(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 43900, 1408] NotebookOptionsPosition[ 37350, 1212] NotebookOutlinePosition[ 37730, 1229] CellTagsIndexPosition[ 37687, 1226] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lesson 1 Introduction to Mathematica", "Section"], Cell[CellGroupData[{ Cell["Mathematica Basics", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is a Computer Algebra System (CAS). It can do simple calculations, just \ like a calculator, but also symbolic manipulations, such as differentiating a \ function. The program consists of two parts :\n \t\[FilledSmallCircle] the \ front - end with the notebook interface\n\t\[FilledSmallCircle] the kernel \ which does the actual computations. \n \n You will interact mainly with \ the front - end part of ", StyleBox["Mathematica", FontSlant->"Italic"], " by typing mathematical expressions and text into a notebook. To get \ started, launch ", StyleBox["Mathematica", FontSlant->"Italic"], ". Usually, a new notebook will open automatically. If this is not the case, \ open a new notebook by selecting ", StyleBox["File\[RightArrow]New", "Input"], " from the menu.\n \n Once you have the notebook available, you can \ start typing your input. As you enter an expression, a cell bracket will \ appear on the right side of the notebook. You will learn more about these \ cell brackets later; for now just ignore them. For example, to add ", StyleBox["2", "Input"], " and ", StyleBox["2", "Input"], ", type" }], "Text", CellChangeTimes->{{3.459625579514392*^9, 3.459625620033966*^9}, { 3.459792791119804*^9, 3.459792832736339*^9}, {3.462810882367262*^9, 3.462810884068953*^9}}], Cell[BoxData[ RowBox[{"2", "+", "2"}]], "Input", CellChangeTimes->{{3.4596256240986233`*^9, 3.459625625205921*^9}}], Cell[TextData[{ "To get an answer from ", StyleBox["Mathematica", FontSlant->"Italic"], ", you need to send this input to the kernel for evaluation. You do this by \ clicking the cell somewhere (putting the cursor in that cell) and then \ pressing the action key(s): either\n \n \t\[FilledSmallCircle] \[ShiftKey]\ \[ReturnKey] (press Return while holding down the Shift key)\n\t\ \[FilledSmallCircle] \[EnterKey] (press Enter on the numerical \ keypad).\n \nWhen you use the action key(s), an ", StyleBox["In[#]:=", "Input"], " tag will appear in front of what you have typed. The answer will be \ displayed in a separate cell labeled ", StyleBox["Out[#]=", "Input"], " :" }], "Text", CellChangeTimes->{{3.459625639644989*^9, 3.459625834168606*^9}, { 3.459625866545046*^9, 3.459625867156002*^9}, {3.4596259266776667`*^9, 3.459625950478436*^9}, {3.459626067956252*^9, 3.459626085683475*^9}, 3.459626466246029*^9, {3.459628269952589*^9, 3.459628322483542*^9}, { 3.4597928377522993`*^9, 3.45979283964077*^9}}], Cell[BoxData["4"], "Output", CellChangeTimes->{3.45962595838277*^9}], Cell[TextData[{ "Try this yourself! To open a new cell, click on a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook at the top, bottom, or between two existing cells so that a \ horizontal line appears across the notebook. " }], "Text", CellChangeTimes->{{3.459626109868507*^9, 3.459626115768647*^9}, { 3.4596262566299973`*^9, 3.459626372885036*^9}}], Cell[TextData[{ "You may have been a bit surprised that it took ", StyleBox["Mathematica", FontSlant->"Italic"], " a very long time to determine that ", StyleBox["2+2=4", "Input"], ". This effect always happens with the first evaluation in a session, as the \ whole kernel has to be started up before any computation can take place. When \ you opened the notebook, only the front-end part of ", StyleBox["Mathematica", FontSlant->"Italic"], " was activated. You will see that subsequent computations will be much \ faster.