Floor and ceiling functions

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).Lemmermeyer, pp. 10, 23. This remained the standarde.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation. Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's. 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⌉}}.Iverson, p. 12.Higham, p. 25. (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.[http://www.mathwords.com/f/floor_function.htm Mathwords: Floor Function].[http://www.mathwords.com/c/ceiling_function.htm Mathwords: Ceiling Function]

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⌋}}Graham, Knuth, & Patashnik, p. 70.

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

Given real numbers x and y, integers m and n and the set of integers $\backslash mathbb\{Z\}$, floor and ceiling may be defined by the equations

:$\backslash lfloor\; x\; \backslash rfloor=\backslash max\; \backslash \{m\backslash in\backslash mathbb\{Z\}\backslash mid\; m\backslash le\; x\backslash \},$

:$\backslash lceil\; x\; \backslash rceil=\backslash min\; \backslash \{n\backslash in\backslash mathbb\{Z\}\backslash mid\; n\backslash ge\; x\backslash \}.$

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

:

where $\backslash lfloor\; x\; \backslash rfloor\; =\; m$ and $\backslash lceil\; x\; \backslash rceil\; =\; n$ may also be taken as the definition of floor and ceiling.

These formulas can be used to simplify expressions involving floors and ceilings.Graham, Knuth, & Patashink, Ch. 3

:$$

\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}

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.

:$$

\begin{align}

x

n

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}

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

:$$

\begin{align}

\lfloor x+n \rfloor &= \lfloor x \rfloor+n,\\

\lceil x+n \rceil &= \lceil x \rceil+n,\\

\{ x+n \} &= \{ x \}.

\end{align}

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

:$\backslash begin\{align\}$

\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}

Both floor and ceiling functions are monotonically non-decreasing functions:

:$$

\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}

It is clear from the definitions that

:$\backslash lfloor\; x\; \backslash rfloor\; \backslash le\; \backslash lceil\; x\; \backslash rceil,$ with equality if and only if x is an integer, i.e.

:$\backslash lceil\; x\; \backslash rceil\; -\; \backslash lfloor\; x\; \backslash rfloor\; =\; \backslash begin\{cases\}$

0&\mbox{ if } x\in \mathbb{Z}\\

1&\mbox{ if } x\not\in \mathbb{Z}

\end{cases}

In fact, for integers n, both floor and ceiling functions are the identity:

:$\backslash lfloor\; n\; \backslash rfloor\; =\; \backslash lceil\; n\; \backslash rceil\; =\; n.$

Negating the argument switches floor and ceiling and changes the sign:

:$\backslash begin\{align\}$

\lfloor x \rfloor +\lceil -x \rceil &= 0 \\

-\lfloor x \rfloor &= \lceil -x \rceil \\

-\lceil x \rceil &= \lfloor -x \rfloor

\end{align}

and:

:$\backslash lfloor\; x\; \backslash rfloor\; +\; \backslash lfloor\; -x\; \backslash rfloor\; =\; \backslash begin\{cases\}$

0&\text{if } x\in \mathbb{Z}\\

-1&\text{if } x\not\in \mathbb{Z},

\end{cases}

:$\backslash lceil\; x\; \backslash rceil\; +\; \backslash lceil\; -x\; \backslash rceil\; =\; \backslash begin\{cases\}$

0&\text{if } x\in \mathbb{Z}\\

1&\text{if } x\not\in \mathbb{Z}.

\end{cases}

Negating the argument complements the fractional part:

:$\backslash \{\; x\; \backslash \}\; +\; \backslash \{\; -x\; \backslash \}\; =\; \backslash begin\{cases\}$

0&\text{if } x\in \mathbb{Z}\\

1&\text{if } x\not\in \mathbb{Z}.

