Ordered Pair
An ordered pair is a set of two numbers written in the form (x, y) that gives the exact location of a point on the coordinate plane.
Definition
An ordered pair is two numbers inside parentheses, separated by a comma, like $(3, 5)$: the first number ($x$) is the horizontal displacement from the origin along the x-axis, and the second ($y$) is the vertical displacement along the y-axis. The pair is "ordered" because swapping the values generally produces a different point, an idea that extends to ordered triples $(x, y, z)$ in 3-D space and n-tuples in higher dimensions. Formally, an ordered pair $(a, b)$ is defined using the Kuratowski definition as the set $\{\{a\}, \{a, b\}\}$, ensuring $(a, b) = (c, d)$ if and only if $a = c$ and $b = d$; this set-theoretic definition is the foundation of the Cartesian product $A \times B = \{(a, b) : a \in A, b \in B\}$.
Example
To plot $(4, 2)$: start at the origin, move $4$ spaces right, then $2$ spaces up; to plot $(-3, 5)$: move $3$ spaces left, then $5$ spaces up. The point $(-5, 3)$ is $5$ units left and $3$ units up from the origin, placing it in Quadrant II, while $(3, -5)$ is $3$ units right and $5$ units down, placing it in Quadrant IV, distinct locations even though the same digits are used. The Cartesian product $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$ is the set of all ordered pairs of real numbers, forming the coordinate plane, with relations and functions defined as subsets of Cartesian products.
Key Insight
The word "ordered" is key: $(3, 5)$ and $(5, 3)$ are different points on the grid, with the first number always the x-coordinate and the second always the y-coordinate. The formalization of ordered pairs by Hausdorff and Kuratowski in the early 20th century was crucial for building mathematics on a rigorous set-theoretic foundation, showing that even seemingly primitive notions can be reduced to pure set theory.