(* 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. 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In other words, we can think of ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " as literally being a vector, the vector with the given magnitude (\ \[OpenCurlyDoubleQuote]length\[CloseCurlyDoubleQuote]) and direction that we \ have just drawn." }], "Text", CellChangeTimes->{{3.6279115455053587`*^9, 3.627911631770093*^9}, { 3.627911678185623*^9, 3.62791167818779*^9}, {3.6279117645895157`*^9, 3.627911916702344*^9}, {3.627913146198986*^9, 3.627913171051347*^9}, { 3.628005583252339*^9, 3.6280056247462187`*^9}, {3.62800566561415*^9, 3.628005667417716*^9}, {3.62800716292312*^9, 3.628007164357918*^9}, { 3.628876329161037*^9, 3.628876330419393*^9}, {3.647877590589396*^9, 3.647877591722166*^9}}], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], ", the most basic way to draw a vector is through the combined use of ", 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["Arrow", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Arrow.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Arrow.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], ", as illustrated in the following line of code." }], "Text", CellChangeTimes->{{3.627913204776683*^9, 3.627913230438877*^9}, { 3.627913273170944*^9, 3.627913293213957*^9}, {3.6279133702384653`*^9, 3.6279133711635647`*^9}, {3.627916129367601*^9, 3.62791613037134*^9}, 3.630421111910212*^9}], Cell[BoxData[ RowBox[{"Graphics", "[", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "3"}], "}"}]}], "}"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.6279133043455153`*^9, 3.627913317068246*^9}}], Cell[TextData[{ "The lists ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"4", ",", "3"}], "}"}], TraditionalForm]]], " represent the base point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "2"}], ")"}], TraditionalForm]]], " and terminal point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"4", ",", "3"}], ")"}], TraditionalForm]]], " of the vector. 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For instance, what if you travel from a point ", StyleBox["P", FontSlant->"Italic"], " to a point ", StyleBox["Q", FontSlant->"Italic"], " and then to a point ", StyleBox["R ", FontSlant->"Italic"], "and then let ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " represent the corresponding \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["displacement vectors", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Displacement_%28vector%29"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Displacement_%28vector%29"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote]? What about if you are swimming at a nonzero \ angle with the straight current of a straight river, where ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " represents your constant velocity relative to the river and ", StyleBox["w", FontWeight->"Bold"], " represents the constant velocity of the current relative to the river \ banks? 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That is, it seems that defining ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}], "+", RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", " ", "\[ImaginaryI]"}]}], ")"}]}], TraditionalForm]]], " to be ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", "c"}], ")"}], "+", RowBox[{ RowBox[{"(", RowBox[{"b", "+", "d"}], ")"}], "\[ImaginaryI]"}]}], TraditionalForm]]], " might be a good idea. Indeed, algebraically-speaking, this is part of \ what is necessary to make the set of all complex numbers \ \[DoubleStruckCapitalC] into an ", StyleBox[ButtonBox["algebraic structure", BaseStyle->"Hyperlink", ButtonData->{"v", None}], FontColor->RGBColor[0, 0, 1]], " called a ", StyleBox[ButtonBox["field", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Field_(mathematics)"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Field_(mathematics)"], FontColor->RGBColor[0, 0, 1]], " (", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " to be confused with a ", StyleBox[ButtonBox["vector field", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Vector_field"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Vector_field"], FontColor->RGBColor[0, 0, 1]], "). The other necessary part to make \[DoubleStruckCapitalC] a field is to \ define multiplication in a \ \[OpenCurlyDoubleQuote]good\[CloseCurlyDoubleQuote] way. 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The number ", Cell[BoxData[ FormBox[ RowBox[{"1", "=", RowBox[{"1", "+", RowBox[{"0", " ", "\[ImaginaryI]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " will be the ", StyleBox[ButtonBox["multiplicative identity", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/1_(number)"], None}, ButtonNote->"https://en.wikipedia.org/wiki/1_(number)"], FontColor->RGBColor[0, 0, 1]], " and it will also turn out that nonzero complex numbers will have \ multiplicative inverses (at the moment, we\[CloseCurlyQuote]ll leave it to \ you to figure out what the ", StyleBox[ButtonBox["multiplicative inverse", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Multiplicative_inverse"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Multiplicative_inverse"], FontColor->RGBColor[0, 0, 1]], " (reciprocal) of a nonzero number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " will be, if you wish)." }], "Text", CellChangeTimes->{{3.6280071912716093`*^9, 3.628007299753134*^9}, { 3.628007354281349*^9, 3.628007367346219*^9}, {3.628007407844302*^9, 3.628007407847151*^9}, {3.628007446639553*^9, 3.6280076827574883`*^9}, { 3.62801356187967*^9, 3.628013585133748*^9}, {3.6280136220839252`*^9, 3.6280136947301903`*^9}, {3.628013816376335*^9, 3.628013816379263*^9}, { 3.6280138581331663`*^9, 3.628013858137542*^9}, {3.628013939863043*^9, 3.628013939866047*^9}, {3.628013972008925*^9, 3.6280139720129213`*^9}, { 3.628014030659895*^9, 3.628014030662787*^9}, {3.628876820023786*^9, 3.628876823674608*^9}, {3.647879671241399*^9, 3.647879680006049*^9}, { 3.6478797259907227`*^9, 3.64787992678592*^9}, {3.647880012097993*^9, 3.6478800146154737`*^9}}], Cell[TextData[{ "To more fully justify this definition of addition of complex numbers as a \ good idea, we need to explore what it means geometrically. 