Old Kingdom hieroglyphic numbers were defined by one, two, three, or more symbols in a cursive algorithm. Egyptian math was based 10. Nearby Babylonian math was base 60 also used a cursive algorithm. Both cultures rounded off cursive infinite series numbers.
The Egyptian algorithm was binary and rounded off to 6-terms. Babylonians wrote inverse rational number tables (1/p) that rounded off to the nearest composite denominator, i.e. 1/91 was recorded as 1/90, rounded up in this case. Egyptian and Babylonian scribes weights and measures rounded off rational numbers in algebra, geometry and weights and measures problems before 2050 BCE.
Babylonian scribes used the rounded off cursive system for its entire history.
The modified Middle Kingdom (MK) numeration system replaced the Old Kingdom rounded off binary system (after 2050 BCE). The innovative hieratic system exactly solved everyday problems by ciphering each counting number onto one sound symbol. The numeration system recorded rational numbers in exact unit fraction series whenever possible. One exception set pi = 256/81, rounded up (256/81 - 22/7) an error of almost 10/567 (an issue indirectly discussed in RMP 38, a round container volume problem).
A line was drawn over each numeral symbol that denoted unit fractions. MK scribes scaled 1/p, 2/p, ..., n/p rational numbers followed an aspect of Babylonian inverse 1/p tables. Egyptian scaled 2/91 by LCM 70 to 140/6370 such that (91 + 49)/6370 = 1/70 + 1/130, exactly. Hieratic arithmetic generally found the best least common multiples (LCM) m that scaled n/p to to mn/mp. Numerators mn were replaced by red auxiliary numbers (the best divisors of mp) that best found concise and exact unit fraction series.
The innovative Egyptian fraction system corrected algebra, geometry and weights and measures round off errors made in the Old Kingdom whenever possible. Middle Kingdom scribes accurately scaled rational number measurements of grain (in hekat units), in square containers, and made wage payments to classes of workers in a finite arithmetic system.
In RMP 36 Ahmes converted 2/53, 3/53, 5/53, and 15/53 to unit fraction series by LCMs 30, 20, 12 and 4, respectively. Red auxiliary numerators (53+ 5 + 2) and (53 + 4 + 2 + 1) equal to 60 defined the primary 2/n table construction method. When n/p could not be converted by one LCM, i.e. 30/53 and 28/97, two problems (n-2)/p + 2/p defined Ahmes second conversion method. Ahmes proved (2/53 + 3/53 + 5/53 + 15/53 + 28/53) hekat equaled one hekat defined a third conversion method.
Scholars reported fragmented info when decoding Ahmes' arithmetic methods for 100 years. Ahmes' hard-to-read arithmetic operations used finite (scaled) rational numbers, algebraic geometry formulas, arithmetic progressions, geometric progressions, and scaled weights and measures units. 21th century scholars are reporting unified LCM m roles and abstract aspects of several ancient formulas. Scholars are adding back missing scribal steps and other hard-to-read methods an approach that was not available to 19th and 20th centuries. Abstract foundations of Egyptian fraction mathematics are being reported in the 21st century.
Ahmes' primary 2/n table construction method was decoded in 2005. Second and third rational number conversion methods were decoded in 2010. The three rational number conversions methods were:
(1) The primary 2/n table construction method scaled rational number (n/p) by LCM m to mn/mp. The best divisors of denominator mp were summed to numerator mn. Each red number was divided by mp calculated a unit fraction . The sum of unit fractions equaled the initial rational number (n/p).
(2) The secondary rational number method solved otherwise impossible n/p conversions that replaced:
n/p with (n -2)/p + n/p
Examples: Ahmes did not convert 30/53 or 28/97 to one unit fraction series by one LCM scaling factor. Ahmes replaced 30/53 with 28/53 + 2/53 and scaled 28/53 by LCM 2 and 2/53 by LCM 30 (in RMP 36); and replaced 28/97 with 26/97 + 2/97 and scaled 26/97 by LCM 4 and 2/97 by LCM 56 (in RMP 31), and wrote out a combined unit fraction series. This method points out an important use of 2/n tables.
(3) A third class of rational number conversion method scaled unities such as:
2/53 + 3/53 + 5/53 + 15/53 + 28/53 = 53/53 = 1 (unity)
by a virtual n/p table. This class of n/p tables were reported by Greeks, Hellenes and Coptic scribes, pointing out an important historical method. A Coptic Akhmim Papyrus reported a family of n/p tables, n/3, ..., n/33 that extended Ahmes 53/53 method.
Returning to the Middle Kingdom, RMP 36 rational number 2/53 was scaled by LCM 30 to 60/1590 within the complete sentence:
2/53 = 60/1590= (53 + 5 + 2 )/1590 = 1/30 + 1/318 + 1/795
with (53 + 5 + 2), recorded in red, and making clear the 2/n table's primary construction method.
Rational numbers n/p also were scaled to 320 ro hekat units in RMP 35-38.RMP 66. added a second theoretical number tool to the scribal math tool kit.
Egyptian scribes scaled rational numbers to solve four area and volume formulas in RMP 41, 42, 43, 44, 45, 46, 47, the Kahun Papyrus, and MMP 10. Geometry was algebraic in area (A) of a circle formulas, and volume (V) of round granary formulas. In RMP 41 and RMP 42, in the area of a circle formula: pi(R^2), R was replaced by D/2, and pi was replaced by 256/81.
Step one: A = (256/8)1((D/2)^2) = (64/81)D^2
Step two A =[(8/9) D]^2 cubits^2 (algebraic geometry formula 1.0)
RMP 41: d = 9 used algebraic geometry formula 1,0 in this shorthand
Step one: A^1/2 = (9 - 9/9) = 8;
Step two: A = 8 x 8 = 64 cubits^2.
In Moscow Mathematical Papryrus 10 (MMP 10) considered the same formula per:
A = [(8/9)D]^2 cubits^2 used 300 years before Ahmes.
RMP 42: D = 10, height (H) = 10; applied formulas
A = [(8/9)D]^2 and
Step one: A = (10 - 10/9) = 80/9;
Step 2: V = (80/9) x (80/9) x 10 = 64000/81 = (790 + 10/81)cubit^3
Step 3: V= 1/2 x (790 + 10/81) = (395 + 5/81) + (790 + 10/81) = (1185 + 10/81)khar (in formula 1.2).
V = (H)[(8/9)D]^2 cubits^3 (algebraic geometry formula 1.1)
V = [(3/2)((H)[(8/9)=D]^2] khar (algebraic geometry formula 1.2)
In RMP 43 Ahmes may have scaled both sides of (formula 1.2) 3/2 obtaining
V = (2/3)(H)[(4/3)]^2 Khar (geometry formula 1.3) =
Robins-Shute (1987) suggested, formula 1.2 morphed in formula 1.3 by scribes obtaining:
V = (3/2)((H)[(8/9)D]^2 khar = (32/27)(D^2)(H) khar = (2/3)(H)[(4/3)]^2 khar = 113 2/3 1/9
Formula 1.3 was used by scribe of the Kahun Papyrus 1750 years earlier with 1-hekat quotients and 1-ro remainders. In RMP 43 Ahmes divided (113 + 2/3 + 1/9 ) khar by 20 and obtained ( 22 + 1/2 + 1/4 + 1/64 + 1/180) 4-hekat . Ahmes converted 1/180 of 4-hekat to 100/180 = 5/9(64/64) = 320/576 = (1/2 + 1/32 + 1/64)4-hekat + [(25/9) 4-ro = (2 + 1/2 + 1/4 + 1/36)4-ro].
Two-part binary quotient 1-hekat and 1-ro remainder statements appear in the AWT, RMP 35, 80, 81, 82 and 83. in total over 60 times. Scholars agree that 20 khar divided by 20 was scaled to one (1) 4 x 100-hekat in RMP 41, 42, 43, 44, 45, 46 and 47, creating 4-hekat quotients. Scholar disagree by reporting 1-ro remainders rather than 4-ro remainders (required by the scribal algebra).
In other words, one (1) khar was equal to 20 hekat based on the length of the cubit and setting one hekat to 4800 ccm, 1/10 of a hekat (hin) to 480 ccm, and 1/320 of a hekat (ro) to 15 ccm.
One under valued two-part 1-hekat quotient and 1-ro remainder data is RMP 83. RMP 83 includes interesting bird feeding rates. The modern scaling of1-hekat to 4800 ccm, 1/hin to 480 ccm and 1-ro to 15 ccm, leave open minor issues of 4-ro, 3-ro and 20 scaled to 60, 40 and 30 ccm reported in RMP 47 and other problems.
The 100-quadruple hekat symbols are fairly read as 400-hekat units. The quadruple hekats symbol appears over 10 times in the RMP. Not one RMP 4-hekat quotient problem can be accurately translated containing 1-ro remainders. Ahmes shorthand style mentioned 4-hekat quotients and implied 4-ro remainders without mentioning the 4-ro term. Only 1-ro remainders seem to be recorded.
Ahmes discussed binary hekat quotient and remainders following the Akhmim Wooden Tablet two-part Q/64 + (5R/n)ro structure. Griffith noticed the RMP 41, 42, 43, 44, 45, 46 and 47 scaling problem. Spalinger (1990) placed the 400-hekat, 4-hekat and 4-ro issue into academic 'limbo'. Spalinger suggested Griffith's approach: read RMP problems in context by considering narratives and formulas to determine scribal values of 400-hekat, 4- hekat, and 4-ro symbols is a valid approach.
Code breakers are parsing rational numbers scaled in pre-800 AD Egyptian, Greek and Arab texts, and post 800 AD texts. These two topics assist in drawing a clear picture of 3,600 year unit fraction math. The earliest scribes scaled rational numbers n/p by LCM m by multiplication and addition operations. Fibonacci in the post-800 AD era scaled (n/p - m) = (mn -p)/mp set (mn -p) = 1 by a subtraction operation, the first of seven rules. When (mn -p) could not be set to 1, ie (4/13 - 1/4) = 3/52 , a second LCM m was chosen, (3/52 - 1/18) = 2/936 = 1/468, defining a seventh rule that was an analogous to Ahmes' second rule that replaced n/p by (n -2)/p + 2/p.
II Old Kingdom Eye of Horus numeration errors were corrected in the Middle Kingdom.
The Old Kingdom (OK) Eye of Horus notation defined
one (1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... (A)
as an infinite series. The binary notation rounded off remainders to 6-terms threw away up to a 1/64 unit.
After 2050 BCE an improved Middle Kingdom (MK) numeration system wrote rational numbers in finite series. Students of the MK system corrected OK round off errors. MK scribes scaled rational numbers n/p to quotients and exact remainders by several methods, one being:
one (1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 10 ro (B)
in grain weights and measures applications. The word ro meant 1/320 of a hekat of grain.
