The expression $(($OPTIND - 1)) in the last graphics utility example shows another way that the shell can do integer arithmetic. As you might guess, the shell interprets words surrounded by $(( and )) as arithmetic expressions.  Variables in arithmetic expressions do not need to be preceded by dollar signs, though it is not wrong to do so.
Arithmetic expressions are evaluated inside double quotes, like tildes, variables, and command substitutions. We’re finally in a position to state the definitive rule about quoting strings: When in doubt, enclose a string in single quotes, unless it contains tildes or any expression involving a dollar sign, in which case you should use double quotes.
For example, the date command on System V-derived versions of UNIX accepts arguments that tell it how to format its output. The argument +%j tells it to print the day of the year, i.e., the number of days since December 31st of the previous year.
We can use +%j to print a little holiday anticipation message:
echo "Only $(( (365-$(date +%j)) / 7 )) weeks until the New Year"
We’ll show where this fits in the overall scheme of command-line processing in Chapter 7.
The arithmetic expression feature is built into bash’s syntax, and was available in the Bourne shell (most versions) only through the external command expr. Thus it is yet another example of a desirable feature provided by an external command being better integrated into the shell. getopts, as we have already ...