tag:blogger.com,1999:blog-69699602013-08-28T23:05:16.168-07:00Egyptian Math HistoryEgyptian fraction arithmetic appeared in the Middle Kingdom. After 2,000 BCE the Akhmim Wooden Tablet, the Reisner Papyrus, EMLR and RMP used the same remainder arithmetic. Egyptian division was inverse to Egyptian multiplication. Egyptian fraction weights and measures units methods diffused to Greek, Hellene, Coptic, Arabs and medieval eras. Ahmes' RMP 36 red auxiliary numbers scaled 2/53 imply that 2/n tables were constructed by the same methodology.milo gardnernoreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6969960.post-79455461704923353732008-11-12T08:10:00.000-08:002012-01-26T10:04:08.014-08:00Least Common Multiples, modern and ancient<a href="http://www.dafont.com/hieratic-numerals.font">Ciphered Egyptian numerals</a> were used within hieratic script after 2050 BCE, with zero written as sfr for accounting and other purposes. One of the oldest texts was the <a href="http://en.wikipedia.org/wiki/Akhmim_wooden_tablets">Akhmim Wooden Tablet</a> (AWT). To translate hieratic Egyptian mathematics and hieratic unit fraction arithmetic to modern base 10 fractions missing mental steps must be added back. In the AWT missing steps summed to an initial (64/64) hekat unity. To prove that the correct missing scribal steps are outlined a doubling check method that follows scribal shorthand proof steps must be introduced. The five AWT division of (64/64) by 3, 7, 10, 11 and 13 answers were multiplied by initial divisors and returned (64/64) five times. Hana Vymazalova published the (64/64) hekat unity in 2002.<br /><br />Incomplete translations were offered by 20th century math historians that did not report the (64/64) hekat unity, and subtle scribal math facts. Clearly 20th century math historians stressed language observations <a href="http://en.wikipedia.org/wiki/transliteration_of_ancient_Egyptian">transliterations</a> correctly reporting ancient arithmetic symbols without reporting subtle mathematical subjects beginning with hard-to-read <a href="http://reisnerpapyri.blogspot.com/">Reisner Papyrus</a> <a href="http://independent.academia.edu/MiloGardner/Papers/623827/Egyptian_Fractions_Unit_Fractions_Hekats_and_Wages_-_an_Update">, EMLR </a>and <a href="http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html">Ahmes 2/n table</a> construction methods. <br /><br />For example, A.B Chace in 1927 concluded that a translation of the RMP was complete. Chace indirectly discussed Vymazalova's hekat unity approach in <a href="http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html">Ahmes' bird (hekat) feeding problem </a>(RMP 87). Ahmes' data was garbled so that no modern scholar can decipher the intended facts, per Chace's view. <br /><br />One RMP entry suggests that a least common multiple (LCM) unity was common as Ahmes thought and wrote out<br /><br />3/11 (6/6) - 18/66 = (11 + 6 + 1)/66 = 1/6 + 1/11 + 1/66<br /><br />with (6/6) being a unity that scaled 3/11 to a solvable vulgar fraction, recorded (11 + 6 + 1) in red, and omitted the order initial details.<br /><br />As readers of Egyptian math history are aware Ahmes' red auxiliary numbers was an idea associated with LCMs. The precise nature of scribal LCMs has come to light.<br /><br />There are four ways to discuss Egyptian proofs, taken from<br /><br /><a href="http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html">http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html</a> ,<br /><br /><br />that 20th century scholars had not considered are summarized by:<br /><br />1. Read/translate RMP 38, that solved the problem:<br /><br />a. 10 hekat equals 3200 ro times 7/22 equals 101 9/11<br /><br />b. proving 101 9/11 times 22/7 equals 3200 ro<br /><br />c. with a footnote that 35/11 times 1/10 = 35/110 = 7/22<br /><br />2. Read/translate RMP 47, that solved the problem<br /><br />a. 100 hekat equals 6400/64 hekat divided by 70<br /><br />b. proving (91/64)hekat + 150/70 ro was the answer<br /><br />3. Read/translate RMP 66 problem<br /><br />a. 10 hekat times 32o ro equals 3200 ro, divided by 365, the number of days per year<br /><br />b. 3200/365 equals 8 + 280/365 with<br /><br />c. a duplation proof for the quotient 8, by writing<br /><br />365 1<br />730 2<br />1460 4<br />2920 8<br /><br />and the remainder (3200 - 2920) equals 280 by<br /><br />(243 1/3 + 36 1/2 + 1/6)/365 = 280/365<br /><br />4. RMP 82 an began with a hekat unity, (64/64), divided by 29 divisors n in the range 1/64 < n < 64, as RMP 83 divided (64/64), divided by 6, 20 , and 40. A final Russian entry generally used an ancient aliquot fraction or ratio idea (Russian terms for red auxiliary numbers) without identifying a hypothetical use of ancient LCMs per: <a href="http://eom.springer.de/A/a013260.htm" target="_blank">http://eom.springer.de/A/a013260.htm</a><br /><br />Note the straight forward modern and ancient arithmetic that easily fills in Ahmes use of red auxiliary LCMs per:<br /><br /><a href="http://rmprectotable.blogspot.com/" target="_blank">http://rmprectotable.blogspot.com/</a><br /><br />and,<br /><br /><a href="http://en.wikipedia.org/wiki/Red_auxiliary_numbers" target="_blank">http://en.wikipedia.org/wiki/Red_auxiliary_numbers</a>milo gardnernoreply@blogger.comtag:blogger.com,1999:blog-6969960.post-1147264147963144682006-05-10T05:07:00.000-07:002012-01-26T09:50:02.796-08:00History of Egyptian FractionsModern scholars have encountered difficulties finding the context from which Egyptian fractions were first created. Egyptologists assumes that the all of the origins of the unit fraction system began in the Old Kingdom. Aspects of it did, solving an Old Kingdom binary round-off problem. Math historians tend to read the Middle Kingdom math texts as independent documents, allowing the mathematical information considered by the personal views of researchers to determine the direction of the research. How can personalized and conflicting points of view be resolved? <br /><br />One way is to show that Old Kingdom binary number numeration system was built upon a numeration system that rounding off to six binary fraction terms. Old Kingdom scribes worked to resolve round off problem. Only in the Middle Kingdom were the binary round off problems corrected. A finite rational number system burst onto the Middle Kingdom stage. Scribes, such as Ahmes, used a new cipher numerations system to record Egyptian fraction mathematics, beginning with modern-like arithmetic definitions of addition, subtraction, multiplication and division, hidden beneath the "<a href="http://www.google.com/search?hl=&q=russian+peasant+multiplication&sourceid=navclient-ff&rlz=1B3GGGL_en___US345&ie=UTF-8&aq=0&oq=russian">Russian Peasant</a>" type of Old Kingdom algorithmic multiplication operation.<br /><br />In other words, jumping to the present era, the formal research of reading the most important Egyptian mathematical document, the Rhind Mathematical Papyrus (RMP), began in 1879, with the mulling over of the 'best' meaning and 'first' use of the 2/nth table. One of the most prepared of the modern mathematicians that have taken time to research this topic is Kevin Brown. Brown <a href="http://www.mathpages.com/home/kmath340/kmath340.htm">published his work </a>on the internet around 1995. Like those those mathematicians that had gone before him, Brown inspected several elements of the 2/nth table. Concerning the critical element of the 'first' uses of Egyptian fractions, Brown seeing many exact mathematical statements in the 2/nth table, he offered the suggestion that the ancient scribes may have been gamblers<br />a point of view that is historically incorrect. <br /><br />Egyptian fraction math was created as a means to<a href="http://planetmath.org/encyclopedia/EconomicContextOfEgyptianFractions.html"> decentralize the Egyptian economy</a>. <br /><br />The majority of the earlier scholars offered less insightful conclusions concerning the first 'uses' of Egyptian fractions. For example, the interesting 2/nth table patterns, as used to exactly solve many of the 87 RMP problems, have not been formally resolved in academic journal debates.<br /><br />Traditional math history scholars have failed to parse Ahmes 2/n table construction methods for several reasons, the main barrier being the consideration of the additive aspects of the unit fraction series. Well motivated scholars muddle the essence of <a href="http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html">Ahmes 87 problems</a>, by not reading every line of Ahmes rational number scaling method --- described in<a href="http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html"> RMP 36 </a>and RMP 37.milo gardnernoreply@blogger.comtag:blogger.com,1999:blog-6969960.post-1135011398732493992005-12-19T08:39:00.000-08:002009-01-14T08:30:38.100-08:00Vulgar fractions and 2/nth tables<a href="http://en.wikipedia.org/wiki/user:milogardner">by: Milo Gardner</a> <a href="http://milorgardner.blogspot.com/2008/08/milo-gardner-personal-info.html">Bio</a><br /><br />INTRODUCTION<br />Ancient Egypt was an unusually well run 'top down' controlled culture. Royal edicts controlled many aspects of Egyptian daily life. For example, prior to 2,050 BCE edicts were primarily written and controlled in <a href="http://en.wikipedia.org/wiki/Eye_of_Horus">Horus-Eye arithmetic</a>. After 2,050 BCE edicts were primarily written and controlled in <a href="http://planetmath.org/encyclopedia/EgyptianFraction2.html">Egyptian fraction arithmetic</a>. Edits controlled land, its unit productivity, and the number of labor hours needed to achievement expected outputs. Edicts also controlled the manufacture, and distribution of beer, bread, and other products. Generally, edicts controlled the production and distribution of state-run and privately-run inventories.<br /><br />Control systems were unitized, with upgrades to Old Kingdom (OK) binary methods taking place during the Middle Kingdom (MK). That is, Horus-Eye infinite series standards were replaced by finite Egyptian fraction units during the MK. Hints of the binary Horus-Eye arithmetic remained in use during the MK through quotients. MK remainders were written in Egyptian fractions, scaled to several units.<br /><br />The Egyptian fraction notation dominated Egyptian arithmetic. Scribes had modified an Old Kingdom binary nation by writing Horus-Eye quotients, and scaled Egyptian fractions. One remainder was scaled to 1/320 of a (hekat). The Old Kingdom 1/320 unit, named ro., was scaled a hekat volume unit into: hin (1/10), a unit dja (1/64), and other hekat sub-units.