Congruence function: Representations through equivalent functions (formula 04.06.27.0015)

Mod

$Mathematica Notation$

Integer Functions

Mod[m,n]

Representations through equivalent functions

With related functions

With Ceiling

For complex arguments

http://functions.wolfram.com/04.06.27.0015.01

Input Form

Mod[m, n] == m - n Ceiling[m/n] + n /; !Element[Re[m/n], Integers] && Element[Im[m/n], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List["m", "-", RowBox[List["n", " ", RowBox[List["Ceiling", "[", FractionBox["m", "n"], "]"]]]], "+", "n"]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List[RowBox[List["Re", "[", FractionBox["m", "n"], "]"]], "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["Im", "[", FractionBox["m", "n"], "]"]], "\[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> <semantics> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`m </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ⩵ </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ⌈ </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ⌉ </mo> </mrow> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </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> <rem /> <ci> FE`Conversion`Private`m </ci> <ci> $CellContext`n </ci> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ceiling /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <ci> n </ci> </apply> </apply> <apply> <and /> <apply> <notin /> <apply> <real /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <in /> <apply> <imaginary /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Mod", "[", RowBox[List["m_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["m", "-", RowBox[List["n", " ", RowBox[List["Ceiling", "[", FractionBox["m", "n"], "]"]]]], "+", "n"]], "/;", RowBox[List[RowBox[List["!", RowBox[List[RowBox[List["Re", "[", FractionBox["m", "n"], "]"]], "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["Im", "[", FractionBox["m", "n"], "]"]], "\[Element]", "Integers"]]]]]]]]]]

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

2007-05-02