Integer part: Representations through equivalent functions (formula 04.04.27.0024)

IntegerPart

$Mathematica Notation$

Integer Functions

IntegerPart[z]

Representations through equivalent functions

With related functions

With Mod

For real arguments

http://functions.wolfram.com/04.04.27.0024.01

Input Form

IntegerPart[x] == x + 1 - Mod[x, 1] - Sign[CharacteristicFunction[x, Integers] + UnitStep[x]] /; Element[x, Reals]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["IntegerPart", "[", "x", "]"]], "\[Equal]", RowBox[List["x", "+", "1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], "-", RowBox[List["Sign", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List["x", ",", "Integers"]], "]"]], "+", RowBox[List["UnitStep", "[", "x", "]"]]]], "]"]]]]]], "/;", RowBox[List["x", "\[Element]", "Reals"]]]]]]

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> <mi> int </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mi> x </mi> <mo> + </mo> <mn> 1 </mn> <mo> - </mo> <semantics> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> - </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> θ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> int </ci> <ci> x </ci> </apply> <apply> <plus /> <ci> x </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Sign </ci> <apply> <plus /> <apply> <apply> <ci> Subscript </ci> <ci> χ </ci> <ci> ℤ </ci> </apply> <ci> x </ci> </apply> <apply> <ci> θ </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["IntegerPart", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["x", "+", "1", "-", RowBox[List["Mod", "[", RowBox[List["x", ",", "1"]], "]"]], "-", RowBox[List["Sign", "[", RowBox[List[RowBox[List["CharacteristicFunction", "[", RowBox[List["x", ",", "Integers"]], "]"]], "+", RowBox[List["UnitStep", "[", "x", "]"]]]], "]"]]]], "/;", RowBox[List["x", "\[Element]", "Reals"]]]]]]]]

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

2007-05-02