# Definition:Coordinate System/Coordinate

## Definition

Let $\sequence {a_n}$ 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 $\sequence {a_n}$.

### 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.

## 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.

## 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.