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The shell allows arithmetic expressions to be evaluated, as one of
the shell expansions or by using the `((`

compound command, the
`let`

builtin, or the `-i` option to the `declare`

builtin.

Evaluation is done in fixed-width integers with no check for overflow, though division by 0 is trapped and flagged as an error. The operators and their precedence, associativity, and values are the same as in the C language. The following list of operators is grouped into levels of equal-precedence operators. The levels are listed in order of decreasing precedence.

`id`++`id`--variable post-increment and post-decrement

`++`

`id`--`id`variable pre-increment and pre-decrement

`- +`

unary minus and plus

`! ~`

logical and bitwise negation

`**`

exponentiation

`* / %`

multiplication, division, remainder

`+ -`

addition, subtraction

`<< >>`

left and right bitwise shifts

`<= >= < >`

comparison

`== !=`

equality and inequality

`&`

bitwise AND

`^`

bitwise exclusive OR

`|`

bitwise OR

`&&`

logical AND

`||`

logical OR

`expr ? expr : expr`

conditional operator

`= *= /= %= += -= <<= >>= &= ^= |=`

assignment

`expr1 , expr2`

comma

Shell variables are allowed as operands; parameter expansion is
performed before the expression is evaluated.
Within an expression, shell variables may also be referenced by name
without using the parameter expansion syntax.
A shell variable that is null or unset evaluates to 0 when referenced
by name without using the parameter expansion syntax.
The value of a variable is evaluated as an arithmetic expression
when it is referenced, or when a variable which has been given the
`integer` attribute using ‘`declare -i`’ is assigned a value.
A null value evaluates to 0.
A shell variable need not have its `integer` attribute turned on
to be used in an expression.

Integer constants follow the C language definition, without suffixes or
character constants.
Constants with a leading 0 are interpreted as octal numbers.
A leading ‘`0x`’ or ‘`0X`’ denotes hexadecimal. Otherwise,
numbers take the form [`base``#`

]`n`, where the optional `base`
is a decimal number between 2 and 64 representing the arithmetic
base, and `n` is a number in that base.
If `base``#`

is omitted, then base 10 is used.
When specifying `n`,
if a non-digit is required,
the digits greater than 9 are represented by the lowercase letters,
the uppercase letters, ‘`@`’, and ‘`_`’, in that order.
If `base` is less than or equal to 36, lowercase and uppercase
letters may be used interchangeably to represent numbers between 10
and 35.

Operators are evaluated in order of precedence. Sub-expressions in parentheses are evaluated first and may override the precedence rules above.

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