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bash: Shell Arithmetic

 
 6.5 Shell Arithmetic
 ====================
 
 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.
 
    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.  Sub-expressions in
 parentheses are evaluated first and may override the precedence rules
 above.
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