The Egyptian mathematical system was decimal and expressed any number by combining the hieroglyphic signs for the numbers 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000. The system (with some exceptions, Roero 1994: 32) was additive: the number 7, for instance, was indicated by seven signs each meaning 1; the number 40 by four signs each meaning 10; a number such as 253 was expressed by two signs indicating 100, plus five indicating 10, plus three indicating 1; and so on.
Beside integers (whole numbers), the Egyptians made large use of unit fractions, that is, fractions with 1 as a nominator and any number as a denominator (with the exception of |). However, they would never use the same fraction twice, that is, 2 would never be expressed as I +1, but preferred strings of unit fractions listed in decreasing order: |, for instance, may be expressed as 1 + yj, but also as + Y17 + 210 or as 1 +1 + yy0 and by six more combinations of three unit fractions each (cf. Gillings 1972: 53-4). The scribes took full advantage of the existence of more than one combination and generally chose the most ‘‘convenient’’: for instance, even numbers were definitely preferred to odd numbers, and short sequences were favoured over long series of unit fractions (Gillings 1972: 49). The most convenient results might be listed in tables and thus made available to the scribes who had to perform quick calculations.
The general preference for even numbers finds a clear explanation in the mechanism underlying the majority of ancient Egyptian mathematical operations: halving, doubling, taking 2/3 of a quantity, multiplying and dividing by 10 were basic actions of many mathematical procedures. The list of the results of 2 divided by the odd numbers from 3 to 101 (which occupies the entire recto of the Rhind papyrus) corresponds, in fact, to the doubling of unit fractions with odd denominators, that clearly posed more problems to the scribes than any even number (Clagett 1999: 18; Ritter 2000: 126-7 and 129).
Problem texts provided solutions for specific cases, but could also be consulted as general examples on how to approach similar issues (Clagett 1999: 94). Starting from the data, the scribes solved these problems by following a defined sequence of steps, in which each result was employed in the following operation (Imhausen 2003). The study of their algorithmic structure helps understanding similarities and differences with the later development of Egyptian mathematics (Imhausen 2002: 158-9): whereas some aspects clearly evolved over the centuries (Silverman 1975), others remained a constant down to the Graeco-Roman Period (Rihll, ch. 22).
World History
