Definition:Euclidean Metric/Real Number Plane

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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 René 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.