(* 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[ 120924, 3840] NotebookOptionsPosition[ 101552, 3321] NotebookOutlinePosition[ 112623, 3577] CellTagsIndexPosition[ 112520, 3571] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.488930259078125*^9, 3.488930259484375*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["Birth of complex numbers: cubic equations", "Subtitle", CellChangeTimes->{ 3.48969559478125*^9, {3.490551533328125*^9, 3.49055153875*^9}}, TextAlignment->Center, TextJustification->0], Cell["14 September 2010", "Subsubtitle", CellChangeTimes->{{3.48893028171875*^9, 3.48893030315625*^9}, { 3.489583451984375*^9, 3.489583455796875*^9}, 3.49028881771875*^9, { 3.4905515529375*^9, 3.490551553125*^9}, 3.490875430171875*^9, { 3.491158425140625*^9, 3.491158435296875*^9}, 3.491217414875*^9, { 3.49193520925*^9, 3.491935209796875*^9}, {3.493460611234375*^9, 3.493460614703125*^9}}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2006\[Dash]2010 by Murray Eisenberg. All rights \ reserved.\ \>", "Text", "SmallText", CellChangeTimes->{{3.488930265421875*^9, 3.488930270109375*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Prerequisites", "Section", CellChangeTimes->{ 3.48969559478125*^9, {3.491157896171875*^9, 3.4911578986875*^9}}, CellID->143077100], Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematica", FontSlant->"Italic"]], "Subsection", CellChangeTimes->{ 3.48969559478125*^9, {3.491157903546875*^9, 3.491157907171875*^9}}, CellID->127073981], Cell[TextData[{ "Aside from having a working ", StyleBox["Mathematica", FontSlant->"Italic"], " system at your disposal and knowing how to type input, how to evaluate an \ Input cell, and how to navigate around a notebook, there are really no \ prerequisites.\nIn fact, working through this notebook is a good way to learn \ some ", StyleBox["Mathematica", FontSlant->"Italic"], " basics and even some more advanced ", StyleBox["Mathematica", FontSlant->"Italic"], " techniques." }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.491157910984375*^9, 3.491157928234375*^9}, { 3.491158020265625*^9, 3.491158128125*^9}, {3.491158162796875*^9, 3.491158177421875*^9}, {3.491217366828125*^9, 3.49121736778125*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->20739139], Cell[TextData[{ "All the input shown uses the display form ", StyleBox["\[ImaginaryI]", FontFamily->"Courier"], " obtained by typing ", StyleBox["\[EscapeKey]", FontSize->16], StyleBox["ii", FontFamily->"Courier"], StyleBox["\[EscapeKey]", FontSize->16], ", but everything would work identically if you typed ", StyleBox["I", FontFamily->"Courier"], " instead." }], "Text", CellChangeTimes->{3.48969559478125*^9, 3.49115815171875*^9}, CellID->83290015], Cell[TextData[{ "David Park\[CloseCurlyQuote]s ", StyleBox["Presentations", FontSlant->"Italic"], " application is ", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " needed for this notebook." }], "Text", CellChangeTimes->{3.48969559478125*^9, 3.491217372484375*^9}, CellID->171441541] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{ 3.48969559478125*^9, {3.49115821378125*^9, 3.491158215890625*^9}}, CellID->195536689], Cell[TextData[{ "Aside from basic algebra and geometry, there are no particular mathematical \ prerequites. You should know the quadratic formula for solving a quadratic \ equation. It\[CloseCurlyQuote]s helpful if you remember the Binomial Formula \ for expanding a power ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "n"], TraditionalForm]], FormatType->"TraditionalForm"], ", but the relevant case here is ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "3"}], TraditionalForm]], FormatType->"TraditionalForm"], ", and the formula is given explicitly in that case." }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.491158227984375*^9, 3.491158362609375*^9}}, CellID->79531803] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["The problem", "Section", ShowGroupOpener->True, CellChangeTimes->{3.48969559478125*^9, 3.490268524109375*^9}, CellID->344771388], Cell["The problem is to solve the general cubic equation:", "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.4894307534375*^9, 3.48943075971875*^9}, { 3.4897525208125*^9, 3.489752523796875*^9}}], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"a", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"b", " ", "x"}], "+", "c"}], " ", "=", "0"}], TraditionalForm]]] }], "EmphasisText", CellChangeTimes->{{3.48893045246875*^9, 3.488930463546875*^9}, { 3.489084584109375*^9, 3.4890845849375*^9}, {3.48943076375*^9, 3.4894307645*^9}, {3.48958448490625*^9, 3.4895844930625*^9}, 3.489592029390625*^9}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{"cubic", "=", RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"a", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"b", " ", "x"}], "+", "c"}]}]], "Input", CellChangeTimes->{{3.48893043359375*^9, 3.48893044603125*^9}, 3.48932826753125*^9, {3.48940764425*^9, 3.489407666203125*^9}, { 3.48958450065625*^9, 3.489584507234375*^9}}, CellLabel->"In[1]:="], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can solve it directly:" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489084490515625*^9, 3.4890845025*^9}, { 3.48908458771875*^9, 3.489084604734375*^9}, {3.489328027546875*^9, 3.48932802775*^9}}], Cell[BoxData[ RowBox[{"x", "/.", RowBox[{"Solve", "[", RowBox[{ RowBox[{"cubic", "\[Equal]", "0"}], ",", "x"}], "]"}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.489084508671875*^9, 3.4890845423125*^9}, { 3.48908458771875*^9, 3.4890846089375*^9}, 3.489328013*^9, { 3.489328272734375*^9, 3.48932829190625*^9}, {3.48940765240625*^9, 3.4894076554375*^9}}, CellLabel->"In[2]:="], Cell["(How did it do that?)", "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489084639203125*^9, 3.489084643140625*^9}, { 3.489584689625*^9, 3.489584690859375*^9}}], Cell[TextData[{ "The output from ", StyleBox["Mathematica", FontSlant->"Italic"], " above is a list of three solutions. Notice that these solutions seem to \ involve non-real complex numbers\[LongDash]numbers of the form ", Cell[BoxData[ FormBox[ RowBox[{"\[Alpha]", "+", RowBox[{"\[Beta]", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " where ", Cell[BoxData[ FormBox["\[Alpha]", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\[Beta]", TraditionalForm]]], " are real. As you may know, a cubic equation has three \ solutions\[LongDash]either three real solutions or else one real solution and \ a pair of non-real complex-conjugate solutions. So for particular \ coefficients ", Cell[BoxData[ FormBox[ RowBox[{"a", ",", " ", "b", ",", " ", "c"}], TraditionalForm]]], ", even the solutions above that explicitly involve the complex number ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]]], " actually simplify to real numbers." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.48958453703125*^9, 3.4895846263125*^9}, { 3.4895847085625*^9, 3.489584886734375*^9}, {3.48958492009375*^9, 3.489585056140625*^9}, {3.48958509490625*^9, 3.489585099734375*^9}, { 3.4897530921875*^9, 3.48975310075*^9}, {3.49021377446875*^9, 3.49021380475*^9}, {3.490550413578125*^9, 3.490550450578125*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " The cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"5", SuperscriptBox["x", "2"]}], "+", RowBox[{"5", "x"}], "+", "3"}], "=", "0"}], TraditionalForm]]], " has ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "3"}], TraditionalForm]]], " as one of its solutions. Without using the ", StyleBox["Mathematica", FontSlant->"Italic"], " solution, above, find the other two. (", StyleBox["Hint", FontSlant->"Italic"], ": If you have one solution ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " of a cubic equation, you may find the others by dividing the cubic \ polynomial by ", Cell[BoxData[ FormBox[ RowBox[{"x", "-", "r"}], TraditionalForm]]], " and then applying the quadratic formula to the resulting quadratic.)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.48975254321875*^9, 3.489752555859375*^9}, {3.4897526316875*^9, 3.489752806234375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " (a) The cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"12", "x"}], "+", "16"}], "=", "0"}], TraditionalForm]]], " has ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{"-", "4"}]}], TraditionalForm]]], " as one of its solutions. Find the others.\n(b) The cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"6", SuperscriptBox["x", "2"]}], "+", RowBox[{"12", "x"}], "-", "8"}], "=", "0"}], TraditionalForm]]], " has ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "2"}], TraditionalForm]]], " as one of its solutions. Find the others.\n(", StyleBox["Rhetorical question", FontSlant->"Italic"], ": How must the assertion above, \[OpenCurlyDoubleQuote]a cubic equation has \ three solutions\[CloseCurlyDoubleQuote] be interpreted?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.489752848796875*^9, 3.489752905828125*^9}, {3.4897529445625*^9, 3.489753080546875*^9}, { 3.48975311203125*^9, 3.48975314434375*^9}, {3.48975318434375*^9, 3.489753209734375*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find all solutions of the cubic equation:\n\t(a) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"5", SuperscriptBox["x", "2"]}], "+", RowBox[{"5", "x"}], "+", "3"}], "=", "0"}], TraditionalForm]]], ".\n\t(b) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"4", SuperscriptBox["x", "2"]}], "+", RowBox[{"14", "x"}], "-", "20"}], "=", "0"}], TraditionalForm]]], ".\n\t(c) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"4", SuperscriptBox["x", "3"]}], "-", RowBox[{"16", SuperscriptBox["x", "2"]}], "+", RowBox[{"4", "x"}], "+", "24"}], "=", "0"}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.489753241859375*^9, 3.489753364796875*^9}, {3.489753413265625*^9, 3.489753442265625*^9}, { 3.489753488578125*^9, 3.489753563953125*^9}}, ParagraphSpacing->{0.5, 0.}] }, Closed]], Cell[CellGroupData[{ Cell["Strategy", "Section", ShowGroupOpener->True, CellChangeTimes->{ 3.48940428696875*^9, {3.489440641234375*^9, 3.48944064296875*^9}, { 3.48958351565625*^9, 3.489583519421875*^9}, 3.49026852415625*^9, { 3.490288601453125*^9, 3.490288611828125*^9}, 3.4902888510625*^9}, CellID->355727], Cell["\<\ The strategy for finding a solution of a cubic equation considered here is:\ \>", "Text", CellChangeTimes->{ 3.48893078675*^9, 3.489073349265625*^9, {3.4890843748125*^9, 3.489084383015625*^9}, {3.48940784828125*^9, 3.489407848453125*^9}, { 3.48958351565625*^9, 3.48958352959375*^9}, 3.489583580859375*^9, { 3.4895839323125*^9, 3.4895839373125*^9}, {3.490288556*^9, 3.49028859665625*^9}, {3.49028867325*^9, 3.49028867459375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "\t\[FilledSmallCircle] reduce the cubic to a \ \[OpenCurlyDoubleQuote]depressed cubic\[CloseCurlyDoubleQuote]\[LongDash]one \ with no quadratic term\[LongDash]by making a linear substitution (Cardano\ \[CloseCurlyQuote]s method);\n\t\[FilledSmallCircle] obtain one solution ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " of the depressed cubic from a formula of del Ferro and Tartaglia;\n\t\ \[FilledSmallCircle] when that solution ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " involves ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]]], ", use ", "Bombelli", "\[CloseCurlyQuote]s method to obtain a real solution; \n\t\ \[FilledSmallCircle] find the corresponding root of the original cubic by \ reversing the linear substitution; and\n\t\[FilledSmallCircle] find the other \ two roots of the cubic by factoring out ", Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"x", "-", "r"}], TraditionalForm]]]]], TraditionalForm]]], " and applying the quadratic formula." }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489084392734375*^9, 3.48908440846875*^9}, { 3.489084803078125*^9, 3.489084803328125*^9}, {3.48940781325*^9, 3.489408005703125*^9}, {3.48958351565625*^9, 3.489583554171875*^9}, { 3.489583599484375*^9, 3.489583690625*^9}, {3.489583972171875*^9, 3.489584113484375*^9}, {3.490288628*^9, 3.490288646171875*^9}, { 3.490288689390625*^9, 3.490288748078125*^9}, {3.490289014984375*^9, 3.490289051890625*^9}, {3.49028908728125*^9, 3.490289146484375*^9}, { 3.490289684171875*^9, 3.490289704359375*^9}, {3.49055045875*^9, 3.490550463765625*^9}, 3.491935387046875*^9}, ParagraphSpacing->{0.5, 0.}] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Cardano\[CloseCurlyQuote]s method: reduction of cubic to depressed cubic\ \>", "Section", ShowGroupOpener->True, CellChangeTimes->{ 3.48969559478125*^9, {3.490213867078125*^9, 3.49021387953125*^9}, { 3.49021396778125*^9, 3.49021398075*^9}, {3.4902158315625*^9, 3.49021583553125*^9}, 3.490268524328125*^9}, CellID->104439264], Cell[TextData[{ "The following method for changing the form of a cubic was described by \ Girolamo Cardano in his book ", StyleBox["Ars Magna", FontSlant->"Italic"], ", 1545, but invented at the end of the 14th century by some unknown \ mathematician." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.4894385360625*^9, 3.489438667546875*^9}, { 3.489589259609375*^9, 3.489589280171875*^9}, {3.48959112059375*^9, 3.489591122671875*^9}, {3.489591169296875*^9, 3.489591171*^9}, { 3.489591680109375*^9, 3.48959168028125*^9}, {3.4902138940625*^9, 3.490213959171875*^9}}], Cell[TextData[{ StyleBox["Cardano\[CloseCurlyQuote]s method:", FontWeight->"Bold", FontSlant->"Italic"], " Make the linear substitution \n\t", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Rule]", RowBox[{"x", "-", RowBox[{ FractionBox["1", "3"], "a"}]}]}], TraditionalForm]]] }], "EmphasisText", CellChangeTimes->{{3.4889304876875*^9, 3.488930505625*^9}, { 3.48940770196875*^9, 3.48940771*^9}, {3.48958512459375*^9, 3.489585128296875*^9}, {3.48959113471875*^9, 3.489591146796875*^9}, { 3.489591833140625*^9, 3.489591840515625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{"cubic", "\[IndentingNewLine]", RowBox[{"depressed", "=", RowBox[{"cubic", "/.", " ", RowBox[{"x", "\[Rule]", RowBox[{"x", "-", RowBox[{ FractionBox["1", "3"], "a"}]}]}]}]}]}], "Input", CellChangeTimes->{{3.488930515796875*^9, 3.488930523796875*^9}, { 3.489328307609375*^9, 3.489328309125*^9}, 3.4894077135*^9, 3.489585132375*^9}, CellLabel->"In[3]:="], Cell[TextData[{ "Collect coefficients of the powers of ", StyleBox["x", FontFamily->"Courier", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ RowBox[{"Collect", "[", RowBox[{"depressed", ",", "x"}], "]"}]], "Input", CellChangeTimes->{3.48893053821875*^9}, CellLabel->"In[5]:="], Cell[TextData[{ "That cubic, which has no ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " term, is said to be \[OpenCurlyDoubleQuote]depressed\ \[CloseCurlyDoubleQuote]. Write it in the form:" }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489591235015625*^9, 3.48959124640625*^9}, { 3.48959134575*^9, 3.489591387625*^9}, {3.48959155784375*^9, 3.489591585953125*^9}}], Cell[TextData[{ StyleBox["Depressed cubic", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[":", FontSlant->"Italic"], "\t\t\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], TraditionalForm]]] }], "EmphasisText", CellChangeTimes->{ 3.48940428696875*^9, {3.4895912644375*^9, 3.489591286015625*^9}, { 3.489591824578125*^9, 3.48959185115625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "(With the coefficients of the depressed cubic written as multiples ", Cell[BoxData[ FormBox[ RowBox[{"3", "p"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"2", "q"}], TraditionalForm]]], ", subsequent formulas become simpler, as you will see.)" }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489589996515625*^9, 3.4895901431875*^9}, { 3.4895915278125*^9, 3.489591547671875*^9}, {3.489591885171875*^9, 3.489591940421875*^9}}], Cell[TextData[{ "Express the values of ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]]], " in the depressed cubic in terms of ", Cell[BoxData[ FormBox[ RowBox[{"a", ",", "b", ",", "c"}], TraditionalForm]]], " by comparing the corresponding coefficients:" }], "Text", CellChangeTimes->{{3.48893055775*^9, 3.48893061384375*^9}, { 3.48908429646875*^9, 3.48908429984375*^9}, {3.48932835709375*^9, 3.489328357765625*^9}, {3.4895851743125*^9, 3.4895852308125*^9}, { 3.4895891041875*^9, 3.4895891099375*^9}, {3.489589390296875*^9, 3.489589448765625*^9}, {3.489589556875*^9, 3.489589561421875*^9}, { 3.489589631890625*^9, 3.48958963515625*^9}, {3.4895898984375*^9, 3.48958989896875*^9}, {3.489589948234375*^9, 3.48958994934375*^9}, { 3.489591445546875*^9, 3.48959149884375*^9}, 3.48959195634375*^9}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[{ RowBox[{ RowBox[{"niceDepressed", "=", RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"CoefficientList", "[", RowBox[{"niceDepressed", ",", "x"}], "]"}], "\[IndentingNewLine]", RowBox[{"CoefficientList", "[", RowBox[{"depressed", ",", "x"}], "]"}]}], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.4895896505625*^9, 3.489589664875*^9}, { 3.490216143875*^9, 3.490216153234375*^9}}, CellLabel->"In[6]:="], Cell[TextData[{ "Thus\n\t", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"p", "=", RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox["a", "2"], "9"]}], "+", FractionBox["b", "3"]}]}], StripOnInput->False, FractionBoxOptions->{AllowScriptLevelChange->False}], TraditionalForm]]], ",\t\t", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"-", FractionBox[ SuperscriptBox["a", "3"], "27"]}], "-", FractionBox[ RowBox[{"a", " ", "b"}], "6"], "+", FractionBox["c", "2"]}], StripOnInput->False, FractionBoxOptions->{AllowScriptLevelChange->False}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.48958968015625*^9, 3.4895897490625*^9}, { 3.48958978034375*^9, 3.489589892609375*^9}, {3.49055074903125*^9, 3.490550809390625*^9}, {3.490550858640625*^9, 3.49055087690625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " By hand, calculate the depressed cubic obtained by Cardano\ \[CloseCurlyQuote]s method for the cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"6", SuperscriptBox["x", "2"]}], "-", RowBox[{"5", "x"}], "+", "11"}], "=", "0"}], TraditionalForm]]], ". Repeat for the cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"12", SuperscriptBox["x", "2"]}], "+", RowBox[{"2", "x"}], "-", "4"}], "=", "0"}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.489092331421875*^9}, {3.4894077291875*^9, 3.489407741390625*^9}, {3.489585246*^9, 3.489585250078125*^9}, 3.4895901590625*^9, {3.48959026365625*^9, 3.48959027596875*^9}, { 3.48959052796875*^9, 3.489590593328125*^9}, {3.489595099984375*^9, 3.48959511328125*^9}, {3.489682342953125*^9, 3.48968236465625*^9}, { 3.48968249271875*^9, 3.489682503078125*^9}, {3.48969955053125*^9, 3.489699554921875*^9}}, CellTags->"exercise"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Suppose you had used Cardano\[CloseCurlyQuote]s method to obtain the \ depressed cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"8", "x"}], "+", "5"}], TraditionalForm]]], " from a cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"a", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"b", " ", "x"}], "+", "c"}], TraditionalForm]]], ". If ", Cell[BoxData[ FormBox[ RowBox[{"a", "=", RowBox[{"-", "9"}]}], TraditionalForm]]], ", what was the original cubic?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.489699673609375*^9, 3.489699771265625*^9}, {3.489699846015625*^9, 3.48969986909375*^9}, { 3.489700029703125*^9, 3.48970006421875*^9}, {3.48970019109375*^9, 3.48970024284375*^9}, {3.4897003364375*^9, 3.489700342546875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Suppose you had used Cardano\[CloseCurlyQuote]s method to obtain a \ depressed cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], TraditionalForm]]], " from a cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"a", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"b", " ", "x"}], "+", "c"}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox[ RowBox[{"a", "=", RowBox[{"-", "9"}]}], TraditionalForm]]], ". One of the roots of the depressed cubic is ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "5"}], TraditionalForm]]], ". What is the corresponding root of the original cubic?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.489699673609375*^9, 3.489699771265625*^9}, {3.489699846015625*^9, 3.48969986909375*^9}, { 3.489700029703125*^9, 3.48970006421875*^9}, {3.48970019109375*^9, 3.48970024284375*^9}, 3.48970032409375*^9, {3.48970035665625*^9, 3.48970047196875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Suppose you wrote the depressed cubic in the form ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"\[Beta]", " ", "x"}], "+", "\[Gamma]"}], TraditionalForm]]], ", without the coefficient multipliers of 3 and 2. Express the coefficients ", Cell[BoxData[ FormBox["\[Beta]", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\[Gamma]", TraditionalForm]]], " in terms of the original cubic\[CloseCurlyQuote]s coefficients ", Cell[BoxData[ FormBox[ RowBox[{"a", ",", "b", ",", "c"}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.489092331421875*^9}, {3.4894077291875*^9, 3.489407741390625*^9}, {3.489585246*^9, 3.489585250078125*^9}, { 3.4895901590625*^9, 3.489590245984375*^9}, 3.489590525921875*^9, { 3.489590613671875*^9, 3.489590630984375*^9}, {3.4896995645625*^9, 3.489699574921875*^9}, 3.490281370328125*^9}, CellTags->"exercise"], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " The linear substitution used was ", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Rule]", RowBox[{"x", "-", RowBox[{ FractionBox["1", "3"], "a"}]}]}], TraditionalForm]]], ". Among all possible linear substitutions ", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Rule]", RowBox[{"x", "-", "cst"}]}], TraditionalForm]]], ", why use ", Cell[BoxData[ FormBox[ RowBox[{"cst", "=", RowBox[{ FractionBox["1", "3"], "a"}]}], TraditionalForm]]], "?" }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.489092331421875*^9}, {3.4894077291875*^9, 3.489407741390625*^9}, {3.489585246*^9, 3.489585250078125*^9}, 3.4895901590625*^9, {3.48959026365625*^9, 3.48959027596875*^9}, 3.489590514984375*^9, {3.4896995845625*^9, 3.489699595046875*^9}}, CellTags->"exercise"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ The del Ferro\[Dash]Tartaglia formula\ \>", "Section", ShowGroupOpener->True, CellChangeTimes->{ 3.48940428696875*^9, 3.48940776928125*^9, {3.4894400239375*^9, 3.489440026296875*^9}, {3.4897013893125*^9, 3.489701393109375*^9}, { 3.490214105953125*^9, 3.49021411315625*^9}, 3.490268524390625*^9, 3.490289168625*^9, {3.490301637359375*^9, 3.490301638359375*^9}}, CellID->10228179], Cell[TextData[{ "In 1515 Scipione del Ferro discovered a formula, which he kept secret, for \ finding a root of a depressed cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q", " "}]}], TraditionalForm]]], " in terms of ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]]], ". In 1530 Niccol\[OGrave] Fontana, aka \[OpenCurlyDoubleQuote]Tartaglia\ \[CloseCurlyDoubleQuote], revealed the same formula to Cardano." }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.488931150765625*^9, 3.48893116215625*^9}, { 3.489439852671875*^9, 3.4894398766875*^9}, {3.489439915859375*^9, 3.48943997259375*^9}, {3.489585258546875*^9, 3.489585263953125*^9}, { 3.48959033296875*^9, 3.489590339203125*^9}, {3.489590386703125*^9, 3.489590398*^9}, {3.4895909973125*^9, 3.489591026140625*^9}, { 3.490214032625*^9, 3.49021409596875*^9}, {3.490215497859375*^9, 3.490215574671875*^9}, {3.490550551640625*^9, 3.490550562984375*^9}}], Cell[TextData[{ "In ", StyleBox["Mathematica,", FontSlant->"Italic"], " the formula is given by the function definition:" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.488931150765625*^9, 3.48893116215625*^9}, { 3.489439852671875*^9, 3.48943985565625*^9}, {3.4895903610625*^9, 3.48959038134375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"delFerroTartagliaRoot", "[", RowBox[{"p_", ",", "q_"}], "]"}], ":=", "\[IndentingNewLine]", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "q"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], ")"}], RowBox[{"1", "/", "3"}]], "+", SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "q"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}]]}], ")"}], RowBox[{"1", "/", "3"}]]}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.488931169046875*^9, 3.488931222546875*^9}, { 3.4895852704375*^9, 3.48958530825*^9}, {3.48959031734375*^9, 3.489590325421875*^9}, {3.489590419140625*^9, 3.48959042525*^9}}, CellLabel->"In[9]:="], Cell["For example:", "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489408138953125*^9, 3.489408141546875*^9}}], Cell[BoxData[ RowBox[{"delFerroTartagliaRoot", "[", RowBox[{ RowBox[{"-", "5"}], ",", RowBox[{"-", "2"}]}], "]"}]], "Input", CellChangeTimes->{ 3.48893078675*^9, 3.4889312261875*^9, {3.48940817040625*^9, 3.48940818546875*^9}, {3.489590462359375*^9, 3.48959046584375*^9}}, CellLabel->"In[10]:="], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Use the del Ferro-Tartaglia formula by hand to calculate a root of the \ depressed cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"9", "x"}], "+", "8"}], TraditionalForm]]], ". (You may leave your answer in a form involving cube-roots.) Check your \ answer against the result of using the ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", Cell[BoxData[ FormBox[ StyleBox["delFerroTartagliaRoot", FontFamily->"Courier"], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.489092331421875*^9}, {3.4894077291875*^9, 3.489407741390625*^9}, {3.489585246*^9, 3.489585250078125*^9}, 3.4895901590625*^9, {3.48959026365625*^9, 3.48959027596875*^9}, { 3.48959048225*^9, 3.48959048578125*^9}, {3.48959064303125*^9, 3.489590709859375*^9}, {3.489592065890625*^9, 3.489592146390625*^9}, { 3.48970050709375*^9, 3.489700515671875*^9}, {3.489701456265625*^9, 3.489701527109375*^9}, {3.489701730765625*^9, 3.489701741265625*^9}, { 3.489702118171875*^9, 3.489702126453125*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Apply Cardano\[CloseCurlyQuote]s method and then the del Ferro-Tartaglia \ formula to find a root of the cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"15", SuperscriptBox["x", "2"]}], "+", RowBox[{"57", "x"}], "+", "27"}], TraditionalForm]]], ". (You may leave your answer in a form involving cube-roots.) " }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.489092331421875*^9}, {3.4894077291875*^9, 3.489407741390625*^9}, {3.489585246*^9, 3.489585250078125*^9}, 3.4895901590625*^9, {3.48959026365625*^9, 3.48959027596875*^9}, { 3.48959048225*^9, 3.48959048578125*^9}, {3.48959064303125*^9, 3.489590709859375*^9}, {3.489590930984375*^9, 3.48959097253125*^9}, { 3.489700528640625*^9, 3.489700542703125*^9}, 3.489701534609375*^9, { 3.489702545328125*^9, 3.489702567*^9}, {3.49021486784375*^9, 3.490214868859375*^9}, 3.49021506996875*^9, {3.4902156496875*^9, 3.49021567553125*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Suppose you wrote the depressed cubic in the form ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"\[Beta]", " ", "x"}], "+", "\[Gamma]"}], TraditionalForm]]], ", without the coefficient multipliers of 3 and 2. What now would the del \ Ferro-Tartaglia formula for a root be, in terms of ", Cell[BoxData[ FormBox["\[Beta]", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["\[Gamma]", TraditionalForm]]], "?\n(Rhetorical question: Do you see now why the del Ferro-Tartaglia formula \ is simpler when the multipliers are included in the coefficients of the \ depressed cubic?)" }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.489092331421875*^9}, {3.4894077291875*^9, 3.489407741390625*^9}, {3.489585246*^9, 3.489585250078125*^9}, 3.4895901590625*^9, {3.48959026365625*^9, 3.48959027596875*^9}, { 3.48959048225*^9, 3.48959048578125*^9}, {3.48959064303125*^9, 3.489590905875*^9}, {3.489592192828125*^9, 3.489592197265625*^9}, { 3.489700556734375*^9, 3.48970057134375*^9}, {3.48970154928125*^9, 3.4897015614375*^9}, {3.490301806515625*^9, 3.490301807671875*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "The del Ferro Tartaglia formula for solving a depressed cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], "=", "0"}], TraditionalForm]]], " can readily be converted into formulas for solving cubic equations of the \ form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}]}], "=", RowBox[{"2", "q"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"2", "q"}]}], "=", RowBox[{"3", "p", " ", "x"}]}], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "=", RowBox[{ RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}]}], TraditionalForm]]], ". We mention this because, in the work of del Ferro and Tartaglia, ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]]], " had to be positive numbers: like most European mathematicians of their \ time, they did not accept the notion of negative numbers." }], "SmallText", CellChangeTimes->{ 3.48940428696875*^9, {3.489440397828125*^9, 3.489440557390625*^9}, { 3.489440596671875*^9, 3.48944076084375*^9}, {3.489441259296875*^9, 3.489441338*^9}, {3.4895922629375*^9, 3.48959232203125*^9}}], Cell[TextData[{ "How did del Ferro and Tartaglia devise their formula? We don\ \[CloseCurlyQuote]t know, but here\[CloseCurlyQuote]s a possible way, which \ uses reasoning by analogy.\nAs they knew, as as you can readily check,\n\t", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{ SqrtBox[ RowBox[{"a", "+", SqrtBox["b"]}]], "+", SqrtBox[ RowBox[{"a", "-", SqrtBox["b"]}]]}]}], TraditionalForm]]], "\nis a solution of the quadratic equation\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "=", RowBox[{ RowBox[{"2", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "-", "b"}]]}], "+", RowBox[{"2", "a"}]}]}], TraditionalForm]]], ".\nwhen ", Cell[BoxData[ FormBox[ RowBox[{"a", ">", SqrtBox["b"]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"b", ">", "0"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.4894510655625*^9, 3.489451095140625*^9}, { 3.489525471015625*^9, 3.48952557265625*^9}, {3.489525604875*^9, 3.489525748453125*^9}, {3.489526008515625*^9, 3.489526069421875*^9}, { 3.489526112109375*^9, 3.489526125203125*^9}, {3.48952637015625*^9, 3.4895263786875*^9}, {3.489526671859375*^9, 3.489526925578125*^9}, { 3.489592349859375*^9, 3.489592351796875*^9}, {3.489592691375*^9, 3.48959272525*^9}, 3.48959278240625*^9, {3.489593488671875*^9, 3.489593489828125*^9}, {3.490215287484375*^9, 3.490215293640625*^9}, { 3.49055091303125*^9, 3.4905509165625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Verify the statement made above about the solution of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "=", RowBox[{ RowBox[{"2", SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "-", "b"}]]}], "+", RowBox[{"2", "a"}]}]}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48940428696875*^9, {3.48952695884375*^9, 3.48952698175*^9}, { 3.48959237734375*^9, 3.48959237821875*^9}, 3.489592789*^9, { 3.4895934935625*^9, 3.4895934949375*^9}, {3.4897005945625*^9, 3.489700603765625*^9}}], Cell[TextData[{ "Then perhaps, by analogy,\n\t", Cell[BoxData[ FormBox[ RowBox[{"x", "=", RowBox[{ RadicalBox[ RowBox[{"a", "+", SqrtBox["b"]}], "3"], "+", RadicalBox[ RowBox[{"a", "-", SqrtBox["b"]}], "3"]}]}], TraditionalForm]]], "\nis a solution of the cubic equation\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "=", RowBox[{ RowBox[{"3", RowBox[{"(", RadicalBox[ RowBox[{ SuperscriptBox["a", "2"], "-", "b"}], "3"], ")"}], "x"}], "+", RowBox[{"2", RowBox[{"a", "."}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.4894510655625*^9, 3.489451095140625*^9}, { 3.489525471015625*^9, 3.48952557265625*^9}, {3.489525604875*^9, 3.489525748453125*^9}, {3.489526008515625*^9, 3.489526069421875*^9}, { 3.489526112109375*^9, 3.489526125203125*^9}, {3.48952637015625*^9, 3.4895263786875*^9}, {3.489526671859375*^9, 3.489526925578125*^9}, { 3.489592349859375*^9, 3.489592351796875*^9}, {3.489592691375*^9, 3.48959272525*^9}, {3.48959278240625*^9, 3.4895928034375*^9}, { 3.489593504859375*^9, 3.489593506203125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "If in the latter, cubic, equation you take ", Cell[BoxData[ FormBox[ RowBox[{"p", "=", RowBox[{"-", RadicalBox[ RowBox[{ SuperscriptBox["a", "2"], "-", "b"}], "3"]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"q", "=", RowBox[{"-", "a"}]}], TraditionalForm]]], ", then ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}], "=", "b"}], TraditionalForm]]], ". Thus the guess for the solution of this cubic equation is indeed what the \ del Ferro-Tartaglia formula gives." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.4894510655625*^9, 3.489451095140625*^9}, { 3.489525471015625*^9, 3.48952557265625*^9}, {3.489525604875*^9, 3.489525748453125*^9}, {3.489526008515625*^9, 3.489526069421875*^9}, { 3.489526112109375*^9, 3.489526125203125*^9}, {3.48952637015625*^9, 3.4895263786875*^9}, {3.489526671859375*^9, 3.489526925578125*^9}, { 3.489592349859375*^9, 3.489592351796875*^9}, {3.489592691375*^9, 3.48959272525*^9}, {3.48959278240625*^9, 3.4895928034375*^9}, { 3.48959351415625*^9, 3.489593515640625*^9}, {3.4895942736875*^9, 3.489594319453125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Verify the statement made above about the solution of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "=", RowBox[{ RowBox[{"3", RowBox[{"(", RadicalBox[ RowBox[{ SuperscriptBox["a", "2"], "-", "b"}], "3"], ")"}], "x"}], "+", RowBox[{"2", "a"}]}]}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48940428696875*^9, {3.48952695884375*^9, 3.48952698175*^9}, { 3.48959237734375*^9, 3.48959237821875*^9}, 3.489592789*^9, { 3.48959345615625*^9, 3.48959347996875*^9}, {3.489593518875*^9, 3.489593526765625*^9}, {3.489594173390625*^9, 3.489594174515625*^9}, { 3.4897006130625*^9, 3.489700623859375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell["A paradox", "Section", CellChangeTimes->{ 3.48940428696875*^9, {3.489417011609375*^9, 3.489417016*^9}, 3.4902685245*^9}, CellID->62156397], Cell[CellGroupData[{ Cell[TextData[{ "Fact 1: a depressed cubic ", StyleBox["always", FontSlant->"Italic"], " has a ", StyleBox["real", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " solution." }], "Subsection", CellChangeTimes->{ 3.48893078675*^9, {3.489084833828125*^9, 3.489084891*^9}, { 3.48909298084375*^9, 3.4890930280625*^9}, {3.489093058203125*^9, 3.489093072578125*^9}, {3.4890932911875*^9, 3.489093305578125*^9}, 3.4894170259375*^9, 3.490268524890625*^9, {3.490289508140625*^9, 3.490289516734375*^9}}, ParagraphSpacing->{0.5, 0}, CellID->238688122], Cell[TextData[{ "In fact, \n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], "=", "0"}], TraditionalForm]]], "\nis equivalent to\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "=", RowBox[{ RowBox[{ RowBox[{"-", "3"}], "p", " ", "x"}], "-", RowBox[{"2", "q"}]}]}], TraditionalForm]]], ",\nand the graph of the cube function \n\t", Cell[BoxData[ FormBox[ RowBox[{"y", "=", SuperscriptBox["x", "3"]}], TraditionalForm]]], " \n", StyleBox["always", FontSlant->"Italic"], " intersects the line \n\t", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{ RowBox[{ RowBox[{"-", "3"}], "p", " ", "x"}], "-", RowBox[{"2", "q"}]}]}], TraditionalForm]]], " \nin at least one point, no matter what ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]]], " are!" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489084833828125*^9, 3.489084891*^9}, { 3.48909298084375*^9, 3.4890930280625*^9}, {3.489093058203125*^9, 3.489093072578125*^9}, {3.4890932911875*^9, 3.489093310453125*^9}, 3.4894170259375*^9}, ParagraphSpacing->{0.5, 0}], Cell["The following plots provide evidence to support Fact 1.", "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489093096234375*^9, 3.489093111984375*^9}, { 3.48909321240625*^9, 3.489093238015625*^9}, 3.4894170259375*^9}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox["x", "3"], ",", RowBox[{ RowBox[{"15", "x"}], "+", "4"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "5"}], ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{ 3.488930170796875*^9, 3.4894170259375*^9, {3.490550926953125*^9, 3.49055095375*^9}, {3.49055103296875*^9, 3.49055103959375*^9}}, CellLabel->"In[17]:="], Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox["x", "3"], ",", RowBox[{ RowBox[{ RowBox[{"-", "3"}], "p", " ", "x"}], "-", RowBox[{"2", "q"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "8"}], ",", "8"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "500"}], ",", "500"}], "}"}]}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Red", ",", "Blue"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}]}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"p", ",", RowBox[{"-", "20"}], ",", "20", ",", RowBox[{"Appearance", "\[Rule]", "\"\\""}]}], "}"}], ",", RowBox[{"{", RowBox[{"q", ",", RowBox[{"-", "20"}], ",", "20", ",", RowBox[{"Appearance", "\[Rule]", "\"\\""}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{ 3.488930170796875*^9, {3.48908492490625*^9, 3.489085031015625*^9}, { 3.489085384625*^9, 3.48908545640625*^9}, {3.489085511890625*^9, 3.489085520109375*^9}, {3.489085560828125*^9, 3.489085613703125*^9}, { 3.4890905588125*^9, 3.48909056675*^9}, {3.48909063628125*^9, 3.489090660890625*^9}, {3.48909069728125*^9, 3.48909078859375*^9}, { 3.48909084315625*^9, 3.489090897328125*^9}, {3.4890909376875*^9, 3.489090993296875*^9}, {3.48909102465625*^9, 3.48909105065625*^9}, { 3.489091081328125*^9, 3.48909108584375*^9}, {3.48909115759375*^9, 3.48909120196875*^9}, {3.4890912326875*^9, 3.489091258296875*^9}, { 3.48909129965625*^9, 3.489091405921875*^9}, {3.48909151428125*^9, 3.48909151890625*^9}, {3.489091558890625*^9, 3.489091561484375*^9}, { 3.48909166525*^9, 3.489091675296875*^9}, {3.48909172496875*^9, 3.489091742265625*^9}, {3.48909179003125*^9, 3.489091812078125*^9}, { 3.48909191353125*^9, 3.48909198053125*^9}, {3.48909251159375*^9, 3.4890925415*^9}, {3.48909263734375*^9, 3.489092658515625*^9}, 3.4894170259375*^9}, CellLabel->"In[12]:="], Cell["\<\ Move the sliders for p and q in the output above to see where the two curves \ meet.