Egyptian Mathematical Leather Roll

2007-01-08T11:00:00.000-08:00

INTRODUCTION
The EMLR dates to the 1850 BC to 2000 BCE period of the Egyptian Middle Kingdom. The leather scroll, housed in the British Museum from 1864 to the present, was not softened and unrolled until 1927. Scholars in the 1927 under reported the majority of the text's arithmetical relationships.

Early scholars under valued the text by only reporting its additive aspects. Scholars from the 20th century did not consider potential uses of p and q as prime numbers, and other facts are needed to be added back to EMLR and the Rhind Mathematical Papyrus (RMP) 2/n table calculations were updated in a 2011 paper.

The EMLR student scribe used seven scaling of 1/p and 1/pq ( 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 10/10), facts outlined in 2004. An eighth two-phase 25/25 and 6/6 method scaled 1/8 and 1/16 to out-of-order unit fraction series, hinting at the scribal method also used in RMP 36.

Several EMLR and RMP connections are easily seen. The EMLR student was apparently asked to convert 26 rational numbers to unit fraction series as fast as as possible without considering optimization issues. Four binary numbers, 1/2, 1/4, 1/8, and 16 were converted, several more than once. For example 1/8 was converted three times, once by multiplying by 3/3 obtaining 3/24 = (2 + 1)/24 = 1/12 + 1/24. In total eight multiples: 2, 3, 4, 5, 6, 7, 10, and 25 converted 22 unique unit fractions to non-elegant Egyptian fraction series.

A least common multiple (LCM), written as m/m, conversion method is decoded from the EMLR by asking what was the simplest method that the 'student' scribe used? A 2008 connection to the well defined RMP LCM (m/m) method simplifies a six year old study of the EMLR.

Given that the RMP 2/n table, and 84 problems, generally applied m/m LCMs as scaling factors before writing red auxiliary additive facts, with RMP 36 being the most detailed example, the EMLR student was being introduced to the world if proto-number theory. Ahmes, in his version of proto-number theory converted 2/pq by selecting (p + 1), a multiple, to write optimzed Egyptian fraction series.

For example, 2/21 was raised to (3 + 1)/(3+ 1), or 4/4, writing:

2/21*(4/4) = 8/84 = (6 + 2)/84 = 1/14 + 1/42.

The EMLR proto-number theory preceeds the RMP's proto-number theory by following a clear pattern.

DECODING SPECIFICS
The EMLR contains eight multiple categories (A -H). Each category begins with one multiple: 2, 3, 4, 5, 6, 7, 10, or 25.

A. Unit fractions was decomposed into repeating unit fractions, generally an unacceptable means of writing Egyptian fraction series. An exposure to Red auxiliary multiples 2/2 and 3/3, wrote out identical unit fraction, as the RMP had written: 2/3 = 1/3 + 1/3 (line 7), possibly as a teaching tool, further discussing like cases:

2. 1/5*(2/2) = 2/10 = 1/10 + 1/10 (line 4)

3. 1/3*(2/2) = 2/6= 1/6 + 1/6 (line 5)

4. 1/2*(3/3) = 3/6= 1/6 + 1/6 + 1/6 (line 6)

The first EMLR lesson restated four rational numbers with multiples. The multiples created non-Egyptian fractions. This lesson shows that a unit fraction, multiplied by a multiple m, factor into m unit fractions.

The second EMLR lesson converted 22 unit fractions by multiple of 3, 4, 5, 6, 7, 10 and 25 . Answers were written in not-so-elegant Egyptian fraction series. Hence, optimal Egyptian fractions were not taught in this course.

A summary of the remaining six categories (B -H) follows:

B. Multiple of 3 (10 to 11 questions)

There are 10 questions, and maybe an 11th, that the student was asked to use a multiple of 3, restating a unit fraction by 1/n* (3/3) = 3/(3n) = (2 + 1)/3n. The student created not-so- elegant Egyptian fraction series by assuming a least common multiple such that: 1/6, 1/8, 1/10, 1/12, 1/14, 1/16, 1/20, 1/30, 1/32, and 1/64 were converted by three-steps (implied by the EMLR to RMP class of Egyptian fraction series):

5. 1/3*(3/3) = 3/9 = (2 + 1)/9 = 1/4 + 1/12 (line 3)

6. 1/6*(3/3) = 3/18 = (2 + 1)/18 = 1/9 + 1/18 (line 11)