<br /><br />Scribal arithmetic after 2,050 BCE double checked observed weights and measures units by first creating a theoretical data system. One purpose of MK weights and measures systems assisted in the management and control of Pharaoh and absentee landlord agricultural products, and other commodity inventories.<br /><br />Scribal mathematics included several theoretical building blocks, from which observed measurements were made. Control units were created from abstract models, one being a commodity based money system. The double entry accounting methods double checked inputs, outputs and wages as implemented on <a href="http://en.wikipedia.org/wiki/Heqanakht_papyri">two large farms</a>, documented in <a href="http://www3.interscience.wiley.com/journal/120118222/abstract?CRETRY=1&SRETRY=0">2002</a> and <a href="http://eh.net/bookreviews/library/0868">2004</a>.<br /><br />A 2002 analysis of this absentee landlord's set of accounts showed that the system allowed estate owners' control " ... by converting the concrete into an abstract theoretical value via the use of â€˜monies of accountâ€™ to attain a measure of economic and social reciprocity." Pharaoh and other absentee landlords, through out the Ancient Near East, used versions of the theoretical first and practical measurement system second for three thousand years as noted when considering the 2,000 BCE Egyptian political <a href="http://planetmath.org/encyclopedia/EconomicContextOfEgyptianFractions.html">economy </a>.<br /><br />The hekat controlled volume inventories. Measurements of grain, beer and other volumes were double checked (in accuracy) by applying a theoretically expected value against an actual inventory number. On another level, double checking was implemented by theoretical calculating pi, a rounded off numbers to a traditional binary 256/81 value, thereby estimating of grain, beer and other volumes hekat within circular containers by any other approximation. The hekat use of pi contained a poor approximation, compared to modern estimates, a point that has been noted by scholars. Yet, beyond pi scribal rational number calculations were exactly written in Egyptian fraction remainder arithmetic. It is important to note that the MK Egyptian fraction arithmetic contained the first system of mathematical proofs, a point that has not been stressed by scholars until recently.<br /><br />EXACT EGYPTIAN FRACTION ARITHMETIC<br />Academic studies of the 4,000 year old Egyptian fraction arithmetic began during the 1880's. Studies first stressed the 1650 BCE Rhind Mathematical Papyrus (RMP), and its 84 problems introduced historians to 2/n tables. A copy of the RMP and its recto <a href="http://en.wikipedia.org/wiki/RMP_2/n_table">2/n table</a> had been pirated out of England in 1873, and printed in Germany in 1877, an event that jump-started academic studies in two European countries, England and Germany. Proposed RMP and 2/n mathematical methods consumed academic debates for over 120 years. Scholars debated fragmented aspects of infinite Horus-Eye binary series compared to finite Egyptian fraction arithmetic by stressing additive methods until the 1930's. Very few scribal hints of abstract math had been suggested by applying additive analytical methods before 1930. Algorithmic, and other suspected abstract aspects, of the Egyptian texts began to be discussed after 1990, especially after 2002, heating up a once stale debate. Two texts, the <a href="http://wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll">Egyptian Mathematical Leather Roll</a> (EMLR), and the <a href="http://wikipedia.org/wiki/Akhmim_Wooden_Tablet">Akhmim Wooden Tablet</a> (AWT) were integrated into scholar discussions after 2002.<br /><br />The EMLR, circa 1950 BCE, records theoretical and practical methods to convert 26 1/p and 1/pq rational numbers to Egyptian fraction series as an introduction to advanced <span style="TEXT-DECORATION: underline"></span><a href="http://rmprectotable.blogspot.com/">RMP 2/n table</a><a href="http://en.wikipedia.org/wiki/RMP_2/n_table"> </a>methods by uses of red auxiliary least common multiples. The AWT, circa 1950 BCE, records a theoretical/ practical method to partiion a hekat by 3, 7, 10, 11 and 13 , but the theoretical use of (64/64) was likely created before a practical application into binary quotients and scaled Egyptian fraction remainders was formalized in the AWT. Modern scholars began to study the AWT in 1906, outlining five division problems of a hekat unity, and five proofs of the hekat unity (64/64), into a scaled 1/320 unit named ro, a task that did not point out the hekat unity (64/64) until 2002 (by Hana Vymazalova). The binary quotient and scaled Egyptian fraction remainder form of division was not published until 2006 (by Milo Gardner).<br /><br />TRANSLATING THE MATHEMATICAL TEXTS<br />The <a href="http://rmprectotable.blogspot.com/">RMP</a>, along with the <a href="http://emlr.blogspot.com/">Egyptian Mathematical Leather Roll</a>, (EMLR) were donated to the British museum in 1864 by the family of Henry Rhind. The Egyptian fraction debate began after an unauthorized copy of the RMP was published in Germany in 1877. The British Museum did not discuss the RMP or unroll or translate the EMLR until 1927. The EMLR's 1927 translation was limited to an additive view, and therefore little of historical importance was reported until 2002. In 2007, a deeper view of the EMLR was connected to the RMP 2/n table, as well by to the 1202 AD Liber Abaci.<br /><br />To begin as historians had done in 1879, the Rhind Mathematical Papyrus (RMP) was first read. It began with a<a href="http://www.mathpages.com/home/kmath340/kmath340.htm"> 2/n table</a> that was built by selecting nearly optimal red auxiliary LCM's (following RMP 21,22, and 23 practice problems). The RMP's numeration, arithmetic, and algebraic data contains controversial aspects since the red auxiliary aspect and the role of RMP 21, 22 and 23, was not published until 2008 (by Milo Gardner). The RMP 2/nth table and its 84 problems had been read in conflicting ways causing scholarly confusion over 120 or more years. Conflicting views continue to merge into a unified body of knowledge that stressed LCMs. That is, scribal mathematics is finally being reported as the scribes thought of the sub ject, increasingly free from scholarly transliteration errors, suppositions, and conjectures.<br /><br />Controversies that limited a fair reading of the RMP can be segmented into four classes of assumptions. Assumption one (1) defines a use, or non-use, of vulgar fractions as intermediate steps in the 2/n table. Egyptian fr Assumption two (2) defines a use, or non-use. of additive arithmetic in the entirety of the RMP. Assumption three (3) defines a use, or non-use, of p, and q, as prime numbers in the 2/n table. Finally, assumption four (4) defines a use of theory followed by practical math in the 2/n table and not practical math followed by theory in the RMP and other Middle Kingdom mathematical texts.<br /><br />The first problem in reading the RMP is that Ahmes did not identify 2/n table conversion methods. The same is true for the majority of the RMP's 84 problems. Several of Ahmes' 84 problem, such as RMP 81 and problems related thereto, have been grossly garbled by the scholarly community. Misreadings have taken place over the last 120 years by introducing four classes of conflicting assumptions: (1) vulgar fractions, (2) additive arithmetic, (3) general use of prime numbers, and (4) a theoretical first form of mathematics.<br /><br />Assumption one (1) parses Ahmes' intermediate and final 2/n table answers as written in hard to read Egyptian (unit) fractions. The 2/n table and data in all Middle Kingdom mathematical texts includes quotients and remainders, a point that has confused scholars. Early scholars tended to discuss only one-side of the two-sided quotient and remainder story line, even in the 2/n table. Quotients and remainders have been decoded as complete sentences by one-part and two-part hekat partitioning methods, thereby reducing the confusion related to four classes of conflicting assumptions.<br /><br />Assumption two (2) tends to correct early views that Ahmes arithmetic operated only on an additive level. Scholars had taken a conservatiove additive position as a guide, hoping that it would coincide with Ahmes and other scribal points of view. Today, it is clear that non-additive aspects of RMP problems are needed to assist in translating the hard-to-read problems, beginning from Ahmes' point of view. That is, tt is clear that Ahmes had not exclusively used additive methods.<br /><br />Assumption three (3) discusses a single multiple method to convert EMLR, RMP 2/n table vulgar fractions. The multiple method employed a LCM 'red auxiliary' technique to concise writing unit fraction series, using p and q as prime numbers.<br /><br />Assumption four (4) discusses the AWT and other mathematical texts that began with a theoretical idea. In the AWT's case, it began with the definition of a hekat unity as (64/64). Early scholars had commenced an analysis of the majority of Middle Kingdom mathematical texts from a practical point of view, almost always missing theoretical elements. That is, early scholars were unable to parse theoretical ideas from the mathematical texts since practical blinders had limited their view. This blog discusses several theoretical ideas that were applied in practical ways in several mathematical texts.<br /><br />Another arithmetic controversy came into focus in 2002 with the publication of scribal remainder arithmetic than contained the use of a hekat unity valued at 64/64. The <a href="http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet">Akhmim Wooden Tablet </a>provided the focus with the scribal use of 64/64 to exactly partition a hekat by 3, 7, 10, 11 and 13 into binary quotients and exact Egyptian fraction remainders scaled to 1/320 hekat units.<br /><br />Despite resolving several arithmetic controversies, additive, algorithmic, and abstract scholars are widely separated in 2008. The three groups appear to have filled in missing scribal shorthand steps, or ignored certain issues altogether, in ways that support their respective positions,. The three groups, by their actions, and inaction, have placed subtle barriers between their positions, and therefore seem not to motivated to enter into formal debates with anyone outside of their respective group. Hopefully, future debates will take place between the three groups and thereby resolve the major differences that separate their respective academic positions.