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Floor and ceiling functions

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Nearest integers from a number

Floor and ceiling functions

Floor function

Ceiling function

Inmathematics, thefloor function is thefunction that takes as input areal numberx, and gives as output the greatestinteger less than or equal tox, denoted⌊x⌋ orfloor(x). Similarly, theceiling function mapsx to the least integer greater than or equal tox, denoted⌈x⌉ orceil(x).^[1]

For example, for floor:⌊2.4⌋ = 2,⌊−2.4⌋ = −3, and for ceiling:⌈2.4⌉ = 3, and⌈−2.4⌉ = −2.

The floor ofx is also called theintegral part,integer part,greatest integer, orentier ofx, and was historically denoted[x] (among other notations).^[2] However, the same term,integer part, is also used fortruncation towards zero, which differs from the floor function for negative numbers.

For an integern,⌊n⌋ = ⌈n⌉ =n.

Althoughfloor(x + 1) andceil(x) produce graphs that appear exactly alike, they are not the same when the value ofx is an exact integer. For example, whenx = 2.0001,⌊2.0001 + 1⌋ = ⌈2.0001⌉ = 3. However, ifx = 2, then⌊2 + 1⌋ = 3, while⌈2⌉ = 2.

Examples
x	Floor⌊x⌋	Ceiling⌈x⌉	Fractional part{x}
2	2	2	0
2.0001	2	3	0.0001
2.4	2	3	0.4
2.9	2	3	0.9
2.999	2	3	0.999
−2.7	−3	−2	0.3
−2	−2	−2	0

Notation

Theintegral part orinteger part of a number (partie entière in the original) was first defined in 1798 byAdrien-Marie Legendre in his proof of theLegendre's formula.

Carl Friedrich Gauss introduced the square bracket notation[x] in his third proof ofquadratic reciprocity (1808).^[3] This remained the standard^[4] in mathematics untilKenneth E. Iverson introduced, in his 1962 bookA Programming Language, the names "floor" and "ceiling" and the corresponding notations⌊x⌋ and⌈x⌉.^[5]^[6] (Iverson used square brackets for a different purpose, theIverson 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⟦x⟧ are used for floor, and reversed brackets⟧x⟦ or]x[ for ceiling.^[7]^[8]

Thefractional part is thesawtooth function, denoted by{x} for realx and defined by the formula

{x} =x − ⌊x⌋^[9]

For allx,

0 ≤ {x} < 1.

These characters are provided in Unicode:

U+2308 ⌈LEFT CEILING (&lceil;, &LeftCeiling;)
U+2309 ⌉RIGHT CEILING (&rceil;, &RightCeiling;)
U+230A ⌊LEFT FLOOR (&LeftFloor;, &lfloor;)
U+230B ⌋RIGHT FLOOR (&rfloor;, &RightFloor;)

In theLaTeX typesetting system, these symbols can be specified with the\lceil, \rceil, \lfloor, and\rfloor commands in math mode. LaTeX has supported UTF-8 since 2018, so the Unicode characters can now be used directly.^[10] Larger versions are\left\lceil, \right\rceil, \left\lfloor, and\right\rfloor.

Definition and properties

Given real numbersx andy, integersm andn and the set ofintegers $\mathbb {Z}$ , floor and ceiling may be defined by the equations

\lfloor x\rfloor =\max\{m\in \mathbb {Z} \mid m\leq x\},

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

Since there is exactly one integer in ahalf-open interval of length one, for any real numberx, there are unique integersm andn satisfying the equation

x-1<m\leq x\leq n<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.^[11]

{\begin{alignedat}{3}\lfloor x\rfloor &=m\ \ &&{\mbox{ if and only if }}&m&\leq x<m+1,\\\lceil x\rceil &=n&&{\mbox{ if and only if }}&\ \ n-1&<x\leq n,\\\lfloor x\rfloor &=m&&{\mbox{ if and only if }}&x-1&<m\leq x,\\\lceil x\rceil &=n&&{\mbox{ if and only if }}&x&\leq n<x+1.\end{alignedat}}

In the language oforder theory, the floor function is aresiduated mapping, that is, part of aGalois 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 an integern 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 are never true ifn is not an integer; however, for everyx andy, the following inequalities hold:

{\begin{aligned}\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{aligned}}

Monotonicity

Both floor and ceiling functions aremonotonically non-decreasing functions:

{\begin{aligned}x_{1}\leq x_{2}&\Rightarrow \lfloor x_{1}\rfloor \leq \lfloor x_{2}\rfloor ,\\x_{1}\leq x_{2}&\Rightarrow \lceil x_{1}\rceil \leq \lceil x_{2}\rceil .\end{aligned}}

Relations among the functions

It is clear from the definitions that

\lfloor x\rfloor \leq \lceil x\rceil ,

with equality if and only ifx 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}}

In fact, for integersn, both floor and ceiling functions are theidentity:

\lfloor n\rfloor =\lceil n\rceil =n.

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

{\begin{aligned}\lfloor x\rfloor +\lceil -x\rceil &=0\\-\lfloor x\rfloor &=\lceil -x\rceil \\-\lceil x\rceil &=\lfloor -x\rfloor \end{aligned}}

and:

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

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

Negating the argument complements the fractional part:

\{x\}+\{-x\}={\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 areidempotent:

{\begin{aligned}{\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{aligned}}

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

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

due to the identity property for integers.

Quotients

Ifm andn are integers andn ≠ 0,

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

Ifn is positive^[12]

\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 .

Ifm is positive^[13]

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

Form = 2 these imply

n=\left\lfloor {\frac {n{\vphantom {1}}}{2}}\right\rfloor +\left\lceil {\frac {n{\vphantom {1}}}{2}}\right\rceil .

More generally,^[14] for positivem (SeeHermite's identity)

\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 .

The following can be used to convert floors to ceilings and vice versa (withm being positive)^[15]

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

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

For allm andn strictly positive integers:^[16]

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

which, for positive andcoprimem andn, reduces to

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

and similarly for the ceiling and fractional part functions (still for positive andcoprimem andn),

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

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

Since the right-hand side of the general case is symmetrical inm andn, this implies that

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

More generally, ifm andn are positive,

{\begin{aligned}&\left\lfloor {\frac {x{\vphantom {1}}}{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{\vphantom {1}}}{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{aligned}}

This is sometimes called areciprocity law.^[17]

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

m\leq \left\lfloor {\frac {x}{n}}\right\rfloor \iff n\leq \left\lfloor {\frac {x}{m}}\right\rfloor \iff n\leq {\frac {\lfloor x\rfloor }{m}}.

Similarly,

m\geq \left\lceil {\frac {x}{n}}\right\rceil \iff n\geq \left\lceil {\frac {x}{m}}\right\rceil \iff n\geq {\frac {\lceil x\rceil }{m}}.

Indeed,

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

keeping in mind that ${\textstyle \lfloor x/n\rfloor ={\bigl \lfloor }\lfloor x\rfloor /n{\bigr \rfloor }.}$ The second equivalence involving the ceiling function can be proved similarly.

Nested divisions

For positive integern, and arbitrary real numbersm,x:^[18]

\left\lfloor {\frac {\lfloor x/m\rfloor }{n}}\right\rfloor =\left\lfloor {\frac {x}{mn}}\right\rfloor

\left\lceil {\frac {\lceil x/m\rceil }{n}}\right\rceil =\left\lceil {\frac {x}{mn}}\right\rceil .

Continuity and series expansions

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

$\lfloor x\rfloor$ isupper semi-continuous and $\lceil x\rceil$ and $\{x\}$ are lower semi-continuous.

Since none of the functions discussed in this article are continuous, none of them have apower series expansion. Since floor and ceiling are not periodic, they do not have uniformly convergentFourier series expansions. The fractional part function has Fourier series expansion^[19] $\{x\}={\frac {1}{2}}-{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}$ forx 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: fory fixed andx a multiple ofy the Fourier series given converges toy/2, rather than tox mod y = 0. At points of continuity the series converges to the true value.

Using the formula $\lfloor x\rfloor =x-\{x\}$ gives $\lfloor x\rfloor =x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}$ forx not an integer.

Applications

Mod operator

For an integerx and a positive integery, themodulo operation, denoted byx mody, gives the value of the remainder whenx is divided byy. This definition can be extended to realx andy,y ≠ 0, by the formula

x{\bmod {y}}=x-y\left\lfloor {\frac {x}{y}}\right\rfloor .

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

ify is positive,

0\leq x{\bmod {y}}<y,

and ify is negative,

0\geq x{\bmod {y}}>y.

