Big O Unit Notation

Big O Unit Notation is a variant of Big O notation that is used when you want to express a notion of order of magnitude. You can usually differentiate these two concepts by the fact that Big O Notation is expressed in terms of functions whereas Big O Unit Notation is expressed in terms of units.

One writes that some process or action takes $O(milliseconds)$ or is of length $O(kilometres)$ to mean that it is in $[0.5,100)$ when represented in that unit.

Formal Usage

Perhaps more formally: A measurement or quantity is said to be $O(unit)$ if:

1. The quantity can be expressed in terms of the given unit.
2. When expressed in that unit, the numerical value falls within the range $[0.5,100)$.
3. It is the largest such unit that satisfies conditions 1 and 2.

Examples:

1. A 100 m race is $O(seconds)$
2. A marathon is run in $O(hours)$
3. A Starbucks is $O(meters)$ away from me
4. San Jose is $O(kilometers)$ away from San Francisco

Colloquial Usage

Colloquial usage weakens the bounds and allows for representing with a multiplier such as $O(100km)$

Conversion to Other Bases

In other bases, we may use $[1,b*b]$ as our limits.