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Mathematics in Ancient Egypt

A Contextual History

PRINCETON UNIVERSITY PRESS

The Invention of Writing and Number Notation

At the beginning of any development of more complicated or advanced mathematical techniques stands the invention of scripts and number systems — at least in those instances where (written) evidence of mathematics is extant. In Mesopotamia, the origins of literacy and numeracy have been shown to be closely linked and to result from the needs of accounting. Although some of the evidence from Egypt supports a similar claim, the situation seems to be more complex as far as the uses of writing and numeracy are concerned. The source material available for tracing the invention and early development of writing comes, as is often the case with ancient Egyptian artifacts, on the one hand from funerary contexts (tombs) and on the other hand from temples; evidence from early settlements, where these developments presumably at least partly took place, is lacking. The funerary context of the evidence is presumably due to the body of source material in ancient Egypt. The deserts of Egypt, where tombs and temples were located, provided excellent conditions for the preservation of artifacts; hence, practically all the ancient papyri (not only Egyptian but also Greek) originate from Egypt. However, settlements and towns, the places where life and — presumably — the invention and further development of writing happened, were located then as now in the proximity of the Nile, which provided the necessary water. Therefore, ancient Egyptian settlements are often buried under their modern successors and cannot be excavated; in addition, their proximity to water renders unlikely the survival of any organic artifacts from periods several millennia before our present time.

It is estimated that writing was invented in Egypt at the end of the fourth millennium BCE. Among the predynastic elite tombs, tomb U-j (assigned to King Scorpion around 3200 BCE) holds a significant place. Within the twelve rooms of the structure, the earliest evidence of hieroglyphic writing from ancient Egypt was discovered. Two types of objects with inscriptions were found in tomb U-j. On the one hand, almost 200 labels made from bone, ivory, or — exceptionally — stone and on the other hand, ceramic vessels. The labels are mostly approximately 1.5 by 2 cm and originate from the tomb U-j and its proximity. They all include a hole, which presumably served to attach them to objects that have since perished. This type of object is also known from later times (since King Narmer), where it is often used to record deliveries of oil. Similar labels have also been found at Naqada. The ceramic vessels contribute approximately 125 inscriptions, which can be assigned to several groups, among which the group with scorpions (60 attestations) is the largest. While the inscriptions on the ceramic vessels consist of one or two large pictorial signs (e.g., a scorpion), inscriptions on the labels can be divided into two groups, either pictorial signs or (combinations of) abstract signs, which are interpreted as representations of numbers. Based on the regularity of the signs, it is assumed that the evidence found in tomb U-j is not the first writing ever achieved in Egypt but rather the product of a previous evolution.

If we presume that some basic assumptions about the earliest written objects from tomb U-j are correct, that is, if the abstract signs were indeed the representation of numbers or quantities and the labels were attached to some goods about which they held information (e.g., the indication of their quantities or their origin or owner) then, like in Mesopotamia, the earliest writing is linked to administrative needs, and the invention of a quantity/numerical notation along with the invention of script is almost a necessary consequence.

However, the situation in Egypt may be more complex. Along with these administrative records, there are also early attestations of Egyptian writing of a representative character, such as, for example, the notation of the name of the king written in a rectangular enclosure called serekh. Writing in Egypt, therefore, was not only used by the elite for their administrative needs, but it was also recognized as a tool to represent and display power. For each of these functions, a preferred writing surface and, along with it, a preferred way of writing developed. While the representation and display of power were well served by hard stone surfaces, into which an immensely durable inscription was chiseled that proved to last extremely well over the millennia, the daily necessities of administration required a writing that could be executed fast. Along with the hieroglyphic writing system, therefore, a second style of writing was developed, which used abbreviated signs that could also be ligatured to make them even more efficient and which was written with a brush and ink onto papyrus or ostraca (pottery or stone shards). This second type of writing is called hieratic and shows a much greater variation in its form over time as well as in the comparison of individual scribal hands. On a rather rudimentary level, one can perhaps compare the hieroglyphic writing to our printed script and the hieratic writing to handwriting.

The Egyptian Number System

The number system used in ancient Egypt can be described in modern terminology as a decimal system without positional (place-value) notation. The basis of the Egyptian number system was 10 (hence decimal system), but unlike our decimal place-value notation using the ten numerics 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, in which the absolute value is determined by its position within the number (e.g., in the number 125, the absolute value of 1 is 1 × 10, the absolute value of 2 is 2 × 102, and the absolute value of 5 is 5 × 101), the Egyptian system used individual symbols for each power of 100, as shown in table 1.

