Actually, you wouldnt necessarily feel it if it was rotten. My wisdom tooth did just exactly what your tooth has done back in Oct. The reason it chipped off is that I had a cavity (that I never did feel) and it weakened the tooth. My whole tooth them became infected, and it really only hurt me for about 2 days. I just had it removed this morning.
My other wisdom tooth was rotten a few years back and I didnt feel any pain (and therefore didnt get anything done about it) for well over a year...not to smart. It chipped many times and I never had any pain.
Bottom line...if your tooth broke its because there is something wrong. You need to get to a dentist before further complications arise. If it isnt a wisdom tooth then there may be a good chance that the tooth can be saved if you get in and get to it before it rots too much. If it is a wisdom tooth or it is infected you want it pulled out before you run into serious medical complications like a blood infection or heart trouble. Rotten teeth are nothing to kid around with.
If it was rotten - you'd feel it, trust me.
You need to get the broken piece removed and built up with a filling to preserve your natural tooth. Otherwise it will go rotten AND cost you a ton of money.