Definition:Usual Metric

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Definition

Let $X$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $X^n$ be a cartesian space on $X$.

The usual metric on $X^n$ is the Euclidean metric on $X^n$:


Real Numbers

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

$\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$

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


Rational Numbers

Definition:Euclidean Metric/Rational Space

Complex Plane

The Euclidean metric on $\C$ is defined as:

$\displaystyle \forall z_1, z_2 \in \C: d \left({z_1, z_2}\right) := \left\vert{z_1 - z_2}\right\vert$

where $\left\vert{z_1 - z_2}\right\vert$ denotes the modulus of $z_1 - z_2$.