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As far as I know, these expressions are correct:

  • В двадцатом веке
  • В век второй мировой войны

Why does the first expression use the prepositional case, whereas the second expression uses the accusative case?

  • If век should be treated as 'age' (figurative)- answered by @shabunc (e.g. В век науки и механики - In the age of science, etc.) the expression В век второй мировой войны is a bit awkward (because it's not figurative). – alexsms Dec 24 '18 at 12:19
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"В веке" is for literal period designation, like in "В 15 веке нарождающийся буржуазный класс всё более настойчиво заявлял о своих интересах".

"В век" is figurative, it's not that there's a century we are talking about, it's like "the age", "в эпоху", like in "мы живём в век инноваций" which can be translated as "we live in the age of innovations".

Now imagine yourself asking a question why age there's "in" in the phrase "in the age of innovations". Well, it's of because this is how in English specific relations between objects is designated.

And next thing crucial for understanding of what's going on is to realize that there's no such thing like one to one correspondences between languages. Because modern English has no cases some of constructions with same preposition can be translated in some other language (including, by the way, Old English) into two constructions with different cases, two constructions with the same case and different prepositions, two constructions with no prepositions at all etc.

It something that just should be memoized - when you talk figuratively, it's, "в век" - аnd that's the way it is.

  • One two one correspondence --> one-to-one correspondence. – KCd Dec 23 '18 at 19:23
  • @KCd types happens - feel free to edit whenever you see something wrong, thank you for noticing! – shabunc Dec 23 '18 at 19:43
  • ещё опечатка: types --> typos. 😀 – KCd Dec 23 '18 at 21:11

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