Actually, we refer to the years 1800 - 18 99 as the eighteen hundreds (1800s) , the years 1700 - 17 99 as the seventeen hundreds (1700s) , and the years 1600 - 16 99 as the sixteen hundreds (1600s) . And so on. So I'm not sure why you think that "the nineteen hundreds" should refer to only 10 years rather than to the entire century (1900 - 1999).
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Yankeepeople have used "the aughts" for the first decade of the 21st century.Here in the UK, the name "noughties" was proposed, but it has not achieved "proper word" status, and is, as far as I'm aware, only ever used light-heartedly.