Coordinate Plane
The coordinate plane is a two-dimensional surface formed by a horizontal x-axis and vertical y-axis intersecting at the origin, used to locate points.
Definition
The coordinate plane is a flat grid formed by two number lines crossing at a right angle, the horizontal x-axis and vertical y-axis, meeting at the origin $(0,0)$ and dividing the plane into four quadrants; every point has an address called an ordered pair $(x, y)$. Formally, the Cartesian plane is $\mathbb{R}^2$, the set of all ordered pairs of real numbers, equipped with the standard Euclidean metric $d((x_1,y_1),(x_2,y_2)) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$; it is a complete metric space, a vector space over $\mathbb{R}$, and the fundamental setting for Euclidean plane geometry. Linear transformations of $\mathbb{R}^2$ (rotations, reflections, scalings) are represented by $2 \times 2$ matrices, with composition of transformations corresponding to matrix multiplication.
Example
Think of a city map where streets run east-west and avenues run north-south: to find point $(3, 4)$ you go $3$ spaces right and $4$ spaces up from the center. The equation $y = 2x + 1$ can be graphed by plotting points such as $(0,1)$, $(1,3)$, $(2,5)$, $(-1,-1)$, showing all $(x, y)$ pairs that satisfy the equation and connecting algebra to visual geometry.
Key Insight
The coordinate plane was invented by Rene Descartes, which is why it is also called the Cartesian plane; before Descartes, algebra and geometry were entirely separate fields, and now every geometric shape can be described with equations while every equation can be given a visual shape. The coordinate plane is a special case of a manifold, a space that is locally Euclidean, and its generalization to $n$ dimensions ($\mathbb{R}^n$) and to curved spaces (Riemannian manifolds) is the foundation of differential geometry and general relativity.