(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 42443, 1432] NotebookOptionsPosition[ 36683, 1260] NotebookOutlinePosition[ 37029, 1275] CellTagsIndexPosition[ 36986, 1272] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.488930259078125*^9, 3.488930259484375*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell[TextData[{ "Homework 1: \nsolved with the aid of ", StyleBox["Mathematica", FontSlant->"Italic"] }], "Subtitle", CellChangeTimes->{{3.493911040546875*^9, 3.4939110668125*^9}, { 3.493911117203125*^9, 3.493911122328125*^9}, {3.493929022328125*^9, 3.49392902875*^9}}, TextAlignment->Center], Cell["21 September 2010", "Subsubtitle", CellChangeTimes->{{3.48893028171875*^9, 3.48893030315625*^9}, { 3.489583451984375*^9, 3.489583455796875*^9}, 3.49028881771875*^9, { 3.4905515529375*^9, 3.490551553125*^9}, 3.490875430171875*^9, { 3.491158425140625*^9, 3.491158435296875*^9}, 3.491217414875*^9, { 3.49193520925*^9, 3.491935209796875*^9}, {3.493460611234375*^9, 3.493460614703125*^9}, {3.49410359903125*^9, 3.494103599265625*^9}}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2010 by Murray Eisenberg. All rights reserved.\ \>", "Text", "SmallText", CellChangeTimes->{{3.488930265421875*^9, 3.488930270109375*^9}, 3.493911107296875*^9}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["2", "Section", CellChangeTimes->{3.493910818296875*^9}], Cell[CellGroupData[{ Cell["(a)", "Subsection", CellChangeTimes->{{3.4939108231875*^9, 3.49391082375*^9}}], Cell["The given depressed cubic is:", "Text", CellChangeTimes->{{3.49391362078125*^9, 3.49391362596875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"depressed", "=", RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"87", "x"}], "-", "130"}]}], ";"}]], "Input", CellChangeTimes->{{3.4939136013125*^9, 3.493913618109375*^9}}], Cell["The del Ferro-Tartaglia formula is realized by:", "Text", CellChangeTimes->{{3.493910846875*^9, 3.4939108604375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"delFerroTartagliaRoot", "[", RowBox[{"p_", ",", "q_"}], "]"}], ":=", "\[IndentingNewLine]", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "q"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], ")"}], RowBox[{"1", "/", "3"}]], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "q"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], ")"}], RowBox[{"1", "/", "3"}]]}]}]], "Input"], Cell[CellGroupData[{ Cell[TextData[{ "Find ", Cell[BoxData[ FormBox["p", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]], FormatType->"TraditionalForm"] }], "Subsubsection", CellChangeTimes->{{3.493915273125*^9, 3.493915279484375*^9}}], Cell[TextData[{ "That assumes that the depressed cubic is written in the form ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"87", "x"}], "-", "130"}], TraditionalForm]], FormatType->"TraditionalForm"], ", so first find ", Cell[BoxData[ FormBox["p", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]], FormatType->"TraditionalForm"], ":" }], "Text", CellChangeTimes->{{3.493910888984375*^9, 3.49391093859375*^9}, { 3.4939136396875*^9, 3.49391367209375*^9}, {3.49391380440625*^9, 3.49391380871875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"threep", ",", "twoq"}], "}"}], "=", RowBox[{"{", RowBox[{ RowBox[{"Coefficient", "[", RowBox[{"depressed", ",", "x", ",", "1"}], "]"}], ",", RowBox[{"Coefficient", "[", RowBox[{"depressed", ",", "x", ",", "0"}], "]"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.493913682765625*^9, 3.49391370759375*^9}, { 3.493913745484375*^9, 3.493913793328125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"p", ",", "q"}], "}"}], "=", RowBox[{ RowBox[{"{", RowBox[{"p", ",", "q"}], "}"}], "/.", RowBox[{"First", "@", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"3", "p"}], "\[Equal]", "threep"}], ",", RowBox[{ RowBox[{"2", "q"}], "\[Equal]", "twoq"}]}], "]"}]}]}]}]], "Input", CellChangeTimes->{{3.49391381415625*^9, 3.49391388571875*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Use the del Ferro-Tartaglia formula", "Subsubsection", CellChangeTimes->{{3.493915289890625*^9, 