In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less ...
Floorandceilingfunctions
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Jumptonavigation
Jumptosearch
Mathematicalfunctionsroundinganumbertothetwoclosestintegers
Forotheruses,seeFloor(disambiguation)andCeiling(disambiguation).
FloorandceilingfunctionsFloorfunctionCeilingfunction
Inmathematicsandcomputerscience,thefloorfunctionisthefunctionthattakesasinputarealnumberx,andgivesasoutputthegreatestintegerlessthanorequaltox,denotedfloor(x)or⌊x⌋.Similarly,theceilingfunctionmapsxtotheleastintegergreaterthanorequaltox,denotedceil(x)or⌈x⌉.[1]
Forexample,⌊2.4⌋=2,⌊−2.4⌋=−3,⌈2.4⌉=3,and⌈−2.4⌉=−2.
Theintegralpartorintegerpartofx,oftendenoted[x]isusuallydefinedasthe⌊x⌋ifxisnonnegative,and⌈x⌉otherwise.Forexample,[2.4]=2and[−2.4]=−2.Theoperationoftruncationgeneralizesthistoaspecifiednumberofdigits:truncationtozerosignificantdigitsisthesameastheintegerpart.
Someauthorsdefinetheintegerpartasthefloorregardlessofthesignofx,usingavarietyofnotationsforthis.[2]
Fornaninteger,⌊n⌋=⌈n⌉=[n]=n.
Examples
x
Floor⌊x⌋
Ceiling⌈x⌉
Fractionalpart{x}
2
2
2
0
2.4
2
3
0.4
2.9
2
3
0.9
−2.7
−3
−2
0.3
−2
−2
−2
0
Contents
1Notation
2Definitionandproperties
2.1Equivalences
2.2Relationsamongthefunctions
2.3Quotients
2.4Nesteddivisions
2.5Continuityandseriesexpansions
3Applications
3.1Modoperator
3.2Quadraticreciprocity
3.3Rounding
3.4Numberofdigits
3.5Factorsoffactorials
3.6Beattysequence
3.7Euler'sconstant(γ)
3.8Riemannzetafunction(ζ)
3.9Formulasforprimenumbers
3.10Solvedproblems
3.11Unsolvedproblem
4Computerimplementations
5Seealso
6Notes
7References
8Externallinks
Notation[edit]
Theintegralpartorintegerpartofanumber(partieentièreintheoriginal)wasfirstdefinedin1798byAdrien-MarieLegendreinhisproofoftheLegendre'sformula.
CarlFriedrichGaussintroducedthesquarebracketnotation[x]inhisthirdproofofquadraticreciprocity(1808).[3]Thisremainedthestandard[4]inmathematicsuntilKennethE.Iversonintroduced,inhis1962bookAProgrammingLanguage,thenames"floor"and"ceiling"andthecorrespondingnotations⌊x⌋and⌈x⌉.[5][6]Bothnotationsarenowusedinmathematics,[7]althoughIverson'snotationwillbefollowedinthisarticle.
Insomesources,boldfaceordoublebrackets⟦x⟧areusedforfloor,andreversedbrackets⟧x⟦or]x[forceiling.[8][9]Sometimes[x]istakentomeantheround-toward-zerofunction.[citationneeded]
Thefractionalpartisthesawtoothfunction,denotedby{x}forrealxanddefinedbytheformula
{x}=x-⌊x⌋[10]
Forallx,
0≤{x}<1
ThesecharactersareprovidedinUnicode:
U+2308⌈LEFTCEILING(HTML ⌈ ·⌈,⌈)
U+2309⌉RIGHTCEILING(HTML ⌉ ·⌉,⌉)
U+230A⌊LEFTFLOOR(HTML ⌊ ·⌊,⌊)
U+230B⌋RIGHTFLOOR(HTML ⌋ ·⌋,⌋)
IntheLaTeXtypesettingsystem,thesesymbolscanbespecifiedwiththe\lfloor,\rfloor,\lceiland\rceilcommandsinmathmode,andextendedinsizeusing\left\lfloor,\right\rfloor,\left\lceiland\right\rceilasneeded.
