(* 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[ 155608, 4654] NotebookOptionsPosition[ 134724, 4128] NotebookOutlinePosition[ 146459, 4403] CellTagsIndexPosition[ 146131, 4391] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.466600156640625*^9, 3.466600159546875*^9}, { 3.4909829860625*^9, 3.4909829899375*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell[TextData[{ "Complex ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th roots" }], "Subtitle", TextAlignment->Center, TextJustification->0], Cell["14 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, 3.49121791565625*^9, 3.4912404999375*^9, {3.49199880075*^9, 3.491998801203125*^9}, {3.49346065771875*^9, 3.49346066703125*^9}}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2004\[Dash]2010 by Murray Eisenberg. All rights \ reserved.\ \>", "SmallText", CellChangeTimes->{ 3.466600163421875*^9, 3.4909841678125*^9, {3.492015127140625*^9, 3.49201512759375*^9}}, TextAlignment->Center, TextJustification->0], Cell[TextData[{ "When you open this notebook, you should see a pop-up window asking whether \ you want to evaluate the Initialization Cells. You should select ", StyleBox["Yes", FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Plain"], "." }], "EmphasisText", CellChangeTimes->{ 3.490633836984375*^9, {3.49089844703125*^9, 3.490898447765625*^9}}], Cell[TextData[{ StyleBox["Explanation", FontSlant->"Italic"], ": In the section \"Initialization\" below, there's an Input cell \ consisting of the expression\n\t", StyleBox["<"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nand I made that cell into an \"initialization cell\". When you answered \ yes, that cell was evaluated automatically, and this in effect loaded the ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " application packages." }], "SmallText", CellChangeTimes->{{3.490633867953125*^9, 3.490633922609375*^9}, { 3.490650173484375*^9, 3.49065017828125*^9}, {3.490907105859375*^9, 3.490907106171875*^9}}, ParagraphSpacing->{0.7, 0}], Cell[TextData[{ "This notebook discusses the ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " different ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th roots of a nonzero complex number and, in particular, of \"unity\"\ \[LongDash]the complex number 1. The motivating examples concern cube roots." }], "Text", CellChangeTimes->{{3.491955035125*^9, 3.491955161375*^9}}], Cell[CellGroupData[{ Cell["Where did the cell brackets go?", "Section", CellChangeTimes->{{3.493421635484375*^9, 3.493421640578125*^9}}], Cell[TextData[{ "When you look at this notebook, and many others involving the ", StyleBox["Presentations", FontSlant->"Italic"], " add-on, you may be surprised to see no cell brackets or group brackets \ along the right-hand edge of the window." }], "Text", CellChangeTimes->{{3.493421645875*^9, 3.493421711453125*^9}, 3.493460746703125*^9}], Cell["\<\ Actually, the brackets are there! They are just hidden until you move the \ mouse toward the right boundary of the notebook window. Try it!\ \>", "Text", CellChangeTimes->{{3.493421696078125*^9, 3.49342173990625*^9}}], Cell["\<\ The reason this happens is because of a special \"style sheet\" that has been \ applied to the notebooks, including this one. 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That is, for a given ", Cell[BoxData[ FormBox[ RowBox[{"w", "\[NotEqual]", "0"}], TraditionalForm]]], ", the equation ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "n"], "=", "w"}], TraditionalForm]]], " has ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " different solutions ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ". (In fact, as you shall see in this notebook, the roots can be calculated \ explicitly.)" }], "Text", CellChangeTimes->{ 3.46660019940625*^9, {3.466616217953125*^9, 3.46661623696875*^9}, { 3.466616287375*^9, 3.466616289046875*^9}, {3.49121760946875*^9, 3.491217699296875*^9}, {3.491217815609375*^9, 3.49121785934375*^9}}, ParagraphSpacing->{1, 0}], Cell[TextData[{ "This is the case, in particular, when ", Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"w", "=", "1"}], TraditionalForm]]]]], TraditionalForm]]], ". In this case, the ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " different values of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " are called the", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ StyleBox["n", FontWeight->"Bold"], TraditionalForm]], FontColor->RGBColor[0, 0, 1]], StyleBox["th roots of unity", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellChangeTimes->{ 3.46660019940625*^9, {3.466616217953125*^9, 3.46661623696875*^9}, { 3.466616287375*^9, 3.466616289046875*^9}, {3.49121760946875*^9, 3.49121775928125*^9}, {3.491217806453125*^9, 3.49121780740625*^9}}, ParagraphSpacing->{1, 0}], Cell[TextData[{ "This notebook shows how to use ", StyleBox["Mathematica", FontSlant->"Italic"], " to calculate such roots as well as how to visualize them geometrically. It \ also includes material about expressing complex roots of unity in \"polar \ form\"." }], "Text", CellChangeTimes->{3.46661629325*^9}], Cell["The motivating examples concern cube roots.", "Text", CellChangeTimes->{{3.491955263515625*^9, 3.491955273484375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Prerequisites", "Section", CellChangeTimes->{{3.466616366328125*^9, 3.46661637828125*^9}, { 3.466616604453125*^9, 3.466616607390625*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematica", FontSlant->"Italic"]], "Subsection", CellChangeTimes->{{3.466616614109375*^9, 3.466616621015625*^9}}], Cell[TextData[{ "Most of this notebook requires David Park's ", StyleBox["Mathematica", FontSlant->"Italic"], " add-on application ", StyleBox["Presentations", FontSlant->"Italic"], ". See the Math 421 notebook ", StyleBox["AboutPresentations.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.46661644090625*^9, 3.466616483484375*^9}, { 3.466616542578125*^9, 3.466616591375*^9}, {3.4666167129375*^9, 3.466616714296875*^9}, {3.490983043921875*^9, 3.490983044875*^9}, { 3.490994209359375*^9, 3.49099423265625*^9}}], Cell[TextData[{ "Before using ", StyleBox["Presentations,", FontSlant->"Italic"], " you need to load its packages. That ", ButtonBox["initialization is done below", BaseStyle->"Hyperlink", ButtonData->"load Presentations"], ", just before ", StyleBox["Presentations", FontSlant->"Italic"], " is first needed." }], "Text", CellChangeTimes->{{3.46661644090625*^9, 3.466616483484375*^9}, { 3.466616542578125*^9, 3.466616591375*^9}, {3.4666167129375*^9, 3.46661679353125*^9}, {3.4666168744375*^9, 3.466616901171875*^9}, { 3.4909830825*^9, 3.4909830910625*^9}, {3.49099426165625*^9, 3.49099432440625*^9}, {3.491217542921875*^9, 3.4912175525625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "Before or while using ", StyleBox["Presentations", FontSlant->"Italic"], " with this notebook, you may find helpful the Math 421 notebook ", StyleBox["DrawingComplexObjects.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.490994361625*^9, 3.49099443996875*^9}}], Cell[TextData[{ "The ", StyleBox["PresentationsPalette", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " is useful for accessing the documentation and for forming templates of ", StyleBox["Presentations", FontSlant->"Italic"], " functions. (For graphics here, you may also find useful the separate ", StyleBox["ComplexGraphics", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " palette. ) On ", StyleBox["PresentationsPalette", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], ":\n\t\[FilledSmallCircle] templates of functions to create complex graphics \ objects are in ", StyleBox["Drawing \[RightPointer] Complex Graphics \[RightPointer] \ Primitives", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], "; and\n\t\[FilledSmallCircle] templates of functions to create a drawing of \ complex graphics objects are in ", StyleBox["Drawing \[RightPointer] DrawingPaper \[RightPointer] Graphics", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " (the one you\[CloseCurlyQuote]ll want is ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ")." }], "Text", CellChangeTimes->{{3.491402670890625*^9, 3.49140273925*^9}, { 3.49140277784375*^9, 3.49140288615625*^9}, {3.491403685546875*^9, 3.491403769828125*^9}, {3.491996702921875*^9, 3.491996703921875*^9}}, ParagraphSpacing->{0.5, 0}] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{{3.46661662978125*^9, 3.466616631109375*^9}}], Cell["\<\ You should already know the basics of the algebra of complex numbers\ \[LongDash]how to add and multiply them\[LongDash]and the representation of \ complex numbers by points in the plane. \ \>", "Text", CellChangeTimes->{{3.466616634859375*^9, 3.46661670978125*^9}, 3.49346077421875*^9}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Initialization", "Section", ShowGroupOpener->True, CellChangeTimes->{{3.4906339376875*^9, 3.49063394653125*^9}, { 3.490638317984375*^9, 3.49063833975*^9}, {3.490735161125*^9, 3.490735165125*^9}, 3.490984367984375*^9}, CellTags->"initialization"], Cell[TextData[{ "When you opened this notebook, it should have prompted you whether you want \ to evaluate Initialization Cells. You should have answered \ \[OpenCurlyDoubleQuote]yes.\[CloseCurlyDoubleQuote]\nCheck now whether you \ did so by looking at the Input cell that follows. At its left it should have \ a label such as ", StyleBox["In[1]", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], ".\nIf you do not see such a label, then ", StyleBox["evaluate the following Input cell now", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ". 