Egyptian Fractions: Unit Fractions, Hekats and Wages - an Update more

EGYPTIAN FRACTIONS: UNIT FRACTIONS, HEKATS, AND WAGES—AN UPDATE Milo Gardner ABSTRACT The present paper updates three ancient Egyptian fraction texts, the Akhmim Wooden Tablet, the Egyptian Mathematical Leather Roll, and the Rhind Mathematical Papyrus. The three hieratic texts were written in the Egyptian Middle Kingdom era (2050 BCE to 1550 BCE), a time of innovation. The paper demonstrates that neglected scribal number theory was written in finite arithmetic. The finite arithmetic scaled rational numbers and commodities to exact unit fraction series. Modern scholars transliterated certain hieratic shorthand notes into hieroglyphic. Other original scribal notes were improperly transliterated by modern scholars. This lack of clarity confused the historical record. Despite omissions, scholars showed that the scribes used algebraic methods, common denominators, progressions, inverses, and proportions that calculated areas, quotients, remainders, solutions to second degree equations, slopes, and volumes. The update of three texts repairs scribal shorthand notes. By including missing information, complete ancient arithmetic sentences are written and appreciated in modern arithmetic. The three updated hieratic texts reveal previously unreported scribal skills, properties of ancient number theory, and aspects of ancient economic life that were enhanced by Egyptian fraction innovations. INTRODUCTION Egyptian fraction arithmetic was closely linked to Old Kingdom mathematics. Old Kingdom numeration, and weights and measures systems were cursive and binary. The hieroglyphic numeration system rounded off infinite series representations of rational numbers to six terms. The Horus-Eye weights and measures system overlooked a 1/64 unit rounding error within a 6-term finite series (Gillings 1972). 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 Note that to exactly sum to unity (1), a 1/64 unit must to be added to the Eye of Horus series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64 = 1 The binary aspect of the Old Kingdom weights and measures was used in balance beam valuations of commodities, business transactions, and higher math. A significant change in writing binary unit fraction series as Eye of Horus series took place in the Middle Kingdom. The inexact Eye of Horus series was replaced after 2050 BCE with exact series. Scribes thereafter introduced exact Egyptian fraction innovations. One innovation demonstrated finite arithmetic within balanced algebraic statements that converted rational number n/p to exact unit fraction series. For example, 4/7 was written as 1/2 + 1/14 (Silverman 1975). Middle Kingdom arithmetic written in Egyptian fraction series corrected Old Kingdom weights and measure rounding errors. Scribes corrected binary measurements and balanced algebraic statements by including the missing portion of the 1/64 of a unit or portion thereof (Gardner 2006). Other exacting innovations created finite units of measure (Gardner 2008b). Scholars, from1860 to the present, transliterated hieratic shorthand notes into fragmented sentences. Transliterations converted hieratic script to hieroglyphs and these transliterations were converted into modern arithmetic and sentences. The two-step translation process revealed algebraic methods, common denominators, progressions, inverses, primes, and proportions. Scribes found areas, proofs, quotients, remainders, second-degree equation solutions, slopes, and volumes (Belluck 2010, Gillings 1972). The deeper aspects of hieratic shorthand were not translated well enough to expose subtle scribal innovations. Scholars worked hard to explain concise 2/n tables and unit fraction statements that relied on the 2/n tables (Peet 1923, Chace 1927). The Egyptian fraction historical record under-reports scribal methods that included rational numbers, least common multiples, common divisors, and red auxiliary numbers (Gardner 2011). Poor 20th century transliterations led to inconsistent translations. This chapter begins to correct the Egyptian fraction historical record by updating six introductory Egyptian Mathematical Leather Roll (EMLR) rules and asks the question, "How 2 were the best unit fraction series selected by scribes?" Proposed answers validate additional scribal innovations, and scribal arithmetic skills that implemented each innovation. The scope of this chapter is limited to early number theory and volume topics reported in the EMLR, the Rhind Mathematical Papyrus (RMP), and the Akhmim Wooden Tablet (AWT). Other related metrology topics are not discussed. A future paper may be dedicated to the weighing of bread, gold, silver, tin, and commodities in the unit called ―debens.