3.49391530209375*^9}}], Cell[TextData[{ "(If you just plug in those values as arguments to delFerroTartaglia, \ unfortunately ", StyleBox["Mathematica", FontSlant->"Italic"], " does all the calculations for you\[Ellipsis]" }], "Text", CellChangeTimes->{{3.493911216421875*^9, 3.493911246203125*^9}, { 3.4939118406875*^9, 3.49391186409375*^9}}], Cell[BoxData[ RowBox[{"delFerroTartagliaRoot", "[", RowBox[{"p", ",", "q"}], "]"}]], "Input", CellChangeTimes->{{3.49391119390625*^9, 3.4939111955625*^9}}], Cell[TextData[{ StyleBox["\[Ellipsis]", FontSlant->"Italic"], "so that you never get a chance to apply Bombelli's method! In order to \ actually use Bombelli's method, you have to proceed more slowly.)" }], "Text", CellChangeTimes->{{3.493911250921875*^9, 3.493911270640625*^9}, { 3.49391130121875*^9, 3.49391132753125*^9}, {3.493911849171875*^9, 3.493911860703125*^9}}], Cell[BoxData[ RowBox[{"sqrt", "=", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}]], "Input", CellChangeTimes->{{3.493911339796875*^9, 3.4939113451875*^9}}], Cell[BoxData[ RowBox[{"sums", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "q"}], "+", "sqrt"}], ",", RowBox[{ RowBox[{"-", "q"}], "+", "sqrt"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.493911356765625*^9, 3.49391137571875*^9}}], Cell["\<\ (If you just take the cube-roots of those to get the terms of the del \ Ferro-Tartaglia formula\[Ellipsis]\ \>", "Text", CellChangeTimes->{{3.493911398953125*^9, 3.49391141440625*^9}, { 3.49391147115625*^9, 3.493911488078125*^9}, 3.493911824765625*^9}], Cell[BoxData[ RowBox[{" ", RadicalBox["sums", "3"]}]], "Input", CellChangeTimes->{{3.493911395828125*^9, 3.493911458609375*^9}}], Cell[TextData[{ "\[Ellipsis]then you see again that ", StyleBox["Mathematica", FontSlant->"Italic"], " has done all the work for you.) " }], "Text", CellChangeTimes->{{3.493911493109375*^9, 3.49391151034375*^9}, 3.49391183128125*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Use Bombelli's method", "Subsubsection", CellChangeTimes->{{3.4939153183125*^9, 3.4939153281875*^9}}], Cell[TextData[{ "With Bombelli, ", StyleBox["assume", FontWeight->"Bold"], " that ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "q"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], ")"}], RowBox[{"1", "/", "3"}]], "=", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "v"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "q"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], ")"}], RowBox[{"1", "/", "3"}]], "=", RowBox[{"u", "-", RowBox[{"\[ImaginaryI]", " ", "v"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ", for some ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], ". Equivalently, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", "q"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], "=", SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "v"}]}], ")"}], "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", "q"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], "=", SuperscriptBox[ RowBox[{"(", RowBox[{"u", "-", RowBox[{"\[ImaginaryI]", " ", "v"}]}], ")"}], "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49391151809375*^9, 3.493911583828125*^9}, { 3.493911970203125*^9, 3.49391198609375*^9}, {3.493912154453125*^9, 3.493912275359375*^9}}], Cell["Work with the first equation.", "Text", CellChangeTimes->{{3.493912284078125*^9, 3.493912295015625*^9}}], Cell[BoxData[ RowBox[{"theCube", "=", RowBox[{"ComplexExpand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "v"}]}], ")"}], "3"], "]"}]}]], "Input", CellChangeTimes->{{3.493911789796875*^9, 3.49391179884375*^9}}], Cell[BoxData[ RowBox[{"cubeParts", "=", RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "theCube", "]"}], ",", RowBox[{"Im", "[", "theCube", "]"}]}], "}"}], "]"}]}]], "Input"], Cell[BoxData[ RowBox[{"eqns", "=", RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "@", "theCube"}], ",", RowBox[{"Im", "@", "theCube"}]}], "}"}], "==", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"First", "@", "sums"}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"First", "@", "sums"}], "]"}]}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4939120401875*^9, 