The beginnings of mathematics and of writing are closely intertwined. It may even be that counting and keeping records of amounts led to the invention of writing. One place to view rare examples of early hieroglyphs is in the Egyptian Museum of Antiquities in Cairo. (The Grand Egyptian Museum, located in Giza, Egypt, is slated to open in 2022.) When the author visited Egypt on an MAA Study Tour in 2009, photographs were not allowed inside museums and certain other places, such as tombs in the Valley of the Kings. So, the images of early hieroglyphs below are from the internet.

The centralized government of Egypt, with the pharaoh at its head, had a need for counting numbers. Among other things, the bureaucracy kept track of supplies, soldiers, taxes owed, and taxes collected. Pictured in Figure 2 are inventory tags—claimed by some to contain the earliest hieroglyphs [Mattessich 2002, p. 196]. Notice the tally marks.

Figure 2. Very early hieroglyphs (ca 3700–3050 BCE) on inventory tags found in a tomb in Abydos. Retrieved from https://web.archive.org/web/20120415143131/http://www.dainst.org/en/project/abydos?ft=all.

Figure 3 displays another of the earliest examples of Egyptian hieroglyphs, the Narmer Palette, located in the Egyptian Museum of Antiquities in Cairo, Egypt. The palette tells the story of Narmer unifying Lower and Upper Egypt and dates to about 3100 BCE.

Figure 3. The Narmer Palette (ca 3200–3000 BCE) which contains early hieroglyphs. Egyptian Museum of Antiquities.

If we zoom in on the upper right corner of the side of the palette pictured on the right in Figure 3, we see a falcon, representing the pharaoh, above six papyrus plants, holding a person by a nose ring. The papyrus plant may be an early hieroglyph for 1000, which then evolved into the lotus plant hieroglyph. So this portion of the palette may be stating that Narmer took 6000 captives [Brier 2016].

Figure 4. 6000 in early hieroglyphs on the Narmer Palette. Detail from Figure 3.

Counting numbers were a crucial part of the centralized government of ancient Egypt, since the bureaucracy needed to keep records of many things, including inventories, workers, time, and taxes. As mentioned above, the Egyptians used an additive decimal system with symbols for the powers of ten from 0 to 6, in which like symbols were grouped together in descending order, and often stacked for aesthetics. If the writing was in columns, the hieroglyphs were to be read from top to bottom. If written horizontally, the hieroglyphs were meant to be read toward faces. So, they might be read left to right or right to left. Typically, the number followed the noun (usually singular) it was counting.

Figure 5: Numeric hieroglyphs in the Louvre (Annals of Thutmose III, 1479–1425 BCE, Karnak Temple).

The example in Figure 5 is from the Louvre and has the writing in columns. Notice the calf at the top of the picture is facing to the right. Thus, the hieroglyphs are read from top to bottom and, within each line, from right to left. The number pictured is composed of four lotus flowers (4000), six stacked coils of rope (600), two hobbles (20), and two tally marks (2), namely 4622. Since the number follows “calf,” it translates as 4622 calves.

The next example is from a papyrus in the Egyptian museum in Turin, Italy, and it also has the writing in columns. From the picture on the left, we see that the person and bird figures face right. So, although the direction does not actually matter in this particular example, the hieroglyphs on each line are read right to left. With four coils of rope (400), five hobbles (50), and three tally marks (3), the number is 453.

Figure 6. Hieroglyphs, read top to bottom and right to left on each line, on a papyrus (1076–722 BCE) in Turin, Italy.
The image on the right shows a close-up of the number 453.

The final examples of counting numbers in this section are from the Karnak Temple (begun around 2000 BCE with expansions through about 30 BCE) in Luxor, Egypt. These hieroglyphs appear on the same wall, one not far above the other. They are read top to bottom and right to left within each row. The number pictured on the left in Figure 7 is 4820, written with four lotus flowers (4000), eight coils of rope (800), and two hobbles (20). The image on the right displays 6823.

