Counting, Computing, and the Representation of Numbers

Abstract

How easy it is to manipulate numbers depends in part on how they are represented visually. In this paper several ancient systems for representing numbers are compared with the Arabic system, which is used throughout the world today. It is suggested that the Arabic system is a superior vehicle for computing, largely because of the compactness and extensibility of its notation, and that these features have been bought at the cost of greater abstractness. Numbers in the Arabic system bear a less obvious relationship to the quantities they represent than do numbers in many earlier systems. Moreover, the elementary arithmetic operations of addition and subtraction are also more abstract; in some of the earlier systems the addition of two numbers is similar in an obvious way to the addition of two sets of objects, and the correspondence between subtraction with numbers and the subtraction of one set of objects from another is also relatively direct. The greater abstractness of the Arabic system may make it somewhat more difficult to learn and may obscure the basis for such elementary arithmetic operations as carrying and borrowing. The power of the system lies in the fact that once it has been learned, it is the most efficient of any system yet developed for representing and manipulating quantities of all magnitudes.