Definition:Coordinate System/Coordinate

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Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$.

Let $\displaystyle x \in G: x = \sum_{k \mathop = 1}^n \lambda_k a_k$.

The scalars $\lambda_1, \lambda_2, \ldots, \lambda_n$ can be referred to as the coordinates of $x$ relative to $\left \langle {a_n} \right \rangle$.

Elements of Ordered Pair

Let $\left({a, b}\right)$ be an ordered pair.

The following terminology is used:

$a$ is called the first coordinate
$b$ is called the second coordinate.

This definition is compatible with the equivalent definition in the context of Cartesian coordinates.

Linguistic Note

It's an awkward word coordinate. It really needs a hyphen in it to emphasise its pronunciation (loosely and commonly: coe-wordinate), and indeed, some authors spell it co-ordinate. However, this makes it look unwieldy.

An older spelling puts a diaeresis indication symbol on the second "o": coördinate. But this is considered archaic nowadays and few sources still use it.

Historical Note

The words coordinate and coordinates entered the mathematical mainstream via the works of Gottfried Wilhelm von Leibniz, who may well have coined them.