Figure 7. Counting numbers on a wall in the Karnak Temple (ca 2000–30 BCE) of Luxor, Egypt.
The image on the left shows 4820 while the image on the right shows 6823.

Fractions in ancient Egypt were almost exclusively unit fractions. According to Annette Imhausen [2016, p. 52], “The Egyptian concept of fractions, that is, parts of a whole, was fundamentally different from our modern understanding.” The notation that was used to signify a fraction—a mouth hieroglyph representing “part”—corresponds with this viewpoint and the use of unit fractions. The rare exceptions to unit fractions include special symbols for \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{1}{4}\), and \(\frac{3}{4}\).

The pictures below demonstrate the unit fractions \(\frac{1}{6}\), \(\frac{1}{16}\), and \(\frac{1}{120}\), respectively.

        
Figure 8. Fractions on temple walls: \(\frac{1}{6}\) (Edfu, 237–57 BCE), \(\frac{1}{16}\) (Kom Ombu, 180–47 BCE),
\(\frac{1}{120}\) (Kom Ombu, 180–47 BCE).

 Figure 9. Fractions on a cubit rod (1327–1295 BCE) in the Louvre.
Notice the special hieroglyph for \(\frac{1}{2}\) on the far right.

A fraction such as \(\frac{13}{16}\), which is not a unit fraction, would be written as a sequence of unit fractions written in decreasing order of denominators, which, when added together, would sum to \(\frac{13}{16}\), such as \(\frac{1}{2}\) \(\frac{1}{4}\) \(\frac{1}{16}\). Figure 10 gives an example to show that expressing non-unit fractions in this way is not unique. The special symbol for \(\frac{1}{2}\) is in the image on the left in Figure 10, followed by \(\frac{1}{3}\) to represent the summed fraction \(\frac{5}{6}\). In the image on the right in Figure 10, the special symbol for \(\frac{2}{3}\) (Ptolemaic version) is followed by \(\frac{1}{6}\) to form another representation of \(\frac{5}{6}\).

Figure 10. Non-unit fractions on a wall in the Edfu Temple (237–57 BCE).
Notice the special hieroglyph for \(\frac{1}{2}\) in the left image and \(\frac{2}{3}\) in the right image.
Thus, the image on the left represents \(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\), while the image on the right also represents \(\frac{5}{6}\), but as \(\frac{2}{3} + \frac{1}{6}\).

Egyptian hieroglyph numerals were also used in recording time. The Egyptian annual calendar consisted of five extra days and twelve months (with each month having three weeks and each week containing ten days) for a total of 365 days. The year was also broken down into three seasons of four months each: inundation called Akhet (when the Nile annually flooded depositing fertile top soil, June to September); emergence or growing season called Peret (October to February); and harvest season called Shemu (March to May) [Brier 2016].

Figure 11 is from a calendar at the Kom Ombo Temple. The circle on the right of each panel is the sun disk determinative to indicate that the writing presents a date. From the top, we see 27, 28, 29 (the circle with the line across represents 9), the 30th day of the last month of the harvest season, the first day of the first month of inundation, 2, 3, and so on.

The Egyptians did not keep a running count of the years. As each new pharaoh began his or her reign, the year count would reset. So, a particular date would be given as the year of the reign of the current pharaoh, the month of the season, and the day.

Figure 11. Portion of a calendar showing hieroglyph numerals from the Kom Ombo Temple (180–47 BCE).

The Egyptian hieroglyphic script consists of over 1000 characters representing either a phonetic sound, an ideogram for a specific concept, or a determinative at the end of a word to clarify its meaning. Not surprisingly, there are some challenges to reading hieroglyphs. For example, there is no punctuation to separate sentences or clauses, although comprehension is aided by knowing that verbs occur at the beginnings of sentences. Although determinatives follow some words, there are nonetheless cases where it is difficult to tell when one word ends and the next begins. Variations can occur among the symbols, since they were created by various scribes or carvers over four millennia. Also, there are places where the hieroglyphs have been damaged. In addition, there are symbols which have different meanings depending on the context. We will discuss several such hieroglyphs related to mathematical notation.

