Is it possible to include numerical values or constants within brackets(time complexity)? I am currently trying to learn a bit more about the theoretical side of computer science and I have stumbled upon this example I found online:


  $5n^2 + 5n = O(0.5n^2)$


I do have basic understanding of how time complexity works but it is the only example I have found that has a number attached to the $n$ (right side). Does it mean that the person writing that article made a mistake or it is possible to include values this way (like in the definition: $0 ≤ f(n) ≤ cg(n)$ where $c = 0.5$ in this case)?

Sorry if I missed an obvious aspect of this concept but I try learning this stuff on my own and have nobody apart from the internet to ask for help. :)
What you read is slightly unusual, but absolutely correct. Follow the definition of Big-O literally: It means there is a c > 0 such that for large n, f(n) &lt;= c * 0.5 * n^2. It’s exactly the same as O(n^2), except that the constant c is different (twice as big will work). 

In your example, for n >= 5 we have 5n &lt;= n^2 and therefore 5n^2 + 5n &lt;= 6n^2. So f(n) &lt;= 12 * 0.5 n^2 and f(n) &lt;= 6*n^2, c=12 instead of c=6.