Editing Floor and ceiling functions

{{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
|}

==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>.

==Definition and properties==
Given real numbers ''x'' and ''y'', integers ''m'' and ''n'' and the set of [[integer]]s <math>\mathbb{Z}</math>, floor and ceiling may be defined by the equations

:<math> \lfloor x \rfloor=\max \{m\in\mathbb{Z}\mid m\le x\},</math>

:<math> \lceil x \rceil=\min \{n\in\mathbb{Z}\mid n\ge x\}.</math>

Since there is exactly one integer in a [[half-open interval]] of length one, for any real number ''x'', there are unique integers ''m'' and ''n'' satisfying the equation
:<math>x-1<m\le x \le n <x+1.</math>

where <math>\lfloor x \rfloor = m</math>&nbsp;and <math>\lceil x \rceil = n</math>&nbsp;may also be taken as the definition of floor and ceiling.

===Equivalences===
These formulas can be used to simplify expressions involving floors and ceilings.<ref>Graham, Knuth, & Patashink, Ch. 3</ref>

:<math>
\begin{alignat}{3}
\lfloor x \rfloor &= m \ \ &&\mbox{ if and only if } &m &\le x < m+1,\\
\lceil x \rceil &= n     &&\mbox{ if and only if } &\ \ n -1 &< x \le n,\\

\lfloor x \rfloor &= m   &&\mbox{ if and only if } &x-1 &< m \le x,\\
\lceil x \rceil &= n    &&\mbox{ if and only if }  &x &\le n < x+1.
\end{alignat}
</math>

In the language of [[order theory]], the floor function is a [[residuated mapping]], that is, part of a [[Galois connection]]: it is the upper adjoint of the function that embeds the integers into the reals.

:<math>
\begin{align}
x<n &\;\;\mbox{ if and only if } &\lfloor x \rfloor &< n, \\
n<x &\;\;\mbox{ if and only if } &n &< \lceil x \rceil, \\
x\le n &\;\;\mbox{ if and only if } &\lceil  x \rceil &\le n, \\
n\le x &\;\;\mbox{ if and only if } &n &\le \lfloor x \rfloor.
\end{align}
</math>

These formulas show how adding an integer {{mvar|n}} to the arguments affects the functions:

:<math>
\begin{align}
\lfloor x+n \rfloor &= \lfloor x \rfloor+n,\\
\lceil x+n \rceil &= \lceil x \rceil+n,\\
\{ x+n \} &= \{ x \}.
\end{align}
</math>

The above are never true if {{mvar|n}} is not an integer; however, for every {{mvar|x}} and {{mvar|y}}, the following inequalities hold:

:<math>\begin{align}
\lfloor x \rfloor + \lfloor y \rfloor &\leq \lfloor x + y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor + 1,\\[3mu]
\lceil x \rceil + \lceil y \rceil -1 &\leq \lceil x + y \rceil \leq \lceil x \rceil + \lceil y \rceil.
\end{align}</math>

=== Monotonicity ===
Both floor and ceiling functions are [[Monotonic function|monotonically non-decreasing functions]]:

:<math>
\begin{align}
x_{1} \le x_{2} &\Rightarrow \lfloor x_{1} \rfloor \le \lfloor x_{2} \rfloor, \\
x_{1} \le x_{2} &\Rightarrow \lceil  x_{1} \rceil  \le \lceil  x_{2} \rceil.  
\end{align}
</math>

===Relations among the functions===
It is clear from the definitions that
:<math>\lfloor x \rfloor \le \lceil x \rceil,</math>  with equality if and only if ''x'' is an integer, i.e.
:<math>\lceil x \rceil - \lfloor x \rfloor = \begin{cases}
0&\mbox{ if } x\in \mathbb{Z}\\
1&\mbox{ if } x\not\in \mathbb{Z}
\end{cases}</math>

In fact, for integers ''n'', both floor and ceiling functions are the [[identity function|identity]]:
:<math>\lfloor n \rfloor = \lceil n \rceil = n.</math>

Negating the argument switches floor and ceiling and changes the sign:
:<math> \begin{align}
\lfloor x \rfloor +\lceil -x \rceil &= 0 \\
-\lfloor x \rfloor &= \lceil  -x \rceil \\
-\lceil  x \rceil  &= \lfloor -x \rfloor
\end{align}
</math>

and:
:<math>\lfloor x \rfloor + \lfloor -x \rfloor = \begin{cases}
0&\text{if } x\in \mathbb{Z}\\
-1&\text{if } x\not\in \mathbb{Z},
\end{cases}</math>