First, the mouth symbol that is used to denote a unit fraction has several other meanings. It is part of the hieroglyphic alphabet representing the letter “R”, and it is also used as a stand-alone word meaning “to” a place (there is another word for “to” a person) or “by”. Later, in Figure 14, the mouth symbol is used both to denote a unit fraction and as “by”.

Also in Figure 14, we see a hieroglyph that looks like the side of a statue base used to mean “at” or “from” or “side”, while a smaller version represents \(\frac{1}{2}\). (Compare with Figure 38.)

The coiled rope symbol representing 100 is also used as a replacement for the quail chick hieroglyph representing the letter “W” or the sound “oo”.

While the stroke can represent the number one, it appears much more often as a determinative following a hieroglyph to indicate it is being used as an ideogram, i.e., a face hieroglyph followed by a stroke means either “face” or something related to a face. The determinative stroke is typically shorter than the stroke for the number 1, as can be seen in Figure 12.

Figure 12. Example of a stroke used as a determinative (twice on the left)
and as a numeral (7 on the right) on a wall in the Edfu Temple (237–57 BCE).

The number in Figure 13 could easily be mistakenly read as 1,333,331, instead of correctly as 1,333,330. Upon closer examination, one sees the stroke is not part of the numeral but is actually an ideogram determinative following the heart hieroglyph.

Figure 13. Example of a stroke used as a determinative and not a numeral on a wall in the Edfu Temple (237–57 BCE).

Three strokes also commonly occur in hieroglyphic text to denote that the preceding word is a plural. Figure 18 shows an example of three strokes indicating a word is plural. Two slanted strokes are also used to denote a noun is plural, specifically in the case of two objects.

Most people would expect to find Egyptian hieroglyphs in ancient temple ruins in Egypt. However, Egyptian hieroglyphs can also be found in museums around the world. Many world history and art museums have Egyptian collections, in which hieroglyphs can be found on pieces of walls, sarcophagi, stelae, vessels, papyri, statues, jewelry, and other objects. In fact, one way to apply this article with students is to explore those collections during in-person or virtual field trips.

The following pages highlight artifacts marked with hieroglyphic numerals that are held in five locations visited by the author:

• Edfu Temple

• Kom Ombo Temple

• British Museum
  • Rosetta Stone
  • Rhind Mathematical Papyrus
  • Stela of Merymose’s Campaign Against the Nubians

• Louvre
  • Annals of Thutmose III
  • Cubit Rod
  • Sculptures of Scribes and Examples of Scribal Tools

• Museo Egizio in Turin, Italy
  • Cubit Rods
  • Amduat Papyrus

Images of each location retrieved from Wikimedia Commons.

Continue to Edfu Temple page.

Instructors may want to create assignments that involve hieroglyph numerals not available in photographs. Of course, one can draw hieroglyphs by hand as the Egyptian scribes did. However, there is an Egyptian hieroglyph word processor available, called JSesh (Sesh is the Egyptian word for scribe). JSesh can be freely downloaded. The hieroglyphs are organized using a system widely employed by Egyptologists and created by Sir Alan H. Gardiner (1879–1963), who wrote the classic middle Egyptian hieroglyph text Egyptian Grammar. The hieroglyphs are classified into 25 categories, such as: man and his occupations, mammals, birds, temple furniture and sacred emblems, and strokes, along with an unclassified section and three other sections—tall narrow signs, low narrow signs, and broad narrow signs. Unfortunately, there is not a section on numerals. The JSesh hieroglyphs can be copied into other programs such as Word or saved as image files. The table below contains a selection of mathematical hieroglyphs as jpg files that can be copied or saved as images on a hard drive to make it easier for readers to access them for classroom use.

