All mathematical procedures seem to have been based on the underlying processes of addition and subtraction. Multiplication was carried out by adding number to itself the requisite number of times, and division consisted of subtracting a number until an indivisible remainder was left.Multiplication tables do not seem to have been used, although multiplica-tion or division by ten was a simple and standard process.
Procedures for dealing with fractions were based on the addition and subtraction of unit fraction. With the exception of the special cases of ⅔ and very rarely ¾, the Egyptians did not use multiples of fractions, but only the single unit fraction. For example, 1/5 was written as r5 "the fifth part", and 1/6 as r6 "the sixth part". Where modern mathematics would write 11/30, Egyptians would write r5 r6 – that is, 1/5 + 1/6.
The surviving mathematical papyri show only calculations that would have been used in practical applications – for example, methods for calculating the areas and volumes of a variety of shapes, including triangles and cylinders. (This is not to say that the Egyptians had no concept of numbers in the abstract, simply that there is no surviving evidence for this.) An Egyptian would calculate how long it would take so many men to build a brick ramp because he expected to supervise me building ramps. In on text, an army scribe, Amenemope, is challenged to calculate the number of men needed to transport an obelisk of given size from the quarries, to erect a colossus in a given time, calculate the rations necessary to feed men digging a lake, and to arrange stores for a major military expedition to Syria.
There is an evident concern in the papyri with formulae and ratios for practical application. A ratio pesu was used as a measure to define the quantities of bread or beer made from a single unit of grain. The term seked defines the slope of a pyramid as a ratio of lateral displacement against rise. There was a formula for calculating the area of a circle, on the basis of taking the diameter, subtracting 1/9, and squaring the result. A circular granary nine cubits in diameter is theus calculated to have a base area of 64 square cubits, instead of the correct answer, 63.64 square cubits.
Unlike the Greeks, the Egyptians seem not to have been interested in producing proofs of mathematical formulae, but they do show a highly sophisticated manipulation of numbers. In the context of applied accounting – that is, the accurate measurement of commodities, areas and volumes absolutely – minor errors were not significant. It is, in fact, fairly typical of surviving accounts papyri dealing with consumables to include errors in arithmetic, with totals that show a minor shrinkage in the sum of the parts, to the advantage of the accountant .
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