Big-o notation confused

Question

In error analysis we say the error is of order $\epsilon$ if the error is less than or equal constant multiple of $\epsilon$.

The question is: what is the benefit we have if we can multiply by any constant we choose?

Answer 1

The point is that you usually don't care about the constant. If $\epsilon = 10^{-10}$, it's more important to know whether your error bounded by $8 \epsilon$ or $8 \epsilon^2$ than it is to know whether your error is bounded by $8\epsilon$ or $\pi^2 \epsilon$...