# 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.

## Sources

- 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

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