:<math>\lceil x \rceil + \lceil -x \rceil = \begin{cases}
0&\text{if } x\in \mathbb{Z}\\
1&\text{if } x\not\in \mathbb{Z}.
\end{cases}</math>

Negating the argument complements the fractional part:

:<math>\{ x \} +  \{ -x \} = \begin{cases}
0&\text{if } x\in \mathbb{Z}\\
1&\text{if } x\not\in \mathbb{Z}.
\end{cases}</math>

The floor, ceiling, and fractional part functions are [[Idempotence|idempotent]]:

:<math>
\begin{align}
\big\lfloor \lfloor x \rfloor \big\rfloor &= \lfloor x \rfloor, \\
\big\lceil \lceil x \rceil \big\rceil &= \lceil x \rceil, \\
\big\{ \{ x \} \big\} &= \{ x \}.
\end{align}
</math>

The result of nested floor or ceiling functions is the innermost function:
:<math>
\begin{align}
\big\lfloor \lceil x \rceil \big\rfloor &= \lceil x \rceil, \\
\big\lceil \lfloor x \rfloor \big\rceil &= \lfloor x \rfloor
\end{align}
</math>
due to the identity property for integers.

===Quotients===
If ''m'' and ''n'' are integers and ''n'' ≠  0,
:<math>0 \le \left\{ \frac{m}{n} \right\} \le 1-\frac{1}{|n|}.</math>

If ''n'' is positive<ref>Graham, Knuth, & Patashnik, p. 73</ref>
:<math>\left\lfloor\frac{x+m}{n}\right\rfloor = \left\lfloor\frac{\lfloor x\rfloor +m}{n}\right\rfloor,
</math>

:<math>\left\lceil\frac{x+m}{n}\right\rceil = \left\lceil\frac{\lceil x\rceil +m}{n}\right\rceil.
</math>

If ''m'' is positive<ref>Graham, Knuth, & Patashnik, p. 85</ref>

:<math>n=\left\lceil\frac{n\vphantom1}{m}\right\rceil + \left\lceil\frac{n-1}{m}\right\rceil +\dots+\left\lceil\frac{n-m+1}{m}\right\rceil,
</math>

:<math>n=\left\lfloor\frac{n\vphantom1}{m}\right\rfloor + \left\lfloor\frac{n+1}{m}\right\rfloor +\dots+\left\lfloor\frac{n+m-1}{m}\right\rfloor.
</math>

For ''m'' = 2 these imply

:<math>n= \left\lfloor \frac{n\vphantom1}{2}\right \rfloor + \left\lceil\frac{n\vphantom1}{2}\right \rceil.</math>

More generally,<ref>Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15</ref> for positive ''m'' (See [[Hermite's identity]])

:<math>\lceil mx \rceil =\left\lceil x\right\rceil  + \left\lceil x-\frac{1}{m}\right\rceil +\dots+\left\lceil x-\frac{m-1}{m}\right\rceil,
</math>

:<math>\lfloor mx \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x+\frac{1}{m}\right\rfloor +\dots+\left\lfloor x+\frac{m-1}{m}\right\rfloor.
</math>

The following can be used to convert floors to ceilings and vice versa (with ''m'' being positive)<ref>Graham, Knuth, & Patashnik, Ex. 3.12</ref>

:<math>\left\lceil \frac{n\vphantom1}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor = \left\lfloor \frac{n - 1}{m} \right\rfloor + 1, </math>

:<math>\left\lfloor \frac{n\vphantom1}{m} \right\rfloor = \left\lceil \frac{n-m+1}{m} \right\rceil = \left\lceil \frac{n + 1}{m} \right\rceil - 1, </math>

For all ''m'' and ''n'' strictly positive integers:<ref>Graham, Knuth, & Patashnik, p. 94.</ref>

:<math>\sum_{k = 1}^{n - 1} \left\lfloor \frac{k m}{n} \right\rfloor = \frac{(m - 1)(n - 1)+\gcd(m,n)-1}2,</math>

which, for positive and [[coprime]]  ''m'' and ''n'', reduces to

:<math>\sum_{k=1}^{n-1} \left\lfloor \frac{km}{n} \right\rfloor = \tfrac{1}{2}(m - 1)(n - 1) ,</math>

and similarly for the ceiling and fractional part functions (still for positive and [[coprime]]  ''m'' and ''n''),

:<math>\sum_{k=1}^{n-1} \left\lceil \frac{km}{n} \right\rceil = \tfrac{1}{2}(m + 1)(n - 1),</math>

