Low Level Fraud

I dislike stores that ask you to carry a loyalty card so that you can save on their products. I carry a very small wallet, so I never keep these cards. But I always want to save money so I sign up for the cards and keep them by my keys at the door, but not once have I remembered to grab them when I head to the store. The clerks are always helpful, saying they can look up my card based on my phone number, but when I sign up for their loyalty cards I never want to give them my personal information, so I make up numbers. It's led to some awkward exchanges when they look at the number I wrote down and recognize the area code and say, "Hey! You're from Nashville." I've never been to Nashville, and here they've caught me trying to save $2 on a pack of batteries. I wonder if by lying to save money I'm committing fraud to the tune of $2? I suppose that would be against the law.