Cartesian coordinate system

Summary

A Cartesian coordinate system in the Euclidean plane is a choice of a unit of length and of two perpendicular directed lines, the axes. The axes meet at a point $O$, the origin. Each point $P$ then has a unique pair $(x, y)$ of real numbers, its coordinates. The numbers $x$ and $y$ are the signed distances from the origin to the perpendicular projections of $P$ on the two axes. In a Euclidean space of dimension $n$, a Cartesian coordinate system has $n$ mutually perpendicular axes through one origin, and each point has $n$ coordinates. The map from a point to its coordinates is a bijection onto $\mathbb{R}^n$ that keeps distances.

Remarks

In three dimensions, the usual names of the coordinates are $x$, $y$ and $z$. The two axes of the plane divide it into four quadrants. The three coordinate planes of space divide it into eight octants. Descartes and Fermat developed the idea in the 17th century.

Explanations

L1 human

A Cartesian coordinate system is like the grid of a map. You choose a starting point and two perpendicular directions. Then you describe each point by how far you go along each direction. For example, "3 to the right and 2 up" is the point $(3, 2)$.