\end{cases}

The floor, ceiling, and fractional part functions are idempotent:

:$$

\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}

The result of nested floor or ceiling functions is the innermost function:

:$$

\begin{align}

\big\lfloor \lceil x \rceil \big\rfloor &= \lceil x \rceil, \\

\big\lceil \lfloor x \rfloor \big\rceil &= \lfloor x \rfloor

\end{align}

due to the identity property for integers.

If m and n are integers and n ≠ 0,

:$0\; \backslash le\; \backslash left\backslash \{\; \backslash frac\{m\}\{n\}\; \backslash right\backslash \}\; \backslash le\; 1-\backslash frac\{1\}$

If n is positiveGraham, Knuth, & Patashnik, p. 73

:$\backslash left\backslash lfloor\backslash frac\{x+m\}\{n\}\backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\backslash frac\{\backslash lfloor\; x\backslash rfloor\; +m\}\{n\}\backslash right\backslash rfloor,$

:$\backslash left\backslash lceil\backslash frac\{x+m\}\{n\}\backslash right\backslash rceil\; =\; \backslash left\backslash lceil\backslash frac\{\backslash lceil\; x\backslash rceil\; +m\}\{n\}\backslash right\backslash rceil.$

If m is positiveGraham, Knuth, & Patashnik, p. 85

:$n=\backslash left\backslash lceil\backslash frac\{n\backslash vphantom1\}\{m\}\backslash right\backslash rceil\; +\; \backslash left\backslash lceil\backslash frac\{n-1\}\{m\}\backslash right\backslash rceil\; +\backslash dots+\backslash left\backslash lceil\backslash frac\{n-m+1\}\{m\}\backslash right\backslash rceil,$

:$n=\backslash left\backslash lfloor\backslash frac\{n\backslash vphantom1\}\{m\}\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\backslash frac\{n+1\}\{m\}\backslash right\backslash rfloor\; +\backslash dots+\backslash left\backslash lfloor\backslash frac\{n+m-1\}\{m\}\backslash right\backslash rfloor.$

For m = 2 these imply

:$n=\; \backslash left\backslash lfloor\; \backslash frac\{n\backslash vphantom1\}\{2\}\backslash right\; \backslash rfloor\; +\; \backslash left\backslash lceil\backslash frac\{n\backslash vphantom1\}\{2\}\backslash right\; \backslash rceil.$

More generally,Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15 for positive m (See Hermite's identity)

:$\backslash lceil\; mx\; \backslash rceil\; =\backslash left\backslash lceil\; x\backslash right\backslash rceil\; +\; \backslash left\backslash lceil\; x-\backslash frac\{1\}\{m\}\backslash right\backslash rceil\; +\backslash dots+\backslash left\backslash lceil\; x-\backslash frac\{m-1\}\{m\}\backslash right\backslash rceil,$

:$\backslash lfloor\; mx\; \backslash rfloor=\backslash left\backslash lfloor\; x\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\; x+\backslash frac\{1\}\{m\}\backslash right\backslash rfloor\; +\backslash dots+\backslash left\backslash lfloor\; x+\backslash frac\{m-1\}\{m\}\backslash right\backslash rfloor.$

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

:$\backslash left\backslash lceil\; \backslash frac\{n\backslash vphantom1\}\{m\}\; \backslash right\backslash rceil\; =\; \backslash left\backslash lfloor\; \backslash frac\{n+m-1\}\{m\}\; \backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\; \backslash frac\{n\; -\; 1\}\{m\}\; \backslash right\backslash rfloor\; +\; 1,$

:$\backslash left\backslash lfloor\; \backslash frac\{n\backslash vphantom1\}\{m\}\; \backslash right\backslash rfloor\; =\; \backslash left\backslash lceil\; \backslash frac\{n-m+1\}\{m\}\; \backslash right\backslash rceil\; =\; \backslash left\backslash lceil\; \backslash frac\{n\; +\; 1\}\{m\}\; \backslash right\backslash rceil\; -\; 1,$

For all m and n strictly positive integers:Graham, Knuth, & Patashnik, p. 94.

