Bases, positions, and computations

Ancient sources attest to the introduction of two families of numeration systems using place-value notations. In such systems, the base and its powers are always represented using position. The existence of a base of this kind is reflected by the repetitive character of the algorithms drawing on the place-value notations to execute operations—specifically, multiplicative operations. The key point for which this article argues is that these two families of place-value numeration systems are of a fundamentally different nature, and that the nature of each is revealed by the way algorithms executing multiplicative operations rely on the base. The algorithms associated with Old-Babylonian sexagesimal place-value notations brought into play the number-theoretical properties of the base (its divisibility properties, if you will) as well as features of the sequence of digits writing the number that the choice of the base made prominent (like the terminal digits of its representation and the related divisibility properties of the number). Other types of computation are independent of the value of the base and could be applied to numbers written using any base. The computations’ iterative character is thus different. To compute a reciprocal, scribes in Old-Babylonian schools relied upon the last digits of the sexagesimal expansion of a number to identify regular divisors (not always written with a single digit) and multiply by their inverse; iterating this step yielded the result, factor by factor. In Chinese sources, by contrast, multiplications and divisions using decimal place-value notation yielded quotients, digit by digit. The part played by the base, and hence its meaning, differed significantly.