(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 45627, 1576] NotebookOptionsPosition[ 37303, 1347] NotebookOutlinePosition[ 40693, 1434] CellTagsIndexPosition[ 40650, 1431] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "Math 421 ", StyleBox["\[FilledSmallCircle]", FontSlant->"Plain"], " Fall 2010" }], "Subsubtitle", CellChangeTimes->{{3.493070131289755*^9, 3.49307013430538*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Introduction\[LongDash]Part 3" }], "Subtitle", TextAlignment->Center], Cell["10 September 2010", "Subsubtitle", CellChangeTimes->{{3.492979416828125*^9, 3.49297941990625*^9}, 3.493068416664755*^9, {3.493111999421875*^9, 3.493111999703125*^9}}, TextAlignment->Center], Cell["\<\ Copyright \[Copyright] 2010 by Murray Eisenberg. All rights reserved.\ \>", "SmallText", TextAlignment->Center, TextJustification->0], Cell[TextData[StyleBox["Your name(s):\n\n", FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell["What else can Mathematica do?", "Section"], Cell[TextData[{ "Already you have seen some examples of numerical, symbolic, and graphical \ work with ", StyleBox["Mathematica", FontSlant->"Italic"], ". Here are a few more. ", StyleBox["Try to predict the output before evaluating each input cell.", FontWeight->"Bold", FontSlant->"Italic"] }], "Text", CellChangeTimes->{{3.493070182039755*^9, 3.49307019596163*^9}}], Cell[CellGroupData[{ Cell["Simplifying expressions", "Subsection"], Cell["\<\ Mathematica has only a limited number of rules automatically applied to \ expressions you evaluate. For example:\ \>", "Text", CellChangeTimes->{3.493070208071005*^9}], Cell[BoxData[ RowBox[{"Log", "[", RowBox[{"2", " ", RowBox[{"Exp", "[", "5", "]"}]}], "]"}]], "Input"], Cell[TextData[{ "You can undoubtedly see another, possibly simpler form for this. (Of course \ simplicity is in the eye of the beholder.) Often you can something more \ closely resembling the \"simple\" form you expected by using the ", StyleBox["Simplify", FontFamily->"Courier"], " function:" }], "Text", CellChangeTimes->{{3.49307021639913*^9, 3.493070220883505*^9}}], Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{"Log", "[", RowBox[{"2", " ", RowBox[{"Exp", "[", "5", "]"}]}], "]"}], "]"}]], "Input"], Cell["Here's another one:", "Text"], Cell[BoxData[ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "^", "2"}], " ", "-", " ", "9"}], ")"}], "/", RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], "]"}]], "Input"], Cell[TextData[{ "To simplify that result, you could use ", StyleBox["Simplify", FontFamily->"Courier"], " by going back to the start of the line, typing ", StyleBox["Sqrt[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", then going to the end of the line and typing the closing bracket ", StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ".\nBut its quicker to use ", StyleBox["postfix notation", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ":, with ", StyleBox["//", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". To do that, after the end of the line, type:\n\t", StyleBox["//Simplify", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{{3.49307024346163*^9, 3.49307038186788*^9}, { 3.493070435133505*^9, 3.493070475946005*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Exercise.", FontWeight->"Bold"], " Do that in the preceding Input cell. Then evaluate the modified cell." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.493070387821005*^9, 3.49307041071163*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Working with lists", "Subsection", CellChangeTimes->{3.49307125586788*^9}], Cell[CellGroupData[{ Cell["Lists", "Subsubsection", CellChangeTimes->{{3.49307100536788*^9, 3.493071006039755*^9}, 3.49307125586788*^9}], Cell[TextData[{ "Several objects may be combined into a ", StyleBox["list", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", which is indicated with ", StyleBox["braces", FontWeight->"Bold"], " ", StyleBox["{", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and whose items are separated by commas. For example:" }], "Text", CellChangeTimes->{{3.493070519196005*^9, 3.49307056202413*^9}, 3.49307125586788*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ FractionBox["15", "30"], ",", " ", "\[Pi]", ",", " ", "\"\\"", ",", FractionBox[ RowBox[{"3", " ", "+", " ", RowBox[{"4", "I"}]}], RowBox[{"5", " ", "+", " ", RowBox[{"6", "I"}]}]]}], "}"}]], "Input", CellChangeTimes->{3.493070616946005*^9, 3.49307125586788*^9}], Cell["Two more examples:", "Text", CellChangeTimes->{{3.49307059064913*^9, 3.49307059496163*^9}, 3.49307125586788*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", " ", RowBox[{"b", " ", "I"}]}], ")"}], "3"], ",", " ", RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", " ", RowBox[{"b", " ", "I"}]}], ")"}], "3"], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.49307060596163*^9, 3.493070611477255*^9}, 3.49307125586788*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"z", " ", "=", " ", RowBox[{"3", " ", "+", " ", RowBox[{"4", "I"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "z", "]"}], ",", " ", RowBox[{"Im", "[", "z", "]"}]}], "}"}]}], "Input", CellChangeTimes->{3.49307125586788*^9}], Cell[TextData[{ "A quick way to form a list of the positive integers 1, 2, \[Ellipsis]", Cell[BoxData[ FormBox["n", TraditionalForm]]], " for some positive integer ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " is to use the ", StyleBox["Range", FontFamily->"Courier", FontWeight->"Bold"], " function:" }], "Text", CellChangeTimes->{3.49307125586788*^9}], Cell[BoxData[ RowBox[{"Range", "[", "10", "]"}]], "Input", CellChangeTimes->{3.49307125586788*^9}], Cell[TextData[{ "Then to form the list ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ SuperscriptBox["1", "2"], ",", " ", SuperscriptBox["2", "2"], ",", " ", SuperscriptBox["3", "2"], ",", " ", "\[Ellipsis]", ",", " ", SuperscriptBox["10", "2"]}], "}"}], TraditionalForm]]], " of squares of those numbers, you could do the following:" }], "Text", CellChangeTimes->{{3.493070660664755*^9, 3.49307072952413*^9}, 3.49307125586788*^9}], Cell[BoxData[ SuperscriptBox[ RowBox[{"Range", "[", "10", "]"}], "2"]], "Input", CellChangeTimes->{{3.493070686008505*^9, 3.49307073189913*^9}, 3.49307125586788*^9}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Using ", StyleBox["Table", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " for lists" }], "Subsubsection", CellChangeTimes->{{3.49307102439913*^9, 3.493071029758505*^9}, 3.49307125586788*^9}], Cell[TextData[{ "Another way to form the list ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ SuperscriptBox["1", "2"], ",", " ", SuperscriptBox["2", "2"], ",", " ", SuperscriptBox["3", "2"], ",", " ", "\[Ellipsis]", ",", " ", SuperscriptBox["10", "2"]}], "}"}], TraditionalForm]]], " is to use a ", StyleBox["Table", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " expression: " }], "Text", CellChangeTimes->{{3.49307104289913*^9, 3.49307106805538*^9}, 3.49307125586788*^9}], Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ SuperscriptBox["j", "2"], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.49307082586788*^9, 3.49307083430538*^9}, 3.49307125586788*^9}], Cell[TextData[{ "Notice that the \"iterator\" expression ", StyleBox["{j,1,10}", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " is itself another list!" }], "Text", CellChangeTimes->{{3.493071553727255*^9, 3.49307156849288*^9}}], Cell[TextData[{ "Of course the preceding way, with ", StyleBox["Table", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and the \"counter\" ", StyleBox["j", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " (or some other letter), is longer to type than ", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox[ RowBox[{"Range", "[", "10", "]"}], "2"], FontFamily->"Courier"], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.493071090539755*^9, 3.493071180352255*^9}, 3.49307125586788*^9, {3.49307158914913*^9, 3.49307159664913*^9}, 3.49307372614913*^9}], Cell["To add up those squares:", "Text", CellChangeTimes->{{3.49307073905538*^9, 3.49307074505538*^9}, 3.493070850727255*^9, 3.49307125586788*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ SuperscriptBox["j", "2"], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.49307087846163*^9, 3.493070887008505*^9}, 3.49307125586788*^9}], Cell[BoxData["385"], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{{3.493071227977255*^9, 3.493071255883505*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", RowBox[{"Table", "[", RowBox[{ SuperscriptBox["j", "2"], ",", RowBox[{"{", RowBox[{"j", ",", "1", ",", "10"}], "}"}]}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.49307121746163*^9, 