Prove that G is an Abelian Group

Let G = {x in Q : 0≤x<1}. Define the operation on G:
a•b = a+b if 0≤a+b<1, a+b-1 if a+b≥1
Prove that (G,*) is an Abelian group.

Attempt: (commutativity was easy). For associativity I got a+b≥1 as the last condition for (a•b)•c but b+c≥1 for a•(b•c). How do I fix this?
For inverses (-a) and a±1 are all out of bound so what else can be used as the inverse (the identity is 0)?