Coordinate Pair
Pre-AlgebraA 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 is two numbers written in parentheses and separated by a comma: (x, y). The first number tells you how far to move left or right, and the second tells you how far to move up or down on a graph.
Example
The coordinate pair $(3, 5)$ means: start at the origin, move $3$ units right, then $5$ units up. That is your point.
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."
Definition
A coordinate pair (ordered pair) $(x, y)$ locates a point in the Cartesian plane. The $x$-value (abscissa) gives horizontal position; the $y$-value (ordinate) gives vertical position. The origin $(0, 0)$ is the reference point.
Example
The solution pairs of $y = 3x$ include $(0, 0)$, $(1, 3)$, $(2, 6)$, and $(-1, -3)$. Each pair is a point on the graph of the equation.
Key Insight
Every two-variable equation corresponds to a set of coordinate pairs that, when plotted, form a curve. The equation and the graph are two representations of the same mathematical relationship.
Definition
An ordered pair $(a, b)$ is formally defined as $\{\{a\}, \{a, b\}\}$ (Kuratowski's definition in set theory), ensuring that $(a, b) \neq (b, a)$ when $a \neq b$. The Cartesian product $A \times B = \{(a, b) \mid a \in A, b \in B\}$ generalizes to $n$-tuples and is the basis for defining functions as sets of ordered pairs.
Example
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
Defining ordered pairs via sets allows all of mathematics to be built from sets alone. Functions, relations, Cartesian products, and the entire foundation of mathematics reduce to set-theoretic constructs built from ordered pairs.