Numbers are crucial to communicating in writing in any field. Numbered centuries are a particularly common usage. In the social sciences, American Psychological Association style dictates specific circumstances for when to use numerals and when to spell out the numbers as words; the general rules apply to centuries.
Use numerals for most centuries, because the "Publication Manual of the American Psychological Association, 6th Edition" calls for using numerals for numbers higher than nine. For example, "Many people dispute whether the millennium ended with the 20th century, but the year 2000 is generally agreed, at least, to have begun the 21st century."
Spell out the names of centuries if the number is smaller than 10. For example, "Vikings began their most vigorous period of raiding in the eighth century." One exception to these guidelines calls for spelling out any numerals, including centuries, which begin a sentence, because you must not start a sentence with a numeral. The APA handbook recommends, however, that you avoid this situation by reworking your sentence. The opening, “Fifteenth-century scientists believed,” could become “In the 15th century, scientists;” alternatively, you could construct a compound sentence so that “15th” follows a semicolon.
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