Monday, December 19, 2005

Vulgar fractions and 2/nth tables

by: Milo Gardner Bio

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 Horus-Eye arithmetic. After 2,050 BCE edicts were primarily written and controlled in Egyptian fraction arithmetic. 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.

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.

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.

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.

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 two large farms, documented in 2002 and 2004.

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 ‘monies of account’ 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 economy .

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.

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 2/n table 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 Egyptian Mathematical Leather Roll (EMLR), and the Akhmim Wooden Tablet (AWT) were integrated into scholar discussions after 2002.

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 RMP 2/n table 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).

The RMP, along with the Egyptian Mathematical Leather Roll, (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.

To begin as historians had done in 1879, the Rhind Mathematical Papyrus (RMP) was first read. It began with a 2/n table 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.

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.

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.

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.

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.

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.

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.

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 Akhmim Wooden Tablet 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.

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.

Additive scholars discuss the Reisner Papyrus, 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.

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.

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 2/n table. 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.

The algorithmic group includes Annette Imhausen, 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.

Theoretical number theory scholars began to publish after 2002. This group reports that the RMP 2/n table and its 84 problems use remainder arithmetic 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 Akhmim Wooden Tablet, Reisner Papyrus, and the EMLR. 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.

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.

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:

(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

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.

The exact scribal remainder arithmetic method allowed MK state-run inventories to be measured and controlled by the hekat volume unit, and its sub-units. The hekat 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.

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.

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.

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.

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.

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. Pope Sylvester II assisted in spreading the numerals and continuing the Egyptian fraction system to Europe in 999. The 1202 Liber Abaci 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.

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.

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 base 10 decimals, 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Gardner's paper, (updated in 2007) 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.

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.

Silger reports seven methods in the 1202 AD Liber Abaci 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.

Vymazalova's paper may be the most important. She indirectly reports an abstract basis of AWT 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.

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 Eye of Horus 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 fixed-point and floating-point arithmetic, were not as accurate as MK Egyptian arithmetic (that exactly wrote rational numbers into Egyptian fraction series in several situations).

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.

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:

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64.

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.

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.

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.

Revised, or updated, one-part numbers were written into vulgar fractions and remainder arithmetic, as recorded in the 1800 BCE Reisner Papyrus, 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 RMP 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.

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 Akhmim Wooden Tablet (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.

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".

1. Why Study Egyptian Mathematics?

a. Ahmes Papyrus, an update

b. Math Forum discussions

2. RMP 2/n Table (Wikipedia)

3. EMLR (Wikipedia)

4. EMLR (Planetmath)

5. Hultsch-Bruins Method (Planetmath)

6. Egyptian fractions (Planetmath)

7. Kahun Papyrus

8. Liber Abaci (Planetmath)

9. Liber Abaci (Blog)

10.Red Auxiliary Numbers (Wikipedia), (Planetmath)

Monday, February 07, 2005

Egyptian Math History

Ciphered Egyptian numerals 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 Akhmim Wooden Tablet (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.

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 transliterations correctly reporting ancient arithmetic symbols without reporting subtle mathematical subjects beginning with hard-to-read Reisner Papyrus , EMLR and Ahmes 2/n table construction methods.

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 Ahmes' bird (hekat) feeding problem (RMP 87). Ahmes' data was garbled so that no modern scholar can decipher the intended facts, per Chace's view.

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.

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.

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.

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.

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.

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.

For example, RMP # 32 states in modern terms

x + (1/2 + 1/4)x = 2 or

x + 7/12x = 2, or

19/12 x = 2, or

x = 24/19 = 1 + 5/19

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:

Find 5/19 by first subtracting 1/12 or

5/19 - 1/12 = (41)/(12*19)

= (38 + 2 + 1)/(12*19)

= 1/6 + 1/114 + 1/224, or

the final answer

1 + 1/6 + 1/12 + 1/114 + 1/224

Ahmes' choice of 1/12 followed the 2/19 2/nth table suggestion, where:

2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114

as also used in 2/95, where 2/19 x 1/5 = 1/5 (1/12 + 1/76 + 1/114)

as Wilbur Knorr, in 1982, as published in the journal Historia Mathematica, and others at other times.

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).

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.

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.

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.

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.

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.

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.

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 .

This blog focuses on clearing up a few of the fragmented facts, one being the remainder arithmetic cited in the RMP and other texts.

For example, RMP 62 states that

100/13 = 7 + 2/3 + 1/39

But what does that mean? The Reisner Papyrus will be shown to have written:

n/10 = Q + R

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:

8/n1 = 1/2 + 1/4 + 1/20 = 8/10

48/n2 = 4 + 1/2 + 1/4 + 1/20 = 4 + 8/10

18/x3 = 1 + 1/2 + 1/10 = 1 + 6/10

64/x4 = 6 + 1/4 + 1/10 + 1/20 = 6 + 4/10

36/x5 = 3 + 1/2 + 1/10 = 3 + 6/10

such that x1 = x2 = x3 = x4 = x5 = 10

That is,

100/13 = 7 + 9/13 or 7 + 2/3 + 1/39

with 7 being a quotient and 9/13 being a remainder that was converted to 2/3 + 1/39.

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.

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.

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.

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,
or other hekat situations.

Future study is required to detail the scribal division methods that were comparable, if any, to two forms of hekat remainder arithmetic (scribal division).

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.

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.

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.

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:

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.

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.

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.

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,

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.

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'.

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?

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.

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.

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.

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?

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.

A closely related arithmetic form states, that by adding: 1/2, 1/20 and 3/40 using the common divisor 1/40, or

(20 + 2 + 3)/40 = 5/8th of a hekat of grain, as Ahmes

the requested to know, how much did seven birds eat in one day?

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.

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.

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.

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.

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.

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.

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.

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:

(64/64)/n = Q/64 + (R/n)*(1/64)

such that, Ahmes' remainder (R/n)(1/64) was multiplied by 5/5 such that

(64/64)/n = Q/64 + (5R/n)(1/320)

and replaced 1/320 with ro

a final scaled quotient and scaled Egyptian fraction remainder was written as

(64/64)/n = Q/64 + (5R/n)*ro.

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

320/n ro

and generally,

m/n 'unit' wrote quotient and unscaled remainders, such as

m = 10 = hin, was written as 10/n hin

m = 64 = dja, was written as 64/n dja

and so forth.