We can define the Lie algebra of a matrix lie group $G$ via the following: $$ Lie(G) = [ X \in M(n,\mathbb{C} \mid e^{t X} \in G \quad \forall t \in \mathbb{R}] $$ The group $U(1)$, complex numbers with modulus $1$, may be written as: $$ [ e^{i \theta} \mid \theta \in \mathbb{R} ] $$
Then it seems to me that $ Lie(U(1)) = [ r \in \mathbb{C} \mid e^{t r} = e^{i \theta} \quad \forall t \in \mathbb{R} ] = i\mathbb{R} $. Both online sources and textbooks state, however, that $Lie(U(1)) = \mathbb{R}$. Is there an error in my calculation, or am I misunderstanding something more fundamental?