Floor And Ceiling Functions

In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or .

For example, , , , and .

Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part.

For an integer, .

Notation

The ''integral part'' or ''integer part'' of a number ( 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 in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth E. Iverson introduced, in his 1962 book ''A Programming Language'', the names "floor" and "ceiling" and the corresponding notations and . (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 are used for floor, and reversed brackets or for ceiling. Sometimes is taken to mean the round-toward-zero function.

The fractional part is the sawtooth function, denoted by for real and defined by the formula
:

For all ''x'',
:.

These characters are provided in Unicode:

*
*
*
*

In the LaTeX typesetting system, these symbols can be specified with the and commands in math mode, and extended in size using \left\lfloor, \right\rfloor, \left\lceil and \right\rceil as needed.

Definition and properties

Given real numbers ''x'' and ''y'', integers ''k'', ''m'', ''n'' and the set of integers $\mathbb$, floor and ceiling may be defined by the equations

:$\lfloor x \rfloor=\max \,$

:$\lceil x \rceil=\min \.$

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
:x-1

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

Equivalences

These formulas can be used to simplify expressions involving floors and ceilings.

:$\begin \lfloor x \rfloor = m &\;\;\mbox &m &\le x < m+1,\\ \lceil x \rceil = n &\;\;\mbox &n -1 &< x \le n,\\ \lfloor x \rfloor = m &\;\;\mbox &x-1 &< m \le x,\\ \lceil x \rceil = n &\;\;\mbox &x &\le n < x+1. \end$

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
x

These formulas show how adding integers to the arguments affects the functions:

:$\begin \lfloor x+n \rfloor &= \lfloor x \rfloor+n,\\ \lceil x+n \rceil &= \lceil x \rceil+n,\\ \ &= \. \end$

The above are never true if ''n'' is not an integer; however, for every and , the following inequalities hold:

:$\begin \lfloor x \rfloor + \lfloor y \rfloor &\leq \lfloor x + y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor + 1,\\ \lceil x \rceil + \lceil y \rceil -1 &\leq \lceil x + y \rceil \leq \lceil x \rceil + \lceil y \rceil. \end$

Monotonicity

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

:$\begin x_ \le x_ &\Rightarrow \lfloor x_ \rfloor \le \lfloor x_ \rfloor, \\ x_ \le x_ &\Rightarrow \lceil x_ \rceil \le \lceil x_ \rceil. \end$

Relations among the functions

It is clear from the definitions that
:$\lfloor x \rfloor \le \lceil x \rceil,$   with equality if and only if ''x'' is an integer, i.e.
:$\lceil x \rceil - \lfloor x \rfloor = \begin 0&\mbox x\in \mathbb\\ 1&\mbox x\not\in \mathbb \end$

In fact, for integers ''n'', both floor and ceiling functions are the identity:
:$\lfloor n \rfloor = \lceil n \rceil = n.$

Negating the argument switches floor and ceiling and changes the sign:
:$\begin \lfloor x \rfloor +\lceil -x \rceil &= 0 \\ -\lfloor x \rfloor &= \lceil -x \rceil \\ -\lceil x \rceil &= \lfloor -x \rfloor \end$

and:
:$\lfloor x \rfloor + \lfloor -x \rfloor = \begin 0&\text x\in \mathbb\\ -1&\text x\not\in \mathbb, \end$

:$\lceil x \rceil + \lceil -x \rceil = \begin 0&\text x\in \mathbb\\ 1&\text x\not\in \mathbb. \end$

Negating the argument complements the fractional part:

:$\ + \ = \begin 0&\text x\in \mathbb\\ 1&\text x\not\in \mathbb. \end$

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

:$\begin \Big\lfloor \lfloor x \rfloor \Big\rfloor &= \lfloor x \rfloor, \\ \Big\lceil \lceil x \rceil \Big\rceil &= \lceil x \rceil, \\ \Big\ &= \. \end$

The result of nested floor or ceiling functions is the innermost function:
:$\begin \Big\lfloor \lceil x \rceil \Big\rfloor &= \lceil x \rceil, \\ \Big\lceil \lfloor x \rfloor \Big\rceil &= \lfloor x \rfloor \end$
due to the identity property for integers.

