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.


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.

  • 1
    $\begingroup$ About the last sentence: This is only if $5\not\mid m$, right? $\endgroup$ – joriki Apr 1 at 6:34
  • $\begingroup$ @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. $\endgroup$ – Jose Arnaldo Bebita-Dris Apr 1 at 6:58
  • $\begingroup$ @joriki, I have added your condition to the question to make it more explicit. $\endgroup$ – Jose Arnaldo Bebita-Dris Apr 1 at 7:00
  • $\begingroup$ 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. $\endgroup$ – Jose Arnaldo Bebita-Dris Apr 1 at 9:13
  • $\begingroup$ 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. $\endgroup$ – 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.

| cite | improve this answer | |
  • $\begingroup$ Thank you for this answer, @BrianHopkins! Just what I needed. +1! =) $\endgroup$ – Jose Arnaldo Bebita-Dris Apr 10 at 18:47
  • 1
    $\begingroup$ +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 $\int Kdx=Kx+C.$ $\endgroup$ – 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.

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.