Zero-based Numbering

Zero-based numbering is numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday circumstances. Under zero-based numbering, the initial element is sometimes termed the zeroth element, rather than the first element; zeroth is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth element. There is not wide agreement regarding the correctness of using zero as an ordinal (nor regarding use of the term zeroth) as it creates ambiguity for all subsequent elements of the sequence when lacking context.

Numbering sequences starting at 0 is quite common in mathematics, in particular in combinatorics. In computer science, array indices also often start at 0, so computer programmers might use zeroth in situations where others might use first, and so forth. In some mathematical contexts, zero-based numbering can be used without confusion, when ordinal forms have well established meaning with an obvious candidate to come before first; for instance a zeroth derivative of a function is the function itself, obtained by differentiating zero times. Such usage corresponds to naming an element not properly belonging to the sequence but preceding it: the zeroth derivative is not really a derivative at all.