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| {{For||Floor (disambiguation)|Ceiling (disambiguation)}}
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| {{Use dmy dates|date=July 2013}}
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| {{multiple image
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| | align = right
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| | direction = vertical
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| | header = Floor and ceiling functions
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| | width = 200
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| | image1 = Floor function.svg
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| | caption1 = Floor function
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| | image2 = Ceiling function.svg
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| | caption2 = Ceiling function
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| }}
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| | |
| In [[mathematics]] and [[computer science]], the '''floor''' and '''ceiling''' [[function (mathematics)|function]]s map a [[real number]] to the largest previous or the smallest following [[integer]], respectively. More precisely, floor(''x'') = <math>\lfloor x\rfloor </math> is the largest integer not greater than ''x'' and ceiling(''x'') = <math> \lceil x \rceil</math> is the smallest integer not less than ''x''.<ref>Graham, Knuth, & Patashnik, Ch. 3.1</ref>
| |
| | |
| ==Notation==
| |
| [[Carl Friedrich Gauss]] introduced the square bracket notation <math>[ x]</math> for the floor function in his third proof of [[quadratic reciprocity]] (1808).<ref>Lemmermeyer, pp. 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 the names "floor" and "ceiling" and the corresponding notations <math>\lfloor x\rfloor </math> and <math>\lceil x \rceil </math> in his 1962 book ''A Programming Language''.<ref>Iverson, p. 12.</ref><ref>Higham, p. 25.</ref> Both notations are now used in mathematics;<ref>See the Wolfram MathWorld article.</ref> this article follows Iverson.
| |
| | |
| The floor function is also called the '''greatest integer''' or '''entier''' (French for "integer") function, and its value at ''x'' is called the '''integral part''' or '''integer part''' of ''x''; for negative values of ''x'' the latter terms are sometimes instead taken to be the value of the ''ceiling'' function, i.e., the value of ''x'' rounded to an integer towards 0. The language [[APL (programming language)|APL]] uses <code>⌊x</code>; other computer languages commonly use notations like <code>entier(x)</code> ([[Algol programming language|Algol]]), <code>INT(x)</code> ([[BASIC]]), or <code>floor(x)</code>([[C (programming language)|C]], [[C++]], [[R (programming language)|R]], and [[Python (programming language)|Python]]).<ref>Sullivan, p. 86.</ref> In mathematics, it can also be written with boldface or double brackets <math>[\![x]\!]</math>.<ref>[http://www.mathwords.com/f/floor_function.htm Mathwords: Floor Function].</ref>
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| | |
| The ceiling function is usually denoted by <code>ceil(x)</code> or <code>ceiling(x)</code> in non-APL computer languages that have a notation for this function. The [[J (programming language)|J Programming Language]], a follow on to APL that is designed to use standard keyboard symbols, uses <code>>.</code> for ceiling and <code><.</code> for floor.<ref>{{cite web|title=Vocabulary|work=J Language|url=http://www.jsoftware.com/help/dictionary/vocabul.htm|accessdate=6 September 2011}}</ref> In mathematics, there is another notation with reversed boldface or double brackets <math>]\!]x[\![</math> or just using normal reversed brackets <nowiki>]</nowiki>''x''<nowiki>[</nowiki>.<ref>[http://www.mathwords.com/c/ceiling_function.htm Mathwords: Ceiling Function]</ref>
| |
| | |
| The '''fractional part''' [[sawtooth function]], denoted by <math>\{x\}</math> for real ''x'', is defined by the formula<ref>Graham, Knuth, & Patashnik, p. 70.</ref>
| |
| :<math>\{x\} = x -\lfloor x\rfloor.</math>
| |
| | |
| For all ''x'',
| |
| :<math>0\le\{x\}<1.\;</math>
| |
| | |
| ===Examples===
| |
| {| class="wikitable"
| |
| ! Sample value ''x''
| |
| ! Floor <math>\lfloor x\rfloor</math>
| |
| ! Ceiling <math>\lceil x\rceil</math>
| |
| ! Fractional part <math> \{ x \} </math>
| |
| |-
| |
| ! 12/5 = 2.4
| |
| | 2
| |
| | 3
| |
| | 2/5 = 0.4
| |
| |-
| |
| ! 2.7
| |
| | 2
| |
| | 3
| |
| | 0.7
| |
| |-
| |
| ! −2.7
| |
| | −3
| |
| | −2
| |
| | 0.3
| |
| |-
| |
| ! −2
| |
| | −2
| |
| | −2
| |
| | 0
| |
| |}
| |
| | |
| ===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. HTML 4.0 uses the same names: ''&lfloor;'', ''&rfloor;'', ''&lceil;'', and ''&rceil;''. [[Unicode]] contains [[codepoint]]s for these symbols at <code>U+2308</code>–<code>U+230B</code>: ⌈''x''⌉, ⌊''x''⌋.
