(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 55477, 2017]*) (*NotebookOutlinePosition[ 56406, 2048]*) (* CellTagsIndexPosition[ 56362, 2044]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Arithmetic", "Title", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True], Cell[TextData[{ "Sean Mauch\nsean@caltech.edu\n", ButtonBox["http://www.its.caltech.edu/~sean", ButtonData:>{ URL[ "http://www.its.caltech.edu/~sean"], None}, ButtonStyle->"Hyperlink"], "\nThis work is distributed under the GNU FDL. See ", ButtonBox["license.nb ", ButtonData:>{"license.nb", None}, ButtonStyle->"Hyperlink"], "for details." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "Welcome to ", StyleBox["Mathematica", FontSlant->"Italic"], "! This is a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks contain text, graphics, and commands to execute. You can write \ in your own commands and text anywhere within the notebook." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " notebooks are grouped by cells. Cells are shown by the brackets on the \ far right. The outermost, (rightmost), bracket encloses the entire notebook. \ Each block of text, input and graphic has its own cell." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["\<\ The list of titles preceded by black boxes below are the sections \ of this notebook. Each section is enclosed in a cell. Cells that contain \ multiple other cells, (like section cells and notebook cells), can be either \ open or closed. Now all of the section cells, (except for this one), are \ closed. When a cell is closed, the enclosing bracket has an arrow at the \ bottom. \ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["\<\ Now we are going to open the section on Arithmetic Operators. \ Position the cursor over the bracket that contains the section. The cursor \ will turn into a left arrow. Align the vertical line on the cursor with the \ section bracket and double click the left mouse button. This will open the \ cell and you will see the contents of the section. The bracket containing \ the section is now highlighted. \ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData[ "Now to close the section cell: Again position the cursor so the vertical \ line is over the section bracket and double click. The section should now be \ closed. Another method of toggling between open and closed sections is to \ select the section bracket and press \[ControlKey]\[LeftModified]'\ \[RightModified]."], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["\<\ Work through each of the following sections in order. When you are \ done with a section it will probably be convenient for you to close it. This \ enables you to move around the notebook more efficiently as the list of \ closed section cells gives you a table of contents of the notebook.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Arithmetic Operators"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can do arithmetic just like you are used to doing on a calculator. Click \ the cursor anywhere in the following arithmetic expression and then press \ \[ShiftKey]\[LeftModified]\[ReturnKey]\[RightModified]. The bracket on the \ right-hand side of the cell will expand as the expression is being evaluated. \ When the ", StyleBox["Mathematica", FontSlant->"Italic"], " kernel has evaluated the expression it will print an \"In[1]:=\" in front \ of the input line and print an \"Out[1]:=\" followed by the value of the \ expression. The first command that you execute in a ", StyleBox["Mathematica", FontSlant->"Italic"], " session takes longer than normal, because the kernel must be loaded \ before any input can be processed." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["3.14159 + 2.71828 - 69 / 47 * 42^2", "Input", AspectRatioFixed->True], Cell[TextData[ "To type your own inputs first position the cursor anywhere in the notebook \ between cells. The cursor will become a short, horizontal bracket. When you \ click there, a horizontal line across the page will appear. Now you can type \ your expression and press \[ShiftKey]\[LeftModified]\[ReturnKey]\ \[RightModified] when you want to evaluate it. You can also move around the \ notebook with the arrow keys."], "Text", TextJustification->1], Cell[TextData[{ "The ^ symbol raises numbers to powers. In ", StyleBox["Mathematica", FontSlant->"Italic"], ", a space denotes multiplication. Thus" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["1 2 3", "Input", AspectRatioFixed->True], Cell["Is the same as", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["1*2*3", "Input", AspectRatioFixed->True], Cell["\<\ The precedence , (or order of evaluation), of the five arithmetic \ operators is as follows:\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {"^", "power"}, {"/", "division"}, {"*", "multiplication"}, {\(+-\), \(addition\ and\ \(subtraction . \)\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "In other words: When ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluates an expression, it first all the exponentiations, (^), then it \ does the divisions and multiplications and finally the additions and \ subtractions. Thus the first expression we entered is interpretted as" }], "Text", Evaluatable->False, TextJustification->1, AspectRatioFixed->True], Cell["(3.14159 + 2.71828) - ((69 / 47) * (42^2))", "Input", AspectRatioFixed->True], Cell["\<\ The associativity of the power operator is right-to-left while the \ rest of the arithmetic operators associate from left to right. This means \ that the expressions\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["2^3^4", "Input", AspectRatioFixed->True], Cell["2/3/4", "Input", AspectRatioFixed->True], Cell["2-3-4", "Input", AspectRatioFixed->True], Cell[TextData["are evaluated as"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["2^(3^4)", "Input", AspectRatioFixed->True], Cell["(2/3)/4", "Input", AspectRatioFixed->True], Cell["(2-3)-4", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Built-In Constants", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has a number of built-in constants. Note that all the names of these \ constants begin with a capital letter." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {"Pi", \(3.14159 ... \)}, {"E", \(The\ base\ of\ the\ natural\ logarithm . \ \ 2.718281828 ... \)}, {"Degree", \(\[Pi]\/180\)}, {"I", \(\@\(-1\)\)}, {"Infinity", \(lim\_\(x \[Rule] \(0\^+\)\)\((1\/x)\)\)}, {"ComplexInfinity", \(lim\_\(z \[Rule] 0\)\((1\/z)\), \ z \[Element] \[DoubleStruckCapitalC]\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ You can use constants in expressions exactly as you would \ numbers.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(Pi\)], "Input"], Cell["E^(I Pi)", "Input", AspectRatioFixed->True], Cell[BoxData[ \(\((1 + I)\) \((1 - I)\)\)], "Input"], Cell[BoxData[ \(\((1 + I)\)^\((\(-1\))\)\)], "Input"], Cell[BoxData[ \(E^\((\(-Infinity\))\)\)], "Input"], Cell["1 / Infinity", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Exercise 1", "Subsubsection"], Cell["\<\ a) Try to divide a number by zero. b) Find the difference and quotient of Infinity and Infinity.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Exact and Approximate Numbers", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "There are two basic types of numbers in ", StyleBox["Mathematica", FontSlant->"Italic"], ", exact and approximate numbers. ", StyleBox["Mathematica", FontSlant->"Italic"], " does symbolic arithmetic when working with exact numbers. Integers, \ rational numbers and built-in constants are exact numbers." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(123456789^4\)], "Input"], Cell[BoxData[ \(1/2 + 1/3 + 1/5 + 1/7\)], "Input"], Cell[BoxData[ \(2\ Pi\)], "Input"], Cell[BoxData[ \(E^\((I\ Pi)\)\)], "Input"], Cell[TextData[{ "Approximate numbers have a decimal point. Calculations with these numbers \ are done with floating point arithmetic. By default, floating point numbers \ have about 16 significant digits in internal memory and six digits are \ displayed in the answer. Thus the default floating point numbers in ", StyleBox["Mathematica", FontSlant->"Italic"], " correspond to double precision numbers in the C programming language." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["1.23 * 4.56", "Input", AspectRatioFixed->True], Cell["1.23 / 4.56", "Input", AspectRatioFixed->True], Cell[TextData[{ "You can see precision of floating point numbers by checking the value of \ $MachinePrecision, (which is another constant in ", StyleBox["Mathematica", FontSlant->"Italic"], ")." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \($MachinePrecision\)], "Input"], Cell[TextData[{ "If you want to see more or fewer digits of an answer you can change the \ default output format with ", StyleBox["NumberForm[]", FontWeight->"Bold"], ". The first argument of this function is the expression to be evaluated, \ and the second is the number of digits to be displayed." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(1.23\)], "Input"], Cell[TextData[{ "Without a second argument, ", StyleBox["NumberForm[]", FontWeight->"Bold"], " will give you the default output form of an expression." }], "Text", TextJustification->1], Cell[BoxData[ \(NumberForm[1.23]\)], "Input"], Cell[TextData[{ "You can tell ", StyleBox["Mathematica", FontSlant->"Italic"], " to show you more or fewer digits." }], "Text", TextJustification->1], Cell[BoxData[ \(NumberForm[1.23, 2]\)], "Input"], Cell[BoxData[ \(NumberForm[1.23, 16]\)], "Input"], Cell[TextData[{ "The approximate real number in the next input is only known to 16 digits. \ ", StyleBox["Mathematica", FontSlant->"Italic"], " will not show you more digits than it knows." }], "Text", TextJustification->1], Cell[BoxData[ \(NumberForm[1.23, 200]\)], "Input"], Cell[TextData[{ "From the above input we see that the floating point representation of 1.23 \ is not the same as the exact number, ", Cell[BoxData[ \(TraditionalForm\`123/100\)]], ". " }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData[{ "If you want ", StyleBox["Mathematica", FontSlant->"Italic"], " to treat an integer as a real number, put a decimal point after the \ number." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(19^19/13^13\)], "Input"], Cell["19.^19./13.^13. ", "Input", AspectRatioFixed->True], Cell["\<\ If any part of an expression is approximate, (uses floating point \ numbers), then the entire expression will be evaluated approximately. \ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["1./2+1/3+1/5 ", "Input", AspectRatioFixed->True], Cell[TextData[{ "By default, any floating point number smaller than ", Cell[BoxData[ \(TraditionalForm\`10\^\(-5\)\)]], " or at least as large as ", Cell[BoxData[ \(TraditionalForm\`10\^6\)]], " will be displayed in scientific notation." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["13.0^10", "Input", AspectRatioFixed->True], Cell["13.0^-10", "Input", AspectRatioFixed->True], Cell["You can enter numbers in scientific notation in two ways:", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["1.23 10^20", "Input", AspectRatioFixed->True], Cell[BoxData[ \(1.23*^20\)], "Input"], Cell[TextData[{ "You can force an output to be displayed in scientific notation with the ", StyleBox["ScientificForm[]", FontWeight->"Bold"], " function." }], "Text", TextJustification->1], Cell[BoxData[ \(ScientificForm[1. /3. ]\)], "Input"], Cell[BoxData[ \(ScientificForm[1. /3. , 12]\)], "Input"], Cell["\<\ Likewise, you can specify that an output should be displayed in \ engineering form. (The exponent will be a multiple of 3.)\ \>", "Text", TextJustification->1], Cell[BoxData[ \(EngineeringForm[1. /3. , 12]\)], "Input"], Cell[TextData[{ "Finally, if you want to supress scientific notation, use the ", StyleBox["AccountingForm[]", FontWeight->"Bold"], " function." }], "Text", TextJustification->1], Cell[BoxData[ \(AccountingForm[13. ^10, 12]\)], "Input"], Cell[CellGroupData[{ Cell["Exercise 2", "Subsubsection"], Cell["\<\ a) Calculate the value of Pi to 16 digits. b) Explain the outputs of the following commands.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(E^2. - E^2\)], "Input"], Cell[BoxData[ \(E^\((2.0 I\ Pi)\)\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["The Numerical Function", "Section", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ You can get a numerical approximation to an exact expression by \ using the numerical function.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["N[1/2+1/3+1/5]", "Input", AspectRatioFixed->True], Cell["N[19^19/13^13]", "Input", AspectRatioFixed->True], Cell["\<\ Note that the argument of the numerical function is enclosed in \ square brackets instead of parenthesis. You can also use the numerical \ function to evaluate an expression to any degree of precision. (The \ precision of an approximate number is defined as the number of digits in the \ number.) The following calculations are evaluated with 30 and 40 decimal \ digits of precision, respectively.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["N[1/2+1/3+1/5,30]", "Input", AspectRatioFixed->True], Cell["N[19^19/13^13,40]", "Input", AspectRatioFixed->True], Cell[TextData[{ "You can see the precision of an expression with the ", StyleBox["Precision[]", FontWeight->"Bold"], " function. If you enter an approximate number with no more that 16 \ digits, ", StyleBox["Mathematica", FontSlant->"Italic"], " will store it as a machine precision number." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(Precision[1.23]\)], "Input"], Cell[TextData[{ "If you enter an approximate number with ", Cell[BoxData[ \(TraditionalForm\`n > 16\)]], " digits, then ", StyleBox["Mathematica", FontSlant->"Italic"], " will store it as an ", Cell[BoxData[ \(TraditionalForm\`n\)]], " digit precision number." }], "Text"], Cell[BoxData[ \(1.3333333333333333333333\)], "Input"], Cell[BoxData[ \(Precision[1.3333333333333333333333]\)], "Input"], Cell["Note that the precision of an exact number is infinite.", "Text"], Cell[BoxData[ \(Precision[Pi]\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " gives different answers for the following inputs." }], "Text"], Cell[BoxData[ \(N[E^\((1/2)\), 40]\)], "Input"], Cell[BoxData[ \(N[E^0.5, 40]\)], "Input"], Cell[TextData[{ "This is because ", Cell[BoxData[ \(TraditionalForm\`E\^\(1/2\)\)]], " is an exact number; ", StyleBox["Mathematica", FontSlant->"Italic"], " can calculate an approximation to any degree of precision. However, ", Cell[BoxData[ \(TraditionalForm\`0.5\)]], " is a machine precision number, (16 digits), thus you can only determine ", Cell[BoxData[ \(TraditionalForm\`E\^0.5\)]], " to machine precision." }], "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell["Exercise 3", "Subsubsection"], Cell["Calculate the value of Pi to 100 digits.", "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Built-In Functions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " provides all of the functions that you are familiar with from algebra, \ finite math and trigonometry. All of the ", StyleBox["Mathematica", FontSlant->"Italic"], " built-in functions begin with a capital letter When using standard form, \ the argument of a function is enclosed in square brackets. " }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Trigonometry"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " supplies the following trigonometric functions:" }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {\(Exp[x]\), \(E\^x\)}, {\(Log[x]\), \(The\ natural\ \(logarithm . \)\)}, {\(Log[b, \ x]\), \(The\ logarithm\ base\ \(b . \)\)}, {\(Sin[x], \ Cos[x], \ Tan[x]\), \(The\ argument\ is\ in\ \(radians . \)\)}, {\(ArcSin[x], \ ArcCos[x], \ ArcTan[x], \ ArcTan[x, \ y]\), \(The\ result\ is\ given\ in\ \(radians . \)\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " is able to evaluate many trigonometric expressions exactly." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Exp[Log[1]]", "Input", AspectRatioFixed->True], Cell["Log[3,81]", "Input", AspectRatioFixed->True], Cell["Sin[Pi/2]", "Input", AspectRatioFixed->True], Cell["Sin[30 Degree] ", "Input", AspectRatioFixed->True], Cell["ArcTan[1]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["ArcTan[x,y]", "Input"], " returns the inverse tangent of ", Cell[BoxData[ \(TraditionalForm\`y/x\)]], " taking into account which quadrant the point ", Cell[BoxData[ \(TraditionalForm\`\((x, y)\)\)]], " is in." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["ArcTan[-1,-1]", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Arithmetic and Algebra"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " provides the following functions from arithmetic and algebra." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {\(Sqrt[x]\), \(The\ square\ \(root . \)\)}, {\(Factorial[x], \ \(x!\)\), \(The\ factorial\ \(function . \)\)}, {\(Abs[x]\), \(The\ absolute\ value\ of\ \(x . \)\)}, {\(Round[x], \ Floor[x], \ Ceiling[x]\), \(Functions\ for\ rounding\ real\ numbers\ to\ \(integers . \)\)}, {\(Mod[n, \ m]\), \(The\ modulus\ \(function . \)\)}, {\(Random[]\), \(Returns\ a\ pseudo - random\ real\ number\ between\ 0\ and\ 1. \)}, {\(Max[x, \ y, \ ... ], \ Min[x, \ y, \ ... ]\), RowBox[{"Maximum", " ", "and", " ", "minimum", " ", "of", " ", StyleBox["a", FontSlant->"Italic"], " ", "sequence", " ", "of", " ", \(arguments . \)}]}, {\(FactorInteger[n]\), \(Returns\ a\ list\ of\ the\ factors\ of\ an\ \(integer . \)\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "The ", StyleBox["Sqrt[]", "Input"], " function returns the positive square root of a complex number. More \ precisely, it return a number whose argument is in the range ", Cell[BoxData[ \(TraditionalForm\`\((\(-\[Pi]\)/2, \[Pi]/2]\)\)]], "." }], "Text", Evaluatable->False, TextJustification->1, AspectRatioFixed->True], Cell["Sqrt[25]", "Input", AspectRatioFixed->True], Cell["Sqrt[-25]", "Input", AspectRatioFixed->True], Cell[TextData[{ "For positive integral arguments, ", Cell[BoxData[ \(TraditionalForm\`n\)]], ", ", StyleBox["Factorial[n]", "Input"], " returns ", Cell[BoxData[ \(TraditionalForm \`\(n \((n - 1)\) \((n - 2)\) ... \) \((2)\) \((1)\)\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Factorial[5]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Another syntax for the factorial of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is ", Cell[BoxData[ \(TraditionalForm\`\(n!\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["100!", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Abs[x]", "Input"], " returns the absolute value of ", Cell[BoxData[ \(TraditionalForm\`x\)]], " for real-valued ", Cell[BoxData[ \(TraditionalForm\`x\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Abs[-23.4]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Round[x]", "Input"], " returns the integer closest to the real number ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". For half-integral values it rounds down." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Round[7.53]", "Input", AspectRatioFixed->True], Cell["Round[8.5]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Floor[x]", "Input"], " returns the greatest integer less than or equal to ", Cell[BoxData[ \(TraditionalForm\`x\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Floor[8.7]", "Input", AspectRatioFixed->True], Cell["Floor[-23.4]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Ceiling[x]", "Input"], " returns the smallest integer greater than or equal to ", Cell[BoxData[ \(TraditionalForm\`x\)]], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Ceiling[13.6]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mod[n,m]", "Input"], " returns the remainder when ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is divided by ", Cell[BoxData[ \(TraditionalForm\`m\)]], " using integer division. The result has the same sign as ", Cell[BoxData[ \(TraditionalForm\`m\)]], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Mod[12,5]", "Input", AspectRatioFixed->True], Cell["Mod[12,-5]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Random[]", "Input"], " returns a uniformly distributed, pseudo-random real number between 0 and \ 1. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Random[]", "Input", AspectRatioFixed->True], Cell[TextData[{ "You can also specify that ", StyleBox["Random[]", "Input"], " return an integer or a complex number. Below is a random integer between \ 0 and 1 inclusive and a random complex number where both the real and \ imaginary parts are between 0 and 1." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Random[Integer]", "Input", AspectRatioFixed->True], Cell["Random[Complex]", "Input", AspectRatioFixed->True], Cell["\<\ Additionally you can specify the range of the random numbers by \ giving a lower and upper bound in the form \"{lower, upper}\" as the second \ argument. If you only give a single number, num, for the second argument \ then the range is {0, num}. Here is a random integer between 5 and 25 \ inclusive and a random real number between 0 and 11.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Random[Integer,{5,25}]", "Input", AspectRatioFixed->True], Cell["Random[Real,11]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Max[x,y,\[Ellipsis]]", "Input"], " returns the largest of the arguments. Likewise, ", StyleBox["Min[x,y,\[Ellipsis]]", "Input"], " returns the minimum of its arguments." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Max[5,3,7,2,4]", "Input", AspectRatioFixed->True], Cell["Min[5,3,7,2,4]", "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["FactorInteger[n]", "Input"], " returns a list of the factors of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " together with their multiplicities. Below we factor the integers:\n\t", Cell[BoxData[ \(TraditionalForm\`1996\ = \ 2\^2\ \[Times]\ 499\)]], "\n\t", Cell[BoxData[ \(TraditionalForm \`123456789\ = \ 3\^2\ \[Times]\ 3607\ \[Times]\ 3803. \)]] }], "Text", Evaluatable->False, TextJustification->1, AspectRatioFixed->True], Cell["FactorInteger[1996]", "Input", AspectRatioFixed->True], Cell["FactorInteger[123456789]", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Exercise 4", "Subsubsection"], Cell[TextData[ "Find the greatest common divisor and least common multiple of 123456789 and \ 987654321."