Cartesian axes pin every point with signed distances
In the plane each point gets a unique pair of real coordinates—signed distances to two fixed perpendicular oriented axes meeting at the origin (0, 0). Three mutually perpendicular planes extend the idea to space, and n coordinates locate points in n-dimensional Euclidean space. Descartes's seventeenth-century invention let geometry speak algebra.
Named for René Descartes, who published the idea in 1637 while living in the Netherlands, Cartesian coordinates revolutionized mathematics by expressing geometric problems through algebra and calculus. Curves become equations in the coordinates—for example a radius-2 circle centered at the origin is the set of points whose x and y satisfy a fixed quadratic relation. The system founds analytic geometry and illuminates linear algebra, complex analysis, differential geometry, multivariate calculus, and applied fields from astronomy to engineering. Pierre de Fermat discovered related ideas independently, including three dimensions, without publishing; Nicole Oresme used similar constructions earlier. Both Descartes and Fermat first used a single reference axis; paired axes entered after Frans van Schooten's 1649 Latin translation of La Géométrie and student commentaries.
The coordinate plane underwrote Newton's and Leibniz's calculus and later generalized toward vector spaces. Polar, spherical, and cylindrical systems followed. An affine line with Cartesian coordinates is a number line: every real labels a point. Two degrees of freedom choose the system—often by assigning zero and one to two points, or fixing an origin and a unit length. Translations and scalings become addition and multiplication. In two dimensions an ordered pair of perpendicular axes, a shared unit length, and orientations define the system. Dropping perpendiculars from a point P to the axes reads abscissa and ordinate, written (x, y); the origin is (0, 0). Mathematics usually draws the first axis rightward and the second upward, though some graphics flip the ordinate down.
A Euclidean plane with such axes is a Cartesian plane, hosting unit circle, unit square, and unit hyperbola as canonical figures. The axes cut four quadrants; the all-positive region is usually the first. Distances to the axes are absolute values of the coordinates. In three dimensions three pairwise perpendicular axes through an origin yield coordinates as signed distances to the coordinate planes, written (x, y, z), dividing space into eight octants. Standard names include abscissa, ordinate, and applicate. The right-hand rule orients the third axis so the turn from the first to the second looks counterclockwise from (0, 0, 1).
Points of the plane identify with the Cartesian product of the reals with itself; n-space with n-tuples of reals. Oblique systems allow non-perpendicular axes or unequal units, changing distance and angle formulas. Parenthetical comma-separated notation and end-of-alphabet letters for unknowns remain conventional across mathematics, physics, and engineering.
Source: Cartesian coordinate system
