The Arithmetic used to Solve an Ancient Horus-Eye Problem

The Arithmetic used to Solve of an Ancient Horus-Eye Problem By: Milo Gardner ABSTRACT Five Egyptian Middle Kingdom two-part statements have been found to contain overlooked historical information. The raw two-part statements had puzzled scholars for 80-130 years. The puzzling statements have recently been translated into modern base 10 fractions in a manner that discloses overlooked scribal arithmetic definitions. The historical information also shows that an ancient Horus-Eye problem had been solved. Apparently scribes had exactly divided a hekat (a volume unit) by rational numbers between 1/64 and 64. Thirty-six 1650 BC two-part statements created from the divisions confirm that an ancient Horus-Eye problem had been solved 350 years earlier. Scribes went on and converted the two-part statements to one-part statements. Two of the one-part statements, a hin unit and a ro unit, are analyzed in a manner that reveals secondary scribal methods. The parent two-part statements reveal that arithmetic definitions had played a theoretical role in ancient Egyptian mathematics. 1. THE HORUS-EYE PROBLEM 1.1. Egyptian unit fraction arithmetic was written in two scripts—hieroglyphic and hieratic. Scribes writing at any time period wrote Horus-Eye numbers two ways. Whenever a binary number was less than one its hieroglyphic version followed the rule 1 = ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64. This form created a single set of no more than six Horus-Eye fractions. The resultant stand-alone series was truncated after the sixth term, creating a rounded off number, where up to 1/64 was thrown away creating an inexactness problem for ancient scribes. When scribes worked with hekat data they divided the volume unit by rational numbers ranging from 1/64 to 64. 1.2. An innovative form of Horus-Eye arithmetic was devised by 2,000 BC. The method solved the ancient roundoff problem. Scribes formally wrote Old Kingdom Horus-Eye numbers in Middle Kingdom hieratic script by adding a correctional Egyptian fraction number without applying the traditional round-off method. 1.3. To detail a mythic aspect of the Horus-Eye round off problem a step back to the dawn of Old Kingdom numerals when hieroglyphic numbers can be made. Early hieroglyphic numerals were mapped one number-tomany symbols in Roman numeral type of structure. Near the earliest recorded date a myth emerged according to Spell 17 of the Book of the Dead. The myth reported that the splintered wedjat eye of the falcon-headed Horus was eaten by Seth and restored to life by the ibis-headed god Thoth. From that point on Thoth became the mythic originator of Egyptian mathematics (Robins & Shute 1987:14). Thoth, the recorder of the scales of truth, married Maat and had eight children, the most important being Amun. 1.4. By the Middle Kingdom hieratic script improved the methodology of writing Egyptian numerals by minimizing the number of symbols needed to express each number. Hieratic numeration reported an innovation that mapped one number-to-one Egyptian symbol. Scholars have titled the innovation ciphered numerals (Boyer 195:12). Fractions were written with a line over each ciphered numeral. 1.5. Starting 1000 to 1500 years after the birth of hieratic numeration, one of Egypt’s trading partners, Greece, and their scribes adopted the hieratic method of writing numerals. Greeks ciphered their numerals and fractions onto Ionian and Dorian alphabets, with ½ written as beta’, and so forth (Eves 1961:14). 1.6. To present readable unit fractions in the present paper, a modified Greek notation, with n being a modern number, e.g., ½ = 2’, is used. That is, Horus-Eye numbers are written in this paper as 1 = 2’ 4’ 8’ 16’ 32’ 64’, with the addition sign (+) omitted between fractions, though the addition sign (+) is included whenever scribal two-part statements or modern fractions are reported. 1.7. The purpose of this paper is to show that the Horus-Eye problem was exactly solved when rational number remainders were written as Egyptian fractions and added to Horus-Eye quotients under limited circumstances. An analysis of the two-part numbers recorded in the Akhmim Wooden Tablet and the RMP represent the core thesis of this paper. 1.8. A second aspect of the Horus-Eye problem’s solution is documented in the EMLR. The EMLR scribe, and other scribes writing in the AWT and RMP are proposed to have applied abstract definitions of three numbers. The proposed use of abstract number methods facilitated the solution of the Horus-Eye problem. 