What is the relation between mathematical approximation of Big O notation and pratical one in programming

Question

I have a question about asymptotic complexity, specifically Big O notation.

Mathematically speaking: we have a function T(n) it's output : the amount of time taken by our algorithm.

for example T(n) = 1000 + 10n . We choose a simple function F(n) = n , and for some large natural number N and constant C , T(n) <= C.F(n) which is equivalent to that T(n) belonging O(f(n)) ( sets of functions which T one of them dominated by C.F(n) ).

My specific question: I didn't get the point of upper bounds , where from a specific input N , all T(n) points are upper bounded by F(n) points .

What's the relation between that mathematical approximation and worst-best case complexity in computer science and Big O notation

Answer 1

After reading more about the topic. This mathematical proof is just an approximation to explain that our function T(n) [ which it's output number of instructions or steps our algorithm spend to do some specific job ] is upper bounded by the equivalent function C.F(n) where C is some specific big constant > 0 . so our upper bound refers to the worst-case complexity , that our algorithm can reach ( and can reach because mathematically T(n) and F(n) intersect in some specific big N > 0 which become a supremum ( upper bound bellong to our function) ) . so in resume , it's an approximation to prove that the worst case complexity is an upper bound ( or supremum ) that our algorithm can reach .