A Cognitive Taxonomy of Numeration Systems

Jiajie Zhang
Department of Psychology
The Ohio State University
Columbus, OH 43210-1222
zhang@canyon.psy.ohio-state.edu

Donald. A. Norman
Apple Computer, Inc.
1 Infinite Loop, MS 301-3G
Cupertino, CA 95014
dnorman@apple.com

Abstract

In this paper, we study the representational properties of numeration systems. We argue that numeration systems are distributed representations?representations that are distributed across the internal mind and the external environment. We analyze number representations at four levels: dimensionality, dimensional representations, bases, and symbol representations. The representational properties at these four levels determine the representational efficiencies of numeration systems and the performance of numeric tasks. From this hierarchical structure, we derive a cognitive taxonomy that can classify most numeration systems.

Introduction

We all know that Arabic numerals are more efficient than Roman numerals for calculation (e.g., 73 ? 27 is easier than LXXIII ? XXVII), even though they both represent the same entities? numbers. This representational effect, the effect that different representations of a common abstract structure can cause different behaviors, is a cognitive phenomenon. However, early studies of numeration systems focused on their historical and mathematical aspects (e.g., Flegg, 1983; Ifrah, 1987). Recently, the cognitive properties of numeration systems have been analyzed (e.g., Nickerson, 1988; Norman, in press; Zhang, 1992). In this paper, we analyze number representa-

This research was supported by a grant to Donald Norman and Edwin Hutchins from the Ames Research Center of the National Aeronautics & Space Agency, Grant NCC 2-591 in the Aviation Safety/Automation Program, technical monitor, Everett Palmer. Additional support was provided by funds from Apple Computer Company and Digital Equipment Corporation to the Affiliates of Cognitive Science at UCSD.

tions under different numeration systems. From this analysis, we derive a cognitive taxonomy that can classify most of the numeration systems that have been invented across the world. This taxonomy is the basis for the study of the representational effect of numeration systems.

Dimensionality of Numeration Systems

1 D Systems

Numeration systems can be analyzed in terms of their dimensions. One of the simplest ways to represent numbers is to use stones: one stone for one, two stones for two, and so on. This StoneCounting system has only one dimension: the quantity of stones. The Body-Counting system used by Torres islanders is another one dimensional system, where the single dimension is represented by the positions of different body parts (e.g., fingers, wrists, etc.). One dimensional systems are denoted as 1 D in this paper.

1?1 D Systems

Many numeration systems have two dimensions: one base dimension and one power dimension. The power dimension decomposes a number into hierarchical groups on a base. The Arabic system is a two dimensional system (Table 1) with a base dimension represented by the shapes of the ten digits (0, 1, 2, ..., 9) and a power dimension represented by positions of the digits with a base ten. For example, the middle 4 in 447 has value 4 on the base dimension and position 1 (counting from the rightmost digit, starting from zero) on the power dimension. The actual value it represents is forty (the product of its values on the