\n \n The ", StyleBox["In[#]", "Input"], " tag has another important use - namely to tell you whether you did indeed \ send your input to the kernel. A common mistake is to just press the \ \[ReturnKey] key without the \[ShiftKey] key, which results only in the \ creation of a new line. Thus, if you do not see the ", StyleBox["In[#]", "Input"], " tag, Mathematica does not know about your definitions, computations, etc. \ The same result, namely no ", StyleBox["In[#]", "Input"], " tag and consequently no computation, can occur when the notebook is not \ the active window. You may have pressed the correct action key(s), but if the \ notebook is not the active window, nothing happens. " }], "Text", CellChangeTimes->{{3.459626490019355*^9, 3.459626512694038*^9}, { 3.4596265681566477`*^9, 3.459626587629064*^9}, {3.459627080440673*^9, 3.45962708188967*^9}, {3.4597928443643303`*^9, 3.459792851849986*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}], Cell[TextData[{ "1) Open up a new cell and calculate ", StyleBox["123+321", "Input"], "." }], "Text", CellChangeTimes->{{3.414942819421444*^9, 3.414942831794449*^9}, { 3.415530360755663*^9, 3.415530376742628*^9}, {3.447348674060577*^9, 3.447348683931106*^9}, {3.4473487206224823`*^9, 3.447348761678961*^9}, { 3.447348929848113*^9, 3.447348953861364*^9}, {3.459627041359744*^9, 3.4596270422011747`*^9}}], Cell["2) What happens if you evaluate the cell below?", "Text", CellChangeTimes->{{3.459626913177559*^9, 3.45962695656695*^9}}], Cell[BoxData[{ RowBox[{"123", "+", "123"}], "\[IndentingNewLine]", RowBox[{"1", "+", "2", "+", "3"}]}], "Input", CellChangeTimes->{{3.45962696177804*^9, 3.459626970044342*^9}}], Cell["How was this cell entered?", "Text", CellChangeTimes->{{3.4596270953174953`*^9, 3.45962710477096*^9}}], Cell["\<\ 3) What happens when you evaluate a cell containing text (such as this cell)?\ \ \>", "Text", CellChangeTimes->{{3.4599645385094233`*^9, 3.459964579731533*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Assignment Statements", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.4596280886636763`*^9, 3.4596280976844482`*^9}}], Cell["Start a new cell and type ", "Text"], Cell[BoxData[{ RowBox[{"x", "=", "2"}], "\n", RowBox[{"y", "=", RowBox[{"x", "+", "3"}]}]}], "Input"], Cell[TextData[{ "This will assign a value of ", StyleBox["2", "Input"], " to the symbol or variable ", StyleBox["x", "Input"], ". In the future, every time ", StyleBox["x", "Input"], " is used, it will have the value ", StyleBox["2", "Input"], ". Thus ", StyleBox["y", "Input"], " should be assigned the value ", StyleBox["5", "Input"], ". Try it out - your result should look like this :" }], "Text", CellChangeTimes->{{3.459628176073681*^9, 3.4596281868580923`*^9}, { 3.4596283959213657`*^9, 3.459628397099461*^9}, {3.459793354582438*^9, 3.459793358769224*^9}}], Cell[BoxData["2"], "Output", CellChangeTimes->{3.4596282307472*^9}], Cell[BoxData["5"], "Output", CellChangeTimes->{3.459628230757563*^9}], Cell[TextData[{ "There is one answer for each expression. First, we get the result for the \ value of ", StyleBox["x", "Input"], ", namely ", StyleBox["2", "Input"], ". Then, the value of ", StyleBox["y", "Input"], " is displayed, which is ", StyleBox["5", "Input"], ", as expected. \n\nSometimes, you may want to assign a value to a variable, \ but do not want to see the result (as you know what it is or it is really, \ really big). To suppress the output, type a ", StyleBox[";", "Input", FontWeight->"Bold"], " after the input. " }], "Text", CellChangeTimes->{{3.4597992171617603`*^9, 3.4597992280885553`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"z", "=", "5"}], ";"}], "\n", RowBox[{ RowBox[{"x", "=", RowBox[{"2", "z"}]}], ";"}]}], "Input"], Cell[TextData[{ "Using the action key(s) produces no output. However, since the cell has an \ ", StyleBox["In[#]", "Input"], " tag, we know that the assignments and computations were executed. You can \ check the value of the variable ", StyleBox["x", "Input"], " by just typing its name, then pressing the action key(s). " }], "Text"], Cell[BoxData["x"], "Input"], Cell[TextData[{ "Note that the new assignment, ", StyleBox["x=2z", "Input"], ", has over-written the previous assignment, ", StyleBox["x=2", "Input"], ". We can check which is the more recent assignment by looking at the \ numbers of the ", StyleBox["In[#]", "Input"], " tag. The higher number indicates the more recent assignment. Note that it \ does not matter where in the notebook you write an expression - ", StyleBox["Mathematica", FontSlant->"Italic"], " keeps track of the order in which the expressions were ", StyleBox["evaluated", FontWeight->"Bold"], "!!!!" }], "Text", CellChangeTimes->{{3.4596285076972923`*^9, 3.459628525615664*^9}}], Cell["\<\ We will have more to say about assignment statements in Lesson ?