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By definition, a ", StyleBox["cube root of unity", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " is a solution of the equation \n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "3"], "=", "1"}], TraditionalForm]]], ". \nSurely ", StyleBox["Mathematica", FontSlant->"Italic"], " can solve this equation directly. Try it:" }], "Text", CellChangeTimes->{{3.466600207890625*^9, 3.466600210921875*^9}, { 3.46661630978125*^9, 3.466616311125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ SuperscriptBox["z", "3"], "\[Equal]", "1"}], ",", "z"}], "]"}]], "Input", CellChangeTimes->{{3.4666163154375*^9, 3.466616324796875*^9}}], Cell[TextData[{ "In the output, the purported cube roots are given by ", StyleBox["rules", FontWeight->"Bold"], ". Here are the cube roots themselves, obtained by successively replacing ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " by what each of the rules prescribes:" }], "Text", CellChangeTimes->{{3.491217975*^9, 3.49121801259375*^9}, { 3.49124181659375*^9, 3.491241836546875*^9}, {3.4919547988125*^9, 3.49195480871875*^9}}], Cell[BoxData[ RowBox[{"z", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ SuperscriptBox["z", "3"], "\[Equal]", "1"}], ",", "z"}], "]"}]}]], "Input", CellChangeTimes->{{3.491218015453125*^9, 3.491218026015625*^9}}], Cell[TextData[{ "That does not seem especially useful! What are the solutions, really\ \[LongDash]in ", "Cartesian", " ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " form? Use ", StyleBox["ComplexExpand", FontFamily->"Courier"], " on the result of ", StyleBox["Solve", FontFamily->"Courier"], ":" }], "Text", CellChangeTimes->{{3.46660021375*^9, 3.4666002166875*^9}, { 3.49121803778125*^9, 3.49121804821875*^9}, {3.491218177328125*^9, 3.4912181906875*^9}, 3.49251897990625*^9}], Cell[BoxData[ RowBox[{"cubeRootsUnity", "=", RowBox[{"ComplexExpand", "[", RowBox[{"z", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ SuperscriptBox["z", "3"], "\[Equal]", "1"}], ",", "z"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.466616328109375*^9, 3.46661634290625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Verify that the cube of each of these numbers really is 1." }], "Exercise", CellChangeTimes->{{3.490994598265625*^9, 3.490994617703125*^9}, { 3.490994688890625*^9, 3.490994706953125*^9}}], Cell[TextData[{ "How can you be sure that there are no more than just these three roots? \ This is a consequence of the Fundamental Theorem of Algebra. (See the \ notebook ", StyleBox["FactorTheorem.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".) " }], "Text", CellChangeTimes->{{3.466600222234375*^9, 3.466600224859375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find, similarly, the fourth roots of unity. Verify that the fourth power \ of each really is 1." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491218146046875*^9, 3.491218166828125*^9}, 3.491218339875*^9, {3.49201314946875*^9, 3.492013158375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Visualizing the cube roots of unity", "Section"], Cell["\<\ The graphics functions introduced below are used here just for visualizing \ the cube roots of unity. But they can be used more generally to visualize \ diverse sets of complex numbers.\ \>", "Text", CellChangeTimes->{ 3.466600234140625*^9, {3.49121837996875*^9, 3.491218384125*^9}}], Cell[TextData[{ "You'll need a number of computational and graphics functions that are not \ already built-in objects in ", StyleBox["Mathematica", FontSlant->"Italic"], ". They are defined in David Park's ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " application." }], "Text", CellChangeTimes->{{3.458998754796875*^9, 3.458998768875*^9}, { 3.466600191203125*^9, 3.46660019325*^9}, 3.46660023690625*^9, { 3.466601196234375*^9, 3.466601198109375*^9}, {3.491218409265625*^9, 3.491218416875*^9}, 3.491218495359375*^9, 3.491955451421875*^9}], Cell[TextData[{ "When you opened this notebook, did you answer \"yes\" when asked whether to \ evaluate Initialization Cells?\nIf not, or if you are unsure, then evaluate \ the following Input cell now.Did you evaluate the preceding cell to load ", StyleBox["Presentations", FontSlant->"Italic"], "? If not, do so now. Otherwise, the rest of this notebook will not work \ properly!" }], "EmphasisText", CellChangeTimes->{{3.4912418788125*^9, 3.491241920375*^9}, { 3.491955382890625*^9, 3.491955433390625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"<<", "Presentations`"}]], "Input", CellChangeTimes->{{3.458998746203125*^9, 3.458998798375*^9}, { 3.491218397953125*^9, 3.491218406609375*^9}}, CellTags->"load Presentations"], Cell[CellGroupData[{ Cell["Drawing roots of unity as points in the plane", "Subsection", CellChangeTimes->{{3.49126030559375*^9, 3.49126030665625*^9}}], Cell[TextData[{ "Below is the drawing we shall eventually produce. The ", ButtonBox["code to create the drawing", BaseStyle->"Hyperlink", ButtonData->"finalCodeCubeRoots"], " in this final form appears later. We shall develop the code step-by-step." }], "Text", CellChangeTimes->{{3.491997977203125*^9, 3.491998052*^9}, { 3.491998108640625*^9, 3.491998155203125*^9}}], Cell[TextData[Cell[BoxData[ GraphicsBox[ {RGBColor[0.689993, 0.089999, 0.119999], PointSize[ Large], {PointBox[{1, 0}], PointBox[NCache[{ Rational[-1, 2], Rational[-1, 2] 3^Rational[1, 2]}, {-0.5, -0.8660254037844386}]], PointBox[NCache[{Rational[-1, 2], Rational[1, 2] 3^Rational[1, 2]}, {-0.5, 0.8660254037844386}]]}, {GrayLevel[0], InsetBox["1", {0.99, 0.2}], InsetBox[ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "-", FractionBox[ RowBox[{"\[ImaginaryI]", " ", SqrtBox["3"]}], "2"]}], {-0.51, -0.6660254037844386}], InsetBox[ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "+", FractionBox[ RowBox[{"\[ImaginaryI]", " ", SqrtBox["3"]}], "2"]}], {-0.51, 1.0660254037844386`}]}}, Axes->True, AxesLabel->{ FormBox["\"Real\"", TraditionalForm], FormBox["\"Imaginary\"", TraditionalForm]}, Background->RGBColor[0.941206, 0.972503, 1.], ImageSize->252., PlotLabel->FormBox[ StyleBox["\"The cube roots of unity\"", 14, Bold, StripOnInput -> False], TraditionalForm], PlotRange->{{-1.25, 1.25}, {-1.25, 1.25}}, PlotRangeClipping->True, Ticks->{{{-1, FormBox[ RowBox[{"-", "1"}], TraditionalForm]}, { NCache[ Rational[-1, 2], -0.5], FormBox[ RowBox[{"-", FractionBox["1", "2"]}], TraditionalForm]}, {0, FormBox["0", TraditionalForm]}, { NCache[ Rational[1, 2], 0.5], FormBox[ FractionBox["1", "2"], TraditionalForm]}, {1, FormBox["1", TraditionalForm]}}, {{-1, FormBox[ RowBox[{"-", "1"}], TraditionalForm]}, { NCache[ Rational[-1, 2], -0.5], FormBox[ RowBox[{"-", FractionBox["1", "2"]}], TraditionalForm]}, {0, FormBox["0", TraditionalForm]}, { NCache[ Rational[1, 2], 0.5], FormBox[ FractionBox["1", "2"], TraditionalForm]}, {1, FormBox["1", TraditionalForm]}}}]], CellChangeTimes->{{3.49126466128125*^9, 3.4912646955*^9}, 3.491997559234375*^9, {3.49199794021875*^9, 3.491997946390625*^9}, { 3.492000352890625*^9, 3.4920003684375*^9}, 3.49200039921875*^9, 3.492000450890625*^9, 3.492000760625*^9}]], "Text", Editable->False, Deletable->False, CellChangeTimes->{{3.492000868375*^9, 3.4920008930625*^9}}, TextAlignment->Center], Cell[TextData[{ "The immediate aim is to convert each of the cube roots of unity into an ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], "-coordinate pair and then surround it with the ", StyleBox["Mathematica", FontSlant->"Italic"], " graphics primitive ", StyleBox["Point", FontFamily->"Courier"], " to produce a graphics object capable of being displayed. The ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ComplexPoint", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " does both of those things at once:" }], "Text", CellChangeTimes->{ 3.466600243109375*^9, {3.491218470765625*^9, 3.491218484453125*^9}, { 3.4912601996875*^9, 3.491260271046875*^9}, 3.492014501*^9}], Cell["\<\ For example, among the cube roots of unity found earlier\[Ellipsis]\ \>", "Text", CellChangeTimes->{{3.4912185405*^9, 3.491218558734375*^9}, { 3.491955487421875*^9, 3.49195549753125*^9}}], Cell[BoxData["cubeRootsUnity"], "Input", CellChangeTimes->{{3.49195550053125*^9, 3.491955503484375*^9}}], Cell["the third of the three cube roots of unity found earlier is:", "Text", CellChangeTimes->{{3.4912185405*^9, 3.491218558734375*^9}, { 3.491955487421875*^9, 3.49195549753125*^9}}], Cell[BoxData[ RowBox[{"cubeRootsUnity", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}]], "Input"], Cell["Make that cube root a graphics object:", "Text", CellChangeTimes->{{3.4912185820625*^9, 3.491218597515625*^9}, { 3.491260178234375*^9, 3.49126019178125*^9}, 3.49126028684375*^9, 3.492014505859375*^9}], Cell[BoxData[ RowBox[{"ComplexPoint", "[", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}], "]"}]], "Input", CellChangeTimes->{{3.4666002758125*^9, 3.466600279046875*^9}}], Cell[TextData[{ "Make a drawing of that cube root of unity in the complex plane by using the \ ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.49124162484375*^9, 3.491241630375*^9}, { 3.49124193221875*^9, 3.491241939421875*^9}, {3.49124203575*^9, 3.491242036046875*^9}, {3.49124207696875*^9, 3.491242080890625*^9}, { 3.491260295796875*^9, 3.49126032334375*^9}}], Cell[BoxData[ RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Set", " ", "color", " ", "and", " ", "point", " ", "size", " ", "for", " ", "plotting", " ", "point"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Red", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Draw", " ", "points"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ComplexPoint", "[", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}], "]"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{ 