User:Virginia-American/Sandbox/Floor and Ceiling

The floor function

The ceiling function

In mathematics and computer science, the floor and ceiling functions map a real number to the next smallest or next largest integer. More precisely, floor(x) is the largest integer not greater than x and ceiling(x) is the smallest integer not less than x.

This article also discusses two related functions, the fractional part function and the mod operator.

Synonyms and Notations

Gauss introduced the square bracket notation [x] for the floor function in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Iverson introduced the names "floor" and "ceiling" and the corresponding notations $\lfloor x\rfloor$ and $\lceil x\rceil$ in his 1962 programming language APL. Both notations are now used in mathematics; this article follows Iverson.

The floor function is also called the greatest integer or entier (French for "integer") function, and the floor of a nonnegative x may be called the integral part or integral value of x. Computer languages (other than APL) commonly use ENTIER(x) (Algol), floor(x), or int(x) (C and C++).

The ceiling function is usually denoted by ceil(x) or ceiling(x) in non-APL computer languages; there is no common alternative to Iverson's $\lceil x\rceil$ in mathematics.

Fractional part

The fractional part function, denoted by $\{x\}$ for real x is defined by the formula

$\{x\}=x-\lfloor x\rfloor .$

For all x,

$0\leq \{x\}<1.\;$

If x is positive, the floor of x is simply x with everything to the right of the decimal point replaced with 0, and the fractional part is x with everything to the left of the decimal point replaced with 0.

mod operator

The mod operator, denoted by x mod y for real x and y, y ≠ 0, is defined by the formula

$x{\mbox{ mod }}y=x-y\left\lfloor {\frac {x}{y}}\right\rfloor .$

x mod y is always between 0 and y; i.e.

if y is positive,

$0\leq x{\mbox{ mod }}y<y,$

and if y is negative,

$0\geq x{\mbox{ mod }}y>y.$

If x is an integer and y is a positive integer,

$(x{\mbox{ mod }}y)\equiv x{\pmod {y}}.$

Examples

	−2.7	−2	12/5	2.7
$\lfloor \;\rfloor$	−3	−2	2	2
$\lceil \;\rceil$	−2	−2	3	3
$\{\;\}$	0.3	0	2/5 = 0.4	0.7

Formulas

$\mathbb {Z}$ is the set of integers (positive, negative, and zero).
 
In these formulas, x and y are real numbers and k, m, and n are integers.

Definitions

Floor and ceiling may be defined by the set equations

$\lfloor x\rfloor =\max \,\{n\in \mathbb {Z} \mid n\leq x\}.$

$\lceil x\rceil =\min \,\{n\in \mathbb {Z} \mid n\geq x\}.$

Since there is exactly one integer in a half-open interval of length one, for any real x there are unique integers m and n such that

$x-1<m\leq x\leq n<x+1.\;$

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

As stated above,

$\{x\}=x-\lfloor x\rfloor ,$

$x{\mbox{ mod }}y=x-y\left\lfloor {\frac {x}{y}}\right\rfloor .$

Equivalences

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

${\begin{aligned}\lfloor x\rfloor =n&\;\;{\mbox{ if and only if }}&n&\leq x<n+1,\\\lceil x\rceil =n&\;\;{\mbox{ if and only if }}&n-1&<x<n,\\\lfloor x\rfloor =n&\;\;{\mbox{ if and only if }}&x-1&<n\leq x,\\\lceil x\rceil =n&\;\;{\mbox{ if and only if }}&x&\leq n<x+1.\end{aligned}}$

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.

${\begin{aligned}x<n&\;\;{\mbox{ if and only if }}&\lfloor x\rfloor &<n,\\n<x&\;\;{\mbox{ if and only if }}&n&<\lceil x\rceil ,\\x\leq n&\;\;{\mbox{ if and only if }}&\lceil x\rceil &\leq n,\\n\leq x&\;\;{\mbox{ if and only if }}&n&\leq \lfloor x\rfloor .\end{aligned}}$

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

${\begin{aligned}\lfloor x+n\rfloor &=\lfloor x\rfloor +n,\\\lceil x+n\rceil &=\lceil x\rceil +n,\\\{x+n\}&=\{x\}.\end{aligned}}$

The above may or may not be true for if n is not an integer, but we do have:

$\lfloor x\rfloor +\lfloor y\rfloor \leq \lfloor x+y\rfloor \leq \lfloor x\rfloor +\lfloor y\rfloor +1.$

Relations among the functions

It is clear from the definitions that

$\lfloor x\rfloor \leq \lceil x\rceil ,$   with equality if and only if x is an integer, i.e.

$\lceil x\rceil -\lfloor x\rfloor ={\begin{cases}0&{\mbox{ if }}x\in \mathbb {Z} \\1&{\mbox{ if }}x\not \in \mathbb {Z} \end{cases}}$

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

$\lfloor x\rfloor +\lceil -x\rceil =0,$   i.e.

