Zero-based numbering: Difference between revisions

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{{Refimprove|date=March 2010}}

'''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 [[0th0 (number)|''zeroth'']] element, rather than the ''first'' element; ''zeroth'' is a [[word coinage|coined]] [[ordinal number (linguistics)|ordinal number]] corresponding to the number [[0 (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 data structure|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 [[derivative|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. However, just as the ''first derivative'' precedes the ''second derivative'', so also does the ''zeroth derivative'' (or the original function itself) precede the ''first derivative''.

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A third property is that a range is more elegantly expressed as the half-open [[Interval (mathematics)|interval]], [0,''n''), as opposed to the closed interval, [1,''n'']. Empty ranges, which often occur in algorithms, are tricky to express with a closed interval without resorting to obtuse conventions like [1,0]. This half-open convention may avoid [[off-by-one error]]s or [[fencepost error]]s. On the other hand, often the repeat count ''n'' is calculated in advance, making the use of counting from 0 to ''n''−1 (inclusive) less intuitive.

This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest index of ''n'' objects will be {{nowrap|''n'' − 1}} and referred to the ''n''th element. For this reason, the first element is often referred to as the ''[[zeroth]]'' element to avoid confusion.

===Disadvantages===