There is no year Zero. The year of the supposed birth was the year 1CE. the year before that was 1 BCE.
Part of the reason why one doesnt use BC and AD is because the monk who did the calculation made a mistake. It was a tough piece of work because he was after the fact trying to make sense of all the partial years where the end of one emperors life was the same year as the first year of the next emperor. We are still finding out emperors who ruled briefly, or ruled while another emperor ruled.
By using BCE and CE you date it back to the Herod the Great. The gospel record relates the birth to Herod the great (4 BC) in Matthew, to the great census (6AD) with Mark and John not taking a position.
Part of the reason why one doesn't use BC and AD is because the monk who did the calculation made a mistake. It's even worse than that. If it were important to know the exact date of Jesus' birth there would be some Biblical authority on it. There isn't any mention of the year or the day.
His birthday isn't important. That's a "So what?", everyone has a birthday.
He's different. He rose from the dead. That's unique. AND The Bible records the exact day and provides enough information to extract the year.
Our calendar is off by about 33 years and 4 months...