<br /><br />Additive scholars discuss the <a href="http://en.wikipedia.org/wiki/Reisner_Papyrus">Reisner Papyrus</a>, and the RMP, with several scholars mentioning quotients, and remainders as disconnected bits of information. Additive scholars do not require quotients and remainders to be unifying aspects of the Reisner Papyrus division by 10 problems, or any of the RMP division by 10 problems. This group, de facto headed by Peet, has dominated ancient Egyptian math publishing since the 1920's. Gillings, 1972, and Robins-Shute, 1987, are also members of this group. Overall, this group concludes that Egyptian fraction arithmetic contained no abstract aspects, only practical arithmetical statements.<br /><br />Algorithmic scholars began to publish their findings in the 1990s. Jim Ritter's group explores Old Kingdom intuitive algorithms that were built into Horus-Eye fractions. Given that Western mathematics' abstract algorithm was first published around 800 AD, by Arab mathematicians, an older intuitive algorithmic arithmetic may have dominated the Egyptian Old Kingdom's numeration system. This group suggests that Babylonian numeration, and Babylonian weights and measures, units may have provided models and an intellectual context to Horus-Eye and hieratic numeration, as well Pharaoh's control units before and after 2,000 BCE.<br /><br />Old Kingdom's cursive arithmetic may, or may not, have been integrated into Egyptian Middle Kingdom Egyptian fraction arithmetic. It is well understood that the pre-2000 BCE Babylonian intuitive algorithmic methods dominated infinite series base 60 numeration during the Egyptian Middle Kingdom. Yet, it is important to note that Demotic script documents Egyptian contact with Babylonians after 1500 BCE. Given hundreds of years of contact Babylonian mathematicians never altered its infinite series, rounded off, form of base 60 system to take on the exact finite features of Egyptian fraction arithmetic and its <a href="http://www.mathpages.com/home/kmath340/kmath340.htm">2/n table</a>. Overall, this group concludes that Old Kingdom algorithmic arithmetic methods contained only practical statements, with only hints of number theory. That is, hieratic Egyptian fractions arithmetic, more than likely, did not contain formal abstract arithmetic.<br /><br />The algorithmic group includes<a href="http://www.annetteimhausen.com/resume.html"> Annette Imhausen</a>, a well known Egyptologist. The group stresses that the RMP and the other mathematical texts are insufficient to fairly decode Egyptian MK arithmetical foundations, hence translations of the mathematical texts into modern base 10 decimal arithmetic are not required. This group looks elsewhere to provide language aspects of the topic, as background, for the competing history of math groups to consider. This group searches outside the traditional Egyptian mathematical texts in a manner that stress hieroglyphic language and iconic foundations of hieroglyphic and hieratic arithmetic symbols. Another way to view algorithmic scholars is that clues are searched out that may allow intuitive algorithms and intuitive Old Kingdom algorithmic precepts to form the basis of Egyptian fraction arithmetic, with hieratic writing only being a cursive form of hieroglyphic writing. Hence this group suggests that hieratic arithmetic is contained in hieroglyphic arithmetic, and that neither numeration system contains formalized abstract methods.<br /><br />Theoretical number theory scholars began to publish after 2002. This group reports that the RMP 2/n table and its 84 problems use <a href="http://planetmath.org/encyclopedia/RemainderArithmetic.html">remainder arithmetic</a> that are built upon quotients and exact remainders. The remainder arithmetic proposal emerged after 2004. The view suggests that scribal remainder arithmetic generally converted vulgar fractions to quotients, written several ways, and exact remainders written as Egyptian fraction series. The oldest texts are the <a href="http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet">Akhmim Wooden Tablet,</a> <a href="http://en.wikipedia.org/wiki/Reisner_Papyrus">Reisner Papyrus</a>, and the <a href="http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll">EMLR</a>. The proposed theoretical aspects of the Egyptian fraction texts considers (fragments of) scribal shorthand to contain vulgar fractions as intermediate steps. The AWT scribe, for example, is shown to have applied rigorous statements and proofs, possibly the first in Western mathematics, to solve five division problems in ways that are easily translated into modern base 10 arithmetic. This group concludes that Egyptian fraction methods contain several abstract arithmetic features.<br /><br />Two features of the abstract methods are two-part remainder arithmetic, and one-part remainder arithmetic. Both arithmetic systems used quotients, and exact Egyptian fraction remainders. The two-part statements, scaled to 1/320 of a hekat, were converted into one-part practical statements, as needed. One-part statements were scaled to sub-units of the hekat, and other measures, were easily converted to two-part statements, as needed. This group suggests that the remainder arithmetic aspect of the Egyptian fraction system had replaced (superseded) the Old Kingdom hieroglyphic (binary) system as Pharaoh's primary arithmetic and control system by 2,000 BCE.<br /><br />Hana Vymazalova, a graduate student from Charles U., Prague, was the first scholar to publish the Akhmim Wooden Tablet's hekat unity (64/64) aspect of this topic, thereby completing a task initiated by Daressy in 1906. It is clear that Vymazalova's discovery showed that a hekat unity (64/64) was divided by any rational number upto 64, writing out quotient and scaled remainder answers. For example let the divisor be 3, the following was written:<br /><br />(64/64)/3=21/64 + (5/3)1/320=(16 + 4 + 1)/64 + (5/3) ro=1/4 + 1/16 + 1/64 + ( 1 + 2/3) ro<br /><br />The medical texts and Ahmes included the use of a generalizied one-part m/n quotients and remainders that allowed divisors to exceed 64 to any size, within a 1/10, 1/64, 1/320 and other scaling values. For example, for 1/10 hin units, Ahmes used the relationship 10/n hin, or for n = 3, 10/3 hin = (3 + 1/3) hin.<br /><br />The exact scribal remainder arithmetic method allowed MK state-run inventories to be measured and controlled by the<a href="http://en.wikipedia.org/wiki/Hekat_%28volume_unit%29"> hekat </a>volume unit, and its sub-units. The<a href="http://en.wikipedia.org/wiki/hekat_%28volume_unit%29"> hekat</a> was divided into two-part numbers and one-part numbers . The Akhmim Wooden Tablet describes two-part number statements and proofs relating to the division a hekat, and its hekat unity (64/64) by divsiors 3, 7, 10, 11 and 13. The medical texts, reported by Tanja Pommerening, in 2002, newly described the dja, and other hekat sub-units, as one-part numbers scaled to revised MK and NK units.<br /><br />Scribes followed the Pharaoh's bidding, and therefore properly ciphered rational numbers into Egyptian numbers, and its four arithmetic operations, addition, subtraction, multiplication and division, as required. Scribal base 10 arithmetic had been improved to exact number during the MK, thereby replacing the Old Kingdom round-off based binary methods. The MK scribes created exact remainder techniques that accurately controlled the distribution of the Pharaoh's inventories. Several elegant aspects of the 2/n table math had not been recorded in the Old Kingdom, though early fragments of the earliest unit fractions, written in binary fractions, have been noticed by scholars (Silverman being one). By 2002 AD upgrades of MK arithmetic and measurement systems began to be published.<br /><br />Several aspects of the abstract MK system continued in use during late ancient Near East (ANE) era. In several ways major features of the MK Egyptian fraction system were continuously used for over 3,000 years. Greek, Hellene, Arab, Islamic, and Medieval businessmen from Italy, Pisa and Venice, and their resident mathematicians used the Egyptian fraction system, each making minor modifications to the Pharaonic system.<br /><br />That is, within the medieval Mediterranean trading area the ancient Egyptian fraction notation functioned as a unifying aspect. The Egyptian fraction system allowed well-defined trading systems to be widely used in ther ANE, from at least the time of Alexander the Great, much as Pharaoh had first created his internal and external control systems years earlier.<br /><br />The medieval Egyptian fraction trading unit system was slowly replaced by base 10 decimals and its trading units. Arab mathematicians around 800 AD, first added Hindu base 10 numerals (replacing ciphered numerals), changing few of the Egyptian fraction units and methods. After the close of the Silk Road, and the flowering of European mathematicians around 1600 AD, the base 10 decimal units and its new form of numeration began to take root.<br /><br />The Egyptian fraction trading units, and its arithmetic, had dominated the ANE from at least the time of Alexander, and the Greeks. The Egyptian fraction unitized trading units slowly faded after the close of the Silk Road, and the fall of the Byzantine empire's capital in 1454 AD. Base 10 numerals, and the Egyptian fraction trading units, had been introduced by Arab mathematicians after 800 AD. <a href="http://en.wikipedia.org/wiki/Pope_Sylvester_II">Pope Sylvester II</a> assisted in spreading the numerals and continuing the Egyptian fraction system to Europe in 999. The 1202 <a href="http://en.wikipedia.org/wiki/Liber_Abaci">Liber Abaci </a>documents the 600 year added life of Egyptian fractions for everyday arithmetic (using lattice multiplication rather than the duplation form of Egyptian multiplication) and for Europe's trading units.<br /><br />At this point is may be interesting to update the average math history book summary of zero that says that a practical zero first arrived in Germany around 1200 AD and slowly was added as a theoretical place-holder in our base 10 decimal system, as finalized in 1585 AD (Stevins, and some say Napier). Such a summary omits the Greek, Babylonian and Egyptian practical use of a zero, with the Greek symbol, topped by two dots, found its way to India, and around 800 AD returned to the Arab world, and documented by Fibonacci in the 1202 AD Liber Abaci. Zero was well known and used in a wide array of Ancient Near East cultures, and used positionally in Mesoamerica, well before Europcentric folks wished ownship of the theoretical properties of zero just 400 years ago.