:<math>\sum_{k=1}^{n-1} \left\{ \frac{km}{n} \right\} = \tfrac{1}{2}(n - 1).</math>


Since the right-hand side of the general case is symmetrical in ''m'' and ''n'', this implies that

:<math>\left\lfloor \frac{m\vphantom1}{n} \right \rfloor + \left\lfloor \frac{2m}{n} \right \rfloor + \dots + \left\lfloor \frac{(n-1)m}{n} \right \rfloor =
\left\lfloor \frac{n\vphantom1}{m} \right \rfloor + \left\lfloor \frac{2n}{m} \right \rfloor + \dots + \left\lfloor \frac{(m-1)n}{m} \right \rfloor.
</math>

More generally, if ''m'' and ''n'' are positive,

:<math>\begin{align}
&\left\lfloor \frac{x\vphantom1}{n} \right \rfloor +
\left\lfloor \frac{m+x}{n} \right \rfloor +
\left\lfloor \frac{2m+x}{n} \right \rfloor +
\dots +
\left\lfloor \frac{(n-1)m+x}{n} \right \rfloor\\[5mu]
=
&\left\lfloor \frac{x\vphantom1}{m} \right \rfloor +
\left\lfloor \frac{n+x}{m} \right \rfloor +
\left\lfloor \frac{2n+x}{m} \right \rfloor +
\cdots +
\left\lfloor \frac{(m-1)n+x}{m} \right \rfloor.
\end{align}
</math>

This is sometimes called a [[#Quadratic reciprocity|reciprocity law]].<ref>Graham, Knuth, & Patashnik, p. 94</ref>

Division by positive integers gives rise to an interesting and sometimes useful property. Assuming <math>m,n >0</math>,

:<math> m \leq \left\lfloor \frac{x}{n} \right \rfloor \iff n \leq  \left\lfloor \frac{x}{m} \right \rfloor \iff n \leq  \frac{ \lfloor x \rfloor }{m}. </math>

Similarly,

:<math> m \geq \left\lceil \frac{x}{n} \right \rceil \iff n \geq  \left\lceil \frac{x}{m} \right \rceil \iff n \geq  \frac{ \lceil x \rceil }{m}. </math>

Indeed,

:<math> m \leq \left\lfloor \frac{x}{n} \right \rfloor \implies m \leq \frac{x}{n} \implies n \leq \frac{x}{m} 
\implies n \leq \left \lfloor \frac{x}{m}\right \rfloor \implies \ldots \implies m \leq \left\lfloor \frac{x}{n} \right \rfloor,</math> 
keeping in mind that <math display=inline> \left\lfloor \frac{x}{n} \right\rfloor = \left\lfloor \frac{\lfloor x \rfloor}{n} \right\rfloor.</math>
The second equivalence involving the ceiling function can be proved similarly.

===Nested divisions===
For a positive integer ''n'', and arbitrary real numbers ''m'' and ''x'':<ref>Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with {{sfrac|''x''|''m''}} as input and the division by ''n'' as function</ref>

: <math>\begin{align}
\left\lfloor \frac{\left\lfloor \frac{x}{m} \right\rfloor}{n} \right\rfloor &= \left\lfloor \frac{x}{mn} \right\rfloor \\[4px]
\left\lceil  \frac{\left\lceil  \frac{x}{m} \right\rceil }{n} \right\rceil  &= \left\lceil  \frac{x}{mn} \right\rceil.
\end{align}</math>

===Continuity and series expansions===
None of the functions discussed in this article are [[continuous function|continuous]], but all are [[piecewise linear function|piecewise linear]]: the functions <math>\lfloor x \rfloor</math>, <math>\lceil x \rceil</math>, and <math>\{ x\}</math> have discontinuities at the integers.

<math>\lfloor x \rfloor</math> is [[semi-continuity|upper semi-continuous]] and <math>\lceil x \rceil</math> and <math>\{ x\}</math> are lower semi-continuous.

Since none of the functions discussed in this article are continuous, none of them have a [[power series]] expansion. Since floor and ceiling are not periodic, they do not have uniformly convergent [[Fourier series]] expansions.  The fractional part function has Fourier series expansion<ref>Titchmarsh, p. 15, Eq. 2.1.7</ref>
<math display="block">
\{x\}= \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^\infty
\frac{\sin(2 \pi k x)} {k}
</math>
for {{mvar|x}} not an integer.

At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for ''y'' fixed and ''x'' a multiple of ''y'' the Fourier series given converges to ''y''/2, rather than to ''x''&nbsp;mod&nbsp;''y''&nbsp;=&nbsp;0. At points of continuity the series converges to the true value.