Definitionandproperties[edit]
Givenrealnumbersxandy,integersk,m,nandthesetofintegers
Z
{\displaystyle\mathbb{Z}}
,floorandceilingmaybedefinedbytheequations
⌊
x
⌋
=
max
{
m
∈
Z
∣
m
≤
x
}
,
{\displaystyle\lfloorx\rfloor=\max\{m\in\mathbb{Z}\midm\leqx\},}
⌈
x
⌉
=
min
{
n
∈
Z
∣
n
≥
x
}
.
{\displaystyle\lceilx\rceil=\min\{n\in\mathbb{Z}\midn\geqx\}.}
Sincethereisexactlyoneintegerinahalf-openintervaloflengthone,foranyrealnumberx,thereareuniqueintegersmandnsatisfyingtheequation
x
−
1
<
m
≤
x
≤
n
<
x
+
1.
{\displaystylex-1
y
.
{\displaystyle0\geqx{\bmod{y}}>y.}
Quadraticreciprocity[edit]
Gauss'sthirdproofofquadraticreciprocity,asmodifiedbyEisenstein,hastwobasicsteps.[20][21]
Letpandqbedistinctpositiveoddprimenumbers,andlet
m
=
p
−
1
2
,
{\displaystylem={\frac{p-1}{2}},}
n
=
q
−
1
2
.
{\displaystylen={\frac{q-1}{2}}.}
First,Gauss'slemmaisusedtoshowthattheLegendresymbolsaregivenby
(
q
p
)
=
(
−
1
)
⌊
q
p
⌋
+
⌊
2
q
p
⌋
+
⋯
+
⌊
m
q
p
⌋
{\displaystyle\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
(
p
q
)
=
(
−
1
)
⌊
p
q
⌋
+
⌊
2
p
q
⌋
+
⋯
+
⌊
n
p
q
⌋
.
{\displaystyle\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}.}
Thesecondstepistouseageometricargumenttoshowthat
⌊
q
p
⌋
+
⌊
2
q
p
⌋
+
⋯
+
⌊
m
q
p
⌋
+
⌊
p
q
⌋
+
⌊
2
p
q
⌋
+
⋯
+
⌊
n
p
q
⌋
=
m
n
.
{\displaystyle\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.}
Combiningtheseformulasgivesquadraticreciprocityintheform
(
p
q
)
(
q
p
)
=
(
−
1
)
m
n
=
(
−
1
)
p
−
1
2
q
−
1
2
.
{\displaystyle\left({\frac{p}{q}}\right)\left({\frac{q}{p}}\right)=(-1)^{mn}=(-1)^{{\frac{p-1}{2}}{\frac{q-1}{2}}}.}
Thereareformulasthatusefloortoexpressthequadraticcharacterofsmallnumbersmododdprimesp:[22]
(
2
p
)
=
(
−
1
)
⌊
p
+
1
4
⌋
,
{\displaystyle\left({\frac{2}{p}}\right)=(-1)^{\left\lfloor{\frac{p+1}{4}}\right\rfloor},}
(
3
p
)
=
(
−
1
)
⌊
p
+
1
6
⌋
.
{\displaystyle\left({\frac{3}{p}}\right)=(-1)^{\left\lfloor{\frac{p+1}{6}}\right\rfloor}.}
Rounding[edit]
Foranarbitraryrealnumber
x
{\displaystylex}
,rounding
x
{\displaystylex}
tothenearestintegerwithtiebreakingtowardspositiveinfinityisgivenby
rpi
(
x
)
=
⌊
x
+
1
2
⌋
=
⌈
⌊
2
x
⌋
2
⌉
{\displaystyle{\text{rpi}}(x)=\left\lfloorx+{\tfrac{1}{2}}\right\rfloor=\left\lceil{\tfrac{\lfloor2x\rfloor}{2}}\right\rceil}
;roundingtowardsnegativeinfinityisgivenas
rni
(
x
)
=
⌈
x
−
1
2
⌉
=
⌊
⌈
2
x
⌉
2
⌋
{\displaystyle{\text{rni}}(x)=\left\lceilx-{\tfrac{1}{2}}\right\rceil=\left\lfloor{\tfrac{\lceil2x\rceil}{2}}\right\rfloor}
.
