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- Fix typo.
- $<$ used for less than.

    1: \label{Tolerance}
    2: A \textbf{tolerance\index{tolerance}} parameter determines how closely
    3: the system will require the student's answer to be in order to count it correct.
    4: The tolerance will default to zero if it is not defined.
    5: The tolerance parameter should always be defined for a numerical problem 
    6: unless you are certain only integer answers are generated from your script and 
    7: you want students to reply with exactly that integer.
    8: 
    9: If the computer answer is a floating point number, the tolerance should not 
   10: be zero. Computers can only approximate computations involving real numbers. For instance,
   11: a computer's {[}decimal{]} answer to the simple problem $\frac{1}{3}$ is
   12: {}``0.33333333333333331''. It \emph{should} be an infinite series of 3's,
   13: and there certainly shouldn't be a {}``1'' in the answer, but no computer
   14: can represent an infinitely long, infinitely detailed real number. Therefore,
   15: for any problem where the answer is not an integer, you \emph{need}
   16: to allow a tolerance factor, or the students will find it nearly impossible
   17: to exactly match the computer's idea of the answer. You may find the
   18: default tolerance too large for some problems, so adjust as appropriate. 
   19: 
   20: There are three kinds of tolerance. For some answer $A$ and a tolerance $T$,
   21: 
   22: \begin{enumerate}
   23: \item an \textbf{Absolute} tolerance\index{absolute tolerance}\index{tolerance, absolute}
   24: will take anything in the range $A\pm T$. So if $A=10$ and $T=2$, then
   25: anything between 8 and 12 is acceptable.
   26:  Any number in the tolerance field \emph{without} a \textbf{\%} symbol is
   27: an absolute tolerance.
   28: \item a \textbf{Relative} tolerance\index{relative tolerance}\index{tolerance, relative}
   29: will take anything in the range $A\pm aT$, where \emph{T} is interpreted
   30: as a percentage/100. Any number in the tolerance field \emph{followed by} a \textbf{\%}
   31: symbol is a relative tolerance. For example, $a=10$ and $t=10\%$ will accept
   32: anything between 9 and 11. 
   33: 
   34: \item a tolerance that is a calculated variable (identified by \$ sign as
   35: the first character). For example, if an answer is $\$X$,and for a student
   36: possible values range from $-\$X1$ to $+\$X1$, you could choose $T =
   37: \$tolerance = \$2X1/100;$ acceptable answers would then be from
   38: $\$X-\$tolerance$ to $\$X+\$tolerance$. (This is especially useful when answers
   39: close to zero are possible for some students)
   40: 
   41: \end{enumerate}
   42: 
   43: Some care is necessary when setting the display format of the computer answer. 
   44: Before testing the tolerance, LON-CAPA converts the computer answer,
   45: as generated in the script block, according to the
   46: format attribute in the numericalresponse tag.
   47: 
   48: Next, the formatted comptuer answer is "graded" relative to the significant 
   49: figures parameter, if it is set (see section \ref{Significant_Figures}. 
   50: If that test was passed, then a numerical comparison of the Computer's
   51: answer is made with the range of values:
   52: 
   53: (\$computerAnswer - \$tolerance) $<$ \$formattedcomputerAnswer $<$
   54: (\$computerAnswer + \$tolerance)
   55: 
   56: If the \$formattedcomputerAnswer satisfies the permitted range, then
   57: "correct" is returned for the computer answer. It is good idea to test multiple
   58: randomizations to make sure that your tolerance is compatible with the display format.
   59: 
   60: