Editing Floor and ceiling functions (section)

==Notation==
The ''integral part'' or ''integer part'' of a number ({{lang|fr|partie entière}} in the original) was first defined in 1798 by [[Adrien-Marie Legendre]] in his proof of the [[Legendre's formula]].

[[Carl Friedrich Gauss]] introduced the square bracket notation {{math|[''x'']}} in his third proof of [[quadratic reciprocity]] (1808).<ref>Lemmermeyer, pp.&nbsp;10, 23.</ref> This remained the standard<ref>e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation. Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's.</ref> in mathematics until [[Kenneth E. Iverson]] introduced, in his 1962 book ''A Programming Language'', the names "floor" and "ceiling" and the corresponding notations {{math|⌊''x''⌋}} and {{math|⌈''x''⌉}}.<ref>Iverson, p.&nbsp;12.</ref><ref>Higham, p.&nbsp;25.</ref> (Iverson used square brackets for a different purpose, the [[Iverson bracket]] notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.

In some sources, boldface or double brackets {{math|⟦''x''⟧}} are used for floor, and reversed brackets {{math|⟧''x''⟦}} or {{math|]''x''[}} for ceiling.<ref>[http://www.mathwords.com/f/floor_function.htm Mathwords: Floor Function].</ref><ref>[http://www.mathwords.com/c/ceiling_function.htm Mathwords: Ceiling Function]</ref>

The [[fractional part]] is the [[sawtooth function]], denoted by {{math|{''x''} }} for real {{mvar|x}} and defined by the formula
:{{math|1={''x''} = ''x'' − ⌊''x''⌋}}<ref>Graham, Knuth, & Patashnik, p.&nbsp;70.</ref>

For all ''x'',
:{{math|1=0 ≤ {''x''} < 1}}.

These characters are provided in Unicode: 

* {{unichar|2308|LEFT CEILING|html=}}
* {{unichar|2309|RIGHT CEILING|html=}}
* {{unichar|230A|LEFT FLOOR|html=}}
* {{unichar|230B|RIGHT FLOOR|html=}}

In the [[LaTeX]] typesetting system, these symbols can be specified with the <code>{{mono|\lceil, \rceil, \lfloor, }}</code> and <code>{{mono|\rfloor}}</code> commands in math mode. LaTeX has supported UTF-8 since 2018, so the Unicode characters can now be used directly.<ref>{{cite web |url=https://www.latex-project.org/news/latex2e-news/ltnews28.pdf |title=LaTeX News, Issue&nbsp;28 |date=Apr 2018 |publisher=The LaTeX Project |format=PDF; 379&nbsp;KB |access-date=2024-07-27}}</ref> Larger versions are<code>{{mono|\left\lceil, \right\rceil, \left\lfloor,}}</code> and <code>{{mono|\right\rfloor}}</code>.