3.4589988103125*^9, {3.46660034296875*^9, 3.4666003444375*^9}, { 3.466600411640625*^9, 3.46660044153125*^9}, {3.466600489765625*^9, 3.466600513234375*^9}, 3.46660062778125*^9, {3.466613689328125*^9, 3.466613696078125*^9}, {3.491218884234375*^9, 3.49121890546875*^9}, { 3.491218956*^9, 3.491218959796875*^9}, {3.491241999203125*^9, 3.491242012671875*^9}, {3.4912420434375*^9, 3.49124207134375*^9}, 3.491252315375*^9}], Cell["\<\ To draw all three roots, you could turn them into graphics objects \ one-by-one, by forming:\ \>", "Text", CellChangeTimes->{{3.491218616875*^9, 3.491218624375*^9}, { 3.491242108296875*^9, 3.4912421669375*^9}, {3.49126034678125*^9, 3.49126037190625*^9}}], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"ComplexPoint", "[", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.466600252890625*^9, 3.4666002723125*^9}, 3.491218633734375*^9}], Cell[TextData[{ "A more direct, \"functional\", way to accomplish the same thing is to use ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s ", StyleBox["Map", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " function:" }], "Text", CellChangeTimes->{{3.49121867640625*^9, 3.49121867996875*^9}, 3.491218786765625*^9, {3.49124134115625*^9, 3.491241352390625*^9}, { 3.491241970265625*^9, 3.491241970796875*^9}}], Cell[BoxData[ RowBox[{"Map", "[", RowBox[{"ComplexPoint", ",", "cubeRootsUnity"}], "]"}]], "Input", CellChangeTimes->{3.466600286671875*^9}], Cell["And a terser way to accomplish the same thing is:", "Text", CellChangeTimes->{{3.49121868503125*^9, 3.491218722515625*^9}, { 3.4912188078125*^9, 3.4912188146875*^9}, {3.49121886915625*^9, 3.49121887484375*^9}}], Cell[BoxData[ RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]], "Input", CellChangeTimes->{{3.49121872646875*^9, 3.491218733453125*^9}}], Cell[TextData[{ "It is often a good idea to use such an abbreviated form for ", StyleBox["Map", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". especially when the ", StyleBox["Map[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["func", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["lis", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression, with its brackets, would be nested inside another expression\ \[CloseCurlyQuote]s brackets." }], "Text", CellChangeTimes->{{3.491218740953125*^9, 3.49121877025*^9}, 3.491241373828125*^9, {3.491241437546875*^9, 3.491241517828125*^9}, { 3.4912421823125*^9, 3.491242187203125*^9}, {3.4919972970625*^9, 3.49199734275*^9}, 3.49201450884375*^9}], Cell["\<\ Here is a drawing of all three cube roots of unity in the complex plane:\ \>", "Text", CellChangeTimes->{{3.49195556578125*^9, 3.49195558184375*^9}}], Cell[BoxData[ RowBox[{"Draw2D", "[", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "Set", " ", "color", " ", "and", " ", "point", " ", "size", " ", "for", " ", "plotting", " ", "point"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"Red", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Draw", " ", "points"}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{ 3.4589988103125*^9, {3.46660034296875*^9, 3.4666003444375*^9}, { 3.466600411640625*^9, 3.46660044153125*^9}, {3.466600489765625*^9, 3.466600513234375*^9}, 3.46660062778125*^9, {3.466613689328125*^9, 3.466613696078125*^9}, {3.491218884234375*^9, 3.49121890546875*^9}, { 3.491218956*^9, 3.491218959796875*^9}, {3.491252324375*^9, 3.491252344296875*^9}, 3.4912524313125*^9}], Cell[TextData[{ "The preceding input cell included ", StyleBox["comments", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], ", enclosed in ", StyleBox["(*", FontFamily->"Courier"], " ", StyleBox["\[Ellipsis]", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " ", StyleBox["*)", FontFamily->"Courier"], " pairs, just to help you understand what the various expressions do that \ constitute the entries in the (list) argument to ", StyleBox["Draw2D", FontFamily->"Courier"], ". Ordinarily such \[OpenCurlyDoubleQuote]obvious\[CloseCurlyDoubleQuote] \ comments are not needed. (", StyleBox["Mathematica", FontSlant->"Italic"], " documentation is more usually written in text cells.) \nBut it is still \ helpful to break the entire expression into separate lines to show successive \ ", StyleBox["graphics directives", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "\[LongDash]color and point size specifications, for example\[LongDash]and ", StyleBox["graphics ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["primitives", FontFamily->"Times", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["\[LongDash]", FontFamily->"Times"], StyleBox["ComplexPoints", FontFamily->"Courier"], ", etc.\[LongDash]to which those directives apply. Here's the same \ expression as before, but with the comments now omitted:" }], "Text", CellChangeTimes->{{3.46660055390625*^9, 3.46660057871875*^9}, { 3.491218981703125*^9, 3.491218988890625*^9}, {3.491252394875*^9, 3.491252418296875*^9}, {3.491955615015625*^9, 3.491955618578125*^9}, { 3.49199738290625*^9, 3.49199738421875*^9}}, ParagraphSpacing->{1, 0}], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Red", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.49121901165625*^9, 3.491219022359375*^9}, {3.491252358046875*^9, 3.49125236296875*^9}}], Cell[TextData[{ "Notice that the entire argument to the function ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is a list (with a ", StyleBox["{}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " pair). That list is followed by some options, such as ", StyleBox["Axes\[Rule]True", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", that affect the overall drawing." }], "Text", CellChangeTimes->{{3.46660064771875*^9, 3.466600652375*^9}, { 3.4912190413125*^9, 3.49121908778125*^9}, {3.49125237621875*^9, 3.491252384359375*^9}}], Cell[TextData[{ "Other named alternatives for the ", StyleBox["PointSize", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " directive are ", StyleBox["Medium", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Small", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", and ", StyleBox["Tiny", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". For more precise control, you can specify the point size by a number, \ which specifies the size as a fraction of the total width of the drawing." }], "Text", CellChangeTimes->{{3.46661370528125*^9, 3.46661386153125*^9}, { 3.466613943921875*^9, 3.466613947234375*^9}, {3.491219105328125*^9, 3.491219141*^9}, {3.491219378265625*^9, 3.491219405484375*^9}, { 3.491219576640625*^9, 3.491219577359375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Experiment with the effects of the following:\n\t(a) Changing the \ argument to ", StyleBox["PointSize", FontFamily->"Courier"], "\n\t(b) Changing, the color from Red to something else pretty. ", StyleBox["Note:", FontSlant->"Italic"], " The named colors ", StyleBox["Red", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Blue", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Green,Orange", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Yellow,Purple,Brown", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Cyan", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Magenta", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", ", StyleBox["Pink", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", among others, are built-in and immediately available. \n\tAdditional \ named colors are available. Use the ", StyleBox["Color Schemes", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " palette to find them. Using the ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["Legacy", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", you may easily access the ", StyleBox["Named", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " colors in the ", StyleBox["Legacy", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " group. For example\n\t(c) Try changing the color to ", StyleBox["Legacy[IndianRed]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". So as to avoid nested brackets, you can use instead the prefix form, ", StyleBox["Legacy@IndianRed", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".\nMore information about selecting colors appears in notebook ", StyleBox["DrawingComplexObjects.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491219153109375*^9, 3.491219169171875*^9}, {3.49121920065625*^9, 3.491219221515625*^9}, { 3.491219287375*^9, 3.491219328984375*^9}, {3.49121955884375*^9, 3.4912196318125*^9}, {3.491219673828125*^9, 3.49121976775*^9}, { 3.491252462359375*^9, 3.491252640328125*^9}, 3.49199743546875*^9, 3.492014516953125*^9}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["The drawings so far are rather large. To ", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["c", FontFamily->"Courier"], "hange the overall size of a drawing, include in ", StyleBox["Draw2D", FontFamily->"Courier"], " an option of the form\n\t", StyleBox["ImageSize\[Rule]72\[Times]", FontFamily->"Courier"], StyleBox["inches", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], "\nwhere ", StyleBox["inches", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " is the number of inches you want for the width of the plot. (", StyleBox["Mathematica", FontSlant->"Italic"], " measures the width in \"printer's points\". There are 72 printer's points \ to the inch. So setting ImageSize to ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"3.5", "\[Times]", "72"}], FontFamily->"Courier"], TraditionalForm]]], ", for example, sets it to 3.5 inches." }], "Text", CellChangeTimes->{{3.46660125503125*^9, 3.466601270203125*^9}, { 3.4666139619375*^9, 3.466613976453125*^9}, {3.49121979334375*^9, 3.491219828953125*^9}, {3.49124224890625*^9, 