Definition:Euclidean Metric/Complex Plane

Definition

Let $\C$ be the complex plane.

The Euclidean metric on $\C$ is defined as:

$\forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$

where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.

Also known as

The Euclidean metric is also known as the Euclidean distance.

Some sources call it the product metric.

Some sources refer to it as the Cartesian distance or Cartesian metric, for René Descartes.

The Euclidean metric is sometimes also referred to as the usual metric.

Also see

• Definition:Euclidean Metric on Real Vector Space

• Complex Plane is Metric Space

• Results about the Euclidean metric can be found here.

Source of Name

This entry was named for Euclid.

Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean space, Euclidean topology and Euclidean norm.

They bear that name because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.4$