問問老土

問問老土，我拿到offer已經上班了，我是否應該給我的老板的老板發個e-mail告知我很高興and start work， I know he must know he approved to hire me, but he will not know when I started. 因為他在外州當是是電話面試的我，is this necessary? is my current boss will be not happy if he knows. thanks