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Arithmetic expansion allows the evaluation of an arithmetic expression and the substitution of the result. The format for arithmetic expansion is:

$((expression))

The `expression` undergoes the same expansions
as if it were within double quotes,
but double quote characters in `expression` are not treated specially
and are removed.
All tokens in the expression undergo parameter and variable expansion,
command substitution, and quote removal.
The result is treated as the arithmetic expression to be evaluated.
Arithmetic expansions may be nested.

The evaluation is performed according to the rules listed below (see Shell Arithmetic). If the expression is invalid, Bash prints a message indicating failure to the standard error and no substitution occurs.