(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 10.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1064, 20] NotebookDataLength[ 76096, 2049] NotebookOptionsPosition[ 73391, 1948] NotebookOutlinePosition[ 73780, 1965] CellTagsIndexPosition[ 73737, 1962] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Complex Number Addition and the Complex (", StyleBox[ButtonBox["Argand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/ArgandDiagram.html"], None}, ButtonNote->"http://mathworld.wolfram.com/ArgandDiagram.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[0, 0, 1]], ") Plane, Activity 1" }], "Title", CellChangeTimes->{{3.6276539583894243`*^9, 3.627653977623808*^9}, { 3.6276540216568527`*^9, 3.6276540406690397`*^9}, {3.627660012060422*^9, 3.6276600131435823`*^9}, {3.6280071117279673`*^9, 3.6280071130225677`*^9}, { 3.630425059429665*^9, 3.630425061817275*^9}}], Cell[CellGroupData[{ Cell["Learning Goals:", "Subchapter", CellChangeTimes->{{3.6294708824182262`*^9, 3.6294708852158613`*^9}, { 3.6294709184784937`*^9, 3.6294711641254807`*^9}, {3.629471214221171*^9, 3.629471286589877*^9}, {3.678540619338686*^9, 3.678540619597479*^9}}], Cell[TextData[{ "1) See that one \[OpenCurlyDoubleQuote]natural\[CloseCurlyDoubleQuote] view \ of complex numbers as linear \[OpenCurlyDoubleQuote]polynomials\ \[CloseCurlyDoubleQuote], with real number coefficients, in the ", StyleBox[ButtonBox["indeterminate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Indeterminate_(variable)"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Indeterminate_(variable)"], FontVariations->{"Underline"->True}, FontColor->RGBColor[0, 0, 1]], " \[ImaginaryI] (forgetting, for the moment, that ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ImaginaryI]", "2"], "=", RowBox[{"-", "1"}]}], TraditionalForm]]], ") leads to \n\t(a) a definition of complex number addition that is \ reasonable algebraically and\n\t(b) a reasonable geometric interpretation of \ complex numbers as points in a plane, analogous to the geometric \ interpretation of real numbers as points on a line.\n\t\n2) Learn basic \ applications of the ", StyleBox["Mathematica", FontSlant->"Italic"], " functions ", StyleBox[ButtonBox["ComplexExpand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/ComplexExpand.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/ComplexExpand.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["ListPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/ListPlot.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/ListPlot.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["Graphics", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Graphics.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Graphics.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["Table", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Table.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Table.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", and ", StyleBox[ButtonBox["Show", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Show.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Show.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " in this setting, as well as various ", StyleBox["Mathematica", FontSlant->"Italic"], " formatting options" }], "Subsection", CellChangeTimes->{{3.629471283194976*^9, 3.629471449316176*^9}, { 3.629471491689554*^9, 3.629471495629924*^9}, 3.629471704802578*^9, { 3.6294717887322683`*^9, 3.629471791315475*^9}, 3.629471883527823*^9, { 3.630073557592288*^9, 3.630073561685176*^9}, {3.6478701952830887`*^9, 3.647870200985578*^9}, {3.647870236097437*^9, 3.6478702457649117`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Prerequisites:", "Subchapter", CellChangeTimes->{{3.6294708824182262`*^9, 3.6294708852158613`*^9}, { 3.6294709184784937`*^9, 3.6294711641254807`*^9}, {3.629471214221171*^9, 3.629471286589877*^9}, {3.62947861284763*^9, 3.629478616140943*^9}, 3.647877110832814*^9}], Cell["\<\ 1) Some comfort with abstract algebra, though not necessarily at the level of \ a course in abstract algebra 2) A basic familiarity with complex numbers in precalculus or calculus is \ helpful 3) Understanding of the way points are described in a rectangular (Cartesian) \ coordinate system\ \>", "Subsection", CellChangeTimes->{{3.629471283194976*^9, 3.629471449316176*^9}, { 3.629471491689554*^9, 3.629471495629924*^9}, 3.629471704802578*^9, { 3.6294717887322683`*^9, 3.629471791315475*^9}, 3.629471883527823*^9, { 3.629478639313504*^9, 3.629478679493908*^9}, {3.629478720658692*^9, 3.629478816037221*^9}, {3.647953675967849*^9, 3.6479536776528473`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Introduction: ", "Subchapter", CellChangeTimes->{{3.6294708824182262`*^9, 3.6294708852158613`*^9}, { 3.6294709184784937`*^9, 3.6294711641254807`*^9}, {3.629471214221171*^9, 3.629471286589877*^9}, {3.6294714662738123`*^9, 3.629471468263523*^9}, { 3.630073613440137*^9, 3.630073624429462*^9}}], Cell[TextData[{ "The ", StyleBox[ButtonBox["arithmetic", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Arithmetic"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Arithmetic"], FontColor->RGBColor[0, 0, 1]], " of ", StyleBox[ButtonBox["complex numbers", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Complex_number"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Complex_number"], FontColor->RGBColor[0, 0, 1]], ", which leads to deeper subject of ", StyleBox[ButtonBox["complex analysis", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Complex_analysis"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Complex_analysis"], FontColor->RGBColor[0, 0, 1]], " (calculus-related ideas and tools for understanding ", StyleBox[ButtonBox["complex functions", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Complex-valued_function"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Complex-valued_function"], FontColor->RGBColor[0, 0, 1]], "), can be thought about and worked through in ", StyleBox[ButtonBox["purely symbolic rule-based (formal) ways", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Formalism_(mathematics)"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Formalism_(mathematics)"], FontColor->RGBColor[0, 0, 1]], ". However, as with many areas of mathematics, the geometry of ", StyleBox[ButtonBox["complex arithmetic", BaseStyle->"Hyperlink", ButtonData->{ URL["http://betterexplained.com/articles/intuitive-arithmetic-with-\ complex-numbers/"], None}, ButtonNote-> "http://betterexplained.com/articles/intuitive-arithmetic-with-complex-\ numbers/"], FontColor->RGBColor[0, 0, 1]], " gives the symbolic ", StyleBox[ButtonBox["mathematical statements of fact", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Theorem"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Theorem"], FontColor->RGBColor[0, 0, 1]], " richer meaning, leads to new insights, and informs possible ", StyleBox[ButtonBox["real-world applications", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathigon.org/mathigon_org/panorama/"], None}, ButtonNote->"http://mathigon.org/mathigon_org/panorama/"], FontColor->RGBColor[0, 0, 1]], ". This module, \[OpenCurlyDoubleQuote]Complex Number Addition and the \ Complex (", StyleBox[ButtonBox["Argand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/ArgandDiagram.html"], None}, ButtonNote->"http://mathworld.wolfram.com/ArgandDiagram.html"], FontColor->RGBColor[0, 0, 1]], ") Plane\[CloseCurlyDoubleQuote], provides an introduction to the marriage \ of the symbolic and the geometric within the study of complex arithmetic, \ providing a solid foundation for work in ", StyleBox[ButtonBox["complex analysis", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Complex_analysis"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Complex_analysis"], FontColor->RGBColor[0, 0, 1]], ". Activity 1 of this module focuses on the basic correspondence between a \ given complex number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " and the point ", StyleBox["z", FontSlant->"Italic"], " in the plane