Let $U_1=\{z\in\mathbf{C}\mid |z|=1\}\subset \mathbf{C}^*$, where $\mathbf{C}^*$ are the complex numbers exluding $0$ under multiplication.
I am asked to prove that $\boxed{\mathbf{C}^*\cong U_1\times \mathbf{R}^+}$.
Here comes the ambiguity. In some footnote, my lecture notes define $\mathbf{R}^+$ to be the real numbers under addition, while I find this definition ambiguous since various texts define $\mathbf{R}^+$ to be the positive real numbers under multiplication.
Which of the two is $\mathbf{R}^+$ in this particular question? I think it should be the positive real numbers under multiplication, since there is an obvious isomorphism $z\mapsto (z/|z|,|z|)$.
I can also apply some theorem in my lecture notes which states that if $H_1,H_2\subset G$ are subgroups and $H_1 H_2=G$, $H_1$ and $H_2$ commute and $H_1\cap H_2=\{e\}$, then $G\cong H_1\times H_2$. But in the context of $\mathbf{R}^+$ meaning the real numbers under addition, the last condition is not met since the intersection contains both $-1$ and $1$.
Two arguments in favor of $\mathbf{R}^+$ meaning the positive reals under multiplication. Who can give a definite answer to this confusion?
It's the real numbers greater than $0$ under multiplication. Besides, under addition they don't even form a group. And your obvious isomorphism is a correct answer.