Definition:Euclidean Metric/Real Number Plane

Definition

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

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

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

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

Also known as

The Euclidean metric is 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 fora specific construct.

Also see

• Definition:Euclidean Metric/Real Vector 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

• 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line
• 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.2$: Metric Spaces
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{III}$: Pythagoras' Theorem
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2.2$: Examples: Examples $2.2.3$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: 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): Entry: Euclidean plane