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1.2       bowersj2    1: \label{Tolerance}
1.1       bowersj2    2: A \textbf{tolerance\index{tolerance}} parameter determines how closely
1.4     ! lira        3: the system will require the student's answer to be in order to count it correct. 
        !             4: The tolerance parameter should always be defined for a numerical problem.
1.1       bowersj2    5: 
                      6: For technical reasons, it is almost never a good idea to set this parameter
                      7: to zero. Computers can only approximate computations involving real numbers. For instance,
                      8: a computer's {[}decimal{]} answer to the simple problem $\frac{1}{3}$ is
                      9: {}``0.33333333333333331''. It \emph{should} be an infinite series of 3's,
                     10: and there certainly shouldn't be a {}``1'' in the answer, but no computer
                     11: can represent an infinitely long, infinitely detailed real number. Therefore,
                     12: for any problem where the answer is not a small integer, you \emph{need}
                     13: to allow a tolerance factor, or the students will find it nearly impossible
                     14: to exactly match the computers idea of the answer. You may find the
                     15: default too large for some problems. 
                     16: 
                     17: There are
1.3       albertel   18: three kinds of tolerance. For some answer $A$ and a tolerance $T$,
1.1       bowersj2   19: 
                     20: \begin{enumerate}
                     21: \item an \textbf{Absolute} tolerance\index{absolute tolerance}\index{tolerance, absolute}
1.3       albertel   22: will take anything in the range $A\pm T$. So if $A=10$ and $T=2$, then
1.1       bowersj2   23: anything between 8 and 12 is acceptable.
                     24:  Any number in the tolerance field \emph{without} a \textbf{\%} symbol is
                     25: an absolute tolerance.
                     26: \item a \textbf{Relative} tolerance\index{relative tolerance}\index{tolerance, relative}
1.3       albertel   27: will take anything in the range $A\pm aT$, where \emph{T} is interpreted
                     28: as a percentage/100. Any number in the tolerance field \emph{followed by} a \textbf{\%}
1.1       bowersj2   29: symbol is a relative tolerance. For example, $a=10$ and $t=10\%$ will accept
                     30: anything between 9 and 11. 
1.3       albertel   31: 
                     32: \item a tolerance that is a calculated variable (identified by \$ sign as
                     33: the first character). For example, if an answer is $\$X$,and for a student
                     34: possible values range from $-\$X1$ to $+\$X1$, you could choose $T =
                     35: \$tolerance = \$2X1/100;$ acceptable answers would then be from
                     36: $\$X-\$tolerance$ to $\$X+\$tolerance$. (This is especially useful when answers
                     37: close to zero are possible for some students)
                     38: 
                     39: \end{enumerate}