Does multiplication take unit time?

Question

I have the following problem

Under what circumstances can multiplication be regarded as a unit time operation?

But I thought multiplication is always considered to be taking unit time. Was I wrong?

Answer 1

It depends on what N is. If N is the number of bits in an arbitrarily large number, then as the number of bits increases, it takes longer to compute the product. However, in most programming languages and applications, the size of numbers are capped to some reasonable number of bits (usually 32 or 64). In hardware, these numbers are multiplied in one step that does not depend on the size of the number.

When the number of bits is a fixed number, like 32, then it doesn't make sense to talk about asymptotic complexity, and you can treat multiplication like an O(1) operation in terms of whatever algorithm you're looking at. When can become arbitrarily large, like with Java's BigInteger class, then multiplication depends on the size of those numbers, as does the memory required to store them.