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" # 5 * 5 * 5 4 echo "z = $z" # z = 125 
modulo, or mod (returns the remainder of an integer division operation)
bash$ expr 5 % 3 2 
This operator finds use in, among other things, generating numbers within a specific range (see Example 911 and Example 915) and formatting program output (see Example 2716 and Example A6). It can even be used to generate prime numbers, (see Example A15). Modulo turns up surprisingly often in 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=85 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 dividend=$1 # Arbitrary assignment. 36 divisor=$2 #! It doesn't matter which of the two is larger. 37 # Why not? 38 39 remainder=1 # If an uninitialized variable is used inside 40 #+ test brackets, an error message results. 41 42 until [ "$remainder" eq 0 ] 43 do # ^^^^^^^^^^ Must be previously initialized! 44 let "remainder = $dividend % $divisor" 45 dividend=$divisor # Now repeat with 2 smallest numbers. 46 divisor=$remainder 47 done # Euclid's algorithm 48 49 } # Last $dividend is the gcd. 50 51 52 gcd $1 $2 53 54 echo; echo "GCD of $1 and $2 = $dividend"; echo 55 56 57 # Exercises : 58 #  59 # 1) Check commandline arguments to make sure they are integers, 60 #+ and exit the script with an appropriate error message if not. 61 # 2) Rewrite the gcd () function to use local variables. 62 63 exit 0 
plusequal (increment variable by a constant) [1]
let "var += 5" results in var being incremented by 5.
minusequal (decrement variable by a constant)
timesequal (multiply variable by a constant)
let "var *= 4" results in var being multiplied by 4.
slashequal (divide variable by a constant)
modequal (remainder of dividing variable by a constant)
Arithmetic operators often occur in an expr or let expression.
Example 82. Using Arithmetic Operations
1 #!/bin/bash 2 # Counting to 11 in 10 different ways. 3 4 n=1; echo n "$n " 5 6 let "n = $n + 1" # let "n = n + 1" also works. 7 echo n "$n " 8 9 10 : $((n = $n + 1)) 11 # ":" necessary because otherwise Bash attempts 12 #+ to interpret "$((n = $n + 1))" as a command. 13 echo n "$n " 14 15 (( n = n + 1 )) 16 # A simpler alternative to the method above. 17 # Thanks, David Lombard, for pointing this out. 18 echo n "$n " 19 20 n=$(($n + 1)) 21 echo n "$n " 22 23 : $[ n = $n + 1 ] 24 # ":" necessary because otherwise Bash attempts 25 #+ to interpret "$[ n = $n + 1 ]" as a command. 26 # Works even if "n" was initialized as a string. 27 echo n "$n " 28 29 n=$[ $n + 1 ] 30 # Works even if "n" was initialized as a string. 31 #* Avoid this type of construct, since it is obsolete and nonportable. 32 # Thanks, Stephane Chazelas. 33 echo n "$n " 34 35 # Now for Cstyle increment operators. 36 # Thanks, Frank Wang, for pointing this out. 37 38 let "n++" # let "++n" also works. 39 echo n "$n " 40 41 (( n++ )) # (( ++n )) also works. 42 echo n "$n " 43 44 : $(( n++ )) # : $(( ++n )) also works. 45 echo n "$n " 46 47 : $[ n++ ] # : $[ ++n ] also works 48 echo n "$n " 49 50 echo 51 52 exit 0 
Integer variables in older versions of Bash were signed long (32bit) integers, in the range of 2147483648 to 2147483647. An operation that took a variable outside these limits gave 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.
Use bc in scripts that that need floating point calculations or math library functions. 
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 provide direct access to system hardware. However, see vladz's ingenious use of bitwise operators in his base64.sh (Example A54) script.
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 NOT
bitwise XOR
bitwise XORequal
NOT
1 if [ ! f $FILENAME ] 2 then 3 ... 
AND
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 inside brackets
7 #+ of [ ... ] construct. 
&& may also be used, depending on context, in an and list to concatenate commands. 
OR
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  $condition2 ]] # Also works.
6 # Note that  operator not permitted inside brackets
7 #+ of a [ ... ] construct. 
Bash tests the exit status of each statement linked with a logical operator. 
Example 83. Compound Condition Tests Using && and 
1 #!/bin/bash 2 3 a=24 4 b=47 5 6 if [ "$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 # Note: if [[ $a eq 24 && $b eq 24 ]] works. 18 # The doublebracket iftest is more flexible 19 #+ than the singlebracket version. 20 # (The "&&" has a different meaning in line 17 than in line 6.) 21 # Thanks, Stephane Chazelas, for pointing this out. 22 23 24 if [ "$a" eq 98 ]  [ "$b" eq 47 ] 25 then 26 echo "Test #2 succeeds." 27 else 28 echo "Test #2 fails." 29 fi 30 31 32 # The a and o options provide 33 #+ an alternative compound condition test. 34 # Thanks to Patrick Callahan for pointing this out. 35 36 37 if [ "$a" eq 24 a "$b" eq 47 ] 38 then 39 echo "Test #3 succeeds." 40 else 41 echo "Test #3 fails." 42 fi 43 44 45 if [ "$a" eq 98 o "$b" eq 47 ] 46 then 47 echo "Test #4 succeeds." 48 else 49 echo "Test #4 fails." 50 fi 51 52 53 a=rhino 54 b=crocodile 55 if [ "$a" = rhino ] && [ "$b" = crocodile ] 56 then 57 echo "Test #5 succeeds." 58 else 59 echo "Test #5 fails." 60 fi 61 62 exit 0 
The && and  operators also find use in an arithmetic context.
bash$ 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. [2]
1 let "t1 = ((5 + 3, 7  1, 15  4))" 2 echo "t1 = $t1" ^^^^^^ # t1 = 11 3 # Here t1 is set to the result of the last operation. Why? 4 5 let "t2 = ((a = 9, 15 / 3))" # Set "a" and calculate "t2". 6 echo "t2 = $t2 a = $a" # t2 = 5 a = 9 
The comma operator finds use mainly in for loops. See Example 1113.
[1]  In a different context, += can serve as a string concatenation operator. This can be useful for modifying environmental variables. 
[2]  Side effects are, of course, unintended  and usually undesirable  consequences. 