:$\backslash sum\_\{k\; =\; 1\}^\{n\; -\; 1\}\; \backslash left\backslash lfloor\; \backslash frac\{k\; m\}\{n\}\; \backslash right\backslash rfloor\; =\; \backslash frac\{(m\; -\; 1)(n\; -\; 1)+\backslash gcd(m,n)-1\}2,$

which, for positive and coprime m and n, reduces to

:$\backslash sum\_\{k=1\}^\{n-1\}\; \backslash left\backslash lfloor\; \backslash frac\{km\}\{n\}\; \backslash right\backslash rfloor\; =\; \backslash tfrac\{1\}\{2\}(m\; -\; 1)(n\; -\; 1)\; ,$

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

:$\backslash sum\_\{k=1\}^\{n-1\}\; \backslash left\backslash lceil\; \backslash frac\{km\}\{n\}\; \backslash right\backslash rceil\; =\; \backslash tfrac\{1\}\{2\}(m\; +\; 1)(n\; -\; 1),$

:$\backslash sum\_\{k=1\}^\{n-1\}\; \backslash left\backslash \{\; \backslash frac\{km\}\{n\}\; \backslash right\backslash \}\; =\; \backslash tfrac\{1\}\{2\}(n\; -\; 1).$

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

:$\backslash left\backslash lfloor\; \backslash frac\{m\backslash vphantom1\}\{n\}\; \backslash right\; \backslash rfloor\; +\; \backslash left\backslash lfloor\; \backslash frac\{2m\}\{n\}\; \backslash right\; \backslash rfloor\; +\; \backslash dots\; +\; \backslash left\backslash lfloor\; \backslash frac\{(n-1)m\}\{n\}\; \backslash right\; \backslash rfloor\; =$

\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.

More generally, if m and n are positive,

:$\backslash begin\{align\}$

&\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}

This is sometimes called a reciprocity law.Graham, Knuth, & Patashnik, p. 94

Division by positive integers gives rise to an interesting and sometimes useful property. Assuming $m,n\; >0$,

:$m\; \backslash leq\; \backslash left\backslash lfloor\; \backslash frac\{x\}\{n\}\; \backslash right\; \backslash rfloor\; \backslash iff\; n\; \backslash leq\; \backslash left\backslash lfloor\; \backslash frac\{x\}\{m\}\; \backslash right\; \backslash rfloor\; \backslash iff\; n\; \backslash leq\; \backslash frac\{\; \backslash lfloor\; x\; \backslash rfloor\; \}\{m\}.$

Similarly,

:$m\; \backslash geq\; \backslash left\backslash lceil\; \backslash frac\{x\}\{n\}\; \backslash right\; \backslash rceil\; \backslash iff\; n\; \backslash geq\; \backslash left\backslash lceil\; \backslash frac\{x\}\{m\}\; \backslash right\; \backslash rceil\; \backslash iff\; n\; \backslash geq\; \backslash frac\{\; \backslash lceil\; x\; \backslash rceil\; \}\{m\}.$

Indeed,

:$m\; \backslash leq\; \backslash left\backslash lfloor\; \backslash frac\{x\}\{n\}\; \backslash right\; \backslash rfloor\; \backslash implies\; m\; \backslash leq\; \backslash frac\{x\}\{n\}\; \backslash implies\; n\; \backslash leq\; \backslash frac\{x\}\{m\}$

\implies n \leq \left \lfloor \frac{x}{m}\right \rfloor \implies \ldots \implies m \leq \left\lfloor \frac{x}{n} \right \rfloor,

keeping in mind that $\backslash left\backslash lfloor\; \backslash frac\{x\}\{n\}\; \backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\; \backslash frac\{\backslash lfloor\; x\; \backslash rfloor\}\{n\}\; \backslash right\backslash rfloor.$

The second equivalence involving the ceiling function can be proved similarly.

