Definition:Euclidean Metric/Real Number Line

Definition

Consider the Euclidean space $\struct {\R^n, d}$.

On the real number line, the Euclidean metric can be seen to degenerate to:

$\map d {x, y} := \sqrt {\paren {x - y}^2} = \size {x - y}$

where $\size {x - y}$ denotes the absolute value of $x - y$.

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:Absolute Value

• Results about the Euclidean metric can be found here.

• Results about Euclidean spaces can be found here.

• Results about the real number line with 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

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: The Definition
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $28$. Euclidean Topology
• 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): One: Metric Spaces: $1$: Open and Closed Sets