<br /><br />Modern base 10 decimal notation, and its use of zero as an exponent, fully replaced the traditional Egyptian factions units and its Liber Abaci trading units after 1586AD. In 1585 Stevins published two rigorous definitions of<a href="http://en.wikipedia.org/wiki/Simon_Stevin#Decimal_fractions"> base 10 decimals</a>, one for science, and another for business. In addition, new decimal trading units were developed. Our modern base 10 decimal system was formally approved by the Paris Academy in 1586. Slightly before 1586, and increasingly after 1600, new decimal trading units began to be used through out Europe. The new trading units bore only vague relationships to their unit fraction predecessors.<br /><br />The newer base 10 decimal unit system grew into our modern metric system, with the algorithm functioning as the unifying principle. In the 1500's Europe used a range competing local trading units, like pounds and yards in England, for local and international trade. Slowly. several competing local trade units were written into decimals, with metrics being used in trade. To eliminate the differences between local trade units a simplified system of decimal metrics emerged. Of course, England and the USA still use many of their local units, such as pounds and yards for its business and everyday transactions, while formally using decimal metrics for its scientific purposes.<br /><br />VULGAR FRACTION CONVERSION MARKERS<br />Readable 'markers' had been left by Egyptian MK scribes. End-product arithmetic markers were written (ciphered) into base 10 Egyptian fractions. The Egyptian fraction data stressed remainders, preceded by quotients. Hard to read vulgar fractions, denoted, and implied, by cryptic scribal shorthand has been the beginning number in solving many scribal problems. That is, scribal quotients and Egyptian fraction answers had been written into two-part and one-part numbers, two forms of remainder arithmetic, using vulgar fractions as an intermediate step.<br /><br />One class of markers was abstract, theoretically writing two-part numbers. Abstract methods had been often associated with the hekat, exactly writing remainders as Egyptian fractions, scaled to 1/320 units. A second class of markers can be seen as practical, distributing the Pharaoh's inventories based on hekat measuring devises. The devises were sized from barrels to cups, scaled by Egyptian fractions to the jar (20), oipe (4), hekat (1), hin (1/10), dja (1/64) and ro (1/320), all hekat units. The system's daily input/output was double checked by theoretically based two-part numbers, one-part numbers, and other methods.<br /><br />Egyptian fraction two-part remainders were translated into one-part numbers by using the system's unitary value, divided by a divisor n. Example: hinu ,1/10th hekat, two-part problems began with (64/64)/10. The 1/10 unit was further partitioned into n one-part units. The translation to one-part statements was achieved by writing 10/n hin statements as an Egyptian fraction series followed by the word hin . Note that one-part numbers allowed vulgar fractions to be converted into Egyptian fractions at different points in each remainder arithmetic definition. It should be stressed that the purpose and scope of the Egyptian fraction mathematics facilitated the daily input/output control of inventories by double checking the practical records. Two forms of abstract methodologies, two-part numbers and one-part numbers provided the intellectual basis for the Pharaoh's practical hekat measurements.<br /><br />SEPARATING ONE-PART AND TWO-PART NUMBERS<br />The two scribal remainder arithmetic systems had looked alike to scholars until 2002. Prior to 2002 scholars had confused the two Egyptian fraction statements Scholars had attempted to translate both classes of data as single additive statements. Very little of either the one-part, or two-part statements, were constructed by additive methods. One-part and two-part statements were written by quotients and remainders, but the Egyptian fractions were used differently, and were denoted by unique markers.<br /><br />For example, the two-part hin, (64/64)/10, was written as 1/16 + 1/32 + 2 ro,. The equal one-part, 10/10, was written as 1 hin. RMP 80 includes a table of 29 equivalent ro and hin series.<br /><br />Generally, the AWT can be read partitioning a hekat unity (64/64) by a divisor n, such that Q/64 plus (5R/n)ro as rigorously defined by a quotient and remainder. The quotient, Q/64. was written as a binary fraction series, and the remainder, 5R/n, was written as and Egyptian fraction scaled to a 1/320th of a hekat.<br /><br />The hin unit, 10/n hin, was written by quotients and remainders as well, with the quotient being an integer. The remainder was written as an Egyptian fraction series, scaled to the 1/10th hekat unit.<br /><br />There were at least two additional on-part units, the dja and the ro. The dja, written as 64/n dja, was written by quotients and remainders, with the quotients being an integer. The remainder was written as an Egyptian fraction series, scaled to the 1/64th hekat unit.<br /><br />The ro, written as 320/n ro, was also written by quotients and remainders in a manner that had confused scholars. The simple ro statements had also used quotients as integers, and remainders as Egyptian fractions scaled to the 1/320th. Scholars had misread the ro information when compared to the two-part ro statement, confusing the translation of two-part numbers, like (64/64)/10's longer form 1/16 + 1/32 + 2 ro. The equivalent 320/10 one-part statement was written as 32 ro.<br /><br /><br />IN 2002 FOUR PAPERS CHANGED THE EGYPTIAN FRACTION DEBATE<br />Gardner, Pommerening, Silger and Vymazalova published aspects of the 130 year old Egyptian fraction debate. Taken together the four papers show that MK scribes wrote Egyptian fractions series from standard conversion methods that had resisted change for over 3,000 years.<br /><br />Gardner's paper, (<a href="http://emlr.blogspot.com/">updated in 2007</a>) shows that the EMLR and the RMP raised 1/p, 1/pq, and 2/pq to multiples, often to (p + 1)/(p + 1) in the RMP. For example 2/21, factored as 1/3 x 2/7, was raised to (3 + 1)/(3 + 1) x 2/21 = 4/4 x 2/21 = 8/84 thereby allowed an elegant Egyptian fraction series (6 + 2)/84 = 1/14 + 1/42 to be reported in the RMP 2/n table. The paper also describes several other basic methods that were commonly used by the EMLR and the RMP scribes.<br /><br />Pommerening's paper reports the hekat dja unit 'healed" the 1/64 Horus-Eye number, the subject of the AWT. Pemmerening shows that the oipe replaced the hekat as the internal control unit in the Egyptian New Kingdom. The oipe then became Egypt's external trading unit, thereby spreading the Egyptian fraction trading system and units across the Ancient Near East.<br /><br />Silger reports seven methods in the 1202 AD <a href="http://liberabaci.blogspot.com/">Liber Abaci </a>that converted vulgar fractions to Egyptian fraction series. Silger's data shows that five of these methos date to the EMLR and the RMP. One medieval conversion method was used to convert 22 of 26 EMLR series, and all but three RMP 2/pq conversions. The first 124 pages of the 500 page Liber Abaci high-lites the fact that five Egyptian fraction conversion methods had been continuously used for over 3,000 years, most likely transmitted through trading units.<br /><br />Vymazalova's paper may be the most important. She indirectly reports an abstract basis of<span style="TEXT-DECORATION: underline"> </span><a href="http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet">AWT</a> two-part numbers, most likely the first exact Egyptian control method. The AWT data reports that five two-part numbers were returned to unity, its beginning value, 64/64.<br /><br />Today, 2007, it is clear that Middle Kingdom scribes always wrote arithmetic data into exact Egyptian fraction notations, one-part numbers and two-part numbers. As background, Old Kingdom <a href="http://en.wikipedia.org/wiki/Eye_of_Horus">Eye of Horus</a> binary numeration had rounded off to six-term series in an inaccurate manner, compared to 20th century round-off methods. Interestingly, the relatively more accurate 20th century computers, that used <a href="http://en.wikipedia.org/wiki/Fixed-point_arithmetic">fixed-point</a> and <a href="http://en.wikipedia.org/wiki/floating-point_arithmetic">floating-point</a> arithmetic, were not as accurate as MK Egyptian arithmetic (that exactly wrote rational numbers into Egyptian fraction series in several situations).<br /><br />Stated another way modern computers round-off rational numbers as decimal and other notations whenever prime number denominators , i.e. 1/3, 1/7, 1/11 and 1/13, are involved. It is important to note MK scribes did not round-off rational numbers in any hieratic document. MK scribes only rounded off irrational numbers, and higher order numbers like pi, a policy improved upon by Greeks.<br /><br />THE OLD KINGDOM ROUND-OFF PROBLEM<br />Expanding the pertinent issue MK numeration, and their two methods, both methods had eliminated rational number round-off errors. As background, prior to 2,000 BCE, Horus-Eye binary numbers had been truncated to 6-terms, a very short limit, defining:<br /><br />1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64.<br /><br />The inaccurate rounded off class of number generally threw away a 7th term, upto 1/64. This class of number round-off and created a huge scientific problem. During the Old Kingdom the Horus-Eye binary 6-term system had inaccurately reported large and small numbers, i.e. whenever the number contained prime number denominators.<br /><br />The most elementary Egyptian method may have first corrected aspects of the Horus-Eye errors was the one-part number. An early form of this method may have existed in the Old Kingdom, though no text has been found to confirm that hypothesis.. What is clear, and validated, is found in MK texts. The well known Old Kingdom binary round-off errors had been eliminated by a Middle Kingdom innovation.<br /><br />TWO-PART NUMBERS SOLVED THE HORUS-EYE PROBLEM<br />The discovery that most likely solved the Horus-Eye problem was made around 2,000 BCE. A new class of two-part number was rigorously reported in the Akhmim Wooden Tablet (AWT). The Old Kingdom's binary notation had been exactly corrected by the AWT discovery.