# Definition:Coordinate System/Coordinate

## Definition

Let $\sequence {a_n}$ be a coordinate system of a unitary $R$-module $G$.

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

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 known as

**Coordinates of $x$ relative to $\sequence {a_n}$** are also known as **coordinates of $x$ with respect to $\sequence {a_n}$**.

## 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

- 1921: C.E. Weatherburn:
*Elementary Vector Analysis*... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Components of a Vector: $7$. The unit vectors $\mathbf i$, $\mathbf j$, $\mathbf k$ - 1964: Steven A. Gaal:
*Point Set Topology*: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 27$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 35$. Coordinates - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates: Fermat

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