Using the formula <math>\lfloor x\rfloor = x - \{x\}</math> gives
<math display="block">
\lfloor x\rfloor = x - \frac{1}{2} + \frac{1}{\pi}  \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}
</math> 
for {{mvar|x}} not an integer.

==Applications==

===Mod operator===

For an integer ''x'' and a positive integer ''y'', the [[modulo operation]], denoted by ''x'' mod ''y'', gives the value of the remainder when ''x'' is divided by ''y''.  This definition can be extended to real ''x'' and ''y'', ''y'' ≠  0, by the formula

:<math>x \bmod y = x-y\left\lfloor \frac{x}{y}\right\rfloor.</math>

Then it follows from the definition of floor function that this extended operation satisfies many natural properties.  Notably, ''x'' mod ''y'' is always between 0 and ''y'', i.e.,

if ''y'' is positive,
:<math>0 \le x \bmod y <y,</math>
and if ''y'' is negative,
:<math>0 \ge x \bmod y >y.</math>

===Quadratic reciprocity===
Gauss's third proof of [[quadratic reciprocity]], as modified by Eisenstein, has two basic steps.<ref>Lemmermeyer, § 1.4, Ex. 1.32–1.33</ref><ref>Hardy & Wright, §§ 6.11–6.13</ref>

Let ''p'' and ''q'' be distinct positive odd prime numbers, and let <math>m = \tfrac12(p - 1),</math>  <math>n = \tfrac12(q - 1).</math>

First, [[Gauss's lemma (number theory)|Gauss's lemma]] is used to show that the [[Legendre symbol]]s are given by

:<math>\begin{align}
\left(\frac{q}{p}\right)
&= (-1)^{\left\lfloor\frac{q}{p}\right\rfloor + \left\lfloor\frac{2q}{p}\right\rfloor +
   \dots + \left\lfloor\frac{mq}{p}\right\rfloor }, \\[5mu]
\left(\frac{p}{q}\right)
&= (-1)^{\left\lfloor\frac{p}{q}\right\rfloor + \left\lfloor\frac{2p}{q}\right\rfloor + 
   \dots + \left\lfloor\frac{np}{q}\right\rfloor }.
\end{align}</math>

The second step is to use a [[Geometric series|geometric]] argument to show that

:<math>\left\lfloor\frac{q}{p}\right\rfloor +\left\lfloor\frac{2q}{p}\right\rfloor +\dots +\left\lfloor\frac{mq}{p}\right\rfloor

+\left\lfloor\frac{p}{q}\right\rfloor +\left\lfloor\frac{2p}{q}\right\rfloor +\dots +\left\lfloor\frac{np}{q}\right\rfloor

= mn.
</math>

Combining these formulas gives quadratic reciprocity in the form

:<math>\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{mn}=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>

There are formulas that use floor to express the quadratic character of small numbers mod odd primes ''p'':<ref>Lemmermeyer, p. 25</ref>
:<math>\begin{align}
\left(\frac{2}{p}\right) &= (-1)^{\left\lfloor\frac{p+1}{4}\right\rfloor}, \\[5mu]
\left(\frac{3}{p}\right) &= (-1)^{\left\lfloor\frac{p+1}{6}\right\rfloor}.
\end{align}</math>

===Rounding===
For an arbitrary real number <math>x</math>, [[rounding]] <math>x</math> to the nearest integer with [[Rounding#Tie-breaking|tie breaking]] towards positive infinity is given by
:<math>\text{rpi}(x)=\left\lfloor x+\tfrac{1}{2}\right\rfloor = \left\lceil \tfrac12\lfloor 2x \rfloor \right\rceil;</math>
rounding towards negative infinity is given as
:<math>\text{rni}(x)=\left\lceil x-\tfrac{1}{2}\right\rceil = \left\lfloor \tfrac12 \lceil 2x \rceil \right\rfloor.</math>

If tie-breaking is away from 0, then the rounding function is
:<math>\text{ri}(x) = \sgn(x)\left\lfloor|x|+\tfrac{1}{2}\right\rfloor</math>
(where <math>\sgn</math> is the [[sign function]]), and [[Rounding#Rounding half to even|rounding towards even]] can be expressed with the more cumbersome
:<math>\lfloor x\rceil=\left\lfloor x+\tfrac{1}{2}\right\rfloor+\left\lceil\tfrac14(2x-1)\right\rceil-\left\lfloor\tfrac14(2x-1)\right\rfloor-1,</math>
which is the above expression for rounding towards positive infinity <math>\text{rpi}(x)</math> minus an [[integer|integrality]] [[indicator function|indicator]] for <math>\tfrac14(2x-1)</math>.

