tag:blogger.com,1999:blog-90731112013-08-28T21:08:26.852-07:00Akhmim Wooden TabletThe AWT partitioned a hekat unity (64/64) five times. The unity was multiplied by 1/3, 1/7, 1/10, 1/11 and 1/13. Five binary hekat quotients (Q/64) plus 1/320 of a hekat (ro) scaled remainders were recorded. The five two-part [(Q/64) + (5R/n)(ro)] answers were proved by multiplying 3, 7, 10, 11 and 13 and returning (64/64) hekat unity five times.Milo R. Gardnerhttp://www.blogger.com/profile/15284868993340980422noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-9073111.post-1099957591113841472005-06-22T15:10:00.000-07:002011-09-23T05:49:59.431-07:00Akhmim Wooden Tablet (AWT)INTRODUCTION<br />-----------------------<br /><br />The Akhmim Wooden Tablet may date to 2,000 BCE, 12th dynasty, or as late as 15th dynasty. The tablet is housed in the Cairo, Egypt Museum. It is 46.5 x 26n cm in size and mentions 27 servant names, an unknown king's name (citing the 8th year of his reign), five division calculations, one of which was repeated four times and five proofs. The document was reported in 1901 and analyzed and published in 1906 by Georges Daressy. Daressy indicated five divisions by 3, 7, 10, 11 and 13, and wrote out exact 1/p unit fraction series, and validated three of the five proofs. <br /><br />Daressy discussed the AWT in terms of binary fractions and minimized aspects of the five Egyptian fraction series. Typos and other errors muddled the 1/11 and 1/13th multiplications. Exactness was not identified in the scribal proof for the 1/11 and 1/13 cases. Daressy cited the cubit-cubit rather than the hekat, the actual AWT context (Peet's main complaint). <br /><br />However, Daressy's cubit view was consistent with a binary fraction remainder arithmetic, and exact partitions. Peet did not identify several scribal arithmetic facts. Daressy's approach properly analyzed AWT data that included binary fractions and scaled remainders. <br /><br />A small number of scholars worked on the 1/10th of a hekat (a volume unit) named hin, hinu, or henu scaled a linear cubit to a cubit-cubit-cubit within a hekat unity (64/64) such that 1/320 of a hekat was named ro<br /><br />(64/64)/10 hekat = (6/64 + 4/640)hekat = <br /><br />(4 + 2)/64 hekat + 20/10 ro =<br /><br />(1/16 + 1/32)hekat + 2ro = <br /><br />1 hin<br /><br />Ahmes in Rhind Mathematical Papyrus (RMP) 81 used 29 binary hekat quotients + scaled ro remainders by following the theoretical statement<br /><br />(64/64)/n = Q/64 + (5R/n)ro <br /><br />with Q = Quotient, R = Remainder, and<br /><br />n limited to the range 1/64 < n < 64. The AWT and RMP two-part statements used one part statements. For example: 10/n hin simply meant a hekat was scaled to a 1/10 unit named hin was the limit scholarly discussions. Peet in 1923 incompletely discussed the AWT's binary fractions statements that conflicted with Daressy's earlier work. Peet reported 1/3, 1/7, 1/10, 1/11, 1/13 multiplication aspects of the problems and only stressed the 1/320 ro unit. Peet had not reported the meta (64/64)/n division that reported five binary quotient plus a 1/320 remainder answers and proofs. Peet under reported the scribal context and modern translation of the AWT n = 11 and 13 cases, correctly reported by: a. (64/64)/11 hekat = (5/64 + 9/704)hekat = (4 + 1)/64 hekat + (45/11)ro = (1/16 + 1/64)hekat + (4 + 1/11)ro b. (64/64)/13 hekat = (4/64 + 12/832)hekat = 4/64 hekat + (60/13)ro = 1/16 hekat + (4 + 8/13)ro with 8/13 scaled by LCM 2 to 16/26 = (13 + 2 + 1)/26 = 1/2 + 1/13 + 1/26 recorded as: (64/64)/13 hekat = 1/16 hekat + (4 + 1/2 + 1/13 + 1/26)ro It 80 years for Vymazalova to correctly report the proof context of the AWT arithmetic story. Hana Vymazalova reported the proof side of the five two-part statements that returned (64/64) five times when multiplied by the initial divisors. Vymazalova did not challenge Peet's calculation views of the five AWT binary quotient and 1/320 remainder answers, points that were corrected in <a href=""http://independent.academia.edu/MiloGardner/Papers/163573/The_Arithmetic_used_to_Solve_an_Ancient_Horus-Eye_Problem"">2006</a>.<br /><br />Daressy's 1906 review of the AWT's data garbled the n = 11 and n = 13 proofs thereby confusing Peet and later researchers. In 2002 by Hana Vymazalova corrected Daressy's two proof errors. Vymazalova's corrections and other meta points were published in <a href="http://independent.academia.edu/MiloGardner/Papers/163573/The_Arithmetic_used_to_Solve_an_Ancient_Horus-Eye_Problem">2006 </a>and <a href="http://independent.academia.edu/MiloGardner/Papers/623827/Egyptian_Fractions_Unit_Fractions_Hekats_and_Wages_-_an_Update">2011</a>.