In other situations MK scribes "healed" the 2/64 remainder by scaling 2/64 by LCM 5 to 10/320. Ahmes used the finite notation over 40 times restating
10/320 = (8 + 2)/320 = 1/40 + 1/160 (C)
and, one (1) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/40 + 1/160 (D)
in binary quotients (Q) and scaled remainders (R) statements. A hekat unity (64/64) was divided by n such
that (64/64)/n/ = Q/64 + (5R/n)ro (E)
Divisor n was any rational number n within limited to 1/64 < n < 64
RMP 82 cited 29 hin units, type E, in column 1 all converted to ro units, type in column 2. Additive 20th century scholars misreported RMP 82 data by garbling the scribal definitions.
In RMP 36 Ahmes recorded the identity 53/53 to prove the identity:
one (1) = 28/53 + 15/53 + 5/53 + 3/53 + 2/53 = 53/53 (G)
III MIDDLE KINGDOM TEXTS AND DECODING ISSUES
The 1825 BCE Kahun Papyrus (KP), and the 1650 BCE Rhind Mathematical Papyrus (Ahmes Papyrus) began with 2/n tables. The two texts solved 112 problems within a finite arithmetic system. Additional problems were reported in the 1900 BCE Akhmim Wooden Tablet, Berlin Papyrus, EMLR, Reisner Papyrus, and the Moscow Mathematical Papyrus. The 3,500 to 4,000 year old mathematics expose meta (theoretical) and practical applications. Each level double checked the other level, steps that scribes often reported by hard-to-read shorthand notes.
In 1800 BCE the students of the Egyptian Mathematical Leather Roll (EMLR) scaled 1/3, 1/5, 1/6, 1/7, 1/8, ..., 1/64 to finite unit fraction series by non-optimal methods. Ahmes, an advanced student showed off 2/n tables skills by selecting 14 LCMs m that scaled 2/3, 2/5, 2/7, ..., 2/21, ... 2/101 to concise unit fraction series. The finite system scaled n/p by LCM m to mn/mp.
RMP 36 scaled 3/53, 5/53, 15/53 and 28/53 to 60/53m by LCM m = 20, 12, 4 and 2, respectively. Ahmes selected divisors of denominator 53m that summed to numerator 60 by (53 + 4 + 2 + 1). Ahmes stressed the best divisors of 53m by recording the smaller ones in red.
Trivalent logic of languages (i.e. Aymara) offers a new analysis method outside of bivalent language limitations. Specifics of trivalent logic include hieratic rational numbers. In 2006 the trivalent view of 2/n tables went outside the traditional additive bivalent box, adding back scribal initial and intermediate steps. The updated data suggests that rational numbers n/p were generally scaled by LCM m (m/m) thought of as: mn/mp. The 'best' divisors of denominators mp were written in red that summed to numerators mn that allowed concise (and exact) unit fraction series to be recorded.
Ahmes detailed these red number numerators in RMP 36. The 320 ro hekat context converted 2/53, 3/53, 5/53, 15/53 and 28/53 and wrote one (1) = 2/53 + 3/53 + 5/53 + 15/53 + 28/53.
Ahmes red number numerators scaled 28/53 by LCM 2 to
56/106 = (53 + 2 + 1)/106 = 1/2 + 1/53 + 1/106.
The final unit fraction series was written from right to left without a (+) sign. Ciphered hieratic "letters" 106, 53 and 2 were denoted as unit fractions by lines drawn over each "letter".
It is important to note that 30/53 can not be solved by LCM 2 or any LCM. Ahmes replaced 30/53 with 28/53 + 2/53. Ahmes scaled 28/53 by LCM 2 and 2/53 by LCM 30 a pair of calculations that serious students may wish to duplicate. Ahmes' math tool kit was adaptable in ways that scholars need to appreciate.
In RMP 31, Ames could not find an LCM that solved 28/97. Again, Ahmes choose 28/87= 26/97 + 2/97, two solvable rational numbers, scaled by LCM 4 and 56, respectively, confirming that when n/p could not be converted an alternative method replaced n/p with (n-2)/p + 2/p, thereby establishing a second purpose of the 2/n table. The first purpose of the 2/n table proved Ahmes' arithmetic competency.
The secondary purposes of the 2/n table solved otherwise impossible n/p conversions by substituting
n/p with (n -2)/p + 2/p
and finding appropriate LCMs for (n -2)/p and 2/p and calculated combined or separate unit fraction series.
RMP 37 exposed two additional aspects of red auxiliary numbers. Ahmes aligned an addive set ot numerators that summed to 72 below its respective unit fraction. The scribe converted 1/4 and 1/8 by scaling each rational number by LCM 72 to 72/288 and 72/576. For example, 1/8 was (8 + 36 + 18 + 9 + 1)/576 = 1/72 + 1/16 + 1/32 + 1/64 + 1/576, with 8 placed below 1/72, and 36 below 1/16, and so forth. Secondly, RMP 36 and RMP 37 calculated fractional red numbers, meaning that duplation proofs proved answers without considering the LCM involved.
Weights and measures units and sub-units were scaled in balanced sentences as 2/n numbers were scaled in balanced sentences. For example, RMP 69 scaled 80 loaves of bread made from 3 + 1/2 hekat of grain to a pesu unit. The pesu unit divided 80 loaves of bread by 3 1/2 hekat and created a rational number distribution method. In RMP 69 the pesu method tracked one loaf made from 14 ro (14/320 of a hekat), or 7/160 of a hekat. The proportional pesu method was used in two Berlin Papyrus problems that solved x^2 + y^2 = 100 and x^2 + y^2 = 400, by applying a geometric analogy. Variables x and y were stated as proportional to one another in the ratio of 1: 3/4 and 2: 3/2, seemingly equal proportions. The scribe solved for 2x in the second problem. Schack-Schackenburg reported RMP 69's use of a proportional method in the context of the Berlin Papyrus proportional method years ago. Clagett footnoted the Schack-Schackenbery suggestion, but misreported Ahmes' pesu and Berlin Papyrus inversse division operation by concluding 'single false position' was present (which surely it was not).
Several Egyptian fraction arithmetic operations, and embedded math methods were muddled by 20th century history of math scholars. One misreported arithmetic operation was scribal division.Math historians reported raw data without reporting subtle aspects of ancient scribal notes. Fair readings RMP 69 in the context of the Berlin Papyrus offers a clear example. Scholars tended to lock themselves inside of modern bivalent language boxes by not adding back missing ancient raw data as the scribal shorthand notes required.
The 20th century additive Egyptian fraction discussions minimally added back missing scribal data. Raw scribal ciphered data remained uncorrected on several levels for 123 years, from 1879 to 2002. Peet and the 1920s scholars discussed need for 'ab initio' considerations to decode Ahmes raw data. Peet nor his colleagues offered appropriate corrective suggestions to resolve the 'ab initio' issue.
Peet, Chace, Gillings, et al, frequently suggested that hard-to-read scribal data often included scribal arithmetic and 'copying' errors. Ahmes recorded few typos, much fewer than suggested by scholars. Math historians tended not to question incomplete transliterations that suggested Ahmes made many typos.
Proposed errors mentioned by Peet, et al, contained non-additive data that were not attempted to be decoded. After 2001, math historians compared classifications of Ahmes papyrus mathematics with other hieratic texts, by adding back 'ab initio' LCMs and red auxiliary numbers in the EMLR and the RMP.
For example, a scribal division method contained division answers inverted and multiplied in RMP 35-38, plus RMP 66 . Embedded scribal operations anticipated the modern invert and multiply rule, thereby refuting the long held 'single false position' division suggestion put forward by Peet, et al.
Scribal multiplication and division, as inverse operations was the focal point of RMP 38. RMP 38 calculated 7320 ro times 7/22 and reached 101 9/11 recorded as a unit fraction seried. Ahmes' proof multiplied the unit fraction answer by 22/7 and returned 320 ro, with additional scribal details reported n RMP 35-38, plus RMP 66
The Egyptian fraction body of math knowledge solved modern-like discrete arithmetic, algebra, geometry, arithmetic progressions, algebra,geometry, and weights and measures problems by several scribal methods. The arithmetic accuracy of scribal answers were often proven by an Old Kingdom multiplication method that followed a doubling pattern.The Old Egyptian doubling multiplication pattern was misreported by 20th century scholars as the only multiplication method that was available to Middle Kingdom scribes.
Early scholars were wedded to the idea that Egyptian math was only additive, and that a dual idea of scaling of rational numbers was not rigorously investigated. For example, the invert and multiply forms of multiplication and division reported in RMP 35-38, plus RMP 66 were often overlooked by 19th and 20th century scholars. Generally scribal number theory was not recognized by early scholars, beginning with selections of scaling factors m/m that scaled rational numbers n/p to mn/mp and concise unit fraction series 51 times in a 2/n table, and many other times in Ahmes' 87problems.
IV MODERN ERA DECODING ISSUES
German scholars gained access to the hieratic RMP text in 1879. Scholars debated the RMP and hieratic math texts by publishing transliterations of the fragmented Egyptian fraction texts. At the dawn of the 20th century, German scholar F. Hultsch (1895) analyzed RMP unit fraction statements by a non-additive aliquot part method. Hultsch's view was independently confirmed by Bruins (1944). The Hultsch-Bruins method detailed the use of aliquot parts (divisors of highly composite numbers) in creating the unit fraction series in the RMP 2/n table. The details of the selection and use of the highly composite numbers, LCMs remained an unresolved math history issue for 100 years. The dominant 20th century additive community chose to not investigate the Hultsch-Bruins fragmented suggestion.
Western tradition scholars in the 19th and 20th centuries suggested that an Old Kingdom binary method, a Middle Kingdom proof method, was the primary multiplication method. The primary Middle Kingdom multiplication had been hidden in 2/n tables, algebra problems, geometry problems, arithmetic progression problems, and weights and measures problems, a method that was not decoded until the 21st century AD.
Ciphered unit fraction data recorded Ancient Near East wage payments and other math problems in localized math systems for 2,850 years. Greeks, for example, used two numeration systems, one ciphered numerals onto the Ionian alphabet, and a second onto the Doric alphabet. Greek ciphered numerals were replaced in the Ancient Near East by Hindu numerals around 800 AD. Hindu-Islamic numerals recorded unit fraction math for another 650 years in Europe. European unit fractions were discontinued after 1454 AD when the Liber Abaci lost favor. Two base 10 decimal books, one for business and one for science, used few unit fractions ideas after the books were approved by the Paris Academy in 1585 AD. Unit fraction math lasted another 850 years in the Arab world. The last Arab unit fraction text was written in 1637 AD. Arab unit fraction methods disappeared from modern Arabic script during the 17th century.
