# Which is the proper way of writing about the 20th century? Should 20th be written out? Should there be a hyphen between the two words? Should either or both words be capitalized?

One correct way is that you used in the question. It is correct to use the number and add the ending of ordinal number (sometimes it is written with upper index ${20}^{t h}$). It is also correct to write the number as a word. It is unnecessary to capitalize any word in this expression. It is not a proper name.