Ceiling function: Representations through equivalent functions (formula 04.02.27.0023)

Ceiling

$Mathematica Notation$

Integer Functions

Ceiling[z]

Representations through equivalent functions

With related functions

With FractionalPart

For real arguments

http://functions.wolfram.com/04.02.27.0023.01

Input Form

Ceiling[x] == x - FractionalPart[x] + 1 /; Element[x, Reals] && x > 0 && !Element[x, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Ceiling", "[", "x", "]"]], "\[Equal]", RowBox[List["x", "-", RowBox[List["FractionalPart", "[", "x", "]"]], "+", "1"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["x", "\[Element]", "Integers"]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ⌈ </mo> <mi> x </mi> <mo> ⌉ </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> frac </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> x </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ceiling /> <ci> x </ci> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> FractionalPart </ci> <ci> x </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> <apply> <notin /> <ci> x </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Ceiling", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["x", "-", RowBox[List["FractionalPart", "[", "x", "]"]], "+", "1"]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "0"]], "&&", RowBox[List["!", RowBox[List["x", "\[Element]", "Integers"]]]]]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2007-05-02