7. 1/8 *(3/3) = 3/24 = (2 + 1)/24 = 1/12 + 1/24 (line 13)

8. 1/10*(3/3) = 3/30 = (2 + 1)/30 = 1/15 + 1/30 (line 24)

9. 1/12 *(3/3) = 3/36 = ( 2 + 1)/36 = 1/18 + 1/36 (line 20)

10. 1/14*(3/3) = 3/42 = (2 + 1)/42 = 1/21 + 1/42 (line 21)

11. 1/16*(3/3) = 3/48 = (2 + 1)/48 = 1/24 + 1/48 (line 19)

12. 1/20*(3/3) = 3/60 = (2 + 1)/60 = 1/30 + 1/60 (line 23)

13. 1/30*(3/3) = 3/90 = (2 + 1)/90 = 1/45 + 1/90 (line 22)

14. 1/32*(3/3) = 3/96 = (2 + 1)/96 = 1/48 + 1/96 (line 25)

15. 1/64*(3/3) = 3/192 = (2 + 1)/192 = 1/96 + 1/192 (line 26)

and possibly,

16. 1/13*(3/3) = 3/39 = (2 + 1)/39 = 2/39 + 1/39 = 1/39 + 1/26 + 1/78

since the RMP reports 2/39 = 1/26 + 1/78

But was the EMLR student exposed to RMP methods?

If not, going on to line 17 and the error 1/13 = 3/49, the student may have assumed 3, 6, 8 or another red reference number. Whatever the student's beginning point, the EMLR answer:

1/13 = 1/28 + 1/49 + 1/196 did not balance.

Since the first term in any RMP series 'gave away" the LCM , LCM 28 was considered per:

1/13 x (28/28) = 28/364 = (13 + 7 + 4 + 2 + 1)/364 = 1/28 + 1/52 + 1/91 + 1/182 + 2/364

which required 2/182 = 1/91 + 1/91

and required 2/91 to be solved ... Ahmes could used LCM 60 and recorded

2/91 = 120/5460 = (91 + 13 + 6)/5460 = 1/60 + 1/420 + 1/90

However Ahmes found a better series per LCM 70

http://rmprectotable.blogspot.com/ wrting

2/91 (70/70) = 140/6370 = (91 + 49)/6370 = 1/70 + 1/30

Note the link 2/35 solved by LCM 30 per

2/35(30/30) = 60/1050 = (35 + 25)/1050 = 1/30 + 1/42

My view is that EMLR was likely an introduction to the RMP 2/n tables ... and not a link to OK aloebraic arithmetic as proposed by others (Stefan M., for example).

In summary: If I would have been the EMLR teacher an impossible 1/13-type problem would have suggested LCM 28. LCM 6 would have solved the problem. but wthout knowing the LCM and red auxiliary number method EMLR students would have been introduced the next level of scribal arithmetic, 2/n tables and RANs in an interesting manner.

Recalling Gillings suggested the identity 1= 1/2 + 1/3 + 1/6 method (written as LCM 6), calculated by:

1/13*(6/6) = 6/78 = ( 3 + 2 + 1)/78 = 1/26 1/39 + 1/78

a deeper lesson may have been involved.

The EMLR student may not have been expected to solve this problem. In a related problem Ahmes solved this problem by red LCM 14:

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

or, in by algebraic relationship used by Fibonacci, 2,900 years later:

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

Note that the EMLR student could have selected LCM 28:

1/13*(28/28) = 28/364 = (14 + 13 + 1)/364 = 1/26 + 1/28 + 1/364

another easy to grasp approach.

Ahmes used the 1/13-type conversion method four times. Ahmes modified the method an additional 20 times to solve larger n/p conversions. Fibonacci included the method as one of his first of seven conversation methods in 1202 AD writing the Liber Abaci.

Ahmes converted 2/97 by a related LCM 56. Advanced EMLR students may have converted 1/13 by using the reference number by, as Ahmes may have also solved this problem:

1/13*(56/56) = 56/(13*56) = (28 + 13+ 8 + 7)/(13*56) = 1/26 + 1/56 + 1/91 + 1/104

That is, had LCM 14, 28 or 56 been suggested the associated data would have been new to the EMLR student. Note the (n + 1) form of its aliquot (additive) parts. The form solved 2/5, 2/7, 2/11, 2/23, with a modified form solving 2/97 in the RMP. In conclusion the student may been expected to be confused.

