One of the great challenges in the history of science is avoiding anachronism, which notation and computational techniques can so easily conceal. Fractions are a case in point. The modern notation a/b combines multiplication and its inverse operation, division, in a single expression. The upper world—the numerator—is the realm of multipliers, while the lower world—the denominator—is the realm of divisors.
Since multiplication and division were not formally defined in Mesopotamia or Egypt, attributing to these cultures a notation for fractions consisting of a numerator and a denominator is wholly anachronistic. Nevertheless, fractional parts of measurements—volumes among the Sumerians and lengths among the Egyptians—were commonly handled in everyday problems. The Egyptian method for calculating these fractions has also been the subject of numerous historical studies.
Fractions of history --------------------
Mesopotamian science preceded Egyptian science by several centuries and was far more advanced in both computational techniques and abstraction. Without the aid of our modern notation, the Sumerians could solve linear and quadratic equations. While this nation of traders was drawn to calculation by the demands of accounting, Egyptian farmers were more concerned with geometry, since the land register had to be re-established after every flooding of the Nile. Assisted by surveyors, the scribes had to recalculate the areas of land covered by the precious silt so that they could be redistributed among the landowners.
The study of numbers—and mathematics more generally—was certainly introduced into Egypt by the Mesopotamians and regularly developed by them. Yet despite the extensive contact between the two civilizations, each seems to have retained its own numerical culture. The Egyptians used a decimal system without positional notation. Each power of 10 was represented by a particular symbol, repeated as many times as necessary to express the desired number. The earliest evidence of Egyptian computational techniques is found in the famous Rhind papyrus (see box).