‖ UPDATING METHODOLOGY Egyptian fraction texts are reported in historical context free from modern mathematical metaphors. To achieve readability in modern arithmetic, scribal shorthand conventions are replaced by a seldom-used scribal longhand convention. For example, the EMLR began with: 1/8 = 1/10 + 1/40 The simplest possible method (Occam’s razor) created an equality likely included LCMs and red auxiliary numbers. A scribal longhand convention reported in RMP 36 and RMP 37, scaled 1/8 by LCM 5 to 5/40. The best divisors of 40 were selected by the scribe that summed to numerator 5. Only 4 + 1 was available in this case. RMP 36 would have recorded 4 + 1 in red. Applying the seldom used RMP longhand convention to the EMLR, line 1 can be re-written as: 1/8(5/5) = 5/40 = (4 + 1)/40 = 1/10 + 1/40. EGYPTIAN MATHEMATICAL LEATHER ROLL AND THE RHIND MATHEMATICAL PAPYRUS This update focuses on the EMLR and early number theory that connect to the RMP 2/n table (Gardner 2002). The EMLR converted 17 rational numbers to 26 unit fraction series. Scribal errors muddled three of the series. Six 1/2n rational numbers—1/2, 1/4, 1/8, 1/16, 1/32, and 1/64—and ten other rational numbers—1/7, 1/9, 1/10, 1/11, 1/13, 1/14, 1/15, 1/20, and 1/30— were converted to unit fraction series. One other trivial case reported 1/6 +1/6 = 1/3. Eight of the 17 rational numbers appeared twice, and one rational number, 1/8, appeared thrice times reporting three different unit fraction series. The 1/13 line was unreadable. A nine year old paper suggested that the EMLR was encoded by six rules: 1/2n = (1/A)(A/2n) 1/p = (1/A)(A/p) (Rule 1.0) (Rule 2.0) 3 1/pq = (1/A)(A/pq) 1/8 = (1/25)(25/8) = 25/200 = 1/25 + 17/200 17/200 = 1/15 + 1/75 + 1/200 1/8 = 1/25 + 1/15 + 1/75 + 1/200 (Rule 3.0) (Rule 4.0) (Rule 5.0) (Rule 6.0) The approach erroneously suggested that a modern 1/p = (1/ A)/(A/p) method was used by the EMLR student scribe (Gardner 2002). This update proposes to repair the EMLR historical record by showing that the EMLR scaled rational numbers 1/p and 1/pq by seven LCMs that applied six rules: (1/2n)(m/m) = m/2mn (1/p)(m/m) = m/mp (1/pq)(m/m) = mp/mpq Both: (1/8) = (1/25)(25/8) = 25/200 = 1/25 + 17/200 (1/16) = (1/25)(25/16) = 25/400 = 1/50 + 17/400 Both: (17/200)(6/6) = 102/1200 = 1/15 + 1/75 + 1/200 (17/400) (6/6) = 102/2400 = 1/30 + 1/150 + 1/400 Both: 1/8 = 1/25 + 1/15 + 1/75 + 1/200 1/16 = 1/50 + 1/30 + 1/150 + 1/400 The EMLR cited two out-of-order series without demonstrating a calculation method. Line 8 of the EMLR converted 1/8 to 1/25 + 1/15 + 1/75 + 1/200. Searching for the scribal calculation method, Rule 4.1 scaled 1/8 by LCM 25 to 25/200, which subtracted 1/25 obtained 17/200. Rule 5.1 scaled 17/200 by LCM 6 to the unit fraction series 1/15 + 1/75 + 1/200. Rule 6.1 reported the total. Line 9 of the EMLR converted 1/16 to 1/50 + 1/30 + 1/150 + 1/400. Rule 4.1 scaled 1/16 by LCM 25 to 25/400 and subtracted 1/50, which obtained 17/400. Rule 5.1 scaled 17/400 by LCM 6 and obtained a unit fraction series. Rule 6.1 shows the total. Seven LCMs in total were used to scale EMLR rational numbers1/ 2n, 1/p, and 1/pq. Final EMLR series were sometimes scaled to awkward unit fraction series. 4 (Rule 1.1) (Rule 2.1) (Rule 3.1) (Rule 4.1) (Rule 5.1) (Rule 6.1) But, how were the best unit fraction series selected by the student scribe? To discuss the best unit fraction series question, another modern splitting proposal recommended: 1/(ab) = [1/(a + b)](1/a + 1/b) Using the example: 1/(4)(7) = (1/11)(4 + 7) = 1/44 + 1/77 a recent proposal asked if the EMLR student understood the rule (Malkevich 2011)? The seven LCMs scaled rational numbers 1/2n, 1/p, and 1/pq to sometimes awkward unit fraction series. Restated EMLR statements include LCMs and red auxiliary numbers show that the student converted 1/8, the thrice repeated rational number, by two single LCMs—3 and 5— and a pair of LCM—25 and 6—per: 1. 1/8(3/3) = 3/24 = (2 + 1)/24 = 1/12 + 1/24 2. 