Coordinate plane

Mathematical statement

In a specified Cartesian coordinate system, a point has the ordered coordinates (x,y). The x-axis satisfies y=0 and the y-axis satisfies x=0, and they meet at (0,0). Away from the axes, the four quadrants are determined by the signs of x and y.

Representative formula (exact TeX)

P(x,y)

Assumptions

• An origin, perpendicular oriented x- and y-axes, and a unit on each axis are fixed in the plane.
• Coordinates are ordered pairs of real numbers and points are compared within the same coordinate system; this entry does not infer distance or figure size.

Symbol scope

• P — representative point label
• x — horizontal coordinate
• y — vertical coordinate

Intuition, definition, and misconceptions

The first coordinate is read along the x-axis direction and the second along the y-axis direction. Swapping the coordinates generally changes the point.

When both coordinates are nonzero, (+,+), (-,+), (-,-), and (+,-) identify quadrants I, II, III, and IV respectively. A point on an axis is not in a quadrant, and the origin lies on both axes.

Within one coordinate system, two points are equal exactly when their corresponding coordinates are equal. Coordinate values are explicit data and must not be guessed from the scale of a diagram.

Common mistakes include placing an axis point or the origin in a quadrant or treating an ordered pair as an unordered set.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real number — Both coordinate axes are labelled by real-number values.
• Points, lines, and planes — Points in the plane and the two coordinate axes provide the geometric setting.

Verification boundary

Roles and signs of explicit integer coordinates can be recomputed, but the Euclidean distance formula, coordinate transformations, and general geometric theorems have separate obligations.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind ordered pairs of real numbers, a specified coordinate system, and exact axis and quadrant tests; handle distance through a separate semantic bridge.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.