The History of Mathematics: A Brief Course

130 6. CALCULATION

for the result of dividing something in two parts. This linguistic peculiarity sug-
gests that doubling is psychologically different from applying the general concept
of multiplying in the special case when the multiplier is 2.
Next consider the absence of what we would call fractions. The closest Egyptian
equivalent to a fraction is what we called a part. For example, what we refer
to nowadays as the fraction j would be referred to as "the seventh part." This
language conveys the image of a thing divided into seven equal parts arranged in
a row and the seventh (and last) one being chosen. For that reason, according to
van der Waerden (1963), there can be only one seventh part, namely the last one;
there would be no way of expressing what we call the fraction |. An exception was
the fraction that we call |, which occurs constantly in the Ahmose Papyrus. There
was a special symbol meaning "the two parts" out of three. In general, however,
the Egyptians used only parts, which in our way of thinking are unit fractions, that
is, fractions whose numerator is 1. Our familiarity with fractions in general makes
it difficult to see what the fuss is about when the author asks what must be added
to the two parts and the fifteenth part in order to make a whole (Problem 21 of
the papyrus). If this problem is stated in modern notation, it merely asks for the
value of 1 — (-JTT + 3), and of course, we get the answer immediately, expressing it
as -pr. Both this process and the answer would have been foreign to the Egyptian,
whose solution is described below.
To understand the Egyptians, we shall try to imitate their way of writing down
a problem. On the other hand, we would be at a great disadvantage if our desire
for authenticity led us to try to solve the entire problem using their notation. The
best compromise seems to be to use our symbols for the whole numbers and express
a part by the corresponding whole number with a bar over it. Thus, the fifth part
will be written 5, the thirteenth part by Ú3, and so on. For "the two parts" (|) we
shall use a double bar, that is, 3.

1.1. Multiplication and division. Since the only operation other than addition
and subtraction of integers (which are performed automatically without comment)
is doubling, the problem that we would describe as "multiplying 11 by 19" would
have been written out as follows:

19 1 *

38 2 *

76 4

152 8 *

Result 209 11

Inspection of this process shows its justification. The rows are kept strictly in
proportion by doubling each time. The final result can be stated by comparing
the first and last rows: 19 is to 1 as 209 is to 11. The rows in the right-hand
column that must be added in order to obtain 11 are marked with an asterisk, and
the corresponding entries in the left-hand column are then added to obtain 209.
In this way any two positive integers can easily be multiplied. The only problem
that arises is to decide how many rows to write down and which rows to mark
with an asterisk. But that problem is easily solved. You stop creating rows when
the next entry in the right-hand column would be bigger than the number you are
multiplying by (in this case 11). You then mark your last row with an asterisk,
subtract the entry in its right-hand column (8) from 11 (getting a remainder of 3),