\ \>", "Text", CellChangeTimes->{{3.45979977607617*^9, 3.459799794622388*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Addition and Subtraction", "Subsection", CellChangeTimes->{{3.459793220353554*^9, 3.459793227441167*^9}}], Cell["Here are some examples for addition and subtraction:", "Text", CellChangeTimes->{3.459793236498806*^9}], Cell[BoxData[ RowBox[{"x", "+", "y"}]], "Input"], Cell[BoxData[ RowBox[{"w", "-", "3"}]], "Input"], Cell[TextData[{ "Note that since no value was assigned to ", StyleBox["w", "Input"], " before, ", StyleBox["Mathematica", FontSlant->"Italic"], " returns the expression unevaluated. It changes the order to list numbers \ first, then symbols. " }], "Text", CellChangeTimes->{3.4597936118076878`*^9}], Cell[BoxData[ RowBox[{"c", "-", "b", "+", "a"}]], "Input", CellChangeTimes->{{3.459793476034354*^9, 3.4597934895410748`*^9}}], Cell[TextData[{ "Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " has its own preferred way of presenting expressions - in this example, ", StyleBox["Mathematica", FontSlant->"Italic"], " re-orders the given expression. But, since the variables ", StyleBox["a", "Input"], ", ", StyleBox["b", "Input"], " and ", StyleBox["c", "Input"], " have no values so far, ", StyleBox["Mathematica", FontSlant->"Italic"], " does not add or subtract anything." }], "Text", CellChangeTimes->{{3.459793498430191*^9, 3.4597935789206343`*^9}, { 3.459793646747922*^9, 3.459793719080798*^9}, 3.459799816361601*^9, { 3.459964663471758*^9, 3.459964668593211*^9}, 3.460219661833542*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Multiplication", "Subsection", CellChangeTimes->{{3.459793596483337*^9, 3.459793600887309*^9}}], Cell["Now let's look at multiplication:", "Text"], Cell[BoxData[ RowBox[{"3", "*", "4"}]], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " accepts the ", StyleBox["*", "Input"], " (\[ShiftKey] -8) or a blank space for multiplication. If a number and a \ symbol are combined without a blank space in between, then the interpretation \ depends on the order:" }], "Text", CellChangeTimes->{{3.459792911274325*^9, 3.4597929131916*^9}}], Cell[BoxData[ RowBox[{"3", "z"}]], "Input"], Cell[BoxData["z3"], "Input"], Cell[TextData[{ "The first expression is interpreted as ", StyleBox["3*z", "Input"], ", and since ", StyleBox["z=5", "Input"], ", the result is ", StyleBox["15", "Input"], ". However, the second expression, ", StyleBox["z3", "Input"], ", is interpreted as a new variable with name ", StyleBox["z3", "Input"], ". Thus, use ", StyleBox["*", "Input"], " to make sure that the result is what you want." }], "Text", CellChangeTimes->{{3.459791101694387*^9, 3.459791104329748*^9}, { 3.4597913949508038`*^9, 3.459791418177133*^9}}], Cell[TextData[{ "The fact that ", StyleBox["z3", "Input"], " is single variable rather than a product is useful. It is common to use \ this symbol in place of symbol with subscript: ", Cell[BoxData[ FormBox[ SubscriptBox["z", "3"], TraditionalForm]]], ". Thus, in ", StyleBox["Mathematica", FontSlant->"Italic"], ", a short sequence of numbers may use the variables", StyleBox[" x1", "Input"], ", ", StyleBox["x2", "Input"], ", ", StyleBox["x3", "Input"], ", ", StyleBox["x4,", "Input"], " rather than ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], ",", " ", SubscriptBox["x", "2"], ",", " ", SubscriptBox["x", "3"], ",", " ", RowBox[{ SubscriptBox["x", "4"], StyleBox[".", "Text"]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.459792155347205*^9, 3.4597922488768806`*^9}, { 3.459792283552998*^9, 3.459792313702794*^9}, {3.4597929432183037`*^9, 3.4597930743074303`*^9}}], Cell[TextData[{ "WARNING: The most common