I am confused to why the zero vector in $X/M$ is the coset $0+M = M$, because $(x+M)+(0+M)=(x+0)+M =x + M$? Why is the zero vector not just $0$ since $(x+M)+(0)=(x+0)+M =x + M$?
I tried to find a concrete simple numbers example for quotient space and its zero vector but cannot find one so any example would be highly appreciated.