whose ", StyleBox[ButtonBox["rectangular (Cartesian) coordinates", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Cartesian_coordinate_system"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Cartesian_coordinate_system"], FontColor->RGBColor[0, 0, 1]], " are ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], ". When this correspondence is specified and the axes are labeled with the \ words \[OpenCurlyDoubleQuote]real\[CloseCurlyDoubleQuote] (horizontal) and \ \[OpenCurlyDoubleQuote]imaginary\[CloseCurlyDoubleQuote] (vertical), the \ plane is then called \[OpenCurlyDoubleQuote]the\[CloseCurlyDoubleQuote] ", StyleBox[ButtonBox["complex plane", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Complex_plane"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Complex_plane"], FontColor->RGBColor[0, 0, 1]], ", or sometimes the ", StyleBox[ButtonBox["Argand plane", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/ArgandPlane.html"], None}, ButtonNote->"http://mathworld.wolfram.com/ArgandPlane.html"], FontColor->RGBColor[0, 0, 1]], ", in honor of the Swiss mathematician ", StyleBox[ButtonBox["Jean-Robert Argand", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Jean-Robert_Argand"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Jean-Robert_Argand"], FontColor->RGBColor[0, 0, 1]], "." }], "Text", CellChangeTimes->{{3.6304183274365187`*^9, 3.630418632279633*^9}, { 3.630418697250352*^9, 3.630418701773437*^9}, {3.6304187693186607`*^9, 3.630418769321042*^9}, {3.63041883960098*^9, 3.630418901440366*^9}, { 3.630418952732256*^9, 3.630418965072918*^9}, {3.630418996273047*^9, 3.6304190678720512`*^9}, {3.630419411193425*^9, 3.630419648870229*^9}, { 3.630419693480446*^9, 3.630419693483626*^9}, {3.647870451615673*^9, 3.647870471492469*^9}, {3.647870639638309*^9, 3.647870698250945*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Content:", "Subchapter", CellChangeTimes->{{3.6294708824182262`*^9, 3.6294708852158613`*^9}, { 3.6294709184784937`*^9, 3.6294711641254807`*^9}, {3.629471214221171*^9, 3.629471286589877*^9}, {3.6294714662738123`*^9, 3.629471468263523*^9}}], Cell[TextData[{ "Complex numbers can be viewed in purely algebraic and symbolic ways, which \ is where we start in this first activity of a learning module on complex \ addition and the complex plane. However, our main goal in this learning \ module is to see that there are natural ", StyleBox["geometric", FontSlant->"Italic"], " ways of viewing complex numbers, and that this mode of thought is a \ powerful tool for understanding and problem-solving. This is most typically \ done with either ", StyleBox[ButtonBox["rectangular", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Cartesian_coordinate_system"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Cartesian_coordinate_system"], FontColor->RGBColor[0, 0, 1]], " or ", StyleBox[ButtonBox["polar", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Polar_coordinate_system"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Polar_coordinate_system"], FontColor->RGBColor[0, 0, 1]], " coordinates, and you will need to become very familiar with translating \ between these two coordinate systems if you want to be successful in \ understanding and using complex numbers and complex functions." }], "Text", CellChangeTimes->{{3.630073667779408*^9, 3.630073914845625*^9}, { 3.647870717155521*^9, 3.6478707944513903`*^9}}], Cell[TextData[{ StyleBox[ButtonBox["Algebraic expressions", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.themathpage.com/alg/algebraic-expressions.htm"], None}, ButtonNote->"http://www.themathpage.com/alg/algebraic-expressions.htm"], FontColor->RGBColor[0, 0, 1]], " are combinations of symbols, such as ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"6", SuperscriptBox["x", "2"]}], "+", RowBox[{"5", "x", " ", "y"}], "+", SuperscriptBox["y", "3"], "+", RowBox[{ FractionBox["2", "3"], "x"}], "+", "z", "+", RowBox[{"3", "z"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["z", "5"], "-", RowBox[{"11", "y", " ", "x"}], "+", RowBox[{"2", "x"}], "+", RowBox[{"x", "(", RowBox[{"4", "+", "x"}], ")"}]}], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SuperscriptBox["x", "2"], "y"], "+", FractionBox[ SuperscriptBox["x", "3"], "x"]}], TraditionalForm]]], ", that combine numbers and variables using operations from arithmetic, such \ as addition, subtraction, multiplication, and division \[LongDash] other \ operations that are sometimes allowed include things such as square roots, \ cube roots, and more exotic kinds of mathematical objects. The variables can \ be thought of as pure symbols with no meaning, but we usually think of them \ as representing numbers, and we usually are interested in determining what \ happens to the values of these expressions as the variables are allowed to \ change, or \[OpenCurlyDoubleQuote]vary\[CloseCurlyDoubleQuote]. In other \ words, we typically want to think of them as defining ", StyleBox[ButtonBox["functions", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Function_(mathematics)"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Function_(mathematics)"], FontColor->RGBColor[0, 0, 1]], " and then use our mathematical knowledge and tools to analyze the behavior \ of those functions. In mathematics classes, this is often a purely mental \ exercise, but it forms the foundation for nearly all advanced applications of \ mathematics, including in the physical sciences, the social sciences, \ computer science, medicine, engineering, finance, and statistics." }], "Text", CellChangeTimes->{{3.6276544638910522`*^9, 3.627654921618792*^9}, { 3.627654953987253*^9, 3.627655134176111*^9}, {3.627655194703266*^9, 3.627655209837347*^9}, {3.627655250230563*^9, 3.627655279668812*^9}, { 3.6276553116190243`*^9, 3.627655331808749*^9}, {3.6276553628027782`*^9, 3.62765536305539*^9}, {3.627655438567178*^9, 3.627655438570181*^9}, { 3.6276554959195538`*^9, 3.627655511174876*^9}, {3.6276556418487177`*^9, 3.627655674424116*^9}, {3.627655722822695*^9, 3.627655747228941*^9}, { 3.6276557879147882`*^9, 3.627655793753824*^9}, {3.627655844144212*^9, 3.6276558869076223`*^9}, {3.627656993685939*^9, 3.627656995104754*^9}, { 3.627659375723448*^9, 3.627659377855976*^9}, {3.627907927584279*^9, 3.627908029154234*^9}, {3.628874881334854*^9, 3.6288749391920156`*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Discussion 1:", FontWeight->"Bold"], " How should the expressions ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], ",", RowBox[{ RowBox[{"6", SuperscriptBox["x", "2"]}], "+", RowBox[{"5", "x", " ", "y"}], "+", SuperscriptBox["y", "3"], "+", RowBox[{ FractionBox["2", "3"], "x"}], "-", "z", "+", RowBox[{"3", "z"}]}], ",", RowBox[{ FractionBox["z", "5"], "-", RowBox[{"11", "y", " ", "x"}], "+", RowBox[{"2", "x"}], "+", RowBox[{"x", "(", RowBox[{"4", "+", "x"}], ")"}]}], ",", RowBox[{ RowBox[{"and", FractionBox[ SuperscriptBox["x", "2"], "y"]}], "+", FractionBox[ SuperscriptBox["x", "3"], "x"]}]}], TraditionalForm]]], " be simplified, if at all? How should arithmetic (addition, subtraction, \ multiplication, and division) be performed and simplification be done? (", StyleBox["Hints:", FontWeight->"Bold"], " recall what \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["like terms", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.mathsisfun.com/algebra/like-terms.html"], None}, ButtonNote->"http://www.mathsisfun.com/algebra/like-terms.html"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote] and \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["common denominators", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.mathsisfun.com/numbers/common-denominator.html"], None}, ButtonNote->"http://www.mathsisfun.com/numbers/common-denominator.html"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote] are. Also recall the ", StyleBox[ButtonBox["distributive property", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Distributive_property"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Distributive_property"], FontColor->RGBColor[0, 