Use a thought experiment.
Posit a situation in which mankind has decided to implement universal calendar reform. Beginning tomorrow, we are arbitrarily calling it New Year’s Day, January 1, the first day of the First Century, NC (New Calendar).
Is it year 0 (zero), or is it year 1 (one)?
If it’s year 1, the first year of the second century will be 101. If on the other hand, it’s year 0, the first year of the second century will be 100.
If it’s year 1, the first year of the third century will be 201. If on the other hand, it’s year 0, the first year of the third century will be 200.
If it’s year 1, the first year of the fourth century will be 301. If on the other hand, it’s year 0, the first year of the fourth century will be 300.
If it’s year 1, the first year of the fifth century will be 401. If on the other hand, it’s year 0, the first year of the fifth century will be 400.
<SNIP>
If it’s year 1, the first year of the twenty-first century will be 2001. If on the other hand, it’s year 0, the first year of the twenty-first century will be 2000.
Do you want to live in a world in which a theoretical calendar can boldly display “This is the Year 0”? I don’t want to. And I maintain that I don’t actually live in such a world.
Therefore, the 21st century began on January 1, 2001. Q.E.D.