<br />The AWT reported a well-defined system of weights and measures arithmetic from an Old Kingdom inexact system to an exact Middle Kingdom rational number system. <br /><br />The meta context in which Peet properly identified the 1/320 ro aspect of the AWT was missed until Hana Vyamazalova's 2002 paper. Scholars for almost 100 years did not connect the AWT partition method to RMP 81 and 29 data points, and over 30 additional two-part partitions of a (64/64) hekat unity discussed in the RMP.<br /><br /><br />CONTENTS OF THE AWT<br />------------------------------<br />Translated to our modern base 10, the AWT simply states that unity (64/64th) was divided by 3, 7, 10, 11 and 13 following a general rule of division, as is clearly read by:<br /><br />(64/64)/n = Q/64 + (5R/n)ro<br /><br />with Q = quotient, and R = Remainder<br /><br />writing the initial problem in modern base 10 fractions.<br /><br />Middle Kingdom scribes used this pattern, by easily writing the quotient term into a Horus-Eye series, for example (64/64)/3 = 21/64 (Q). Several scholars have seen this portion, but become foggy with respect to the remainder, one (1) in the case of n = 3.<br /><br />The 2nd portion, the handling of the remainder, has been grossly confused by scholars. One reason can be excused since R/(n*64) was 'encoded' by scribes replacing 1/64th with equivalent 5/320 remainders. This allowed student scribes to add Q and R values as one number.<br /><br />To the average scholar seeing an Egyptian fraction series, actually (5R/n) followed by ro (long known to be 1/320) did not 'feel' like a remainder component. However, Ro, clearly used as a common divisor, can also have been seen by Ahmes as n LCM, or even a GCD. Whatever ro's meaning to Ahmes, its details was left to modern code breakers. <br /><br />Only a few scholars opnenly 'scratched' their respective heads when seeing AWT and RMP two-part data. Two recent scholars: Robins-Shute saw (6400/64)/70 one of 10 RMP 47 problems. Robins-Shute did not report the two-part Q/64 + (5R/70)ro expression in a 1987 RMP book. Robins-Shute fairly reported products and remainders contained in the data, a form of scratching of their respective heads.<br /><br />An earlier motivated scholar was Chace. In a 1927 RMP book, RMP 83 reported three data sets for (64/64) divided by n = 6, 20, and 40. The bird-feeding rates made little sense to him, suggesting that Ahmes had garbled the data. Chace mentioned that Ahmes left no clues to on this matter, a personal point that is obviously incorrect when reading the AWT in its broader context.<br /><br />Factually, Ahmes and the AWT scribes created<br /><br />Q/64 +(5R/n)(1/320)statements, over 60 times following the same theoretical style.<br /><br />For none mathematicians reading, it may be best to refer to the modern base 10 version of the 4,000 year old arithmetic notation (shorthand), and then, say a few days later, actually attempt write in the ancient 2-part notation. That is, clearly think in our modern base 10 for a few days, before trying to think and write as a 4,000 year old scribe.<br /><br />There are four trees being discussed here, each suggesting confusion, unless care is taken. They are: (1) the Horus-Eye notation, followed by a hekat, written in the first half of the expression, (2) Hieratic Egyptian fraction notion, followed by ro in the second half of the expression, (3) the Egyptian fraction series represented only the rational number (5*R/n), noting n in the denominator, allowing it to grow to any size, thereby allowing an exact computation, every time, and (4)the word ro, as 1/320th was factored from the remainder term.<br /><br />Clearly ro was a minor term, in the expression, possibly only a common divisor factor, used to add the Horus-Eye and Egyptian fraction series together, allowing a proof to be quickly performed, as was listed five times in the AWT (and not at all in the RMP).<br /><br />It is therefore recommended that novice readers of this blog do not allow one or two of the different types of trees that you run into, to be confusing. Look for the forest of each type of notation, stated as clearly as your education allows. Then and only then try to read and work with the inner workings of the AWT, such as an individual tree. The work on a complete (64/64)/n division problem as the 2,000 BC student scribe was trained.