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

: $\backslash begin\{align\}$

\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}

None of the functions discussed in this article are continuous, but all are piecewise linear: the functions $\backslash lfloor\; x\; \backslash rfloor$, $\backslash lceil\; x\; \backslash rceil$, and $\backslash \{\; x\backslash \}$ have discontinuities at the integers.

$\backslash lfloor\; x\; \backslash rfloor$ is upper semi-continuous and $\backslash lceil\; x\; \backslash rceil$ and $\backslash \{\; x\backslash \}$ 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 expansionTitchmarsh, p. 15, Eq. 2.1.7

\{x\}= \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^\infty

\frac{\sin(2 \pi k x)} {k}

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 mod y = 0. At points of continuity the series converges to the true value.

Using the formula $\backslash lfloor\; x\backslash rfloor\; =\; x\; -\; \backslash \{x\backslash \}$ gives

\lfloor x\rfloor = x - \frac{1}{2} + \frac{1}{\pi} \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}

for {{mvar|x}} not an integer.

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

:$x\; \backslash bmod\; y\; =\; x-y\backslash left\backslash lfloor\; \backslash frac\{x\}\{y\}\backslash right\backslash rfloor.$

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,

:

and if y is negative,

:$0\; \backslash ge\; x\; \backslash bmod\; y\; >y.$

Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.Lemmermeyer, § 1.4, Ex. 1.32–1.33Hardy & Wright, §§ 6.11–6.13

Let p and q be distinct positive odd prime numbers, and let $m\; =\; \backslash tfrac12(p\; -\; 1),$ $n\; =\; \backslash tfrac12(q\; -\; 1).$

First, Gauss's lemma is used to show that the Legendre symbols are given by

:$\backslash begin\{align\}$

\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}

The second step is to use a geometric argument to show that

:$\backslash left\backslash lfloor\backslash frac\{q\}\{p\}\backslash right\backslash rfloor\; +\backslash left\backslash lfloor\backslash frac\{2q\}\{p\}\backslash right\backslash rfloor\; +\backslash dots\; +\backslash left\backslash lfloor\backslash frac\{mq\}\{p\}\backslash right\backslash rfloor$

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

= mn.

Combining these formulas gives quadratic reciprocity in the form

:$\backslash left(\backslash frac\{p\}\{q\}\backslash right)\; \backslash left(\backslash frac\{q\}\{p\}\backslash right)\; =\; (-1)^\{mn\}=(-1)^\{\backslash frac\{p-1\}\{2\}\backslash frac\{q-1\}\{2\}\}.$

There are formulas that use floor to express the quadratic character of small numbers mod odd primes p:Lemmermeyer, p. 25

:$\backslash begin\{align\}$

\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}

For an arbitrary real number $x$, rounding $x$ to the nearest integer with tie breaking towards positive infinity is given by

:$\backslash text\{rpi\}(x)=\backslash left\backslash lfloor\; x+\backslash tfrac\{1\}\{2\}\backslash right\backslash rfloor\; =\; \backslash left\backslash lceil\; \backslash tfrac12\backslash lfloor\; 2x\; \backslash rfloor\; \backslash right\backslash rceil;$

rounding towards negative infinity is given as

:$\backslash text\{rni\}(x)=\backslash left\backslash lceil\; x-\backslash tfrac\{1\}\{2\}\backslash right\backslash rceil\; =\; \backslash left\backslash lfloor\; \backslash tfrac12\; \backslash lceil\; 2x\; \backslash rceil\; \backslash right\backslash rfloor.$

If tie-breaking is away from 0, then the rounding function is

:$\backslash text\{ri\}(x)\; =\; \backslash sgn(x)\backslash left\backslash lfloor|x|+\backslash tfrac\{1\}\{2\}\backslash right\backslash rfloor$

(where $\backslash sgn$ is the sign function), and rounding towards even can be expressed with the more cumbersome

:$\backslash lfloor\; x\backslash rceil=\backslash left\backslash lfloor\; x+\backslash tfrac\{1\}\{2\}\backslash right\backslash rfloor+\backslash left\backslash lceil\backslash tfrac14(2x-1)\backslash right\backslash rceil-\backslash left\backslash lfloor\backslash tfrac14(2x-1)\backslash right\backslash rfloor-1,$

which is the above expression for rounding towards positive infinity $\backslash text\{rpi\}(x)$ minus an integrality indicator for $\backslash tfrac14(2x-1)$.