mistake made by new ", StyleBox["Mathematica", FontSlant->"Italic"], " users is to assume that ", StyleBox["xy", "Input"], " is ", StyleBox["x", "Input"], " times ", StyleBox["y", "Input"], ". It is not! ", StyleBox["xy", "Input"], " is a variable name. Thus you can evaluate" }], "Text", CellChangeTimes->{{3.459791457493709*^9, 3.4597915102069387`*^9}, { 3.45979163069882*^9, 3.459791759667755*^9}, {3.459791836554723*^9, 3.459791868105472*^9}, 3.4597930905684223`*^9}], Cell[BoxData[ RowBox[{"xy", "=", "37"}]], "Input", CellChangeTimes->{{3.459791855987174*^9, 3.459791858670259*^9}}], Cell[TextData[{ "and the symbol ", StyleBox["xy ", "Input"], "will have the value ", StyleBox["37", "Input"], ". If you really want ", StyleBox["x", "Input"], " times ", StyleBox["y", "Input"], ", you need to insert a \[SpaceKey] character or a ", StyleBox["*", "Input"], " between the ", StyleBox["x", "Input"], " and ", StyleBox["y", "Input"], ":" }], "Text", CellChangeTimes->{{3.459791873559285*^9, 3.459791899135255*^9}, { 3.459792378229158*^9, 3.459792471612481*^9}, {3.459793108241365*^9, 3.459793112201796*^9}, {3.459797242010037*^9, 3.459797248870914*^9}}], Cell[BoxData[{ RowBox[{"x", " ", "y"}], "\[IndentingNewLine]", RowBox[{"x", "*", "y"}]}], "Input", CellChangeTimes->{{3.459792474316751*^9, 3.459792481106682*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}], Cell[TextData[{ "1) What happens if you type ", StyleBox["2 \[SpaceKey] 3", "Input"], " into a cell and evaluate it? " }], "Text", CellChangeTimes->{{3.4597919945900383`*^9, 3.459792056995644*^9}}], Cell[TextData[{ "2) Does ", StyleBox["Mathematica", FontSlant->"Italic"], " do the right thing when combining addition and multiplication? Check that \ ", StyleBox["1+2*3=7", "Input"], ", not ", StyleBox["9", "Input"], "." }], "Text", CellChangeTimes->{{3.459626996378853*^9, 3.459626997233926*^9}, { 3.45962705480154*^9, 3.459627064233761*^9}, {3.4596271132479486`*^9, 3.459627212786786*^9}, 3.459627271925597*^9, {3.459792111092297*^9, 3.459792112150128*^9}, 3.4597972214780407`*^9}], Cell[TextData[{ "3) Check that parentheses work as they should by calculating ", StyleBox["(1+2)*3 ", "Input"], "and ", StyleBox["1+(2*3)", "Input"], "." }], "Text", CellChangeTimes->{{3.4596276660921173`*^9, 3.4596277607516117`*^9}, 3.4596278406769953`*^9, 3.4596279007965508`*^9, 3.459792117330449*^9, { 3.4597927258613157`*^9, 3.4597927565306177`*^9}}], Cell[TextData[{ "4) What does ", StyleBox["Mathematica", FontSlant->"Italic"], " do with ", StyleBox["(a+b)(c+d)", "Input"], "? What does ", StyleBox["Mathematica", FontSlant->"Italic"], " do with ", StyleBox["(4+1)(2+3)", "Input"], "? (In Lesson 4 we will see how to expand, factor and simplify expressions \ like ", StyleBox["(a+b)(c+d)", "Input"], ".)" }], "Text", CellChangeTimes->{{3.4597957300214453`*^9, 3.4597957947047863`*^9}, { 3.459795887751952*^9, 3.459795922464432*^9}, {3.459795952960786*^9, 3.459796004937544*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Division and Decimal Approximations", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.459796026927599*^9, 3.459796028233349*^9}, { 3.459797011106838*^9, 3.4597970186213284`*^9}}], Cell[TextData[{ "Division is fairly straightforward and illustrates a very important feature \ of the software: ", StyleBox["Mathematica", FontSlant->"Italic"], " never gives approximate results unless specifically requested. In \ particular, a fraction will not be converted to a decimal value unless you \ ask for it." }], "Text", CellChangeTimes->{{3.459791101694387*^9, 3.459791104329748*^9}, { 3.459791398863002*^9, 3.45979140265104*^9}, {3.4597960710359488`*^9, 3.459796169921709*^9}, 3.4597962120275908`*^9}], Cell[BoxData[ RowBox[{"3", "/", "11"}]], "Input", CellChangeTimes->{{3.459796238661091*^9, 3.4597962388193197`*^9}}], Cell[BoxData[ RowBox[{"60", "/", "220"}]], "Input", CellChangeTimes->{{3.4597961820877247`*^9, 3.4597961876760063`*^9}, { 3.45979624686687*^9, 3.4597962474751587`*^9}}], Cell["\<\ To get a decimal approximation for this fraction, you can use the following:\ \>", "Text", CellChangeTimes->{{3.459796262942134*^9, 3.459796273048379*^9}, { 3.459796523179991*^9, 3.459796526006894*^9}, {3.4597967435125637`*^9, 3.459796744160174*^9}}], Cell[BoxData[ RowBox[{"N", "[", RowBox[{"3", "/", "11"}], "]"}]], "Input", CellChangeTimes->{{3.459796279669243*^9, 3.4597962797956257`*^9}}], Cell[TextData[{ StyleBox["N", "Input"], " stands for numerical value and is a built-in function. Notice the square \ brackets around the fraction. This is our first example of a ", StyleBox["Mathematica", FontSlant->"Italic"], " function and the associated function syntax. " }], "Text", CellChangeTimes->{{3.459796415435192*^9, 3.4597964750236187`*^9}, 3.459796550448004*^9, {3.459964774495226*^9, 3.45996477662421*^9}}], Cell[TextData[{ "The fact that there is a decimal point in the output of this evaluation \ means that it is an approximate result. ", StyleBox["Mathematica", FontSlant->"Italic"], " makes great efforts to ensure that, so far as possible, no accuracy is \ lost in its calculations. The number ", StyleBox["1", "Input"], " in the input or output represents the number ", StyleBox["1", "Input"], " exactly, whereas ", StyleBox["1.0", "Input"], " represents an approximation (by default, with 6 significant places of \ accuracy). For example," }], "Text", CellChangeTimes->{{3.459796564317237*^9, 3.459796675159041*^9}, { 3.4597967943039227`*^9, 3.459796863238612*^9}, {3.459796899585738*^9, 3.45979695866185*^9}, 3.4597970493587847`*^9}], Cell[BoxData[ RowBox[{"3.0", "/", "11"}]], "Input", CellChangeTimes->{{3.459796919184123*^9, 3.459796922090947*^9}}], Cell[BoxData[ RowBox[{"3", "/", "11."}]], "Input", CellChangeTimes->{{3.459796967860428*^9, 3.459796971120841*^9}}], Cell[BoxData[ RowBox[{"3.0000", "/", "11.0000"}]], "Input", CellChangeTimes->{{3.4597969811561193`*^9, 3.459796993648284*^9}}], Cell[TextData[{ "If you want more accuracy, you can add a number of significant places to \ the ", StyleBox["N", "Input"], " function:" }], "Text", CellChangeTimes->{{3.459797081162859*^9, 3.459797134885429*^9}, 3.460219684370322*^9}], Cell[BoxData[ RowBox[{"N", "[", RowBox[{ RowBox[{"3", "/", "11"}], ",", " ", "25"}], "]"}]], "Input", CellChangeTimes->{{3.459797142096936*^9, 3.4597971553287888`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.4597974023393373`*^9, 3.459797404805896*^9}}], Cell[TextData[{ "1) Calculate ", StyleBox["100/10/2", "Input"], ". Is the result the same as ", StyleBox["(100/10)/2", "Input"], " or ", StyleBox["100/(10/2)", "Input"], "?" }], "Text", CellChangeTimes->{{3.459627583797243*^9, 3.459627635860958*^9}, 3.45962776401709*^9, {3.459627909872384*^9, 3.4596279648953543`*^9}, 3.459797413029327*^9}], Cell[TextData[{ "2) Evaluate ", StyleBox["N[3.0/11,25]", "Input"], ". Do you get 25 places of accuracy?" }], "Text", CellChangeTimes->{{3.4597974199307756`*^9, 3.459797479722907*^9}, 3.460219688730023*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Exponentiation", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.459797529587755*^9, 3.459797534624146*^9}, 3.4602196960310287`*^9}], Cell[TextData[{ "The last basic operation is exponentiation. To enter an exponent, use the \ ", StyleBox["^", "Input"], " (=\[ShiftKey] 6) key." }], "Text"], Cell[BoxData[ RowBox[{"2", "^", "3"}]], "Input"], Cell[BoxData[ RowBox[{"x", "^", "w"}]], "Input", CellChangeTimes->{{3.459798167220241*^9, 3.4597981693920593`*^9}}], Cell[BoxData[ RowBox[{"8", "^", RowBox[{"(", RowBox[{"1", "/", "3"}], ")"}]}]], "Input"], Cell[TextData[{ "Be careful to use parentheses where necessary to group expressions, as ", StyleBox["Mathematica", FontSlant->"Italic"], " follows the rules about the order of operations:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"8", "^", "1"}], "/", "3"}]], "Input"], Cell[TextData[{ "This was interpreted as ", StyleBox["8^1", "Input"], " divided by ", StyleBox["3", "Input"], "!!!" }], "Text", CellChangeTimes->{{3.459964895486888*^9, 3.459964901864849*^9}}], Cell[TextData[{ "Here is another context where ", StyleBox["Mathematica", FontSlant->"Italic"], "'s rule about never making