If you write a novel set in the future, do you call it the 35th Century? 35th century? Thirty-fifth Century? Thirty-Fifth Century? What about this century?
The short answer is that thirty-fifth century is almost always correct. Exceptions include if it is at the beginning of a sentence or part of a proper name. If it is part of a proper name, there are no set rules. It will depend on whoever created the proper name.
For example, you may see places with these kinds of names:
Twenty-first Century Innovations
Twenty-First Century Electronics
Twenty-First-Century Customs
21st Century Fox
Century 21 Investments
New Year Rings in 21st Century (A newspaper title)
When in doubt, use all lower case and write out all numbers: twenty-first century or thirty-fifth century.
Decades
There can also be confusion about how to write decades. There are two options: spell it out or express it with numerals:
the sixties
the 1960s
If you use the numerals, do not add an apostrophe between the numerals and the letter s. And leave the decade lower case unless it is part of a title or proper name.
It can be confusing to refer to the first two decades of a century.
If you want to talk about the first decade of the century, you can’t just write the 1900s. Some readers would think you’re referring to the entire century. So, should you use 1900 to 1909? Or maybe 1900-1910?
The second century is tricky, too. You can’t just reference the “teens” of a century. That doesn’t include the years ending in 10, 11, and 12.
It may be better to add a few more words for clarity. So, you could write: “There were many industrial advancements in the decade 1900-1909.”
Related: Commas in dates: writing a date correctly
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