2. METHODOLOGY 2.1. This paper presents a new approach that exposes glimpses of the earliest solution to the Horus-Eye problem, one that introduces modern fractions as a decoding key. The approach references forty-one two-part numbers by presenting a forward and reverse analysis, beginning or ending with modern fractions. The results of the analyses are summarized in Appendices I-II. The forward and reverse analyses are proposed to have filled gaps left by scholars publishing on this topic 80–130 years ago. 2.2. The proposed Horus-Eye gaps had occurred because modern scholars had commonly read only segments of the ancient two-part numbers, and other fractional data, usually the additive portions. For example, only the ro aspect of the five problem Akhmim Wooden tablet had been reported by most scholars (Peet, 1923). Scholars therefore had only read an aspect of the AWT, missing its beginning- ending use of a hekat unity (64/64) = 1, and its ancient arithmetic method. Peet, for example, had not seen the critical role of the hekat unity and the related contextual definitions of scribal subtraction and division. The additive scholarly position related to the AWT remained unchanged until 2002 when the beginning and ending use of the hekat unity fact was published (Vymazalova 2002). 2.3. Once the scribal use of a hekat unity was reported scholars began to rigorously read and find missing hieratic arithmetic facts. By 2006, several omitted facts were pointed out and the related missing scribal arithmetic began to be resolved. 2.4. A second area of scribal techniques that are being considered to added back to the scribal body of arithmetic knowledge are its definitions and uses of abstract numbers – potential facts that have appeared in three hieratic texts the AWT, EMLR and RMP. 3. BACKGROUND 3.1. The current scholarly view is that lowly additive forms of arithmetic were primarily used to write record inventories in ancient times when written in either Horus-Eye or Egyptian fraction numeration. Stated another way, scholars have intentionally or unintentional, avoided controversial issues by omitting the confusing lines of texts and suggesting that Old Kingdom and Middle Kingdom mathematical methods had most likely been based exclusively in additive forms of arithmetic (Silverman 1997:94). 3.2. What was the scope of Egyptian arithmetic when its hard to read two-part fragments of data, and other omitted lines, are added back into the body of knowledge of scribal mathematics? This paper proposes that abstract forms of arithmetic had been in use by 2,000 BCE. The proposed abstractions include definitions of numbers built around alternative rational number statements, especially number one (1), the hekat unity, and conversions of 1/p, 1/pq, per the EMLR. In addition, abstract definitions of number are found in 2/p, 2/pq, and by implication n/p and n/pq, when these vulgar fractions were converted to Egyptian fraction series per the RMP and other texts. 3.3. To Middle Kingdom scribal rational number statements were intended to be exact, except when higher order numbers like pi and the square root of two were introduced by approximation methods. The structure of an innovative form of rational numbers had employed both Horus-Eye binary numbers and Egyptian fraction numbers. 3.4. It may be important to note that scholars had spotted the use of finite series in Old Kingdom Egyptian fractions, written in hieroglyphic (Silverman 1975:248). It is unknown, at this time, if rounding off was connected to binary arithmetic. 3.5. What is known concerning Middle Kingdom Egyptian fractions is that it systemically applied rational numbers in ways that included several hard to parse numerical twists. One of the more difficult twists introduced Egyptian fraction remainders reported in the RMP 2/nth table, and other texts. Considering discussions related to these Egyptian fraction twists, professional scholars have, over the last 80-130 years, often been confused concerning the ‘original intent’ and scope of the 2/nth tables. Scholars had prepared conservative additive papers often considering Egyptian fraction mathematics as marked by ‘intellectual decline’ (Neugebauer, 1962). 3.6. However, Middle Kingdom arithmetic contained binary numbers and Egyptian fractions that were generally allowed to contain any size of denominator, clues that marked ‘intellectual advancement’. The Akhmim Wooden Tablet, the Egyptian Mathematical Leather Roll (EMLR) and the Rhind Mathematical Papyrus (RMP) detail snap-shots of large sized Egyptian fraction data. Data from the AWT reveal that five fuzzy twopart patterns had commenced with the hekat unity statement (64/64)/n. 3.6. Data from the EMLR show that four of its 26 lines converted 1/pq rational numbers to Egyptian fraction series by applying a method that may include the use of abstract numbers (Gardner: 2002). 