Congruence function: Representations through equivalent functions (formula 04.06.27.0009)

Mod

$Mathematica Notation$

Integer Functions

Mod[m,n]

Representations through equivalent functions

With related functions

With Round

For real arguments

http://functions.wolfram.com/04.06.27.0009.01

Input Form

Mod[m, n] == m - n Round[m/n - 1/2] - n /; Element[(m + n)/(2 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["Round", "[", RowBox[List[FractionBox["m", "n"], "-", FractionBox["1", "2"]]], "]"]]]], "-", "n"]]]], "/;", RowBox[List[FractionBox[RowBox[List["m", "+", "n"]], RowBox[List["2", "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> <mi> n </mi> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌉ </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <mi> n </mi> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌉ </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </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["Round", "[", RowBox[List[FractionBox["m", "n"], "-", FractionBox["1", "2"]]], "]"]]]], "-", "n"]], "/;", RowBox[List[FractionBox[RowBox[List["m", "+", "n"]], RowBox[List["2", " ", "n"]]], "\[Element]", "Integers"]]]]]]]]

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

2007-05-02