Although there is no information about the choice of the individual signs for the respective values, some of them seem plausible choices. The most basic, the simple stroke to represent a unit, is used not only in Egypt but also in a variety of other cultures, possibly originating from marks on a tally stick. The sign for 10 was interpreted by Gardiner as a "hobble for cattle, without the cross bar," presumably because its variant with the cross bar (Gardiner V19) occurs as ideogram or determinative in the Egyptian word mdt ("stable"), the sign without the cross bar is (according to Gardiner) attested as a phonogram in the plural mdwt ("stables"). Why this sign was chosen to represent the number 10 can only be speculated — maybe 10 animals tied to this hobble? The sign to represent 100, a measuring rope, possibly points to a measuring rope that was used to measure the length of a field, which may have had a standard length of 100 cubits. The white lotus plant, the phonogram for [??], is often depicted in Egyptian representations as a plant in a landscape, as an ornament, and often with symbolic meaning in ritual or religious actions. Possibly its frequency (apparently it can still be found in the Egyptian canals today) within the landscape may have resulted in its use as a symbol for the number 1000. The hieroglyph of a finger as 10,000 is plausible only if the Egyptian construction, similar to the modern English construction, used a concept of a "bundle" of thousands, that is, "ten thousand," and hence the idea of 10 fingers, each of value 1000, makes the use of the finger somewhat plausible. The sign for 100,000, a tadpole, probably does not need much explanation for anyone who has ever seen a lake with frog spawn and the multitude of tadpoles developing from it. Finally, the largest of the Egyptian number symbols, representing 1,000,000, is the seated god Heh. Heh was one of the eight primeval gods, supposed to carry the heaven under the earth. His depiction is often found on temple walls, vases, and jewelry, where he was supposed to grant millions of years of life.

Each of these signs was written as often as required to represent the number, with groupings of individual signs arranged symmetrically. For example, to write the number 845, the Egyptian scribe would have placed eight times the sign V1 (arranged as a group of four times the sign V1 over another group of four times the sign V1), followed by four times the sign V20 (arranged in pairs of two), followed by five times the sign Z1 (arranged in a group of three placed over a group of two): [??]

A special writing was used up to the Sixth Dynasty in hieroglyphic representations of numbers for indicating multiple thousands. Several lotus flowers were depicted not as individual groups of signs but rather as a bush of flowers.

This style of number writing did not require a symbol for 0 — where our number system employs the 0 to indicate an empty place, the Egyptian notation would simply lack the respective sign altogether. Thus, in the writing of the number 100, our 0 indicating lack of tens and units is indicated in the Egyptian system by the absence of these signs. Therefore, the Egyptian number system does not require a symbol for 0. The modern 0 also fulfills some other functions; for example, it also indicates if a balance has come to 0 or the result if, for instance, 4 is subtracted from 4. In these cases, Egyptian writing used the hieroglyph [??] (F35), the phonogram nfr. Nfr has the basic meaning "good" or even "perfect" and thus may have indicated the lack of a remainder in these cases. Nfr is also used as a negative word in combination with verb forms and nominal sentences; hence this may also have been the origin of the usage of nfr, indicating the lack of "something." Another symbol used where we might employ 0 was [??] (D35), "arms in gesture of negation," the ideogram for jw, the relative adjective meaning "which is not" to indicate that something was not there.

The hieratic writing of numbers shows, as does hieratic writing in general, simplified sign forms. It also used a set of regular ligatures to write the combination of groups of identical signs, as required by numbers such as 9, 40, and others. Thus individual ligatures existed for each series 1–9 in the various powers of 10 (1, 2, 3, 4, ...; 10, 20, 30, 40, ...), which were written from the highest to the lowest to represent the respective number.

According to Sethe, the sign for one million, [??] was no longer in use during the Middle Kingdom, and a multiplicative way of number notation for large numbers was developed, in which a multiplicative factor was put under the sign for larger numbers; for example, in papyrus UC 32161, the number 925,157 is written as [??] Multiplicative writings are attested for 100.000 and 10.000 (Middle Kingdom and later), rarely also for 1000 (Bilgai stele, end of Dynasty 19).

Uses of Numbers and Their Contexts in Predynastic and Early Dynastic Times

The earliest attested Egyptian number recording originates, as was described previously, from the tomb U-j in Abydos. However, the ivory and bone tags that are supposed to carry numbers on them do not yet show the Egyptian number system described in the previous section. There seem to have been only three types of signs used in writing a number: a vertical stroke, a horizontal stroke, and a rope (see figure 1).

