Why did the Egyptians not represent $2/3$ as a sum of unit fractions in the Rhind papyrus?

They had a special symbol for 2/3, presumably because of frequent use, so there was no need to work out its representation. See chapter 7 of Annette Imhausen's Mathematics in Ancient Egypt: A Contextual History, Princeton University Press, 2016.


The first fractions Ancient Egyptians used were $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{4}$ and $\frac{3}{4}$. They used special words for these natural fractions.

As techniques of calculation developed unit fractions were introduced. Notation was concise except previously established symbols for natural fractions. Since $\frac{2}{3}$ and $\frac{3}{4}$ already had designators, these weren't broken down into unit fractions. Egyptians just continued using them in the old way. Nevertheless, after some time, $\frac{3}{4}$ began to be expressed like other fractions, but somehow $\frac{2}{3}$ remained a exception.

I based my answer Chapter I. The Egyptians from Waerden, B. L. van der. (1988). Science awakening. Dordrecht, The Netherlands: Kluwer Academic Publishers and https://www.bibalex.org.

P.S. It's a curious thing to see what we are talking about. Ancient Egyptians wrote unit fractions by placing ovals (not "1") above numbers. Below regular $\frac{1}{12}$.

1/12

And here are the exceptional $\frac{1}{2}$, $\frac{2}{3}$, $\frac{1}{4}$, $\frac{3}{4}$ and normalized $\frac{3}{4}$ respectively.

1/2, 2/3, 1/4, 3/4, 3/4 normalized

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Conjectures

Perfect Numbers

Math History

Divisor Sum

Egyptian Fractions

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