Arithmetic expansion evaluates an arithmetic expression and substitutes the result. The format for arithmetic expansion is:
$(( expression ))
The expression undergoes the same expansions as if it were within double quotes, but unescaped 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. Since the way Bash handles double quotes can potentially result in empty strings, arithmetic expansion treats those as expressions that evaluate to 0. 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, does not perform the substitution, and does not execute the command associated with the expansion.