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It appears that we have created a ", StyleBox[ButtonBox["parallelogram", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Parallelogram"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Parallelogram"], FontColor->RGBColor[0, 0, 1]], ". Thinking in terms of vectors, it appears that if we let ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " be a vector pointing from ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", "0"}], ")"}], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"4", ",", "1"}], ")"}], TraditionalForm]]], " and let ", Cell[BoxData[ FormBox[ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " be a vector pointing from ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", "0"}], ")"}], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "2"}], ")"}], TraditionalForm]]], ", then after an appropriate translation, ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " will point from ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "2"}], ")"}], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"5", ",", "3"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " will point from ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"4", ",", "1"}], ")"}], TraditionalForm]]], " to ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"5", ",", "3"}], ")"}], TraditionalForm]]], ". This can be confirmed visually with the next line of code (make sure the \ preceding line of code has been entered before entering this line so that the \ variable ", StyleBox["DotPlot", FontWeight->"Bold"], " has a \[OpenCurlyDoubleQuote]value\[CloseCurlyDoubleQuote])." }], "Text", CellChangeTimes->{{3.628014471381733*^9, 3.628014849064849*^9}, { 3.6288769242177896`*^9, 3.628876925500538*^9}, {3.647880285922982*^9, 3.6478803024170027`*^9}}], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"DotPlot", ",", RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", ".01", "]"}], ",", "Brown", ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "1"}], "}"}]}], "}"}], "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "3"}], "}"}]}], "}"}], "]"}], ",", "Orange", ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}], "}"}], "]"}], ",", RowBox[{"Arrow", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"4", ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "3"}], "}"}]}], "}"}], "]"}]}], "}"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.6280148250998297`*^9, 3.628014939732933*^9}, { 3.647880315187766*^9, 3.6478803192961884`*^9}}], Cell[TextData[{ "In fact, we can \[OpenCurlyDoubleQuote]quantify\[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " by describing their ", StyleBox[ButtonBox["horizontal and vertical displacements", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.physicsclassroom.com/class/vectors/Lesson-2/Horizontal-\ and-Vertical-Displacement"], None}, ButtonNote-> "http://www.physicsclassroom.com/class/vectors/Lesson-2/Horizontal-and-\ Vertical-Displacement"], FontColor->RGBColor[0, 0, 1]], "; any vector is completely determined, within a given rectangular \ coordinate system, by how much it points to the ", StyleBox["right", FontSlant->"Italic"], " (", StyleBox["positive", FontWeight->"Bold", FontSlant->"Italic"], " horizontal displacement) or ", StyleBox["left", FontSlant->"Italic"], " (", StyleBox["negative", FontWeight->"Bold", FontSlant->"Italic"], " horizontal displacement) and by how much it points ", StyleBox["upward", FontSlant->"Italic"], " (", StyleBox["positive", FontWeight->"Bold", FontSlant->"Italic"], " vertical displacement) or ", StyleBox["downward", FontSlant->"Italic"], " (", StyleBox["negative", FontWeight->"Bold", FontSlant->"Italic"], " vertical displacement). For the example above, ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " has a horizontal displacement of 4 and a vertical displacement of 1; while \ ", Cell[BoxData[ FormBox[ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " has a horizontal displacement of 1 and a vertical displacement of 2. It\ \[CloseCurlyQuote]s no accident that these numbers match the coordinates of \ the points ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"4", ",", "1"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "2"}], ")"}], TraditionalForm]]], ", respectively; and it\[CloseCurlyQuote]s also no accident that these \ numbers match the real and imaginary parts of the complex numbers ", Cell[BoxData[ FormBox[ RowBox[{"4", "+", "\[ImaginaryI]"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], ", respectively. Finally, note that we can also obtain these displacements \ by subtracting the coordinates of ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "2"}], ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"4", ",", "1"}], ")"}], TraditionalForm]]], ", respectively, from the point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"5", ",", "3"}], ")"}], TraditionalForm]]], ". This is essentially a geometric ", StyleBox[ButtonBox["proof", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Mathematical_proof"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Mathematical_proof"], FontColor->RGBColor[0, 0, 1]], " that the figure in the picture is a ", StyleBox[ButtonBox["parallelogram", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Parallelogram"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Parallelogram"], FontColor->RGBColor[0, 0, 1]], ", and it leads us to write ", Cell[BoxData[ FormBox[ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], TraditionalForm]]], " in a new way. One common notation used for these vectors is ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["v", FontWeight->"Bold", FontSlant->"Plain"], "=", RowBox[{"\[LeftAngleBracket]", RowBox[{"4", ",", "1"}], "\[RightAngleBracket]"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["w", FontWeight->"Bold", FontSlant->"Plain"], "=", RowBox[{"\[LeftAngleBracket]", RowBox[{"1", ",", "2"}], "\[RightAngleBracket]"}]}], TraditionalForm]]], ", and this is the notation we will use. In physics, the standard way to \ represent these vectors is in terms of the \[OpenCurlyDoubleQuote]", StyleBox[ButtonBox["standard unit vectors", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Unit_vector#Cartesian_coordinates"], None}, ButtonNote-> "http://en.wikipedia.org/wiki/Unit_vector#Cartesian_coordinates"], FontColor->RGBColor[0, 0, 1]], "\[CloseCurlyDoubleQuote]. We will ", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " use that notation, because of the obvious confusion that could arise when \ using it within the study of complex numbers through the use of the \ \[OpenCurlyDoubleQuote]i\[CloseCurlyDoubleQuote] in that notation (also \ called \[OpenCurlyDoubleQuote]i hat\[CloseCurlyDoubleQuote] and written ", Cell[BoxData[ FormBox[ OverscriptBox["i", "^"], TraditionalForm]]], ")." }], "Text", CellChangeTimes->{{3.6280149709829082`*^9, 3.62801513939699*^9}, { 3.628015220407031*^9, 3.628015220409556*^9}, {3.628015290772086*^9, 3.628015607410637*^9}, {3.628015710386314*^9, 3.628015744694153*^9}, { 3.628015775660411*^9, 3.628015796492584*^9}, {3.628015830467173*^9, 3.628015988904243*^9}, {3.6304211768106937`*^9, 3.6304211896416187`*^9}}], Cell[TextData[{ "What we have just done, for the given example, is confirmed that complex \ number addition also satisfies the parallelogram law, just as vector addition \ does. In fact, the ", StyleBox[ButtonBox["extended view of vector addition", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.virtualnerd.com/intro-physics/vectors/addition-\ subtraction/add-subtract-multiple-vectors/add-head-tail-three-vector"], None}, ButtonNote-> "http://www.virtualnerd.com/intro-physics/vectors/addition-subtraction/add-\ subtract-multiple-vectors/add-head-tail-three-vector"], FontColor->RGBColor[0, 0, 1]], " works when we add three or more complex numbers. We will therefore want \ visualize complex numbers as vectors when we add them." }], "Text", CellChangeTimes->{{3.6280160200716467`*^9, 3.6280161701999598`*^9}, { 3.647880485020097*^9, 3.64788048502352*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" Exercise 2:", FontWeight->"Bold"], " Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to visually illustrate the sum of the complex numbers ", Cell[BoxData[ FormBox[ RowBox[{"3", "+", RowBox[{"2", "\[ImaginaryI]"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", "5"}], "+", RowBox[{"4", "\[ImaginaryI]"}]}], TraditionalForm]]], " as equivalent to the addition of the vectors ", Cell[BoxData[ FormBox[ RowBox[{"\[LeftAngleBracket]", RowBox[{"3", ",", "2"}], "\[RightAngleBracket]"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"\[LeftAngleBracket]", RowBox[{ RowBox[{"-", "5"}], ",", "4"}], "\[RightAngleBracket]"}], TraditionalForm]]], ". What is different about the parallelogram so generated compared to the \ parallelogram created with the example above? Use the ", StyleBox[ButtonBox["distance formula", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.purplemath.com/modules/distform.htm"], None}, ButtonNote->"http://www.purplemath.com/modules/distform.htm"], FontColor->RGBColor[0, 0, 1]], " to confirm that opposing sides of this parallelogram have the same length \ and use facts from trigonometry, such as the ", StyleBox[ButtonBox["Law of Cosines", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.mathsisfun.com/algebra/trig-cosine-law.html"], None}, ButtonNote->"http://www.mathsisfun.com/algebra/trig-cosine-law.html"], FontColor->RGBColor[0, 0, 1]], ", to confirm that opposing angles of this parallelogram have the same \ measure." }], "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}, { 3.627915943542495*^9, 3.627915953032187*^9}, {3.62791602506989*^9, 3.627916047611404*^9}, {3.627916089879258*^9, 3.6279161105186167`*^9}, 3.6280059612857428`*^9, {3.628016222252342*^9, 3.628016460612976*^9}, { 3.628016498207847*^9, 3.6280164982123003`*^9}, {3.6280165432046423`*^9, 3.6280165480001173`*^9}, {3.628016599170849*^9, 3.628016599174406*^9}, { 3.6288770389034233`*^9, 3.628877045706953*^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}, {3.6280162311838093`*^9, 3.628016231340686*^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}}], 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[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}, { 3.628016233684417*^9, 3.6280162337884083`*^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[{ "In Activity 3 of this learning module on complex addition and the complex \ plane, we will use ", StyleBox["Mathematica", FontSlant->"Italic"], "\[CloseCurlyQuote]s ", StyleBox[ButtonBox["Manipulate", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/Manipulate.html"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/Manipulate.html"], FontVariations->{"Underline"->True}, FontColor->RGBColor[1, 0.5, 0]], " function to create dynamic (interactive) versions of these diagrams. 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