1/8(5/5) = 5/40 = (4 + 1)/40 = 1/10 + 1/40 3. 1/8(25/25) = (25/200) a. 25/200 = (8 +17)/200 = (1/25 + 17/200) b. 17/200(6/6) = 102/1200 = (80 + 16 + 6)/1200 = 1/15 + 1/75 + 1/200 c. 1/8 = 1/25 + 1/15 + 1/75 + 1/200 (line 8) (line 13) (line 1) (proposed rule 7.0) The 26 restated EMLR statements used seven LCMs—2, 3, 5, 6, 7, 10, and 25. Two 1/8 and 1/16 sentences used four LCMs—3, 5, 25, and 6—and the remaining 24 sentences used six LCMs—2, 3, 5, 6, 7, and 10 (Gardner 2007). To answer the proposed question, based on the EMLR scaling of 1/14 by LCM 3 consider: 1/14(3/3) = 3/42 = (2 + 1)/42 = 1/21 + 1/42 and the EMLR-like series: 1/28(3/3) = 3/84 = (2 + 1)/84 = 1/42 + 1/84 Suggests the EMLR student would have scaled 1/28 by LCM 3. Moreover, the EMLR student would have considered LCM 6, 8, 10, 12, and 14, such that: 1. 1/28(6/6) = 6/168 = (3 + 2 + 1)/168 = 1/56 + 1/84 + 1/168 2. 1/28(8/8) = 8/224 = (7 + 1)/224 = 1/32 + 1/224 (line 21) 5 3. 1/28(10/10) = 10/280 = (7 + 2 + 1)/224 = 1/40 + 1/140 + 1/280 4. 1/28(12/12) = 12/336 = (7 + 3 + 2)/336 = 1/49 + 1/112 + 1/168 5. 1/28(14/14) = (14/392) = (7 + 4 + 2 + 1)/392 = 1/56 + 1/98 + 1/196 + 1/292 6. 1/28(16/16) = 16/448 = (8 + 7 + 1)/448 = 1/56 + 1/64 + 1/448 Note that alternate EMLR-like series’ last-term denominators are larger than the LCM 3 series last term denominator. Going on, was the best EMLR series the one with smallest last-term denominator? Or, was the best EMLR series the one with shortest series and the smallest first-term denominator? The question of the best EMLR conversion of 1/n and 1/p to a unit fraction series can be answered by discussing the best RMP 2/n table series (Gardner 2002, 2009b). To begin, 87 RMP problems used three rational number conversion methods: 1. 2/n(m/m)= 2m/mn 2. n/p = (n -2)/p + n/p 3. p/p = (numerators summed to p)/p (Rule 8.0) (Rule 9.0) (Rule 10.0) The 2/n table scaled 2/3, 2/5, 2/7 to 2/101 to the best unit fraction series (Gardner 2008a) The 51 rational numbers were scaled by 15 LCMs—2, 3, 4, 6, 8, 10, 12, 20, 24, 30, 36, 40, 56, 60, and 70—followed by 28 sets of red auxiliary numbers; facts are included in Appendix II. Red auxiliary numbers did not appear in the majority of the RMP’s shorthand notes. As scholars reported, red auxiliary numbers were cited in RMP 7 through RMP 20 as completion problems (Gillings 1972). This paper suggests that scribal constructions of the 2/n table must include LCMs and red auxiliary numbers. By considering the best LCMs and the best red auxiliary numbers presented in RMP 36 and 37, wider views of 2/n table calculations and scribal skills are offered. Scholarly reviews of scribal shorthand notes suggest that LCMs and red auxiliary numbers were not understood beyond RMP 7 to RMP 20 completion problems (Gillings 1972). Appendix II is written in scribal longhand that cites the best LCMs and the best red auxiliary numbers. The scribal longhand offers an additional scribal innovation. Rule 8.0 scaled n/p by LCM m to mn/mp that defined the scribal longhand innovation. The best divisors of denominator mp were summed to numerator mn. Each red auxiliary number was 6 divided by mp that calculated a unit fraction. The finite sum of the unit fractions equaled the initial rational number n/p. A breakdown of 28 sets of red auxiliary numbers report one set, (3 + 1), that appeared 16 times; a second set, (5 + 1), that appeared four times; a third set, (7 + 1), that appeared three times; and a fourth and fifth set, (11+ 1) and (19 + 3 + 2), that each appeared twice. Twentythree (23) sets of red number numbers appeared once. The 23 single red number series followed Rule 1 were summed to numerator 2m. Rule 8.0 did not work for 30/53, 28/97, and other rational numbers. The second method (Rule 9.0) solved otherwise impossible n/p conversions. The method was a parallel to the EMLR lines that scaled 1/8 and 1/16 by LCM 25 and LCM 6. Rule 8.0 replaced n/p with (n -2)/p + 2/p and solved (n -2)/p by one LCM with the best red auxiliary numbers, and 2/p by a second LCM with the red auxiliary number series (Gardner 2009a). Ahmes provided two proofs in RMP 36 and RMP 31. The first replaced 30/53 with 28/53 + 2/53, and solved 28/53 by LCM 2 and 2/53 by LCM 30 (in RMP 36). The second proof replaced 28/97 with 26/97 + 2/97, and solved 26/97 by LCM 4 and 2/97 by LCM 56 (in RMP 31) (Gardner 2009a). Rule 9.0 points