An Ancient Egyptian Mathematical Photo Album: Fractions

Author(s):

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 $\frac{2}{3} + \frac{1}{6}$.