Additive and Multiplicative Groups of Numbers

In the settings of Groups: Standing Notation and Complex Analysis: Standing Notation, let $\exp:\mathbb{C}\to\mathbb{C}$ be the complex exponential function, and let $(0,\infty)=\{x\in\mathbb{R}:x>0\}$ be the ray.

(i) Additive groups. Each of $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$, with addition, is an abelian group, with identity $0$ and with $-x$ the inverse of $x$.

(ii) Multiplicative groups. Each of $\mathbb{Q}\setminus\{0\}$, $\mathbb{R}\setminus\{0\}$, $(0,\infty)$ and $\mathbb{C}\setminus\{0\}$, with multiplication, is an abelian group, with identity $1$ and with $1/x$ the inverse of $x$.

(iii) The circle group. The unit circle $\{z\in\mathbb{C}:|z|=1\}$ is a subgroup of the group $\mathbb{C}\setminus\{0\}$ of (ii).

(iv) The exponential. The function $\mathbb{R}\to(0,\infty)$, $x\mapsto\exp_{\mathbb{R}}(x)$, is an isomorphism from the additive group $\mathbb{R}$ of (i) onto the multiplicative group $(0,\infty)$ of (ii), and the function $\mathbb{C}\to\mathbb{C}\setminus\{0\}$, $z\mapsto\exp z$, is a homomorphism from the additive group $\mathbb{C}$ of (i) to the multiplicative group $\mathbb{C}\setminus\{0\}$ of (ii):

$$\exp_{\mathbb{R}}(x+y)=\exp_{\mathbb{R}}(x)\exp_{\mathbb{R}}(y),\qquad \exp(z+w)=\exp z\,\exp w\qquad(x,y\in\mathbb{R},\ z,w\in\mathbb{C}).$$