$\lfloor x\rfloor +\lfloor -x\rfloor ={\begin{cases}0&{\mbox{ if }}x\in \mathbb {Z} \\-1&{\mbox{ if }}x\not \in \mathbb {Z} ,\end{cases}}$

$\lceil x\rceil +\lceil -x\rceil ={\begin{cases}0&{\mbox{ if }}x\in \mathbb {Z} \\1&{\mbox{ if }}x\not \in \mathbb {Z} .\end{cases}}$

Negating the argument complements the fractional part:

$\{x\}+\{-x\}={\begin{cases}0&{\mbox{ if }}x\in \mathbb {Z} \\1&{\mbox{ if }}x\not \in \mathbb {Z} .\end{cases}}$

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

${\begin{aligned}\lfloor \lfloor x\rfloor \rfloor &=\lfloor x\rfloor ,\\\lceil \lceil x\rceil \rceil &=\lceil x\rceil ,\\\{\{x\}\}&=\{x\}.\end{aligned}}$

In fact, since for integers n   $\lfloor n\rfloor =\lceil n\rceil =n,$

${\begin{aligned}\lfloor \lceil x\rceil \rfloor &=\lceil x\rceil ,\\\lceil \lfloor x\rfloor \rceil &=\lfloor x\rfloor .\end{aligned}}$

For fixed y, x mod y is idempotent:

$(x{\mbox{ mod }}y){\mbox{ mod }}y=x{\mbox{ mod }}y.\;$

Also, from the definitions,

$\{x\}=x{\mbox{ mod }}1.\;$

Quotients

If n ≠ 0,

$0\leq \left\{{\frac {m}{n}}\right\}\leq 1-{\frac {1}{|n|}}.$

If n is positive

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

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

If m is positive

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

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

For m = 2 these imply

$\lfloor k/2\rfloor +\lceil k/2\rceil =k$.

More generally, for positive m

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

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

These can be used to convert floors to ceilings and vice-versa

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

$\left\lceil {\frac {n}{m}}\right\rceil =\left\lfloor {\frac {n+m-1}{m}}\right\rfloor .$

If m and n are positive and coprime, then

$\sum _{i=1}^{n-1}\lfloor im/n\rfloor =(m-1)(n-1)/2.$

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

$\left\lfloor {\frac {m}{n}}\right\rfloor +\left\lfloor {\frac {2m}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m}{n}}\right\rfloor =\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {2n}{m}}\right\rfloor +\dots +\left\lfloor {\frac {(m-1)n}{m}}\right\rfloor .$

More generally, if m and n are positive,

${\begin{aligned}&\left\lfloor {\frac {x}{n}}\right\rfloor +\left\lfloor {\frac {m+x}{n}}\right\rfloor +\left\lfloor {\frac {2m+x}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m+x}{n}}\right\rfloor \\=&\left\lfloor {\frac {x}{m}}\right\rfloor +\left\lfloor {\frac {n+x}{m}}\right\rfloor +\left\lfloor {\frac {2n+x}{m}}\right\rfloor +\dots +\left\lfloor {\frac {(m-1)n+x}{m}}\right\rfloor .\end{aligned}}$

This is sometimes called a reciprocity law.

Continuity

None of the functions discussed in this article are continuous, but all are piecewise linear. $\lfloor x\rfloor$  and $\lceil x\rceil$ are piecewise constant functions, with discontinuites at the integers. $\{x\}$ also has discontinuites at the integers, and   $x{\mbox{ mod }}y$ as a function of x for fixed y is discontinuous at multiples of y.

$\lfloor x\rfloor$  is upper semi-continuous and  $\lceil x\rceil$  and $\{x\}\;$  are lower semi-continuous. x mod y is lower semicontinuous for positive y and upper semi-continuous for negative y.

Series expansions

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 Fourier expnsions.

x mod y for fixed y has the Fourier series expansion

$x{\mbox{ mod }}y={\frac {y}{2}}-{\frac {y}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin({\frac {2\pi kx}{y}})}{k}}.$

in particular {x} = x mod 1 is given by

$\{x\}={\frac {1}{2}}-{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.$

At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right. For x mod y, y fixed, the Fourier series converges to y/2 at multiples of y. At points of continuity the series converges to the true value.

