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Cell[CellGroupData[{
Cell["Operator Precedence", "Section"],
Cell[TextData[{
"This woksheet is designed to accompany Chapter 3 of Introduction to \
Scientific Programming: Computational Problem Solving Using ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" and C by Joseph L. Zachary. In it, we will use ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to explore the concept of operator precedence."
}], "Text"],
Cell[CellGroupData[{
Cell["Multiplication/Division vs. Addition/Subtraction", "Subsection"],
Cell[TextData[{
"Examine the ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" expression below. Try to predict what its value will be before you ask ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to evaluate it. There seem to be two possibilities. If the addition is \
performed first, the result will be 4/2, which is 2. If the division is \
performed first the result will be 5/2. Which is it? Try it out and see."
}], "Text"],
Cell[BoxData[
\(\(\ 1 + 3/2\)\)], "Input"],
Cell["\<\
The second possibility was correct: the division was performed \
first. We say that division takes precedence over addition.\
\>", "Text"],
Cell[TextData[{
"Now consider this expression. What are two reasonable possible values? \
Which do you think ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" will find?"
}], "Text"],
Cell[BoxData[
\(2 - 3*4\)], "Input"],
Cell["\<\
Here, the multiplication was performed prior to the subtraction. \
(Had the subtraction been performed first, the answer would have been -4.) \
We say that multiplication takes precedence over addition.\
\>", "Text"],
Cell["\<\
Now try the following experiments:
(1) In the first expression above, replace the + operator with a - operator \
and reevaluate. Which operator takes precedence, subtraction or division?
(2) In the second expression above, replace the - operator with a + operator \
and reevaluate. Which operator takes precedence, addition or multiplication?\
\
\>", "Text"],
Cell[TextData[{
"Your experiments should reveal that in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" (as in most other programming languages) multiplication and division take \
precedence over addition and subtraction. We say that multiplication and \
division are at a higher level of precedence than are addition and \
subtraction."
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Multiplication vs. Division and Addition vs. Subtraction.", "Subsection"],
Cell["\<\
What happens when an expression involves more than operator at \
the same level of precedence? (For example, more than one multiplication and \
division or more than one addition and subtraction.)\
\>", "Text"],
Cell["What would be two possible values for this expression?", "Text"],
Cell[BoxData[
\(2/3*4\)], "Input"],
Cell["\<\
The division was performed first (to obtain 2/3), and the result \
was then multiplied by 4 (to obtain 8/3). Had the multiplication been \
performed before the division, the result would have been 1/6.\
\>", "Text"],
Cell["\<\
This does not mean that multiplication takes precedence over \
division, however. In the absence of parentheses, multiplication and \
division are performed left to right. We say that multiplication and \
division are left associative.\
\>", "Text"],
Cell["\<\
Try to predict the outcome of each of the following calculations \
before you try it. If your prediction turns out to be incorrect, be sure to \
figure out where the flaw in your reasoning occurred.\
\>", "Text"],
Cell[BoxData[
\(2*3/4\)], "Input"],
Cell[BoxData[
\(\(2/3\)/4\)], "Input"],
Cell[BoxData[
\(4/3*2/5\)], "Input"],
Cell[BoxData[
\(6*3/4*5\)], "Input"],
Cell["\<\
Addition and subtraction are also left associative. In the absence \
of parentheses, addition and multiplication is performed from left to right. \
In rational arithmetic (at least in the absence of overlow) this does not \
make any difference, but in floating-point arithmetic it can become \
important. \
\>", "Text"],
Cell["\<\
The following two expressions are mathematically identical; they \
differ only in the fact that their last two terms have been interchanged. \
Evaluate them.\
\>", "Text"],
Cell[BoxData[
