Wikipedia says that
The 21st century began on January 1, 2001 and will end on December 31, 2100.
The first decade of 21st century is the 2000s. Here Wikipedia says that
The 2000s was a decade that began on January 1, 2000, and ended on December 31, 2009.
I don't understand. If the 21st century starts from 2001, then how can it be that the first decade of the 21st century starts from 2000?
A decade is simply a time span of 10 years and a century a span of 100 years. The start dates of each are determined by how they are being used.
The first decade of 21st century is 2000s
I think that statement is where the problem lies. Strictly speaking it isn't true.
Under the Gregorian calendar, the 21st Century started on January 1, 2001. The date is fixed by the fact that there was no year zero. So every century within that calendar started on January 1, XX01.
As far as decades within centuries, it's a matter of how they are being used. Most people would interpret the "2000s" as being January 1, 2000 to December 31, 2009 because they would interpret 2010s to be from January 1, 2010 and ending on December 31, 2019, etc.
However, if you were talking specifically about the first decade of the 21st Century you'd be discussing January 1, 2001 to December 31, 2010.
You're conflating mathematical precision with linguistic labeling.
The first decade starts on January 1 of a year ending with a "0".
The problem stems from the fact that when our dating system was created people commonly used the reigns of kings and leaders to describe events. We are now in the second year of the reign of Trump. In the 6th year of the reign of Obama the Russians invaded Crimea.
The same thinking was used by to create the current calendar. We are now in the Year of our Lord (Anno Domini) 2018. We changed AD to CE but we kept the numbering system.
Of course, in such a numbering system there is no year 0, hence causing our problem when we try to conform our naming system with more precise mathematical rules.
Maybe this would help. Imagine that you have 2100 bottles of beer in 21 cases, with 100 bottles per case. From which case does the 2000th bottle come from - the 20th or the 21st? What about the 2001st bottle?
Dates in the Gregorian calendar use Natural numbers (i.e., 1,2,3,4,...) and not Integer numbers (i.e., ...-2,-1,0,1,2,...) The dates in a Month are numbered 1 to 31 in most months, 1 to 30 in some other months and 1 to 28 or 1 to 29 in February, the 2nd month of the year. The months themselves are numbered 1st January, 2nd February,... up to 12th December (and months are not numbered 0 to 11.) The centuries are numbered 1st, 2nd, ..., 21st. The years in a century are numbered 1 to 100 so year 100 is the last year of the 1st century and year 1900 is the last year of the 19th century and year 2000 is the last year in the 20th century. The years in a Decade are numbered 1 to 10 (and not numbered from 0 to 9.) So the first decade of the 21st century contains years 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 and 2010. So year 2010 is the last year of the first decade of the twenty-first century. Roman numerals have historically been used to represent year numbers. Roman numerals do not have a representation for zero so Roman numerals are useful for representing natural numbers but less useful for representing integers. You might say MM/XII/XXXI is the last day of the 20th century where MM is the year, XII is the month and XXXI is the day-of-the-month.
