Abs[Quotient[m, n]] == Sqrt[Floor[(Im[m] Im[n] + Re[m] Re[n])/(Im[n]^2 + Re[n]^2)]^2 + Floor[((-Im[n]) Re[m] + Im[m] Re[n])/(Im[n]^2 + Re[n]^2)]^2]

Cell[BoxData[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]], "]"]], "\[Equal]", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["Im", "[", "m", "]"]], " ", RowBox[List["Im", "[", "n", "]"]]]], "+", RowBox[List[RowBox[List["Re", "[", "m", "]"]], " ", RowBox[List["Re", "[", "n", "]"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["Im", "[", "n", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Re", "[", "n", "]"]], "2"]]]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Im", "[", "n", "]"]]]], " ", RowBox[List["Re", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["Im", "[", "m", "]"]], " ", RowBox[List["Re", "[", "n", "]"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["Im", "[", "n", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Re", "[", "n", "]"]], "2"]]]], "]"]], "2"]]], " ", ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#10869; </mo> <msqrt> <mrow> <msup> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msup> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <abs /> <apply> <ci> quotient </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <real /> <ci> m </ci> </apply> <apply> <real /> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <imaginary /> <ci> m </ci> </apply> <apply> <imaginary /> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <imaginary /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <imaginary /> <ci> m </ci> </apply> <apply> <real /> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <imaginary /> <ci> n </ci> </apply> <apply> <real /> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <imaginary /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <real /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Abs", "[", RowBox[List["Quotient", "[", RowBox[List["m_", ",", "n_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["Im", "[", "m", "]"]], " ", RowBox[List["Im", "[", "n", "]"]]]], "+", RowBox[List[RowBox[List["Re", "[", "m", "]"]], " ", RowBox[List["Re", "[", "n", "]"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["Im", "[", "n", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Re", "[", "n", "]"]], "2"]]]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Im", "[", "n", "]"]]]], " ", RowBox[List["Re", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["Im", "[", "m", "]"]], " ", RowBox[List["Re", "[", "n", "]"]]]]]], RowBox[List[SuperscriptBox[RowBox[List["Im", "[", "n", "]"]], "2"], "+", SuperscriptBox[RowBox[List["Re", "[", "n", "]"]], "2"]]]], "]"]], "2"]]]]]]]]

2001-10-29