Monday, July 21, 2008

RMP 2/n table

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

INTRODUCTION
Breaking the RMP 2/n table code has been an important project for historians. By decoding scribal 2/n table methods scholars are learning 1650 BCE and earlier arithmetic methods. In 2006 and 2011 scribal thinking was broken and confirmed by demonstrating that rational numbers n/p were always scaled to mn/mp. The conclusion was reached by decoding every line and shorthand note recorded by the scribe Ahmes in RMP 36. The attested scribal  methodology selected  the best least common multiple (LCM) m before red auxiliary numbers, divisors of denominators mp, were recorded in concise unit fraction series.

During the 19th, 20th and 21st centuries the hard-to-read scribal shorthand was required to be parsed by looking for omitted initial and intermediate steps. Peet applied the Latin name "ab initio" to denote required and proposed studies, the last of which may be the above 2006 and 2011 studies.

Scribal shorthand did not reveal pertinent scribal details nor needed decoding keys easily.. The hieratic unit fraction system was proposed to anticipate modern finite arithmetic by F. Hultsch in 1895. Hultsch's suggestion was confirmed by EM Bruins in 1944, but usually ignored by the larger scholarly community. Several 20th century scholars also proposed finite arithmetic and number theory as decoding keys to read aspects of  scribal arithmetic. For example, Kevin Brown in 1995 proposed consistent scribal number theory patterns without pointing out the needed ancient details.

By 2006, attested historical details began to appear. The actual scribal facts were parsed by adding back actual missing initial and intermediate factual steps mentioned in RMP 36. Fully decoding RMP 36  and other RMP problems in historical context have exposed previously unknown scribal encoding keys.

By 2011, using the new decoding keys it is clear that a LCM m method was used by Ahmes that facilitated conversions of  rational number n/p in a multiplication context. A formal paper was posted on-line that may soon end the 1879 to 2011 debate.The paper describes hieratic arithmetic that scaled n/p by m/m to mn/mp in steps that allowed Ahmes to inspect the divisors of denominator mp for the purpose of listing the best set of divisors that summed to numerator mn. Red numbers were used by Ahmes to denote the importance of the additive set of divisors that summed to mn. Ahmes then wrote out concise unit fraction series representations of rational numbers n/p within a scribal method that considered number theory (that were close to F. Hultsch's  117 year old, and Kevin Brown's 17 year old suggestions).

Scribes like Ahmes outlined the information in once hard-to-read shorthand notes.  Ahmes' notes revealed that LCM m method selected the best divisors of denominator mp summed in red auxiliary numbers to mn. Breaking the 2/n table code is allowing all 87 RMP problems to be decoded as well as other Middle Kingdom mathematical text problems.

In retrospect, the resultant Egyptian fraction system was reported in 2006 and 2011 by showing that concise unit fraction series had 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 Mathematical 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
Egyptian fraction series were created from rational fractions that applied versions of optimizing red auxiliary numbers fo 500 of its 2850 year life. One red auxiliary number example is recorded in RMP 36.  Ahmes converted 2/53, 3/53 5/15, 15/53 and 28/53 to sum 53/53 hekat = 1 hekat. .

Historians have parsed aspects of Ahmes rational number arithmetic by working backwards. Oddly, personalized hieratic Egyptian fraction patterns dominated Egyptian math by considering unproductive and dead end methods..

For example, Kevin Gong and Kevin Brown reported aspects of ancient scribal patterns by applying modern mathematics. Kevin Gong's 1992 analysis demonstrated that 5/23 could have been scaled to 12/12 rather than the correct scribal scaling of 6/6 as Ahmes' method reported  30/138 = (23 + 6+1)/138 = 1/6 + 1/23 + 1/138. Kevin Brown's 1995 analysis  went beyond Kevin Gong's work by demonstrating that scribal unit fraction patterns were consistent without reporting the meaning and purposes of scribal red auxiliary numbers and other shorthand notes.

One confirming 800 AD text is the Akhmim Papyrus, a Greek text that demonstrated n/p tables from n/3, n/5, n/7,   n/33 conversions to unit fraction series. In the AP's n/17, n/19 and n/23 tables several rational number conversions did not follow Ahmes main 2/n method, seen as:

n/p = 2/p + (n -p)/np

The AP scaled 11/23 by first replacing 11/23 by  6/23 + 7/23. Second, the AP scribe considered:

a. 6/23 (4/4) = (23 + 1)/92 = 1/4 + 1/92
b. 7/23(4/4) = (23 + 4 + 1) = 1/4 + 1/23 + 1/92
c. 1123 = 1/2 + 1/23 + 1/46

The non-2/n table method was used in several  n/17, n/19, n/23 and other n/p table entries.

Another methodology appeared in an 800 year old data base.  Fibonacci re-wrote Ahmes adn Greek unit fraction methods in an algorithm and subtraction context.  demonstrated 2/n-like tables without applying red auxiliary numbers.f  Fibonacci's 1202 AD Liber Abaci (LA) used algorithmic aspects of Ahmes non-algorithm methods. The first 126 pages of the Fibonacci's book covered algorithm conversions of n/p by scaling by LCM m  to awkward unit fraction series within a subtraction context.  Fibonacci's algorithms summarize seven medieval unit fraction methods. Four of the methods indirectly date to the time of Ahmes. Ahmes used a three-step conversion method without applying an algorithm. Four of Fibonacci's seven algorithms were known to Ahmes as non-algorithms (finite arithmetic).  For example, Ahmes first step, the selection of  a multiplication scaled n/p by LCM to mn/mp was reported by Fibonacci as a subtraction step that considered (n/p - 1/m) = (mn -p)/mp, with (mn -p) set to 1 as often as possible.

CONCLUSION
Middle Kingdom  2/n table rules were applied  by Ahmes to 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. Milo Gardner: Egyptian Fraction:s Unit Fractions, Hekats and Wages, an update, 2011
 

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

6. Kevin Gong, 1992 UC Berkeley paper ..

7. Heinz L├╝neburg, Leonardi Pisani Liber Abbaci oder Lesevergn├╝gen eines Mathematikers, Mannheim: B. I. Wissenschaftsverlag , 1993.

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

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

10. 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.

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

12. 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)