0, 1]], " and the ", StyleBox[ButtonBox["rules for exponents", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.purplemath.com/modules/exponent.htm"], None}, ButtonNote->"http://www.purplemath.com/modules/exponent.htm"], FontColor->RGBColor[0, 0, 1]], "). Write down some sample computations and simplifications using the given \ expressions. How does this work correspond to the properties of the \ arithmetic of the ", StyleBox[ButtonBox["real numbers", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Real_number"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Real_number"], FontColor->RGBColor[0, 0, 1]], "? For example, are you using the ", StyleBox[ButtonBox["commutative property", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Commutative_property"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Commutative_property"], FontColor->RGBColor[0, 0, 1]], " for addition or multiplication anywhere?" }], "Subsection", CellChangeTimes->{{3.6276557036467533`*^9, 3.627655717913577*^9}, { 3.627655764618641*^9, 3.62765590921062*^9}, {3.627656765938553*^9, 3.627656864463704*^9}, {3.627656948349906*^9, 3.627656997576027*^9}, { 3.627657059639831*^9, 3.6276571369713497`*^9}, {3.627657178490571*^9, 3.627657203292487*^9}, {3.6276573617503653`*^9, 3.627657361906416*^9}, { 3.627657492593141*^9, 3.6276575163946*^9}, {3.627659369469182*^9, 3.627659372254017*^9}, {3.6279081214851007`*^9, 3.627908138201469*^9}, { 3.627908171701682*^9, 3.6279082835834637`*^9}, {3.627908355561986*^9, 3.627908496036009*^9}, {3.62790879589528*^9, 3.627908795898261*^9}, { 3.647871039181404*^9, 3.647871044573051*^9}}], Cell[CellGroupData[{ Cell["Response 1: ", "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659516682569*^9, 3.627659516839205*^9}}], Cell[TextData[StyleBox["(You can type your thoughts and answers here \ formatted in text mode)", FontWeight->"Bold", FontColor->RGBColor[0, 0.67, 0]]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659535820641*^9, 3.627659537608328*^9}, {3.627659670483754*^9, 3.627659682423826*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Grader/Instructor Response 1: ", "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659516682569*^9, 3.627659516839205*^9}, {3.627659556506875*^9, 3.62765955845679*^9}, {3.627659617632988*^9, 3.62765961929755*^9}}], Cell[TextData[StyleBox["(The grader/instructor will give you feedback about \ your work here)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659535820641*^9, 3.627659578217204*^9}, {3.627659621472638*^9, 3.627659622910438*^9}, {3.628006013730137*^9, 3.6280060155901546`*^9}, { 3.628877211175973*^9, 3.628877213714664*^9}}], Cell["\<\ The main point of Discussion 1 is that we do indeed often treat expressions \ as if the variables represent numbers; and indeed, we often use properties of \ real numbers to simplify these expressions and arithmetic combinations of \ these expressions. If you are at a point in your mathematical education \ where you can do such operations without much thought, that is a sign that \ you have internalized much of your pre-college mathematics well. \ \>", "Text", CellChangeTimes->{{3.627658719107088*^9, 3.627658784867366*^9}, { 3.627658889349979*^9, 3.62765892162006*^9}, {3.6276589710657177`*^9, 3.62765897108265*^9}, {3.627659002581394*^9, 3.627659053400085*^9}, { 3.627659092197673*^9, 3.627659092200029*^9}, {3.627659131019465*^9, 3.627659159258582*^9}, {3.627659194021532*^9, 3.627659236063088*^9}, { 3.6276592849170923`*^9, 3.6276592929284286`*^9}, {3.627659401231691*^9, 3.627659404516089*^9}, {3.627908540015504*^9, 3.6279086290492496`*^9}, { 3.628875035777446*^9, 3.628875036539165*^9}, {3.647871072116899*^9, 3.647871157026165*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can do purely symbolic arithmetic, based on these assumptions, though \ sometimes we must use the built-in symbolic manipulation functions ", StyleBox[ButtonBox["Expand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Expand.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Expand.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["Simplify", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Simplify.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Simplify.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["FullSimplify", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/FullSimplify.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/FullSimplify.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["Together", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Together.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Together.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["Apart", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Apart.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Apart.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", or ", StyleBox[ButtonBox["Factor", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Factor.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Factor.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " to get the answer into a form that we want. Enter each of the following \ three lines of code to see examples of this." }], "Text", CellChangeTimes->{ 3.647871166278544*^9, {3.647871305866301*^9, 3.647871325965828*^9}}], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"x", "^", "2"}], "y"], "+", FractionBox[ RowBox[{"x", "^", "3"}], "x"]}]], "Input", CellChangeTimes->{{3.647871290674178*^9, 3.647871296191863*^9}}], Cell[BoxData[ RowBox[{"Together", "[", RowBox[{ FractionBox[ RowBox[{"x", "^", "2"}], "y"], "+", FractionBox[ RowBox[{"x", "^", "3"}], "x"]}], "]"}]], "Input", CellChangeTimes->{{3.627659394788508*^9, 3.6276594187173862`*^9}, { 3.647871246404592*^9, 3.647871253870778*^9}}], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"x", "^", "2"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "/", "y"}], "+", RowBox[{ RowBox[{"x", "^", "3"}], "/", "x"}]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.627662600656457*^9, 3.627662624758122*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" Exercise 1:", FontWeight->"Bold"], " Experiment with the arithmetic operations ", Cell[BoxData[ FormBox[ RowBox[{"+", RowBox[{",", RowBox[{"-", RowBox[{",", RowBox[{"*", RowBox[{",", "/"}]}]}]}]}]}], TraditionalForm]]], " and the built-in symbolic manipulation functions mentioned in the \ preceding paragraph on the expressions in Discussion 1 to determine what \ these ", StyleBox["Mathematica", FontSlant->"Italic"], " functions do. Check ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s answers by hand. Are ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s answers always in the form you would leave your answers?" }], "Subsection", CellChangeTimes->{{3.6276557036467533`*^9, 3.627655717913577*^9}, { 3.627655764618641*^9, 3.62765590921062*^9}, {3.627656765938553*^9, 3.627656864463704*^9}, {3.627656948349906*^9, 3.627656997576027*^9}, { 3.627657059639831*^9, 3.6276571369713497`*^9}, {3.627657178490571*^9, 3.627657203292487*^9}, {3.6276573617503653`*^9, 3.627657361906416*^9}, { 3.627657492593141*^9, 3.6276575163946*^9}, {3.627659108756077*^9, 3.627659190362417*^9}, {3.627908677348465*^9, 3.6279087177368393`*^9}, { 3.62887512976138*^9, 3.628875179245981*^9}, {3.628875231758624*^9, 3.628875233478917*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Work 1: " }], "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659434447322*^9, 3.627659440180417*^9}, {3.627659519063675*^9, 3.6276595191998262`*^9}}], Cell[TextData[{ StyleBox["(Enter your code under this cell when ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" is in \[OpenCurlyDoubleQuote]Input mode\[CloseCurlyDoubleQuote] \ \[LongDash] make sure a horizontal line is showing before you start typing)", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659460201036*^9, 3.627659504551834*^9}}, FontColor->RGBColor[0, 0.67, 0]], Cell[TextData[StyleBox["(You can type your thoughts and answers here \ formatted in text mode)", FontWeight->"Bold"]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659535820641*^9, 