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Science                                         16*9
Two-thirds seems to have been a primary concept, and the
Egyptian could write down f of a number without calculation.
One-third was obtained by halving f, which was regarded as
'the two parts' of a length divided in three parts, J being 'the
third (and last) part'. (Cf. Genesis xlvii. 24.) Even now, we
say 'three parts full', where division into four parts is tacitly
assumed. It should be noted that f is, in fact, equivalent to ^,
and that, as for i|, both the Greeks and the Romans had a
special word for it. (Cf. German—anderthalb.)
With such a cumbrous system of fractional notation, calcula-
tion was a lengthy process, frequently involving the use of very
small fractions, e.g. 5^5 occurs. There was a danger, too, that a
number of fractions might be set down without realizing that
they would readily combine to form one or more simpler
The system of unit fractions survived long after the use of
mixed fractions had become general. It is found, with the same
exceptional treatment of f, in the papyrus of Akhmim, written
in Greek about A.D. 600. Modern Stock Exchange quotations
in Cairo are still often given in the same form, e.g. 98 £ ^
^ in excess of J conveys a clearer meaning than }|.
It is not difficult to see how the fractional notation originated
in practical problems dealing with division of food and other
commodities. The word 'division5 originally meant partition
in two. Suppose 5 loaves are to be divided among 6 persons—
an actual problem. The primitive method, still in use in remote
parts of the world to-day, is to divide each loaf in half, and give
one half to each person. The remaining 4 half loaves are again
divided. Of the 8 portions, I is distributed to each individual,
leaving 2 quarter loaves over. Each of these is divided in three,
giving one portion to each person, who has thus received i J ^
of a loaf. To the Egyptian, this was a complicated process,
because of the limitations of his notation. To us, who can write
{, it presents no difficulty.