<br /><br />Revised, or updated, one-part numbers were written into vulgar fractions and remainder arithmetic, as recorded in the 1800 BCE <a href="http://en.wikipedia.org/wiki/Reisner_Papyrus">Reisner Papyrus</a>, and other texts. The Reisner reported the digging rates of workers in units of ten by quotients and remainders. The first six problems in the <a href="http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus">RMP</a> confirm this manpower management system by dividing other work outputs by 10, and writing out quotient and remainder answers. One implication may be that 10 hour work days were reduced to hourly efficiency reports. A second implication may be that 10 workers were assigned to each daily task; hence, dividing the daily output by 10 created a daily worker efficiency rate.<br /><br />The most complicated, and therefore, the most interesting innovation was the two-part number. The oldest text to report this method is the 2000-1850 BCE <a href="http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet">Akhmim Wooden Tablet</a> (AWT). The AWT scaled five Egyptian fraction remainders to an exact 1/320 unit called ro. The AWT method created an innovative two-part number where quotients were written as binary fractions, and remainders were written as Egyptian fractions followed by the word ro. The two-part number calculated exact units of 1/320 of a hekat remainders, thereby solving the Old Kingdom's Horus-Eye round-off problem.<br /><br />Formally, the AWT partitioned a hekat unity was written as (64/64), and partitioned by five divisors 3, 7, 10, 11 and 13. At other times in the MK, i.e. the RMP, other divisors were taken from the range 1/64 < 5 =" 5/40" 40 =" 1/10" p =" 3" 1 =" 7," 4 =" 8/84" 84 =" 1/14".<br /><br />1.<a href="http://planetmath.org/encyclopedia/WhyStudyEgyptianFractionMathematics.html"> Why Study Egyptian Mathematics</a>?<br /><br />a.<a href="http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html"> Ahmes Papyrus, an update</a><br /><br />b. <a href="http://mathforum.org/kb/thread.jspa?threadID=1876268&tstart=0">Math Forum discussions</a><br /><br />2. <a href="http://en.wikipedia.org/wiki/RMP_2/n_table">RMP 2/n Table</a> (Wikipedia)<br /><br />3. <a href="http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll">EMLR</a> (Wikipedia)<br /><br />4. <a href="http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html">EMLR</a> (Planetmath)<br /><br />5. <a href="http://planetmath.org/encyclopedia/EgyptianMath3.html">Hultsch-Bruins Method</a> (Planetmath)<br /><br />6. <a href="http://planetmath.org/encyclopedia/EgyptianFraction2.html">Egyptian fractions</a> (Planetmath)<br /><br />7. <a href="http://en.wikipedia.org/wiki/Kahun_Papyrus">Kahun Papyrus</a><br /><br />8. <a href="http://planetmath.org/encyclopedia/LiberAbaci.html">Liber Abaci</a> (Planetmath)<br /><br />9. <a href="http://liberabaci.blogspot.com/">Liber Abaci</a> (Blog)<br /><br />10.<a href="http://en.wikipedia.org/wiki/Red_auxiliary_numbers">Red Auxiliary Numbers</a> (Wikipedia), (<a href="http://planetmath.org/encyclopedia/FirstLCMMethodRedAuxiliaryNumbers.html">Planetmath</a>)<br /><br /><a href="http://en.wikipedia.org/wiki/User:Milogardner"><br /></a>milo gardnernoreply@blogger.comtag:blogger.com,1999:blog-6969960.post-1107792449929107842005-02-07T08:07:00.000-08:002012-01-26T09:48:53.605-08:00Egyptian Math History<a href="http://www.dafont.com/hieratic-numerals.font">Ciphered Egyptian numerals</a> were used within hieratic script after 2050 BCE, with zero written as sfr for accounting and other purposes. One of the oldest texts was the <a href="http://en.wikipedia.org/wiki/Akhmim_wooden_tablets">Akhmim Wooden Tablet</a> (AWT). To translate hieratic Egyptian mathematics and hieratic unit fraction arithmetic to modern base 10 fractions missing mental steps must be added back. In the AWT missing steps summed to an initial (64/64) hekat unity. To prove that the correct missing scribal steps are outlined a doubling check method that follows scribal shorthand proof steps must be introduced. The five AWT division of (64/64) by 3, 7, 10, 11 and 13 answers were multiplied by initial divisors and returned (64/64) five times. Hana Vymazalova published the (64/64) hekat unity in 2002.<br /><br />Incomplete translations were offered by 20th century math historians that did not report the (64/64) hekat unity, and subtle scribal math facts. Clearly 20th century math historians stressed language observations <a href="http://en.wikipedia.org/wiki/transliteration_of_ancient_Egyptian">transliterations</a> correctly reporting ancient arithmetic symbols without reporting subtle mathematical subjects beginning with hard-to-read <a href="http://reisnerpapyri.blogspot.com/">Reisner Papyrus</a> <a href="http://independent.academia.edu/MiloGardner/Papers/623827/Egyptian_Fractions_Unit_Fractions_Hekats_and_Wages_-_an_Update">, EMLR </a>and <a href="http://planetmath.org/encyclopedia/RMP36AndThe2nTable.html">Ahmes 2/n table</a> construction methods. <br /><br />For example, A.B Chace in 1927 concluded that a translation of the RMP was complete. Chace indirectly discussed Vymazalova's hekat unity approach in <a href="http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html">Ahmes' bird (hekat) feeding problem </a>(RMP 87). Ahmes' data was garbled so that no modern scholar can decipher the intended facts, per Chace's view. <br /><br />Reading only the RMP problems and answers, and considering no other texts, Chace's view was correct for his time period. However, considering the Akhmim Wooden Tablet, translated by a Charles U, Prague grad student, in 2002 and other texts at later times, Ahmes' hekat 'feeding rates' is clearly6 read and confirmed, thereby updating Chace's excellent 1927 transliteration in major ways.<br /><br />In other words, traditional language-type transliterations had poorly detailed scribal number facts written when considering all four arithmetic operations. Scribal subtraction, and division operations had not been fully understood by the early researchers, especially 1920's scholars.<br /><br />Modern decoding keys are needed to convert all four of the scribal arithmetic operations into modern arithmetic operations. Scribal subtraction, and division, decoding keys became available after 2002, facts that are not well known. Beginning in 2002, the new scribal subtraction and division decoding keys has allowed big chunks of once unreadable hieratic data to be translated into modern base 10 fractions (for the first time). These post-2002 two-way translations have made the 1920's and other transliteration views of the RMP and other texts obsolete, in several important respects, exposing additional aspects of Ahmes and other scribe's mathematics.<br /><br />Translation efforts published before 2002 had been unfair and incomplete in reporting scribal arithmetic and algebra exclusively within additive arithmetic. Traditional scholars had omitted the hard-to-read aspects of the ancient arithmetic operations that had not been additive in scope. Scholarly omissions had introduced additive 'transliterations' of the numerical information, thereby being unfair to the ancient scribal math in serious ways.<br /><br />Seen as modern translation gaps, post 2002 readings of the ancient arithmetic methods, especially subtraction and division operations had begun anew. Several new decoding keys had been pin-pointed related to scribal subtraction and division. The new decoding keys were not additive. Two-way abstract and practical streets had been built that details rigorous modern arithmetic versions of the newly decoded aspects of the ancient hieratic arithmetic.<br /><br />One of the subtraction and division keys was the scribal use vulgar fractions. Scholars had suggested that scribal division consisted only of unit fraction terms, thereby eliminating the need for scholars to consider vulgar fractions. However, vulgar fractions appear in almost every RMP problem, hidden away in the often unreported (or under reported) scribal shorthand, sometimes explicitly, and sometimes implicitly.<br /><br />For example, RMP # 32 states in modern terms<br /><br />x + (1/2 + 1/4)x = 2 or<br /><br />x + 7/12x = 2, or<br /><br />19/12 x = 2, or<br /><br />x = 24/19 = 1 + 5/19<br /><br />Ahmes' shorthand notes, often fragmented in reporting its details, worked with the 5/19 vulgar fraction. Noting his 1 + 1/6 + 1/12 + 1/114 + 1/224 answer, it appears generalized, but not optimized. As a Hultsch-Bruin method, Ahmes did the following:<br /><br />Find 5/19 by first subtracting 1/12 or<br /><br />5/19 - 1/12 = (41)/(12*19)<br /><br />= (38 + 2 + 1)/(12*19)<br /><br />= 1/6 + 1/114 + 1/224, or<br /><br />the final answer<br /><br />1 + 1/6 + 1/12 + 1/114 + 1/224<br /><br />Ahmes' choice of 1/12 followed the 2/19 2/nth table suggestion, where:<br /><br />2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114<br /><br />as also used in 2/95, where 2/19 x 1/5 = 1/5 (1/12 + 1/76 + 1/114)<br /><br />as Wilbur Knorr, in 1982, as published in the journal Historia Mathematica, and others at other times.<br /><br />Had Ahmes 'thought of another number' like 1/4, a more optional series 1/4 + 1/76 would have been found (as the 400 AD Akhmim Papyrus, from the Coptic era includes in its n/19 table).<br /><br />Gillings and other authors, i.e. like Robins-Shute, had avoided discussing Ahmes' shorthand that revealed the general use of vulgar fractions . Ahmes's shorthand notes are fragmentary, for sure, but facts that point in the remainder direction when intermediate vulgar fractions were converted by Ahmes, and other scribes, to Egyptian fraction series.<br /><br />This blog details a range of specific textual ways that scribal shorthand had been hard to read. This researcher has been lucky in several areas. Several others had assisted over the last 20 years.<br /><br />In summary ad hoc relationships have corrected several scholarly oversights over the last few years. Sadly, gaps still exist in reading several ancient texts. As a consequence the oldest Egyptrian number system, and its Horus-Eye and Egyptian fraction arithmetic methods, will continue to be poorly read by only applying language transliteration rules. Additional ancient arithmetic rules, parsed from the available texts, are needed.<br /><br />Special care needs to be taken when reading hieratic arithmetic and its fragmented scribal shorthand. One new new method to achieve this goal is documented on this blog. Choose whatever name that fits this new discovered method. One way is to build two-way streets, working within ancient and modern remainder arithmetic. Final translations should not be offered until the ancient and modern remainder arithmetic are doubled checked.