and then doubled to get 1⁄5.[1] The reciprocals of other numbers were sometimes used as multipliers, when the numbers themselves had appeared in previous multiplications that could be transformed into the multiplications desired.[2] In particular, they often used the fact that the reciprocal of a number multiplying the number itself gives 1.
Egyptian division might be described as a second kind of multiplication, where the multiplicand and product were given to find the multiplier.[3] In the first kind of multiplication, the multiplier, being given, can be made up as a combination of the multipliers that were generally used, and the corresponding combination of products would be the required product. When it was the product that was given along with the multiplicand, various multipliers would be tried, 2, 10, and combinations of these numbers, or combinations of the fractions ⅔, ½, and 1⁄10, and from the products thus obtained the Egyptians would endeavor to make up the entire given product.[4] When they succeeded in doing this the corresponding combination of multipliers would be the required multiplier. But they were not always able to get the given product at once in this way, and in such cases the complete solution of the problem involved three steps: (a) multiplications from which selected products would make a sum less than the required product but nearly equal to it; (b) determination of the remainder that must be added to this sum to make the complete product; and (c) determination of the multiplier or multipliers necessary to produce this remainder. The multipliers used in the first and third steps made up the required multiplier. The second step was called completion and will be explained below. For the third step they had
- ↑ Peet, page 19, footnote 1. Examples occur in Problems 21, 22, and 35.
- ↑ Thus in Problem 34, 4 times 1 ½ ¼ is equal to 7, and therefore ½ of 1½ ¼ is equal to ¼. There are some striking illustrations in Problem 38. See also Problems 33 and 70.
- ↑ The usual form of expression was “Operate on (Make, Make the operation on) . . . for the finding of” . . . (Problems 21, 22, 26, 30, 56, etc., see Literal Translation). In the table at the beginning of the papyrus the Egyptian says several times, “Call 2 out of” some number, and this form of expression is used also in Problems 35, 37, 38, 63, 66, and 67. But the point of view is the same, the idea being to get 2 as a result of operating on the number (see Gunn, page 124), and the process is the same as when the former expression is used. In Problems 1-6 and in 65 the author speaks of “making” a certain number of loaves “for” a certain number of men, and in 54 and 55 he uses the expression “Take away a certain area from a certain number of fields,” that is, by taking an equal amount from each field (see page 96), but these are forms of expression used for the statement of the problems and not technical expressions for any kind of division. The expression “Call 2 out of” . . . should be particularly noticed, for we should naturally take the numbers the other way. We say that “3 will go into 6 twice” and we can imagine some one saying “Call 3 out of 6,” but the Egyptian says “Call 6 out of 3.”
- ↑ This happens in Problems 34-38, also in 69. See notes to 34 and 37.
