The Distance on the Complex Numbers

In the setting of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers, for $z\in\mathbb{C}$ write $|z|$ for the modulus of $z$, a real number; for real $z$ this is the absolute value of $z$ by The Modulus of a Complex Number §agreement. Thus $|z-w|\in\mathbb{R}$ for all $z,w\in\mathbb{C}$, so the map below exists and is unique by Sets and Maps: Ordinary Notation §maps.

¶ The distance on $\mathbb{C}$ is the map $d_{\mathbb{C}}:\mathbb{C}\times\mathbb{C}\to\mathbb{R}$ with

$$d_{\mathbb{C}}(z,w)=|z-w|\qquad\text{for all }z,w\in\mathbb{C}.$$