Iftie-breakingisawayfrom0,thentheroundingfunctionis
ri
(
x
)
=
sgn
(
x
)
⌊
|
x
|
+
1
2
⌋
{\displaystyle{\text{ri}}(x)=\operatorname{sgn}(x)\left\lfloor|x|+{\tfrac{1}{2}}\right\rfloor}
,androundingtowardsevencanbeexpressedwiththemorecumbersome
⌊
x
⌉
=
⌊
x
+
1
2
⌋
+
⌈
2
x
−
1
4
⌉
−
⌊
2
x
−
1
4
⌋
−
1
{\displaystyle\lfloorx\rceil=\left\lfloorx+{\tfrac{1}{2}}\right\rfloor+\left\lceil{\tfrac{2x-1}{4}}\right\rceil-\left\lfloor{\tfrac{2x-1}{4}}\right\rfloor-1}
,whichistheaboveexpressionforroundingtowardspositiveinfinity
rpi
(
x
)
{\displaystyle{\text{rpi}}(x)}
minusanintegralityindicatorfor
2
x
−
1
4
{\displaystyle{\tfrac{2x-1}{4}}}
.
Numberofdigits[edit]
Thenumberofdigitsinbasebofapositiveintegerkis
⌊
log
b
k
⌋
+
1
=
⌈
log
b
(
k
+
1
)
⌉
.
{\displaystyle\lfloor\log_{b}{k}\rfloor+1=\lceil\log_{b}{(k+1)}\rceil.}
Factorsoffactorials[edit]
Letnbeapositiveintegerandpapositiveprimenumber.Theexponentofthehighestpowerofpthatdividesn!isgivenbyaversionofLegendre'sformula[23]
⌊
n
p
⌋
+
⌊
n
p
2
⌋
+
⌊
n
p
3
⌋
+
⋯
=
n
−
∑
k
a
k
p
−
1
{\displaystyle\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
n
=
∑
k
a
k
p
k
{\textstylen=\sum_{k}a_{k}p^{k}}
isthewayofwritingninbasep.Thisisafinitesum,sincethefloorsarezerowhenpk>n.
Beattysequence[edit]
TheBeattysequenceshowshoweverypositiveirrationalnumbergivesrisetoapartitionofthenaturalnumbersintotwosequencesviathefloorfunction.[24]
Euler'sconstant(γ)[edit]
ThereareformulasforEuler'sconstantγ=0.5772156649...thatinvolvethefloorandceiling,e.g.[25]
γ
=
∫
1
∞
(
1
⌊
x
⌋
−
1
x
)
d
x
,
{\displaystyle\gamma=\int_{1}^{\infty}\left({1\over\lfloorx\rfloor}-{1\overx}\right)\,dx,}
γ
=
lim
n
→
∞
1
n
∑
k
=
1
n
(
⌈
n
k
⌉
−
n
k
)
,
{\displaystyle\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
γ
=
∑
k
=
2
∞
(
−
1
)
k
⌊
log
2
k
⌋
k
=
1
2
−
1
3
+
2
(
1
4
−
1
5
+
1
6
−
1
7
)
+
3
(
1
8
−
⋯
−
1
15
)
+
⋯
{\displaystyle\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}
Riemannzetafunction(ζ)[edit]
ThefractionalpartfunctionalsoshowsupinintegralrepresentationsoftheRiemannzetafunction.Itisstraightforwardtoprove(usingintegrationbyparts)[26]thatif
ϕ
(
x
)
{\displaystyle\phi(x)}
isanyfunctionwithacontinuousderivativeintheclosedinterval[a,b],
∑
a
<
n
≤
b
ϕ
(
n
)
=
∫
a
b
ϕ
(
x
)
d
x
+
∫
a
b
(
{
x
}
−
1
2
)
ϕ
′
(
x
)
d
x
+
(
{
a
}
−
1
2
)
ϕ
(
a
)
−
(
{
b
}
−
1
2
)
ϕ
(
b
)
.
{\displaystyle\sum_{a1,definetherealnumberαbythesum
α
=
∑
m
=
1
∞
p
m
r
−
m
2
.
{\displaystyle\alpha=\sum_{m=1}^{\infty}p_{m}r^{-m^{2}}.}
Then[30]
p
n
=
⌊
r
n
2
α
⌋
−
r
2
n
−
1
⌊
r
(
n
−
1
)
2
α
⌋
.
