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\section{Graphing Calculators and Computers}

Numerical analysis may be described as the study of errors introduced by
using\emph{\ floating point} arithmetic (\emph{round-off errors}) and by
using a finite number of terms when an infinite number of terms are required
for exactness (\emph{truncation errors}). Floating point is a data type that
is machine dependent. It is important to understand that a floating point
number is neither rational nor irrational; indeed, each floating-point
number represents an infinite number of possible rational numbers and an
infinite number of possible irrational numbers. (An \emph{irrational number}
is a real number such as $\sqrt{2}$ and $\pi $ that cannot be expressed as a
quotient of integers.)

Computer algebra systems use what is called \emph{infinite precision} or 
\emph{extended precision} to represent integers and rationals exactly.
(Since computers are necessarily finite, there is really no such thing as
`infinite precision'. The term `extended precision' is probably a better
term. Sooner or later, any computer attempting to perform sufficiently
complicated extended-precision computations will run out of memory.)

Numbers such as $\sin 1$, $\sqrt{2}$, and $\pi $ are additional examples of
numbers that are represented exactly. These numbers cannot be further
`simplified'. Numerical evaluation leads to the following approximations. 
\begin{eqnarray*}
\sin 1 &=&.\,84147\,09848\,07896\,50665\,25023 \\
\sqrt{2} &=&1.\,41421\,35623\,73095\,04880\,1689 \\
\pi &=&3.\,14159\,26535\,89793\,23846\,2643
\end{eqnarray*}
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"XNPEU";}}]{Margin Hint}{%
Choose \textsf{Maple + Settings}, \textsf{Engine Parameters} page.
\par
Set \textsf{Digits Used in Computations} and \textsf{Digits Used in Display}
both to $25$. Choose \textsf{OK}.
\par
Choose \textsf{Maple + Evaluate Numerically}}

Notice that the approximations are broken into blocks of length 5 decimal
digits in order to make them more readable. The numbers on the left are
exact, while the numbers on the right are merely approximations. In
particular, $\left( \sqrt{2}\right) ^2=\allowbreak 2$ exactly whereas 
\[
1.\,41421\,35623\,73095\,04880\,1689^2=2.\,00000\,00000\,00000\,00000\,0001 
\]
The remarkable fact is that modern applications of mathematics in fields
such as cryptology require absolute precision. A single missing bit would
completely scramble a message and make it unrecoverable (even to users with
authorized passwords).

The following example illustrates typical products in extended precision. If 
$a$ requires $n$ BYTES and $b$ requires $m$ BYTES of memory, then the
product requires $n+m$ BYTES, roughly proportional to the length of the
decimal representation. 
\[
93478934923678895\times 37893267207=\allowbreak
354\,22222\,59288\,72851\,97014\,96265 
\]
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With the insertion point in the expression $93478934923678895\times
37893267207$, choose \textsf{Evaluate}}

Another place where the finite nature of computers is evident is with
plotting. The two pictures below are versions of the same graph, with the
plot style on the left being \textsf{Point} and on the right \textsf{Line}.
Line plots are created by connecting the points with a smooth curve. You can
see that the system concentrates the plotted points in the regions where the
graph has the greatest curvature.

\begin{center}
\FRAME{itbpF}{1.8005in}{1.1995in}{0in}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 1.8005in;height 1.1995in;depth 0in;display
"PICT";function \TEXUX{$\sin x$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "box";pointplot TRUE;xmin
"-5";xmax "5";xviewmin "-5";xviewmax "5";yviewmin "-1.04";yviewmax
"1.041";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}\quad \FRAME{itbpF}{1.8005in}{%
1.1995in}{0in}{}{}{}{\special{language "Scientific Word";type
"MAPLEPLOT";width 1.8005in;height 1.1995in;depth 0in;display
"PICT";function \TEXUX{$\sin x$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "box";xmin "-5";xmax
"5";xviewmin "-5";xviewmax "5";yviewmin "-1.04";yviewmax
"1.041";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

Graphing calculators utilize similar principles for computing and graphing,
and if you own a graphing calculator then you can use it to do many of the
activities in this book. Other activities require symbol manipulation and
extended-precision arithmetic and may be beyond the reach of your calculator.

Since \textsl{Scientific Notebook} encourages you to do mathematics using
ordinary words and symbols, it will help you understand the underlying
mathematical principles. When you go back to paper and pencil or graphing
calculators, your approach to mathematical problem solving should be the
same. Think of \textsl{Scientific Notebook} as a faithful assistant who
knows how to spell words and perform messy calculations, but that
understands nothing about setting up problems and interpreting the solution.
Working together you will become a mathematical problem-solving monster team.

\end{document}
