Advanced BashScripting Guide: A complete guide to shell scripting, using Bash  

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Initializing or changing the value of a variable
Allpurpose assignment operator, which works for both arithmetic and string assignments.
var=27 category=minerals # No spaces allowed after the "=". 
Do not confuse the "=" assignment operator with the = test operator.

plus
minus
multiplication
division
exponentiation
# Bash, version 2.02, introduced the "**" exponentiation operator. let "z=5**3" echo "z = $z" # z = 125 
modulo, or mod (returns the remainder of an integer division operation)
bash$ echo `expr 5 % 3` 2 
This operator finds use in, among other things, generating numbers within a specific range (see Example 918 and Example 919) and formatting program output (see Example 266). It can even be used to generate prime numbers, (see Example A11).
"plusequal" (increment variable by a constant)
let "var += 5" results in var being incremented by 5.
"minusequal" (decrement variable by a constant)
"timesequal" (multiply variable by a constant)
let "var *= 4" results in var being multiplied by 4.
"slashequal" (divide variable by a constant)
"modequal" (remainder of dividing variable by a constant)
Arithmetic operators often occur in an expr or let expression.
Example 81. Using Arithmetic Operations
#!/bin/bash # Counting to 6 in 5 different ways. n=1; echo n "$n " let "n = $n + 1" # let "n = n + 1" also works. echo n "$n " : $((n = $n + 1)) # ":" necessary because otherwise Bash attempts #+ to interpret "$((n = $n + 1))" as a command. echo n "$n " n=$(($n + 1)) echo n "$n " : $[ n = $n + 1 ] # ":" necessary because otherwise Bash attempts #+ to interpret "$((n = $n + 1))" as a command. # Works even if "n" was initialized as a string. echo n "$n " n=$[ $n + 1 ] # Works even if "n" was initialized as a string. #* Avoid this type of construct, since it is obsolete and nonportable. echo n "$n "; echo # Thanks, Stephane Chazelas. exit 0 
Integer variables in Bash are actually signed long (32bit) integers, in the range of 2147483648 to 2147483647. An operation that takes a variable outside these limits will give an erroneous result.

Bash does not understand floating point arithmetic. It treats numbers containing a decimal point as strings.

bitwise operators. The bitwise operators seldom make an appearance in shell scripts. Their chief use seems to be manipulating and testing values read from ports or sockets. "Bit flipping" is more relevant to compiled languages, such as C and C++, which run fast enough to permit its use on the fly.
bitwise left shift (multiplies by 2 for each shift position)
"leftshiftequal"
let "var <<= 2" results in var leftshifted 2 bits (multiplied by 4)
bitwise right shift (divides by 2 for each shift position)
"rightshiftequal" (inverse of <<=)
bitwise and
"bitwise andequal"
bitwise OR
"bitwise ORequal"
bitwise negate
bitwise NOT
bitwise XOR
"bitwise XORequal"
and (logical)
if [ $condition1 ] && [ $condition2 ] # Same as: if [ $condition1 a $condition2 ] # Returns true if both condition1 and condition2 hold true... if [[ $condition1 && $condition2 ]] # Also works. # Note that && operator not permitted within [ ... ] construct. 
&& may also, depending on context, be used in an and list to concatenate commands. 
or (logical)
if [ $condition1 ]  [ $condition2 ] # Same as: if [ $condition1 o $condition2 ] # Returns true if either condition1 or condition2 holds true... if [[ $condition1  $condition2 ]] # Also works. # Note that  operator not permitted within [ ... ] construct. 
Bash tests the exit status of each statement linked with a logical operator. 
Example 82. Compound Condition Tests Using && and 
#!/bin/bash a=24 b=47 if [ "$a" eq 24 ] && [ "$b" eq 47 ] then echo "Test #1 succeeds." else echo "Test #1 fails." fi # ERROR: if [ "$a" eq 24 && "$b" eq 47 ] # attempts to execute ' [ "$a" eq 24 ' # and fails to finding matching ']'. # # if [[ $a eq 24 && $b eq 24 ]] works # (The "&&" has a different meaning in line 17 than in line 6.) # Thanks, Stephane Chazelas. if [ "$a" eq 98 ]  [ "$b" eq 47 ] then echo "Test #2 succeeds." else echo "Test #2 fails." fi # The a and o options provide #+ an alternative compound condition test. # Thanks to Patrick Callahan for pointing this out. if [ "$a" eq 24 a "$b" eq 47 ] then echo "Test #3 succeeds." else echo "Test #3 fails." fi if [ "$a" eq 98 o "$b" eq 47 ] then echo "Test #4 succeeds." else echo "Test #4 fails." fi a=rhino b=crocodile if [ "$a" = rhino ] && [ "$b" = crocodile ] then echo "Test #5 succeeds." else echo "Test #5 fails." fi exit 0 
The && and  operators also find use in an arithmetic context.
bash$ echo $(( 1 && 2 )) $((3 && 0)) $((4  0)) $((0  0)) 1 0 1 0 
comma operator
The comma operator chains together two or more arithmetic operations. All the operations are evaluated (with possible side effects, but only the last operation is returned.
let "t1 = ((5 + 3, 7  1, 15  4))" echo "t1 = $t1" # t1 = 11 let "t2 = ((a = 9, 15 / 3))" # Set "a" and calculate "t2". echo "t2 = $t2 a = $a" # t2 = 5 a = 9 
The comma operator finds use mainly in for loops. See Example 1011.