Given the limits of sixth-century math, the monk had no choice. He had to start the first millennium in A.D. 1 and continue it through the year 1000. And the third will begin on Jan. 1, 2001. So the sticklers are correct. The rest of us are suffering from premature congratulation.
Nature knows no centuries, no millennia.
It's basic astronomy. Thus, the day and the year. All attempts to wrestle the lunar cycle onto the solar-year template eventually fail. In other words, they're arbitrary. Because early humans started counting on their fingers, we naturally count by 10s.