Примерно половина этих книг написана иностранными авторами.

the book

Breakdown of Примерно половина этих книг написана иностранными авторами.

• иностранный: foreign
• эти: these
• написать: to write

Questions & Answers about Примерно половина этих книг написана иностранными авторами.

Why is it половина, not половину or some other case?

Половина is in the nominative case because it is the grammatical subject of the sentence.

• Примерно половина этих книг = approximately half of these books (this whole phrase is the subject).
• The verb / predicate написана agrees with половина, so половина must be nominative.

If половина were an object (for example, after a verb or a preposition), you could see other cases, like accusative половину, but here it is clearly the subject:
(Что?) Примерно половина этих книг (что сделала?) написана…

Why is it этих книг and not эти книги?

Because after половина (and most words meaning a quantity or part of something), Russian uses the genitive case for the thing being measured.

Structure here:

• половина чего? — этих книг (genitive plural: of these books)

So:

• половина этих книг = half of these books
• эти книги would be nominative plural (these books) and would not correctly show the idea of half of something.

This is the same pattern as:

• много книг (a lot of books) — genitive plural
• стакан воды (a glass of water) — genitive singular
• кусок хлеба (a piece of bread) — genitive singular