EGYPTIAN ARITHMETIC
Numbers and Operations
We have very few traces of Egyptian arithmetic from a date earlier than the original sources of the Rhind papyrus. But before this date there was a long period of Egyptian civilization, and we may well believe that for more than a thousand years there had been a slow development of the elaborate system before us.
The Egyptians apparently conceived of two kinds of numbers, an ascending series from 1 to 1,000,000 of numbers that we call integers, and a corresponding descending series consisting of ⅔ and reciprocal numbers or unit fractions.[1]
For integers they had
1. A well-defined decimal notation, without, however, the positional device that renders our modern notation so simple;
2. A thorough understanding of the four arithmetical operations,—addition, subtraction, multiplication, and division.
Addition and subtraction were easily accomplished. Direct multiplication by integers was generally confined to the multipliers 2 and 10; that is, in cases where the process was explicit, and it may be presumed that in all cases[2] they multiplied by repeated doubling, or occasionally multiplying by 10, and adding the products formed from multipliers that would make up the given multiplier. This is not very different from our modern way, where we multiply by the unit figure of our multiplier, and then by the ten figure, and so on, and then add the re-
- ↑ These are the same as our fractions with numerator 1, and it will be convenient, for the most part, to speak of them as fractions. They are sometimes called fundamental fractions (Stammbrüchen). When the Egyptians wish to express a number that is not a single unit fraction, they use a combination of such fractions. Thus for our 13⁄4, they say 1 1⁄2 1⁄4; for 2 times 1⁄5, 1⁄3 1⁄15, and so on. Such an expression we may speak of as a fractional number or fractional expression; it was understood that the fractions, or the whole number and fractions, were to be added, and I shall always write them as do the Egyptians without any sign of addition, just as we write 1½. A given number could be expressed as a sum of unit fractions, or as the sum of a whole number and such fractions, in an infinite number of ways, and sometimes the expression for a given number is varied, but the fractions of an expression must all be different fractions, and very rarely were they allowed to have a sum equal to or greater than 1.
- ↑ Many times the details of the multiplication are omitted in the papyrus and we cannot say with certainty what was done in every case, but it is fair to make inferences if we take into account all of the cases in which the details are given.