3.491242266484375*^9}, { 3.492000533265625*^9, 3.4920005668125*^9}, {3.49200067221875*^9, 3.492000677015625*^9}, {3.49200071384375*^9, 3.49200071428125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Red", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.466601292125*^9, 3.466601303078125*^9}, 3.491219840984375*^9, { 3.491219899640625*^9, 3.491219906203125*^9}, {3.491242275953125*^9, 3.49124227725*^9}, {3.49125265184375*^9, 3.49125265225*^9}, { 3.4920005143125*^9, 3.49200051509375*^9}, {3.49200068153125*^9, 3.49200069278125*^9}}], Cell[TextData[{ "It would be a good idea to label the entire drawing with a caption. The ", StyleBox["PlotLabel", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option is used for that:" }], "Text", CellChangeTimes->{{3.4912210075*^9, 3.49122104946875*^9}}], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Red", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.466601292125*^9, 3.466601303078125*^9}, 3.491219840984375*^9, { 3.491219899640625*^9, 3.491219906203125*^9}, {3.491221071171875*^9, 3.49122108584375*^9}, {3.491252664671875*^9, 3.491252670046875*^9}, { 3.492000508890625*^9, 3.492000509484375*^9}, {3.492000697828125*^9, 3.492000698203125*^9}}], Cell[TextData[{ "To annotate the display, you may include text at any locations by invoking \ the ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " function ", StyleBox["ComplexText", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". This function is used in the form\n\t", StyleBox["ComplexText[", FontFamily->"Courier"], StyleBox["txt", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["z", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"], "\nwhere ", StyleBox["txt", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " is the text you want to display and ", StyleBox["z", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " is the complex number in the plane where (the center of) the text is to be \ displayed." }], "Text", CellChangeTimes->{{3.4666119384375*^9, 3.46661196540625*^9}, { 3.491219871703125*^9, 3.4912198773125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "For example, create a label for the root ", Cell[BoxData[ FormBox[ RowBox[{"1", "=", RowBox[{"1", "+", RowBox[{"0", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " of unity like this\[Ellipsis]" }], "Text", CellChangeTimes->{{3.4666119716875*^9, 3.466611974234375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"ComplexText", "[", RowBox[{ RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "]"}]], "Input", CellChangeTimes->{{3.466611984140625*^9, 3.4666119919375*^9}, { 3.491252716453125*^9, 3.491252722640625*^9}}], Cell["\<\ \[Ellipsis]and include that label like this:\ \>", "Text"], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Cyan", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}], ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", RowBox[{"ComplexText", "[", RowBox[{ RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "]"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.466601292125*^9, 3.466601303078125*^9}, {3.466612016*^9, 3.46661204425*^9}, {3.466612106859375*^9, 3.46661210884375*^9}, { 3.46661227490625*^9, 3.46661227565625*^9}, {3.466612488078125*^9, 3.466612520609375*^9}, {3.491219925125*^9, 3.49121999990625*^9}, { 3.49122049665625*^9, 3.491220500734375*^9}, {3.491221104546875*^9, 3.491221109921875*^9}, {3.4912527541875*^9, 3.49125276065625*^9}, { 3.49125280034375*^9, 3.4912528403125*^9}, 3.492000181265625*^9, { 3.4920005028125*^9, 3.492000503421875*^9}, 3.4920007066875*^9}], Cell[TextData[{ "The point color there was changed to ", StyleBox["Cyan", FontFamily->"Courier"], " so you can see that, unfortunately, the label was displayed directly over \ the point. Ordinarily you will want to use some \"offset\" to move the text \ away from the point\[Ellipsis]" }], "Text", CellChangeTimes->{{3.466612282421875*^9, 3.46661232546875*^9}, { 3.4666125253125*^9, 3.46661254971875*^9}, {3.49122048696875*^9, 3.491220489734375*^9}, {3.4912528505625*^9, 3.49125286359375*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"\[CapitalDelta]z", "=", RowBox[{ RowBox[{"-", "0.08"}], "+", RowBox[{"0.2", "\[ImaginaryI]"}]}]}], ";"}], "\n", RowBox[{"ComplexText", "[", RowBox[{ RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", " ", RowBox[{ RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], " ", "+", "\[CapitalDelta]z"}]}], "]"}]}], "Input", CellChangeTimes->{{3.4666121284375*^9, 3.466612142609375*^9}, { 3.46661300559375*^9, 3.466613010765625*^9}, {3.491220420921875*^9, 3.491220421703125*^9}, 3.4912204543125*^9, {3.49122050721875*^9, 3.491220540171875*^9}, 3.491220741453125*^9, {3.49124236884375*^9, 3.491242377734375*^9}, {3.4912522465*^9, 3.4912522500625*^9}, 3.49200024375*^9, 3.492000296796875*^9, 3.49200034328125*^9, 3.4920003910625*^9, 3.492000424015625*^9, {3.492000597734375*^9, 3.492000616578125*^9}}], Cell[TextData[{ "\[Ellipsis]and include that in the ", StyleBox["ComplexText", FontFamily->"Courier New"], " item of the ", StyleBox["Draw2D", FontFamily->"Courier New"], StyleBox[" expression, where the original ", FontFamily->"Times New Roman"], StyleBox["Red", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[" point color has been restored:", FontFamily->"Times New Roman"] }], "Text", CellChangeTimes->{{3.466612373421875*^9, 3.466612424171875*^9}, { 3.491252915484375*^9, 3.49125292865625*^9}}], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Red", ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}], ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", RowBox[{"ComplexText", "[", RowBox[{ RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", RowBox[{ RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "+", "\[CapitalDelta]z"}]}], "]"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.466601292125*^9, 3.466601303078125*^9}, {3.466612016*^9, 3.46661204425*^9}, {3.466612106859375*^9, 3.46661210884375*^9}, { 3.466612222609375*^9, 3.466612255296875*^9}, 3.466612563609375*^9, { 3.491220026734375*^9, 3.491220036609375*^9}, {3.4912211249375*^9, 3.4912211315625*^9}, {3.491242385078125*^9, 3.491242388015625*^9}, { 3.49125226165625*^9, 3.491252285328125*^9}, {3.491252897640625*^9, 3.491252898328125*^9}, {3.491252937921875*^9, 3.491252938296875*^9}, 3.492000190953125*^9, {3.4920004955*^9, 3.492000496171875*^9}, 3.492000722453125*^9}], Cell[TextData[{ "Form labels for all three points at once by using ", StyleBox["Map", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", like this:" }], "Text", CellChangeTimes->{{3.491220085625*^9, 3.4912201018125*^9}, { 3.491242397859375*^9, 3.491242399046875*^9}}], Cell[BoxData[ RowBox[{"Map", "[", RowBox[{ RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], ",", "cubeRootsUnity"}], "]"}]], "Input", CellChangeTimes->{{3.46661244015625*^9, 3.466612451671875*^9}, { 3.46661302625*^9, 3.46661302665625*^9}, {3.491242405265625*^9, 3.491242411734375*^9}, {3.491252950234375*^9, 3.4912529531875*^9}}], Cell[TextData[{ "In the preceding Input cell, the expression ", StyleBox["ComplexText[#,#+\[CapitalDelta]z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "& is a ", StyleBox["pure function", FontWeight->"Bold"], ", sometimes called an \[OpenCurlyDoubleQuote]anonymous function.\ \[CloseCurlyDoubleQuote]. It\[CloseCurlyQuote]s a function that has no name \ but that you can use \[OpenCurlyDoubleQuote]on the \ fly\[CloseCurlyDoubleQuote]\[LongDash]without having to make a separate \ definition of a named function such as: \n", StyleBox["myText[cmplxNum_]:=ComplexText[cmplxNum,cmplxNum+\[CapitalDelta]z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nNotice the ampersand ", StyleBox["&", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " at the end of ", StyleBox["ComplexText[#,#+\[CapitalDelta]z&", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as well as the use of the Slot symbol ", StyleBox["#", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " there. When you use this pure function, the Slot symbol ", StyleBox["#", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " gets replaced with an actual value." }], "SmallText", CellChangeTimes->{{3.49124058290625*^9, 3.491240998078125*^9}, { 3.491241029125*^9, 3.4912410420625*^9}, {3.491241106625*^9, 3.49124114890625*^9}, {3.49124246946875*^9, 3.491242476625*^9}, { 3.491259537203125*^9, 3.49125959775*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "Do the same thing, but more succinctly, with the ", StyleBox["/@", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " abbreviation for using ", StyleBox["Map", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.491220114234375*^9, 3.491220137546875*^9}, { 3.491252967671875*^9, 3.491252983890625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], "/@", "cubeRootsUnity"}]], "Input", CellChangeTimes->{{3.49122007390625*^9, 3.49122008046875*^9}, { 3.49122035140625*^9, 3.49122035453125*^9}, {3.49124250115625*^9, 3.49124251253125*^9}, {3.491252988*^9, 3.49125298971875*^9}}], Cell[TextData[{ "Then here's the display of the three cube roots of unity with each labeled \ with its value (and the point color changed to the ", StyleBox["Legacy", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " color ", StyleBox["IndianRed", FontFamily->"Courier"], "):" }], "Text", CellChangeTimes->{{3.491252998671875*^9, 3.4912530111875*^9}}], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Legacy", "@", "IndianRed"}], ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"Map", "[", RowBox[{"ComplexPoint", ",", "cubeRootsUnity"}], "]"}], ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], "/@", " ", "cubeRootsUnity"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.466601292125*^9, 