As mentioned in the introduction, most of the photographs, except for those noted otherwise (specifically Figures 2–4, 15, and 40), were taken by the author. The author’s photographs can be freely downloaded for classroom use under a Creative Common Attribution 4.0 International license.

Instructors may wish to use the photographs in presentations to illustrate authentic Egyptian hieroglyphs. Seeing actual images of hieroglyphs can have a powerful impact on student learning. The author has observed how these photographs grab the interest of the students, bringing the mathematics to life and motivating students to learn more. The photos can also be used in assessments in which students are asked to determine the number being represented.

In addition to History of Mathematics courses, use of the photos would be appropriate in any class that includes the study of numeration systems, such as number theory or mathematics courses for pre-service teachers. Comparing and contrasting the Egyptian hieroglyph numerals with Hindu-Arabic numerals affords students the opportunity to think about place value and what it means to utilize a decimal (base ten) system. Since students have used the Hindu-Arabic numeration system for many years, it may be challenging for them to notice its properties. Seeing a decimal system that does not use place value may lead students to reflect on what it means to have a place-value system and on how using one affects arithmetical algorithms. The Egyptian numeration system could also be compared and contrasted with the Roman numeration system, which is still used today in some analog clocks, the release dates of movies, and the number of Super Bowl games.

The photographs can also be used to supplement activities involving Egyptian mathematics. David Reimer’s book, Count Like an Egyptian: A Hands-On Introduction to Ancient Mathematics [2014], is a good source for interesting problems for students. The author of this article has shown pictures of Egyptian fractions before students complete a group activity based on the “Fractions” chapter of Reimer’s book, especially pages 16–17. Students are divided into groups of two or three and given a paper plate, a plastic knife, and three slices of bread (each slice representing a loaf of bread). The students, assuming the role of work supervisors, are challenged to divide the three slices of bread among five workers in such a way that the workers receive obviously equal portions with at least some pieces of bread large enough to enjoy with honey. The latter condition is not met if a group wants to divide each slice of bread into five equal pieces, with each worker receiving three, as in the top row in Figure 42. This manner of thinking corresponds to a modern view of the fraction \(\frac{3}{5}\) as “three divided by five.” A solution which relates to how Egyptians may have viewed fractions as a sum of unit fractions is shown on the bottom row of Figure 42, where each worker would receive \(\frac{1}{2}\) and \(\frac{1}{5}\) of a \(\frac{1}{2}\), or \(\frac{1}{10}\).

Figure 42. A demonstration of two ways to “equally” divide three slices of bread among five people.
With the method in the top row, each person would receive three strips of bread.
With the method in the bottom row, each person would receive one half and one-tenth of a slice.

The author has also used some of the photographs, including those of scribes and scribal tools, in an “Egyptian Scribal School” activity with students at the secondary and undergraduate levels. The setting is a scribal school, with each student provided with a clipboard, paper, and fine point black marker to simulate a board, papyrus, and brush with ink. In addition to the photos of scribes and scribal tools, the students are shown some photos of hieroglyphs, including Figure 1, which shows six of the Egyptian numeric hieroglyphs, and Figure 6, a papyrus with hieroglyphs. (The author has a framed poster-size print of Figure 1, which she prominently displays in the classroom during the activity.) The students then practice writing each of the numeric hieroglyphs several times before solving arithmetic problems using Egyptian hieroglyphs and methods similar to those in Chapter 1 of Count Like an Egyptian [Reimer 2014].

An introduction to ancient Egyptian mathematics, including the numeration system, benefits students. For example, by contrasting the Egyptian numeration system with the Hindu-Arabic system, pre-service teachers can gain a deeper understanding and appreciation of our decimal place-value system. Unit fractions are a great example of the importance of notation. The purpose of this article was to provide photographs of genuine Egyptian hieroglyphic numerals to motivate students and to make the historical use of numerals more real for them. Images were found in Egyptian temples and museums around the world. We hope that readers will be inspired to be on the hunt for Egyptian numerals themselves as they travel and visit museums with ancient Egyptian collections.

