A tale of two names
Some individuals’ traces can survive centuries or millennia because their name remains attached to an artefact.
The Scottish antiquarian and archaeologist Alexander Henry Rhind and the Egyptian scribe Ahmes are two names that have been dragged till modern times by a papyrus.
The papyrus is called Rhind’s papyrus or Ahmesh’s (it also has the more aseptic name of pBM 10057 and pBM 10058, the pair of initials of the two parts of the artefact given by the British Museum) and its fame derives from the fact that it is one of the oldest and most complete examples of an Egyptian mathematical manual.
Rhind was an archeologist with a short but very interesting life that apparently ended in Italy at the age of 30. He is the one who bought it in Luxor in 1858 (it had been found in Thebes in the ruins of Ramesseum) and Ahmesh is the scribe who transcribed it around 1650 BC during the reign of Aphophis (fifth ruler of the 15th dynasty) from a lost text produced during the reign of King Amenemhat III (12th dynasty, 1860–1814 BC).
As can be seen, even a tiny fragment of this story induces interest and curiosity. And if one appreciates the fractions, the text does not disappoint.
The text is a true technical mathematical manual. The topics covered are:
- Arithmetic of fractions (the topic of this post)
- Practical problems of geometry
- Problems of measurement and progressions
Today the content may not seem esoteric and mystical (perhaps), although it is interesting to note that the section on arithmetics is entitled “Directions for Knowing All Dark Things”.
For the Egyptians who lived when the text was drafted, fractions were not represented as it happens in modernity by the ratio of two integers n/m, but by a sum of unit fractions (with numerator 1) strictly different from each other (with the addition to the restricted club of the fraction ⅔, which was indicated by a special symbol).
These are therefore Egyptian fractions:
Despite the apparent limitation of the representation, all normal fractions (hence rationals) can be represented in this form.
Even though it is evident today that an infinite number of such representations is possible, the mathematicians of ancient Egypt seem to have been more interested in optimising the form of a fraction to meet particular practical uses.
Numbers and the Egyptians
From the outset, it seems clear from the sources that the Egyptians knew and knew how to write the series of integers from 1 to 10,000,000 (thus a representation of positive integers) and fractions defined by the reciprocals of integers with the addition of the fraction ⅔ .
The writing was not positional, but the number was defined by the sum of the symbols.
Addition and subtraction were carried out without problems, but direct multiplication and division were only carried out for 2 and 10. In different cases procedures similar to modern ones were iterated. Precisely in the Rhind papyrus it is shown how to perform division by successive multiplications of the divisor.
The notations that have come down to us are those in Hieratic (as in the case of the papyrus cited above) and those in hieroglyphic.
In Hieroglyphic the graphic elements used were:
Thus, to write the number 23312, the symbols were juxtaposed in an iterated manner in order to represent, by their sum, the final value:
Since the writing is not positional, the two representations (from right and left) are equivalent.
Fractions defined by the reciprocals of integers had a very easy representation. The number x of the denominator was drawn under the following symbol:
So, for example:
The exception of the fraction ⅔ was represented by the symbol:
It is easy to see that the Egyptian method also conceals a hierarchical representation.
The hierarchy is given by the size of the fractions that are added together and is pervasive in both human thought and mathematics.
You can try to represent more and more precisely an element x with a sum of known objects, knowing, assuming or believing one knows (a vast theme that could easily lead up to quasi-modernity, with Godel’s incompleteness theorem) that there is an equality of the substance of x with the sum, regardless of the representations.
Hierarchy is not essential but is historically preferred. The first term of Egyptian fractions (or the last, if the order is reversed) is a good first approximation, as is the first term of a series of functions.
Thus, returning to the initial examples:
The first line makes it clear that ⅖ is approximately ⅓.
In the second line ⅔ is a very crude approximation of 9/10 which already becomes good with the addition of ⅕ .
The motivations that led to this type of representation are not entirely clear, the millennia tend to obscure motivations that are perhaps far away from those assumed in modernity.
There are explanations that point to practical, linguistic, semiotic, evolutionary and esoteric motivations.
A simple practical reason that is often cited is the following: if it is necessary to divide k objects into n parts (5 Egyptian cakes to be divided equally among 8 scribes), the Egyptian representation reduces the number of fragments (interestingly, the comparison is always made with the modern method, which, it is assumed, was not known to the Egyptians). The advantage would be that reducing the number of fragments reduces the number of cuts needed, which, depending on the problem, could be costly in their production or difficult.
Thus, using the example of the 5 cakes for the 8 scribes:
⅝ = ½ + ⅛
It is possible to divide in half (½) the five cakes (10 pieces) and give one half of a cake to each scribe. This leaves 2 half cakes to be divided into eight parts (so just divide each part into four) and to be divided equally. In total you therefore have 8 larger pieces and 8 smaller pieces.
Using the fraction ⅝ directly, it would be necessary to divide the 5 cakes into eight pieces each (resulting in 40 pieces) and deliver 5 pieces of cake to each scribe.
The Egyptian slices would then be more inviting and the crumbs reduced.
With respect to esoteric motives, it would certainly be interesting to explore the related topic of the Eye of Horus and Pi in a dedicated post.
Specific fractions are associated with the parts of Horus’ eye in the figure. In one of the episodes of the dispute between Horus and Set, the latter succeeds in tearing the left eye of Horus and breaking it into 64 parts. The god Thoth picks up and repairs the eye, but leaves Set 1/64th of it by hiding it.
The story is articulated, but there are several hypotheses of connections with Pi. One is the fact that 1/64 multiplied by 2 eyes and 100 gives the result 3.125, which is the Babylonian approximation of Pi.
It should also be noted that the Rhind papyrus is also connected in a non-explicit way to Pi through the enigmatic problem 48.
Example of technique
Constructing an Egyptian fraction is not always easy, and the manual is dedicated to techniques that achieve this.
Since Hieratic is not known to most people, we recommend for details the ancient (but modern if compared to the papyrus) Chace, Arnold Buffum; et al. (1927). The Rhind Mathematical Papyrus Volume I .
However, there are more or less modern methods to achieve this for particular classes of fractions and also a general (but not optimal since the solution often leads to very large denominators) algorithm.
Fractions that are already 1/n can be written as a sum in the following way:
1/n = 1/(n+1) + 1/n(n+1)
So, for example:
1/11 = 1/12 + 1/11*12 = 1/12 + 1/132
For fractions of type 2/n there are several decompositions cited in the literature. One of the most interesting is:
2/n = 1/n + 1/2n + 1/3n + 1/6n
which thus somehow links the numerator number 2 with the sequence 1,2,3,6.
For fractions of a more general type, a greedy algorithm is known (first described by Fibonacci — https://en.wikipedia.org/wiki/Greedy_algorithm_for_Egyptian_fractions) which does not have the same poetry as the Egyptian methods, but which has the advantage of always working and being easily implementable from an algorithmic point of view.
For those interested in learning more about the subject, here are some interesting links:
Are you interested in the topic and potential of mathematical models?
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