3.466601303078125*^9}, {3.466612016*^9, 3.46661204425*^9}, {3.466612106859375*^9, 3.46661210884375*^9}, { 3.466612222609375*^9, 3.466612255296875*^9}, {3.466612576203125*^9, 3.466612604*^9}, {3.4912201664375*^9, 3.491220175546875*^9}, { 3.4912203426875*^9, 3.49122034590625*^9}, {3.491221138015625*^9, 3.49122113978125*^9}, {3.49124252184375*^9, 3.49124254328125*^9}, { 3.49125302334375*^9, 3.491253025578125*^9}, 3.49200019840625*^9, { 3.49200048796875*^9, 3.4920004886875*^9}, 3.492000728265625*^9}], Cell[TextData[{ StyleBox["Oops! O", FontSlant->"Italic"], "ne of the text labels is clipped at the top, and two are clipped at the \ left edge of the drawing. What's more, the whole plot has the origin \ off-center. To fix all this at the same time, include the ", StyleBox["PlotRange", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " option to ", StyleBox["Draw2D", FontFamily->"Courier"], ". The value of ", StyleBox["PlotRange", FontFamily->"Courier"], " specifies the graphics \"window\" (just like for a calculator plot) and is \ used in the form:\n\t", StyleBox["{{", FontFamily->"Courier"], StyleBox["xmin", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["xmax", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["},{", FontFamily->"Courier"], StyleBox["ymin", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["ymax", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["}}", FontFamily->"Courier"], "\nYou'll often need to do some figuring and some experimenting to adjust \ the value for ", StyleBox["PlotRange", FontFamily->"Courier"], " so as to provide appropriate dimensions. (Good graphics takes work\ \[LongDash]no matter how good the software creating it!)" }], "Text", CellChangeTimes->{{3.46661261584375*^9, 3.466612724453125*^9}, { 3.46661404303125*^9, 3.46661404703125*^9}, {3.491220201421875*^9, 3.491220252625*^9}, {3.491220800859375*^9, 3.491220801171875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Legacy", "@", "IndianRed"}], ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}], ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], ")"}], "/@", " ", "cubeRootsUnity"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466600583640625*^9, 3.46660063884375*^9}, { 3.466601292125*^9, 3.466601303078125*^9}, {3.466612016*^9, 3.46661204425*^9}, {3.466612106859375*^9, 3.46661210884375*^9}, { 3.466612222609375*^9, 3.466612255296875*^9}, {3.466612576203125*^9, 3.466612604*^9}, {3.46661274140625*^9, 3.466612758703125*^9}, { 3.46661290421875*^9, 3.46661293221875*^9}, {3.491220280625*^9, 3.491220334625*^9}, {3.49122062028125*^9, 3.491220621171875*^9}, { 3.4912211454375*^9, 3.491221220875*^9}, {3.491253038421875*^9, 3.491253056578125*^9}, 3.492000211875*^9, {3.492000435140625*^9, 3.49200043546875*^9}, 3.49200048184375*^9, {3.49200062815625*^9, 3.4920006540625*^9}}], Cell["\<\ That\[CloseCurlyQuote]s looking really nice now.\ \>", "Text", CellChangeTimes->{{3.49122082165625*^9, 3.49122083075*^9}}], Cell[TextData[{ "When the entire ", StyleBox["Draw2D", FontFamily->"Courier"], " expression such as the preceding one gets too long for your comfort, you \ may want to define the graphics objects separately and then reference those \ names within the argument of the ", StyleBox["Draw2D", FontFamily->"Courier"], " expression. For example:" }], "Text", CellChangeTimes->{{3.46661294840625*^9, 3.46661297415625*^9}, { 3.4912208859375*^9, 3.491220901359375*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"pts", "=", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[CapitalDelta]z", "=", RowBox[{ RowBox[{"-", "0.01"}], "+", RowBox[{"0.2", "\[ImaginaryI]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ptLabels", "=", RowBox[{ RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], "/@", " ", "cubeRootsUnity"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Legacy", "@", "IndianRed"}], ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", "pts", ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", "ptLabels"}], "\[IndentingNewLine]", "}"}], ",", " ", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", "\"\\""}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.46661297703125*^9, 3.466612993984375*^9}, { 3.4666130485*^9, 3.466613186515625*^9}, {3.466615349328125*^9, 3.466615351140625*^9}, 3.46661612446875*^9, {3.49122090890625*^9, 3.491220971578125*^9}, {3.491221155578125*^9, 3.491221157375*^9}, 3.4912212358125*^9, {3.49125967328125*^9, 3.49125974071875*^9}, 3.49200022459375*^9, {3.492000441484375*^9, 3.4920004418125*^9}, 3.49200047521875*^9, 3.4920007370625*^9}], Cell[TextData[{ "Perhaps two further refinements: (1) Make the drawing label stand out \ better by using a larger font size and bold face; for that, use the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Style", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". (2) Change the background color of the drawing so as to make the ", StyleBox["whole", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " thing stand out better against the notebook background; for that use the \ option ", StyleBox["Background", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{{3.4912212724375*^9, 3.491221342984375*^9}, { 3.4912214738125*^9, 3.49122148634375*^9}, {3.49125976003125*^9, 3.49125978403125*^9}, {3.49126400621875*^9, 3.491264128046875*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"pts", "=", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[CapitalDelta]z", "=", RowBox[{ RowBox[{"-", "0.01"}], "+", RowBox[{"0.2", "\[ImaginaryI]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ptLabels", "=", RowBox[{ RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], "/@", " ", "cubeRootsUnity"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Legacy", "@", "IndianRed"}], ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", "pts", ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", "ptLabels"}], "\[IndentingNewLine]", "}"}], ",", " ", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "14", ",", "Bold"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", RowBox[{"Legacy", "@", "AliceBlue"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.46661297703125*^9, 3.466612993984375*^9}, { 3.4666130485*^9, 3.466613186515625*^9}, {3.466615349328125*^9, 3.466615351140625*^9}, 3.46661612446875*^9, {3.49122090890625*^9, 3.491220971578125*^9}, {3.491221155578125*^9, 3.491221157375*^9}, 3.4912212358125*^9, {3.491221351515625*^9, 3.491221374703125*^9}, { 3.49122150746875*^9, 3.49122151265625*^9}, 3.491221621109375*^9, { 3.49125979165625*^9, 3.49125989396875*^9}, {3.491264146015625*^9, 3.491264177125*^9}, {3.4912642324375*^9, 3.49126425290625*^9}, { 3.4912643050625*^9, 3.491264311015625*^9}, {3.491264442265625*^9, 3.49126444653125*^9}, {3.492000377796875*^9, 3.4920003785625*^9}, 3.4920007421875*^9}], Cell[TextData[{ "In this example, it\[CloseCurlyQuote]s probably not useful to have all the \ shorter tick marks on the axes between the ones at ", Cell[BoxData[ FormBox[ RowBox[{"-", "1.0"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"-", "0.5"}], TraditionalForm]]], ", 0, 0.5, and 1.0. Change that by directly specifying where ticks should go \ with the ", StyleBox["Ticks", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option:" }], "Text", CellChangeTimes->{{3.491264503015625*^9, 3.491264633484375*^9}, { 3.491997520671875*^9, 3.4919975505625*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"pts", "=", RowBox[{"ComplexPoint", "/@", "cubeRootsUnity"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"\[CapitalDelta]z", "=", RowBox[{ RowBox[{"-", "0.01"}], "+", RowBox[{"0.2", "\[ImaginaryI]"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"ptLabels", "=", RowBox[{ RowBox[{ RowBox[{"ComplexText", "[", RowBox[{"#", ",", RowBox[{"#", "+", "\[CapitalDelta]z"}]}], "]"}], "&"}], "/@", " ", "cubeRootsUnity"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Legacy", "@", "IndianRed"}], ",", RowBox[{"PointSize", "[", "Large", "]"}], ",", "\[IndentingNewLine]", "pts", ",", "\[IndentingNewLine]", "Black", ",", "\[IndentingNewLine]", "ptLabels"}], "\[IndentingNewLine]", "}"}], ",", " ", "\[IndentingNewLine]", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.25"}], ",", "1.25"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Ticks", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", RowBox[{ RowBox[{"-", "1"}], "/", "2"}], ",", "0", ",", RowBox[{"1", "/", "2"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", RowBox[{ RowBox[{"-", "1"}], "/", "2"}], ",", "0", ",", RowBox[{"1", "/", "2"}], ",", "1"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "14", ",", "Bold"}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"Background", "\[Rule]", RowBox[{"Legacy", "@", "AliceBlue"}]}], ",", "\[IndentingNewLine]", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3.5", " ", "72"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.46661297703125*^9, 3.466612993984375*^9}, { 3.4666130485*^9, 3.466613186515625*^9}, {3.466615349328125*^9, 3.466615351140625*^9}, 3.46661612446875*^9, {3.49122090890625*^9, 3.491220971578125*^9}, {3.491221155578125*^9, 3.491221157375*^9}, 3.4912212358125*^9, {3.491221351515625*^9, 3.491221374703125*^9}, { 3.49122150746875*^9, 3.49122151265625*^9}, 3.491221621109375*^9, { 3.49125979165625*^9, 3.49125989396875*^9}, {3.491264146015625*^9, 3.491264177125*^9}, {3.4912642324375*^9, 3.49126425290625*^9}, { 3.4912643050625*^9, 3.491264311015625*^9}, {3.491264442265625*^9, 3.49126444653125*^9}, {3.491264648875*^9, 3.4912646903125*^9}, { 3.491997938671875*^9, 3.49199794596875*^9}, {3.4920003566875*^9, 3.49200036790625*^9}, 