| |
| | |
| ==Definition and properties==
| |
| In the following formulas, ''x'' and ''y'' are real numbers, ''k'', ''m'', and ''n'' are integers, and <math>\mathbb{Z}</math> is the set of [[integer]]s (positive, negative, and zero).
| |
| | |
| Floor and ceiling may be defined by the set 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 ''x'' there are unique integers ''m'' and ''n'' satisfying
| |
| :<math>x-1<m\le x \le n <x+1.\;</math>
| |
| | |
| Then <math>\lfloor x \rfloor = m\;</math> and <math>\;\lceil x \rceil = n\;</math>
| |
| 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{align}
| |
| \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{align}
| |
| </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 integers to the arguments affect 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 not necessarily true if ''n'' is not an integer; however:
| |
| | |
| :<math>\begin{align}
| |
| &\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{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, since for integers ''n'':
| |
| :<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&\mbox{ if } x\in \mathbb{Z}\\
| |
| -1&\mbox{ if } x\not\in \mathbb{Z},
| |
| \end{cases}</math>
| |
| | |
| :<math>\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}</math>
| |
| | |
| Negating the argument complements the fractional part:
| |
| | |
| :<math>\{ x \} + \{ -x \} = \begin{cases}
| |
| 0&\mbox{ if } x\in \mathbb{Z}\\
| |
| 1&\mbox{ if } x\not\in \mathbb{Z}.
| |
| \end{cases}</math>
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| | |
| 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 \}. \\
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| \end{align}
| |
| </math>
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| | |
| 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>
| |
| | |
| For fixed ''y'', ''x'' mod ''y'' is idempotent:
| |
| :<math>(x \,\bmod\, y) \,\bmod\, y = x \,\bmod\, y.\;</math>
| |
| | |
| Also, from the definitions,
| |
| :<math>\{x\}= x \,\bmod\, 1.\;</math>
| |
| | |
| ===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}{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}{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}{2}\right \rfloor + \left\lceil\frac{n}{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 (''m'' positive)<ref>Graham, Knuth, & Patashnik, Ex. 3.12</ref>
| |
| | |
| :<math>\left\lceil \frac{n}{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}{m} \right\rfloor = \left\lceil \frac{n-m+1}{m} \right\rceil = \left\lceil \frac{n + 1}{m} \right\rceil - 1, </math>
| |
| | |
| If ''m'' and ''n'' are positive and [[coprime]], then
| |
| :<math>\sum_{i=1}^{n-1} \left\lfloor \frac{im}{n} \right\rfloor = \frac{1}{2}(m - 1)(n - 1).</math>
| |
| | |
| Since the right-hand side is symmetrical in ''m'' and ''n'', this implies that
| |
| | |
| :<math>\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.
| |
| </math>
| |
| | |
| More generally, if ''m'' and ''n'' are positive,
| |
| | |
| :<math>\begin{align}
| |
| &\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{align}
| |
| </math>
| |
| | |
| This is sometimes called a [[#Quadratic reciprocity|reciprocity law]].<ref>Graham, Knuth, & Patashnik, p. 94</ref>
| |
| | |
| ===Nested divisions===
| |
| For positive integers ''m'',''n'', and arbitrary real number ''x'':
| |
| | |
| : <math> \left\lfloor \frac{\lfloor x/m\rfloor}{n} \right\rfloor = \left\lfloor \frac{x}{mn} \right\rfloor </math>
| |
| | |
| : <math> \left\lceil \frac{\lceil x/m\rceil}{n} \right\rceil = \left\lceil \frac{x}{mn} \right\rceil </math>
| |
| | |
| ===Continuity===
| |
| None of the functions discussed in this article are [[continuous function|continuous]], but all are [[piecewise linear function|piecewise linear]]. <math>\lfloor x \rfloor</math> and <math>\lceil x \rceil</math> are piecewise [[constant function]]s, with discontinuites at the integers. <math>\{ x\}</math> also has discontinuites at the integers, and <math>x \,\bmod\, y</math> as a function of ''x'' for fixed ''y'' is discontinuous at multiples of ''y''.
