(Remember this is a pedantic point about math, not history.)
which do you choose, how would you make sense of that ?
For me it's easy.
The answer is that the Christians did not invent the calendar year. The Calendar year had already been well-established and the Christians were not in a position to make radical changes to the structure of the entire Calendar. Obviously if they could then they would have designed their newly invented Calendar with Jesus being born on January 1st, (just by arbitrary definition of their new Calendar
But clearly they could not do such a drastic thing.
However what already was a common practice was for various cultures to NUMBER the years however they saw fit. So apparently you could number the years however you wanted to, as long as you didn't mess with the overall structure of the calendar which was actually designed around the rotation of the earth on its axis (giving rise to the days), the orbit of the earth around the sun (giving rise to the year), and the orbit of the moon around the earth (giving rise to the "moonths", or months).
So all of that was already carved in "Celestial Stone". The only thing left open to arbitrary choice would be how you decide to actually number the years. When you started counting years wouldn't change the calendar much at all. But it could confuse a lot of people who might have already been numbering them differently.
So since this is all the Christians could do they would most likely pick the year in which Jesus was born as the FIRST "Year of Our Lord". And so the year Jesus was born would become year 1 A.D. It wouldn't matter that Christ's birth wasn't in perfect sync with the calendar new year. The year in which he was born could still be arbitrarily numbered as 1 A.D.
Anything prior to that year would then be considered to be B.C.
There is no "mathematical
" problem here because the Birth of Jesus isn't really defining the calendar. The year in which he was born is simply being used to start numbering the years from that year forward. And of course, the years B.C. would be retroactively renumbered backward in time.
Keep in mind that in the beginning this was just a Christian convention. Obviously after Roman made Christianity the official religion of the state, this kind of numbering of the calendar years became officially formalized in state commerce. And of course after that it just grew until it became a standard for anyone who wanted to do commerce with Rome, or anyone else who was doing commerce with Rome.
To be honest, I'm no expert in precisely how this all unfolded historically. But I'm pretty sure my main points are correct.
So all this amounted to was a change in how the years were arbitrarily numbered. It wasn't a complete reinvention of the calendar. The calendar is based on the motions of the earth and moon, not on the birth of Jesus.
This is also why they can say that Jesus was born on December 25th instead of January 1st without any problem. It was widely understood that they didn't re-invent the calendar. They simply chose a new arbitrary numbering system for the years based on the birth of their demigod Christ.
So it's not a calendar based on the birth of Christ. It's just a religion choosing to number the years based on the birth of Christ. They didn't change the actual calendar.
Edited to add:
By the way, this would mean that technically Jesus was born on December 25th, 1 A.D. (by Christian convention)
That's just the way it would work out. Keep in mind also that all of this calendar stuff wasn't done for many decades after the death of Jesus. So it's hard to know how the Jews had been numbering years at the time Jesus might have actually been born,. But they most certainly woiuldn't have been calling it year 1 A.D.
I would think that some historians must know how the Jews were numbering years back in those days. To be honest I have never even thought about this. But you can be certain that the Jews weren't calling the year Jesus was born in 1 A.D. They necessarily had to be calling it something else. They had to be using a different numbering system back then to count the years.
Now that you have me thinking about this I am inspired to look into it, just out of curiosity. Surely historians who have studied these time periods would know what calendar system was being used at the time of the birth of Jesus and how the Jews had been counting the years at that time. Surely they had calendars back then. They had commerce, surely they were keeping track of dates.