Egyptian Mathematical Leather Roll
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The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathematical Papyrus but the former was not chemically softened and unrolled until 1927 (Scott, Hall 1927).
The writing consists of Middle Kingdom hieratic characters were written right to left. There were 26 rational numbers listed. Each rational number was followed by its equivalent Egyptian fraction series.
There were ten Eye of Horus numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted to Egyptian fractions. There were seven other even rational numbers converted to Egyptian fractions: 1/6 (twice–but wrong once), 1/10, 1/12, 1/14, 1/20 and 1/30. Finally, there were nine odd rational numbers converted to Egyptian fractions: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13 and 1/15.
The British Museum examiners found no introduction or description of how and why the equivalent unit fraction series were computed (Gillings 1981: 456-457). A series of equivalent unit fractions is associated with the fractions 1/3, 1/4, 1/8 and 1/16. There was a trivial error associated with the unit fraction series total of 1/15. It was listed as 1/6. A serious error was associated with the rational number 1/13, a problem that the examiners did not resolve in 1927.
The British Museum Quarterly (1927) naively reported the chemical analysis to be more interesting than the roll's additive contents.
The Middle Kingdom Egyptian fraction restatements of the older binary fractions are best seen as corrections to the older Horus-Eye, or Eye of Horus, numeration system. The earlier Horus-Eye arithmetic had employed an infinite series numeration system using binary fractions, in all situations similar to our present decimals, a method that nearly always rounded off, (Ore 1944: 331-325). Note the Horus-Eye definition for one (1): 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + … with the last term 1/64th being thrown away, (Gillings 1972: 210).
Because the Middle Kingdom arithmetic was written in hard to read unit fraction series, modern researchers had frequently minimized the EMLR’s significance. One minimalist report stated that the Horus-Eye binary fraction system was superior to the Egyptian fraction notation, a false conclusion. The EMLR and RMP data demonstrated ways to convert rational numbers to exact unit fraction series, methods that were unavailable in the Horus-Eye notation.
[edit] Chronology
The following chronology shows several milestones that marked the recent progress toward reporting a clearer understanding of the EMLR's contents.
1895 – Hultsch suggested that all RMP 2/p series were coded by an algebraic identity, using a parameter A (Hultsch 1895).
1927 – Glanville prematurely concluded that EMLR arithmetic was purely additive (Glanville 1927).
1929 – Vogel reported the EMLR to be more important, though it contains only 26 unit fraction series (Vogel 1929)
1950 – Bruins independently confirmed Hultsch’s RMP 2/p analysis (Bruins 1950)
1972 – Gillings found solutions to an easier problem, the 2/pq series of the RMP (Gillings 1972: 95-96).
1982 – Knorr identified the RMP fractions 2/35, 2/91 and 2/95 as exceptions to the 2/pq problem (Knorr 1982).
1990s – A multiple method was used in the EMLR to find 1/p and 1/pq unit fraction series. The multiple was improved in the RMP, written as (p + 1), and was used to convert 21 of 24 Ahmes’ 2/(pq) series. Ahmes’ multiple method has been reported as an algebraic identity: 2/(pq) = 1/A x A/(pq), with A = (p + 1). However, the multiple form, also written as (p + 1), is seen in a simplier form by the example 2/21: (3+ 1)/(3+ 1)x 2/21 = 2/(2+1)x(3+ 1)/21. The elegant or optimal 2/21 series was written: 1/2 x (1/7 + 1/21) = 1/14 + 1/42 as Ahmes reported.
There are five categories (a–e) that may summarize the EMLR’s 26 unit fraction series. Three are identities (a, b, c) and one (d) is a possible remainder. The firsr four categories can been seen a multiples of the initial rational number. The first four categories have been understood in additive arithmetic terms since 1927. However, the ideas of multiples and other non-additive conversion methods were not explored in 1927.
Going on to the fifth method (e), it has been seen as an algebraic identity. Or, it too may have been a simple multiple in the eyes of the ancient scribe. If so, the EMLR may be fairly be seen as a body of knowledge that converted 26 rational numbers, and by implication, any rational number, using multiples of 2,3,4, 5, 7 and 25, plus using an identity 1 = 1/2 + 1/3 + 1/6, generally find not-so elegant Egyptian fractions.
To analyze each of the five categories, the following information is offered.
a. Four rational numbers used the identity 1 = 1/2 + 1/2 was written as 1/n = 1/(2n) + 1/(2n). As a multiple of 2, or 1 = 2/2 = (1 + 1)/2 = 1/2 + 1/2.
b. Ten rational numbers use the identity 1/2 = 1/3 + 1/6 were written as 1/(2p) = 1/p x (1/3 + 1/6). As multiple of 3, or 1/2p = 1/2p x 3/3 = (2 + 1)/6p = 1/3p + 1/6p.
c. Four rational numbers use the identity 1 = 1/2 + 1/3 + 1/6 were written as 1/p = 1/p x (1/2 + 1/3 + 1/6) = 1/2p + 1/3p + 1/6p.
d. Three rationals used a remainder 1/p-1/(p + 1) = 1/p x (p + 1) were written as 1/p = 1/(p +1) + 1/p x (p + 1). Several multiples can also used this method, thus subtraction was not a requirement.
e. Five rational numbers may have used an advanced algebraic identity method 1/(pq) = 1/A x A/(pq), or a simple multiple method, setting the multiple to the variable (p + 1).
For example, the EMLR student set p = 1, q = 8, A = 25 such that: 1/8 = 1/25 x 25/8 = 1/5 x 25/40 = 1/5 x 5/8, with 5/8 = 1/5 + 1/3 + 1/15 + 1/40
may have been mentally computed by:
5/8 – 1/5 = (25 – 8)/40 17/40 – 1/3 = (51 – 40)/120 11/120 = (8 + 3)/120 = 1/5 x (1/5 + 1/3 + 1/15 + 1/40) = 1/25 + 1/15 + 1/75 + 1/200
Interestingly, the EMLR's four values for A ( 2, 3, 4, 5, 7 and 25), or stated as multiples of 2, 3, 4, 5, 7, and 25, did not create elegant fraction series in a manner comparable to the RMP 2/nth table’s entries. The EMLR's partitioning A's are therefore considered to be a RMP training technique, a method that leads to the RMP (p + 1) multiple method. The serious error 1/13th may have been resolved by methods c, d or e. The error was related to a failed attempt to apply method e, or the teacher of the EMLR student may have intentionally asked a problem that the student was unprepared to answer.
The EMLR has been proven to be a student’s introduction to the innovative Egyptian fraction numeration system. The improved Middle Kingdom arithmetic converted any rational number, after employing a factoring process, to elegant and exact unit fraction series. This conclusion was difficult to reach for a number of reasons, the greatest one being the small collection of surviving fragments of Middle Kingdom texts. Seen in its most basic terms, the EMLR was of one two elementary texts written during the 2000BC to 1650 BC period. The second text was the Reisner Papyrus.
[edit] Sources
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