Typical examples:
num=$((num1 + num2))
num=$(($num1 + $num2)) # Also works
num=$((num1 + 2 + 3)) # ...
num=$[num1+num2] # Old, deprecated arithmetic expression syntax
num=`expr $num1 + $num2`# Whitespace for expr is important
# this is only needed for really old systems
Other examples:
((i++))
count=$[count+1]