# Developers Planet

coders February 2016

### Big O notation of a constant

I calculate my runtime complexity to be 4, what is the Big O notation of this?

For example if my runtime complexity is 4 + n then its Big O = O(n).

dfri February 2016

Let's look loosely at the definition of what we mean by `f(n) is in O(g(n))`:

`f(n)` is in `O(g(n))` means that `c · g(n)` is an upper bound on `f(n)`. Thus there exists some constant `c` such that `f(n) ≤ c · g(n)` holds for sufficiently large `n` (i.e. , `n ≥ n0` for some constant `n0`).

You can treat a constant function just as any other function, w.r.t. analysing its asymptotic behaviour using e.g. big-O notation.

``````f(n) = 4
g(n) = 1

f(n) ≤ c · g(n) = c · 1, for c ≥ 4 and for all n           (*)

(*) with e.g. n0=0 and c=4 => f(n) is in O(1)
``````

Note: as Ctx notes in the comments below, `O(1)` (or e.g. `O(n)`) describes a set of functions, so to be fully correct, `f` should be described to be in `O(1)` (`f ∈ O(n)`, `f`:s set membership in `O(1)`), rather than "`f(n)` being in `O(1)`". You can, however, probably expect to see the less rigorous version "`f(n)` is in `O(1)`" (or some `O(g(n))`) just as frequently at the web, at least outside of the scope of scientific articles.