3.627659537608328*^9}, {3.627659670483754*^9, 3.627659682423826*^9}}, FontColor->RGBColor[0, 0.67, 0]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Grader/Instructor ", StyleBox["Mathematica", FontSlant->"Italic"], " Assessment 1: " }], "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659516682569*^9, 3.627659516839205*^9}, {3.627659556506875*^9, 3.62765955845679*^9}, {3.627659617632988*^9, 3.627659649751532*^9}}], Cell[TextData[StyleBox["(The grader/instructor will give you feedback about \ your work here)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659535820641*^9, 3.627659578217204*^9}, {3.627659621472638*^9, 3.627659622910438*^9}, {3.628006013730137*^9, 3.6280060155901546`*^9}, { 3.628877211175973*^9, 3.628877213714664*^9}}], Cell[TextData[{ "Initially, on a purely symbolic level, any ", StyleBox[ButtonBox["complex number", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Complex_number"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Complex_number"], FontColor->RGBColor[0, 0, 1]], ", such as ", Cell[BoxData[ FormBox[ RowBox[{"4", "+", RowBox[{"3", "\[ImaginaryI]"}]}], TraditionalForm]]], ", can be thought of as just an expression in the \ \[OpenCurlyDoubleQuote]variable\[CloseCurlyDoubleQuote] \[ImaginaryI], \ completely ignoring the fact that ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ImaginaryI]", "2"], "=", RowBox[{"-", "1"}]}], TraditionalForm]]], ". The meaning of the word \[OpenCurlyDoubleQuote]variable\ \[CloseCurlyDoubleQuote] here is a bit of a misnomer, however. We don\ \[CloseCurlyQuote]t want to allow \[ImaginaryI] to take on different values. \ It might be better to use a different word for the role that \[ImaginaryI] \ plays here; mathematicians often use the word \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["indeterminate", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Indeterminate_%28variable%29"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Indeterminate_%28variable%29"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote] instead." }], "Text", CellChangeTimes->{{3.6276568889840937`*^9, 3.6276569418882113`*^9}, { 3.627657673966083*^9, 3.627657690980295*^9}, {3.627658432152196*^9, 3.627658538113209*^9}, {3.627658585055251*^9, 3.62765861342761*^9}, { 3.627659706344949*^9, 3.6276597065415277`*^9}, {3.627908764043284*^9, 3.627908764045835*^9}, {3.62790881820674*^9, 3.6279088304672832`*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Discussion 2:", FontWeight->"Bold"], " Based on the view of a complex number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], ", where ", StyleBox["a", FontSlant->"Italic"], " and ", StyleBox["b", FontSlant->"Italic"], " are ", StyleBox[ButtonBox["real numbers", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/RealNumber.html"], None}, ButtonNote->"http://mathworld.wolfram.com/RealNumber.html"], FontColor->RGBColor[0, 0, 1]], ", as just an expression in the indeterminate \[ImaginaryI], describe how \ the addition of two or more complex numbers should be done. Give examples. \ Write down a general formula for the sum of two arbitrary complex numbers, ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"c", "+", RowBox[{"d", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], ", where ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], ", and ", StyleBox["d", FontSlant->"Italic"], " are arbitrary real numbers (often written ", Cell[BoxData[ FormBox[ RowBox[{"a", ",", "b", ",", "c", ",", RowBox[{"d", "\[Element]", "\[DoubleStruckCapitalR]"}]}], TraditionalForm]]], " and read \[OpenCurlyDoubleQuote]", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], ", and ", StyleBox["d", FontSlant->"Italic"], " are all elements of the set of real numbers \[DoubleStruckCapitalR]\ \[CloseCurlyDoubleQuote]). Does your definition of complex number addition \ satisfy the ", StyleBox[ButtonBox["associative property", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Associative_property"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Associative_property"], FontColor->RGBColor[0, 0, 1]], "? The ", StyleBox[ButtonBox["commutative property", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Commutative_property"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Commutative_property"], FontColor->RGBColor[0, 0, 1]], "? Why or why not? Base your answer to this last question on related facts \ about real number arithmetic. What role does the complex number ", Cell[BoxData[ FormBox[ RowBox[{"0", "+", RowBox[{"0", "\[ImaginaryI]"}]}], TraditionalForm]]], " play? For a given complex number, ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], ", what can you say about the related complex number ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"-", "a"}], ")"}], "+", RowBox[{ RowBox[{"(", RowBox[{"-", "b"}], ")"}], "\[ImaginaryI]"}]}], TraditionalForm]]], "? Is the set of real numbers \[DoubleStruckCapitalR] a subset of the set \ of complex numbers \[DoubleStruckCapitalC]? If not, is there a \ \[OpenCurlyDoubleQuote]natural\[CloseCurlyDoubleQuote] way to \ \[OpenCurlyDoubleQuote]associate\[CloseCurlyDoubleQuote] \ \[DoubleStruckCapitalR] with some subset of \[DoubleStruckCapitalC]. Are \ there other \[OpenCurlyDoubleQuote]natural ways\[CloseCurlyDoubleQuote] to \ create such as association? Is one association superior to the others? Are \ these even good questions to ask in the first place?" }], "Subsection", CellChangeTimes->CompressedData[" 1:eJxTTMoPSmViYGAQAWIQbbLpi5MJ+2tHNpu9LiD6xenJXiA6be+yGBC9gGPf cRC96hrfFRCdZ9R9D0RfyNv0BETXV5a8AdE+ovLfQPShTf6/QTRfTOY/EO1g ystsCqT3B/PwgehTk/cJgWgX08NgOm1GqCSI/rcWQq8/V6wOol/m7zQD0U88 TtiC6JQ1J51BtFqfhD/YnNUQWiNdPBhEx/89HwuiDXM/PVgPpHdlXX4IojdK /HsBorPddV6C6PAYg4PtnK8dy5tOgunv664cBdGXN0gdA9FffkWcAdFtMilg +orNdJ/Duq8dW+9Vx4BoHX6Dj0eB9ItIGzANACl3krw= "]], Cell[CellGroupData[{ Cell["Response 2: ", "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.6276595224736834`*^9, 3.627659522575448*^9}}], Cell[TextData[StyleBox["(You can type your thoughts and answers here \ formatted in text mode)", FontWeight->"Bold", FontColor->RGBColor[0, 0.67, 0]]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659540480051*^9, 3.6276595418563213`*^9}, {3.627659689579788*^9, 3.62765968981131*^9}, 3.627660031273603*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Grader/Instructor Response 2: ", "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659516682569*^9, 3.627659516839205*^9}, {3.627659556506875*^9, 3.62765955845679*^9}, {3.627659617632988*^9, 3.62765961929755*^9}, { 3.627660043703747*^9, 3.627660043838567*^9}}], Cell[TextData[StyleBox["(The grader/instructor will give you feedback about \ your work here)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659535820641*^9, 3.627659578217204*^9}, {3.627659621472638*^9, 3.627659622910438*^9}, {3.628006013730137*^9, 3.6280060155901546`*^9}, { 3.628877211175973*^9, 3.628877213714664*^9}}], Cell[TextData[{ "A capital ", StyleBox[ButtonBox["I", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/I.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/I.html"], FontWeight->"Bold", FontColor->RGBColor[1, 0.5, 0]], " is the most basic way to represent the imaginary unit \[ImaginaryI] in \ input/output mode in ", StyleBox["Mathematica", FontSlant->"Italic"], ". ", StyleBox[ButtonBox["ComplexExpand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/ComplexExpand.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/ComplexExpand.