Seen from the long term point of view, Greeks and/or Arabs modified the finite Egyptian Middle Kingdom (m/m) scaled unit fraction mn/mp notation to a subtraction (n/p - 1/m) to (mn - p)/mp context. Arabs after 800 AD and Fibonacci scaled n/p by (n/p - 1/m) = (mn - p)/mp, with (mn -p) = 1 by an algorithmic method. 20th century scholars minimally added back missing scribal number theory fragments not noticing the unifying LCM m played from 800 AD to the end of the use of unit fraction arithmetic.
Base 10 decimals appeared in Europe (1585 AD) and modern Arabic script in the Arab world (17th century) marking the death of finite unit fraction arithmetic. Difficult scaled rational numbers were discussed by Fibonacci in a 7th distinction (Liber Abaci), (2002 Sigler translation). A second subtraction step allowed the 2,800 older Egyptian conversion method to additively select divisors of denominator mp to summed numerators, creating unit fraction series in an ancient 2/n table manner, a set of facts that J.J. Sylvester misread in 1891 as the greedy algorithm, an ancient Egyptian point that math history scholars in the 1920s translated as the 'method of single false position'.
Several aspects of early translation translation efforts omitted missing scribal initial and intermediate statements. For example, for about 80 years (from 1920 to 2005) scholars, such as Marshall Clagett's 1999 treatise, and Joran Friberg's "Unexpected links between Egyptian and Babylonian Mathematics" 2005 treatise, created valid transliterations of MK data that were limited to additive assumptions. Misleading sets of additive assumptions required scribal arithmetic to be additive in scope.
In general, 20th century scholars, including Clagett, Friberg and Gillings, omitted discussions of non-additive Akhmim Wooden Tablet (AWT) data, and confirming data in the Moscow Mathematical Papyrus (MMP 20), and over 40 RMP problems. Only Gillings added rhetorical algebra, alternando [ (y/x) =(q/p)] and dividendo [(y -x)/x = (q -p)/p], in RMP 73 and RMP 75 as proportional mixes of beer formula raw materials, and a harmonic mean in preparing sacrificial bread in RMP 76 and MMP 21.
For example, the non-additive AWT's five two-part division answers were discussed by Ahmes 29 times in RMP 83, and additional times other problems. Clagett, Friberg, et al, muddled the details and the scope of non-additive information by not adding back missing initial and intermediate steps.
Five AWT calculations and proofs applied a scaled method that began and ended with unity (64/64) as Ahmes cited in RMP 82 and RMP 83 and elsewhere. In one column of RMP 82, unscaled equivalent 1/10 and 1/320 hekat units, hin and ro, were listed as 10/n hin and 320/n ro. The scaled AWT binary quotient and scaled remainders answers were returned to unity (64/64). To understand Ahmes' and the AWT scribe's scaled notation, the division method:
(64/64)/n = Q/64 + (5R/n)ro
with Q a quotient, and R a remainder, needs to be fully parsed.
All five AWT answers were returned to unity (64/64) by being multiplied by the initial divisor n, a point confirmed by Hana Vymazalova in 2002. Vymazalova's (64/64) conclusion corrected two major errors reported by Georges Daressy's 1906 paper on the proof side of the topic.
An additive assumption had fuzzily reported several classes of scribal arithmetic statements, muddling omitted scribal initial and intermediate information. For example, scholars confused the traditional Old Kingdom (OK) duplation proof as the primary MK multiplication operation. The MK proof aspect of Ahmes 87 problems did apply a specialized OK duplation method for returning unit fraction answers to initial rational number statements. The additive assumption misreported the primary MK multiplication and division operations, and the mathematical facts contained in MK algebra, arithmetic progressions, and weights and measures problems, calculations and proofs.
Ahmes' primary multiplication operation looked more like our modern arithmetic operation than the OK duplation multiplication operation. Rigorous decoding efforts meant to create clear translations of MK numerical information were left to 21st century scholars. For example, Ahmes' primary MK multiplication operation was hidden in initial and intermediate arithmetic statements, algebraic statements, arithmetic progression statements, geometry statements, and other data, a fact that has been rigorously parsed in the 21st century in the above situations.
Concerning Ahmes' algebra problems, RMP 24-34 and cubit problems, RMP 35-38, the primary MK multiplication operation was parsed by functioning as our modern multiplication operation functions, inverse to division, and with division inverse to multiplication. Occam's Razor assisted in parsing to the once hard-to-read raw algebra data.
Ahmes' arithmetic progression statements and geometry statements were cleared up in the context of RMP and KP problems. The arithmetic proportion statements were cleared up in RMP 39, 40,64 and the Kahun Papyrus, and geometry statements were cleared up in RMP 53 thru RMP 59.
Looking in a rear view mirror, 20th century transliterations guessed at non-additive mathematical arithmetic operations recorded in the RMP and other hieratic math texts. As translations, the transliterations assumed that the MK data was additive without offering alternative translations of non-additive data. For over 100 years scholars reported fragmented transliterated statements that misclassified the scope and arithmetic building blocks of Egyptian mathematics.
Gaps in transliterated hieratic arithmetic statements, such as marked by RMP 36, RMP 53, RMP 82, and RMP 83 were caused by hard-to-read scribal shorthand notations. Scholars, sometimes short on self-knowledge, mentioned possible scribal arithmetic mistakes, rather than reporting the data as hard-to-read hieratic data, which it surely was.
For example, Chace and his 1927 team of scholars fairly transliterated the RMP, making few errors. When Chace and his team ran into unclear data fragments, such as RMP 36, RMP 53, and RMP 83 (the bird-feeding rate problem). The information in RMP 36, RMP 53. and RMP 83 was fairly classified, from an additive point of view, as unreadable.
Chace and his team did not suspect the theoretical scope of the scribal 2/n table, scribal arithmetic operations and weights and measure unity definitions and applications that lay hidden in the unread RMP 36, RMP 53, and RMP 83 raw data.
After 2001 AD, the 3,500 to 4,000 year old Egyptian mathematical text problems have been corrected by stripping away unit fractions series by replacing equivalent rational numbers written in non-additive and finite ways. Several decoding methods have cleared up the majority of finite arithmetic, and discrete algebra, geometry, and weights and measures fuzziness placing. For example, the Old Kingdom duplation multiplication operation was proven to to have been only a Middle Kingdom proof method, and the not the primary MK multiplication operation.
Another example is outlined by Ahmes' bird-feeding rate problem. The problem was directly solved in 2005 in the context of 30 other MK problems. Fuzziness, once reported in the RMP, KP, AWT and other MK texts, was replaced by clear initial, intermediate, final and proof statements using MK Egyptian (and Greek, Arab and medieval) arithmetic operations. The nearly complete 2001 to 2009 translations are allowing the majority of the Middle Kingdom math problems to be decoded in the context of one or more closely related Middle Kingdom problems. Clear translations are written within updated arithmetic operations that look and act like modern arithmetic operations, a major surprise that history of science and math scholars are digesting.
The KP and the RMP contain additional examples of closely related 2/n tables and arithmetic progression (A.P.) problems that have been fairly translated after 2001. Improved translations of the 2/n tables and A.P. problems, at times, report surprising ancient rational number (vulgar fraction) calculations. The KP and RMP 2/n table and four arithmetic progression problems can now be fully read. Overall, scribal division reported in the texts, assumed by 1920s scholars to have only followed a single false position method, are also fully translated in RMP 38 as inverse to multiplication. In addition, scribal multiplication was inverse to division, anticipating medieval and later era's arithmetic operation definitions.
V RECENT TRANSLATION ISSUES
At the beginning of the 21st century AD, RMP 2/n table initial calculations LCMs began to be exposed by stripping away Ahmes unit fraction series and replacing initial, intermediate, and final rational number calculations. Ahmes' 2/n table construction methods hinted at the use of LCMs. IAhmes' intermediate red auxiliary number calculation steps in RMP 36 the LCMs clearly defined this long misunderstood topic. Divisors of the LCMs, aliquot parts in RMP 36 were consistent with Hultsch-Bruins aliquot part discussion of the RMP 2/n table's construction method of over 100 years earlier.
In 2009, parsing RMP 35-38, plus RMP 38 unexpected 2/n table facts further expose red number numerators in RMP 36. Ahmes' red numbers, and attested 2/n table methods that Ahmes used to convert 30/53 by solving 2/53 + 28/53 with numerators written in red. Ahmes earlier converted 28/97 to a unit fraction series by converting 2/97 + 26/97 (in RMP 31) with numerators written in red. The red numbers defined an intermediate step that was not directly reported in 2/n table calculations. Red number numerators also appeared in RMP 14-34, though direct connections to the 2/n table, and RMP 36, took over 100 years to decode.
Conversions of 30/53, 28/53, 15/53, 5/53, 3/53 and 2/53 in RMP 36 now show that aliquot part numerators were written in red in the same manner as RMP 13-34.The red numbers outlined an ancient rule that assisted scribes to convert difficult rational numbers. The rule allowed n/p rational numbers to consider n/p = 2/p + (n-2)/p prior to conversion. Ahmes converted 2/p, and (n- 2)/p by selecting an LCM (m) for 2/p annd (n-2)/p, before applying the Hultsch-Bruins aliquot part red number step. For example, to convert 30/53. Ahmes selected m = (30/30), and (2/2), to convert 2/53 and 28/53, respectively, with green omitted by Ahmes, such that:
30/53 = 2/53*(30/30) + 28/53*(2/2) = 60/1590 + 56/106
= (53 + 5 + 2)/1590 + (53+ 2 + 1)/106
= 1/30 + 1/318 + 1/795 + 1/2 + 1/53 + 1/106
= 30/53 = 1/2 + 1/30 + 1/53 + 1/106 + 1/318 + 1/795
As RMP 31 solved x = 14 + 28/97 selecting m= (56/56), and (4/4), for 2/97 and 26/97, respectively, such that:
28/97 = 2/97*(56/56) + 26/97*(4/4) = 112/(56*97) + 104/(4*97)
= (97 + 8 + 7)/(56*97) + (97 + 4 + 2 + 1)/(4*97)
= 1/56 + 1/679 + 1/776 + 1/4 + 1/97 + 1/194 + 1/388
meant that Ahmes final answer:
14 28/97 = 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776
was written in the reverse direction, thereby following an interesting rational number conversion method that used 2/n table members.
