Coordinate Pair
A coordinate pair (also called an ordered pair) is a pair of numbers (x, y) that specifies a unique location in the coordinate plane.
Formula
(x, y)
Definition
A coordinate pair, also called an ordered pair, is two numbers written in parentheses and separated by a comma, $(x, y)$, that locate a point in the Cartesian plane: the first number (the $x$-value, or abscissa) tells you how far to move left or right, and the second (the $y$-value, or ordinate) tells you how far to move up or down, with the origin $(0, 0)$ as the reference point. Formally, an ordered pair $(a, b)$ is defined in set theory as $\{\{a\}, \{a, b\}\}$ (Kuratowski's definition), ensuring $(a, b) \neq (b, a)$ when $a \neq b$; the Cartesian product $A \times B = \{(a, b) \mid a \in A, b \in B\}$ generalizes this to $n$-tuples and is the basis for defining functions as sets of ordered pairs.
Example
The coordinate pair $(3, 5)$ means: start at the origin, move $3$ units right, then $5$ units up. The solution pairs of $y = 3x$ include $(0, 0)$, $(1, 3)$, $(2, 6)$, and $(-1, -3)$, each a point on the graph of the equation. A function $f: \mathbb{R} \to \mathbb{R}$ can be defined as the set of ordered pairs $\{(x, f(x)) \mid x \in \mathbb{R}\}$; for $f(x) = x^2$, this is $\{(x, x^2) \mid x \in \mathbb{R}\}$, a parabola in $\mathbb{R}^2$.
Key Insight
Order matters: $(3, 5)$ is a different location than $(5, 3)$, just like "3rd floor, room 5" is a different place than "5th floor, room 3." Every two-variable equation corresponds to a set of coordinate pairs that, when plotted, form a curve, and defining ordered pairs via sets allows all of mathematics, functions, relations, Cartesian products, and more, to be built from sets alone.