Generating A Denominator With More Terminal Nines
An additional technique is available to find another A.F. with a smaller divisor. By a judicious choice of the multiplier one can produce an equivalent fraction with more nines in the denominator.
When F = 1/7 = 7/49, then the A.F. = 0.7/5; We use a multiplier of 7 to produce an equivalent fraction, 7/49. Look at the denominator of the equivalent fraction, 49. Consider the 4. To build the 4 to a nine we need to add 5 in the tens place. The 5th multiple of 7 ends in 5, so we may use 5 tens or 57 as the multiplier.
F = 1/7 = 57/399, and the A.F. = 0.57/4 (dividing in bundles of two digits).
We may proceed likewise, until we have F = 1/7 = 142,857/999,999 giving a bundle of six nines and an A.F. = 0.142857/1. This means we have the repeating decimal digit set on sight because when we divide this bundle of six digits (0.142857) by one and we have no remainder and this bundle repeats.
Read more about this topic: Auxiliary Fraction
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