What years are in th 20th century?

There was no year 0 AD. The first century began with year 1. One hundred years later was 100 AD. Therefore, unless the first century had only 99 years, 101 AD would have

been the first year of the second century, By logical induction, the 20 century began with

1901 and lasted through 2000. The 21st century began with 2001.