3.7. Data from RMP 47 show four two-part patterns, RMP 80, show 29 patterns, RMP 83 three patterns and the AWT show five patterns. Taken together, these forty-one two-part patterns have often been omitted or erroneously read by scholars. When discussing these two-part patterns scholars have often proposed an additive connection, omitting discussions of potential abstract contents. 3.9. Read in terms of the Horus-eye problem, data from the AWT, EMLR and the RMP draw a clear scribal math road map. Taking all three potential abstract definitions into account, scribes were plausibly not additive in all applications. . 3.10. The suggestion presented in this paper is that the singular practical view of additive Egyptian arithmetic was an incomplete view. The additive view apparently has caused the majority of the scholarly confusion. This paper exposes an innovative form of arithmetic that tends to resolve the several areas of confusion. The innovative arithmetic had been hidden in forty-one AWT and RMP patterns. 3.11. Five example statements are formally parsed in the AWT aspect of this paper, with an additional thirty-six examples summarized in Appendices I, II. The innovative arithmetic solved central aspects of the Horus-Eye problem by providing duality, implementing abstract and practical numbers. Two-way processed data cited in Appendix I summarize the hard to read Middle Kingdom patterns. The newly found arithmetic may have been added to the scribal tool kit as early as the Old Kingdom, but no later than the beginning of the Middle Kingdom. 3.12. Dualism is also inferred by several versions of mythic characters. One is Anpu, Anubis, and Yinepu, names of one monitor of the scales of truth. The myth is dated to the Old Kingdom. Thoth stood by and recorded the results of scales, in the temple and elsewhere. Egyptian mythic characters hint at a connection to the balance beam, with the results recorded by Thoth. Versions of Egyptian mythology may assist in placing the older practical arithmetic alongside the proposed theoretical based arithmetic. 3.13. The present paper proposes that the newly found scribal remainder arithmetic may have been first applied to solve a balance beam problem, the exact weighing of any item with binary weights. Assuming that the balance beam problem had been solved, scribes may have easily applied that knowledge to solve the HorusEye problem, by adding corrective Egyptian fraction remainders to binary series quotients. 4. EXACT DIVISIONS OF A HEKAT, AN ANCIENT METHOD 4.1. Forty-one two-part statements, taken from three hieratic texts, followed a simple but innovative arithmetic structure. A hekat unity (64/64) was substituted for the number one (1). The hekat unity was then divided, or partitioned, by a divisor (n), in terms of a quotient (Q) and remainder ®. Readers are introduced to this arithmetic by first translating its structure in modern decimal fractions, as shown by line 4.1.1. (The symbol * means multiplication has been inserted). 4.1.1 (64/64)/n = Q/64 + R/(n*64) or, when the divisor n = p/q is a fraction try, 4.1.2 (64/64)*(p/q) = Q/64 + R/((p/q)*64) 4.2. A modified version of this remainder arithmetic is provided by Ahmes’ method of calculating a hin unit, 1/10 of a hekat, as cited 29 times in RMP 80, 4.2.1 (640/64)/10 = 64/64 = 1 hin or, converting a hekat unit into a hin unit, of n size, Ahmes shortened his notation, such that: hin data was directly written by: 4.2.2 10/n hin 4.3. Ahmes wrote his second version of this ancient remainder arithmetic by selecting divisors of n (to not exceed 64) to achieve a special form of hekat partitioning. This method included the use of a scaling factor named ro. The method is best stated as: 4.3.1 (64/64)/n = Q/64 + (5*R/n)ro or, when n = p/q, a fraction, try 4.3.2. (64/64)*(p/q) = Q/64 + (5R*(p/q))ro Example: divide a hekat unit by n = 15, and also Ahmes’ equivalent hinu unit as reported in RMP 80 (cited on line 4.3.1): hekat division, or 4.3.3 (64/64)/15 = 4/64 + (20/15)*ro = 1/16 + (1 1/3)*ro and converted to hin data per line 4.2.1; 4.3.4 10/15 = 2/3 hin 4.4. Ahmes used ro within hekat division, omitted ro when writing hin units. Both sets of numbers were placed in a table of equivalent 29 times in RMP 80. This table confirms that the scaling factor ro had been built for a0 special purpose, and that ro was not needed in all hekat division situations. 