out an important use of 2/n tables, a method that converted difficult rational numbers by solving two rational numbers by Rule 8.0 LCMs and red auxiliary numbers. The third RMP rule (Rule 10.0) partitioned unity (1) into vulgar fractions such as 53/53 into: 53/53 = 2/53 + 3/53 + 5/53 + 15/53 + 28/53 Rule 10.0 was mentioned in RMP 36. Ahmes partitioned the identity 53/53 into six Rule 8.0 unit fraction series: 53/53 = (2 + 3 + 5 + 15 + 28)/53 = 2/53 + 3/53 + 5/53 + 15/53 + 28/53 Five LCMs—30, 20, 12, 4, and 2—and three sets of red auxiliary numbers—(53 + 5 + 2), (53 + 4 + 2 + 1), and (53 + 2 + 1)—scaled the five n/53 rational numbers per: 2/53 by LCM 30 = 60/1590 = (53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795 3/53 by LCM 20 = 60/1060 = (53 + 4 + 2 + 1)/1060 = 1/20 + 1/265 + 1/530 + 1/1060 5/53 by LCM 12 = 60/636 = (53 + 4 + 2 + 1)/636 = 1/12 + 1/159 + 1/106 + 1/212 7 15/53 by LCM 4 = 60/212 = (53 + 4 + 2 + 1)/212 = 1/4 + 1/53 + 1/106 + 1/212 The importance of the rule 10.0 offers a fail-safe third conversion method that always scaled rational number n/p to a concise unit fraction series (Gardner 2009b). The method defined a virtual table. Virtual unity tables were used as an alternative to the second RMP n/p conversion rule. Focusing on RMP 36, 2/53 was scribal long hand scaled by LCM 30 to 60/1590 within the balanced sentence: 2/53 = 60/1590= (53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795 Divisors of numerator 60 = (53 + 5 + 2) were recorded in red. Red made it clear that the best divisors had been selected. The first RMP conversion rule was used 24 times in the EMLR. Egyptian scribes tried to write the best unit fractions series that were available. Rephrasing scribal notes begins to parse the scribal skills topics. Considering Ahmes’ best 2/35 and 2/91 series by using an EMLR-type rule reports: 1. 2/35 = 1/30 + 1/42 a. per proposed rule 7.0 a = 5 an d b = 7 such that b. 2/35 = (1/6)(1/5 + 1/7) = 1/30 + 1/42 2. 2/91 = 1/70 + 1/130 a. per proposed rule 7.0 a = 7 and b = 13 such that b. 2/91 = (1/10)(1/7 + 1/13) = 1/70 + 1/130 Ahmes’ 2/n table data reported 2/35 = 1/30 + 1/42 2/91 = 1/70 + 1/30 There is little hard evidence to refute a claim that proposed rule 7.0 was used in the RMP 2/n table. One theme offers a contrary view that Ahmes may have scaled 2/35 by LCM 30, and 2/91 by LCM 70. A small fragment of 2/35 shorthand data mentions 6/210. Had 12/210 been 8 mentioned, full agreement with 2/35(6/6) = 12/210 reported in Appendix II would have closed this issue. Since the issue is open to a small degree, readers are free to choose. Of course, had proposed rule 7/0 been known to Ahmes, 2/99 = [(9 + 11)(1/9 + 1/11) = 1/90 + 1/110 may have been the best 2/99 unit fraction series. Yet, Ahmes recorded: 2/99 = 1/66 + 1/198 It appears that LCM 3 was considered. AKHMIM WOODEN TABLET AND THE RHIND MATHEMATICAL PAPYRUS Egyptian scribes scaled rational numbers n/p to unit fraction series within practical statements. One hekat of grain was scaled to 64/64 of a hekat, and one hekat of grain was scaled to 320 ro—1/320 a hekat. Thus, 64/64 hekat and 320 ro both meant 1 hekat. In RMP 36, (3/53)ro was scaled by LCM 20 to (1/20 + 1/265 + 1/530 + 1/1060) hekat Scribal rational numbers, LCMs, red auxiliary numbers, 2/n tables, algebra, geometry, unit fraction series, and hekat (volume) units jump-started Middle Kingdom finite arithmetic. A developing economy was the beneficiary. Scribes created finite quotient and exact remainders data, combining theoretical methods that precisely valued commodities. Commodities, including beer and bread, were economically allocated for a range of purposes. One allocation paid predetermined wages to a diverse labor force. The AWT reports five exact divisions of a hekat by a quotient and scaled remainder method. The AWT detailed a hekat unity 64/64 divided by 3, 7, 10, 11, and 13 using: (64/64)/n = Q/64 + (5R/n)ro Example: (64/64)3 = 21/64 + (5/3)(1/320) = (16 + 4 + 1)/64 + (5/3)ro = (1/4 + 1/16 + 1/64)hekat + (1 + 2/3)ro Each answer was proven by inverting the divisor and multiplying: Example: [(1/4 + 1/16 + 1/64)hekat + (1 + 2/3)ro] times 3 = 9 [21/64 + 5/3(1/320)] hekat times 3 = (63/64 + 1/64)hekat = 1 hekat Surviving records report that absentee landlords grew grain and flax for clothing. The hekat was used from the field to the commodities consumed as