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An Ancient Egyptian Mathematical Photo Album: Fractions

Cynthia J. Huffman (Pittsburg State University)


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.

        Hieroglyph for one-sixth. Hieroglyph for one-sixteenth. Hieroglyph for 1/120.
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).

Cubit rod marked in fractions.
 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 \(\frac{5}{6}\).

Hieroglyphs for fractions from Edfu Temple. Hieroglyphs for a sum of fractions from Edfu Temple.
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 \(\frac{2}{3} + \frac{1}{6}\).

Cynthia J. Huffman (Pittsburg State University), "An Ancient Egyptian Mathematical Photo Album: Fractions," Convergence (April 2022)