Definition:Big-O Notation

Definition
Let $f$ and $g$ be real functions.

The statement:
 * $f = \mathcal O \left({g}\right)$

is equivalent to the statement:
 * $\displaystyle \exists \alpha \in \R: \alpha \ge 0 : \lim_{x \to \infty} \frac{f \left({x}\right)}{g \left({x}\right)} = \alpha$

This statement is voiced $f$ is big-O of $g$ or simply $f$ is big-O $g$.

Also denoted as
In number theory, sometimes the notation $f \ll g$ is used to mean $f = \mathcal O \left({g}\right)$.

This is clearer for estimates leading to typographically complex error terms.

Some sources use an ordinary $O$:
 * $f = O \left({g}\right)$