wages. Workers were paid at standard Middle Kingdom levels, from two to eight hekats a month. During flood years, low crop yields managed pay rates to be proportionally reduced (Ezzamel 2002). The RMP and the Kahun Papyrus include three arithmetic progression allocations (Gardner 2008c). Several Middle Kingdom texts included pesu and hekat calculations. The RMP converted hekats into different strengths of pesu beer and bread by applying an inverse proportion (Clagett 1999). Distributions of commodities were achieved by arithmetic proportions discussed in RMP 40 and RMP 64 (Gardner 2008c). Scribal longhand included rational numbers, least common multiples, red auxiliary numbers, and other innovations. Scribal multiplication and division sentences reveal inverse operations with ancient number theory properties. Longhand volume sentences reveal that the hekat was scaled to 64/64, 320 ro, and a pesu unit (Gardner 2006). Pesu sentences show that an inverse arithmetic method scaled strengths of loaves of bread, jugs of beer, and other commodities (Gillings 1972). The improved Egyptian fraction numeration, and weights and measures assisted Pharaoh and absentee landlords to control granary inventories. Longhand sentences can be read in modern arithmetic. Unit fraction data included practical valuations of commodities double-checked inventories and allocations as wages. Farm productions and commodity inventories were often decentralized. Productions of bread, beer, and other grain-based product inventories were monitored. Inventory withdrawals were scaled in grain units to pay wages (Ezzamel 2002). Scribal algebraic geometry created linear cubits, square cubits, volume khar unit, and other hekat unit formulas. A cubit times a squared-cubit was transcribed as a cubit-cubit (Peet 1923, Gillings 1972). The cubit-cubit contained 3/2 khar. The khar contained 20 hekat. A hekat contained close to 4800 cm3 when translated into modern metrics. Theoretical and practical cubit measurements of cubit-cubits, khar, 400-hekat, 100-hekat, 4-hekat, 2-hekat, 1-hekat, 4-ro, 2-ro, and 1-ro report hekat sub-divisions (Gardner 2011). Scribes converted rational numbers to concise unit fraction series for several purposes. To value commodities, scribes usually scaled the hekat three ways. The first replaced one hekat with 10 64/64 and applied a multiplier, 1/n, that created a finite quotient Q/64 plus a remainder (5R/n)ro. This class of hekat substitution was used in the AWT and over 40 times in the RMP (Gardner 2006). The second hekat form replaced one hekat with 320 ro. The 320 ro form created a multiplier 1/n and rational number quotients plus unit fraction remainder answers. Answers were doublechecked by an inverse arithmetic operation (Gardner 2009a). In RMP 38, a multiplier 7/22 reported: 320 ro times 7/22 = 2240/22 = (101 + 9/11)ro. Rational number 9/11 scaled to a unit fraction series by LCM 4 such that: 9/11 x (4/4) = 36/44 = (22 + 11 + 2 + 1)/44 = (1/2 + 1/4 + 1/22 + 1/44) Ahmes recorded (101 + 1/2+ 1/4 + 1/22 + 1/44)ro by writing ciphered sound symbols, denoted as fractions by placing a line over the symbol, writing from right to left, omitting plus (+) signs, not making it clear that that 101 was included. Scribes recorded weights and measures in double-entry bookkeeping systems that made scribes accountants and mathematicians. Inventory control answers were proven. In RMP 38, Ahmes proved (101 + 1/2 + 1/4 + 1/22 + 1/44)ro by multiplying by 22/7, the inverse of the 7/22 multiplier such that: (101 + 1/2 + 1/4 + 1/22 + 1/44) (22/7)ro = (2204 + 22 + 11 + 2 +1)/7 ro = 320 ro In returning 320 ro, Ahmes commented that an exact hekat had been found. The scribal multiplication and division operations were inverse operations. Transliterations incorrectly reported duplation multiplication and single false position division as historical inverse operations (Chace et al. 1927, Clagett 1999, Peet 1923). Scribal division was based on a wellknown definition, invert the divisor n to 1/n and multiply. Longhand scribal arithmetic reported multiplication and division as inverse operations to one another. To prove a multiplication answer, scribes inverted multiplicand n to 1/n and multiplied. To prove a division answer, scribes inverted divisors n to 1/n and multiplied. Practical hekat measurement units were recorded in wage payments and