Integer arithmetic and command status

The idea

Arithmetic expansion $((expression)) produces a numeric value as text, while the arithmetic command ((expression)) has an exit status based on whether its result is nonzero. The command succeeds for a nonzero result and fails for zero. This can surprise code that uses post-increment: ((n++)) evaluates to the old value, so incrementing an initial zero produces a failure status even though the variable changes.

Bash arithmetic uses fixed-width integers and does not provide arbitrary-precision overflow protection. It also interprets certain numeric spellings specially, including leading-zero forms. Validate expected decimal input before converting it, and keep numeric strings within a stated supported range. Do not insert arbitrary user text into arithmetic expressions as if arithmetic evaluation were a general safe parser for untrusted syntax.

n=0
if ((n++)); then printf 'true\n'; else printf 'false\n'; fi
printf '%s\n' "$n"

false
1

Post-increment yields the old zero as the command result, so the conditional fails while n still increases.