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Notebook[{
Cell["Solving Equations One Step at a Time", "Title"],
Cell["\<\
How Greek mythology and modern mathematics come to the rescue and allow us to \
do really neat things.\
\>", "Subtitle"],
Cell[CellGroupData[{
Cell[TextData[{
"Using the ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" Solve Routine. Nice but we would rather do it ourselves, thank you."
}], "Section"],
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" has powerful routines for solving equations. For regular equations we can \
use Solve. Here is an equation."
}], "Text"],
Cell[BoxData[
\(eqn = 3 x + 4 y/b \[Equal] 6 \((x - y)\)\)], "Input"],
Cell[TextData[{
"Solve will automatically solve for one of the variables. Suppose we wish \
to solve for ",
Cell[BoxData[
\(y\)]],
"..."
}], "Text"],
Cell[BoxData[
\(Solve[eqn, y]\)], "Input"],
Cell["\<\
Everything has been done automatically. But suppose we wish to solve the \
equation ourselves. Suppose we want to manipulate the equation step by step \
to see how it is done. There are many cases where we may wish to manipulate \
an equation, rather than formally solve it. How do we do that?\
\>", "Text"],
Cell[TextData[{
"Well, in order to do that using ",
StyleBox["Mathematica",
FontSlant->"Italic"],
", and not somehow having it done for us automatically, we have to learn \
about two new kinds of operations in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". These operations may at first seem strange. But each really does a very \
simple thing. Furthermore they are each powerful operations and so well worth \
learning. Furthermore they are really 20th century mathematics. Most high \
school students are plodding along with 17th century mathematics. So join the \
in crowd!"
}], "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Pure Functions - The first really really in operation", "Section"],
Cell[TextData[{
"A regular function like ",
Cell[BoxData[
\(Sin[x]\)]],
" has a name, Sin, and an argument, x. We put in an x and ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" grinds away and gives us an answer. In fact, calculating ",
Cell[BoxData[
\(Sin[x]\)]],
" is fairly complicated and most of us wouldn't want to know the details. "
}], "Text"],
Cell[TextData[{
"A pure function is all action. It has no name. Its argument is only \
indicated in the most symbolic of ways. Pure functions were invented when \
mathematicians realized that in many cases what is actually done is the \
important thing. In ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" pure functions are useful when we just want to do something and not go to \
all the bother of setting up a definition for the action. So suppose we want \
to add 1 to something. We can do it this way with a pure function..."
}], "Text"],
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StyleBox[\(# + 1 &\),
FontColor->RGBColor[0, 0, 1]], "[", "x", "]"}]], "Input"],
Cell["\<\
The # sign stands for whatever is going to be acted upon. When we follow the \
pure function with [x], then it is x that is acted upon. All pure functions \
have to end with an &. Suppose we want to multiply something by 3...\
\>", "Text"],
Cell[BoxData[
RowBox[{
StyleBox[\(3 # &\),
FontColor->RGBColor[0, 0, 1]], "[", "x", "]"}]], "Input"],
Cell["Suppose we want to square something and add 3...", "Text"],
Cell[BoxData[
RowBox[{
StyleBox[\(#\^2 + 3 &\),
FontColor->RGBColor[0, 0, 1]], "[", "x", "]"}]], "Input"],
Cell["\<\
Don't forget the &! (I am always forgetting the &.) This is what happens when \
you forget it...\
\>", "Text"],
Cell[BoxData[
RowBox[{
StyleBox[\(#\^2\),
FontColor->RGBColor[0, 0, 1]],
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FontColor->RGBColor[0, 0, 1]],
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Cell[TextData[{
"You get a completely useless result. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" absolutely needs the & to know that it is a pure function."
