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. This article argues that these two families of place-value numeration systems are of a fundamentally different nature, and that their respective nature 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) and features of the sequence of digits writing the number that the choice of the base made prominent (like the last digits of its representation). Other types of computation were 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 and multiply by their inverse; the result was yielded factor by factor. By contrast, multiplications and divisions using decimal place-value notation in Chinese sources dealt with numbers digit by digit. The part played by the base, and hence its meaning, differed deeply. This article is part of the theme issue ‘A solid base for scaling up: the structure of numeration systems’.