The Rhind Mathematical Papyrus (RMP) is an Ancient Egyptian hieratic text that has been dated to 1650 BCE. The papyrus has been housed in the British Museum since 1863. The math text contains a series of equalities called the "2/n table" which represents about a third of the papyrus. The remainder of the papyrus contains 87 loosely grouped arithmetic problems.
The RMP reported fragmented initial, intermediate, final, and proof calculations. In the 2/n table and the majority of the 87 problems, fragmented shorthand notes were misunderstood by Marshall Clagett, Ancient Egyptian Science, Vol III, 1999 and many scholars beginning in 1879. Clagett's 1999 transliteration of Middle Kingdom math texts minimally added back initial and intermediate facts by considering a 1920s additive suggestion .
The 2/n table contained 51 representations of odd denominator rational numbers from 2/3, 2/5, ..., 2/101 recorded by concise unit fraction series. The data was fragmented in ways that confused scholars for 120 years. Scholar proposed to decode scribal shorthand calculations by personalized methods that minimally added back missing scribal data by applying additive suggestions.
This blog systematically adds back additive and non-additive data to expose a wider view of Ahmes' 2/n table theoretical and practical calculations. Added back data returns Ahmes' two red auxiliary number methods by confirming every line of RMP 36 against each other line.
RMP 36: A 2/n TABLE and EMLR DECODING DOOR
A. Background: Ahmes, the RMP scribe, solved 87 problems by applying 2/n table arithmetic and arithmetic, algebraic, arithmetic proportion and geometric methods. All scribal methods were poorly decoded during the 20th century and the beginning of the 21st century. Several under reported 2/n table methods were detailed in RMP 36, and published on-line in 2008 . RMP 36 and RMP 37 disclosed additional scribal red number facts in 2010.
RMP transliteration and translation problems are corrected by decoding fragmented red auxiliary number data from RMP 36 and applying the knowledge to the 2/n table. Red numbers were not recorded in Ahmes' 2/n table nor in the 150 year older Kahun Papyrus 2/n table . However, red numbers were included in several of Ahmes' 87 problems, consistently in RMP 7-20, that exposed sufficient details in RMP 36 to decode every entry in the RMP 2/n table.
RMP 36 was decoded in 2008 that omitted several lines of Ahmes' shorthand notes. A complete decoding of RMP 36 was obtained in 2010 by reading every line of the text. Ahmes' red number method outlined a rational number conversion method that converted five rational numbers 2/53, 3/53, 5/53, 15/53 and 28/53 t summed to a unity value of 53/53, a third n/p conversion method.
Concerning the 2/n table, Ahmes missing shorthand notes exposed 2/53 details adding back in blue. Ahmes' complete info reports 2/53 converted by LCM 30 by:
2/53 (30/30) = 60/1590 = (53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795
There was a second and third red auxiliary number method recorded in RMP 36.The second method converted 30/53 to a unit fraction series by substituting known 28/53 and 2/53 unit fraction series. Note that 30/53 could not be converted by an easy to find LCM. The smallest LCM 2 obtained 60/53 = (53 + 2 + 1)/53 = 56 /53, an incorrect answer. Yet, 28/56 can use LCM 2, and 2/53 can use LCM 30, exposing a method that Ahmes used in RMP 31 to convert 28/97 by substituting 26/97 with LCM 4 + 2/97 with LCM 56.
The third n/p conversion method created virtual n/p tables, one being
2/53 + 3/53 + 5/53 + 15/53 + 28/53 = 53/53 = 1
Any set of unity sums, 3/3, 5/5, 7/7, 9/9, 11/11, ..., 53/53,... p/p defined the third method.
B. RMP 36 solved a rhetorical algebra problem with in a hekat volume unit (the primary MK economic unit). The algebra problem included a discussion of one hekat per:
3x + x/3 + x/5 = 1 hekat
((45 + 5 + 3)x)/15 = 1 hekat
x = 53/15 = 3 + 8/15 = 3 + (5 + 3)/15 = 3 + 1/3 + 1/5
Ahmes' arithmetic details went beyond finding x in terms of a hekat. Ahmes generally reported critical abstract details of his 2/n table construction method.
