Cost of shifting a number I was wondering what would be a time complexity of shifting a binary or a decimal number?
For example: 0011, when I shift it left I get 0001.

I was thinking that the time complexity is $\Theta(n)$, because we have to shift every number.

I need it to calculate $m*{b^\frac{n}{2}}$, where $m$ is some number 
and $b$ might be 2 (in case of binary) or 10 (decimal).
Therefore instead of multiplication, I just shift to multiply, and therefore complexity is $\Theta(n)$.

Am I correct or is my thinking wrong?
Time complexity is not god-given but is rather defined by the model of computation. To take a simple example, what is the time complexity needed for adding two numbers? In the RAM machine model, this operations takes constant time, as long as the numbers are not "too big". The rationale is that modern hardware implements addition as a fast machine instruction; if the numbers are not "too big" then they fit into single registers. The same reasoning holds for shifts. You have to be careful, however, that the size of the relevant quantities doesn't blow up.