Floor and ceiling functions - Wikipedia

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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 FromWikipedia,thefreeencyclopedia 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 ⌈ ·⌈,&LeftCeiling;) U+2309⌉RIGHTCEILING(HTML ⌉ ·⌉,&RightCeiling;) U+230A⌊LEFTFLOOR(HTML ⌊ ·&LeftFloor;,⌊) U+230B⌋RIGHTFLOOR(HTML ⌋ ·⌋,&RightFloor;) 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. 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