Sumerian Number System

By about 3000 BC, the Sumerians were drawing images of counters on clay tablets. Each type of goods was represented by a specific symbols, and the symbol was repeated to indicate multiple quantities. Two grain marks represent two units of grain or three oil marks represented three jars of oil. There were two problems with the system:
  • each item counted had to have its own symbol, leading eventually to a proliferation of symbols which had to be memorized
  • the system worked well with small quantities, but large amounts were often represented inaccurately
In the late 4th millennium large temple complexes, like the temple of Inanna at Uruk, became centers of considerable economic activity, dealing in large quantities of goods and labor. To keep up with the increased economy of the temples the first great innovation was to separate the quantity of the good from the symbol for the good so that the representation of three units of grain was, first, a symbol for 3, followed by a symbol for 'grain-unit.'

At the beginning, the symbol for 'three' was determined by the good being measured. While we use the same number signs regardless of what is being measured, the Sumerians had a great number of signs, about 60 of them, grouped into a dozen or so measuring systems. These systems are like the conversions we use, as an example: 12 inches in a foot and 3 feet in a yard. The Sumerian conversion factors, however, were all simple fractions of 60.

In this sexagesimal system, a single object, like sheep or cow or fish, was represented by a small cone. Ten cones equaled one small circle, six small circles equaled one big cone, ten big cones equaled was a big cone with a circle inside it, six of those was a large circle and ten large circles was given by a large circle with a small circle inside. Thus, a large circle with a small one inside represents 10x6x10x6x10 or 36000 base units. Later, when the stylus became the writing instrument of choice the vertical wedge became the circle and the wedge on an angle became the cone. When measuring grain, the conversion factors were 5, 10, 3, and 10, so that the largest unit, a large cone containing a small circle, was worth 10x3x10x5=1500 of the small units.

The culmination of the Sumerian system happened some time in the Ur III period, at the end of the 3rd millennium, when the sexagesimal place value system was developed. The number of signs was reduced to just two: a vertical wedge derived from the small cone often used for the base unit, and a corner wedge, derived from the small circle. The corner wedge had a value of ten vertical wedges. In the sexagesimal counting system described above, the next size unit was the large cone, worth six circles.

In the place value system, this unit was denoted by only 2 signs, the same-sized vertical wedge as the base unit, and it was worth six corner wedges. Now the pair of symbols could be repeated in an indefinitely larger alternating series of corner and vertical wedges, always keeping the same conversion factors of 10 and 6. Unfortunately this meant that a vertical wedge could be equal either 1 or 60 (6x10), or 3600 (60x60), and so on. It's value was determined by its place. The sexagesimal place-value system greatly facilitated calculations, but, eventually the final quanitity had to be translated back into the older system of units.


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