Rounding a real number $x$ to the nearest integer value forms a very basic type of quantizer – a uniform one. A typical (mid-tread) uniform quantizer with a quantization step size equal to some value $\backslash Delta$ can be expressed as

:$Q(x)\; =\; \backslash Delta\; \backslash cdot\; \backslash left\backslash lfloor\; \backslash frac\{x\}\{\backslash Delta\}\; +\; \backslash frac\{1\}\{2\}\; \backslash right\backslash rfloor$,

The number of digits in base b of a positive integer k is

:$\backslash lfloor\; \backslash log\_\{b\}\{k\}\; \backslash rfloor\; +\; 1\; =\; \backslash lceil\; \backslash log\_\{b\}\{(k+1)\}\; \backslash rceil\; .$

The number of possible strings of arbitrary length that doesn't use any character twice is given by{{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.){{Better source needed|date=February 2022}}

:$(n)\_0\; +\; \backslash cdots\; +\; (n)\_n\; =\; \backslash lfloor\; e\; n!\; \backslash rfloor$

where:

• {{math|n}} > 0 is the number of letters in the alphabet (e.g., 26 in English)
• the falling factorial $(n)\_k\; =\; n(n-1)\backslash cdots(n-k+1)$ 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.

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 formulaHardy & Wright, Th. 416

:$\backslash left\backslash lfloor\backslash frac\{n\}\{p\}\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\backslash frac\{n\}\{p^2\}\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\backslash frac\{n\}\{p^3\}\backslash right\backslash rfloor\; +\; \backslash dots\; =\; \backslash frac\{n-\backslash sum\_\{k\}a\_k\}\{p-1\}$

where $n\; =\; \backslash sum\_\{k\}a\_kp^k$ is the way of writing n in base p. This is a finite sum, since the floors are zero when p^k > n.

The Beatty sequence shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.Graham, Knuth, & Patashnik, pp. 77–78

There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.These formulas are from the Wikipedia article Euler's constant, which has many more.

:$\backslash gamma\; =\backslash int\_1^\backslash infty\backslash left(\{1\backslash over\backslash lfloor\; x\backslash rfloor\}-\{1\backslash over\; x\}\backslash right)\backslash ,dx,$

:$\backslash gamma\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{n\}\; \backslash sum\_\{k=1\}^n\; \backslash left(\; \backslash left\; \backslash lceil\; \backslash frac\{n\}\{k\}\; \backslash right\; \backslash rceil\; -\; \backslash frac\{n\}\{k\}\; \backslash right),$

and

:$$

\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

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

:

\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).

Letting $\backslash varphi(n)\; =\; n^\{-s\}$ for real part of s greater than 1 and letting a and b be integers, and letting b approach infinity gives

:$\backslash zeta(s)\; =\; s\backslash int\_1^\backslash infty\backslash frac\{\backslash frac12-\backslash \{x\backslash \}\}\{x^\{s+1\}\}\backslash ,dx\; +\; \backslash frac\{1\}\{s-1\}\; +\; \backslash frac\; 1\; 2.$

This formula is valid for all s with real part greater than −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.Titchmarsh, pp.14–15

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

:$\backslash zeta(s)=s\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-\backslash sigma\backslash omega\}(\backslash lfloor\; e^\backslash omega\backslash rfloor\; -\; e^\backslash omega)e^\{-it\backslash omega\}\backslash ,d\backslash omega.$