4.5. One view of the special purpose the word ro is that the remainder term written was an Egyptian fraction series (5R/n), used on line 3.0, was relatively small in size. Note that the smaller vulgar fraction (5R/n) was easier to convert to an Egyptian fraction series, compared to the alternative remainder term (R/(n*64)), had line 1.0 rules been used. 4.6. It should be stressed that the remainder 1/64 term (from line1.0) was replaced by its equivalent value 5/320 to create line 3.0. The 5/320 fraction was factored into (5R/n) and1/320, and 1/320 was replaced by the word ro. These two substitutions converted the remainder term R/(n*64) to (5*R/n)*ro. This odd-looking remainder term had not been recognized nor understood by early scholars that had only considered additive arithmetic. 4.7. One additional reason for this oversight was that scholars had not rigorously translated all of the scribal unit fraction data into modern base 10 fractions. Modern fractions, as Daressy began to write in 1906, therefore are not only needed, but they are required to open the simplest available door to completely decode the scribal arithmetic. 4.8. A contrary view says that a ro was only a volume unit (Peet 1923:93). This point is partially valid, as discussed above. Peet’s position strongly disagreed with Daressy’s emerging view of remainder arithmetic. It is shown that Peet and Daressy both had used a bottom-line approach. Both scholars had not translated all of the top onehalf of the scribal two-part statements (Peet 1923:92–93). Peet and Daressy were both, therefore, unable to see the scribal remainder arithmetic contained in the five AWT problems. 4.9. The forward processed data cites remainder facts that are confirmed by reverse processed data (Gardner 2005). Scholars should apply Sarton’s Occam Razor and its simplicity standard to add the two-part data (Q + R), a step that always reports a simple divisor n, or p/q (Sarton: 1927). This (Q + R) fraction represents the divisor n used on line 3.0, or the fraction p/q used on line 3.1. 4.10. An additional scribal problem is implied by the scope on line 3.0. How had scribes handled divisors larger than 64 when the numerator was not increased? This issue can be studied by investigating the potential specialcase uses of scribal remainder arithmetic. One potential area of investigation is found in Papyrus Ebers data. The so-called medical texts are reported as containing divisors larger than 64 within two dyadic systems. The first system used a Horus-Eye number stopping at the 1/64 term as used on line 3.0. The second number contained a modified Horus-Eye series, one that extended to the 1/128 term. The second dyadic system was used to define a dja hekat measurement unit (Pommerening 2002:2). 4.11. Stated another way, the Egyptian fraction system itself may have been the by-product of the scribal innovation of remainder arithmetic. A proof is suggested by the fact that prime numbers p and q may have appeared early, dividing a weight by a divisor n, and writing the rational number remainder as used in the remainder term R/(n*64). Had this simple Egyptian fraction series been used, for example, within an Old Kingdom balance beam discussion, it may have been the first solution to the Horus-Eye problem. Scribes would then have had over 500 years to develop ro a special-case hekat common divisor as cited in the AWT. A second proof is detailed in RMP 80 and its 29 examples of linked hekat and hin division as summarized in Appendix II. 4.12. A third abstract number proof is found in the Egyptian Mathematical Leather Roll (EMLR). The EMLR remainder arithmetic method appeared four to five times. The method assisted in the conversion of rational numbers 1/p and 1/pq to exact and concise Egyptian fraction series (Gardner 2002:123–24). 4.13. A fourth proof is suggested by the phrase “partial products and remainders” when several RMP problems were recently discussed (Robins & Shute 1987:18). Scholars have mentioned fragments of partial products, quotients and remainders in the RMP, and other texts like the Reisner Papyri for years. But deeper discussions on these topics have been rare. 4.14. This paper proposes that disjointed scholarly views of quotients and remainders can be fairly unified under a remainder arithmetic label. 4.15. In summary, scribes had developed more one special-case solution to the Horus- Eye problem. At present three abstract proofs are associated with two-part remainder arithmetic facts. The two-part remainder arithmetic data are reported as complete arithmetic statements by connecting the facts to modern fractions. 