Page:The Rhind Mathematical Papyrus, Volume I.pdf/20

sults. In most of the multiplications all through the papyrus the author checks those multipliers that he is to use.^[1] Division was performed by successive multiplication of the divisor until the dividend was obtained.^[2]

For the numbers of the descending series they had:

1. A notation which for reciprocal numbers was nearly like their notation for integers, these numbers being distinguished from integers by having a dot in hieratic and the sign

in hieroglyphic written over them, except that the first three had special signs in hieratic, and the first, ½, in hieroglyphic also;

2. Special devices for addition and subtraction, because it was necessary to express the result using only integers and different unit fractions (see page 3, footnote 1); and for multiplication, because, apparently the only fractions that they could use as direct multipliers were ⅔. ½, and ¹⁄₁₀.

Addition and subtraction will be explained below. In multiplication they generally took ⅔ and then halved to get ⅓,^[3] and they took ¹⁄₁₀

1. For the multiplication of 19 by 6 the Egyptians would say,

   		1	19
   	\	2	38
   	\	4	76
   Total,		6	114,

   the two multipliers which make up 6 being checked; and for 12 times 23,

   		1	23
   	\	10	230
   	\	2	46
   Total		12	276.

   See page 52.

   In Problem 79, 2801 is multiplied by 7 in the same way,

   1	2801
   2	5602
   4	11204
   Total	19607.

2. Dr. Reisner tells me that when the modern Egyptian peasant wishes to divide things of some sort among a number of persons he first distributes a certain number, say 5, to each, and then perhaps 2, and so on.
3. Peet, page 20. Examples are in Problems 25, 29, 32, 38, 42, 43, and 67; besides 8 and 16—20, and many times in the table at the beginning. ⅓ is given without any ⅔ in Problems 28, 29, 32, 39, 42 (line 8 of the multiplication), and several times in 35—37. We may suppose that in these cases the author did not intend to put in the work, but only the result. Gunn says (page 125) that “The Egyptian, like everyone else, had ultimately no way of arriving at ⅔ but via ⅓.” This is not quite true, for, whatever ⅔ meant to him, he knew for a fact that it is the reciprocal of 1½ (see, for example, page 75, footnote, also Gunn, page 129), and it was possible for him to make a table for 1½ and read it backwards, just as he sometimes reads backwards the table for the division of 2 by odd numbers (page 20, footnote 2). A number not found in his table (for example, a number that is not a multiple of 3) could be obtained by putting other numbers together, using ⅔ for ⅔ of 1, and 1⅓ for ⅔ of 2. This is explained by Gunn himself on page 125.