approximations is apparent:" }], "Text", CellChangeTimes->{{3.459798239110353*^9, 3.4597982953930597`*^9}}], Cell[BoxData[ RowBox[{"8", "^", RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}]}]], "Input", CellChangeTimes->{{3.4597982158237333`*^9, 3.459798219260048*^9}}], Cell[BoxData[ RowBox[{"N", "[", RowBox[{"8", "^", RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.45979830874366*^9, 3.459798314524427*^9}}], Cell[BoxData[ RowBox[{"N", "[", RowBox[{ RowBox[{"8", "^", RowBox[{"(", RowBox[{"1", "/", "2"}], ")"}]}], ",", "25"}], "]"}]], "Input", CellChangeTimes->{{3.45979830874366*^9, 3.459798322845386*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.459965632102285*^9, 3.459965634210628*^9}}], Cell[TextData[{ "1) Calculate ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["123", "123"], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.459965651625599*^9, 3.459965686563156*^9}}], Cell[TextData[{ "2) Calculate ", Cell[BoxData[ FormBox[ SuperscriptBox["2", "0.1"], TraditionalForm]]], " to 20 decimal places. Not quite as easy as it looks!" }], "Text", CellChangeTimes->{{3.459965704602277*^9, 3.4599658497334757`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Built-in Constants", "Subsection", CellFrame->{{0, 0}, {0, 2}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has many built-in constants. Here are some that you will encounter. Note \ that their names all start with a capital letter. They are also displayed as \ a symbol, not as a numerical constant. To get their numerical value (with 6 \ significant digits by default), use the function ", StyleBox["N[ ]", "Input"], ".\n\nThe quantity \[Pi] shows up in the formulas for circumference and area \ of a circle, as well as in trigonometric functions:" }], "Text"], Cell[BoxData["Pi"], "Input"], Cell[BoxData[ RowBox[{"N", "[", "Pi", "]"}]], "Input"], Cell[TextData[{ "Euler's constant ", StyleBox["e", FontWeight->"Bold"], " plays a prominent role in exponential functions:" }], "Text"], Cell[BoxData["E"], "Input"], Cell[BoxData[ RowBox[{"N", "[", "E", "]"}]], "Input"], Cell[TextData[{ "The constant ", StyleBox["i", FontSlant->"Italic"], " is used to define complex numbers: ", StyleBox["i", FontSlant->"Italic"], " = ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"-", "1"}]], TraditionalForm]]], ". This constant may show up in some answers, as ", StyleBox["Mathematica", FontSlant->"Italic"], " computes answers over the complex numbers." }], "Text"], Cell[BoxData["I"], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Practice", "Subsection"]], "Subsubsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.4599660406133432`*^9, 3.459966058690249*^9}}], Cell[TextData[{ "1) Which number is bigger: ", Cell[BoxData[ FormBox[ SuperscriptBox["\[Pi]", "\[ExponentialE]"], TraditionalForm]]], "or ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", "\[Pi]"], TraditionalForm]]], "?" }], "Text", CellChangeTimes->{{3.459966076531096*^9, 3.459966095468425*^9}, { 3.4599661290525723`*^9, 3.459966155612987*^9}, 3.4599663038683853`*^9}], Cell["\<\ 2) Calculate \[Pi] to 100 decimal places.\ \>", "Text", CellChangeTimes->{{3.414942819421444*^9, 3.414942831794449*^9}, { 3.415530360755663*^9, 3.415530376742628*^9}, {3.447348674060577*^9, 3.447348683931106*^9}, {3.4473487206224823`*^9, 3.447348761678961*^9}, { 3.447348929848113*^9, 3.447348953861364*^9}, {3.447349068217382*^9, 3.447349075293099*^9}, {3.4473491369417562`*^9, 3.447349207435671*^9}, { 3.447349288845924*^9, 3.44734930673006*^9}, {3.447349381035655*^9, 3.447349435958561*^9}, {3.4599662830613728`*^9, 3.459966287699099*^9}}], Cell[TextData[{ "3) Calculate ", Cell[BoxData[ FormBox[ SuperscriptBox["\[ExponentialE]", "\[ImaginaryI]\[Pi]"], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.414942819421444*^9, 3.414942831794449*^9}, { 3.415530360755663*^9, 3.415530376742628*^9}, {3.447348674060577*^9, 3.447348683931106*^9}, {3.4473487206224823`*^9, 3.447348761678961*^9}, { 3.447348929848113*^9, 3.447348953861364*^9}, {3.447349068217382*^9, 3.447349075293099*^9}, {3.4473491369417562`*^9, 