No. it's not necessary unless it forms part of the proper noun. But twentieth can also be written in numeral.
Examples:
20th Century Fox
Twentieth Century ( title of the movie directed by Howard Hawks)
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No, it is mid to late twentieth century
Both are equally acceptable, equally used and equally understood
Yes.
It is capitalized at the beginning of the sentence or when it forms part of the proper noun. Examples: The 2nd century is considered part of the Classical era. The Second Century Initiative is celebrating its anniversary.
S. Hurok was a famous impresario of the twentieth century.
Yes, "Twenty-First Century" should be capitalized because it refers to a specific time period.
No. Numbers in a normal sentence are not capitalized. Example: "My birthday is on the twentieth of January."
On the Twentieth Century was created in 1978.
No, it is mid to late twentieth century
The Twentieth Century Approaches was created in 1986.
The ISBN of The Unfinished Twentieth Century is 9781859847800.
The Myth of the Twentieth Century was created in 1930.
Twentieth Century - typeface - was created in 1937.
The Hoax of the Twentieth Century was created in 1976.
Twentieth Century Eightball was created in 2002.
Paris in the Twentieth Century was created in 1994.
The duration of The Twentieth Century Approaches is 1.7 hours.