Knowing a little more of the 2/n tables and rational number conversion methods, let's go onto Ahmes' initial, intermediate, and final algebra and weights and measure calculaitons. Newly parsed RMP facts are fairly summarized by:
A. scribal addition:
1. (n/p + m/q) = (nq + pm)/pq
2: (1/p + 1/q) = (p + q)/pq
3: (1/p + 1/np) = (n + 1)/np
B. scribal subtraction: (n/p - m/q) =(nq -pm)/pq
C. scribal multiplication:
1.[(n/p) * (m/q)] = (nm)/(pq)
Stripped away were Ahmes unit fractions created from the rule
[(n/p)*(m/m)] = (nm)/(mp) = (mpi + ... + mpj)/nm
with (mpi, ..., mpj) being optimized divisors of denominator mm
2. (n/pq x (m/m) = nm/mpq)
Stripped away were the (q + mpq1 + ... + mpqi)/mpq, with q and mpq optimized divisors of mpq, Ahmes' conversion steps that computed his final Egyptian fraction answers.
3. Old Kingdom duplation data and methods were not stripped away.
D. scribal division:
1.(nm/pq)/(m/q) = (nm/pq) x (q/m) = n/p.
2. (nm/pq)/10 = (nm/pq)* (1/10) = Q + R
with the Q = quotient and R = remainder statements not stripped away,
3. (64/64)/n = Q/64 + (5R/n)* (1/320)
with Q (quotient) and R (remainder) info defining 1900 BCE to 1650 BCE raw data. Updated RMP classifications of RMP arithmetic of the data are required to replace outdated 20th century classifications to translate the ancient data into modern arithmetic. For example, Don Allen (linked on McTutor) offered outdated classifications that were discussed in terms of (true), (true-and false), or (false). One agreement with Allen's views cites Middle Kingdom, and later, Egyptian math remaining uniform over time. Egyptian arithmetic, weights and measures, algebraic, and geometry formulas structure an early chapter of a 3,600-year story line also remainded uniform over time. Six major false point disagreements, and two minor true-and false disagreements, introduce Wikipedia, Planetmath, and other Egyptian data bases.
A second response updates 21 century classifications of Ahmes' Middle Kingdom proto-number theory methods by stripping away the Egyptian fraction notaiton. Ahmes arithmetic is validated with modern looking arithmetic applications written within red auxiliary and algebraic formulas. One consequence of the updated 21st century classifications is that 20th century "false position" and greedy algorithm suppositions have been discarded from Ahmes' algebraic formula tool kit. Robins-Shute anticipated 21st century classifications by removing the false position supposition with a 1987 book on the RMP (printed by the British Museum).
A third response takes an Occam's Razor view by testing 20th century AD classifications of Ahmes' mathematics. Testing acknowledging 20 century scholars partially understood Ahmes writing style. Ahmes did not state problems, nor cite initial calculations. Scholars understood Ahmes' shorthand often began with answers, hiding initial calculations. Occam's Razor methods reconstruct Ahmes' often missing initial calculations have been translated into ancient and rational number classifications for the reader.
IV NEW (2st Century) CLASSIFICATIONS
A. The RMP 2/n table data selected optimized series applying 14 red auxiliary reference numbers as multipliers. Reference numbers were placed in a Middle Kingdom formula. The formula converted 2/n and vulgar fractions elsewhere in the RMP by applying modern multiplication and division operations within an ancient aliquot part methodology.
Stated in modern terms: the aliquot part methodology converted n/p or n/pq to an optimized Egyptian fraction series. Ahmes mentally, or writing elsewhere, selected a red reference number m, as an element in the ancient formula. The formula considered the divisors of mp, cited as: mp, m1, ..., m, p, p1, ...,pn) such that an optimized (but not always optimal) set of mp divisors added to mn. Optimizing required the selection the largest divisor as its last term, as noted below:
n/p *(m/m) = mn/mp = (best set of additive mp divisors = mn)/mp
For example, in RMP 31 the vulgar fraction 28/97 was solved by separately converting 26/97 + 2/97 by selecting 4 for 26/97 and 56 for 2/97, meaning that:
26/97 *(4/4) = 104/388 = (97 + 4 + 2 + 1)/284 = 1/4 + 1/97 + 1/194 + 1/388
had pondered divisors of 97 and 4: 97, 4, 2, 1 (selecting them all), and
2/97*(56/56) = 112/(56*97) = (97 + 8 + 7)/(56*97)
= 1/56 + 1/679 + 1/776
had pondered divisors of 97 and 56: 97, 56, 28, 14, 8, 7, 4, 2, 1 (selecting: 97, 8, 7)
and writing the 28/97 answer as:
1/4 + 1/56 + 1/97 + 1/194 + 1/388 + 1/679 + 1/776
calculated an optimized unit fraction series (elegant to use Silger's 2002 phrase).
Fibonacci listed seven versions of the formula on pages 123-124 of the Liber Abaci (Sigler's 2002 translation). Ahmes knew at least five forms of the formula, facts that are easily validated by the reader.
B. Problems 1-6: division by 10 by using quotient and exact remainders. The method was used for every Egyptian fraction and weights and measures problem by first selecting a red reference number.
6/10*(1/1) = (5 + 1)/10 = 1/2 + 1/10
7/10 *(3/3)= 21/30 = (20 + 1)/30 = 2/3 + 1/30
8/10*(3/3) = 24/30 = (20 + 3 + 1)/30 = 2/3 + 1/10 + 1/30
RMP 6. (the green can be added by the reader)
9/10*(3/3) = 27/30 = (20 + 6 + 1)/30 = 2/3 + 1/5 + 1/30
Ahmes' shorthand notes began with the answers. RMP 6 began with:
2/3 + 1/5 + 1/30
and offered doubling information within a multplication context. What did this information mean?
1. 2/3 1/5 1/30 = 9/10
2. 1 2/3 1/10 1/30 = 9/5
3. 3 1/2 1/10 = 18/5
4. 7 1/5 = 36/5
Many scholars have suggested an Old Kingdom multiplication method. Ahmes' data was intended to prove that the 9/10 unit fraction series must be be returned to unity by multiplying by the inverse of the answer, in this case 9/10 times 10/9 = 90/90 = 1.
Robins-Shute, 1987, summarized RMP 1-6 garbled information by saying: " No working is included in the RMP to show how any of the identities (answers) were obtained. On the other hand, in all cases, including the examples where the dividends are 1 and 2 , proofs working back from the answer to the problem are given in full."
Two additional steps are needed to complete the proof as math is defined today, namely:
5. 7 1/2 plus 2/3 1/10 1/30 = 36/5 + 9/5 = 45/5 (or 90/10)
6. 45/5 times 1/9 = 5/5, obtaining the desired unity. (or 90/10* 1/9 = 10/10).
The two missing steps offers a minor point of disagreement with Robins-Shute, with respect to the proof, and a major disagreement with respect to the initial calculation.
Concerning Peet, Chase, and Gillings' views of the problem, none had suggested an inverse proof, but rather all suggested an outline of an initial calculation. Hence, two major disagreements are attached to Peet, Chace and Gillings' views of RMP 1-6.
C. Problems 7- 34: Arithmetic and algebraic completion problems, with optimized red reference number answers, eliminating false position in all problems as suggested by Robins-Shute 1987.
1. The entire RMP 7-24 list of problems, including RMP 21, 22 and 23, employed 4-steps that stressed aliquot parts as numerators to solve arithmetic and algebraic problems by writing out optimized Egyptian fraction series.
2. Given that Gillings and Robins-Shute disagree on the status of RMP 21-23, let's begin with its completion problems, a theme that touched all RMP 7-23 problems.
21. complete the series 1/5 + 1/15 + x = 1
LCM 15 was used, says Robins-Shute, gave these as Ahmes' steps
a. 15*(2/3 + 1/15) = 10 + 1 = 11
b. 15 - 11 = 4
c. 15*(1/5 + 1/15) = 3 + 1 = 4
d. 15*(1/3 + 15) + (1/5 + 1/15) = 11 + 4 = 15
e. (2/3 + 1/15) + (1/5 + 1/15) = 1 (proof)
Considering 10 + 1 = 11 as numerators, the following rational number logic took place:
Ahmes' calculations used red, thought at times the color was omitted.
a. Find vulgar fraction: (2/3 + 1/5) = (10 + 1)/15 = 11/15
b. Find missing vulgar fraction: 15/15 - 11/15 = 4/15 = (3 + 1)/15 = 1/5 + 1/15
c. Ahmes' proof: (2/3 + 1/5) + (1/5 + 1/15) = 1
22. Complete 2/3 + 1/30 + x = 1 : find an LCM (30) ans: x = 1/5 + 1/10,
Robins-Shute mentions "ab initio" related to Ahmes' steps
a. 30*(2/3 + 1/30) = 20 + 1 = 21
b. 30-21 = 9
c. 30*(1/5 + 1/10) = 6 + 3 = 9
d. 30*(2/3 + 1/30) + (1/5 + 1/10) = 21 + 9 = 30
e. (2/3 + 1/30) + (1/5 + 1/10) = 1 (proof)
Ahmes' calculation did not use red.
a. Find vulgar fraction: (2/3 + 1/20) = (20 + 1)/30 = 21/30
b. Find missing vulgar fraction: 30/30 - 21/30 = 9/30 = (6 + 3)/30 = 1/5 + 1/10
c. Ahmes' proof: (2/3 + 1/5) + (1/5 + 1/15) = 1
Note: The aliquot parts of 30: 15, 10, 6, 5, 3, 2, 1 optimally found (6 + 3 )/30 = (1/5 + 1/10)
rejecting (6 + 2 + 1) and ( 5 + 3 + 1) alternatives, a minor rule presented in the 2/n table.
23. complete the series 1/4 + 1/8 + 1/10 + 1/30 + 1/45 + x = 2/3
Optimally find a series with LCM 45 (ans. 1/9 + 1/40 )
a. Find vulgar fraction: 1/4 + 1/8 + 1/10 + 1/30 + 1/45 + x = 2/3 (multiply by 45)
b. 45/4 + 45/8 + 45/10 + 45/30 + 45/45 + 45x = 45*(2/3)
c. 11 + 1/4 + 5 + 1/2 + 1/8 + 1 + 1/2 + 45x = 30
d. 45x = 30 - (23 1/2 + 1/4 + 1/8 )= 6 + 1/8
e. 45x = 49/8,
f. x = 49/360 = (40 + 9) = 1/9 + 1/40
Ahmes omitted the last two steps, by only writing the answer 1/9 + 1/40. Gillings did not comment on the 49/360 aspect of the last step, thereby maintaining his 'ab initio' posture.
