Editing Floor and ceiling functions (section)

{{Short description|Nearest integers from a number}}
{{Use dmy dates|date=May 2023}}
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}}

In [[mathematics]], the '''floor function''' is the [[function (mathematics)|function]] that takes as input a [[real number]] {{mvar|x}}, and gives as output the greatest [[integer]] less than or equal to {{mvar|x}}, denoted {{math|⌊''x''⌋}} or {{math|floor(''x'')}}.  Similarly, the '''ceiling function''' maps {{mvar|x}} to the least integer greater than or equal to {{math|''x''}}, denoted {{math|⌈''x''⌉}} or {{math|ceil(''x'')}}.<ref>Graham, Knuth, & Patashnik, Ch. 3.1</ref>

For example, for floor: {{math|⌊2.4⌋ {{=}} 2}}, {{math|⌊&minus;2.4⌋ {{=}} &minus;3}}, and for ceiling: {{math|⌈2.4⌉ {{=}} 3}}, and {{math|⌈&minus;2.4⌉ {{=}} &minus;2}}.  

The floor of {{mvar|x}} is also called the '''integral part''', '''integer part''', '''greatest integer''', or '''entier''' of {{mvar|x}}, and was historically denoted {{math|[''x'']}} (among other notations).<ref>
1) Luke Heaton, ''A Brief History of Mathematical Thought'', 2015, {{isbn|1472117158}} (n.p.)
<br/>2) Albert A. Blank ''et al.'', ''Calculus: Differential Calculus'', 1968, p. 259
<br/>3) John W. Warris, Horst Stocker, ''Handbook of mathematics and computational science'', 1998, {{isbn|0387947469}}, p. 151
</ref> However, the same term, ''integer part'', is also used for [[truncation]] towards zero, which differs from the floor function for negative numbers.

For an integer {{mvar|n}}, {{math|⌊''n''⌋ {{=}} ⌈''n''⌉ {{=}} ''n''}}.

Although {{math|floor(''x'' + 1)}}  and {{math|ceil(''x'')}} produce graphs that appear exactly alike, they are not the same when the value of {{mvar|x}} is an exact integer. For example, when {{math|''x'' {{=}} 2.0001}}, {{math|⌊2.0001 + 1⌋ {{=}} ⌈2.0001⌉ {{=}} 3}}. However, if {{math|''x'' {{=}} 2}}, then {{math|⌊2 + 1⌋ {{=}} 3}}, while  {{math|⌈2⌉ {{=}} 2}}. 

{| class="wikitable" title
|+Examples
! ''x''
! Floor {{math|⌊''x''⌋}}
! Ceiling {{math|⌈''x''⌉}}
! [[Fractional part]] {{math|{''x''} }}
|-
!  2
|   2
| 2
| 0
|-
!  2.0001
|   2
| 3
| 0.0001
|-
!  2.4
|   2
| 3
| 0.4
|-
! 2.9
|   2
| 3
| 0.9
|-
!  2.999
|   2
| 3
| 0.999
|-
! &minus;2.7
| &minus;3
| &minus;2
| 0.3
|-
! &minus;2
| &minus;2
| &minus;2
| 0
|}