In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.Crandall & Pomerance, p. 391

The floor function appears in several formulas characterizing prime numbers. For example, since

$$\backslash left\backslash lfloor\backslash frac\{n\}\{m\}\; \backslash right\backslash rfloor\; -\backslash left\backslash lfloor\backslash frac\{n-1\}\{m\}\backslash right\backslash rfloor\; =\; \backslash begin\{cases\}$$

1 &\text{if } m \text{ divides } n \\

0 &\text{otherwise},

\end{cases}

it follows that a positive integer n is a prime if and only ifCrandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced with n. An equivalent condition is n > 1 is prime if and only if

$$\backslash sum\_\{m=1\}^\{\backslash lfloor\; \backslash sqrt\; n\; \backslash rfloor\}\; \backslash left(\backslash left\backslash lfloor\backslash frac\{n\}\{m\}\backslash right\backslash rfloor-\backslash left\backslash lfloor\backslash frac\{n-1\}\{m\}\backslash right\backslash rfloor\backslash right)\; =\; 1.$$

:$\backslash sum\_\{m=1\}^\backslash infty\; \backslash left(\backslash left\backslash lfloor\backslash frac\{n\}\{m\}\backslash right\backslash rfloor-\backslash left\backslash lfloor\backslash frac\{n-1\}\{m\}\backslash right\backslash rfloor\backslash right)\; =\; 2.$

One may also give formulas for producing the prime numbers. For example, let p_n be the n-th prime, and for any integer r > 1, define the real number α by the sum

:$\backslash alpha\; =\; \backslash sum\_\{m=1\}^\backslash infty\; p\_m\; r^\{-m^2\}.$

ThenHardy & Wright, § 22.3

:$p\_n\; =\; \backslash left\backslash lfloor\; r^\{n^2\}\backslash alpha\; \backslash right\backslash rfloor\; -\; r^\{2n-1\}\backslash left\backslash lfloor\; r^\{(n-1)^2\}\backslash alpha\backslash right\backslash rfloor.$

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

:$\backslash left\backslash lfloor\; \backslash theta^3\; \backslash right\backslash rfloor,\; \backslash left\backslash lfloor\; \backslash theta^9\; \backslash right\backslash rfloor,\; \backslash left\backslash lfloor\; \backslash theta^\{27\}\; \backslash right\backslash rfloor,\; \backslash dots$

are all prime.Ribenboim, p. 186

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

:$\backslash left\backslash lfloor\; 2^\backslash omega\backslash right\backslash rfloor,\; \backslash left\backslash lfloor\; 2^\{2^\backslash omega\}\; \backslash right\backslash rfloor,\; \backslash left\backslash lfloor\; 2^\{2^\{2^\backslash omega\}\}\; \backslash right\backslash rfloor,\; \backslash dots$

are all prime.

Let {{pi}}(x) be the number of primes less than or equal to x. It is a straightforward deduction from Wilson's theorem thatRibenboim, p. 181

:$\backslash pi(n)\; =\; \backslash sum\_\{j=2\}^n\backslash Biggl\backslash lfloor\backslash frac\{(j-1)!+1\}\{j\}\; -\; \backslash left\backslash lfloor\backslash frac\{(j-1)!\}\{j\}\backslash right\backslash rfloor\backslash Biggr\backslash rfloor.$

Also, if n ≥ 2,Crandall & Pomerance, Ex. 1.4, p. 46

:$\backslash pi(n)\; =\; \backslash sum\_\{j=2\}^n\; \backslash left\backslash lfloor\; \backslash frac\{1\}\; \{\backslash displaystyle\backslash sum\_\{k=2\}^j\backslash left\backslash lfloor\backslash left\backslash lfloor\backslash frac\{j\}\{k\}\backslash right\backslash rfloor\backslash frac\{k\}\{j\}\; \backslash right\backslash rfloor\}\; \backslash right\backslash rfloor.$

None of the formulas in this section are of any practical use.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 ... "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."