{\displaystylep_{n}=\left\lfloorr^{n^{2}}\alpha\right\rfloor-r^{2n-1}\left\lfloorr^{(n-1)^{2}}\alpha\right\rfloor.}
Asimilarresultisthatthereisanumberθ=1.3064...(Mills'constant)withthepropertythat
⌊
θ
3
⌋
,
⌊
θ
9
⌋
,
⌊
θ
27
⌋
,
…
{\displaystyle\left\lfloor\theta^{3}\right\rfloor,\left\lfloor\theta^{9}\right\rfloor,\left\lfloor\theta^{27}\right\rfloor,\dots}
areallprime.[31]
Thereisalsoanumberω=1.9287800...withthepropertythat
⌊
2
ω
⌋
,
⌊
2
2
ω
⌋
,
⌊
2
2
2
ω
⌋
,
…
{\displaystyle\left\lfloor2^{\omega}\right\rfloor,\left\lfloor2^{2^{\omega}}\right\rfloor,\left\lfloor2^{2^{2^{\omega}}}\right\rfloor,\dots}
areallprime.[31]
Letπ(x)bethenumberofprimeslessthanorequaltox.ItisastraightforwarddeductionfromWilson'stheoremthat[32]
π
(
n
)
=
∑
j
=
2
n
⌊
(
j
−
1
)
!
+
1
j
−
⌊
(
j
−
1
)
!
j
⌋
⌋
.
{\displaystyle\pi(n)=\sum_{j=2}^{n}\left\lfloor{\frac{(j-1)!+1}{j}}-\left\lfloor{\frac{(j-1)!}{j}}\right\rfloor\right\rfloor.}
Also,ifn≥2,[33]
π
(
n
)
=
∑
j
=
2
n
⌊
1
∑
k
=
2
j
⌊
⌊
j
k
⌋
k
j
⌋
⌋
.
{\displaystyle\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.}
Noneoftheformulasinthissectionareofanypracticaluse.[34][35]
Solvedproblems[edit]
RamanujansubmittedtheseproblemstotheJournaloftheIndianMathematicalSociety.[36]
Ifnisapositiveinteger,provethat
⌊
n
3
⌋
+
⌊
n
+
2
6
⌋
+
⌊
n
+
4
6
⌋
=
⌊
n
2
⌋
+
⌊
n
+
3
6
⌋
,
{\displaystyle\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,}
⌊
1
2
+
n
+
1
2
⌋
=
⌊
1
2
+
n
+
1
4
⌋
,
{\displaystyle\left\lfloor{\tfrac{1}{2}}+{\sqrt{n+{\tfrac{1}{2}}}}\right\rfloor=\left\lfloor{\tfrac{1}{2}}+{\sqrt{n+{\tfrac{1}{4}}}}\right\rfloor,}
⌊
n
+
n
+
1
⌋
=
⌊
4
n
+
2
⌋
.
{\displaystyle\left\lfloor{\sqrt{n}}+{\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+2}}\right\rfloor.}
Unsolvedproblem[edit]
ThestudyofWaring'sproblemhasledtoanunsolvedproblem:
Arethereanypositiveintegersk≥6suchthat[37]
3
k
−
2
k
⌊
(
3
2
)
k
⌋
>
2
k
−
⌊
(
3
2
)
k
⌋
−
2
{\displaystyle3^{k}-2^{k}\left\lfloor\left({\tfrac{3}{2}}\right)^{k}\right\rfloor>2^{k}-\left\lfloor\left({\tfrac{3}{2}}\right)^{k}\right\rfloor-2}
?
Mahler[38]hasprovedtherecanonlybeafinitenumberofsuchk;noneareknown.
Computerimplementations[edit]
Intfunctionfromfloating-pointconversioninC
Inmostprogramminglanguages,thesimplestmethodtoconvertafloatingpointnumbertoanintegerdoesnotdofloororceiling,buttruncation.Thereasonforthisishistorical,asthefirstmachinesusedones'complementandtruncationwassimplertoimplement(floorissimplerintwo'scomplement).FORTRANwasdefinedtorequirethisbehaviorandthusalmostallprocessorsimplementconversionthisway.Someconsiderthistobeanunfortunatehistoricaldesigndecisionthathasledtobugshandlingnegativeoffsetsandgraphicsonthenegativesideoftheorigin.[citationneeded]
Abit-wiseright-shiftofasignedinteger
x
{\displaystylex}
by
n
{\displaystylen}
isthesameas
⌊
x
2
n
⌋
{\displaystyle\left\lfloor{\frac{x}{2^{n}}}\right\rfloor}
.Divisionbyapowerof2isoftenwrittenasaright-shift,notforoptimizationasmightbeassumed,butbecausethefloorofnegativeresultsisrequired.Assumingsuchshiftsare"prematureoptimization"andreplacingthemwithdivisioncanbreaksoftware.[citationneeded]
Manyprogramminglanguages(includingC,C++,[39][40]C#,[41][42]Java,[43][44]PHP,[45][46]R,[47]andPython[48])providestandardfunctionsforfloorandceiling,usuallycalledfloorandceil,orlesscommonlyceiling.[49]ThelanguageAPLuses⌊xforfloor.TheJProgrammingLanguage,afollow-ontoAPLthatisdesignedtousestandardkeyboardsymbols,uses<.forfloorand>.forceiling.[50]
ALGOLusesentierforfloor.
