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 fixedwidth 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 equalprecedence operators. The levels are listed in order of decreasing precedence.
id++ id
variable postincrement and postdecrement
++id id
variable preincrement and predecrement
 +
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.
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, 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. Subexpressions in parentheses are evaluated first and may override the precedence rules above.
Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
Licensed under the GNU Free Documentation License.
https://www.gnu.org/software/bash/manual/html_node/ShellArithmetic.html