Definition:Distance/Points

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Definition

Metric Space

Let $\left({X, d}\right)$ be a metric space.

The mapping $d: A \times A \to \R$ is referred to as a distance function on $A$ or simply distance.


Real Numbers

Let $x, y \in \R$ be real numbers.

Let $\size {x - y}$ be the absolute value of $x - y$.


Then the function $d: \R^2 \to \R$:

$\map d {x, y} = \size {x - y}$

is called the distance between $x$ and $y$.


Complex Numbers

Let $x, y \in \C$ be complex numbers.

Let $\cmod {x - y}$ be the complex modulus of $x - y$.


Then the function $d: \C^2 \to \R$:

$\map d {x, y} = \cmod {x - y}$

is called the distance between $x$ and $y$.