Is big O notation additive? For example, I have one program that requires $O(i)$ time complexity, and a second program requires $O(j)$ time complexity. Would the total time complexity be $O(i+j)$? And why?
If $f(n) = O(A(n))$ and $g(n) = O(B(n))$ then $f(n) + g(n) = O(A(n) + B(n)) = O(\max(A(n),B(n))$. These are standard properties of big O, which follow almost directly from the definitions.