# 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 $\tuple {a, b}$ 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 coordinate systems.

## Also denoted as

It is usual to use the subscript technique to denote the coordinates where $n$ is large or unspecified:

$\tuple {x_1, x_2, \ldots, x_n}$

However, note that some texts (often in the fields of physics and mechanics) prefer to use superscripts:

$\tuple {x^1, x^2, \ldots, x^n}$

While this notation is documented here, its use is not endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$ because:

there exists the all too likely subsequent confusion with notation for powers
one of the philosophical tenets of $\mathsf{Pr} \infty \mathsf{fWiki}$ is to present a system of notatiion that is as completely consistent as possible.

## 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.

## Sources

WARNING: This link is broken. Amend the page to use {{KhanAcademySecure}} and check that it links to the appropriate page.