C. The second bit of information that Ahmes provided was a new form of multiplication and division calculation that inverted 15/53 to 53/15, a property of modern division:
with 30, the least common multiple (LCM) used in scaling 2/53,
and 1060, the greatest common divisor (GCD) used in solving 3/53 after LCM 20 scaled 60/1060
Ahmes' division operation inverted the rational number divisor, and multiplied. This well-known modern property of division was also reported in RMP 38.
D. Ahmes reported 15/53(4/4)= 60/212 =(53 + 4 + 2 + 1)/212 by writing:
(1/4 + 1/53 + 1/106 + 1/212) hekat
Clagett and 20th century scholars oddly agreed that 'single false position' was the only multiplication operation used by Ahmes. From a 'single false position' point of view, Ahmes' selection of the number 30 at the beginning of RMP 36, was thought to have been arbitrarily picked from Ahmes' head. Clearly Ahmes selected 30 from the 2/n table and a well calculated use of LCM 30, and GCD 106.
The 2/n table's complete 2/53 2/n table-type calculation by the explicit calculation of 3/53 per:
3/53 *(20/20) = 60/1060 = (53 + 4 + 2 + 1)/1060 =1/20 + 1/265 + 1/530 + 1/1060 and
(60/1060 - 1/20) = 7/1060 = (4 + 2 + 1)/1060 = 1/265 + 1/530 + 1/1060
E. Ahmes converted 2/53, 5/53, 15/53 and 28/53 by implicitly following the 3/53 conversion method. Note that Ahmes' 3/53 calculation was incomplete in minor ways.
To fill in the 2/53, 5/53, 15/53 and 28/53 conversions to unit fraction series a color blue data element denotes the adding back of Ahmes' missing raw data.
1. 2/53*(30/30)= 60/1590 = (53+ 5 + 2)/1590=1/30+1/318+ 1/795 and
(60/1590 - 1/30) = 7/1590 = (5 + 2)/1590 = 1/318+ 1/795
2. 5/53*(12/12)= 60/636=(53+4+2+1)/636 = 1/12 + 1/159 + 1/318 + 1/63
3. 15/53*(4/4) =60/212=(53+4+2+1)/212=1/4+1/53+1/106+1/212 and
(60/212 - 1/4) = 7/212 = (4 + 2 + 1)/212 = 1/53 + 1/106 + 1/212
4. 28/53*(2/2)= 56/106 = (53+2+1)/106= 1/2 + 1/53 + 1/06 and
(56/106 - 1/2) = 7/106 = (2 + 1)/106 = 1/53 + 1/06
5. sum: 28/53 + 15/53 + 5/53 + 3/5 + 2/53 = 53/53=one hekat (unity)
Note that the rational number 30/53 was not converted. Readers may verify this fact by testing LCM 2 that obtains 60/106. Parsing 60 following the 2/53, 3/53, 5/15 and 15/53 examples reports 53 + 2 + 1 = 56, not reaching 60. Hence 4/106 = 2/53 was missing. Clearly, the difficult 30/53 conversion to a unit fraction series must add back a missing 2/53 by some method.
Ahmes' method to convert 30/53 was used in RMP 31. RMP 31 solved 28/97 by solving two problems 26/97 and 2/97, and combining the two sets of unit fraction series into one unit fraction series. In RMP 36 Ahmes replaced 30/53 with 28/53 + 2/53, though he did not combine the unit fractions into one series. Ahmes offered an obvious method for solving the difficult 30/53 and 28/97 conversions. Ahmes hi-lights one of the major purposes of the 2/n table. The 2/n table offered a handy set of 2/n series that assisted conversions of difficult rational numbers n/p by substituting (n -2)/p + 2/p before recording a unit fraction series.
Ahmes used the (n -2)/p + 2/p substitution method when difficult rational number conversions were met. In RMP 36 Ahmes met 30/53 and solved the problem by substituting 28/53 + 2/53. In RMP 31 Ahmes solved 28/97 by substituting 26/97 + 2/97 and solving two separate problems.
F. In summary, in the RMP's 87 problems Ahmes rarely calculated complete beginning, intermediate, calculations or answers. Complete proofs were often not present as well, leading math historians up several blind alleys.
To fairly parse Middle Kingdom arithmetic explicit details are required. RMP 35-38 and RMP 66 are examples of 21st century parsed RMP arithmetic.
In RMP 36, the proof side of the discussion was fairly outlined by Marshall Clagett and the Egyptology community. However duplation was misread as Ahmes' primary and only multiplication operation. Many 19th and 20th century scholars falsely concluded that 'single false position was Ahmes' division operation. No where in RMP 36, or any RMP problem, does a single division operation look or act like 'single false position', even in fragmented details.
"Single false position" was an 800 AD Arab method for finding roots of second degree equations. J.J. Sylvester in 1891 may have been the first scholar to inappropriately suggest that 'single false position' was likely an Egyptian arithmetic operation. Sylvester had read Fibonacci's the Liber Abaci's 1202 AD's 7th distinction, that solved a difficult rational number conversion, as Ahmes had done, by applying a two step method. Sylvester misread the medieval arithmetic as using a greedy n-step algorithm method. Fibonacci and Ahmes used a common two-step method to convert difficult rational numbers. Ahmes used no algorithms, or any arithmetic operation that was comparable to the medieval 'single false position' algebraic method. Interestingly Ahmes used a subtraction context (60/212 - 1/4) = 7/212, (60/106 - 1/2) = 7/106 and (60/1590 - 1/30) = 7/1590 within a rational number conversion method that dominated Fibonacci's unit fraction arithmetic.
Ahmes' red number method was only implicitlu used in 2/n tables. Ahmes shorthand omissions caused scholars to guess, often badly, and muddle the historical record. Ahmes' actual 2/n table construction details were unexposed until 2006, and published on-line in 2008 when RMP 36 was read, line-by-line.
A majority of 19th and 20th century Egyptologists correctly identified duplation proofs without directly showing the details of scribal arithmetic calculations, Scholars, as a group, falsely concluded, without citing direct evidence, that 'single false position' division was Ahmes division operation.
To explicitly describe Ahmes' multiplication and division operations, RMP 36 has been parsed in terms of red auxiliary numbers. Confirmation of RMP 36 tentative results have been obtained from RMP 35, RMP 37, RMP 38 and RMP 66.
In the context of parsing Ahmes uses of of red auxiliary numbers surprising arithmetic facts were discovered. In 2008 RMP 36 and related 2/n table construction methods explicitly spelled solutions to a critical problem. Rational numbers n/p were scaled to mn/mp by selecting "optimized, but not optimal" m/m LCMs. Ahmes calculated one red number example in RMP 36, 2/53, and associated four rational number conversions (3/53, 5/53, 15/53, and 28/53).
That is, calculated five n/53 unit fraction series. Stated in meta terms Ahmes scaled n/53 by (m/m) to mn/53m. Unit fraction series were calculated by selecting the divisors of 53m, (usually 4 + 2 + 1) that added to mn, by denoting selections in red only in the 2/53 case.
Related Egyptian fraction method also 'healed' an Old Kingdom "Eye of Horus" binary numeration problem that wrote:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 10 ro
Middle Kingdom scribes reported the remainder 2/64 as 10/320 in hekat problems The rational number 1/320 was named ro in about 40 RMP problems.
In other texts, scribes scaled 2/64 to 10/320 and (8 + 2)/320 = 1/40 + 1/160 writing a complete statement
1 = 1/2 + 1/4+ 1/16 + 1/32 + 1/40 + 1/160
and in RMP 36 by:
one hekat (unity)= 28/53 + 15/53+ 5/53+ 3/53 + 2/53 =53/53
as Middle Kingdom scribes wrote arithmetic statements in unity statements, solving a once impossible "Eye of Horus" problem in the 2/n table. Ahmes also solved 87 RMP problems by writing optimized, but not optimal, unit fraction series within an innovative form of finite arithmetic.
Conclusion: RMP 36 opens a decoding door to Ahmes' uses of red auxiliary numbers. In addition a general 2/n table rule that difficult rational numbers that could not be associated with a scaling factor were solved by replacing n/p with (n- 2)/p + 2/p, as 30/53 solved 28/53 + 2/53 with scaling factors (2/2) and (56/56). The scribal scaling method was introduced to scribal students in the EMLR, and improved upon in 2/n tables cited in the RMP and Kahun Papyrus.
Reference: Hibeh Papyrus 300 BCE - 270 BCE.