double-parenthesis

Arithmetic evaluation

$ (( x = 5 + 3 ))

$ (( count++ ))

$ (( x > 10 )) && echo "big"

$ result=$(( a * b ))

$ (( result = (a + b) * c / d ))

((expression))
$((expression))

(( )) is the shell's arithmetic evaluation construct. It evaluates mathematical expressions and returns success (0) if the result is non-zero, failure (1) if zero.
Inside ((...)), variables don't need the $ prefix and whitespace is ignored. Standard C-style operators work as expected.
$((...)) performs arithmetic expansion, substituting the result. ((...)) evaluates without substitution (for side effects like assignment).

$ # Evaluation (for conditionals/assignments)
(( x = 5 ))
(( x > 3 )) && echo "yes"

# Expansion (to get the value)
echo "Result: $(( 5 + 3 ))"

Arithmetic: +, -, *, /, % (modulo), ** (exponent)
Assignment: =, +=, -=, *=, /=, %=
Comparison: ==, !=, <, >, <=, >=
Logical: && (and), || (or), ! (not)
Bitwise: &, |, ^, ~, <<, >>
Increment/Decrement: ++, --
Ternary: condition ? true : false

((...)) only handles integers. For floating-point, use bc or awk.
Division is integer division: $(( 5 / 2 )) gives 2, not 2.5.
Return value logic is inverted from typical boolean: (( 0 )) returns failure (exit 1), (( 1 )) returns success (exit 0).
This is a bash/zsh feature, not POSIX. Use $((...)) for better portability.

bash(1), bc(1), expr(1), let(1)