Quadratic reciprocity

Gauss's third proof ofquadratic reciprocity, as modified by Eisenstein, has two basic steps.^[20]^[21]

Letp andq be distinct positive odd prime numbers, and let $m={\tfrac {1}{2}}(p-1),$ $n={\tfrac {1}{2}}(q-1).$

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

{\begin{aligned}\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{aligned}}

The second step is to use ageometric 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 primesp:^[22]

{\begin{aligned}\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{aligned}}

Rounding

For an arbitrary real number $x {\displaystyle x}$ ,rounding $x {\displaystyle x}$ to the nearest integer withtie breaking towards positive infinity is given by ${\text{rpi}}(x)=\left\lfloor x+{\tfrac {1}{2}}\right\rfloor =\left\lceil {\tfrac {1}{2}}\lfloor 2x\rfloor \right\rceil$ ; rounding towards negative infinity is given as ${\text{rni}}(x)=\left\lceil x-{\tfrac {1}{2}}\right\rceil =\left\lfloor {\tfrac {1}{2}}\lceil 2x\rceil \right\rfloor$ .

If tie-breaking is away from 0, then the rounding function is ${\text{ri}}(x)=\operatorname {sgn}(x)\left\lfloor |x|+{\tfrac {1}{2}}\right\rfloor$ (seesign function), androunding towards even can be expressed with the more cumbersome $\lfloor x\rceil =\left\lfloor x+{\tfrac {1}{2}}\right\rfloor +{\bigl \lceil }{\tfrac {1}{4}}(2x-1){\bigr \rceil }-{\bigl \lfloor }{\tfrac {1}{4}}(2x-1){\bigr \rfloor }-1$ , which is the above expression for rounding towards positive infinity ${\text{rpi}}(x)$ minus anintegrality indicator for ${\tfrac {1}{4}}(2x-1)$ .

Rounding areal number $x {\displaystyle x}$ to the nearest integer value forms a very basic type ofquantizer – auniform one. A typical (mid-tread) uniform quantizer with a quantizationstep size equal to some value $\Delta$ can be expressed as

Q(x)=\Delta \cdot \left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor

,

Number of digits

The number of digits inbaseb of a positive integerk is

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

Number of strings without repeated characters

The number of possiblestrings of arbitrary length that doesn't use any character twice is given by^[23]^{[better source needed]}

(n)_{0}+\cdots +(n)_{n}=\lfloor en!\rfloor

where:

n > 0 is the number of letters in the alphabet (e.g., 26 inEnglish)
thefalling factorial $(n)_{k}=n(n-1)\cdots (n-k+1)$ denotes the number of strings of lengthk that don't use any character twice.
n! denotes thefactorial ofn
e = 2.718... isEuler's number

Forn = 26, this comes out to 1096259850353149530222034277.

Factors of factorials

Letn be a positive integer andp a positive prime number. The exponent of the highest power ofp that dividesn! is given by a version ofLegendre's formula^[24]

\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}}

where ${\textstyle n=\sum _{k}a_{k}p^{k}}$ is the way of writingn in basep. This is a finite sum, since the floors are zero whenp^k >n.

Beatty sequence

TheBeatty sequence shows how every positiveirrational number gives rise to a partition of thenatural numbers into two sequences via the floor function.^[25]

Euler's constant (γ)

There are formulas forEuler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.^[26]

\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}}-\cdots -{\tfrac {1}{15}}\right)+\cdots

Riemann zeta function (ζ)

The fractional part function also shows up in integral representations of theRiemann zeta function. It is straightforward to prove (using integration by parts)^[27] that if $\varphi (x)$ is any function with a continuous derivative in the closed interval [a,b],

\sum _{a<n\leq b}\varphi (n)=\int _{a}^{b}\varphi (x)\,dx+\int _{a}^{b}\left(\{x\}-{\tfrac {1}{2}}\right)\varphi '(x)\,dx+\left(\{a\}-{\tfrac {1}{2}}\right)\varphi (a)-\left(\{b\}-{\tfrac {1}{2}}\right)\varphi (b).