InMicrosoftExcelthefloorfunctionisimplementedasINT(whichroundsdownratherthantowardzero).[51]ThecommandFLOORinearlierversionswouldroundtowardzero,effectivelytheoppositeofwhat"int"and"floor"doinotherlanguages.Since2010FLOORhasbeenfixedtorounddown,withextraargumentsthatcanreproducepreviousbehavior.[52]TheOpenDocumentfileformat,asusedbyOpenOffice.org,Libreofficeandothers,usesthesamefunctionnames;INTdoesfloor[53]andFLOORhasathirdargumentthatcanmakeitroundtowardzero.[54]
Seealso[edit]
Bracket(mathematics)
Integer-valuedfunction
Stepfunction
Notes[edit]
^Graham,Knuth,&Patashnik,Ch.3.1
^
1)LukeHeaton,ABriefHistoryofMathematicalThought,2015,ISBN 1472117158(n.p.)
2)AlbertA.Blanketal.,Calculus:DifferentialCalculus,1968,p.259
3)JohnW.Warris,HorstStocker,Handbookofmathematicsandcomputationalscience,1998,ISBN 0387947469,p.151
^Lemmermeyer,pp. 10,23.
^e.g.Cassels,Hardy&Wright,andRibenboimuseGauss'snotation,Graham,Knuth&Patashnik,andCrandall&PomeranceuseIverson's.
^Iverson,p. 12.
^Higham,p. 25.
^SeetheWolframMathWorldarticle.
^Mathwords:FloorFunction.
^Mathwords:CeilingFunction
^Graham,Knuth,&Patashnik,p. 70.
^Graham,Knuth,&Patashink,Ch.3
^Graham,Knuth,&Patashnik,p.73
^Graham,Knuth,&Patashnik,p.85
^Graham,Knuth,&Patashnik,p.85andEx.3.15
^Graham,Knuth,&Patashnik,Ex.3.12
^J.E.blazek,CombinatoiredeN-modulesdeCatalan,Master'sthesis,page17.
^Graham,Knuth,&Patashnik,p.94
^Graham,Knuth,&Patashnik,p.71,applytheorem3.10withx/masinputandthedivisionbynasfunction
^Titchmarsh,p.15,Eq.2.1.7
^Lemmermeyer,§1.4,Ex.1.32–1.33
^Hardy&Wright,§§6.11–6.13
^Lemmermeyer,p.25
^Hardy&Wright,Th.416
^Graham,Knuth,&Patashnik,pp.77–78
^TheseformulasarefromtheWikipediaarticleEuler'sconstant,whichhasmanymore.
^Titchmarsh,p.13
^Titchmarsh,pp.14–15
^Crandall&Pomerance,p.391
^Crandall&Pomerance,Ex.1.3,p.46.Theinfiniteupperlimitofthesumcanbereplacedwithn.Anequivalentconditionisn > 1isprimeifandonlyif
∑
m
=
1
⌊
n
⌋
(
⌊
n
m
⌋
−
⌊
n
−
1
m
⌋
)
=
1
{\displaystyle\sum_{m=1}^{\lfloor{\sqrt{n}}\rfloor}\left(\left\lfloor{\frac{n}{m}}\right\rfloor-\left\lfloor{\frac{n-1}{m}}\right\rfloor\right)=1}
.
^Hardy&Wright,§22.3
^abRibenboim,p.186
^Ribenboim,p.181
^Crandall&Pomerance,Ex.1.4,p.46
^Ribenboim,p.180saysthat"Despitethenilpracticalvalueoftheformulas...[they]mayhavesomerelevancetologicianswhowishtounderstandclearlyhowvariouspartsofarithmeticmaybededucedfromdifferentaxiomatzations..."