3.49200075990625*^9}, CellTags->"finalCodeCubeRoots"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Having to specify the coordinates explicitly like that twice, in a nested \ list, is unpleasant. To create an equally-spaced list of numbers, there\ \[CloseCurlyQuote]s an alternative, with the function ", StyleBox["Range", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". To figure out how ", StyleBox["Range", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " works, read its documentation and try each of the following:\n\t", StyleBox["Range[10]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\n\t", StyleBox["Range[0,10]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\n\t", StyleBox["Range[0,10,2]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\n\t", StyleBox["Range[-10,10,5]\n", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["\t", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Range[-10,10,1/5]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\nNow redo the previous exercise, but using a suitable ", StyleBox["Range", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " expression in place of each of the lists ", StyleBox["{-1,-1/2,0,1/2,1}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in the value of the ", StyleBox["Ticks", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.4912647719375*^9, 3.491265054609375*^9}, {3.491997578125*^9, 3.49199765246875*^9}, { 3.491997686921875*^9, 3.49199774640625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " You can change the style of ", StyleBox["all", FontSlant->"Italic"], " text appearing in a drawing\[LongDash]including point labels, axes labels, \ and even axes tick marks\[LongDash]by using the ", StyleBox["Draw2D", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option ", StyleBox["BaseStyle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".\n(a) Try, for example, using a ", StyleBox["BaseStyle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " value for font size 13 points and bold face.\nYou may still override the ", StyleBox["BaseStyle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " for individual kinds of text in a drawing. Thus you may adjust the style \ for just the point labels or just the axes labels, by using ", StyleBox["Style", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". But to override the ", StyleBox["BaseStyle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " for tick labels, you need to use the ", StyleBox["TicksStyle", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option.\n(b) Make the label for the whole drawing italic as well as bold." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491221974578125*^9, 3.491222074546875*^9}, {3.491222122765625*^9, 3.491222347390625*^9}, { 3.491222379140625*^9, 3.491222445890625*^9}, {3.491259913859375*^9, 3.4912599728125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". (a) Change the axes labels from \[OpenCurlyDoubleQuote]Real\ \[CloseCurlyDoubleQuote] and \[OpenCurlyDoubleQuote]Imaginary\ \[CloseCurlyDoubleQuote] to \[OpenCurlyDoubleQuote]x\[CloseCurlyDoubleQuote] \ and \[OpenCurlyDoubleQuote]y\[CloseCurlyDoubleQuote].\nLook carefully at the \ result. You should see that those letters appear in the drawing upright \ rather than Italic. Single-letter mathematical names are traditionally set in \ italic.\n(b) Make the axes labels appear in italic, as ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], ".\n(c) Just as the labels for the cube roots of unity were automatically \ typeset in a traditional mathematical format (using square-root signs and a \ horizontal fraction line), if you use ", StyleBox["AxesLabel\[Rule]{x,y}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " with ", StyleBox["no", FontSlant->"Italic"], " quotes around the ", StyleBox["x", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and the ", StyleBox["y", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", then you\[CloseCurlyQuote]ll automatically get italic ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " in the drawing. Try it.\n(d) Why might it be a bad idea to do what you \ tried in (c)? (", StyleBox["Hint", FontSlant->"Italic"], ": Suppose ", StyleBox["x", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " was a name to which you had already assigned some value.)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49126041409375*^9, 3.49126077528125*^9}, 3.491260824203125*^9, {3.491260904515625*^9, 3.491260925859375*^9}, 3.49201453246875*^9}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Draw the fourth roots of unity in the complex plane. Include nice point \ labels and an overall label for the drawing." }], "Exercise", CellChangeTimes->{{3.491221518078125*^9, 3.491221569375*^9}, 3.49122198421875*^9, {3.49126002421875*^9, 3.49126003878125*^9}, { 3.491260082765625*^9, 3.49126012878125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Plotting roots of unity as vertices of a triangle", "Subsection"], Cell[TextData[{ "The relevant graphics primitive from ", StyleBox["Presentations", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " here is ", StyleBox["ComplexLine", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". An expression of the form\n\t", StyleBox["ComplexLine[{", FontFamily->"Courier"], StyleBox["z1", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["z2", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier"], "\nrepresents the line segment from the complex number ", StyleBox["z1", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " to the complex number ", StyleBox["z2.", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], " More generally, an expression of the form\n\t", StyleBox["ComplexLine[{", FontFamily->"Courier"], StyleBox["z1", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["z2", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["\[Ellipsis]", FontFamily->"Times"], StyleBox[",", FontFamily->"Courier"], StyleBox["zn}", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"], "\nrepresents the \"broken line\"\[LongDash]the polygonal \ curve\[LongDash]with vertices the complex numbers ", StyleBox["z1,z2", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"Underline"->False}], StyleBox["\[Ellipsis]", FontFamily->"Times"], StyleBox[",zn", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], ". For example:" }], "Text", CellChangeTimes->{{3.466613201703125*^9, 3.46661320403125*^9}, { 3.466613249484375*^9, 3.46661328253125*^9}, 3.466613323328125*^9, { 3.49122249165625*^9, 3.491222535171875*^9}, {3.49122257265625*^9, 3.491222606859375*^9}, 3.491998236015625*^9}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"ComplexLine", "[", "cubeRootsUnity", "]"}]], "Input"], Cell[TextData[{ "The result of ", StyleBox["ComplexLine[{", FontFamily->"Courier"], StyleBox["z1", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["z2", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",", FontFamily->"Courier"], StyleBox["\[Ellipsis]", FontFamily->"Times"], StyleBox[",", FontFamily->"Courier"], StyleBox["zn", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["}]", FontFamily->"Courier"], " is the same as that of\n\t", StyleBox["Line[{", FontFamily->"Courier"], "ToCoordinates", StyleBox["[", FontFamily->"Courier"], StyleBox["z1", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["],", FontFamily->"Courier"], "ToCoordinates", StyleBox["[", FontFamily->"Courier"], StyleBox["z2", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["],", FontFamily->"Courier"], StyleBox["\[Ellipsis]", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", ", FontFamily->"Courier"], "ToCoordinates", StyleBox["[", FontFamily->"Courier"], StyleBox["zn", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]}]", FontFamily->"Courier"], "\nwhere the ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ToCoordinates", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " converts a complex number to the corresponding list of its real and \ imaginary parts. The function ", StyleBox["Line", FontFamily->"Courier"], StyleBox[" is a ", FontFamily->"Times"], "built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " graphics primitive." }], "SmallText", CellChangeTimes->{{3.46661344746875*^9, 3.46661350609375*^9}, 3.466616407421875*^9, {3.491222617359375*^9, 3.491222623140625*^9}, { 3.491222668921875*^9, 3.49122268321875*^9}}, ParagraphSpacing->{0.5, 0}], Cell["Here is the plot of that polygonal line:", "Text"], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Blue", ",", "\[IndentingNewLine]", RowBox[{"ComplexLine", "[", "cubeRootsUnity", "]"}]}], "\[IndentingNewLine]", "}"}], ",", "\[IndentingNewLine]", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"ImageSize", "\[Rule]", RowBox[{"3", " ", "72"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.466613520515625*^9, 3.466613564328125*^9}, 3.46661534328125*^9, 3.491222719546875*^9, {3.491260959609375*^9, 3.49126099046875*^9}}], Cell["\<\ To close up such a polygonal curve to form a polygon, you need to append the \ first point to the end of the list. 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" }], "Text", CellChangeTimes->{{3.466614394125*^9, 3.46661439509375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find two other arguments of ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "3"], TraditionalForm]]], ", one positive and the other negative." }], "Exercise", CellChangeTimes->{ 3.466614402875*^9, {3.492013091140625*^9, 3.492013099703125*^9}}, CellTags->"otherArgsOmega3"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find an argument for each of the other two cube roots of unity." }], "Exercise", CellChangeTimes->{ 3.46661440625*^9, {3.492013102609375*^9, 3.492013111875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " An ", ButtonBox["earlier exercise", BaseStyle->"Hyperlink", ButtonData:>"cube roots triangle"], " asked you to say, from what you saw in the graphics display, what kind of \ triangle it is that has the cube roots of unity as its vertices. 