# Definition:Euclidean Metric/Complex Plane

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## 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 RenĂ© Descartes.

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

## Also see

- 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 $2.2.4$