FWIW, both "½ książek" and "½ książki" are valid Polish, but they mean different things:

- "½ książek" == "połowa książek" == "half of the books", and refers to half of some set, e.g. "połowa książek spadła z półki", "half of the books fell off the shelf". It gets tricky when the set turns out to contain only one book, but that's not spelling/grammar issue anymore.

- "½ książki" == "pół książki" or "połowa książki" == "half of the book", as in "pierwszap połowa tej książki jest nudna", "the first half of this book is boring". There's some difference between "pół" and "połowa" that makes them not interchangeable in most cases, but I don't feel confident enough to articulate a rule here.