Editing Zero-based numbering (section)

=== Numerical properties ===
With zero-based numbering, a range can be expressed as the half-open [[Interval (mathematics)|interval]], {{math|[0, ''n'')}}, as opposed to the closed interval, {{math|[1, ''n'']}}. Empty ranges, which often occur in algorithms, are tricky to express with a closed interval without resorting to obtuse conventions like {{math|[1, 0]}}. Because of this property, zero-based indexing potentially reduces [[Off-by-one error|off-by-one]] and [[fencepost error]]s.<ref name="dijkstra">{{cite web |url=https://www.cs.utexas.edu/users/EWD/transcriptions/EWD08xx/EWD831.html |title=Why numbering should start at zero (EWD 831) |last=Dijkstra |first=Edsger Wybe |author-link=Edsger W. Dijkstra |date=May 2, 2008 |work=E. W. Dijkstra Archive |publisher=[[University of Texas at Austin]] |access-date=2011-03-16}}</ref> On the other hand, the repeat count {{mvar|n}} is calculated in advance, making the use of counting from 0 to {{math|''n'' − 1}} (inclusive) less intuitive. Some authors prefer one-based indexing, as it corresponds more closely to how entities are indexed in other contexts.<ref>Programming Microsoft® Visual C# 2005 by Donis Marshall.</ref>

Another property of this convention is in the use of [[modular arithmetic]] as implemented in modern computers. Usually, the [[modulo operation|modulo function]] maps any integer modulo {{mvar|N}} to one of the numbers {{math|0, 1, 2, ..., ''N'' − 1}}, where {{math|''N'' ≥ 1}}. Because of this, many formulas in algorithms (such as that for calculating hash table indices) can be elegantly expressed in code using the modulo operation when array indices start at zero.

Pointer operations can also be expressed more elegantly on a zero-based index due to the underlying address/offset logic mentioned above. To illustrate, suppose {{mvar|a}} is the [[memory address]] of the first element of an array, and {{mvar|i}} is the index of the desired element. To compute the address of the desired element, if the index numbers count from 1, the desired address is computed by this expression:

: <math>a + s \times (i-1),</math>

where {{mvar|s}} is the size of each element. In contrast, if the index numbers count from 0, the expression becomes

: <math>a + s \times i.</math>

This simpler expression is more efficient to compute at [[run time (program lifecycle phase)|run time]].

However, a language wishing to index arrays from 1 could adopt the convention that every array address is represented by {{math|1=''a''′ = ''a'' – ''s''}}; that is, rather than using the address of the first array element, such a language would use the address of a fictitious element located immediately before the first actual element. The indexing expression for a 1-based index would then be

: <math>a' + s \times i.</math>

Hence, the efficiency benefit at run time of zero-based indexing is not inherent, but is an artifact of the decision to represent an array with the address of its first element rather than the address of the fictitious zeroth element. However, the address of that fictitious element could very well be the address of some other item in memory not related to the array.

