Remainder arithmetic formally appeared 4,000 years ago. The Akhmim Wooden Tablet (AWT), circa 1950 BCE, theoretically defined a remainder arithmetic application within a weights and measures context. Prior to 2,000 BCE, remainder arithmetic had only been intuitively reported in Egyptian calendars. The AWT scribe partitioned a volume unit, named hekat, into cubit-cubit units. Hekat sub-units were created by dividing (64/64) by rational numbers n. The hekat definition, at first, limited divisors n to 1/64 < n < 64. Answers were written in two-part binary quotients, and scaled Egyptian fraction remainders.

Later, a simplified second remainder arithmetic method appeared. The 1650 BCE Rhind Mathematical Papyrus (RMP) allowed divisors n to be of any size within m/n units. The second RMP method reports one-part integer quotients, and non-scaled Egyptian fraction remainders. For example, to write 1/10 units of a hekat numerator m was set to 10, or 10/n hin. RMP 81 lists a table of 29 divisors between 1/64 < n < 64. The table lists scaled (mod 5) 1/320 (ro) units created from (64/64)/n, and equivalent unscaled 10/n hin, and 320/n ro, units created from 10/n hin, and 320/n ro.

Scholars began to report the scaled (64/64)/n divisions in 2002, and its connections to the unscaled m/n remainder arithmetic method in 2005.

The AWT's five scaled examples are sufficient to define the oldest known remainder arithmetic method. The AWT scribe divided a hekat unity, written as: divided by integers 3, 7, 10, 11, and 13. Answers were converted to two-part binary quotients, and Egyptian fraction remainders. The two-part answers were proven. Proof was achieved by multiplying the quotient and remainder answer by the initial divisor (3, 7, 10, 11 or 13). The AWT scribe returned the answers to the initial hekat unity. Hana Vymazalova published the proof side of this set of facts in 2002, with others reporting the AWT's remainder arithmetic properties in 2005.

The first scholar to unscramble the AWT's arithmetic was G. Daressy in 1901. In 1906, Daressy published the AWT text line by line, garbling two division problems. Daressy correctly reported three hekat divisions: divisors 3, 7, and 10. Daressy improperly reported scribal divisors 11, and 13 facts. In retrospect, typographical errors had been introduced. All five hekat unity divisions remained an unresolved issue until corrected by Hana Vymazalova in 2002.

Hana Vymazalova, a Charles U., Prague grad student published the five AWT hekat division problems by correcting Daressy's two 1906 errors. Vymazalova's paper proved that all five two-part answers were returned to a hekat unity . Vymazalova's paper did not suggest an abstract context. Yet, the she shows that the AWT scribe, and other MK scribes, recorded binary quotients, and scaled Egyptian fraction remainders, in an innovative manner.

Vymazalova's 2002 paper may be found on the web by using Google Scholar, as found in 2004. For those that are interested, the AWT scribe's original hieratic information is included as an appendix to Vymazalova's paper.

Concerning the second remainder arithmetic method, over 2,000 Middle and New Kingdom (NK) medical prescriptions were written in a method. Several NK units had been garbled by translators, much as Daressy had garbled two AWT answers. In 2002, a German graduate student, Tanja Pommerening, published a dja hekat unit properly scaled to 1/64 of a hekat, correcting a unit that had confused modern physicians trying to reconstitute ancient Egyptian medicines.

Dr. Tanja Pommerening's 2005 PhD correctly discussed the dja as a healing unit of a broken Horus-Eye series. The healing definition was implied by hieratic symbols. Pommerening's definition depicted the dja unit as a hekat sub-unit equal to 1/64 of a hekat. The dja was only written in a one-part remainder arithmetic system. A second definition (not reported by Pommerening) depicted the two-part remainder arithmetic units was implicit in Vymazalova's paper. A third definition (not reported by Pommerening, or Vymazalova) corrects Old Kingdom Horus-Eye series errors, that had generally thrown away 1/64 binary units. Corrects 'healed' both remainder arithmetic systems by introducing exact Egyptian fraction unit measurements.

To review the one-part method, its practical hinu unit was seen very early by scholars. Hinu units had been properly understood as 1/10th of a hekat for over 100 years. However, theoretical two-part ro units, contrasted to one-part hinu units inRMP 81 had not been accurately reported by scholars. Scholars had not scaled the abstract aspects of the hinu until 2005 per: http://www.mathorigins.com/image In easy to read terms:

1. 1/16 + 1/32 + 2/320 = 32/320 = 1/10

with ro = 1/320, and hin = 1/10

allowed ancient scribes to write

2. (1/16 + 1/32)hekat + 2ro = 32 ro = hin

as cited 29 times in RMP 81, scaled to various n values. Gillings' view of one-part RMP hin unit was listed in a table as hin units equivalent to ro sub-units. Gillings had stressed the practical aspects of the one-part hinu and the two-part ro units. Vymazalova, and Pommerening's were first to begin to correct abstract arithmetic oversights related to hin, and ro units. Prior to 2002 math historians and Egyptolists had not proven a theoretical basis of two-part or one-part units. Cudos to Vymazalova and Pommerening for opening an abstract arithmetic door to these important remainder arithmetic units of measures.

Pommerening's paper stressed one-part data. Despite not reviewing RMP two-part data as theoretical in scope, Pommerening informally agrees that RMP two-part hinu and ro data are consistently connected to her views of one-part dja, and other hekat sub-units.