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " can be used to confirm or refute your ideas in Discussion 2 above." }], "Text", CellChangeTimes->{{3.627663667163701*^9, 3.627663676151565*^9}, { 3.627663731420498*^9, 3.627663890126082*^9}, {3.6279089626557817`*^9, 3.627908962658252*^9}, 3.630424737403521*^9, {3.6478714722728863`*^9, 3.647871476151133*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"(", RowBox[{"3", "+", RowBox[{"2", "I"}]}], ")"}], "+", RowBox[{"(", RowBox[{"4", "+", RowBox[{"2", "I"}]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.6276637615327377`*^9, 3.6276637745392113`*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", "*", "I"}]}], ")"}], "+", RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", "*", "I"}]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.627663801358947*^9, 3.627663812637382*^9}, { 3.627663905397036*^9, 3.627663906547554*^9}}], Cell[TextData[{ "Do you have any ideas about how the multiplication of two arbitrary complex \ numbers ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"c", "+", RowBox[{"d", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " should be done? Make a conjecture and then see what ", StyleBox["Mathematica", FontSlant->"Italic"], " gives for the answer." }], "Text", CellChangeTimes->{{3.628875374472515*^9, 3.628875428486012*^9}, { 3.628875507403345*^9, 3.62887551187906*^9}, {3.647871499489142*^9, 3.6478715036541147`*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", "*", "I"}]}], ")"}], "*", RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", "*", "I"}]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.6288754299334087`*^9, 3.628875442054089*^9}}], Cell[TextData[{ "You can also check that the ", StyleBox[ButtonBox["distributive property", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Distributive_property"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Distributive_property"], FontColor->RGBColor[0, 0, 1]], " works in this setting. The following cell contains two lines of input \ code that will generate two outputs. " }], "Text", CellChangeTimes->{{3.628875449318366*^9, 3.628875486047296*^9}, { 3.647871518440497*^9, 3.647871532316657*^9}}], Cell[BoxData[{ RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", "*", "I"}]}], ")"}], "*", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", "*", "I"}]}], ")"}], "+", RowBox[{"(", RowBox[{"e", "+", RowBox[{"f", "*", "I"}]}], ")"}]}], ")"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"ComplexExpand", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", "*", "I"}]}], ")"}], "*", RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", "*", "I"}]}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", "*", "I"}]}], ")"}], "*", RowBox[{"(", RowBox[{"e", "+", RowBox[{"f", "*", "I"}]}], ")"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.627663907582281*^9, 3.627663947751939*^9}}], Cell[TextData[{ "Since a complex number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " is uniquely determined by the values of ", StyleBox["a", FontSlant->"Italic"], " and ", StyleBox["b", FontSlant->"Italic"], ", as well as their role in this expression \[LongDash] next to the \ \[ImaginaryI] or not \[LongDash] it is natural to associate the complex \ number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " (which can also be represented as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"b", " ", "\[ImaginaryI]"}], "+", "a"}], TraditionalForm]]], ") with the ", StyleBox[ButtonBox["ordered pair", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Ordered_pair"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Ordered_pair"], FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " (which should ", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " be written as ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"b", ",", "a"}], ")"}], TraditionalForm]]], "). When we impose a ", StyleBox[ButtonBox["Cartesian (rectangular) coordinate system", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Cartesian_coordinate_system"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Cartesian_coordinate_system"], FontColor->RGBColor[0, 0, 1]], " on a plane, we can use these ordered pairs to visualize the complex \ numbers as points in this plane in a standard way. The \ \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["real part", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/RealPart.html"], None}, ButtonNote->"http://mathworld.wolfram.com/RealPart.html"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox["a", TraditionalForm]]], ", of ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " \[LongDash]the first coordinate of ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " \[LongDash] specifies the horizontal (right/left) perpendicular \ displacement, with respect to the vertical axis, of the number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " as a point in the plane; with positive displacement being to the right and \ negative displacement being to the left. The \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["imaginary part", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/ImaginaryPart.html"], None}, ButtonNote->"http://mathworld.wolfram.com/ImaginaryPart.html"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox["b", TraditionalForm]]], ", of ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " \[LongDash]the second coordinate of ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " \[LongDash] specifies the vertical (up/down) perpendicular displacement, \ with respect to the horizontal axis, of the number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " as a point in the plane; with positive displacement being upward and \ negative displacement being downward. When a complex number is plotted in \ the given plane in this way, we call this plane \[OpenCurlyDoubleQuote]the\ \[CloseCurlyDoubleQuote] ", StyleBox[ButtonBox["Complex (Argand) Plane", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/ComplexPlane.html"], None}, ButtonNote->"http://mathworld.wolfram.com/ComplexPlane.html"], FontColor->RGBColor[0, 0, 1]], "; we label the horizontal axis to be the ", StyleBox[ButtonBox["real axis", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/RealAxis.html"], None}, ButtonNote->"http://mathworld.wolfram.com/RealAxis.html"], FontColor->RGBColor[0, 0, 1]], " and we label the vertical axis to be the ", StyleBox[ButtonBox["imaginary axis", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/ImaginaryAxis.html"], None}, ButtonNote->"http://mathworld.wolfram.com/ImaginaryAxis.html"], FontColor->RGBColor[0, 0, 1]], ". These things can just be thought of as labels, and no philosophical \ objections about the deeper nature of what we are doing need be sustained. \ Questions about applicability to ", StyleBox[ButtonBox["the real world", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Reality"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Reality"], FontColor->RGBColor[0, 0, 1]], ", however, do need to eventually be addressed." }], "Text", CellChangeTimes->{{3.627657978735114*^9, 3.627658120845636*^9}, { 3.627658184513383*^9, 3.6276581936991663`*^9}, {3.6276587041979923`*^9, 3.6276587076181173`*^9}, {3.627659734862486*^9, 3.627659777052169*^9}, { 3.627659812917594*^9, 3.627659991349291*^9}, {3.627660114240464*^9, 3.6276601168740396`*^9}, {3.6276604155072823`*^9, 3.627660508140432*^9}, { 3.627660555373166*^9, 3.627660555388114*^9}, {3.627660591367168*^9, 3.627660682357828*^9}, {3.627660715624648*^9, 3.627660755190234*^9}, { 3.627660828604656*^9, 3.627660828608768*^9}, {3.62790903462843*^9, 3.627909111181098*^9}, {3.627909156536913*^9, 3.6279092056316338`*^9}, { 3.627911654057898*^9, 3.627911657414344*^9}, {3.628875592642117*^9, 3.6288755929093437`*^9}, {3.647871826150104*^9, 3.647871835498371*^9}}], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], ", the ordered pair ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " is typically represented as a \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["list", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/List.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/List.html"], FontColor->RGBColor[1, 0.5, 0]], "\[CloseCurlyDoubleQuote] with two entries, or elements: ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"a", ",", "b"}], "}"}], TraditionalForm]]], " (see a summary page of many manipulations that can be done, in a very \ general setting, with all kinds of ", StyleBox[ButtonBox["lists in Mathematica", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/guide/ListManipulation.html"], None}, ButtonNote-> "http://reference.wolfram.com/language/guide/ListManipulation.html"], FontColor->RGBColor[1, 0.5, 0]], "). These points can then be plotted with ", StyleBox[ButtonBox["ListPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/ListPlot.