Initializing or changing the value of a variable
Allpurpose assignment operator, which works for both arithmetic and string assignments.
1 var=27 2 category=minerals # No spaces allowed after the "=". 
Do not confuse the "=" assignment operator with the = test operator.

plus
minus
multiplication
division
exponentiation
1 # Bash, version 2.02, introduced the "**" exponentiation operator. 2 3 let "z=5**3" 4 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 923 and Example 926) and formatting program output (see Example 2615 and Example A7). It can even be used to generate prime numbers, (see Example A17). Modulo turns up surprisingly often in various numerical recipes.
Example 81. Greatest common divisor
1 #!/bin/bash 2 # gcd.sh: greatest common divisor 3 # Uses Euclid's algorithm 4 5 # The "greatest common divisor" (gcd) of two integers 6 #+ is the largest integer that will divide both, leaving no remainder. 7 8 # Euclid's algorithm uses successive division. 9 # In each pass, 10 #+ dividend < divisor 11 #+ divisor < remainder 12 #+ until remainder = 0. 13 #+ The gcd = dividend, on the final pass. 14 # 15 # For an excellent discussion of Euclid's algorithm, see 16 # Jim Loy's site, http://www.jimloy.com/number/euclids.htm. 17 18 19 #  20 # Argument check 21 ARGS=2 22 E_BADARGS=65 23 24 if [ $# ne "$ARGS" ] 25 then 26 echo "Usage: `basename $0` firstnumber secondnumber" 27 exit $E_BADARGS 28 fi 29 #  30 31 32 gcd () 33 { 34 35 # Arbitrary assignment. 36 dividend=$1 # It does not matter 37 divisor=$2 #+ which of the two is larger. 38 # Why? 39 40 remainder=1 # If uninitialized variable used in loop, 41 #+ it results in an error message 42 #+ on first pass through loop. 43 44 until [ "$remainder" eq 0 ] 45 do 46 let "remainder = $dividend % $divisor" 47 dividend=$divisor # Now repeat with 2 smallest numbers. 48 divisor=$remainder 49 done # Euclid's algorithm 50 51 } # Last $dividend is the gcd. 52 53 54 gcd $1 $2 55 56 echo; echo "GCD of $1 and $2 = $dividend"; echo 57 58 59 # Exercise : 60 #  61 # Check commandline arguments to make sure they are integers, 62 #+ and exit the script with an appropriate error message if not. 63 64 exit 0 
"plusequal" (increment variable by a constant)
let "var += 5" results in var being incremented by 5.
"minusequal" (decrement variable by a constant)
"timesequal" (multiply variable by a constant)
let "var *= 4" results in var being multiplied by 4.
"slashequal" (divide variable by a constant)
"modequal" (remainder of dividing variable by a constant)
Arithmetic operators often occur in an expr or let expression.
Example 82. Using Arithmetic Operations
1 #!/bin/bash 2 # Counting to 6 in 5 different ways. 3 4 n=1; echo n "$n " 5 6 let "n = $n + 1" # let "n = n + 1" also works. 7 echo n "$n " 8 9 : $((n = $n + 1)) 10 # ":" necessary because otherwise Bash attempts 11 #+ to interpret "$((n = $n + 1))" as a command. 12 echo n "$n " 13 14 n=$(($n + 1)) 15 echo n "$n " 16 17 : $[ n = $n + 1 ] 18 # ":" necessary because otherwise Bash attempts 19 #+ to interpret "$[ n = $n + 1 ]" as a command. 20 # Works even if "n" was initialized as a string. 21 echo n "$n " 22 23 n=$[ $n + 1 ] 24 # Works even if "n" was initialized as a string. 25 #* Avoid this type of construct, since it is obsolete and nonportable. 26 echo n "$n "; echo 27 28 # Thanks, Stephane Chazelas. 29 30 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.
As of version 2.05b, Bash supports 64bit integers. 
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)
1 if [ $condition1 ] && [ $condition2 ] 2 # Same as: if [ $condition1 a $condition2 ] 3 # Returns true if both condition1 and condition2 hold true... 4 5 if [[ $condition1 && $condition2 ]] # Also works. 6 # Note that && operator not permitted within [ ... ] construct. 
&& may also, depending on context, be used in an and list to concatenate commands. 
or (logical)
1 if [ $condition1 ]  [ $condition2 ] 2 # Same as: if [ $condition1 o $condition2 ] 3 # Returns true if either condition1 or condition2 holds true... 4 5 if [[ $condition1  $condition2 ]] # Also works. 6 # Note that  operator not permitted within [ ... ] construct. 
Bash tests the exit status of each statement linked with a logical operator. 
Example 83. Compound Condition Tests Using && and 
1 #!/bin/bash 2 3 a=24 4 b=47 5 6 if [ "$a" eq 24 ] && [ "$b" eq 47 ] 7 then 8 echo "Test #1 succeeds." 9 else 10 echo "Test #1 fails." 11 fi 12 13 # ERROR: if [ "$a" eq 24 && "$b" eq 47 ] 14 # attempts to execute ' [ "$a" eq 24 ' 15 # and fails to finding matching ']'. 16 # 17 # if [[ $a eq 24 && $b eq 24 ]] works 18 # (The "&&" has a different meaning in line 17 than in line 6.) 19 # Thanks, Stephane Chazelas. 20 21 22 if [ "$a" eq 98 ]  [ "$b" eq 47 ] 23 then 24 echo "Test #2 succeeds." 25 else 26 echo "Test #2 fails." 27 fi 28 29 30 # The a and o options provide 31 #+ an alternative compound condition test. 32 # Thanks to Patrick Callahan for pointing this out. 33 34 35 if [ "$a" eq 24 a "$b" eq 47 ] 36 then 37 echo "Test #3 succeeds." 38 else 39 echo "Test #3 fails." 40 fi 41 42 43 if [ "$a" eq 98 o "$b" eq 47 ] 44 then 45 echo "Test #4 succeeds." 46 else 47 echo "Test #4 fails." 48 fi 49 50 51 a=rhino 52 b=crocodile 53 if [ "$a" = rhino ] && [ "$b" = crocodile ] 54 then 55 echo "Test #5 succeeds." 56 else 57 echo "Test #5 fails." 58 fi 59 60 exit 0 
The && and  operators also find use in an arithmetic context.
bash$ 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.
1 let "t1 = ((5 + 3, 7  1, 15  4))" 2 echo "t1 = $t1" # t1 = 11 3 4 let "t2 = ((a = 9, 15 / 3))" # Set "a" and calculate "t2". 5 echo "t2 = $t2 a = $a" # t2 = 5 a = 9 
The comma operator finds use mainly in for loops. See Example 1012.