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Postfix Form"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ In the standard form of using functions, (the form we have been \ using so far), the function name is followed by the arguments to the function \ enclosed in square brackets. In the Postfix form, the function argument is \ followed by two backslashes and then the function name. The following two \ inputs are equivalent.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Sin[1]", "Input", AspectRatioFixed->True], Cell["1//Sin", "Input", AspectRatioFixed->True], Cell[TextData[{ "The postfix operator has very low precendence so ", StyleBox["Pi+3//Sqrt ", "Input"], "is interpretted as ", StyleBox["Sqrt[Pi+3] ", "Input"], "and not ", StyleBox["Pi+Sqrt[3].", "Input"] }], "Text", Evaluatable->False, TextJustification->1, AspectRatioFixed->True], Cell["Pi+3//Sqrt", "Input", AspectRatioFixed->True], Cell[TextData[ "It is common to use the postfix form for the numerical function."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Sqrt[3+Pi]//N", "Input", AspectRatioFixed->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Exact Expressions"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will not approximate exact expressions unless you explicitly tell it to." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Sin[1]", "Input", AspectRatioFixed->True], Cell[TextData[{ "is the exact value of the sine of 1. If you want an approximation of ", StyleBox["Sin[1]", "Input"], " type one of the following:" }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["N[Sin[1]]", "Input", AspectRatioFixed->True], Cell["Sin[1.0]", "Input", AspectRatioFixed->True], Cell[TextData["Likewise, an expression like"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["1/2+Sqrt[1/2]", "Input", AspectRatioFixed->True], Cell[TextData["will be left in its exact form."], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Mathematica Help", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "If you want to get the definition of a constant or function in ", StyleBox["Mathematica", FontSlant->"Italic"], ", just type a question mark followed by the name of the function or \ constant." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["?Max", "Input", AspectRatioFixed->True], Cell["?Pi", "Input", AspectRatioFixed->True], Cell[TextData[{ "You can use the asterisk as a wildcard to match any string of letters in \ the name. The following command returns all of the ", StyleBox["Mathematica", FontSlant->"Italic"], " expressions that start with \"Mod\"." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["?Mod*", "Input", AspectRatioFixed->True], Cell[TextData[{ "This query returns all of the ", StyleBox["Mathematica", FontSlant->"Italic"], " expressions that contain the string \"Mod\"." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["?*Mod*", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Referring to Previous Input and Output"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "You can access the result of previous output with ", StyleBox["%", "Input"], ". ", StyleBox["%%", "Input"], " refers to the result before the last, ", StyleBox["%%%", "Input"], " refers to the third previous result, etc." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["1/2+Sqrt[1/2]", "Input", AspectRatioFixed->True], Cell[TextData["%^2"], "Input", AspectRatioFixed->True], Cell[TextData["Sqrt[%%]"], "Input", AspectRatioFixed->True], Cell[TextData["%%%^3"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Out[n]", "Input"], " refers to the ", Cell[BoxData[ \(TraditionalForm\`n\^th\)]], " output. ", StyleBox["In[n]", "Input"], " refers to the ", Cell[BoxData[ \(TraditionalForm\`n\^th\)]], " input. An alternate notation for ", StyleBox["Out[n]", "Input"], " is ", StyleBox["%n", "Input"], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData[{ "You can copy the previous input by selecting Input\[Rule]Copy Input From \ Above or by pressing \[ControlKey]\[LeftModified]l\[RightModified]. (That's \ the letter l and not the number 1). You can copy the previous output into a \ new input by selecting Input\[Rule]Copy Output from Above or by pressing \ \[ControlKey]\[LeftModified]\[ShiftKey]\[LeftModified]l\[RightModified]\ \[RightModified]. This is handy when you want to edit an output. Below I \ have calculated an approximation of ", Cell[BoxData[ \(TraditionalForm\`\@13\^17\)]], " by copying the output from above." }], "Text", TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(13^17\)], "Input"], Cell[BoxData[ \(8650415919381337933\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(8650415919381337933 // Sqrt\)], "Input"], Cell[BoxData[ \(815730721\ \@13\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(815730721\ \@13 // N\)], "Input"], Cell[BoxData[ \(2.94115894153671`*^9\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["User-Defined Constants"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will let you define your own constants as follows." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["x=1/2", "Input", AspectRatioFixed->True], Cell["y=Sqrt[1/2]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Anywhere you use ", Cell[BoxData[ \(TraditionalForm\`x\)]], " or ", Cell[BoxData[ \(TraditionalForm\`y\)]], " in an input expression, ", StyleBox["Mathematica", FontSlant->"Italic"], " will substitute the value of the constant, just as it would for built-in \ constants like ", StyleBox["Pi", "Input"], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["z=x+y", "Input", AspectRatioFixed->True], Cell[TextData[{ "Note that changing the value of ", Cell[BoxData[ \(TraditionalForm\`x\)]], " now will not change the value of ", Cell[BoxData[ \(TraditionalForm\`z\)]], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["x=42", "Input", AspectRatioFixed->True], Cell[TextData["z"], "Input", AspectRatioFixed->True], Cell[TextData[{ "In order to get rid of the definition of a variable, use the ", StyleBox["Clear[]", "Input"], " function. The ", StyleBox["Clear[]", "Input"], " function can take any number of arguments." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Clear[x,y,z]", "Input", AspectRatioFixed->True], Cell[TextData[{ "Now ", Cell[BoxData[ \(TraditionalForm\`x\)]], " is no longer a constant. Inputting ", Cell[BoxData[ \(TraditionalForm\`x\)]], " returns ", Cell[BoxData[ \(TraditionalForm\`x\)]], " as a formal variable." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[TextData["x"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Recall that assignment associates right to left so the following statement \ is interpretted as ", Cell[BoxData[ \(TraditionalForm\`x = \((y = 42)\)\)]], " and has the effect of assigning the value of 42 to both ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`y\)]], "." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["x=y=42", "Input", AspectRatioFixed->True], Cell[TextData["x"], "Input", AspectRatioFixed->True], Cell[TextData[{ "You can use ", StyleBox["Mathematica", FontSlant->"Italic"], " help to get information about a user-defined constant. Below ", StyleBox["Mathematica", FontSlant->"Italic"], " says that ", Cell[BoxData[ \(TraditionalForm\`x\)]], " is a globally defined constant with value 42." }], "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["?x", "Input", AspectRatioFixed->True], Cell["\<\ Since built-in constants begin with a capital letter, it is best to \ have user-defined constants start with a lower-case letter so there is no \ confusion. Also, it is best to clear the definitions of user-defined \ constants when you are done with them so you don't accidentally use their \ definitions in subsequent calculations.\ \>", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell["Clear[x,y]", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Aborting a Calculation", "Section"], Cell[TextData[{ "If ", StyleBox["Mathematica", FontSlant->"Italic"], " is taking longer than you are willing to wait to perform a calculation, ", Cell[BoxData[ \(TraditionalForm\`you\)]], " can abort the command. Execute the command below that will try to \ calculate the ", Cell[BoxData[ \(TraditionalForm\`\((10\^10)\)\^th\)]], " prime number. " }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(Prime[10^10]\)], "Input"], Cell[TextData[ "You can abort this calculation either by pressing \[AltKey]\[LeftModified].\ \[RightModified] or clicking on the cell bracket and then selecting Kernel\ \[Rule]Abort Evaluation."], "Text", TextAlignment->Left, TextJustification->1] }, Closed]], Cell[CellGroupData[{ Cell["Typsetting Input", "Section"], Cell[TextData[{ "You've probably noticed by now that ", StyleBox["Mathematica", FontSlant->"Italic"], " output looks better than ", StyleBox["Mathematica", FontSlant->"Italic"], " input. For example:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(\((1 + x)\)^2/\((y + z)\)\)], "Input"], Cell["\<\ The input is one-dimensional. It consists of a sequence of \ characters written in a line. The output is two dimensional. It can have \ superscripts, subscripts and fractions.\ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell["Greek Letters", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " will allow you to make the input look spiffy. For starters, you can type \ any Greek letter. Below this cell type \[EscapeKey]a\[EscapeKey]. When you \ hit the escape key you will see the character, \[AliasDelimiter]. Note that \ as soon as you finish typing the second escape key the three letter sequence \ turns into \[Alpha], the Greek letter alpha. Now hit \[ShiftKey]\ \[LeftModified]\[ReturnKey]\[RightModified] and note that \[Alpha] is also \ output." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " treats Greek letters exactly as it treats English letters. You can use \ them in input and expressions exactly as you would other letters. Below is a \ table showing how to type some Greek letters." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {"\[Alpha]", "\[EscapeKey]a\[EscapeKey]", "\[Nu]", "\[EscapeKey]n\[EscapeKey]", "\[CapitalGamma]", "\[EscapeKey]G\[EscapeKey]"}, {"\[Beta]", "\[EscapeKey]b\[EscapeKey]", "\[Xi]", "\[EscapeKey]x\[EscapeKey]", "\[CapitalDelta]", "\[EscapeKey]D\[EscapeKey]"}, {"\[Gamma]", "\[EscapeKey]g\[EscapeKey]", "\[Pi]", "\[EscapeKey]p\[EscapeKey]", "\[CapitalTheta]", "\[EscapeKey]Th\[EscapeKey]"}, {"\[Delta]", "\[EscapeKey]d\[EscapeKey]", "\[Rho]", "\[EscapeKey]r\[EscapeKey]", "\[CapitalLambda]", "\[EscapeKey]L\[EscapeKey]"}, {"\[Epsilon]", "\[EscapeKey]e\[EscapeKey]", "\[Sigma]", "\[EscapeKey]s\[EscapeKey]", "\[CapitalPi]", "\[EscapeKey]P\[EscapeKey]"}, {"\[Zeta]", "\[EscapeKey]z\[EscapeKey]", "\[Tau]", "\[EscapeKey]t\[EscapeKey]", "\[CapitalSigma]", "\[EscapeKey]S\[EscapeKey]"}, {"\[Eta]", "\[EscapeKey]eta\[EscapeKey]", "\[Phi]", "\[EscapeKey]phi\[EscapeKey]", "\[CapitalUpsilon]", "\[EscapeKey]Ui\[EscapeKey]"}, {"\[Theta]", "\[EscapeKey]th\[EscapeKey]", "\[CurlyPhi]", "\[EscapeKey]cphi\[EscapeKey]", "\[CapitalPhi]", "\[EscapeKey]Phi\[EscapeKey]"}, {"\[Kappa]", "\[EscapeKey]k\[EscapeKey]", "\[Chi]", "\[EscapeKey]c\[EscapeKey]", "\[CapitalChi]", "\[EscapeKey]C\[EscapeKey]"}, {"\[Lambda]", "\[EscapeKey]l\[EscapeKey]", "\[Psi]", "\[EscapeKey]y\[EscapeKey]", "\[CapitalPsi]", "\[EscapeKey]Y\[EscapeKey]"}, {"\[Mu]", "\[EscapeKey]m\[EscapeKey]", "\[Omega]", "\[EscapeKey]o\[EscapeKey]", "\[CapitalOmega]", "\[EscapeKey]O\[EscapeKey]"} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "Try