I'm reading A first Course on Logic, (Hedman).
An algorithm is said to be polynomial-time if there is some number $k$ so that, given any input of size n, the algorithm reaches it's conclusion in fewer than $n^k$ steps.
What is this number k? When it says "there's a number $k$", I'm thinking that any number could fill the $k$ and this would make the concept irrelevant.