| |
| | |
| <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. ''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 uniformly convergent [[Fourier series]] expansions.
| |
| | |
| ''x'' mod ''y'' for fixed ''y'' has the Fourier series expansion<ref>Titchmarsh, p. 15, Eq. 2.1.7</ref>
| |
| | |
| :<math>x \,\bmod\, y = \frac{y}{2} - \frac{y}{\pi} \sum_{k=1}^\infty
| |
| \frac{\sin\left(\frac{2 \pi k x}{y}\right)} {k}\qquad\mbox{for }x\mbox{ not a multiple of }y.
| |
| </math>
| |
| | |
| in particular {''x''} = ''x'' mod 1 is given by
| |
| | |
| :<math>\{x\}= \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^\infty
| |
| \frac{\sin(2 \pi k x)} {k}\qquad\mbox{for }x\mbox{ not an integer}.
| |
| </math>
| |
| | |
| 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 {x} = x − floor(x), floor(x) = x − {x} gives
| |
| | |
| :<math>\lfloor x\rfloor = x - \frac{1}{2} + \frac{1}{\pi} \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}\qquad\mbox{for }x\mbox{ not an integer}.</math>
| |
| | |
| ==Applications==
| |
| | |
| ===Mod operator===
| |
| | |
| The '''[[mod operator]]''', denoted by ''x'' mod ''y'' for real ''x'' and ''y'', ''y'' ≠ 0, can be defined by the formula
| |
| | |
| :<math>x \,\bmod\, y = x-y\left\lfloor \frac{x}{y}\right\rfloor.</math>
| |
| | |
| ''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>
| |
| | |
| If ''x'' is an integer and ''y'' is a positive integer,
| |
| :<math>(x \,\bmod\, y) \equiv x \pmod{y}.</math>
| |
| | |
| ''x'' mod ''y'' for a fixed ''y'' is a '''[[sawtooth function]]'''.
| |
| | |
| ===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 = \frac{p - 1}{2},\;\; n = \frac{q - 1}{2}.</math>
| |
| | |
| First, [[Gauss's lemma (number theory)|Gauss's lemma]] is used to show that the [[Legendre symbol]]s are given by
| |
| | |
| :<math>\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 }
| |
| </math>
| |
| | |
| and
| |
| :<math>\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 }.
| |
| </math>
| |
| | |
| The second step is to use a 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>\left(\frac{2}{p}\right) = (-1)^{\left\lfloor\frac{p+1}{4}\right\rfloor},</math>
| |
| | |
| :<math>\left(\frac{3}{p}\right) = (-1)^{\left\lfloor\frac{p+1}{6}\right\rfloor}.</math>
| |
| | |
| ===Rounding===
| |
| The ordinary [[rounding]] of the positive number ''x'' to the nearest integer can be expressed as <math>\lfloor x + 0.5\rfloor.</math> The ordinary [[rounding]] of the negative number ''x'' to the nearest integer can be expressed as <math>\lceil x - 0.5\rceil.</math>
| |
| | |
| ===Truncation===
| |
| The [[truncation]] of a nonnegative number is given by <math>\lfloor x\rfloor.</math> The truncation of a nonpositive number is given by <math>\lceil x \rceil</math>.
| |
| | |
| The truncation of any real number can be given by: <math>\sgn(x) \lfloor |x| \rfloor</math>, where sgn(x) is the [[sign function]].
| |
| | |
| ===Number of digits===
| |
| The number of digits in [[base (exponentiation)|base]] ''b'' of a positive integer ''k'' is
| |
| ::<math>\lfloor \log_{b}{k} \rfloor + 1 = \lceil \log_{b}{(k+1)} \rceil ,</math>
| |
| with the right side of the equation also holding true for <math> k = 0 </math>.