5. MODERN FRACTIONS, A TWO-WAY WINDOW TO THE PAST 5.1. By 2,000 BCE, two-part arithmetic statements had appeared. But what did the statements mean? And how can the statements are fully read in modern terms? The two-part statements consisted of a Horus-Eye series (the first part) plus an Egyptian fraction series followed by the word ro (the second part). The present paper shows that the two-part statements represented an innovative form of arithmetic with Egyptian fractions representing exact corrections to the binary fractions. The two-part statements are fully read by modern fractions by adding the first part (Q) and the second part ®. The sum (Q + R) finds the initial divisor, as listed in Appendix I, in its far right column. 5.2. At least two Middle Kingdom texts, the AWT and RMP, include these two- part arithmetic statements. The first five AWT patterns were first reported over 100 years ago. Typographical errors clouded the readability of two of the statements, divisors 11 and 13 (Daressy 1901). Daressy had only found exact three arithmetic statements, per divisors 3, 7 and 10 (Daressy 1906:64). 5.3. A few years later scholars, thinking in terms of practical arithmetic, had lost the thread of Daressy’s potential remainder arithmetic. Beginning in the 1920s, scholars had been misguided in four ways when attempting to translate the two-part data into modern arithmetic. 5.3.1 5.3.2. 5.3.3. Scholars had under valued the data and reported only a ro volume unit (Peet 1923:93). Scholars had concluded that the data could not be read (Chace 1927:114). Scholars had ignored Ahmes’ data and stressed a lower feature, the hin volume unit—1/10 of a hekat, without finding Ahmes remainder arithmetic (Gillings 1972:250); 5.3.4 Scholars had unnecessarily corrected Ahmes’ data, partially destroying the data’s original meaning (Robins & Shute 1987 :40). 5.4. The typographical problems encountered by Daressy in 1901 and 1906 were independently corrected in 2002 (Vymazalova 2002). The corrected data exactly divided a hekat by 3, 7, 10, 11, and 13, respectively, as given by five two-part statements: 5.4.1 5.4.2 5.4.3 5.4.4. 5.4.5. (4’ 16’ 64’) hekat + (1 2/3) ro (8’ 64’) hekat + (2 7’) ro (16’ 32’) hekat + 2 ro (16’ 64’) hekat + (4 5’) ro 16’ hekat + (4 2’ 13’ 26’) ro (division by 3) (division by 7) (division by 10) (division by 11) (division by 13) 5.5. For an unknown reason, the AWT listed line 5.4.2 and its complete calculation was cited four times. The repetitions included calculating line 5.4.2 and multiplying the statement by seven. Each calculation had therefore found a full hekat for all five two-part statements, as summarized in Appendix I. 5.6. Vymazalova (2002: 27-42) reported another important aspect of the calculation. She showed that a hekat unity had been written as (64/64). Without a hekat unity, the scribe may not have been able to create the two-part statements. 5.7. A second scholar reported additional two-part data mentioned by Ahmes in RMP 83 (Chace 1927:114). Chace concluded that Ahmes had left no clue to read these two-part statements. The data cites three groupings of birds, each group eating unknown daily amounts of grain, reported as: 5.7.1 5.7.2 5.7.3 Three birds each ate (8’ 32’) hekat + (3 1/3) ro One bird at (32’ 64’) hekat + 1 ro Three birds each ate (64’) hekat + 3 ro (divison by 6) (divison by 20) (division by 40) 5.8. Noting that (8’ 32) hekat + (3 1/3)ro = 1/6; (32’ 64’ hekat) + 1 ro = 1/20; and 64’ hekat + 3ro = 1/40, as detailed in Appendix I, Ahmes rewrote the two-part data into singular ro units. Ahmes added the three 1/6 portions, (53 1/3 + 53/13 + 53 1/3) and multiplied by ro obtaining 160 ro. Ahmes then added the 1/20 and the three 1/40 portions obtaining (16 + 8 + 8 + 8), and multiplied by ro obtaining 40 ro. Finally, 160 + 40 were added, with 200 ro being Ahmes’ final answer. Ahmes‘ answer had summed the units to 200 ro, or rewritten into hekat terms 200/320 or, 5/8 of a hekat. 5.9. Stated as modern fractions, ro data shows that it was created from the 320/n relationship, with the divisor n set to 3, 20 and 40 in RMP 83. Modern fractions, the first fraction 1/6 can be added three times 1/20 added once, and 1/40 can be added three times. Summed they total 5/8 of a hekat, per: 5.9.1 5.9.2 5.9.3 5.9.4 (1/6 + 1/6 + 1/6) = 20/40 = 1/2 1/20 = 2/40 = 1/20 (1/40 + 1/40 + 1/40) = 3/40, such that (20 + 2 + 3)/40 = 25/40 = 5/8 of a hekat 5.10. It should be noted that modern fractions select 1/40 as the least common divisor by avoiding the larger 1/120 fraction and even the larger 1/320 greatest common divisor as selected by Ahmes. This paper proposes that Ahmes solved RMP 83 by skipping over the two alternative common divisors for two reasons: 5.10.1 5.10.2 Ahmes may have reserved least common divisors to assist in finding optimal 2/nth table series. Ahmes may have solved the problem by selecting a traditional greatest common divisor. 