Using expr, $(()), (())

1. For arithmetic, expr is archaic. Don't use it.*

2. $((...)) and ((...)) are very similar. Both do only integer calculations. The difference is that $((...)) returns the result of the calculation and ((...)) does not. Thus $((...)) is useful in echo statements:

   $ a=2; b=3; echo $((a*b))
   6
   

   ((...)) is useful when you want to assign a variable or set an exit code:

   $ a=3; b=3; ((a==b)) && echo yes
   yes
   

3. If you want floating point calculations, use bc or awk:

   $ echo '4.7/3.14' | bc -l
   1.49681528662420382165
   
   $ awk 'BEGIN{print 4.7/3.14}'
   1.49682
   

*As an aside, expr remains useful for string handling when globs are not good enough and a POSIX method is needed to handle regular expressions.

expr is old, but it does have one limited use I can think of. Say you want to search a string. If you want to stay POSIX with grep, you need to use a pipe:

if echo november | grep nov 
then
  : do something
fi

expr can do this without a pipe:

if expr november : nov
then
  : do something
fi

the only catch is expr works with anchored strings, so if you want to match after the beginning you need to change the REGEXP:

if expr november : '.*ber'
then
  : do something
fi

Regarding (( )), this construct is not POSIX, so should be avoided.

Regarding $(( )), you do not have to include the dollar sign:

$ fo=1
$ go=2
$ echo $((fo + go))
3