Congruence function: Representations through equivalent functions (formula 04.06.27.0003)

Mod

$Mathematica Notation$

Integer Functions

Mod[m,n]

Representations through equivalent functions

With related functions

With Round

For complex arguments

http://functions.wolfram.com/04.06.27.0003.01

Input Form

Mod[m, n] == m + n (Round[(1 + I)/2 - m/n] - I CharacteristicFunction[(1/2) (1 + Im[m/n]), Integers] - CharacteristicFunction[(1/2) (1 + Re[m/n]), Integers])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Mod", "[", RowBox[List["m", ",", "n"]], "]"]], "\[Equal]", RowBox[List["m", "+", RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["Round", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], RowBox[List[" ", "2"]]], "-", FractionBox["m", "n"]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Im", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List["1", "+", RowBox[List["Re", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]], ")"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> $CellContext`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> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> </mrow> <mo> ⌉ </mo> </mrow> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <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> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> $CellContext`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> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> </mrow> <mo> ⌉ </mo> </mrow> <mo> - </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> χ </mi> <mi> ℤ </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <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> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </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["m", "+", RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["Round", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], "-", FractionBox["m", "n"]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Im", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]], "-", RowBox[List["CharacteristicFunction", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Re", "[", FractionBox["m", "n"], "]"]]]], ")"]]]], ",", "Integers"]], "]"]]]], ")"]]]]]]]]]]

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

2001-10-29