I'm afraid you are incorrect, with an ADSL connection (standard broadband internet) even with your router left on your IP address can change every few days. Your ISP (internet service provider) leases you an IP address for a certain period of time, when that time period runs out if your router is still on it will automatically ask for another IP address, you may get the same one, or you may get a different one, again with a time limit, and so on, the time limit depends on your ISP. I presume you can negotiate with your ISP to have a constant IP address, but I'm not sure about this.