Why isn't $(\Bbb R, \cdot)$ a subgroup of nonzero complex numbers under multiplication?

Question

I'm doing an exercise from my textbook of abstract algebra class. And the book answer says that $(\Bbb R,\cdot)$ is not a subgroup of nonzero complex numbers under multiplication. But I thought:

1. The set of $\Bbb R$ does have $1$, which is the identity of nonzero complex numbers under multiplication.
2. $\Bbb R$ is closed under multiplication.
3. $\Bbb R$ does have inverse for each of its member.

Any hints or pointing out my mistakes are very appreciated!

Answer 1

Since $\mathbb R$ isn't even a subset of $\mathbb{C}\setminus\{0\}$, it cannot be a subgroup.