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(The second argument to Simplify indicates that the \ simplification is to be done using the ", StyleBox["assumption", FontWeight->"Bold"], " that ", StyleBox["r\[ThinSpace]>\[ThinSpace]0", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".)" }], "Text", CellChangeTimes->{{3.466614422015625*^9, 3.46661444959375*^9}, { 3.492013297421875*^9, 3.492013343875*^9}, {3.492013387390625*^9, 3.492013398109375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"z", ",", "r", ",", "\[Theta]"}], "]"}], "\[IndentingNewLine]", RowBox[{"z", "=", RowBox[{ RowBox[{"r", " ", RowBox[{"Cos", "[", "\[Theta]", "]"}]}], "+", RowBox[{"\[ImaginaryI]", " ", "r", " ", RowBox[{"Sin", "[", "\[Theta]", "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"Abs", "[", "z", "]"}], "]"}], ",", RowBox[{"r", ">", "0"}]}], " ", "]"}], "\[IndentingNewLine]", RowBox[{"Clear", "[", RowBox[{"z", ",", "r", ",", "\[Theta]"}], "]"}]}], "Input", CellChangeTimes->{{3.466614452953125*^9, 3.466614510078125*^9}, { 3.49201328128125*^9, 3.492013292546875*^9}}], Cell[TextData[{ "Because ", Cell[BoxData[ FormBox["cos", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["sin", TraditionalForm]]], " are periodic, a single point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " in the plane has infinitely many different polar coordinates. But when ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"a", ",", " ", "b"}], ")"}], " ", "\[NotEqual]", " ", RowBox[{"(", RowBox[{"0", ",", " ", "0"}], ")"}]}], TraditionalForm]]], ", that is, when the complex number ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", "=", RowBox[{ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], "\[NotEqual]", "0"}]}], TraditionalForm]]], ", then the point has ", StyleBox["unique", FontSlant->"Italic"], " polar coordinates ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]]], " for which\n\t", Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{ RowBox[{"r", " ", "cos", " ", "\[Theta]"}], "+", RowBox[{"\[ImaginaryI]", " ", "r", " ", "sin", " ", "\[Theta]"}]}]}], TraditionalForm]]], ",\t\t", Cell[BoxData[ FormBox[ RowBox[{"r", ">", "0"}], TraditionalForm]]], ", \t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "\[Pi]"}], "<", "\[Theta]", "\[LessEqual]", "\[Pi]"}], TraditionalForm]]], ".\nThis unique ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]]], " in the half-open, half-closed interval ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "\[Pi]"}], ",", "\[Pi]"}], "]"}], TraditionalForm]]], " is called the", StyleBox[" ", FontWeight->"Bold"], StyleBox["principal argument", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.466614520953125*^9, 3.466614587796875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "The ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Arg", FontFamily->"Courier", FontWeight->"Bold"], " finds the principal argument. For example:" }], "Text"], Cell[BoxData[ RowBox[{"Arg", "[", SubscriptBox["\[Omega]", "3"], "]"}]], "Input", CellChangeTimes->{{3.49201346834375*^9, 3.49201347109375*^9}}], Cell[TextData[{ "Thus the principal argument of ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Omega]", "3"], "=", RowBox[{ RowBox[{ RowBox[{"-", "1"}], "/", "2"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{ SqrtBox["3"], "/", "2"}]}]}]}], TraditionalForm]]], " is the number ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", "=", RowBox[{"2", RowBox[{"\[Pi]", "/", "3"}]}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.466614593921875*^9, 3.466614608484375*^9}, { 3.4920134799375*^9, 3.492013486515625*^9}}], Cell[TextData[{ StyleBox["\[WarningSign]", FontSize->24, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->RGBColor[1, 1, 0]], " ", StyleBox["Caution:", FontSlant->"Italic"], " You may be accustomed to selecting polar coordinates \[Theta] with ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "\[Theta]", " ", "<", RowBox[{"2", "\[Pi]"}]}], TraditionalForm]]], ". However, in complex analysis ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "\[Pi]"}], "<", "\[Theta]", "\[LessEqual]", "\[Pi]"}], TraditionalForm]]], " is the convention usually adopted for the principal argument; that is the \ convention used by ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text", CellChangeTimes->{{3.466614612296875*^9, 3.466614642078125*^9}, { 3.492013237765625*^9, 3.492013260765625*^9}, {3.492015870234375*^9, 3.49201587628125*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Check that ", StyleBox["neither", FontSlant->"Italic"], " of the other two arguments \[Theta] of ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "3"], TraditionalForm]]], " that you found in Exercise ", CounterBox["Exercise", "otherArgsOmega3"], " satisfies the condition ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "\[Pi]"}], "<", "\[Theta]", "\[LessEqual]", "\[Pi]"}], TraditionalForm]]], " required to be the principal argument." }], "Exercise", CellChangeTimes->{{3.4666146474375*^9, 3.466614661515625*^9}, { 3.49201299878125*^9, 3.492013008078125*^9}, {3.492013551140625*^9, 3.49201355375*^9}, 3.492013628265625*^9}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find the principal arguments of the other two of the three cube roots of \ unity." }], "Exercise", CellChangeTimes->{ 3.466614670125*^9, {3.492012952390625*^9, 3.4920129621875*^9}}], Cell[TextData[StyleBox["Please do not proceed to the next section until you \ have done the preceding exercise!", FontWeight->"Bold", FontSlant->"Italic"]], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["The primitive cube root of unity", "Section"], Cell["\<\ The principal arguments of the three cube roots of unity are:\ \>", "Text", ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"Arg", "[", "cubeRootsUnity", "]"}]], "Input"], Cell[TextData[{ "The smallest strictly positive one of these three arguments is the third \ one, ", Cell[BoxData[ FormBox[ RowBox[{"2", RowBox[{"\[Pi]", "/", "3"}]}], TraditionalForm]]], ". And that is the principal argument of \n\t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Omega]", "3"], "=", RowBox[{ RowBox[{"cos", " ", "2", RowBox[{"\[Pi]", "/", "3"}]}], "+", RowBox[{"\[ImaginaryI]", " ", "sin", " ", "2", RowBox[{"\[DoubledPi]", "/", "3"}]}]}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.46661468471875*^9, 3.46661468990625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "This root with the smallest strictly positive principal argument is called \ the ", StyleBox["primitive", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " cube root of unity." }], "Text"], Cell[TextData[{ "According to the theory of ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th roots of unity:" }], "Text", CellChangeTimes->{ 3.4666146968125*^9, {3.492013873171875*^9, 3.492013874609375*^9}}], Cell["\<\ The set of all three cube roots of unity consists of the powers of the \ primitive cube root of unity.\ \>", "EmphasisText", CellChangeTimes->{ 3.4666146968125*^9, {3.492013873171875*^9, 3.492013895625*^9}}], Cell[TextData[{ " Here are those three powers of ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "3"], TraditionalForm]]], ":" }], "Text", CellChangeTimes->{ 3.4666146968125*^9, {3.492013873171875*^9, 3.492013895625*^9}}], Cell[BoxData[ RowBox[{"cubeRoots", "=", RowBox[{"Expand", "[", SuperscriptBox[ SubscriptBox["\[Omega]", "3"], RowBox[{"{", RowBox[{"0", ",", " ", "1", ",", " ", "2"}], "}"}]], "]"}]}]], "Input", CellChangeTimes->{ 3.466614701125*^9, {3.492013690171875*^9, 3.492013693140625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Verify that the set of three cube roots of unity appearing in the list ", StyleBox["cubeRootsUnity", FontFamily->"Courier New"], " is exactly the same as the set of powers of the primitive cube root of \ unity appearing in the list ", StyleBox["cubeRoots", FontFamily->"Courier New"], ". Do this by evaluating a single equation in ", StyleBox["Mathematica", FontSlant->"Italic"], " that has result ", StyleBox["True", FontFamily->"Courier New"], ". (", StyleBox["Hint:", FontSlant->"Italic"], "The difficulty is that ", StyleBox["Mathematica", FontSlant->"Italic"], " lists give their entries in some particular order, whereas a set need not \ have any particular order for its members. Try the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Sort", FontFamily->"Courier New"], ".)" }], "Exercise", CellChangeTimes->{{3.46661470725*^9, 3.466614720859375*^9}, { 3.492012903203125*^9, 3.492012920078125*^9}, 3.492013840515625*^9}] }, Closed]], Cell[CellGroupData[{ Cell["Roots of other complex numbers", "Section"], Cell[TextData[{ StyleBox["Example:", FontWeight->"Bold"], " Find the cube roots of ", Cell[BoxData[ FormBox[ RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.466614737625*^9, 3.466614743796875*^9}, { 3.492012780984375*^9, 3.49201278234375*^9}}], Cell[TextData[{ "One way to find these cube roots is simply to solve the corresponding cubic \ equation ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "3"], "\[Equal]", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], TraditionalForm]]], ", as usual using ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " so as to obtain actual complex numbers in ", Cell[BoxData[ FormBox[ RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], TraditionalForm]]], " form:" }], "Text", CellChangeTimes->{{3.466614749015625*^9, 