| |
| | |
| ===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<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>n = \sum_{k}a_kp^k</math> is the way of writing ''n'' in base ''p''. Note that 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 - \dots - \tfrac1{15}\right) + \dots
| |
| </math>
| |
| | |
| ===Riemann 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 φ(''x'') is any function with a continuous derivative in the closed interval [''a'', ''b''],
| |
| | |
| :<math>{ \sum_{a<n\le b}\phi(n) =
| |
| \int_a^b\phi(x) dx +
| |
| \int_a^b\left(\{x\}-\tfrac12\right)\phi'(x) dx +
| |
| \left(\{a\}-\tfrac12\right)\phi(a) -
| |
| \left(\{b\}-\tfrac12\right)\phi(b). }
| |
| </math>
| |
| | |
| Letting φ(''n'') = ''n''<sup>−s</sup> 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} + \frac12.
| |
| </math>
| |
| | |
| 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.<ref>Titchmarsh, pp.14–15</ref>
| |
| | |
| For ''s'' = σ + ''i t'' in the critical strip (i.e. 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===
| |
| ''n'' is a prime if and only if<ref>Crandall & Pomerance, Ex. 1.3, p. 46</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>
| |
| | |
| Let ''r'' > 1 be an integer, ''p''<sub>''n''</sub> be the ''n''<sup>th</sup> prime, and define
| |
| | |
| :<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>
| |
| | |
| 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"/>
| |
| | |
| π(''x'') is 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\left\lfloor\frac{(j-1)!+1}{j} - \left\lfloor\frac{(j-1)!}{j}\right\rfloor\right\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}{\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 is 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 problem===
| |
| [[Ramanujan]] submitted this problem to the ''Journal of the Indian Mathematical Society''.<ref>Ramanujan, Question 723, ''Papers'' p. 332</ref>
| |
| | |
| If ''n'' is a positive integer, prove that
| |
| | |
| (i) <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>
| |
| | |
| (ii) <math>\left\lfloor\tfrac12 + \sqrt{n+\tfrac12}\right\rfloor = \left\lfloor\tfrac12 + \sqrt{n+\tfrac14}\right\rfloor,
| |
| </math>
| |
| | |
| (iii) <math>\left\lfloor\sqrt{n}+ \sqrt{n+1}\right\rfloor = \left\lfloor \sqrt{4n+2}\right\rfloor.
| |
| </math>
| |
| | |
| ===Unsolved problem===
| |
| The study of [[Waring's problem]] has led to an unsolved problem:
| |
| | |
| Are there any positive integers ''k'', ''k'' ≥ 6, such that<ref>Hardy & Wright, p. 337</ref>
| |
| | |
| :<math>3^k-2^k\left\lfloor \left(\tfrac32\right)^k \right\rfloor > 2^k-\left\lfloor \left(\tfrac32\right)^k \right\rfloor -2\;\;?</math>
| |
| | |
| [[Kurt Mahler|Mahler]]<ref>Mahler, K. ''On the fractional parts of the powers of a rational number II'', 1957, Mathematika, '''4''', pages 122-124</ref> has proved there can only be a finite number of such ''k''; none are known.
| |
| | |
| ==Computer implementations==
| |
| [[File:Int function.svg|thumb|right|Int function from floating-point conversion]]
| |
| Many programming languages (including [[C (programming language)|C]], [[C++]],<ref>{{cite web
| |
| | title=C++ reference of <code>floor</code> function
| |
| | url=http://cppreference.com/wiki/numeric/c/floor
| |
| | accessdate=5 December 2010}}
| |
| </ref><ref>{{cite web
| |
| | title=C++ reference of <code>ceil</code> function
| |
| | url=http://php.net/manual/function.floor.php
| |
| | accessdate=5 December 2010}}
| |
| </ref> [[PHP]],<ref>{{cite web
| |
| | title=PHP manual for <code>ceil</code> function
| |
| | url=http://php.net/manual/function.ceil.php
| |
| | accessdate=18 July 2013}}
| |
| </ref><ref>{{cite web
| |
| | title=PHP manual for <code>floor</code> function
| |
| | url=http://php.net/manual/function.floor.php
| |
| | accessdate=18 July 2013}}
| |
| </ref> and [[Python (programming language)|Python]]<ref>{{cite web
| |
| | title=Python manual for <code>math</code> module
| |
| | url=http://docs.python.org/2/library/math.html
| |
| | accessdate=18 July 2013}}
| |
| </ref>) provide standard functions for floor and ceiling.
| |
| ===Spreadsheet software===
| |
| {{Refimprove|date=August 2008}}
| |
| | |
| Most [[spreadsheet]] programs support some form of a <code>ceiling</code> 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. For example, <code>ceiling(2, 3)</code> rounds 2 up to the nearest multiple of 3, giving 3. The definition of what "round up" means, however, differs from program to program.