}], "Text"],
Cell["\<\
Notice that a pure function is very much like a name of a function. It stands \
in the same place that you would usually have a name. You can think of pure \
functions as throw-away functions. Use them once and throw them away.\
\>", "Text"],
Cell["\<\
So now you have mastered one of the significant pieces of 20th century \
mathematics. Let's go on to the next piece.\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Map - The second really really in operation", "Section"],
Cell["\<\
The second operation we have to learn about is Map. Suppose we want to do \
something to a whole bunch of stuff. We could do it to each thing \
individually, but if we are going to always be doing the same thing, why not \
do it all at once. Suppose you have collected a number of items in mystuff \
and you want to take the Sin of each of the items. We can do it my \
\"mapping\" Sin onto the items.\
\>", "Text"],
Cell[BoxData[
RowBox[{
StyleBox["Map",
FontColor->RGBColor[1, 0, 0]], "[", \(Sin,
mystuff[x, y, 0, \[Pi]/2, 90 \[Degree]]\), "]"}]], "Input"],
Cell[TextData[{
"Notice that Sin was applied to each item in mystuff. This is such a useful \
operation, and such an efficient way of doing things, that ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" has a special notation that can also be used. We can do the same thing as \
above this way."
}], "Text"],
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RowBox[{"Sin",
StyleBox["/@",
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Cell[TextData[{
"The symbols ",
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\( /@ \)]],
" stands for Map. It tells ",
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FontSlant->"Italic"],
" to Map the function which precedes ",
Cell[BoxData[
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" onto the items in the thing that follows ",
Cell[BoxData[
\( /@ \)]],
"."
}], "Text"],
Cell["\<\
Did I say that Map was efficient? It is really efficient! Map will do things \
many times faster than more mundane ways of doing things. So, if you can \
think in terms of using Map, when you get to some real number crunching \
problems you will be way ahead of the game. \
\>", "Text"],
Cell["\<\
Now you know another great piece of 20th century mathematics. And since we \
also know about pure functions let's put them together. Suppose we want to \
square a number of items...\
\>", "Text"],
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Cell["\<\
So now we have all the tools we need, pure functions and Map, for solving \
equations step by step.\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["At last - Solving an Equation", "Section"],
Cell["Here is an equation...", "Text"],
Cell[BoxData[
\(y + x \[Equal] 3\)], "Input"],
Cell[TextData[{
"What do we do when we solve an equation? The usual procedure is to perform \
the same operation on each side of the equation. We add the same thing to \
each side, subtract the same thing from each side, multiply both sides by the \
same thing or divide both sides by the same thing. Each operation transforms \
the equation and we continue until we get what we want. In ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" it is often useful to look at the FullForm of an expression. So let's \
look at the full form of our equation."
}], "Text"],
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Cell["\<\
Comparing to the example in Map, we see that Equal is like mystuff. The two \
sides of the equation are like the items that were in mystuff. If we want to \
do the same action on both sides of the equation, then we can Map that action \
onto the equation. Lastly, the action we want to do can be specified with a \
pure function. So if we want to solve for y, then we want to subtract x from \
each side of the equation...\
\>", "Text"],
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StyleBox[\(# - x &\),
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Cell["\<\
The parenthesis around the equation were necessary because otherwise the \
function is mapped onto y which doesn't even have any parts. Here is another \
way to do it. It shows the equation at each step of solving. Here there is \
only the initial equation and one step. Remember that % always stands for the \
output of the last expression that was evaluated.\
\>", "Text"],
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StyleBox[\(# - x &\),
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Cell["\<\
And now we are really going to unleash the power! Watch out below!\
\>", "Section"],
Cell["\<\
This was the equation used as an example in the first section.\
\>", "Text"],
Cell[BoxData[
\(eqn = 3 x + 4 y/b \[Equal] 6 \((x - y)\)\)], "Input"],
Cell["\<\
Let's solve this equation for y. We want to get y by itself on the left hand \
side and everything else on the right hand side. So let's start by \
subtracting 3x from each side.\
\>", "Text"],
Cell[BoxData[
RowBox[{
StyleBox[\(# - 3 x &\),
FontColor->RGBColor[0, 0, 1]],
StyleBox["/@",
FontColor->RGBColor[1, 0, 0]], "eqn"}]], "Input"],
Cell["\<\
Now it looks like we are going to have to expand out the left hand side.\
\>", "Text"],
Cell[BoxData[
RowBox[{
StyleBox["Expand",
FontColor->RGBColor[0, 0, 1]],
StyleBox["/@",
FontColor->RGBColor[1, 0, 0]], "%"}]], "Input"],
Cell[TextData[{
"Wait! Why didn't we have to use an & here? Because Expand is not a pure \
function that we made up, but the name of an existing function in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". Remember that a pure function just takes the place of a name of a \
function. So wherever we might use a pure function we could also use the name \
of an existing defined function. Notice that ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" automatically simplified. Now we have to add 6y to each side."