Letting $\varphi (n)=n^{-s}$ forreal part ofs greater than 1 and lettinga andb be integers, and lettingb 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 alls with real part greater than −1, (excepts = 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.^[28]

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

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

In 1947van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.^[29]

Formulas for prime numbers

The floor function appears in several formulas characterizing prime numbers. For example, since ${\textstyle {\bigl \lfloor }{\frac {n}{m}}{\bigr \rfloor }-{\bigl \lfloor }{\frac {n-1}{m}}{\bigr \rfloor }}$ is equal to 1 ifm dividesn, and to 0 otherwise, it follows that a positive integern is a primeif and only if^[30]

\sum _{m=1}^{\infty }\left({\biggl \lfloor }{\frac {n}{m}}{\biggr \rfloor }-{\biggl \lfloor }{\frac {n-1}{m}}{\biggr \rfloor }\right)=2.

One may also give formulas for producing the prime numbers. For example, letp_n be then-th prime, and for any integerr > 1, define the real numberα by the sum

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

Then^[31]

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

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

\left\lfloor \theta ^{3}\right\rfloor ,\left\lfloor \theta ^{9}\right\rfloor ,\left\lfloor \theta ^{27}\right\rfloor ,\dots

are all prime.^[32]

There is also a numberω = 1.9287800... with the property that

\left\lfloor 2^{\omega }\right\rfloor ,\left\lfloor 2^{2^{\omega }}\right\rfloor ,\left\lfloor 2^{2^{2^{\omega }}}\right\rfloor ,\dots

are all prime.^[32]

Letπ(x) be the number of primes less than or equal tox. It is a straightforward deduction fromWilson's theorem that^[33]

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

Also, ifn ≥ 2,^[34]

\pi (n)=\sum _{j=2}^{n}{\Biggl \lfloor }1\,{\bigg /}\ {\sum _{k=2}^{j}\left\lfloor \left\lfloor {\frac {j}{k}}\right\rfloor {\frac {k}{j}}\right\rfloor }{\Biggr \rfloor }.

None of the formulas in this section are of any practical use.^[35]^[36]

Solved problems

Ramanujan submitted these problems to theJournal of the Indian Mathematical Society.^[37]

Ifn is a positive integer, prove that

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

Some generalizations to the above floor function identities have been proven.^[38]

Unsolved problem

The study ofWaring's problem has led to an unsolved problem:

Are there any positive integersk ≥ 6 such that^[39]

3^{k}-2^{k}{\Bigl \lfloor }{\bigl (}{\tfrac {3}{2}}{\bigr )}^{k}{\Bigr \rfloor }>2^{k}-{\Bigl \lfloor }{\bigl (}{\tfrac {3}{2}}{\bigr )}^{k}{\Bigr \rfloor }-2\ ?

Mahler has proved there can only be a finite number of suchk; none are known.^[40]

Computer implementations

Int function from floating-point conversion inC

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 usedones' complement and truncation was simpler to implement (floor is simpler intwo'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.^{[citation needed]}

Anarithmetic right-shift of a signed integer $x {\displaystyle x}$ by $n {\displaystyle n}$ is the same as $\left\lfloor x/2^{n}\right\rfloor$ . 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.^{[citation needed]}

Many programming languages (includingC,C++,^[41]^[42]C#,^[43]^[44]Java,^[45]^[46]Julia,^[47]PHP,^[48]^[49]R,^[50] andPython^[51]) provide standard functions for floor and ceiling, usually calledfloor andceil, or less commonlyceiling.^[52] The languageAPL uses⌊x for floor. TheJ Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses<. for floor and>. for ceiling.^[53]ALGOL usesentier for floor.

InMicrosoft Excel the functionINT rounds down rather than toward zero,^[54] whileFLOOR rounds toward zero, the opposite of what "int" and "floor" do in other languages. Since 2010FLOOR has been changed to error if the number is negative.^[55] TheOpenDocument file format, as used byOpenOffice.org,Libreoffice and others,INT^[56] andFLOOR both do floor, andFLOOR has a third argument to reproduce Excel's earlier behavior.^[57]

