, subtraction, multiplication, and division are defined as follows: These definitions agree in every case with the definitions given above; only the notation is different. Alternatively, instead