Let:
$a = (1011011)_2 = (1123)_4$
There two ways to solve it:
Convert the number in base 2 to base 10 then to base $4$.
Consider $4 = 2^2$ and group each of two numbers in base $2$ to one in base $4$.
I can easily convert when the source base in less than then destiny base but how could I do the inverse?
Said:
$b = (1123)_4 = (x)_2$
I tried to do successive divisions grouping each two numbers in base $4$ as the source base is greater than destiny but I get different results, that can be seen:
$(1123)_4 = (11)_4 (23)_4$
$(11)_4 \Rightarrow (1011)_2$
$(23)_4 \Rightarrow (10111)_2$
$(1123)_4 \nRightarrow (1011 10111)_2$
I am doing the successive divisions correctly but it seems I am getting one bit more than expected.
How to correctly do this?
I think each single digit in base 4 corresponds to 2 digits in base 2 (possibly 0-filled):
$$(1123)_4 = (\underbrace{01}_1 \underbrace{01}_1 \underbrace{10}_2 \underbrace{11}_3)_2$$
One usually omits leading zeroes in the final result, so you have $(1011011)_2$.