Modern rational number version:
a. 1/4 + 1/8 + 1/10 + 1/30 + 1/45 = (90 + 45 + 36 + 12 + 8 )/360= 191/360
b. Find vulgar fraction:
c. aliquot parts of 360: 180, 90 45, 40, 20, 15, 9, 5, 3, 2, 1 (find 49)
d. Find missing vulgar fraction: 49/360 = (40 + 9)/360 = 1/9 + 1/40
e. (1/4 + 1/8 + 1/10 + 1/30 + 1/45) + (1/9 + 1/40)
Robins-Shute and Gillings appropriately cite the EMLR as a related document, though neither specified the optimized LCM, red auxiliary, details used in the majority of RMP problems. The EMLR introduced non-optimal LCMs. LCM 7 converted 1/4 and 1/8 in RMP 15,
6. 9/10 = 2/3 + 1/5 + 1/30 (1.e. Ahmes' initial calculation)
9/10*(3/3) = 27/30 = (20 + 6 + 1)/30 = 2/3+ 1/5 + 1/30
as the other algebra problems collected x's and solved in a modern manner:
24. x + (1/7)x = 19
(8/7)x = 19,
x = 133/8 = 16 + 5/8 = 16 + (4 + 1)/8 = 16 + 1/2 + 1/8
25. x + (1/2)x = 16;
(3/2) x = 16,
x = 32/3 = 10 + 2/3
26. x + (1/4)x = 15
(7/4)x = 15
x = 60/7 = 8 + 4/7 = 8 + 4/7(4/4) = 8 + (16/28) = 8 + (14 + 2)/28 = 8 + 1/2 + 1/14
27. x + (1/5)x = 21
(6/5)x = 21
x = 105/6 = 17 + 1/2
28. (2/3)x - (1/3)y = 10; (2/3)y = 10
29. a solution method, not a problem
1 +1/4 + 1/10 = 13 1/2
that scholars classify as a diversion.
31. x + (2/3 + 1/2 + 1/7)x = 33
(97/42)x = 33
x = 1386/97 = 14 + (28/97)
x =14 + (2/97)(56/56) + (26/97)(4/4) = 112/5432 + 104/388 =
14 + (97 + 8 + 7)/5432 + (97 + 4 + 2 + 1)/388
32. x + x/3 + x/4 = 2
(19/12)x = 2
x = 24/19 = 1 +5/19(4/4) == 1 + 20/76 = 1 + (19 + 1)/76 = 1 + 1/4 + 1/76
Ahmes discussed x + x/3 + x/4 = 2, by considering
(144 + 144/3 + 1449/4)x = 288
x = 1 + 1/6 + 1/12+ 1/114 + 1/288
citing in shorthand notes:
1. 1 + 1/3 + 1/2 to get 2
\2/3 1 1/3 152
\1/3 1/2 1/16 76
\1/6 1/4 1/72 38
\1/12 1/8 1/144 19
\1/228 1/144 1
\1/114 1/72 2
a scribal longhand answer (that Ahmes did not mention) ... clearly Ahmes algebra scaled
x = 1 + 5/19 = 1 + 30/288 = 1 + (38 + 19 + 2 +1)/288 = 1 + 1/6 + 1/12 + 1/114 + 1/288
33. x + (2/3 +1/2 + 1/7)x = 37
(97/42)x = 37
x = 1554/97 = 16 + 2/97
with 2/97 (56/56) = 112/5432 = (97 + 8 + 7)/56432
34. x + (1/2 + 1/4)x = 10
(7/4)x = 10
x= 40/7 = 5 + 5/7 (4/4) = 5 + 20/28 = 5 + (14 + 4 + 2)/28 = 5 + 1/2 + 1/7 + 1/14
D. RMP 35- 38, 66
Five problems mirror a 1900 BCE hekat division method. Ahmes began with a hekat and replaced one hekat by 320 ro.
In RMP 35 Ahmes multiplied 320 ro by 3/10 obtaining 96 ro.
3/10*320 = 96
6/10*320 = 192
1/10*320 = 32
sum: 10/10 = 320 ro
In RMP 36 Ahmes began by solving the algebraic statement
3x + (1/3)x + 1/5(x) = 1 hekat,
by LCM 15 such that:
(45x + 5x + 3x)/15 = 1,
(53/15)x = 1,
53x = 15, and
x = 15/53 hekat
as algebra was solved by Greeks, Arabs, in the medieval world, and in the modern era.
As additional supporting evidence, Ahmes also converted 2/53, 3/53, 5/53, 15/53, 28/53 and 30/53 using LCMs with red auxiliary numerators to partition every rational number in an optimized manner.
The exception was 30/53 which converted 2/53*(30/39) + 28/53*(2/2) as the 2/n table was constructed. For example 2/53*(30/30) = 60/(30*53) = (53 + 5 + 2)/(30*53) = 1/30 + 1/318 + 1/795, with 5 + 2 written in red.
Ahmes combined RMP 18-23 completion lessons with RMP 24 - 34 algebra lessons to find x = 15/53 hekat.
Ahmes went on to convert 15/53 to a unit fraction series by considering:
(15/53)*(4/4) = 60/212 = (53 + 4 + 2 + 1)/212= (1/4 + 1/53 + 1/106 + 1/212) hekat.
Ahmes also converted 2/53, 3/53, 5/53, 28/53, and 30/53 to unit fraction series by 2/n table red auxiliary numbers within two proofs.
1. The first 2/n table proof:
a. 15/53*(4/4) = 60/212= (53 + 4 + 2 + 1)/212 = 1/4 + 1/53 + 1/106 + 1/212.
b. (15/53)*2 = 30/53 = 2/53 + 28/53= (2/53)*(30/30) + (28/53)*(2/2) = 1/53 + 1/318 + 795 + 1/2 + 1/53 + 1/106
c. 5/53 = (5/53)*(12/12) = (53 + 4 + 2 + 1)/(12*53)= 1/12 + 1/159 + 1/318 + 1/636
d. 3/53 = (5/53)*(20/20) = (53 + 4 + 2 + 1)/(20*53)= 1/20 + 265 + 1/530 + 1/1060
e. sum: 15/53 + 30/53 + 5/53 + 3/53 = 53/53 = one hekat (unity)
2. The second proof discussed 2/53, 3/53, 5/53, 15/53, 28/53 and 30/53) as parts of a hekat in terms of red auxiliary numbers, and other points, such as"
a. (20 + 10 + 5) scaled 15/53 = (3/53)*5 = 3/53 - 1/20) = (4 + 2 + 1)/212]*5= (20 + 10 + 5)
b. (35 + 1/3) + (3 + 1/3) + (1 + 2/3) + 20 + 10 scaled 28/53 + 2/53 = 30/53
c. (88 + 1/3) + (6 + 2/3) + (3 + 1/3) + (1+ 2/3) scaled 5/53= (5/53)*(12/12) = (53 + 4 + 2 + 1)/636
d. 53 + 4 + 2 + 1 scaled (3/53) by (3/53)*(20/20)= 60/1060 = (53 + 4 + 2 + 1)/1060
e. Each part of 15/53 = (1/4 + 1/53 + 1/106 + 1/212)hekat, namely 3/53 and 5/53 are multiples of 15/53.
The second proof converted 2/53, 3/53, 15/5, 28/53, and 30/53, with 30/53 equal to 2/53 + 28/53 pointing out:
3/53 + 5/53 + 15/53 + 30/53 = 53/53 = one hekat (unity)
A related unity aspect of the problems was mentioned by Peet with 45/53 + 5/53 + 3/53 = 1 hekat. Ahmes' two proofs contained proto-number theory facts not mentioned by Peet or Chace. For example, red auxiliary numbers show that proto-number theory was one central RMP 36 fact.
The proto-number fact allowed n/pq to be converted by solving for 2/pq + (n -2)/pq, when needed (as also discussed in RMP 31 converting 28/97 by 2/97 + 26/97).
A second RMP 36 fact showed that Ahmes' used multiplication and division as inverse operations (as discussed in RMP 24-34).
In RMP 37 Ahmes playfully converted 320*(1/180) = 64/3, 320*(1/360) = 64/72, 320*(1/720) = 64/144, 320*(1/1440) = 64/288 and 320*(1/2880) = 64/576 to unit fraction series such that the unity sum (64/72 + 64/576 = 1 would have meaning.
Ahmes added 1/4 = 72/288 = (9 + 18 + 24 + 3 + 8 + 1 + 8 + 1)/288, with the additive numerators recorded in red ink aligned below an EMLR-like non-optimal (1/32 + 1/16 + 1/12 + 1/96 + 1/36 + 1/288 + 1/36 + 1/288) series (that meant that red integers were inverse proofs of unit fractions). Ahmes also recorded 1/8 as 72/576 with (8 + 36 + 18 + 9 + 1)/576 recorded in red below (1/72 + 1/16 + 1/32 + 1/64 + 1/576), ending a playful problem and proof.
In RMP 38 two rational numbers, (35/11)/10 = 35/110 = 7/22, were multiplied 320, by a doubling ' method citing:
1. Initial calculation
(320 ro)*(35/11) = (320 ro)*(2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10 = 101 + 9/11 ro
(101 9/11 ro) was multiplied by 22/7, and returned one hekat , 320 ro. This class of hekat calculation infers that the traditional Old Kingdom pi value of 256/81 was corrected by considering : " ... that (7/22) and (22/7) were shown and proved to be inverses, and that the AE scribes were skilled and aware of the natural inverse operations of multiplication and division. In effect, the AE were adept at finding reciprocals" (Bruce Friedman)!
In RMP 66 Ahmes divided 10 hekat of fat, written as 3200 ro, by 365, the civil calendar number of days in the year, finding 8 + 280/365, or 8 + 2/3 + 1/10 + 1/2190.
Proof: each of the unit fraction answers were multiplied by 365, citing
8 * 365 = 2920
2/3 * 365 = 243 1/3
1/10 * 365 = 36 1/2
1/2190*365 = 1/6
sum: 3200 ro
RMP 35-38 and RMP 66 data and methods mirrored the 1900 BCE Akhmim Wooden Tablet arithmetic where a hekat was written as a unity (64/64). In the AWT a hekat unity was divided by 3, 7, 10, 11 and 13. All five binary quotient and scaled remainder answers were returned to one hekat. In RMP 35- 38 and 66 multiplied by the initial divisor proving that one hekat was 320 ro, and RMP 66 returned 3200 ro, also meaning that multiplication and division operations were inverse to one another, a property of modern arithmetic operations.
e. Problems 39, 40, 62-68, Arithmetic Progression Problems
RMP 40 solved for the smallest term, x1, in a 5-term series that summed to 100. Formula 1.0, from Wikipedia's Kahun Papyrus analysis, found x5 in a five-term series that summed to 60, with a share difference of 5 1/2, per:
x5 = 11/2(1/2)(5 -1) + 60/5 = 11 + 12 = 23.
Ahmes found: x3 = 60/5 = 12, x1 = 1, x2 = 6 1/2, and x4 = 16 1/2.
