# Definition:Cartesian Product/Coordinate

## Definition

Let $\ds \prod_{i \mathop \in I} S_i$ be a cartesian product.

Let $j \in I$, and let $s = \sequence {s_i}_{i \mathop \in I} \in \ds \prod_{i \mathop \in I} S_i$.

Then $s_j$ is called the **$j$th coordinate of $s$**.

If the indexing set $I$ consists of ordinary numbers $1, 2, \ldots, n$, one speaks about, for example, the **first**, **second**, or **$n$th coordinate**.

For an element $\tuple {s, t} \in S \times T$ of a binary cartesian product, $s$ is the **first coordinate**, and $t$ is the **second coordinate**.

## 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 notation that is as completely consistent as possible.

## 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

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 9$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 5$: Cartesian Products - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Order