Rounding a [[real number]] <math>x</math> to the nearest integer value forms a very basic type of [[Quantization (signal processing)|quantizer]] – a ''uniform'' one.  A typical (''mid-tread'') uniform quantizer with a quantization ''step size'' equal to some value <math>\Delta</math> can be expressed as

:<math>Q(x) = \Delta \cdot  \left\lfloor \frac{x}{\Delta} + \frac{1}{2} \right\rfloor</math>,

===Number of digits===
The number of digits in [[radix|base]] ''b'' of a positive integer ''k'' is
:<math>\lfloor \log_{b}{k} \rfloor + 1 = \lceil \log_{b}{(k+1)} \rceil .</math>

===Number of strings without repeated characters===
The number of possible [[String (computer science)|strings]] of arbitrary length that doesn't use any character twice is given by<ref>{{OEIS el |1=A000522 |2=Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.}} (See Formulas.)</ref>{{Better source needed|date=February 2022}} 

:<math>(n)_0 + \cdots + (n)_n = \lfloor e n! \rfloor</math>

where:

* {{math|''n''}} > 0 is the number of letters in the alphabet (e.g., 26 in [[English language|English]])
* the [[falling factorial]] <math>(n)_k = n(n-1)\cdots(n-k+1)</math> denotes the number of strings of length {{math|''k''}} that don't use any character twice.
* {{math|''n''}}! denotes the [[factorial]] of {{math|''n''}}
* {{math|''e''}} = 2.718... is [[Euler's number]]

For {{math|''n''}} = 26, this comes out to 1096259850353149530222034277.

===Factors of factorials===
Let ''n'' be a positive integer and ''p'' a positive prime number. The exponent of the highest power of ''p'' that divides ''n''! is given by a version of [[Legendre's formula]]<ref>Hardy & Wright, Th. 416</ref>

:<math>\left\lfloor\frac{n}{p}\right\rfloor + \left\lfloor\frac{n}{p^2}\right\rfloor + \left\lfloor\frac{n}{p^3}\right\rfloor + \dots = \frac{n-\sum_{k}a_k}{p-1}</math>

where <math display="inline">n = \sum_{k}a_kp^k</math> is the way of writing ''n'' in base ''p''. This is a finite sum, since the floors are zero when ''p''<sup>''k''</sup> > ''n''.

===Beatty sequence===
The [[Beatty sequence]] shows how every positive [[irrational number]] gives rise to a partition of the [[natural number]]s into two sequences via the floor function.<ref>Graham, Knuth, & Patashnik, pp. 77–78</ref>

===Euler's constant (γ)===
There are formulas for [[Euler–Mascheroni constant|Euler's constant]] γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.<ref>These formulas are from the Wikipedia article [[Euler–Mascheroni constant|Euler's constant]], which has many more.</ref>

:<math>\gamma =\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx,</math>

:<math>\gamma = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left( \left \lceil \frac{n}{k} \right \rceil - \frac{n}{k} \right),</math>

and

:<math>
 \gamma = \sum_{k=2}^\infty (-1)^k \frac{ \left \lfloor \log_2 k \right \rfloor}{k}
  = \tfrac12-\tfrac13
  + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right)
  + 3\left(\tfrac18 - \cdots - \tfrac1{15}\right) + \cdots
</math>

===Riemann zeta function (ζ)===
The fractional part function also shows up in integral representations of the [[Riemann zeta function]]. It is straightforward to prove (using integration by parts)<ref>Titchmarsh, p. 13</ref> that if <math>\varphi(x)</math> is any function with a continuous derivative in the closed interval [''a'', ''b''],

:<math>\sum_{a<n\le b}\varphi(n) =
\int_a^b\varphi(x) \, dx +
\int_a^b\left(\{x\}-\tfrac12\right)\varphi'(x) \, dx +
\left(\{a\}-\tfrac12\right)\varphi(a) -
\left(\{b\}-\tfrac12\right)\varphi(b).
</math>

Letting <math>\varphi(n) = n^{-s}</math> for [[real part]] of ''s'' greater than 1 and letting ''a'' and ''b'' be integers, and letting ''b'' approach infinity gives

:<math>\zeta(s) = s\int_1^\infty\frac{\frac12-\{x\}}{x^{s+1}}\,dx + \frac{1}{s-1} + \frac 1 2.</math>

