And what we're talking about is that it doesn't work that way
Wait wtf was I thinking...?
Anyways so here's what happens.
If your computer isn't configured with a static ip, then it requests one from your router's DHCP server. But it holds on to this IP for a specified lease time (several hours to a day at least), and automatically re-requests the same IP before the lease is over. Which brings up the question: how long are you leaving the router unplugged for, and why?
Either way, set up a static IP on your
computer, not your router/modem (Why would you not have done this already?)