Coordinate Plane: What It Is and How It Works

Two Lines That Meet

The coordinate plane is the plane formed by two coordinate axes perpendicular to each other, as Cuemath puts it. The horizontal axis is called the x-axis, the vertical one the y-axis; the point where the two axes meet is the origin, with coordinates (0, 0).

According to Treccani, to find the coordinates of a point P you draw lines parallel to the axes: the lengths of the two segments that form, measured from the origin, are the point’s two coordinates. This creates a correspondence between every point in the plane and every pair of real numbers.

Abscissa and Ordinate

Treccani calls the first number of a coordinate pair the abscissa and the second the ordinate: in practice, x is always the abscissa and y is always the ordinate. The line along which the abscissa is measured is the x-axis; the other is the y-axis. Cuemath defines the distance symmetrically: x is the point’s distance from the y-axis, y is its distance from the x-axis.

Practical example: to plot the point (3, 2), start from the origin, move 3 units along the x-axis, then 2 units upward, parallel to the y-axis. The point you land on has an x-coordinate of 3 and a y-coordinate of 2.

The Four Quadrants

The axes divide the plane into four infinite regions, the quadrants, numbered I through IV starting from the top-right one and moving counterclockwise, according to Math.net. The origin belongs to none of the four. Each quadrant has a fixed combination of signs:

For example, whenever x is negative and y is positive, the point always sits in the second quadrant: the signs alone tell you which of the four zones to look in.

Where the Name “Cartesian” Comes From

The name traces back to René Descartes, who according to MacTutor published La Géométrie in Leiden in 1637, the work that gave rise to what is now called Cartesian geometry. The term traces back to the Latinized form of Descartes’ name, “Cartesius.” MacTutor notes, however, that some of the treatise’s ideas may derive from earlier work by the medieval mathematician Nicole Oresme, even though that work shows no sign of linking algebra and geometry; and that the English mathematician John Wallis argued forcefully that those ideas belonged to Thomas Harriot, a historical dispute that MacTutor reports without endorsing. The method’s attribution to Descartes stands, alongside the debts to earlier work that MacTutor itself points out.

What It’s Used for, Beyond Plotting Points

The coordinate plane alone already makes it possible to place known geometric shapes and describe their properties with numbers: the theorems and postulates of Euclidean geometry, for instance, still hold for triangles drawn on the plane, and the proportions from Thales’ theorem can be checked by reading the coordinates of the points involved. Once the coordinate plane is used to describe lines and curves with equations, it becomes analytic geometry, a step beyond simply reading a point.

Two triangles drawn on the plane, with vertices at different points, can be compared using the triangle similarity theorems: coordinates make it possible to calculate exactly the measurements those theorems require, without having to draw everything by hand.