^Hardy&Wright,pp.344—345"Anyoneoftheseformulas(oranysimilarone)wouldattainadifferentstatusiftheexactvalueofthenumberα...couldbeexpressedindependentlyoftheprimes.Thereseemsnolikelihoodofthis,butitcannotberuledoutasentirelyimpossible."
^Ramanujan,Question723,Papersp.332
^Hardy&Wright,p.337
^Mahler,K.OnthefractionalpartsofthepowersofarationalnumberII,1957,Mathematika,4,pages122–124
^"C++referenceoffloorfunction".Retrieved5December2010.
^"C++referenceofceilfunction".Retrieved5December2010.
^dotnet-bot."Math.FloorMethod(System)".docs.microsoft.com.Retrieved28November2019.
^dotnet-bot."Math.CeilingMethod(System)".docs.microsoft.com.Retrieved28November2019.
^"Math(JavaSE9&JDK9)".docs.oracle.com.Retrieved20November2018.
^"Math(JavaSE9&JDK9)".docs.oracle.com.Retrieved20November2018.
^"PHPmanualforceilfunction".Retrieved18July2013.
^"PHPmanualforfloorfunction".Retrieved18July2013.
^"R:RoundingofNumbers".
^"Pythonmanualformathmodule".Retrieved18July2013.
^Sullivan,p. 86.
^"Vocabulary".JLanguage.Retrieved6September2011.
^"INTfunction".Retrieved29October2021.
^"FLOORfunction".Retrieved29October2021.
^"Documentation/HowTos/Calc:INTfunction".Retrieved29October2021.
^"Documentation/HowTos/Calc:FLOORfunction".Retrieved29October2021.
References[edit]
J.W.S.Cassels(1957),AnintroductiontoDiophantineapproximation,CambridgeTractsinMathematicsandMathematicalPhysics,45,CambridgeUniversityPress
Crandall,Richard;Pomerance,Carl(2001),PrimeNumbers:AComputationalPerspective,NewYork:Springer,ISBN 0-387-94777-9
Graham,RonaldL.;Knuth,DonaldE.;Patashnik,Oren(1994),ConcreteMathematics,ReadingMa.:Addison-Wesley,ISBN 0-201-55802-5
Hardy,G.H.;Wright,E.M.(1980),AnIntroductiontotheTheoryofNumbers(Fifthedition),Oxford:OxfordUniversityPress,ISBN 978-0-19-853171-5
NicholasJ.Higham,Handbookofwritingforthemathematicalsciences,SIAM.ISBN 0-89871-420-6,p. 25
ISO/IEC.ISO/IEC9899::1999(E):Programminglanguages—C(2nded),1999;Section6.3.1.4,p. 43.
Iverson,KennethE.(1962),AProgrammingLanguage,Wiley
Lemmermeyer,Franz(2000),ReciprocityLaws:fromEulertoEisenstein,Berlin:Springer,ISBN 3-540-66957-4
Ramanujan,Srinivasa(2000),CollectedPapers,ProvidenceRI:AMS/Chelsea,ISBN 978-0-8218-2076-6
Ribenboim,Paulo(1996),TheNewBookofPrimeNumberRecords,NewYork:Springer,ISBN 0-387-94457-5
MichaelSullivan.Precalculus,8thedition,p. 86
Titchmarsh,EdwardCharles;Heath-Brown,DavidRodney("Roger")(1986),TheTheoryoftheRiemannZeta-function(2nd ed.),Oxford:OxfordU.P.,ISBN 0-19-853369-1
Externallinks[edit]
WikimediaCommonshasmediarelatedtoFloorandceilingfunctions.
"Floorfunction",EncyclopediaofMathematics,EMSPress,2001[1994]
ŠtefanPorubský,"Integerroundingfunctions",InteractiveInformationPortalforAlgorithmicMathematics,InstituteofComputerScienceoftheCzechAcademyofSciences,Prague,CzechRepublic,retrieved24October2008
Weisstein,EricW."FloorFunction".MathWorld.
Weisstein,EricW."CeilingFunction".MathWorld.
Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Floor_and_ceiling_functions&oldid=1061612626"
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