Quotients

If ''m'' and ''n'' are integers and ''n'' ≠ 0,
:$0 \le \left \ \le 1-\frac.$

If ''n'' is a positive integer
:$\left\lfloor\frac\right\rfloor = \left\lfloor\frac\right\rfloor,$

:$\left\lceil\frac\right\rceil = \left\lceil\frac\right\rceil.$

If ''m'' is positive

:$n=\left\lceil\frac\right\rceil + \left\lceil\frac\right\rceil +\dots+\left\lceil\frac\right\rceil,$

:$n=\left\lfloor\frac\right\rfloor + \left\lfloor\frac\right\rfloor +\dots+\left\lfloor\frac\right\rfloor.$

For ''m'' = 2 these imply

:$n= \left\lfloor \frac\right \rfloor + \left\lceil\frac\right \rceil.$

More generally, for positive ''m'' (See Hermite's identity)

:$\lceil mx \rceil =\left\lceil x\right\rceil + \left\lceil x-\frac\right\rceil +\dots+\left\lceil x-\frac\right\rceil,$

:$\lfloor mx \rfloor=\left\lfloor x\right\rfloor + \left\lfloor x+\frac\right\rfloor +\dots+\left\lfloor x+\frac\right\rfloor.$

The following can be used to convert floors to ceilings and vice versa (''m'' positive)

:$\left\lceil \frac \right\rceil = \left\lfloor \frac \right\rfloor = \left\lfloor \frac \right\rfloor + 1,$

:$\left\lfloor \frac \right\rfloor = \left\lceil \frac \right\rceil = \left\lceil \frac \right\rceil - 1,$

For all ''m'' and ''n'' strictly positive integers:

:$\sum_^ \left\lfloor \frac \right\rfloor = \frac2,$

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

:$\sum_^ \left\lfloor \frac \right\rfloor = \frac(m - 1)(n - 1) ,$

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

:$\sum_^ \left\lceil \frac \right\rceil = \frac(m + 1)(n - 1),$

:$\sum_^ \left\ = \frac(n - 1).$

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

:$\left\lfloor \frac \right \rfloor + \left\lfloor \frac \right \rfloor + \dots + \left\lfloor \frac \right \rfloor = \left\lfloor \frac \right \rfloor + \left\lfloor \frac \right \rfloor + \dots + \left\lfloor \frac \right \rfloor.$

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

:$\begin &\left\lfloor \frac \right \rfloor + \left\lfloor \frac \right \rfloor + \left\lfloor \frac \right \rfloor + \dots + \left\lfloor \frac \right \rfloor\\= &\left\lfloor \frac \right \rfloor + \left\lfloor \frac \right \rfloor + \left\lfloor \frac \right \rfloor + \cdots + \left\lfloor \frac \right \rfloor. \end$

This is sometimes called a reciprocity law.

Nested divisions

For positive integer ''n'', and arbitrary real numbers ''m'',''x'':

: $\left\lfloor \frac \right\rfloor = \left\lfloor \frac \right\rfloor$

: $\left\lceil \frac \right\rceil = \left\lceil \frac \right\rceil.$

Continuity and series expansions

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

$\lfloor x \rfloor$  is upper semi-continuous and  $\lceil x \rceil$  and $\$  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

:$\= \frac - \frac \sum_^\infty \frac$
for 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 floor(x) = x − gives

:$\lfloor x\rfloor = x - \frac + \frac \sum_^\infty \frac$
for 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

:$x \bmod y = x-y\left\lfloor \frac\right\rfloor.$

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,
:0 \le x \bmod y
and if ''y'' is negative,
:$0 \ge x \bmod y >y.$

Quadratic reciprocity

Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.

Let ''p'' and ''q'' be distinct positive odd prime numbers, and let
:$m = \frac,$ $n = \frac.$

First, Gauss's lemma is used to show that the