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.