<br /><br />Have patience, each of the ideas are simple, seen separately. Taking in the set of anjcient ideas at once causes problems for many people. Please avoid shortcuts, especially the ones that Ahmes himself practiced, until you discover the foundations of Ahmes' arithmetic, such as the ancient methods that created 2/n tables.<br /><br />Peet, Gillings and other may have taken a couple of modern shortcuts, missing an ancient tree or two.<br /><br />Begin at the beginning of each of the AWT problems, and compare your beginning, middle and end work with the same type of problem written out in the RMP (#47, 81, 83 are the best examples). Then work to the end of each AWT problem, doing your own work, every step of the way.<br /><br />You will be rewarded. Spend the necessary time to work through more than one ancient problem as scribes solved it, using all of the old tools.<br /><br />BACKGROUND<br />-------------------<br />Thomas E. Peet, 1923, partially repeated an analysis of the AWT by showing several connections between Egyptian math and the practical experiences of an ancient Egyptian scribe that used two numeration systems, Horus-Eye and the Egyptian fractions (cited in the RMP). Peet muddled where one system ended and where the other system began by only detailing additive aspects of the two numeration systems, missing the exact Egyptian division features using 5ro as a partitioning idea, using numerators and denominators = 320 in an interesting way. Peet also prematurely concluded that the Egyptian division was only an inverse of the Egyptian multiplication operation.<br /><br />Peet did not directly discuss Egyptian division, as a general operation, as confirmed by the AWT examples. However, contrary to Gillings and Robins-Shute, Peet did seem to compute with 5ro, 4ro, 3ro, 2 ro and ro, but only from a limited view of the AWT student. Peet was slightly myopic, asking few meta questions, such as: were all of the student's divisions required to be exact? More importantly, no comparisons of Peet's view of the AWT were made to the RMP and its 84 problems. At least ten RMP problems, 36-43, and 81-82, have been misread with respect to ro, suggesting it was a weights and measures unit. Ro was actually connected to a generalized partitioning role, as closely related to other exact partitioning methods cited in the RMP, and other Middle Kingdom mathematical texts.<br /><br />Scholars are, of course, free to explore these issues on their own, commenting on the actual mix of Egyptian mathematics that meets a few of the standards that are deduced from the AWT, and its interesting set of division methodologies.<br /><br />Peet, and two later scholars, Gillings, 1972, and Robin-Shute, 1987, show that all the three scholars prematurely concluded in independent analyses that MK ro data (from the RMP and AWT) only meant 1/320 of a hekat, and no more. Not one of the three scholars grasped Ahmes central fact, that quotients and scaled ro remainders defined a weights and measures unit by beginning with a hekat unity (64/64), and dividing by any rational number less than 64.<br /><br />Returning to Peet, and his analysis will be shown in the next few paragraphs. He apparently made serious errors with respect to ro and its relationship to Egyptian division, as was vividly declared in the AWT hekat and 1/64 divided by 1/n context. Peet did not see ro's actual association with remainders, though he mentioned remainders from time to time. It is clear that 64 times 5, an early form of mod 5 in the R/3 term, included the use of numerators and denominators, or, by example, let the divisor of 64/64 be n, then<br /><br />Q + R/3<br /><br />appeared in two shorthand forms, the first being<br /><br />(1) (64/64)/n = Q/64 + R/(n*64).<br /><br />(2) (64/64)/n = Q/64 + (5*R/n)* 1/320, with ro = 1/320<br /><br />Ahmes used the second form. The first form is the manner in which modern mathematicians read this type of information.<br /><br />One of the most exciting aspects of the AWT is that the rational number (5*R/n) was easily converted to an Egyptian fraction series. It not known if this was the<br />first generalized used of Egyptian fractions.<br /><br />Silverman's 1975 point, that Egyptian fractions were found in the Old Kingdom, may mean that the balance beam problem was solved by an Old Kingdom scribe, noting:<br /><br />R/(n*64)<br /><br />with the Egyptian fractions series being either R/n or R/(n*64).<br /><br />Egyptian remainder arithmetic research continues. Thanks to the ancient scribes for leaving red flags raised by the AWT so that the simple five division problems opens a major door to decoding Egyptian fraction math from any era prior to 800 AD.