# Definition:Euclidean Metric/Real Number Line

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## Definition

Consider the Euclidean space $\left({\R^n, d}\right)$.

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

- $d \left({x, y}\right) := \sqrt {\left({x - y}\right)^2} = \left|{x - y}\right|$

where $\left|{x - y}\right|$ denotes the absolute value of $x - y$.

## Also known as

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

## Also see

- Results about
**Euclidean spaces**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 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

- 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): $\text{III}$: The Definition - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 28$ - 1999: Theodore W. Gamelin and Robert Everist Greene:
*Introduction to Topology*(2nd ed.) ... (previous) ... (next): $\S 1.1$: Open and Closed Sets