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/ListPlot.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ". For instance, we can, as done in ", StyleBox[ButtonBox["Figure [1] (Ch. 1.I .1, p. 2) of Tristan Needham\ \[CloseCurlyQuote]s Visual Complex Analysis", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/\ 0198534469/ref=sr_1_1?s=books&ie=UTF8&qid=1419886831&sr=1-1&keywords=visual+\ complex+analysis"], None}, ButtonNote-> "http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/\ 0198534469/ref=sr_1_1?s=books&ie=UTF8&qid=1419886831&sr=1-1&keywords=visual+\ complex+analysis"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0, 1]], ", plot the complex numbers ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"4", "=", RowBox[{"4", "+", RowBox[{"0", " ", "\[ImaginaryI]"}]}]}], ",", " ", RowBox[{"4", "+", RowBox[{"3", "\[ImaginaryI]"}]}], ",", " ", RowBox[{ RowBox[{"3", "\[ImaginaryI]"}], "=", RowBox[{"0", "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"-", "7"}], "+", "\[ImaginaryI]"}], "=", RowBox[{ RowBox[{"(", RowBox[{"-", "7"}], ")"}], "+", RowBox[{"1", " ", "\[ImaginaryI]"}]}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", "\[ImaginaryI]"}]}], "=", RowBox[{ RowBox[{"(", RowBox[{"-", "2"}], ")"}], "+", RowBox[{ RowBox[{"(", RowBox[{"-", "3"}], ")"}], "\[ImaginaryI]"}]}]}], ",", " ", RowBox[{ RowBox[{ RowBox[{"and", " ", "2"}], "-", RowBox[{"2", "\[ImaginaryI]"}]}], "=", RowBox[{"2", "+", RowBox[{ RowBox[{"(", RowBox[{"-", "2"}], ")"}], "\[ImaginaryI]"}]}]}]}], TraditionalForm]]], " using the code to follow. For efficiency\[CloseCurlyQuote]s sake, we put \ these lists (of two elements each) in a \[OpenCurlyDoubleQuote]list of lists\ \[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "7"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", RowBox[{"-", "3"}]}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "2"}]}], "}"}]}], "}"}], TraditionalForm]]], " (with six total elements) in order to plot them all at once. Note also \ that the options ", StyleBox[ButtonBox["PlotStyle", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/PlotStyle.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/PlotStyle.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", ", StyleBox[ButtonBox["PlotRange", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/PlotRange.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/PlotRange.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", and ", StyleBox[ButtonBox["AxesLabel", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/AxesLabel.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/AxesLabel.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], StyleBox[",", FontColor->RGBColor[1, 0.5, 0]], " can be used to, respectively, make the dots black and larger than normal, \ choose the plotting window, and label the axes. Note also the use of \ \[OpenCurlyDoubleQuote]", StyleBox["InitialFigure1", FontWeight->"Bold"], " =\[CloseCurlyDoubleQuote] at the beginning of the line will store the \ resulting output graph in a variable we have named ", StyleBox["InitialFigure1", FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{{3.62766008659522*^9, 3.627660103138002*^9}, { 3.6276601397144814`*^9, 3.627660188929512*^9}, {3.627660277491293*^9, 3.6276604093848667`*^9}, {3.627660846294976*^9, 3.627660894643497*^9}, { 3.627660931852092*^9, 3.6276610814852037`*^9}, {3.627661117605917*^9, 3.62766122600981*^9}, {3.627661421563168*^9, 3.627661606099361*^9}, { 3.627661670184834*^9, 3.627661688721341*^9}, {3.628875650810205*^9, 3.628875650841794*^9}, 3.630424764039959*^9}], Cell[BoxData[ RowBox[{"InitialFigure1", "=", RowBox[{"ListPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "7"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", RowBox[{"-", "3"}]}], "}"}], ",", RowBox[{"{", RowBox[{"2", ",", RowBox[{"-", "2"}]}], "}"}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Black", ",", RowBox[{"PointSize", "[", ".02", "]"}]}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "9"}], ",", "9"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", "4"}], "}"}]}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.6276610889420233`*^9, 3.6276611061661463`*^9}, { 3.6276611940880632`*^9, 3.627661260925015*^9}, {3.6276613034521437`*^9, 3.627661317608073*^9}, {3.6276613793788347`*^9, 3.6276614154510317`*^9}, { 3.627661554097768*^9, 3.627661556202138*^9}, {3.627661590985866*^9, 3.627661591738063*^9}}], Cell[TextData[{ "After the preceding line has been entered, the graph can be displayed again \ by just entering its name ", StyleBox["InitialFigure1", FontWeight->"Bold"], ".", StyleBox[" ", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.6276616146676903`*^9, 3.627661656154912*^9}, { 3.627909319633389*^9, 3.627909324176015*^9}, {3.6478727465528708`*^9, 3.6478727513189898`*^9}}], Cell[BoxData["InitialFigure1"], "Input", CellChangeTimes->{{3.627661657728503*^9, 3.6276616593720093`*^9}, { 3.6288757064887753`*^9, 3.628875709406294*^9}, {3.64787273930554*^9, 3.647872741171612*^9}}], Cell[TextData[{ "The preceding line of code will not produce the desired output after we \ quit ", StyleBox[ButtonBox["Mathematica\[CloseCurlyQuote]s kernel", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Mathematica#Interface"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Mathematica#Interface"], FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], ", either through the ", StyleBox["Evaluation menu", FontWeight->"Bold", FontSlant->"Italic"], " or by quitting the program. " }], "Text", CellChangeTimes->{{3.627909329553458*^9, 3.627909362585091*^9}, { 3.6279094139787893`*^9, 3.627909413980949*^9}, {3.628875726757264*^9, 3.628875754995224*^9}, {3.629040652187572*^9, 3.6290406679557247`*^9}}], Cell[TextData[{ "Another option, ", StyleBox[ButtonBox["GridLines", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/GridLines.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/GridLines.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", can be used to create a standard rectangular grid in the background of \ the picture. The function ", StyleBox[ButtonBox["Show", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Show.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Show.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " combines the graphic outputs into one output and the option ", StyleBox[ButtonBox["AspectRatio", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/AspectRatio.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/AspectRatio.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " can be used to make the scales on the axes the same." }], "Text", CellChangeTimes->{{3.62766171755234*^9, 3.627661783360642*^9}, { 3.6276618855495157`*^9, 3.6276619385686483`*^9}, {3.627661971621619*^9, 3.6276619995052967`*^9}, 3.6304248651261787`*^9, {3.6478728232134647`*^9, 3.647872843099244*^9}, {3.647872877207528*^9, 3.647872894653553*^9}, { 3.647873071276359*^9, 3.6478730716092577`*^9}}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"InitialFigure1", ",", RowBox[{"GridLines", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "9"}], ",", RowBox[{"-", "8"}], ",", RowBox[{"-", "7"}], ",", RowBox[{"-", "6"}], ",", RowBox[{"-", "5"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "6", ",", "7", ",", "8", ",", "9"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "4"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "1", ",", "2", ",", "3", ",", "4"}], "}"}]}], "}"}]}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input", CellChangeTimes->{{3.6276617867672367`*^9, 3.627661879246334*^9}, { 3.627661941781152*^9, 3.627661945291995*^9}, {3.6478728458154373`*^9, 3.647872849508155*^9}, 3.647873074524427*^9}], Cell[TextData[{ "When there are many grid-lines, the locations of the grid-lines can be \ generated more efficiently with ", StyleBox[ButtonBox["Table", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Table.