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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Graphing Calculators and Computers">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Graphing Calculators and Computers\label%
{Graphing calculators and computers}}

\label{Extended precision}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 examples of numbers
that are represented exactly only by symbols. Numerical evaluation leads to
approximations such as the following: 
\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*}
\label{Floating point}Machines represent these numbers in \textbf{floating
point}, a data type that is machine dependent. A floating point number is
neither rational nor irrational; each floating-point number represents an
infinite number of possible rational numbers and an infinite number of
possible irrational numbers. (An \textbf{irrational number} is a real number
such as $\sqrt{2}$ and $\pi $ that cannot be expressed as a quotient of
integers.)

The finite nature of computers is evident 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.

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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
computations you can do with computer algebra systems. Some of your work
could require symbol manipulation and extended-precision arithmetic that are
beyond the reach of your calculator.

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

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