3.49307122436788*^9}, 3.493071255883505*^9}], Cell[BoxData["385"], "Output", GeneratedCell->False, CellAutoOverwrite->False, CellChangeTimes->{3.493071225789755*^9, 3.493071255883505*^9}] }, Closed]], Cell[BoxData[ RowBox[{"Total", "[", SuperscriptBox[ RowBox[{"Range", "[", "10", "]"}], "2"], "]"}]], "Input", CellChangeTimes->{{3.493070746946005*^9, 3.49307075293038*^9}, 3.493071255883505*^9}] }, Closed]], Cell[CellGroupData[{ Cell["Indexing and parts of lists", "Subsubsection", CellChangeTimes->{{3.49307140921163*^9, 3.493071413196005*^9}}], Cell[TextData[{ "Individual entries in a list can be extracted by ", StyleBox["indexing", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], ", as follows:" }], "Text", CellChangeTimes->{3.493071255883505*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"w", "=", RowBox[{"5", "+", RowBox[{"6", "I"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"theParts", "=", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "w", "]"}], ",", RowBox[{"Im", "[", "w", "]"}]}], "}"}]}], "\n", RowBox[{"theParts", "[", RowBox[{"[", "2", "]"}], "]"}]}], "Input", CellChangeTimes->{ 3.493071255883505*^9, {3.49307142411788*^9, 3.49307142468038*^9}, { 3.49307147611788*^9, 3.493071485539755*^9}}], Cell[TextData[{ "For indexing, you may prefer to type ", StyleBox["\[EscapeKey]", Magnification->1.5], StyleBox["[[", FontFamily->"Courier"], StyleBox["\[EscapeKey]", Magnification->1.5], " and ", StyleBox["\[EscapeKey]", Magnification->1.5], StyleBox["]]", FontFamily->"Courier"], StyleBox["\[EscapeKey]", Magnification->1.5], " to get more distinctive double brackets:" }], "Text", CellChangeTimes->{3.493071255883505*^9}], Cell[BoxData[ RowBox[{"theParts", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]], "Input", CellChangeTimes->{3.493071255883505*^9}], Cell[TextData[{ StyleBox["Exercise.", FontWeight->"Bold"], " (a) Type and evaluate an expression that creates the list {1 + \ \[ImaginaryI], 2 + 2\[ImaginaryI], 3 + 3\[ImaginaryI], \[Ellipsis], 6 + 6\ \[ImaginaryI]} and assigns it to the name ", StyleBox["nums", FontFamily->"Courier"], ". [", StyleBox["Hint:", FontSlant->"Italic"], " Recall that the shortcut for ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]]], " is ", StyleBox["\[EscapeKey]", Magnification->1.5], StyleBox["ii", FontFamily->"Courier"], StyleBox["\[EscapeKey]", Magnification->1.5], " (with a ", StyleBox["double", FontSlant->"Italic"], " ", StyleBox["i", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ").]\n(b) Then use indexing to extract the third entry in ", StyleBox["nums", FontFamily->"Courier"], ". \n(c) Finally, type another expression that uses indexing to extract \ both the first and third entry in ", StyleBox["nums", FontFamily->"Courier"], " together in the form of a list with two entries." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.493071502196005*^9, 3.49307151274288*^9}, 3.49307160864913*^9}, ParagraphSpacing->{0.5, 0}], Cell["\<\ For some parts of a list, it's easier to use functions such as First and Last \ instead of indexing:\ \>", "Text", CellChangeTimes->{{3.493071646102255*^9, 3.493071679289755*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"w", "=", RowBox[{"5", "+", RowBox[{"6", "I"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"theParts", "=", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "w", "]"}], ",", RowBox[{"Im", "[", "w", "]"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"First", "[", "theParts", "]"}]}], "Input", CellChangeTimes->{{3.49307169014913*^9, 3.49307169577413*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Numerical values", "Subsection"], Cell[TextData[{ "As you have already seen, ", StyleBox["Mathematica", FontSlant->"Italic"], " can work with exact rational and irrational quantities. When you want to \ convert such an exact quantity, say one involving ", Cell[BoxData[ FormBox[ SqrtBox["2"], TraditionalForm]]], ", into its numerical equivalent, use the function ", StyleBox["N", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ". (So don't ever try to name something N yourself!)" }], "Text", CellChangeTimes->{{3.493071711133505*^9, 3.49307174586788*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ SqrtBox["2"], "+", RowBox[{"E", " ", "I"}]}], ",", RowBox[{"N", "[", RowBox[{ SqrtBox["2"], "+", RowBox[{"E", " ", "I"}]}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.493071751664755*^9, 3.49307179943038*^9}}], Cell[TextData[{ "By default, ", StyleBox["Mathematica", FontSlant->"Italic"], " displays only 6 significant digits, but you can ask for more if you start \ with exact numbers:" }], "Text"], Cell[BoxData[ RowBox[{"N", "[", RowBox[{ RowBox[{ SqrtBox["2"], "+", RowBox[{"E", " ", "I"}]}], ",", "40"}], "]"}]], "Input", CellChangeTimes->{{3.493071773258505*^9, 3.49307178036788*^9}, { 3.49307181127413*^9, 3.493071812102255*^9}}], Cell[TextData[{ "If, though, your input includes even one floating-point (approximate) \ number including a decimal point, then ", StyleBox["Mathematica", FontSlant->"Italic"], " will only give meaningful results to machine precision:" }], "Text"], Cell[BoxData[ RowBox[{"N", "[", RowBox[{ SqrtBox["2.0"], " ", "+", " ", RowBox[{"E", " ", "I"}]}], "]"}]], "Input", CellChangeTimes->{{3.49307182258663*^9, 3.49307182383663*^9}}], Cell["Predict the result of evaluating the following cell:", "Text"], Cell[BoxData[ RowBox[{"N", "[", RowBox[{ RowBox[{ SqrtBox["2."], "+", RowBox[{ SqrtBox["3"], "I"}]}], ",", "12"}], "]"}]], "Input", CellChangeTimes->{{3.493071837196005*^9, 3.493071859633505*^9}, { 3.49307189661788*^9, 3.49307196377413*^9}}], Cell[TextData[{ "Are your surprised at the result? Did you expect to see 12 digits? \ Actually, ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["did", FontVariations->{"Underline"->True}], " approximate ", Cell[BoxData[ FormBox[ RowBox[{ SqrtBox["2."], " ", "+", RowBox[{ SqrtBox["3"], "\[ImaginaryI]"}]}], TraditionalForm]]], " to that many digits, but it still displayed the result with the default 6 \ digits precision.\nTo see all 12 digits, wrap the expression inside the ", StyleBox["NumberForm", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1]], " function:" }], "Text", CellChangeTimes->{{3.493071872821005*^9, 3.493072036977255*^9}}], Cell[BoxData[ RowBox[{"NumberForm", "[", RowBox[{ RowBox[{"N", "[", RowBox[{ RowBox[{ SqrtBox["2."], " ", "+", " ", RowBox[{ SqrtBox["3"], "I"}]}], ",", " ", "12"}], "]"}], ",", " ", "12"}], "]"}]], "Input", CellChangeTimes->{{3.49307205271163*^9, 3.493072058852255*^9}}], Cell[TextData[{ "High precision arithmetic is possible with ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Text", CellChangeTimes->{3.49307210189913*^9}], Cell[TextData[{ StyleBox["Exercise.", FontWeight->"Bold"], " Create a cell below with an expression that will give 100 digits of ", Cell[BoxData[ FormBox["\[Pi]", TraditionalForm]]], ", then evaluate that cell to see the result." }], "Exercise", CellChangeTimes->{{3.49307210189913*^9, 3.49307213305538*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["Solving equations", "Subsection"], Cell[TextData[{ StyleBox["\[WarningSign]", Magnification->2, FontColor->RGBColor[1, 0, 0]], " ", StyleBox["Equations are formed with ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["two", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" equal signs", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "!" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"2", "+", "2"}], "==", "4"}]], "Input", CellChangeTimes->{{3.493072166883505*^9, 3.493072170164755*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"2", "+", "3"}], "\[Equal]", "4"}]], "Input", CellChangeTimes->{{3.493072172352255*^9, 3.49307217483663*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "z", "]"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"z", RowBox[{"(", RowBox[{"z", "+", "z"}], ")"}]}], "==", RowBox[{"2", " ", SuperscriptBox["z", "2"]}]}]], "Input", CellChangeTimes->{{3.49307218018038*^9, 3.493072183352255*^9}}], Cell[TextData[{ StyleBox["To solve an equation", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " use the ", StyleBox["Solve", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " function:" }], "Text"], Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "z"}], "-", "7"}], "==", RowBox[{ FractionBox["8", "5"], "+", RowBox[{"2", "I"}]}]}], ",", "z"}], "]"}]], "Input", CellChangeTimes->{{3.49307220043038*^9, 3.49307220830538*^9}}], Cell[TextData[{ "Surprised at the output? It does not directly give you the solution, but \ rather a ", StyleBox["replacement rule", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " , with ", StyleBox["\[Rule]", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ", for the solve variable. (Moreover, the rule is in the form of a \ brace-enclosed list; more about that later.)" }], "Text", CellChangeTimes->{{3.49307221314913*^9, 3.493072244664755*^9}, { 3.49307373918038*^9, 3.493073740946005*^9}}], Cell[TextData[{ StyleBox["To convert a replacement rule into an actual value", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" use the form ", FontWeight->"Bold"], StyleBox["expr/.rule", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"] }], "EmphasisText", CellFrame->True, CellChangeTimes->{{3.493072282102255*^9, 3.493072283977255*^9}}], Cell[BoxData[ RowBox[{"z", " ", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "z"}], "-", "7"}], "==", RowBox[{ FractionBox["8", "5"], "+", RowBox[{"2", "I"}]}]}], ",", "z"}], "]"}]}]], "Input", CellChangeTimes->{{3.493072297008505*^9, 3.493072300789755*^9}}], Cell[TextData[{ "Better yet, anticipate having to do this and assign the replacement rule \ result of the ", StyleBox["Solve", FontFamily->"Courier"], " to a name: " }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"mysoln", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "z"}], "-", "7"}], "==", RowBox[{ FractionBox["8", "5"], "+", RowBox[{"2", "I"}]}]}], ",", "z"}], "]"}]}], ";"}], "\n", RowBox[{"z", "/.", "mysoln"}]}], "Input", CellChangeTimes->{{3.493072312321005*^9, 3.49307234293038*^9}}], Cell[TextData[{ "There is still one nasty thing about the answer\[LongDash]it's a list, \ albeit with a single entry. And that's because the result of ", StyleBox["Solve", FontFamily->"Courier"], " was a one-entry list, and that entry is itself a list. To get rid of the \ outer list braces, you can index with ", StyleBox["[[1]]", FontFamily->"Courier"], " or use in this case the equivalent function ", StyleBox["First", FontFamily->"Courier"], ":" }], "Text", CellChangeTimes->{{3.49307235399288*^9, 3.493072361446005*^9}}], Cell[BoxData[{ RowBox[{"solns", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "z"}], "-", "7"}], "==", RowBox[{ FractionBox["8", "5"], "+", RowBox[{"2", "I"}]}]}], ",", "z"}], "]"}]}], "\n", RowBox[{"First", "[", "solns", "]"}], "\[IndentingNewLine]", RowBox[{"z", "/.", RowBox[{"First", "[", "solns", "]"}]}]}], "Input", CellChangeTimes->{{3.493072384914755*^9, 3.49307242586788*^9}}], Cell[TextData[{ "Instead of ", StyleBox["First[solns]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", you could use the ", StyleBox["prefix", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " (composition) form ", StyleBox["First@solns", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{{3.493072441789755*^9, 3.49307249905538*^9}}], Cell[BoxData[ RowBox[{"First", "@", "solns"}]], "Input", CellChangeTimes->{{3.49307250096163*^9, 3.49307250355538*^9}}], Cell["\<\ And then you could do the whole thing of finding the solution, as a number as \ follows:\ \>", "Text", CellChangeTimes->{{3.493072441789755*^9, 3.49307249905538*^9}, { 3.493072540383505*^9, 3.493072558321005*^9}}], Cell[BoxData[ RowBox[{"z", "/.", RowBox[{"First", "@", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", "z"}], "-", "7"}], "==", RowBox[{ FractionBox["8", "5"], "+", RowBox[{"2", "I"}]}]}], ",", "z"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.49307255996163*^9, 3.49307257968038*^9}}], Cell["\<\ Of course equations may have more than one solution\[LongDash]especially when \ the solutions may be non-real complex numbers:\ \>", "Text"], Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["z", "2"], "+", "1"}], "\[Equal]", "0"}], ",", "z"}], "]"}]], "Input", CellChangeTimes->{{3.493072594571005*^9, 3.49307259855538*^9}}], Cell["\<\ So there is a list of two solutions, just shown however in the form of a list \ of rules. 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For example:" }], "Text", CellChangeTimes->{{3.49311094821875*^9, 3.493110966640625*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"3", RowBox[{"Sin", "[", RowBox[{"2", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"4", "\[Pi]"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.493110968296875*^9, 3.493110999515625*^9}, 3.493111303734375*^9}], Cell[TextData[{ "In any expression where you have a constant, such as the multiplier ", StyleBox["2", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " above, you may change it to a \"parameter\"\[LongDash]a variable quantity \ on which the result depends\[LongDash]and then manipulate the result by \ varying that parameter. 