Editing Zero-based numbering (section)

=== Origin ===
[[Martin Richards (computer scientist)|Martin Richards]], creator of the [[BCPL]] language (a precursor of [[C (programming language)|C]]), designed arrays initiating at 0 as the natural position to start accessing the array contents in the language, since the value of a [[pointer (computer programming)|pointer]] ''p'' used as an address accesses the position {{nowrap|''p'' + 0}} in memory.<ref>{{cite book |url=http://cm.bell-labs.com/cm/cs/who/dmr/bcpl.pdf |title=The BCPL Reference Manual |publisher=Massachusetts Institute of Technology |author=Martin Richards |year=1967 |page=11}}</ref><ref name="mikehoye">{{cite web |url=http://exple.tive.org/blarg/2013/10/22/ |title=Cita{{not a typo|tion Nee}}ded |access-date=28 January 2014 |author=Mike Hoye}}</ref> BCPL was first compiled for the [[IBM&nbsp;7094]]; the language introduced no [[Run time (program lifecycle phase)|run-time]] [[indirection lookup]]s, so the indirection optimization provided by these arrays was done at compile time.<ref name="mikehoye"/> The optimization was nevertheless important.<ref name="mikehoye"/><ref>{{cite web |url=https://www.multicians.org/thvv/7094.html |title=The IBM 7094 and CTSS |date=1995 |access-date=28 January 2014 |author=Tom Van Vleck}}</ref>

In 1982 [[Edsger W. Dijkstra]] in his pertinent note ''Why numbering should start at zero''<ref name="dijkstra"/> argued that arrays subscripts should start at zero as the latter being the most [[natural number]]. Discussing possible designs of array ranges by enclosing them in a chained inequality, combining sharp and standard inequalities to four possibilities, demonstrating that to his conviction zero-based arrays are best represented by non-overlapping index ranges, which start at zero, alluding to [[Interval (mathematics)#Definitions|open, half-open and closed intervals]] as with the real numbers. Dijkstra's criteria for preferring this convention are in detail that it represents empty sequences in a more natural way {{nowrap|(''a'' ≤ ''i'' < ''a'' ?)}} than closed "intervals" ({{nowrap|''a'' ≤ ''i'' ≤ (''a'' − 1) ?}}), and that with half-open "intervals" of naturals, the length of a sub-sequence equals the upper minus the lower bound ({{nowrap|''a'' ≤ ''i'' < ''b''}} gives {{nowrap|(''b'' − ''a'')}} possible values for ''i'', with ''a'', ''b'', ''i'' all integers).