5.11. A third scholar reported two-part data also cited Ahmes and RMP 47 (Robins & Shute 1987:40). Their book reported “curious” data. The data divided 100 hekat by 70. This problem was solved by beginning with the vulgar fraction 100/70, with Ahmes’ two-part answer taking the two-part form: 5.11.1 (1 4’ 8’ 32’ 64’) hekat + (2 14’ 21’ 42’) ro. Referencing Appendix I, Ahmes had divided 100 hekat by 70, by following these steps: 5.11.2 5.11.3 5.11.4 (6400/64)/70 = 91/64 + ((5*30)/70)ro (64 + 16 + 8 + 2 + 1)/64 + (150/70)ro (1 4’ 8’ 32’ 64’) + (2 7’)ro 5.11.41 with 7’ = 1/7*(1/2 + 1/3 + 1/6) as 7’ 9’ 11’ and 13’ were calculated in the EMLR, and as 101’ was calculated in the RMP. 6. FINAL COMMENTS 6.1. Egyptian record keeping was intended to be accurate. Working towards perfection, and eliminating round off, the Horus-Eye problem was exactly solved during the Middle Kingdom in practical and theoretical situations. The clear use of three abstract number definitions and uses in the AWT, 1 = 64/64, 1/p in the EMLR, and 2/p in the RMP, may have appeared earlier than 2000 BC. One potential source, possibly 2700 BC, may have allowed scribes to find a solution to a balance beam problem. Such a potential solution may have added a remainder correction fraction to Horus-Eye numbers. If a balance beam solution had been abstractly developed another form of unity statement may be found. The potential balance beam solution may have used along side awkward looking binary fractions comparable to those reported by Daressy’s in his analysis of the AWT. The present author suggests that the AWT may be the first known text to use a hekat unity and the word ro within the context of hekat division. The hekat unity and ro innovations are proposed to have been applied to rigorously solve the Horus-Eye problem by at least 2,000 BC. 6.2. Daressy and Peet had reported different sides of the same two-sided AWT subject. Both scholars missed subtle connections to modern fractions. Taken as a whole Daressy’s remainder arithmetic reported exact divisions. Peet’s important contribution was the ro unit. The two-part remainder arithmetic, containing Daressy and Peet’s views, may have given birth to the system of Egyptian fraction as remainders. 6.3 Whatever the final judgment may be with respect to the innovative scribal arithmetic connecting Horus-Eye fractions to Egyptian fractions, by myth or by number, additional study is needed. Scholars are urged to resolve related scribal methods used beyond dividing a hekat unity by any rational number n The study of heket subunits, oipe, dja and so forth, may offer additional scribal remainder arithmetic subjects for discussion. 6.4 Researchers may be able to locate scribal applications that include additional abstract number definitions. Potential unity statement may be found that connect to the balance beam or other problems. Potential unity statements, or other relationships to the hekat, may be found associated within 2,000 pieces of medical text hekat sub-units. This can be done by analyzing the oipe, dja, and other one-part statements. 6.5 Finding a fourth, fifth or even higher definition and use of abstract numbers may reveal additional Old Kingdom and Middle Kingdom forms of remainder arithmetic. At that time, a deepening of the scholarly understanding of scribal arithmetic certainly would take place, intellectually and historically, proceeding well beyond this introductory paper. APPENDIX 1 The following Table 1 and Table 2 information details an innovative form of two-part number. The ancient Egyptian method has been dated to 2,000 BCE. Scribes developed it to generally divide a hekat volume unit by certain rational numbers n. The Middle Kingdom division method introduced the use of a hekat unity, (64/64) = 1. This two-part division method divided (64/64) by n, creating a quotient (Q) in the first-part and a remainder (R) in the second-part. Quotients were written as binary series and its fractions were limited to 1/2 to 1/64. Remainders were composed of vulgar fractions (5R/n) and converted to an Egyptian fraction series followed by a scaling value of 1/320, the later term named by scribes as ro. The two-part division notation was a special case measurement system. It was used when scribes worked with the hekat volume unit. When scribes worked with small units, beginning with a hin, 1/10 of a hekat, a one-part scaled vulgar fraction was written in unit fractions .The hin and