What is the point of a base system? Denotation, computation and the non-integers

As researchers and as laypeople, we inhabit worlds replete with numerals, numerals structured by a base system. We rely on exponentiation and powers of the base to describe these systems. This research habit, along with our familiar experience of numerals as expressing precision, produces a conceptual blind spot regarding their development, structure and purpose. In examining the protracted historical developments of the decimal base system for non-integers-which (i) emerged hundreds of years after the integer positional system, (ii) relied on tools developed in different domains separate from concerns of precise denotation, and (iii) were adopted owing to contingent computational benefits-we see that this blind spot obscures our understanding of base, exponentiation and the non-integers. The usage of exponentiation in the structure of 'base' and positional systems emerged as a result of this extension rather than through the explication of this structure. Although lingua francas are important in research, we must be sensitive to their history, especially when their structure and purpose in contemporary contexts may differ from their historical and developmental contexts.This article is part of the theme issue 'A solid base for scaling up: the structure of numeration systems'.