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Table.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", which is the basic list-generating command in ", StyleBox["Mathematica", FontSlant->"Italic"], " when the elements of a list have a \[OpenCurlyDoubleQuote]formula\ \[CloseCurlyDoubleQuote]. For example, ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "9"}], ",", "9"}], "}"}]}], "]"}], FontWeight->"Bold"], TraditionalForm]]], " generates the list ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{"-", "9"}], ",", RowBox[{"-", "8"}], ",", RowBox[{"-", "7"}], ",", RowBox[{"-", "6"}], ",", RowBox[{"-", "5"}], ",", RowBox[{"-", "4"}], ",", RowBox[{"-", "3"}], ",", RowBox[{"-", "2"}], ",", RowBox[{"-", "1"}], ",", "0", ",", "1", ",", "2", ",", "3", ",", "4", ",", "5", ",", "6", ",", "7", ",", "8", ",", "9"}], "}"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.6276620140095253`*^9, 3.627662105209556*^9}, { 3.6276621769175787`*^9, 3.627662198061448*^9}, 3.63042482162609*^9, { 3.647872922834806*^9, 3.6478729261592712`*^9}, {3.647877042590454*^9, 3.6478770446862087`*^9}}], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "9"}], ",", "9"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.627662062043561*^9, 3.627662066087792*^9}}], Cell[TextData[{ "As a related example, ", Cell[BoxData[ FormBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{"i", "^", "2"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "9"}], ",", "9"}], "}"}]}], "]"}], TraditionalForm]], FontWeight->"Bold"], " makes a list of the squares of the numbers in the preceding list." }], "Text", CellChangeTimes->{{3.62766270160417*^9, 3.6276627227788773`*^9}}], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"i", "^", "2"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "9"}], ",", "9"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.627662062043561*^9, 3.627662066087792*^9}, { 3.627662728214615*^9, 3.6276627284442883`*^9}}], Cell[TextData[{ "Note that lists in ", StyleBox["Mathematica", FontSlant->"Italic"], " are not the same as sets in Mathematics. The preceding list is ", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " equivalent to the ", StyleBox["mathematical set", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ "0", ",", "1", ",", "4", ",", "9", ",", "16", ",", "25", ",", "36", ",", "49", ",", "64", ",", "81"}], "}"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.647872947727357*^9, 3.647872992172153*^9}}], Cell[TextData[{ StyleBox[ButtonBox["Table", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Table.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Table.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " is very flexible. It can also generate, for example, lists of points," }], "Text", CellChangeTimes->{{3.627662756518363*^9, 3.6276627737982597`*^9}, { 3.627662825098092*^9, 3.627662875864808*^9}, {3.627909529035808*^9, 3.62790952935911*^9}, 3.630424872285877*^9}], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"i", "^", "3"}], ",", RowBox[{"i", "^", "4"}]}], "}"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "5"}], ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.62766277689592*^9, 3.62766283773356*^9}}], Cell["or even lists of graphs.", "Text", CellChangeTimes->{{3.627909484395495*^9, 3.627909488496395*^9}, { 3.627909531191312*^9, 3.627909531471408*^9}}], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"x", "^", RowBox[{"(", "i", ")"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "1"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", ".5", ",", "2", ",", ".5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.6279094904566317`*^9, 3.627909518653182*^9}}], Cell["\<\ We now have a way to create the graph of the complex numbers in the complex \ plane more efficiently (especially through the use of \ \[OpenCurlyDoubleQuote]Copy & Paste\[CloseCurlyDoubleQuote] to save typing \ time).\ \>", "Text", CellChangeTimes->{{3.627662107856578*^9, 3.627662128007699*^9}, { 3.6276626633089943`*^9, 3.627662666456585*^9}, {3.627909550275042*^9, 3.6279095578159103`*^9}, {3.64787304426075*^9, 3.647873046809783*^9}, 3.647876987698955*^9}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"InitialFigure1", ",", RowBox[{"GridLines", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "9"}], ",", "9"}], "}"}]}], "]"}], ",", RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}]}], "]"}]}], "}"}]}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input", CellChangeTimes->{{3.6276617867672367`*^9, 3.627661879246334*^9}, { 3.627661941781152*^9, 3.627661945291995*^9}, {3.6276621378577127`*^9, 3.627662143073347*^9}, {3.647873038748006*^9, 3.647873039039154*^9}, { 3.6478730817879963`*^9, 3.647873082045033*^9}}], Cell[TextData[{ "Finally, we may combine the built-in functions ", StyleBox[ButtonBox["Graphics", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Graphics.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Graphics.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " and ", StyleBox[ButtonBox["Text", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Text.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Text.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " to label these points in a way that will be useful, not just for producing \ a nice static output picture, but also for producing animations where the \ labeling of a point can move with the point while the animation is occurring. \ We have also included the option ", StyleBox[ButtonBox["AxesStyle", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/AxesStyle.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/AxesStyle.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " and the graphics directive ", StyleBox[ButtonBox["Thick", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Thick.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Thick.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " to make the axes thicker." }], "Text", CellChangeTimes->{{3.627663989023061*^9, 3.627664081159441*^9}, { 3.627664191856192*^9, 3.627664216475559*^9}, {3.628875945121029*^9, 3.628875990285302*^9}, 3.630424789051283*^9, {3.6478730624608107`*^9, 3.647873107451933*^9}}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"InitialFigure1", ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Text", "[", RowBox[{"4", ",", RowBox[{"{", RowBox[{"4.5", ",", ".5"}], "}"}]}], "]"}], ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"4", "+", RowBox[{"3", "\[ImaginaryI]"}]}], ",", RowBox[{"{", RowBox[{"4.5", ",", "3.5"}], "}"}]}], "]"}], ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"3", "\[ImaginaryI]"}], ",", RowBox[{"{", RowBox[{".5", ",", "3.5"}], "}"}]}], "]"}], ",", RowBox[{"Text", "[", RowBox[{ RowBox[{ RowBox[{"-", "7"}], "+", "\[ImaginaryI]"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "7.3"}], ",", "1.5"}], "}"}]}], "]"}], ",", RowBox[{"Text", "[", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", "\[ImaginaryI]"}]}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "2.5"}], ",", RowBox[{"-", "2.5"}]}], "}"}]}], "]"}], ",", RowBox[{"Text", "[", RowBox[{ RowBox[{"2", "-", RowBox[{"2", "\[ImaginaryI]"}]}], ",", RowBox[{"{", RowBox[{"2.5", ",", RowBox[{"-", "1.5"}]}], "}"}]}], "]"}]}], "}"}], "]"}], ",", RowBox[{"GridLines", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "9"}], ",", "9"}], "}"}]}], "]"}], ",", RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"-", "4"}], ",", "4"}], "}"}]}], "]"}]}], "}"}]}], ",", RowBox[{"AxesStyle", "\[Rule]", "Thick"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}]}], "]"}]], "Input", CellChangeTimes->{{3.6276617867672367`*^9, 3.627661879246334*^9}, { 3.627661941781152*^9, 3.627661945291995*^9}, {3.6276621378577127`*^9, 3.627662143073347*^9}, {3.627664106625504*^9, 3.627664175947159*^9}, { 3.6276642464009123`*^9, 3.627664332907702*^9}, {3.628875850707377*^9, 3.628875887398748*^9}, {3.62887592620961*^9, 3.628875929241864*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" Exercise 2:", FontWeight->"Bold"], " Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to make a plot, in the complex plane, of the points ", Cell[BoxData[ FormBox[ RowBox[{"2", "-", RowBox[{"3", "\[ImaginaryI]"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "5"}], "+", RowBox[{"8", "\[ImaginaryI]"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "4"}], "\[ImaginaryI]"}], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox["10", TraditionalForm]]], ". Make sure your axes are labeled appropriately, your window is chosen \ well, your points are larger than the default (and make them colored ", StyleBox[ButtonBox["Red", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Red.