This formula is valid for all ''s'' with real part greater than &minus;1, (except ''s'' = 1, where there is a pole) and combined with the Fourier expansion for {''x''} can be used to extend the zeta function to the entire complex plane and to prove its functional equation.<ref>Titchmarsh, pp.14–15</ref>

For ''s'' = ''σ'' + ''it'' in the critical strip 0 < ''σ'' < 1,

:<math>\zeta(s)=s\int_{-\infty}^\infty e^{-\sigma\omega}(\lfloor e^\omega\rfloor - e^\omega)e^{-it\omega}\,d\omega.</math>

In 1947 [[Balthasar van der Pol|van der Pol]] used this representation to construct an analogue computer for finding roots of the zeta function.<ref>Crandall & Pomerance, p. 391</ref>

===Formulas for prime numbers===
The floor function appears in several formulas characterizing prime numbers.  For example, since
<math display=block>\left\lfloor\frac{n}{m} \right\rfloor -\left\lfloor\frac{n-1}{m}\right\rfloor = \begin{cases}
1 &\text{if } m \text{ divides } n \\
0 &\text{otherwise},
\end{cases}</math>
it follows that a positive integer ''n'' is a prime [[if and only if]]<ref>Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced with ''n''. An equivalent condition is ''n''&nbsp;&gt;&nbsp;1 is prime if and only if
<math display=block>\sum_{m=1}^{\lfloor \sqrt n \rfloor} \left(\left\lfloor\frac{n}{m}\right\rfloor-\left\lfloor\frac{n-1}{m}\right\rfloor\right) = 1.</math></ref>

:<math>\sum_{m=1}^\infty \left(\left\lfloor\frac{n}{m}\right\rfloor-\left\lfloor\frac{n-1}{m}\right\rfloor\right) = 2.</math>

One may also give formulas for producing the prime numbers.  For example, let ''p''<sub>''n''</sub> be the ''n''-th prime, and for any integer ''r''&nbsp;>&nbsp;1, define the real number ''α'' by the sum

:<math>\alpha = \sum_{m=1}^\infty p_m r^{-m^2}.</math>

Then<ref>Hardy & Wright, § 22.3</ref>

:<math>p_n = \left\lfloor r^{n^2}\alpha \right\rfloor - r^{2n-1}\left\lfloor r^{(n-1)^2}\alpha\right\rfloor.</math>

A similar result is that there is a number ''θ'' = 1.3064... ([[Mills' constant]]) with the property that

:<math>\left\lfloor \theta^3 \right\rfloor, \left\lfloor \theta^9 \right\rfloor, \left\lfloor \theta^{27} \right\rfloor, \dots</math>

are all prime.<ref name="Ribenboim, p. 186">Ribenboim, p. 186</ref>

There is also a number ''ω'' = 1.9287800... with the property that

:<math>\left\lfloor 2^\omega\right\rfloor, \left\lfloor 2^{2^\omega} \right\rfloor, \left\lfloor 2^{2^{2^\omega}} \right\rfloor, \dots</math>

are all prime.<ref name="Ribenboim, p. 186"/>

Let {{pi}}(''x'') be [[Prime-counting function|the number of primes less than or equal to ''x'']]. It is a straightforward deduction from [[Wilson's theorem]] that<ref>Ribenboim, p. 181</ref>

:<math>\pi(n) = \sum_{j=2}^n\Biggl\lfloor\frac{(j-1)!+1}{j} - \left\lfloor\frac{(j-1)!}{j}\right\rfloor\Biggr\rfloor.</math>

Also, if ''n'' ≥ 2,<ref>Crandall & Pomerance, Ex. 1.4, p. 46</ref>

:<math>\pi(n) = \sum_{j=2}^n \left\lfloor \frac{1} {\displaystyle\sum_{k=2}^j\left\lfloor\left\lfloor\frac{j}{k}\right\rfloor\frac{k}{j} \right\rfloor} \right\rfloor.</math>

None of the formulas in this section are of any practical use.<ref>Ribenboim, p. 180 says that "Despite the nil practical value of the formulas ... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different axiomatzations ... "</ref><ref>Hardy & Wright, pp. 344—345 "Any one of these formulas (or any similar one) would attain a different status if the exact value of the number α ... could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."</ref>

===Solved problems===
[[Ramanujan]] submitted these problems to the ''Journal of the Indian Mathematical Society''.<ref>Ramanujan, Question 723, ''Papers'' p. 332</ref>