<br /><br />Daressy in 1906 discussed Egyptian fraction math as remainder based. Daressy saw the exact divisions of some unit labeled in cubit units. Going beyond Daressy's analysis, a vivid clue to the Old Kingdom manner of partitioning cubit units, the hekat was partitioned in a manner that defined Egyptian fraction math for 2,800 years, ending in <a href="http://planetmath.org/encyclopedia/HibehPapyrus.html">800 AD</a>. <br /><br />A required meta view of the AWT included <a href="http://planetmath.org/encyclopedia/EconomicContextOfEgyptianFractions.html">Egyptian economy</a> and absentee landlords and Pharaohs that used the Egyptian fraction math for practical applications.<br /><br />author: <a href="http://en.wikipedia.org/wiki/User:Milogardner">Milo Gardner<br /></a>Milo R. Gardnerhttp://www.blogger.com/profile/15284868993340980422noreply@blogger.comtag:blogger.com,1999:blog-9073111.post-1119199643000046622005-06-19T09:25:00.000-07:002006-06-29T14:48:49.646-07:00Forward and Reverse Engineering AWT & RMP dataNine data points will be forward and reverse engineered as ancient<br />scribes would have recognized. Five data points are found in the<br />Akhmim Wooden Tablet, 1/3, 1/7, 1/10, 1/11 and 1/13, and four<br />are found in the RMP, 1/6, 1/20, 1/40 and 100/70, following the<br />modern remainder arithmetic form:<br /><br />(64/64)/n = Q/64 + R/(n*64)<br /><br />with Q = quotient and R = remainder.<br /><br />For example, AWT 1 divides 1/3 of a hekat as:<br /><br />(64/64)/3 = 21/64 + 1/(3*64),<br /><br />then the scribe converted the remainder data (often mentally) into<br />a fixed common divisor 1/320, a number that scribes named ro.<br />To better explain the scibal method of compulation, which may<br />have looked like this to the scribe:<br /><br />(64/64)/3 = (4' 16' 64')hekat + ( 1 3")ro, consider the following,<br /><br />A. Forward computation: follow a remainder arithmetic structure,<br />computing a quotient (Q) and a remainder (R), a two-part statement:<br /><br />1. AWT 1:<br /><br />A Foward Engineeriing<br /><br />(64/64)/3 = 21/64 (Q) + 1/(3*64)(R)<br /> = (16 + 4 + 1)/64) + (5/5)*1/192)<br /> = (4' 16' 64') hekat + 5/960<br /> = (4' 16' 64') hekat + (5/3) ro<br /> = (4' 16' 64') hekat + (1 3") ro<br /> <br /><br />Note, the 2nd remainder step multiplied 5/5 * 1/(3*64) = 5/(3*320)<br />thereby creating one common divisor multiple, 1/320, for all five<br />AWT solutions.<br /><br />B. Reverse engineering computation: begin with (4' 16' 64') hekat<br />+ (1 2")ro and find the original problem. First, it must be noted that<br />2' = 32/64. 4' = 16/64, 8' = 8/64, 16' = 4/64, 32' = 2/64 and<br />64' = 1/64, or:<br /><br />(1) Quotient: (4' 16' 64') = (16 + 4 + 1)/64 = 21/64<br /><br /><br />(2) Remaidner, Egyptian fraction: ( 1 2/3) = 5/3<br /><br />(3) Remainder, including ro: 5/3 ro = 5/(3*320)<br /> = 5/960<br /> = 1/192<br /><br />C: Double check the result consists of taking a Q value like 21/64<br />and an R values, such as 1/192 and adding them:<br /><br />21/64 + 1/192 = (63 + 1)/192 = 64/192 = 1/3.<br /><br />2. RMP 83 - 1<br /><br />A. Forward Engineering<br /><br />(64/64)/6 = 10/64 + 4/(6*64)<br /> = (8 + 2)/64 + (20/6 * 1/320)<br /> = (8' 32') hekat + ( 3 3') ro<br /><br />B. Reverse Engineering<br /><br />(1) Quotient part (8' 32') = (8 + 2)/64 = 10/64<br /><br />(2) Remainder part (3 3') ro = 10/3 * 1/320 = 10/960 = 1/96<br /><br />C. Proof ( Q + R):<br /><br /> 10/64 + 1/96 = (30 + 2)/192 = 32/192 = 1/6<br /><br />which means that B (2) actually should read:<br /> (3 3') ro = 20/6 * 1/320 = 20/1920 = 1/96<br /><br />3. AWT 2<br /><br />A. Forward Engineering<br /><br />(64/64)/7 = 9/64 + 1/(7*64)<br /> = (8 + 1)/64 + (5/7)* 1/320<br /> = (8' 64') hekat + (2' 7' 14') ro<br /><br />B. Reverse Engineereing<br /><br />(1) quotient part: (8' 64') = (8 + 1)/64 = 9/64<br /><br />(2) remainder part: convert Egyptian fraction<br /><br /> 2' 14' 28' = (7 + 2 + 1)/14 = 10/14 = 5/7, or<br /><br />(3) remainder part including ro : 5/7 ro = 5/(2240) = 1/448<br /><br />C.Proof: (Q + R):<br /><br /> 9/64 + 1/448 = (63 + 1)/448 = 64/448 = 1/7<br /><br />4. AWT 3<br /><br />A. Forward Engineering<br /><br />(64/64)/10 = 6/64 + 4/(10*64)<br /> = (4 + 2)/64 + 20/(10)* 1/320<br /> = (16' 32') hekat + 2 ro<br /><br />B. Reverse Engineering<br /><br /> (1) quotient: (16' 32') = (4 + 2)/64 = 6/64<br /><br /> (2) remainder: 2 ro = 2/320 = 1/160<br /><br /> C. Proof (Q + R):<br /><br /> 6/64 + 1/160 = (30 + 2)/320 = 32/320 = 1/10<br /><br /><br />5. AWT 4<br /><br />A. Forward Engineering<br /><br />(64/64)/11 = 5/64 + 9/(11*64)<br /><br /> = (4 + 1)/64 + (45/11)* 1/320<br /><br /> = (16' 64') hekat + (4 11') ro<br /><br />B. Reverse Engineering<br /><br />(1) quotient: (16' 64') = (4 + 1)/64 = 5/64<br /><br />(2) remainder, Egyptian fraction: (4 11') = ( 44 + 1)/11 = 45/11<br /><br />(3) remainder including ro: 45/11 ro = 45/11 * 1/320 =<br /> 45/(11*320) = 9/(11*64) = 9/704<br /><br /> C. Proof: (Q + R):<br /><br />5/64 + 9/704 = (55 + 9)/704= 64/704 = 1/11<br /><br />6. AWT 5<br /><br />A. Forward Engineering<br /><br />(64/64)/13 = 4/64 + 12/(13*64)<br /> = 16' + (60/13)* 1/320<br /> = 16' hekat + (4 8/13 ) ro<br /> = 16' hekat + (4 2' 13' 26') ro<br /><br />B. Reverse Engineering<br /><br /> (1) quotient: 16' = 4/64<br /><br /> (2) remainder, Egyptian fraction:<br /><br /> 4 2' 13' 26' = (104 + 13 + 2 + 1)/26<br /> = 120/26 = 60/13<br /> (3) remainder including ro<br /><br /> (60/13)*ro = 60/(13*320) = 12/(13*64)<br /><br /> C. Proof: (Q + R):<br /><br />4/64 + 12/(13*64)= (52 + 12)/(13*64)<br />= 64/(13*64) = 1/13<br /> <br />7. AWT 83 - 2<br /><br />A. Forward Engineering<br /><br />(64/64)/20 = 3/64 + 4/(20*64)<br /> = (2 + 1)/64 + (20/20)*1/320<br /> = (32' 64') hekat + 1 ro<br /><br />B. Reverse Engineering<br /><br />(1) quotient: (32' 64') = (2 + 1)/64 = 3/64<br /><br />(2) remainder, Egyptian fraction 1, insufficient data<br /><br />(3) remainder including ro : 1 ro = 1/320<br /><br /> C. Proof: (Q + R):<br /><br /> 3/64 + 1/320= (15 + 1)/320<br /> = 16/320 = 1/20<br /><br />which means the Egyptian fraction step B (2) should be 20/20,<br />or 4/(20*64)<br /><br />8. RMP 83 - 3<br /><br />A. Forward Engineering<br /><br />(64/64)/40 = 1/64 + 24/(40*64)<br /> = 64' + (120/40)*1/320<br /> = 64' hekat + 3 ro<br /><br />B. Reverse Engineering<br /><br /> (1) quotient: 64' = 1/64<br /><br /> (2) remainder, Egyptian fraction 3, insufficient data<br /><br /> (3) remainder including ro: 3 ro = 3/320<br /><br />C. Proof (Q + R):<br /><br /> 1/64 + 3/320 = (5 + 3)/320= 8/320 = 1/40<br /><br />which means that the Egyptian fraction was 120/40, or<br />24/(40*64)<br /><br />9 . RMP 47<br /><br />A. Forward Engineering<br /><br /> (6400/64)/70 = 91/64 + 30/(70*64)<br /> = (64 + 16 + 8 + 2 + 1)/64<br /> + (150/70)*1/320<br /><br /> = (1 4' 8' 32' 64')hekat<br /> + (2 7') ro<br /><br />B. Reverse Engineering:<br /><br /> (1) Quotient: (1 4' 8' 32' 64') = (64 + 16 + 8 + 2 + 1)/64<br /> = 91/64<br /> <br /> (2) Remainder, Egyptian fraction: (2 7') = 15/7<br /><br /> (3) Remainder including ro: (15/7) ro = 15/(7*320)<br /> = 3/(7*64)<br /><br />C. Proof: (Q + R)<br /><br /> 91/64 + 3/(7*64) = (637 + 3)/(7*64)<br /> = 640/(7*64)<br /> = 10/7<br /><br /><br />The 100/70 fact seems hidden. However, the point that is found<br />in the scribal narrative, meaning the initial problem was 100/70.<br /><br />In summary the forward and reverse processes work equally well<br />for the 33 additional data points found in RMP 69 and 81. Oddly<br />scholars reporting RMP 81 only cited hin data. A hin was 1/10th<br />of a hekat, and implementing forward and reverse engineering takes<br />the hin fact into account, while exactly scaling to 1/3200th in the<br />remainder term as scribes first calculated the data.Milo R. Gardnerhttp://www.blogger.com/profile/15284868993340980422noreply@blogger.comtag:blogger.com,1999:blog-9073111.post-1100816643788670942004-10-18T14:05:00.000-07:002007-04-07T12:34:16.836-07:00Egyptian Arithmetic (Appendix)INTRODUCTION: Defining any arithmetic system, at any time,<br />or anywhere, requires the use of numbers. In the Egyptian<br />case, Old Kingdom hieroglyphic numbers were many to one,<br />such as listing the symbol one(1) four times to show that the<br />number four (4) was being listed. This numeral system was<br />superceded by ciphered numerals at the beginning of the Middle<br />Kingdom, about 2200 BC. The ciphered Hieratic numerals began<br />with one (1), and proceeded to over one million, as was the case<br />forthe its predecessor. Hieratic numbers were represented within a<br />base 10 system, mapping each number onto an alphabetical symbol.<br />To save space the actual listing of numbers versus letters<br />will not be cited here.