}], "Text"],
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FontColor->RGBColor[0, 0, 1]],
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Cell[TextData[{
"Now we need to collect the y terms. We can do that by taking the regular \
",
StyleBox["Mathematica",
FontSlant->"Italic"],
" Collect function and turning it into a pure function. (We don't actually \
have to collect y terms on the right hand side, but it does no harm.)"
}], "Text"],
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StyleBox[\(Collect[#, y] &\),
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Cell[TextData[{
"Now we need to divide by ",
Cell[BoxData[
\(\((6 + 4/b)\)\)]],
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}], "Text"],
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StyleBox[\(#/\((6 + 4/b)\) &\),
FontColor->RGBColor[0, 0, 1]],
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Cell["This is the answer, but perhaps you wish to \"simplify\" it.", "Text"],
Cell[BoxData[
RowBox[{
StyleBox["Simplify",
FontColor->RGBColor[0, 0, 1]],
StyleBox["/@",
FontColor->RGBColor[1, 0, 0]], "%"}]], "Input"],
Cell["\<\
Instead of doing the work in multiple cells, it is nice to do all the work in \
one cell. You can reevaluate as you add each step. And the % always refers to \
just what you intend because everything is initialized when you reevaluate \
the cell. Here the color coding is dropped because by now you should know \
what the various parts of each expression do.\
\>", "Text"],
Cell[BoxData[{
\(eqn\), "\[IndentingNewLine]",
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Cell[TextData[{
"Doesn't it all look very arcane? But once you see that each step is just a \
pure function, or function name, mapped onto each side of the equation, \
transforming the equation, there is nothing mysterious at all. Now you can \
perform operations on equations, step by step, and solve things ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" might not be so good at, such as manipulating inequalities."
}], "Text"],
Cell[TextData[{
"Now cast your eye over the last set of outputs. What do you see? A whole \
bunch of equations? No, they are all the same equation! They are all equally \
valid. They are all true. They all contain exactly the same information. Try \
to think of them all as ",
StyleBox["one",
FontWeight->"Bold"],
" object that can take many different forms."
}], "Text"],
Cell["\<\
Here is a story. In Greek mythology, Proteus was \"The Old Man of the Sea\". \
Proteus was very wise. If you had a question, Proteus could give you the \
answer. But Proteus was ornery. He didn't necessarily want to answer people's \
questions. So, when he was approached, he would try to escape. He could \
change his form. He would change into a goat thinking people would never ask \
a question of a goat. Then he would change into a lion thinking to scare \
people off. Then he would change into a snake thinking to slip away. Then he \
might change into a river, or a cloud, or the wind. But if the person held on \
tight, and never let Proteus escape, eventually Proteus would have to yield \
and give the answer.\
\>", "Text"],
Cell[TextData[{
"So think of the equation as being Proteus. Proteus is actually doing us a \
favor by changing form. Because ",
StyleBox["the form is the answer",
FontWeight->"Bold"],
". If we can grab hold of Proteus, and make him change his form the way we \
want, then we will eventually get the form we desire. We change the form \
until we get the one that we think is \"nicest\". In our case we thought that \
the \"nicest\" form was one that had y on one side of the equation and \
everything else on the other side. But there may be reasons to pick some \
other form. This is a common theme in mathematics. Often we will have some \
object that can take many forms. We want to learn how to force it into a form \
which we think is \"nicest\", perhaps a form that is most useful, or a form \
that is most beautiful."
}], "Text"],
Cell["\<\
So now you have not only learned some old mythology but some very modern \
mathematics which can be applied to many kinds of problems. Leave that 17th \
century mathematics to the kiddies.\
\>", "Text"]
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