Ramanujan submitted these problems to the Journal of the Indian Mathematical Society.Ramanujan, Question 723, Papers p. 332

If n is a positive integer, prove that

1. $\backslash left\backslash lfloor\backslash tfrac\{n\}\{3\}\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\backslash tfrac\{n+2\}\{6\}\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\backslash tfrac\{n+4\}\{6\}\backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\backslash tfrac\{n\}\{2\}\backslash right\backslash rfloor\; +\; \backslash left\backslash lfloor\backslash tfrac\{n+3\}\{6\}\backslash right\backslash rfloor,$
2. $\backslash left\backslash lfloor\backslash tfrac12\; +\; \backslash sqrt\{n+\backslash tfrac12\}\backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\backslash tfrac12\; +\; \backslash sqrt\{n+\backslash tfrac14\}\backslash right\backslash rfloor,$
3. $\backslash left\backslash lfloor\backslash sqrt\{n\}+\; \backslash sqrt\{n+1\}\backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\; \backslash sqrt\{4n+2\}\backslash right\backslash rfloor.$

The study of Waring's problem has led to an unsolved problem:

Are there any positive integers k ≥ 6 such thatHardy & Wright, p. 337

:$3^k-2^k\backslash Bigl\backslash lfloor\; \backslash bigl(\backslash tfrac\; 3\; 2\backslash bigr)^k\; \backslash Bigr\backslash rfloor\; >\; 2^k-\backslash Bigl\backslash lfloor\; \backslash bigl(\backslash tfrac\; 3\; 2\backslash bigr)^k\; \backslash Bigr\backslash rfloor\; -2\; \backslash \; ?$

Mahler has proved there can only be a finite number of such k; none are known.{{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}}

File:Int function.svg]]

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 right-shift of a signed integer $x$ by $n$ is the same as $\backslash left\backslash lfloor\; \backslash tfrac\{x\}\{2^n\}\; \backslash right\backslash rfloor$. 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, C++,{{cite web

| title=C++ reference of floor function

| url=http://en.cppreference.com/w/cpp/numeric/math/floor

| access-date=5 December 2010}}

{{cite web

| title=C++ reference of ceil function

| url=http://en.cppreference.com/w/cpp/numeric/math/ceil

| access-date=5 December 2010}}

C#,{{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}}{{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}} Java,{{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}}{{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}}

Julia,{{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}}

PHP,{{cite web

| title=PHP manual for ceil function

| url=http://php.net/manual/function.ceil.php

| access-date=18 July 2013}}

{{cite web

| title=PHP manual for floor function

| url=http://php.net/manual/function.floor.php

| access-date=18 July 2013}}

R,{{Cite web | url=https://stat.ethz.ch/R-manual/R-devel/library/base/html/Round.html | title=R: Rounding of Numbers}} and Python{{cite web

| title=Python manual for math module

| url=https://docs.python.org/2/library/math.html

| access-date=18 July 2013}}

) provide standard functions for floor and ceiling, usually called floor and ceil, or less commonly ceiling.Sullivan, p. 86. The language APL uses ⌊x for floor. The J Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses <. for floor and >. for ceiling.{{cite web|title=Vocabulary|work=J Language|url=http://www.jsoftware.com/help/dictionary/vocabul.htm|access-date=6 September 2011}}

ALGOL usesentier for floor.

In Microsoft Excel the function INT rounds down rather than toward zero,{{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}} while FLOOR rounds toward zero, the opposite of what "int" and "floor" do in other languages. Since 2010 FLOOR has been changed to error if the number is negative.{{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}} The OpenDocument file format, as used by OpenOffice.org, Libreoffice and others, INT{{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}} and FLOOR both do floor, and FLOOR has a third argument to reproduce Excel's earlier behavior.{{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}}

• Bracket (mathematics)
• Integer-valued function
• Step function
• Modulo operation

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{{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