<br /><br />Zero itself was known and used for many situations, in the Old and<br />New Kingdom. However, zero did not appear in either of the<br />Egyptian numeration systems, nor any Ancient Near East numeration<br />system.The first use of a numeration based zero, outside of Europe<br />and its base 10 application, was in ancient Mexico, and its fully<br />positional base 20 numeration system (the later being bases 4 and<br />5, not a pure base 20 system, counting only 0-19). More on zero<br />later. It is a topic worthy of study, noting that many books have<br />been written on the subject. I too may add a blog in the near future.<br /><br />But, getting back to Western numeration, after number itself was<br />defined, following the Greek tradition, Europeans began to accept<br />Vedic- Arabic base 10 numerals, around 900 AD, though zero was<br />not formalized within a numeration system in that wave of diffusion.<br />Around 1200 AD, zero did arrive in Europe, first in Germany, and<br />then it began to formalized in our base 10 decimal positional<br />system and its four functions of arithmetic, addition, subtraction,<br />multiplication and division, as we know them today. Given this<br />incomplete history, other math subjects need to be brought into<br />the discussion.<br /><br />THE BASE 10 DECIMAL EXAMPLE<br /><br />Stevins wrote two books in 1585 AD, one for business and one<br />for science, that may have first formalized base 10 decimals, using<br />zero as a place holder. Addition began with the use of rational<br />numbers, usually a counting number, or integer, a, b, c, and d,<br />all unequal such that:<br /><br />1. addition: a + b = c<br /><br />Addition was straight forward, when all the numbers involved<br />were > 1. However, Greeks and many cultures had trouble<br />accepting the general use of a negative number, so even here<br />problems had arisen in certain cultures. Stevin's work resolved<br />a range of issues with his definition of our base 10 decimal system.<br /><br />2. subtraction: a - b = d<br /><br />Subtraction was apparently straight forward when a > b. However,<br />when a remainder involved a fraction, a translation to base 10<br />decimals took place, defining a method of round off was sometimes<br />required. It should be recalled that Arabic-Vedic history had been<br />closely linked to the Babylonian infinite series numeration, with<br />several of base 60 thinking ending up in Stevin's analysis and<br />final product. Citing an extreme Babylonian case, subtraction base<br />remainders, stated as fractions, were rounded off when certain<br />situations took place. For Babylon, one of their cases cited<br />d = 1/91. Round off too place such that only multiples of 2, 3 and 5<br />were used as denominators, usually rounding off to 1/90. Stevins<br />view of base 10 round off was not than complicated. We round off<br />today based on a predetermined number of significant digits, so<br />accuracy has been left to the user, to the businessman and to the<br />scientist.<br /><br />3. Multiplication: a x b = ab = e<br /><br />Multiplication was seen as repeated addition, such as<br />5 x 7 = 35 as, 5 x 1 = 5, added seven time, or 7 x 1 = 7,<br />added together 5 times. Tables have been memorized<br />over the years, the easiest ones being based on prime<br />numbers.<br /><br />4. Division: a divided by n meant = q + r/n (but from this point<br />a general notation of A/n = Q + R/n will be used, following the<br />shorthand notation presented in three Middle Kingdom<br />mathematical texts, <a href="http://akhmimwoodentablet.blogspot.com/">Akhmim Wooden Tablet, 2000 BCE:</a><br /><a href="http://reisnerpapyri.blogspot.com/">Reisner Papyri, 1800 BCE</a> and the <a href="http://egyptianmath.blogspot.com/">RMP or Ahmes<br />Papyrus, 1650 BCE.</a> These texts defined R = remainder,<br />somtimes allowing 0, an integer or a fraction, preceded<br />by Q = quotient.<br /><br />A rigorous definition of the factor and remainder theorem, a method<br />that is reported only in terms of algebra, will not be offered at this<br />time. At some future point the pure arithmetic arithmetic aspects<br />will be compared to the ancient versions of Egyptian two-part<br />numbers. However, to fill a logical hole, I offer Oystein Ore's<br />analysis on this general history of arithmetic (number theory)<br />subject, as his history of number theory developed in Greece and<br />elsewhere, based on LCM's, GCD's, aliquot parts and a<br />range of other ideas.<br /><br />Concerning the sophistication of Egypt arithmetic or any cultural<br />view of arithmetic, the manner in which remainders were handed<br />offers significant insight into the depth their mathematical thinking.<br />To understand Egypt's Middle Kingdom arithmetic, a close look<br />at its small set of mathematical texts need to take place.