# Definition:Euclidean Metric/Real Number Plane

## Definition

Let $\R^2$ be the real number plane.

The **Euclidean metric** on $\R^2$ is defined as:

- $\ds \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.

## 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**.

The **real number plane with the Euclidean metric** is also known as the **Euclidean plane**, but in the field of abstract geometry that term is used for a specific construct.

## 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

- 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Pythagoras' Theorem - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Examples $2.2.3$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Euclidean plane** - 1999: Theodore W. Gamelin and Robert Everist Greene:
*Introduction to Topology*(2nd ed.) ... (previous) ... (next):**One**: Metric Spaces: $1$: Open and Closed Sets - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Euclidean plane**