| |
| | |
| Until Excel 2010, [[Microsoft Excel]]'s <code>ceiling</code> function was incorrect for negative arguments; ceiling(-4.5) was -5.
| |
| . This has followed through to the [[Office Open XML]] file format. The correct ceiling function can be implemented using "<code>-INT(-''value'')</code>". Excel 2010 now follows the standard definition.<ref>But the online help provided in 2010 does not reflect this behavior.</ref>
| |
| | |
| The [[OpenDocument]] file format, as used by [[OpenOffice.org]] and others, follows the mathematical definition of ceiling for its <code>ceiling</code> function, with an optional parameter for Excel compatibility. For example, <code>CEILING(-4.5)</code> returns −4.
| |
| | |
| ==See also==
| |
| * [[Nearest integer function]]
| |
| * [[Truncation]], a similar function
| |
| * [[Step function]]
| |
| | |
| ==Notes==
| |
| {{reflist|30em}}
| |
| | |
| ==References==
| |
| *{{Citation
| |
| | author=J.W.S. Cassels
| |
| | title=An introduction to Diophantine approximation
| |
| | series=Cambridge Tracts in Mathematics and Mathematical Physics
| |
| | volume=45
| |
| | publisher=[[Cambridge University Press]]
| |
| | year=1957}}
| |
| *{{Citation
| |
| | last1 = Crandall | first1 = Richard
| |
| | last2 = Pomerance | first2 = Carl
| |
| | title = Prime Numbers: A Computational Perspective
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = New York
| |
| | year = 2001
| |
| | isbn = 0-387-94777-9}}
| |
| *{{Citation
| |
| | last1 = Graham | first1 = Ronald L.
| |
| | last2 = Knuth | first2 = Donald E.
| |
| | last3 = Patashnik | first3 = Oren
| |
| | title = Concrete Mathematics
| |
| | publisher = Addison-Wesley
| |
| | location = Reading Ma.
| |
| | year = 1994
| |
| | isbn = 0-201-55802-5}}
| |
| *{{Citation
| |
| | last1 = Hardy | first1 = G. H.
| |
| | last2 = Wright | first2 = E. M.
| |
| | title = An Introduction to the Theory of Numbers (Fifth edition)
| |
| | publisher = [[Oxford University Press]]
| |
| | location = Oxford
| |
| | year = 1980
| |
| | isbn = 978-0-19-853171-5}}
| |
| *Nicholas J. Higham, ''Handbook of writing for the mathematical sciences'', SIAM. ISBN 0-89871-420-6, p. 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. 43.
| |
| *{{Citation
| |
| | last1 = Iverson | first1 = Kenneth E.
| |
| | title = A Programming Language
| |
| |publisher = Wiley
| |
| | year = 1962}}
| |
| *{{Citation
| |
| | last1 = Lemmermeyer | first1 = Franz
| |
| | title = Reciprocity Laws: from Euler to Eisenstein
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = Berlin
| |
| | year = 2000
| |
| | isbn = 3-540-66957-4}}
| |
| *{{Citation
| |
| | last1 = Ramanujan | first1 = Srinivasa
| |
| | title = Collected Papers
| |
| | publisher = AMS / Chelsea
| |
| | location = Providence RI
| |
| | year = 2000
| |
| | isbn = 978-0-8218-2076-6}}
| |
| *{{Citation
| |
| | last1 = Ribenboim | first1 = Paulo
| |
| | title = The New Book of Prime Number Records
| |
| | publisher = Springer
| |
| | location = New York
| |
| | year = 1996
| |
| | isbn = 0-387-94457-5}}
| |
| *Michael Sullivan. ''Precalculus'', 8th edition, p. 86
| |
| *{{Citation
| |
| | last1 = Titchmarsh | first1 = Edward Charles
| |
| | last2 = Heath-Brown | first2 = David Rodney ("Roger")
| |
| | title = The Theory of the Riemann Zeta-function
| |
| | publisher = Oxford U. P.
| |
| | edition = 2nd
| |
| | location = Oxford
| |
| | year = 1986
| |
| | isbn = 0-19-853369-1}}
| |
| | |
| ==External links==
| |
| {{commonscat|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]]
| |