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Viewing Version
49
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'Egyptian Mathematical Leather Roll'
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| Title of object: |
Egyptian Mathematical Leather Roll |
| Canonical Name: |
EgyptianMathematicalLeatherRoll2 |
| Type: |
Definition |
| Created on: |
2007-12-06 16:12:45 |
| Modified on: |
2008-07-27 21:35:21 |
| Classification: |
msc:01A16 |
| Keywords: |
rational numbers |
| Defines: |
Egyptian fractions |
Preamble:
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Content:
The Egyptian Mathematical Leather Roll (\PMlinkexternal{EMLR}{http://emlr.blogspot.com}) dates to the 1850 BC to 2000 BCE period of the Egyptian Middle Kingdom. The leather scroll has been housed in the British Museum from 1864 to the present. The leather roll was not softened and unrolled until 1927. Scholars initially had correctly reported the the text's practical arithmetical relationships, but overlooked theoretical considerations.
The text is an important historical document, recording 26 lines of hieratic era data. The data defines one scribal method that converted 1/p and 1/pq to non-optimal Egyptian fraction series. Early 1930's scholars had considered the EMLR data as likely being decoding door to understand RMP 2/n table construction methods. Today, scholars understand the scribal use of theoretical p, and q, prime numbers, as well as the EMLR student method that wrote 22 unit fractions, 1/8 three times, 1/7 two times, and 1/16 two times, in total writing out 26 different Egyptian fraction series.
Concerning details, all 26 were multiplied by one of eight(red auxiliary) multiples: 2, 3, 4, 5, 6, 7, 10, and 25. Each conversion obtained a non-optimal Egyptian fraction series.
The EMLR student used one multiple 25 times, two multiples once, to compute the non-optimal Egyptian fraction series. For example, 1/8 was initially converted by three multiples. The first two 3, and 5 obtained non-optimal Egyptian fraction series by:
1/8*(3/3) = 3/24 = (2 + 1)/24= 1/12 + 1/24,
1/8*(5/5) = 5/40 = (4 + 1))/40 = 1/10 + 1/40, and
Two multiples, 25 and 6, were likely used in the following manner:
multiple one: 1/8*(25/25) = 25/200 = (17 + 8)/200 = 17/200 + 1/25
multiple two: 17/200*(6/6) = 102/1200 = (80 + 16 + 6)/200 = (1/15 + 1/75 + 1/200
final answer: 1/25 + 1/15 + 1/75 + 1/200
Maintaining the out-order series implied that a two-phase method had been applied. Scholars had reported until 2008 that another two step method likely decreased the denominator by:
multiple one: 1/8*(25/25) = 25/200 = (24 + 1)/200 = 24/200 + 1/200;
factor by 1/5: 24/200 = 1/5*(3/5);
multiple 3:1/5*[3/5*(3/3) = 9/15 = (5 + 3 + 1)/15)] = 1/15 + 1/25 + 1/75;
final answer: 1/15 + 1/25 + 1/75 + 1/200,
with an listing the order of the series unit fractions was changed to
1/25 + 1/15 + 1/75 + 1/200
to indicate that a two-to-three step method had been employed.
More likely was that a two-phase multiple 26, 6 method was used by the student to increase the size of the denominator.
Continuing to a general use of method 6, the multiple was used four additional times to convert 1/7, 1/9, 1/11 and 1/15 by:
1/7*(6/6)= 6/42 =(3 + 2+ 1))/42 = 1/14 + 1/21 + 1/42,
1/9*(6/6)= 6/54 = (1/2 + 1/3 + 1/6)/54 = 1/18 + 1/27 + 1/54,
1/11*(6/6)= 6/66 =(1/2 + 1/3 + 1/6)/66 = 1/22 + 1/33 + 1/66,
1/15*(6/6)= 6/90 = (1/2 + 1/3 + 1/6)/90 = 1/30 + 1/45 + 1/90.
The EMLR student scribal converted
1/13 = 1/28 + 1/49 + 1/96 = 3/49.
an error, rather than 3/39. The error may infer that a multiple 3 had been improperly applied. Two other multiples, 8 and 14, may have been expected (by the student's instructor). The RMP used multiple 8, 200 years later. However, if multiple 14 had been applied the form [(n - 1) + 1] would have computed:
1/13*(14/14) = (13 + 1)/182 = 1/14 + 1/182
a pattern reported several times in the \PMlinkexternal{RMP 2/n tables}{http://rmprectotable.blogspot.com}.
Overall, the EMLR reported 22 unit fractions, prime and composite denominators, converted to Egyptian fractions by using eight multiples: 2, 3, 4, 5, 6, 7, 10 and 25. A broader narrative describing the EMLR is the found on \PMlinkexternal{Egyptian Mathematical Leather Roll}{http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll}.
It should be noted that advanced Egyptian scribes would have converted 2/n, n/p, and n/pq to short, and concise unit fraction series. In other words, advanced scribes generally converted small and large rational numbers to optimal Egyptian fraction series. Hence, Egyptian fraction classes introduced students to non-optimal versions of the Egyptian fraction conversion methods that were closely related to optimal \PMlinkexternal{RMP 2/n tables}{http://rmprectotable.blogspot.com} methods.
In summary, the EMLR was a leather roll containing 26 Egyptian fraction series. The text likely was produced by a student scribe interested in learning 2/n table construction methods. The EMLR used a single multiple method. Multiples were introduced by applying 2, 3, 4, 5, 6, 7, 10 and 25, to binary and prime rational numbers by creating non-optimal unit fraction series. In one case a two-phase pair of multiples was likely used. An error in the EMLR incorrectly converted 1/13. The error may have shown that the student was not ready to graduate.
\begin{thebibliography}{6}
\bibitem{1} Milo Gardner, \emph{The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences}, Hindustan Book Company, 2002.
\bibitem{2}Milo Gardner, \emph{"Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures}, Springer, 2005
\bibitem{3} Richard J Gillings, \emph{The Egyptian Mathematical Leather Roll}, Australian Journal of Science 24 pgs 339-344, 1962.
\bibitem{4} Richard J Gillings, \emph{The Egyptian Mathematical Leather Roll}, Archive for History of Exact Sciences pgs 291-298, 1974
\bibitem{5} Richard J Gillings, \emph{The Egyptian Mathematical Leather Roll Line 8, how did the Scribe do it?}, Historia Mathematica pgs 456-457, 1981.
\bibitem{6}S.R.K Glanville, \emph{"Mathematical Leather Roll in the British Museum"}, Journal of Egyptian Archaeology pgs 232-8, 1927
\end{thebibliography}
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