Fractional part: Representations through equivalent functions (formula 04.05.27.0025)

FractionalPart

$Mathematica Notation$

Integer Functions

FractionalPart[z]

Representations through equivalent functions

With related functions

With Mod

For complex arguments

http://functions.wolfram.com/04.05.27.0025.01

Input Form

FractionalPart[z] == Mod[z, 1] /; (Re[z] >= 0 && Im[z] >= 0) || Element[z, Integers] || Element[I z, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FractionalPart", "[", "z", "]"]], "\[Equal]", RowBox[List["Mod", "[", RowBox[List["z", ",", "1"]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List["z", "\[Element]", "Integers"]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[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> <mi> frac </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <semantics> <mrow> <mi> z </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> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≥ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≥ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <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> <ci> FractionalPart </ci> <ci> z </ci> </apply> <apply> <rem /> <ci> $CellContext`z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <geq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <geq /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <in /> <ci> z </ci> <integers /> </apply> <apply> <in /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FractionalPart", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Mod", "[", RowBox[List["z", ",", "1"]], "]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]], ")"]], "||", RowBox[List["z", "\[Element]", "Integers"]], "||", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Element]", "Integers"]]]]]]]]]]

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

2007-05-02