The 5-term sum to 100 was solved without finding is share difference 9 1/6. Ahmes computed x2, x3 x4 and x5 using the fact that arithmetic proportions contain paired members. In this case, x3 = 100/5 = 20, and x1 + x5 = x2 + x4 = 40, with x1 = 1 2/3 solved the problem (1 2/3 + x5 =40, x5 = 38 1/3, and so forth).
Ahmes allocated precious metal, loaves of bread between mean, division of fat, grain and cattle, the cattle cited as tribute by the same arithmetic proportion formula:
xn = (d/2)(n -1) + S/n, (formula 1.0)
with xn (largest term), d (difference between shares), n (number of terms in series), and S (sum of all terms in the series). Ahmes was usually given three of the four variables ( xn, d, n and S) and solved for the fourth variable. In RMP 40 Ahmes was given two variables, with a third available as a 1/7 relationship to the sum of the first two-terms to the final three-terms, in a five-term series. Ahmes created a proportional 5-term series that set S = x2 + x3 + x4 of the unknown five-term series, as mentioned above.
RMP 41, 42, 43, 48 and 50 volume of a cylinder and area of a circle problems
Ahmes solved the most difficult problem first, proving A = [(8/9)D]^2 and Volume (V) in terms of pi = 256/81, radius (R) by (D/2) . In RMP 41 diameter (D) = 9 and, RMP 42 D = 10 AND heighth 10 of the cylinder, and the Area (A), D =9 was reportedd
RMP 41: (9 - 9/9) = (8 x 8) = 64 cubits^2
RMP 42: (10 - 10/9) = (64/9 x 64/9) = 6400/81 = 790 10/81 cubit^3
1. Area (A) = [(8/9)D]^2 cubits^2
2. Volume (V) = ([(8/9)D]^2)h = 79 10/81 cubit^3
3. V = (79 10/81) x 3/2 = 1185 15/81 khar
and RMP 43 Ahnes used the Kahun Papyrus used Volume formula scaled by 4/3 to obtain a khar unit by
V = (2/3)(H)[(4/3)D]^2 khar on line 2
4096/9 = 455 1/9 on line 3,
1/20 of 455 1/9= 22 1/2 + 1/4 + 1/180 (not 1/45) on line 4
4090/180 = (22 + 1/2 + 1/4 + 1/180) hekat on line 5
writing the remainder 1/180 of 100 hekat unit to a hekat unit by
100/180 = (5/9) and
(5/9)(64/64) = 320/576 =
(288 + 18 + 9)/576 + [5/288(5/5) = (25/9)ro)
(1/2 + 1/32 + 1/64)hekat + (2 + 7/9) ro, with
Ahmes recorded 7/9 ro as (1/2 + 1/4 + 1/36)ro
CC and khar units scales to hekat units are under review. Schack-Schnackenberg and other scholars that worked with RMP 41, 42 and 43 CC, khar and hekat raw data need to be closely reviewed.
RMP 47, 81- 83 (Division of a hekat problems, 36 examples solved in 2005) are also discussed on the math forum.
RMP 47 includes a complex example dividing 100 hekat, written as (6400/64) by 70. The quotient 91/64 and remainder 30/64 were written in binary (Horus-Eye)(64 + 16 + 8 + 2 + 1)/64 units and scaled 1/320, (150/70)ro unit, quotient 2 and remainder 1/7, respectively. Robins-Shute offered the best understanding of the problem found one correct final answer while garbling the beginning and intermediate steps.
Ahmes mixed two 400-hekat multiplicaitons by n= 1/10 and 1/20, with eight multiplications by 1/30, 1/40, 1/50,1/ 60, 1/70, 1/80, 1/90 and 1/100. Table A details all ten as 400-hekat multiplications into 4-hekat and 4-ro two-part answers (that Ahmes would have written). Table B details all ten as 100-hekat multiplications into 1-hekat and 1-ro tw-part answers with the last eight recorded by Ahmes.
A. 4 x (6400/64) x (1/n) = (Q/64) 4-hekat + (5R/n)4-ro
- 1/10 gives 10 4-hekat
- 1/20 gives 5 4-hekat
- 1/30 gives (3 1/4 1/16 1/64) 4-hekat + (1 2/3)4-ro
- 1/40 gives (2 1/2) 4-hekat
- 1/50 gives 2 4-hekat
- 1/60 gives (1 1/2 1/8 1/32) 4-hekat + (3 1/3)4- ro
- 1/70 gives (1 1/4 1/8 1/32 1/64) 4-hekat + (2 1/14 1/21 1/42) 4-ro
- 1/80 gives (1 1/4) 4-hekat
- 1/90 gives (1 1/16 1/32 1/64) 4-hekat (1/2 1/18)4-ro
- 1/100 gives 1 4-hekat( or,
- 1/10 gives 10 hekat
- 1/20 gives 5 hekat
- 1/30 gives (3 1/4 1/16 1/64) hekat + (1 2/3)ro
- 1/40 gives (2 1/2) hekat
- 1/50 gives 2 hekat
- 1/60 gives (1 1/2 1/8 1/32) hekat + (3 1/3)ro
- 1/70 gives (1 1/4 1/8 1/32 1/64) hekat + (2 1/14 1/21 1/42)ro
- 1/80 gives (1 1/4) hekat
- 1/90 gives (1 1/16 1/32 1/64) hekat (1/2 1/18)ro
- 1/100 gives 1 hekat(
Ahmes reported khar divided by 20 (or multiplied by 1/20) into 400 hekat units by two volume formulas. The 400 hekat and 100-hekat initial divisions byt rational numbers have been fairly translated into one hekat into 4800 ccm, 1/10 of a hekat (hin) into 480 ccm, 4-ro into 60 ccm, and 1-ro = 15 ccm by scholars. Scholars often muddled the scribal 4-hekat and 4-ro intermediate details, a topic that has been corrected above.
RMP 48 offered a 'squaring the circle method that' began with diameter (d) =9, reduced to 80^ 1/2, effectively reporting pi = 22/7 as was also implied in RMP 38.
RMP 53 reports three areas, two of triangles, 45/8 setat, and 63/8 setat, and a third area, 1/10 of 11/8 mh plus 10 mh = 91 mh.
RMP 53, 54, 55
As an aide to reading RMP 53, RMP 54 reported 7/10 of a setat divided by 2, 5 and 10; proved that setat land was restated to 1/8 setats by
7/10 setat = 5/8 setat + 7 1/2 COL
14/10 setat = 11/8 setat + 2 1/2 COL
28/10 setat = 11/4 setat + 5 COL
56/8 setat = 11/2 setat + 10 COL
with a COL (cubits of land) = mh = 1/100 setat
That is, RMP 53's third area:
reported 1/8 setats by cosidering 190/8 mh times 1/100 = 190/800 setat
= 19/80 setat = (10 + 9)/80 setat = 1/8 setat + 11 1/4 mh
RMP 56 to RMP 60 are slope (Seked) problems of a pyramid.
RMP 62, 66, and 69-78 (Robins-Shute adds RMP 82-84 to a "Fair value/exhange, pesu and feeding" category).
Ahmes recorded interesting hekat and ro problem. He often presented complete answers without showing initial calculations. His binary proofs were sometimes complete, but, most often they were not. RMP 66, for example, divided 10 hekat of fat written as 3200 ro by 365, the number of days in the year. Ahmes found the theoretical/expected daily usage, (8 + 2/3 + 1/10 + 1/2190)ro by showing one intermediate step, 8 + 280/365, taken from his shorthand:
3200/365 = 8 + (280/365)= 8+ (246 1/3 + 36 1/2 + 1/6)/365
= (8 + 2/3 + 1/10 + 1/2190) ro used per day.
The calculation was used to manage inventory items, fat being one. The calculation allowed actual daily, weekly, and monthly usages to be compared against expected usages.
In RMP 69-78 two-level, and higher level inventory controls of grain and other products were reported by a pesu unit. RMP 69 offers and interesting introduction to scribal proportional thinking, a mathematical tool also used in the Berlin Papyrus, as noted by Schack-Schackenburg, to solve two second degree equations using a geometry analogy. The pesu was an arithmetic inverse to the amount of grain in the product. On a macro level about 1/3 of grain was used for bread, and about 2/3 was used for several types of beers, including one with dates as an ingredient.
For example RMP 71 showed the second level use of a pesu. Beer was made in besha unit, 1/2 of a hekat of grain. 1/4 of the besha (1/8 of a hekat) was poured off, and replace by water. The pesu calculation began with a besha composed of 1/2 a hekat such that:
1/2 - 1/8 = 3/8 was returned to a hekat unit by multiplying by 8/3 or
3/8 x 8/3 = 1,
with 8/3 = 2 2/3 recorded as the pesu. This calculation is confirmed in Moscow Mathematical Papyrus (MMP) problems 12 and 16 by two strengths of beer, the first confirming the second level 2 2/3 pesu made from a besha.
RMP 72 (from Scott Williams, the U. of Buffalo )
Rhind papyrus Problem 72. 100 loaves of pesu10 are to be exchanged for a certain number of loaves of pesu 45. What is the number?
The pesu number of a bread determined its strength in inverse order; in particular, pesu 10 is stronger/better than pesu 45 with the number determining some kind of percent of something undesired. Thus the problem reads in our modern sense as "What is (45/10)100?" Obviously its 450. Here's the egyptian solution:
First find the excess of 45 over 10. You get 35. Divide 35 by 10. You get 3 + 1/2. Now multiply 3 + 1/2.
You get 350. Ad 100 to 350, and get 450.
The above info is algebraically correct since Ahmes was thinking rhetorically, the algebra of his era. Generally, Ahmes asked " Suppose we have x loaves of pesu p and y loaves of x loaves of pesu p is to be exchanged for pesu q".
A small number of RMP pesu problems began with 15 or 16 hekats, and removed 5 hekats for making bread, 100 or 200 loaves of 10 pesu or 20 pesu. The remaining hekats, 10 and 11, were used for one or more beers, a methodology that was repeated in the MMP.
Gillings (Mathematics in the TIME OF THE PHARAOHS) showed that alternando and dividendo from modern algebra were used in RMP 73 and 75 (for proportionally mixing beer ingredients). The modern algebraic methods were also used in the MMP. Another special formula, a harmonic mean, mixed sacrificial bread in RMP 76, a point confirmed by Gillings by MMP 21.
RMP 79 (from Scott Williams). There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all the enumerated things
|heads of barley|
|hekats of barley|
RMP 82 cites 29 examples of one hekat, written as 64/64, divided by 29 rational numbers n in the range 1/64 bird-Feeding rate problem was not read by Chace, nor by other 20th century scholars. Today the Akhmim Wooden Tablet method allows a clear view of a hekat, written as 64/64, divided by 6, 20 and 40 (by multiplying by 1/6, 1/20 and 1/40) calculating 5/8 of a hekat eaten by all birds in one day.
