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version 1.2, 2002/07/18 15:52:27 version 1.5, 2013/07/08 03:29:27
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 \label{Tolerance}  \label{Tolerance}
 A \textbf{tolerance\index{tolerance}} parameter determines how closely  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.  the system will require the student's answer to be in order to count it correct.
   The tolerance will default to zero if it is not defined.
   The tolerance parameter should always be defined for a numerical problem 
   unless you are certain only integer answers are generated from your script and 
   you want students to reply with exactly that integer.
   
   If the computer answer is a floating point number, the tolerance should not 
 For technical reasons, it is almost never a good idea to set this parameter  be zero. Computers can only approximate computations involving real numbers. For instance,
 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  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,  {}``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  and there certainly shouldn't be a {}``1'' in the answer, but no computer
 can represent an infinitely long, infinitely detailed real number. Therefore,  can represent an infinitely long, infinitely detailed real number. Therefore,
 for any problem where the answer is not a small integer, you \emph{need}  for any problem where the answer is not an integer, you \emph{need}
 to allow a tolerance factor, or the students will find it nearly impossible  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  to exactly match the computer's idea of the answer. You may find the
 default too large for some problems.   default tolerance too large for some problems, so adjust as appropriate. 
   
 There are  There are three kinds of tolerance. For some answer $A$ and a tolerance $T$,
 two kinds of tolerance. For some answer $a$ and a tolerance $t$,  
   
 \begin{enumerate}  \begin{enumerate}
 \item an \textbf{Absolute} tolerance\index{absolute tolerance}\index{tolerance, absolute}  \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  will take anything in the range $A\pm T$. So if $A=10$ and $T=2$, then
 anything between 8 and 12 is acceptable.  anything between 8 and 12 is acceptable.
  Any number in the tolerance field \emph{without} a \textbf{\%} symbol is   Any number in the tolerance field \emph{without} a \textbf{\%} symbol is
 an absolute tolerance.  an absolute tolerance.
 \item a \textbf{Relative} tolerance\index{relative tolerance}\index{tolerance, relative}  \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  will take anything in the range $A\pm aT$, where \emph{T} is interpreted
 as a percentage. Any number in the tolerance field \emph{followed by} a \textbf{\%}  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  symbol is a relative tolerance. For example, $a=10$ and $t=10\%$ will accept
 anything between 9 and 11.   anything between 9 and 11. 
 \end{enumerate}  
   
   
   \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}
   
   Some care is necessary when setting the display format of the computer answer. 
   Before testing the tolerance, LON-CAPA converts the computer answer,
   as generated in the script block, according to the
   format attribute in the numericalresponse tag.
   
   Next, the formatted compuer answer is "graded" relative to the significant 
   figures parameter, if it is set (see section \ref{Significant_Figures}. 
   If that test was passed, then a numerical comparison of the Computer's
   answer is made with the range of values:
   
   (\$computerAnswer - \$tolerance) < \$formattedcomputerAnswer <
   (\$computerAnswer + \$tolerance)
   
   If the \$formattedcomputerAnswer satisfies the permitted range, then
   "correct" is returned for the computer answer. It is good idea to test multiple
   randomizations to make sure that your tolerance is compatible with the display format.
   
   
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  Added in v.1.5

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