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| ``Unit Fractions''
by milogardner on 2006-03-28 10:13:31 |
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| Scribal unit fractions, as recorded in the Rhind
Mathematical Papyrus 2/nth table table used
several methods to write its 2/p and 2/pq series.
For example, Ahmes, the RMP scribe wrote his
2/71 series used the Hultsch-Bruins method, known
in the modern era since 1895 (due to the work of
F. Hultsch, and independently confirmed 50 years
later by E.M. Bruins).
The scribe first looked for the inverse composite numbers 1/A
within the range from p/2 to p, with those being 1/36, 1/38,
1/40, ..., 1/70. Each was tested, with an optimal series being
selected by the use of LCM considerations, as noted in the
RMP by red auxiliary numbers. To compute the 2/71 conversion
to an Egyptian fraction series, Ahmes used the following steps:
1. Subtract 1/40 from 2/71
2/71 - 1/40 = (80 - 71)/(40*71)
2. Inspect the numerator of the remainder 9/(40*71),
nine, as a sum of divisors of 40, those being
20, 10, 8, 5, 4, 2, 1: or select between 8 + 1
and 5 + 4.
3. Using the red auxiliary number rule, choosing the
alternative with the largest last term, 5 + 4 was
selected, meaning that
4. 2/71 = 1/40 + (5 + 4)/(40*71)
= 1/40 + 1/568 + 1/710
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