VI UPDATED 21ST CENTURY AHMES PAPYRUS CLASSIFICATIONS
Ahmes' red auxiliary number operations are consistent with modern arithmetic formulas. Validation of the use of modern arithmetic operations requires rigorous reviews of every formula and math operation employed by Ahmes.
Compared to a 21st century classification of Ahmes math operations, Allen's traditional classifications (listed 1.0 to 11.0) inadequately explain Ahmes' 2/n table methods, and arithmetic operations and functions contained in Ahmes' algebraic and geometry formulas.
1.0 Allen's first point "Egyptian mathematics remained remarkably uniform throughout time".
Analysis: But what were the unifying elements of Egyptian fraction mathematics? About 3,600 years of Egyptian fraction academic use ended around 1600 AD with the rise of base 10 decimals, published by Stevin. Previously, unit fraction number theory building blocks: positive prime numbers (including 1), rational numbers, LCMs and modern arithmetic operations were hidden in medieval Latin texts, Arab language texts, Greek script, and hieratic (Egyptian) texts. Decoding Egyptian arithmetic operations has been difficult. Distorted representations were created by unintended 20th century barriers. One barrier suggested that Ahmes' arithmetic was only additive in scope.
It is clear that Ahmes wrote incomplete proofs. Scholarly decodings of Ahmes' mathematics have created muddled additive translations for about 100 years. For example, Ahmes' red auxiliary reference numbers were used to calculate the 2/n table. A meta red reference number formula was not published until 2008. Previously, scholars reported aspects of the formula, missing the larger formula. Gillings and Robins-Shute only reported the multiplier. The complete 2008 translation shows a four-step method to convert a vulgart fractions n/pq:
1. select a red reference multiplier m.
2. multiply the vulgar fraction by the multiplier n/pq *(m/m) = (mn/mpq)
3. replace the numerator (mn) by a set of additive aliquot parts of the denominator (mpq)
4. divide aliquot parts by mpq to write optimized unit fraction series.
Scholars stressed the additive aspects of step 3, the aliquot parts, by only reporting a doubling method. That is, scholars did not report Ahmes' contextual method: selections of LCMs m, multiplication steps, selections of aliquot parts, and division steps.
Scholars fairly reported that Ahmes did not state formulas. Several scholars reported that Ahmes shorthand proofs omitted logical steps, and critical facts. Recreating Ahmes initial calculations and his formulas may have been completed in 2008.
Ahmes often had answers in hand. Inverse logic allowed Ahmes to prove answers arithmetically accurate. Ahmes did not prove formulas. For example, red auxiliary numbers show that a formulas assisted to write out optimized Egyptian fraction series. Few intermediate steps were detailed by scholars prior to 2008.
F. Hultsch in 1895 pointed out an aliquot part aspect, a point of view that was confirmed by E.M. Bruins in 1944. It would take 64 years to publish broad understandings of the red reference numbers. Several points of view assisted in parsing Ahmes Papyrus' 2/n table construction method. Hultsch's point of vew was overly abstract. But it pointed out an important feature of Ahmes red auxilinary method.
Additional decoding techniques have parsed the majority of Ahmes' original formulas. In 2008 red reference numbers were documented in the 2/n table, and applied in RMP 7-38 arithmetic and algebra problems.
For example, Ahmes red auxiliary calculation of 2/7 in the 2/n table used four (4) as its red auxiliary number. But how was the reference number selected to optimized unit fraction series that represented 2/7?
Ahmes provided this information:
2/7*(4/4) = 8/28 = (7+ 1)/28 = 1/4 + 1/28
followed by an initial calculation that Ostracon No. 53 alternatively selected six (6) as its red auxiliary reference number reporting:
2/7*(6/6) = 12/42 = (7 + 3 + 2)/42 = 1/6 + 1/14 + 1/21
Ahmes did not use a plus (+). A plus (+) sign, and (*) for multiplication, will be used for the convenience of new readers of the hieratic numeration convention.
Gillings and others had muddled Ahmes' proofs as 'ab initio' calculations. On page 87 "Mathematics in the Time of the Pharaohs" Gillings says:
" 2/7 = 1/4 1/28
The interesting thing about this ostracon is that the decomposition calculated on its is the unexpected 3-term one
2/7 = 1/6 1/14 1/21
and the author's technique in deriving it throws light upon the Egyptian method of using a reference number together with red auxiliaries when adding fractions. The decomposition is
1.1 1 1/7
1.3 1/2 1/6 1/14 1/21 (but the data cites information taken from an undefined source)
1.4 3 1/2 1 1/2 1
1.5 4 1/2 1/14
1.6 10 1/2 1 1/2
Whoever inscribed the ostracon was doing just what the scribe of the RMP did in problems 28, 32, 32 and several others. The red beneath the 2/7 means "Take 3 as a multiple of 3, to give the reference number 21." He then multiplied the 2 (of line 3) by his multiplier to give 6 the reference number which he then partitioned 3 1/2, 1 1/2 and 1, each of which divides the reference number 21 in integers, and wrote out in red (line 4),"
Gillings had not noticed the proof aspect of this data, with the answer given at the beginning of the steps, nor other closely related shorthand RMP proofs.
2.0 It was built around addition.
Analysis: Egyptian fraction mathematics was built around four hieratic arithmetic operations. Three of operations had been poorly translated to modern base 10 rational number operations. Traditional transliteration methods stressed additive considerations thereby missing major chunks of data. That is, Egyptian fraction addition, subtraction, multiplication, and division had been incorrectly translated to modern base 10 rational numbers. Only additive fragments had been transliterated, garbling Ahmes' attested arithmetic based formulas and other methods. The remaining hieratic arithmetic operations, especially multiplication and division, decode and confirm many of Ahmes' formulas.
The traditional, and muddled, additive premise allowed 20th century translators to avoid discussing Ahmes' use of modern addition, subtraction, multiplication, and division operations. To correct Ahmes use of addition, subtraction, multiplication and division 20th century scholars oversights must be brought into the discussion.
For example, Ahmes' subtraction operation, decoded from 2/n table patterns, subtracted an optimized LCM, written as 1/m, from 2/n, following the relationship:
(2/n - 1/m) = (2A - n)/(mn) (formula 1.0)
solving for (2A -n) in terms of the divisors of mn (m, n1, n2, ...)
Ahmes infrequently used m as an aliquot part to write: m + n1 + n2 + ... = (2A -n).
Fibonacci used seven subtraction versions of formula 1.0 in the Liber Abaci that converted p/q
(n/pq - 1/m) = (mn - pq)/(mpq) (formula 1.1)
thereby always used Ahmes's 1/m by taking second and third subtractions to solve (mn -pq). To understand Fibonacci, his seven distinctions need to be reviewed closely.
Ahmes subtraction proofs were often implied, but in use before Greek and Arabs passed the arithmetic operations to Fibonacci's seven major subtraction distinctions.
Note the modern 'feel' of both subtraction formulas (1.0 and 1.1). The consistency is rooted in our modern arithmetic operations. For example, Ahmes used modern multiplication, addition, and division operations to define his red auxiliary method. Fibonacci applied seven subtraction formulas by formally inserting cross multiplication and division to solve formula 1.1.
At this point, does the question: Is mathematics discovered or invented", apply to Ahmes? I think that it does since our four arithmetic operations were used by Ahmes, Fibonacci and Stevin well before modern algebraic number theory types rigorously defined them.
In other words, Ahmes' number theory considered n and A as rational numbers and positive prime numbers, as the modern fundamental theorem of arithmetic defines them. Yes, Ahmes worked in a well defined domain of four arithmetic operations, three of which (subtraction, multiplication and division) had been hidden in an Old Kingdom doubling proof (thought for 100 years to define Ahmes multiplication method). It is clear to 21st century code breakers that Ahmes' initial multiplication operation chose an optimized 1/m first partition, which was an LCM, and selected aliquot parts of divisors to create optimized Egyptian fraction answers.
Ahmes' optimized 4-step method in RMP 7-34. Gillings discussed RMP 7- 20 as arithmetic in scope, RMP 24-34 as algebraic, by adding an incorrect false position step (as several scholars had done before him). Robins-Shute proved in 1987 that false position was a false scholarly false supposition.
Sylvester in 1891 misreported Fibonacci's seventh subtraction method as an algorithm.
The entire RMP 7-24 list of problems employed the 4-step method. The method stressed the 3rd step's aliquot part numerators to write out optimized, but not always optimal, Egyptian fraction series, a point seen by Hultsch in 1895, and confirmed by Bruins in 1944.
3.0 Little theoretical contributions were evident. Only the slightest of abstraction is evident.
Analysis: A hekat unity, written as (64/64), was pointed out by Hana Vymazalova, a Charles U., Prague, graduate student, in 2002. In 2005, the theoretical (64/64) theoretical idea was used to decode exact Akhmim Wooden Tablet (AWT)'s five division statements. In total, the hekat unity idea reveals the formula:
(64/64)/n = Q/64 + (5R/n)*(1/320),
with n limited to the range 1/64. A quotient and remainder rule was used in the AWT, and over 40 times in the RMP. The quotient and exact remainder decoding pattern emerges within the first six problems of the RMP. Previously only arithmetic formulas were implied by the 2/n table. The RMPs fist six problem define a division by 10 formula. Its quotient and exact remainder scope is confirmed by the Reisner Papyrus.
4.0 It was substantially practical.
Analysis: Validating more than one exact formula, the theoretical side of Egyptian fraction mathematics quickly comes into clear view. One summary of the newly exposed theoretical formulas states that inexact Old Kingdom binary fraction methods were exactly correctly by finite Egyptian fraction arithmetic methods. A second summary says, beyond a round-off formula for pi, written as 256/81, all finite rational number formulas and their arithmetic operations were exact.
5.0 The texts were for students.
response: (true, and false)
Analysis: The EMLR was clearly a student document. The RMP 2/n table and 87 problems were both student and professional scribe teaching documents.
6.0 No "principles" are evident, neither are there laws, theorems, axioms postulates or demonstrations; the problems of the papyri are examples from which the student would generalize to the actual problem at hand.
response: (true, and false)
Analysis: The identification of exact formulas, and empowering theoretical principles, quickly emerge. For example, the least common multiple (LCM) formula , practiced in RMP 24-27, tests a scribal student's ability to find optimal LCMs, marked by red auxiliary numbers. Note that RMP 24-27 did not use a medieval form of false position, an idea misreported by Sylvester in 1891, by misreading the Liber Abaci's first 124 pages.
7.0 The papyri were probably not written for self-study.
Analysis: The beginning six RMP problems defines a division by 10 method of quotient and remainder division that was used for all remaining RMP problems. The divison by 10 method also was the focus of the Reisner Papyrus. Once understanding division by 10 problems, students would have easily advanced to second classification problems. Close mentoring of each RMP classification was not required, as it surely was required in the EMLR. The under valued EMLR, is an introduction to non-optimal LCMs, as well as a prerequisite to related RMP topics.
