It is common to suppose that in ancient Egypt all fractions other than 2/3 were expressed as sums, where
each element was a unit fraction. Their only writing of
fractions was 2/3 or something with one (1) in the numerator. However,
other readings of the texts show that the issue is more complex. See
Egyptian Math,
Math Forum, Milo Gardner and Letters by Milo Gardner. The general idea
of using unit fractions appears in several sources. Both many examples
of ordinary fractions represented as sums of unit fractions and the
analysis by Milo Gardner about how they could have been found are at SUNY Buffalo, Scott W. Williams, Professor of Mathematics
Detailed examples of unit fractions based on Gillings (see below) are at:
Shorter ... and Visually
(HOWEVER, Gardner expressly indicates that they also used ordinary
fractions.)
Principles How They Reckoned - A Number With An
Overbar Signifies The Fraction One Divided By That Numeral = 1/5
The problem of dividing nine loaves of bread among ten people can be
solved in a seemingly-fair way ... or as the Egyptians would have done
it. [Here dividing ... among means giving equal portions of the nine
loaves to all ten people involved. The task also requires distributing the
entire initial amount of bread.]
We would say that each person is satisfied with exactly the same
quantity of bread. Hence nine would receive nine-tenths a loaf. However
that is a substantial hunk, one that would likely be envied by the tenth
person who receives nine one-tenth loaves.
Egyptian mathematics used only unit fractions
(numerators must be one). Their solution to dividing items equally among
individuals has merit even today. The key observation is that their
results diplomatically distress none.
To solve the problem as they would have, recognize that thirds, fifths,
ninths, tenths,
fifteenths, and thirtieths are all fractions with denominators that are
factors of the product of nine and ten. (A simplified version of their
approach eliminates fifths, thirds and ninths. It assigns a-many tenths,
b-fifteenths, and c-thirtieths, to be added, then multiplied by ten to yield nine:
[a(1/10) + b(1/15) +c(1/30)]10 = 9
Contemporary notation would rewrite this as:
a + b(2/3) +c(1/3) = 9
We allow nonunity numerators, so chosing a four, b three and
c nine satisfies the relationship. Then:
This is the amount of one loaf that can be given to each of the ten
people. [Clearly ten times 27/30 is simply nine.])
Why the Egyptians would only work with unit fractions is not known.
Neither is their rational for allowing a single exception: 2/3, for
which they had a separate symbol. Nevertheless they tabulated enormous
representations akin to what we would write as:
2/15 = 1/30 + 1/10 [ = (1 + 3)/30]
Framing the problem as in the words above it is fairly simple to derive
this acceptable division:
2/3 + 1/5 + 1/30 [ = (20 + 6 + 1)/30]
Hence take the nine loaves and create nine 2/3 of a loaf pieces. Store
two of the remaining 1/3 pieces: they will be used to compose the tenth
2/3 sized major unit. Divide one of the remaining seven 1/3 pieces into
tenths, yielding the required ten 1/30 units. There are now six 1/3
pieces. We require ten 1/5 units. But 1/3 is 10/30 and that fact can be used.
Just apportion each 10 into a 6 and 4. We immediately get six of size
6/30 = 1/5. Combine the remaining parts to find [6(4) = 24]/30 and cut
that quantity in fourths to complete the fair division.
Improved Unit Fraction Partitions
Paul Stelling read the above and "wondered about the solution that satisfies the
following conditions:
a) all portions are identical (not just in size, but also in makeup;
b) each portion has the minimum number of pieces possible; and
c) subject to a) and b), the size of the smallest piece is maximized.
d) [alternative to c)] subject to a) and b), the number of cuts is minimized.
The solution that I came up with that meets this criteria is to divide 5 loaves
into halves, and the remainder into fifths, so that each portion consists of
1/2 + 1/5 + 1/5.
It is easy to see (by examining alternatives) that this solution satisfies all
the conditions."
More About the Principles of the Egyptian Fractions
"... the ancient Egyptians(') ... mathematics was based on two very
elementary concepts. ... their complete knowledge of the twice-times
table, and ... their ability to find two-thirds of any
number, whether integral or fractional." [Gillings, op. cit., p. 3].
"If one-fifth ... was cut off from one end of a loaf of bread, the
Egyptian did not seem to think of what was left as being 4/5 of the
original loaf, but rather as the loaf reduced; if a fifth of this
reduced loaf was cut off, the two "fifths" thus removed would of course
be unequal." ... "the scribes ... carefully avoided writing
for 2/5 and wrote instead "
[For further information please see Gillings, Richard J., Mathematics in the Time of the
Pharaohs, Cambridge, MA: The MIT Press, 1972; NY: Dover Publications,
Inc., 1982, ISBN 0-486-24315-X.]
= 1/5
"