"Why are additions known to be cheaper than multiplications?"
In contexts pertaining to algebraic complexity theory, this statement is often cited. Can someone elaborate on this? I don't understand the reason behind multiplications being more expensive than additions.
Thank you!
On
This a very good question, and the answer is surprinsingly an open problem in Theoretical Computer Science. In a nutshell: one can do addition in linear time, and the best known upper bound for multiplication is (approximately) $O(n\log n)$. But there is no proof that multiplication cannot be done in linear time.
You should look at Is there a proof that addition is faster than multiplication? for more details.
Essentially, because in order to perform a multiplication, you need many additions:
$$ M \times N = \underbrace{M + \ldots + M}_{N \text{ times}} $$ for $M,N$ integers.