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Red.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], "), the grid-lines are included, the scales on the axes are the same, the \ axes are thick, and the points are labeled. Experiment with your code to \ strive to make it as efficient as possible (that is, as short as possible \ \[LongDash] and possibly use just one ", StyleBox[ButtonBox["cell", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/tutorial/WorkingWithCells.\ html"], None}, ButtonNote-> "http://reference.wolfram.com/language/tutorial/WorkingWithCells.html"], FontColor->RGBColor[1, 0.5, 0]], " of code rather than multiple cells). Note that a semicolon \ \[OpenCurlyDoubleQuote];\[CloseCurlyDoubleQuote] will suppress output in ", StyleBox["Mathematica", FontSlant->"Italic"], " but can also be used to separate distinct lines of code within the same \ cell if you wish." }], "Subsection", CellChangeTimes->{{3.6276557036467533`*^9, 3.627655717913577*^9}, { 3.627655764618641*^9, 3.62765590921062*^9}, {3.627656765938553*^9, 3.627656864463704*^9}, {3.627656948349906*^9, 3.627656997576027*^9}, { 3.627657059639831*^9, 3.6276571369713497`*^9}, {3.627657178490571*^9, 3.627657203292487*^9}, {3.6276573617503653`*^9, 3.627657361906416*^9}, { 3.627657492593141*^9, 3.6276575163946*^9}, {3.627659108756077*^9, 3.627659190362417*^9}, {3.627662239262556*^9, 3.6276624566615953`*^9}, { 3.6276624914328527`*^9, 3.627662499510579*^9}, {3.627664347055848*^9, 3.627664356094934*^9}, {3.627909630592648*^9, 3.627909643219245*^9}, { 3.62790969635893*^9, 3.627909739165676*^9}, {3.6279097708285637`*^9, 3.627909788142371*^9}, {3.628876008395164*^9, 3.628876060054185*^9}, { 3.647873166593781*^9, 3.647873166600049*^9}, 3.647877045776937*^9}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Work 2: " }], "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659434447322*^9, 3.627659440180417*^9}, {3.627659519063675*^9, 3.6276595191998262`*^9}, 3.627662240565363*^9}], Cell[TextData[{ StyleBox["(Enter your code under this cell when ", FontWeight->"Bold", FontColor->RGBColor[0, 0.67, 0]], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0.67, 0]], StyleBox[" is in \[OpenCurlyDoubleQuote]Input mode\[CloseCurlyDoubleQuote] \ \[LongDash] make sure a horizontal line is showing before you start typing)", FontWeight->"Bold", FontColor->RGBColor[0, 0.67, 0]] }], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659460201036*^9, 3.627659504551834*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Grader/Instructor ", StyleBox["Mathematica", FontSlant->"Italic"], " Assessment 2: " }], "Subsubsection", CellChangeTimes->{{3.627658284055971*^9, 3.627658317587431*^9}, { 3.627659516682569*^9, 3.627659516839205*^9}, {3.627659556506875*^9, 3.62765955845679*^9}, {3.627659617632988*^9, 3.627659649751532*^9}, 3.6276622421979313`*^9}], Cell[TextData[StyleBox["(The grader/instructor will give you feedback about \ your work here)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.6276582897121067`*^9, 3.627658333401252*^9}, { 3.627659535820641*^9, 3.627659578217204*^9}, {3.627659621472638*^9, 3.627659622910438*^9}, {3.628006013730137*^9, 3.6280060155901546`*^9}, { 3.628877211175973*^9, 3.628877213714664*^9}}], Cell["\<\ In Activity 2 of this learning module on complex number addition and the \ complex plane, we will consider the geometric importance of this view for \ complex number addition.\ \>", "Text", CellChangeTimes->{{3.6478765286404448`*^9, 3.647876597620336*^9}, 3.647877562470521*^9}] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{1212, 673}, WindowMargins->{{13, Automatic}, {51, Automatic}}, Magnification:>1.25 Inherited, FrontEndVersion->"10.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (December 4, \ 2014)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1486, 35, 645, 14, 183, "Title"], Cell[CellGroupData[{ Cell[2156, 53, 255, 3, 78, "Subchapter"], Cell[2414, 58, 3098, 74, 266, "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[5549, 137, 281, 4, 78, "Subchapter"], Cell[5833, 143, 682, 14, 175, "Subsection"] }, Open ]], Cell[CellGroupData[{ Cell[6552, 162, 305, 4, 78, "Subchapter"], Cell[6860, 168, 5478, 130, 227, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[12375, 303, 250, 3, 188, "Subchapter"], Cell[12628, 308, 1368, 28, 321, "Text"], Cell[13999, 338, 3220, 70, 596, "Text"], Cell[CellGroupData[{ Cell[17244, 412, 3634, 87, 781, "Subsection"], Cell[CellGroupData[{ Cell[20903, 503, 152, 2, 127, "Subsubsection"], Cell[21058, 507, 327, 6, 110, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[21422, 518, 263, 3, 127, "Subsubsection"], Cell[21688, 523, 420, 7, 110, "Text"], Cell[22111, 532, 1078, 16, 331, "Text"], Cell[23192, 550, 2250, 57, 257, "Text"], Cell[25445, 609, 205, 6, 188, "Input"], Cell[25653, 617, 295, 8, 188, "Input"], Cell[25951, 627, 329, 10, 108, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[26329, 643, 1437, 36, 445, "Subsection"], Cell[CellGroupData[{ Cell[27791, 683, 264, 7, 127, "Subsubsection"], Cell[28058, 692, 502, 12, 183, "Text"], Cell[28563, 706, 327, 6, 110, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[28927, 717, 336, 8, 127, "Subsubsection"], Cell[29266, 727, 420, 7, 110, "Text"], Cell[29689, 736, 1754, 38, 335, "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[31492, 780, 3967, 115, 1183, "Subsection"], Cell[CellGroupData[{ Cell[35484, 899, 154, 2, 127, "Subsubsection"], Cell[35641, 903, 352, 6, 110, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[36030, 914, 312, 4, 127, "Subsubsection"], Cell[36345, 920, 420, 7, 110, "Text"], Cell[36768, 929, 1086, 27, 183, "Text"], Cell[37857, 958, 299, 9, 108, "Input"], Cell[38159, 969, 354, 10, 108, "Input"], Cell[38516, 981, 648, 19, 183, "Text"], Cell[39167, 1002, 307, 9, 108, "Input"], Cell[39477, 1013, 550, 12, 183, "Text"], Cell[40030, 1027, 896, 31, 171, "Input"], Cell[40929, 1060, 5885, 152, 908, "Text"], Cell[46817, 1214, 5897, 158, 620, "Text"], Cell[52717, 1374, 1496, 43, 171, "Input"], Cell[54216, 1419, 401, 11, 110, "Text"], Cell[54620, 1432, 207, 3, 108, "Input"], Cell[54830, 1437, 758, 18, 183, "Text"], Cell[55591, 1457, 1513, 32, 183, "Text"], Cell[57107, 1491, 1112, 28, 235, "Input"], Cell[58222, 1521, 1620, 45, 257, "Text"], Cell[59845, 1568, 227, 6, 102, "Input"], Cell[60075, 1576, 433, 13, 110, "Text"], Cell[60511, 1591, 302, 8, 102, "Input"], Cell[60816, 1601, 617, 21, 183, "Text"], Cell[61436, 1624, 579, 12, 110, "Text"], Cell[62018, 1638, 326, 10, 102, "Input"], Cell[62347, 1650, 155, 2, 110, "Text"], Cell[62505, 1654, 393, 11, 102, "Input"], Cell[62901, 1667, 482, 9, 183, "Text"], Cell[63386, 1678, 823, 20, 108, "Input"], Cell[64212, 1700, 1835, 41, 257, "Text"], Cell[66050, 1743, 2289, 63, 297, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[68388, 1812, 2940, 72, 722, "Subsection"], Cell[CellGroupData[{ Cell[71353, 1888, 288, 7, 127, "Subsubsection"], Cell[71644, 1897, 573, 14, 183, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[72254, 1916, 366, 9, 127, "Subsubsection"], Cell[72623, 1927, 420, 7, 110, "Text"], Cell[73046, 1936, 293, 6, 183, "Text"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *) (* NotebookSignature 3uppdXyfxT4cQCgEVS6yxTxv *)