Definition:Euclidean Metric/Real Number Line

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

 * Definition:Absolute Value

Bear in mind that Euclid himself did not in fact conceive of the Euclidean metric. It is called that because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.