3.4939121371875*^9}}], Cell[TextData[{ "(Of course ", StyleBox["Mathematica", FontSlant->"Italic"], " can solve these equations fo ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], "\[Ellipsis]" }], "Text", CellChangeTimes->{{3.493912143390625*^9, 3.493912149359375*^9}, { 3.493912311703125*^9, 3.49391234115625*^9}}], Cell[BoxData[ RowBox[{"First", "@", RowBox[{"Solve", "[", RowBox[{"eqns", ",", RowBox[{"{", RowBox[{"u", ",", "v"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.493912343015625*^9, 3.49391236421875*^9}}], Cell["\<\ \[Ellipsis] the point of the problem is to carry out Bombelli's reasoning in \ its entirety.)\ \>", "Text", CellChangeTimes->{{3.493912143390625*^9, 3.493912149359375*^9}, { 3.493912311703125*^9, 3.49391234115625*^9}, {3.493912371578125*^9, 3.493912385625*^9}}], Cell[TextData[{ "With Bombelli, ", StyleBox["assume", FontWeight->"Bold"], " that ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], " are positive integers. Factor the left-hand sides of the pair of equations:" }], "Text", CellChangeTimes->{{3.493912389984375*^9, 3.493912441125*^9}}], Cell[BoxData[ RowBox[{"factoredEqns", "=", RowBox[{"Factor", "[", RowBox[{"Thread", "[", "eqns", "]"}], "]"}]}]], "Input", CellChangeTimes->{{3.4939124440625*^9, 3.49391246378125*^9}, { 3.4939125468125*^9, 3.493912551546875*^9}}], Cell[TextData[{ "Now reason as in the solution that did not apply ", StyleBox["Mathematica", FontSlant->"Italic"], " at all. The divisors of 65 are:" }], "Text", CellChangeTimes->{{3.49391247496875*^9, 3.493912499546875*^9}, { 3.49391337128125*^9, 3.4939133811875*^9}}], Cell[BoxData[ RowBox[{"divisors", "=", RowBox[{"Divisors", "[", "65", "]"}]}]], "Input", CellChangeTimes->{{3.493912517375*^9, 3.493912522390625*^9}, { 3.493912799734375*^9, 3.493912802515625*^9}}], Cell["Try these in the first of the two equations.equations:", "Text", CellChangeTimes->{{3.493912533515625*^9, 3.493912558140625*^9}, { 3.493912891296875*^9, 3.493912912578125*^9}, {3.49391338703125*^9, 3.4939133874375*^9}}], Cell[BoxData[ RowBox[{"firstEqn", "=", RowBox[{"First", "[", "factoredEqns", "]"}]}]], "Input", CellChangeTimes->{{3.493912918453125*^9, 3.493912943328125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Thread", "[", RowBox[{"firstEqn", "/.", RowBox[{"u", "\[Rule]", "divisors"}]}], "]"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.4939129478125*^9, 3.49391298265625*^9}}], Cell[TextData[{ "For a positive integer ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], ", clearly all these equations have no solutions except the second one, \ where the values of ", Cell[BoxData[ FormBox["u", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]], FormatType->"TraditionalForm"], " are:" }], "Text", CellChangeTimes->{{3.493912994671875*^9, 3.4939130594375*^9}, { 3.493913175453125*^9, 3.4939131861875*^9}, {3.4939132326875*^9, 3.493913237703125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"uValue", ",", "vValue"}], "}"}], "=", RowBox[{"{", RowBox[{ RowBox[{ "divisors", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", "2"}], "}"}]}]], "Input", CellChangeTimes->{{3.493913188484375*^9, 3.49391320309375*^9}, { 3.493913240515625*^9, 3.4939132609375*^9}}], Cell[TextData[{ "Check this ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " satisfies the second equation :" }], "Text", CellChangeTimes->{{3.493913074859375*^9, 3.493913093671875*^9}, { 3.493913271609375*^9, 3.49391328853125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Last", "[", "factoredEqns", "]"}], "/.", RowBox[{"{", RowBox[{ RowBox[{"u", "\[Rule]", "uValue"}], ",", RowBox[{"v", "\[Rule]", "vValue"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.493913097546875*^9, 3.493913140625*^9}, { 3.493913227203125*^9, 3.4939132280625*^9}}], Cell[TextData[{ "You could now also check that this ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " also satisfies the equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", "q"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], "=", SuperscriptBox[ RowBox[{"(", RowBox[{"u", "-", RowBox[{"\[ImaginaryI]", " ", "v"}]}], ")"}], "3"]}], TraditionalForm]], FormatType->"TraditionalForm"], ", but that will be automatic." }], "Text", CellChangeTimes->{{3.49391314625*^9, 3.49391314978125*^9}, { 3.49391330003125*^9, 3.493913314515625*^9}, {3.493913421484375*^9, 3.49391343059375*^9}}], Cell["\<\ By the theory, one solution of the original depressed cubic will therefore be:\ \>", "Text", CellChangeTimes->{{3.493913485*^9, 3.493913498796875*^9}}], Cell[BoxData[ RowBox[{"r1", "=", RowBox[{"2", "uValue"}]}]], "Input", CellChangeTimes->{{3.493913514609375*^9, 3.493913526015625*^9}}], Cell["\<\ (You could plug that into the cubic to check your work, but the theory \ guarantees it is a solution!)\ \>", "Text", CellChangeTimes->{{3.49391354278125*^9, 3.493913568796875*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Find the other two roots", "Subsubsection", CellChangeTimes->{{3.49391535321875*^9, 3.4939153559375*^9}}], Cell[TextData[{ "Find the other two solutions of the depressed cubic as follows. The linear \ polynomial ", Cell[BoxData[ FormBox[ RowBox[{"x", "-", "10"}], TraditionalForm]], FormatType->"TraditionalForm"], " is a factor of the given cubic. The quotient of the cubic by that linear \ factor is:" }], "Text", CellChangeTimes->{{3.493913582890625*^9, 3.49391358796875*^9}, { 3.49391390790625*^9, 3.4939139816875*^9}}], Cell[BoxData[ RowBox[{"quadratic", "=", RowBox[{"PolynomialQuotient", "[", RowBox[{"depressed", ",", RowBox[{"x", "-", "r1"}], ",", "x"}], "]"}]}]], "Input", CellChangeTimes->{{3.493913984484375*^9, 3.493914001859375*^9}, { 3.49391406403125*^9, 3.4939140655625*^9}}], Cell[TextData[{ "Finally, find the roots of that directly with ", StyleBox["Mathematica", FontSlant->"Italic"], " \[Ellipsis]" }], "Text", CellChangeTimes->{{3.49391400709375*^9, 3.49391405525*^9}}], Cell[BoxData[ RowBox[{"others", "=", RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{"quadratic", "==", "0"}], ",", "x"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4939140575625*^9, 3.49391408459375*^9}}], Cell["\<\ \[Ellipsis]or by using the quadratic formula:\ \>", "Text", CellChangeTimes->{{3.49391400709375*^9, 3.49391405525*^9}, { 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Instead \ of using ", StyleBox["Replace", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " ", StyleBox["(/.", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ") with any ", StyleBox["Rule", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " (", StyleBox["\[Rule]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "), it uses a so-called ", StyleBox["upvalue", FontWeight->"Bold"], " of a function. For a function ", StyleBox["g", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", an upvalue tells how some other function ", StyleBox["f", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " affects ", StyleBox["g", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " when ", StyleBox["g[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["expr", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", for some expression ", StyleBox["expr", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], ", is the argument to f. And such an upvalue is assigned by evaluating an \ expression of the form:\n\t", StyleBox["g /: f[g[expr_]\[ThinSpace]:=\[ThinSpace]\[Ellipsis]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.494102218140625*^9, 3.494102390921875*^9}, { 3.494102483265625*^9, 3.49410254665625*^9}, {3.494102588671875*^9, 3.4941026763125*^9}, {3.49410298965625*^9, 3.494102992796875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "Instead of ", StyleBox["Abs", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["Conjugate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", use expressions ", StyleBox["abs", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["conj", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", so