According to multiple definitions and my math professor, problems in $NP$ can be rewritten to a problem in $NPC$, including problems in $P$. Why can I solve a $P$ problem in polynomial time, but can't solve the $NPC$ version in polynomial time. Are the two problems not the same? Did I create a different problem by rewriting it?
The problem in my train of thought is probably that I don't understand how rewriting a problem works. Right now, I thought it was like 2 + 2 = 3 + 1, but it looks like it is something like 2 + 2 = e/pi, making the problem much harder.
