Round Numbers revisited. A new fuzzy approach

A round number was characterized by Hardy and Ramanujan as being one that has a considerable number of comparatively small prime factors. This description contains words such as ‘considerable’ and ‘comparatively small’ that have vague meaning. This paper deals with fuzzy models for grasping the imprecision attached to these words. The paper starts by introducing Hardy and Ramanujan’s measure of roundness, which is based on the number of prime factors into which a natural number is decomposed, as well as Goguen’s fuzzy logic approach based on the analysis of the terms ‘considerable’ and ‘comparatively small’. The proposal by Trillas et al. for supplying an order of roundness is also discussed. Next, two new models for representing ‘considerable’ and ‘comparatively small’ terms are provided. These models grasp certain properties attributed by Hardy and Ramanujan to round numbers better than the Goguen’s model, as they are very rare. Finally, a model for defining a more adequate order of roundness than that of Trillas et al. is provided.

keywords: Round number, Fuzzy models, Order