\ \>", "SmallText", CellChangeTimes->{ 3.48893078675*^9, {3.489092582625*^9, 3.48909262309375*^9}, 3.4890926756875*^9, 3.4894170259375*^9}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Explain in detail why, in fact, the curve ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", SuperscriptBox["x", "3"]}], TraditionalForm]]], " must actually intersect the line ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{ RowBox[{ RowBox[{"-", "3"}], "p", " ", "x"}], "-", RowBox[{"2", "q"}]}]}], TraditionalForm]]], " in at least one point ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], ", no matter what the values of ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["q", TraditionalForm]]], " are." }], "Exercise", CellChangeTimes->{{3.488930637625*^9, 3.488930671546875*^9}, 3.48893113778125*^9, {3.489084235*^9, 3.489084242203125*^9}, { 3.48909233084375*^9, 3.48909242915625*^9}, 3.489092775359375*^9, { 3.489093092703125*^9, 3.489093120328125*^9}, {3.489328464875*^9, 3.489328467125*^9}, {3.48940804290625*^9, 3.489408060234375*^9}, 3.4894170259375*^9, {3.4897006606875*^9, 3.489700669171875*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Fact 2: del Ferro-Tartaglia formula involves ", StyleBox["square-roots of negative numbers", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], " if ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["q", "2"], "<", RowBox[{"-", SuperscriptBox["p", "3"]}]}], TraditionalForm]]], "." }], "Subsection", Editable->True, Selectable->True, Deletable->True, CellOpen->True, CellChangeTimes->{ 3.48893078675*^9, {3.489092125015625*^9, 3.48909212546875*^9}, 3.489092168125*^9, {3.48909228634375*^9, 3.489092288890625*^9}, { 3.489093123640625*^9, 3.489093137453125*^9}, {3.4890933443125*^9, 3.489093352*^9}, {3.48909339784375*^9, 3.489093406203125*^9}, 3.4894170769375*^9, 3.4902685249375*^9, 3.4902889665*^9, { 3.490289199953125*^9, 3.49028920009375*^9}, {3.490289520359375*^9, 3.4902895276875*^9}}, ParagraphSpacing->{0.5, 0}, CellID->195307992], Cell["In fact, recall the formula is the value:", "Text", Editable->True, Selectable->True, Deletable->True, CellOpen->True, CellChangeTimes->{ 3.48893078675*^9, {3.489093422578125*^9, 3.48909345075*^9}, { 3.4890935139375*^9, 3.489093530515625*^9}, {3.48909357046875*^9, 3.48909358178125*^9}, {3.4894170769375*^9, 3.4894171039375*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[BoxData[ RowBox[{"delFerroTartagliaRoot", "[", RowBox[{"p", ",", "q"}], "]"}]], "Input", Editable->True, Selectable->True, Deletable->True, CellOpen->True, CellChangeTimes->{3.48893078675*^9, 3.489093566515625*^9, 3.4894170769375*^9, 3.49021573003125*^9}, CellLabel->"In[13]:="], Cell[TextData[{ "As Cardano noticed, the square-root here may be that of a negative number. \ This is the case, for example, when the depressed cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], TraditionalForm]]], " has ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"p", "=", RowBox[{"-", "5"}]}], ",", " ", RowBox[{"q", "=", RowBox[{"-", "2"}]}]}], TraditionalForm]]], ":" }], "Text", Editable->True, Selectable->True, Deletable->True, CellOpen->True, CellChangeTimes->{ 3.48940428696875*^9, {3.489417116703125*^9, 3.489417123171875*^9}, { 3.489417153484375*^9, 3.489417184375*^9}, {3.490215890453125*^9, 3.490215923734375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}], "/.", RowBox[{"{", " ", RowBox[{ RowBox[{"p", "\[Rule]", RowBox[{"-", "5"}]}], ",", RowBox[{"q", "\[Rule]", RowBox[{"-", "2"}]}]}], "}"}]}]], "Input", Editable->True, Selectable->True, Deletable->True, CellOpen->True, CellChangeTimes->{ 3.48893078675*^9, {3.489092878421875*^9, 3.489092899125*^9}, { 3.489267150453125*^9, 3.48926716090625*^9}, 3.489417147703125*^9}, CellLabel->"In[14]:="] }, Closed]], Cell[CellGroupData[{ Cell["Resolving the paradox", "Subsection", CellChangeTimes->{ 3.48969559478125*^9, {3.490289385375*^9, 3.490289390140625*^9}}, CellID->58437946], Cell["\<\ Bombelli\[CloseCurlyQuote]s method, explained next, resolves the paradox: it \ changes the form of the root provided by the del Ferro-Tartaglia formula so \ as to see it is actually real.\ \>", "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490289241953125*^9, 3.490289360578125*^9}, { 3.490289393921875*^9, 3.490289428109375*^9}, 3.491935387140625*^9}, CellID->148481078] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Bombelli\[CloseCurlyQuote]s method", "Section", ShowGroupOpener->True, CellChangeTimes->{ 3.48969559478125*^9, {3.490215783875*^9, 3.49021579640625*^9}, { 3.490215960921875*^9, 3.490215992359375*^9}, 3.49026852509375*^9, { 3.490276109609375*^9, 3.490276116203125*^9}, {3.490288991140625*^9, 3.490288994953125*^9}, 3.490289187140625*^9, 3.49193538715625*^9}, CellID->54838321], Cell[TextData[{ "In his book ", StyleBox["L\[CloseCurlyQuote]algebra,", FontSlant->"Italic"], " 1572, Rafael ", "Bombelli", " looked at the depressed cubic ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", "p", " ", "x"}], "+", RowBox[{"2", "q"}]}], TraditionalForm]]], " when ", Cell[BoxData[ FormBox[ RowBox[{"p", "=", RowBox[{"-", "5"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"q", "=", RowBox[{"-", "2"}]}], TraditionalForm]]], ": " }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489417231453125*^9, 3.489417287921875*^9}, { 3.490215758234375*^9, 3.490215780171875*^9}, {3.4902160126875*^9, 3.490216015546875*^9}, {3.490288409890625*^9, 3.490288410296875*^9}, 3.491935387171875*^9}], Cell[TextData[Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"15", "x"}], "-", "4"}], "=", "0"}], TraditionalForm]]]]], TraditionalForm]]]], "EmphasisText", CellChangeTimes->{ 3.48893078675*^9, {3.489093650125*^9, 3.489093666859375*^9}, { 3.490275999984375*^9, 3.4902760658125*^9}}], Cell[BoxData[ RowBox[{"example", "=", RowBox[{"niceDepressed", "/.", RowBox[{"{", RowBox[{ RowBox[{"p", "\[Rule]", RowBox[{"-", "5"}]}], ",", RowBox[{"q", "\[Rule]", RowBox[{"-", "2"}]}]}], "}"}]}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.4893285335*^9, 3.489328571296875*^9}, { 3.489328619203125*^9, 3.4893286204375*^9}, {3.49021603646875*^9, 3.4902160379375*^9}, {3.490216182375*^9, 3.490216195375*^9}}, CellLabel->"In[15]:="], Cell[TextData[{ "Of course, there ", StyleBox["must", FontSlant->"Italic"], " be at least one solution\[LongDash]the cubic ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", SuperscriptBox["x", "3"]}], TraditionalForm]]], " and the line ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{ RowBox[{"15", " ", "x"}], "+", "4"}]}], TraditionalForm]]], " must intersect:" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.48909246753125*^9, 3.4890924815625*^9}, { 3.48909272334375*^9, 3.48909273478125*^9}, {3.4893284971875*^9, 3.489328503125*^9}, {3.4894173110625*^9, 3.48941731834375*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ SuperscriptBox["x", "3"], ",", RowBox[{ RowBox[{"15", " ", "x"}], "+", "4"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "7"}], ",", "7"}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Red", ",", "Blue"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{3.488930171*^9}, CellLabel->"In[16]:="], Cell[TextData[{ "In fact, the plot suggests that ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "4"}], TraditionalForm]]], " is a solution. Verify that it really is:" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489092748640625*^9, 3.489092749828125*^9}, { 3.48932851034375*^9, 3.489328522234375*^9}, {3.489418174765625*^9, 3.48941819325*^9}}], Cell[BoxData[ RowBox[{"example", "/.", RowBox[{"x", "\[Rule]", "4"}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.489092753234375*^9, 3.489092756140625*^9}, { 3.4893286263125*^9, 3.48932862721875*^9}}, CellLabel->"In[17]:="], Cell["\<\ That graphical approach is not what Bombelli used. Instead, he used an \ algebraic approach.\ \>", "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.4902815458125*^9, 3.490281637546875*^9}, 3.49030182375*^9, 3.491935387203125*^9}, CellID->204387247], Cell[CellGroupData[{ Cell["Bombelli's \"wild thought\"", "Subsection", ShowGroupOpener->True, CellChangeTimes->{3.48969559478125*^9, 3.490268525140625*^9, 3.49193538721875*^9}, ParagraphSpacing->{0.5, 0}, CellID->198090096], Cell["The del Ferro-Tartaglia formula for a root is:", "Text", CellChangeTimes->{ 3.48893078675*^9, {3.4890927961875*^9, 3.48909281246875*^9}, { 3.489092867796875*^9, 3.489092870609375*^9}, {3.4890929173125*^9, 3.48909291846875*^9}, {3.489093969578125*^9, 3.489093970171875*^9}, { 3.489417387171875*^9, 3.489417395203125*^9}}], Cell[BoxData[ RowBox[{"delFerroTartagliaRoot", "[", RowBox[{"p", ",", "q"}], "]"}]], "Input", CellChangeTimes->{ 3.48893078675*^9, 3.489092792375*^9, {3.490274955640625*^9, 3.490274956359375*^9}}, CellLabel->"In[18]:="], Cell["\<\ And in the example, the quantity under the square-root signs is:\ \>", "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489092851375*^9, 3.48909285575*^9}, { 3.4890940379375*^9, 3.489094043171875*^9}, {3.4894174090625*^9, 3.489417443828125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[BoxData[ RowBox[{ RowBox[{ SuperscriptBox["p", "3"], "+", SuperscriptBox["q", "2"]}], "/.", RowBox[{"{", " ", RowBox[{ RowBox[{"p", "\[Rule]", RowBox[{"-", "5"}]}], ",", RowBox[{"q", "\[Rule]", RowBox[{"-", "2"}]}]}], "}"}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.489092878421875*^9, 3.489092899125*^9}, { 3.489267150453125*^9, 3.48926716090625*^9}}, CellLabel->"In[19]:="], Cell[TextData[{ "So the del Ferro-Tartaglia solution in this example is:\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RadicalBox[ RowBox[{"2", "+", FormBox[ SqrtBox[ RowBox[{"-", "121"}]], TraditionalForm]}], "3"], "+", RadicalBox[ RowBox[{"2", "-", FormBox[ SqrtBox[ RowBox[{"-", "121"}]], TraditionalForm]}], "3"]}], "=", RowBox[{ RadicalBox[ RowBox[{"2", "+", RowBox[{"11", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}], "3"], "+", RadicalBox[ RowBox[{"2", "-", RowBox[{"11", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}], "3"]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.489092910296875*^9, 3.4890929145*^9}, { 3.48909430040625*^9, 3.489094378390625*^9}, {3.48941745525*^9, 3.489417531953125*^9}, 3.489417585390625*^9}, ParagraphSpacing->{1., 0}], Cell[TextData[{ "Bombelli", "\[CloseCurlyQuote]s \[OpenCurlyDoubleQuote]wild thought\ \[CloseCurlyDoubleQuote] was the following. ", StyleBox["Assume", FontSlant->"Italic"], " there are numbers ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " with: " }], "Text", CellChangeTimes->{{3.490226596953125*^9, 3.49022667971875*^9}, { 3.49022689825*^9, 3.490226995015625*^9}, {3.490271889609375*^9, 3.490271894953125*^9}, 3.49027500425*^9, {3.490275046109375*^9, 3.4902750664375*^9}, 3.49193538725*^9}, CellID->927876621], Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RadicalBox[ RowBox[{"2", " ", "+", " ", RowBox[{"11", " ", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}], "3"], "=", " ", RowBox[{"m", " ", "+", " ", RowBox[{"n", " ", SqrtBox[ RowBox[{"-", "1"}]]}]}]}], ",", " ", RowBox[{ RadicalBox[ RowBox[{"2", " ", "-", " ", RowBox[{"11", " ", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}], "3"], " ", "=", " ", RowBox[{"m", " ", "-", " ", RowBox[{"n", SqrtBox[ RowBox[{"-", "1"}]]}]}]}]}], TraditionalForm]]], ".