A representational analysis of numeration systems.

This article explores the representational structures of numeration systems and the cognitive factors of the representational effect in numerical tasks, focusing on external representations and their interactions with internal representations. Numeration systems are analyzed at four levels: dimensionally, dimensional representations, bases, and symbol representations. The representational properties at these levels affect the processes of numerical tasks in different ways and are responsible for different aspects of the representational effect. This hierarchical structure is also a cognitive taxonomy that can classify nearly all numeration systems that have been invented across the world. Multiplication is selected as an example to demonstrate that complex numerical tasks require the interwoven processing of information distributed across internal and external representations. Finally, a model of distributed numerical cognition is proposed and an answer to the question of why Arabic numerals are so special is provided.