Definition:Euclidean Metric/Real Number Plane

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

The Euclidean metric on $\R^2$ is defined as:
 * $\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2}$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.

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

Also see

 * Definition:Euclidean Metric/Real Vector Space

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.