Reduce Infinite Decimals to Vulgar Fractions

These are repeating decimals where blocks of digits after the decimal place repeat ad infinitum. A standard notation is to draw a line over the repeating block of digits. For example, in his section on reducing fractions to decimals Macdonald had written  as  which I mentioned was a repeating decimal  etc. This is written compactly as .

To obtain the fraction, write out the digits of the decimal until the numbers begin to repeat (318). Subtract the nonrepeating digits from this number to get the numerator in the fraction (318 – 3 = 315). The denominator is comprised of a string of 9’s where the length of the string equals the number of repeating digits, followed by a string of ciphers (0’s) where the length of this string equals the number of nonrepeating digits (990). Then reduce the resulting fraction to a simple fraction ().

Examples:

(1) Reduce  to a vulgar fraction.

 23562 235 23327

Macdonald gives the result  without further explanation or possible simplification. It turns out that 23327 and 99000 are relatively prime since their greatest common divisor is 1. This is verified with the algorithm to find the greatest common divisor.

99000 = 4 × 23327 + 5692

23327 = 4 × 5692 + 559

5692 = 10 × 559 + 102

559 = 5 × 102 + 49

102 = 2 × 49 + 4

49 = 12 × 4 + 1

4 = 4 × 1

(2) Reduce  to a vulgar fraction.

 740384615 740 740383875

The answer is . This reduces to , which Macdonald provides without further demonstration. The reduction is obtained through the usual algorithm which shows that the greatest common divisor between numerator and denominator is 9615375.