<br /><br />MIDDLE KINGDOM ARITHMETIC<br /><br />The four arithmetic operations will be defined in reverse order,<br />division, multiplication, subtraction, and addition. In this<br />manner the interesting aspects of division can be high-lighted,<br />revealing a few of the complexities of Egyptian thinking. As a<br />basic point, Egyptian math tended to be limited to the domain of<br />positive rational numbers, with only a small number of exceptions.<br />For the purposes of this discussion, only positive rational numbers<br />will be considered.<br /><br />1. Division<br /><br />The AWT lits five problems of the division of unity (64/64) by 3, 7, 10,<br />11 and 13. A shorthand notation was developed that we can recognize<br />as following our modern base 10 structure, as given by:<br /><br />(64/64)/n = Q/64 + R/64.<br /><br />A simplier Reisner and RMP version says<br /><br />p/n or pq/n = Q + R/n<br /><br />with p and q being primes, without the additional hekat<br />unity substitution.<br /><br />Divisors n were easily read in exact ways for both forms of<br />arithmetic when remainders were converted to vulgar fractions<br />and finally to an Egyptian fraction series.<br /><br />This shorthand notations is explained by one division example, let n = 3,<br />then Q = 21, R= 1 such that a final answer in our modern base 10 would<br />have been:<br /><br />64/64 divided by 3 = 21/64 + 1/(3*64) = 320/960<br /><br />This form of thinking lies at the center of the discussion of<br />the AWT and its five division problems. The most complicated<br />division was dividing a fraction by a fraction, in these cases<br />1/64 divided by 1/n (with n < 64). The Akhmim Wooden Tablet<br />lists five of these divisions. The division process was simplified<br />by a mental process that first divided 64 by n, producing a two<br />part answer, and quotient (Q) and a remainder(R). Suspecting<br />that the history of Egyptian numeration was wished to be closely<br />associated with the Old Kingdom, the Q portion was stated as<br />a Horus-Eye series. R was stated in the Middle Kingdom mode<br />of numeration, Hieratic fractions.<br /><br />In the case of 64/3, Q = 21, such that the question, which<br />divisors of 64 (1, 2, 4, 8, 16, 32, 64) added up to 21 was asked?<br />To an ancient scribe 16 + 4 + 1 achieved 21, meaning that 1/64th<br />fractional inverses of 16, 4 and 1, were easily written out as<br />(1/4 + 1/16 + 1/64) hekat.<br /><br />Concerning the 64/3 remainder, R = 1, meant that 1/3 our simple<br />answer of 1/(3*64) = 1/192 was not written. What was written<br />can be explained by 1/64 being replaced by 5/320, such that<br />1/3 x 5/320 = 5/960, or (1 + 2/3) ro was written as the remainder<br />portion, or totally,<br /><br />(1/4 + 1/16 + 1/64) hekat (1 + 2/3)ro.<br /><br />Continuing with . the 64/3 case, 1/192 was written out as 5/3 ro or<br />(1 + 2/3)ro using Hieratic script, since ro = 320, which also meant<br />that:<br /><br />(5*Q*n + 5*R)<br /><br />was the numerator, which always added up to 320, or generally<br /><br />320/(n*320) = 1/n<br /><br />In that way 1/n was multiplied by n, as the AWT did with its<br />five division problems, thereby finding and confirming that<br />a 100% accurate hekat was found in each case.<br /><br />The AWT data is confirmed in the Rhind Mathematical Papyrus,<br />or the Ahmes Papyrus, if you prefer, by ro being used several<br />times, always mistranlated by historians, for one myopic reason or<br />another. Strange but true.<br /><br />2. Multiplication: repeated additions can be used to describe the<br />operation, as an introduction. Duplation tables were listed for one<br />of the numbers being multiplied such that the final answer was picked<br />from the table. At this time, a rigorous discussion of multiplication<br />will not be offered.<br /><br />The AWT proved each of its 1/3, 1/7, 1/10, 1/11, and 1/13 valuations<br />by multiplying by 3, 7, 10, 11 and 13, respectively. Examples taken<br />from the AWT will be listed at a later time.<br /><br />3. Subtraction: This operation can be seen as a reminder, much as<br />the division operation calculates a remainder. Seen as a remainder,<br />Egytians did not allow round off, except when irrational numbers like<br />pi were being discussed, no matter how small the fractions that were<br />involved. The exact remainder for all rational number substractions<br />sets Egyptian method at a higher level than its Babylonian counter<br />parts. Five examples follow, as variations of the AWT theme,<br />beginning with unity (1), and subtracting: 1/3, 1/7, 1/10, 1/11 and 1/13.