RMP 2/n table

BREAKING the RMP 2/n Table Code
Author: Milo Gardner Bio

INTRODUCTION
Breaking the RMP 2/n table code is central task for historians that wish to learn Egyptian mathematics as scribes understood and recorded the subject. Egyptian scribes wrote rational numbers in two ways. The first way anticipated our modern rational numbers, facts that scribes omitted in their documents. The second way converted the rational numbers to equivalent unit fractions within optimized but not always optimal series.

The first, and second ways, are recorded in the Kahun Papyrus and the Rhind Mathematical Papyrus, by a 2/n table and 87 problems, especially RMP 36.

The Kahun and Ahmes Papyri solved arithmetic progression problems by considering aspects of rational number statements written into unit fraction series. Egyptian fraction series were used in intermediate and final answers as a notation. In RMP 36 Ahmes specifically showed that his 2/n table was written by applying a LCM multiplication method with red numbers. The red numbers selected the optimizing divisors of the LCM that were used to partition the rational number. In RMP 36 the algebraic equation:

3x + x/3 + x/5 = 1 hekat was solved by selecting the LCM 15 such that:

(45x + 5x + 3x)/15 = 1,

53x = 15,

x = 15/53

as well as converting 2/53, 3/53, 5/53, 15/53 and 28/53 by exposing additive red numerators.

For example, following the 2/n table method, Ahmes converted 2/53 by selecting 30/30 mentally considering 60/1590 before optimally writing (53 + 5 + 2) which meant

(53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795

the series reported in the 2/n table.

Another key 2/n table point is that 15/53 was converted by selecting 4 as an identity LCM (4/4), and considering the divisors of 4: 4, 2 and 1. Ahmes complete conversion follows:

15/53*(4/4) = 60/219 = (53 + 4 + 2 + 1)/212 = 1/4 + 1/53 + 1/106 + 1/212

with optimizing numerators written in red.

Ahmes' RMP 2/n table , points out that a general use of red auxiliary multiple. A 350 year older text, the EMLR details 26 closely related conversions of 1/p and 1/pq by selecting non-optimal multiples (as a RMP practice method).

Following a Middle Kingdom tradition, texts published in the Ancient Near East continued for 3,400 years publishing optimized and elegant Egyptian fraction series. The selection of the multiple will be denoted by the number m and (m/m). First-steps, second-steps, and third-steps were personalized by each scribe. Ahmes' 2/n table rules reported alternative multiples, compared to the Kahun 2/n table, MMP and EMLR. Alternative multiples also were reported in Greek, Ancient Near East, and medieval texts.

Ahmes, and other scribes, first converted vulgar fractions by multiplying 2/n fractions by red auxiliary numbers (m/m). Two additional steps were used by Ahmes when 2/n table information was fully translated into modern arithmetic. Ahmes' arithmetic partitioned numerator 2m into additive integers, a fact pointed out in the 1920's.

Ahmes' 2/n conversions created a numerator 2m, a denominator mn, a set of additive 2m integers, and an optimized Egyptian fraction series. Ahmes method is symbolically written as Rule one, a fact published in 2005.

Rule One:

2/n = 2/n*(m/m) = 2m/mn= (mn1 + ... + mni)/(mn) = 1/a + 1/a1(n) + ...+ 1/ai(n)

where:

1. 2m was additively partitioned into i integers: (mn1 + mn2 + ... + mni) = 2m,

based on mn being factored into mn1, mn2, ..., mni and other primes and composites

such that:

2. mn1, mn2, ..., mni each divide mn

and,

3. m = a, in most cases, when for m > 8


and,

4. As an observation a1, ..., ai were always divisors of the denominator a of the first partition 1/a

example: 2/7 = 2/7*(4/4) = 8/28 = (7 + 1)/28 = 1/4 + 1/28

where:

2m = 8, with additive parts (7 + 1) = 8

and,

a = 4, with 4 and n: being divisors of 28 (and all a1, a2, ... denominators).

The second-step, 2m/mn (8/28) has been omitted. The omission provides a sense of Ahmes' red auxiliary shorthand, a major decoding barrier. The decoded data therefore leaves work for readers, as Ahmes did himself, to compute 2m and confirm the numerator's additive parts that adds to 2m.