using a few Greek letters in some expressions. In ", StyleBox["Mathematica", FontSlant->"Italic"], " the letter \[Pi] stands for the mathematical constant, Pi:" }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(N[\[Pi]]\)], "Input"], Cell[TextData[{ "This is the only Greek letter in ", StyleBox["Mathematica", FontSlant->"Italic"], " that has a built-in value." }], "Text", TextAlignment->Left, TextJustification->1] }, Closed]], Cell[CellGroupData[{ Cell["Built-In Constants", "Subsection"], Cell[TextData[{ "Some of the built-in constants in ", StyleBox["Mathematica", FontSlant->"Italic"], " have spiffy typeset forms." }], "Text"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {"Character", "Input", "Meaning"}, {"\[Pi]", "\[EscapeKey]p\[EscapeKey]", "Pi"}, {"\[ExponentialE]", "\[EscapeKey]ee\[EscapeKey]", "E"}, {"\[Degree]", "\[EscapeKey]deg\[EscapeKey]", "Degree"}, {"\[ImaginaryI]", "\[EscapeKey]ii\[EscapeKey]", "I"}, {"\[Infinity]", "\[EscapeKey]inf\[EscapeKey]", "Infinity"} }, ColumnAlignments->{Left}, RowLines->{True, False}, ColumnLines->True], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ Below is an typeset input. Try a few inputs of your own using \ these constants.\ \>", "Text", TextJustification->1], Cell[BoxData[ \(\[ExponentialE]\^\(\[ImaginaryI]\ \[Pi]\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Two-Dimensional Input", "Subsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " interprets" }], "Text"], Cell[BoxData[ \(x^3\)], "Input"], Cell[TextData[{ "as ", Cell[BoxData[ \(TraditionalForm\`x\)]], " to the power 3. You can type the superscript as input with the character \ sequence\n\tx \[ControlKey]\[LeftModified]6\[RightModified] 3 \[ControlKey]\ \[LeftModified]\[SpaceKey]\[RightModified]\nType this below and then \ \[ShiftKey]\[LeftModified]\[ReturnKey]\[RightModified]. The \[ControlKey]\ \[LeftModified]6\[RightModified] places you in superscript mode. (Note that \ \[ControlKey]\[LeftModified]6\[RightModified] makes sense as \[ShiftKey]\ \[LeftModified]6\[RightModified] would type a ^.) The \[ControlKey]\ \[LeftModified]\[SpaceKey]\[RightModified] returns you to normal mode." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[ "\[ControlKey]\[LeftModified]-\[RightModified] puts you in subscript mode. \ (This is analogous to \[ShiftKey]\[LeftModified]-\[RightModified] making _.) \ \[ControlKey]\[LeftModified]5\[RightModified] toggles between superscript and \ subscript mode."], "Text", TextAlignment->Left, TextJustification->1], Cell["\<\ Here are a few examples of using subscripts and superscripts that \ you can try typing as input.\ \>", "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ { \(\(x\ \[ControlKey]\[LeftModified]6\[RightModified]\ 2\ \[ControlKey]\[LeftModified]5]\)\ n\), \(x\_n\%2\)}, { \(x\ \[ControlKey]\[LeftModified]6\[RightModified]\ 3\ \[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified]\ + \ y\ \[ControlKey]\[LeftModified] - \[RightModified]\ 2\), \(x\^3 + y\_2\)}, { \(x\ \[ControlKey]\[LeftModified]6\[RightModified]\ y\ \[ControlKey]\[LeftModified]6\[RightModified]\ z\), \(x\^\(y\^z\)\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->1], Cell[TextData[ "You can create fractions with \[ControlKey]\[LeftModified]/\[RightModified]. \ For example, you can try:"], "Text", TextAlignment->Left], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {\(x\ \[ControlKey]\[LeftModified]/\[RightModified]\ y\), \(x\/y\)}, {\(x\ \[ControlKey]\[LeftModified]/\[RightModified]\ x + y\), \(x\/\(x + y\)\)}, { \(x\ \[ControlKey]\[LeftModified]/\[RightModified]\ y\ \[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified] + z \), \(x\/y + z\)}, { \(x + y\ \[ControlKey]\[LeftModified] . \[RightModified]\ \[ControlKey]\[LeftModified] . \[RightModified]\ \[ControlKey]\[LeftModified]/\[RightModified]\ z\), \(\(x + y\)\/z\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center], Cell[TextData[{ "In the last example, pressing \ \[ControlKey]\[LeftModified].\[RightModified] twice highlights \"", Cell[BoxData[ \(TraditionalForm\`x + y\)]], "\". Pressing \[ControlKey]\[LeftModified].\[RightModified] selects the \ next larger subexpression. You could also produce the last example by typing \ ", Cell[BoxData[ \(TraditionalForm\`x + y\)]], ", highlighting this text by clicking and dragging the mouse and then \ typing ", Cell[BoxData[ \(TraditionalForm \`\[ControlKey]\[LeftModified]/\[RightModified]\ z\)]], "." }], "Text"], Cell[TextData[ "You can create square roots with \[ControlKey]\[LeftModified]2\ \[RightModified]. Try the following examples."], "Text"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {\(\[ControlKey]\[LeftModified]2\[RightModified]\ x\ + \ y\), \(\@\(x + y\)\)}, { \(\[ControlKey]\[LeftModified]2\[RightModified]\ x\ \[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified]\ + \ y\), \(\@x + y\)}, { \(\[ControlKey]\[LeftModified]2\[RightModified]\ x\ \[ControlKey]\[LeftModified]5\[RightModified]\ 3\ \[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified]\ + \ y\), \(\@x\%3 + y\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ The following table summarizes the control key sequences we have \ learned so far.