If ''n'' is a positive integer, prove that
<ol type="i">
<li> <math>\left\lfloor\tfrac{n}{3}\right\rfloor + \left\lfloor\tfrac{n+2}{6}\right\rfloor + \left\lfloor\tfrac{n+4}{6}\right\rfloor = \left\lfloor\tfrac{n}{2}\right\rfloor + \left\lfloor\tfrac{n+3}{6}\right\rfloor,</math></li>

<li> <math>\left\lfloor\tfrac12 + \sqrt{n+\tfrac12}\right\rfloor = \left\lfloor\tfrac12 + \sqrt{n+\tfrac14}\right\rfloor,</math></li>

<li> <math>\left\lfloor\sqrt{n}+ \sqrt{n+1}\right\rfloor = \left\lfloor \sqrt{4n+2}\right\rfloor.</math></li>
</ol>Some generalizations to the above floor function identities have been proven.<ref>{{Cite journal |last1=Somu |first1=Sai Teja |last2=Kukla |first2=Andrzej |title=On some generalizations to floor function identities of Ramanujan |url=http://math.colgate.edu/~integers/w33/w33.pdf |journal=Integers |year=2022 |volume=22|arxiv=2109.03680 }}</ref>

===Unsolved problem===
The study of [[Waring's problem]] has led to an unsolved problem:

Are there any positive integers ''k'' ≥ 6 such that<ref>Hardy & Wright, p. 337</ref>

:<math>3^k-2^k\Bigl\lfloor \bigl(\tfrac 3 2\bigr)^k \Bigr\rfloor  > 2^k-\Bigl\lfloor \bigl(\tfrac 3 2\bigr)^k \Bigr\rfloor -2 \ ?</math> 

[[Kurt Mahler|Mahler]] has proved there can only be a finite number of such ''k''; none are known.<ref>{{cite journal
| last1=Mahler | first1=Kurt | authorlink1=Kurt Mahler
| title=On the fractional parts of the powers of a rational number II
| date=1957
| journal=[[Mathematika]]
| volume=4
| issue=2 | pages=122–124
| doi=10.1112/S0025579300001170}}</ref>

==Computer implementations==
[[File:Int function.svg|thumb|right|<code>int</code> function from floating-point conversion in [[C (programming language)|C]]]]
In most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation. The reason for this is historical, as the first machines used [[ones' complement]] and truncation was simpler to implement (floor is simpler in [[two's complement]]). [[FORTRAN]] was defined to require this behavior and thus almost all processors implement conversion this way. Some consider this to be an unfortunate historical design decision that has led to bugs handling negative offsets and graphics on the negative side of the origin.{{cn|date=November 2018}}

An [[arithmetic shift|arithmetic right-shift]] of a signed integer <math>x</math> by <math>n</math> is the same as <math>\left\lfloor \tfrac{x}{2^n} \right\rfloor</math>. Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is required. Assuming such shifts are "premature optimization" and replacing them with division can break software.{{cn|date=March 2019}}

Many programming languages (including [[C (programming language)|C]], [[C++]],<ref>{{cite web
 | title=C++ reference of <code>floor</code> function
 | url=http://en.cppreference.com/w/cpp/numeric/math/floor
 | access-date=5 December 2010}}
</ref><ref>{{cite web
 | title=C++ reference of <code>ceil</code> function
 | url=http://en.cppreference.com/w/cpp/numeric/math/ceil
 | access-date=5 December 2010}}
</ref> [[C Sharp (programming language)|C#]],<ref>{{Cite web|url=https://docs.microsoft.com/en-us/dotnet/api/system.math.floor|title=Math.Floor Method (System)|last=dotnet-bot|website=docs.microsoft.com|language=en-us|access-date=28 November 2019}}</ref><ref>{{Cite web|url=https://docs.microsoft.com/en-us/dotnet/api/system.math.ceiling|title=Math.Ceiling Method (System)|last=dotnet-bot|website=docs.microsoft.com|language=en-us|access-date=28 November 2019}}</ref>  [[Java (programming language)|Java]],<ref>{{Cite web|url=https://docs.oracle.com/javase/9/docs/api/java/lang/Math.html#floor-double-|title=Math (Java SE 9 & JDK 9 )|website=docs.oracle.com|language=en|access-date=20 November 2018}}</ref><ref>{{Cite web|url=https://docs.oracle.com/javase/9/docs/api/java/lang/Math.html#ceil-double-|title=Math (Java SE 9 & JDK 9 )|website=docs.oracle.com|language=en|access-date=20 November 2018}}</ref>
[[Julia (programming language)|Julia]],<ref>{{Cite web|url=https://docs.julialang.org/en/v1/base/math/|title=Math (Julia v1.10)|website=docs.julialang.org/en/v1/|language=en|access-date=4 September 2024}}</ref>
[[PHP]],<ref>{{cite web
 | title=PHP manual for <code>ceil</code> function
 | url=http://php.net/manual/function.ceil.php
 | access-date=18 July 2013}}
</ref><ref>{{cite web
 | title=PHP manual for <code>floor</code> function
 | url=http://php.net/manual/function.floor.php
 | access-date=18 July 2013}}
</ref> [[R (programming language)|R]],<ref>{{Cite web | url=https://stat.ethz.ch/R-manual/R-devel/library/base/html/Round.html | title=R: Rounding of Numbers}}</ref> and [[Python (programming language)|Python]]<ref>{{cite web
 | title=Python manual for <code>math</code> module
 | url=https://docs.python.org/2/library/math.html
 | access-date=18 July 2013}}
</ref>) provide standard functions for floor and ceiling, usually called <code>floor</code> and <code>ceil</code>, or less commonly <code>ceiling</code>.<ref>Sullivan, p.&nbsp;86.</ref> The language [[APL (programming language)|APL]] uses <code>⌊x</code> for floor.  The [[J (programming language)|J Programming Language]], a follow-on to APL that is designed to use standard keyboard symbols, uses <code><.</code> for floor and <code>>.</code> for ceiling.<ref>{{cite web|title=Vocabulary|work=J Language|url=http://www.jsoftware.com/help/dictionary/vocabul.htm|access-date=6 September 2011}}</ref>
[[ALGOL]] uses<code>entier</code> for floor.