implemented management controls in finite Egyptian fraction quotients and remainders (Gillings 1972, Simpson 1973). Scribes precisely scaled grain to bread, beer, and other products in Egyptian 11 fraction statements, an Appendix III topic. Pesu and besha, and des-jugs sub-units further scaled grain products for practical wage distributions. One RMP problem reported 5 hekats of grain produced 200 loaves of bread. The balance of the hekats—10 hekat— produced different pesu strengths of beer. Each hekat produced one, two, and three types of beer—labeled from 8/3 pesu to 6 pesu—denoted by grain content by an inverse relationship to the product. A product with 3/8 hekat of grain reported 8/3 pesu, enhancing everyday Middle Kingdom life by another Egyptian fraction innovation (Gardner 2009b). CONCLUSIONS This update of EMLR and AWT papers applies a scribal longhand convention. A student EMLR scribe converted 1/p and 1/pq rational numbers to 26 unit fraction series by six introductory rules. Ahmes converted 2/p and 2/n to the 50 best unit fraction series by three advanced rules. Scribes scaled rational numbers n/p and n/pq to unit fraction series and selected the best LCMs with the best red auxiliary numbers. Scribal longhand exposes early number theory. The number theory was finite and algebraic. Middle Kingdom scribes recorded unity objects in unit fraction series. Four unities were associated with the hekat. The unities assisted scribes in calculating wages paid in bread, beer, meat and other commodities. Wage payments were audited in unit fraction innovations (unities), summarized by: 1. 1 hekat = 64/64 hekat 2. 1 hekat = 320 ro 3. 1/n hekat = n pesu, and n hekat = 1/n pesu The fourth RMP unity innovation was based in Rule 10.0 p/p hekat = (1/p + 2/p + 3/p + … + n/p)hekat Scribes reported scribal multiplication and division that anticipated the modern definition of division—invert the divisor and multiply. ACKNOWLEDGMENTS 12 This chapter is dedicated to the memory of Professor B.S. Yadav. Professor Yadav edited EMLR and AWT papers that provide this paper’s predicate. Pam Belluck, Bruce Friedman, and Cora Dillard provided themes that made this paper possible. Pam Belluck, a New York Times writer, captured the oldest known Egyptian puzzles in everyday language. Bruce Friedman improved early number theory discussions that reported puzzle solutions in everyday language. Cora Dillard provided important editing for the enjoyment of the reader. 13 REFERENCES Belluck, P. (2010) Math puzzles’ oldest ancestors took form on Egyptian Papyrus, New York Times, Science Section, Dec. 6, 2010. Boyer, C .B., A History of Mathematics, New York, John Wiley & Sons, 1967 Chace, A. Bull, L., Manning, H.P., and Archibald, R.C. (1927) The Rhind Mathematical Papyrus. 2 Volumes. Oberlin, OH, Mathematical Association of America. Clagett, M. (1999) Ancient Egyptian Sciences A Source Book, Volume III. Ancient Egyptian Mathematics. Philadelphia, PA, American Philosophical Society. Daressy, G. (1901) Catelogue General des Antiquities Egyptiennes du Musee du Caire, Ostraca 25001–25385. Ezzamel, M. (2002) Accounting for private estates and the household in the 20th century BC Middle Kingdom. Abacus 38 (2), 235–262. Gardner, M.R. (2002) The Egyptian Mathematical Leather Roll, attested short term and long term. In: History of the Mathematical Sciences, Grattan-Guiness, I. and Yadav, B.S., eds. New Delhi, India, Hindustan Book Agency, pp 119–134. Note that you had two Ga05 references listed that were not the same. I used only one, so make sure that I used the one you intended (other one is below this ref list). Gardner, M. Mathematical Roll of Egypt (2005) Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. New York, NY, Springer. Gardner, M. (2006) An ancient Egyptian problem and its innovative arithmetic solution. Ganita Bharati: Bulletin of the Indian Society for the History of Mathematics 28 (1–2), 157–173. Gardner, M. (2008a) Breaking the 2/n table Code. http://rmprectotable.blogspot.com/. Accessed 6 April 2011. Gardner, M. (2008b) Economic context of Egyptian fractions. http://planetmath.org/encyclopedia/EconomicContextOfEgyptianFractions.html-Berlin Papyrus.html. Accessed 6 April 2011. Gardner, M. (2007) Egyptian Mathematical Leather Roll. http://emlr.blogspot.com/. Accessed 6 April 2011. Gardner, M. (2008c) Kahun Papyrus and arithmetic progressions. http://planetmath.org/encyclopedia/KahunPapyrusAndArithmeticProgressions.html. Accessed 6 April 2011. 