Complex Plane is Metric Space

Theorem
Let $\C$ be the set of all complex numbers.

Let $d: \C \times \C \to \R$ be the function defined as:
 * $\map d {z_1, z_2} = \size {z_1 - z_2}$

where $\size z$ is the modulus of $z$.

Then $d$ is a metric on $\C$ and so $\struct {\C, d}$ is a metric space.

Proof
Let $z_1 = x_1 + i y_1, z_2 = x_2 + i y_2$.

From the definition of modulus:


 * $\size {z_1 - z_2} = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$

This is the euclidean metric on the real number plane.

This is shown in Euclidean Metric on Real Vector Space is Metric to be a metric.

Thus the complex plane is a 2-dimensional Euclidean space.