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Not everybody agreed if the Third Millennium actually began on January 1st in the year 2000 AD, despite the fact that many millennium celebrations took place around the world. Many people think that we have just moved into the third millennium this January 1st, 2001.
What is agreed upon is that a millennium is an interval of 1000 years and a century is an interval of 100 years. However, because there is no such thing as year zero, from the start of this era an interval of 1 year had only passed at the end of the year named 1 AD - not at the beginning of the year. This is easier to understand if you think about the first century. 100 years will only have elapsed at the end of the year 100 AD. You wouldn't expect the second century to begin until after the first the century was over, which would be at the end of the year 100 AD. It is therefore clear that 2000 years will only have elapsed when we arrive at midnight on 31 December 2000. So the 3rd Millennium and the 21st Century actually start at the same moment, which is zero hour UTC (commonly known as GMT) on January 1st 2001.
Scientists, especially astronomers, are interested in how time is measured. There are web sites to visit observatories like the one in Greenwich, England, where you can find more information about the start of the millennium and time in general (http://www.rog.nmm.ac.uk/leaflets/new_mill.html ) and the history of time measuring tools (http://physics.nist.gov/GenInt/Time/time.html ).