It's quite a simple argument that leads you to think the millennium is actually in 2001, which is why it's wrong. It's only half the argument. Look:
In the fifth century (or thereabouts), a monk called Dionysus was told to conjure up a calendar. Naturally, it began with the birth of Jesus, but unnaturally, Jesus was apparently born in the year one, which is why the new millennium starts in 2001. Hmm? Wrong. We don't know when Jesus was born accurately enough to pin it down to the nearest five years, so calling it year one instead of year zero has little effect in reality. Which is why, for mathematical simplicity (not to mention correctness), the old Roman calendar of Dionysus is considered a total work of fiction. For simplicity's sake, we say Jesus was born in the year zero as we know it, and since we don't know any better, nobody can say it's wrong.
Thank you.