This example illustrates what I mean by “applying a fractional expression to some particular number.”
Sometimes the Egyptian wishes to use this method with an expression involving a whole number as well as fractions. Thus in the first multiplication of Problem 32 of the Rhind Papyrus the author desires to select partial products that will add up to 2. Regarding all of them as referring to a group of 144 things of some kind, assuming perhaps that he has several such groups, he finds the values of these partial products as applied to 144 and seeks to make up in this way the number of things in two groups, that is, 288. The multiplicand, 1 ⅓ ¼, makes one whole group and ⅓ and ¼ of another, or 228 things in all. The next expression, 11⁄18, makes 152 things, and so for the others.
The examples in the papyrus seem to indicate that there was no definite rule for determining what number would be most convenient to take. Naturally it would be desirable to take a number for which it was easy to find the parts indicated by the given fractions. The number taken is often the largest number whose reciprocal is among them. Thus in many cases the parts are not all whole numbers, but are whole numbers and simple fractions. In Problem 33 one of the parts is[1] 13 1⁄2 1⁄4 1⁄14 1⁄28.
This method of applying an expression to a particular number was continued for many generations and is found in the papyrus of Akhmîm already mentioned. Rodet (1882, page 37) calls the number taken a bloc extractif out of which these fractions are drawn. His idea may be somewhat like that which I have expressed, but his explanation seems a little abstract, and the use of a technical term, while very convenient, makes the process seem more improbable for so ancient a people. Hultsch (1895) has a theory still more formidable. He supposes that the Egyptian introduces an auxiliary unit. In the example above he would say that the Egyptian, for purposes of addition, makes 1⁄105 a new unit, in terms of which the given fractions become whole numbers that can be readily added. Peet (page 18) considers that the question is merely one of notation, that the Egyptian really did have the conception of a fraction with numerator greater than 1 but no notation for
- ↑ The problems in which there is evidence of the use of this method are 7-20, 21-23, 31-34, 36-38, and 76, and the case of 2 divided by 35 in the first table. In Problem 36 this method is used for a division process, and in the case of 2 divided by 35 for a multiplication, but generally it is used for the addition or subtraction of quantities. In all cases in the papyrus the only fractions that appear as parts of the number taken are 2⁄3, 1⁄3, and fractions whose denominators are powers of 2, except in Problem 33.
