Page:The Rhind Mathematical Papyrus, Volume I.pdf/21

Rh and then doubled to get $1/undefined$. 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. 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. 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 $1/undefined$, and from the products thus obtained the Egyptians would endeavor to make up the entire given product. 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