Floor and Ceiling Functions - Math24.net

Definitions. Let be a real number. The floor function of denoted by or is defined to be the greatest integer that is less than or equal to. The ceiling ... Precalculus Calculus DifferentialEquations FloorandCeilingFunctions Home→Calculus→SetTheory→FloorandCeilingFunctions Definitions Let\(x\)bearealnumber.Thefloorfunctionof\(x,\)denotedby\(\lfloor{x}\rfloor\)or\(\text{floor}\left(x\right),\)isdefinedtobethegreatestintegerthatislessthanorequalto\(x.\) Theceilingfunctionof\(x,\)denotedby\(\lceil{x}\rceil\)or\(\text{ceil}\left(x\right),\)isdefinedtobetheleastintegerthatisgreaterthanorequalto\(x.\) Forexample, \[\lfloor{\pi}\rfloor=3,\;\;\lceil{\pi}\rceil=4,\;\;\lfloor{5}\rfloor=5,\;\;\lceil{5}\rceil=5.\] \[\lfloor{-e}\rfloor=-3,\;\;\lceil{-e}\rceil=-2,\;\;\lfloor{-1}\rfloor=-1,\;\;\lceil{-1}\rceil=-1.\] Itfollowsfromthedefinitionsthatthefloorandceilingfunctionshavetype\(\mathbb{R}\to\mathbb{Z}.\)Formally,forany\(x\in\mathbb{R},\)theycanbedefinedas \[\begin{array}{*{20}{l}}\text{floor:}&{\lfloor{x}\rfloor=\max\left\{{n\in\mathbb{Z}:n\lex}\right\}}\\[1em]\text{ceiling:}&{\lceil{x}\rceil=\min\left\{{n\in\mathbb{Z}:n\lex}\right\}}\end{array}\] GraphsoftheFloorandCeilingFunctions Thefloorandceilingfunctionslooklikeastaircaseandhaveajumpdiscontinuityateachintegerpoint. Figure1. Figure2. PropertiesoftheFloorandCeilingFunctions Therearemanyinterestingandusefulpropertiesinvolvingthefloorandceilingfunctions,someofwhicharelistedbelow.Thenumber\(n\)isassumedtobeaninteger. \(\left\lfloorx\right\rfloor=n\;\text{iff}\;n\lex\ltn+1\) \(\left\lceilx\right\rceil=n\;\text{iff}\;n-1\ltx\len\) \(\left\lfloorx\right\rfloor=n\;\text{iff}\;x-1\ltn\lex\) \(\left\lceilx\right\rceil=n\;\text{iff}\;x\len\ltx+1\) \(\left\lfloor{-x}\right\rfloor=-\left\lceilx\right\rceil\) \(\left\lceil{-x}\right\rceil=-\left\lfloorx\right\rfloor\) \(\left\lfloorx\right\rfloor+\left\lfloor{-x}\right\rfloor\)\(=\left\{{\begin{array}{*{20}{l}} 0&{\text{if}x\in\mathbb{Z}}\\ {-1}&{\text{if}x\notin\mathbb{Z}} \end{array}}\right.\) \(\left\lceilx\right\rceil+\left\lceil{-x}\right\rceil\)\(=\left\{{\begin{array}{*{20}{l}} 0&{\text{if}x\in\mathbb{Z}}\\ 1&{\text{if}x\notin\mathbb{Z}} \end{array}}\right.\) \(\left\lfloor{x+n}\right\rfloor=\left\lfloorx\right\rfloor+n\) \(\left\lceil{x+n}\right\rceil=\left\lceilx\right\rceil+n\) FractionalPartFunction Thefractionalpartofanumber\(x\in\mathbb{R}\)isthedifferencebetween\(x\)andthefloorof\(x:\) \[\left\{x\right\}=x-\left\lfloorx\right\rfloor.\] Forexample, \[\left\{2\right\}=2-\left\lfloor2\right\rfloor=2-2=0,\] \[\left\{{3.51}\right\}=3.51-\left\lfloor{3.51}\right\rfloor=3.51-3=0.51,\] \[\left\{{\frac{7}{3}}\right\}=\frac{7}{3}-\left\lfloor{\frac{7}{3}}\right\rfloor=\frac{7}{3}-2=\frac{1}{3},\] \[\left\{{-5.98}\right\}=-5.98-\left\lfloor{-5.98}\right\rfloor=-5.98-\left({-6}\right)=-5.98+6=0.02\] Thegraphofthefractionalpartfunctionlookslikeasawtoothwave,withaperiodof\(1.\) Figure3. Therangeoffractionalpartfunctionisthehalf-openinterval\(\left[{0,1}\right).\) Someotherpropertiesofthefractionalpartare \(\left\{x\right\}=0\;\text{iff}\;x\in\mathbb{Z}\) \(\left\{{x+n}\right\}=\left\{x\right\},n\in\mathbb{Z}\) \(\left\{x\right\}+\left\{{-x}\right\}\)\(=\left\{{\begin{array}{*{20}{l}} 0&{\text{if}x\in\mathbb{Z}}\\ 1&{\text{if}x\notin\mathbb{Z}} \end{array}}\right.\) SeesolvedproblemsonPage2. RecommendedPages LogicandSetNotation IntroductiontoSets SetOperationsandVennDiagrams SetIdentities CartesianProductofSets BinaryRelations CompositionofRelations Page1 Page2