\t(*)" }], "EmphasisText", CellFrame->True, CellChangeTimes->{ 3.48940428696875*^9, 3.489417790421875*^9, {3.490275080984375*^9, 3.490275083859375*^9}, {3.49027518296875*^9, 3.490275191546875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "Then the sum ", Cell[BoxData[ FormBox[ RowBox[{ RadicalBox[ RowBox[{"2", "+", RowBox[{"11", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}], "3"], "+", RadicalBox[ RowBox[{"2", "-", RowBox[{"11", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}], "3"]}], TraditionalForm]]], "would equal:" }], "Text", CellChangeTimes->{{3.490226695109375*^9, 3.49022678171875*^9}, { 3.49022700846875*^9, 3.490227008828125*^9}, 3.490275111203125*^9, { 3.490276138640625*^9, 3.490276161671875*^9}, {3.490551164171875*^9, 3.49055116478125*^9}}, CellID->135694709], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"m", "+", RowBox[{"n", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "+", RowBox[{"(", RowBox[{"m", "-", RowBox[{"n", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.489094440765625*^9, 3.48909444646875*^9}, 3.490226539375*^9, 3.490275111203125*^9}, ParagraphSpacing->{0.5, 0}, CellLabel->"In[20]:=", CellID->512193855], Cell[TextData[{ "Thus to obtain a (real) solution of the depressed cubic, ", "Bombelli", " just needs to solve (*) for ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ". And that amounts to solving:" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.4890944880625*^9, 3.489094542296875*^9}, { 3.48941832190625*^9, 3.48941832665625*^9}, {3.48941860221875*^9, 3.489418606265625*^9}, 3.4894186418125*^9, 3.4895774535625*^9, 3.489577513765625*^9, 3.490226539375*^9, {3.490226833625*^9, 3.49022689071875*^9}, {3.4902270470625*^9, 3.49022706134375*^9}, { 3.490274227578125*^9, 3.49027424115625*^9}, 3.490275111203125*^9, 3.49193538728125*^9}, ParagraphSpacing->{0.5, 0}, CellID->231773451], Cell[TextData[{ " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", " ", RowBox[{"m", "+", RowBox[{"n", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "3"], "=", RowBox[{"2", "+", RowBox[{"11", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}]}], ",", " ", RowBox[{ SuperscriptBox[ RowBox[{"(", " ", RowBox[{"m", "-", RowBox[{"n", " ", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "3"], "=", RowBox[{"2", "-", RowBox[{"11", " ", FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]}]}]}]}], TraditionalForm]]], "\t\t(**)" }], "EmphasisText", CellChangeTimes->{{3.490274260671875*^9, 3.490274285296875*^9}, 3.4902751311875*^9, {3.49027517078125*^9, 3.490275197515625*^9}}, CellID->508222106], Cell[TextData[{ "How? Use the binomial formula to expand ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", " ", RowBox[{"m", "+", RowBox[{"n", " ", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "3"], TraditionalForm]]], ", ", StyleBox["assuming", FontSlant->"Italic"], ": \n\t\[FilledSmallCircle]", StyleBox[" ", FontWeight->"Bold"], "the usual rules of algebra hold for expressions involving \n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[ImaginaryI]", "=", SqrtBox[ RowBox[{"-", "1"}]]}], TraditionalForm]]], "\n\t\[FilledSmallCircle]", StyleBox[" ", FontWeight->"Bold"], "the special rule \n\t\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ImaginaryI]", "2"], "=", RowBox[{ SuperscriptBox[ RowBox[{"(", SqrtBox[ RowBox[{"-", "1"}]], ")"}], "2"], "=", RowBox[{"-", "1"}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.4894179686875*^9, 3.48941806878125*^9}, 3.490226539375*^9, {3.490227068625*^9, 3.4902270839375*^9}, { 3.490227461375*^9, 3.490227659265625*^9}, {3.490272112796875*^9, 3.490272405296875*^9}, {3.4902736033125*^9, 3.490273606859375*^9}, { 3.4902742895*^9, 3.490274293171875*^9}, {3.490274425859375*^9, 3.49027453478125*^9}, {3.4902752748125*^9, 3.490275329375*^9}, 3.490301826265625*^9, {3.490551198046875*^9, 3.490551198796875*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->148126141], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Use the binomial formula for cubes, ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "3"], "=", RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", SuperscriptBox["x", "2"], "y"}], "+", RowBox[{"3", "x", " ", SuperscriptBox["y", "2"]}], "+", SuperscriptBox["y", "3"]}]}], TraditionalForm]]], ", along with the rules about ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]]], " assumed above, in order to express ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", " ", RowBox[{"m", "+", RowBox[{"n", " ", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "3"], TraditionalForm]]], " in the form ", Cell[BoxData[ FormBox[ RowBox[{"u", "+", RowBox[{"v", SqrtBox[ RowBox[{"-", "1"}]]}]}], TraditionalForm]]], " for real ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], ". Of course ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " each be an expression in terms of ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490275461078125*^9, 3.490275487140625*^9}, {3.490551217734375*^9, 3.490551228671875*^9}}, CellID->62450556], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " already knows the binomial formula as well as the rules assumed for \ algebraic expressions involving the \[OpenCurlyDoubleQuote]imaginary\ \[CloseCurlyDoubleQuote] number ", Cell[BoxData[ FormBox[ RowBox[{"\[ImaginaryI]", "=", SqrtBox[ RowBox[{"-", "1"}]]}], TraditionalForm]]], ":" }], "Text", CellChangeTimes->{{3.49022770709375*^9, 3.49022777953125*^9}, { 3.4902726470625*^9, 3.490272647953125*^9}, {3.49027268746875*^9, 3.490272711265625*^9}, 3.490275564890625*^9, {3.490282664546875*^9, 3.49028267971875*^9}, {3.49055123425*^9, 3.4905512358125*^9}}, CellID->101897756], Cell[BoxData[ RowBox[{"theCube", "=", RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"m", "+", RowBox[{"n", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "3"], "]"}]}]], "Input", CellChangeTimes->{ 3.48893078675*^9, {3.489094583578125*^9, 3.489094586484375*^9}, { 3.489418078015625*^9, 3.48941807965625*^9}, 3.490226539375*^9, 3.49022708796875*^9, 3.490275564890625*^9}, ParagraphSpacing->{0.5, 0}, CellLabel->"In[21]:=", CellID->48744119], Cell[TextData[{ "Separate the \"real part\" from the \[OpenCurlyDoubleQuote]imaginary part\ \[CloseCurlyDoubleQuote] that multiplies ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]]], ":" }], "Text", CellChangeTimes->{ 3.48893078675*^9, {3.4890945958125*^9, 3.4890946035625*^9}, 3.48941809078125*^9, {3.48941878003125*^9, 3.489418832234375*^9}, 3.490226539375*^9, 3.490275564890625*^9}, ParagraphSpacing->{0.5, 0}, CellID->530507026], Cell[BoxData[ RowBox[{"ComplexExpand", "[", "theCube", "]"}]], "Input", CellChangeTimes->{3.490226539375*^9, 3.490275564890625*^9}, ParagraphSpacing->{0.5, 0}, CellLabel->"In[22]:=", CellID->560356888], Cell[TextData[{ "See the ", ButtonBox["Appendix", BaseStyle->"Hyperlink", ButtonData->"appendix1"], " for a discussion of ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "SmallText", CellChangeTimes->{{3.49022710071875*^9, 3.49022712015625*^9}, 3.49027417409375*^9, {3.490274570484375*^9, 3.4902745705*^9}, { 3.490275611921875*^9, 3.490275626015625*^9}, {3.490281931109375*^9, 3.49028195134375*^9}, {3.49055125425*^9, 3.490551271921875*^9}}, CellID->416894146], Cell[TextData[{ "Thus the desired ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " are to satisfy the complex equation:\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["m", "3"], "-", RowBox[{"3", " ", "m", " ", SuperscriptBox["n", "2"]}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", SuperscriptBox["m", "2"], " ", "n"}], "-", SuperscriptBox["n", "3"]}], ")"}]}]}], "=", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.49028254359375*^9, 3.49028259175*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->544007337], Cell["\<\ That left-hand side\[CloseCurlyQuote]s real and imaginary parts are\[Ellipsis]\ \>", "Text", CellChangeTimes->{{3.4902279481875*^9, 3.490227978765625*^9}, { 3.49027460459375*^9, 3.49027460534375*^9}, 3.490275626015625*^9, { 3.49028262390625*^9, 3.49028262915625*^9}}, CellID->853432109], Cell[BoxData[ RowBox[{"cubeParts", "=", RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "theCube", "]"}], ",", RowBox[{"Im", "[", "theCube", "]"}]}], "}"}], "]"}]}]], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.489418954984375*^9, 3.489418972265625*^9}, { 3.48941911184375*^9, 3.48941911215625*^9}, 3.48941914740625*^9, { 3.489419190078125*^9, 3.489419200484375*^9}, 3.490226539375*^9, { 3.490227169296875*^9, 3.4902271715625*^9}, 3.490227988515625*^9, 3.490275626015625*^9, 3.49028385840625*^9}, CellLabel->"In[23]:=", CellID->110691365], Cell["\<\ \[Ellipsis]and according to the first equation of (**), those parts should \ equal\ \>", "Text", CellChangeTimes->{{3.490227992125*^9, 3.49022802090625*^9}, { 3.49027460875*^9, 3.49027462040625*^9}, 3.490275626015625*^9}, CellID->56582438], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}]}], "}"}]], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.489418991390625*^9, 3.489419005671875*^9}, 3.490226539375*^9, {3.490228032203125*^9, 3.4902280611875*^9}, 3.490272746140625*^9, 3.490275626015625*^9}, CellLabel->"In[24]:=", CellID->331101330], Cell["Thus solving the first equation of (**) amounts to solving:", "Text", CellChangeTimes->{{3.490227850296875*^9, 3.49022788615625*^9}, 3.4902280456875*^9, {3.490272779796875*^9, 3.490272780359375*^9}, { 3.49027464896875*^9, 3.490274656453125*^9}, 3.4902756835*^9}, CellID->563007215], Cell[BoxData[ RowBox[{"equations", "=", "\[IndentingNewLine]", RowBox[{"cubeParts", "\[Equal]", RowBox[{"{", RowBox[{"2", ",", "11"}], "}"}]}]}]], "Input", CellChangeTimes->{{3.49022807140625*^9, 3.490228095046875*^9}, { 3.49022828134375*^9, 3.490228282890625*^9}, 3.490273029359375*^9, 3.4902756835*^9}, CellLabel->"In[25]:=", CellID->195723882], Cell["Write that as two separate scalar equations:", "Text", CellChangeTimes->{{3.490228113390625*^9, 3.490228122171875*^9}, 3.49022830440625*^9, {3.490272792578125*^9, 3.4902727929375*^9}, { 3.4902730418125*^9, 3.49027304284375*^9}, {3.490273529515625*^9, 3.490273534296875*^9}, {3.490274676640625*^9, 3.490274689953125*^9}, 3.4902756835*^9, {3.490281440796875*^9, 3.49028144165625*^9}}, CellID->95768003], Cell[BoxData[ RowBox[{"equations", "=", RowBox[{"Thread", "[", "equations", "]"}]}]], "Input", CellChangeTimes->{{3.490228291*^9, 3.490228294359375*^9}, {3.4902283895*^9, 3.4902283909375*^9}, 3.4902756835*^9}, CellLabel->"In[26]:=", CellID->51416592], Cell["Factor the left-hand sides:", "Text", CellChangeTimes->{{3.490228175078125*^9, 3.490228192734375*^9}, 3.4902756835*^9}, CellID->352294015], Cell[BoxData[ RowBox[{"equations", "=", RowBox[{"Factor", "[", "equations", "]"}]}]], "Input", CellChangeTimes->{{3.490228348015625*^9, 3.4902283549375*^9}, { 3.4902283959375*^9, 3.490228403296875*^9}, 3.4902756835*^9}, CellLabel->"In[27]:=", CellID->606236280], Cell[TextData[{ "Now ", "Bombelli", " seeks solutions ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " of the equations that are positive ", StyleBox["integers", FontSlant->"Italic"], ". The process is indicated in the following exercise." }], "Text", CellChangeTimes->{{3.4902282326875*^9, 3.4902282660625*^9}, { 3.49022842371875*^9, 3.490228681609375*^9}, 3.49027284809375*^9, { 3.49027365234375*^9, 3.490273675046875*^9}, {3.490273999*^9, 3.490274000671875*^9}, 3.4902756835*^9, {3.490301994734375*^9, 3.49030201778125*^9}, {3.49055134028125*^9, 3.490551355703125*^9}, 3.491935387328125*^9}, ParagraphSpacing->{0.5, 0.}, CellID->213028755], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " By hand, and without merely guessing, find the integers ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " that are solutions of the pair of equations\n\t", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"{", GridBox[{ { RowBox[{"m", RowBox[{"(", RowBox[{ SuperscriptBox["m", "2"], "-", RowBox[{"3", SuperscriptBox["n", "2"]}]}], ")"}]}], "=", RowBox[{"2", ","}]}, { RowBox[{"n", RowBox[{"(", RowBox[{ RowBox[{"3", SuperscriptBox["m", "2"]}], "-", SuperscriptBox["n", "2"]}]}]}], "=", "11."} }]}]}], TraditionalForm]]], "\n(", StyleBox["Hint.", FontSlant->"Italic"], " The only positive integer factors of 2 are 1 and 2. From the first \ equation, this means that either ", Cell[BoxData[ FormBox[ RowBox[{"m", "=", "1"}], TraditionalForm]]], " or else ", Cell[BoxData[ FormBox[ RowBox[{"m", "=", "2"}], TraditionalForm]]], ". Show why the case ", Cell[BoxData[ FormBox[ RowBox[{"m", "=", "1"}], TraditionalForm]]], " is impossible, so that ", Cell[BoxData[ FormBox[ RowBox[{"m", "=", "2"}], TraditionalForm]]], ". Then use the second equation to determine ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ". Be sure to verify that your ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " actually satisfy both equations.)" }], "Exercise", CellChangeTimes->{{3.49027369715625*^9, 3.490273915390625*^9}, { 3.490273963578125*^9, 3.49027414046875*^9}, {3.490275751484375*^9, 3.490275768875*^9}}, ParagraphSpacing->{0.5, 0.}, SpanMaxSize->Infinity, CellID->332462960], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can find integer solutions of the equations directly:" }], "Text", CellChangeTimes->{{3.490273551703125*^9, 3.490273582859375*^9}, 3.4902756835*^9}, CellID->16106302], Cell[BoxData[ RowBox[{"Reduce", "[", RowBox[{"equations", ",", RowBox[{"{", RowBox[{"m", ",", "n"}], "}"}], ",", "Integers"}], "]"}]], "Input", CellChangeTimes->{{3.49027308159375*^9, 3.490273132609375*^9}, { 3.490273170796875*^9, 3.490273178515625*^9}, {3.490273241609375*^9, 3.4902733098125*^9}, 3.4902756835*^9}, CellLabel->"In[28]:=", CellID->4500715], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Check that the solution found for the first equation in (**) also \ satisfies the second equation." }], "Exercise", CellChangeTimes->{{3.49027470615625*^9, 3.490274739484375*^9}, { 3.49027582940625*^9, 3.490275871359375*^9}}, CellID->542723926], Cell[TextData[{ "Thus ", "Bombelli", "\[CloseCurlyQuote]s solution of the depressed cubic ", Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"15", "x"}], "-", "4"}], "=", "0"}], TraditionalForm]]]]], TraditionalForm]]], " of his example is ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", "4"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490275968109375*^9, 3.49027597528125*^9}, { 3.49027637278125*^9, 3.490276420734375*^9}, {3.490301572703125*^9, 3.490301576828125*^9}, 3.49193538734375*^9}, CellID->248000342], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Go through ", "Bombelli", "\[CloseCurlyQuote]s process of solution for his example, but start with the \ second equation in (**) instead of the first." