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

The following is taken verbatim from the MathWorld Wolfram page on Egyptian fractions:

An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of $$2/n$$ as Egyptian fractions for odd $$n$$ between $$5$$ and $$101$$. ... The unique fraction that the Egyptians did not represent using unit fractions was $$2/3$$ (The Penguin Dictionary of Curious and Interesting Numbers, Wells 1986, p. 29).

Well, I kind of find this surprising, since it is relatively easy to compute $$\frac{2}{3}=\frac{1}{2}+\frac{1}{6},$$ which, according to the cited webpage [equation (4)], can be obtained using the greedy algorithm.

I checked the Wells reference, and it only has the following story to tell:

$$2/3$$ - The uniquely unrepresentative "Egyptian" fraction, since the Egyptians used only unit fractions, with this one exception. All other fractional quantities were expressed as sums of unit fractions.

I know that the answer to this question may be covered in some History of Mathematics book, but I currently do not have the expertise to gauge which authoritative sources to check.

MOTIVATION

It is currently unknown whether there is a number $$m$$ such that $$I(m) = \frac{\sigma(m)}{m} = \frac{5}{3} = 1 + \frac{2}{3},$$ where $$I$$ is the abundancy index and $$\sigma$$ is the sum-of-divisors function. If such a number $$m$$ exists, then $$5m$$ is an odd perfect number, where $$5 \nmid m$$.

For more information on the exact connection between the Egyptian fraction decomposition of $$1$$ with odd denominators, and odd perfect numbers, I refer the interested reader to the following answer to a closely related question by MSE user Thomas Bloom.

• About the last sentence: This is only if $5\not\mid m$, right? – joriki Apr 1 at 6:34
• @joriki, yes it can be proved that if $$I(m)=\frac{\sigma(m)}{m}=\frac{5}{3}=1+\frac{2}{3},$$ then $3 \mid m$ and $5 \nmid m$. In fact, it is easy to show that $m$ would be a square. – Jose Arnaldo Bebita-Dris Apr 1 at 6:58
• @joriki, I have added your condition to the question to make it more explicit. – Jose Arnaldo Bebita-Dris Apr 1 at 7:00
• Curiously, Wikipedia states: The first part of the papyrus is taken up by the $2/n$ table. The fractions $2/n$ for odd $n$ ranging from $3$ to $101$ are expressed as sums of unit fractions. I do wonder whether that $3$ is a typo. – Jose Arnaldo Bebita-Dris Apr 1 at 9:13
• Further down in the Content section of that Wikipedia page: Also bear in mind that the fraction 2/3 is the single exception, used in addition to integers, that Ahmes uses alongside all (positive) rational unit fractions to express Egyptian fractions. – Jose Arnaldo Bebita-Dris Apr 1 at 9:18

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.

• Thank you for this answer, @BrianHopkins! Just what I needed. +1! =) – Jose Arnaldo Bebita-Dris Apr 10 at 18:47
• +1. I was tempted to suggest it may be omitted because 2/3=1/2+1/6 was so well-known, just as a reference manual of indefinite integrals might omit $\int Kdx=Kx+C.$ – DanielWainfleet Apr 11 at 4:02

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 $$\frac{1}{12}$$.

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

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• Thank you for this insightful answer, @CaptchaSamurai! +1. =) – Jose Arnaldo Bebita-Dris Apr 10 at 22:52