3.46661476225*^9}, { 3.492012802296875*^9, 3.492012806796875*^9}}], Cell[BoxData[ RowBox[{"newRoots", "=", RowBox[{"ComplexExpand", "[", RowBox[{"z", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ SuperscriptBox["z", "3"], "\[Equal]", RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}]}], ",", "z"}], "]"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.466614765*^9, 3.46661478671875*^9}, {3.49201282275*^9, 3.492012825890625*^9}}], Cell[TextData[{ "According to the theory, you only need one of the three cube roots of ", Cell[BoxData[ FormBox[ RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], " in order to find them all. Here's the last of the three:" }], "Text", CellChangeTimes->{{3.466614793203125*^9, 3.466614809234375*^9}}], Cell[BoxData[ RowBox[{"\[Zeta]", "=", RowBox[{"Last", "[", "newRoots", "]"}]}]], "Input", CellChangeTimes->{{3.466614812421875*^9, 3.466614813234375*^9}, { 3.492012830703125*^9, 3.4920128320625*^9}}], Cell[TextData[{ "Then all the cube roots of ", Cell[BoxData[ FormBox[ RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], " are, according to the theory, the products of that one cube root and the \ cube roots of unity. And the cube roots of unity are just the powers of the \ primitive cube root of unity ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "3"], TraditionalForm]]], ". So again the three cube roots of ", Cell[BoxData[ FormBox[ RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], " are:" }], "Text", CellChangeTimes->{{3.466614820328125*^9, 3.466614869640625*^9}, { 3.492012837765625*^9, 3.492012841671875*^9}}], Cell[BoxData[ RowBox[{"rootsAgain", "=", RowBox[{"ComplexExpand", "[", RowBox[{"\[Zeta]", " ", SuperscriptBox[ SubscriptBox["\[Omega]", "3"], RowBox[{"{", RowBox[{"0", ",", "1", ",", "2"}], "}"}]]}], "]"}]}]], "Input", CellChangeTimes->{{3.466614857484375*^9, 3.466614860671875*^9}, { 3.4920128635*^9, 3.492012866234375*^9}, {3.4920141696875*^9, 3.49201417084375*^9}}], Cell[TextData[{ "Just to be convinced that the theory really is correct, check that these \ numbers all do have ", Cell[BoxData[ FormBox[ RowBox[{"2", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], " as their cubes:" }], "Text", CellChangeTimes->{{3.466614877640625*^9, 3.466614878390625*^9}, { 3.492014206765625*^9, 3.49201422109375*^9}}], Cell[BoxData[ RowBox[{"Simplify", "[", SuperscriptBox["rootsAgain", "3"], "]"}]], "Input", CellChangeTimes->{{3.466614883890625*^9, 3.466614884828125*^9}, { 3.49201419734375*^9, 3.492014199390625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find the cube roots of ", Cell[BoxData[ FormBox[ RowBox[{"2", "-", RowBox[{"11", "\[ImaginaryI]"}]}], TraditionalForm]]], ". Use one of them together with powers of the cube root of unity to find \ them all again." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.492014233484375*^9, 3.492014256734375*^9}, {3.492014340625*^9, 3.49201438884375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Exploration: fourth, fifth, sixth, \[Ellipsis] roots of unity\ \>", "Section", CellTags->"exploration"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) Repeat for fourth roots of unity the calculations done above for cube \ roots of unity. How many do you obtain?\n", StyleBox["(b)", FontWeight->"Bold"], " Plot the fourth roots of unity as points in the plane and as the vertices \ of some polygon. What do you observe about this polygon?" }], "Exercise", CellChangeTimes->{{3.4666148979375*^9, 3.466614903734375*^9}, { 3.49201119621875*^9, 3.4920112068125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) Repeat for fifth roots of unity the calculations done above for cube \ roots of unity. How many do you obtain?\n(b) Plot the fifth roots of unity as \ points in the plane and as the vertices of some polygon. What do you observe \ about this polygon?" }], "Exercise", CellChangeTimes->{{3.46661490603125*^9, 3.46661490903125*^9}, { 3.492011209890625*^9, 3.492011220203125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) Repeat for sixth roots of unity the calculations done above for cube \ roots of unity. How many do you obtain?\n(b) Plot the sixth roots of unity as \ points in the plane and as the vertices of some polygon. What do you observe \ about this polygon?" }], "Exercise", CellChangeTimes->{{3.4666149195625*^9, 3.46661494703125*^9}, { 3.492011222984375*^9, 3.492011230671875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " What does the evidence obtained so far suggest as to: \n\t(a) how many ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th roots of unity there are; and\n\t(b) the configuration of these ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th roots of unity as points in the complex plane?" }], "Exercise", CellChangeTimes->{ 3.46661494925*^9, {3.49201123403125*^9, 3.4920112425625*^9}, { 3.49201129159375*^9, 3.492011292875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Use ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to create a dynamic drasing of the ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th roots of unity, and with the primitive ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "th root of unity highlighted so as to be distinguished from the others, \ where the control changes the value of ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ", starting at ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "2"}], TraditionalForm]]], ". (For help about ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " with complex graphics, see the section \"Making dynamic drawings by using ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\" in notebook ", StyleBox["DrawingComplexObjects.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".)" }], "Exercise", CellChangeTimes->{{3.49346089253125*^9, 3.49346114753125*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Appendices", "Section", CellChangeTimes->{{3.492006698671875*^9, 3.4920067005625*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Appendix ", CounterBox["Subsection"], ": Plotting roots of unity as vectors from the origin" }], "Subsection", CellChangeTimes->{{3.492006870734375*^9, 3.492006884640625*^9}}], Cell[TextData[{ "Instead of viewing the cube roots of unity just as points, you can also \ represent them by vectors\[LongDash]arrows drawn from the origin to those \ points. The ", StyleBox["Presentations", FontSlant->"Italic"], " graphics function ", StyleBox["ComplexArrow", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " creates such vectors. This function takes as argument a list of two \ complex numbers that prescribe the tail and the head of the arrow, \ respectively." }], "Text", CellChangeTimes->{{3.466614964765625*^9, 3.466614998078125*^9}, { 3.466615489859375*^9, 3.4666155234375*^9}, {3.492000985765625*^9, 3.49200100384375*^9}}], Cell["\<\ Consider, for example, the third cube root of unity (in the order that the \ three cube roots of unity were found)\[Ellipsis]\ \>", "Text"], Cell[BoxData[ RowBox[{"cubeRootsUnity", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}]], "Input"], Cell["\<\ \[Ellipsis]and form the arrow from the origin to that cube root of unity: \ \>", "Text"], Cell[BoxData[ RowBox[{"ComplexArrow", "[", RowBox[{"{", RowBox[{"0", ",", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{ 3.4666150049375*^9, {3.466615531265625*^9, 3.46661553584375*^9}}], Cell[TextData[{ "As you see, ", StyleBox["ComplexArrow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " creates an equivalent object with head ", StyleBox["Arrow", FontFamily->"Courier"], ". The function ", StyleBox["Arrow", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is a built-in ", StyleBox["Mathematica", FontSlant->"Italic"], " object. And what the ", StyleBox["Presentations", FontSlant->"Italic"], " function ", StyleBox["ComplexArrow", FontFamily->"Courier"], StyleBox[" ", FontFamily->"Times"], "does is to apply ", StyleBox["ToCoordinates", FontFamily->"Courier"], " to each of the complex number arguments\[LongDash]here 0 and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", "1"}], "/", "2"}], " ", "+", " ", RowBox[{"\[ImaginaryI]", " ", RowBox[{ SqrtBox["3"], "/", "2"}]}]}], TraditionalForm]]], "\[LongDash]to convert them into the corresponding ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], "-coordinate pairs ", StyleBox["{0,0}", FontFamily->"Courier"], " and ", StyleBox["{-1/2,", FontFamily->"Courier"], Cell[BoxData[ FormBox[ SqrtBox["2"], TraditionalForm]], FontFamily->"Courier"], StyleBox["/2}", FontFamily->"Courier"], " required by ", StyleBox["Arrow", FontFamily->"Courier"], ". In other words,\n\t", StyleBox["ComplexArrow[{0,cubeRootsUnity\[LeftDoubleBracket]3\ \[RightDoubleBracket]}]", FontFamily->"Courier"], "\ngives the same result as\n\t", StyleBox["Arrow[{ToCoordinates[0],ToCoordinates[cubeRootsUnity\ \[LeftDoubleBracket]3\[RightDoubleBracket]]}]", FontFamily->"Courier"] }], "SmallText", CellChangeTimes->{{3.466615035921875*^9, 3.46661513571875*^9}, { 3.46661517175*^9, 3.466615302328125*^9}, {3.466615539984375*^9, 3.466615605171875*^9}, 3.46661641784375*^9, {3.492001043921875*^9, 3.492001073203125*^9}}, ParagraphSpacing->{0.5, 0}], Cell["Plot that vector:", "Text"], Cell[BoxData[ RowBox[{"Draw2D", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"{", "\[IndentingNewLine]", RowBox[{"Blue", ",", RowBox[{"Arrowheads", "[", "Large", "]"}], ",", "\[IndentingNewLine]", RowBox[{"ComplexArrow", "[", RowBox[{"{", RowBox[{"0", ",", RowBox[{ "cubeRootsUnity", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}]}], "}"}], "]"}]}], "\[IndentingNewLine]", "}"}], ",", 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