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49027646921875*^9, 3.490276571703125*^9}, {3.49030159259375*^9, 3.490301595609375*^9}, 3.491935387375*^9}, CellID->227269981], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find ", StyleBox["all", FontSlant->"Italic"], " solutions of ", StyleBox["equations", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " in terms of ", StyleBox["m", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["n", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Use those to find the corresponding values of ", Cell[BoxData[ FormBox[ RowBox[{"2", "m"}], TraditionalForm]]], ". Which of those values are (not necessarily real) solutions of the \ depressed cubic equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"15", "x"}], "-", "4"}], "=", "0"}], TraditionalForm]]], " and which are not?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49028089021875*^9, 3.490280920734375*^9}, {3.4902809949375*^9, 3.490281071296875*^9}, { 3.490281250625*^9, 3.49028130128125*^9}, 3.490281467078125*^9, 3.490281743125*^9}, CellID->273140987], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find a real root of the given depressed cubic equation by applying ", "Bombelli", "\[CloseCurlyQuote]s method to the result from the del Ferro-Tartaglia \ formula:\n\t(a) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"6", "x"}], "+", "4"}], "=", "0"}], TraditionalForm]]], ".\n\t(b) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"102", "x"}], "+", "20"}], "=", "0"}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490300578375*^9, 3.4903007093125*^9}, {3.49030084790625*^9, 3.490300873875*^9}, 3.490300914984375*^9, {3.490301166765625*^9, 3.49030117559375*^9}, 3.491935387390625*^9}, ParagraphSpacing->{0.5, 0.}, CellID->99294381], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find a real root of the given cubic equation by first applying Cardano\ \[CloseCurlyQuote]s method to obtain a depressed cubic and then proceeding as \ in the preceding exercise. Be sure your final answer is a root of the given \ cubic rather than a root of the depressed cubic!\n\t(a) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"6", SuperscriptBox["x", "2"]}], "-", RowBox[{"18", "x"}], "-", "88"}], "=", "0"}], TraditionalForm]]], ".\n\t(b) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"2", SuperscriptBox["x", "2"]}], "-", RowBox[{ RowBox[{"(", RowBox[{"176", "/", "3"}], ")"}], "x"}], "-", RowBox[{"1936", "/", "27"}]}], "=", "0"}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48940428696875*^9, {3.48968161421875*^9, 3.489681850921875*^9}, { 3.48968189203125*^9, 3.489682211578125*^9}, {3.489700737921875*^9, 3.489700746890625*^9}, {3.489702857953125*^9, 3.489703015109375*^9}, 3.490276801453125*^9, 3.49028146415625*^9, 3.490281690578125*^9, 3.490281766*^9, {3.490301155453125*^9, 3.490301293359375*^9}, { 3.490301408796875*^9, 3.490301455609375*^9}, {3.4903015229375*^9, 3.49030154003125*^9}, {3.49055137228125*^9, 3.490551373390625*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->383133609], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Find ", StyleBox["all", FontSlant->"Italic"], " solutions of each of the cubic equations in the preceding exercise.\n", "(", StyleBox["Hint", FontSlant->"Italic"], ": If you have one root ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " of a cubic, you may find the others by dividing the cubic by ", Cell[BoxData[ FormBox[ RowBox[{"x", "-", "r"}], TraditionalForm]]], " and then applying the quadratic formula to the resulting quadratic.)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49030146825*^9, 3.49030152553125*^9}}, CellID->473401276], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " The \"usual rules of algebra\" include such identities as:\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "+", RowBox[{"(", RowBox[{"u", "+", "v"}], ")"}]}], "=", RowBox[{ RowBox[{"(", RowBox[{"x", "+", "u"}], ")"}], "+", RowBox[{"(", RowBox[{"y", "+", "v"}], ")"}]}]}], TraditionalForm]]], ",\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], " ", RowBox[{"(", RowBox[{"u", "+", "v"}], ")"}]}], "=", RowBox[{ RowBox[{"x", " ", "u"}], "+", RowBox[{"y", " ", "v"}], "+", RowBox[{"x", " ", "v"}], "+", RowBox[{"y", " ", "u"}]}]}], TraditionalForm]]], ",\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"k", "(", RowBox[{"x", "+", "y"}], ")"}], "=", RowBox[{ RowBox[{"k", " ", "x"}], "+", RowBox[{"k", " ", "y"}]}]}], TraditionalForm]]], ",\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"k", "(", RowBox[{"x", " ", "y"}], ")"}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"k", " ", "x"}], ")"}], "y"}], "=", RowBox[{"x", "(", RowBox[{"k", " ", "y"}], ")"}]}]}], TraditionalForm]]], ",\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "+", "z"}], "=", RowBox[{"x", "+", RowBox[{"(", RowBox[{"y", "+", "z"}], ")"}]}]}], TraditionalForm]]], ".\nYou know that these identities do hold for real numbers ", Cell[BoxData[ FormBox[ RowBox[{"x", ",", "y", ",", "u", ",", "v", ",", "k", ",", "z"}], TraditionalForm]]], ".\n", StyleBox["Assume", FontSlant->"Italic"], " that such identities also hold for \"complex numbers\"\[LongDash]numbers \ of the form ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " where ", Cell[BoxData[ FormBox["a", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["b", TraditionalForm]]], " are real. And continue to assume that ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ImaginaryI]", "2"], "=", RowBox[{ RowBox[{"\[ImaginaryI]", " ", "\[ImaginaryI]"}], "=", RowBox[{"-", "1"}]}]}], TraditionalForm]]], ". Then put each of the following into the form ", Cell[BoxData[ FormBox[ RowBox[{"u", "+", RowBox[{"\[ImaginaryI]", " ", "v"}]}], TraditionalForm]]], " with ", Cell[BoxData[ FormBox["u", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["v", TraditionalForm]]], " real:\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}], "+", RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", " ", "\[ImaginaryI]"}]}], ")"}]}], TraditionalForm]]], ",\t ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], ")"}], " ", RowBox[{"(", RowBox[{"c", "+", RowBox[{"d", " ", "\[ImaginaryI]"}]}], ")"}]}], TraditionalForm]]] }], "Exercise", CellChangeTimes->{ 3.48893078675*^9, {3.489094680859375*^9, 3.48909468115625*^9}, { 3.489094729140625*^9, 3.489094975953125*^9}, {3.489419271703125*^9, 3.489419274515625*^9}, 3.48957760015625*^9, {3.48970070640625*^9, 3.489700715109375*^9}}, ParagraphSpacing->{0.5, 0.}] }, Closed]], Cell[CellGroupData[{ Cell["The moral", "Subsection", ShowGroupOpener->True, CellChangeTimes->{ 3.48940428696875*^9, {3.4896817164375*^9, 3.48968171765625*^9}, 3.489682298546875*^9, 3.49026852525*^9}, CellID->329223310], Cell[TextData[{ "Square-roots of negative numbers are useful in obtaining ", StyleBox["real", FontSlant->"Italic"], " roots of certain cubic equations." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, 3.489682298546875*^9, {3.49087540546875*^9, 3.4908754168125*^9}}], Cell[TextData[{ "(But what ", StyleBox["are", FontSlant->"Italic"], " such \"complex\" numbers? That's what's next in this course!)" }], "Text", CellChangeTimes->{3.48893078675*^9, 3.48909498759375*^9, 3.489682298546875*^9}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Appendix: real and imaginary parts", "Section", ShowGroupOpener->True, CellChangeTimes->{ 3.48940428696875*^9, {3.489420354453125*^9, 3.48942039984375*^9}, 3.48942203184375*^9, 3.490268525375*^9, {3.490276898859375*^9, 3.49027691996875*^9}, {3.490276969375*^9, 3.490276983515625*^9}, { 3.49028185084375*^9, 3.49028188103125*^9}, {3.490288336921875*^9, 3.490288337625*^9}, 3.490288915578125*^9, {3.490551289984375*^9, 3.490551307921875*^9}}, ParagraphSpacing->{0.5, 0}, CellTags->"appendix1", CellID->233045865], Cell[TextData[{ "To find the real and imaginary parts of ", Cell[BoxData[ FormBox[ RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], TraditionalForm]]], " with ", StyleBox["Mathematica", FontSlant->"Italic"], ", directly use ", StyleBox["Re", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], " and ", StyleBox["Im", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.48941960078125*^9, 3.489419660296875*^9}, { 3.4894201663125*^9, 3.489420187359375*^9}, {3.4902768803125*^9, 3.4902768836875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}]}], "}"}]], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.48941966475*^9, 3.489419685890625*^9}}, CellLabel->"In[29]:="], Cell["\<\ But trying the same thing directly with\[Ellipsis]\ \>", "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489419694546875*^9, 3.489419722140625*^9}}], Cell[BoxData[ RowBox[{"theCube", "=", RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"m", "+", RowBox[{"n", SqrtBox[ RowBox[{"-", "1"}]]}]}], ")"}], "3"], "]"}]}]], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.48941972575*^9, 3.489419732390625*^9}, 3.490551420703125*^9}, CellLabel->"In[30]:="], Cell[TextData[{ "\[Ellipsis]will ", StyleBox["not", FontSlant->"Italic"], " work:" }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.48941973546875*^9, 3.48941974634375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "theCube", "]"}], ",", RowBox[{"Im", "[", "theCube", "]"}]}], "}"}]], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.489419753328125*^9, 3.489419766390625*^9}}, CellLabel->"In[31]:="], Cell[TextData[{ "The reason is that, ", StyleBox["by default, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" regards all symbolic variables representing numbers to be complex", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], "!" }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489419769390625*^9, 3.489419937828125*^9}, { 3.489420016078125*^9, 3.489420017734375*^9}, {3.48942006003125*^9, 3.489420061125*^9}, {3.490280766890625*^9, 3.49028080634375*^9}}], Cell[TextData[{ "You have to explicitly tell ", StyleBox["Mathematica", FontSlant->"Italic"], " when you want all such variables in an expression, instead, to be regarded \ as real. And to do that, you use ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Bold", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489419769390625*^9, 3.489419937828125*^9}, { 3.489420016078125*^9, 3.489420091171875*^9}, {3.490283492734375*^9, 3.49028349321875*^9}}], Cell["For example, as was done earlier:", "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489419947046875*^9, 3.489419955828125*^9}}], Cell[BoxData[ RowBox[{"ComplexExpand", "[", "theCube", "]"}]], "Input", CellChangeTimes->{3.48969559478125*^9, {3.49028359209375*^9, 3.490283598*^9}}, CellLabel->"In[32]:=", CellID->227865933], Cell[BoxData[ RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "theCube", "]"}], ",", RowBox[{"Im", "[", "theCube", "]"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{ 3.48969559478125*^9, {3.490283824625*^9, 3.4902838393125*^9}}, CellLabel->"In[33]:=", CellID->506911647], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Explain why ", Cell[BoxData[ FormBox[Cell[TextData[StyleBox["Re[ComplexExpand[theCube]]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"]]], TraditionalForm]]], " does not give an explicit value (in terms of ", StyleBox["m", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " and ", StyleBox["n", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ") for the real part of ", StyleBox["theCube", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", whereas ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"ComplexExpand", "[", RowBox[{"Re", "[", "theCube", "]"}], "]"}], FontFamily->"Courier"], TraditionalForm]]], " does." }], "Exercise", CellChangeTimes->{ 3.48940428696875*^9, {3.4894192316875*^9, 3.489419318890625*^9}, { 3.489419355875*^9, 3.489419361015625*^9}, 3.489420444953125*^9, 3.4894221330625*^9, 3.4895772629375*^9, 3.48957758871875*^9, { 3.489700691109375*^9, 3.48970069934375*^9}, 