Using the formula {x} = x − floor(x), floor(x) = x − {x} gives

$\lfloor x\rfloor =x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.$

Applications

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

$m={\frac {p-1}{2}},\;\;n={\frac {q-1}{2}}.$

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

$\left({\frac {q}{p}}\right)=(-1)^{\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor }$

and

$\left({\frac {p}{q}}\right)=(-1)^{\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor }.$

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

$\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor +\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor =mn.$

Combining these formulas gives quadratic reciprocity in the form

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

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

$\left({\frac {2}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{4}}\right\rfloor },$

$\left({\frac {3}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{6}}\right\rfloor }.$

Rounding

The ordinary rounding of the positive number x to the nearest integer can be expressed as $\lfloor x+0.5\rfloor .$

Number of digits

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

$\lfloor \log _{b}{k}\rfloor +1.$

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 the formula

$\left\lfloor {\frac {n}{p}}\right\rfloor +\left\lfloor {\frac {n}{p^{2}}}\right\rfloor +\left\lfloor {\frac {n}{p^{3}}}\right\rfloor +\dots$

Note that this is a finite sum, since the floors are zero when p^k > n.

Spectra (Beatty's theorem)

Beatty's theorem shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.

Euler's constant γ

There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.

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

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

and

$\gamma =\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-\dots -{\tfrac {1}{15}}\right)+\dots$

Riemann ζ function

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

$\sum _{a<n\leq b}\phi (n)=\int _{a}^{b}\phi (x)dx+\int _{a}^{b}(\{x\}-{\frac {1}{2}})\phi '(x)dx+(\{a\}-{\frac {1}{2}})\phi (a)-(\{b\}-{\frac {1}{2}})\phi (b)$

Letting φ(n) = n^−s for real part of s greater than 1 and a and b be integers, and letting b approach infinity gives

$\zeta (s)=s\int _{1}^{\infty }{\frac {{\frac {1}{2}}-\{x\}}{x^{s+1}}}\;dx+{\frac {1}{s-1}}+{\frac {1}{2}}.$

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.

Formula for prime numbers

Let r > 1 be an integer, p_n be the n^th prime, and define

$\alpha =\sum _{m=1}^{\infty }p_{m}r^{-m^{2}}.$

Then

$p_{n}=\lfloor r^{n^{2}}\alpha \rfloor -r^{2n-1}\lfloor r^{(n-1)^{2}}\alpha \rfloor .$

Until a way is found to calculate α without using the primes explicitly, this formula is of no practical use.

Unsolved problem

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

Are there any positive integers k such that

$3^{k}-2^{k}\left\lfloor \left({\frac {3}{2}}\right)^{k}\right\rfloor >2^{k}-\left\lfloor \left({\frac {3}{2}}\right)^{k}\right\rfloor -2$

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

Computer implementations

The operator (int) in C

C and related programming languages convert floating point values into integers using the type casting syntax: (int) value. The fractional part is discarded (i.e., the value is truncated toward zero).

Further information: [[:Floating point]]

Spreadsheet software

Most spreadsheet programs support some form of a ceiling function. Although the details differ between programs, most implementations support a second parameter—a multiple of which the given number is to be rounded to. As a typical example, ceiling(2, 3) would round 2 up to the nearest multiple of 3, so this would return 3. The definition of what "round up" means, however, differs from program to program.

Microsoft Excel's ceiling function does not follow the mathematical definition, but rather as with (int) operator in C, it is a mixture of the floor and ceiling function: for x ≥ 0 it returns ceiling(x), and for x < 0 it returns floor(x). This has followed through to the Office Open XML file format. For example, CEILING(-4.5) returns -5. A mathematical ceiling function can be emulated in Excel by using the formula "-INT(-value)" (please note that this is not a general rule, as it depends on Excel's INT function, which behaves differently that most programming languages).

The OpenDocument file format, as used by OpenOffice.org and others, follows the mathematical definition of ceiling for its ceiling function, with an optional parameter for Excel compatibility. For example, CEILING(-4.5) returns -4.

Typesetting

The floor and ceiling function are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing, and, e.g., in the LaTeX typesetting system these symbols can be specified with the \lfloor, \rfloor, \lceil and \rceil commands in math mode. Unicode contains codepoints for these symbols, at U+2308–U+230B: ⌈x⌉, ⌊x⌋.

See also

• Nearest integer function
• Truncation, a similar function

References

• J.W.S. Cassels (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press.

• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete Mthematics, Reading Ma.: Addison-Wesley, ISBN 0-201-55802-5

• Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0198531715

• Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0898714206, p. 25

• ISO/IEC. ISO/IEC 9899::1999(E): Programming languages — C (2nd ed), 1999; Section 6.3.1.4, p. 43.

• Iverson, Kenneth E. (1962), A Programming Language, Wiley

• Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66967-4 {{citation}}: Check |isbn= value: checksum (help)

• Michael Sullivan. Precalculus, 8th edition, p. 86

• Titchmarsh, Edward Charles; Heath-Brown, David Rodney ("Roger") (1986), The Theory of the Riemann Zeta-function (2nd ed.), Oxford: Oxford U. P., ISBN 0-19-853369-1

External links

• Štefan Porubský, "Integer rounding functions", Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, retrieved 10/24/2008

• Floor Function — from Wolfram MathWorld