I've been reading around the area of complexity and arithmetic operations using logic gates; one thing that is confusing me is that \begin{equation}\Theta (n^{2})\end{equation}is quoted as being the complexity for multiplication for iterative adition. But addition of a number requires \begin{equation}log_2(n)\end{equation}operations, 1 for each bit or 8 times that for each nand gate involved in doing this. So it strikes me as obvious that adding that number n times will have a complexity of \begin{equation}n \log_2(n)\end{equation}Which is definitely less than \begin{equation}\Theta (n^{2})\end{equation}So where is this additional factor of\begin{equation}\frac{n}{\log_2(n)}\end{equation}coming from?
Complexity of multiplication
complexity theory;arithmetic
Addition of a number of size $n$ takes time $O(n)$. Don't confuse a number and its encoding size, which is logarithmically smaller.When multiplying an $n$-bit integer $a$ by an $n$-bit integer $b$ using the iterative addition algorithm, you are adding up to $n$ shifted copies of $a$. Each addition costs you $O(n)$ rather than $O(\log n)$. The numbers $a,b$ itself could be as large as $2^n$.