I want to calculate the complexity of multiplying an a-bit number by a b-bit number. Intuitively, this would have complexity $O(ab)$ but I do not know how to show that this is true.
I am also interested in the analogous complexities of division and modular arithmetic in this case so any links to sources such as textbooks or academic papers is greatly appreciated.
You need to perform $ab$ single digit multiplications, and at most $3ab$ single digit additions (just count casually, overcounting anything that is hard to estimate, also, $\min(a,b)$ multi-digit additions). So it's $O(ab)$.