3.490277070171875*^9, { 3.49028070590625*^9, 3.490280710125*^9}, 3.49028325753125*^9, 3.490283895375*^9}, CellID->527559892], Cell[TextData[{ "Sometimes you want to bring in ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " as an \[OpenCurlyDoubleQuote]afterthought\[CloseCurlyDoubleQuote] to the \ main expression, and then you may use the following \ \[OpenCurlyDoubleQuote]postfix\[CloseCurlyDoubleQuote] form of input:" }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490283603515625*^9, 3.490283658703125*^9}, { 3.49028376921875*^9, 3.49028377653125*^9}, {3.490283913453125*^9, 3.490283925515625*^9}}, CellID->321563456], Cell[BoxData[{ RowBox[{"theCube", "//", "ComplexExpand"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "theCube", "]"}], ",", RowBox[{"Im", "[", "theCube", "]"}]}], "}"}], "//", "ComplexExpand"}]}], "Input", CellChangeTimes->{ 3.48969559478125*^9, {3.490283660921875*^9, 3.49028366771875*^9}, { 3.49028393571875*^9, 3.490283936046875*^9}}, CellLabel->"In[34]:=", CellID->107301876], Cell[TextData[{ "Then the desired values of ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " in ", "Bombelli", "\[CloseCurlyQuote]s example are given by:" }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.49028511878125*^9, 3.4902851305*^9}, { 3.4905514625*^9, 3.49055146775*^9}, 3.4919353874375*^9}, CellID->603719737], Cell[BoxData[ RowBox[{"First", "@", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"ComplexExpand", "[", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", "theCube", "]"}], ",", RowBox[{"Im", "[", "theCube", "]"}]}], "}"}], "]"}], "\[Equal]", RowBox[{"{", RowBox[{ RowBox[{"Re", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}], ",", RowBox[{"Im", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"m", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{ 3.48940428696875*^9, {3.48942059825*^9, 3.489420647234375*^9}, { 3.4894211963125*^9, 3.48942119709375*^9}, {3.49028298996875*^9, 3.490283020453125*^9}, {3.49028401609375*^9, 3.49028405378125*^9}}, CellLabel->"In[36]:="], Cell[TextData[{ "(The reason for using ", StyleBox["First", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " like that is to discard all but the first, real, solution.)" }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490284436109375*^9, 3.490284460921875*^9}}, CellID->84087322], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will ", StyleBox["not", FontSlant->"Italic"], " provide actual values for ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " that are solutions of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"m", "+", RowBox[{"n", " ", "\[ImaginaryI]"}]}], ")"}], "3"], "=", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " if you try the following:" }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.49028452671875*^9, 3.490284563703125*^9}, { 3.490284847796875*^9, 3.490284861453125*^9}, {3.49055148315625*^9, 3.490551484109375*^9}}, CellID->53511381], Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"ComplexExpand", "[", "theCube", "]"}], "\[Equal]", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}]}], ",", RowBox[{"{", RowBox[{"m", ",", "n"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{ 3.48969559478125*^9, {3.490284221078125*^9, 3.49028428125*^9}, { 3.490284582640625*^9, 3.490284613671875*^9}}, CellLabel->"In[37]:=", CellID->261589126], Cell[TextData[{ "Rather, as you see, it just expresses one of the variables in ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"m", "+", RowBox[{"n", " ", "\[ImaginaryI]"}]}], ")"}], "3"], "=", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}]}], TraditionalForm]]], " in terms of the other." }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490284630796875*^9, 3.490284658984375*^9}, { 3.49028469384375*^9, 3.490284812234375*^9}, {3.490284879578125*^9, 3.490284900171875*^9}, {3.49028493628125*^9, 3.490284993328125*^9}, 3.490285197078125*^9, {3.490301839671875*^9, 3.490301843921875*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->450839061], Cell[TextData[{ "If you want ", StyleBox["Mathematica", FontSlant->"Italic"], " to determine actual values for ", Cell[BoxData[ FormBox["m", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " (without specifying that they be integers), you need to bring in the \ second equation, ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"m", "-", RowBox[{"n", " ", "\[ImaginaryI]"}]}], ")"}], "3"], "=", RowBox[{"2", "-", RowBox[{"11", "\[ImaginaryI]"}]}]}], TraditionalForm]]], ", too.\nAnd then there is no need to separate each complex equation into a \ pair of real equations, one for real parts and the other for imaginary parts; \ ", StyleBox["Mathematica", FontSlant->"Italic"], " can handle the entire solution in one fell swoop:" }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.490284630796875*^9, 3.490284658984375*^9}, { 3.49028469384375*^9, 3.490284812234375*^9}, {3.490284879578125*^9, 3.490284900171875*^9}, {3.49028493628125*^9, 3.490284993328125*^9}, { 3.490285197078125*^9, 3.490285262015625*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->115202901], Cell[BoxData[ RowBox[{"First", "@", RowBox[{"Solve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"m", "+", RowBox[{"n", " ", "\[ImaginaryI]"}]}], ")"}], "3"], "\[Equal]", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}]}], ",", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"m", "-", RowBox[{"n", " ", "\[ImaginaryI]"}]}], ")"}], "3"], "\[Equal]", RowBox[{"2", "-", RowBox[{"11", "\[ImaginaryI]"}]}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"m", ",", "n"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{ 3.48969559478125*^9, {3.49028274934375*^9, 3.490282835265625*^9}, 3.49028287325*^9, {3.49028491165625*^9, 3.490284914015625*^9}}, CellLabel->"In[38]:=", CellID->829447880], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " The ", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["Conjugate", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " changes a complex number ", Cell[BoxData[ FormBox[ RowBox[{"a", "+", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " into ", Cell[BoxData[ FormBox[ RowBox[{"a", "-", RowBox[{"b", " ", "\[ImaginaryI]"}]}], TraditionalForm]]], " when ", Cell[BoxData[ FormBox["a", TraditionalForm]]], " and ", Cell[BoxData[ FormBox["b", TraditionalForm]]], " are real. For example, input ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"Conjugate", "[", RowBox[{"2", "+", RowBox[{"11", "\[ImaginaryI]"}]}], "]"}], FontFamily->"Courier"], TraditionalForm]]], " gives output ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ StyleBox["2", FontSlant->"Plain"], "-", RowBox[{"11", "\[ImaginaryI]"}]}], FontFamily->"Courier"], TraditionalForm]]], ".\nWhy doesn\[CloseCurlyQuote]t the input\n\t", Cell[BoxData[ FormBox[ RowBox[{"Conjugate", "[", RowBox[{ StyleBox["m", FontSlant->"Plain"], "+", RowBox[{ StyleBox["n", FontSlant->"Plain"], " ", "\[ImaginaryI]"}]}], "]"}], TraditionalForm]], FontFamily->"Courier"], "\ngive output ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ StyleBox["m", FontSlant->"Plain"], "-", RowBox[{ StyleBox["n", FontSlant->"Plain"], " ", "\[ImaginaryI]"}]}], FontFamily->"Courier"], TraditionalForm]]], "? And how can you modify that input so as to obtain output ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ StyleBox["m", FontSlant->"Plain"], "-", RowBox[{ StyleBox["n", FontSlant->"Plain"], " ", "\[ImaginaryI]"}]}], FontFamily->"Courier"], TraditionalForm]]], "?" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490285423171875*^9, 3.490285683859375*^9}, {3.490288134453125*^9, 3.490288168875*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->340743135], Cell[TextData[{ "For more information about ", StyleBox["ComplexExpand", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ", see notebook ", StyleBox["CartesianPolarForms.nb", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.49193516215625*^9, 3.491935181390625*^9}}, CellID->63624785] }, Closed]], Cell[CellGroupData[{ Cell["References", "Section", CellChangeTimes->{ 3.48940428696875*^9, {3.48943885315625*^9, 3.48943885553125*^9}, 3.490268525609375*^9}, CellID->37236245], Cell[TextData[{ "For a summary of the history of solving cubics, see the article \ \[OpenCurlyDoubleQuote]Cubic function\[CloseCurlyDoubleQuote], ", StyleBox["Wikipedia,", FontSlant->"Italic"], " ", ButtonBox["http://en.wikipedia.org/wiki/Cubic_function", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Cubic_function"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Cubic_function"], "." }], "Text", CellChangeTimes->{ 3.48940428696875*^9, {3.489438873640625*^9, 3.489438980328125*^9}}] }, Closed]] }, Open ]] }, WindowSize->{572, 557}, WindowMargins->{{100, Automatic}, {Automatic, 37}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, ShowSelection->True, CreateCellID->True, CellLabelAutoDelete->False, Hyphenation->True, SpellingDictionaries->{"CorrectWords"->{ "Eisenberg", "del", "Ferro", "Tartaglia", "Bombeli\[CloseCurlyQuote]s", "Girolamo", "Ars", "Magna", "Scipione", "Niccol\[OGrave]", "Bombeli"}}, PrivateNotebookOptions -> {"NotebookAuthor" -> ""}, Magnification->1, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->Notebook[{ Cell[ StyleData[StyleDefinitions -> "Default.nb"]], Cell[ CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[ StyleData[All, "Working"], Background -> RGBColor[0.9921568627450981, 0.9607843137254902, 0.9019607843137255]], Cell[ StyleData[All, "Presentation"], Background -> RGBColor[0.9921568627450981, 0.9607843137254902, 0.9019607843137255]], Cell[ StyleData[All, "SlideShow"], Background -> RGBColor[0.9921568627450981, 0.9607843137254902, 0.9019607843137255]], Cell[ StyleData[All, "Condensed"], Background -> RGBColor[0.9921568627450981, 0.9607843137254902, 0.9019607843137255]], Cell[ StyleData[All, "Printout"], Background -> None]}, Closed]], Cell[ CellGroupData[{ Cell["Notebook Options Settings", "Section"], Cell[ StyleData["Notebook"], ShowCellBracket -> Automatic, InputAliases -> {"intt" -> RowBox[{"\[Integral]", RowBox[{"\[SelectionPlaceholder]", RowBox[{"\[DifferentialD]", "\[Placeholder]"}]}]}], "dintt" -> RowBox[{ SubsuperscriptBox[ "\[Integral]", "\[SelectionPlaceholder]", "\[Placeholder]"], RowBox[{"\[Placeholder]", RowBox[{"\[DifferentialD]", "\[Placeholder]"}]}]}], "sumt" -> RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"\[SelectionPlaceholder]", "=", "\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], "prodt" -> RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"\[SelectionPlaceholder]", "=", "\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], "dt" -> RowBox[{ SubscriptBox["\[PartialD]", "\[Placeholder]"], " ", "\[SelectionPlaceholder]"}], "prime" -> "\:02b9", "dprime" -> "\:02ba"}, Magnification -> 1.04]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Title and Section Cells", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Title"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {0, 5}}, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, FontFamily -> "Helvetica", FontSize -> 36, FontWeight -> "Bold", FontColor -> GrayLevel[0], Background -> RGBColor[0.737255, 0.894118, 0.807843]], Cell[ StyleData["Title", "Printout"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {0, 5}}, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, FontFamily -> "Helvetica", FontSize -> 26, FontWeight -> "Bold", FontColor -> GrayLevel[0], Background -> None]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Subtitle"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {0, 0}}, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, FontFamily -> "Helvetica", FontSize -> 24, Background -> RGBColor[0.815686, 0.901961, 0.92549]], Cell[ StyleData["Subtitle", "Printout"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {0, 0}}, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, FontFamily -> "Helvetica", FontSize -> 18, Background -> None]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Chaptertitle"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {0, 0}}, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1170, FontFamily -> "Helvetica", FontSize -> 24, Background -> RGBColor[0.941176, 0.87451, 0.784314]], Cell[ StyleData["Chaptertitle", "Printout"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {0, 0}}, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1170, FontFamily -> "Helvetica", FontSize -> 18, Background -> None]}, Open]], Cell[ StyleData["Subsubtitle"], ShowCellBracket -> False, CellMargins -> {{12, Inherited}, {20, 5}}, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["Section"], CellFrame -> False, ShowGroupOpener -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}], Cell[ StyleData["Subsection"], CellDingbat -> "", ShowGroupOpener -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}], Cell[ StyleData["Subsubsection"], CellDingbat -> "", ShowGroupOpener -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}], Cell[ StyleData["Subsubsubsection"], CellDingbat -> "", ShowGroupOpener -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1360], Cell[ StyleData["Subsubsubsubsection"], ShowGroupOpener -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1370]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Text"], AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, FontSize -> 14], Cell[ StyleData["Text", "Printout"], AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["EvaluatableText", StyleDefinitions -> StyleData["Text"]], Evaluator -> None, Evaluatable -> True, MenuPosition -> 1405], Cell[ StyleData["EvaluatableText", "Printout"], Evaluator -> None, Evaluatable -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1405, FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["EmphasisText", StyleDefinitions -> StyleData["Text"]], CellFrame -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1410, FontFamily -> "Helvetica", FontSize -> 14], Cell[ StyleData["EmphasisText", "Printout"], CellFrame -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1410, FontFamily -> "Helvetica", FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["BlueComments", StyleDefinitions -> StyleData["Text"]], AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1420, FontSize -> 14, Background -> RGBColor[0.941207, 0.972503, 1]], Cell[ StyleData[ "BlueComments", "Printout", StyleDefinitions -> StyleData["Text"]], CellMargins -> {{2, 2}, {6, 6}}, TextJustification -> 0.5, Hyphenation -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1420, FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData[ "Exercise", StyleDefinitions -> StyleData["BlueComments"]], CellFrame -> True, TextJustification -> 0.5, Hyphenation -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, CounterIncrements -> "Exercise", MenuPosition -> 1425, FontSize -> 14, Background -> RGBColor[0.941207, 0.972503, 1]], Cell[ StyleData[ "Exercise", "Printout", StyleDefinitions -> StyleData["Text"]], CellFrame -> True, TextJustification -> 0.5, Hyphenation -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, CounterIncrements -> "Exercise", MenuPosition -> 1425, FontSize -> 10]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["PinkComments", StyleDefinitions -> StyleData["Text"]], AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1430, FontSize -> 14, Background -> RGBColor[1, 0.894102, 0.882399]], Cell[ StyleData[ "PinkComments", "Printout", StyleDefinitions -> StyleData["Text"]], CellMargins -> {{2, 2}, {6, 6}}, TextJustification -> 0.5, Hyphenation -> True, AutoItalicWords -> {"Presentations", "Mathematica", "Tensorial"}, MenuPosition -> 1430, FontSize -> 10]}, Open]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Input and Output Cells", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Input"], FontSize -> 13, FontWeight -> "Bold", Background -> RGBColor[0.976471, 0.909804, 0.815686]], Cell[ StyleData["Input", "Printout"], FontSize -> 10, FontWeight -> "Bold", Background -> None]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Output"], FontSize -> 14, FontWeight -> "Bold"], Cell[ StyleData["Output", "Printout"], FontSize -> 10, FontWeight -> "Bold"]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["MSG"], FontSize -> 14], Cell[ StyleData["MSG", "Printout"], FontSize -> 10]}, Open]]}, Closed]]}, Visible -> False, FrontEndVersion -> "7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions -> "PrivateStylesheetFormatting.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "exercise"->{ Cell[20481, 642, 1363, 38, 89, "Exercise", CellTags->"exercise"], Cell[24162, 760, 1217, 33, 89, "Exercise", CellTags->"exercise"], Cell[25382, 795, 1084, 34, 84, "Exercise", CellTags->"exercise"]}, "appendix1"->{ Cell[85844, 2779, 542, 11, 36, "Section", CellTags->"appendix1", CellID->233045865]} } *) (*CellTagsIndex CellTagsIndex->{ {"exercise", 112173, 3557}, {"appendix1", 112403, 3564} } *) (*NotebookFileOutline Notebook[{ Cell[557, 20, 154, 2, 41, "Subsubtitle"], Cell[CellGroupData[{ Cell[736, 26, 194, 4, 43, "Subtitle"], Cell[933, 32, 454, 8, 41, "Subsubtitle"], Cell[1390, 42, 233, 6, 24, "Text"], Cell[CellGroupData[{ Cell[1648, 52, 140, 3, 66, "Section", CellID->143077100], Cell[CellGroupData[{ Cell[1813, 59, 185, 4, 36, "Subsection", CellID->127073981], Cell[2001, 65, 785, 20, 114, "Text", CellID->20739139], Cell[2789, 87, 473, 17, 53, "Text", CellID->83290015], Cell[3265, 106, 310, 11, 31, "Text", CellID->171441541] }, Closed]], Cell[CellGroupData[{ Cell[3612, 122, 142, 3, 36, "Subsection", CellID->195536689], Cell[3757, 127, 755, 20, 106, "Text", CellID->79531803] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[4561, 153, 137, 3, 36, "Section", CellID->344771388], Cell[4701, 158, 202, 3, 30, "Text"], Cell[4906, 163, 524, 15, 48, "EmphasisText"], Cell[5433, 180, 398, 10, 47, "Input"], Cell[5834, 192, 280, 8, 30, "Text"], Cell[6117, 202, 409, 10, 47, "Input"], Cell[6529, 214, 171, 3, 30, "Text"], Cell[6703, 219, 1356, 33, 144, "Text"], Cell[8062, 254, 1159, 39, 127, "Exercise"], Cell[9224, 295, 1369, 44, 162, "Exercise"], Cell[10596, 341, 1239, 45, 132, "Exercise"] }, Closed]], Cell[CellGroupData[{ Cell[11872, 391, 297, 6, 36, "Section", CellID->355727], Cell[12172, 399, 484, 9, 30, "Text"], Cell[12659, 410, 1741, 38, 234, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[14437, 453, 346, 8, 62, "Section", CellID->104439264], Cell[14786, 463, 593, 13, 68, "Text"], Cell[15382, 478, 591, 16, 80, "EmphasisText"], Cell[15976, 496, 401, 10, 83, "Input"], Cell[16380, 508, 139, 6, 31, "Text"], Cell[16522, 516, 155, 4, 47, "Input"], Cell[16680, 522, 419, 11, 30, "Text"], Cell[17102, 535, 488, 17, 73, "EmphasisText"], Cell[17593, 554, 492, 14, 49, "Text"], Cell[18088, 570, 917, 21, 49, "Text"], Cell[19008, 593, 550, 14, 88, "Input"], Cell[19561, 609, 917, 31, 66, "Text"], Cell[20481, 642, 1363, 38, 89, "Exercise", CellTags->"exercise"], Cell[21847, 682, 1052, 34, 89, "Exercise"], Cell[22902, 718, 1257, 40, 108, "Exercise"], Cell[24162, 760, 1217, 33, 89, "Exercise", CellTags->"exercise"], Cell[25382, 795, 1084, 34, 84, "Exercise", CellTags->"exercise"] }, Closed]], Cell[CellGroupData[{ Cell[26503, 834, 404, 9, 36, "Section", CellID->10228179], Cell[26910, 845, 1086, 25, 68, "Text"], Cell[27999, 872, 320, 9, 30, "Text"], Cell[28322, 883, 869, 27, 86, "Input"], Cell[29194, 912, 118, 2, 30, "Text"], Cell[29315, 916, 313, 8, 47, "Input"], Cell[29631, 926, 1366, 35, 109, "Exercise"], Cell[31000, 963, 1253, 29, 89, "Exercise"], Cell[32256, 994, 1447, 34, 154, "Exercise"], Cell[33706, 1030, 1475, 46, 93, "SmallText"], Cell[35184, 1078, 1604, 46, 190, "Text"], Cell[36791, 1126, 723, 25, 81, "Exercise"], Cell[37517, 1153, 1234, 35, 124, "Text"], Cell[38754, 1190, 1269, 34, 79, "Text"], Cell[40026, 1226, 861, 27, 89, "Exercise"] }, Closed]], Cell[CellGroupData[{ Cell[40924, 1258, 152, 4, 36, "Section", CellID->62156397], Cell[CellGroupData[{ Cell[41101, 1266, 575, 17, 36, "Subsection", CellID->238688122], Cell[41679, 1285, 1301, 47, 246, "Text"], Cell[42983, 1334, 227, 3, 30, "Text"], Cell[43213, 1339, 464, 14, 49, "Input"], Cell[43680, 1355, 2242, 51, 157, "Input"], Cell[45925, 1408, 240, 6, 24, "SmallText"], Cell[46168, 1416, 1221, 39, 89, "Exercise"] }, Closed]], Cell[CellGroupData[{ Cell[47426, 1460, 915, 27, 55, "Subsection", CellOpen->True, CellID->195307992], Cell[48344, 1489, 379, 9, 30, "Text", CellOpen->True], Cell[48726, 1500, 296, 9, 47, "Input", CellOpen->True], Cell[49025, 1511, 769, 26, 68, "Text", CellOpen->True], Cell[49797, 1539, 529, 18, 47, "Input", CellOpen->True] }, Closed]], Cell[CellGroupData[{ Cell[50363, 1562, 149, 3, 36, "Subsection", CellID->58437946], Cell[50515, 1567, 393, 8, 68, "Text", CellID->148481078] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[50957, 1581, 396, 7, 36, "Section", CellID->54838321], Cell[51356, 1590, 812, 29, 49, "Text"], Cell[52171, 1621, 386, 10, 49, "EmphasisText"], Cell[52560, 1633, 489, 13, 47, "Input"], Cell[53052, 1648, 627, 20, 49, "Text"], Cell[53682, 1670, 454, 15, 49, "Input"], Cell[54139, 1687, 365, 10, 30, "Text"], Cell[54507, 1699, 243, 6, 47, "Input"], Cell[54753, 1707, 268, 7, 49, "Text", CellID->204387247], Cell[CellGroupData[{ Cell[55046, 1718, 209, 5, 36, "Subsection", CellID->198090096], Cell[55258, 1725, 339, 5, 30, "Text"], Cell[55600, 1732, 231, 6, 47, "Input"], Cell[55834, 1740, 293, 7, 30, "Text"], Cell[56130, 1749, 436, 14, 47, "Input"], Cell[56569, 1765, 1041, 38, 91, "Text"], Cell[57613, 1805, 600, 18, 49, "Text", CellID->927876621], Cell[58216, 1825, 964, 35, 74, "EmphasisText"], Cell[59183, 1862, 702, 25, 40, "Text", CellID->135694709], Cell[59888, 1889, 461, 17, 55, "Input", CellID->512193855], Cell[60352, 1908, 781, 20, 49, "Text", CellID->231773451], Cell[61136, 1930, 929, 35, 72, "EmphasisText", CellID->508222106], Cell[62068, 1967, 1479, 46, 149, "Text", CellID->148126141], Cell[63550, 2015, 1674, 66, 137, "Exercise", CellID->62450556], Cell[65227, 2083, 672, 17, 52, "Text", CellID->101897756], Cell[65902, 2102, 501, 15, 56, "Input", CellID->48744119], Cell[66406, 2119, 462, 12, 30, "Text", CellID->530507026], Cell[66871, 2133, 206, 5, 47, "Input", CellID->560356888], Cell[67080, 2140, 542, 16, 24, "SmallText", CellID->416894146], Cell[67625, 2158, 791, 27, 60, "Text", CellID->544007337], Cell[68419, 2187, 304, 6, 30, "Text", CellID->853432109], Cell[68726, 2195, 611, 14, 47, "Input", CellID->110691365], Cell[69340, 2211, 254, 6, 30, "Text", CellID->56582438], Cell[69597, 2219, 499, 14, 47, "Input", CellID->331101330], Cell[70099, 2235, 296, 4, 30, "Text", CellID->563007215], Cell[70398, 2241, 369, 9, 68, "Input", CellID->195723882], Cell[70770, 2252, 423, 6, 30, "Text", CellID->95768003], Cell[71196, 2260, 261, 6, 47, "Input", CellID->51416592], Cell[71460, 2268, 150, 3, 30, "Text", CellID->352294015], Cell[71613, 2273, 271, 6, 47, "Input", CellID->606236280], Cell[71887, 2281, 737, 21, 49, "Text", CellID->213028755], Cell[72627, 2304, 1967, 71, 202, "Exercise", CellID->332462960], Cell[74597, 2377, 240, 7, 30, "Text", CellID->16106302], Cell[74840, 2386, 378, 9, 47, "Input", CellID->4500715], Cell[75221, 2397, 410, 13, 70, "Exercise", CellID->542723926], Cell[75634, 2412, 671, 22, 49, "Text", CellID->248000342], Cell[76308, 2436, 496, 15, 70, "Exercise", CellID->227269981], Cell[76807, 2453, 1273, 48, 109, "Exercise", CellID->273140987], Cell[78083, 2503, 987, 33, 124, "Exercise", CellID->99294381], Cell[79073, 2538, 1567, 43, 162, "Exercise", CellID->383133609], Cell[80643, 2583, 771, 27, 127, "Exercise", CellID->473401276], Cell[81417, 2612, 3614, 132, 324, "Exercise"] }, Closed]], Cell[CellGroupData[{ Cell[85068, 2749, 206, 5, 36, "Subsection", CellID->329223310], Cell[85277, 2756, 282, 8, 49, "Text"], Cell[85562, 2766, 233, 7, 30, "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[85844, 2779, 542, 11, 36, "Section", CellTags->"appendix1", CellID->233045865], Cell[86389, 2792, 629, 24, 50, "Text"], Cell[87021, 2818, 360, 11, 47, "Input"], Cell[87384, 2831, 164, 4, 30, "Text"], Cell[87551, 2837, 363, 12, 56, "Input"], Cell[87917, 2851, 187, 7, 30, "Text"], Cell[88107, 2860, 258, 7, 47, "Input"], Cell[88368, 2869, 607, 17, 52, "Text"], Cell[88978, 2888, 516, 15, 69, "Text"], Cell[89497, 2905, 139, 2, 30, "Text"], Cell[89639, 2909, 196, 4, 47, "Input", CellID->227865933], Cell[89838, 2915, 316, 9, 47, "Input", CellID->506911647], Cell[90157, 2926, 1321, 45, 91, "Exercise", CellID->527559892], Cell[91481, 2973, 572, 14, 50, "Text", CellID->321563456], Cell[92056, 2989, 439, 12, 68, "Input", CellID->107301876], Cell[92498, 3003, 401, 14, 30, "Text", CellID->603719737], Cell[92902, 3019, 877, 24, 88, "Input"], Cell[93782, 3045, 317, 10, 50, "Text", CellID->84087322], Cell[94102, 3057, 787, 28, 49, "Text", CellID->53511381], Cell[94892, 3087, 448, 13, 47, "Input", CellID->261589126], Cell[95343, 3102, 729, 19, 49, "Text", CellID->450839061], Cell[96075, 3123, 1185, 34, 133, "Text", CellID->115202901], Cell[97263, 3159, 871, 26, 71, "Input", CellID->829447880], Cell[98137, 3187, 2250, 88, 193, "Exercise", CellID->340743135], Cell[100390, 3277, 397, 15, 51, "Text", CellID->63624785] }, Closed]], Cell[CellGroupData[{ Cell[100824, 3297, 162, 4, 36, "Section", CellID->37236245], Cell[100989, 3303, 535, 14, 49, "Text"] }, Closed]] }, Open ]] } ] *) (* End of internal cache information *)