that we don't let ", StyleBox["Mathematica", FontSlant->"Italic"], " apply rules it knows that would obscure what's involved in the proof:" }], "Text", CellChangeTimes->{{3.49391676803125*^9, 3.493916820859375*^9}, { 3.493916908203125*^9, 3.493916967140625*^9}}], Cell["Express reciprocal in terms of modulus and conjugate:", "Text", CellChangeTimes->{{3.49391697621875*^9, 3.493917011953125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"reciprocal", "[", "z_", "]"}], ":=", RowBox[{ FractionBox["1", SuperscriptBox[ RowBox[{"abs", "[", "z", "]"}], "2"]], RowBox[{"conj", "[", "z", "]"}]}]}]], "Input", CellChangeTimes->{{3.4939168776875*^9, 3.49391689384375*^9}, { 3.49391700584375*^9, 3.493917006609375*^9}, {3.494095216421875*^9, 3.494095218421875*^9}, {3.4940954443125*^9, 3.49409544528125*^9}}], Cell["\<\ Express the definition of complex quotient in terms of reciprocal:\ \>", "Text", CellChangeTimes->{{3.49391607290625*^9, 3.4939160955625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"quotient", "[", RowBox[{"z1_", ",", "z2_"}], "]"}], ":=", RowBox[{"z1", " ", RowBox[{"reciprocal", "[", "z2", "]"}]}]}]], "Input", CellChangeTimes->{{3.493916085859375*^9, 3.493916117890625*^9}}], Cell[TextData[{ "Next, tell ", StyleBox["Mathematica", FontSlant->"Italic"], " that the modulus of a product is the product of the moduli. Owing to the \ way that ", StyleBox["Mathematica", FontSlant->"Italic"], " represents a power, we must also tell ", StyleBox["Mathematica", FontSlant->"Italic"], " separately about the modulus of a power." }], "Text", CellChangeTimes->{{3.494102715390625*^9, 3.494102781625*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"abs", "[", RowBox[{"z_", " ", "w_"}], "]"}], ":=", RowBox[{ RowBox[{"abs", "[", "z", "]"}], RowBox[{"abs", "[", "w", "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"abs", "[", RowBox[{"z_", "^", "n_Integer"}], "]"}], ":=", SuperscriptBox[ RowBox[{"abs", "[", "z", "]"}], "n"]}]}], "Input"], Cell[TextData[{ "Start with arbitrary ", Cell[BoxData[ FormBox[ SubscriptBox["z", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["z", "2"], FontFamily->"Courier"], TraditionalForm]], FormatType->"TraditionalForm"], ":" }], "Text", CellChangeTimes->{{3.494103073375*^9, 3.49410313934375*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"z1", ",", "z2"}], "]"}]], "Input", CellChangeTimes->{{3.493916130859375*^9, 3.49391613396875*^9}}], Cell["Form their quotient:", "Text", CellChangeTimes->{{3.4941035405*^9, 3.494103546*^9}}], Cell[BoxData[ RowBox[{"theQuotient", "=", RowBox[{"quotient", "[", RowBox[{"z1", ",", "z2"}], "]"}]}]], "Input", CellChangeTimes->{{3.493916120921875*^9, 3.493916123953125*^9}, { 3.493921280015625*^9, 3.493921280671875*^9}, {3.4939222159375*^9, 3.493922217625*^9}, {3.494103500796875*^9, 3.494103503625*^9}}], Cell["With just the properties so far:", "Text", CellChangeTimes->{{3.49391703625*^9, 3.493917038234375*^9}, { 3.49410306053125*^9, 3.49410306209375*^9}, {3.494103143140625*^9, 3.4941031511875*^9}}], Cell[BoxData[ RowBox[{"abs", "[", "theQuotient", "]"}]], "Input", CellChangeTimes->{{3.494103181578125*^9, 3.4941031921875*^9}, { 3.494103511359375*^9, 3.494103514015625*^9}}], Cell[TextData[{ "Next, tell ", StyleBox["Mathematica", FontSlant->"Italic"], " about how modulus affects conjugate:" }], "Text", CellChangeTimes->{{3.49410329365625*^9, 3.494103307734375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"abs", "[", RowBox[{"conj", "[", "z_", "]"}], "]"}], ":=", RowBox[{"abs", "[", "z", "]"}]}]], "Input"], Cell["After that:", "Text", CellChangeTimes->{{3.494103324609375*^9, 3.49410332715625*^9}}], Cell[BoxData[ RowBox[{"abs", "[", "theQuotient", "]"}]], "Input", CellChangeTimes->{{3.49410332815625*^9, 3.49410334425*^9}, { 3.494103559265625*^9, 3.494103562234375*^9}}], Cell[TextData[{ "Next, tell ", StyleBox["Mathematica", FontSlant->"Italic"], " that the modulus of a number is necssarily nonnegative, and that the \ modulus of a nonnegative (real) number is that number. 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