Anyone else's babys hurt them when angry?

When do kids start understanding they can hurt you? Our son is going to be 9 months in 2 days and he's been really bad about this lately. Whenever he's upset we'll pick him up and he immediately goes for your face. He's broke skin on 3 people now, plus me, and made my husband bleed today. To add we keep his nails practically filed off because of this. Every single time we pull his hand away shake our heads and say "no" firmly.