It is expected that rigorous studies will become common place resolving subtle issues that had confused scholars for 100 years. That is, scholars will be discussing two-part, and one-part remainder arithmetic, written in practical and abstract ways, as needed Future discussions are expected to point out the earliest practical and theoretical threads that hekat measurement units utilized.

For the present, ANE scholars have chosen to stress one-part practical statements, such as:

2. 10/n hin

example:

and, skip over equivalent abstract two-part form written as

3. 320/n ro

has been written in an several modern of forms by scholars. For example, let n = 10 used a one-part form, written as 320/10 = 32 ro is a valid form.

Increasingly, the abstract form of ancient one-part statements are being properly linked to abstract two-part statements, as noted by 1/64 < n < 64.

That is, (64/64)/n, defined the beginning point of meta two-part statements. When n was set to 10, one-part hin data appeared, a fact that is notg well known. Today only the one-part 10/n hin statements are well understood.

Pommerening and Vymazalova opened two ancient Egyptian abstract arithmetic doors. The scholarly community is increasingly finding ways to report Middle Kingdom scribal 2,000 BCE hekat partitioning statements to Ahmes' one-part and two-part 1650 BCE data bases. Meta views of these hekat measurement units allowed n to partition hekat unities into abstract quotients and remainders.

To review, the older two-part statements allowed scribes to generally partition a hekat unity by any rational number divisors n, such that:

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

where Q was a quotient, and (5R/n)*ro was a Egyptian fraction remainder, scaled to ro, a 1/320th common divisor.

To the Egyptian scribe, practical bowl sized measuring devises assisted in the every day use of this volume measurement system (much as teaspoons, table spoons, ..., pints, quarts and gallons are used today). Pommerening's paper published practical aspects of MK, NK and later measurement artifacts. The practical bowl system allowed various hekat units, like the hin, dja and ro units to be measured per the one-part form, such as:

3. 320/n ro

example: for a hin, 320/10 = 32 ro

proven by: 1 hin = 1/16 + 1/32 + 2ro,

1/16 = 20 ro, 1/32 = 10, ro 2ro = 2 ro totals 32 ro.

Pommerening evaluated the dja as 1/64 a hekat or

64/n dja in the MK (75 cc),

and,

64/n dja (oipe) (300 cc) in the NK.

Clearly, the hin and ro units were also used as independent medical prescription volume units (Eber Paprus et al), as decoded by many scholars.

At present we may be left with reviewing Gillings' data listed in "Mathematics in the Time of the Pharaohs" a hin was scaled to 1/10th of hekat using the following one-part rule (by not mentioning the use of ro).

This was done by Ahmes with 29 hekat to hin examples, using the rule:

10/n hin.

Example: n = 3

10/3 = 3 + 1/3

with only 3 + 1/3 being listed by Ahmes (per Gillings' page 250 translation).

There is more to this story. So let's jump to the most difficult of the five two-part problems, the divisor

and proceed step by step through the (64/64) hekat system. But before doing that, it should also be pointed out that the scribe passed down to Ahmes and other Middle Kingdom scribes a generalized method that exactly divided a hekat as follows:

with the scribe replacing the remainder 1/64 term with 5/320, its equal such that,

with

being replaced by the word ro, and Q = quotient and R = remainder.

Now to the

and,

with the vulgar fraction converted to an Egyptian

fraction by first writing

by the following steps:

The above data allowed the final partition by 13 statement to equal

This two-part number partition is proven to equal . The scribe multiplied his answer by 13, as noted by,

as all five AWT answers were proven, as reported by Vymazalova in 2002.

Ahmes used the (64/64) hekat two-part partitioning system, as did all scribes after 2000 BCE. The later medical scribes also used a simplified one-part variation of the hekat partitioning system. The one-part system was structured by m/n hin, dja and ro sub-units.

The numerators of each m/n 'unit' system have been difficult to confirm. The main problem has been the absence of equivalent raw data. That is, unless raw data links are confirmed to ancient texts, linking unknown m/n units to hekat subunits, i.e, hin, and ro, problems of correctly reading hekat m/n sub-units will continue.

Yet, beyond a small number of 'undecoded' hekat m/n sub-units, clarity of the oldest two-part weights and measures system is well established. To 2,000 BCE and later scribes binary or integer quotients were always followed by scaled, or unscaled Egyptian fraction remainders.

The quotient and remainder based methods partitioned a hekat unity by n, as needed. The AWT and RMP 81 partitioned a hekat with divisors between 1/64 and 64, reaching a limit. Two-part quotients and remainders wrote binary fractions quotients Egyptian fraction remainders scaled to a 1/320 hekat unit. Beyond the divisor n = 64 limit, one-part statements were preferred in scribal 'weights and measures', writing only integer quotients and Egyptian fraction remainders.

REFERENCES

Daressy, G. "Cairo Museum des Antiquities Egyptiennes." Catalogue General Ostraca, Volume No. 25001-25385, 1901.

Daressy, Georges. Calculs Egyptiens du Moyan Empire, Recueil de Travaux Relatifs De La Phioogie et al Archaelogie Egyptiennes Et Assyriennes XXVIII, 1906, pages 62 to 72.

Gardner, Milo. "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, MD Publications, New Delhi, pp 157-173.

Gillings, Richard. "Mathematics in the Time of the Pharaohs" Dover, reprint from, Cambridge, Mass, MIT Press 1972, ISBN 0-486-24315-X, 1972.

Pommerening, Tanja, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass" in studien zur Altagyptischen Kulture, Beiheft, 10, Hamburg, Buske-Verlag, 2005.

Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archiv Orientalai, Charles U., Prague, pp. 27-42, 2002.