3.447349207435671*^9}, { 3.447349288845924*^9, 3.44734930673006*^9}, {3.447349381035655*^9, 3.447349435958561*^9}, 3.459966293789747*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Built-in Functions", "Subsection", CellFrame->{{0, 0}, {0, 2}}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has hundreds of built-in functions. Their names all start with a capital \ letter, and if the name is concatenated from several words, each one is \ capitalized (with no spaces in between). Note also that functions use square \ brackets, not parentheses as used in mathematical textbooks. For example:" }], "Text", CellChangeTimes->{{3.459966367766713*^9, 3.459966380486079*^9}, { 3.459966976153298*^9, 3.459966979737802*^9}}], Cell[BoxData[ RowBox[{"Sqrt", "[", "2", "]"}]], "Input"], Cell[TextData[{ "Again, the exact result is displayed because ", Cell[BoxData[ FormBox[ SqrtBox["2"], TraditionalForm]]], " is an irrational number (i.e., its decimal expansion does not terminate). \ As we have already seen, the same number can also be represented as ", StyleBox["2^(1/2)", "Input"], ". Of course," }], "Text", CellChangeTimes->{{3.459966500955579*^9, 3.45996656876525*^9}, { 3.4599669526018953`*^9, 3.4599669633650312`*^9}, {3.459967756722699*^9, 3.459967759684183*^9}}], Cell[BoxData[ RowBox[{"Sqrt", "[", "9", "]"}]], "Input", CellChangeTimes->{{3.459967762544574*^9, 3.459967770337655*^9}}], Cell["gives an exact answer.", "Text", CellChangeTimes->{{3.459967775652596*^9, 3.459967783066175*^9}}], Cell["Here are some other well known functions:", "Text", CellChangeTimes->{{3.459966460655937*^9, 3.4599664896064367`*^9}}], Cell[BoxData[ RowBox[{"Sin", "[", "1", "]"}]], "Input", CellChangeTimes->{{3.459966575391213*^9, 3.459966577462998*^9}}], Cell[BoxData[ RowBox[{"Cos", "[", "Pi", "]"}]], "Input", CellChangeTimes->{{3.4599665854867573`*^9, 3.459966589909042*^9}}], Cell[BoxData[ RowBox[{"Tan", "[", ".4", "]"}]], "Input", CellChangeTimes->{{3.459966642200074*^9, 3.459966648450371*^9}}], Cell[BoxData[ RowBox[{"ArcTan", "[", "1", "]"}]], "Input", CellChangeTimes->{{3.459966925068864*^9, 3.45996693249545*^9}, { 3.459967079831321*^9, 3.459967080543721*^9}}], Cell[BoxData[ RowBox[{"Exp", "[", "u", "]"}]], "Input", CellChangeTimes->{{3.459966600178349*^9, 3.4599666078831997`*^9}}], Cell[TextData[{ "This is another way of writing ", StyleBox["E^u", "Input"], ", that is, ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "u"], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.4599667140184927`*^9, 3.459966760423716*^9}}], Cell[BoxData[ RowBox[{"Log", "[", "3", "]"}]], "Input", CellChangeTimes->{{3.459966685479797*^9, 3.459966696324244*^9}}], Cell[TextData[{ StyleBox["Log", "Input"], " is the ", StyleBox["Mathematica", FontSlant->"Italic"], " name for the natural logarithm function. ", StyleBox["Log[10,3]", "Input"], " is the logarithm base 10 of 3." }], "Text", CellChangeTimes->{{3.459966782552936*^9, 3.459966814641574*^9}, { 3.4599668657738237`*^9, 3.459966885357141*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Clear", "Subsection", CellChangeTimes->{{3.461178682341413*^9, 3.461178683348691*^9}}], Cell[TextData[{ "A very useful function, after you lose track of what variables have what \ value, is ", StyleBox["Clear", "Input"], "." }], "Text", CellChangeTimes->{{3.459967237229508*^9, 3.459967272615746*^9}, { 3.459967832152649*^9, 3.459967836030126*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}]], "Input", CellChangeTimes->{{3.459967283281513*^9, 3.4599672902646303`*^9}}], Cell[TextData[{ "removes any value that may have been assigned to the variables ", StyleBox["x", "Input"], ", ", StyleBox["y", "Input"], " and ", StyleBox["z", "Input"], "." }], "Text", CellChangeTimes->{{3.4599673141128807`*^9, 3.459967353179707*^9}}], Cell[BoxData[ RowBox[{"x", "=", "4"}]], "Input", CellChangeTimes->{{3.45996736743817*^9, 3.459967369408143*^9}}], Cell[BoxData[ RowBox[{"y", "=", RowBox[{"x", "^", "x"}]}]], "Input", CellChangeTimes->{{3.459967376435753*^9, 