Superficially, the fictitious element doesn't scale well to multidimensional arrays. Indexing multidimensional arrays from zero makes a naive (contiguous) conversion to a linear address space (systematically varying one index after the other) look simpler than when indexing from one. For instance, when mapping the three-dimensional array {{math|A[''P''][''N''][''M'']}} to a linear array {{math|L[''M⋅N⋅P'']}}, both with {{mvar|M ⋅ N ⋅ P}} elements, the index {{mvar|r}} in the linear array to access a specific element with {{math|L[''r''] {{=}} A[''z''][''y''][''x'']}} in zero-based indexing, i.e. {{math|[0 ≤ ''x'' < ''P'']}}, {{math|[0 ≤ ''y'' < ''N'']}}, {{math|[0 ≤ ''z'' < ''M'']}}, and {{math|[0 ≤ ''r'' < ''M ⋅ N ⋅ P'']}}, is calculated by 
:<math>r = z \cdot M \cdot N + y \cdot M + x.</math>
Organizing all arrays with 1-based indices ({{math|[1 ≤ ''x′'' ≤ ''P'']}}, {{math|[1 ≤ ''y′'' ≤ ''N'']}}, {{math|[1 ≤ ''z′'' ≤ ''M'']}}, {{math|[1 ≤ ''r′'' ≤ ''M ⋅ N ⋅ P'']}}), and assuming an analogous arrangement of the elements, gives 
:<math>r' = (z'-1) \cdot M \cdot N + (y'-1) \cdot M + (x'-0)</math>
to access the same element, which arguably looks more complicated. Of course, {{math|''r''′ {{=}} ''r'' + 1,}} since {{math|[''z'' {{=}} ''z''′ – 1],}} {{math|[''y'' {{=}} ''y''′ – 1],}} and {{math|[''x'' {{=}} ''x''′ – 1].}} A simple and everyday-life example is [[positional notation]], which the invention of the zero made possible. In positional notation, tens, hundreds, thousands and all other digits start with zero, only units start at one.<ref>{{cite AV media | url=https://www.khanacademy.org/math/cc-1st-grade-math/cc-1st-place-value | title= Math 1st Grade / Place Value / Number grid | quote=Youtube title: Number grid / Counting / Early Math / Khan Academy | publisher=Khan Academy | author = Sal Khan | access-date = July 28, 2018}}.</ref>

<div><ul>
<li style="display: inline-table;">
{| class="wikitable"
|+ ''Zero''-based indices
|- 
! {{diagonal split header|{{mvar|y}} | {{mvar|x}}}} !! 0 !! 1 !! 2 !! .. !! {{tmath|1=x = x' - 1}} !! .. !! 8 !! 9 
|-
! scope="row" | 0
| {{gray|0}}0 || {{gray|0}}1 || {{gray|0}}2 || || || || {{gray|0}}8 || {{gray|0}}9 
|-
! scope="row" | 1
| 10 || 11 || 12 || || || || 18 || 19 
|-
! scope="row" | 2
| 20 || 21 || 22 || || || || 28 || 29 
|-
! scope="row" | ..
|  ||  ||  || || || || ||  
|-
! scope="row" | {{tmath|1=y = y' - 1}}
|  ||  ||  || || {{tmath|1=y \cdot M + x}} || || ||  
|-
! scope="row" | ..
|  ||  ||  || || || || ||  
|-
! scope="row" | 8
| 80 || 81 || 82 || || || || 88 || 89 
|-
! scope="row" | 9
| 90 || 91 || 92 || || || || 98 || 99
|-
| colspan="9" style="text-align: center;"| The table content represents the index {{mvar|r}}.
|} </li>&emsp;
<li style="display: inline-table;">
{| class="wikitable"|
|+ ''One''-based indices
|-
! {{diagonal split header|{{mvar|y'}} | {{mvar|x'}}}} !! 1 !! 2 !! 3 !! .. !! {{tmath|1=x' = x + 1}} !! .. !! 9 !! 10
|-
! scope="row" | 1
| {{gray|0}}1 || {{gray|0}}2 || {{gray|0}}3 || || || ||  {{gray|0}}9 || 10
|-
! scope="row" | 2
| 11 || 12 || 13 || || || || 19 || 20
|-
! scope="row" | 3
|  21 || 22 || 23 || || || || 29 || 30 
|-
! scope="row" | ..
|  ||  ||  || || || || || 
|-
! scope="row" | {{tmath|1=y' = y + 1}}
|  ||  ||  || || {{tmath|1=(y'-1) \cdot M + x'}} || || || 
|-
! scope="row" | ..
|  ||  ||  || || || || || 
|-
! scope="row" | 9
| 81 || 82 || 83 || || || || 89 || 90
|-
! scope="row" | 10
| 91 || 92 || 93 || || || || 99 || 100
|-
| colspan="9" style="text-align: center;"| The table content represents the index {{mvar|r′}}.
|}</li>
</ul></div>

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 {{mvar|n}} objects will be {{math|''n'' − 1}}, and it refers to the {{mvar|n}}th element. For this reason, the first element is sometimes referred to as the [[array data structure|zeroth]] element, in an attempt to avoid confusion.