Bender and Beller
report that a mixed number system superimposing three binary steps onto a decimal structure was invented long before Leibniz’s description of binary numerals, for easier calculations to measure mixed quantities of yams and other garden products, on the Pacific island of Mangareva. It did not arise on other islands of the Pacific with similar ecologies.
...the mixed counting system in Mangarevan...is unique in that it had three binary steps superposed onto a decimal structure. In showing how these steps affect calculation, our analysis yields important insights for theorizing on numerical cognition: counting systems serve as complex cultural tools for numerical cognition, apparently unwieldy systems may in fact be cognitively advantageous, and such advantageous systems can be—and have been—developed by nonindustrialized societies and in the absence of notational systems. These insights also help to dismiss simple notions of cultural complexity as a homogenous state and emphasize that investigating cultural diversity is not merely an optional extra, but a must.
From their conclusions:
Summarizing our results, it seems clear that superposing a decimal system with binary steps is indeed advantageous, as described by Leibniz because it allows arduous calculations and retrieval of addition facts to be replaced by simple transformations in the binary range. Crucially, these advantages are not vitiated by the downside of longer number representations also anticipated by Leibniz. The decimal basis of the system guarantees that number representations remain relatively compact. The mixed system thus neutralizes the tradeoff associated with base size by combining the benefits of a small base (fewer or no addition facts) with the benefits of a larger base (compact representation). Actually, both of these implications adjust to the constraints on working memory and thereby benefit mental arithmetic: the more compact representation relieves cognitive load in retaining information, and the reduction in addition facts relieves cognitive load in processing. The main cost inflicted by this mixing is an increase in irregularity, which requires additional lexemes and rules and thus affects the ease with which a system is learned and mastered....Apparently, its users favored the benefits of the mixed system over the regularity of the general decimal system, as they developed the mixed system out of the purely decimal system in wide use across Polynesia.
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