14 Gardner, M. (2009a) New and old Ahmes Papyrus classifications." http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html. Accessed 6 April 2011. Gardner, Milo (2009b) RMP 36 and the 2/n table. http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html, Accessed 6 April 2011. [Ga11] Gardner, M. (2011) RMP 47 and the hekat. http://planetmath.org/encyclopedia/RMP47AndTheHekat.html. Accessed 6 April 2011. Gillings, R. (1972) Mathematics in the Time of the Pharaohs. Cambridge, MA: MIT Press. Malkevitch, J. (2011) Egyptian fractions. Accessed 6 April 2011. Neugebauer, O. (1962) The Exact Science in Antiquity, New York, NY, Harper and Row; Reprint Dover, 1969. Peet, T.E. (1923) Arithmetic in the Middle Kingdom. Journal of Egyptian Archaeology 9, 91–99. Pommerening, T. Altagyptische Rezepturen metrologish neu interpriert (2002) Beriche zur wissenschaft-Geschichte 26 (1), 1–16. Robins, Gay and Shute. Charles, The Rhind Mathematical Papyrus: an ancient Egyptian text, (London, British Museum Publications Ltd, London, 1987, Reprint: New York, Dover, 1987). Silverman, D. Fractions in the Abusir Papyri (1975) Journal of Egyptian Archaeology 61, 248– 249. Silverman, D. (1997) Ancient Egypt. New York, NY, Oxford University Press. Simpson, W.K. (1973) The Reisner Papyrus. Journal of Egyptian Archaeology 59, 220-222. Vymazalova, H. (2002) The Wooden Tablets from Cairo: The use of the grain unit hk3t in ancient Egypt. Archiv Orientální 70, 27–42. 15 Appendix I. Egyptian Mathematical Leather Roll (EMLR): adapted from Gardner (2007). 1. (1/8)(5/5) = 5/40 = (4 + 1)/40 = 1/10 + 1/40 2. (1/4)(5/5)= 5/20 = (4 +1)/20 = 1/5 + ½ 3. (1/3)(3/3)) = 3/9 = (2 + 1)/9 = 1/4 + 1/12 4. (1/5)(2/2)) = 2/10 = 1/10 + 1/10 5. (1/3)(2/2) = 2/6 = 1/6 + 1/6 6. (1/2)(3/3) = 3/6 = 1/6 + 1/6 + 1/6 7. 2/3 = 1/3 + 1/3 8. (1/8)(25/25) = 25/200 = (8 +17)/200 = 1/25 + (17/200)(6/6) = 1/25 + (80 + 16 + 6)/1200 = 1/8 = 1/25 + 1/15 + 1/75 + 1/200 9. (1/16)(25/25) = 25/400 = (8 +17)/400 = 1/50 + (17/2400)(6/6) = 1/50 + (80 + 16 + 6)/2400 = 1/16 = 1/50 + 1/30 + 1/150 + 1/400 10. (1/15)(10/10) = 10/150 = (6 + 3 + 1)/150 = 1/25 + 1/50 + 1/150, (1/6 was initial term) 11. (1/6)(3/3) = 3/18 = (2 + 1)/18 = 1/9 + 1/18 12. (1/4)(7/7) = 7/28 = (4 + 2 + 1)/28 = 1/7 + 1/14 + 1/28 13. (1/8)(3/3) = 3/24 = (2 + 1)/24 = 1/12 + 1/24 14. (1/7)(6/6) = 6/42 = (3 + 2 + 1)/42= 1/14 + 1/21 + 1/42 15. (1/9)(6/6) = 6/54 = (3 + 2 + 1)/54 = 1/18 + 1/27 + 1/54 16. (1/11)(6/6) = 6/66 = (3 + 2 + 1)/66= 1/22 + 1/33 + 1/66 17. (1/13)(?) = 1/28 + 1/49 + 1/196 (corrected by?) (1/13) (6/6) = 6/78 = (3 + 2 + 1)/78 =1/26 + 1/39 + 1/78 18. (1/15)(6/6) = 6/90 = (3 + 2+ 1)/90 = 1/30 + 1/45 + 1/90 19. (1/16)(3/3) = 3/48 = (2 + 1)/48 = 1/24 + 1/48 20. (1/12)(3/3) = 3/36 = (2 + 1)/36 = 1/18 + 1/36 21. (1/14)(3/3) = 3/42 = (2 + 1)/42 = 1/21 + 1/42 22. (1/30)(3/3) = 3/90 = (2 + 1)/90 = 1/45 + 1/90 23. (1/20)(3/3) = 3/60 = (2+ 1) 60 = 1/30 + 1/60 16 24. (1/10)(3/3) = 3/30 = (2 + 1)/30 = 1/15 + 1/30 25. (1/32)(3/3) = 3/96 = (2 + 1)/96= 1/48 + 1/96 26. (1/64)(3/3) = 3/192 = (2 + 1)/192 + 1/96 + 1/92 Information in this Appendix is further discussed by Gardner (2002, 2005). 17 Appendix II. Rhind Mathematical Papyrus (RMP) 2/n Table (Gardner 2008a, 2009a, 2009b) 1. 2/3 = 1/3 + 1/3) 2. 2/5(3/3) = 6/15 = (5+ 1) = 1/3 + 1/15 3. 2/7(4/4) = 8/28 = (7 + 1)/28 = 1/4 + 1/28 4. 2/9 (2/2) = 4/18 = (3 + 1)/18 = 1/6 + 1/18 5. 2/11(6/6) = 12/66 = (11 + 1)/66 = 1/6 + 1/66 6. 2/13(8/8) = 16/104 = (13 + 2 + 1)/104 = 1/8 + 1/52 + 1/104 7. 2/15(2/2) = 4/30= (3 + 1)/30 = 1/10 + 1/30 8. 2/17(12/12) = 24/204 = (17 + 4 + 3)/204 = 1/12 + 1/51 + 1/68 9. 2/19(12/12) = 24/228 = (19 + 3 + 2)/228 = 1/12 + 1/76 + 1/114 10. 2/21((2/2) = 4/42 = (3 + 1)/42 = 1/14 + 1/42 11. 2/23(12/12) = 24/276 = (23 +1)/276 = 1/12 1/276 12. 2/25(3/3) = 6/75 = (5 + 1)/75 = 1/15 + 1/75 13. 2/27(2/2) = 4/54 = (3 + 1)/54 = 1/18 + 1/54 14. 2/29(24/24) = 48/696 = (29 + 12 + 4 + 3)/696 = 1/24 + 1/58 + 1/174 + 1/232 15. 2/31(20/20) = 40/1620 = (31 + 5 + 4)/1620 = 1/20 + 1/124 + 1/155 16. 2/33(2/2) = 4/66 (3 + 1)/66 = 1/22 + 1/66 17. 2/35(30/30) = 60/1050 = (35 + 25)/1050 = 1/30 + 1/42 18. 2/37(24/24) = 48/888 = (37 + 8 + 3 )/888 = 1/24 + 1/111 + 1/296 19. 2/39(2/2) = 4/78 = (3 + 1)/78 = 1/26 + 1/78 20. 