Editing Big O notation (section)

=== Infinite asymptotics ===
{{Dark mode invert|image=yes|[[File:comparison computational complexity.svg|thumb|Graphs of functions commonly used in the analysis of algorithms, showing the number of operations <math>N </math> versus input size <math>n </math> for each function]]}}
Big O notation is useful when [[analysis of algorithms|analyzing algorithms]] for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size <math> n</math> might be found to be <math> T(n)=4n^2-2n+2 </math>. As <math> n</math> grows large, the <math> n^2</math> [[Summand|term]] will come to dominate, so that all other terms can be neglected—for instance when <math> n=500</math>, the term <math> 4n^2</math> is 1000 times as large as the <math> 2n</math> term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the [[coefficient]]s become irrelevant if we compare to any other [[Orders of approximation|order]] of expression, such as an expression containing a term <math> n^3</math> or <math> n^4</math>. Even if <math> T(n)=1000000n^2</math>, if <math> U(n)=n^3</math>, the latter will always exceed the former once {{mvar|n}} grows larger than <math> 1000000</math>, ''viz.'' <math display="block"> T(1000000)=1000000^3=U(1000000)</math>. Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either
:<math>T(n)= O(n^2) </math>
or
:<math>T(n) \in O(n^2) </math>
and say that the algorithm has ''order of {{math|n<sup>2</sup>}}'' time complexity. The sign "{{math|1==}}" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "[[#Equals sign|Equals sign]]" discussion below) while the first is considered by some as an [[abuse of notation]].<ref name="clrs3" />