\ \>", "Text"], Cell[TextData[Cell[BoxData[ FormBox[GridBox[{ {"\[ControlKey]\[LeftModified]6\[RightModified]", \(Superscript\ \(position . \)\)}, {\(\[ControlKey]\[LeftModified] - \[RightModified]\), \(Subscript\ \(position . \)\)}, {"\[ControlKey]\[LeftModified]2\[RightModified]", \(Square\ \(root . \)\)}, {"\[ControlKey]\[LeftModified]5\[RightModified]", \(Switch\ between\ superscript\ and\ subscript\ or\ \n \t\tgo\ to\ the\ exponent\ position\ in\ a\ \(root . \)\)}, {\(\[ControlKey]\[LeftModified]/\[RightModified]\), \(Denominator\ in\ a\ \(fraction . \)\)}, {"\[ControlKey]\[LeftModified]\[SpaceKey]\[RightModified]", \(Return\ from\ a\ special\ position\ or\n \t\tmove\ to\ the\ right\ of\ the\ current\ \(structure . \)\)}, {\(\[ControlKey]\[LeftModified] . \[RightModified]\), \(Select\ the\ next\ larger\ \(subexpression . \)\)}, {"\[Rule]", \(Move\ to\ the\ next\ \(character . \)\)}, {\( \[LeftArrow] \ \), \(Move\ to\ the\ previous\ \(character . \)\)} }, ColumnAlignments->{Left}], TraditionalForm]]]], "Text", TextAlignment->Center, TextJustification->0] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Writing Text and Comments", "Section"], Cell[TextData[{ "There are several different styles of cells in ", StyleBox["Mathematica", FontSlant->"Italic"], ". This is a Text cell. The cell above this one is a Section cell. When \ you type a command in ", StyleBox["Mathematica,", FontSlant->"Italic"], " the cell has Input style. If you have a toolbar in this window, the cell \ type is shown in the left side of the toolbar." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "To type text like this, click between two cells or after the last cell in \ the notebook, then either select ", Cell[BoxData[ \(TraditionalForm\`Format \[Rule] \(Style \[Rule] Text\)\)]], ", select Text from the toolbar or press \[Mod1Key]\[LeftModified]7\ \[RightModified]. (On many keyboards, the \[Mod1Key] is labeled with a \ diamond shape.) When you begin typing, a Text style cell will appear." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[ "If you want to write mathematical formulas in a text cell, press \ \[ControlKey]\[LeftModified]9\[RightModified]. A Highlighted box will \ appear. When you are done typing the formula, press \[ControlKey]\ \[LeftModified]0\[RightModified]."], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " also allows you to write comments in inputs, just as one would do when \ programming. Comments are opened with ", Cell[BoxData[ \(TraditionalForm\` (*\)]], " and closed with ", Cell[BoxData[ \(TraditionalForm\`*) \)]], ". Comments are ignored by the kernel and have have no effect on the \ output." }], "Text", TextJustification->1], Cell[BoxData[ \(\( (*\ Calculate\ a\ 40\ digit\ numerical\ approximation\ of\ the\ cube\ root\ of\ \[Pi]\ *) \nN[\@\[Pi]\%3, 40]\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Solutions", "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solution 1", "Subsubsection"], Cell[TextData[{ "a) When you try to divide a number by zero, ", StyleBox["Mathematica", FontSlant->"Italic"], " will issue a warning and the result is the constant, ", StyleBox["ComplexInfinity", "Input"], "." }], "Text", Evaluatable->False, TextJustification->1, AspectRatioFixed->True], Cell["1/0", "Input", AspectRatioFixed->True], Cell[TextData[{ "b) When you try to find the difference and quotient of Infinity and \ Infinity, ", StyleBox["Mathematica", FontSlant->"Italic"], " recognizes that these indeterminate forms are undefined." }], "Text", TextJustification->1], Cell[BoxData[ \(Infinity - Infinity\)], "Input"], Cell[BoxData[ \(Infinity/Infinity\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Solution 2", "Subsubsection"], Cell["a) We calculate the value of Pi to 16 digits.", "Text", Evaluatable->False, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True], Cell[BoxData[ \(NumberForm[1.0\ Pi, 16]\)], "Input"], Cell[TextData[{ "b) The expression ", Cell[BoxData[ \(TraditionalForm\`E^2. \)]], " is calculated with floating point arithmetic while ", Cell[BoxData[ \(TraditionalForm\`E^2\)]], " is an exact number. One should expect the difference of these two \ quantities to only be approximately equal to zero. Likewise, the number ", Cell[BoxData[ \(TraditionalForm\`2.0\ I\ Pi\)]], " is approximate and is not equal to ", Cell[BoxData[ \(TraditionalForm\`2\ I\ Pi\)]], ". The output of ", Cell[BoxData[ \(TraditionalForm\`E^\((2.0\ I\ Pi)\)\)]], " is approximately 1." }], "Text", TextAlignment->Left, TextJustification->1], Cell[BoxData[ \(E^2. - E^2\)], "Input"], Cell[BoxData[ \(E^\((2.0 I\ Pi)\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Solution 3", "Subsubsection"], Cell["N[Pi,100]", "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Solution 4", "Subsubsection"], Cell["FactorInteger[123456789]", "Input", AspectRatioFixed->True], Cell["FactorInteger[987654321]", "Input", AspectRatioFixed->True], Cell["The greatest common divisor is", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["3^2", "Input", AspectRatioFixed->True], Cell["The least common factor is", "Text", Evaluatable->False, AspectRatioFixed->True], Cell["3^2 * 3607 * 3803 * 17^2 * 379721", "Input", AspectRatioFixed->True] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1680}, {0, 963}}, WindowToolbars->"EditBar", CellGrouping->Automatic, WindowSize->{772, 765}, WindowMargins->{{10, Automatic}, {Automatic, 0}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"XAutomaticEncoding", Magnification->1.5 ] (******************************************************************* Cached data follows. 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