How did egyptian mathematical fractions evolve in antiquity?

Question

There are certain explanations on how integers might have evolved, like for example "the wings of a bird to symbolize the number two, clover--leaves three, the legs of an animal four, the fingers on his own hand five."$_1$ Seeing all these, and making experience short--abstract, can be said to have given the numbers$_2$. Egyptian integer number notation can itself be traced back to their notion of marks on tally (for the sign of one), hobble for cattle (for ten), measuring rope (for hundred).$_3$

Similarly what are the explanations given for the evolution of fractions in Egypt, by any of the found evidence? Did division discovery gave raise to fractions or fractions discovery gave raise to division, in Egypt?

Book and journal reading suggestions would also be helpful.

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$_1$Tobias Dnatzig, Number: The Language of Science
$_2$This expression can be found in Hamilton's letter of Sep.16, 1828, in the book Life of Sir William Rowan Hamilton.
$_3$Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History Page No.18-21.

Answer 1

A contibution to a discussion that ended nearly four years may appear quixotic but it is difficult to let stand an answer which, though accurately summing up Boyer's views on the subject, is completely out of date and misleading. The state of the question of the origin and evolution of fractions still today is to be found in the acts of an international conference on the question, published as: Histoire de fractions, fractions d'histoire (Basel: Birkhäuser) 1992 — unfortunately very little has been done in the field since. The specific question of Ancient Egyptian fractions is treated in the editors' introduction, "The earliest fractions" (pp. 3-15) and in the chapter by James Ritter, "Metrology and the prehistory of fractions" (pp. 19-34). (Note that though the bulk of the book is in French, these two chapters are in English).

Summing up rapidly, the earliest Egyptian fractions date back to the earliest period of Egyptian writing and state formation (1st Dynasty, roughly 3000 BCE). They appear first as elements of particular systems of weights and measures but later are integrated into a system of abstract numbers used in doing mathematics. They are indeed "unit fractions" (except for 2/3) but they did not combine to make up our rational fractions p/q; rather, they were conceptualized as unique values, specific parts of a whole, and were written, for the most part, as inverses of integers (a dot placed above the integer), the exceptions being 1/2, 1/3, (2/3), and 1/4 which had special signs, with no relation to the integers 2, 3 and 4. (Remember that Egyptians did not normally write in hieroglyphics but in a cursive form called hieratic; hieroglyphs were reserved for the most part for monumental inscriptions) By the way, this way of writing fractions was adopted by the Ancient Greeks (see "The earliest fractions", cited above, pp. 12-15).

The rest of Boyer's presentation of Egyptian calculations is also wrong but since it is irrelevant to the question of fractions, no point would be served by correcting it here.

Answer 2

According to Carl Boyer, in his text A History of Mathematics, the earliest record we have regarding fractions in ancient Egypt comes from the Ahmes Papyrus, popularly known as the Rhind Papyrus, circa 1650 BCE. The scribe, Ahmes, tells us that it is derived from a prototype of the Middle Kingdom, circa 2000-1800 BCE, and it is possible that some of this knowledge may be handed down from as early as 3000 BCE.

According to Boyer, the general concept of fraction appears to have been an enigma to the Egyptians. They did not regard the general rational fraction $\frac{n}{m}$ as an elementary "thing", but as an uncompleted process.

Where today we think of $\frac{3}{5}$ as a single irreducible fraction, Egyptian scribes thought of it as reducible to sum of the three unit fractions $\frac13$ and $\frac15$ and $\frac1{15}$.

With this in mind, the papyrus begins with two tables; the first expressing fractions of the form $\frac{2}{n}$ for odd $n$ from $5$ to $101$ expressed as the sum unit factions ($\frac{1}{m}$). A second, shorter table follows giving a similar treatment for fractions of the form $\frac{n}{10}$ for $n = 1$ to $9$.

(To complicate things, in addition to unit fractions, the Egyptians also appear to have been comforatble with the fraction $\frac23$, which appears in the tables. For example $\frac9{10}$ is given as $\frac{1}{30} + \frac15 + \frac23$.)

Following the two tables is a list of eighty-four assorted problems which may provide some motivation for the evolution of fractions in Egypt. Typical of these problems is the division of a given number of loaves of bread amongst ten men.

According to Boyes :

The fundamental arithmetic operation in Egypt was addition, and our operations of multiplication and division were performed in Ahmes' day through successive doubling, or "duplication".

This suggests that the concept of fraction preceded the formal concept of division.