In [[Microsoft Excel]] the function <code>INT</code> rounds down rather than toward zero,<ref>{{cite web|url = https://support.microsoft.com/en-us/office/int-function-a6c4af9e-356d-4369-ab6a-cb1fd9d343ef | title = INT function | access-date = 29 October 2021}}</ref> while <code>FLOOR</code> rounds toward zero, the opposite of what "int" and "floor" do in other languages. Since 2010 <code>FLOOR</code> has been changed to error if the number is negative.<ref>{{cite web|url = https://support.microsoft.com/en-us/office/floor-function-14bb497c-24f2-4e04-b327-b0b4de5a8886 | title = FLOOR function | access-date = 29 October 2021}}</ref> The [[OpenDocument]] file format, as used by [[OpenOffice.org]], [[Libreoffice]] and others, <code>INT</code><ref>{{cite web|url = https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_INT_function | title = Documentation/How Tos/Calc: INT function | access-date = 29 October 2021}}</ref> and <code>FLOOR</code> both do floor, and <code>FLOOR</code> has a third argument to reproduce Excel's earlier behavior.<ref>{{cite web|url = https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_FLOOR_function | title = Documentation/How Tos/Calc: FLOOR function | access-date = 29 October 2021}}</ref>

==See also==
* [[Bracket (mathematics)]]
* [[Integer-valued function]]
* [[Step function]]
* [[Modulo operation]]

== Citations ==
{{reflist|30em}}

==References==
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  }}
*Nicholas J. Higham, ''Handbook of writing for the mathematical sciences'', SIAM. {{ISBN|0-89871-420-6}}, p.&nbsp;25
*[[International Organization for Standardization|ISO]]/[[International Electrotechnical Commission|IEC]]. ''ISO/IEC 9899::1999(E): Programming languages — C'' (2nd ed), 1999; Section 6.3.1.4, p.&nbsp;43.
*{{Citation
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*{{Citation
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*{{Citation
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*Michael Sullivan. ''Precalculus'', 8th edition, p.&nbsp;86
*{{Citation
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  | last2 = Heath-Brown | first2 = David Rodney ("Roger")
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  | isbn = 0-19-853369-1}}

==External links==
{{commons category|Floor and ceiling|Floor and ceiling functions}}
* {{springer|title=Floor function|id=p/f130150}}
* Štefan Porubský, [http://www.cs.cas.cz/portal/AlgoMath/NumberTheory/ArithmeticFunctions/IntegerRoundingFunctions.htm "Integer rounding functions"], ''Interactive Information Portal for Algorithmic Mathematics'', Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, retrieved 24 October 2008
* {{MathWorld|urlname=FloorFunction|title=Floor Function}}
* {{MathWorld|urlname=CeilingFunction|title=Ceiling Function}}

{{DEFAULTSORT:Floor And Ceiling Functions}}
[[Category:Special functions]]
[[Category:Mathematical notation]]
[[Category:Unary operations]]