its 10/n scaled vulgar fraction is detailed in Appendix II. Scribes may have used the scaling word ro as a common divisor. Factoring ro from the vulgar fraction 5R/(n*320), reduced the size of vulgar fraction that was required to be converted to an Egyptian fraction. Ro was also used as an independent measurement. Table 1 forward engineers data reported in two texts, the Akhmim Wooden Tablet (AWT), 2000 BCE, and the Rhind Mathematics Papyrus (RMP), 1650 BCE. The red data denotes the scribal information translated by Chace and Gillings. The blue data denotes the omitted steps that Chace, Gillings and others had been unable to decipher . Table 2 reverse engineers the red data with the ‘lost’ scribal steps written in blue. In 2004 AD the reverse decoding method was effectively used to unify the red data following the hekat unity idea that was reported in 2002 AD TABLE 1: Forward engineering: Horus-Eye (Q) + Egyptian fraction, including ro, (R) = rational number (Q + R) Five rational number divisors by 3, 7, 10, and 13 are found in the Akhmim Wooden Tablet. Four divisors, 6, 20, 40 and 70 are found in the RMP. All nine hekat divisions followed the remainder form: (64/64)/n = Q/64 + R/(n*64), with Q = quotient, and R = remainder. For example, AWT 1 divided a hekat unity by 3, with Q = 21 and R = 1 based on the remainder arithmetic facts (64/64)/3 = 21/64 + 1/(3*64). The scribe then converted R = 1/(192), as written in blue, by multiplying the information by 5/5 and factoring out a common divisor 1/320. Scribes named the 1/320 common divsior ro. An Egyptian fraction remainder (5R/n), R4, is found in table 1.To better explain the arithmetic of the n = 3 answer (64/64)/3 = (4' 16' 64') hekat + (1 3") ro consider the following five steps: step 1: (64/64)/3 = Q1 + R1 = 21/64 + 1/(3*64) step 2 Q2 + R2 = (16 + 4 + 1)/64) + (5/5)*1/192) step 3 Q + R3 = (4' 16' 64') hekat + 5/960 setp 4 Q3 + R4 = (4' 16' 64') hekat + (5/3) ro step 5 Q3 + R4 = (4' 16' 64') hekat + (1 3") ro The later, Q3 + R4 data written in unit fractions being the only scribal datea reported by Chace and Gillinbgs. Note that step 2 multiplied the remainder term 1/(3*64) by 5/5, or 5/5 * 1/(3*64) = 5/(3*320). Step 2 and Step 3 show that the common divisor ro, 1/320, had been factored, leaving 5/3, or, ( 3”), or (1 3”) ro as written by the scribe. Text AWT 1 RMP 83 AWT 2 AWT 3 AWT 4 AWT 5 RMP 83 RMP 83 Problem (64/64)/3 (64/64)/6 (64/64)/7 (64/64)/10 (64/64)/11 (64/64)/13 (64/64)/20 (64/64)/40 Q1 21/64 10/64 9/64 6/64 5/64 4/64 3/64 1/64 R1 1/(3*64) 4/(6*64) 1/(7*64) 4/(10*64) 9/(11*64) 12/(13*64) 4/(20*64) 24/(40*64) Q2 (16 + 4 + 1)/64 (8 + 2)/64 (8 + 1)/64 (4 + 2)/64 (4 + 1)/64 4/64 (2 + 1)/64 1/64 R2 5/(3*320) 20/(6*320) 5/(7*320) 20/(10*320) 45/(11`*320) 60//(13*320) 20/(20*320) 120/(40*320) Q3 4’ 16’ 64’ 8’ 32’ 8’ 64’ 16’ 32’ 16’ 64’ 16’ 32’ 64’ 64’ R4 (1 3”) ro (3 3’) ro (2’ 7’ 14’) ro 2 ro (4 4’ 11’ 44’) ro (4 2’ 13’ 26’) ro 1 ro 3 ro Q+R 1/3 1/6 1/7 1/10 1/11 1/13 1/20 1/40 RMP 47 (6400/64)/70 91/64 30/(70*64) (64+16+8+2+1)/64 150/(70*320) 1 4’8’32’64’ (2 7’ ) ro 100/70 TABLE 2: Reverse Engineer: Horus-Eye (Q) + Egyptian Fraction, including ro, (R) = fraction (Q + R) Five division data points 3, 7, 10, 11, 13 are found in the Akhmim Wooden Tablet. Four additonal division data points 6, 20, 40, and 70 are found in the RMP. All nine points follow the theoretical two-part form: (64/64)/n = Q/64 + (R5/n)*ro with Q equaling a quotient and R equaling a remainder. For example, AWT 1 divided a hekat by 7 as: (64/64)/7 = 9/64 + 1/(7*64), then the scribe converted the remainder (often mentally) into a common divisor 1/320, a number that scribes named ro. To better explain the scribal method of computation, which may have looked like this to the scribe: A. (64/64)/7 = (8' 64')hekat + ( 2’ 7’ 14’)ro, consider the following example, which: follows a remainder arithmetic structure, computing a quotient (Q) and a remainder (R), a two-part statement: Begin with (8’ 64’) hekat + (2’ 7’ 14’) ro and find the original problem, or (64/64) times Q3 + R4 . B. First, it should be noted, 2' = 32/64. 