Of these, the vertical stroke and the rope have successors in the later evidence of Egyptian number notations in the writing of units and hundreds, respectively. It is not clear if the horizontal stroke is not simply another way of writing units — especially since in some instances, the proper orientation of the label (which renders the strokes as being either horizontal or vertical) is not certain. Their current orientation is assigned to them based on the orientation of grain of the material. Günter Dreyer suggested that the horizontal stroke may be explained as a possible sign for 10 by assuming that the number tags were meant to record quantities of fabric, which were kept in wooden boxes and have since perished.

The individual tags from the tomb U-j do not mix different numerical symbols. Thus we either find vertical strokes or horizontal strokes but never a combination of the two. Note the symmetric grouping of the signs in tags with vertical and horizontal strokes. The hundreds appear in two versions, either with an additional stroke or only as the curled rope. Rather than interpreting the curled rope and the stroke as standing for 100 + 1, it seems more likely that we have here two kinds of metrological systems. Thus the curled rope alone may represent 100 in one system, and the curled rope with the stroke may stand for 100 within another system.

The number of strokes on some tags exceeds 9; for instance, there are examples of 12 strokes on one tag. As John Baines correctly points out, this is not likely to indicate a base other than 10 for the Egyptian number system. However, it may point to the fact that the "full use" of a decimal system is still on its way.

Similar tags were also found in other cemeteries, e.g. at Naqada (see the examples in figure 2). The Naqada tags are inscribed on the front and the back, with one side carrying the number notation. They show the next stage in the development of Egyptian number notation; several signs are combined to form a number on one tag. Again, the signs are symmetrically grouped. The decimal nonpositional system seems to be in place using the signs for units, tens, and hundreds, as they are known from later sources.

The evidence of tomb U-j suggests that the invention of writing occurred during the process of establishing a social hierarchy and the expansion of the Naqada culture. This expansion may have created the need for an organized administration in which writing was a powerful tool to denote ownership and provenance of goods. This tool seems to have been limited to the elite from the beginning. In consequence, this may have led to the use of writing for representational and legitimatizing purposes. Writing thus helped the unification of the state and its consolidation.

The context of this early evidence is purely funerary; however, it is probably reasonable to assume that the administration of goods displayed in this context was also common and presumably originated from a daily life practice, which was then taken over into the funerary context.

The Egyptian number system was fully developed at the time of King Narmer, as is documented by his ritual macehead, which was found at the temple of the god Horus at Hierakonpolis, the most important predynastic site in the south of Egypt (cf. figure 3).

The object originated from a ceremonial context, as is indicated not only by its findspot but also by its material, soft limestone. The macehead is carefully decorated with the scenes shown in figure 3, depicting King Narmer, who receives a tribute. King Narmer, the central figure toward whom the scenes are oriented, is shown seated on a throne in a shrine. The shrine is elevated and can be reached via a stair of nine steps. King Narmer is shown wearing the red crown of Lower Egypt, he is dressed in a long robe, and he holds a flail. The area of the throne is protected by a vulture, which is hovering above its roof. Under the king, two fan bearers are depicted (possibly to be understood to have been placed to both sides of the king). Another five figures, further subordinates of the king, are shown in two rows behind the throne. Above them, the serekh with the name of the king is shown. In front of the king are three registers with depictions oriented toward the king. In the upper register, four men with standards are shown. The middle register, immediately opposite the seated king, shows another person, who is seated in a litter. Behind the litter, a group of three bearded figures is depicted; the group is framed by two sets of three crescent-shaped signs. The lower register holds the depiction of the tribute, which is presented to King Narmer and which is the earliest representation of the full Egyptian number system, even before the first dynasty. To the right of these three registers is the representation of various animals: an ibis and an enclosure of three antelopes.

The scenes on the Narmer macehead, (and, likewise, those on related objects, such as the Scorpion macehead and the Narmer palette) generally represent the civic, priestly, and military functions of the king within the state. Scholars have tried to link these scenes to specific historic events, but our knowledge of this early period does not permit us to confirm any of these claims beyond a doubt. The scenes have been interpreted several times, resulting in interpretations as a concrete commemoration of the unification of Egypt and the foundation of the Egyptian state, as a celebration of a royal festival, as the celebration of a marriage alliance of Narmer, and as the elaborate writing of a year name. For the purpose of this book, it suffices to state that the scenes are meant to represent King Narmer, who receives a tribute in some ceremonial context.

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Excerpted from Mathematics in Ancient Egypt by Annette Imhausen. Copyright © 2016 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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