Returning to other EMLR decoding issues, LCM 5 allowed the student to solve

1/4 and 1/8:

C. Multiple of 5 (2 questions):

17. 1/4*(5/5)= 5/20 = (4 +1)/20= 1/5 + 1/20 (line 2)

18. 1/8*(5/5)=5/40 = (4+ 1)/40 = 1/10 + 1/40 (line 1)

D. Multiple of 6 ( 4 questions)

19. 1/7*(6/6) = 6/42 = (3 + 2 + 1)/42= 1/14 + 1/21 + 1/42 (line 14)

20. 1/9*(6/6) = 6/54 = (3 + 2 + 1)/54 = 1/18 + 1/27 + 1/54 (line 15)

21. 1/11*(6/6) = 6/66 = (3 + 2 + 1)/66= 1/22 + 1/33 + 1/66 (line 16)

16. 1/13*(6/6) = 6/78 = ( 3 + 2 + 1)/78 = 1/26 1/39 + 1/78 (Gillings' suggestion)

22. 1/15*(6/6) = 6/90 = (3 + 2+ 1)/90 = 1/30 + 1/45 + 1/90 (line 18)

E. Multiple of 7 (one question)

23 . 1/4*(7/7) = 7/28 = (4 + 2 + 1)/28 = 1/7 + 1/14 + 1/28 (line 12)

F. Multiple of 10 (one question)

24. 1/15*(10/10) = 10/150 = ( 6 + 3 + 1)/150 = 1/25 + 1/50 + 1/150 (line 10)

G. Modified multiple of 25 (2 questions)

Alternative one:

Increased denominator by LCM 25, and use LCM 6 (likely reading of the text):

25. 1/8*(25/25) = 25/200 = (8 + 17)/200 = 1/25 + 17/200

with,

17/200*(6/6) = 102/1200 = (80 + 16 + 6)/1200 = 1/15 + 1/75 + 1/200

hence an out-of-order series indicated a 2-phase method writing:

1/8 = 1/25 + 1/15 + 1/75 + 1/200 (line 8)

26. 1/16*(25/25) = 25/400 = (17 + 8)/400 = 1/50 + 17/400

with,

17/400 *(6/6) = 102/2400 = (80 + 16 + 6)/2400 = 1/50 + 1/30 + 1/150 + 1/400 (line 9)

25a . Alternative two (an unlikely reading of the text)

Decrease the denominator by factoring 1/5 in a second step per:

1/8*(25/25) = 25/200 = (24 + 1)/200 = 24/200 + 1/200

factor 1/5 from 24/200 = 1/5 *(24/40) = 1/5*(3/5)

such that.

3/5*(3/3) = 9/15 = (5 + 3 + 1)/15 = 1/3 + 1/5 + 1/15

meant

1/8 = 1/5*(1/3 + 1/5 + 1/15) + 1/200

re-written as

1/8 = 1/25 + 1/15 + 1/75 + 1/200

to show that an out-of-order series indicated two LCMs 25 and 3 had been used.

SUMMARY

The EMLR recorded 26 lines of unit fraction information. To decode one or more scribal methods in which 1/p and 1/pq unit fractions were converted to a unit fraction series, non-optimal LCM m values must be determined. Scribal LCM m values can be seen as scaling factors by adding back initial details. Seen in terms of the RMP 2/n table (concise but not optimal unit fraction series) suggests that the EMLR answer sheet recoded basic test results. The EMLR student data can be seen recording conversions of 1/p and 1/pq to non-optimal unit fraction series as an opportunity to learn RMP 2/n table conversion methods at a later time.

REFERENCES

1. Gillings, Richard J, "Mathematics in the Time of the Pharaohs" Dover books, New York, 1982, ISBN 0-486-24315-X

2. Gardner, Milo R., 'The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term'. in History of the Mathematical Sciences, editors Ivor Grattan-Guiness, B.S. Yadav, Hindustan Book Agency, New Delhi, 2002, ISBN 81-84931-45-3.

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

LINKS:

1. EMLR (Wikipedia)

2. EMLR (Planetmath)

3. Ahmes Papyrus

4. RMP 2/n Table (Wikipedia)

5. Breaking the RMP 2/n Table Code (blog)

6. RMP 36 and the 2/n table (blog)

7. Hultsch-Bruins Method (Planetmath)

author: Milo Gardner