Abstract: To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar — a claim that is widely disputed. An analysis of our models’ elementary processing steps reveals intricate trade-offs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on Roman numerals. While symbolic computation with Roman numerals requires fewer internal resources than with Arabic ones, the large number of needed symbols inflates the number of external processing steps.
Keywords: Numeral systems, mental arithmetic, algorithms, mathematical practice, representation, notation, immediate interactive behavior (IIB)
Related: The cognitive basis of arithmetic | Interactive addition | Thinking by doing | Interactive coin addition | Arithmetic with Arabic vs. Roman numerals | Immediate interactive behavior | A taxonomy of practical vs. theoretical actions