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The difference between the Millennium and year 2000
The 3rd Millennium starts January 1st, year 2001 NOT year 2000 as many people believe, when using the Gregorian calendar.
Year 2000 starts January 1st, year 2000.
The reason why the 3rd Millennium / 21st Century starts in 2001 is because there was no year 0 (or AD 0, 0 BC).
The year before 1 A.D. is defined as year 1 B.C., so year 0 was skipped. (See below. Therefore,
January 1st, year 1 is defined to be the start of the 1st century and the 1st Millennium.
Because one Millennium is 1000 years, the first Millennium ends with year 1000. The next (2nd) Millennium starts 1000 years after the first, that is in year 1+1000 = 1001. And the 3rd one starts 1000 years later than the 2nd: 1001+1000 = 2001. The same procedure could be followed for centuries. See the tables below for more information.
What should we celebrate?
It's no problem celebrating both of them, although probably most people will think year 2000 is the biggest of them.
When going to year 2000, we celebrate that it's a special round number, and in year 2001 we can celebrate a new century, a new Millennium, and that it's passed 2000 years since year 1 started.
Why was there no year 0?
When the present system we use to count years was invented by a scholar called Dionysius Exiguus in the 6th century (and later established around Europe) they used Roman numerals which did not have zero. Therefore 1 BC is the year before AD 1, with no intervening year 0. (Sequence ... 3 BC, 2 BC, 1 BC, AD 1, AD 2, AD 3 ...).
Exceptions - other calendars
There are many other calendars in use than the Gregorian Calendar, e.g. Jewish, Moslem, Buddhist, that started the numbering of years at a different time (initial epoch) than the Julian/Gregorian calendar, and therefore do not celebrate the 3rd Millennium at the same time.