RMP 2/n table

2008-07-21T05:47:00.000-07:00

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

INTRODUCTION
Breaking the RMP 2/n table code was important project for historians to learn Egyptian arithmetic recorded by scribes It is known that scribes scaled rational numbers n/p to mn/mp by selecting the best least common multiple (LCM) m before recording concise unit fraction series. The dominate LCM m method selected the best divisors of denominator mp that summed to numerator mn in the 2/n table and other RMP problems.

The ancient hieratic method anticipated modern finite arithmetic. Scribal shorthand confused math historians for over 100 years. Ancient scribal shorthand omitted initial and intermediate notes thereby confusing historians. Adding back the missing initial and intermediate steps, a clear use of that LCM m method recorded rational numbers in multiplication context.

The resultant Egyptian fraction system reported concise unit fraction series that corrected an Old Kingdom Horus-Eye binary infinite series notation reported as:

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64+ ...

by adding back the missing 1/64 unit.

RMP 2 /n TABLE: DECODED
Ahmes' RMP 2/n table reports a red auxiliary multiple m method. An older text, the EMLR details 26 closely related conversions of 1/p and 1/pq by selecting non-optimal multiple m values. Following a Middle Kingdom tradition, Ancient Near East arithmetic texts continued for 2850 years recorded within a ciphered number context (ended in 800 AD). Less concise and less elegant Egyptian fraction series continued until 1454 AD in Europe, and 1637 AD in Arabic texts that used a subtraction metaphor ( that wrote n/p - 1//m = (mn - p)/mp, with (mn -p) set to unity (1) as often as possible.

Ahmes' selection of LCM m is denoted by LCM m as (m/m). First, second, and third calculation steps were personalized by each Middle Kingdom scribe. Ahmes' 2/n table style reported alternative multiple m selections are easily compared to the Kahun 2/n table, the Moscow Mathematical Papyrys (MMP) and the Egyptian Mathematical Leather Roll (EMLR). Alternative multiples were reported in Greek, Ancient Near East, and medieval texts, as late as 500-800 AD.

Ahmes, and other scribes, 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 began wit\h 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, solved 2/7 by selecting LCM 4 such that:

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

The first and second 2m/mn steps were omitted by Ahmes. Correcting the omissions provides direct evidence of Ahmes' red auxiliary method, decoding barriers uncrossed by 19th and 20th century scholars. Adding back missing data corrects Ahmes fragmented arithmetic as detailed in Ahmes' 2/n table

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*(60/60) = (95 + 15 + 10)/5700 = 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

HEKAT PROBLEMS AND THE 2/n TABLE

Another exact technique reported exact rational numbers in hekat volume unit problems. Hekat problems added back a 1/64 remainder such that:

1 hekat = (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64) hekat + 5 ro

A ro was 1/320 of a hekat such that 5 ro = 5/320 = 1/64 hekat

Ahmes used the binary quotient hekat + ro remainder system over 60 times in the Rhind Mathematrical Papyrus (Ahmes Papyrus). Special cases uses of the method were first recorded in the Akhmim Wooden Tablet (1900 BCE). Special case methods converted rational numbers to equivalent unit fractions series by correcting the binary problem :

a. 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/40 + 1/160

that solved 2/64 = 10/320 = (8 + 2)/320 = 1/40 + 1/160,

and,

b. 1 = 30/53 + 15/53 + 5/53 + 3/53

recorded in RMP 36.

c. Ahmes solved arithmetic and algebra completion problems in RMP 7-34 showing selections of LCM m, leading up to red auxiliary numbers (RAN)s fully reported in RMP 36 within an algebra problem:

1. 1/3 + 1/5 + x = 1

RMP 21, another unity completion problem, also used RANs that solved:

2. x/3 + x/5 = 1

In RMP 36 optimized but not optimal unit fraction series were calculated, once with the assistance of red auxiliary numbers, and several times implicitly by red auxiliary numbers.

A broader range of rational number conversion methods were recorded in the Kahun Papyrus and the Rhind Mathematical Papyrus, by 2/n tables 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 fractions were used in intermediate and final answers as a finite arithmetic notation. In RMP 36 Ahmes specifically showed that his 2/n table was written by a LCM multiplication method with red numbers. The red numbers selected optimizing divisors taken from LCDs. 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, with 1590 being the LCD.

Another key 2/n table point converted 15/53 by selecting 4 as a LCM (4/4) that considered the divisors of 4, 2 and 1 taken from LCD 212. Ahmes complete conversion method 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, and added back data in blue.

ADDITIONAL CODE BREAKING CONSIDERATIONS

It has been shown that Egyptian fraction series were created from rational fractions that applied versions of optimizing red auxiliary numbers for over 2850 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) used aspects of Ahmes conversion methods. 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.

Kevin Gong reports aspects of these issues when 5/23 was scaled to 12/12 rather than the correct 6/6 as Ahmes would have found 30/138 = (23 + 6+1)/138 = 1/6 + 1/23 + 1/138 .

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 within a finite numeration system that "de facto" replaced the Old Kingdom's infinite series system. A second narrative reports Ahmes' red auxiliary method . Ahmes 2/n table method converted rational numbers n/p and n/pq to optimized, but not optimal, unit fraction series. All the RMP's 87 problems applied 2/n table rules, the color coded red auxiliary numbers being the most prominent. The clearest example of 2/n table rules were recorded in RMP 36 within a hekat algebra context. On close inspection, adding back missing data, RMP 36 was more a proto-number theory problem than a weights and measures algebra problem.

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. Kevin Gong, 1992 UC Berkeley paper ..

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

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

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

9. 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.

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

11. 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)

Greek era text:

12. Hibeh Papyrus (Planetmath)

Breaking the RMP 2/n Table Code

RMP 2/n table