# What does n with an index 0 mean in defenition of Theta/Big Omega/ Big-oh notation?

We define theta notation as follows: $$\Theta(g(n))$$ = {f(n): there are exist $$c_1, c_2$$ > 0 and $$n_0$$ such that 0 $$\leq$$ $$c_1$$g(n) $$\leq$$ f(n) $$\leq$$ $$c_2$$g(n) for all n > $$n_0$$}.

I found an illustration of this statement on Wikipedia, but I cannot figure out what exactly point $$n_0$$ means. How should we choose $$n_0$$ in practice? If I found, how should I use it?

It just means that the condition $$0 \le c_1 g(n) \le f(n) \le c_2 g(n)$$ only needs to hold for large enough integers.
This allows you to ignore all integers that are smaller than $$n_0$$ so you can just pick $$n_0$$ as large as needed for the inequalities to hold.
Notice that you do not need to pick $$n_0$$ as the smallest integer with that property, so you might as well pick one that makes proving the inequalities as easy as possible.
• For strictly positive functions on the integers domain (such as functions representing running time), the smallest $n_0$ is actually 0: just choose a constant $c$ big enough to overcome the first $n_0$ values. So I think its important to also note that $n_0$ isn't always needed, and when it isn't it can just be ommited altogether :) (to save your precious time from writing that $n_0$) Jul 8 '21 at 11:07