3.459967381427793*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "x", "]"}]], "Input", CellChangeTimes->{{3.45996738566645*^9, 3.4599673901323843`*^9}}], Cell[BoxData["x"], "Input", CellChangeTimes->{3.4599674388921547`*^9}], Cell[BoxData[ RowBox[{"y", "=", RowBox[{"x", "^", "x"}]}]], "Input", CellChangeTimes->{{3.459967393120844*^9, 3.459967406428214*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "y", "]"}]], "Input", CellChangeTimes->{{3.45996741078615*^9, 3.45996741295392*^9}}], Cell[BoxData["y"], "Input", CellChangeTimes->{3.459967419385044*^9}], Cell["\<\ To clear all definitions that you have made use the following:\ \>", "Text", CellChangeTimes->{{3.461178408983086*^9, 3.4611784433879967`*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.461095191027028*^9, 3.461095199611408*^9}, 3.461095287965748*^9}], Cell[TextData[{ "Since one of the most common problems when starting out with ", StyleBox["Mathematica", FontSlant->"Italic"], " is losing track of which variables have been assigned values, we will \ sometimes put this command at the beginning of these lessons. " }], "Text", CellChangeTimes->{{3.461178458767337*^9, 3.461178596345893*^9}, 3.461178650837016*^9, {3.4628112045061283`*^9, 3.4628112341998568`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Postfix Notation", "Subsection", CellChangeTimes->{{3.46117869527355*^9, 3.461178699969893*^9}}], Cell[TextData[{ "A useful alternative function syntax is postfix notation - instead of ", StyleBox["f[x]", "Input"], ", write ", StyleBox["x//f", "Input"], ". Here are some examples." }], "Text", CellChangeTimes->{{3.460215981469836*^9, 3.4602160834930687`*^9}}], Cell[BoxData[ RowBox[{"Pi", "//", "N"}]], "Input", CellChangeTimes->{{3.4602160919361353`*^9, 3.4602160971815653`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Pi", "^", "2"}], "/", "6"}]], "Input", CellChangeTimes->{{3.460216444733753*^9, 3.460216451267672*^9}}], Cell[BoxData[ RowBox[{"%", "//", "N"}]], "Input", CellChangeTimes->{{3.460216455483795*^9, 3.460216457323893*^9}}], Cell[BoxData[ RowBox[{"3", "//", "Exp"}]], "Input", CellChangeTimes->{{3.4602163797445087`*^9, 3.4602163908351088`*^9}}], Cell[BoxData[ RowBox[{"x", "//", "Clear"}]], "Input", CellChangeTimes->{{3.460216398654698*^9, 3.460216400991736*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Practice", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.459967120400589*^9, 3.459967122791553*^9}}], Cell[TextData[{ "1) Calculate ", Cell[BoxData[ FormBox[ SqrtBox[ SuperscriptBox["x", "2"]], TraditionalForm]]], "and (", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{ SqrtBox["x"], ")"}], "2"], TraditionalForm]]], "( assuming that ", StyleBox["x", FontSlant->"Italic"], " has not been assigned a value. Use ", StyleBox["Clear[x]", "Input"], " to be sure.)." }], "Text", CellChangeTimes->{{3.4599671272402573`*^9, 3.459967166968575*^9}, { 3.4599675138715553`*^9, 3.459967609354484*^9}, {3.459967937954493*^9, 3.459967965686331*^9}, {3.462811268192885*^9, 3.4628112778969603`*^9}, { 3.462811338872448*^9, 3.462811341833713*^9}}], Cell[TextData[{ "2) Calculate ", StyleBox["tan(", "InlineFormula"], "\[Pi]", StyleBox["/3).", "InlineFormula"] }], "Text", CellChangeTimes->{{3.459967862463379*^9, 3.4599678642094507`*^9}, { 3.459967982431273*^9, 3.459967985878909*^9}, {3.4599680260067177`*^9, 3.4599680536362047`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Homework", "Subsection", CellFrame->{{0, 0}, {0, 2}}, CellChangeTimes->{{3.459968176088738*^9, 3.459968183858399*^9}}], Cell[TextData[{ "1) ", "Evaluate the following expressions using ", StyleBox["Mathematica", FontSlant->"Italic"], ". Explain the difference between the pairs of inputs:" }], "Text", CellChangeTimes->{{3.45996923141405*^9, 3.459969234078657*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"3", "x"}], " ", "+", " ", RowBox[{"x", " ", "7"}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"3", "x"}], " ", "+", " ", "x7"}]], "Input"], Cell[TextData[{ "2) Evaluate the following expressions using ", StyleBox["Mathematica", FontSlant->"Italic"], ". 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