3 gallons is the minimum for a single betta, so that's ok, but absolutely no more tankmates can be added. Fish don't appreciate friends when it cramps their already tight space and fouls up the water.
What temperature is your house consistently set at? If it's not high enough to keep the water temperature around 77-80, I'd agree that you need a heater. Also, please make sure you're doing regular weekly water changes. 3 gallon tanks have their water parameters get out of whack so quickly (especially with no live plants) that you want to make sure things are staying clean.
I recommend getting a water test kit (drops kit, not a strip test kit since those are extremely inaccurate) and getting in the habit of testing regularly (daily would be ideal until you get a sense for how quickly the water fouls up). Any time the nitrates go over 25 or so, it's a good idea to do a water change. If you haven't cycled your tank, you'll want to check ammonia and nitrites as well until those read as zero consistently. If either of those creep up at all, you'll need a water change to keep from poisoning the betta.