4' = 16/64, 8' = 8/64, 16' = 4/64, 32' = 2/64 and 64' = 1/64, or: (1) quotient: (8’ 64’) = (8 + 1)/64 = 9/64 (2) Remainder: Egyptian fraction (2’ 7’ 14’) = 5/7, since the remainder including ro = (2’ 7’ 14’ )ro= (5/7) ro = 5/(7*320) = 5/2240 = 1/448 C: Double check the result consists of taking a Q value like 8/64 and an R values, such as 1/448 and adding them: 9/64 + 1/448 = (63 + 1)/448 = 64/448 = 1/7. Text AWT 1 RMP 83 AWT 2 AWT 3 AWT 4 AWT 5 RMP 83 RMP 83 RMP 47 Q3 4’ 16’ 64’ 8’ 32’ 8’ 64’ 16’ 32’ 16’ 64’ 16’ 32’ 64’ 64’ Q2 (16 + 4 + 1)/64 (8 + 2)/64 (8 + 1)/64 (4 + 2)/64 (4 + 1)/64 4/64 (2 + 1)/64 1/64 Q1 21/64 10/64 9/64 6/64 5/64 4/64 3/64 1/64 R4 (1 3” ) ro (3 3’) ro (2’ 7’ 14’) ro 2 ro (4 11’) ro (4 2’ 13’ 26’) ro 1 ro 3 ro (2 7’ ) ro R3 (5/3) ro (10/3) ro (5/7) ro (20/10) ro (45/11) ro (60/13) ro ( 20/20) ro R2 5/960 10/960 5/2440 20/3200 45/3520 60/4160 20/6400 R1 1/192 4/384 1/448 4/640 9/704 12/832 4/12800 24/25600 30/4480 Q+ R 1/3 1/6 1/7 1/10 1/11 1/13 1/20 1/40 100/70 (120/40) ro 120/12800 (150/70) ro 150/22400 1 4’8’ 32’ 64’ (64+16+ 8 + 2 +1) /64 91/64 Appendix II, RMP 80 DATA The history of mathematics first introduced two-part numbers in the modern era by C.F. Gauss in 1796. Gauss wrote complex numbers a + bi as an innovative number that was effectively used to solve the fundamental theorem of algebra problem, thereby naming and exactly finding n roots in nth degree equations. Another earlier form of two-part number was used five times in the Akhmim Wooden Tablet, documented to 2,000 BC in ancient Egyptian. Later scribes like Ahmes, writing in 1650 BCE, generally outlined two-part numbers linking binary and Egyptian fraction series, followed by a scaling factor named ro 36 times to solve hekat problems. The ‘earliest’ two-part number facilitated exact solutions of finding volume measure’s rational number divisor. This innovative two-part number created a quotient (Q) in the first-part, and a remainder (R) in the second-part. Scribes had used a scaling factor 1/320 of a hekat, when dividing a hekat unity (64/64) by a number n no larger than 64. The data in Part I forward engineers 16 scribal red data numbers reported in RMP 80, citing the Ahmes entries. The last entry n = 64, was the highest possible n. Note that five entries did not calculate a remainder, the R terms are vacant five times. The Part II raw data is best seen as converting 13 rational numbers beginning with 3/4 and ending with 49/64 to create 13 two-part hekat numbers. The Part II data contrasts hekat two-part numbers to its equal hinu one-part number. Note that all 13 entries do not calculate a remainder, hence the R terms (R1, R4) are vacant. \ PART1: Forward engineering: Horus-Eye (Q) + Egyptian fraction, including ro, letting n = 2/3 to 64, and converting hekat twopart numbers to one-part hin number by using the relationship 10/n Problem (64/64)/n Q1 Q2 Q3 R1 R2 R4 Hinu System Hin= 10/n Hin One-part number (64/64)*2/3 42/64 (64/64)/2 (64/64)/3 (64/64)/4 (64/64)/5 (64/64)/6 (64/64)/8 (64/64)/10 (64/64)/15 32/64 21/64 16/64 12/64 10/64 8/64 6/64 4/64 (32 + 8 +2 )/64 32/64 (16 + 4 + 1)/64 16/64 (8 + 4)/64 (8 + 2)/64 8/64 (4 + 2)/64 4/64 2’ 8’ 32’ 2’ 4’ 16’ 64’ 4’ 8’ 16’ 8’ 32’ 8’ 16’ 32’ 16’ 2/(3*64) (10/3)*1/320 (3 3’)ro (9 + 1)/3 10/2 (6 3”) hin 5 hin ( 3 3’) hin 2 2’ hon 2 hin (1 3”) hin 1 4’ hin 1 hin 3” hin 1/(3*64) (5/3)*1/320 (1 3”)ro (9 + 1)/3 (8 + 2)/4 4/(5*64) 4/(6*64) (20/5)*1/320 (20/6)*1/320 4 ro (3 3” )ro 10/5 (6 + 4)/6 (8 +2)/8 4/(10*64) 4/(15*64) (20/10)*1/320 (20/15)*1/320 2 ro (1 3’) ro 10/10 10/15 (64/64)/16 (64/64)/20 (64/64)/30 (64/64)/32 (64/64)/40 (64/64)/60 (64/64)/64 4/64 3/64 2/64 2/64 1/64 1/64 1/64 4/64 (2 +1)/64 2/64 2/64 1/64 1/64 1/64 16’ 32’ 64’ 32’ 32’ 64’ 64’ 64’ 24/(40*64) 4/(60*64) (120/40)1/320 (20/60)*1/320 3 ro 3’ ro 4/(20*64) 4/(30*64) (20/20)*1/320 (20/30)*1/320 1 ro 3” ro (8 + 2)/16 10/20 10/30 (8 + 2)/32 10/40 10/60 (8 + 2)//64 2’ 8’ hin 2’ hin 3’ hin 4’ 16’ hin 4’ hin 6’ hin 8’ 32’ hin Part 2: Forward Engineer: Horus-Eye (Q) + Egyptian Fraction, including ro, (R) = fraction (Q + R) Hinu System Divisor Q1 Q2 Q3 R1 R4 Hinu Missing Step Hin One-part number System 10/n (64/64)*(3/.4) (64/64)*(5/8) (64/64)*(9/16) (64/64)*(17/32) (64/64)*(33/64) (64/64)*3/8) (64/64)*(5/16) (64/64)*(9/32) (64/64)*(17/64) (64/64)*(7/8) (64/64)/*13/16) (64/64)*(25/32) (64/64)*(49/64) 3/4 5/8 9/16 17/32 33/64 3/8 5/16 9/32 17/64 7/8 13/16 25/32 49/64 (2 + 1)/4 (4 + 1)/8 (8 + 1)/16 (16 + 1)/32 (32 + 1)/64 (2 + 1)/8 (4 + 1)/16 (8 + 1)/32 (16 + 1)/64 (4 + 2 + 1)/8 (8 + 4 + 1)/16 (16 + 8 + 1)/32 (32 + 16 +1)/64 2’ 4’ 2’ 8’ 2’ 16’ 2’ 32’ 2’ 64’ 4’ 8’ 4’ 16’ 4’ 32’ 4’ 64 2’ 4’ 8’ 2’ 4’ 16’ 2’4’ 32’ 2’ 4’ 64’ (10*3)/4 (10*5)/8 (10*9)/16 (10*17)/32 (10*33)/64 (10*3)/8 (10*5)/16 (10*9)/32 (10*17)/64 (10*7)/8 (10*13)/16 (10*25)/32 (10*49)/64 ( (28 + 2)/4 (48 +2)//8 80 + 8 + 2)/16 7 2’ 6 4’ 5 2’ 8’ 5 4’ 16’ 5 8’ 32’ 3 2’ 4’ 3 8’ 2 2’ 4’ 16’ 2 2’ 8’ 32’ 8 2’ 4’ 8 8’ 7 2’ 4’ 16’ 7 2’ 8’ 32’ ( 160 + * + 2)/32 (320+ 8 + 2)/64 (24 + 4 + 2)/8 (48 + 2)/16 (64+ 16 + 8 + 2)/32 (128 +32 +8 +2)/64 (64+ 4 + 2)/8 (128 +2)/16 (224 + 16 + 8 + 2) (448+ 32+ 8 + 2)/64 REFERENCES Boyer, Carl B. 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