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