See also

Citations

^Graham, Knuth, & Patashnik, Ch. 3.1
^1) Luke Heaton,A Brief History of Mathematical Thought, 2015,ISBN 1472117158 (n.p.)
2) Albert A. Blanket al.,Calculus: Differential Calculus, 1968, p. 259
3) John W. Warris, Horst Stocker,Handbook of mathematics and computational science, 1998,ISBN 0387947469, p. 151
^Lemmermeyer, pp. 10, 23.
^e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation. Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's.
^Iverson, p. 12.
^Higham, p. 25.
^Mathwords: Floor Function.
^Mathwords: Ceiling Function
^Graham, Knuth, & Patashnik, p. 70.
^"LaTeX News, Issue 28"(PDF; 379 KB). The LaTeX Project. April 2018. Retrieved27 July 2024.
^Graham, Knuth, & Patashink, Ch. 3
^Graham, Knuth, & Patashnik, p. 73
^Graham, Knuth, & Patashnik, p. 85
^Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15
^Graham, Knuth, & Patashnik, Ex. 3.12
^Graham, Knuth, & Patashnik, p. 94.
^Graham, Knuth, & Patashnik, p. 94
^Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with x/m as input and the division by n as function
^Titchmarsh, p. 15, Eq. 2.1.7
^Lemmermeyer, § 1.4, Ex. 1.32–1.33
^Hardy & Wright, §§ 6.11–6.13
^Lemmermeyer, p. 25
^OEIS sequence A000522 (Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.) (See Formulas.)
^Hardy & Wright, Th. 416
^Graham, Knuth, & Patashnik, pp. 77–78
^These formulas are from the Wikipedia articleEuler's constant, which has many more.
^Titchmarsh, p. 13
^Titchmarsh, pp.14–15
^Crandall & Pomerance, p. 391
^Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced withn. An equivalent condition isn > 1 is prime if and only if ${\textstyle \sum _{m=1}^{\lfloor {\sqrt {n}}\rfloor }{\bigl (}{\bigl \lfloor }{\frac {n}{m}}{\bigr \rfloor }-{\bigl \lfloor }{\frac {n-1}{m}}{\bigr \rfloor }{\bigr )}=1}$ .
^Hardy & Wright, § 22.3
^^a ^bRibenboim, p. 186
^Ribenboim, p. 181
^Crandall & Pomerance, Ex. 1.4, p. 46
^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, Question 723,Papers p. 332
^Somu, Sai Teja; Kukla, Andrzej (2022)."On some generalizations to floor function identities of Ramanujan"(PDF).Integers.22.arXiv:2109.03680.
^Hardy & Wright, p. 337
^Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II".Mathematika.4 (2):122–124.doi:10.1112/S0025579300001170.
^"C++ reference offloor function". Retrieved5 December 2010.
^"C++ reference ofceil function". Retrieved5 December 2010.
^dotnet-bot."Math.Floor Method (System)".docs.microsoft.com. Retrieved28 November 2019.
^dotnet-bot."Math.Ceiling Method (System)".docs.microsoft.com. Retrieved28 November 2019.
^"Math (Java SE 9 & JDK 9 )".docs.oracle.com. Retrieved20 November 2018.
^"Math (Java SE 9 & JDK 9 )".docs.oracle.com. Retrieved20 November 2018.
^"Math (Julia v1.10)".docs.julialang.org/en/v1/. Retrieved4 September 2024.
^"PHP manual forceil function". Retrieved18 July 2013.
^"PHP manual forfloor function". Retrieved18 July 2013.
^"R: Rounding of Numbers".
^"Python manual formath module". Retrieved18 July 2013.
^Sullivan, p. 86.
^"Vocabulary".J Language. Retrieved6 September 2011.
^"INT function". Retrieved29 October 2021.
^"FLOOR function". Retrieved29 October 2021.
^"Documentation/How Tos/Calc: INT function". Retrieved29 October 2021.
^"Documentation/How Tos/Calc: FLOOR function". Retrieved29 October 2021.

References

J.W.S. Cassels (1957),An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45,Cambridge University Press
Crandall, Richard; Pomerance, Carl (2001),Prime Numbers: A Computational Perspective, New York:Springer,ISBN 0-387-94777-9
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994),Concrete Mathematics, 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 978-0-19-853171-5
Nicholas J. Higham,Handbook of writing for the mathematical sciences, SIAM.ISBN 0-89871-420-6, 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 3-540-66957-4
Ramanujan, Srinivasa (2000),Collected Papers, Providence RI: AMS / Chelsea,ISBN 978-0-8218-2076-6
Ribenboim, Paulo (1996),The New Book of Prime Number Records, New York: Springer,ISBN 0-387-94457-5
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

Wikimedia Commons has media related toFloor and ceiling functions.

"Floor function",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
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 24 October 2008
Weisstein, Eric W."Floor Function".MathWorld.
Weisstein, Eric W."Ceiling Function".MathWorld.

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