“Eighteen hundreds,” “sixteen hundreds” and so forth are not exactly errors; the problem is that they are used almost exclusively by people who are nervous about saying “nineteenth century” when, after all, the years in that century begin with the number eighteen.
This should be simple: few people are unclear about the fact that this is the twenty-first century even though our dates begin with twenty. For most dates you can just add one to the third digit from the right in a year and you’ve got the number of its century. It took a hundred years to get to the year 100, so the next hundred years, which are named “101,” “102,” etc. were in the second century.
This also works BC. The four hundreds BC are the fifth century BC. Using phrases like “eighteen hundreds” is a signal to your readers that you are weak in math and history alike.
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