Big O Magnitude Notation

Like Big O Unit Notation, Big O Magnitude Notation refers to a notation used to express orders of magnitude.

Formal Definition for Functions

A function $f(n)=O(10^{k})$ for some fixed $k\in \mathbb {Z+}$ if $\exists n_{0}\in \mathbb {R+}$ such that $\forall n\geq n_{0}:10^{k}\leq f(n)<10^{k+1}$

Formal Definition for Random Variables

A random variable $X=O(10^{k})$ if $P[10^{k}\leq X<10^{k+1}]=1$.