<br /><br />This blog shows that ancient words and numbers are only modern snapshots of ancient arithmetic statements. To fully translate a snapshot of an arithmetic statement, and find the ancient context, fragmented arithmetic shorthand must be read using rigorous methods. Filling in a scribal gap at one's leisure, skipping ones that are not readily understood, had commonly omitted major segments of the scribal methods, one being the use of remainder arithmetic.<br /><br />To resolve a few of the fragmented aspects of the ancient texts, wider translations of the scribal math shorthand has been developed that covers added aspects of the available data. One technique translates the ancient data into modern base 10 fractions as a bilingual text. The new methods may be seen as comparable to the Rosetta Stone's use of phonetic symbols to link the third language's decoding to the first two. The new method uses quotients and remainders, often written within common divisors, like 1/320 of a hekat.<br /><br />Math and language scholars had often taken an aspect of a text, and offered interpretations outside of the original context. When attempting to translate MK mathematics and its metrology, translations have been claimed to be accurate and complete. Later translators found that the early translators had 'translated what they could', and had omitted important fragmented facts .<br /><br />This blog focuses on clearing up a few of the fragmented facts, one being the remainder arithmetic cited in the RMP and other texts.<br /><br />For example, RMP 62 states that<br /><br />100/13 = 7 + 2/3 + 1/39<br /><br />But what does that mean? The <a href="http://reisnerpapyri.blogspot.com/">Reisner Papyrus</a> will be shown to have written:<br /><br />n/10 = Q + R<br /><br />where Q was a quotient and R was a remainder, first stated as vulgar fraction, that was easily converted to an Egyptian fraction series. The Reisner cites:<br /><br />8/n1 = 1/2 + 1/4 + 1/20 = 8/10<br /><br />48/n2 = 4 + 1/2 + 1/4 + 1/20 = 4 + 8/10<br /><br />18/x3 = 1 + 1/2 + 1/10 = 1 + 6/10<br /><br />64/x4 = 6 + 1/4 + 1/10 + 1/20 = 6 + 4/10<br /><br />36/x5 = 3 + 1/2 + 1/10 = 3 + 6/10<br /><br />such that x1 = x2 = x3 = x4 = x5 = 10<br /><br /><br />That is,<br /><br />100/13 = 7 + 9/13 or 7 + 2/3 + 1/39<br /><br />with 7 being a quotient and 9/13 being a remainder that was converted to 2/3 + 1/39.<br /><br />Additional proof is provided by the word ro, a term often read in metrology as 1/320th of a hekat. The word ro has a broader meaning including rest, remainder and related ideas that connect to the 'healing' of Horus-Eye arithmetic.<br /><br />Ro was defined in at least two different situations. By reading the hekat division operation, cited in the Akhmim Wooden Tablet and the RMP, ro was used as a remainder scaling factor. Specifically Middle Kingdom scribes wrote (64/64)/n = 64/n + (5R/n)ro in the AWT and RMP, and (320/n)ro in the RMP and the medical texts.<br /><br />On an arithmetic level the word ro also meant 'common divisor' as well scaling factor. Ro was linked to binary fractions and Egyptian fraction series, points that scholars have been slowly appreciating since 2002.<br /><br />Stated in other terms, scholars have reported ancient math data in fragmentary ways beginning in the 1880's. By 1923 scholars like Peet began to conclude that their Egyptian fraction decoding work was complete. Peet concluded that the ro idea had only equaled 1/320th of a hekat and, by implication, could not have been re-valued within another context, such as partitioning a cubit,<br />or other hekat situations.<br /><br />Future study is required to detail the scribal division methods that were comparable, if any, to two forms of hekat remainder arithmetic (scribal division).<br /><br />Peet's analysis stressed the idea of Egyptian division, as a limiting factor on how and why the Akhmim Wooden Tablet scribe partitioned a hekat (unity) by 3, 7, 10, 11 and 13. Peet did not go into detail to show how the AWT scribe proved his answers or seek comparative cubit or hekat partitioning methods, as reported by Masse and Gewiche.<br /><br />Peet was not the only scholar that introduced misconceptions by skipping over otherwise 'unreadable' fragments of texts. Peet created a set of unproven additive conclusions concerning the scope and content of Egyptian mathematics. Scribal subtraction and scribal division was not well reported by Peet, especially related to Daressy's 1906 reading of the Akhmim Wooden Tablet.<br /><br />Considering original texts, often assumed to only additive, Egyptian fraction based arithmetic several texts had often been unfairly read in trivial ways. It will be shown that arithmetic texts containi9ng closely related Horus-Eye and Egyptian fraction data had not been fairly read by Peet and other early scholars.<br /><br />Over 40 RMP examples of binary and Egyptian fraction data had been ignored, or grossly misread, by early scholars. The following data sets seemed not fit the standard scholarly additive paradigm:<br /><br /><a href="http://www.mathorigins.com/image%20grid/awta.htm">http://www.mathorigins.com/image%20grid/awta.htm</a><br /><br /><a href="http://akhmimwoodentablet.blogspot.com/">http://akhmimwoodentablet.blogspot.com/</a><br /><br />Only recently, 2002 to be precise, and increasingly for the next five nears, these 40 data points been parsed and reviewed. The data was not additive as Peet had concluded in 1923.<br /><br />This blog analyzes 40 data points and updates an ancient Egyptian division operation. One modern name for a 4,000 year old division method is remainder arithmetic.<br /><br />This blog also offers a rigorous method that may allow experts and amateurs to double check their steps involved in translating Egyptian mathematical texts to modern base 10 fractions.<br /><br />Continuing to step back, scholars, had often worked alone. Scholars had unfairly thought that a rigorous ancient text translation to modern base 10 fractions was not necessary. The step is necessary,<br /><br />Frances I. Griffith in 1891 sensed that ro meant 'greatest common measure' within the RMP. Others scholars working in other texts, Daressy in 1906 with the Akhmim Wooden Tablet sensed other interesting arithmetic features. Daressy sensed exactness in all but the 1/11 and 1/13 cases, where scribal typographical errors had hindered his analysis. Yet, Daressy cited three examples that showed three divisions of a hekat unity and three proofs that found (64/64) , clues that were not followed up by Peet.<br /><br />Interestingly, by 1923 Peet tried to close off debate related to the highest form of Egyptian division operations by attempting to refute Daressy's 1906 analysis. Sadly, the majority of 1920's scholars had tried to stay within a singular view of Middle Kingdom arithmetical texts, an additive one. Early scholars, therefore, had picked out readable aspects from certain scribal shorthand statements, skipping over otherwise 'unreadable aspects'.<br /><br />Oddly, by 1923 scholars agreed on the additive aspects of the mathematical texts. But were all the Egyptian mathematical texts only additive or subtractive in scope, as David Silverman (and others) continue to suggest in 1994?<br /><br />It will be shown that math historians had not completely reported Egyptian mathematical facts. This blog will show that Egyptologists and math historians had created only additive methodologies that had not generally linked the original scribal base 10 data to modern base 10 equivalents. Fair 'decoding' systems are needed.<br /><br />Corrective oversights continue to be difficult. Formal corrective methodologies have only recently been published. One corrective paper was published in India in 2006. The paper decodes hekat units that involved divisors of 64 or smaller.<br /><br />Therefore, these newly published decoding keys confirm that additive scribal arithmetic had not, in fact, dominated Middle Kingdom mathematics as had been assumed for over 100 years.<br /><br />Proof that non-additive arithmetic had dominated scribal arithmetic lies in several examples, beginning with the Rhind Mathematical Papyrus. The RMP lists several beginning type problems that our modern 4-6th graders would recognize, data that appears to have been additive, but actually was not. One problem is #83, where three classes of birds were fed three different amounts, 1/6th of a hekat of grain for three birds, 1/20th for one bird, and 1/40th for each of three birds. Ahmes, the RMP scribe, then asks: how much grain did all seven birds eat in one day?<br /><br />A modern discussion of this trivial problem requires the addition of 1/6, 1/6, 1/6, 1/20, 1/40, 1/40 and 1/40, Note that an easy to reach common multiple (1/120) is required to solve this problem if no modification in procedure is introduced. Today, kids might improve the ease of working the ancient problem by first adding the three 1/6 fractions, obtaining 1/2. Then they could add 1/20 and 3/40. In this manner only (1/40) is required to be used as a common divisor. This is a basic form of modern logic overlay (a bilingual text, to refer to a cryptanalysis standard) can be restated several ways.<br /><br />A closely related arithmetic form states, that by adding: 1/2, 1/20 and 3/40 using the common divisor 1/40, or<br /><br />(20 + 2 + 3)/40 = 5/8th of a hekat of grain, as Ahmes<br /><br />the requested to know, how much did seven birds eat in one day?<br /><br />Ahmes performed this same arithmetical task, but chose another interesting value for his common divisor, 1/320. Ahmes wrote his fractional divisors of a hekat, his feeding rates, in terms of a larger one that he named ro, one that was commonly seen in several additional RMP problems, #35-38, 47, and 81.<br /><br />To understand Ahmes' reason for selected 1/320th for a class of problems, any one of which could have used a smaller common divisor, let us examine Ahmes used of the word ro.<br /><br />Stated in terms of ro (common divisor) units, Ahmes first added up 53 1/3ro, 53 1/3 ro, 53 1/3 ro, reaching 160 ro. Ahmes then added 16 ro, 8 ro, 8 ro and 8 ro, reaching 200 ro, or 200/320 = 5/8th of hekat. Trivial right? I think not.<br /><br />At this point several non-trivial points should be hi-lighted. Why did Ahmes apparently chose only one common divisor ro, 1/320th of a hekat, when divisors of a hekat unity was smaller than 64? One clue is given by the idea of common divisor. A common divisor was needed for to find a remainder related to a divisor n, where n less than or equal to 64. A second clue is given by ro being used in another situation 32o/n ro, a statement that appears in the bird feeding problem.