RMP 2 /n TABLE: DECODED

2/3 = 1/3 + 1/3 = 2/3 followed an EMLR 1/3 = 1/6 + 1/6 'rule'.
2/5 = 2/5*(3/3) = (5+ 1)/15 = 1/3 + 1/15
2/7 = 2/7*(4/4) = (7 + 1)/28 = 1/4 + 1/28
2/9 = 2/9*(2/2) = (3 + 1)/18 = 1/6 + 1/18
2/11 = 2/11*(6/6) = (11 + 1)/66 = 1/6 + 1/66
2/13 = 2/13*(8/8) = (13 + 2 + 1)/104 = 1/8 + 1/52 + 1/104
2/15 = 2/15*(2/2) = (3 + 1)/30 = 1/10 + 1/30
2/17 = 2/17*(12/12) = (17 + 4 + 3)/204 = 1/12 + 1/51 + 1/68
2/19 = 2/17*(12/12) = (19 + 3 + 2)/228 = 1/12 + 1/76 + 1/114
2/21 = 2/21*(2/2) = (3 + 1)/42 = 1/14 + 1/42
2/23 = 2/23*(12/12) =(23 +1)/276 = 1/12 1/276
2/25 = 2/25*(3/3) = (5 + 1)/75 = 1/15 + 1/75
2/27 = 2/27*(2/2) = (3 + 1)/54 = 1/18 + 1/54
2/29 = 2/29*(24/24)= (29 + 12 + 4 + 3)/696 = 1/24 + 1/58 + 1/174 + 1/232
2/31 = 2/31*(20/20) = (31 + 5 + 4)/1620 = 1/20 + 1/124 + 1/155
2/33 = 2/33*(2/2) = (3 + 1)/66 = 1/22 + 1/66
2/35 = 2/35*(30/30) = (35 + 25)/1050 = 1/30 + 1/42
2/37 = 2/37*(24/24) = ( 37 + 8 + 3 )/888 = 1/24 + 1/111 + 1/296
2/39 = 2/39*(2/2)= (3 + 1)/78 = 1/26 + 1/78
2/41 = 2/41*(24/24)= (41 + 4 + 3 )/984 = 1/24 + 1/246 + 1/328
2/43 = 2/43*(42/42)=(43 + 21 + 14 + 6)/1806 = 1/42 + 1/86 + 1/129 + 1/301
2/45 = 2/45*(2/2)= ( 3 + 1)/90 = 1/30 + 1/90
2/47 = 2/47*(30/30)= (47 + 10 + 3)/1410 = 1/30 + 1/141 + 1/470
2/49 = 2/49*(4/4)= (7 + 1)/196 = 1/28 + 1/196
2/51 = 2/51*(2/2) = (3 + 1)/102 = 1/34 + 1/102
2/53 = 2/53*(30/30)= (53 + 5 + 2 )/1590 = 1/30 + 1/318 + 1/795
2/55 = 2/55(6/6) = (11 + 1)/330 = 1/30 + 1/330
2/57 = 2/57*(2/2) = (3 + 1)/114 = 1/38 + 1/114
2/59 = 2/59*(36/36) =(59 + 9 + 4) /2124 = 1/36 + 1/236 + 1/531
2/61 = 2/61*(40/40) =(61 + 10 + 5 + 4)/2440 = 1/40 + 244 + 1/488 + 1/610
2/63 = 2/63*(2/2)= (3 + 1)/126 = 1/42 + 1/126
2/65 = 2/65*(3/3)= (5 + 1)/195 = 1/39 + 1/195
2/67 = 2/67*(40/40)=(67 + 8 +5 )/2680 = 1/40 + 1/335 + 1/536
2/69 = 2/69*(2/2)= (3 + 1)/138 = 1/46 +1/138
2/71 = 2/71*(40/40)= (71+ 5 + 4)2840 = 1/40 + 1/568 + 1/710
2/73 = 2/73*(60/60)=(73 + 20 + 15 + 12)/4380 = 1/60 + 1/219 + 1/292 + 1/365
2/75 = 2/75*(2/2)= (3 +1)/150 = 1/50 + 1/75
2/77 = 2/77*(4/4)= (7 + 1)/388 = 1/44 + 1/308
2/79 = 2/79*(60/60)=(79 + 20 + 15 + 6 )/4740 = 1/60 + 237 + 1/316 + 1/790
2/81 = 2/81*(2/2)= (3 + 1)/162 = 1/54 + 1/162
2/83 = 2/83* (60/60)=(83+ 15 + 12 +10)/4980 = 1/60 + 1/332 + 1/415 + 1/498
2/85 =2/85*(3/3)= (5 + 1)/255 = 1/51 + 1/255
2/87 = 2/87*(2/2)= (3 + 1)/174 = 1/58 + 1/74
2/89 = 2/89*(60/60)=(89 + 15 +10 + 6)/5340 = 1/60 + 1/356 + 1/534 + 1/890
2/91 = 2/91*(70/70)= (91 + 49)/6370 = 1/70 + 1/130
2/93 = 2/93*(2/2)= (3 + 1)/186 = 1/62 + 1/186
2/95 = 2/95*(12/12)= (19 + 3 + 2)/1140 = 1/60 + 1/380 + 1/570
2/97 = 2/97*(56/56)= (97+ 8 + 7 )/5432 = 1/56 + 1/679 + 1/776
2/99 = 2/99*(2/2) = (3 + 1)/198 = 1/66 + 1/198
2/101 = 2/101*(6/6)= (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606


OBSERVATIONS

1. When n is a prime > 99

a. The EMLR cited: m = 6 was used to convert 1/101, as Ahmes used m = 6 to convert 2/101 and likely all larger 2/p prime denominators.

b. By implication (since Ahmes did not mention this class of detail) when n was composite, m was selected in terms of its largest prime number. Taking two examples, n = 19 and n = 95 (5*19), Ahmes used m = 12.

