`(( ))` is an arithmetic evaluation block. Its content has to be an arithmetic expression. Arithmetic expressions don't require `$` before simple variable names

`(( ))` is an arithmetic evaluation block. Its content has to be an arithmetic expression. Arithmetic expressions don't require `$` before simple variable names and some more complex expressions like array indexing. `(( i + 1 ))` will evaluate the result of adding one to `$i` and then throw away the result. It doesn't do anything useful (other than having a different exit code depending on whether the expression evaluated to 0 or not). `$(( ... ))` evaluates the expression and then returns its value. ie `i=$(( i + 1 ))` will increment `$i`, just like `(( i++ ))` did.