<br /><br />The common divisors used to solve the seven bird eating grain problem makes sense in ancient and modern base 10 fractions. In Ahmes' use, his common divisors were large, all multiples of 1/320. The multiple of 1/320 common divisor was repeated in several other hekat division problems. One other is the Akhmim Wooden Tablet and five division problem. There are an additional 29 cases in the RMP. The 29 RMP data points were also written out in terms of a hin unit, so Ahmes was well versed in using common divisors and other forms of fractions.<br /><br />This blog reports that Ahmes used remainder arithmetic to exactly partition a hekat a volume unit. Vulgar fractions can be seen as converted into quotients and exact Egyptian fraction remainders. The remainders, scaled to ro units, can be seen as representing 'healed' Horus-Eye units.<br /><br />The 'healed' Horus-Eye quotients were given Egyptian fraction remainders that included 1/320 ro common divisors. That is, whenever a hekat unity, 64/64, was divided by any prime number, like 3, 7, 11, and 13, or a none-prime number, a remainder term was created that included the possible use of ro.<br /><br />Ahmes and a mentor writing in the Akhmim Wooden Tablets substituted the fraction 5/320 for the 1/64th fraction that appeared in the remainder arithmetic term, defined by:<br /><br />(64/64)/n = Q/64 + (R/n)*(1/64)<br /><br />such that, Ahmes' remainder (R/n)(1/64) was multiplied by 5/5 such that<br /><br />(64/64)/n = Q/64 + (5R/n)(1/320)<br /><br />and replaced 1/320 with ro<br /><br />a final scaled quotient and scaled Egyptian fraction remainder was written as<br /><br />(64/64)/n = Q/64 + (5R/n)*ro.<br /><br />Further, it will be shown that when n > 64, a second form of remainder arithmetic was used. Ro was not often used in the Papyrus Ebers and other medical texts., most often in the form<br /><br />320/n ro<br /><br />and generally,<br /><br />m/n 'unit' wrote quotient and unscaled remainders, such as<br /><br />m = 10 = hin, was written as 10/n hin<br /><br />m = 64 = dja, was written as 64/n dja<br /><br />and so forth.milo gardnernoreply@blogger.comtag:blogger.com,1999:blog-6969960.post-1089501241351747382004-07-16T16:09:00.000-07:002009-01-05T10:33:50.816-08:00MiloGardner<a href="http://en.wikipedia.org/wiki/user:milogardner">This blog page summarizes my personal information</a>. Born to <a href="http://reagardner.blogspot.com/">Rea and Pansy Gardner</a> on 1370 Richins Avenue, <a href="http://maps.google.com/maps?f=q&hl=en&ie=UTF8&ll=37.09024,-95.712891&spn=47.167389,92.8125&z=4&om=1&layer=c&utm_campaign=en&utm_source=en-ha-na-us-google-svn&utm_medium=ha">Gridley</a>, CA in 1938 (the house, canals, and town are viewable on google street maps). I am 11th child of a 7 boy 4 girl family. One girl, a twin of Colin, Coleen, died at birth. My growing up years were economically easier than depression era siblings. Rea and Pansy were high school grads. Rea, born in 1891, attended some college, but poor financing killed his dream of graduating, the same situation that happened to his father, <a href="http://serenusgardner.blogspot.com/">Sy</a>, 30 years earlier. In his free time Rea was a sports enthusiast, checkers champion, geographical historian, and religious philosopher, considering astronomy and cosmology. Rea followed several of his father's life interests, i.e. county checkers champs, political geography, loving the medieval era, the <a href="http://en.wikipedia.org/wiki/Silk_Road">Silk Road</a>,<a href="http://en.wikipedia.org/wiki/Charlemagne"> Charlemagne</a>, and <a href="http://en.wikipedia.org/wiki/Ancient_economic_thought">economic links</a> that empowered the <a href="http://en.wikipedia.org/wiki/Age_of_Enlightenment">European Enlightenmen</a>t. Pansy was the musician (piano), an interest passed down to one son (Jack, clarinet), and all three daughters through singing and piano. Rea and Pansy's youngest three sons graduated with degrees in Agriculture, Physics and Mathematics, respectively. All seven sons may have graduated from college except for War War II. Three daughters attended two years of college, a high level for the early 1950's.<br /><br />Reading came early for myself, enjoying Prince Valiant and the comics before the first grade. School added westerns, and <a href="http://en.wikipedia.org/wiki/United_States_Exploring_Expedition">local facts</a> (i.e.,<a href="http://books.google.com/books?hl=en&id=THEFAAAAQAAJ&dq=wilkes+expedition,+1838&printsec=frontcover&source=web&ots=NIHmR83JlT&sig=6FPMAhv7LOQtQFxXUpOUbAQZDdA&sa=X&oi=book_result&resnum=2&ct=result"> Wilkes Expedition</a> and <a href="http://books.google.com/books?hl=en&id=fHFHAAAAIAAJ&dq=mining+camps,+shinn&printsec=frontcover&source=web&ots=XjQHCduZJf&sig=huqfBJ5gzR5HLSt4HbZ-o3M-skA&sa=X&oi=book_result&resnum=1&ct=result">Shinn's "Mining Camps"_</a>) of pre-Gold Rush and scholarly documented Gold Rush people and events. Reviews of native Americans and Wild West people and myths also included<a href="http://www.library.ucsf.edu/collres/archives/hist/ishi/"> Ishi</a>. who walked into nearby <a href="http://www.ucpress.edu/books/pages/1064001.php">Oroville in 1911</a>, <a href="http://www.historybuff.com/library/refhickok.html">Wild Bill Hickok</a>, and Kit Carson type books. From our backyard we dug up Indian grinding bowls and baskets, and in the evenings read books of the period.<br /><br />As a farm boy, serious work was not required until age ten. From that point on I dreamed of another life, not enjoying dairy, custom hay cutting, drying, chopping, and blowing into barns, at 50% the cost of baled hay. My father also grew row crops and rice on his 90 acres, and 200 acres of rented land. Revenue and segmented costs per acre dominated discussions. Many hours were spent driving tractors, plowing, and discing, sometimes all night, to avoid the heat of the day.<br /><br />My social conscience developed early in labor and interpersonal areas. Gridley was the home of a migrant labor camp. Weekly new "Okies" came to school, often with chips on their shoulders. To make friends playground/recess fights of some sort would take place. Winning the majority of those fights, I made many friends, almost never fighting with that person again. School administrators allowed the fighting, as well as allowing many of its insensitive teachers to discriminate against the poorly dressed kids in ways that made my heart and conscience cry.<br /><br />Sports were an early passion. Grammar school championships in flag football, basketball and softball, were great fun. I once broke a leg at half-time of a high school football game, playing tackle football in the back of the end zone. My cheerleader sister was surprised to see an ambulance carrying off her 8th grade brother. I missed the championship flag football game, which we won anyway. High school was sports filled. I improved on my 880 and 1320 times (though no where near one of my brother (Vere)'s times, who once held a national class B record in the 1320). High school basketball was played on D, C, B, and A teams, as was the case in the weight/height/age based teams of that era, winning D and C championships. Baseball, and track were played two years, and football one year, winning a couple of southern league 880 championships. During all times of the year pick up games were played on local sports fields, especially basketball and baseball at the outdoor facilities.<br /><br />Movie going was enjoyed frequently, often twice a week. Friday nights and Saturday matinees opened my eyes to fantasy, as well as the wider news of the world, and US sports. Given that my hometown had one theater, it was the main source of entertainment. I was often first in line. Once an older brother shook me hard when I emulated Gentleman Jim Corbett's boxing skills (at age 5 or 6). This brotherly act showed his lack of appreciation of fantasy as a motivating aspect of life that many, like myself, consider when setting high standards and goals in life.<br /><br />During the summer swimming was big. Behind my home was an <a href="http://maps.google.com/maps?f=q&hl=en&ie=UTF8&ll=37.09024,-95.712891&spn=47.167389,92.8125&z=4&om=1&layer=c&utm_campaign=en&utm_source=en-ha-na-us-google-svn&utm_medium=ha">irrigation canal</a>. The kids from south end of Gridley swam there seven days a week. There were two other canal swimming holes in our area, places that the kids from all over town reached on bicycles. The town also had a swimming pool, and tennis courts.<br /><br />Math became a special high school interest, as well as Spanish, chess and the science side of life. History was fun, but my school offered only US history. Taking college entrance exams, achieving only a 50 percentile score, and being under financed, I decided to work a couple of years before going to college.<br /><br />A summer AT&SF railroad fireman job in hot Needles, Barstow CA, and Seligman, AZ, following my mother's family trade, further focused my attention to better working conditions. Work for work's sake was a bore, so I dreamed of going to Europe via the US Army.<br /><br />Three years were spent in the US Army. Processed in at the Oakland Army Terminal, a meningitis outbreak at Fort Ord, ended a two week stay, riding the train to rainy Ft. Lewis, WA for basic training. We all worried about Quemoy, Matsu, and the revolt in Hungary, spots that we thought that a few of us would end up. Advanced training was spent in the Boston area, Ft. Devens, seeing professional sports for the first time, at Fenway (Ted Williams ) and the Boston Garden (Bill Russell, Bob Cousy). Finishing 3rd in my class, beaten by two college grads, I became a cryptanalyst. I had qualified for the Army Language School, in Monterey, an assignment that would have brought me home for Christmas. But code breaking was for me. It applied my chess, math, and language interests. Code breaking continues as a retirement hobby. Code breaking allows an outside-in view of life, providing enjoyable and productive perspectives to the common inside-out view of life. Several classes of unsolved problems have been 'broken', or nearly broken, by working in ad hoc teams put together on the internet.<br /><br />Earning a desired European assignment two years were spent in Bavaria, between Munich and Salzburg, Austria. Three of us GI's owned a 1939 Opel. It was used to travel as far away as the French Riviera, Switzerland and Vienna. Our favorite spot was Salzburg, located 60 miles away in a neutral country where military uniforms were not allowed. We felt like civilians almost every weekend. Train trips were taken to Italy (Venice to Rome), France (its Riviera, and Paris), and Holland, during tulip time. A two month side trip to Lebanon, thanks to Pres. Eisenhower widened my interests to include Arabic, and several the cultures of the Middle East. Tent cities were set up in biblical olive orchards. Housing soon moved to the beach outside of Beirut. Daily swims were relaxing. But my time away from California soon ended.