Centuries are designated with ordinal numbers (fourth, fifth, 20th), and individual years are designated with cardinal numbers (45, 1492, 2016).
The division into BC and AD is purely historical and arbitrary. Jesus was not even born in 1 AD. We now recognize that Christ was born about four years Before Christ. But as long as we have a starting base for our years and centuries, I don’t care if it is the birth date of Christ or Cleopatra or Jack the Ripper.
What seems to throw people is that years like 1900 are the LAST year of the 19th century, and 1901 is the first year of the 20th century, with 2000 being the last year of the century that began in 1901.
Also confusing is that every century seems to have the wrong ordinal number. 1939 is part of the 20th century and Columbus sailed in 1492,so he sailed in the 15th century.
This is not just convention. It based on the reality of numbers.
Think about it like this. Imagine I am paying you $20. But I am doing it all in pennies, so I am giving you 2000 pennies. And every penny has a number from 1 to 2000 painted on it.
I start giving you the first dollar (the first century in other words). I begin with penny 1 and keep giving you the pennies one at at time. When have I given you the first dollar? When I give you penny 99? NOOOO! When I give you penny 100. So penny 100 belongs in the first dollar, right?
Now I start giving you the second dollar. The first penny of the second dollar is penny 101. The last penny of the second dollar (i.e. the second century)
So we just keep going like this. Finally I give you penny 1900. It is the last penny of the 19th dollar (the 19th century).
And finally the last dollar. It begins with penny 1901, and ends with… penny 2000, the last penny of the 20th dollar (20th century).
Does that make more sense?