\label{Tolerance}
A \textbf{tolerance\index{tolerance}} parameter determines how closely
the system will require the student's answer to be in order to count it correct.


For technical reasons, it is almost never a good idea to set this parameter
to zero. Computers can only approximate computations involving real numbers. For instance,
a computer's {[}decimal{]} answer to the simple problem $\frac{1}{3}$ is
{}``0.33333333333333331''. It \emph{should} be an infinite series of 3's,
and there certainly shouldn't be a {}``1'' in the answer, but no computer
can represent an infinitely long, infinitely detailed real number. Therefore,
for any problem where the answer is not a small integer, you \emph{need}
to allow a tolerance factor, or the students will find it nearly impossible
to exactly match the computers idea of the answer. You may find the
default too large for some problems. 

There are
three kinds of tolerance. For some answer $A$ and a tolerance $T$,

\begin{enumerate}
\item an \textbf{Absolute} tolerance\index{absolute tolerance}\index{tolerance, absolute}
will take anything in the range $A\pm T$. So if $A=10$ and $T=2$, then
anything between 8 and 12 is acceptable.
 Any number in the tolerance field \emph{without} a \textbf{\%} symbol is
an absolute tolerance.
\item a \textbf{Relative} tolerance\index{relative tolerance}\index{tolerance, relative}
will take anything in the range $A\pm aT$, where \emph{T} is interpreted
as a percentage/100. Any number in the tolerance field \emph{followed by} a \textbf{\%}
symbol is a relative tolerance. For example, $a=10$ and $t=10\%$ will accept
anything between 9 and 11. 

\item a tolerance that is a calculated variable (identified by \$ sign as
the first character). For example, if an answer is $\$X$,and for a student
possible values range from $-\$X1$ to $+\$X1$, you could choose $T =
\$tolerance = \$2X1/100;$ acceptable answers would then be from
$\$X-\$tolerance$ to $\$X+\$tolerance$. (This is especially useful when answers
close to zero are possible for some students)

\end{enumerate}