<br /><br />Before going home, a couple words on my European sports activities might be of interest. Company level basketball and fast pitch softball were played. The softball brought two championships, beating nearby Special Forces and other nearby military units. We had a guy that threw in the 90 mph range, I was often his catcher. Three of us softballers tried out for an all-ASA (my Army unit) baseball team to play in the larger European military league. One made it. I soon choose to end my enlistment. After an Atlantic boat ride, seeing the Statue of Liberty, Ft. Dix, and throwing away my last 10 cent pack of cigarettes overboard, I never smoked again. A discharge at the Oakland Army Terminal was obtained in 1959, in time to start the fall semester of college.<br /><br />College entrance tests obtained 90 percentile results, a level that motivated my college career. An undergrad math degree was earned. Ample electives allowed the history of economic thought, from the Ancient Near East uses of money, to European advances in trade, to the economic systems that preceded Capitalism, and final our modern economic systems were studied.<br /><br />Post-college work began in aerospace, Vandenberg AFB and missile range issues, learning NASA definitions of mathematical astronomy. Bertha and I dated for four years. We were married in 1965. She is the <a href="http://share.shutterfly.com/action/welcome?sid=8AYsmzhi0ZNGOz&emid=sharview&linkid=link2">second oldest</a> of a 9 boy, 5 girl family. She was born in <a href="http://share.shutterfly.com/action/welcome?sid=8AYsmzhi0ZNGN_&emid=sharview&linkid=link2">Durango, Mexico</a>, moving to Gridley in 1955, the year of the <a href="http://americahurrah.com/Flood55/YubaCity.htm">big flood</a>. <a href="http://share.shutterfly.com/action/welcome?sid=8AYsmzhi0ZNGNu&emid=sharview&linkid=link2%20"> Bertha</a> and I enjoyed southern Cal for five years, especially beaches, Disneyland, Dodger and Angels baseball games. Three years were spent in LA, working for Rockwell in Anaheim, Downey and Fullerton. Three evenings/week were spent at Cal State Fullerton grad school.<br /><br />Completing an MBA degree allowed a career change to take place. Bertha I and two children moved to northern California to be near our families. In northern California public health, the medical field and public service were enjoyed. After 10 years my wife's family restaurant business called. As the corporate CFO, a splitting of a seven brother chain of stores to a three brother operation was enjoyed.<br /><br />Bertha and I are the parents of three children. Tommy and Michelle arrived in Downey, and Anaheim, respectively. Missy arrived in Yuba City. Raising the kids was great fun. Tommy loved baseball. Michelle loved softball and basketball. Missy loved softball, basketball and volleyball . Completive softball was enjoyed by attending three national tournaments finishing 2nd, 5th and 9th. Missy played center field at Cal Poly (SLO), making all tournament in the NCAA Western Regionals, her last completive softball day. Today, Missy is a registered civil engineer. Michelle is a corp. tax collector for the State of California.<br /><br />In 1987 Tommy passed on as a passenger in a tragic accident. Since the accident, half-time work has been spent in local libraries studying a range of under reported, and unsolved math history issues. The first topic, the history of zero, mostly a medieval topic reported in my college days. The actual history of zero reports its use in 2,000 BCE Egypt, 2,000BCE Babylon and 600 BCE Mesoamerica. Several projects have centered on analyzing, and decoding under reported texts, mostly from the ancient Near East. Military code breaking skills were brought in to analyze several fragmented Egyptian and Mesoamerican data bases. At times, projects have suggested upgrades to a data base by freshly linking to a closely related data base.<br /><br />Three professional papers have been published. Two papers describe scribal methods used in the Egyptian Mathematical Leather Roll (EMLR), and the Akhmim Wooden Tablet, each written around 2,000 BCE. The papers were published in India in 2002, and 2006. A condensed version of the EMLR paper was published in the Germany by Springer: Encyclopedia of the Non Western History of Science, and Medicine, 2005.<br /><br />The Akhmim Wooden Tablet was published by Ganita Bharati, Bulletin of the Indian Society for the History of Mathematics, V ol 28 (2006), Numbers 1-2. The topic, the two-part numbers used in the<a href="http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet"> Akhmim Wooden Tablet </a>and the<a href="http://en.wikipedia.org/wiki/RMP_2/n_table"> RMP </a>. The exact quotient and scaled Egyptian fraction remainder method solved an Old Kingdom binary round-off problem. MK scribes partitioned a volume hekat unity (64/64) by divisors 3, 7, 10, 11, and 13. For example, divisor 13 set Q = 4 and R = 12; 4/64 + (60/13)*(1/320); and 1/4 + (4 + 1/2 + 1/13 + 1/26)*1/320.<br /><br />The AWT is the first text that reported the exact partitioning method, a method often used in the RMP and the medical texts.<br /><br />Online publications consist of:<br /><br />A. Egypt<br /><br />1.<a href="http://planetmath.org/encyclopedia/WhyStudyEgyptianFractionMathematics.html"> Why Study Egyptian Mathematics</a>?<br /><br />2. <a href="http://akhmimwoodentablet.blogspot.com/"><span style="text-decoration: underline;">Akhmim Woodent Tablet</span></a> (blog) ; <a href="http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet">(Wikipedia</a>); <a href="http://mathworld.wolfram.com/AkhmimWoodenTablet.html">(Mathworld</a>)<br /><br />3. <a href="http://planetmath.org/encyclopedia/AhmesBirdFeedingRateMethod.html">Ahmes 'bird-feeding rate problem</a> (Planetmath)<br /><br />4. <a href="http://planetmath.org/encyclopedia/EconomicContextOfEgyptianFractions.html">Economic Context of Egyptian Fractions</a> (Planetmath)<br /><br />5. <a href="http://emlr.blogspot.com/">Egyptian Mathematical Leather Roll</a> (blog); (<a href="http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll">Wikipedia</a>); (<a href="http://mathworld.wolfram.com/EgyptianMathematicalLeatherRoll.html">Mathworld</a>); (<a href="http://planetmath.org/encyclopedia/EgyptianMathematicalLeatherRoll2.html">PlanetMath</a>)<br /><br />6. <a href="http://planetmath.org/encyclopedia/EgyptianFraction2.html">Egyptian fractions</a> (Planetmath),<a href="http://historyofegyptianfractions.blogspot.com/"> History of Egyptian fractions</a> (blog)<br /><br />7. <a href="http://planetmath.org/?op=getobj&from=objects&id=7824">Egyptian Fractions, Hultsch-Bruins Method</a> (Planetmath)<br /><br />8. <a href="http://en.wikipedia.org/wiki/Heqanakht_papyri">Heqanakh Papyri</a> (Wikipedia)<br /><br />9. <a href="http://en.wikipedia.org/wiki/Kahun_Papyrus">Kahun</a> <a href="http://en.wikipedia.org/wiki/Kahun_Papyrus">Papyrus </a>(Wikipedia)<br /><br />10. <a href="http://reisnerpapyri.blogspot.com/"> <span style="text-decoration: underline;">Reisner Papyri</span></a> (blog); (<a href="http://en.wikipedia.org/wiki/Reisner_Papyrus">Wikipedia</a>)<br /><br />11. <a href="http://en.wikipedia.org/wiki/Red_auxiliary_numbers">Red Auxiliary Numbers</a> (Wikipedia), (<a href="http://planetmath.org/encyclopedia/FirstLCMMethodRedAuxiliaryNumbers.html">Planetmath</a>)<br /><br />12. <a href="http://rmprectotable.blogspot.com/">RMP 2/n table</a>, (blog) <a href="http://en.wikipedia.org/wiki/RMP_2/n_table">Wikipedia</a><br /><br />13. <a href="http://planetmath.org/?op=getobj&from=objects&id=7783"> Remainder Arithmetic</a> (Planetmath)<br /><br />14. <a href="http://planetmath.org/?op=getobj&from=objects&id=7819">Remainder Arithmetic vs Egyptian Fractions</a> (Planetmath)<br /><br /><br />B. Medieval<br /><br />15. <a href="http://planetmath.org/encyclopedia/ArabicNumerals.html">Arabic Numerals</a> (Planetmath)<br /><br />16. <a href="http://liberabaci.blogspot.com/">Liber Abaci</a> (blog); <a href="http://planetmath.org/encyclopedia/LiberAbaci.html">(Planetmath)</a><br /><br />C. Mesoamerica<br /><br />17. <a href="http://aztecsurveying.blogspot.com/">Aztec Surveying</a> (blog)<br /><br />18. <a href="http://planetmath.org/encyclopedia/MayanMath2.html">Mayan Math</a> (Planetmath)<br /><br />D. Other (Astronomy)<br /><br />19. <a href="http://planetmath.org/encyclopedia/AcanoALunarCalendarMethod.html">Acano Lunar Calendar Method</a> (Planetmath)<br /><br />Today, our family has grown to five grand children. Michelle has three boys, and Missy two girls. The oldest boy, <a href="http://share.shutterfly.com/action/welcome?sid=8AYsmzhi0ZNGNd&emid=sharview&linkid=link2">Chris</a> has earned a 2nd degree black belt in karate, and is a college student. Erik loves football, (middle linebacker, and fullback: gaining 191 yards in one game). Two years ago Erik's team won a section title in DIII football, nearly attaining a second section title last year. Rugby is Erik's first sport. Last year's team won 11 games in a row, winning a West Coast B tournament on Stanford U.'s field. This year Erik was a rugby all star, finishing 4th in a national 20 and under tournament in Colorado. He'll play football his senior year with rugby team mates.<br /><br />My other three grandchildren are small, so their sports and other interests will be made known years from now. Adam (age 7) plays baseball, and is in a public Montessori school. Avery (age 6) plays soccer, improving greatly this year, scoring five goals in one-half of one game, while learning to read and write in kindergarten. Hannah (4) begins soccer (with Dad as coach). <a href="http://photos.walmart.com/shareethumbnailshare/AlbumID=25976529/albumcount=1/p=24311207086765162/l=10233701/g=15031865/cobrandOid=1011/otsc=SYE/otsi=SALB/pns/share/p=24311207086765162/l=10233701/g=15031865/cobrandOid=1011/otsc=SYE/otsi=SALB">Family</a> and sports fill my time, leaving plenty of time for hobbies, and<a href="http://photos.walmart.com/shareethumbnailshare/AlbumID=29938015/albumcount=1/p=329261214512350076/l=10535519/g=15031865/cobrandOid=1011/otsc=SYE/otsi=SALB/pns/share/p=329261214512350076/l=10535519/g=15031865/cobrandOid=1011/otsc=SYE/otsi=SALB"> family vacations</a>.milo gardnernoreply@blogger.com