<br /><br />The Hultsch-Bruins 2/p method will be used, as first re-discovered in<br />1895, as easily generalized to the n/p case:<br /><br />1. 1 - 1/3 = 2/3<br /><br />(The EMLR lists 1/3 + 1/3, but usually 2/3 was considered prime,<br />and not reduced to lower partitions)<br /><br />2. 1 - 1/7 = 6/7<br /><br />= 1/2 + (12 - 7)/14, since 5 can not be found amidst the divisors of 14,<br /><br />= 1/2 + 1/3 + (15 - 14)/(3*14)<br /><br />= 1/2 + 1/3 + 1/42<br /><br />(check 2/3 for the first partition, an interest series)<br /><br />3. 1 - 1/10 = 9/10<br /><br />= 1/2 + (18 - 10)/20, with 8 being found amidst the divisors of 20<br /><br />= 1/2 + (5 + 2 + 1)/20<br /><br />= 1/2 + 1/4 + 1/10 + 1/20<br /><br />( try 2/3 as the first partition, it is also interesting)<br /><br />4. 1 - 1/11 = 10/11<br />= 1/2 + (20- 11)/22, with 20-11 = 9 not amidst the divsors of 22, therefore<br />= 1/2 + 1/3 + (27 - 22)/66, with 27 - 22 = 5 admidst divisors of 66, such that<br />= 1/2 + 1/3 + (3 + 2)/66<br />= 1/2 + 1/3 + 1/22 + 1/33<br /><br />5. 1 - 1/13 = 12/13<br />= 1/2 + (24 - 13)/26, with 11 not being amidst the divisors of 26, therefore<br />= 1/2 + 1/3 + (33 - 26)/72, with 7 being amidst the divisors of 75, such that<br />= 1/3 + 1/3 + 1/13 + 1/39<br /><br />(of course 2/3 can be a first partition, revealing 2/3 + 1/4 + 1/156, as an<br />alternative)<br /><br />4. Addition: a + b stated as unit fraction series were added, with<br />ease, allowing for the understanding of the general conversion<br />of n/p and n/pq tables of unit fraction series, all providing<br />guidance to each scribe, to know the range of the expected<br />final answers.<br /><br />Conclusion: Completely reading Egyptian mathematical texts has<br />always been a difficult task. The primary reason is that scribes left<br />only brief outlines of their work, and therefore modern scholar,<br />until very recently, have not fully translate Middle Kingdom data<br />fully into modern base 10 fractions. This ancient hieratic shorthand<br />has been called many things, including only a form of hieroglyphc<br />math from an older era. Clearly, hieratic shorthand contained at<br />least two forms of remainder arithmetic, as modern researchers<br />have been detailing in 2005. This researcher has followed faint<br />clues by fully translating data into modern base 10 fractions, by<br />finding and explaining the scribal conversion used in the missing<br />vulgar fraction step. This extra work has been necessary since<br />scribes usually thought of in a form of mental arithmetic, and<br />when they did not, scribal notes were very brief.<br /><br />In addition it had been shown that the word ro, 1/320, also meant<br />common divisor. Common divisors is a concept long over looked,<br />one that sheds new light on this 80- 120 year old question.<br /><br />Irrespective of anyone's particular view of a particular ancient <br />problem, a clear translation to modern base 10 decimal fractions must <br />take place at some point. Fuzziness aside, to achieve that goal, as <br />best as anyone is able, all of the knowable arithmetic steps <br />mentioned and omitted in the ancient texts must be discussed, and <br />compared to the modern base 10 decimal fraction version of the <br />problem. 'No stone or text should be left unturned or unread', <br />is one way to summarize the operational aspect of this task. <br />Skipping over one ancient or modern arithmetic issue, or<br />another, has caused the great confusion that still exists in <br />the modern reading of the ancient mathematical texts. However, <br />given a little patience and humility, a better reading will be <br />appearing over the next few years. Adding a vulgar fraction step <br />to both forms of remainder arithmetic has greatly assisted in <br />translating scribal arithmetic into our modern base 10 arithmetic.<br /><br />There is still additional work to do in reading Middle Kingdom<br />mathematical texts. But 2005 has been a productive year.<br /><br />June 2006 has been even more productive, noted by the obvious proof<br />that 10/n hin was used in 29 RMP 80 cases, a method that could have<br />been easily extended to 320/n ro, and any other volume unit. The <br />problems with proving the exact value of a dja, oipe, or any other <br />infrequently used volume measure is to directly link it to a known <br />value, hopefully the hekat itself.Milo R. Gardnerhttp://www.blogger.com/profile/15284868993340980422noreply@blogger.com