\(1\ - \ 1\ + \ 10.^\(-20\)\)], "Input"],
Cell[BoxData[
\(\(\ 1\ + \ 10.^\(-20\)\ - \ 1\)\)], "Input"],
Cell["The answers are not the same! Why?", "Text"],
Cell["\<\
In the first expression, the subtraction is performed first and the \
result is zero. When zero is added to the number 10^-20, we end up with the \
result 10^-20.\
\>", "Text"],
Cell["\<\
In the second expression, the addition is performed first. With \
only 16 digits of mantissa, however, the number 10^-20 is the lost in the \
roundoff and the result is 1. The subtraction thus yields a result of zero. \
(This example shows that floating-point arithmetic need not satisfy the \
associative rule.)\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Exponentiation.", "Subsection"],
Cell["\<\
What about the precedence of the exponentiation operator? See \
if you can figure out the rule by evaluating the following expressions:\
\>",
"Text"],
Cell[BoxData[
\(4*3^2\)], "Input"],
Cell[BoxData[
\(1 + 3^2\)], "Input"],
Cell[BoxData[
\(5/2^3\)], "Input"],
Cell[BoxData[
\(3/10^3*7\)], "Input"],
Cell["\<\
Notice that in each of these examples, the exponentiation was \
performed first. This is indeed the way that it works: exponentiation has \
higher precedence than multiplication and division, which in turn have higher \
precedence than addition and multiplication.\
\>", "Text"],
Cell["Now try this expression.", "Text"],
Cell[BoxData[
\(2^\(3^4\)\)], "Input"],
Cell["\<\
The exponentiation operator is right-associative. Thus, the \
expression above is equivalent to\
\>", "Text"],
Cell[BoxData[
\(2^\((3^4)\)\)], "Input"],
Cell["and not", "Text"],
Cell[BoxData[
\(\((2^3)\)^4\)], "Input"]
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Cell[CellGroupData[{
Cell["Parentheses", "Subsection"],
Cell[TextData[{
"You can override ",
StyleBox["Mathematica",
FontSlant->"Italic"],
"'s precedence rules by using parentheses to group expressions in the order \
that you wish ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to evaluate them. Here are some examples where we use parentheses to \
force a different evaluation order than ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" would otherwise use."
}], "Text"],
Cell[BoxData[
\(2*\((3 + 4)\)\)], "Input"],
Cell[BoxData[
\(2^\((3*4)\)\)], "Input"],
Cell[BoxData[
\(2/\((3/3)\)\)], "Input"],
Cell[TextData[{
"If you have any doubt, use parentheses to make sure ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" executes your expressions in the order in which you intend."
}], "Text"]
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Cell[CellGroupData[{
Cell["Summary", "Subsection"],
Cell["There are five arithmetic operators: ^, *, /, +, and -.", "Text"],
Cell["\<\
Exponentiation (^) is at the highest level of precedence, \
multiplication (*) and division (/) are at a lower level of precedence, and \
addition (+) and subtraction (-) are at an even lower level of \
precedence.\
\>", "Text"],
Cell["\<\
In the absence of parentheses, operators at a higher level of \
precedence are performed before operators at a lower level of \
precedence.\
\>", "Text"],
Cell["\<\
In the absence of parentheses, multiplication and division are \
performed from left to right, as are addition and subtraction.\
\>", "Text"],
Cell["\<\
Parentheses can be used to explicitly control the order of \
evaluation of an expression.\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Exercises", "Subsection"],
Cell["\<\
1. Evaluate the following expression. \
\>", "Text"],
Cell[BoxData[
\(\(\ 2\ - \ 2^3*2\)\)], "Input"],
Cell["\<\
Add one set of parentheses so that the expression evalutes to -12. \
Now move your parentheses so that the expression evaluates to -62. Move your \
parentheses one last time to make the expression evaluate to 0.\
\>", "Text"],
Cell["2. Evaluate the following expression. ", "Text"],
Cell[BoxData[
\(\(\ 2/3*4^2\)\)], "Input"],
Cell["\<\
Add two sets of parentheses that leave the value of the expression \
unchanged. (Your parentheses may enclose neither the entire expression nor a \
single number.)\
\>", "Text"],
Cell["3. Repeat exercise 2 for this expression.", "Text"],
Cell[BoxData[
\(1 + 2/3*4 + 5\)], "Input"]
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