The Egyptian Mathematical Leather Roll (EMLR) 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 document, recording 26 lines of 22 rational number conversions. The data defines a scribal method that non-optimally converted 1/p and 1/pq to unit fraction series. Early (1927-1930's) scholars considered the EMLR as a possible decoding door to RMP 2/n table construction methods, but did not guess the scribal method. The EMLR method converted 22 rational numbers, 1/8 three times, 1/7 two times, and 1/16 two times, in total writing 26 different non-optimized Egyptian fraction series following a 2/n table method by selecting a LCM, m, written as m/m.

Concerning all 26 initial EMLR rational numbers each was multiplied by one of eight LCMs: 2, 3, 4, 5, 6, 7, 10, and 25 written as 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 10/10 and 25/25. Each non-optimal conversion obtained a unit fraction series.

The EMLR student used one multiple 24 times, and two multiples twice, to compute 26 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 created an out-of-order series:

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

a modifiied method that RMP 31 and RMP 36 used by parsing 28/97 into 2/97 + 26/97 and 30/53 into 2/53 + 28/97 by using two LCMs, in RMP 31 56/56 to convert 2/97, and 4/4 to convert 26/97; and in RMP 36 30/30 to convert 2/53 and 2/2 to convert 28/53.

Prior to 2008 it was suggested that another two step method had 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 the out-of-order unit fractions denoting an unknown two-phase method.

Detailing additional multiple 6 examples, it 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.

Interestingly, the EMLR student wrote out an error

1/13 = 1/28 + 1/49 + 1/96 = 3/49.

rather than the proper 3/39.

Three multiples 3, 8, and 14, may have been expected (by the student's instructor). The RMP used multiple 8, 200 years later than the EMLR, hence it may have been expected. However, if multiple 14 was used a (n + 1 pattern would have coincided with four RMP 2/n Table patterns. The (n + 1) pattern apparently was generalized in the RMP in the form:

2/p = 2/p*[(n + 1)/(n + 1)] = (n + 1)/np,

as the EMLR studen may have been expected to write:

1/13*(14/14) = (13 + 1)/182 = 1/14 + 1/182

with n = 13.

More importantly, the (n + 1) pattern was modified 20 additional times in the RMP 2/n table to convert large prime 2/p denominators.

It should be noted that the modified RMP (n + 1) pattern was reported 3,000 years later as one of Fibonacci's seven Liber Abaci methods.

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, iuncluding a two phase multiples 25 and 6 method (to convert 1/8 and 1/16). A broader narrative describing the EMLR is the found on Egyptian Mathematical Leather Roll.

It should also 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 RMP 2/n tables 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 LCM scaling method before writing out non-optimal unit fraction series. LCM multiples, 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 10/10 and 25/25 were multiplied by the 26 rational numbers to create non-optimal unit fraction series. In one case a two-phase pair of multiples, 25/25 and 6/6, was used to convert 1/8 and 1/16. An error in the EMLR incorrectly converted 1/13. The error shows that the student may not have been not ready to graduate. Or, as likely, the student was thrown a curve ball as an introduction to the advanced 2/n table class.

Bibliography

1

Milo Gardner, The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences, Hindustan Book Company, 2002.

2

Milo Gardner, "Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, 2005

3

Richard J Gillings, The Egyptian Mathematical Leather Roll, Australian Journal of Science 24 pgs 339-344, 1962.

4

Richard J Gillings, The Egyptian Mathematical Leather Roll, Archive for History of Exact Sciences pgs 291-298, 1974

5

Richard J Gillings, The Egyptian Mathematical Leather Roll Line 8, how did the Scribe do it?, Historia Mathematica pgs 456-457, 1981.

6

S.R.K Glanville, "Mathematical Leather Roll in the British Museum", Journal of Egyptian Archaeology pgs 232-8, 1927