Definition:Cartesian Product/Coordinate

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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": co├Ârdinate.

But this is considered archaic nowadays and few sources still use it.