2. When m > 8

a. mn1 = n

b. a = m

3. The 2/n table method was extended to n/p and n/pq tables. A Coptic Akhmim Papyrus
included over-analyzed n/17 and n/19 data. Applying Occam's Razor the Coptic multiples ae simplified by considering:

a. 2/19 x 10/10 = (19 +1)/190 = 10' 190' (Greek notation 1/n = n'):
b. m = 60 to convert 3/19, additive parts (76 + 57 + 20 + 15 + 12)/1140
c. m = 6 to convert 4/19,
d. m = 4 to convert 5/19.
e. m = 6 to convert 6/19
f. m= 6 to convert 7/19
g. m = 30 to convert 8/19

ADDITIONAL CODE BREAKING CONSIDERATIONS

It has been shown that Egyptian fraction series were created from rational fractions that applied optimizing red auxiliary numbers for over 3,400 years. One set of examples were recorded in RMP 36, that converted 15/53, 30/53, 5/53 and 3/53.

Historians had parsed aspects of Ahmes rational number arithmetic by working backwards. Historians also considered personalized Egyptian fraction patterns by considering unproductive analytical methods, thereby muddling data translation.

One modern test for historians is to confirm their conclusions is to find the simplest forward engineered method. Early scholarly methods pointed out muddled translations that suggested scribal errors had taken place, when no errors were made.

By applying Occam's Razor muddled translations have been sifted. Needles in hay stacks have been found. For example, red auxiliary numbers represent one needle in the RMP 2/n table decoding effort.

CONFIRMATION

One confirming 800 year old method corrects modern analytical errors of misreading 2/n tables. Fibonacci's 1202 AD Liber Abaci (LA) defines the confirming method. The first 126 pages of the LA covers conversion and factoring examples. The examples summarize seven Egyptian fraction conversion methods. Four of the conversion methods date to the time of Ahmes. It has v been shown that Ahmes used a three-step conversion method. Four of Fibonacci's seven conversion methods may have been known to Ahmes. For example, Ahmes first step, the selection of a multplication multiple was reported by Fibonacci as a subtraction step. The selection of Ahmes' red auxiliary numbers were reported by Fibonacci without color coding.

CONCLUSION

Ahmes' 2/n table rules were applied to several of the 87 problems. Ahmes understood aspects of the fundamental theorem of arithmetic that uniquely factored integers into prime numbers. Ahmes did not use algorithms. Fragments of a proposed single false position algorithm had been improperly reported for 100 years. Single false position was not used in RMP 31, RMP 36, nor any rational number conversion problems cited in Ahmes' 87 problems when considering Occam's Razor, the simplest method is the historical method.

Ahmes' traditional red auxiliary multiples offered a simpler approach. In RMP 31 Ahmes solved 28/97 = 2/97 + 26/97, as RMP 36 solved 30/53 = 2/53 + 28/53. Note that 2/97 and 2/53 series were reported in the 2/n table, hence Ahmes was consistent in selecting 2/n conversions.

Concerning a wider view of Egyptian arithmetic, several narratives connect Middle Kingdom numeration to the Old Kingdom binary numeration system. One narrative covers Middle Kingdom arithmetic defining a finite numeration system that "de facto" replaced the Old Kingdom's infinite series numeration system. A second narrative reports Ahmes' red auxiliary first-step, used in 2/n tables, converting rational numbers n/p and n/pq to optimized, but not optimal, unit fraction series. Several of the RMP's 87 problems applied 2/n table rules, the color coded red auxiliary numbers being the most prominent. The clearest examples fo 2/n table rules are recorded in RMP 36 within a once hard-to-read hekat partitioning context.


BIBLIOGRAPHY

1. Mahmoud Ezzamel, Accounting for Private Estates and the Household in the 20th Century BC Middle Kingdom, Abacus Vol 38 pp 235-263, 2002

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

3. Milo Gardner, An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati, MD Publications Pvt Ltd, 2006.

4. Richard Gillings, Mathematics in the Time of the Pharaohs, Dover Books, 1992.

5. Heinz Lüneburg, Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Mannheim: B. I. Wissenschaftsverlag , 1993.

6. Oystein Ore, Number Theory and its History, McGraw-Hill, 1948.

7. T.E. Peet, Arithmetic in the Middle Kingdom, Journal Egyptian Archeology, 1923.

8. Tanja Pommerening, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass, Buske-Verlag, 2005.

9. L.E. Sigler, Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation, Springer, 2002.

10. Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.

LINKS:

1. Ahmes Papyus (blog)

2. Egyptian fractions (Planetmath)

3. EMLR (Wikipedia)

4. EMLR (Planetmath)

5. Hultsch-Bruins Method (Planetmath)

6. Kahun Papyrus (Wikipedia)

7. Liber Abaci (Planetmath)

8. Liber Abaci (Blog)

9. RMP 2/n Table (Wikipedia)

10. RMP 35-38, Plus RMP 66 (Planetmath)

11. RMP 36 and the 2/n table (Planetmath)