Magazine article UNESCO Courier

Prime Numbers

Magazine article UNESCO Courier

Prime Numbers

Article excerpt

MATHEMATICS and writing have a close, symbiotic relationship. Indeed, recent archaeological discoveries have shown that it was the need t6 measure, divide and distribute the material wealth of societies that gave birth to the first writing systems.

For a society to develop a mathematics that goes beyond simple counting, a material support of some kind is essential. Without writing, the limitations of human memory are such that only a certain degree of numerical sophistication can be achieved.

By allowing us to follow the development of two writing systems, one in southern Mesopotamia towards the middle of the fourth millennium BC, and the other in the area around Susa in Iran, slightly later, archaeological discoveries in the last few decades have shown that the inverse is equally true. For a society to develop writing, material needs and, in particular, the need for record-keeping, are central.

In these societies the material support was Clay, which is virtually indestructible, and the first documents are accounts. Cuneiform (wedge-shaped) Mesopotamian writing in particular was to know great success over the succeeding 3,000 years. Used to write not only the original Sumerian and Akkadian, but also Hittite, Elamite, Hurrian and many other languages of the ancient Near East, it died out only at the beginning of our era.

Meanwhile, an independent civilization developed rapidly in Egypt towards the end of the fourth millennium. Here the situation concerning writing is less clear. First of all, the material support, for other than monumental inscriptions, was principally papyrus, the reed-like plant growing along the Nile and in the Delta, and, to a lesser extent, other perishable materials. Egypt has thus yielded fewer documents than Mesopotamia by a factor of thousands.

Number systems The third millennium marks, for both the Mesopotamian and the Egyptian civilizations, the gradual emergence of an abstract concept of number. Originally each number is attached to a given system of units. The "four" of "four sheep" and "four measures of grain" are not, for example, written with the same symbol.

Equally, the different systems of units are not connected among themselves. Measures of area, for instance, have no simple relationship to measures of length since the connection between the two-that area could be calculated from the product of a length and a width-was not yet operational.

But the very practice of writing things down, yielding permanent records of measures, opens up the possibility of observing regularities and patterns. This possibility was taken advantage of in these two societies over a period of about 1,000 years so that by the end of the third millennium Egyptian and Sumerian scribes had learned how to calculate areas and volumes from lengths, how to divide rations among workers, how to calculate the time necessary for a given ob of work from volumes, numbers of men, and work rates. Next our evidence shows how a new level of abstraction was reached in which the concept of number became more and more detached from its metrological context.

By the beginning of the second millennium both civilizations had succeeded in developing equally abstract systems of numeration, though they had chosen different paths to represent their numbers. The Egyptians, like the vast majority of modern societies, had a written number system based on ten; that is, one counts up to nine of each unit before moving up to the next higher unit-after nine "ones" comes a "ten", after nine "tens" comes a "hundred", etc. But, unlike modern systems, their writing of numbers is additive", that is there are separate signs for units, tens, and hundreds, and these are repeated as necessary.

The Mesopotamians used a base of sixty for their mathematical calculations, and they developed the first known system of position. The signs for numbers repeat after fifty-nine, the actual value being determined by the position of the digit in the number as a whole. …

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