Akademie der Wissenschaften zu Göttingen. 2022. Edfu Database. Accessed 1 August 2021. https://adw-goe.de/en/forschung/abgeschlossene-forschungsprojekte/akademienprogramm/edfu-projekt/die-datenbanken-des-edfu-projekts/edfu-datenbank/.

Brier, Bob. 2016. Decoding the Secrets of Egyptian Hieroglyphs. The Teaching Company.

Budge, E.A. Wallis. 1905. Translation of the Rosetta Stone. In The Nile, Notes for Travellers in Egypt, 199–211. 9th ed. London: Thos. Cook and Son.

Childers, Christopher, and Cynthia Huffman. 2015. Napoleon and Mathematics: A Case Study of the Interplay Between Mathematics and History. The Midwest Quarterly 56(3): 209–216.

Coppens, Filip, and Hana Vymazalová. 2010. Medicine, Mathematics and Magic Unite in a Scene from the Temple of Kom Ombo (KO 950). Anthropologie 48(2): 127–132.

Deutsches Archäologisches Institut. 2011. Abydos – Umm el-Qaab. Accessed 1 August 2021. https://web.archive.org/web/20120415143131/http://www.dainst.org/en/project/abydos?ft=all.

Gardiner, Alan H. 1973. Egyptian Grammar: Being an Introduction to the Study of Hieroglyphs. 7th ed. Oxford: Griffith Institute.

Gaudard, François. 2010. Ptolemaic Hieroglyphs. In Visible Language: Inventions of Writing in the Ancient Middle East and Beyond, edited by Christopher Woods, with Emily Teeter and Geoff Emberling, 173–175. Oriental Institute Museum Publications No. 32. Chicago: The Oriental Institute.

Imhausen, Annette. 2016. Mathematics in Ancient Egypt: A Contextual History. Princeton: Princeton University Press.

Louvre. n.d. Accessed 1 August 2021. https://www.louvre.fr/en.

Mattessich, Richard. 2002. The Oldest Writings, and Inventory Tags of Egypt. The Accounting Historians Journal. 29(1): 195–208.

Ministry of Tourism and Antiquities. 2019. The Egyptian Museum. Accessed 1 August 2021. https://egymonuments.gov.eg/en/museums/egyptian-museum.

Museo Egizio, Torino. 2018. Accessed 1 Aug. 2021. https://www.museoegizio.it/en/.

Reimer, David. 2014. Count Like an Egyptian. Princeton: Princeton University Press.

Rosmorduc, Serge. 2014. JSesh Documentation. Last modified 13 January 2022. Accessed 26 July 2021. http://jseshdoc.qenherkhopeshef.org.

Trustees of the British Museum. 2022. The British Museum. Accessed 1 August 2021. https://www.britishmuseum.org/.

Wilkinson, Richard H. 2000. The Complete Temples of Ancient Egypt. London: Thames & Hudson.

Wilkinson, Toby, trans. and intro. 2017. Writings from Ancient Egypt. London: Penguin Group USA.

Acknowledgements

The author is indebted to the editors and anonymous referees for help in improving the article. Many of the author’s photos in this article were taken on MAA study tours. Special thanks to Lisa Kolbe for initiating the MAA Study Tour program, along with Victor Katz and the planning committee. Additional thanks to mathematics professor and Egyptologist Jim Ritter for sharing his knowledge and expertise during the MAA tour to Egypt, and to all those who participated in the wonderful educational study trip experiences from 2003 to 2013. The author cherishes the special friendships formed while sharing the thrill of viewing objects and sites of mathematical importance in person.

About the Author

Cynthia J. Huffman is a University Professor in the Department of Mathematics at Pittsburg State University. She has always been interested in history of mathematics but her interest was especially sparked by participation in several of the MAA Study Tours, beginning with one to Egypt in 2009. Her research areas include computational commutative algebra and history of mathematics. Dr. Huffman is a handbell soloist and has a black belt in Chinese Kenpo karate.