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Below, instead of just varying the frequency ", StyleBox["m", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", vary also the amplitude as well:" }], "Text", CellChangeTimes->{{3.493111230703125*^9, 3.493111316234375*^9}}], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"a", " ", RowBox[{"Sin", "[", RowBox[{"m", " ", "x"}], "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"4", "\[Pi]"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "3"}], ",", "3"}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"a", ",", "0.5", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"m", ",", "1", ",", "5", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.493110968296875*^9, 3.493110999515625*^9}, { 3.49311113503125*^9, 3.493111167765625*^9}, {3.49311130803125*^9, 3.49311137525*^9}}], Cell[TextData[{ "The ", StyleBox["PlotRange", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option in ", StyleBox["Plot", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " was used so as to keep the dimensions the scale the same on the vertical \ axis as the amplitude varied; without it, the scale would change." }], "SmallText", CellChangeTimes->{{3.493111394515625*^9, 3.49311145953125*^9}}], Cell[TextData[{ StyleBox["Exercise.", FontWeight->"Bold"], " Move both sliders in the preceding output." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.493111181125*^9, 3.493111215578125*^9}, {3.49311127190625*^9, 3.49311127296875*^9}, { 3.493111489328125*^9, 3.493111503484375*^9}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " does not restrict you to manipulating plots. You may ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " any expression with a variable in it. Start, for example, with the \ following:" }], "Text", CellChangeTimes->{{3.4931115170625*^9, 3.49311160575*^9}}], Cell[BoxData[ RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "3"], "]"}]], "Input", CellChangeTimes->{{3.49311160846875*^9, 3.493111646671875*^9}}], Cell[TextData[{ "Now change the constant power ", StyleBox["3", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " there to a variable and ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " it:" }], "Text", CellChangeTimes->{{3.49311164965625*^9, 3.493111662921875*^9}}], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "n"], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "8", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.49311157821875*^9, 3.493111590328125*^9}, { 3.49311167275*^9, 3.4931116925*^9}}], Cell[TextData[{ StyleBox["Exercise.", FontWeight->"Bold"], " You could \"multiply out\" ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], "2"], TraditionalForm]], FormatType->"TraditionalForm"], " by using ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":\n\t", StyleBox["ComplexExpand[", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "\[ImaginaryI]"}], ")"}], "2"], TraditionalForm]], FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\n(a) Do that in a new Input cell below.\n(b) Change the power ", StyleBox["2", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " to a variable, say ", StyleBox["n", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", and ", StyleBox["Manipulate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " that expression. 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You have completed this brief introduction to ", StyleBox["Mathematica.", FontSlant->"Italic"], " \nNow you're ready to start using ", StyleBox["Mathematica", FontSlant->"Italic"], " to learn topics in this course.\nGo to the ", Cell[BoxData[ FormBox[ ButtonBox[ RowBox[{ "Math", " ", "421", " ", "web", " ", "site", " ", "Files", " ", "page"}], BaseStyle->"Hyperlink", ButtonData->{ URL["http://blogs.umass.edu/math421-murray/files/"], None}, ButtonNote->"http://blogs.umass.edu/math421-murray/files/"], TraditionalForm]], FormatType->"TraditionalForm"], " again and open the notebook ", StyleBox["nthRoots.nb", FontFamily->"Courier", FontWeight->"Bold"], ". 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