As a word of warning, Microsoft no longer supply the wireless PC adapter - even those on Amazon that claim to be official Microsoft products are cheap knock offs. As evidenced by the reviews, for some people these adapters work, for some they don't.
Personally I bought the wired version rather than spend extra money on a wireless controller that would end up being more trouble than its worth. If wireless were a requirement I'd have bought a different controller that still had official PC wireless support from the manufacturer.