2/41(24/24) = 48/984 = (41 + 4 + 3)/984 = 1/24 + 1/246 + 1/328 21. 2/43(42/42) = 84/1806 = (43 + 21 + 14 + 6)/1806 = 1/42 + 1/86 + 1/129 + 1/301 22. 2/45(2/2) = 4/90 = ( 3 + 1)/90 = 1/30 + 1/90 23. 2/47(30/30) = 60/1410 = (47 + 10 + 3)/1410 = 1/30 + 1/141 + 1/470 24. 2/49(4/4) = 8/196 = (7 + 1)/196 = 1/28 + 1/196 25. 2/51(2/2) = 4/102 = (3 + 1)/102 = 1/34 + 1/102 26. 2/53(30/30)= 60/1590 = (53 + 5 + 2 )/1590 = 1/30 + 1/318 + 1/795 27. 2/55(6/6) = 12/330 = (11 + 1)/330 = 1/30 + 1/330 28. 2/57(2/2) = 4/114= (3 + 1)/114 = 1/38 + 1/114 18 29. 2/59(36/36) = 72/2124= (59 + 9 + 4) /2124 = 1/36 + 1/236 + 1/531 30. 2/61(40/40) = 80/2440 = (61 10 + 5 + 4)/2440 = 1/40 + 244 + 1/488 + 1/610 31. 2/63(2/2) = 4/126 = (3 + 1)/126 = 1/42 + 1/126 32. 2/65(3/3) = 6/195 = (5 + 1)/195 = 1/39 + 1/195 33. 2/67(40/40) = 80/2680 = (67 + 8 +5 )/2680 = 1/40 + 1/335 + 1/536 34. 2/69(2/2) = 4/138 = (3 + 1)/138 = 1/46 +1/138 35. 2/71(40/40) = 80/2840 = (71+ 5 + 4)2840 = 1/40 + 1/568 + 1/710 36. 2/73(60/60) = 120/4380 = (73 + 20 + 15 + 12)/4380 = 1/60 + 1/219 + 1/292 + 1/365 37. 2/75(2/2) = 4/150 = (3 +1)/150 = 1/50 + 1/150 38. 2/77(4/4) = 8/388 = (7 + 1)/388 = 1/44 + 1/308 39. 2/79(60/60)= 120/4740 =(79 + 20 + 15 + 6 )/4740 = 1/60 + 237 + 1/316 + 1/790 40. 2/81(2/2)= 4/162 = (3 + 1)/162 = 1/54 + 1/162 41. 2/83(60/60) = 120/4980 = (83+ 15 + 12 +10)/4980 = 1/60 + 1/332 + 1/415 + 1/498 42. 2/85(3/3) = 6/255 = (5 + 1)/255 = 1/51 + 1/255 43. 2/87(2/2) = 4/174 = (3 + 1)/174 = 1/58 + 1/74 44. 2/89 = (60/60) = 120/5340 = (89 + 15 +10 + 6)/5340 = 1/60 + 1/356 + 1/534 + 1/890 45. 2/91(70/70) = 140/6370= (91 + 49)/6370 = 1/70 + 1/130 46. 2/93(2/2) = 4/186 = (3 + 1)/186 = 1/62 + 1/186 47. 2/95(60/60) = 120/5700 = (95 + 15 + 10)/5700 = 1/60 + 1/380 + 1/570 48. 2/97(56/56) = 112/5432= (97+ 8 + 7 )/5432 = 1/56 + 1/679 + 1/776 49. 2/99 (2/2) = 4/198 = (3 + 1)/198 = 1/66 + 1/198 50. 2/101(6/6) = 12/606 = (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606 19 Appendix III. Akhmim Wooden Tablet (AWT) and the Rhind Mathematical Papyrus (RMP) The AWT was written in 1925 BCE. The text was transliterated in 1906 (Daressy 1906). An improved transliteration was published in 2002 (Vymazalova 2002). One hekat was scaled by 1/3, 1/7, 1/10, 1/10, and 1/13 to strings of binary quotients and remainders that were returned to a hekat unity written as (64/64). The Rhind Mathematical Papyrus (RMP), a 1650 BCE text, used a related quotient and remainder method over 60 times (Gardner 2006). The binary quotients plus scaled 1/320 of a hekat(ro) remainder was first reported to the modern era by the Akhmim Wooden Tablet (AWT). The scaled ro remainder was implicitly amended by Ahmes, in his own shorthand notes, to include 2-ro, 3-ro, and 4-ro remainders in RMP 47 (Gardner 2011). Early in the 20th century, Ahmes' bird-feeding problem (RMP 83) was reported by unclear additive patterns (Chace et al. 1927). Ahmes listed seven grain (hekat) portions within the AWT’s hekat unity (64/64) divided by divisor n quotient and scaled remainder pattern, asking how much grain did the seven birds eat in one day, and how much did all the birds eat in 1, 10, 20, and 30 days. Corrected AWT quotient and remainder patterns report RMP 83 by: 1. 2 geese and a crane each ate (1/8 + 1/32 hekat + (3 + 1/3) ro 2. a set-duck ate (1/32 + 1/64) hekat + 1 ro, and 3. a set-goose, dove, and quail each ate (1/64) hekat + 3 ro Ahmes reported seven portions of grain recorded within a hekat unity (64/64)—a scribal context was misunderstood by scholars such as Peet (1923), Chace et al. (1927), Clagett (1999), Gillings (1982), Neugebauer (1962), Pommerening (2002), and Robins (1987). Ahmes reported 1/6 of a hekat (three times), 1/20 of a hekat (once), and 1/40 of a hekat (three times such that: (3/6 + 1/20 + 3/40) hekat (20/40 + 2/40 + 3/40) hekat (25/40 = 5/8) of a hekat (of grain) was eaten by seven birds in one day. Vulgar fractions were scaled by LCMs and red auxiliary numbers before unit fraction answers (Gardner 2008a). 20 The bird feeding method was extended to wage valuations by pesu and other methods (Gardner 2008c). About the author: Milo Gardner has been a military cryptanalyst and is a mathematician. He has been decoding ancient texts for 25 years, Egyptian texts for 20 years, and lives in the Sacramento area of California. Mailing address: 7255 Sumter Drive, Fair Oaks, CA 95628 USA. Email: milogardner@yahoo.com. 21