Mastering EMLR conversion rules, ancient and modern advanced students can preceed to RMP 2/n table issues, and a use of exact quotient and remainder arithmetic in 87 problems.
8.0 No doubt there was a teacher present to assist the student learning the examples and then giving ``exercises" for the student to solve.
response: (true and false)
Analysis: Ahmes' teachers/mentors were likely available for the 26 practice EMLR problems. Mastering conversions of EMLR 1/p and 1/pq rational numbers, by non-optimal LCMs, AND the RMP's simple notation, described in RMP 7-20, introduce RMP 21-23 and the use of optimal LCMs to solve algebra problems. It is not clear, but likely, that Ahmes independently studied the early classes of the 84 problems, possibly RMP 1-34 Of course, there are later classes of RMP problems, RMP 35-84, that may have required the assistance of mentor scribes.
9.0 There seems to be no clear differentiation between the concepts of exactness and approximate.
Analysis: The whole body of Egyptian fraction mathematics clearly separated the Old Kingdom's infinite series arithmetic (Horus-Eye binary fractions) from Middle Kingdom finite arithmetic (Egyptian fractions). Of all of Allen's poor evaluations, this topic may mark Allen's most serious "error".
10.0 Elementary congruencies were used only for mensuration.
Analysis: The scope of Egyptian weights and measures was marked by two classes of exact units. The first unit was marked by the divisions of a hekat unity value (64/64) by divisors n. The first class of unit limited n to the range of 1/64. A Catala language summary of Ahmes classifications is found on Wikipedia. An English translation of Gillings' 1972 classifications are substituted. This blog, therefore, has compared, and corrected Gillings' 1972 incomplete views while also correcting Allen's views on the topic. Robins-Shute's 1987 views, and other 20 century views, including Chace, 1927 and Peet 1923, are mentioned as notes in Gilling's traditional additive view.
VII OUTDATED 20TH CENTURY AHMES MATH CLASSIFICATIONS, Gillings 1972
A. Problems 1 to 6 : Division by 10 problems
Unclear connections to the Reisner Papyrus were made
B. Problems 7 to 20: multiplication of fractions
These red auxiliary 'multipliers defined the selection and use of red reference numbers.
Gillings compares these problems to EMLR identities, and not much more. Robins-Shute
addes problems 21 to 23 into this category, since 7-23 can be seen as completion problems.
C. Problems: 21 to 23: Completion problems and the Red Auxiliaries
Two of the three red reference numbers used in these three problems were identified, with RMP 23's use of 360 in his calculation was confused by a use of 45 to explain all butt two of the answere's last unit fraction terms.
D. Problems 24 to 27: Equations solved by false position
E. Problems 28 to 29: Think of a number
F. Problems 30 to 34: Complex linear equations solved by division
G. Problems 35 to 38: Complex linear equations solved by false position
H. Problems 39 to 40, 62-68, 79: Arithmetic progressions
I. Problems 41 to 46: Volume formula
J. Problem 47: Table of Horus-Eye fractions
K. Problems 48 to 55: Formula for triangles, rectangles, trapezoids and circles
L. Problems 56 to 60: Volume and slope formulas for pyramids
M. Problem 61: Table of 2/3 unit fractions
N. Problems 69 to 78: PESU problems
O. Problems 79, 86, 87: Recreational mathematics
P. Problems 80 to 81: Binary related tables of Hekat and Hinu measures
Q. Problems 82- 84: Other hekat unit topics
R. Problem 85: Unknown script
S. Summary: Gillings accepted the often repeated suggestion that Ahmes and Egyptians had not developed formal mathematics with statements and proofs. Gilling, at times, disagreed with Ahmes' harshest modern critics. But Gilling's conclusions misidentified Ahmes shorthand duplation proofs following other 20th century scholarly 'ab initio' incorrect suggestions.
The RMP duplation proof method was first an Old Kingdom 'doubling' multiplication method. Gillings fairly mentioned aspects of Ahmes shorthand proofs related to RMP 4, 23, 28, 41, 43, 44, 46, 55, 61b, and 66. Overall, Gillings had not identified Ahmes' 'ab initio' arithmetic operations, such as aliquot parts, that assisted the conversion of vulgar fractions p/q to optimized quotient and exact unit fraction remainders.
The exact remainders can be seen as causing stomach ulcers to 20th century Egyptian fraction researchers. For most of the 20th century stomach ulcers were treated as an imbalance of acids. A contrary position, long thought to be impossible, that bacteria cause ulcers, was affirmed in 1984. Johns Hopkins University President Dr. William Brody gave a valid overview of an "outside the box" pedagogical position that is necessary in science. Dr. Brody's "Uncommon Sense & Innovation" offers an approach that has been needed to fully read Egyptian mathematics. New classifications have recovered over 40 examples of Ahmes exact units of measures, all empowered by proto-number theory, the later long thought to have been impossible. Ahmes' exact unit method were applied in Egyptian medicine, creating a pertinent parallel, connecting ancient Egyptian medical measurements to modern scientific methods.
One 21st century analysis of Ahmes 87 problems strips away hard-to-read Middle Kingdom Egyptian fraction notation. Scholars in the 20th century transliterated hieratic data within a muddled additive unit fraction context without decoding and translating subtle non-additive facts. Stripped of confusing unit fraction answers Ahmes' raw data fragments reveal easy-to-read initial, intermediate, and final calculations written within a non-duplation multiplication method. Ahmes multiplication method looked more like our modern arithmetic definition than the method reported by 20th century scholars in duplation proofs.
A trivalent Aymara syntax first solved the general translation of European languages problem by capturing embedded bivalent meanings in language one before creating final language two translations. Clear Egyptian math text translations to modern arithmetic are achieved by following a parallel path. Captured Egyptian fraction data adds back initial and intermediate statements, assisted in translating once fragmented raw hieratic data into readable modern arithmetic statements.
The cross cultural method emerged in 1990s language studies of Aymara, a Bolivian language. Aymara applied a trivalent syntax that captured bivalent data without losing embedded meanings. Clear translations of hieratic texts are achieved by directly, indirectly, and by hearsay solving a well known, but seldom solved 'excluded middle' translation problem. Historically, Esperanto, an artificial bivalent language failed to solve the hieratic excluded middle problem. Aymara and artificial trivalent languages solved several ancient translation problems.
Weights and measures units used in the RMP and the Middle Kingdom contained several classes of errors. The most serious errors are connected to 20th century scholars not recognizing and utilizing the theoretical and practical properties of the MK Egyptian fraction system. Theoretical weights and measures, arithmetic, algebra and geometry 'initial calculations' assisted scribes in double checking the accuracy of 'practical' unit sizes. Interdisciplinary teams are being established to fully translate/decode the Ebers Papyrus, a medical text, and related texts in a double checked manner, as scribes understood them.
Considering the 3,700 year life of the Egyptian fraction system, few theoretical changes in the system were introduced from 2,000 BCE to 800 AD and 1454 AD, though notational changes were implemented by Greeks, Arabs and others. Arab scribes replaced ciphered hieratic and Greek numerals with base 10 numerals while substituting a subtraction method (n/p - 1/m) for Ahmes' 2/n table written with a multiplication and aliquot part method (n/p*(m/m)= nm/pm). Fibonacci reported the medieval version of the Egyptian system in the Liber Abaci. With the fall of the Byzantine Empire in 1454 AD, the Liber Abaci fell out of general use, as did the Egyptian fraction system, after 1637 AD, reported by the last (known) Arabic text.
By 1600 AD, Stevin's 1585 AD decimal system, written within an Arab algorithm, erased the majority of the ancient building blocks of Egyptian fraction arithmetic, algebra, geometry, and weights and measures. Today, reminders of the greatness of Egyptian fraction mathematics, last used in an Arabic 1637 AD text, can be inspected via fragments of ancient aliquot part methods, arithmetic proportions, algebra , and geometric series reported in over 200 ancient problems, written in a dozen ancient texts. The richest text to read is the RMP and its 87 problems.
IX On-line publications, with several citing academic references:
1. Why Study Egyptian Mathematics?
i.e. Ahmes Papyrus(blog); (Planetmath)
2. Akhmim Wooden Tablet (blog) ; (Wikipedia); (Mathworld)
3. Ahmes Bird-feeding rate problem (Planetmath)
4. Berlin Papyrus (Planetmath), (Wikpedia)
5. Economic Context of Egyptian Fractions (Planetmath)
6. Egyptian Mathematical Leather Roll (blog); (Wikipedia); (Mathworld); (PlanetMath)
7. Egyptian fractions (Planetmath), History of Egyptian fractions (blog), Vulgar fractions (blog)
8. Egyptian Fractions, Hultsch-Bruins Method (Planetmath)
9. Egyptian Geometry (Planetmath)
10. Egyptian multiplication and division, Wikipedia
11. Heqanakh Papyri (Wikipedia)
12. Kahun Papyrus (Wikipedia), (Planetmath)
13. Least Common Multiples, modern and ancient (blog)
14. Red Auxiliary Numbers (Wikipedia), (Planetmath)
15. Reisner Papyri (blog); (Wikipedia)
16. RMP 2/n table, (blog) Wikipedia, RMP 36 (Planetmath, Open University forum)
17. RMP 47 and the hekat (Planetmath)
18. RMP 35-38 and RMP 66, RMP 36 (Planetmath, Open University forum)
19. RMP 53-55 (blog)
20. RMP 69 and the Berlin Papyrus (Planetmath)
21. Remainder Arithmetic (Planetmath)
22. Remainder Arithmetic vs Egyptian Fractions (Planetmath)
23. Corrective comments to a Feb. 2010 BBC radio program #17 "The Rhind Mathematical Papyrus" in a"History of the World in 100 Objects" series are also available on a BBC discussion group. The 15 minute BBC broadcast interviewed staff of the British Museum and a Babylonian academic in a manner that degraded Egyptian arithmetic into additive algorithms. The actual RMP arithmetic, algebra, geomety and weights and measures contained early number theory that followed 250 to 350 year older Akhmim Wooden Tablet, Egyptian Mathematical Leather Roll, and Kahun Papyrus traditions.
24. Webinar, a one hour video response to the Feb. 2010 BBC program, was recorded on July, 2010 was offered to the math education community.
25. Oldest Puzzle, per the New York Times
B. Classical Greek
26. Plato's Mathematics (Planetmath), Math Forum
27. Archimedes' Calculus (Planetmath)
28. Hibeh Papyrus (Planetmath)
29. Arabic Numerals (Planetmath)
30. Liber Abaci (blog); (Planetmath)