By your logic, memory access takes time Ω(n^(1/3)) because memory cells need to occupy physical space and we live in a three-dimensional reality, so the time to

By your logic, memory access takes time Ω(n^(1/3)) because memory cells need to occupy physical space and we live in a three-dimensional reality, so the time to access a memory cell must necessarily increase with the physical distance to that memory cell. But using a model that made such assumptions would be needlessly complex, so nobody does it. (Incidentally, there was a post on HN that argued this a few years ago, and the comments were full of people who felt the need to ignore the reasoning behind the assumptions backing commonly-used machine models, much like it is happening in this thread.) Arbitrarily large numbers, represented in binary, certainly take time that is linear in the length of their representation to add. Nobody's arguing with that. But using a model that assumes bitwise operations would be needlessly complex, so outside of some really specialized contexts, nobody does it. > You're just wrong about this. Yes, you can absolutely say "assuming storage of all numbers is O(1), then the algorithm is O(1)" (essentially what OP said) but that is not a default assumption I don't know the basis on which you argue that the word RAM model isn't the default assumption, but I feel confident in claiming that outside of some niche communities, it most certainly is. > and it's not accurate to reality (since actually storing numbers requires O(log n) of space). That's why it's called a model. We use models because reality is too messy to work with, and we usually don't care about the exact details. Just ask a physicist, they make simplifying assumptions all the time because without them, they'd be getting absolutely nowhere. It's pretty much the same in algorithmics.